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https://www.movingbeyondthepage.com/online/samplelesson.aspx?unitID=386&lessonPartID=231133 | 1,718,383,079,000,000,000 | text/html | crawl-data/CC-MAIN-2024-26/segments/1718198861567.95/warc/CC-MAIN-20240614141929-20240614171929-00658.warc.gz | 815,350,007 | 5,104 | # Lesson 1: Numbers to 10,000 Review
## Getting Started
### Questions to Explore
• How does place value work?
• How can we use place value to name, create, and compare numbers to a million?
• How do we write and speak in mathematical language?
### Facts and Definitions
• Ones place: position on the right in a number
• Tens place: position to the left of the ones place
• Hundreds place: position to the left of the tens place
• Thousands place: position to the left of the hundreds place
• Expanded form: a way to write a number that shows the value of each digit (for example, 4,529=4000+500+20+9)
• Expanded notation: also called super expanded form, expresses a number by showing each digit multiplied by its place value. For example, 4,532 in expanded notation is (4×1,000)+(5×100)+(3×10)+(2×1)
### Skills
• Read and write multi-digit whole numbers using base-ten numerals, number names, and expanded form
• Order and compare multi-digit numbers
### Materials
• deck of playing cards (kit)
• fine point dry-erase markers (kit)
• index cards (kit)
• Interactive Notebook
• whiteboard (kit)
### Introduction
Materials: fine point dry-erase markers (kit), whiteboard (kit)
Write 6,482 on the whiteboard, and pose the following questions:
• How many digits does this number have? (4)
• How do we say this number? (six thousand four hundred eighty-two — remind your child as needed that we don't say "and" in between the numbers)
• What digit is in the ones place? (2)
• What digit is in the tens place? (8)
• What digit is in the hundreds place? (4)
• What digit is in the thousands place? (6)
• What is the value of 6 in this number? (6,000 — repeat for each digit)
• What number is 10 more than this number? (6,492)
• What number is 10 less than this number? (6,472)
• What number is 100 more than this number? (6,582)
• What number is 100 less than this number? (6,382)
Now, write 3,871 on the whiteboard, and ask your child to say the number aloud. He should say, "Three thousand eight hundred seventy-one." Ask, "Can you recall another way we can show large numbers?" As needed, remind your child that we also use expanded form to show numbers. Tell him to write 3,871 in expanded form on the whiteboard (3,000+800+70+1). Help him as needed by asking him the value of each digit in the number, such as, "What is the value of the 3 in the thousands place?" (3,000) Repeat this process for the numbers 8,902 (8,000+900+2) and 1,996 (1,000+900+90+6).
Tell your child that expanded notation (sometimes called super expanded form) is another way to represent a number. Tell him that it is similar to expanded form but goes one step further to show each digit multiplied by its place value. Show 3,871 in expanded form again on the whiteboard (3,000+800+70+1), and then show him the expanded notation underneath:
Tell your child that this method provides a clear way of showing the value of each place in the number. Explain that the parentheses help the numbers stay organized. Ask him to try this new form with 1,996; write out the expanded form as needed (1,000+900+90+6). His answer should be as follows:
Without erasing the previous example, show the expanded form for 8,902 (8,000+900+2). Tell your child that in expanded form, the zero isn't included but that it is in expanded notation since expanded notation shows the value of each place. Ask your child to try to write the expanded notation of 8,902. Remind him that he should take each digit and multiply it by its place value. If he gets stuck when he gets to the 0, ask him what place the digit is in (tens place), so he should multiply the digit (0) by 10. The answer is (8×1,000)+(9×100)+(0×10)+(2×1).
NOTE: In Lesson 3, your child will learn about placing commas in numbers. If he already knows how to put commas in 4-digit numbers, he is free to do so in this lesson. | 989 | 3,848 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.75 | 5 | CC-MAIN-2024-26 | latest | en | 0.853423 |
https://www.cfd-online.com/Forums/main/167030-oscillating-residuals-steady-flow-problem.html | 1,718,438,118,000,000,000 | text/html | crawl-data/CC-MAIN-2024-26/segments/1718198861584.65/warc/CC-MAIN-20240615062230-20240615092230-00752.warc.gz | 631,806,093 | 17,078 | # Oscillating residuals in steady flow problem
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February 22, 2016, 06:08
Oscillating residuals in steady flow problem
#1
New Member
mohit
Join Date: Jun 2015
Location: Trivandrum
Posts: 13
Rep Power: 10
Hello,
I am recently trying to run a transonic flow simulation in cfd++ and have come across this phenomenon of oscillating residuals, even when the problem is specified as steady flow. I'm not able to interpret that and also not able to extract the data as it changes with changing residual. For your reference, htte snapshots of two of the cases have been attached herewith. Does anyone has faced such a problem or do anyone knows how to deal with such a case? Any kind of help would be appreciable.
Mohit
Attached Images
res_160gp.jpg (71.2 KB, 126 views) res_41gp.jpg (72.2 KB, 67 views)
February 23, 2016, 12:38 #2 Senior Member Matt Join Date: Aug 2014 Posts: 947 Rep Power: 17 Have you considered the possibility that your flow isn't steady state? If you analyze an unsteady flow assuming it's steady your convergence can often behave as you have shown. If you are using an implicit solver, an iterative approach is used to advance the solution from a starting state (initialization) to a final (converged) state. This is true if the solution is one step in a transient problem or a final steady-state result. You can think of the iterations as a pseudo temporal progression that will still allow non-steady state behavior to develop.
February 23, 2016, 14:30
#3
New Member
mohit
Join Date: Jun 2015
Location: Trivandrum
Posts: 13
Rep Power: 10
Quote:
Originally Posted by MBdonCFD Have you considered the possibility that your flow isn't steady state? If you analyze an unsteady flow assuming it's steady your convergence can often behave as you have shown. If you are using an implicit solver, an iterative approach is used to advance the solution from a starting state (initialization) to a final (converged) state. This is true if the solution is one step in a transient problem or a final steady-state result. You can think of the iterations as a pseudo temporal progression that will still allow non-steady state behavior to develop.
Hello Matt Sir,
First of all, thanks for your quick reply. As you mentioned flow may be an unsteady state since I'm simulating flow over payload fairing in transonic regime of M=0.9. But I would also like to mention that I have specified the simulation to be steady state in cfd++ i.e. the flow is supposed to be in steady state.
I would be grateful to you if you could please elaborate the meaning of solution being one step in a transient problem and the term pseudo temporal regression. What would you recommend that could be done in such a case?
Can you describe the significance of Courant-Freidrich-Lewy (CFL) number in a steady flow simulation? | 689 | 2,879 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.9375 | 3 | CC-MAIN-2024-26 | latest | en | 0.9458 |
http://oeis.org/A209487 | 1,600,537,867,000,000,000 | text/html | crawl-data/CC-MAIN-2020-40/segments/1600400192783.34/warc/CC-MAIN-20200919173334-20200919203334-00785.warc.gz | 103,063,163 | 3,992 | The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.
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A209487 Number of 6-bead necklaces labeled with numbers -n..n allowing reversal, with sum zero and avoiding the patterns z z+1 z+2 and z z-1 z-2. 1
11, 136, 731, 2606, 7179, 16660, 34233, 64220, 112263, 185506, 292759, 444680, 653957, 935472, 1306483, 1786806, 2398979, 3168444, 4123729, 5296612, 6722303, 8439626, 10491183, 12923536, 15787389, 19137752, 23034123, 27540670, 32726395 (list; graph; refs; listen; history; text; internal format)
OFFSET 1,1 COMMENTS Row 6 of A209485. LINKS R. H. Hardin, Table of n, a(n) for n = 1..210 Index entries for linear recurrences with constant coefficients, signature (5, -10, 11, -10, 11, -10, 5, -1). FORMULA Empirical: a(n) = 5*a(n-1) - 10*a(n-2) + 11*a(n-3) - 10*a(n-4) + 11*a(n-5) - 10*a(n-6) + 5*a(n-7) - a(n-8) for n > 9. EXAMPLE Some solutions for n=8: -6 -8 -6 -7 -8 -7 -7 -8 -8 -4 -5 -6 -8 -8 -6 -8 -1 -3 -6 -6 -2 0 -5 0 -4 -3 3 -2 1 -2 -3 3 2 -1 0 6 6 6 2 -2 -5 6 -2 3 -4 7 -4 -4 -2 -1 -3 2 -1 -7 5 -3 8 -3 -4 6 -3 -4 4 6 6 6 7 4 6 6 6 5 8 -4 2 -3 8 4 1 -1 1 7 8 1 -1 2 -1 8 1 8 6 2 6 3 8 4 MATHEMATICA Join[{11}, LinearRecurrence[{5, -10, 11, -10, 11, -10, 5, -1}, {136, 731, 2606, 7179, 16660, 34233, 64220, 112263}, 40]] (* Harvey P. Dale, Feb 04 2013 *) CROSSREFS Sequence in context: A250460 A157773 A336180 * A227101 A024143 A057718 Adjacent sequences: A209484 A209485 A209486 * A209488 A209489 A209490 KEYWORD nonn AUTHOR R. H. Hardin, Mar 09 2012 STATUS approved
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Last modified September 19 13:25 EDT 2020. Contains 337178 sequences. (Running on oeis4.) | 963 | 2,094 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.5 | 4 | CC-MAIN-2020-40 | latest | en | 0.30275 |
https://www.studycrow.com/questions/quantitative-aptitude/races-and-games | 1,560,747,964,000,000,000 | text/html | crawl-data/CC-MAIN-2019-26/segments/1560627998376.42/warc/CC-MAIN-20190617043021-20190617065021-00312.warc.gz | 904,542,233 | 9,090 | # Races and Games
1
In a 100 m race, A can give B 10 m and C 28 m. In the same race B can give C:
2
A and B take part in 100 m race. A runs at 5 kmph. A gives B a start of 8 m and still beats him by 8 seconds. The speed of B is:
3
In a 500 m race, the ratio of the speeds of two contestants A and B is 3 : 4. A has a start of 140 m. Then, A wins by:
4
In a 100 m race, A beats B by 10 m and C by 13 m. In a race of 180 m, B will beat C by:
5
At a game of billiards, A can give B 15 points in 60 and A can give C to 20 points in 60. How many points can B give C in a game of 90? | 207 | 587 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.03125 | 3 | CC-MAIN-2019-26 | longest | en | 0.943347 |
https://dissertationpapers.net/unit-5-short-answers-17-music-world-has-nine-employees-the/ | 1,631,826,639,000,000,000 | text/html | crawl-data/CC-MAIN-2021-39/segments/1631780053759.24/warc/CC-MAIN-20210916204111-20210916234111-00024.warc.gz | 265,465,870 | 13,048 | # Unit 5 -short answers 17. music world has nine employees. the
17. Music World has nine employees. The employees are paid weekly, with overtime after 40 hours per week. The overtime rate is 1 1/2 times the regular pay rate. Payroll information for the week ending June 12th is as follows:
18. Scoot Rentals pays employees an hourly wage or a salary plus commission based on rental revenue. Hourly wage employees can earn overtime. The overtime rate for hours worked over 40 in a week is 1 1/2 times more that the regular hourly rate of pay.
Instructions:
For each of the following employees, determine the total gross pay for the pay period.
John
Earns an hourly wage of \$7.20
Worked 45 hours this week
Stan
Receives a salary of \$300 per week, plus a 2% commission on rental revenue
Had rental revenue of \$760 this week
Ryan
Earns an hourly wage of \$8.50
Worked 40 hours this week
Janice
Receives a salary of \$200 per week plus 2% commission
Had rental revenue of \$1,235 this week.
19. Sandy Dry Cleaners has four employees. They are paid on a weekly basis, with overtime paid for hours worked over 40 hours in a week. The overtime rate is 1 1/2 times the regular rate of pay. The payroll information is as follows:
During the week ending January 9th, James worked 39 hours, Gibson worked 41 hours, and David and John each worked 36 hours.
Instructions:
a. Prepare a payroll register for the week ending January 9th. The date of payment is October 9. List employees in alphabetical order by last name. Use the tables on page 10, section 2, of Unit 5 to determine the Federal Income Tax withholding. The rate for the State Income Tax is 2%. Compute the Social Security Tax at 6.2% and Medicare Tax at 1.45%. Union members pay weekly dues of \$4.50. Both Gibson and John had \$6.75 deducted for health and hospital insurance.
b. Total the amount columns. Subtract total deductions from total earnings. Does the result equal the sum of the Net Pay column? If not, find and correct any error(s) in the payroll register.
20. David & Co. has seven employees who are paid weekly. For hourly wage employees, overtime is paid at 1 1/2 times more than the regular rate of pay, for hours worked over 40 in a week.
Mary, the office manager, is paid a salary of \$375.00 per week plus a bonus of 3 % of all revenue over \$6,000 per week. Stan, an office assistant, is paid a salary of \$250.00 per week plus 5% of all telephone sales made in the office.
David, the office secretary, is paid a salary of \$230.00 per week. Justin and William, placement workers, are paid an hourly wage of \$8.95. Fernando is also a placement worker but is paid a commission of \$35.00 for every job placement completed. Ryan, a part-time maintenance worker, is paid \$6.75 per hour. For the week ending October 24th, the office recorded the following payroll information:
Total office sales for the week were \$8,420.00.
Justin worked for 38 1/2 hours.
William worked for 41 1/4 hours.
Phone sales for the week were \$1,375.00.
Ryan worked a total of 23 hours.
Instructions:
Calculate the gross earnings for the workers at David & Co. for the week ending October 24th. Analyze and identify the employee who had the highest gross earnings.
21. David & Co. pays its employee twice a month. Employee earnings and tax amounts for the pay period ending December 31st are:
Instructions:
1. Prepare the general journal records for the payment of the payroll.
2. Post the payroll transaction to the general ledger.
3. Compute the payroll tax expense forms and journalize the entry to record the employer’s payroll taxes using these rates:
a. Social Security, 6.2%
b. State Unemployment, 2%
c. Medicare, 1.45%
d. Federal unemployment, 0.8%
e. No employee has reached the taxable earnings limit
4. Post the entry to the general journal.
5. Journalize and post the entries for the payment of payroll liabilities.
6. Analyze and calculate the employer’s total payroll-related expense for the pay period. | 971 | 3,992 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.75 | 3 | CC-MAIN-2021-39 | latest | en | 0.934969 |
https://us-issues.com/2016/01/18/using-silver-dollars-part-3/ | 1,582,856,156,000,000,000 | text/html | crawl-data/CC-MAIN-2020-10/segments/1581875146940.95/warc/CC-MAIN-20200228012313-20200228042313-00348.warc.gz | 587,156,247 | 26,106 | # Using Silver Dollars – Part 3
By Bob Shapiro
Using Liberty Eagle Silver Dollars in everyday commerce carries several advantages. In Part 1, I detailed an Exchange mechanism, as well as a way to store the Silver Dollars for later use. In Part 2 of this series, we began to see the advantages as they related to a business’ sales and taxable profits.
Today, I’d like to show how employees may be paid in a mix of Paper and Silver Dollars in a way that benefits both the employee and the employer. (The pay needs to be a mix of Paper and Plastic because of the Minimum Wage. Most states have a higher Minimum than the federal rate of \$7.50 per hour – in my home state of MA, the rate is \$9.00.)
Here’s how it would work. Suppose you had an employee working 40 hours at \$25 an hour, or \$1000 for the week. As an existing employee, he may be offered perhaps a 5% premium – an incentive add-on encouraging him to participate, bringing his weekly all-paper pay to \$1050. (You may or may not choose to offer an incentive to new employees.)
You would tell my company (let’s call it Cambi Money Services) the person’s name, etc, that the hours worked was 40, the incentive rate, and the pay amount. Cambi would calculate how much that comes to after tax, and that amount would be the target purchasing power. Cambi then would use the hours and the state Minimum Wage to calculate the total Paper plus Silver Dollars (eg. 40 x \$9 = \$360), and the after tax amount.
Using the two after tax amounts, and the Exchange mechanism, we would tell you to use your regular payroll provider, with \$360 as the amount, and a Direct Deposit amount to Cambi (in Paper) which would go into a Silver account with us.
Cambi would bill you for the Paper Dollars needed to change the Direct Deposit amount into Silver. Cambi would act as both a Payroll Advisory and as a Depository. Let’s look at an example for a married worker with 4 exemptions and an Exchange Rate of 20:1.
Nominal Pay \$1000
Incentive 5%
Pre-Tax Total \$1050
After-Tax Target \$858.48
Hours 40
State Minimum \$9.00
Pre-Tax Total \$360
After-Tax Amount \$325.66
Paper Difference \$532.82
Direct Deposit \$ 28.04
Silver Deposited \$ 28.0432
Exchange Rate 20:1
P.P. Of Silver \$560.86
Paper Pay Amount\$297.62
Employer Billed
P.P. Make Up \$532.82
Exchange Fee-2% \$ 10.66 (on amount exchanged)
Payroll Fee-0.5% \$ 5.25 (on pre-tax total)
Total Bill \$548.73
Employer Cost
Employee Pay \$360.00
Cambi Bill \$548.73
FICA Match \$ 27.54
Total Cost \$936.27
Previous Pay \$1000
FICA Match \$ 76.50
Total \$1076.50
Amount Saved \$ 140.23
Percent Saved 13.0%
The fee charged by Cambi to you is for the advice on the payroll amount (\$360) and on the Direct Deposit amount, and for the Account handling, so that the Dollar amount of the Direct Deposit is made as a Silver Dollar deposit. Cambi’s charge is not pay for the employee, but just a fee for services rendered to you.
So, how much profit value can an employer add to the bottom line by using the Payroll Services? Let’s assume a \$10 Million sales business in MA, with \$1 Million gross profit and \$1 Million payroll averaging as in the example above.
The profit becomes around \$600,000 after tax. The 13% you saved comes to about \$130,000, or a profit bump over 21½%. How much growth can your business achieve with this much more in bottom line profit? How many more jobs can you create? | 899 | 3,653 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.640625 | 3 | CC-MAIN-2020-10 | latest | en | 0.962888 |
https://the-algorithms.com/algorithm/ant-colony-optimization-algorithms | 1,723,098,874,000,000,000 | text/html | crawl-data/CC-MAIN-2024-33/segments/1722640723918.41/warc/CC-MAIN-20240808062406-20240808092406-00871.warc.gz | 435,652,322 | 21,324 | #### Ant Colony Optimization Algorithms
I
```"""
Use an ant colony optimization algorithm to solve the travelling salesman problem (TSP)
"Given a list of cities and the distances between each pair of cities, what is the
shortest possible route that visits each city exactly once and returns to the origin
city?"
https://en.wikipedia.org/wiki/Ant_colony_optimization_algorithms
https://en.wikipedia.org/wiki/Travelling_salesman_problem
Author: Clark
"""
import copy
import random
cities = {
0: [0, 0],
1: [0, 5],
2: [3, 8],
3: [8, 10],
4: [12, 8],
5: [12, 4],
6: [8, 0],
7: [6, 2],
}
def main(
cities: dict[int, list[int]],
ants_num: int,
iterations_num: int,
pheromone_evaporation: float,
alpha: float,
beta: float,
q: float, # Pheromone system parameters Q, which is a constant
) -> tuple[list[int], float]:
"""
Ant colony algorithm main function
>>> main(cities=cities, ants_num=10, iterations_num=20,
... pheromone_evaporation=0.7, alpha=1.0, beta=5.0, q=10)
([0, 1, 2, 3, 4, 5, 6, 7, 0], 37.909778143828696)
>>> main(cities={0: [0, 0], 1: [2, 2]}, ants_num=5, iterations_num=5,
... pheromone_evaporation=0.7, alpha=1.0, beta=5.0, q=10)
([0, 1, 0], 5.656854249492381)
>>> main(cities={0: [0, 0], 1: [2, 2], 4: [4, 4]}, ants_num=5, iterations_num=5,
... pheromone_evaporation=0.7, alpha=1.0, beta=5.0, q=10)
Traceback (most recent call last):
...
IndexError: list index out of range
>>> main(cities={}, ants_num=5, iterations_num=5,
... pheromone_evaporation=0.7, alpha=1.0, beta=5.0, q=10)
Traceback (most recent call last):
...
StopIteration
>>> main(cities={0: [0, 0], 1: [2, 2]}, ants_num=0, iterations_num=5,
... pheromone_evaporation=0.7, alpha=1.0, beta=5.0, q=10)
([], inf)
>>> main(cities={0: [0, 0], 1: [2, 2]}, ants_num=5, iterations_num=0,
... pheromone_evaporation=0.7, alpha=1.0, beta=5.0, q=10)
([], inf)
>>> main(cities={0: [0, 0], 1: [2, 2]}, ants_num=5, iterations_num=5,
... pheromone_evaporation=1, alpha=1.0, beta=5.0, q=10)
([0, 1, 0], 5.656854249492381)
>>> main(cities={0: [0, 0], 1: [2, 2]}, ants_num=5, iterations_num=5,
... pheromone_evaporation=0, alpha=1.0, beta=5.0, q=10)
([0, 1, 0], 5.656854249492381)
"""
# Initialize the pheromone matrix
cities_num = len(cities)
pheromone = [[1.0] * cities_num] * cities_num
best_path: list[int] = []
best_distance = float("inf")
for _ in range(iterations_num):
ants_route = []
for _ in range(ants_num):
unvisited_cities = copy.deepcopy(cities)
current_city = {next(iter(cities.keys())): next(iter(cities.values()))}
del unvisited_cities[next(iter(current_city.keys()))]
ant_route = [next(iter(current_city.keys()))]
while unvisited_cities:
current_city, unvisited_cities = city_select(
pheromone, current_city, unvisited_cities, alpha, beta
)
ant_route.append(next(iter(current_city.keys())))
ant_route.append(0)
ants_route.append(ant_route)
pheromone, best_path, best_distance = pheromone_update(
pheromone,
cities,
pheromone_evaporation,
ants_route,
q,
best_path,
best_distance,
)
return best_path, best_distance
def distance(city1: list[int], city2: list[int]) -> float:
"""
Calculate the distance between two coordinate points
>>> distance([0, 0], [3, 4] )
5.0
>>> distance([0, 0], [-3, 4] )
5.0
>>> distance([0, 0], [-3, -4] )
5.0
"""
return (((city1[0] - city2[0]) ** 2) + ((city1[1] - city2[1]) ** 2)) ** 0.5
def pheromone_update(
pheromone: list[list[float]],
cities: dict[int, list[int]],
pheromone_evaporation: float,
ants_route: list[list[int]],
q: float, # Pheromone system parameters Q, which is a constant
best_path: list[int],
best_distance: float,
) -> tuple[list[list[float]], list[int], float]:
"""
Update pheromones on the route and update the best route
>>>
>>> pheromone_update(pheromone=[[1.0, 1.0], [1.0, 1.0]],
... cities={0: [0,0], 1: [2,2]}, pheromone_evaporation=0.7,
... ants_route=[[0, 1, 0]], q=10, best_path=[],
... best_distance=float("inf"))
([[0.7, 4.235533905932737], [4.235533905932737, 0.7]], [0, 1, 0], 5.656854249492381)
>>> pheromone_update(pheromone=[],
... cities={0: [0,0], 1: [2,2]}, pheromone_evaporation=0.7,
... ants_route=[[0, 1, 0]], q=10, best_path=[],
... best_distance=float("inf"))
Traceback (most recent call last):
...
IndexError: list index out of range
>>> pheromone_update(pheromone=[[1.0, 1.0], [1.0, 1.0]],
... cities={}, pheromone_evaporation=0.7,
... ants_route=[[0, 1, 0]], q=10, best_path=[],
... best_distance=float("inf"))
Traceback (most recent call last):
...
KeyError: 0
"""
for a in range(len(cities)): # Update the volatilization of pheromone on all routes
for b in range(len(cities)):
pheromone[a][b] *= pheromone_evaporation
for ant_route in ants_route:
total_distance = 0.0
for i in range(len(ant_route) - 1): # Calculate total distance
total_distance += distance(cities[ant_route[i]], cities[ant_route[i + 1]])
delta_pheromone = q / total_distance
for i in range(len(ant_route) - 1): # Update pheromones
pheromone[ant_route[i]][ant_route[i + 1]] += delta_pheromone
pheromone[ant_route[i + 1]][ant_route[i]] = pheromone[ant_route[i]][
ant_route[i + 1]
]
if total_distance < best_distance:
best_path = ant_route
best_distance = total_distance
return pheromone, best_path, best_distance
def city_select(
pheromone: list[list[float]],
current_city: dict[int, list[int]],
unvisited_cities: dict[int, list[int]],
alpha: float,
beta: float,
) -> tuple[dict[int, list[int]], dict[int, list[int]]]:
"""
Choose the next city for ants
>>> city_select(pheromone=[[1.0, 1.0], [1.0, 1.0]], current_city={0: [0, 0]},
... unvisited_cities={1: [2, 2]}, alpha=1.0, beta=5.0)
({1: [2, 2]}, {})
>>> city_select(pheromone=[], current_city={0: [0,0]},
... unvisited_cities={1: [2, 2]}, alpha=1.0, beta=5.0)
Traceback (most recent call last):
...
IndexError: list index out of range
>>> city_select(pheromone=[[1.0, 1.0], [1.0, 1.0]], current_city={},
... unvisited_cities={1: [2, 2]}, alpha=1.0, beta=5.0)
Traceback (most recent call last):
...
StopIteration
>>> city_select(pheromone=[[1.0, 1.0], [1.0, 1.0]], current_city={0: [0, 0]},
... unvisited_cities={}, alpha=1.0, beta=5.0)
Traceback (most recent call last):
...
IndexError: list index out of range
"""
probabilities = []
for city in unvisited_cities:
city_distance = distance(
unvisited_cities[city], next(iter(current_city.values()))
)
probability = (pheromone[city][next(iter(current_city.keys()))] ** alpha) * (
(1 / city_distance) ** beta
)
probabilities.append(probability)
chosen_city_i = random.choices(
list(unvisited_cities.keys()), weights=probabilities
)[0]
chosen_city = {chosen_city_i: unvisited_cities[chosen_city_i]}
del unvisited_cities[next(iter(chosen_city.keys()))]
return chosen_city, unvisited_cities
if __name__ == "__main__":
best_path, best_distance = main(
cities=cities,
ants_num=10,
iterations_num=20,
pheromone_evaporation=0.7,
alpha=1.0,
beta=5.0,
q=10,
)
print(f"{best_path = }")
print(f"{best_distance = }")
``` | 2,361 | 7,088 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.59375 | 4 | CC-MAIN-2024-33 | latest | en | 0.493279 |
http://archive.advancedfootballanalytics.com/2013_05_01_archive.html | 1,490,242,888,000,000,000 | text/html | crawl-data/CC-MAIN-2017-13/segments/1490218186774.43/warc/CC-MAIN-20170322212946-00489-ip-10-233-31-227.ec2.internal.warc.gz | 21,461,999 | 17,870 | ## Exploring the Causes of a Sack Pt. 1
A guest post by David Giller. Born and raised in Swampscott, MA, David attended Vanderbilt University where he was the starting longsnapper for the Commodores. He graduated summa cum laude with a degree in economics/corporate finance. David currently works as a business analyst for a Bain Capital Ventures portfolio company.
I would first like to thank Brian for his suggestion to post my study here in an effort to spark some interesting conversation and obtain some valuable takeaways. My post contains the results of a recent study I put together which focuses on the causes of sacks in NFL games. Although it is fairly detailed, I believe there are still areas of further development, some of which have been explored in an appendix to this initial study and will be coming in the second installment of this post.
As a disclaimer, the number of sacks were provided from an official source; however, the timing of the sacks, count of offensive blockers/defensive rushers was determined from my individual film study.The full piece is attached in a link, however, I have included some highlights below.
## Point / Counterpoint on Rodgers' Extension
Today we're going to try a new format here at ANS--a debate between me and myself on the market value of Aaron Rodgers' recent contract extension. Rodgers recently signed a deal adding 5 years to his current contract. This will pay him roughly \$21M per season over the next 3 years. See if you can figure out which Brian has the right idea and why they get different results.
Brian 1: Rodgers' new deal is a fantastic bargain. He's one of the truly elite QBs in the league today, and guys like that don't grow on trees. But more scientifically, just look at this super scatterplot I made of all veteran/free-agent QBs. The chart plots Expected Points Added (EPA) per Game versus adjusted salary cap hit. Both measures are averaged over the veteran periods of each player's contracts. I added an Ordinary Least Squares (OLS) best-fit regression line to illustrate my point (r=0.46, p=0.002).
Rodgers' production, measured by his career average Expected Points Added (EPA) per game is far higher than the trend line says would be worth his \$21M/yr cost. The vertical distance between his new contract numbers, \$21M/yr and about 11 EPA/G illustrates the surplus performance the Packers will likely get from Rodgers.
(This plot includes for all free-agent or veteran extensions since 2006. Cap figures are averaged for each player's career and, to account for cap inflation, are adjusted for overall league cap ceiling by season. Only seasons with 7 or more starts were included.) | 569 | 2,679 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.828125 | 3 | CC-MAIN-2017-13 | longest | en | 0.966607 |
https://entranceindia.com/tag/keam-engineering-physics-syllabus/ | 1,680,107,844,000,000,000 | text/html | crawl-data/CC-MAIN-2023-14/segments/1679296949009.11/warc/CC-MAIN-20230329151629-20230329181629-00227.warc.gz | 270,532,080 | 28,163 | ## KEAM – Engineering 2014 Mathematics Syllabus
UNIT I: ALGEBRA
SETS, RELATIONS AND FUNCTIONS
Sets and their Representations: Finite and Infinite sets; Empty set; Equal sets; Subsets; Power set; Universal set; Venn Diagrams; Complement of a set; Operations on Sets (Union, Intersection and Difference of Set); Applications of sets: Ordered Pairs, Cartesian Product of Two sets; Relations: Domain, Co-domain and Range: Functions: into, on to, one – one in to, one-one on to Functions; Constant Function; Identity Function; composition of Functions; Invertible Functions; Binary Operations.
Complex Numbers
Complex Numbers in the form a + i b ; Real and Imaginary Parts of a complex Number; Complex Conjugate, Argand Diagram, Representation of Complex Number as a point in the plane; Modulus and Argument of a Complex Number; Algebra of Complex Numbers; Triangle Inequality; Polar Representation of a Complex Number.
Solution of a Quadratic Equation in the Complex Number System by (i) Factorization (ii) Using Formula; Relation between Roots and Coefficients; Nature of Roots; Formation of Quadratic Equations with given Roots; Equations Reducible to Quadratic Forms.
Sequences and Series
Sequence and Examples of Finite and Infinite Sequences; Arithmetic Progression (A..P): First Term, Common Difference, nth Term and sum of n terms of an A.P.; Arithmetic Mean (A.M); Insertion of Arithmetic Means between any Two given Numbers; Geometric Progression (G.P): first Term, Common Ratio and nth term, Sum to n Terms, Geometric Mean (G.M); Insertion of Geometric Means between any two given Numbers.
Permutations, Combinations, Binomial Theorem and Mathematical Induction
Fundamental Principle of Counting; The Factorial Notation; Permutation as an Arrangement; Meaning of P(n, r); Combination: Meaning of C(n,r); Applications of Permutations and Combinations. Statement of Binomial Theorem; Proof of Binomial Theorem for positive integral Exponent using Principle of Mathematical Induction and also by combinatorial Method; General and Middle Terms in Binomial Expansions; Properties of Binomial Coefficients; Binomial Theorem for any Index (without proof); Application of Binomial Theorem. The Principle of Mathematical Induction, simple Applications.
Matrices and Determinants
Concept of a Matrix; Types of Matrices; Equality of Matrices (only real entries may be considered): Operations of Addition, Scalar Multiplication and Multiplication of Matrices; Statement of Important Results on operations of Matrices and their Verifications by Numerical Problem only; Determinant of a Square Matrix; Minors and Cofactors; singular and non-singular Matrices; Applications of Determinants in (i) finding the Area of a Triangle (ii) solving a system of Linear Equations (Cramer’s Rule); Transpose, Adjoint and Inverse of a Matrix; Consistency and Inconsistency of a system of Linear Equations; Solving System of Linear Equations in Two or Three variables using Inverse of a Matrix (only up to 3X3 Determinants and Matrices should be considered).
Linear Inequations
Solutions of Linear Inequation in one variable and its Graphical Representation; solution of system of Linear Inequations in one variable; Graphical solutions of Linear inequations in two variables; solutions of system of Linear Inequations in two variables.
Mathematical Logic and Boolean Algebra
Statements; use of Venn Diagram in Logic; Negation Operation; Basic Logical Connectives and Compound Statements including their Negations.
UNIT II : TRIGONOMETRY
Trigonometric functions and Inverse Trigonometric functions
Degree measures and Radian measure of positive and negative angles; relation between degree measure and radian measure, definition of trigonometric functions with the help of a unit circle, periodic functions, concept of periodicity of trigonom etric functions, value of trigonometric functions of x for
x = 0, π/6, π / 4, π /3, π / 2, π , π, 3π /2 , 2 π ; trigonometric functions of sum and difference of numbers.
UNIT III: GEOMETRY
Cartesian System of Rectangular Co ordinates
Cartesian system of co ordinates in a plane, Distance formula, Centroid and incentre, Area of a triangle, condition for the collinearity of three points in a plane, Slope of line, parallel and perpendicular lines, intercepts of a line on the co ordinate axes, Locus and its equation.
Lines and Family of lines
Various forms of equations of a line parallel to axes, slope-intercept form, The Slope point form, Intercept form, Normal form, General form, Intersection of lines. Equation of bisectors of angle between two lines, Angles between two lines, condition for concurrency of three lines, Distance of a point from a line,
Equations of family of lines through the intersection of two lines.
Circles and Family of circles
Standard form of the equation of a circle General form of the equation of a circle, its radius and center, Equation of the circle in the parametric form.
Conic sections
Sections of a cone. Equations of conic sections [Parabola, Ellipse and Hyperbola] in standard form.
Vectors
Vectors and scalars, Magnitude and Direction of a vector, Types of vectors (Equal vectors, unit vector, Zero vector). Position vector of a point, Localized and free vectors, parallel and collinear vectors, Negative of a vector, components of a vector, Addition of vectors, multiplication of a vector by a scalar, position vector of point dividing a line segment in a given ratio, Application of vectors in geometry. Scalar product of two vectors, projection of a vector on a line, vector product of two vectors.
Three Dimensional Geometry
Coordinate axes and coordinate planes in three dimensional space, coordinate of a point in space, distance between two points, section formula, direction cosines, and direction ratios of a line joining two points, projection of the join of two points on a given line, Angle between two lines whose direction ratios are given, Cartesian and vector equation of a line through (i) a point and parallel to a given vector (ii) through two points, Collinearity of three points, coplanar and skew lines, Shortest distance between two lines, Condition for the intersection of two lines, Carterian and vector equation of a plane (i) When the normal vector and the distance of the plane from the origin is given (ii) passing though a point and perpendicular to a given vector (iii) Passing through a point and parallel to two given lines through the intersection of two other planes (iv) containing two lines (v) passing through three points, Angle between (i) two lines (ii) two planes (iii) a line and a plane, Condition of coplanarity of two lines in vector and Cartesian form, length of perpendicular of a point from a plane by both vector and Cartesian methods.
UNIT IV: STATISTICS
Statistics and probability
Mean deviation for ungrouped data, variance for grouped an ungrouped data, standard deviation. Random experiments and sample space, Events as subset of a sample space, occurrence of an event, sure and impossible events, Exhaustive events, Algebra of events, Meaning of equality likely outcomes, mutually exclusive events. Probability of an event; Theorems on probability; Addition rule, Multiplication rule, Independent experiments and events. Finding P (A or B), P (A and B), random variables, Probability distribution of a random variable.
UNIT V : CALCULUS
Functions, Limits and continuity
Concept of a real function; its domain and range; Modulus Function, Greatest integer function: Signum functions; Trigonometric functions and inverse trigonometric functions and their graphs; composite functions, Inverse of a function. Limits at Infinity and infinity limits; continuity of a function at a point, over an open/ closed interval; Sum, Product and quotient of continuous functions; Continuity of special functions- Polynomial, Trigonometric, exponential, Logarithmic and Inverse trigonometric functions.
Differentiation
Derivative of a function; its geometrical and physical significance; Relationship between continuity and differentiability; Derivatives of polynomial, basic trigonometric , exponential, logarithmic and inverse trigonometric functions from first principles; derivatives of sum, difference, product and quotient of functions ; derivatives of polynomial, trigonometric, exponential, logarithmic, inverse trigonometric and implicit functions; Logarithmic differentiation; derivatives of functions expressed in parametric form; chain rule and differentiation by substitution; Derivatives of Second order.
Application of Derivatives
Rate of change of quantities; Tangents and Normals; increasing and decreasing functions and sign of the derivatives; maxima and minima; Greatest and least values; Rolle’s theorem and Mean value theorem; Approximation by differentials.
Indefinite Integrals
Integration as inverse of differentiation; properties of integrals; Integrals involving algebraic, trigonometric, exponential and logarithmic functions; Integration by substitution; Integration by parts;
Definite Integrals
Definite integral as limit of a sum; Fundamental theorems of integral calculus without proof); Evaluation of definite integrals by substitution and by using the following properties. Application of definite integrals in finding areas bounded by a curve, circle, parabola and ellipse in standard form between two ordinates and x-axis; Area between two curves, line and circle; line and parabola: line and ellipse.
Differential Equations
Definition; order and degree; general and particular solutions of a differential equation; formation of differential equations whose general solution is given; solution of differential equations by method of Separation of variables; Homogeneous differential equations of first order and their solutions; Solution of
linear differential equations of the type P(x) y Q(x).
Back to Syllabus
## KEAM – Engineering 2014 Chemistry Syllabus
UNIT 1: BASIC CONCEPTS AND ATOMIC STRUCTURE
Laws of chemical combination: Law of conservation of mass. Law of definite proportion. Law of multiple proportions. Gay-Lussac’s law of combining volumes. Dalton’s atomic theory. Mole concept. Atomic, molecular and molar masses. Chemical equations. Balancing and calculation based on chemical equations.
Atomic structure: Fundamental particles. Rutherford model of atom. Nature of electromagnetic radiation. Emission spectrum of hydrogen atom. Bohr model of hydrogen atom. Drawbacks of Bohr model. Dual nature of matter and radiation. de Broglie relation. Uncertainty principle. Wave function (mention only). Atomic orbitals and their shapes (s, p and d orbitals only). Quantum numbers. Electronic configurations of elements. Pauli’s exclusion principle. Hund’s rule. Aufbau principle.
UNIT 2: BONDING AND MOLECULAR STRUCTURE
Kossel and Lewis approach of bonding. Ionic bond, covalent character of ionic bond, Lattice energy. Born-Haber cycle. Covalent bond. Lewis structure of covalent bond. Concept of orbital overlap. VSEPR theory and geometry of molecules. Polarity of covalent bond. Valence bond theory and hybridization (sp,sp, sp2 ,sp3,dsp2,d2sp3 and sp3d). Resonance. Molecular orbital method. Bond order. Molecular orbital diagrams of homodiatomic molecules. Bond strength and magnetic behaviour. Hydrogen bond. Coordinate bond. Metallic bond.
UNIT 3: STATES OF MATTER
Gaseous state: Boyle’s law. Charles’ law. Avogadro’s hypothesis. Graham’s law of diffusion. Absolute scale of temperature. Ideal gas equation. Gas constant and its values. Dalton’s law of partial pressure. Aqueous tension. Kinetic theory of gases. Deviation of real gases from ideal behaviour. Inter molecular interaction, van der Waals equation. Liquefaction of gases. Critical temperature.
Liquid state: Properties of liquids. Vapour pressure and boiling point. Surface tension. Viscosity.
Solid state: Types of solids (ionic, covalent and molecular). Space lattice and unit cells. Cubic crystal systems. Close packing. Different voids (tetrahedral and octahedral only). Density calculations. Point defects (Frenkel and Schottky). Electrical properties of solids. Conductors, semiconductors and insulators. Piezoelectric and pyroelectric crystals. Magnetic properties of solids. Diamagnetic, paramagnetic, ferromagnetic, antiferromagnetic and ferrimagnetic substances.
UNIT 4: PERIODIC PROPERTIES OF ELEMENTS AND HYDROGEN
Classification of elements: Mendeleev’s periodic table. Atomic number and modern periodic law. Long form of periodic table. Electronic configurations of elements and their position in the periodic table. Classification into s-, p-, d- and f-block elements. Periodic properties: Ionization energy, electron affinity, atomic radii, valence and electro negativity.
Hydrogen: Position in the periodic table, occurrence, isolation, preparation (including commercial), properties, reactions and uses. Isotopes of hydrogen.
Hydrides: Molecular, saline and interstitial hydrides.
Water: Structure of water molecule and its aggregates. Physical and chemical properties of water. Hard and soft water. Removal of hardness. Preparation and uses of heavy water: Liquid hydrogen as fuel.
UNIT 5: S-BLOCK ELEMENTS AND PRINCIPLES OF METALLURGY
Alkali metals: Occurrence, electronic configuration, trends in atomic and physical properties (ionization energy, atomic radii and ionic radii), electrode potential, and reactions with oxygen, hydrogen, halogens and liquid ammonia. Oxides, hydroxides and halides.
Alkaline earth metals: Occurrence, electronic configuration, trends in atomic and physical properties, electrode potential, and reactions with oxygen, hydrogen and halogens. Oxides, hydroxides, halides and sulphides. Anomalous properties of lithium and beryllium.
Compounds of s-block elements: Large scale preparation of NaOH and Na2CO3, their properties and uses. Preparation and properties of CaO, Ca(OH)2, Plaster of Paris and MgSO4. Industrial uses of lime, limestone and cement.
Principles of metallurgy: Occurrence of metals. Concentration of ores. General principles of extraction of metals from ore. Thermodynamic and electro chemical principles of metallurgy. Refining of metals. Extraction of zinc, aluminium, iron and copper.
UNIT 6: P-BLOCK ELEMENTS
General characteristics of p-block elements: atomic and physical properties. Oxidation states. Trends in chemical reactivity of Groups 13, 14, 15, 16 and 17 elements.
Boron: Occurrence, isolation, physical and chemical properties. Borax and boric acid. Boron hydrides. Structure of diborane. Uses of boron and its compounds. Carbon: Allotropes, properties, Oxides of Carbon. Nitrogen: Terrestrial abundance and distribution, isolation, properties and chemical reactivity.
Ammonia: Haber process of manufacture, properties and uses. Nitric acid: Ostwald process of manufacture and important uses. Oxides of nitrogen: Preparation and structures (skeletal only).
Oxygen: Terrestrial abundance, isolation, properties and chemical reactivity.
Oxides: Acidic, basic and amphoteric oxides. Preparation, structure, properties and uses of ozone and hydrogen peroxide.
Silica: Different forms and uses. Structures of silicates. Silicones, Zeolites, Uses of Silicon Tetra Chloride.
Phosphorus: Production, allotropes and phosphine. Preparation and structures of PCl3, PCl5, oxyacids of phosphorus. Comparison of halides and hydrides of Group 15 elements. Sulphur: Production, allotropes, oxides and halides, Oxoacids of Sulphur (structure only).
Sulphuric acid: Manufacture, properties and uses. Comparison of oxides, halides and hydrides of Group 17 elements, Oxoacids of halogens (structure only), hydrides and oxides of chlorine. Interhalogen compounds.
Group 18 elements: Occurrence, isolation, atomic and physical properties, uses.
Compounds of xenon: Preparation of fluorides and oxides, and their reactions with water.
UNIT 7: D-BLOCK AND F-BLOCK ELEMENTS
d-Block elements: Electronic configuration and general characteristics. Metallic properties, ionization energy, electrode potential, oxidation states, ionic radii, catalytic properties, coloured ions, complex formation, magnetic properties, interstitial compounds and alloys. Preparation and properties of KMnO4, K2Cr2O7 .
f-Block elements: Lanthanides: Occurrence, electronic configuration and oxidation states. Lanthanide contraction. Uses.
Actinides: Occurrence, electronic configuration and comparison with lanthanides.
UNIT 8: THERMODYNAMICS
System and surrounding: Types of systems. Types of processes. Intensive and extensive properties. State functions and path functions. Reversible and irreversible processes.
First law of thermodynamics: Internal energy and enthalpy. Application of first law of thermodynamics. Enthalpy changes during phase transition. Enthalpy changes in chemical reactions. Standard enthalpy of formation. Hess’s law of constant heat summation and numerical problems. Heat capacity and specific heat.
Second law of thermodynamics: Entropy and Gibbs free energy. Free energy change and chemical equilibrium. Criteria for spontaneity.
UNIT 9: CHEMICAL EQUILIBRIUM
Physical and chemical equilibria: Dynamic nature of equilibrium. Equilibria involving physical changes (solid-liquid, liquid-gas, dissolution of solids in liquids and dissolution of gases in liquids). General characteristics of equilibria involving physical processes.
Equilibria involving chemical systems: Law of chemical equilibrium. Magnitude of equilibrium constant. Numerical problems. Effect of changing conditions of systems at equilibrium (changes of concentration, temperature and pressure). Effect of catalyst. The Le Chatelier principle and its applications. Relationship between Kp and Kc. Ionic equilibrium. Ionization of weak and strong electrolytes. Concepts of acids and bases: Those of Arrhenius, Bronsted-Lowry and Lewis. Acid-base equilibrium. Ionization of water. pH scale. Salt hydrolysis. Solubility product. Common ion effect. Buffer action and buffer solutions.
UNIT 10: SOLUTIONS
Types of solutions: Different concentration terms (normality, molarity, molality, mole fraction and mass percentage). Solubility of gases and solids. Vapour pressure of solutions and Raoult’s law. Deviation from Raoult’s law. Colligative properties: Lowering of vapour pressure, elevation in boiling point, depression in freezing point and osmotic pressure. Ideal and non-ideal solutions. Determination of molecular mass. Abnormal molecular mass. The van’t Hoff factor and related numerical problems.
UNIT 11: REDOX REACTIONS AND ELECTROCHEMISTRY
Oxidation and reduction: Electron transfer concept. Oxidation number.
Balancing equations of redox reactions: Oxidation number method and ion electron method (half reaction method).
Faraday’s laws of electrolysis: Quantitative aspects. Electrolytic conduction. Conductance. Molar conductance. Kohlrausch’s law and its applications. Electrode potential and electromotive force (e.m.f.). Reference electrode (SHE only). Electrolytic and Galvanic cells. Daniel cell. The Nernst equation. Free energy and e.m.f. Primary and secondary cells. Fuel cell (H2-O2 only).
Corrosion and its prevention: Electrochemical theory of rusting of iron. Methods of prevention of corrosion. Galvanization and cathodic protection.
UNIT 12: CHEMICAL KINETICS
Rate of reaction. Average and instantaneous rates. Rate expressions. Rate constant. Rate law. Order and molecularity. Integrated rate law expressions for zero and first order reactions and their derivations. Units of rate constant. Half life period. Temperature dependence of rate constant. Arrhenius equation. Activation energy, Collision Theory (Elementary theory) and related numerical problems. Elementary and complex reactions with examples.
UNIT 13: SURFACE CHEMISTRY
Adsorption: Physical and chemical adsorption. Factors affecting adsorption. Effect of pressure. Freundlisch adsorption isotherm. Catalysis. Enzymes.
Zeolites Colloids: Colloids and suspensions. Dispersion medium and dispersed phase.
Types of colloids: Lyophobic, lyophilic, multimolecular, macromolecular and associated colloids. Preparation, properties and protection of colloids. Gold number.
Hardy Schulze rule. Emulsions.
UNIT 14: COORDINATION COMPOUNDS AND ORGANOMETALLICS
Ligand. Coordination number. IUPAC nomenclature of coordination compounds mononuclear, Isomerism in coordination compounds. Geometrical, optical and structural isomerism. Bonding in coordination compounds. Werner’s coordination theory. Valence bond approach. Hybridization and geometry. Magnetic properties of octahedral, tetrahedral and square planar complexes. Introduction to crystal field theory. Splitting of d orbitals in octahedral and tetrahedral fields (qualitative only). Importance of coordination compounds in qualitative analysis and biological systems such as chlorophyll, hemoglobin and vitamin B12 (structures not included).
UNIT 15: BASIC PRINCIPLES, PURIFICATION AND CHARACTERIZATION OF ORGANIC COMPOUNDS
Distinction between organic and inorganic compounds. Tetra valence of carbon. Catenation. Hybridization (sp, sp2 and sp3). Shapes of simple molecules. General introduction to naming of organic compounds. Trivial names and IUPAC nomenclature. Illustrations with examples. Structural isomerism. Examples of functional groups containing oxygen, hydrogen, sulphur and halogens.
Purification of carbon compounds: Filtration, crystallization, sublimation, distillation, differential extraction and chromatography (column and paper only).
Qualitative analysis: Detection of carbon, hydrogen, nitrogen and halogens.
Quantitative analysis: Estimation of carbon, hydrogen, nitrogen, sulphur, phosphorus and halogens (principles only), and related numerical problems. Calculation of empirical and molecular formulae.
UNIT 16: HYDROCARBONS
Classification of hydrocarbons. Alkanes and cycloalkanes: Nomenclature and conformation of ethane. 3D structures and 2D projections (Sawhorse and Newman). Alkenes and alkynes: Nomenclature. Geometrical isomerism in alkenes. Stability of alkenes. General methods of preparation. Physical and chemical properties. Markownikoff’s rule. Peroxide effect. Acidic character of alkynes. Polymerization reactions of dienes.
Aromatic hydrocarbons: Nomenclature. Isomerism. Benzene and its homologues. Structure of Benzene. Resonance. Delocalisation in benzene. Concept of aromaticity (an elementary idea). Chemical reactions of benzene. Polynuclear hydrocarbons and their toxicity.
UNIT 17: ORGANIC REACTION MECHANISM
Electronic displacement in a covalent bond: Inductive, electromeric, resonance and hyperconjugation effects. Fission of a covalent bond. Free radicals, electrophiles, nucleophiles, carbocations and carbanions.
Common types of organic reactions: Substitution, addition, elimination and rearrangement reactions. Illustrations with examples. Mechanism of electrophilic addition reactions in alkenes. Concept of delocalisation of electrons. Mechanism of electrophilic substitution reactions. Directive influence of substituents and their effect on reactivity (in benzene ring only).
UNIT 18: STEREOCHEMISTRY
Stereoisomerism: Geometrical isomerism and optical isomerism. Specific rotation. Chirality and chiral objects. Chiral molecules. Configuration and Fischer projections. Asymmetric carbon. Elements of symmetry. Compounds containing one chiral center. Enantiomers. Racemic form. Racemization. Compounds containing two chiral centers. Diastereo isomers. Meso form. Resolution.
UNIT 19: ORGANIC COMPOUNDS WITH FUNCTIONAL GROUPS CONTAINING HALOGENS
Haloalkanes and haloarenes: Nomenclature and general methods of preparation. Physical properties. Nature of C-X bond in haloalkanes and haloarenes. Chemical properties and uses of chloromethane and chlorobenzene. Polyhalogen compounds: Preparation and properties of chloroform and iodoform. Uses of some commercially important compounds (chloroform, iodoform, DDT, BHC and freon).
UNIT 20: ORGANIC COMPOUNDS WITH FUNCTIONAL GROUPS CONTAINING OXYGEN
Alcohols: Nomenclature. Important methods of preparation (from aldehydes, ketones, alkyl halides and hydration of alkenes). Manufacture of ethanol from molasses. Physical and chemical properties. Reactions with alkali metals and acids. Formation of alkenes, ethers and esters. Reactions with PX3, PX5, SOCl2. Oxidation of alcohols. Dehydrogenation.
Phenols: Nomenclature. Preparation of phenol (from sodium benzenesulphonate, benzene diazoniumchloride and chlorobenzene). Physical and chemical properties of phenol. Acidity of phenol. Action of phenol with FeCl3. Bromination, sulphonation and nitration of phenol.
Ethers: Nomenclature. Methods of preparation (from alcohols and alkyl halides). Williamson’s synthesis. Physical and chemical properties. Formation of peroxides. Actions with HI, HF and H2SO4. Some commercially important compounds: Methanol, ethanol (fermentation).
Aldehydes and ketones: Nomenclature. Electronic structure of carbonyl group. Methods of preparation (from alcohols, acid chlorides, ozonolysis of alkenes and hydration of alkynes). Friedel-Crafts acylation for acetophenone. General properties (physical and chemical) of aldehydes and ketones. Formation of paraldehyde and metaldehyde. Addition of NaHSO3, NH3 and its derivatives, Grignard reagent, HCN and alcohols. Oxidation reactions with Tollen’s reagent and Fehling’s solution. Oxidation of ketones. Reduction with LiAlH4. Clemmensen reduction. Wolff- Kischner reduction. Aldol condensation. Cannizzaro reaction.
Carboxylic acid: Nomenclature. Electronic structure of –COOH. Methods of Preparation (from alcohols, aldehydes, ketones, alkyl benzenes and hydrolysis of cyanide). Physical properties. Effects of substituents on acid strength. Chemical reactions.
UNIT 21: ORGANIC COMPOUNDS WITH FUNCTIONAL GROUPS CONTAINING NITROGEN
Amines: Nomenclature. Primary, secondary and tertiary amines. Methods of preparation. Physical properties. Basic nature. Chemical reaction. Separation of primary, secondary and tertiary amines. Cyanides and isocyanides. Diazonium salts. Preparation and chemical reactions of benzene diazoniumchloride in synthetic organic chemistry.
UNIT 22: POLYMERS AND BIOMOLECULES
Polymers: Classification. Addition and condensation polymerization. Copolymerization. Natural rubber and vulcanization. Synthetic rubbers. Condensation polymers. Biopolymers. Biodegradable polymers. Some commercially important polymers: Polyethene, polystyrene, PVC, Teflon, PAN, BUNA-N, BUNA-S, neoprene, Terylene, glyptal, nylon-6, nylon-66 and Bakelite.
Biomolecules: Classification of carbohydrates. Structure and properties of glucose. Reducing and nonreducing
sugars: Properties of sucrose, maltose and lactose (structures not included).
Polysaccharides: Properties of starch and cellulose. Proteins: Amino acids. Zwitterions. Peptide bond. Polypeptides. Primary, secondary and tertiary structures of protein. Denaturation of proteins. Enzymes. Nucleic acids. Types of nucleic acids. DNA and RNA, and their chemical composition. Primary structure of DNA. Double helix.
Vitamins: Classification and functions in biosystems.
UNIT 23: ENVIRONMENTAL CHEMISTRY AND CHEMISTRY IN EVERY DAY LIFE
Soil, water and air pollutions. Ozone layer. Smog. Acid rain. Green house effect and global warming. Industrial air pollution. Importance of green chemistry.
Chemicals in medicine and health care. Drug-target interaction, Analgesics, tranquillizers, antiseptics, antacids, antihistamines, antibiotics, disinfectants, antifertility drugs, chemicals in food, preservatives, artificial sweetening agents, antioxidants and edible colours, cleansing agents, soaps and synthetic detergents, antimicrobials.
Back to Syllabus
## KEAM – Engineering 2014 Physics Syllabus
UNIT 1: INTRODUCTION AND MEASUREMENT
Physics – Scope and excitement; Physics in relation to science, society and technology – inventions, names of scientists and their fields, nobel prize winners and topics, current developments in physical sciences and related technology. Units for measurement – systems of units, S .I units, conversion from other
systems to S.I units. Fundamental and derived units. Measurement of length, mass and time, least count in measuring instruments (eg. vernier calipers, screw gauge etc), Dimensional analysis and applications, order of magnitude, accuracy and errors in measurement, random and instrumental errors, significant figures and rounding off principles.
UNIT 2 : DESCRIPTION OF MOTION IN ONE DIMENSION
Objects in motion in one dimension – Motion in a straight line, uniform motion – its graphical representation and formulae; speed and velocity – instantaneous velocity; ideas of relative velocity with expressions and graphical representations; Uniformly accelerated motion, position – time graph, velocity – time graph and formulae. Elementary ideas of calculus – differentiation and integration – applications to motion.
UNIT 3 : DESCRIPTION OF MOTION IN TWO AND THREE DIMENSIONS
Vectors and scalars, vectors in two and three dimensions, unit vector, addition and multiplication, resolution of vector in a plane, rectangular components , scalar and vector products. Motion in two dimensions – projectile motion, ideas of uniform circular motion, linear and angular velocity, relation
between centripetal acceleration and angular speed.
UNIT 4 : LAWS OF MOTION
Force and inertia, first law of motion, momentum, second law of motion, forces in nature, impulse, third law of motion, conservation of linear momentum, examples of variable mass situation, rocket propulsion, equilibrium of concurrent forces. Static and kinetic friction, laws of friction, rolling friction, lubrication. Inertial and non-inertial frames (elementary ideas); Dynamics of uniform circular motion – centripetal and centrifugal forces, examples :
banking of curves and centrifuge.
UNIT 5 : WORK, ENERGY AND POWER
Work done by a constant force and by a variable force, units of work – Energy – kinetic and potential forms, power, work-energy theorem. Elastic and inelastic collisions in one and two dimensions. Gravitational potential energy and its conversion to kinetic energy, spring constant, potential energy of a
spring, Different forms of energy, mass – energy equivalence (elementary ideas), conservation of energy, conservative and non-conservative forces.
UNIT 6: MOTION OF SYSTEM OF PARTICLES AND RIGID BODY ROTATION
Centre of mass of a two particle system, generalisation to N particles, momentum conservation and center of mass motion, applications to some familiar systems, center of mass of rigid body. Moment of a force, torque, angular momentum, physical meaning of angular momentum, conservation of angular momentum with some examples, eg. planetary motion. Equilibrium of rigid bodies, rigid body rotation and equation of rotational motion, comparison of linear and rotational motions, moment of inertia and its physical significance, radius of gyration, parallel and perpendicular axes theorems (statements only), moment of inertia of circular ring and disc, cylinder rolling without slipping.
UNIT 7 : GRAVITATION
Universal law of gravitation, gravitational constant (G) and acceleration due to gravity (g), weight and gravitation, variation of g with altitude, latitude, depth and rotation of earth. Mass of earth, gravitational potential energy near the surface of the earth, gravitational potential, escape velocity, orbital velocity of satellite, weightlessness, motion of geostationary and polar satellites, statement of Kepler’s laws of planetary motion, proof of second and third laws, relation between inertial and gravitational masses.
UNIT 8 : MECHANICS OF SOLIDS AND FLUIDS
Solids : Hooke’s law, stress – strain relationships, Youngs modulus, bulk modulus, shear modulus of rigidity, some practical examples. Fluids : Pressure due to fluid column, Pascal’s law and its applications (hydraulic lift and hydraulic brakes), effect of gravity on fluid pressure, Buoyancy, laws of floatation and Archimedes principles, atmospheric pressure. Surface energy and surface tension, angle of contact, examples of drops and babbles, capillary rise, detergents and surface tension, viscosity, sphere falling through a liquid column, Stokes law, streamline flow, Reynold’s number, equation of continuity, Bernoulli’s theorum and applications.
UNIT 9 : HEAT AND THERMODYNAMICS
Kinetic theory of gases, assumptions, concept of pressure, kinetic energy and temperature, mean-rms and most probable speed, degrees of freedom, statement of law of equipartition of energy, concept of mean free path and Avogadros’ number Thermal equilibrium and temperatures, zeroth law of thermodynamics, Heat-work and internal energy, Thermal expansion – thermometry. First law of thermodynamics and examples, specific heat, specific
heat of gases at constant volume and constant pressure, specific heat of solids, Dulong and Petit’s law. Thermodynamical variables and equation of state, phase diagrams, ideal gas equation, isothermal and adiabatic processes, reversible and irreversible processes, Carnot engines, refrigerators and heat pumps, efficiency and coefficient performance of heat engines , ideas of second law of thermodynamics with practical applications. Thermal radiation – Stefan-Boltzmann law, Newton’s law of cooling.
UNIT 10 : OSCILLATIONS
Periodic motion – period, frequency, displacement as a function of time and periodic functions; Simple harmonic motion (S.H.M) and its equation, uniform circular motion and simple harmonic motion, oscillations of a spring, restoring force and force constant, energy in simple harmonic motion, kinetic and
potential energies, simple pendulum – derivation of expression for the period; forced and damped oscillations and resonance (qualitative ideas only), coupled oscillations.
UNIT 11: WAVES
Longitudinal and transverse waves, wave motion, displacement relation for a progressive wave, speed of a traveling wave, principle of superposition of waves, reflection of waves, standing waves in strings and pipes, fundamental mode and harmonics, beats, Doppler effect of sound with applications.
UNIT 12: ELECTROSTATICS
Frictional electricity; Properties of electric charges – conservation, additivity and quantisation. Coulomb’s law – Forces between two point electric charges, Forces between multiple electric charges; Superposition principle and continuous charge distribution. Electric field and its physical significance, electric field due to a point charge, electric field lines; Electric dipole, electric field due to a dipole and behavior and dipole in a uniform electric field. Electric potential-physical meaning, potential difference, electric potential due to a point charge, a dipole and system of charges; Equipotential surfaces, Electrical
potential energy of a system of point charges, electric dipoles in an electrostatic field. Electric flux, statement of Gauss’ theorem-its application to find field due to an infinitely long straight wire, uniformly charged infinite plane sheet and uniformly charged thin spherical shell. Conductors and insulatorspresence of free charges and bound charges; Dielectrics and electric polarization, general concept of a capacitor and capacitance, combination of capacitors in series and in parallel, energy stored in a capacitor, capacitance of a parallel plate capacitor with and without dielectric medium between the
plates, Van de Graff generator.
UNIT 13: CURRENT ELECTRICITY
Electric current, flow of electric charges in a metallic conductor, drift velocity and mobility, their relation with electric current; Ohm’s law, electrical resistance, V-I characteristics, limitations of Ohm’s law, electrical resistivity and conductivity, classification of materials in terms of conductivity;
Superconductivity (elementary idea); Carbon resistors, colour code for carbon resistors; combination of resistances – series and parallel. Temperature dependence of resistance. Internal resistance of a cell, Potential difference and emf of a cell, combination of cells in series and in parallel. Kirchoff’s lawsillustration by simple applications, Wheatstone bridge and its applications, Meter bridge. Potentiometer – principle and applications to measure potential difference, comparison of emf of two cells and determination of internal resistance of a cell. Electric power, thermal effects of current and Joule’s law; Chemical effects of current, Faraday’s laws of electrolysis, Electro-chemical cells.
UNIT 14: MAGNETIC EFFECT OF CURRENT AND MAGNETISM
Concept of a magnetic field, Oersted’s experiment, Biot-Savart’s law, magnetic field due to an infinitely long current carrying straight wire and a circular loop, Ampere’s circuital law and its applications to straight and toroidal solenoids. Force on a moving charge in a uniform magnetic field, cyclotron. Force on current carrying conductor and torque on current loop in magnetic fields, force between two parallel current carrying conductors, definition of the ampere. Moving coil galvanometer and its conversion into ammeter and voltmeter. Current loop as a magnetic dipole, magnetic moment, torque on a magnetic dipole in a uniform magnetic field, Lines of force in magnetic field. Comparison of a bar magnet and solenoid. Earth’s magnetic field and magnetic elements, vibration magnetometer. Para, dia and ferromagnetic substances with examples. Electromagnets and permanent magnets.
UNIT 15: ELECTROMAGNETIC INDUCTION AND ALTERNATING CURRENT
Electromagnetic induction, Faraday’s laws, Induced e.m.f. and current, Lenz’s law, Eddy currents, self and mutual inductance. Alternating current, peak and rms value of alternating current/voltage, reactance and impedance, L.C. oscillations, LCR series circuit. (Phasor diagram), Resonant circuits and Q-factor; power in A.C. circuits, wattless current. AC generator and Transformer.
UNIT 16: ELECTROMAGNETIC WAVES
Properties of electromagnetic waves and Maxwell’s contributions (qualitative ideas), Hertz’s experiments, Electromagnetic spectrum (different regions and applications), propagation of electromagnetic waves in earth’s atmosphere.
UNIT 17: OPTICS
Reflection in mirrors, refraction of light, total internal reflection and its applications, spherical lenses, thin lens formula, lens maker’s formula; Magnification, Power of a lens, combination of thin lenses in contact; Refraction and dispersion of light due to a prism, Scattering of light, Blue colour of the sky and appearance of the sun at sunrise and sunset. Optical instruments, Compound microscope, astronomical telescope (refraction and reflection type) and their magnifying powers. Wave front and Huygen’s principle. Reflection and refraction of plane wave at a plane surface using wave fronts (qualitative idea); Interference-Young’s double slit experiment and expression for fringe width, coherent sources and sustained interference of light; Diffraction due to a single slit, width of central maximum, difference between interference and diffraction, resolving power of microscope and telescope; Polarisation, plane
polarised light, Brewster’s law, Use of polarised light and polaroids.
UNIT 18: DUAL NATURE OF MATTER AND RADIATIONS
Photoelectric effect, Einstein photoelectric equation – particle nature light, photo-cell, Matter waves – wave nature of particles. De Broglie relation, Davisson and Germer experiment.
UNIT 19: ATOMIC NUCLEUS
Alpha particle scattering experiment, size of the nucleus – composition of the nucleus – protons and neutrons. Nuclear instability – Radioactivity-Alpha, Beta and Gamma particle/rays and their properties, radio- active decay laws, Simple explanation of -decay, -decay and decay; mass-energy relation,
mass defect, Binding energy per nucleon and its variation with mass number. Nature of nuclear forces, nuclear reactions, nuclear fission, nuclear reactors and their uses; nuclear fusion, elementary ideas of energy production in stars.
UNIT 20: SOLIDS AND SEMICONDUCTOR DEVICES
Energy bands in solids (qualitative ideas only), difference between metals, insulators and semi-conductors using band theory; Intrinsic and extrinsic semi-conductors, p-n junction, Semi-conductor diodecharacteristics forward and reverse bias, diode as a rectifier, solar cell, photo-diode, zener diode as a
voltage regulator; Junction transistor, characteristics of a transistor; Transistor as an amplifier (common emitter configuration) and oscillator; Logic gates (OR, AND, NOT, NAND, NOR); Elementary ideas about integrated circuits.
UNIT 21: PRINCIPLES OF COMMUNICATIONS
Elementary idea of analog and digital communication; Need for modulation, amplitude, frequency and pulse modulation; Elementary ideas about demodulation, Data transmission and retrieval, Fax and Modem. (basic principles) Space communications – Ground wave, space wave and sky wave propagation, satellite communications.
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## KEAM – Engineering 2014 Syllabus
© Copyright Entrance India - Engineering and Medical Entrance Exams in India | Website Maintained by Firewall Firm - IT Monteur | 9,292 | 41,161 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.375 | 3 | CC-MAIN-2023-14 | latest | en | 0.776923 |
http://www.edupristine.com/discuss/viewtopic.php?f=12&t=510 | 1,521,851,469,000,000,000 | text/html | crawl-data/CC-MAIN-2018-13/segments/1521257649508.48/warc/CC-MAIN-20180323235620-20180324015620-00237.warc.gz | 365,926,077 | 6,660 | ## VaR
Finance Junkie
Posts: 99
Joined: Sat Apr 07, 2012 10:24 am
### VaR
Pls explain:
Consider the following single bond of \$10 million, a modified duration of 3.6 yrs and annualized yield of 2% and annual standard deviation of 3%; Using the duration method and assuming that the daily return on the bond position is independently identically normally distributed, calculate the 10 day holding period VaR of the position with a 99% confidence interval, assuming there are 252 days in a year.
a. 334,186
b. 699, 000
c. 139240
d. 144840
Pristine Solution: The correct answer is 334,186. VAR = \$10,000,000* 0.02*3.6* [sqrt10/ (sqrt252)]* 2.33 = \$334,186
Why they have multiplied with annualized yield instead of annual std deviation??
Tags:
content.pristine
Finance Junkie
Posts: 356
Joined: Wed Apr 11, 2012 11:26 am
### Re: VaR
Hi Suresh,
This question deals with bonds. The measure of risk in bonds is given by Duration * change in yields.. The standard deviation here is not relevant. In fact, it is wrong
Standard deviation should be Duration * Change in yields.
Remember, VaR for a Bond Portfolio is:
z * Market Value * Duration * Change in yields for the period..
Here, since annual yield change is given, then it needs to be unannualized.
Hope this helps
Finance Junkie
Posts: 99
Joined: Sat Apr 07, 2012 10:24 am
Thanks! | 378 | 1,346 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.1875 | 3 | CC-MAIN-2018-13 | latest | en | 0.865068 |
https://runyoncanyon-losangeles.com/blog/how-do-you-wrap-an-angle-in-matlab/ | 1,723,033,968,000,000,000 | text/html | crawl-data/CC-MAIN-2024-33/segments/1722640694449.36/warc/CC-MAIN-20240807111957-20240807141957-00840.warc.gz | 376,267,521 | 10,119 | ## How do you wrap an angle in Matlab?
Wrap Angles to Pi Radians lambda = [-2*pi -pi-0.1 -pi -2.8 3.1 pi pi+1 2*pi]; Wrap the angles to the range [- π , π ] radians. Specify a second list of angles, and wrap them. lambda2 = -10:0.1:10; lambda2Wrapped = wrapToPi(lambda2);
How do you shift phase angle in Matlab?
Apply Phase Shift to Matrix Unwrap the phase angles by first comparing the elements columnwise. Specify the dim argument as 1. Use the default jump threshold π by specifying the second argument as [] . To shift phase angles by rows instead of by columns, specify dim as 2 instead of 1.
What happens if the wrapping angle is greater than 180?
A Snub Pulley is therefore always necessary if the angle needs to be greater than 180 deg. A greater angle of wrap provided more grip area and therefore increases the tension in the belt.
### What is the difference between phase and angle in Matlab?
1 Answer. At first, ANGLE command is from MATLAB core, PHASE from system identification toolbox. ANGLE command always give result in range [-pi, pi]. phase(X) command will give answer greater than pi for the second value (difference between phases values should be less than pi).
What is wrapping angle?
Wrap Angle is a measurement in degrees of the diameter of a sensor roller that the tensioned material contacts. If the angle around the load cell roll changes, as with the changing diameter of an unwinding or rewinding roll of material, the effective force on the load cell changes.
How does Matlab calculate phase?
Create a signal that consists of two sinusoids of frequencies 15 Hz and 40 Hz. The first sinusoid has a phase of – π / 4 , and the second has a phase of π / 2 . Sample the signal at 100 Hz for one second. fs = 100; t = 0:1/fs:1-1/fs; x = cos(2*pi*15*t – pi/4) – sin(2*pi*40*t);
#### What angle is more than 180 but less than 360?
Reflex Angle
Reflex Angle – An angle greater than 180 degrees and less than 360 degrees.
Which angle is greater than 180 degrees but less than 360 degrees?
Reflex Angles
A reflex angle is the larger angle, it’s always more than 180° (half a circle) but less than 360° (A full circle).
How does Matlab calculate phase difference?
The present code is a Matlab function that provides a measurement of the phase difference between two signals. The measurement is based on Discrete Fourier Transform (DFT) and Maximum Likelihood (ML) estimation of the signals’ initial phases. The method is highly noise resistive.
## Is angle of lap same as angle of contact?
Angle of Lap : The angle of lap is defined as the angle subtended by the portion of the belt which is in contact at the pulley surface of the pulley.
How to shift phase angles in MATLAB unwrap?
Description. Q = unwrap (P) unwraps the radian phase angles in a vector P. Whenever the jump between consecutive angles is greater than or equal to π radians, unwrap shifts the angles by adding multiples of ±2 π until the jump is less than π. If P is a matrix, unwrap operates columnwise.
How to shift the phase curve in unwrap?
Plot the phase curve. Use unwrap to shift the phase angle using the default jump threshold radians. Plot the shifted phase curve. Both jumps are shifted since they are greater than the jump threshold radians. Now shift the phase angle using a jump threshold of 5 radians. Plot the shifted phase curve.
### How to unwrap phase angles in a vector p?
Q = unwrap (P) unwraps the radian phase angles in a vector P. Whenever the jump between consecutive angles is greater than or equal to π radians, unwrap shifts the angles by adding multiples of ±2 π until the jump is less than π. If P is a matrix, unwrap operates columnwise.
What is the difference between phase and angle in MATLAB?
ANGLE Phase angle. ANGLE (H) returns the phase angles, in radians, of a matrix with complex elements. Class support for input X: float: double, single | 909 | 3,891 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.546875 | 4 | CC-MAIN-2024-33 | latest | en | 0.830533 |
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# When I type in an equation, the graph is fine, but the table goes up by a strange scale with decimals. How do I get my table to go up by a scale of 1?
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1 Answer
### Polynomial equations
This calculator cannot do algebra so it does not handle polynomials. It does not have an EQN (equation) solving mode so you cannot enter an expression set = 0 and solve it for an unknown variable X.
It does have a table Mode where you can enter an expression y= F(X) and generate a table of values. If in the expression F(x) a numerical coefficient is a fraction just enter it as a fraction, possibly enclosed in parentheses. If it is a decimal number just enter it as a decimal number. Converting it to a fraction is not going to generate any more precision in the result. You lose precision when you go from fraction to decimal, but you do not make a number more precise by converting it from a decimal representation to a fraction.
Nov 16, 2013 | Casio fx-300ES Calculator
2 Answers
### I am trying to graph a least-squares line on my TI-84 Plus. I have the numbers correctly plugged into my Stat Edit tables, and I am able to access my linear equation, and the first time I entered my graph...
After you enter the data draw the SCATTER plot first.
When you perform the (linear) regression analysis LineReg, save the regression equation to a function, for example Y1
The syntax is
LineReg L1, L2, Y1 ENTER
From the Stat>CALC menu press LineReg
The command echoes on command line
Press [2nd][1] to enter the L1 list on command line
Press [,] to separate
Press [2nd] [2] to enter L2 on the command line
Press [,] the comma again
Press [VARS][>][Y-Vars][1~Function][1:Y1] [ENTER]
Press [GRAPH] to have the regression line equation drawn
on the same screen as the Scatter Plot you drew before.
Sep 13, 2011 | Texas Instruments TI-84 Plus Calculator
1 Answer
### I am using the TI Nspire Cas Graphics Calculator. I have calculated a linear regression that has been copied to f1. When I go to graph the function it is displayed in f1 as 100.x+2000. when the function...
Scientific mode is a display mode. It concerns the display of results. If the number are too small or too large they will be displayed in Scientific mode, because the screen does not have an unlimited number of digit positions.
Regarding the dots. A number followed by a radix mark (decimal mark) is considered to be a floating number, while the same number without a decimal mark is considered an integer (whole number).
The CAS is able to do symbolic and exact calculations, so that distinction between floting and integer numbers is important. Sometimes, to force a result to be displayed in its decimal representation all it takes is to append a decimal mark at the end of an integer.
Hope it answers your queation.
Apr 23, 2011 | Texas Instruments TI-Nspire Graphic...
3 Answers
### Fibonacci formula won't graph, but will show table
Fibonacci sequence is the series of numbers 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ...
At first type Fibonacci equation in Y editor with condition x>=0 and then graphing it. See captured images with enlarged form of graph
Apr 11, 2010 | Texas Instruments TI-84 Plus Silver...
1 Answer
### I am trying to see 2 different tables on my table screen from 2 different equations, but it when i go to the screen, instead of showing 2 different tables, it only shows one of them. I also noticed that...
Hi,
In the Y= equation editor you can enter two functions, and graph them both or calculate a table for both. However, the two functions will be evaluated at the same values of X.
Rather than trying to guess what is happening, I will show you how to get your two tables.
If you have used one the applications Inequalz or Transform it si probably still stuck in memory, and you must unload it from the RAM before recovering the usual calculator behavio(u)r
1. Press the [APPS] button, find Inequalz or Transform, point to it and click [ENTER].
2. You will be asked to Continue or Quit the application.
3. Press the number corresponding to Quit (usually 2).
4. Press [2nd][Y=] (STAT PLOT). Press [4:PlotsOff][ENTER]
5. Calculator replies Done.
Open the [Y=] equation editor. Type in your equations. Press [ENTER] at the end of each equation. Verify that the [=] signs after Y1, and Y2 remain highlighted when cursor is moved away from them.
Press [2nd][GRAPH] (Table) to see your 2 tables y1 and y2 calculated at the same X.
Hope it helps.
Thank you for rating the solution.
Dec 10, 2009 | Texas Instruments TI-84 Plus Calculator
1 Answer
### Graphing
i had the same problem .... just go to apps and go all the way down transfrm and click it then click the uninstall button !!!
Apr 01, 2009 | Texas Instruments TI-84 Plus Calculator
1 Answer
### No Y's in Table
The tables will not say Y specifically, they only say L1, L2, L3, etc. UNLESS you type in an equation in the "Y=" button menu. When you type an equation in on that page, and then hit "2nd" "GRAPH", you will get X and Y values for that equation.
Jan 19, 2009 | Texas Instruments TI-83 Plus Calculator
1 Answer
### No graph appears?
did you make sure the domain, range, and scale settings are correct?
Oct 29, 2008 | Texas Instruments TI-84 Plus Calculator
4 Answers
### How to graph linear equation in two variables
I am not quite sure what you are asking...
(1) if asking for graphing y= -2/3x+2, then press [diamond] [F1: Y=] type in .2/3x +2, see graph by [diamond][graph]
(2) if asking for graphing y= -2/3x+a, where a=3x, then first archive a=3x in Home screen by [home], type 3x [sto] a, then go back to y= menu, and type -2/3x+a in Y1
(3) if asking for graphing an equation that is dependent on 2 variables at the same time, use x-y-z axis (aka, change mode to graphing 3-D by [mode]graph [5:3D]) then enter desired equation in y= menu using Z function in terms of x and y aka z(x,y)= ...)
May 18, 2008 | Texas Instruments TI-89 Calculator
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Loading... | 1,851 | 7,266 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.75 | 3 | CC-MAIN-2017-47 | latest | en | 0.899418 |
https://community.carbide3d.com/t/how-to-draw-a-star-carbide-create/16022 | 1,569,175,907,000,000,000 | text/html | crawl-data/CC-MAIN-2019-39/segments/1568514575627.91/warc/CC-MAIN-20190922180536-20190922202536-00453.warc.gz | 434,276,719 | 4,887 | # How to draw a star --- Carbide Create
While Carbide Create has a number of stars available, some regular stars are not provided:
Fortunately, it’s pretty straight-forward to draw a regular star:
Draw a circle which is the size of the desired star:
Draw a rectangle which is taller than the radius of the circle and duplicate it:
Rotate each angle by either positive or negative:
360° ÷ number of sides/points (5) ÷ 4 == 18° (for a 5 pointed star)
and drag them into alignment at the top center node of the circle:
Draw in a rectangle which is left-right centered and top-aligned with the circle:
Select a rotated rectangle and the drawn square and Boolean subtract:
Draw a square and position its top at the center of the circle and shift click on the shape and the square and Boolean subtract:
Duplicate a triangle:
Drag it on top of the original:
Select one triangle and the circle:
Rotate them by
360° ÷ number of sides/points (5) == 72° (for a 5 pointed star)
Repeat for each remaining point/arm.
Select the triangles:
Boolean Union everything together:
4 Likes | 241 | 1,086 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.9375 | 3 | CC-MAIN-2019-39 | latest | en | 0.899332 |
https://www.jiskha.com/display.cgi?id=1290142366 | 1,532,284,494,000,000,000 | text/html | crawl-data/CC-MAIN-2018-30/segments/1531676593438.33/warc/CC-MAIN-20180722174538-20180722194538-00627.warc.gz | 900,557,780 | 3,670 | # research and evaluation
posted by sue
Give examples of both discrete and continuous data? What makes them different? Is the measurement of money continuous or discrete? Explain. Can discrete data be analyzed using the normal distribution? Why or why not?
1. SraJMcGin
Please try some of the following links:
http://search.yahoo.com/search?fr=mcafee&p=discrete+%26+continuous+data
Sra
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More Similar Questions | 523 | 2,262 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.46875 | 3 | CC-MAIN-2018-30 | latest | en | 0.906555 |
https://studylib.net/doc/5724893/arithmetic-sequences-finding-the-nth-term | 1,600,773,042,000,000,000 | text/html | crawl-data/CC-MAIN-2020-40/segments/1600400205950.35/warc/CC-MAIN-20200922094539-20200922124539-00457.warc.gz | 627,229,913 | 12,973 | # Arithmetic Sequences Finding the nth Term
### Arithmetic Sequences
• • A pattern where all numbers are related by the same common difference.
• The common difference must be an addition or subtraction constant.
• The common difference can be used to predict future numbers in the pattern.
Ex. 4, 7, 10, 13, ___, ___, ___ The common difference in this pattern is
+3
. Based on this information, you can say that the next 3 terms will be
16, 19, and 22
.
Ex. -1, -5, -9, ___, ___, ___ The common difference in this pattern is
-4
. Based on this information, you can say that the next 3 terms will be
-13, -17, and -21
.
### Finding the nth Term
• If you want to find a term in an arithmetic sequence that is far into the pattern, there is a formula to use.
a n = a 1 + (n – 1)(d) a n =
the answer term you are looking for in the sequence
a 1 =
the first term in the sequence
n =
the ordinal number term you are looking for in the sequence
d =
the common difference Ex. 23, 18, 13, 8, … find the 63 rd term
a n = 23 + (63 – 1)(-5) a n = 23 + 62(-5) a n = 23 + (-310) = -287
### Practice Problems
1.
11, 13, 15, 17, … Find the 85 th term 2.
25, 22, 19, 16, … Find the 50 th term 3. a 1 = -15 d = +4 Find the 71 st term 4. a n = 255 d = +3 a 1 = 36 Find n | 407 | 1,281 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.5625 | 5 | CC-MAIN-2020-40 | longest | en | 0.895269 |
https://www.kidsacademy.mobi/lesson/use-drawings-to-help-solve-3314864/ | 1,719,125,700,000,000,000 | text/html | crawl-data/CC-MAIN-2024-26/segments/1718198862464.38/warc/CC-MAIN-20240623064523-20240623094523-00739.warc.gz | 737,056,876 | 33,034 | # Logic and Early Math Lesson - Use Drawings to Help Solve, Subtraction Story Problems (#'s 1-5), Preschool
In the "Use Drawings to Help Solve" lesson, preschool students will dive into the captivating world of subtraction story problems with numbers 1-5. This lesson is an integral part of the Operations and Algebraic Thinking Within 5 unit, designed to introduce young learners to the foundational concepts of mathematics in a fun and engaging way. Through hands-on activities like the Car Shop Subtraction Worksheet and the What to Mail? Worksheet, students will learn how to visualize and solve subtraction problems by using drawings as a problem-solving tool.
Understanding subtraction and developing algebraic thinking from an early age is crucial for building a solid mathematical foundation. It enhances cognitive abilities, improves problem-solving skills, and fosters logical thinking. By learning to use drawings to solve subtraction problems, students not only grasp the concept of 'taking away' in a tangible manner but also begin to appreciate the power of visual aids in simplifying complex problems. This lesson is designed to spark curiosity, encourage creativity, and boost confidence in young learners as they embark on their mathematical journey.
Estimated classroom time: 6 min
Chapter: Subtraction Story Problems (#'s 1-5)
Unit: Operations and Algebraic Thinking Within 5
Click on any activity below to start learning.
1st
3:00 min
Car Shop Subtraction Worksheet
worksheet
2nd
3:00 min
What to Mail? Worksheet
worksheet
Share your lesson with students by clicking:
• ### Activity 1 / Car Shop Subtraction Worksheet
Addition and subtraction of numbers should not be too difficult for your kids as long as you give them enough exercises to practice with, and also guide them through carefully. In this worksheet, there are two simple equations. Ask your kids to study the picture and read the short word problems aloud. Then, help them solve the problems and then find and check the correct answer from the options provided.
• ### Activity 2 / What to Mail? Worksheet
Do your kids know what postmen do? These professionals deliver mail from the post office to everybody these letters are addressed to. Word problems are a kind of mathematical problems which is represented in sentence form. This means that there is an equation that needs to be deciphered from within that sentence. In this worksheet, your kids will need to examine the picture and carefully read the word problem, then figure out the correct answers to each question. | 508 | 2,562 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.34375 | 3 | CC-MAIN-2024-26 | latest | en | 0.921431 |
https://mathfilefoldergames.com/2023/10/05/visualizing-equivalent-fractions-a-hands-on-exploration/ | 1,701,969,460,000,000,000 | text/html | crawl-data/CC-MAIN-2023-50/segments/1700679100677.45/warc/CC-MAIN-20231207153748-20231207183748-00119.warc.gz | 422,248,556 | 23,633 | # Visualizing Equivalent Fractions: A Hands-On Exploration
Hello fellow educators!
In a world where math often seems abstract and distant to many of our young learners, the task of bringing it to life becomes imperative. Today, I’d like to introduce you to a delightful hands-on activity that not only makes math tangible but also addresses a crucial concept: equivalent fractions. Ready? Let’s dive in!
### The Power of Visualization
As teachers, we’ve all seen that ‘aha!’ moment when a child’s face lights up with understanding. This activity, “Visualizing Equivalent Fractions,” is designed to provoke just that. By providing a tactile and visual experience, students can literally see and touch the mathematical relationships they’re learning about. This method not only deepens understanding but also makes the learning experience more engaging and memorable.
### Gameplay Instructions
1. Materials Required: For each student, you will need:
• A blank piece of white copy paper.
• Colored pencils or markers.
• A math journal or notebook.
2. Setting the Stage: Begin by discussing fractions and what it means for two fractions to be equivalent. You might ask, “If I told you 2/4 is the same as 1/2, what would you say?”
3. Step-by-Step Guide:
• Distribute a blank piece of white copy paper to each student.
• Direct students to fold the paper horizontally in half.
• Have them open the paper and color one half using colored pencils or markers.
• Next, instruct them to fold the paper into fourths by folding it in half and then in half again.
• Upon opening the paper, discuss and reinforce the concept: How many fourths make up the colored half? (Answer: 2/4)
• Have them record this observation (2/4 = 1/2) in their math journal.
• Continue this process by folding the paper into eighths and sixteenths, discussing and recording findings each time.
4. Challenge Round: Once students are comfortable with the basic concept, challenge them to identify other equivalent fractions from their folded paper. This can be an exciting treasure hunt for mathematical relationships!
### Accommodations and Modifications
1. For Students with Visual Impairments: Use paper with different textures or temperatures (like thermal paper) to distinguish between colored and uncolored sections.
2. For Students with Fine Motor Challenges: Pre-fold the papers, or use larger sheets to make folding easier. Alternatively, use fraction tiles or fraction circles as a substitute.
3. Differentiation: For advanced learners, introduce fractions that are not powers of two, like thirds or fifths, to further challenge their understanding. Conversely, for students needing more foundational support, begin with simpler visuals like fraction circles or pie charts before moving to the folding activity.
### Gameplay Scenarios and Examples
1. Scenario One: Maria folds her paper into eighths and notices that four of those eighths are colored. She excitedly shares, “So, 4/8 is the same as 1/2!”
2. Scenario Two: Alex, an advanced student, decides to fold his paper into twelfths. He discovers that 6/12 is equivalent to 1/2 and eagerly jots down his findings.
3. Scenario Three: Jayden struggles with the folding but, using fraction tiles, discovers the relationship between 1/2 and 2/4, feeling a sense of accomplishment.
### Conclusion and Reflection
Once the activity is complete, have a class discussion. Let students share their discoveries and realizations. Encourage them to reflect on their learning journey in their math journals. This reflection not only reinforces the day’s lesson but also helps them internalize the concept of equivalent fractions.
### Link to Common Core State Standards (CCSS)
This activity aligns with the following Common Core State Standards:
• CCSS.MATH.CONTENT.3.NF.A.3: Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size.
• CCSS.MATH.CONTENT.4.NF.A.1: Explain why a fraction a/b is equivalent to a fraction (n×a)/(n×b) by using visual fraction models.
Incorporating hands-on activities in the math classroom can transform abstract concepts into tangible, understandable, and fun learning experiences. “Visualizing Equivalent Fractions” is just one tool in our vast teaching arsenal. Let’s continue sharing and learning together for the betterment of our students!
Happy teaching!
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This site uses Akismet to reduce spam. Learn how your comment data is processed. | 1,241 | 5,886 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.65625 | 5 | CC-MAIN-2023-50 | latest | en | 0.920288 |
http://www.pharmacy-tech-study.com/mgkghr.html | 1,487,711,970,000,000,000 | text/html | crawl-data/CC-MAIN-2017-09/segments/1487501170839.31/warc/CC-MAIN-20170219104610-00405-ip-10-171-10-108.ec2.internal.warc.gz | 559,182,547 | 30,029 | ## MG.KG/HR
by ebony
(austin,tx)
I'm having trouble solving this problem. Looks easy, but maybe I'm trying to put too much info to solve. Okay here it goes:
The dose of a drug for a patient is 0.8 mg/kg/hr for 12 hours. How many ml of an IV solution containing 800 mg per 100 ml would be required for a 242 lb. male over this 12 hour period.
A. 96mL
B. 132mL
C. 264mL
D. None of the above
## Comments for MG.KG/HR
Solution by: Anonymous B. 132mL242lbs/2.2lbs/kg=110kg0.8mg/kg/hr*12hrs=9.6mg/kg/12hrs110kg*9.6mg/kg/12hrs=1056mg/12hrs1056mg------*100mL=132mL800mg
Hmm? by: Anonymous What?! Where did 96mg come from?
MG.KG/HR by: Anonymous This is the formula0.8/kg/hr X 110kg X 100ml 800mgThat calculation gives you 11 which you multiply by 12hrs which gives you 132.
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Tutorials Top Drugs Exams Forums Misc. Contact [?] Subscribe To This Site | 287 | 859 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.171875 | 3 | CC-MAIN-2017-09 | latest | en | 0.880959 |
https://math.stackexchange.com/questions/3999593/unifying-the-connections-between-the-trigonometric-and-hyperbolic-functions | 1,721,447,489,000,000,000 | text/html | crawl-data/CC-MAIN-2024-30/segments/1720763514981.25/warc/CC-MAIN-20240720021925-20240720051925-00686.warc.gz | 359,655,356 | 45,167 | # Unifying the connections between the trigonometric and hyperbolic functions
There are many, many connections between the trigonometric and hyperbolic functions, some of which are listed here. It is probably too optimistic to expect that a single insight could explain all of these connections, but is there a holistic way of seeing the parallels between $$\sin$$ and $$\sinh$$, $$\cos$$ and $$\cosh$$? Can all of these seemingly disparate connections be shown to be essentially the same, or at least very similar?
Geometric connections
• Sine and cosine parameterise the unit circle $$x^2+y^2=1$$, just as hyperbolic sine and cosine parameterise the 'unit hyperbola' $$x^2-y^2=1$$. Both circles and hyperbolas are conic sections.
• The sector of the circle connecting the points $$(0,0)$$, $$(1,0)$$, and $$(\cos t,\sin t)$$ has an area of $$t/2$$. The region of the hyperbola connecting the points $$(0,0)$$, $$(1,0)$$, and $$(\cosh t,\sinh t)$$ has an area of $$t/2$$. This can even be used to define the hyperbolic functions geometrically, and many authors do the same with the trigonometric functions.
• Sine and hyperbolic sine are odd, whereas cosine and hyperbolic cosine are even. But sine and cosine are periodic functions, unlike the hyperbolic counterparts.
• The analogue of the identity $$\cos^2x+\sin^2x \equiv 1$$ is $$\cosh^2x-\sinh^2x \equiv 1$$. The compound angle formulae are almost identical to their hyperbolic counterparts, save for a pesky minus sign: \begin{align} \sin(x+y) &= \sin(x)\cos(y)+\cos(x)\sin(y) \\ \sinh(x+y) &= \sinh(x)\cosh(y) + \cosh(x)\sinh(y) \\[4pt] \cos(x+y) &= \cos(x)\cos(y) \color{red}{-} \sin(x)\sin(y) \\ \cosh(x+y) &= \cosh(x)\cosh(y) \color{blue}{+} \sinh(x)\sinh(y) \, . \end{align}
• In general, given a trigonometric identity, it is (usually) possible to find a corresponding hyperbolic identity using Osborn's rule: replace every occurrence of $$\cos$$ with $$\cosh$$; replace every occurrence of $$\sin$$ by $$\sinh$$, but negate the product of two $$\sinh$$ terms.
Analytic connections
• $$\sin$$ is the unique solution to the initial value problem \begin{align} f''(x) &= \color{red}{-}f(x) \\ f'(0) &= 1 \\ f(0) &= 0 \, , \end{align} and the corresponding initial value problem for $$\sinh$$ is the same, except $$f''(x) = \color{blue}{+}f(x)$$.
• Likewise, the initial value problem for $$\cos$$ is \begin{align} f''(x) &= \color{red}{-}f(x) \\ f'(0) &= 0 \\ f(0) &= 1 \, , \end{align} and again we see a mysterious sign change for $$\cosh$$: $$f''(x) = \color{blue}{+}f(x)$$.
• It follows that the higher-order derivatives of $$\sin$$ and $$\sinh$$ form periodic sequences.
• If we solve the initial value problems shown above, we obtain the exponential forms of all $$4$$ functions: \begin{align} \sin x &= \frac{e^{\color{green}{i}x}-e^{-\color{green}{i}x}}{2\color{green}{i}} \quad{} \cos x = \frac{e^{\color{\green}{i}x}+ e^{-\color{green}{i}x}}{2} \\[3pt] \sinh x &= \frac{e^{x}-e^{-x}}{2} \quad{} \cosh x = \frac{e^x + e^{-x}}{2} \, . \end{align}
• All $$4$$ functions are analytic, and their Taylor series bear a striking resemblance to each other: \begin{align} \sin x &= x \color{red}{-} \frac{x^3}{3!} + \frac{x^5}{5!} \color{red}{-} \frac{x^7}{7!} + \ldots \\[4pt] \sinh x &= x \color{blue}{+} \frac{x^3}{3!} + \frac{x^5}{5!} \color{blue}{+} \frac{x^7}{7!} + \ldots \\[4pt] \cos x &= 1 \color{red}{-} \frac{x^2}{2!} + \frac{x^4}{4!} \color{red}{-} \frac{x^6}{6!} + \ldots \\[4pt] \cosh x &= 1 \color{blue}{+} \frac{x^2}{2!} + \frac{x^4}{4!} \color{blue}{+} \frac{x^6}{6!} + \ldots \end{align}
• And Euler's formula $$e^{ix} = \cos x + i \sin x$$ is replaced by the underwhelming $$e^x = \cosh x + \sinh x \, .$$
• The connection of both circular and hyperbolic trigonometric functions with Jacobi elliptic functions is worth mentioning. Commented Jan 25, 2021 at 19:08
• @Joe: "Deep" Pythagoras thm $\cos c/R = \cos a/R \cos b/R \rightarrow \cosh c/R = \cosh a/R \cosh b/R$. Circular trigonometry "mirrors" hyperbolic trigonometry and was termed " pan-geometry " in Roberto Bonola's non-Euclidean hyperbolic geometry text book.... etc. Commented Jan 25, 2021 at 19:21
• Joe, you may be interested in these identities: \begin{align} \arctan x=-i~\text{artanh} ~ix\\ \arcsin x=-i~\text{arsinh}~ix\\ \end{align} and so on for the rest of them. Commented Jan 25, 2021 at 20:12
• @A-LevelStudent Thank you, I hadn't seen these before. Since the trigonometric functions are arguably exponential functions in disguise, I would suspect this has something to do with logarithms.
– Joe
Commented Jan 25, 2021 at 20:26
• They're just all combinations of exponential functions. That explains everything there is to explain about their connections. Commented Jan 31, 2021 at 21:35
I my opinion I think that when you know
$$\sin(x) = \frac{e^{ix}-e^{-ix}}{2i} ~~~~~~~~~~~~~ \cos(x) = \frac{e^{ix} + e^{-ix}}{2}$$
you can derive all the circular trigonometric identities.
If you add the "Wick" transformation $$x \to ix$$ then you will step into the hyperbolic world, with all the consequent identities.
$$\sin(ix) = \frac{e^{-x} - e^{x}}{2i} = i\frac{e^x - e^{-x}}{2} = i\sinh(x) ~~~~~ \longrightarrow ~~~~~ \sinh(x) = \frac{e^x + e^{-x}}{2}$$
And similarly for $$\cosh(x)$$.
• Good illustration indeed! Commented Jan 25, 2021 at 19:33
• @mrs Much obliged! Commented Jan 25, 2021 at 20:31
• @Turing Hi. I accepted Yves' answer but I did find yours very useful as well. The identity $\sin(ix)=i\sinh x$ does shine some light on things.
– Joe
Commented Feb 2, 2021 at 19:41
• @Joe Don't worry, you can accept the answer you like the most! I am very glad you found mine useful too! :D Commented Feb 2, 2021 at 19:46
Yes; in fact they can all be traced to the same point, and it is a point that you brought up in your question. One important thing to note is that in Math there can be many equivalent definitions of concepts that can lead to each other, for instance one can define $$\pi$$ as the ratio of circumference to diameter; or area to radius squared. Both are equivalent.
Similarly we can define these functions geometrically,
And from that obtain differential equations that describe the functions, and from that obtain their exponential definitions; this can be used to unify all of the points you brought up, the Pythagorean Identity, sum of angles, Taylor Series, and exponential definition.
Alternatively we could have defined these functions in terms of their exponential definitions (As @Turing suggests), and from that obtain the Pythagorean Identity, sum of angles, Taylor Series, and geometric definition.
The latter is much easier, but the former is how we came to understanding this family of functions historically.
• Great graphics!! Commented Jan 25, 2021 at 19:20
The trigonometric functions are related to the imaginary exponential $$e^{iv}=\cos v+i\sin v$$.
The hyperbolic functions are related to the real exponential $$e^{u}=\cosh u+\sinh u$$.
They both are special cases of the complex exponential $$e^z=e^{u+iv}$$.
The associated conics are obtained by
$$1=e^{iv}e^{-iv}=(\cos v+i\sin v)(\cos v-i\sin v)=\cos^2u-i^2\sin^2u,$$
$$1=e^{v}e^{-v}=(\cosh v+\sinh v)(\cosh v-\sinh v)=\cosh^2u-\sinh^2u.$$
A unifying ODE is
$$z''''=z$$ giving the characteristic polynomial $$\omega^4=1,\\\omega=\pm1,\pm i$$
and the solution
$$z=a\cos t+b\sin t+c\cosh t+d\sinh t.$$
Just think
$$\text{real}\leftrightarrow\text{imaginary}.$$
• Now I regret saying that the identity $e^x = \cosh x + \sinh x$ is underwhelming ;-)
– Joe
Commented Feb 2, 2021 at 19:51
The connection between the trigonometric and hyperbolic functions becomes more intimate when one introduces split-complex numbers: the numbers of the form $$a+bj$$, where $$j^2=1$$ and $$j\ne \pm 1$$.
Using Taylor series you can easily find that $$e^{jx} = \cosh x + j\sinh x , \quad \sinh x = \frac{e^{jx}-e^{-jx}}{2j}, \quad \cosh x = \frac{e^{jx} + e^{-jx}}{2}.$$
One can understand $$\cos x$$ and $$\sin x$$ as the real and imaginary parts of the complex exponential $$e^{ix}$$. Likewise, $$\cosh x$$ and $$\sinh x$$ are the real and 'imaginary' parts of the split-complex exponential $$e^{jx}$$.
The analogy can be summarized as follows: \begin{align} & &&\textbf{Complex numbers} && \textbf{Split-complex numbers} \\ &\text{Imaginary unit:} && i^2 = -1, && j^2 = 1, \\ &\text{Numbers:} && z = a+bi, && z = a+bj, \\ &\text{Real part:} && \mathfrak{R}(a+bi)=a, && \mathfrak{R}(a+bj)=a, \\ &\text{Conjugation:} && z=a+bi\mapsto \bar{z}=a-bi, && z=a+bj\mapsto \bar{z}=a-bj, \\ &\text{Sesquilinear form:} && \langle z|w\rangle= z\bar{w}, && \langle z|w\rangle= z\bar{w} , \\ &\text{Real bilinear form:} && \mathfrak{R}(z\bar{w})\quad \text{(positive definite)}, && \mathfrak{R}(z\bar{w}) \quad\text{(of indefinite signature)} , \\ &\text{Other real bilinear form:} && \mathfrak{R}(zw)\quad \text{(of indefinite signature)}, && \mathfrak{R}(zw) \quad\text{(positive definite)} , \\ &\text{Squared modulus:} && |z|^2=\langle z|z\rangle = a^2+b^2, && |z|^2=\langle z|z\rangle = a^2-b^2, \\ &\text{Euler's formula:} && e^{ix} = \cos x + i \sin x, && e^{jx} = \cosh x + j \sinh x, \\ &\text{Pythagorean theorem:} && |e^{ix}| = \cos^2 x + \sin^2 x =1, && |e^{jx}| = \cosh^2 x - \sinh^2 x = 1, \\ &\text{Geometric meaning:} && \text{Unit circle,} && \text{Unit hyperbola (right branch)}. \\ \end{align}
The connection is in the correspondence: $$\begin{pmatrix} \text{complex} \; i \\ \sin, \; \cos \end{pmatrix} \leftrightarrow \begin{pmatrix} \text{split-complex} \; j \\ \sinh, \; \cosh \end{pmatrix}.$$
One can use imagination to say that $$\sin$$, $$\cos$$ and complex numbers come from the elliptic world, while $$\sinh$$, $$\cosh$$ and split-complex numbers are their counterparts from the hyperbolic world.
• An observation: the most natural topology on $\mathbb{R}[J]$, $J=i$ or $j$, the one you use to define convergence of sequences, is induced by $\mathfrak{R}(z\bar{w})$ in the case of $\mathbb{C}=\mathbb{R}[i]$, and by $\mathfrak{R}(zw)$ in the case of $\mathbb{R}[j]$ . Commented May 28 at 11:15
Closed relation of the trigonometrical and hyperbolic functions can be demonstrated on the example of the Chebyshev Polyhomials of the First Kind $$T_n(x) = \begin{cases} \cos(n\arccos x),\;\text{if}\;|x|\le 1\\ \cosh(n\text{ arccosh } x),\;\text{otherwize}. \end{cases}$$ This example shows that hyperbolic functions can be suitable addition to the trigonometrical ones in the real analysis.
The Jacobi elliptic functions, in particular the sine and cosine of the amplitude are also connected to the hyperbolic trigonometric functions using the Gudermannian function which relates the circular functions and hyperbolic functions without explicitly using complex numbers. More precisely, if the parameter $$\,m=0\,$$ then $$\text{sn}(x,0) = \sin(x)\quad \text{ and } \quad \text{cn}(x,0) = \cos(x)$$ while if $$\,m=1\,$$ then $$\text{sn}(x,1) = \tanh(x)\quad \text{ and } \quad \text{cn}(x,1) = \text{sech}(x).$$ The Gudermannian function links the two cases with the identity $$\sin(\text{gd}(x)) = \tanh(x) \quad \text{ and } \quad \tan(\text{gd}(x)) = \sinh(x)$$ and the related identity $$\cos(\text{gd}(x)) = \text{sech}(x) \quad \text{ and } \quad \sec(\text{gd}(x)) = \cosh(x).$$ In some sense, the Jacobi functions generalize the circular and hyperbolic trigonometric functions. Note the fundamental identity $$\text{sn}(x,m)^2 + \text{cn}(x,m)^2 = 1.$$ Whem $$\,m=0\,$$ this becomes $$\sin(x)^2+\cos(x)^2=1$$ and when $$\,m=1\,$$ this becomes $$\tanh(x)^2+\text{sech}(x)^2=1$$ which is equivalent to $$\cosh(x)^2-\sinh(x)^2=1.$$ | 3,851 | 11,612 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 81, "wp-katex-eq": 0, "align": 7, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.9375 | 4 | CC-MAIN-2024-30 | latest | en | 0.813153 |
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September 4, 2013, 03:05
transient conduction in a wall
#1
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this the problem. steam inlet from below the drum with temp 380 C, mass flow 0,4 kg/s with int condition 60 C (for wall and fluid), steam injected for 460 min. air inside the drum with temp. 60 C. here is the step what i've done.
gambit :
1.import model 2D
2. create face 1 for solid and create face 2 for liquid
3. mesh face for face 1 and face 2 together
4. create countinuum type face 1 solid and face 2 liquid selected
5. creat BC inlet at buttom line and symetry on left edge
6. export mesh
fluent 6 :
2. scale
2. define model solver - unsteady
3. define mdl multiphase- eularian
4. define mdl energy - check
5. define mdl visc - k-epsilon
6. define material - vapor water (solid with rho=7850, cp =450, K = 40)
7. define phase- phase 1 (air) phase 2 (vapor)
8. define OC - gravity (-9.81)
9. define BC- inlet (phase 2, 0,4kg/s, temp 380 C) (phase 1, 0 kg/s, temp 60 C) there is wall 1 and wall 01:shadows appear at edge between solid and luqiud, BC without outlet
10. solve-control solution-ok
11. solve initialize- inlet
12. adapt region-insert number of x/y max/min - adapt - mark
13. solve initialize - patch (phase 2, liquid , value=0) (mixture, solid, value 60 C) this i want to make theres no vapor inside the drum and int temp. of wall is 60 C
14. solve monitor residual - plot
15. iterate - time step size (1) number of TS (76000 s) max iteration/TS (2)
16. iterate
there're all the steps friends, but the result temperature of the wall is not as i expected, its about 250 C and thats should be around 350 C. Please tell me what i missed. is that because i used fluent 6 ? i really need your help
thank you very much
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Last edited by chabib; September 4, 2013 at 05:18.
September 4, 2013, 04:22 #2 Member Join Date: Jul 2011 Posts: 72 Blog Entries: 1 Rep Power: 10 do you have specific question or you want us to solve your problem
September 4, 2013, 05:10 #3 New Member chabib muhammad Join Date: Aug 2013 Posts: 2 Rep Power: 0 dear centurion2011 i just want to share my problem why i have different wall temperature from numerical result with measured result. i hope some one can explain to me. i write the steps because i thing there something wrong in my steps or something else. i am still learning using Fluent. sorry if my question too general or say something wrong in this forum regards
September 4, 2013, 05:20 #4 Member Join Date: Jul 2011 Posts: 72 Blog Entries: 1 Rep Power: 10 maybe you should start with steady solution and then progress toward unsteady case. For more info on unsteady problems, go to following link Hope this will help you a bit __________________ I'M NOT A GYNECOLOGIST BUT I'LL TAKE A LOOK.
September 4, 2013, 07:49
#5
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duri
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Quote:
Originally Posted by chabib 10. solve-control solution-ok 11. solve initialize- inlet 12. adapt region-insert number of x/y max/min - adapt - mark
Since you used default solver settings. Check whether it is second order or higher. And don't use mesh adaption near boundary (if any), better to create a fine mesh than mesh adaption. k-epsilon turbulence model uses wall function which is not suitable for heat transfer applications try sst model.
September 5, 2013, 05:55
#6
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Daniele
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Quote:
Originally Posted by chabib 15. iterate - time step size (1) number of TS (76000 s) max iteration/TS (2)
Are you sure you are getting convergence with only 2 iterations per time step?
Daniele
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1. Nov 1, 2016
### Frank Castle
As I understand it, the proper time, $\tau$, between to events in spacetime is defined in terms of the spacetime interval $ds^{2}=\eta_{\mu\nu}dx^{\mu}dx^{\nu}$, such that $$d\tau =\sqrt{-ds^{2}}$$ (where we are using the "mostly +" signature with $c=1$).
Now, for time-like intervals, for which $ds^{2}<0$, it is clear that proper time is well-defined since the quantity $\sqrt{-ds^{2}}$ is positive, and furthermore, one can always find a frame in which the two events occur at the same point in space, such that one can construct a worldline connecting the two events, along which an observer can travel, at rest with respect to both events, such that $d\tau =\sqrt{-ds^{2}}=dt$.
However, why is it the case that for space-like, $ds^{2}>0$, and light-like intervals, $ds^{2}=0$, the notion of proper time is undefined (or perhaps ill-defined)?
For the space-like case, I get that heuristically, one cannot construct a path between the two events along which an observer can travel and so in this sense proper time is meaningless, since a worldline connecting the events does not exist and so no clock can pass through both events. However, can this be seen purely by examining the definition of proper time in terms of the spacetime interval? Is it simply that the quantity $\sqrt{-ds^{2}}$ will become imaginary and so clearly cannot be used to represent any physical time interval?
Likewise, for a light-like interval, only a beam of light can pass between both events and since there is no rest frame for light one cannot construct a frame in which a clock is at rest with respect to the beam and passes through both events. However, purely in terms of the spacetime interval, is it simply because the quantity $\sqrt{-ds^{2}}$ equals $0$, and so the notion of proper time is ill-defined since there is no invertible map between reference frames (here I'm thinking in terms of time dilation, $t =\gamma\tau$ and so for a light-like interval, $\gamma\rightarrow\infty$ meaning that the inverse relation $\tau =\frac{t}{\gamma}$ is ill-defined)?!
2. Nov 1, 2016
### Maxila
I'm trying to understand exactly what you are asking because you appear to answer the question when you say:
Since you already know physically a time could not pass for a light like or space like interval, what are you trying to ask?
3. Nov 1, 2016
### Staff: Mentor
Because it's only defined for timelike intervals.
As Maxila said, I think you need to clarify exactly what you are asking. The answer to your question as you ask it is trivial, as above.
4. Nov 1, 2016
### Frank Castle
I was wondering if one can argue directly from the definition of proper time in terms of the metric, i.e. $$d\tau =\sqrt{-ds^{2}}$$ that proper time is undefined for space-like and light-like (null) intervals?
For space-like intervals we have that $ds^{2}>0$ and so the quantity $\sqrt{-ds^{2}}$ becomes imaginary and therefore cannot be used to describe any physical passage of time, hence proper time for space-like intervals is undefined. However, proper distance $dl^{2}=\sqrt{ds^{2}}$ makes sense for a space-like interval, since $\sqrt{ds^{2}}>0$ and therefore can be used to describe a physical distance. Hence one can parametrise space-like paths with proper distance, but not proper time.
For time-like intervals we have that $ds^{2}=0$, however, I can't see how one can argue that proper time for a null interval is undefined directly from this?!
I have seen people claim that "photons experience no proper time", however, this doesn't sound right to me, since a photon does not have a rest frame (which follows directly from the postulate that light travels at the speed of light $c$ for all observers, hence, there cannot exist any frame in which light travels at less that $c$, ergo there cannot exist a rest frame for photons). Consequently, one cannot construct a reference frame in which a clock is at rest with respect to a photon along its worldline and hence proper time simply cannot be defined for null spacetime intervals.
5. Nov 1, 2016
### Staff: Mentor
The definition you give only applies to timelike intervals, by definition. The interval $ds^2$ is obviously well-defined for all three cases (timelike, null, and spacelike), but the term "proper time" is only used to refer to it, by definition, for the timelike case.
You need to take a step back and think carefully about what you are asking. So far you have only asked about terminology, not physics. "Proper time" is an ordinary language term, and what kinds of intervals it applies to is a question about words, not physics. What question about physics are you asking?
6. Nov 1, 2016
### Staff: Mentor
You are making an easy question harder than it needs to be.
The convention is that we use the phrase "proper time" to describe timelike spacetime intervals, so the answer to the question in the title is "because that's what the convention is". You could as usefully ask why male horses aren't mares.
7. Nov 1, 2016
### Maxila
Again I think you answer your own question, possibly without realizing it? Yes, we can extrapolate physics for a space like interval; however those physics cannot affect two events with such an interval since nothing can travel faster than light, including gravity. In other words, how could you attribute a time between two events where no observation or physical interaction is possible. It would be infinite which is nonsensical in terms of a physical time.
8. Nov 1, 2016
### Frank Castle
I think I may be over thinking things. Apologies for going round in circles a bit.
Is there an intuitive reason for why proper-time is defined in terms of the space-time interval as $d\tau =\sqrt{-ds^{2}}$? I get that it is in part from the requirement that it is a Lorentz invariant quantity, but does the definition also follow from the fact that the proper-time of an object is the time as measured in the rest frame of that object, and so the corresponding space-time interval in this frame is given by $ds^{2}=-d\tau^{2}$.
Why can't one define a notion of proper time for null space-time intervals? Is it simply because there is no rest frame for particles travelling at the speed of light?
9. Nov 1, 2016
### jbriggs444
You can define whatever you like in any way you like. No meaning is obliged to flow from the resulting definition. Others have asked whether there is a physical question here rather than a question about terminology and definitions.
10. Nov 1, 2016
### Frank Castle
It's just that I've had people tell me before that "time stands still" for photons since the space-time interval is null in this case. To make this statement they are obviously assuming the definition of proper time in terms of the space-time interval (since then $d\tau =0$), which then conflicts with the definition being in terms of timelike intervals. Physically, is it even meaningful to quantify a "proper-time" (a photons "own time") for null intervals?
11. Nov 1, 2016
### Staff: Mentor
It's not so much that you're over-thinking as that you're thinking about words instead of physics. Again: what physics are you asking about? If you think the term "proper time" implies some particular physics, what?
12. Nov 1, 2016
### Staff: Mentor
Yes, and they are wrong.
This is still a question about words, not physics; but I think I see at least one question about physics that you are groping towards. Reading this forum FAQ might help:
13. Nov 1, 2016
### Frank Castle
Up until now I've thought of proper time of an object as a physical quantity, measuring the elapsed time in the objects rest frame, or in other words, it is the the time measured by a clock that travels along the worldline of the object.
I get that no rest frame exists for a particle travelling along a null path, so given the physical interpretation I have phrased in the sentence above for proper time, is this the reason why it is not meaningful to define such a quantity, since it is impossible for a clock to measure time along the worldline of a photon?!
Last edited: Nov 1, 2016
14. Nov 1, 2016
### Staff: Mentor
Yes, and this is the same as the definition you gave earlier, since in the mathematical model the arc length along a timelike curve is what represents this physical quantity.
What quantity? It's certainly meaningful to define the arc length along a null curve; it's just that the definition (the interval) gives a result of zero, regardless of which points on the null curve we pick. This fact makes "proper time" an unsuitable term for describing such an arc length, but that's still a matter of choice of words, not physics.
This is getting closer to the physics. A more precise statement would be that, in order to build a clock, you need more than just one light beam; in the simplest kind of clock, a light clock, you also need a pair of mirrors, traveling on timelike worldlines. (There are more complicated ways to construct a series of timelike separated points by looking at the intersections of light rays traveling in different directions, but we'll leave that aside here.)
But the physical difference between timelike intervals and null intervals involves more than just clocks. Objects traveling on timelike worldlines can have different relative speeds in different frames. Objects traveling on null worldlines have the same speed in all frames; but that doesn't mean they're unaffected--light rays, for example, change their frequency/wavelength (relativistic Doppler shift). So changing frames (Lorentz transformation) has fundamentally different effects on the two kinds of objects. (And spacelike intervals are different from both: a Lorentz transformation can change a spacelike interval from "future-directed" to "past-directed"--or even make it neither, make the two events it connects simultaneous--but it can't do that to timelike or null intervals.)
15. Nov 1, 2016
### Frank Castle
Is this why one can't physically measure time along a null geodesic since even the most basic of clocks requires time to be measured along a timelike geodesic?!
Would what be the correct argument for why the statement that a photon "experiences no time" is incorrect?
16. Nov 1, 2016
### Staff: Mentor
I think these questions have already been answered. I have described the physics.
17. Nov 1, 2016
### Mister T
You already answered this when you said that a clock can't co-move with a photon.
The notion that photons experience no time is poorly stated, but it does serve an explanatory purpose. For example, before neutrino oscillations were confirmed experimentally it was thought that perhaps neutrinos are massless and hence travel at speed c. One great mystery was that of the missing solar neutrinos. Detectors pick up only a fraction of what should be there, given the proposed reactions that produce the known temperature and energy output of the sun. In other words, the nuclear reactions needed to account for the sun's temperature and energy output should produce more electron neutrinos than we detect. If neutrinos are massless and travel at speed c then there's no way for them to change, on their journey from the sun to Earth, from one type to another because they can't, loosely speaking, experience time.
That is at best a loose way to put it, and at worst a misleading and erroneous way to put it. I've heard it stated that way, though, by a Nobel laureate speaking to a group of physicists, with of course the added caveat that it's "loose". I suppose a correct way to state it is that if neutrinos are massless they travel on null geodesics and hence can't oscillate. We now know that they are massive, they do oscillate, and their speed is less than c. But they don't have a mass eigenstate.
18. Nov 2, 2016
### SiennaTheGr8
You could just as well ask why we don't define the proper distance in terms of the spacetime interval.
For entirely practical reasons, we define proper time as the time measured by a co-moving inertial observer. Then we find that $ds = \sqrt{|(c \, d \tau)^2|}$ for timelike intervals.
Likewise, we define the proper distance $\sigma$ between two events as their spatial separation as measured by an inertial observer for whom they occur simultaneously. That leads to $ds = \sqrt{|(d \sigma)^2|}$ for spacelike intervals.
What if we dropped the "for [timelike/spacelike] intervals" caveats? Well, then we'd basically have three different names for the same thing. What's the point? Proper time and proper distance are useful concepts only insofar as they aren't redundant with each other or with the spacetime interval. | 2,934 | 12,706 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.984375 | 3 | CC-MAIN-2018-30 | latest | en | 0.954299 |
http://stackoverflow.com/questions/3844048/ceiling-to-nearest-50/3844140 | 1,394,453,175,000,000,000 | text/html | crawl-data/CC-MAIN-2014-10/segments/1394010776091/warc/CC-MAIN-20140305091256-00064-ip-10-183-142-35.ec2.internal.warc.gz | 179,057,542 | 16,093 | # Ceiling to Nearest 50
I can round the elements of `A` to the nearest integers greater than or equal to `A`
``````ceil(A)
``````
But what about if I want to round it to the nearest 50 greater than or equal to `A`?
For example, given the following `A` array,
``````A=[24, 35, 78, 101, 199];
``````
A subroutine should return the following
``````B=Subroutine(A)=[50, 50, 100, 150, 200];
``````
-
You could just divide by 50, take ceil(), and multiply by 50 again:
`````` octave:1> A=[24, 35, 78, 101, 199];
octave:2> ceil(A)
ans =
24 35 78 101 199
octave:3> 50*(ceil(A/50.))
ans =
50 50 100 150 200
``````
-
Note that this could conceivably introduce floating-point rounding errors, for large values in A. – Piet Delport Oct 3 '10 at 17:53
Meh, so get rid of the decimal point after the 50 in the division. Either the numbers are already floating-point values, in which case that issue has already come up elsewhere in the code, or the numbers are integers, in which case just get rid of the decimal point and now you're doing integer division and addition, in which case no floating point issues. So I don't see the problem here. – Jonathan Dursi Oct 3 '10 at 22:29
The problem is that the results will be incorrect. :-) For example, try `77777777777777777` as input: this method gives the incorrect result `77777777777777792`, while the modulus method correctly gives `77777777777777800`. – Piet Delport Oct 4 '10 at 22:03
Yes, and so does your method: "a = 77777777777777777; a + mod(-a,50)" also gives 77777777777777792. You may want go look at wikipedia's article on machine epsilon (en.wikipedia.org/wiki/Machine_epsilon) and note that you are never going to have numerical operations produce correct results in the 16th decimal place of a floating point number even in double precision. And of course, with (say) 64bit ints, either approach gives the correct result, because 50*ceil(a/50) in integer arithmetic is precisely the same thing as a+mod(-a,50). – Jonathan Dursi Oct 4 '10 at 22:23
@Piet and Jonathan: I tested both your answers in MATLAB (what the question asks about) and I got the exact same result from each. However, testing the output with a number like `77777777777777777` doesn't make much sense, since this is a larger integer than what a double (the default MATLAB type) can hold. You'd need a 64-bit integer to handle that number. – gnovice Oct 5 '10 at 13:59
An easy way is to just add each number's complement modulo 50:
``````octave> A = [24, 35, 78, 101, 199]
octave> mod(-A, 50) # Complement (mod 50)
ans =
26 15 22 49 1
octave> A + mod(-A, 50) # Sum to "next higher" zero (mod 50)
ans =
50 50 100 150 200
octave> A - mod(A, 50) # Can also sum to "next lower" zero (mod 50)
ans =
0 0 50 100 150
``````
(Note that this only depends on integer arithmetic, which avoids errors due to floating-point rounding.)
- | 872 | 2,927 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.484375 | 3 | CC-MAIN-2014-10 | latest | en | 0.829233 |
https://atcoder.jp/contests/abc168/submissions/13402510 | 1,590,758,504,000,000,000 | text/html | crawl-data/CC-MAIN-2020-24/segments/1590347404857.23/warc/CC-MAIN-20200529121120-20200529151120-00283.warc.gz | 263,455,611 | 7,313 | Contest Duration: ~ (local time) (100 minutes) Back to Home
Submission #13402510
Source Code Expand
Copy
```# Studying https://atcoder.jp/contests/abc168/tasks/abc168_f
N,M = gets.split.map(&:to_i)
GW = 経線.size*2+3
GH = 緯線.size*2+3
GI = lambda{|横,縦|
i = 経線.bsearch_index{|_| 横<=_ }||経線.size # 線分の端点の座標に対応する
j = 緯線.bsearch_index{|_| 縦<=_ }||緯線.size
i1 = 経線[i] != 横 ? 1 : 0 # 線分間の座標に対応するための補正
j1 = 緯線[j] != 縦 ? 1 : 0
next (2+2*j-j1)*GW+(2+2*i-i1)
}
Grid = lambda{|g|
縦線.each{|a,b,c|
GI[c,a].step(GI[c,b], GW){|i|
g[i] = '0'
}
}
i = 0
GH.times{
g[i] = g[i+GW-1] = '0'
i += GW
}
横線.each{|d,e,f|
GI[e,d].upto(GI[f,d]){|i|
g[i] = '0'
}
}
0.upto(GW-1){|i|
g[i] = g[-1-i] = '0'
}
return g.to_i(2)
}['1'*(GW*GH)]
Area = lambda{|a|
(GW+1).upto(GW+GW-2){|i|
a[i] = a[-1-i] = Float::INFINITY
}
i = GW+1
(GH-2).times{
a[i] = a[i+GW-3] = Float::INFINITY
i += GW
}
heights = 緯線.each_cons(2).map{|_1,_2| _2-_1 }
widths = 経線.each_cons(2).map{|_1,_2| _2-_1 }
j = (GW+1)*3
heights.each{|h|
i = 0
widths.each{|w|
a[j+i] = h*w
i += 2
}
j += GW+GW
}
return a
}[[0]*(GW*GH)]
cow = 1 << GW*GH-1-GI[0,0]
while 0
cow＋ = (cow | cow>>1 | cow<<1 | cow>>GW | cow<<GW) & Grid
break if cow == cow＋
cow = cow＋
end
area = 0
until cow == 0
a = cow ^ cow-1
area += Area[-a.bit_length]
cow &= ~a
end
puts(area.finite? ? area : 'INF')
```
#### Submission Info
Submission Time 2020-05-19 19:49:53+0900 F - . (Single Dot) ds14050 Ruby (2.7.1) 0 1726 Byte WA 3315 ms 202824 KB
#### Compile Error
```./Main.rb:53: warning: `_1' is reserved for numbered parameter; consider another name
./Main.rb:53: warning: `_2' is reserved for numbered parameter; consider another name
./Main.rb:54: warning: `_1' is reserved for numbered parameter; consider another name
./Main.rb:54: warning: `_2' is reserved for numbered parameter; consider another name
./Main.rb:73: warning: literal in condition
```
#### Test Cases
Set Name Score / Max Score Test Cases
Sample 0 / 0 sample_01.txt, sample_02.txt
Subtask1 0 / 600 sample_01.txt, sample_02.txt, sub1_01.txt, sub1_02.txt, sub1_03.txt, sub1_04.txt, sub1_05.txt, sub1_06.txt, sub1_07.txt, sub1_08.txt, sub1_09.txt, sub1_10.txt, sub1_11.txt, sub1_12.txt, sub1_13.txt, sub1_14.txt, sub1_15.txt, sub1_16.txt, sub1_17.txt, sub1_18.txt, sub1_19.txt, sub1_20.txt, sub1_21.txt, sub1_22.txt, sub1_23.txt, sub1_24.txt, sub1_25.txt, sub1_26.txt, sub1_27.txt, sub1_28.txt, sub1_29.txt, sub1_30.txt, sub1_31.txt, sub1_32.txt, sub1_33.txt, sub1_34.txt, sub1_35.txt, sub1_36.txt, sub1_37.txt, sub1_38.txt, sub1_39.txt, sub1_40.txt, sub1_41.txt, sub1_42.txt, sub1_43.txt, sub1_44.txt, sub1_45.txt, sub1_46.txt, sub1_47.txt, sub1_48.txt, sub1_49.txt, sub1_50.txt, sub1_51.txt, sub1_52.txt, sub1_53.txt, sub1_54.txt, sub1_55.txt, sub1_56.txt, sub1_57.txt, sub1_58.txt, sub1_59.txt, sub1_60.txt, sub1_61.txt, sub1_62.txt, sub1_63.txt, sub1_64.txt, sub1_65.txt, sub1_66.txt, sub1_67.txt, sub1_68.txt, sub1_69.txt, sub1_70.txt, sub1_71.txt, sub1_72.txt, sub1_73.txt, sub1_74.txt, sub1_75.txt, sub1_76.txt, sub1_77.txt, sub1_78.txt, sub1_79.txt, sub1_80.txt, sub1_81.txt, sub1_82.txt, sub1_83.txt, sub1_84.txt, sub1_85.txt, sub1_86.txt, sub1_87.txt, sub1_88.txt, sub1_89.txt, sub1_90.txt, sub1_91.txt, sub1_92.txt, sub1_93.txt, sub1_94.txt
Case Name Status Exec Time Memory
sample_01.txt 52 ms 14416 KB
sample_02.txt 53 ms 14400 KB
sub1_01.txt 274 ms 107488 KB
sub1_02.txt 268 ms 104904 KB
sub1_03.txt 109 ms 63288 KB
sub1_04.txt 109 ms 67568 KB
sub1_05.txt 1911 ms 167256 KB
sub1_06.txt 272 ms 106504 KB
sub1_07.txt 135 ms 58784 KB
sub1_08.txt 65 ms 18604 KB
sub1_09.txt 98 ms 24528 KB
sub1_10.txt 291 ms 59332 KB
sub1_11.txt 57 ms 15044 KB
sub1_12.txt 55 ms 14416 KB
sub1_13.txt 52 ms 14176 KB
sub1_14.txt 56 ms 14172 KB
sub1_15.txt 53 ms 14300 KB
sub1_16.txt 53 ms 14276 KB
sub1_17.txt 1716 ms 174100 KB
sub1_18.txt 3314 ms 179648 KB
sub1_19.txt 3314 ms 179636 KB
sub1_20.txt 3314 ms 179988 KB
sub1_21.txt 3314 ms 173212 KB
sub1_22.txt 3315 ms 193832 KB
sub1_23.txt 3314 ms 174844 KB
sub1_24.txt 3314 ms 181384 KB
sub1_25.txt 3314 ms 179364 KB
sub1_26.txt 3314 ms 178912 KB
sub1_27.txt 3314 ms 173780 KB
sub1_28.txt 3313 ms 156640 KB
sub1_29.txt 3315 ms 187212 KB
sub1_30.txt 3314 ms 161892 KB
sub1_31.txt 277 ms 111868 KB
sub1_32.txt 3312 ms 123936 KB
sub1_33.txt 3314 ms 166952 KB
sub1_34.txt 3314 ms 170532 KB
sub1_35.txt 3312 ms 117704 KB
sub1_36.txt 3314 ms 187832 KB
sub1_37.txt 3315 ms 193136 KB
sub1_38.txt 3312 ms 169952 KB
sub1_39.txt 799 ms 88716 KB
sub1_40.txt 3315 ms 192996 KB
sub1_41.txt 3313 ms 170608 KB
sub1_42.txt 51 ms 14492 KB
sub1_43.txt 56 ms 14412 KB
sub1_44.txt 52 ms 14372 KB
sub1_45.txt 55 ms 14328 KB
sub1_46.txt 52 ms 14360 KB
sub1_47.txt 491 ms 101032 KB
sub1_48.txt 665 ms 108868 KB
sub1_49.txt 375 ms 65828 KB
sub1_50.txt 160 ms 79836 KB
sub1_51.txt 780 ms 109604 KB
sub1_52.txt 824 ms 96572 KB
sub1_53.txt 600 ms 83900 KB
sub1_54.txt 753 ms 85384 KB
sub1_55.txt 3314 ms 177656 KB
sub1_56.txt 3315 ms 193140 KB
sub1_57.txt 3314 ms 168860 KB
sub1_58.txt 3314 ms 174196 KB
sub1_59.txt 3315 ms 191904 KB
sub1_60.txt 3315 ms 202824 KB
sub1_61.txt 1178 ms 158684 KB
sub1_62.txt 3315 ms 191812 KB
sub1_63.txt 66 ms 19476 KB
sub1_64.txt 3313 ms 153872 KB
sub1_65.txt 86 ms 45504 KB
sub1_66.txt 3314 ms 173992 KB
sub1_67.txt 1247 ms 134764 KB
sub1_68.txt 3313 ms 168808 KB
sub1_69.txt 3313 ms 176092 KB
sub1_70.txt 3313 ms 170672 KB
sub1_71.txt 52 ms 14304 KB
sub1_72.txt 56 ms 14552 KB
sub1_73.txt 51 ms 14348 KB
sub1_74.txt 227 ms 27864 KB
sub1_75.txt 96 ms 17572 KB
sub1_76.txt 547 ms 54336 KB
sub1_77.txt 62 ms 19468 KB
sub1_78.txt 64 ms 19380 KB
sub1_79.txt 75 ms 23788 KB
sub1_80.txt 79 ms 23932 KB
sub1_81.txt 78 ms 24228 KB
sub1_82.txt 92 ms 26504 KB
sub1_83.txt 53 ms 14232 KB
sub1_84.txt 75 ms 21672 KB
sub1_85.txt 91 ms 23900 KB
sub1_86.txt 52 ms 14452 KB
sub1_87.txt 54 ms 14512 KB
sub1_88.txt 53 ms 14244 KB
sub1_89.txt 412 ms 54088 KB
sub1_90.txt 408 ms 54144 KB
sub1_91.txt 429 ms 53884 KB
sub1_92.txt 411 ms 53972 KB
sub1_93.txt 427 ms 54024 KB
sub1_94.txt 52 ms 14308 KB | 2,700 | 6,006 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.953125 | 3 | CC-MAIN-2020-24 | latest | en | 0.533631 |
https://www.r-bloggers.com/2010/04/significant-figures-in-r-and-rounding/ | 1,632,253,082,000,000,000 | text/html | crawl-data/CC-MAIN-2021-39/segments/1631780057227.73/warc/CC-MAIN-20210921191451-20210921221451-00044.warc.gz | 994,965,561 | 23,896 | [This article was first published on Taking the Pith Out of Performance, and kindly contributed to R-bloggers]. (You can report issue about the content on this page here)
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This is a follow-on to my previous post about determining significant digits, or sigdigs, in performance and capacity planning calculations. Once we know how to do that, inevitably we will be faced with rounding the result of a calculation to the least number of sigdigs. Whereas the signif() function in R suffered from truncating trailing info-zeros in measured values, when it comes to rounding, signif shines. Better yet, it agrees with the Algorithm 3.2 in my GCaP book. Let’s see how well it does.
Example (Old Rule): Consider the number
7.245
which, from the previous post, we know has 4 sigdigs and now, we wish to round it to 3 sigdigs. The 3rd sigdig is ‘4’ and the 4th sigdig (the one we plan to drop) is ‘5’. According the old-school rule, the ‘5’ tells us we should round the preceding ‘4’ up, i.e., increment the ‘4’ to a ‘5’ and indeed, that is what Excel does. The function ROUND(7.245,2) produces:
7.25
The second argument in ROUND (the ‘2’) refers the 2nd decimal place, rather than the number of rounded sigdigs.
The old-school rule is now considered to be biased by the odd-even parity of nearby digits. As explained in Chapter 3, the new-school rule takes that effect into account. Suppose we have a generic number nn..nnXYZ and we want to round it to the position where the digit X now sits. How do we do it? In pseudocode, the new rule (Algorithm 3.2) tells us how:
a. Scan from left to right and examine digit Y
b. If Y < 5 then goto (i)
c. If Y > 5 then set X = X + 1 and goto (i)
d. If Y == 5 then examine Z
e. If Z >= 1 then set Y = Y + 1 and goto (a)
f. If Z is blank or a string of zeros then
g. Examine the parity of X (odd/even)
h. If X is odd then set X = X + 1
i. Drop Y and all trailing digits
I have also implemented this algorithm in Perl.
Example (New Rule): Round 7.245 to 3 sigdigs using the new rule. Start by rewriting the number without the decimal point. To make the steps clearer, I’ll write three rows with the first row enumerating the position of each digit of our number; shown in the second row:
1 2 3 4 5
7 2 4 5 _
_ _ x y z
The third row indicates the alignment of the X, Y and Z labels in the rounding algorithm with X in the 3rd position because we want to round to 3 sigdigs.
From step (a) Y equals 5 and step (d) says to look at Z, which is blank. Since it’s blank, step (g) says look at the parity of X = 4, which is even. Finally, step (i) tells us to drop everything from digit Y to the right. The result of rounding to 3 sigdigs in this way is therefore:
7.24
which is different from the Excel result. Let’s compare signif in R with Algorithm 3.2:
> signif(7.245,3)
[1] 7.24
which is in agreement with the new rule, in this case. Unlike Excel ROUND, the second argument in signif is the number of sigdigs to be displayed. Let’s try some other examples in R.
Example (R signif): The test numbers in this table
Number SD Algor
1 62.53470 4 62.53
2 3.78721 3 3.79
3 726.83500 5 726.84
4 24.85140 3 24.90
show the value to be rounded in the first column, the number of rounded significant digits (SD) in the second column, and the rounded result (Algor) obtained by applying Algorithm 3.2.
The following R code appends the value obtained with signif
for(i in 1:dim(tabl)[1]) {
tablstr<-sprintf("%dt%8.4ft%dt%6.2ft%6.2fn",
i, tabl$Number[i], tabl$SD[i], tabl$Algor[i], signif(tabl$Number[i],tabl\$SD[i]))
cat(tablstr)
}
to the Rsn column in this table:
Number SD Algor Rsn
1 62.5347 4 62.53 62.53
2 3.7872 3 3.79 3.79
3 726.8350 5 726.84 726.84
4 24.8514 3 24.90 24.90
Since the Algor and Rsn values agree, it looks like signif incorporates the new rounding rule so, it can be used as is, straight out of the box. Nice. | 1,199 | 4,015 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.90625 | 4 | CC-MAIN-2021-39 | longest | en | 0.903361 |
https://www.datainsightonline.com/post/an-analysis-of-coronavirus-cases-across-the-world | 1,721,088,282,000,000,000 | text/html | crawl-data/CC-MAIN-2024-30/segments/1720763514724.0/warc/CC-MAIN-20240715224905-20240716014905-00078.warc.gz | 638,357,545 | 182,342 | top of page
# An analysis of Coronavirus cases across the world.
The Covid-19 pandemic has changed the world in great ways this year. It has pushed all human beings to focus and unite against one common enemy. It has also changed a lot in the ways we deal with daily life. Many countries have instigated curfews and confinements that have made our time indoors much longer. It is a strange time we live in.
Today I have tried to answer several questions that arose while exploring and combining datasets.
The datasets I have explored are the following :
Which I used for the information it contains on mortality rates from covid-19 around the world.
Which I used for the general information it contains on the countries of the world.
Which I used to determine ethnic diversity in countries.
Which I used for bloodwork results from covid-19 and other patients.
The notebook can be found here: https://github.com/FarahKa/covid_assignment
This blog will follow the order of the answered questions while hopefully contributing insights on the Covid-19 pandemic and our world.
1) Which are the most hit countries, using total deaths?
2) Which are the most hit countries, using total deaths/100000 inhabitants?
3) Are more populous countries hit?
4) What is the correlation between total deaths, habitants and GDP?
5) What is the relationship between total deaths and how diverse a country is?
6) What is the difference between the bloodwork of people with covid and people without covid?
1) Which are the most hit countries, using total deaths?
We start by loading the datasets, then doing primary work to make it possible to merge them. Here we merge the latest coronavirus data with the population info of countries:
```recent.rename(columns = {'location':'Country'}, inplace = True)
recent['Country']=recent['Country'].str.strip()
pop['Country']=pop['Country'].str.strip()
df=pd.merge(recent, pop)
df=df.set_index('Country')```
Which lets us determine the five most hit countries by total death count:
```df=df.sort_values('total_deaths', ascending=False)
print("Five most hit countries by total deaths:")
Five most hit countries by total deaths: Country United States 40682 Italy 23660 Spain 20453 France 19718 United Kingdom 16060
2) Which are the most hit countries, using total deaths/100000 inhabitants?
However it seems unfair to judge smaller countries and bigger countries on the same scale, which takes us to dividing the death count by 100000 inhabitants. The result is as follows:
Five most hit countries by total deaths/100000 inhabitants: Country San Marino 133.328775 Belgium 54.754440 Spain 50.628942 Andorra 50.561088 Italy 40.699418
We can notice the difference in results, though knowing San Marino's proximity to Italy, it is not surprising.
This led me to the following question:
3) Are more populous countries hit by the coronavirus?
The more people, the bigger the risk, right? We tested the hypothesis as such:
```#3) Are more populous countries hit?
print(np.corrcoef(df_nozero.total_deaths, df_nozero.Population))
#it seems there is a slight positive correlation between population and total deaths when we consider all countries
print(np.corrcoef(df2.total_deaths, df2.Population))
#but when we consider the 50 most hit countries, the positive correlation spikes up to around 0.7
sns.scatterplot(df2['total_deaths'], df2['Population'])
plt.xlabel("Total Deaths")
plt.ylabel("Population")
sns.regplot(x='total_deaths', y='Population', data=df2)
plt.show()
```
When we consider the 50 most hit countries, the positive correlation spikes up to around 0.7.
The hypothesis might be correct.
Another question presents itself: Do poorer countries fare worse than richer countries? Or does the Coronavirus have the same kind of impact on countries regardless of GDP?
4) What is the correlation between total deaths, habitants and GDP?
Hypothesis: the lower the GDP, the more poorly a country fares.
```
#Total deaths and GDP: for all countries with deaths:
sns.scatterplot(df_nozero.death_by_pop, df_nozero['GDP (\$ per capita)'])
sns.regplot(df_nozero.death_by_pop, df_nozero['GDP (\$ per capita)'])
plt.xlabel("Total Deaths / 100000 inhabitants")
plt.ylabel("GDP (\$ per capita)")
plt.show()
print(np.corrcoef(df_nozero.death_by_pop, df_nozero['GDP (\$ per capita)']))
```
A correlation of 0.4 is considered weak to moderate. The hypothesis is disproved since the correlation is positive: Countries with bigger GDP seem to be more hit by the covid19 pandemic.
5) What is the relationship between total deaths and ethnic diversity in a country ?
The hypothesis is the following: If a country is more diverse, there would be more traveling to and from it, which would make it more at risk of a big spike of cases.
We have used the Historical Index of Ethnic Fractionalization Dataset. The Index is the probability that two randomly picked individuals from the population pool would be of different ethnicities.
First we have explored the dataset on its own:
```#Which are the most ethnically diverse countries in this dataset?
print("The most ethnically diverse countries in this dataset:")
#And the least diverse?
print("The least ethnically diverse countries in this dataset:")
The results are as follows:
The most ethnically diverse countries in this dataset: EFindex Country Liberia 0.889 Uganda 0.883 Togo 0.880 Nepal 0.860 South Africa 0.856 Chad 0.855 Kenya 0.855 Mali 0.852 Nigeria 0.850 Guinea-Bissau 0.808 The least ethnically diverse countries in this dataset: EFindex Country Japan 0.019 Democratic People's Republic of Korea 0.020 Bangladesh 0.025 Tunisia 0.034 Egypt 0.041 Jordan 0.044 Armenia 0.045 Comoros 0.054 Poland 0.069 Republic of Korea 0.095
We notice the interesting fact that most diverse countries are african.
We then combined the death rates and diversity data to see if there was any correlation. All combinations we did had this kind of appearance:
Which disproves our hypothesis: diversity is uncorrelated with death rate.
The last question comes from the exploration of another dataset containing clinical bloodwork of covid and non-covid patients:
6) What is the difference between the bloodwork of people with covid and people without covid? The attempt is to find the variables in the bloodwork whose mean differs by more than 0.5 between covid patients and other patients.
First we did some exploration by separating sick and healthy patients who were tested for covid 19 (which meant the healthy ones presented the symptoms but did not have the sickness).
```sick=clinical.loc[clinical['sars_cov_2_exam_result'] == 'positive' ]
healthy=clinical.loc[clinical['sars_cov_2_exam_result'] == 'negative' ]
sick.describe()
healthy.describe()
```
Then we searched for variables in the bloodwork whose mean differs by more than 0.5 between covid patients and other patients. patient_age_quantile 1.456424 platelets -0.818720 leukocytes -0.836818 eosinophils -0.558663 monocytes 0.571967 ionized_calcium -0.938920 segmented 0.642866 ferritin 0.641919 pco2_arterial_blood_gas_analysis -0.648399 ph_arterial_blood_gas_analysis 0.630072 po2_arterial_blood_gas_analysis 0.625503 arteiral_fio2 -0.624296 phosphor -0.561041 cto2_arterial_blood_gas_analysis 0.535281 It mignt be useful to consider these variables more closely in the proccess of understanding the disease and diagnosing it.
Insights:
We have learned through this analysis to find new questions to ask about a very important issue, and searching for ways to answer them.
These questions relied mostly on number of deaths which is a variable that does not depend on how much testing there was or whether a country reports accurately.
Hopefully the information gathered can help some understand this disease and where it stems from better.
Thank you for reading and stay safe. | 1,773 | 7,838 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.671875 | 3 | CC-MAIN-2024-30 | latest | en | 0.917721 |
https://forums.ni.com/t5/LabVIEW/Osciloscope-with-NI-9242/m-p/3555891?profile.language=en | 1,569,179,605,000,000,000 | text/html | crawl-data/CC-MAIN-2019-39/segments/1568514575627.91/warc/CC-MAIN-20190922180536-20190922202536-00104.warc.gz | 478,582,965 | 24,065 | # LabVIEW
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## Osciloscope with NI 9242
Dear community, at the moment I am reading the values of voltage curves using NI 9242 module. The whole system is a test bed for synchronous generator which works in the island mode, but it also should be able to connect to a stiff grid (for the sake of experiment). The 9242 is working really good, so I can see frequency fluctuations in the grid - the actual value of voltage frequency is deviating between (approximately) 49.97 and 50.02 Hz (that is in Munich, if you are interested). Because frequency is varying the phase of the signal also varies. So my question is:
what technique should I use to receive a mean value of sine wave curve, and so to have a stable phase and frequency of the grid? Should it be some filter, or smth else? I am a complete rookie in signal processing, so any ideas would help me.
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## Re: Osciloscope with NI 9242
Let's first review some of the terminology and concepts.
1 - When you say you want to measure the mean value of your sine curve you probably mean the 'amplitude' in V or 'level´ in Vrms. The mean value (Vdc) is probably 0 or negligible
2 - Phase is always a relative number. You measure your phase relative to a specific reference that you need to specify. It could for example be the phase of an ideal 50.00 Hz signal at a given time
But I think what you are asking is a VI that can measure the actual average frequency (over a few periods) and the amplitude or level of your signal.
Try to use the Extract Single Tone Information VI in your Signal Processing>>Wfm Measure palette. The VI returns the frequency of the fundamental signal (approx 50 Hz) the amplitude of that fundamental and the phase relative to the beginning of your input signal.
From there you can derive different values of interest. Note that the amplitude is for the fundamental tone of 50 Hz only.
- If you need the RMS value of the fundamental only just divide the amplitude value by SQRT(2)
- If you need the amplitude of your entire waveform the simpleste is to find the min and max value in your data and calculate (Vmax-Vmin)/2. This may be sensitive to noise in your signal. You may consider adding low-pass filtering to remove part of the noise.
- If you need the RMS value of the entire signal (not just the 50 Hz fundamental) you should use the Basic DC-RMS VI in the same palette. Select Hanning window to get a stable result.
The length of the signal you analyze should be long enough to return stable values (except for phase that likely will jump around) but short enough to not be affected by frequency drift. I expect record lengths of 100 ms to 1 s would work fine.
Alain
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(800 Views) | 634 | 2,771 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.953125 | 3 | CC-MAIN-2019-39 | latest | en | 0.899784 |
http://www.chestnut.com/en/eg/6/mathematics/7/2/examples/3/ | 1,477,441,723,000,000,000 | text/html | crawl-data/CC-MAIN-2016-44/segments/1476988720471.17/warc/CC-MAIN-20161020183840-00115-ip-10-171-6-4.ec2.internal.warc.gz | 377,957,198 | 16,468 | # 6.7.2. Geometric Transformations and Translation Transformation
Find the image of the where , , and by translation .
• A, ,
• B, ,
• C, ,
• D, ,
### Example
Find the image of the where , , and by translation .
### Solution
Since the triangle is translated by , we can find the image by adding 3 to each -value and add 2 to each -value.
0
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skipped | 104 | 376 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.765625 | 3 | CC-MAIN-2016-44 | latest | en | 0.821513 |
http://math.stackexchange.com/questions/367917/abi-is-prime-in-mathbbzi-if-and-only-if-a2b2-is-prime-in-mathbb | 1,469,469,205,000,000,000 | text/html | crawl-data/CC-MAIN-2016-30/segments/1469257824337.54/warc/CC-MAIN-20160723071024-00146-ip-10-185-27-174.ec2.internal.warc.gz | 159,519,988 | 20,232 | # $a+bi$ is prime in $\mathbb{Z}[i]$ if and only if $a^2+b^2$ is prime in $\mathbb{Z}$
$a+bi$ is prime in $\mathbb{Z}[i]$ if and only if $a^2+b^2$ is prime in $\mathbb{Z}$.
How can I prove this? Can anybody help me please?
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$a^2+b^2=(a+ib)(a-ib)$ – Mathematician Apr 21 '13 at 4:35
this is obvious.but I could not understand how should I use this result – habuji Apr 21 '13 at 4:40
False. Three is prime in $\mathbb{Z}[i]$, yet $3^2+0^2=9$ is not prime in $\mathbb{Z}$. – Jyrki Lahtonen Apr 21 '13 at 4:43
If the norm is prime, it can't be the product of Norms of two other numbers unless one is a unit. However the converse isn't always true. – Macavity Apr 21 '13 at 4:57
you would need restrictions on your statements, such as both a, b are nonzero, as @Jyrki has shown the hole there. Also, $a^2+b^2$ can not be of the form $4k+3$ (again, as is 3). – Eleven-Eleven Apr 21 '13 at 5:01
This is not true when $b=0$ and $a$ is an ordinary prime of the form $4k+3$. And for the same reason it is not true if $a=0$ and $b$ is an ordinary prime of the form $4k+3$.
Added: If $a^2+b^2$ is an ordinary prime, then $a+bi$ is a Gaussian prime. For suppose that $a+bi=(s+ti)(u+vi)$. By a norm calculation we have $a^2+b^2=(s^2+t^2)(u^2+v^2)$. So since $a^2+b^2$ is prime, one of $s+ti$ or $u+vi$ is a unit.
For the other direction, suppose $a+bi$ is a Gaussian prime, where neither $a$ nor $b$ is equal to $0$. We show that $a^2+b^2$ is an ordinary prime. Proof would be easy if we assume standard results that characterize the Gaussian primes. So we try not to use much machinery.
If $a$ and $b$ are both odd, then $a+bi$ is divisible by $1+i$. Then $a+bi$ is an associate of $1+i$, and $a^2+b^2=2$.
So we may assume that $a$ and $b$ have opposite parities. In that case, $a+bi$ and $a-bi$ are relatively prime. For any common divisor $\delta$ must divide $2a$ and $2b$. Since $a$ and $b$ have opposite parity, any common divisor divides $a$ and $b$, so must have norm $1$ if $a+bi$ are prime.
Now suppose that $p$ is a prime that divides $a^2+b^2$. Then $p$ divides $(a+bi)(a-bi)$. Note that $p$ cannot be a Gaussian prime, else it would divide one of $a+bi$ or $a-bi$,
Let $\pi$ be a Gaussian prime that divides $p$. Then $\pi$ divides one of $a+bi$ or $a-bi$. So $\pi$ must be an associate of one of these, and the conjugate of $\pi$ is an associate of the other. Since $a+bi$ and $a-bi$ arer relatively prime, we conclude that $(a+bi)(a-bi)$ divides $p$, which forces $p=a^2+b^2$.
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Here is something that is true: $a+bi$ is prime in $\def\Z{\Bbb Z}\Z[i]$ implies that $(a+bi)\Z[i]\cap\Z=p\Z$ for some (ordinary) prime $p$ of $\Z$. This is because, (1) since $a+bi$ is irreducible and $\Z[i]$ is a Euclidean and therefore unique factorisation domain, $(a+bi)\Z[i]$ is a prime ideal of $\Z[i]$, so (2) its intersection with $\Z$ is a prime ideal of $\Z$ (as is always true for the intersection of a prime ideal and a subring), and (3) the intersection is not reduced to$~\{0\}$ because $(a+bi)(a-bi)=a^2+b^2\in\Z\setminus\{0\}$. (Here, like in the question, one does not have "if and only if": here the converse fails for $a+bi$ equal or associated to a prime number $p\not\equiv3\pmod4$, such as $p=2$ or $p=5$; then $(a+bi)\Z[i]\cap\Z=p\Z$, but $p$ and therefore $a+bi$ are composite in $\Z[i]$.)
The case $a+bi$ is prime in $\Z[i]$ splits into two subcases. By the above there exists a prime number $p$ and $z\in\Z[i]$ with $p=(a+bi)z$; then $p^2=N(p)=N(a+bi)N(z)$, and either $z$ is non-invertible, in which case $N(a+bi)=p$ and $z=a-bi$, or $z$ is invertible, in which case $a+bi\in\{p,ip,-p,-ip\}$ and it can be shown that $p\equiv3\pmod4$. Indeed, the irreducibility of $p$ in the UFD $\Z[i]$ means that $\Z[i]/p\Z[i]$ is an integral domain (it is a field), so the kernel $(X^2+1)$ of the ring morphism $(\Z/p\Z)[X]\to\Z[i]/p\Z[i]$ sending $X\mapsto i$ is a prime ideal, so $X^2+1\in(\Z/p\Z)[X]$ is irreducible, which excludes both $p=2$ (for which $X^2+1=(X+1)^2$) and $p\equiv1\pmod4$ (in which case $X^2+1=(X+a)(X-a)$ for some element $a$ of order $4$ in the cyclic group $(\Z/p\Z)^\times$ of order $p-1$.
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The first paragraph does not make clear precisely what is a consequence of said UFD property (that prime ideals are preserved under contraction is true in any ring). – Math Gems Apr 21 '13 at 16:20
@MathGems: The "since ... UFD" in the second sentence was awkwardly placed very early to indicate that that it is needed to justify the immediately following "is a prime ideal". And you are right, pulling back a prime ideal through a ring morphism always gives a prime ideal; maybe that is why they are so useful. – Marc van Leeuwen Apr 21 '13 at 16:45
I surmised what you intended. But I fear that those beginning their studies might have more difficulty inferring the intended meaning. Whenever I encounter things like that I leave comments in the hope that the author might improve the exposition to eliminate ambiguities etc. Thankfully many folks do the same for me when I too do likewise. – Math Gems Apr 21 '13 at 17:13
@MathGems: Fine, thank you. Did I succeed in reducing the ambiguity, or if not what would be better? – Marc van Leeuwen Apr 21 '13 at 18:39
Yes, that reads much more clearly. Thanks and +1. – Math Gems Apr 21 '13 at 19:05 | 1,781 | 5,275 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4 | 4 | CC-MAIN-2016-30 | latest | en | 0.90197 |
https://www.doubtnut.com/qna/649503272 | 1,721,528,164,000,000,000 | text/html | crawl-data/CC-MAIN-2024-30/segments/1720763517544.48/warc/CC-MAIN-20240720235600-20240721025600-00229.warc.gz | 652,407,614 | 36,091 | # If a,b,c are respectively the pth,qth,rth ters of an A.P., then ∣∣ ∣ ∣∣ap1bq1cr1∣∣ ∣ ∣∣=
A
1
B
1
C
0
D
pqr
Text Solution
Verified by Experts
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Updated on:21/07/2023
### Knowledge Check
• Question 1 - Select One
## If a,b,c be respectively the pth,qthandrth terms of a H.P., then Δ=∣∣ ∣ ∣∣bccaabpqr111∣∣ ∣ ∣∣ equals
A1
B0
C-1
Dpqr
• Question 2 - Select One
## If a>0,b>0,c>0 are respectively the pth,qth,rth terms of a G.P.,. Then the value of the determinant ∣∣ ∣ ∣∣logap1logbq1logcr1∣∣ ∣ ∣∣ is
A1
B-1
Cabcpqr
D0
• Question 3 - Select One
## If a,b,c are respectively the pth,qthandrth terms of a G.P. show that (q−r)loga+(r−p)logb+(p−q)logc=0.
A0
B1
C1
Dpqr
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how do you use x and y intercepts to graph a line in standard form . | 911 | 4,549 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.875 | 3 | CC-MAIN-2019-13 | latest | en | 0.790712 |
https://homeworkontime.com/2021/08/24/south-university-biostatistical-method-hypothesis-testing-and-inference-hw-hypothesis-testing-and-inferencethis-assignment-focuses-on-estimation-and-hypo/ | 1,669,828,439,000,000,000 | text/html | crawl-data/CC-MAIN-2022-49/segments/1669446710765.76/warc/CC-MAIN-20221130160457-20221130190457-00601.warc.gz | 350,262,848 | 14,927 | # South University Biostatistical Method Hypothesis Testing and Inference HW Hypothesis Testing and Inference This assignment focuses on estimation and hypo
South University Biostatistical Method Hypothesis Testing and Inference HW Hypothesis Testing and Inference
This assignment focuses on estimation and hypothesis testing with one-sample and two-sample inferences.
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The essence of parametric testing is the use of standard normal distribution tables of probabilities. For each exercise, there will be a sample problem that shows how the calculations are done and at least one problem for you to work out.
For the first assignment, you will not need any statistical software. However, you will use a standardized normal distribution table (a z-score table) provided in the course textbook (Table 3—The normal distribution—in the Tables section in APPENDIX) to obtain your responses.
Problem 1: Probability Using Standard Variable z and Normal Distribution Tables
Variables are the things we measure. A hypothesis is a prediction about the relationship between variables. Variables make up the words in a hypothesis.
In the attention-deficit/hyperactivity disorder’s (ADHD’s) hypothetical example provided in the tables below, the research question was: What is the most effective therapy for ADHD? One of the variables is type of therapy. Another variable is change in ADHD-related behavior, given exposure to therapy. You might measure change in the mean seconds of concentration time when children read. This experiment is designed to obtain children’s concentration times while they read a science textbook and to find out whether the therapy used worked on any of the children.
Use the stated µ and σ to calculate probabilities of the standard variable z to get the value of p (up to three decimal places). In addition, respond to the following questions for each pair of parameters:
Which child or children, if any, appeared to come from a significantly different population than the one used in the null hypothesis?
What happens to the “significance” of each child’s data as the data are progressively more dispersed?
In addition to the above, write a formal statement of conclusion for each child in APA style. A report template is provided for submission of your work.
Note: Tables 1 and 2 are practice tables with answers. Tables 3 and 4 are the assignment tables for you to work on.
Table 1 (µ = 100 seconds and σ = 10)
Child Mean seconds of concentration in an experiment of reading z-score
(z = [X – µ]/σ) p-value
1 75 -2.50 0.0
2 81 -1.90 0.0
3 89 -1.10 0.1
4 99 -0.10 0.4
5 115 1.50 0.0
6 127 2.70 0.0
7 138 3.80 <0.0 8 139 3.90 <0.0 9 142 4.20 <0.0 10 148 4.80 <0.0 Table 2 (µ = 100 seconds and σ = 20) Child Mean seconds of concentration in an experiment of reading z-score (z = [X – µ]/σ) p-value 1 75 -1.25 0.1 2 81 -0.95 0.1 3 89 -0.55 0.2 4 99 -0.05 0.4 5 115 0.75 0.2 6 127 1.35 0.0 7 138 1.90 0.0 8 139 1.95 0.0 9 142 2.10 0.0 10 148 2.40 0.0 Table 3 (µ = 100 seconds and σ = 30) Child Mean seconds of concentration in an experiment of reading z-score p-value 1 75 -0.83 2 81 -0.63 3 89 -0.37 4 99 -0.03 5 115 0.50 6 127 0.09 7 138 1.27 8 139 1.30 9 142 1.40 10 148 1.60 Table 4 (µ = 100 seconds and σ = 40) Child Mean seconds of concentration in an experiment of reading z-score p-value 1 75 -0.63 2 81 -0.48 3 89 -0.28 4 99 -0.03 5 115 0.38 6 127 0.68 7 138 0.95 8 139 0.98 9 142 1.05 10 148 1.20 Problem 2: Two-Sample Inferences A two-sample inference deals with dependent and independent inferences. In a two-sample hypothesis testing problem, underlying parameters of two different populations are compared. In a longitudinal (or follow-up) study, the same group of people is followed over time. Two samples are said to be paired when each data point in the first sample is matched and related to a unique data point in the second sample. This problem demonstrates inference from two dependent (follow-up) samples using the data from the hypothetical study of new cases of tuberculosis (TB) before and after the vaccination was done in several geographical areas in a country in sub-Saharan Africa. Conclusion about the null hypothesis is to note the difference between samples. The problem that demonstrates inference from two dependent samples uses hypothetical data from the TB vaccinations and the number of new cases before and after vaccination. Table 5: Cases of TB in Different Geographical Regions Geographical regions Before vaccination After vaccination 1 85 11 2 77 5 3 110 14 4 65 12 5 81 10 6 70 7 7 74 8 8 84 11 9 90 9 10 95 8 Using the Minitab statistical analysis program to enter the data and perform the analysis, complete the following: Construct a one-sided 95% confidence interval for the true difference in population means. Test the null hypothesis that the population means are identical at the 0.05 level of significance In addition, in a Microsoft Word document, provide a written interpretation of your results in APA format. Problem 3: Cross-Sectional Study In a cross-sectional study, the participants are seen at only one point of time. Two samples are said to be independent when the data points in one sample are unrelated to the data points in the second sample. The problem that demonstrates inference from two independent samples will use hypothetical data from the American Association of Poison Control Centers. There are two groups of independent data collected in different regions, which also calls for a t-test. The numbers represent the number of recorded cases of poisoning with chemicals in the homes of 100,000 people in two regions. Table 6: Cases of Poisoning With Chemicals Year Region 1 Region 2 1 150 11 2 160 10 3 132 14 4 110 12 5 85 10 6 45 11 7 123 9 8 180 11 9 143 10 10 150 14 Using the Minitab statistical analysis program to enter the data and perform the analysis, complete the following: Formulate a null and an alternative hypothesis for a two-sided test. Conduct the test at the 0.05 level of significance. In addition, in a Microsoft Word document, provide a written interpretation of your results in APA format.
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## Essay Writing Service
You are welcome to choose your academic level and the type of your paper. Our academic experts will gladly help you with essays, case studies, research papers and other assignments. | 2,078 | 8,495 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.734375 | 4 | CC-MAIN-2022-49 | latest | en | 0.892305 |
http://mathematica.stackexchange.com/questions/28987/finding-intersection-of-two-graphs | 1,469,462,365,000,000,000 | text/html | crawl-data/CC-MAIN-2016-30/segments/1469257824319.59/warc/CC-MAIN-20160723071024-00213-ip-10-185-27-174.ec2.internal.warc.gz | 145,171,776 | 18,686 | # Finding intersection of two graphs
Here is a toy problem:
I have a (directional) line that starts from {2,0} and goes through {1.8, 0.01}. Also I have a parametric plot defined like this: {Sin[u], Sin[2 u]} where 0 <= u <= 2 Pi.
I would like to find the intersection of these two.
I tried something like:
Solve[a*({1.8, .01} - {2, 0}) + {2, 0} == {Sin[u], Sin[2 u]}
&& 0 <= u < 2 Pi && a > 0, {u, a}]
But both Solve and NSolve seem to struggle with this. Findroot is able to find an answer quickly but it gives me only one answer depending on the starting point while I'd like to have all correct answers, four in this case.
Also I see if I redefine the line section so that it has only one intersecting point with the curve (e.g. line starts at {0, 0} and goes through {-0.1 , 0.01}) then Solve is able to find that one answer comfortably!
Any idea how to solve this?
-
Please include working code for both your line and the parametric plot. – Mr.Wizard Jul 22 '13 at 9:40
@Mr.Wizard I think my question boils down to: how can I solve this Solve[a*({1.8, .01} - {2, 0}) + {2, 0} == {Sin[u], Sin[2 u]} && 0 <= u < 2 Pi && a > 0, {u, a}] – user5131 Jul 22 '13 at 9:45
At first, one should appropriately define the system of equations. Instead of machine precission numbers we prefer exact numbers therefore we would define:
{a, b} /. Solve[{ 9/5 a + b == 1/100, 2 a + b == 0}, {a, b}]
{{-(1/20), 1/10}}
Now
pts = {Sin[u], Sin[2 u]} /. Solve[{ x == Sin[u],
Sin[2 u] == -(1/20) x + 1/10, -2 Pi <= u <= 2 Pi},
{u}, {x}, Reals]
{ {Sin[2 ArcTan[Root[1 - 41 #1 + 2 #1^2 + 39 #1^3 + #1^4 &, 1]]],
Sin[2 (2 Pi + 2 ArcTan[ Root[1 - 41 #1 + 2 #1^2 + 39 #1^3 + #1^4 &, 1]])]},
{Sin[ 2 ArcTan[Root[1 - 41 #1 + 2 #1^2 + 39 #1^3 + #1^4 &, 2]]],
Sin[2 (2 Pi + 2 ArcTan[ Root[1 - 41 #1 + 2 #1^2 + 39 #1^3 + #1^4 &, 2]])]},
{Sin[2 ArcTan[Root[1 - 41 #1 + 2 #1^2 + 39 #1^3 + #1^4 &, 3]]],
Sin[4 ArcTan[Root[1 - 41 #1 + 2 #1^2 + 39 #1^3 + #1^4 &, 3]]]},
{Sin[2 ArcTan[Root[1 - 41 #1 + 2 #1^2 + 39 #1^3 + #1^4 &, 4]]],
Sin[4 ArcTan[Root[1 - 41 #1 + 2 #1^2 + 39 #1^3 + #1^4 &, 4]]]} }
N @ %
{{-0.0513515, 0.102568}, {-0.997173, 0.149859},
{0.0488373, 0.0975581}, {0.999687, 0.0500156}}
ParametricPlot[{ {Sin[u], Sin[2 u]},
{u, (-(1/20) u + 1/10) ConditionalExpression[1, -5/4 <= u <= 5/4]}},
{u, -2 Pi, 2 Pi},
Epilog -> {Red, PointSize[0.017], Point[pts]}]
Edit
Your original system works as well if you add the specification domain: Reals (mind different variables here).
Solve[ a ({1.8, .01} - {2, 0}) + {2, 0} == {Sin[u], Sin[2 u]} &&
0 <= u < 2 Pi && a > 0, {u, a}, Reals]
Solve::ratnz: Solve was unable to solve the system with inexact coefficients.
The answer was obtained by solving a corresponding exact system
and numericizing the result. >>
{{u -> 0.0488568, a -> 9.75581}, {u -> 1.54578, a -> 5.00156},
{u -> 3.19297, a -> 10.2568}, {u -> 4.63718, a -> 14.9859}}
alternatively substituting {1.8, .01} by {9/5, 1/100} yields appropriate results without adding domain specification, however in terms or radicals not explicitly real.
Even though 0 <= u < 2 Pi and a > 0 inequalities should restrict variables u and a to the real numbers, nonetheless most likely on the internal processing level there might appear some inconsistency. Therefore one should explicitly add the domain specification.
-
thank you. Adding 'Reals' solved my problem. – user5131 Jul 22 '13 at 12:54 | 1,272 | 3,392 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.703125 | 4 | CC-MAIN-2016-30 | latest | en | 0.815424 |
https://www.technologyreview.com/s/401979/puzzle-corner/ | 1,558,466,861,000,000,000 | text/html | crawl-data/CC-MAIN-2019-22/segments/1558232256546.11/warc/CC-MAIN-20190521182616-20190521204616-00215.warc.gz | 950,411,222 | 32,473 | # Puzzle Corner
Jul 1, 2003
Introduction
It has been a year since I reviewed the criteria used to select solutions for publication. Let me do so now.
As responses to problems arrive, they are simply put together in neat piles, with no regard to their date of arrival or postmark. When it is time for me to write the column in which solutions are to appear, I first weed out erroneous and illegible responses. For difficult problems, this may be enough; the most publishable solution becomes obvious. Usually, however, many responses still remain. I next try to select a solution that supplies an appropriate amount of detail and that includes a minimal number of characters that are hard to set in type. A particularly elegant solution is, of course, preferred, as are contributions from correspondents whose solutions have not previously appeared. I also favor solutions that are neatly written, typed, or sent via e-mail, since these produce fewer typesetting errors.
Problems
J/A 1. Larry Kells reports on an “oddball-sounding remark” he overheard at his bridge club. “That was an awfully risky grand slam you bid! You sure were lucky to get the six-zero trump break you needed to make it.” Naturally he was very curious but never did find out what had occasioned that remark. We are asking for help in solving this mystery.
J/A 2. Ken Rosatto has 12 identical coins and another that looks the same but weighs either more or less. He wants to find the oddball in three weighings using only a balance.
J/A 3. Rocco Giovanniello has a pyramidal variant of the tetrahedral “wink jumping” puzzle we published last year. The pyramid has four square layers. The top layer has one space, the second has four, the third nine, and the fourth 16. Each of the spaces except the top is occupied with a wink; the top is empty. The goal is to eliminate all but one wink by a sequence of 3-D checkerlike jumps.
Speed Department
David E. Brahm reports that a popular game in Naples, FL (at least, between his parents), is a dice game called farkle. A player gets points on his first throw if his dice either a) contain a one or a five, b) form three or more of a kind, or c) form three pairs. Otherwise the roll is a “farkle,” and the turn is over. What is the probability of rolling a farkle if six dice are used?
Solutions
Mar 1. I wonder whether Larry Kells was a Course I major: he is such an expert on bridge(s). Today he reports on an argument that arose at his bridge club; neither side was vulnerable. After South’s weak 2 opening, East-West bid up to 7 and made it for +1,510. Afterward, one of the kibitzers expressed an opinion that a 7 sacrifice would have been worthwhile for North-South. The four players all objected, saying that as long as West led his trump, 7 was obviously doomed to go down seven for 1,700, more than the value of the successful grand slam. Who was right? Here’s the deal:
David Cipolla enjoyed this problem and writes, “The kibitzer is right. With ideal execution by North and South, even after a West lead of the four of trumps, the sacrifice could have gone down only four and not seven as the players argued. The requirement is for North and South to give up a trick that they could win, thereby allowing a cross-ruff strategy to become effective. At trick one, North and South must play the three and two, respectively, losing to West’s four. With West still on lead, he must play and win the next three heart tricks while North (or South) drops three diamonds and South (or North) drops three clubs. Note that East cannot win the first round of hearts even if he did not previously discard the two and so is unable to gain the lead. When West leads his fourth heart, South ruffs and North drops his last diamond. South then plays any diamond with North ruffing. North then plays a club with South ruffing. This continues until North and South are out of clubs or diamonds. They each have one spade at the end of the cross ruff, and either North or South wins that trick. So East and West win only four tricks: one trump trick and three heart tricks. North and South win nine tricks: one heart ruff, four diamond ruffs, three club ruffs, and a trump trick at the end. Down four.
Mar 2. Andrew Russell offers an interesting variant of an old problem. You have nine coins all of equal weight. Some material has been removed from one coin and added to another so that the total weight is unchanged. You have four weighings on a balance scale. Can you find four weighings that permit you to determine the lighter and heavier coin without previous weighings?
Tom Terwilliger sent us the following detailed solution.
Weighing #1 1, 2, 3 vs. 4, 5, 6 Note that 9 is never weighed!
Weighing #2 1, 4, 7 vs. 2, 5, 8
Weighing #3 1, 5 vs. 2, 4
Weighing #4 3, 6 vs. 7, 8
You have 72 possibilities of light and heavy coins (9 x 8). In four weighings you have 81 possible results (34). The trick is to place the proper number of coins on each pan of the scale. Each weighing must divide the above 72 possibilities nearly in thirds, or else there won’t be a unique solution. I started by listing the results of using various numbers of coins in each pan as follows:
One coin in each pan: There are 15 ways each pan can be heavy and 42 ways the scale can balance. If you weigh 1 vs. 2, then either 1 can be heavy (eight possibilities) or 2 can be light (seven possibilities, not counting the case of 1 heavy and 2 light twice). Since seven coins are not weighed, the scale will balance any time both odd coins are in the group of seven (7 x 6). Clearly this is no good, as after this weighing, there are 42 possibilities left if the scale balances, and there are only 27 results possible on the remaining three weighings.
Two coins in each pan: Weigh 1, 2 vs. 3, 4. If 1, 2 is heavy, then the following are possible:
1 heavy 3, 4, 5, 6, 7, 8, 9 light 7 possibilities
(If 2 is light the scale balances)
2 heavy 3, 4, 5, 6, 7, 8, 9 light 7 possibilities
3 light 5, 6, 7, 8, 9 heavy 5 possibilities
(Don’t count 1 heavy and 3 light twice!)
4 light 5, 6, 7, 8, 9 heavy 5 possibilities
Total: 24 possibilities
There are also 24 ways pan 3, 4 can be heavy, and by subtraction 24 ways the scale can balance.
5, 6, 7, 8, 9 containing both odd coins 5 x 4 = 20 possibilities
1 heavy 2 light or the reverse 2 possibilities
3 heavy 4 light or the reverse 2 possibilities
Here, each weighing divides the possibilities exactly in thirds. I then tried to solve this with four weighings, each containing two coins per pan. I wrote a computer program to try all combinations and found that there are none, so I tried three coins in each pan.
Weigh 1, 2, 3 vs. 4, 5, 6. Pan 1, 2, 3 will be heavy if
1, 2, 3 heavy 4, 5, 6, 7, 8, 9 light 6 x 3 = 18 possibilities
7, 8, 9 heavy 4, 5, 6 light 3 x 3 = 9 possibilities
Total: 27 possibilities
The scale will balance any time the two odd coins are in the same group of three, which will happen 6 x 3 = 18 times.
This is not as good as 24/24/24, but since four weighings of two coins each didn’t work, the solution must involve combining two- and three-coin weighings. Now there is an easy way to do two (symmetric) weighings of three coins in each pan, namely the rows and columns of a square:
#1 1, 2, 3 vs. 4, 5, 6 #2 1, 4, 7 vs. 2, 5, 8
There are no more than nine possibilities for each result. In fact, the scale can never balance both times, and there are exactly nine ways each of the other eight results can be obtained:
1, 2, 3 and 1, 4, 7 pans heavy:
1 heavy 5, 6, 8, 9 light 4 possibilities
3 heavy 5, 8 light 2 possibilities
7 heavy 5, 6 light 2 possibilities
9 heavy 5 light 1 possibility
1, 2, 3 heavy and 1, 4, 7 balanced
1 heavy 4, 7 light 2 possibilities
2 heavy 5, 8 light 2 possibilities
3 heavy 6, 9 light 2 possibilities
7 heavy 4 light 1 possibility
8 heavy 5 light 1 possibility
9 heavy 6 light 1 possibility
The only remaining step is to show that there is no solution if you weigh three coins in each pan once and then two coins in each pan three times. This was done easily by modifying the above-mentioned computer program.
Better Late Than Never
Y2002. Avi Ornstein noticed that we mistakenly omitted the parentheses from 10 = (20+0)/2.
2002 Oct 3. Tom Harriman and Gunnar Bergman provided very different proofs from the one given. Note that the trigonometry was omitted from the published solution due to space considerations; it was present in Hess’s solution. Also I just noticed a typo: the correct formula for q is 2 sin-1 (9/16) + 3 sin-1 (1/8).
Other Respondents
P. W. Abrahams, J. Astolfi, D. Bator, H. M. Blume Jr., C. K. Brown, C. Cheng, D. Cipolla, G. Coram, J. Dieffenbach, I. Gershkoff, D. Gross, J. Grossman, J. E. Hardis, T. J. Harriman, R. Hess, H. Hochheiser, J. Hoebel, S. Hsu, S. Kanter, D. King, S. Klein, P. Latham, R. Lax, B. Layton, R. Marks, R. D. Marshall, L. J. Nissim, L. Peters, C. Polansky, J. Rudy, L. Sartori, I. Shalom, G. Steele, C. Swift, L. Villalobos, and G. Waugh.
Proposer’s Solution to Speed Problem
The only way to roll a six-dice farkle is to form two pairs and two singletons from the numbers 2, 3, 4, and 6. There are ( ) = 6 ways to distribute 2/3/4/6 among pairs and singletons and 6 x 5 x ( ) = 180 ways to order them. There are 6 x 180 = 1,018 farkles out of 66 = 46,656 possible rolls, giving a farkle probability of 1,080 / 46,656 2.3%.
Send problems, solutions, and comments to Allan Gottlieb, New York University, 715 Broadway, 7th Floor, New York NY 10003, or to gottlieb@nyu.edu. | 2,681 | 9,610 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.25 | 3 | CC-MAIN-2019-22 | latest | en | 0.975247 |
https://engineering.stackexchange.com/questions/58620/are-there-structural-or-code-reasons-for-choosing-longer-or-shorter-beam-spans | 1,713,597,099,000,000,000 | text/html | crawl-data/CC-MAIN-2024-18/segments/1712296817491.77/warc/CC-MAIN-20240420060257-20240420090257-00608.warc.gz | 208,860,429 | 40,288 | Are there structural (or code) reasons for choosing longer or shorter beam spans (besides bending moments, and ignoring cosmetics)?
When selecting the length of beams for a bridge, the better approach is to choose multiple full-length beams.
Unless there is a compelling reason, one would not choose multiple shorter spans.
An elementary exercise by sketching the bending moment diagrams would verify why the first is superior to the second.
Now suppose that said spans are for a backyard patio. (So far I'm only considering 16-ft long spans of natural wood, but composite/synthetic materials are an option.)
If the span is slightly longer than 16 ft, then one option is to use full-length spans, and splice the remaining part, while of course staggering the cuts. (Here the joists—our beams—are not the subject. We look instead at the structural flooring material.)
Another option is to use 8 ft-long cuts. That would simplify transportation, but let's ignore this convenience factor.
Are there structural (or code) reasons for choosing longer or shorter beam spans (besides the obvious bending moments, and ignoring cosmetics)?
• you should go look at a bridge that's more than a couple dozen feet long, exactly none of them has a single span member. Jan 15 at 7:20
• @TigerGuy Was what I wrote somehow ambiguous regarding this issue? Jan 19 at 2:28
$$\delta=\frac{5\omega L^4}{384EI}$$ Also, A longer beam is heavier and in an earthquake will vibrate with greater amplitude and will likely be cut off by the shear at supports. same as has happened in Los Angeles 1994 Northridge quake. | 348 | 1,597 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 1, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.671875 | 3 | CC-MAIN-2024-18 | longest | en | 0.927431 |
https://calamityjanetheshow.com/tag/decimal-worksheets-for-grade-5-with-answers/ | 1,621,179,053,000,000,000 | text/html | crawl-data/CC-MAIN-2021-21/segments/1620243991224.58/warc/CC-MAIN-20210516140441-20210516170441-00120.warc.gz | 185,787,760 | 15,782 | Calamityjanetheshow
# Decimal Worksheets For Grade 5 With Answers Tags
Time Worksheet Creator
## Time Worksheet Creator
Julietta Fanta Worksheets, 2020-11-17 04:07:31. Writing worksheets is a very important lesson in a student’s life. Worksheet writing is used by the teachers to make the students familiar with the writing scheme and help them to improve their writing skills which they would be requiring throughout their life. Writing worksheets effectively helps the young students a lot and makes them learn to write properly on a sheet of paper so that the reader can read their writing easily and he would not have any difficulty in reading the writings of the student. The worksheets contain various types of exercises for students with the help of which they can improve their writing skills.
Canva Worksheets
## Canva Worksheets
Bailey Sabrina Worksheets, 2020-11-22 07:28:10. Learning about numbers includes recognizing written numbers as well as the quantity those numbers represent. Mathematics worksheets should provide a variety of fun activities that teach your child both numbers and quantity. Look for a variety of different ways to present the same concepts. This aids understanding and prevents boredom. Color-by-Numbers pictures are a fun way to learn about numbers and colors too. The next step is learning to write numbers, and this is where mathematics worksheets become almost a necessity. Unless you have great handwriting, lots of spare time and a fair amount of patience, writing worksheets will help you teach this valuable skill to your child. Dot-to-dot, tracing, following the lines and other writing exercises will help your child learn how to write numbers. A good set of worksheets will include practice sheets with various methods to help your child learn to write numbers.
Create Fill In The Blank Worksheets
## Create Fill In The Blank Worksheets
Ninon Céleste Worksheets, 2020-11-17 04:03:53. Though the students always remain in an education friendly environment at schools, time taken by individuals for becoming skilled at new concepts & things vary from one student to another. The only way out to learn things in a quick time duration is thorough practicing of newly learnt concepts. Thus, parents of such Children reach out to various worksheet providers who provide conversational & innovative study materials, which would interest these kids to practice the lessons taught in schools.
Budget Worksheet Templates
## Budget Worksheet Templates
Mallory Livia Worksheets, 2020-11-17 04:42:02. Homeschool worksheets are a vital part of the student’s homeschool experience. They allow the child to test his or her knowledge, and they offer them a practical application for their learning. Worksheets also, when used properly, provide both the students and parent / tutor immediate feedback as to the child’s progress. This means they can be used to point out areas where the student needs further reinforcement. Homeschool worksheets fortunately will not over-tax your budget. There are many places where you can get them at extremely low costs. In fact, several websites offer printable worksheets for free. You can find worksheets for a wide range of courses--almost any course you want to teach your children. These include spelling, writing, English, history, math, music, geography, and others. They’re also available for nearly all grade levels. There are printable middle school, high school, elementary school, and even pre-school worksheets.
Quality Control Worksheet Template
## Quality Control Worksheet Template
Marcellia Alicia Worksheets, 2020-11-17 04:43:23. The writing of the worksheets can also be categorized based on the focus of the study of a student. The worksheet writing activity helps the students to come up with better and polished sentences. It also helps them to organize the sentence and used much better words in their sentences. It also helps them to increase their vocabulary and it automatically makes them to use better words in their sentences while writing a document. A student can begin his worksheet writing exercises anytime in his student life but first he has to learn how to pronounce the consonants and vowels properly. The initial writing exercises in the worksheet include writing short and simple sentences which can be easily understood by the student himself and also the reader who reads them. These writing exercises help the student a lot with the understanding the meaning of sentences and also help him to improve his reading skills so that he can read more number of sentences in the less required time. Writing activities help them to create sentences with new words each time and hence the repetition of words in their sentences can be eradicated permanently.
Facebook Worksheet Template For Students
## Facebook Worksheet Template For Students
Arlette Inaya Worksheets, 2020-11-17 04:43:41. Worksheets have been used in our day to day lives. More and more people use these to help in teaching and learning a certain task. There are many kinds or worksheets often used in schools nowadays. The common worksheets used in schools are for writing letters and numbers, and connect the dots activities. These are used to teach the students under kindergarten. The letter writing involves alphabets and words. These worksheets illustrate the different strokes that must be used to create a certain letter or number. Aside from this, such worksheets can also illustrate how to draw shapes, and distinguish them from one another. Teachers use printable writing paper sheets. They let their students trace the numbers, letters, words or dots as this is the perfect way for a child to practice the controlled movements of his fingers and wrist. With continued practice or tracing, he will soon be able to write more legibly and clearly.
Budget Worksheets Templates
### Budget Worksheets Templates
Evony Léa Worksheets, 2020-11-17 04:42:43. Learning about numbers includes recognizing written numbers as well as the quantity those numbers represent. Mathematics worksheets should provide a variety of fun activities that teach your child both numbers and quantity. Look for a variety of different ways to present the same concepts. This aids understanding and prevents boredom. Color-by-Numbers pictures are a fun way to learn about numbers and colors too. The next step is learning to write numbers, and this is where mathematics worksheets become almost a necessity. Unless you have great handwriting, lots of spare time and a fair amount of patience, writing worksheets will help you teach this valuable skill to your child. Dot-to-dot, tracing, following the lines and other writing exercises will help your child learn how to write numbers. A good set of worksheets will include practice sheets with various methods to help your child learn to write numbers.
Budget Template Worksheet
#### Budget Template Worksheet
Marian Andrea Worksheets, 2020-11-17 04:43:03. There are many opportunities to teach your child how to count. You probably already have books with numbers and pictures, and you can count things with your child all the time. There are counting games and blocks with numbers on them, wall charts and a wide variety of tools to help you teach your child the basic principles of math. Mathematics worksheets can help you take that initial learning further to introduce the basic principles of math to your child, at a stage in their lives where they are eager to learn and able to absorb new information quickly and easily. By the age of three, your child is ready to move onto mathematics worksheets. This does not mean that you should stop playing counting and number games with your child; it just adds another tool to your toolbox. Worksheets help to bring some structure into a child’s education using a systematic teaching method, particularly important with math, which follows a natural progression.
Template Budget Worksheet
##### Template Budget Worksheet
Alhertine Irina Worksheets, 2020-11-17 04:42:25. It is important to learn letter first. The children must need to know how to write letters in printable form. After that, they can be taught how to write cursive. Writing cursives is not as easy as writing letters in printable form. Remember that kids put more attention on animation. They are more interested on having fun so it is best for a teacher to teach them write letters in a fun way. Teachers may have noticed that when children are just being told on what to do, they may not do it right out of lack of interest. Remember that with the so many worksheets available, choose one that is best suited for a certain lesson. Plan ahead what type of worksheet to use for a given day, depending on what you plan to teach. There are many free worksheets available, especially online, but still the best worksheet is one that you personally draft. This way, you are able to match the level of difficulty of the activity in accordance to the performance level of your own students.
Current Event Worksheet Template
###### Current Event Worksheet Template
Cateline Capucine Worksheets, 2020-11-17 04:44:08. Once downloaded, you can customize the math worksheet to suit your kid. The level of the child in school will determine the look and content of the worksheet. Use the school textbook that your child uses at school as a reference guide to help you in the creation of the math worksheet. This will ensure that the worksheet is totally relevant to the kid and will help the child improve his or her grades in school. The math worksheet is not only for the young children in kindergarten and early primary school; they are also used for tutoring high school and university students to keep the students’ math skills sharp. The sites that offer these worksheets have helped a lot and this resource is now a common thing to use for all kinds and levels of educators. The formats for the worksheets differ according to the level and content of the worksheets. For the young kids it is preferable to have the worksheet in large print, while the older students commonly use the small print ones that are simple and uncluttered.
User Favorite
##### Project Worksheet Template Word
###### Classroom Handout Template
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Any content, trademark/s, or other material that might be found on this site that is not this site property remains the copyright of its respective owner/s. In no way does LocalHost claim ownership or responsibility for such items and you should seek legal consent for any use of such materials from its owner. | 2,121 | 10,746 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.578125 | 3 | CC-MAIN-2021-21 | latest | en | 0.938513 |
http://www.gradesaver.com/textbooks/math/algebra/elementary-and-intermediate-algebra-concepts-and-applications-6th-edition/chapter-4-polynomials-test-chapter-4-page-300/12 | 1,524,257,229,000,000,000 | text/html | crawl-data/CC-MAIN-2018-17/segments/1524125944682.35/warc/CC-MAIN-20180420194306-20180420214306-00380.warc.gz | 429,269,616 | 13,829 | ## Elementary and Intermediate Algebra: Concepts & Applications (6th Edition)
$\dfrac{3y^{5}}{x^5}$
Using the laws of exponents, the given expression, $\dfrac{9x^3y^2}{3x^8y^{-3}} ,$ is equivalent to \begin{array}{l}\require{cancel} \dfrac{\cancel{3}\cdot3x^{3-8}y^{2-(-3)}}{\cancel{3}} \\\\= 3x^{-5}y^{5} \\\\= \dfrac{3y^{5}}{x^5} .\end{array} | 140 | 345 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.15625 | 4 | CC-MAIN-2018-17 | latest | en | 0.532939 |
https://www.jiskha.com/display.cgi?id=1318001847 | 1,503,470,740,000,000,000 | text/html | crawl-data/CC-MAIN-2017-34/segments/1502886117874.26/warc/CC-MAIN-20170823055231-20170823075231-00442.warc.gz | 927,602,474 | 4,086 | # math
posted by .
To advertise their business, Toni and Mandy can either purchase flyers or e-mail addresses. They can print, distribute and pay the postage of 3 million flyers for \$450. Find the unit price for the flyers.
• math -
Divide:
450 / 3,000,000 = unit price
• math -
Divide ... to find unit price.
• math -
Thanks for refreshing my memory, Writeacher.
## Similar Questions
1. ### Education
Hello, I have to make a flyer for my early childhood education program, to promote healthy pregnancies. Does anyone know how to go about making a flyer?
2. ### math
Patrick has a summer job passing out flyers for a store. Each flyer weighs about 0.55 ounce. If he is given a bundle of flyers that weighs 16.5 lbs, how many flyers must he pass out?
3. ### English
what is the meaning on distaff, flyers, spool, and reel?
A company spent \$100.80 to purchcase flyers for their grand opening. The flyers cost between \$0.50 and \$0.70 each. The price was a multiple of \$0.05. The company paid the same amount for each flyer. 5. How many flyers did the company …
A company spent \$100.80 to purchase flyers for their grand opening. The flyers cost between \$0.50 and \$0.70 each. The price was a multiple of \$0.05. The company paid the same amount for each flyer. How many flyers did the company buy?
6. ### Mathematics
Patrick has a summer job passing out flyers for a store. Each flyer weighs about 0.55 ounce. If he is given a bundle of flyers that weighs 16.5 Ibs , how many flyers must he pass out ?
7. ### Math
It cost your office \$620 to mail 500 flyers. How much did it cost to mail each flyer?
8. ### math
Ursula works at a print shop. She uses a printer that can print 12 pages per minute. Yesterday she started printing flyers for an order. Today at 8 a.m. she continued working on the order, and by 9 a.m. she had 420 flyers for the order …
9. ### statistics
A marketing research shows that for every 100 flyers distributed to potential customers, 1.2 customers will be captured. If 10000 flyers are being distributed, find a. the probability that 150 customers are captured b. the expected …
10. ### Math
I'm not sure where to start... Mail order marketing companies have a response rate of 15% to their advertising flyers. If 25 people receive the flyer, what is the probability that 5 people respond to the flyer?
More Similar Questions | 578 | 2,375 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.3125 | 3 | CC-MAIN-2017-34 | latest | en | 0.945062 |
http://mathhelpforum.com/business-math/202099-modified-duration-perpetuity.html | 1,526,918,202,000,000,000 | text/html | crawl-data/CC-MAIN-2018-22/segments/1526794864405.39/warc/CC-MAIN-20180521142238-20180521162238-00231.warc.gz | 190,534,786 | 9,787 | ## Modified duration of a perpetuity.
Find the modified duration of perpetuity immediate paying a level payment of d beginning in five years, at an annual effective interest rate of 10%.
Ok. I know that the modified duration of a level payment perpetuity is $\displaystyle \frac{-p'(i)}{p(i)}=\frac{1}{i^2}$
But since this perpetuity starts in five years, I'm a little confused as to what the answer should be.
If the perpetuity started right now, it's modified duration would be 10. Since it's starting in five years, does that make it's modified duration $\displaystyle 10+5=15$?
Or is the modified duration going to be $\displaystyle \frac{-p'(i)}{p(i)}$ where $\displaystyle p(i)=\frac{d}{i(1+i)^5}$? | 186 | 709 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.765625 | 4 | CC-MAIN-2018-22 | latest | en | 0.929896 |
https://www.physicsforums.com/threads/finding-the-limit.328265/ | 1,529,873,635,000,000,000 | text/html | crawl-data/CC-MAIN-2018-26/segments/1529267867055.95/warc/CC-MAIN-20180624195735-20180624215735-00244.warc.gz | 869,592,275 | 15,315 | # Homework Help: Finding the limit
1. Aug 1, 2009
### azizz
1. The problem statement, all variables and given/known data
For the function:
$$f(i,z) := \frac{1}{z-z_0} \left( \sqrt{g} + \sqrt{\frac{k}{m}} \frac{i-i_0}{z-z_0} \right)$$
we have to find a solution for $$f(i_0,z_0)$$.
2. Relevant equations
$$z_0 = \sqrt{\frac{k}{mg}} i_0$$
$$i_0 = 1$$
3. The attempt at a solution
As you can see the system goes to $$\infty$$ if we let $$z$$ and $$i$$ go to $$z_0$$ and $$i_0$$ at the same time.
So my fist guess was to first let $$i$$ go to $$i_0$$, this gives
$$f(i_0,z) := \frac{1}{z-z_0} \sqrt{g}$$
But again for $$z \rightarrow z_0$$ the function $$f(i_0,z_0)$$ will go to $$\infty$$ (even if I fill in the equation for $$z_0$$ as given above; no cancellations take place).
4. Hints from the professor
I asked my professor for a couple of hints, but Im still unable to solve the problem with this information. Perhaps you can use it?
- it you let both go to its nominal value ($$z_0$$ or $$i_0$$) at the same time then the function will go to $$\infty$$. But there exists a certain trajectory (fixed $$i=i_0$$ and $$z(i) \rightarrow z_0$$ or vv) where the function has a solution.
- so you cannot let $$z$$ and $$i$$ go to $$z_0$$ and $$i_0$$ at the same time. Instead you have to let one of the two go to its nominal value ($$z_0$$ or $$i_0$$) and then let the other go to its nominal value as a function of the other.
Anyone has a suggestion how this problem can be tackled?
Last edited: Aug 1, 2009
2. Aug 1, 2009
### Dick
How about making sure that sqrt(k/m)*(i-i0)/(z-z0) is always equal to -sqrt(g)?
3. Aug 1, 2009
### azizz
Thanks for the fast reply! That might be a solution indeed, because then f(i0,z0)=0. I havent checked this yet, but in the remainder of the exercise I have to let f(i0,z0) vary 50% from its nominal value (that is f(i0,z0)). So I dont expect this to be the solutions, but I will take a look at it.
4. Aug 1, 2009
### Dick
I rather suspect if you are clever about the path, you can make it converge to anything you want. That's what makes the problem not very interesting.
5. Aug 2, 2009
### azizz
I can see f(i0,z0)=0 is a solution indeed. I cant see that other values are solutions as well, but I can imagine your right. However, for the remainder of the exercise I expect only one solution to be the desired one. Is it possible that there exist something like a "best" solution (I doubt this though).
6. Aug 2, 2009
### Dick
Think about solving sqrt(g)+sqrt(k/m)*(i-i0)/(z-z0)=(z-z0)/K for i as a function of z and let i->i0. Wouldn't that give you a limit of K?
7. Aug 2, 2009
### azizz
Im sorry, but I still dont see how this is to be done. First of all, why introduce the new variable K? Isnt that the same as 1/f(i,z)?
Then you say that the limit of K can be found by solving the given equation for i as function of z, that would be
$$i = \sqrt{\frac{m}{k}} (z-z_0) \left( \frac{z-z_0}{K} - \sqrt{g} \right) + i_0$$
So for $$z \rightarrow z_0$$ we get $$i \rightarrow i_0$$, but we knew that already...
Last edited: Aug 2, 2009
8. Aug 2, 2009
### Dick
The point is to make limit of f(i,z) as i->i0, z->z0 equal to 1/K. Where K is pretty much anything we want. Your function f just plain doesn't have a definite limit. I'm not sure I can think of any interesting questions to ask about the 'limit'.
9. Aug 3, 2009
### HallsofIvy
You say "limit" in the title but then you say just "find f(i0,z0)" in the body of you post. That value is, of course, NOT defined. Do you mean to define f(i0,z0) as the limit so that the function is continuous? (And again, as you have been told, that limit does not exist so this is impossible.)
10. Aug 16, 2009
### azizz
yes that is right, the limit does not exist. I misused this term. Indeed Im looking for a value for f(i0,z0) such that the function is continuous at this point. | 1,214 | 3,898 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.875 | 4 | CC-MAIN-2018-26 | latest | en | 0.877015 |
http://study.com/academy/lesson/what-is-debt-ratio-calculation-lesson-quiz.html | 1,508,751,700,000,000,000 | text/html | crawl-data/CC-MAIN-2017-43/segments/1508187825889.47/warc/CC-MAIN-20171023092524-20171023112524-00785.warc.gz | 328,857,713 | 36,584 | # What Is Debt Ratio? - Calculation & Overview
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• 0:02 A Look at Debt Ratios
• 1:02 What Is the Debt Ratio?
• 2:13 Good and Bad Debt Ratios
• 2:55 Examples
• 4:55 Lesson Summary
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Lesson Transcript
Instructor: Rebekiah Hill
Rebekiah has taught college accounting and has a master's in both management and business.
In this lesson, you'll learn what a debt ratio is in accounting. You will also learn how to calculate a company's debt ratio and why it is important to financers and investors.
## A Look at Debt Ratios
Have you ever been to a car dealership to buy a car? If so, then you know that picking out the car is the easy part. All the paperwork that is involved in the buying process can seem to take forever! Potential financers want to know two key things. First and foremost, they want to know how much money that you make. Second, they want to know how much money that you pay out on bills.
Why do you think that they want to know these things? The answer is quite simple. They want to know if you'll be able to pay for the car once you get it. If you have more bills than you have money coming in, then you are not a good candidate for a new car loan. However, if your money coming in each month is greater than the total of all the bills that you owe in a month, then you are exactly what the potential financer is looking for and will more than likely be approved for your loan.
This is the exact scenario that organizations face when seeking financing on new or existing projects. Financial institutions and potential investors rely heavily on something called the debt ratio.
## What Is the Debt Ratio?
In the business world, the term debt ratio refers to the amount of assets that an organization has in relation to the amount of liabilities. Assets are things that a business owns and uses to produce an income. Liabilities are what the business owes. The debt ratio shows how much a company relies on debt to finance company assets. The total liabilities and total assets values can be found on the company's balance sheet. The formula for calculating the debt ratio is:
Debt Ratio = Total Liabilities / Total Assets
Another common term that is seen when discussing the debt ratio is the term equity. In the accounting sense, the term equity is used to describe the funding that an organization gets from selling shares of stock. Selling stock to raise money is different than taking out a bank loan. When a company offers shares of stock for sale, it is allowing people to make investments and become partial owners in the company. In return for the stock purchase, the company may also agree to pay the investor a certain percentage of any yearly profit. This percentage is called a dividend. If the company doesn't make any money, then the shareholder will make no money. Funding received by sales of stock is called equity funding.
## Good and Bad Debt Ratios
In an ideal world, a company would have a debt ratio of 0.5, or 50%. This would mean that the company was funded equally by debt and equity funding. If this company were to decide to seek additional funding for a project from a bank, the bank would look favorably on this debt ratio. Let's say that the company had a debt ratio of 0.3, or 30%. With this debt ratio, the bank would consider the company low risk and would jump at the chance to loan it money. What if the debt ratio was much higher, like 0.8, or 80%? A debt ratio this high would throw up a red flag to the bank. At this level, the company would appear to have most of their assets funded by debt and would be a high risk for the bank.
## Examples
Let's take a minute to look at a few examples of calculating debt ratio.
### Example 1
XYZ Corporation's balance sheet shows a total asset value of \$350,000. The total liabilities shown on the balance sheet are \$125,000. What is the debt ratio?
Debt Ratio = Total Liabilities / Total Assets
Debt Ratio = \$125,000 / \$350,000
Debt Ratio = 0.36, or 36%
XYZ has a low debt ratio and would be a low risk for a potential financing source.
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Did you know… We have over 95 college courses that prepare you to earn credit by exam that is accepted by over 2,000 colleges and universities. You can test out of the first two years of college and save thousands off your degree. Anyone can earn credit-by-exam regardless of age or education level. | 1,102 | 4,933 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.71875 | 3 | CC-MAIN-2017-43 | latest | en | 0.95313 |
http://www.sonoma.edu/users/w/wilsonst/Courses/Math_165/Solns1/P3.html | 1,484,898,304,000,000,000 | text/html | crawl-data/CC-MAIN-2017-04/segments/1484560280801.0/warc/CC-MAIN-20170116095120-00459-ip-10-171-10-70.ec2.internal.warc.gz | 694,122,265 | 2,333 | 3. Students in a class were given two quizzes. The scores are presented in the following table.
Student Quiz 1 Quiz 2 A 5 9 B 4 7 C 2 5 D 6 10 E 3 4
a) Find the least squares regression equation which will be used to predict the scores on the second test as a function of the scores on the first test. i.e., let the scores on the first test be the explanatory variable and let the scores on the second test be the response variable.
b) Find the correlation coefficient r.
c) Use the regression equation to predict the score that student E should have gotten on the second quiz.
d) What is the residual for student E?
a) The explanatory variable, the scores on the first quiz will be our x, and the response variable, the scores on the second quiz will be our y. Your calculator should be able to compute the coefficients in the least squares regression formula. For the calculation, first find the means of the x's and y's
Student x y A 5 9 B 4 7 C 2 5 D 6 10 E 3 4 Totals 20 35
Since there are 5 scores, the means are
mx = 20/5 = 4
my = 35/5 = 7
If the regression equation is
y = ax + b
then
So we perform the following computations.
x y x - mx y - my (x - mx)(y - my) (x - mx)2 5 9 1 2 2 1 4 7 0 0 0 0 2 5 -2 -2 4 4 6 10 2 3 6 4 3 4 -1 -3 3 1 20 35 0 0 15 10
So
a = 15/10 = 1.5
The formula for b is
b = my - amx
= 7 - 1.5(4)
= 7 - 6 = 1
So the equation is
y = 1.5x + 1.
top
b) The formula for the correlation coefficient, r is
So we also need the sum of the squares of the deviations on the ys.
x y x - mx y - my (x - mx)(y - my) (x - mx)2 (y - my) 2 5 9 1 2 2 1 4 4 7 0 0 0 0 0 2 5 -2 -2 4 4 4 6 10 2 3 6 4 9 3 4 -1 -3 3 1 9 20 35 0 0 15 10 26
So
= .93026
top
c) Student E got a 3 on the first quiz, so the regression equation would predict that the score on the second quiz should have been
y = 1.5(3) + 1 = 4.5 + 1 = 5.5
top
d) The student actually got a 4 on the second quiz, so the residual is
4 - 5.5 = -1.5
The student's score on the second quiz was 1.5 points lower than the equation predicted.
top | 727 | 2,049 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.65625 | 5 | CC-MAIN-2017-04 | longest | en | 0.875409 |
https://convertit.com/Go/Maps/Measurement/Converter.ASP?From=olympiad | 1,695,543,004,000,000,000 | text/html | crawl-data/CC-MAIN-2023-40/segments/1695233506623.27/warc/CC-MAIN-20230924055210-20230924085210-00463.warc.gz | 208,394,032 | 5,547 | Search Maps.com
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Conversion Result: ```olympiad = 126227703.901824 second (time) ``` Related Measurements: Try converting from "olympiad" to autumn, calendar yr (calendar year), decade, fortnight, hour, leap yr (leap year), millenium, minute, month, octennial, quindecennial, quinquennial, septennial, sidereal day, sidereal month, sidereal yr (sidereal year), summer, week, winter, yr (year), or any combination of units which equate to "time" and represent time. Sample Conversions: olympiad = 16.29 autumn, .04 century, 1,460.97 day, .4 decade, 4.06 fiscal yr (fiscal year), 104.35 fortnight, 35,063.25 hour, 3.99 leap yr (leap year), 2,103,795.07 minute, 48 month, .44444444 novennial, .5 octennial, .26666667 quindecennial, 1,464.97 sidereal day, 53.47 sidereal month, 4 sidereal yr (sidereal year), 15.72 spring, 49.47 synodic month, 16.41 winter, 4 yr (year).
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# What are 1.75 m in feet? – Conversion, Formula, and More
## 1.75 m in Feet
1.75 m in feet shows you how many feet are equal to 1.75 meters as well as in other units such as miles, inches, yards, centimetres, and kilometres
## How many Feet are 1.75 Meters?
1.75 meters to feet equals 5.74 feet because 1 meter equals roughly 3.28 feet. To convert 1.75 meters to feet, multiply by 3.28.
Other Conversions Feet: 5.74147 Meters: 1.75 Miles 0.00109 inches: 68.89768 Yards: 1.91382 Kilometers: 0.00175 Centimeters: 175.00000
## Meters to Feet Conversion Formula
• [X] ft = 3.2808398950131 × [Y] m
• where [X] is the result in ft and [Y] is the amount of m we want to convert
## 1.75 Meters to Feet Conversion breakdown and explanation
1.75 m to ft conversion result above is displayed in three different forms: as a decimal (which could be rounded), in scientific notation (scientific form, standard index form or standard form in the United Kingdom) and as a fraction (exact result).
• Every display form has its advantages, and in different situations, a particular condition is more convenient than another.
• For example, using scientific notation when working with big numbers is recommended due to more effortless reading and comprehension.
• Usage of fractions is recommended when more precision is needed.
• If we want to calculate how many Feet are 1.75 Meters, we have to multiply 1.75 by 1250 and divide the product by 381. So for 1.75 we have: (1.75 × 1250) ÷ 381 = 2187.5 ÷ 381 = 5.741469816273 Feet
Hence: 1.75 m = 5.741469816273 ft
## Formula
Ft = meters × 3.28084According to the ‘m to feet’ conversion formula, if you want to convert 1.75 (one point seven five) M to Feet, you have to multiply 1.75 by 3.28084.
Complete solution:1.75 m × 3.28084=5.74′If you want to convert 1.75 M to both Feet and Inches parts, then you first have to calculate the whole number part for Feet by rounding 1.75 × 3.28084 fractions down.
And then convert the remainder of the division to Inches by multiplying by 12 (according to the Feet to Inches conversion formula)
Complete solution: ( 1.75 meters × 3.28084 )=5′get the Inches Part((1.75 × 3.28084) – 5′) * 12=(5.741 – 5′) * 12=0.741 * 12=8.9″so the full record will look like5′8.9″
## How to convert Meters (m) to Feet (ft)?
It is simple to convert Meters (m) to Feet (ft). All length conversions are easy to solve. You need to know the ratio from one unit type to another.
To convert from Meters to Feet, all you have to do is multiply by 3.28. It is the ratio of one m to one ft. Converting one unit can often be a helpful way to approach a problem in a new way.
### Other units of length
• Millimetres (mm)
• Centimeters (cm)
• Meters (m)
• Kilometers (km)
• Inches (in)
• Yards (yd)
• Feet (ft)
• Miles (mi)
## How to convert 1.75 m in Feet?
To convert 1.75 M to Feet, you have to multiply 1.75 by 3.2808398950131 since 1 Meter is 3.2808398950131 Feet. The result is the following:
• 1.75 m × 3.2808398950131 = 5.741 ft
• 1.75 m = 5.741 ft
We conclude that one point seven five Meters is equivalent to five point seven four one Feet:
• 1.75 Meters is equal to 5.741 Feet.
## Meters to Feet Table
Meters Feet 1.60 m 5.249 ft 1.61 m 5.282 ft 1.62 m 5.315 ft 1.63 m 5.348 ft 1.64 m 5.381 ft 1.65 m 5.413 ft 1.66 m 5.446 ft 1.67 m 5.479 ft 1.68 m 5.512 ft 1.69 m 5.545 ft 1.70 m 5.577 ft 1.71 m 5.610 ft 1.72 m 5.643 ft 1.73 m 5.676 ft 1.74 m 5.709 ft 1.75 m 5.741 ft 1.76 m 5.774 ft 1.77 m 5.807 ft 1.78 m 5.840 ft 1.79 m 5.873 ft 1.80 m 5.906 ft 1.81 m 5.938 ft 1.82 m 5.971 ft 1.83 m 6.004 ft 1.84 m 6.037 ft 1.85 m 6.070 ft 1.86 m 6.102 ft 1.87 m 6.135 ft 1.88 m 6.168 ft 1.89 m 6.201 ft 1.90 m 6.234 ft
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## 5.433.750 ltda viver lab laboratorio de analises clinicas sao jose do calcado
Viver Lab plays a critical role in the healthcare sector with a steadfast commitment to… Read More
October 3, 2023
## khu nghá»?? dæ°á»¡ng tại johor
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Theapknews.shop Taking good care of our health and appearance is crucial in today’s hectic environment.… Read More
September 27, 2023 | 1,530 | 4,534 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.375 | 4 | CC-MAIN-2023-40 | latest | en | 0.891953 |
https://ec2-18-140-35-61.ap-southeast-1.compute.amazonaws.com/lessons/aljabar-bentuk-05-konsep-dasar-8-smp/ | 1,713,884,166,000,000,000 | text/html | crawl-data/CC-MAIN-2024-18/segments/1712296818711.23/warc/CC-MAIN-20240423130552-20240423160552-00068.warc.gz | 194,415,606 | 11,792 | # Aljabar Bentuk 5
## Konsep Dasar
#### Ekspansi Binomial
###### Segitiga Pascal
Baris pada segitiga pascal menunjukkan koefisien penjabaran $$(a + b)^n$$. Koefisien untuk $$(a + b)^n$$ dapat dibaca pada baris ke$$-(n + 1)$$
Contoh,
• $$\color{purple}(a + b)^2 = a^2 + 2ab + b^2$$ dari baris 1, 2, 1
• $$\color{purple}(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$$ dari baris 1, 3, 3, 1
• $$\color{purple}(a + b)^4 = a^4 + 4a^3b + 6a^2b^2 + 4ab^3 + b^4$$ dari baris 1, 4, 6, 4, 1
• $$\color{purple}(a + b)^5 = a^5 + 5a^4b + 10a^3b^2 + 10a^2b^3 + 5ab^4 + b^5$$ dari baris 1, 5, 10, 10, 5, 1
• $$\color{purple}(a + b)^6 = a^6 + 6a^5b + 15a^4b^2 + 20a^3b^3 + 15a^2b^4 + 6ab^5+b^6$$ dari baris 1, 6, 15, 20, 15, 6, 1
Contoh 1
Jabarkan $$(2x + 3)^3$$
Misal $$a = 2x$$ dan $$b = 3$$
maka,
\begin{equation*}
\begin{split}
(2x + 3)^3&= (a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3\\\\
(2x + 3)^3&= (2x)^3 + 3(2x)^2(3) + 3(2x)(3)^2 + 3^3\\\\
(2x + 3)^3& = 8x^3 + 3(4x^2)(3) + 3(2x)(9) + 27\\\\
(2x + 3)^3& = 8x^3 + 36x^2 + 54x + 27
\end{split}
\end{equation*}
Contoh 2
Jabarkan $$(3m - 5)^5$$
Misal $$a = 3m$$ dan $$b = -5$$
maka,
\begin{equation*}
\begin{split}
(3m - 5)^5&= (a + b)^5 = a^5 + 5a^4b + 10a^3b^2 + 10a^2b^3 + 5ab^4 + b^5\\\\
(3m - 5)^5& = (3m)^5 + 5(3m)^4(-5) + 10(3m)^3(-5)^2 + 10(3m)^2(-5)^3 + 5(3m)(-5)^4 + (-5)^5\\\\
(3m - 5)^5& = 243m^5 + 5(81m^4)(-5) + 10(27m^3)(25) + 10(9m^2)(-125) + 5(3m)(625) - 3125\\\\
(3m - 5)^5& = 243m^5 -2025m^4 + 6750m^3 - 11250m^2 + 9375m - 3125\\\\
\end{split}
\end{equation*} | 909 | 1,523 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.6875 | 5 | CC-MAIN-2024-18 | latest | en | 0.188591 |
http://pari.math.u-bordeaux1.fr/archives/pari-users-0506/msg00051.html | 1,511,232,867,000,000,000 | text/html | crawl-data/CC-MAIN-2017-47/segments/1510934806310.85/warc/CC-MAIN-20171121021058-20171121041058-00789.warc.gz | 228,218,471 | 2,712 | Bill Allombert on Thu, 30 Jun 2005 23:19:03 +0200
Re: Sin(x)
```On Thu, Jun 30, 2005 at 10:16:07PM +0200, Sascha Rissel wrote:
> Thanks for all your quick help.
> I now do understand this slight difference to the expected values.
>
> But why do cheap pocket calculators calculate exact results of sin and cos?
> Their approximation of Pi shouldn't be more precise than that Pari provides.
It is actually slightly less precise. Since Pi is not equal to
3.141592653589793239, sin(3.141592653589793239) should not yield 0.
> Do they round the results for reasons of "usability"?
No, they simply use fixed precision.
When you ask sin(x) with Pi/2<x<3*Pi/2 the natural thing to do is to
substract Pi from x to move x in ]-Pi/2,Pi/2[. However, since they use
fixed precision, Pi has always the same value, so when you ask sin(Pi)
the calculator do Pi-Pi and get 0.
In PARI, the precision is not fixed, and Pi is in fact a function of the
precision. PARI alse compute x-Pi, but to avoid the error on the value of Pi
to affect the result, it computes Pi with an higher precision that the
precision of x. When you do sin(Pi), it computes Pi(28)-Pi(38) which is
very small but not 0.
To check that try:
? \p28
? pi28=Pi
%1 = 3.141592653589793238462643383
? \p38
realprecision = 38 significant digits
? pi38=Pi
%2 = 3.1415926535897932384626433832795028842
? pi38-pi28
%3 = -5.04870979 E-29
? sin(pi28)
%4 = -5.04870979 E-29
However, that exhibit the main limitation with PARI real:
the precision increase only by large increment. In fact
pi38-pi28 has less than 1 bit of accuracy, but 32 bits
are returned nevertheless.
Pocket calculators also use other tricks to return better looking
results (binary-coded decimals, guard digits), but they should not
come into play here.
Cheers,
Bill.
```
• References:
• Sin(x)
• From: Sascha Rissel <s_rissel@web.de>
• Re: Sin(x)
• From: Bill Allombert <allomber@math.u-bordeaux.fr>
• Re: Sin(x)
• From: Sascha Rissel <s_rissel@web.de> | 609 | 1,978 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.28125 | 3 | CC-MAIN-2017-47 | latest | en | 0.815701 |
https://www.physicsforums.com/threads/problem-metrics-and-induced-topologies.90589/ | 1,675,149,588,000,000,000 | text/html | crawl-data/CC-MAIN-2023-06/segments/1674764499845.10/warc/CC-MAIN-20230131055533-20230131085533-00370.warc.gz | 958,532,590 | 14,590 | # Problem : Metrics and Induced Topologies
Homework Helper
The Euclidean metric, d, is defined by:
$$d(x, y) = \left [\sum _{i = 1} ^n |x_i - y_i|^2\right ]^{1/2}$$
Define metrics dp for each p in {1, 2, 3, ...} as follows:
$$d_p(x,y) = \left [\sum _{i = 1} ^n |x_i - y_i|^p\right ]^{1/p}$$
Prove that each dp induces the same topology as the Euclidean metric.
To do this, I want to show that for every $\epsilon > 0$ and for every $x \in \mathbb{R}^n$, there is are $\delta _1,\, \delta _2 > 0$ such that for every $y \in \mathbb{R}^n$:
$$\left [\sum _{i = 1} ^n |x_i - y_i|^2\right ]^{1/2} < \delta _1 \Rightarrow \left [\sum _{i = 1} ^n |x_i - y_i|^p\right ]^{1/p} < \epsilon$$
and
$$\left [\sum _{i = 1} ^n |x_i - y_i|^p\right ]^{1/p} < \delta _2 \Rightarrow \left [\sum _{i = 1} ^n |x_i - y_i|^2\right ]^{1/2} < \epsilon$$
Is this the right way to prove it? Where do I go from here? Induction on n, or p? Or maybe both? Or is there a way to do it without induction? Help would be very much appreciated!
Homework Helper
Actually, the problem I really have to solve is to show that, assuming each dp is a metric, they all induce the usual topology on Rn, and I figured the best way to do this was to show that they induced the same topology as the Euclidean metric since these "metrics" (they might not all be metrics, but the problem says to assume they are) look a lot like the Euclidean metric.
Staff Emeritus
Gold Member
Well, when you don't understand something, draw a picture.
A circle (or an n-sphere, in general) is a characteristic of the Euclidean metric, right? What about these other metrics? | 525 | 1,621 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.796875 | 4 | CC-MAIN-2023-06 | latest | en | 0.909215 |
sourav-64777.medium.com | 1,627,464,326,000,000,000 | text/html | crawl-data/CC-MAIN-2021-31/segments/1627046153709.26/warc/CC-MAIN-20210728092200-20210728122200-00305.warc.gz | 538,927,768 | 33,466 | # Estimating Expectations with Samples
The problem of computing an expectation of a function that takes a random variable as an input crops up in many areas of statistics and machine learning. There are numerous ways of estimating this expectation such as by using Markov Chain Monte Carlo (MCMC) methods or by using variational inference. In this short read, I’ll discuss the idea of using samples to estimate expectations and introduce one such MCMC method — Gibbs sampling — and how it can be used to estimate these expectations. But first, a bit on notation…
We say samples of a random variable x is drawn from a distribution P(X) as x ~ P(x). X is the sample space/domain and x is a random variable and an instance from this sample space. For example, let’s imagine we have an unbiased coin such that it has equal probability of showing heads and tails during a toss. In this case, the sample space, X = {heads, tails}. The probability that x = heads is P(x = heads) = 0.5.
Let’s say, we have a continuous random variable drawn from a unit normal distribution, ie, x ~ N(0, 1), where the mean μ = 0 and the variance σ = 1. Drawing samples from a unit normal distribution is relatively simple and can be easily done in numpy with a simple function call:
Well, that was easy! How about drawing a sample from not a unit normal but a normal distribution with a specific mean μ = 10 and σ = 3?
That is not difficult to do either. We can easily transform a sample, x, drawn from a unit normal to a sample, y, drawn from N(μ, σ) using the transformation:
The examples we’ve seen so far might give the illusion that it is easy to sample from an arbitrary probability density function P. It is easy to sample from distributions such as the normal or the uniform distribution but that’s absolutely not true for any arbitrary distribution P!
The complexity of sampling from an arbitrary distribution comes from the fact that probability distributions over the entire domain (sample space) must sum to 1. Oftentimes what we have access to is the unnormalised density function, T. To obtain a valid distribution P from the unnormalised densities, we need to divide values of T(x) over the entire domain of x, (the domain of x is denoted by X — note the capitalization) by the normalizing factor, Z:
For high-dimensional values of x, computing the normalizing factor becomes intractable because T(x) will need to be evaluated over the entire domain X. More concretely, to compute Z for an n-dimensional samples of x from X:
# Why do we care about sampling from P ?
Let’s say we have a continuous random variable x and a deterministic function f that takes x as an input. We might be interested in computing values such as the mean or the variance of f(x). For instance, the mean can be computed as:
As mentioned earlier, if P is a simple distribution such as the uniform or normal, the mean can be computed analytically. For arbitrary distributions, P, the above computation is intractable.
## From integrals to Monte-Carlo estimates
The idea of Monte-Carlo estimates is that we replace the integral over the domain, X, with N samples of x. The average of the f computed at the samples of x approximate the true expectation. More concretely,
We say mu hat is a Monte-Carlo estimate of mu(without the hat), the true mean . Monte-Carlo estimates are great when it’s intractable to exactly compute a function f as described earlier. The estimates (mu_hat, for example) are themselves random variables now because we evaluate f at a bunch of random variables and average them. However, an attractive property of Monte-Carlo estimates is that they are unbiased estimates, meaning expectation of the estimate is equal to the expectation of the function f:
Also, Monte-Carlo estimates have variance which is O(1/N), meaning the variance reduces with the number of samples in the order of 1/N, where N is the number of samples.
Monte-Carlo estimates are quickly roughly right but need a lot of samples to be very accurate estimates of the true expectation
# Generating samples with MCMC methods
So now we’ve seen the importance of drawing samples from a distribution and how the average of the function f computed at these samples can provide Monte-Carlo estimates of the expectation of the function. Before I go on to describing a method for drawing such independent samples from an arbitrary distribution, let’s discuss a bit about Markov chains.
## Markov Chains
A Markov Chain is a systematic way of drawing samples of a random variable, where the probability of drawing a sample at timestep t is only conditionally dependent on the the random variable drawn in the prior timestep. That is, in the example below P(x4 | x1, x2, x3) = P(x4 | x3).
The idea is that we use this Markov Chain to create a chain of samples of a random variable. But note, these samples aren’t really independent of one another because the sample at timestep t is dependent on the sample at timestep (t — 1)! Yes, but if we run this chain long enough, take all the samples from each timestep and jumble them together before we draw samples from this set, the samples drawn will asymptotically be independent (this is actually what’s called the “mixing” of a Markov Chain)
Okay, so this is where we are so far — we can approximate an expectation of a function f over a random variable x by taking a bunch of independent samples of x, evaluating f and taking its average. That’s the Monte Carlo estimate. Next, we’ve seen that we can run a Markov Chain long enough to generate independent samples of x.
But, how exactly do we create this Markov Chain? There are a few Markov Chain Monte Carlo methods (MCMC) but I’ll discuss one such method of generating a Markov Chain — Gibbs sampling!
Intuition with MH
Let’s say we want samples of x that maximize the unnormalized density T. Remember, it’s actually easy to evaluate T(x) but hard to evaluate P(x) because computing the normalization constant Z is expensive. Let’s say we sample these xs from a proposal distribution q(x’, x_t)
## Gibbs Sampling
Gibbs sampling is an MCMC technique that can help with generating independent samples from a distribution P. Let’s say we have a random variable x, which is a d-dimensional vector where each of the components x^i is an independent random variable. The idea in Gibbs sampling is to generate samples by sweeping through each dimension of x and sample from its conditional distribution with the remaining variables/dimensions fixed to their current values.
According the algorithm, we start with an intial value of the random variable x at iteration 0. These values are sampled from a prior distribution q. Then, at each iteration i, each dimension of the variable x is drawn from a conditional distribution p. We assume it’s easy to draw samples from these conditionals. It’s important to note that we aren’t drawing samples from the full joint P, which as I have discussed is hard — we are only simulating samples from this distribution P by sweeping through each of the dimensions of x at each iteration and drawing the values of each dimension from the conditional. The idea is, if we run this algorithm for enough number of iterations, then take the collect the samples of x from each iteration into a set and then draw a sample from this set, it will simulate drawing from the actual distribution P.
The animation below shows a Gibbs sampler for a bivariate Gaussian. The conditionals of a bivariate Gaussian are also Gaussians, so they’re incredibly easy to sample from! The red crosses are the samples drawn and the contour shows the actual distribution. I intentionally set the initial sample at (15, 15) but as you can see, the sampler quickly starts sampling from the higher densities. Code for this available here.
# Conclusion
That was a brief overview of MCMC methods and how they can be used for approximate inference! We looked at the technique of estimating expectations of a function f that takes a random variable x as an input using Monte Carlo estimates and how drawing samples of x from P can help us with that. These Monte Carlo estimates are great because they are unbiased estimators and the variance of these estimates decays in the order 1/N where N in the number of samples averaged over. Finally, we looked an example of a Markov Chain Monte Carlo technique — Gibbs sampling — and how it can be used to draw independent samples of x from the distribution P(x) by constructing a Markov Chain of samples of the random variable.
# References
1. http://www.mit.edu/~ilkery/papers/GibbsSampling.pdf
2. Information Theory, Inference and Learning Algorithms, David J.C MacKay | 1,851 | 8,686 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.875 | 4 | CC-MAIN-2021-31 | latest | en | 0.920831 |
https://chat.stackexchange.com/transcript/115367/2020/10/22/5-7 | 1,653,220,824,000,000,000 | text/html | crawl-data/CC-MAIN-2022-21/segments/1652662545326.51/warc/CC-MAIN-20220522094818-20220522124818-00635.warc.gz | 219,678,430 | 15,615 | 04:00 - 05:0005:00 - 07:00
5:00 AM
oh missed a step
hm, you still have an x in there - can you get rid of that? (also, it might be worth keeping things in terms of "[stuff] = 0" for Reasons™)
Oh. Ohh
t^2-3t+1?
right!
now, the equation should be much more manageable - how many solutions are there to t² - 3t + 1 = 0 ?
well it's not a perfect square, so two
wait... yup not a perfect square...
I think
that's not the only thing you have to check - it's not a perfect square, so either two or zero. how do you know it's two rather than zero?
5:02 AM
Because there's that +1
and 1=/=zero
sure, but "t² + 1 = 0" has no (real) solutions
what tells you how many solutions a quadratic has?
Oh I have to apply one of those stuff from algebra 2 I forgot
There's a simplified form of the quadratic formula
yep, it's one of those things!
Is b^2-4ac thing available at that level?
you don't need to actually calculate the full thing, but there's part of the quadratic formula that might help...
yes that
5:04 AM
Yes, I think it was that equation
that's what i was hinting at - the discriminant
do you remember what it tells you?
(also, that's an expression, not an equation! an equation is a statement "[stuff] = [other stuff]", and it can be true or false. an expression doesn't have an = sign -- it can't be true or false.)
It tells you whether the graph of a quadratic will have 0,1, or 2 roots
do you remember how it tells you that?
Urm, I would be accidentally setting stuff up. It had to do with b and c though if I remember correctly
5:06 AM
@Bubbler (the other half is about getting it wrong)
not sure what you mean by "I would be accidentally setting stuff up"
but b^2-4ac is the discriminant -- it appears in the quadratic formula as "...±√(b^2-4ac)..."
well okay. If b^2-4ac>0, it has two roots. If b^2-4ac=0, it has exactly one root. If b^-4ac<0, it has no real roots
yep!
Exactly
so, how many roots does "t² - 3t + 1 = 0" have?
5:08 AM
2, because -3^2-4(1)(1)>0
Okay, now let's get to the real question and see if I can deduce it in a similar fashion
It was -3^2, not 3
So
well, there's one more thing to note - you're not quite done
Oh
darn
Nitpick: you should write (-3)^2, not -3^2
(also yes, that ^. we know what you meant but technically exponents come before the negative sign)
you know that there are two values of t that make it 0. but the question asked for values of x -- t was just something we made up to make things easier!
so, for any value of t, how many corresponding values of x are there?
5:10 AM
explain?
Nope
Waitttt
For every value of t, there is an "(x^3)" value of x?
No no
remember, t is just shorthand for x³
so if you know what t is, how many xs are possible that could give you that value of t?
(x)(x)(x). Urm
if you know "x³ is 8", how many possible values of x are there that would end up with that result?
5:12 AM
2
or
The question is not asking for that
Oh
No urgh
the cube root is how you would reverse it
i'm just asking you to count something here though
how many numbers are there whose cube is 8?
what numbers are they?
5:13 AM
2
you've given me one number, but you said there were three of them
(2),(2),(2)
those are the same number
for comparison, if i asked "how many numbers are there whose square is 25", the answer would be "there are two: 5 and -5 both square to 25"
5:14 AM
Oh
i'm asking how many different numbers there are that would give you a result of 8 when you cube them
Just 1
That being 2
yep!
and as you mentioned before, you can do this more generally - no matter what value you have for t, you can always take the cube root and get a value for x
right
(another nitpick: more precisely, there are 3 of them, but two of them are not real)
5:15 AM
so every t value will give you exactly one x value. you know there are 2 t values that work, so...
(yes, i'm ignoring complex numbers at the moment)
There are exactly 2 x values
and there you go! now you're done
Hurray! Now for the real question
What, it isn't the end?
It was -3x^2, not -3x^3
I typoed
5:16 AM
that extra thing didn't actually matter here, but it might in some other scenario where you couldn't get exactly one every time
x^6-3x^2+1=0. Can we apply the same principal. Lemme see
This time, it's t^3-t+1
no quadratics. rats. however, that looks like a special factor thingy
unfortunately, i... don't believe it is
oh darn
craaap more time i need to sleep soon
Whatever, this is genuinely the last question
not sure what specifically they want you to do here - you can do the same technique but it's a bit more painful
(Spoiler using calculus: t^3-t+1=0 has only one real root, and it is negative, so the original equation has zero real roots.)
5:19 AM
There's Descarte Rule of signs still
(it's -3t, not -t)
Oops
Ah right
maybe they expect some form of IVT argument?
yeah that's my guess, but without a graph that seems weird
5:20 AM
3t makes three roots, and you can indeed use Descartes for that
because descartes only gives you the parity of the number of roots of each sign
I interpreted the question as "don't use graph for justification", not "don't use graphs at all"
Idk what they intended
hm that's possible
Use Descartes at t = -2, 0, 1, 2.
That gives - + - + which gives the maximum of 3 roots
Not sure what that means?
5:22 AM
huh? No
How do you use Descartes at a point
also confused
it's x^6-3x^2+1
well, you've "simplified" it to t³-3t+1
Same principle
That means that there are a maximum of 2 roots for positive
5:23 AM
right, so the question is whether there are 2 or 0
And negative
oh wait this argument can work actually
hm no never mind not without a bit more
sorry, go on?
Oh, it's not Descartes
sorry
It can be 2,1, or zero
Wait not one
2 or zero
Forgot about conjugates for a second
agreed
5:25 AM
yeah, i'm guessing that the intended argument is just to actually show there are positive roots by IVT or something? but ew
What's IVT?
intermediate value theorem
not sure what theorems you have 'access' to
Oh wait we learned this theorem but I wasn't paying attention
Yeah, I think I meant IVT
It's like
5:26 AM
Almost ten years since calculus :/
f(a)-f(b)/f(a-b)=c or something??
@Bubbler ah ok yes
@PrinceNorthLæraðr that's MVT
IVT is much simpler
Mean Value Theorem
@Deusovi Oh it's the one where C must be somewhere between a and b
5:27 AM
The gist is that if f(a)<0 and f(b)>0, there is a root between a and b
ah
So... I plug in two numbers?
Basically if f changes sign from a to b, there's a root between a and b
Two numbers to prove one root is there
So if you can find a,b so that f(a) and f(b) have different signs. there's definitely a root between them
5:29 AM
Four numbers to prove three roots are there
what numbers do I pick though?
that's the tricky part
like, you can just graph and look
but i'm not sure how they intend you to find them
Just keep trying the "simplest" numbers you can think of?
Usually the rule of thumb is to try small positive/negative integers
5:30 AM
I'm guessing they don't want you to graph
so 1 and -1
like -2, -1, 0, 1, 2
yeah just trying simple numbers is a good approach when all else fails
you're looking for positive roots though so you want to keep your numbers positive (or 0)
"there's a root between -1 and 1" doesn't help you find positive roots, because it might be -1/2 or something
But if you don't identify the intervals of all three roots, you don't get conclusive results for the original problem
not necessarily bubbler
wdym sign changw?
5:32 AM
we know there's one negative root, so the question is whether there are 0 or 2 positive roots
Ok then
Then the numbers are simpler: try just 0, 1, 2
@PrinceNorthLæraðr Like if at 0, the value is say -20, at 1 the value is 30, so then you know there's a root between 0 and 1
@Deusovi ah, not necessarily
hm?
x^6-3x^2+1 has different values in Descarte Rule of Signs than t^3-3t^1+1
5:34 AM
right now we're just looking at t³-3t+1
@PrinceNorthLæraðr You don't need to apply Descartes to the original equation
the whole point of this is that you're finding where t³-3t+1=0, and then trying to see if the results you get from that give you anything for the original
@Bubbler Well, I kind of feel like that might've been the point, just really badly worded
I just graphed x^6-3x^2+1=0, and it has four roots, 2 negative, 2 positive, which ta-da fits the descarte rules with 2 complex
descartes can't help you very much there - you can't easily prove the number of roots of the original, even using both descartes and IVT
5:36 AM
it fits, but other numbers fit too
descartes only tells you the parity of the number of roots so it's not super helpful
There must be an easier solution- the graph of x^6-3x^2+1 is oddly very symmetrical
It is symmetrical because it is also a polynomial in t=x^2
i.e. all terms have even powers
It looks almost identical to an x^4 graph
right, since all the powers of x are even, it's going to be an even polynomial so it'll be symmetric
Huh didn't know that
Wait that means that descarte could work, no?
Wait no
5:39 AM
you could use descartes, symmetry of even polynomials, and IVT all together
but that sounds like a pain
It's possible the graph could've just looked like an x^2 graph right
with different coefficients, yeah
honestly i'm still not sure what they exactly want here
@Deusovi That sounds way too much for Pre-Calc
"How many real solutions"...
it is just things you know (or at least i assume things that have been taught at one point)
but it's a lot to do all at once
I don't remember IVT being in this unit. It was last unit. And I don't think we learned about even symmetry
ugh, i'm so tired
ivt was prob on this unit actually
5:43 AM
even then you'd still need to get lucky enough to pick points that give you a sign change for ivt
yeah
well, (0,1) worked
but what if it didnt?
right, that's why we were hesitant too
they happen to be small but if they weren't you'd just be out of luck
If a question is intended to use IVT, I bet it is always designed to work with simple numbers
right, but the question is whether IVT was intended
5:46 AM
Honestly I don't think you can prove there are exactly 4 roots without IVT or calculus
oh crap i missed a question
Forget it
It's 11 pm here and I need to get up at 6 am
I'm going to sleep
Thanks everyone for the help!
good night!
Good night!
Well, it's almost done (assuming IVT is allowed) - we've got t^3-3t+1=0 has two or zero positive roots, and we've got one positive root is in (0,1), so it has two positive roots. Then t=x^2, which has two roots per positive t, so the original equation has four roots.
At this point, we can't distinguish mathematics from puzzling.
It feels too hit-or-miss for a homework problem ig
6 mins ago, by Bubbler
If a question is intended to use IVT, I bet it is always designed to work with simple numbers
5:53 AM
Guess so
If you plug in some simple numbers to use IVT and it doesn't work, you can quickly abandon IVT on that equation.
Hmm
But that's kinda quirky problem solving technique after all
(which works only until you get to basic calculus, where you can identify local minima/maxima and etc directly)
The calculus way: Identify t=x^2 substitution. Identify that local minima/maxima are at t=-1 and t=1, and the graph has rotational symmetry around t=0. Plug in the three values to guess the shape of the graph and derive that two roots are positive and one is negative. Plug in to t=x^2 to conclude that there are four real roots.
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https://www.remc.org/mitechkids/5th-grade/ | 1,726,651,186,000,000,000 | text/html | crawl-data/CC-MAIN-2024-38/segments/1725700651886.88/warc/CC-MAIN-20240918064858-20240918094858-00739.warc.gz | 891,984,114 | 31,692 | Under each category is a link to the task card and its description of the activity. Cards may be in multiple categories to assist you in locating an activity to use with your students.
These task cards focus on math activities.
The math learning center is an app and online platform that allows students to use manipulatives virtually. In this activity, students will use virtual manipulatives to add fractions.
Students will learn how to use some basic functions of Google Sheets by using information from Major League Baseball’s official site, mlb.com. Any web browser may be used to access this information.
Classroom Bank Account
This is a virtual online bank for your classroom. The teacher creates and manages a savings account for the students. Each student can actually log in to their account and make withdrawals or deposits.
Classroom Economy
Classroom Economy is a program that teachers can use for free to teach students financial responsibility. This is a classroom economic system where students “rent” their desks, learn the value of earning a paycheck, budget their spending, save money, spend money, make deposits, withdrawals, etc.
Digital Fraction Book
Students have fun creating, sharing and reading their own digital fractions math book created on StoryJumper.
Equal Fractions
The Fraction Equality Lab allows students to experiment with fractions by creating equivalent fractions. Different shapes are used as well as a number line. When the game is played students select a level and match equivalent fractions. The higher the level the more challenging it will be.
Finding the Area and Perimeter
Area Builder is an interactive website that allows students to build shapes to find the area and the perimeter. There are two-parts for students to use, the explore and the game. The first allows for exploration while building, the second is a game that has them figuring out the area and the perimeter. There are 6 levels.
I Know it Math
I Know It is an engaging, interactive math website for elementary students. Teachers create their own classroom and assign questions to either individual students or the whole class. Teachers can also provide hints to the questions if they want. The student is also provided with feedback if they get a question incorrect.
In a Pickle With Math
Math Pickle is a resource filled with math puzzles and games to engage students in problem-solving. Puzzles are organized by grade level and subject.
Legends of Learning
Legends of Learning provides interactive math and the sciences games to students that give them superpowers. The teacher can select games that are connected to current lessons. It is aligned with the common core and NGSS. It is FREE for teachers and students in school.
Operations and Algebraic Thinking
Math Playground provides grade-level appropriate activities for math operations and algebraic thinking with a variety of activities to engage students and strengthen their skills. It can be used in grades one-six.
PhET - Math Fractions
PhET provides fun, free, interactive, research-based simulations for math and science. The simulations are written in HTML5, and can be used online or downloaded to your computer. This is free for all students and teachers. There are hundreds of lesson plans and materials using the simulations.
Study Jams
Scholastic has produced a FREE website full of educational videos and step-by-step slides for students to work on. Study Jams focuses on science and mathematical concepts that are prevalent in third, fourth, and fifth grade.
Survey Says
Students are presented with a scenario surrounding a relevant contemporary public issue on which they can take a stand. Students will work together to create a survey that will then be shared with various groups, in order to analyze the ways in which certain groups perceive a given issue. Once collected, the data will be discussed and analyzed. Finally, students will support their opinion using the data collected.
Learn how to use some of the toolbars in Google Sheets. The students will learn about setting up data, clicking and dragging to use data in a graph with the insert chart tool, and learn how to edit parts of the graph for better understanding.
Visnos Virtual Math Manipulatives
Visnos is an interactive math website that provides many different math tools to allow teachers to show math visually and allow students to explore with the tools.
Would You Rather Math
Would You Rather Math is a website that contains fun math challenges that students can solve. A picture is presented with a math task and a question that begins with, Would you rather?”.
Zearning With Math
Zearn is a math website that is directly connected to the Common Core Standards.
These task cards focus on letter and word recognition, reading activities and writing improvement and research projects.
8 P*ARTS Eduprotocol
Students will master the parts of speech, practice writing, and revising using a visual cue for engagement.
Biblionasium is a digital sharing platform, also known as the ¨Goodreads for Kids¨. It provides a safe social network where students can review, recommend, and rate books. Students can engage with their peers and others to support their reading.
Classroom Book
Boomwriter is an interactive group story writing website. Students will improve their creative story writing skills by collaborating with their classmates. They will read the first chapter and then finish the story.
The Daily News
Students can begin researching and interviewing others on topics for a weekly or monthly newsletter for the school. The students will learn techniques for interviewing as well as become better writers.
Students will have fun using the Pure Random Emoji Generator to practice their writing skills based on the main idea, supporting evidence, and conclusion being randomly generated emojis.
Evaluating Websites HyperDoc
Students will learn how to evaluate a website while using a HyperDoc in a group task. Hyperdocs are interactive digital documents that may easily be edited by a teacher. These multimedia-rich documents contain links for research, videos, images, etc. along with the steps for students to follow and record their work.
Evernote - Absent Students
Evernote is an app used for note-taking, organizing, task management, and archiving. Evernote allows users to create notes, which can be text, drawings, photographs, or saved web content. Students can add to their notes outside of school. This is great because teachers can now send absent students the missed notes for the day. Teachers can simply take a picture of their notes from the board or PowerPoint and save it under today’s date. Then, teachers can share it out to students so they receive the notes right away.
Evernote - Writer's Notebook
Evernote is an app used for note-taking, organizing, task management, and archiving. Evernote allows users to create notes, which can be text, drawings, photographs, or saved web content. Students can add to their notes outside of school.
Express Yourself
Students will do research on a topic, after which they will present their findings in a creative, visual manner. Utilizing information from a variety of resources, students will create an original infographic.
Students seek and provide feedback from classmates using the Comments feature within Google Docs or Slides. Students will then revise their writing and/or presentation based on the feedback received.
Finding the Main Idea
Sometimes finding the main idea of a story can be a challenge. Once the student understands the main idea, they begin to read for understanding. Room Recess provides an activity for students to be actively engaged in reading and finding the main idea.
In this activity, students will read a piece that is uploaded to SeeSaw by the teacher. The teacher may want to have students use the piece again in the year to measure growth.
Infographic Timeline
How to edit and create an infographic may be taught in under 20 minutes, however, students will need additional time to complete research on an assigned topic such as this task where a group of students will collaborate, research and create a timeline of the American Revolution.
Learning to Copy and Paste
Learning to copy and paste is an important skill to learn. Students will learn how to highlight text, copy it, and then paste it into a document. Students will have fun adding text to National Geographic's Funny Fill-ins and then will copy and paste the completed story into a word document.
Letter to a Soldier
Students will review parts of a friendly letter to type a letter to a soldier.
March Madness Sweet 16 Tournament of Books
March is not only Reading Month, but it is also time for March Madness NCAA College Basketball Tournament. The students will review the top 16 books that have circulated throughout your library during the school year, and the students choose the book of the year. You will use the same format as the Sweet 16 NCAA Tournament.
Media Literacy Digital Breakout
Students will complete a digital breakout with a Google site and Google Form to learn about the role that media literacy plays in their ability to understand, analyze, and critically evaluate media messages.
Mentimeter Character Word Cloud
Mentimeter is a free resource teachers can use to create fun and interactive presentations. Teachers will create a word cloud assignment where students will type in adjectives to describe a character from a story.
Students will learn how to use the Explore feature in Google Docs. Students will create a two-column table to organize their research. Finally, students will utilize the citation tool to correctly cite their research.
Practice Spelling With a Puzzle Creator
Puzzel is a free puzzle platform where teachers can create different types of puzzles for students such as crossword puzzles, word searches, memory games, and lots more!
Pronouns for the Intermediate Level
Students will be able to learn about pronouns by watching a Schoolhouse Rock video, Schoolhouse Rock Rufus Xavier Sarsaparilla, with its catchy tune, then see instructional videos on the definition of pronouns and how they are used in sentences. Finally, students will be able to check their knowledge of pronouns through games and online quizzes and apply their knowledge in their own writing.
Publish a Collaborative Research Project With Google Slides
Students will use Google Slides to publish a research project. This project will include links to other slides and information from the internet. Students will also search and add images from the internet and video to support their topic.
Quill Writing
Quill provides 400 free grammar and writing activities for elementary, middle school, and high school. The students take a diagnostic test on the website and then can be assigned activities to help them improve their writing.
Sifting Key Words
Wordsift is a free online tool where teachers and/or students copy text and put relevant vocabulary words into a word cloud. Wordsift “sifts” through the text and looks at more important academic vocabulary and creates word clouds with the key vocabulary.
Spectacular Spellers
VocabularySpellingCity is a spelling and vocabulary website and app that allows the teacher to input spelling lists for students to practice. The site also has premade lists.
Understanding Nouns at the Intermediate Level
Students will be able to learn about nouns by watching a Schoolhouse Rock video, A Noun is a Person, Place or Thing, with its catchy tune, then see instructional videos on the definition of nouns and how they are used in sentences. Finally, students will be able to check their knowledge of nouns through games and online quizzes and apply their knowledge in their own writing.
Visualizing our Goals
We would like our students to read daily to promote good reading habits. In many classrooms students/parents are encouraged or required to keep track of the minutes they read each month.
Students will use Canva to create a goal-tracking sheet for the minutes they’ve read each month.
Wakelet Digital Storytelling
Wakelet is a free platform where teachers and students can save/post links, videos, images, and articles. You can easily share your page with others. Collaborators can also add notes or comments to each item. Students will be telling a story with multimedia to engage their viewers.
Wakelet is a free platform where teachers and students can save/post links, videos, images, and articles. You can easily share your page with others. Collaborators can add notes or comments to each item. Students will learn how to write a newspaper using Wakelet.
Whoooś Reading is a free website that supports students´ reading comprehension and writing development. Students will enter the book they read, take a quiz, answer questions, and write book reviews.
These task cards focus on science topics in the classroom.
Build a Biome
Switch Zoo is a free interactive website that allows students to create their own animals, habitats and biomes. It provides games for students to get involved with their created animal and zoo.
Data Nuggets
Data Nuggets have FREE classroom activities that bring research and authentic data into the classroom. Each “nugget” contains “messy data” at different levels. The student follows the scientific process as they go through the nugget of research. The nugget includes information about the scientist. Students can also create their own nuggets.
Exploring in Minecraft - Farming
Students will create a world on Minecraft. They will learn how to farm and eat on Minecraft while learning how to use the game's tools.
Exploring Scientific Concepts Related to Climate
In this activity, students will explore how climate affects the land, people, and animals around them as well as other regions across the globe. Students will use NASAClimateKids to participate in independent research, ask and answer questions, and record information. Students will relay their findings to the class using Voki by creating an animated video report on the topic.
Great Plant Escape
The Great Plant Escape has students find clues, do experiments, and solve problems as they journey into the world of plants.
Herman the Worm
Herman the Worm is an Illinois University Extension website that teaches about the life of worms through a guided study.
Legends of Learning
Legends of Learning provides interactive math and the sciences games to students that give them superpowers. The teacher can select games that are connected to current lessons. It is aligned with the common core and NGSS. It is FREE for teachers and students in school.
Light Up Learning With Microbits
Students will use knowledge of coding to create a closed circuit. Microbits will be required for this activity.
Paper Circuit Greeting Cards
Students will use knowledge of series circuits to create a closed circuit greeting card.
PhET Science Energy Forms
PhET provides fun, free, interactive, research-based simulations for math and science. The simulations are written in HTML5, and can be used online or downloaded to your computer. This is free for all students and teachers. There are hundreds of lesson plans and materials using the simulations.
Solar System Vocabulary Flash Cards
Science topics such as the Solar System require students
to be able to identify scientific terms by picture just as much as writing down the definition. Students will be creating Solar System Vocabulary Flashcards using Google Slides. Students will be using the Explore tool to search
for definitions and pictures.
Stargazing with Star Chart
Star Chart is an app for your iPad or Android device. Students can learn about constellations by aiming their phones at the night sky. There are additional features that allow students to interact with facts about planets and space discovery.
Study Jams
Scholastic has produced a FREE website full of educational videos and step-by-step slides for students to work on. Study Jams focuses on science and mathematical concepts that are prevalent in third, fourth, and fifth grade.
Wakelet Collaboration Weather Project
Wakelet is a free platform where teachers and students can save/post links, videos, images, and articles. You can easily share your page with others. Collaborators can also add notes or comments to each item. Students will learn about weather and collaborate together using Wakelet to share their research and findings.
These task cards focus on social science topics, geography, mapping and historical events.
A History Remix Using Photos
An engaging way for students to learn about and make comparisons in history is to create remixes using photos and paintings from that time in history. Using the Library of Congress archives and Google images in the public domain, students can download the images and remix them using iPiccy to communicate about that time in history.
Battles of the American Revolution War
Any cause worth fighting for will usually come at a cost. In the fight to gain independence from England, many patriots lost their lives. In this comparison of some battles fought in the Revolutionary War, learn how the outcomes affected the quest for independence.
Ben Franklin and the U.S. Government
Learn about Ben Franklin and how important he was to the development of the United States. Not only was he an important historical figure in the development of the United States Constitution, but he was also a great scientist and inventor.
Bill of Rights WebQuest
Send students on a WebQuest to learn about the key components outlined in the Bill of Rights, their importance, and what makes them necessary. Students will examine and summarize the ten amendments in the United States Bill of Rights, then identify and explain the ten they feel are the most important and why.
Canva World War II Trading Cards
Students will use Canva (canva.com) to incorporate learning they have done about key people of World War II to create trading cards that include pictures, text, and other information about their key person.
Famous Explorer Logos in Canva
Students will finish their learning about explorers by using pictures, text, and other information to create a logo in Canva.
Students will be able to learn about how a bill becomes a law by watching instructional videos and reading articles in MeL (Michigan eLibrary). Finally, students will be able to create an infographic on the process of how a bill becomes a law.
Interactive Maps
National Geographic has an interactive mapmaker website where you can use markers, lines, or shapes to tell your story on MapMaker. You can add in text, photos, and videos with their rich editing tool to customize the map.
Mind Mapping Basics
Mind Mapping is one of the best ways to capture your thoughts. It is great for note-taking, helping you become more creative and solve problems effectively.
Preamble of the Constitution
Students will be able to learn about the preamble of the Constitution of the United States by watching instructional videos and reading articles in MeL (Michigan eLibrary). Students will be able to create a rewrite of the preamble into today’s terminology.
The 13 Colonies
The 13 Colonies is an interactive map that leads the students on a scavenger hunt as they learn about the history and characteristics of the colonies. Students are provided immediate feedback.
The Three Branches of Government
The United States Government is divided into three parts, or branches: the legislative branch, the executive branch, and the judicial branch. Each branch has a different duty, but all three branches must work together. Students learn about the Three Branches of Government and the jobs that they have. They read, play games, and take an interactive quiz.
Wonderopolis
Wonderopolis is a free informational site that asks and answers interesting questions about the world. Wonders of the Day questions are posted daily, and each is designed to get kids to think, talk, and find learning moments in everyday life.
These task cards cover the basic skills a fifth grader should know from mousing to keyboarding to learning how to treat technology devices.
Learning to Copy and Paste
Learning to copy and paste is an important skill to learn. Students will learn how to highlight text, copy it, and then paste it into a document. Students will have fun adding text to National Geographic's Funny Fill-ins and then will copy and paste the completed story into a word document.
Shortcuts
Students enjoy being proficient in using technology shortcuts and teaching the skill to others. Some shortcuts are fairly common across all devices and applications and others are specific to a device. Being familiar with shortcuts helps students transfer the skill to other applications and devices.
Text to Speech
There are FREE amazing tools that will read text to you. These tools help students to listen to what they have written, listen to another person's work, and some will even allow you to transfer the audio to an mp3 player.
These task cards cover apps and programs that can be used across multiple age levels and curriculums. They are primarily how-to task cards on how to use the application in the classroom.
3D Pens
3D pens are an easy way to create a 3D object. You can design from scratch or use a template. Students can develop motor skills and skills in spatial thinking, creativity and imagination, planning and organization, attention to details, and hand-eye coordination.
Assess With Edulastic
Edulastic is a free online assessment platform that provides teachers a customizable and time-saving solution for assigning tests, quizzes, or worksheets to their students. This also has auto-grading and instant data, along with incorporating audio files, videos, and images into your tests. Note: This can be used with Kindergarten- 12th grade.
Assess With PlayPosit
PlayPosit is an interactive video platform that allows the teacher to take videos from different sources and edit them. As teachers add pauses (questions) at regular intervals, it keeps the students actively engaged. Students can also rewind to find the answers.
Basics of Quizlet
Quizlet allows students to study using tools and games. There are thousands of study sets already created, or teachers and students can create their own. When students find a study set, they can choose from different study mode buttons (flashcards, learn, write, spell, test, match, gravity). They can also choose to play live against other students. Students of all ages can use this application.
Blooket Fun
Blooket is an engaging classroom game that allows students to review various classroom skills, concepts, and topics. This task card explains the application and how to use it with students.
Chromebook Superpowers
Using a simple KWL chart on a Google Doc, students can collaborate and acquire shortcuts for using a Chromebook (or just the Chrome browser) more efficiently.
Classkick
Classkick is a free online site that lets teachers create and share assignments. As students work from iPads or Chromebooks, teachers can monitor them and give feedback in real-time. Teachers can also create rosters via class code, enter manually, or use Google Classroom.
Classroomscreen Tools for the Teacher
Teachers are able to use this Classroom Screen in their classroom in a variety of ways. There are 21 widgets to support class activities and help students get to work.
Communicate With Bloomz
Bloomz is a communication platform that helps students succeed in the classroom and at home. It is a great tool to use to stay connected in the school community and send real-time messages to a class or person. Parents and students can view a calendar, view behavior, sign up for conferences or volunteer work, and view photos from both the website or an app. Students will enjoy seeing the petals bloom as they watch behavior points blossom into a flower.
Communicating With Parents Using Klassroom
Klassroom is an app that teachers and parents can download to their phones to easily communicate and engage with each other. Teachers can post assignments, add events, schedule conferences, and upload pictures. This is very similar to Facebook.
Connect With Class Dojo
ClassDojo connects teachers with students and parents to build amazing classroom communities by creating a positive culture. Teachers can encourage students for any skill or value — whether it is working hard, being kind, helping others, or something else.
Padlet is an online post-it-like board that you can use as an easy collaboration platform for students. Padlets can be created by you or students. Once created, a student can share ideas and comment on other students as well. Note: Padlet is no longer free. You can create three for free. Please check the pricing structure.
Create Custom Activities With Wordwall
Wordwall is a free resource for teachers to create interactive and printable activities. These can be played on any web-enabled device. Students can play individually or with each other.
Creating a Graphic Organizer
Lucidchart is a visual platform for creating flowcharts and graphic organizers in real-time. Students under the guidance of teachers can use the free version.
Digital Learning for Kids
e-Learning for Kids is an educational website for kids grades K-5. Lessons are organized by grade level as well as skill topics. Students can easily sort the lessons and then select the lesson they want to work on. They are guided through the lesson with a cartoon video-audio and prompts.
Edpuzzle - Interactive Video Lessons
Edpuzzle allows students to interactively engage with videos through audio notes and questioning. Teachers can use the videos already uploaded, crop, or upload their own. It automatically records students' grades, allows equation options for math, and also allows voice-over. This can be used by students of all ages.
Exploring in Minecraft
Students will create a world on Minecraft. They will eat, explore, and tell the difference between creative, survival, and exploration modes while placing and destroying blocks.
Fast and Curious Eduprotocol
Begin the week’s vocabulary words with a short 10-15 questions round of Quizizz (in live mode). After round one, do a mini-lesson on common mistakes and then give the quiz again to see tremendous growth. Continue reps throughout the week for mastery.
Freckle
Freckle is a FREE website that provides online learning in math, science, social studies, and English language arts. Students take a pretest and then their individualized learning plan begins.
Fun 4 the Brain
Fun4thebrain is a website for students PreK-5 that has many different activities in all content areas for students.
Gimkit
Gimkit is an easy to set up group quiz-based assessment tool which is fast-paced. Teachers do not need to prompt questions. Gimkit also has a monetization component where students build up in-game cash that can then be used to purchase in-game upgrades. You can also paste any Quizlet into Gimkit.
Hooray! Go Formative!
Go Formative! is a web-based tool that allows teachers to formatively assess students on the fly, or at the beginning and end of a lesson. Teachers create the assessment, and students complete the assessment online.
Immersive Reader is a free assistive tech tool for Office 365 that brings assistive technology tools into the Office 365 environment to improve reading and writing. Immersive Reader can improve reading comprehension and increase fluency for English language learners.
Thinglink allows teachers to create visual learning materials and virtual tours, empowers students to work on projects and assignments using text, voice, photos, and video.
Interactive Lessons with Insert Learning
InsertLearning is a Chrome extension that turns websites into interactive lessons. As long as students have a Google login and password, a Chromebook or Chrome extension on a computer, they can log in or be assigned an assignment by the teacher through Google Classroom.
Interactive Lessons With Nearpod
Nearpod is an online site where a teacher can create and import interactive lessons along with online assessments, polls, virtual field trips, etc. Teachers share a live session, students enter a code, and the lesson is synced to all devices.
Kahoot
Kahoot! is a free tool for using technology to administer quizzes, discussions or surveys. It is a game-based classroom response system played by the whole class in real-time. Multiple-choice questions are projected on the screen. Students answer the questions with their smartphone, tablet or computer.
Google Hangouts is a communication platform that includes instant messaging, video chat, SMS and VOIP features. With Google Hangouts, classrooms can connect with each other and collaborate on various topics.
Lino offers the capacity to post sticky notes on a canvas using a web browser. The teacher may create a canvas and share the link with the class. The students do not have to log in to collaborate. Students may collaboratively post color-coded sticky notes, share images, links, etc. to the canvas.
Loom Video Recording
Loom is a free extension to make recordings of Chromebook screen, a combination of a person and the screen, or just a person to create a video with sharing capabilities. Students are able to create recordings to teach others about a subject, an application or website they are using.
Mentimeter Interactive Quiz
Mentimeter is a free resource teachers can use to create fun and interactive presentations. The teacher will create a multiple choice quiz and students will answer each question in real-time.
Popplet is a web-based and app-based tool that allows students to mind map their ideas with user-friendly functions. Students can create brainstorming webs easily and collaboratively. Students learn about United States symbols.
Mystery Doug
Mystery Doug is a FREE weekly video series where students ask questions. The videos are short, approximately five minutes long.
Operations and Algebraic Thinking
Math Playground provides grade-level appropriate activities for math operations and algebraic thinking with a variety of activities to engage students and strengthen their skills. It can be used in grades one-six.
Showbie is an app used by teachers to assign, collect, and review student work. Showbie keeps student work organized by classes and assignments. Students can see their upcoming assignments. Teachers can upload documents for students to access and complete the assignment. Teachers can comment on student’s work.
Pear Deck Interactive Slide Presentation
Pear Deck is an interactive presentation teachers use to enhance engagement in learning. Students will use their Chromebook to follow along with the teacher’s slideshow. Throughout the presentation students will answer a variety of questions, including drawing, dragging, text, number, and multiple choice. Teachers will receive instant feedback and can then discuss this feedback with students.
Presenting With Buncee
Buncee is a presentation tool that allows students to create content that is unique to them and their learning experiences. Buncee offers free public domain images, the ability to embed videos, and upload content, as well as interactive materials, colorful backgrounds, and fun templates. Note: A free Buncee account allows up to five Buncees.
QR codes are everywhere! So what are they all about? Students will be able to create a QR code to share a digital artifact with parents, students and teachers.
Quizalize - Engaging Your Students in Learning
Quizalize is an online platform where teachers can create quizzes to test students’ knowledge and see results in the teacher dashboard. Teachers can also choose from over 12,000 pre-made quizzes. This is great to use as a pre-assessment tool to see which students need intervention or enrichment. Teachers can then use it as a formative assessment to check for understanding. Students can also create their own quizzes to be used by classmates.
Remind - A Communication Tool
Remind is a communication platform that helps students succeed in the classroom and at home. It’s a great tool to use to stay connected in the school community and send real-time messages to a class or person.
Review Game Zone
Review Game Zone is a free online interactive game that helps prepare the students for tests and quizzes. Teacher generated study questions are inserted into the school game to help reinforce classroom learning and test preparation for students.
Smarty Games
Smarty Games is a fun interactive website that helps to develop creativity, visualization, problem-solving skills, math skills and the curiosity for knowledge. It is very easy to understand. It can help students be more successful with their elementary school curriculum. It is kid-friendly, safe and FREE!
Socrative Exit Tickets
Socrative, an online student response system, quickly assesses students with prepared activities or on-the-fly questions to get immediate insight into student understanding. The teacher can use the auto-populated results to determine the best instructional approach to most effectively drive learning.
Storyboarding
Storyboard That is an online storyboard that is like a graphic organizer. It can be used by students and teachers to create storyboards using motion picture, animation, motion graphics, and interactive media.
Sumo Paint
Sumo Paint is a free online paint program and image editor similar to Photoshop. Flash is required for this site and ads are present.
Symbaloo For Educators
SymbalooEDU is a bookmarking website that helps teachers and students organize their favorite websites. This is a resource management tool that can be used to organize lesson plans, add links, web pages, or any other content that would be helpful for your students. You can organize the tiles by subject, grade, or however you want.
Ted Talk Back to School
Ted Talks is a free website that covers a variety of topics. You can use Ted Talks effectively in your classroom through video. This can be used for elementary, middle school, and high school depending on the subject area. It brings together the brightest and best in a broad spectrum of fields, like technology, engineering, music, and more. Ted Talks helps get students thinking critically.
Testmoz - Auto-graded Tests and Quizzes
Testmoz is a web tool that allows teachers to create auto-graded tests and quizzes. You can choose four types of questions - multiple-choice, true/false, multiple response, and fill in the blank. Without an account, you are allowed 50 questions for each test.
Text to Speech
There are FREE amazing tools that will read the text to you. These tools help students to listen to what they have written, listen to another person's work, and some will even allow you to transfer the audio to an mp3 player.
Students will learn more about coding and maker resources. There are activities to better understand computational thinking and how it can be applied when working on projects.
Innovator Eric Curts has coded this marvelous Google Sheets template to create a fantastic game of Battleship. It may be used to introduce students to useful tech skills such as how to share, use of revision history, locate specific cells and enter data.
Characterization Through Coding
Students will apply their knowledge of characterization by coding a sample character’s thought, actions, and words.
Lesson credit to Google CS First Curriculum
Coding Punch Cards
So you have robots...now what? How do you manage them? These coding punch cards can be customized and used to offer students activity choices while managing your learning environment.
When students are learning to code they use often sprites as icons to move around the screen. Piskel is a free online sprite creator for creating animated sprites and pixel art.
Dash Dance Party
Coding with Dash is so much fun! Using the Blockly app, students will be able to utilize block coding to plan, create, and execute a dance for Dash. This task card could be used with grades 3-5. This task card will take two-three class periods.
Whether you build with paper or design something to print in 3D, the simple addition of a USB charged fizzbit vibrating motor can make your creation come alive. A Fizzbit, a product you currently must order from the UK, can easily be attached to a 3D printed object or papercraft character to make them vibrate and move.
Let's Learn to Play Chess
ChessKids is a website that teaches children how to play the game of Chess. It walks them through the game, has tutorials, puzzles and involves the child. It helps to keep them motivated as they are learning the game and improving their skills. It also helps with critical thinking skills.
Gamify Robots
Learn how to use your Robots (e.g., Code and Go Mice, Dash, and Spheros) by gaming. Gaming creates a high-interest, team-building, and problem-solving approach to using the robots, and the games are easily adapted to most robots and many levels.
In a Pickle With Math
Math Pickle is a resource filled with math puzzles and games to engage students in problem-solving. Puzzles are organized by grade level and subject.
Light Up Learning With Microbits
Students will use knowledge of coding to create a closed circuit. Microbits will be required for this activity.
Lightbot Coding
Coding or programming is the way that we as programmers, can tell a computer what to do using instructions that the computer will understand. The instruction for coding can be either in words or icons. Lightbot is a programming puzzle game.
Logic Puzzles
Computational thinking is a powerful skill that students need to learn. A great place to start is puzzles! Students love to solve puzzles and it helps develop reasoning skills useful for programming, computer science, and anything they might do.
Paper Circuit Greeting Cards
Students will use knowledge of series circuits to create a closed circuit greeting card.
Survey Says
Students are presented with a scenario surrounding a relevant contemporary public issue on which they can take a stand. Students will work together to create a survey that will then be shared with various groups, in order to analyze the ways in which certain groups perceive a given issue. Once collected, the data will be discussed and analyzed. Finally, students will support their opinion using the data collected.
Unplugged: What is Computational Thinking?
Computational Thinking and coding go hand-in-hand. But what does it mean to think computationally, and how do we introduce this type of thinking to students? In this unplugged activity, students experience what it’s like to be a programmer while they code another student to complete a drawing.
These task cards assist the student in communicating and presenting creatively in different ways.
Common Good Collaborating
Google Sites are a great platform to communicate both locally & globally. They are easy to set up for your students and easy to link information both individually for your students as well as setting up a group or classroom page.
Communicate With Bloomz
Bloomz is a communication platform that helps students succeed in the classroom and at home. It is a great tool to use to stay connected in the school community and send real-time messages to a class or person. Parents and students can view a calendar, view behavior, sign up for conferences or volunteer work, and view photos from both the website or an app. Students will enjoy seeing the petals bloom as they watch behavior points blossom into a flower.
Communicating With Parents Using Klassroom
Klassroom is an app that teachers and parents can download to their phones to easily communicate and engage with each other. Teachers can post assignments, add events, schedule conferences, and upload pictures. This is very similar to Facebook.
Connect With Class Dojo
ClassDojo connects teachers with students and parents to build amazing classroom communities by creating a positive culture. Teachers can encourage students for any skill or value — whether it is working hard, being kind, helping others, or something else.
Express Yourself
Students will do research on a topic, after which they will present their findings in a creative, visual manner. Utilizing information from a variety of resources, students will create an original infographic.
Infographic Timeline
How to edit and create an infographic may be taught in under 20 minutes, however, students will need additional time to complete research on an assigned topic such as this task where a group of students will collaborate, research and create a timeline of the American Revolution.
Parlay Ideas
Parlay is a written discussion activity that increases student voice and encourages a diversity of perspectives. Students will join discussions, provide constructive feedback, and build on each other's ideas.
Publish a Collaborative Research Project With Google Slides
Students will use Google Slides to publish a research project. This project will include links to other slides and information from the internet. Students will also search and add images from the internet and video to support their topic.
Solving Problems Together
Problems get solved when people work together. Students will choose a problem or issue from their local community that needs a solution. They will create a Google Form that will be used to gather community opinions and perceptions to help create a solution to a local issue.
In this lesson, students will take a story they have written and publish it using Google Slides or PowerPoint. They will be able to insert pictures and speech bubbles to make their story come to life for their audience.
Video Remix
Students love to express themselves with images and videos. As we know there is a plethora of images and video readily available on the Internet for use in remixing other materials to create something new. Students can use public domain images and videos to remix for an assigned school project.
Wakelet Digital Storytelling
Wakelet is a free platform where teachers and students can save/post links, videos, images, and articles. You can easily share your page with others. Collaborators can also add notes or comments to each item. Students will be telling a story with multimedia to engage their viewers.
What's the Issue
Students will create a short screencast that helps present a local or global issue to a wider audience. The screencast will include a voiced narration presenting an idea on how to solve the issue and requesting ideas in return. After considering the feedback received students will create a new screencast to present the ideas.
Working Together to Stop Bullying
Students will obtain multiple viewpoints on the issue of bullying and brainstorm possible solutions, with peers, educators, and parents, via video-conferencing. After collaborating with at least two individuals directly impacted by bullying, the class will create a “No Bullying” campaign based on solutions discussed during the video- conferences and the class discussions.
Ziteboard Collaboration
Ziteboard is a free online whiteboard with real-time collaboration. It can be used for brainstorming, presentation and communicating. Not only can you use tools that are provided, but pictures and voice can also be inserted.
Design Thinking is a process that teachers use to take a structured approach to generate and develop ideas. The task cards in this area help teach the design process.
Roller Coaster Challenge
Students will design a fun and safe roller coaster with at least one loop. Students start by simulating a coaster with Amusement Park Physics. After simulating a coaster and understanding how to design a safe and fun coaster, students use the design process to create a prototype coaster for an amusement park.
Strawbees Egg Drop Challenge
Do the egg drop challenge with Strawbees! Have the students be innovative designers and construct a model for the egg with the Strawbees so when it is dropped, it will not break. The students are learning about force and gravity. Using technology they can gather information to look at how they can develop the best model.
Students will be creating a virtual collage of their summer using canva.com. They will be able to upload pictures of their own or use stock photos on the site, use a background, and include text telling a little bit about their summer through labels.
It is a lot of fun to create music and Incredibox makes it easy using a beatbox technique along with dragging and dropping sound icons on different characters to create music. Students can easily create a music track in seconds and download it as background music in a project.
The Frayer Model is an instructional strategy that helps students learn new concepts by using examples and non-examples. By combining this instructional strategy with Google Draw, teachers may quickly customize and disburse organizers for classroom use and also improve student engagement.
Incredibox Song Composition
The learner will use Incredibox (website or app) to create an a capella song that will accompany a piece of their (creative) writing. Using Incredibox, students will create a mix of seven individuals who sing/beatbox/make sound effects.
Me: A Work of Arts
The learner will use a Google Slide template to create an “All About Me” using photos, artwork, etc. They will then create a song (score) using Incredibox to go along with their slide. When finished, they will present to the class. This could also be done on PowerPoint.
Pop Art Image Editing
Students may create new images in the style of Andy Warhol while learning photo editing skills. The Pixlr photo editor is a free site where students may create or modify images and then download them for use on other projects.
Stop Motion Animation
Your students will love stop-motion animation for movie making. There is a free app for both Android and iOS that your students can use, but there are in-app purchases that may be necessary for any extra features needed for a professional-looking product.
Sumo Paint
Sumo Paint is a free online paint program and image editor similar to Photoshop.
Using Chrome Music Lab to Create Music Digitally
In this lesson, students will become more advanced in their understanding of music by creating their own music with Chrome Music Lab.
You Gotta Have Art
Art inspires students to do well in and out of the classroom. Students that have art may stay in school, be more motivated, have a positive attitude, and it may improve academic performance. Students will “tour” the free website of the Metropolitan Museum of Art and learn about art from around the world.
These task cards have activities to teach students about how to be a good citizen, stay safe on the Internet and prevent bullying.
Becoming a Confident Internet Explorer
Interland is an online game created by Google, that provides learning about digital citizenship through gaming. Internauts are characters in the game. Students practice the skills that they need to be confident online explorers.
Google Scholar Citations provide a simple way for students to keep track of citations to their articles. When students are using Google Docs to write their papers, they can easily link and cite the articles used for references.
Students will learn how to use the Explore feature in Google Docs. Students will create a two-column table to organize their research. Finally, students will utilize the citation tool to correctly cite their research.
Evaluating Websites HyperDoc
Students will learn how to evaluate a website while using a HyperDoc in a group task. Hyperdocs are interactive digital documents that may easily be edited by a teacher. These multimedia-rich documents contain links for research, videos, images, etc. along with the steps for students to follow and record their work.
Media Literacy Digital Breakout
Students will complete a digital breakout with a Google site and Google Form to learn about the role that media literacy plays in their ability to understand, analyze, and critically evaluate media messages.
Properly Citing Sources
Students write an essay expressing their opinion, wherein they choose an influential person and explain why this person should be included within a list of most influential people of all time. In order to ensure that students are respecting the rights of others, they will use an online tool for generating citations to cite sources within a bibliography.
Safe Search
When using either of these two search engines, Google and Bing will filter out explicit and violent content from search results (web sites, images and videos.) Unfortunately, Safe Search is not perfect, but there will be much less unsuitable content and we know students are going to use these search engines when doing research.
Stop the Plagiarism
Students need to learn early on how important it is to create original work. Luckily there are digital tools to help students ensure that when they are doing their research that they are not copying the work of others.
Working Together to Stop Bullying
Students will obtain multiple viewpoints on the issue of bullying and brainstorm possible solutions, with peers, educators, and parents, via video-conferencing. After collaborating with at least two individuals directly impacted by bullying, the class will create a “No Bullying” campaign based on solutions discussed during the video- conferences and the class discussions.
MITECS (Michigan Integrated Technology Competencies for Students): Michigan adopted a state-wide version of the "ISTE Standards for Students" and named them MITECS (Michigan Integrated Technology Competencies for Students) in 2018.
To learn more about MITECS, visit the State of Michigan's TechPlan.org website for definitions and support documents to assist you in the classroom.
Search for task cards by MITECS/ISTE Standards | 9,847 | 50,096 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.796875 | 3 | CC-MAIN-2024-38 | latest | en | 0.92279 |
https://www.eph.com.sg/23-tips/55-heuristics-methods-used-in-mathematics-part-1 | 1,695,598,959,000,000,000 | text/html | crawl-data/CC-MAIN-2023-40/segments/1695233506669.96/warc/CC-MAIN-20230924223409-20230925013409-00017.warc.gz | 830,077,708 | 12,907 | ### Career
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Posted on : 2011-08-10
Part 1
The word ‘heuristic’ is taken directly from the Greek verb, heuriskein which means ‘to discover’. In Mathematics, there are usually different ways to go about solving problem sums. These ways or methods are known as heuristics.
Heuristics can be divided into 4 main types, which will be covered in this 2-part article.
One: Giving a representation
· Pupils can transform word problems into pictorial representations and represent information with a diagram/model. This skill helps pupils to understand the question better when they see the visual representation of the word problems.
· A systematic list should be made for word problems that require pupils to identify patterns such as repeated numbers or a series of events that repeat. This skill helps pupils in identifying patterns easily as the list organises all possible answers systematically.
Example: Michele saved \$150 on the first month. On the second month, she saved \$60 more than the first month. On the third month, she saved \$70 more than the second month. On the fourth month, she saved \$55 more than the third month. How much did she save in four months? Solution: Making a list: 1st month → \$150 2nd month → \$150 + \$60 = \$210 3rd month → \$210 + \$70 = \$280 4th month → \$280 + \$55 = \$335 Total amount saved = \$150 + \$210 + \$280 + \$335 = \$975 She saved \$975 in four month
Two: Making a calculated guess
· The ‘guess and check’ method is used for word problems when certain information is lacking. It requires them to make a guess first and check it, and making subsequent guesses and checks until the correct answer is derived. It is often used together with a systematic list as it helps pupils to narrow down the possibilities within a short time frame.
Example:
Jenny has a total of 7 dogs and parrots. The animals have 20 legs altogether.
How many dogs does she have?
Solution:
Using the ‘guess and check’ method,
Number of dogs Number of legs Number of parrots Number of legs Total number of legs Check 1 1 x 4 = 4 6 6 x 2 = 12 4 + 12 = 16 X 2 2 x 4 = 8 5 5 x 2 = 10 8 + 10 = 18 X 3 3 x 4 = 12 4 4 x 2 = 8 12 + 8 = 20 √
She has 3 dogs.
· The ‘look for patterns’ method is usually used by pupils when they have to identify a certain pattern in a number sequence.
Example:
12 4 5 13 11 9 16 18 16 12 3 X
Solution:
Making a list of possibilities:
12 - 4 + 5 = 13
11 - 9 + 16 = 18
16 - 12 + 3 = 7
The value of X is 7.
Hence by using the systematic list, it is more effective to find the underlying pattern.
Written by Gui Yan Tong
Check out the series "Use of Heuristics in Problem Solving" by EPH and have more practice on the different types of heuristics
Extra!
For more pointers on Model Drawing and Guess and Check, check out these related articles at the Popular Community:
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https://testbook.com/question-answer/given-below-are-two-statements-one-is-labeled-as--5feb2d4fe07704a1e75bfe0d | 1,627,101,708,000,000,000 | text/html | crawl-data/CC-MAIN-2021-31/segments/1627046150129.50/warc/CC-MAIN-20210724032221-20210724062221-00531.warc.gz | 570,950,637 | 30,989 | # Given below are two statements, one is labeled as Assertion A and the other is labeled as Reason RAssertion A: A windmill with proper design considerations can be made to operate at efficiencies in the range of 60% to 65%Reason R: The efficiency of the windmill depends on its design parameters.In light of the above statements, choose the most appropriate answer from the options given below
Free Practice With Testbook Mock Tests
## Options:
1. Both A and R are correct and R is the correct explanation of A
2. Both and R are correct but R is NOT the correct explanation of A
3. A is correct but R is not correct
4. A is not correct but R is correct
### Correct Answer: Option 1 (Solution Below)
This question was previously asked in
Official Paper 2: Held on 24 Sep 2020 Shift 2
## Solution:
Wind power:
Wind power is the generation of electricity from wind. Wind power harvests the primary energy flow of the atmosphere generated from the uneven heating of the Earth’s surface by the Sun. Therefore, wind power is an indirect way to harness solar energy. Wind power is converted to electrical energy by wind turbines.
Turbine Design:
• Wind turbines are designed to maximize the rotor blade radius to maximize power output.
• Larger blades allow the turbine to capture more of the kinetic energy of the wind by moving more air through the rotors.
• However, larger blades require more space and higher wind speeds to operate.
• As a general rule, turbines are spaced out at four times the rotor diameter.
• This distance is necessary to avoid interference between turbines, which decreases the power output.
• The efficiency of the windmill depends on the turbine
• Weather condition, density, and speed of air, etc, also have their effects.
• In 1919, German physicist Albert Betz showed that for a hypothetical ideal wind-energy extraction machine, the fundamental laws of conservation of mass and energy allowed no more than 59.3% of the wind's kinetic energy to be captured.
• This Betz' law limit can be approached by modern turbine designs which reach 70 to 80% of this theoretical limit.
Therefore, Both A and R are correct and R is the correct explanation of A. | 518 | 2,191 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.03125 | 3 | CC-MAIN-2021-31 | latest | en | 0.786432 |
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Weighted Average Challenge
8 - Cloud Apps
Hi Community,
I'm trying to resolve a weighted average challenge, but i can't get to it. Hoping there is help out there.
My table has data as follows:
In a pivot a present the two months side-by-side in the following way:
Additionally to the side-by-side data, I want to calculate how much revenue I potentially could have gotten. Number of devices typically increase, but the MRC (Monthly Recurring Revenue) doesn't necessarily increase in the same scale. Usually price discounts are given with more devices added.
So I want to calculate the "Expected_new_Revenue". I do this by multiplying the ARPU (Average Revenue Per Unit) of the previous month (January) with the Devices of the current month (Feb). From this "Expected_new_Revenue" I now want to derive my "Delta" missing revenue; aka: "This month revenue" - "Expected_new_Revenue".
To make this more clear, here is an excel example. Sisense today presents me the 1,305.68 result; but what I need is the 592.71 as result.
My formular for the "Expected_new_Revenue" is:
``````((
([Total MRC], PREV([Months in Revenue_Date],1))
/
([Total Devices], PREV([Months in Revenue_Date],1))
)
* [Total Devices]
)
``````
And the formular for the "Delta" (missing revenue):
``````[Total MRC] -
((
([Total MRC], PREV([Months in Revenue_Date],1))
/
([Total Devices], PREV([Months in Revenue_Date],1))
)
* [Total Devices]
)
``````
Even if I add the "Customer" in the pivot table, and it shows the right "subvalues" per row, the Grand Total still is the wrong compute.
2 ACCEPTED SOLUTIONS
In this case you need multi-pass aggregation.
You manually set the "Customer" group in the formula and summarize the results.
``````Sum( [Customer],
[Total MRC] -
((
([Total MRC], PREV([Months in Revenue_Date],1))
/
([Total Devices], PREV([Months in Revenue_Date],1))
)
* [Total Devices]
)
)``````
Kind regards,
Angelina
8 - Cloud Apps
OMG, wonderful. You @Angelina_QBeeQ are my rockstar!
4 REPLIES 4
10 - ETL
Hi @mamuewue ,
In Sisense, you can set the rules for calculating the values in the grand total.
By default, it is calculated by the original formula. But you can set it as a sum of values in the table.
Here is the example:
Always here to help,
Angelina from QBeeQ
[email protected]
QBeeQ - Gold Implementation and Development Partner
8 - Cloud Apps
Hi Angelina,
this is very helpful as a first step. I can can compute the number now in the table where I have all customers listed (picture marker 2). However, is there also any way to make the formular "Weighted Average" in the above pivot (picture marker 1), where no customers are listed (and just the sum)?
In this case you need multi-pass aggregation.
You manually set the "Customer" group in the formula and summarize the results.
``````Sum( [Customer],
[Total MRC] -
((
([Total MRC], PREV([Months in Revenue_Date],1))
/
([Total Devices], PREV([Months in Revenue_Date],1))
)
* [Total Devices]
)
)``````
Kind regards,
Angelina
8 - Cloud Apps
OMG, wonderful. You @Angelina_QBeeQ are my rockstar!
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Product Feedback Forum: | 822 | 3,178 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.828125 | 3 | CC-MAIN-2024-10 | latest | en | 0.889175 |
https://physics.stackexchange.com/questions/34346/parton-distribution-functions-plot | 1,721,103,577,000,000,000 | text/html | crawl-data/CC-MAIN-2024-30/segments/1720763514726.17/warc/CC-MAIN-20240716015512-20240716045512-00178.warc.gz | 390,062,382 | 39,236 | # Parton Distribution Functions plot
I was looking at a plot of the parton distribution functions today and had a question. On the y axis, it seems like the value of x f(x) for gluons is greater than one at small x. I was under the impression that parton distribution functions are probability densities and cannot be greater than one. Also, x is a fraction of momentum and can also not be greater than one. Does anyone know why this is?
A probability cannot be greater than 1, but a probability density can be. The parton distribution represents basically the probability per unit momentum fraction, so it can be large over a small region of $x$ without contributing much to the actual probability, $\int f(x)\mathrm{d}x$.
• Actually, I think the integral $\int_0^1 f(x)\mathrm{d}x$ can be interpreted as the average value of the particle number operator, and that average is going to be 1 by definition. (I had to think about it for a while to reconvince myself of this) Commented Aug 17, 2012 at 4:30 | 237 | 1,005 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.734375 | 3 | CC-MAIN-2024-30 | latest | en | 0.957239 |
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# Heuristic for finding elements that appears often together in a big data set
Problem:
I have a list of millions of transactions. Each transaction contains items (eg 'carrots', 'apples') the goal is to generate a list of pair of items that frequently occur together in individual transactions. As far as I can tell doing an exhaustive search isn't feasible.
Solution attempts:
So far I have two ideas. 1) Randomly sample some appropriate fraction of transactions and only check those or 2) count how often each element appears, use that data to calculate how often elements should appear together by chance and use that to modify the estimate from 1.
Any tips, alternative approaches, ready-made solutions or just general reading suggestions are much appreciated.
Edit:
Number of diffent items: 1,000 to 100,000
Memory constraint: A few gigs of ram at the most for a few hours.
Frequency of use: More or less a one off.
Available resources: 20-100 hours of newbie programmer time.
Desired result list format: Pair of items and some measure how often they appear, for the n most frequent pairs.
Distribution of items per transactions: Unknown as of now.
-
How many items do you have are they again in the order of millions? – Ivaylo Strandjev Jan 4 '13 at 14:17
More like 1,000-100,000 – RalleG Jan 4 '13 at 14:26
Definitely an interesting question. I believe both points that you propose are good suggestions and it's worth trying using them. – Ivaylo Strandjev Jan 4 '13 at 14:31
What is the memory constraint? How often do you need the list re-generated? What resources can you devote to this task? How many different items are there in your universe? What part of the list of pairs do you want? What is the distribution of the number of items per transaction? – Deer Hunter Jan 4 '13 at 14:50
Try searching for frequent item set mining. – ElKamina Jan 4 '13 at 16:02
Let the number of transactions be `n`, the number of items be `k`, and the average size of a transaction be `d`.
The naive approach (check pair in all records) will give you `O(k^2 * n * d)` solution, not very optimal indeed. But we can improve it to `O(k*n*d)`, and if we assume uniform distribution of items (i.e. each items repeats on average `O(n*d/k)` times) - we might be able to improve it to `O(d^2 * n + k^2)` (which is much better, since most likely `d << k`).
This can be done by building an inverted index of your transactions, meaning - create a map from the items to the transactions containing them (Creating the index is `O(nd + k)`).
Example, if you have transactions
``````transaction1 = ('apple','grape')
transaction2 = ('apple','banana','mango')
transaction3 = ('grape','mango')
``````
The inverted index will be:
``````'apple' -> [1,2]
'grape' -> [1,3]
'banana' -> [2]
'mango' -> [2,3]
``````
So, after understanding what an inverted index is - here is the guidelines for the solution:
1. Build an inverted index for your data
2. For each item x, iterate all documents it appears in, and build a histogram for all the pairs `(x,y)` such that `y` co-occures with `x`.
3. When you are done, you have a histogram containing k^2 items, which you need to process. This question discusses how to get top-k elements out of an unsorted list.
Complexity analysis:
1. Building an inverted index is `O(nd+k)`
2. Assuming each element repeats in `O(nd/k)` transactions, each iteration takes `O(nd/k * d)` time, and you have `k` iterations in this step, so you get `O(nd^2 + k)` for this step.
3. Processing the list can be done in O(k^2logk) if you want full ordering, or if you just want to print top X elements, it can be done in `O(k^2)`.
Totaling in `O(nd^2 + k^2)` solution to get top-X elements, which is MUCH better then naive approach, assuming `d << k`.
In addition, note that the bottleneck (step 2) can be efficiently parallelized and distributed among threads if needed.
-
Thank you very much for the comprehensive answer! – RalleG Jan 4 '13 at 19:24
@RalleG: You are most welcome. I hope it helps. Note that building an inverted index helps a lot when talking about a lot of collections/documents which contains much fewer items/terms/words - and thus is often used by search engines (just an FYI addition). – amit Jan 4 '13 at 19:26
If the number of items ordered in one purchase is small(<10) do this:
have map of maps(dictionary of dictionaries) : key in the first map is item,
value in the first map is map whose key is second item, value count how many times it appeared in the purchase with first key.
So go through every order and for every pair update map. At the end go through map of maps and look for "big values" in the "second value"
Note: depending on the size and "distribution"of input data you might end up with not enough memory
- | 1,238 | 4,975 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.875 | 3 | CC-MAIN-2016-30 | latest | en | 0.930713 |
https://www.physicsforums.com/threads/conservation-of-angular-momentum-derivation.640269/ | 1,590,374,471,000,000,000 | text/html | crawl-data/CC-MAIN-2020-24/segments/1590347387155.10/warc/CC-MAIN-20200525001747-20200525031747-00057.warc.gz | 891,411,095 | 18,363 | # Conservation of angular momentum derivation
## Main Question or Discussion Point
Confusion over derivation of angular momentum equation
Hello, I'm a little confused over the relation between torque and angular momentum.
When $$L=r×mv$$
$$\frac{dL}{dt}=r×m\frac{dv}{dt}+mv×\frac{dr}{dt}$$
According to Wikipedia,
$$v=\frac{dr}{dt}$$
$$mv×\frac{dr}{dt}=mv×v=0$$
So $$\frac{dL}{dt}=r×m\frac{dv}{dt}=r×F=τ$$
But isn't $$v=r\frac{dθ}{dt}≠\frac{dr}{dt}$$
This is the part which bugs me. I hope someone can clarify it.
Last edited:
Related Classical Physics News on Phys.org
actually $$v_{\bot}=r\frac{dθ}{dt}≠v$$
but i still can't see how this leads to $$v=\frac{dr}{dt}$$
vanhees71
Gold Member
2019 Award
I think there is, first of all, some confusion in your notation. You should clearly distinguish vector and scalar quantities and define you quantities.
In coordinate-free vector notation for the angular momentum of a point particle
$$\vec{L}=m \vec{r} \times \dot{\vec{r}}.$$
Here, $\vec{r}$ is the position vector with respect to an (arbitrary) fixed point. Taking the time derivative you get from Newton's Equation of motion
$$\dot{\vec{L}}=m \dot{\vec{r}} \times \dot{\vec{r}}+m \vec{r} \times \ddot{\vec{r}}=\vec{r} \times \vec{F}=\vec{\tau}.$$
Here, $\vec{\tau}$ is the torque of the total force on the particle measured with respect to the arbitrary fixed origin of your reference frame.
I do not know how you define $\theta$. So I can't say anything about it.
sorry, that's the notation used in Wikipedia.
afraid i don't understand your notation with all those dots.
θ is used in the derivation of angular velocity.
vanhees71
Gold Member
2019 Award
The dots are time derivative, i.e., for any quantity $f$ I use the usual notation
$$\dot{f}:=\frac{\mathrm{d} f}{\mathrm{d} t}.$$
The section on the definition of the quantities in Wikipedia is quite confusing. Forget it. Use the definitions in terms of vectors. Then there shouldn't be any problems. The derivation, I've given is identical to the one in Wikipedia in the Section titled "Proof of the equivalence of definitions".
L=r×p, now to avoid cross only that component of p should be chosen which is perpendicular to r so
L=mr2 dθ/dt
now dL/dt can be obtained just by differentiation of above.
now torque=r×F ,again F perpendicular should be chosen .now it is possible to show that acceleration along θ direction is 1/r d/dt( r2 dθ/dt) which you can prove by yourself or see somewhere else( transverse acceleration)
so torque=m d/dt( r2 dθ/dt) which is of course equal to dL/dt.( this is all two dimension)
to avoid confusion one should use cartesian coordinates rather polar in which it is much simple.
vanhees71
Gold Member
2019 Award
I also don't understand, which one should use this angle. If you want to describe physics in non-cartesian coordinates, better use Hamilton's principle of least action, where such issues become much less complicated.
Of course even there you need to calculate time derivatives of the position vector. Let's look at the case of two-dimensional motion in cylinder coordinates. First of all one must realize that also the basis vectors become position dependent. So let's first derive this dependence by using a cartesian basis first. The position vector reads
$$\vec{r}=r (\vec{e}_x \cos \varphi + \vec{e}_y \sin \varphi).$$
The basis of the cylinder coordinates $(r,\varphi)$ is given by the derivatives with respect to these coordinates (and then normalized). First the non-normalized basis vectors are
$$\vec{b}_r=\partial_r \vec{r}=\vec{e}_x \cos \varphi + \vec{e}_y \sin \varphi, \quad \vec{b}_{\varphi}=r (-\vec{e}_x \sin \varphi + \vec{e}_y \cos \varphi).$$
One finds that both vectors are orthogonal to each other. Thus usually one also normalizes the basis vectors, using
$$\vec{e}_r=\vec{b}_r=\vec{e}_x \cos \varphi + \vec{e}_y \sin \varphi \quad \vec{e}_{\varphi}=(-\vec{e}_x \sin \varphi + \vec{e}_y \cos \varphi).$$
As you see, the new basis vectors depend on the position, and you have
$$\partial_r \vec{e}_r=\partial_r \vec{e}_{\varphi}=0, \quad \partial_{\varphi} \vec{e}_r=\vec{e}_{\varphi}, \quad \partial_{\varphi} \vec{e}_{\varphi}=-\vec{e}_r.$$
Now you can calculate the time derivative of the position vector directly in cylinder coordinates. You have by definition $\vec{r}=r \vec{e}_r$ and thus the velocity is given by
$$\dot{\vec{r}}=\frac{\mathrm{d} \vec{r}}{\mathrm{d} t}=\dot{r} \vec{e}_r + r \dot{\vec{e}}_r=\dot{r} \vec{e}_r + r \dot{\varphi} \vec{e}_{\varphi}.$$
In the last step, I've used the chain rule for deriving the basis vector $\vec{e}_r$, using the above given partial derivatives. Of course you can also use the cartesian coordinates to come to the same result.
The second derivative with respect to time, i.e., the acceleration gives
$$\vec{a}=\dot{\vec{v}} = \ddot{r} \vec{e}_r + \dot{r} \dot{\vec{e}}_r + \frac{\mathrm{d} (r \dot{\varphi})}{\mathrm{d} t} \vec{e}_\varphi + r \dot{\varphi} \dot{\vec{e}}_{\varphi}.$$
Again using the chain rule you get
$$\vec{a}=(\ddot{r} - r \dot{\varphi}^2) \vec{e}_r + (2 \dot{r} \dot{\varphi}+r \ddot{\varphi}) \vec{e}_{\varphi}.$$
Last edited:
$$v=\frac{dr}{dt}$$ This is the definition of the vector on the left via the quantity on the right. Velocity is the rate of change of position.
$$v=r\frac{dθ}{dt}≠\frac{dr}{dt}$$ This makes no sense, because velocity is not equal to the quantity in the middle. The quantity in the middle, via an abuse of notation, might be meaningful only in the particular case of circular motion.
$$v=\frac{dr}{dt}$$ This is the definition of the vector on the left via the quantity on the right. Velocity is the rate of change of position.
$$v=r\frac{dθ}{dt}≠\frac{dr}{dt}$$ This makes no sense, because velocity is not equal to the quantity in the middle. The quantity in the middle, via an abuse of notation, might be meaningful only in the particular case of circular motion.
But for an object undergoing uniform circular motion, its angular momentum is constant, its radius is also constant, so $$\frac{dr}{dt}=0$$
Its tangential velocity would be $$v=r\frac{dθ}{dt}$$
Also its angular momentum is still $$m(v×r)=mvr$$since v is the tangential velocity.
I don't understand how the velocity can be construed as the rate of change of radius.
You are confusing vectors and scalars, speed with velocity and and radius with position. You should start from defining what all those letters you use are.
if you want to treat the components of velocity then use what I have written above otherwise you should better stick to the definition of velocity for it.
vanhees71
Gold Member
2019 Award
But for an object undergoing uniform circular motion, its angular momentum is constant, its radius is also constant, so $$\frac{dr}{dt}=0$$
Its tangential velocity would be $$v=r\frac{dθ}{dt}$$
Also its angular momentum is still $$m(v×r)=mvr$$since v is the tangential velocity.
I don't understand how the velocity can be construed as the rate of change of radius.
Again, in this way you never will get a clear understanding what's going on. You have to clearly distinguish between vector and scalar quantities. That's what the little arrows are good for. In Wikipedia they print vectors bold face. I prefer the arrow notation, because it's better visible.
Look at my last posting. There everything is evaluated in polar coordinates for the motion in a plane. The special case of uniform circular motion around a fixed axis (here the $z$) is given by $\dot{r}=0$ (note, here it's the magnitude of the position vector! That's why there's no arrow). For the velocity from this you get, using the general formulas, derived in my posting
$$\vec{v}=r \dot{\varphi} \vec{e}_{\varphi}.$$
BTW: You should draw all these vectors, appearing in my derivation. Then a lot becomes much more clear and you get a better intuition of what's going on!
unfortunately, your explanation has made it very unclear and counter-intuitive for me.
not to criticize you, its probably because im still a noob when it comes to angular momentum.
maybe next time. | 2,249 | 8,079 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.953125 | 4 | CC-MAIN-2020-24 | latest | en | 0.839331 |
https://zdoc.pub/q-1-q-5-carry-one-mark-each414ee72caaf7f7e86bf7d22d7d7c5c3d71878.html | 1,669,989,487,000,000,000 | text/html | crawl-data/CC-MAIN-2022-49/segments/1669446710902.80/warc/CC-MAIN-20221202114800-20221202144800-00431.warc.gz | 1,163,766,812 | 15,761 | # Q. 1 Q. 5 carry one mark each
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https://number.academy/176 | 1,679,493,295,000,000,000 | text/html | crawl-data/CC-MAIN-2023-14/segments/1679296943809.76/warc/CC-MAIN-20230322114226-20230322144226-00376.warc.gz | 489,416,508 | 13,667 | # Number 176
Number 176 spell 🔊, write in words: one hundred and seventy-six . Ordinal number 176th is said 🔊 and write: one hundred and seventy-sixth. The meaning of the number 176 in Maths: Is it Prime? Factorization and prime factors tree. The square root and cube root of 176. What is 176 in computer science, numerology, codes and images, writing and naming in other languages. Other interesting facts, news related to 176.
## What is 176 in other units
The decimal (Arabic) number 176 converted to a Roman number is CLXXVI. Roman and decimal number conversions.
The number 176 converted to a Mayan number is Decimal and Mayan number conversions.
#### Weight conversion
176 kilograms (kg) = 388.0 pounds (lbs)
176 pounds (lbs) = 79.8 kilograms (kg)
#### Length conversion
176 kilometers (km) equals to 109.361 miles (mi).
176 miles (mi) equals to 283.245 kilometers (km).
176 meters (m) equals to 283.245 feet (ft).
176 feet (ft) equals 53.645 meters (m).
176 centimeters (cm) equals to 69.3 inches (in).
176 inches (in) equals to 447.0 centimeters (cm).
#### Temperature conversion
176° Fahrenheit (°F) equals to 80° Celsius (°C)
176° Celsius (°C) equals to 348.8° Fahrenheit (°F)
#### Power conversion
176 Horsepower (hp) equals to 129.43 kilowatts (kW)
176 kilowatts (kW) equals to 239.33 horsepower (hp)
#### Time conversion
(hours, minutes, seconds, days, weeks)
176 seconds equals to 2 minutes, 56 seconds
176 minutes equals to 2 hours, 56 minutes
### Codes and images of the number 176
Number 176 morse code: .---- --... -....
Sign language for number 176:
Number 176 in braille:
Images of the number
Image (1) of the numberImage (2) of the number
More images, other sizes, codes and colors ...
### Gregorian, Hebrew, Islamic, Persian and Buddhist Year (Calendar)
Gregorian year 176 is Buddhist year 719.
Buddhist year 176 is Gregorian year 367 a. C.
Gregorian year 176 is Islamic year -460 or -459.
Islamic year 176 is Gregorian year 792 or 793.
Gregorian year 176 is Persian year -447 or -446.
Persian year 176 is Gregorian 797 or 798.
Gregorian year 176 is Hebrew year 3936 or 3937.
Hebrew year 176 is Gregorian year 3584 a. C.
The Buddhist calendar is used in Sri Lanka, Cambodia, Laos, Thailand, and Burma. The Persian calendar is the official calendar in Iran and Afghanistan.
## Mathematics of no. 176
### Multiplications
#### Multiplication table of 176
176 multiplied by two equals 352 (176 x 2 = 352).
176 multiplied by three equals 528 (176 x 3 = 528).
176 multiplied by four equals 704 (176 x 4 = 704).
176 multiplied by five equals 880 (176 x 5 = 880).
176 multiplied by six equals 1056 (176 x 6 = 1056).
176 multiplied by seven equals 1232 (176 x 7 = 1232).
176 multiplied by eight equals 1408 (176 x 8 = 1408).
176 multiplied by nine equals 1584 (176 x 9 = 1584).
show multiplications by 6, 7, 8, 9 ...
### Fractions: decimal fraction and common fraction
#### Fraction table of 176
Half of 176 is 88 (176 / 2 = 88).
One third of 176 is 58,6667 (176 / 3 = 58,6667 = 58 2/3).
One quarter of 176 is 44 (176 / 4 = 44).
One fifth of 176 is 35,2 (176 / 5 = 35,2 = 35 1/5).
One sixth of 176 is 29,3333 (176 / 6 = 29,3333 = 29 1/3).
One seventh of 176 is 25,1429 (176 / 7 = 25,1429 = 25 1/7).
One eighth of 176 is 22 (176 / 8 = 22).
One ninth of 176 is 19,5556 (176 / 9 = 19,5556 = 19 5/9).
show fractions by 6, 7, 8, 9 ...
### Calculator
176
#### Is Prime?
The number 176 is not a prime number. The closest prime numbers are 173, 179.
The 176th prime number in order is 1049.
#### Factorization and factors (dividers)
The prime factors of 176 are 2 * 2 * 2 * 2 * 11
The factors of 176 are 1 , 2 , 4 , 8 , 11 , 16 , 22 , 44 , 88 , 176
Total factors 10.
Sum of factors 372 (196).
#### Powers
The second power of 1762 is 30.976.
The third power of 1763 is 5.451.776.
#### Roots
The square root √176 is 13,266499.
The cube root of 3176 is 5,604079.
#### Logarithms
The natural logarithm of No. ln 176 = loge 176 = 5,170484.
The logarithm to base 10 of No. log10 176 = 2,245513.
The Napierian logarithm of No. log1/e 176 = -5,170484.
### Trigonometric functions
The cosine of 176 is 0,997494.
The sine of 176 is 0,070752.
The tangent of 176 is 0,07093.
### Properties of the number 176
More math properties ...
## Number 176 in Computer Science
Code typeCode value
Unix timeUnix time 176 is equal to Thursday Jan. 1, 1970, 12:02:56 a.m. GMT
IPv4, IPv6Number 176 internet address in dotted format v4 0.0.0.176, v6 ::b0
176 Decimal = 10110000 Binary
176 Decimal = 20112 Ternary
176 Decimal = 260 Octal
176 Decimal = B0 Hexadecimal (0xb0 hex)
176 BASE64MTc2
176 MD538af86134b65d0f10fe33d30dd76442e
176 SHA256cba28b89eb859497f544956d64cf2ecf29b76fe2ef7175b33ea59e64293a4461
176 SHA38456fd9c02db569512d43b2dbe739ea9c80a80a947055ef1ef449d66676766ca1ecbb672b353c60a0d1a92230f6423d13b
More SHA codes related to the number 176 ...
If you know something interesting about the 176 number that you did not find on this page, do not hesitate to write us here.
## Numerology 176
### The meaning of the number 7 (seven), numerology 7
Character frequency 7: 1
The number 7 (seven) is the sign of the intellect, thought, psychic analysis, idealism and wisdom. This number first needs to gain self-confidence and to open his/her life and heart to experience trust and openness in the world. And then you can develop or balance the aspects of reflection, meditation, seeking knowledge and knowing.
More about the the number 7 (seven), numerology 7 ...
### The meaning of the number 6 (six), numerology 6
Character frequency 6: 1
The number 6 (six) denotes emotional responsibility, love, understanding and harmonic balance. The person with the personal number 6 must incorporate vision and acceptance in the world. Beauty, tenderness, stable, responsible and understanding exchange, the sense of protection and availability also define the meaning of the number 6 (six).
More about the the number 6 (six), numerology 6 ...
### The meaning of the number 1 (one), numerology 1
Character frequency 1: 1
Number one (1) came to develop or balance creativity, independence, originality, self-reliance and confidence in the world. It reflects power, creative strength, quick mind, drive and ambition. It is the sign of individualistic and aggressive nature.
More about the the number 1 (one), numerology 1 ...
## Interesting facts about the number 176
### Asteroids
• (176) Iduna is asteroid number 176. It was discovered by C. H. F. Peters from Litchfield Observatory, Clinton on 10/14/1877.
### Areas, mountains and surfaces
• The total area of Moresby Island is 1,007 square miles (2,608 square km). Country Canada (British Columbia). 176th largest island in the world.
• The total area of Benin is 44,310 square miles (114,763 km²) and the water area is 176.668 square miles (457.569 km²) and the land area is 44,133 square miles (114,305 km²).
### Distances between cities
• There is a 176 miles (282 km) direct distance between Adana (Turkey) and Homs (Syria).
• There is a 176 miles (282 km) direct distance between Delhi (India) and Ludhiāna (India).
• There is a 176 miles (282 km) direct distance between Saitama (Japan) and Sendai (Japan).
• There is a 176 miles (283 km) direct distance between Saitama (Japan) and Sendai-shi (Japan).
• More distances between cities ...
• There is a 176 miles (283 km) direct distance between Suwon-si (South Korea) and Ulsan (South Korea).
### History and politics
• United Nations Security Council Resolution number 176, adopted 4 October 1962. Admission of Algeria. Resolution text.
### Mathematics
• 176 is the smallest even number is not the sum of two binary number with palindromic binary expansions
• 176 is an octagonal pentagonal number.
## № 176 in other languages
How to say or write the number one hundred and seventy-six in Spanish, German, French and other languages.
Spanish: 🔊 (número 176) ciento setenta y seis German: 🔊 (Nummer 176) einhundertsechsundsiebzig French: 🔊 (nombre 176) cent soixante-seize Portuguese: 🔊 (número 176) cento e setenta e seis Hindi: 🔊 (संख्या 176) एक सौ, छिहत्तर Chinese: 🔊 (数 176) 一百七十六 Arabian: 🔊 (عدد 176) مائةستة و سبعون Czech: 🔊 (číslo 176) sto sedmdesát šest Korean: 🔊 (번호 176) 백칠십육 Danish: 🔊 (nummer 176) ethundrede og seksoghalvfjerds Hebrew: (מספר 176) מאה שבעים ושש Dutch: 🔊 (nummer 176) honderdzesenzeventig Japanese: 🔊 (数 176) 百七十六 Indonesian: 🔊 (jumlah 176) seratus tujuh puluh enam Italian: 🔊 (numero 176) centosettantasei Norwegian: 🔊 (nummer 176) en hundre og sytti-seks Polish: 🔊 (liczba 176) sto siedemdziesiąt sześć Russian: 🔊 (номер 176) сто семьдесят шесть Turkish: 🔊 (numara 176) yüzyetmişaltı Thai: 🔊 (จำนวน 176) หนึ่งร้อยเจ็ดสิบหก Ukrainian: 🔊 (номер 176) сто сiмдесят шiсть Vietnamese: 🔊 (con số 176) một trăm bảy mươi sáu Other languages ...
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If you know something interesting about the number 176 or any other natural number (positive integer), please write to us here or on Facebook. | 2,779 | 9,023 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.828125 | 3 | CC-MAIN-2023-14 | longest | en | 0.689385 |
https://mn.khanacademy.org/math/cc-fifth-grade-math/cc-5th-fractions-topic/tcc-5th-div-unit-frac-by-whole/v/example-diving-a-unit-fraction-by-a-whole-number | 1,638,865,376,000,000,000 | text/html | crawl-data/CC-MAIN-2021-49/segments/1637964363337.27/warc/CC-MAIN-20211207075308-20211207105308-00358.warc.gz | 471,481,119 | 28,522 | If you're seeing this message, it means we're having trouble loading external resources on our website.
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Үндсэн товъёог
# Dividing a unit fraction by a whole number
## Video transcript
- [Instructor] So let's see if we can figure out what 1/3 divided by five is. And I'll give you a hint. Try to draw out 1/3 of a whole and then divide it into five equal sections. Pause this video and try to do that. I always try to work through it together and to help us as I promised or I suggested I guess. We said let's draw 1/3. I will represent a whole by that square right over there. And now let me split up into three equal sections. So this is all hand drawn with the aid of a computer. So it's not going to be perfect. But let's say that that is three equal sections. It's roughly three equal sections. I didn't do it perfectly but you hopefully get the idea. And so 1/3 would be one of those three equal sections. So that's a third right over there that I have just shaded in and I want to divide it into five. I want to divide it by five, I should say. So let's do that. So to divide it by five, I'm gonna divide it into five equal sections. And if I'm dividing that one into five equal sections, let me just divide all of the thirds into five equal sections. I'm essentially just going to make five rows here. One, and I'm gonna eyeball it, so it's going to be approximate. Two, three, and then four and five equal sections. I now split this whole into one, two, three, four, five rows of equal height. Now if I go to my original third and I divide it by five, I would be left with this right over here. But what fraction is this of the whole? Well what I've done now is I've split my whole into 15 equal sections. How do I know that? Well I could count them. One, two, three, four, five, six, seven, eight, nine, 10, 11, 12, 13, 14, 15. Or you could just say look I had one, two, three thirds and now each of those have been split into one, two, three, four, five equal sections. So three times five is 15. So each of these is a 15th and so the 1/3 divided by five is just one of those 15ths. So that right over there is one of those 15ths. So this is going to be equal to 1/15 and we are done. | 639 | 2,321 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.15625 | 4 | CC-MAIN-2021-49 | latest | en | 0.9558 |
http://www.jiskha.com/display.cgi?id=1347630716 | 1,493,613,165,000,000,000 | text/html | crawl-data/CC-MAIN-2017-17/segments/1492917127681.50/warc/CC-MAIN-20170423031207-00473-ip-10-145-167-34.ec2.internal.warc.gz | 561,231,254 | 3,642 | Physics
posted by on .
Runner A is initially 2.4 km west of a flagpole and is running with a constant velocity of 8.4 km/h due east. Runner B is initially 2.2 km east of the flagpole and is running with a constant velocity of 7.4 km/h due west. What will be the distance of the two runners from the flagpole when their paths cross?
Answer: ____ km from the flagpole due (east, west, south or north)
• Physics - ,
D = 2.2 -(-2.4) = 4.6 km = Distance between runners.
Va*t + Vb*t = 4.6.
8.4t + 7.4t = 4.6
15.8t = 4.6
t = 0.291 Hours.
Da = Va*t = 8.4km/h * 0.291h=2.4444km.=
Dist. ran by runner A.
2.4444-2.400=0.044 km=East of flagpole.
Db = Vb*t = 7.4km/h * 0.291h=2.1534 km.=
Dist. ran by runner B.
2.2000 - 2.1534 = 0.0466 km East of
flagpole. | 285 | 752 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.03125 | 4 | CC-MAIN-2017-17 | latest | en | 0.915799 |
https://turningtooneanother.net/2019/12/27/how-do-you-find-displacement-with-average-velocity-and-time/ | 1,660,565,348,000,000,000 | text/html | crawl-data/CC-MAIN-2022-33/segments/1659882572174.8/warc/CC-MAIN-20220815115129-20220815145129-00080.warc.gz | 520,157,903 | 8,534 | # How do you find displacement with average velocity and time?
## How do you find displacement with average velocity and time?
Calculator Use The average velocity of the object is multiplied by the time traveled to find the displacement. The equation x = ½( v + u)t can be manipulated, as shown below, to find any one of the four values if the other three are known.
### Why is average velocity displacement time?
Average velocity is the displacement of an object over time. To find the average speed of an object we divide the distance travelled by the time elapsed. We know that velocity is a vector quantity and average velocity can be found by dividing displacement by time.
What is the formula for calculating displacement?
1. If an object is moving with constant velocity, then.
2. Displacement = velocity x time.
3. If an object is moving with constant acceleration then the equation of third law of motion used to find displacement:
4. S = ut + ½ at²
5. S = v2−u22a.
6. If v = final velocity,
7. u = Initial velocity.
8. s = displacement.
Does velocity need direction?
When evaluating the velocity of an object, one must keep track of direction. It would not be enough to say that an object has a velocity of 55 mi/hr. One must include direction information in order to fully describe the velocity of the object.
## What is unit for velocity?
Since the derivative of the position with respect to time gives the change in position (in metres) divided by the change in time (in seconds), velocity is measured in metres per second (m/s).
### What is the formula for distance and displacement?
Displacement is the distance between two different positions of an object in m motion. So, it depends on the initial position and its final position. Also, displacement is the minimum distance between the starting and final positions….s = s_f – s_i.
s displacement
s_i initial position
s_f final position | 406 | 1,915 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.34375 | 4 | CC-MAIN-2022-33 | latest | en | 0.917046 |
https://www.chegg.com/homework-help/identify-possible-proper-subset-relationships-occur-among-fo-chapter-2.2-problem-13a-solution-9780321828019-exc | 1,563,310,416,000,000,000 | text/html | crawl-data/CC-MAIN-2019-30/segments/1563195524879.8/warc/CC-MAIN-20190716201412-20190716223412-00491.warc.gz | 658,464,630 | 46,664 | # A Problem Solving Approach to Mathematics for Elementary Teachers, Books a la Carte Edition Plus MyMathLab -- Access Card Package (11th Edition) Edit edition Problem 13A from Chapter 2.2: Identify all the possible proper subset relationships that o...
We have solutions for your book!
Chapter: Problem:
Identify all the possible proper subset relationships that occur among the following sets:
A = {3n | n N} B = {6n | n N}
C = {12n | n N}.
Step-by-step solution:
Chapter: Problem:
• Step 1 of 4
Let us consider that a sets , and .
Now will convert the sets to roster form:
The roster form of the set
The roster form of the set
The roster form of the set
• Chapter , Problem is solved.
Corresponding Textbook
A Problem Solving Approach to Mathematics for Elementary Teachers, Books a la Carte Edition Plus MyMathLab -- Access Card Package | 11th Edition
9780321828019ISBN-13: 0321828011ISBN: | 225 | 905 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.0625 | 3 | CC-MAIN-2019-30 | latest | en | 0.786657 |
https://testfunda.com/examprep/learningresources/qod/question-of-the-day.htm?assetid=19e7a5f0-a2bb-408a-a6ec-71a73a6f674d | 1,579,625,373,000,000,000 | text/html | crawl-data/CC-MAIN-2020-05/segments/1579250604849.31/warc/CC-MAIN-20200121162615-20200121191615-00247.warc.gz | 705,384,268 | 15,282 | Alternatively, you can register/login faster using
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QUESTION OF THE DAY
Question of the day
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Question of the Day (02-Jan-20)
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A question is followed by two statements, A and B. Answer the question using the following instructions:
Mark (1) if the question can be answered by using the statement A alone but not by using the statement B alone.
Mark (2) if the question can be answered by using the statement B alone but not by using the statement A alone.
Mark (3) if the question can be answered by using either of the statements alone.
Mark (4) if the question can be answered by using both the statements together but not by either of the statements alone.
Mark (5) if the question cannot be answered on the basis of the two statements.
P and R are two distinct integers and S = P × R
Is 2 the last digit of S?
1. R = P + 2
2. P is an odd prime number
OPTIONS 1) 1 2) 2 3) 3 4) 4 5) 5
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©2008-2020 All rights reserved | 317 | 1,218 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.5625 | 3 | CC-MAIN-2020-05 | longest | en | 0.873663 |
https://developer.mozilla.org/en-US/docs/Web/CSS/transform-function/matrix() | 1,642,621,240,000,000,000 | text/html | crawl-data/CC-MAIN-2022-05/segments/1642320301488.71/warc/CC-MAIN-20220119185232-20220119215232-00423.warc.gz | 262,463,063 | 31,889 | # matrix()
The `matrix()` CSS function defines a homogeneous 2D transformation matrix. Its result is a `<transform-function>` data type.
Note: `matrix(a, b, c, d, tx, ty)` is a shorthand for `matrix3d(a, b, 0, 0, c, d, 0, 0, 0, 0, 1, 0, tx, ty, 0, 1)`.
## Syntax
The `matrix()` function is specified with six values. The constant values are implied and not passed as parameters; the other parameters are described in the column-major order.
``````matrix(a, b, c, d, tx, ty)
``````
### Values
a b c d
Are `<number>`s describing the linear transformation.
tx ty
Are `<number>`s describing the translation to apply.
Cartesian coordinates on ℝ^2 Homogeneous coordinates on ℝℙ^2 Cartesian coordinates on ℝ^3 Homogeneous coordinates on ℝℙ^3
$\left(\begin{array}{cc}a& c\\ b& d\end{array}\right)$ $\left(\begin{array}{ccc}a& c& \mathrm{tx}\\ b& d& \mathrm{ty}\\ 0& 0& 1\end{array}\right)$ $\left(\begin{array}{ccc}a& c& \mathrm{tx}\\ b& d& \mathrm{ty}\\ 0& 0& 1\end{array}\right)$ $\left(\begin{array}{cccc}a& c& 0& \mathrm{tx}\\ b& d& 0& \mathrm{ty}\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right)$
`[a b c d tx ty]`
The values represent the following functions: `matrix( scaleX(), skewY(), skewX(), scaleY(), translateX(), translateY() )`
## Examples
### HTML
``````<div>Normal</div>
<div class="changed">Changed</div>
``````
### CSS
``````div {
width: 80px;
height: 80px;
background-color: skyblue;
}
.changed {
transform: matrix(1, 2, -1, 1, 80, 80);
background-color: pink;
}
``````
## Browser compatibility
BCD tables only load in the browser | 526 | 1,557 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 4, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.015625 | 3 | CC-MAIN-2022-05 | latest | en | 0.382039 |
https://mail.python.org/pipermail/numpy-discussion/2013-May/066478.html | 1,631,806,172,000,000,000 | text/html | crawl-data/CC-MAIN-2021-39/segments/1631780053657.29/warc/CC-MAIN-20210916145123-20210916175123-00556.warc.gz | 451,805,039 | 2,637 | # [Numpy-discussion] "Lists" and "Join" function needed
Bakhtiyor Zokhidov bakhtiyor_zokhidov at mail.ru
Sun May 5 07:39:28 EDT 2013
``` Thanks for the answer in Stackoverflow. I checked it out.
Суббота, 4 мая 2013, 21:24 -04:00 от Warren Weckesser <warren.weckesser at gmail.com>:
>On 5/4/13, Bakhtiyor Zokhidov < bakhtiyor_zokhidov at mail.ru > wrote:
>>
>> Hi,
>> I have the following code which represents intersected point of each cell in
>> the given two points, A(x0,y0) and B(x1,y1).
>>
>> def intersected_points(x0, x1, y0, y1):
>> # slope
>> m = (y1 - y0 )/( x1 - x0)
>> # Boundary of the selected points
>> x_ceil = ceil( min (x0, x1 ))
>> x_floor = floor( max(x0, x1))
>> y_ceil = ceil( min(y0, y1))
>> y_floor = floor( max(y0, y1))
>> # calculate all intersected x coordinate
>> ax = []
>> for x in arange(x_ceil, x_floor + 1, 1):
>> ax.append([ x, m * ( x - x0 ) + y0 ])
>> # calculate all intersected y coordinate
>> for y in arange(y_ceil, y_floor + 1, 1):
>> ax.append([ ( y - y0 ) * ( 1./m ) + x0, y ])
>> return ax
>>
>> Sample values: intersected_points(1.5,4.4,0.5,4.1)
>> Output: [[2.0, 1.1206896551724137], [3.0, 2.3620689655172411], [4.0,
>> 3.6034482758620685], [1.9027777777777779, 1.0], [2.7083333333333335, 2.0],
>> [3.5138888888888893, 3.0], [4.3194444444444446, 4.0]]
>>
>> The output I got is unsorted values, so, for each cell coordinates, where
>> line crosses:
>> BUT, The result I want to get should be something in increased orders
>> like: (x0,y0), (x1,y1), (x2,y2), (x3,y3)
>> where x0, y0 - intial, x1,y1 - final point. Other values are intersected
>> line coordinates!
>>
>> Any answers will be appreciated,
>>
>> -- Bakhtiyor Zokhidov
>
>
>You also asked this question on stackoverflow
>( http://stackoverflow.com/questions/16377826/distance-for-each-intersected-points-of-a-line-in-increased-order-in-2d-coordina ).
> I've posted an answer there.
>
>Warren
>_______________________________________________
>NumPy-Discussion mailing list
>NumPy-Discussion at scipy.org
>http://mail.scipy.org/mailman/listinfo/numpy-discussion
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``` | 774 | 2,254 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.140625 | 3 | CC-MAIN-2021-39 | latest | en | 0.622068 |
http://www.electro-tech-online.com/threads/how-is-output-voltage-calculated-in-lpf.150502/ | 1,513,079,580,000,000,000 | text/html | crawl-data/CC-MAIN-2017-51/segments/1512948516843.8/warc/CC-MAIN-20171212114902-20171212134902-00457.warc.gz | 351,523,172 | 15,644 | # how is output voltage calculated in LPF
Discussion in 'Circuit Simulation & PCB Design' started by ravi17, Mar 28, 2017.
1. ### ravi17New Member
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Hi,
can someone explain why the output voltage of the following circuit is showing near +5V? the input signal is pulse signal of 5V and 0V. After the resistor there should be voltage drop and from my calculation the output voltage should be somewhere near 2.5V for high duration.
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2. ### ericgibbsWell-Known MemberMost Helpful Member
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hi 17,
As there is no resistive load across the cap to discharge the cap, it will charge to 5v, assuming that the 5v source impedance is high.
E
3. ### ravi17New Member
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thanks
edit:
i connected a load resistor of 1Ohm and 10Ohm across capacitor but the voltage there is just micro volt range!
Last edited: Mar 29, 2017
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5. ### PommieWell-Known MemberMost Helpful Member
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You have a voltage divider with a 300k and 100r resistors. Microvolts is what you should get.
Mike.
6. ### ravi17New Member
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thanks mike
one confusion i want to clear up. i assumed and calculated the voltage divider rule by applying reactance of the capacitor Xc = 1/wC which is in parallel the the 300k res(with or without the load). so should the capacitor reactance not be applied in the calculation?
7. ### ericgibbsWell-Known MemberMost Helpful Member
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hi,
What is the frequency of the square wave input.?
E
8. ### MikeMlWell-Known MemberMost Helpful Member
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• Informative x 1
9. ### ravi17New Member
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hi eric, the frequency is 0.5Hz.
Mike, the resistor R2 should be at the load, after the capacitor, i guess, because the R1 and C are designed for LPF for 0.5Hz. correct me if i am wrong. also i didn't understand why cap reactance is not taken into account while calculating the output voltage using voltage divider rule!!
10. ### crutschowWell-Known MemberMost Helpful Member
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Because reactance varies with frequency.
A square-wave is composed of many frequencies in the frequency-domain, as shown by its Fourier transform components, so it does not have a single reactance for the square-wave signal.
So for such signals you normally use a time-domain analysis, not a frequency domain.
You would take reactance into account if you were doing a frequency-domain analysis with a single-frequency sine-wave.
11. ### MikeMlWell-Known MemberMost Helpful Member
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The load R2 is part of a voltage divider with R1. The filter is calculated by first combining R1 and R2 (Rt= R1*R2/(R1+R2) using Thevenin's theorem. The voltage is based on the voltage divider R2/(R1+R2). Finally, you calculate the cut-off frequency by finding where Xc1 = Rt.
The reason I drew the schematic the way I did was to remind you that R2 has to be considered first before worrying about C1.
12. ### MikeMlWell-Known MemberMost Helpful Member
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Back to your original question in post #1:
V1 has a duty-cycle of 50%. What would happen if it had a duty cycle of 10%? What would the average V(out) be then?
• Like x 1
13. ### ericgibbsWell-Known MemberMost Helpful Member
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hi 17,
I did reply much earlier but my post disappeared, possibly due to the site software upgrade.
Using your posted plot, you can see the 2.5V point of the cap charging voltage occurs at 0.22Secs, as you would expect.
E
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32 | 1,168 | 4,105 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.0625 | 3 | CC-MAIN-2017-51 | latest | en | 0.871869 |
https://www.scribd.com/document/376799140/Chap3 | 1,571,532,428,000,000,000 | text/html | crawl-data/CC-MAIN-2019-43/segments/1570986700560.62/warc/CC-MAIN-20191020001515-20191020025015-00320.warc.gz | 1,080,345,096 | 64,315 | You are on page 1of 8
# CHAPTER 3: RANDOM VARIABLES and PROBABILITY DISTRIBUTIONS
## This chapter is all about
1. Another method to study our events of interest
2. Modeling probability by creating catalogs of common distributions and their
applications
Learning Objectives:
(2) MAIN Objectives
1. To grasp the concept of a random variable
2. To grasp the concept of different distribution functions
## *along the way we will also learn
1. To compute probabilities using these distribution functions
1. To compute expectations using these distribution functions
2. To generate moments
OUTLINE
PART ONE: GENERAL
3.1.1 Random Variables
3.1.2 Cumulative Distribution Function
## PART TWO: SPECIFICS
3.2.1 Discrete Random Variables
3.2.2 Probability Mass Distribution (PMF)
## 3.3.1 Continuous Random Variables
3.3.2 Probability Density Distribution (PDF)
## PART THREE: EXPECTATIONS
3.4.1 Expected values of X
3.4.2 Expected valued of g(X)
3.4.3 Variance of X
3.4.4 Properties
I. Properties of Expectations
II. Properties of Variance
PART ONE: GENERAL
## 3.1.1 Random Variables
I. Random Variable
→ a function whose value is a real number that is determined by each
sample point in the sample space achieved through a random experiment
→ notation: uppercase letters *mostly X
**lowercase letters mostly 'x' as one of its values
## Random Variable as a function [ Ω → R ]
→ since a function,
each outcome in the sample space must be mapped to exactly one
real number
Remark:
→ this assures us that the random variable X will have one and only one
realized value, whatever the outcome of the random experiment
## Random Variable as an Event
→ meaning, random variable is a new way of expressing event
## Examples: *to better understand
X≤x
the event containing all sample points that is the value for the random
variable X is less than or equal to x
X>x
the event containing all sample points that is the value for the random
variable is greater than x
a<X<b
the event containing all sample points that is the value for the random
variable X is between a and b
IV. SKILL
1. Translating events to random variables format
1.) Determine what is X
2.) set the relationship based on the given to one of its values
example:
Let X be the number of tails
## A = is the event observing atleast one tail
A = { (T,T), (T,H), (H, T) }
Given that then, X > 0 or X ≥ 1
## VI. Indicator Function
Let Ω be the universal set : sample space
A be the subset of Ω : the event that is a subset of sample space
: one of the partitions of Ω
Indicator Function
→ the function that indicates partitions of the sample space Ω that is, it
indicates the A's
SKILL
1. Translating a piecewise function to an indicator function
example:
{ x , 0<x<1
f(x) = { 2 – x , 1 ≤x≤2
{0 , elsewhere
indicator function: f(x) = xI( 0 ,1 )( x ) +( 2 – x )I[ 1, 2 ](x)
2. Evaluation of indicator function
Given: Indicator Function
known value of x
Find: f(known value of x)
Solution:
1. Plug-in the known value of x to the indicator function
2. delete terms where x is not an element of the interval
3. evaluate
## 3.1.2 CUMULATIVE DISTRIBUTION FUNCTION (CDF) of a random variable X
I. Definitions
→ notation: F(x)
→ a function defined for any real number x as:
F(x) = P( X ≤ x )
→ usage: use to compute for the probability of an even that is
expressed in terms of the random variable
II. Properties
1. 0 ≤ F(x) ≤ 1 since it's equated to a probability
2. It is a nondecreasing function
3. Every random variable will have one and only one CDF
*rather, every known value x of X will have one and only one CDF
III. Templates
(4) Four Main Templates
1. P ( a < X ≤ b ) = FX(b) – FX(a)
2. P ( X ≤ a ) = FX (a)
3. P ( X >a ) = 1- FX (a)
4. P( X = a ) = P ( a- < X ≤ a ) = FX(a) - FX(a-)
*any changes in the inequality in a way of being not equal or equal to will get a
penalty:
ex: P( X < a) = FX (a-)
IV. SKILL
1. Computing the probability through CDF
Given: CDF , probabilities in question
Find: Probabilities
Solution:
1. Translate probabilities in question in the form of the left hand-side of CDF
example: P(X < 1) = FX(1-) [ like the reverse of the given form of it ]
*reduced further to the template if needed
2. Find in the CDF what satisfies it
PART TWO: SPECIFICS
3.2.1 DISCRETE RANDOM VARIABLE
## Discrete Random Space
→ sample space that is finite or countably finite
## Discrete Random Variable
→ a subset that is an event by which is finite or countably finite
## 3.2.2 Probability Mass Function (PMF) of a discrete variable
I. Definitions
→ notation: p(x)
→ a function defined for any real number x as
p(x) = P(X=x)
## → usage: use PMF to compute for the probabilities expressed in terms of X
: use PMF to compute summary measures like the mean and the
standard deviation
mass points
→ the value of the discrete random variable X for which p(x)>0
II. SKILLs
1. Getting Probability through PMF
Given: the random variable X that is an event is known for what it represents
: the sample space
Find: Probability through PMF
Solution:
* Identify the mass points of X / possible values of X
*refers to the range of the function X
*given X and plugged into p(X) what are the possible values then?
* Table
1. columns: possible values of X which are under the heading 'x'
Events associated with X=x
p(x) = P(X=x) *read as-- what is the probability given the event X is x
as seen from the events associated
*Validity
1. p(x) > 0 : where x is a mass point
2. ∑ p(x) = 1
*Probability
1. see in the table the needed probability
3.3.1 Continuous Random Variable
## Continuous Sample Space
→ a sample space that isn't countable or where its sample points cannot
be put in one to one correspondence with counting numbers; mostly intervals
## Continuous Random Variable
→ a subset of that sample space
## 3.3.2 Probability Density Function (PDF) of a continuous random variable X
I. Definition
Probability Density Function (PDF) of a continuous random variable X
→ notation: f(x)
→ a function defined for any real number x and
satisfy the following properties:
1. f(x) > 0 for all x
2. the area below the the whole curve f(x) and above the x-axis is always
equal to 1
3. P( a ≤ X ≤ b ) is the area bounded by the curve f(x), the x-axis and the
lines x=a and x=b
→ usage: to compute probabilities
: to compute summary measures
II. SKILLs
1.
Given: PDF *it's given and not constructed this time
interval of interest
Find: Probability
Solution:
*Validity
1. f(x) > 0 : see each term in the function this time if there will be a negative
domain
2. ∫from negative infinity to infinity f(x) dx =1
*Probability
1. Write the probability function with regards to the probability of interest
2. now from negative infinity to infinity or the given interval,
equate #1 to the rewritten integral with regards to the interval of interest
3. Then arrive at the needed probability
CAUTION:
→ when evaluating the probabilities of events expressed in terms of a
continuuos random variable, it *does not matter whether we are dealing with
the event X < a or X ≤ a since P(X=a) = 0.
therefore, P(X < a) = P(X ≤ a) or the inequality with or without equal is equivalent
PART THREE: EXPECTATIONS
## 3.5.1 Expected Value of X [specific]
I. Definition
Let X be a random variable
Expected Value of X [ or mean of X ]
→ notation: E(X) or μX
defined by:
{ ∑for all m.p xp(x) , if X is discrete
E(X) = { ∫from negative infinity to infinity xfX(x)dx , if continuous
→ E(X) is actually a weighted mean of the values that the random variable
takes on, where each value is weighted by the probability that the random
variable is equal to that value
II. SKILLs
1.
Given: a PMF (the table) or PDF (the indicator function)
Find: E(X)
Solution:
1. Determine if discrete of continuous
2. If discrete, use the E(X) = ∑for all m.p xp(x)
If continuous, use the E(X) = ∫from negative infinity to infinity xfX(x)dx
## 3.5.2 Expected Values of G(X)
I. Definition
Let X be a random variable
## Expected Value of g(x)
→ notation: E(g(x))
{ ∑for all m.p g(x)p(x) , if X is discrete
E(X) = { ∫from negative infinity to infinity g(x)fX(x)dx , if continuous
II. SKILLs
Given: a word problem
with events
Find: what is needed
Solution:
*Construct table
1. One for each event
2. Solve events based on x
*Expectation
1. then use ∑for all m.p xp(x) *this depends on the wording of the target
3.5.3 Variance of X
I. Definition
Let X be a random variable with mean, μ
Variance of X
→ notation: σ2 or Var(X)
σ2 = Var(X) = E(X- μ )2
## *the standard deviation is the positive square root of the variance
→ still a measure of dispersion and the average squared difference between
the value of X and μ.
→ also being mean, is also in terms of expectations
3.5.4 Properties
I. Properties of Expectation
1. (E-μ) = 0
2. E(aX + b) = aE(X) + b
if b=0, E(aX) = aE(X)
if a=0, E(b) = b
## 3. E(X+Y) = E(X) + E(Y)
4. E(X + Y) = E(X) – E(Y)
## II. Properties of Variance
1. Var(aX+b) = a2Var(X)
if b=0, then Var(aX) = a2Var(X)
if a=0, then Var(b) = 0
## 2. Var (X+Y) = Var(X) + Var(Y)
3. Var (X-Y) = Var(X) + Var(Y) *only if X and Y are independent variables
## 4. Var(X) = E(X2) – [E(X)]2
III. SKILLs
1. Evaluate Expectations or Variance
Given: Expectations or Variance
Find: its equivalent real number
Solution:
1. Reduce it to the given forms above
2. Evaluate using the given values | 2,630 | 9,336 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.34375 | 4 | CC-MAIN-2019-43 | latest | en | 0.814559 |
https://www.aqua-calc.com/calculate/mass-molar-concentration/substance/dipotassium-blank-chromate | 1,713,723,282,000,000,000 | text/html | crawl-data/CC-MAIN-2024-18/segments/1712296817790.98/warc/CC-MAIN-20240421163736-20240421193736-00283.warc.gz | 592,244,604 | 9,507 | # Concentration of Dipotassium chromate
## dipotassium chromate: convert between mass and molar concentration
### Molar concentration per milliliter
0.01 mmol/ml 10 µmol/ml 10 000 nmol/ml 10 000 000 pmol/ml
### Molar concentration per deciliter
1 mmol/dl 1 000 µmol/dl 1 000 000 nmol/dl 1 000 000 000 pmol/dl
### Molar concentration per liter
10 mmol/l 10 000 µmol/l 10 000 000 nmol/l 10 000 000 000 pmol/l
### Mass concentration per milliliter
0 g/ml 1.94 mg/ml 1 941.9 µg/ml 1 941 900 ng/ml 1 941 900 000 pg/ml
### Mass concentration per deciliter
0.19 g/dl 194.19 mg/dl 194 190 µg/dl 194 190 000 ng/dl 194 190 000 000 pg/dl
### Mass concentration per liter
1.94 g/l 1 941.9 mg/l 1 941 900 µg/l 1 941 900 000 ng/l 1 941 900 000 000 pg/l
### Equivalent molar concentration per milliliter
0.01 meq/ml 10 µeq/ml 10 000 neq/ml 10 000 000 peq/ml
### Equivalent molar concentration per deciliter
1 meq/dl 1 000 µeq/dl 1 000 000 neq/dl 1 000 000 000 peq/dl
### Equivalent molar concentration per liter
10 meq/l 10 000 µeq/l 10 000 000 neq/l 10 000 000 000 peq/l
• The units of amount of substance (e.g. mole) per milliliter, liter and deciliter are SI units of measurements of molar concentrations.
• The units of molar concentration per deciliter:
• millimole per deciliter [mm/dl], micromole per deciliter [µm/dl], nanomole per deciliter [nm/dl] and picomole per deciliter [pm/dl].
• The units of molar concentration per milliliter:
• millimole per milliliter [mm/ml], micromole per milliliter [µm/ml], nanomole per milliliter [nm/ml] and picomole per milliliter [pm/ml].
• The units of molar concentration per liter:
• millimole per liter [mm/l], micromole per liter [µm/l], nanomole per liter [nm/l] and picomole per liter [pm/l].
• The units of mass per milliliter, liter and deciliter are non-SI units of measurements of mass concentrations still used in many countries.
• The units of mass concentration per deciliter:
• gram per deciliter [g/dl], milligram per deciliter [mg/dl], microgram per deciliter [µg/dl], nanogram per deciliter [ng/dl] and picogram per deciliter [pg/dl].
• The units of mass concentration per milliliter:
• gram per milliliter [g/ml], milligram per milliliter [mg/ml], microgram per milliliter [µg/ml], nanogram per milliliter [ng/ml] and picogram per milliliter [pg/ml].
• The units of mass concentration per liter:
• gram per liter [g/l], milligram per liter [mg/l], microgram per liter [µg/l], nanogram per liter [ng/l] and picogram per liter [pg/l].
• The equivalent per milliliter, liter and deciliter are obsolete, non-SI units of measurements of molar concentrations still used in many countries. An equivalent is the number of moles of an ion in a solution, multiplied by the valence of that ion.
• The units of equivalent concentration per deciliter:
• milliequivalent per deciliter [meq/dl], microequivalent per deciliter [µeq/dl], nanoequivalent per deciliter [neq/dl] and picoequivalent per deciliter [peq/dl].
• The units of equivalent concentration per milliliter:
• milliequivalent per milliliter [meq/ml], microequivalent per milliliter [µeq/ml], nanoequivalent per milliliter [neq/ml] and picoequivalent per milliliter [peq/ml].
• The units of equivalent concentration per liter:
• milliequivalent per liter [meq/l], microequivalent per liter [µeq/l], nanoequivalent per liter [neq/l] and picoequivalent per liter [peq/l].
#### Foods, Nutrients and Calories
AMERICA'S ORIGINAL GRANOLA LOW FAT WITH RAISINS and ALMONDS, UPC: 024300090059 weigh(s) 106 grams per metric cup or 3.5 ounces per US cup, and contain(s) 400 calories per 100 grams (≈3.53 ounces) [ weight to volume | volume to weight | price | density ]
81 foods that contain Vitamin E (label entry primarily). List of these foods starting with the highest contents of Vitamin E (label entry primarily) and the lowest contents of Vitamin E (label entry primarily)
#### Gravels, Substances and Oils
CaribSea, Marine, Aragonite, Bermuda Pink weighs 1 281.5 kg/m³ (80.00143 lb/ft³) with specific gravity of 1.2815 relative to pure water. Calculate how much of this gravel is required to attain a specific depth in a cylindricalquarter cylindrical or in a rectangular shaped aquarium or pond [ weight to volume | volume to weight | price ]
Paving Asphalt Cement, AR-8000 weighs 1 023.32 kg/m³ (63.88378 lb/ft³) [ weight to volume | volume to weight | price | density ]
Volume to weightweight to volume and cost conversions for Refrigerant R-12, liquid (R12) with temperature in the range of -51.12°C (-60.016°F) to 71.12°C (160.016°F)
#### Weights and Measurements
The long ton per cubic foot density measurement unit is used to measure volume in cubic feet in order to estimate weight or mass in long tons
Electric current is a motion of electrically charged particles within conductors or space.
st/ft³ to mg/tbsp conversion table, st/ft³ to mg/tbsp unit converter or convert between all units of density measurement.
#### Calculators
Area of a parallelogram calculator. Definition of a parallelogram | 1,543 | 5,095 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.625 | 3 | CC-MAIN-2024-18 | latest | en | 0.272368 |
https://www.gradesaver.com/textbooks/science/physics/college-physics-7th-edition/chapter-23-mirrors-and-lenses-learning-path-questions-and-exercises-multiple-choice-questions-page-804/2 | 1,701,227,192,000,000,000 | text/html | crawl-data/CC-MAIN-2023-50/segments/1700679100047.66/warc/CC-MAIN-20231129010302-20231129040302-00410.warc.gz | 908,813,775 | 11,957 | ## College Physics (7th Edition)
Plane mirrors form virtual, upright and unmagnified images. The object distance is equal to the absolute value of the image distance $d_{0}=|d_{i}|$ Hence, the answer is (d). | 53 | 208 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.59375 | 3 | CC-MAIN-2023-50 | latest | en | 0.766043 |
http://cnx.org/content/m17061/1.2/ | 1,394,622,685,000,000,000 | application/xhtml+xml | crawl-data/CC-MAIN-2014-10/segments/1394021724521/warc/CC-MAIN-20140305121524-00082-ip-10-183-142-35.ec2.internal.warc.gz | 40,114,621 | 9,102 | # Connexions
You are here: Home » Content » F Distribution and ANOVA: ANOVA Lab
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# F Distribution and ANOVA: ANOVA Lab
Summary: Note: This module is currently under revision, and its content is subject to change. This module is being prepared as part of a statistics textbook that will be available for the Fall 2008 semester.
Note: You are viewing an old version of this document. The latest version is available here.
Class Time:
Names:
## Student Learning Outcome:
• The student will conduct a simple ANOVA test involving three variables.
## Do the Experiment:
• a. Record the price per pound of 8 fruits, 8 vegetables, and 8 breads in your local supermarket.
• b. Explain how you could try to collect the data randomly.
1. Compute the following:
### Fruit:
• a. x¯=___________x¯= size 12{ {overline {x}} } {}___________
• b. sx=___________sx= size 12{s rSub { size 8{x} } } {}___________
• c. n=___________n= size 12{n} {}___________
### Vegetable:
• a. x¯=___________x¯= size 12{ {overline {x}} } {}___________
• b. sx=___________sx= size 12{s rSub { size 8{x} } } {}___________
• c. n=___________n= size 12{n} {}___________
• a. x¯=___________x¯= size 12{ {overline {x}} } {}___________
• b. sx=___________sx= size 12{s rSub { size 8{x} } } {}___________
• c. n=___________n= size 12{n} {}___________
• a. dfnumerator=___________dfnumerator= size 12{ ital "df" left ( ital "num" right )} {}___________
• b. dfdenominator=___________dfdenominator= size 12{ ital "df" left ( ital "denom" right )} {}___________
2. State the approximate distribution for the test.
3. Test statistic: FF size 12{F} {} = ____________________
4. Sketch a graph of this situation. CLEARLY, label and scale the horizontal axis and shade the region(s) corresponding to the p-valuep-value.
5. What is the p-value?
6. Test at α=0.05α=0.05 size 12{α=0 "." "05"} {}.
• a. State your decision. Why did you make that decision?
• b. State your conclusion (write a complete sentence).
• c. Based on the results of your study, is there a need to further investigate any of the food groups’ prices? Why or why not?
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http://www.chegg.com/homework-help/questions-and-answers/two-trains-separate-tracks-move-toward-one-another-train-1-hasa-speed-105-km-h-train-2-spe-q399645 | 1,475,318,008,000,000,000 | text/html | crawl-data/CC-MAIN-2016-40/segments/1474738662705.84/warc/CC-MAIN-20160924173742-00044-ip-10-143-35-109.ec2.internal.warc.gz | 400,097,739 | 14,170 | Two trains on separate tracks move toward one another. Train 1 hasa speed of 105 km/h, train 2 a speedof 55.0 km/h. Train 2 blows its horn,emitting a frequency of 500 Hz. What is the frequency heard by theengineer on train 1? (Assume the speed of sound is 345 m/s.)
Hz | 80 | 268 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.640625 | 3 | CC-MAIN-2016-40 | latest | en | 0.893368 |
https://www.jiskha.com/display.cgi?id=1328158376 | 1,529,361,691,000,000,000 | text/html | crawl-data/CC-MAIN-2018-26/segments/1529267861456.51/warc/CC-MAIN-20180618222556-20180619002556-00189.warc.gz | 842,540,386 | 4,210 | # Chemistry
posted by ELIJAH B
What is the empirical of compound (X) going through the following decomposition reaction?
X = sulfur + oxygen + nitrogen
sulfur = 21 gr
oxygen = 69.70gr
nitrogen = 6.10gr
1. DrBob222
Convert g to moles. moles = grams/molar mass.
Then find the ratio of the elements to each other with the smallest number being 1.00. The easy way to do that is to divide the smallest number by itself followed by dividing all of the other numbers by the same small number. Then round to whole numbers EXCEPT don't round more than 0.1 or so. What I mean by that is that if you come up with 1.5 obviously that number could be multiplied by a whole number (2) to obtain 3.
All of the other numbers would be multiplied by 2 also. Post your workif you get stuck.
2. ELIJAH B
Would the answer to the X = S2O10N
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More Similar Questions | 831 | 3,150 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.9375 | 3 | CC-MAIN-2018-26 | latest | en | 0.90547 |
http://ask.csdn.net/questions/56546 | 1,516,368,083,000,000,000 | text/html | crawl-data/CC-MAIN-2018-05/segments/1516084887981.42/warc/CC-MAIN-20180119125144-20180119145144-00628.warc.gz | 31,257,688 | 13,016 | u010340708 于 2014.02.22 07:59 提问
id temp status
1 1 0
2 1 1
3 2 0
4 2 1
5 2 1
select total.temp ,used.c1,total.c2
from (select temp,count(1) c1 from pm group by temp ) total
left join (select temp,count(1) c2 from pm where status=‘1’ group by temp ) used
on total.temp=used.temp
2个回答
Ouvidia 2014.02.22 17:50
status 为1或者为0 ~~ 那么直接count(status) 不就得到了呀~~?
select temp,count(1),count(status) from pm group by temp;
select temp,count(1),count(1-sign(status-1)*sign(status-1)) from pm group bu temp;
1-sign(status-1)*sign(status-1) 这个是 如果不为1 那么sign-1 一定不是0 不是0的平凡 显然是正的1 所以 1-1 显然就是0 了~~ 最终累计的就是status 为1的了~~
shendixiong 2014.04.04 11:33
`````` select
temp,
count(case when status =‘1’then status end) used ,
count(case when status !=‘1’then status end) total
from pm
group by temp ;
`````` | 328 | 801 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.515625 | 3 | CC-MAIN-2018-05 | latest | en | 0.256961 |
https://practicepaper.in/gate-ec/instructions-of-8085-microprocessor | 1,637,982,873,000,000,000 | text/html | crawl-data/CC-MAIN-2021-49/segments/1637964358078.2/warc/CC-MAIN-20211127013935-20211127043935-00492.warc.gz | 536,913,446 | 21,628 | # Instructions of 8085 Microprocessor
Question 1
The clock frequency of an 8085 microprocessor is 5 MHz. If the time required to execute an instruction is 1.4 $\mu s$, then the number of T-states needed for executing the instruction is
A 1 B 6 C 7 D 8
GATE EC 2017-SET-1 Microprocessors
Question 1 Explanation:
Given than,
$f_{\mathrm{CLK}}=5 \mathrm{MHz}$
Execution time $=1.4 \mu \mathrm{s}$
Execution time $=n(T-\text { state })$
$n=$ number of T-states required to execute the instruction
T- state (or) $T_{\mathrm{CLK}}=\frac{1}{f_{\mathrm{CLK}}}=0.2 \mu \mathrm{s}$
So, $\quad n=\frac{1.4 \mu \mathrm{s}}{T_{\mathrm{CLK}}}=\frac{1.4}{0.2}=7$
Question 2
In an 8085 microprocessor, the contents of the accumulator and the carry flag are A7 (in hex) and 0, respectively. If the instruction RLC is executed, then the contents of the accumulator (in hex) and the carry flag, respectively, will be
A 4E and 0 B 4E and 1 C 4F and 0 D 4F and 1
GATE EC 2016-SET-3 Microprocessors
Question 2 Explanation:
RLC: Rotate accumulator left by 1 bit without carry
Before RLC operation:
\begin{aligned} A &=A 7 H=(10100111)_{2} \\ C Y &=0 \end{aligned}
After RLC operation:
\begin{aligned} A &=(01001111)_{2}=4 \mathrm{FH} \\ \mathrm{CY} &=1 \end{aligned}
Question 3
In an 8085 system, a PUSH operation requires more clock cycles than a POP operation. Which one of the following options is the correct reason for this?
A For POP, the data transceivers remain in the same direction as for instruction fetch (memory to processor), whereas for PUSH their direction B174has to be reversed. B Memory write operations are slower than memory read operations in an 8085 based system. C The stack pointer needs to be pre-decremented before writing registers in a PUSH, whereas a POP operation uses the address already in the stack pointer. D Order of registers has to be interchanged for a PUSH operation, whereas POP uses their natural order.
GATE EC 2016-SET-1 Microprocessors
Question 3 Explanation:
The stack pointer needs to be pre-decremented before wiriting data into stack. For reading data from stack, such pre-decrement or pre-increment operations are not needed, as already stack pointer indicates the address of stack top from where the read operation takes place. Hence, PUSH operation requires more clock cycles than POP operation.
Question 4
In an 8085 microprocessor, which one of the following instructions changes the content of the accumulator?
A MOV B,M B PCHL C RNZ D SBI BEH
GATE EC 2015-SET-2 Microprocessors
Question 4 Explanation:
The only instruction that changes the contents of accumulator is SBI BEH.
Question 5
An 8085 assembly language program is given below. Assume that the carry flag is initially unset. The content of the accumulator after the execution of the program is
A 8 CH B 64 H C 23 H D 15 H
GATE EC 2011 Microprocessors
Question 5 Explanation:
$MVI \;A,07\Rightarrow A=07H=(0000111)_{2}$
\begin{aligned} \text { After } \mathrm{RLC}, \mathrm{A}&=(00001110)_{2}=\mathrm{OEH} \\ \mathrm{MOV} \mathrm{B}, \mathrm{A} &\Rightarrow \mathrm{B} \leftarrow \mathrm{A} \Rightarrow \mathrm{B}=\mathrm{OEH} \\ \mathrm{RLC} \quad &\Rightarrow \mathrm{A}=(00011100)_{2}=1 \mathrm{CH} \\ \mathrm{RLC} \quad &\Rightarrow \mathrm{A}=(00111000)_{2}=38 \mathrm{H} \\ \mathrm{ADDB} &\Rightarrow \mathrm{A} \leftarrow \mathrm{A}+\mathrm{B} \\ &\Rightarrow \mathrm{A}=38 \mathrm{H}+\mathrm{OEH}=46 \mathrm{H} \\ &\qquad=(01000110)_{2} \\ \mathrm{RRC} \quad &\Rightarrow \mathrm{A}=(00100011)_{2}=23 \mathrm{H} \end{aligned}
Question 6
For the 8085 assembly language program given below, the content of the accumulator after the execution of the program is
A 00H B 45H C 67H D E7H
GATE EC 2010 Microprocessors
Question 6 Explanation:
$\mathrm{MVI} \quad \mathrm{A}, 45 \mathrm{H} \Rightarrow \mathrm{A}=45 \mathrm{H}=01000101$
$\mathrm{MOV} \mathrm{B}, \mathrm{A} \Rightarrow \mathrm{B}=45 \mathrm{H}$
$\mathrm{STC} \Rightarrow \mathrm{CY}=1$
$\mathrm{CMC} \Rightarrow \mathrm{CY}=0$
$RAR$
\begin{aligned} A &=00100010 \\ XR A B & \Rightarrow A \leftarrow(00100010) \oplus(01000101)\\ A&=01100111=67 \mathrm{H} \end{aligned}
Therefore, the content of the accumulator after the execution of the program is 67H.
Question 7
An 8085 executes the following instructions
2710 LXI H, 30A0 H
2414 PCHL
All address and constants are in Hex. Let PC be the contents of the program counter and HL be the contents of the HL register pair just after executing PCHL. Which of the following statements is correct ?
A PC = 2715H HL = 30A0H B PC = 30A0H HL = 2715H C PC = 6140H HL = 6140H D PC = 6140H HL = 2715H
GATE EC 2008 Microprocessors
Question 7 Explanation:
\begin{aligned} \mathrm{LXIH}, 30 \mathrm{AOH} & \Rightarrow \mathrm{HL}=30 \mathrm{AOH} \\ \mathrm{DADH} & \Rightarrow \mathrm{HL}=6140 \mathrm{H}\\ &(i.e. 30 \mathrm{AOH}+3 \mathrm{OAOH})\\ \mathrm{PCH} & \Rightarrow \mathrm{PC}=6140 \mathrm{H}\\ \end{aligned}
Therefore, contents are
$P C=6140 H$
$H L=6140 H$
Question 8
Consider an 8085-microprocessor system.
The following program starts at location 0100H.
LXI SP, OOFF
LXI H, 0701
MVI A, 20H
SUB M
If in addition following code exists from 019H onwards.
ORI 40 H
What will be the result in the accumulator after the last instruction is executed ?
A 40 H B 20 H C 60 H D 42 H
GATE EC 2005 Microprocessors
Question 8 Explanation:
$\begin{array}{l} 0109 H \text { ORI } 40 \mathrm{H} \\ 010 \mathrm{BH} \mathrm{ADD} \mathrm{M} \\ \text { Intial: } \mathrm{A}=00 \mathrm{H} \\ 0109 \mathrm{H}: \text { ORI } 40 \mathrm{H} \Rightarrow \mathrm{A} \leftarrow \mathrm{A}(\mathrm{OR}) 40 \mathrm{H}=40 \mathrm{H} \\ 010 \mathrm{BH}: \mathrm{ADDM} \Rightarrow \mathrm{A} \leftarrow \mathrm{A}+\mathrm{M}=40 \mathrm{H}+20 \mathrm{H}=60 \mathrm{H} \\ \therefore \quad \mathrm{A}=60 \mathrm{H} \end{array}$
Question 9
Consider an 8085-microprocessor system.
The following program starts at location 0100H.
LXI SP, OOFF
LXI H, 0701
MVI A, 20H
SUB M
The content of accumulator when the program counter reaches 0109 H is
A 20 H B 02 H C 00 H D FF H
GATE EC 2005 Microprocessors
Question 9 Explanation:
\begin{aligned} 0100H&: LXISP,00FFH\\ 0103H&: LXIH, 0107H\\ 0106H&: MVI A, 20H\\ 0108H&: SUBM \Rightarrow A\leftarrow A- M \end{aligned}
M contains the data of memory location whose
Contents of HL pair $=0107 \mathrm{H}$
Contents of location $0106 \mathrm{H}=$ opcode of MVIA, Data
Contents of location $0107 \mathrm{H}=20 \mathrm{H}$
$\therefore \quad A-M=20 H-20 H=00 H$
Question 10
It is desired to multiply the numbers 0AH by 0BH and store the result in the accumulator. The numbers are available in registers B and C respectively. A part of the 8085 program for this purpose is given below :
MVI A, 00H
LOOP:
------
------
-----
HLT
END
The sequence of instructions to complete the program would be
A JNZ LOOP, ADD B, DCR C B ADD B, JNZ LOOP, DCR C C DCR C, JNZ LOOP, ADD B D ADD B, DCR C, JNZ LOOP
GATE EC 2004 Microprocessors
Question 10 Explanation:
ADD B, OCR C, JNZ LOOP
There are 10 questions to complete. | 2,276 | 7,052 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 26, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.28125 | 3 | CC-MAIN-2021-49 | longest | en | 0.792603 |
http://stackoverflow.com/questions/9052133/number-of-bytes-available-given-address-bus-width-and-various-information | 1,406,163,702,000,000,000 | text/html | crawl-data/CC-MAIN-2014-23/segments/1405997883905.99/warc/CC-MAIN-20140722025803-00100-ip-10-33-131-23.ec2.internal.warc.gz | 357,796,611 | 14,570 | number of bytes available given address bus width and various information
A processor has
• 16 bit data bus
• word contains 2 bytes
Peripherals and memory units will be connected and the entire memory space most likely will be used.
There are quite a few questions and I only ever use the fact that there is a 24 bit address bus.
What is the total number of addressable locations for the system? 2^24
1/4 of the address space is to be used for the peripherals, what is the total number of addresses for peripherals? 2^24/2^2
12/16 of the addresses are to be used for disk addressing, how many? (12/16)2^22
3/4 of the address space are to be used for memorey requirements of RAM and ROM, what is the total number of addresses avaliable? (3/4)2^24
This seems to easy
-
So what is your question? (BTW, that sounds like an 8086). – DarkDust Jan 29 '12 at 9:25
My question is am I doing it right? Is it just a red haring to give us the data bus, the word size and the fact that it is byte addressable? – user796388 Jan 29 '12 at 20:05
1. What is the total number of addressable locations for the system? `2^24 = 16,777,216`
2. 1/4 of the address space is to be used for the peripherals, what is the total number of addresses for peripherals? `2^24/2^2(?) = (1/4)2^24 = 4,194,304`
3. 12/16 of the addresses are to be used for disk addressing, how many? (12/16)2^22 `(12/16)2^24 = (3/4)2^24 = 12,582,912`
4. 3/4 of the address space are to be used for memorey requirements of RAM and ROM, what is the total number of addresses avaliable? `(3/4)2^24 = 12,582,912` (same as 3.) | 474 | 1,575 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.15625 | 3 | CC-MAIN-2014-23 | latest | en | 0.914639 |
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# Solution - On 1-4-2013 JN Ltd had 5,000, 10% Debentures of 100 each outstanding. (i) On 1-4-2014 the company purchased in the open market 2000 of its own debentures for 105 each and cancelled the same immediately. - Redemption of Debentures for Immediate Cancellation - Draw of Lots, Lump Sum and Purchase in the Open Market
ConceptRedemption of Debentures for Immediate Cancellation - Draw of Lots, Lump Sum and Purchase in the Open Market
#### Question
On 1-4-2013 JN Ltd had 5,000, 10% Debentures of 100 each outstanding.
(i) On 1-4-2014 the company purchased in the open market 2000 of its own debentures for 105 each and cancelled the same immediately.
(ii) On 1-4-2015 the company redeemed at par debentures of 1,00,000 by draw of a lot.
(iii) On 28-2-2016 the remaining debentures were purchased for immediate cancellation for 1,97,000
#### Solution
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#### APPEARS IN
2015-2016 (March) Delhi Set 2
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#### Video TutorialsVIEW ALL [2]
Solution for question: On 1-4-2013 JN Ltd had 5,000, 10% Debentures of 100 each outstanding. (i) On 1-4-2014 the company purchased in the open market 2000 of its own debentures for 105 each and cancelled the same immediately. concept: Redemption of Debentures for Immediate Cancellation - Draw of Lots, Lump Sum and Purchase in the Open Market. For the courses CBSE (Arts), CBSE (Commerce), CBSE (Science)
S | 429 | 1,502 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.6875 | 3 | CC-MAIN-2018-13 | latest | en | 0.905241 |
http://forums.wolfram.com/mathgroup/archive/2010/Nov/msg00109.html | 1,718,828,615,000,000,000 | text/html | crawl-data/CC-MAIN-2024-26/segments/1718198861832.94/warc/CC-MAIN-20240619185738-20240619215738-00140.warc.gz | 15,406,540 | 7,846 | How does one fix a Graphics3D display?
• To: mathgroup at smc.vnet.net
• Subject: [mg113642] How does one fix a Graphics3D display?
• From: Joseph Gwinn <joegwinn at comcast.net>
• Date: Fri, 5 Nov 2010 05:14:53 -0500 (EST)
• Organization: Gwinn Instruments
```I'm using a 3dconnexion "SpaceNavigator" 3-D mouse (works like the old
SpaceBall joystick), and am running into an annoying problem. I have a
simple DynamicModule that accepts and integrates the XYZ inputs from the
SpaceNavigator, and uses the integral to move a little sphere around in
a 3D box. So far so good. But I cannot get the box to stay still.
Matheamtica keeps trying to help me by recomputing what it thinks is the
best way to view the scene, which leads to disorienting jumps in the
box and the viewpoint.
So, the question is how to convince Mathematica to keep the box fixed and let
the little sphere fly around within this cage? I have tried many
variations of the View* options, to no avail.
For the record, here is some sample code. You will need some kind of 3D
controller for this code to work.
DynamicModule[{x=0,y=0,z=0},
Graphics3D[
{Dynamic[Text[ToString[{x,y,z}],{0,0,10}]],Sphere[Dynamic[{x,y,z}+=
{1,1,-1}*ControllerState["SpaceNavigator",{"X Axis","Z Axis",
"Y Axis"}]],0.35]},PlotRange->{{-10,+10},{-10,+10},{-10,+10}},
Axes->True,Ticks->False,AxesLabel->{"x","y","z"},ViewPoint->Front,
ViewVertical->{0,0,1},ViewCenter->{0,0,0},ViewVector->{{0,30,0},{0,-1,0}}
]
] | 447 | 1,462 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.65625 | 3 | CC-MAIN-2024-26 | latest | en | 0.826367 |
http://ycharts.com/companies/QBC/return_on_invested_capital | 1,369,276,504,000,000,000 | text/html | crawl-data/CC-MAIN-2013-20/segments/1368702749808/warc/CC-MAIN-20130516111229-00078-ip-10-60-113-184.ec2.internal.warc.gz | 835,362,202 | 44,324 | ### Cubic Energy (QBC)
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## About Return on Invested Capital (ROIC)
ROIC is an acronym for "Return on Invested Capital", and it is a concept that doesn't have a fixed definition. In YCharts, it is the net income that a company earned as a percentage of all of the capital given to a company by shareholders and debt holders. It is a ratio that tries to answer the question: "If I gave \$1 to this company, how much money could the company earn by investing that \$1?" A ROIC of 5% means that the company can return \$0.05 per dollar invested.
ROIC is often considered a more reasonable estimate of managerial performance than Return on Equity (ROE) because it takes into account investments by debt holders, which should be invested to increase net income.
It is also more reasonable than Return on Assets (ROA) because it only assumes that capital which was "invested" into the company can be used to earn income.
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Overall 42nd percentile 4573 of 8005 Sector 26th percentile 302 of 410 in Energy Industry 30th percentile 128 of 183 in Oil & Gas E&P
## QBC Return on Invested Capital Range, Past 5 Years
Minimum Go Pro Sep 2009 Maximum Go Pro Sep 2010 Average Go Pro | 1,019 | 3,039 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.53125 | 3 | CC-MAIN-2013-20 | latest | en | 0.920472 |
https://docs.microsoft.com/ja-jp/dotnet/api/system.numerics.complex.exp?redirectedfrom=MSDN&view=netframework-4.8 | 1,563,598,747,000,000,000 | text/html | crawl-data/CC-MAIN-2019-30/segments/1563195526446.61/warc/CC-MAIN-20190720045157-20190720071157-00423.warc.gz | 374,942,592 | 10,013 | # Complex.Exp(Complex)Complex.Exp(Complex)Complex.Exp(Complex)Complex.Exp(Complex) Method
## 定義
`e` を指定した複素数で累乗した値を返します。Returns `e` raised to the power specified by a complex number.
``````public:
static System::Numerics::Complex Exp(System::Numerics::Complex value);``````
``public static System.Numerics.Complex Exp (System.Numerics.Complex value);``
``static member Exp : System.Numerics.Complex -> System.Numerics.Complex``
``Public Shared Function Exp (value As Complex) As Complex``
#### パラメーター
value
Complex Complex Complex Complex
## 例
``````using System;
using System.Numerics;
public class Example
{
public static void Main()
{
Complex[] values = { new Complex(1.53, 9.26),
new Complex(2.53, -8.12),
new Complex(-2.81, 5.32),
new Complex(-1.09, -3.43),
new Complex(Double.MinValue/2, Double.MinValue/2) };
foreach (Complex value in values)
Console.WriteLine("Exp(Log({0}) = {1}", value,
Complex.Exp(Complex.Log(value)));
}
}
// The example displays the following output:
// Exp(Log((1.53, 9.26)) = (1.53, 9.26)
// Exp(Log((2.53, -8.12)) = (2.53, -8.12)
// Exp(Log((-2.81, 5.32)) = (-2.81, 5.32)
// Exp(Log((-1.09, -3.43)) = (-1.09, -3.43)
// Exp(Log((-8.98846567431158E+307, -8.98846567431158E+307)) = (-8.98846567431161E+307, -8.98846567431161E+307)
``````
``````Imports System.Numerics
Module Example
Public Sub Main()
Dim values() As Complex = { New Complex(1.53, 9.26),
New Complex(2.53, -8.12),
New Complex(-2.81, 5.32),
New Complex(-1.09, -3.43),
New Complex(Double.MinValue/2, Double.MinValue/2) }
For Each value As Complex In values
Console.WriteLine("Exp(Log({0}) = {1}", value,
Complex.Exp(Complex.Log(value)))
Next
End Sub
End Module
' The example displays the following output:
' Exp(Log((1.53, 9.26)) = (1.53, 9.26)
' Exp(Log((2.53, -8.12)) = (2.53, -8.12)
' Exp(Log((-2.81, 5.32)) = (-2.81, 5.32)
' Exp(Log((-1.09, -3.43)) = (-1.09, -3.43)
' Exp(Log((-8.98846567431158E+307, -8.98846567431158E+307)) = (-8.98846567431161E+307, -8.98846567431161E+307)
``````
## 注釈
Exp複素数のメソッドに対応して、Math.Exp実数のメソッド。The Exp method for complex numbers corresponds to the Math.Exp method for real numbers. Exp 逆ですLogします。Exp is the inverse of Log. | 779 | 2,215 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.84375 | 3 | CC-MAIN-2019-30 | latest | en | 0.30646 |
https://plainmath.net/precalculus/67016-proof-of-displaystyle-ln-left | 1,686,359,081,000,000,000 | text/html | crawl-data/CC-MAIN-2023-23/segments/1685224656869.87/warc/CC-MAIN-20230609233952-20230610023952-00232.warc.gz | 490,647,734 | 24,521 | ruililorNokmncf
2022-03-31
Proof of
$-\mathrm{ln}\left(2\mathrm{sin}\left(\frac{x}{2}\right)\right)=\sum _{k=1}^{\mathrm{\infty }}\frac{\mathrm{cos}\left(kx\right)}{k}$
tabido8uvt
As $\mathrm{ln}\left(1-h\right)=-\sum _{r=1}^{\mathrm{\infty }}\frac{{h}^{r}}{r}$
$-\frac{{e}^{ikx}}{k}=\mathrm{ln}\left[1-{e}^{ix}\right]$
$=\mathrm{ln}\left(\frac{{e}^{ix}}{2}\right)+\mathrm{ln}\left({e}^{i\frac{x}{2}}-{e}^{-i\frac{x}{2}}\right)$
$=i\frac{x}{2}+\mathrm{ln}\left(i\cdot 2\frac{\mathrm{sin}x}{2}\right)$
$=i\frac{x}{2}+\mathrm{ln}i+\mathrm{ln}\left(2\frac{\mathrm{sin}x}{2}\right)$
$i={e}^{i\frac{\pi }{2}}⇒\mathrm{ln}\left(i\right)=i\frac{\pi }{2}$
Do you have a similar question? | 318 | 683 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 20, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.625 | 4 | CC-MAIN-2023-23 | latest | en | 0.145819 |
https://www.coursehero.com/file/6881502/02-Hypothesis-testing/ | 1,513,360,921,000,000,000 | text/html | crawl-data/CC-MAIN-2017-51/segments/1512948577018.52/warc/CC-MAIN-20171215172833-20171215194833-00043.warc.gz | 716,364,109 | 24,771 | 02. Hypothesis testing
# 02. Hypothesis testing - Hypothesis testing Terminology •...
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Unformatted text preview: Hypothesis testing Terminology • Null hypothesis – e.g. H0: µ = 1 0 , H0: p = .5 , H0: µ1 = µ2 • Alternative hypothesis – e.g. H1 : µ ≠ 1 0 , H1: p < .5 , H1 : µ1 > µ2 • • • • • • Type I error? Type II error? Significance Level? Critical Region? p-value? Power? 2 Type I and II errors • Type I error – Reject H0 when H0 is true • Type II error – Accept H0 when H0 is false • Probabilities of Type I and II errors should be as small as possible – Fixed sample size • reducing probability of type I error ->increases probability of type II error 3 Hypothesis testing approach – significance level • FIX the maximum allowable probability of Type I error = α, significance level of the test • Test is then arranged to minimize Type II error • Most expensive error should represent Type I error • Type I error can be fixed • Under more control than type II error 4 Hypothesis testing example • Large sample size σ x ~ N (µ , ) n 2 • One sided test – H0:µ = µ0 vs H1:µ > µ0 – e.g. H0:µ = 1 0 vs H1:µ > 1 0 Ｉ Ｉ Ｉ Ｉ Ｉ Ｉ is true σ2 n α µ0 Critical region: Reject H0 µ0 + k x Hypothesis testing approach – significance level • FIX the maximum allowable probability of Type I error = α, significance level of the test • Reject H0 if test statistic lies in the critical region x − µ0 > zα 1 σ n 7 Computer Output (p-value) • p-value is a measure of exactly where the test-statistic lies in the critical region – Smaller p-value -> more significant result • Reject H0 if p-value is less than α p-value measures how significant a result is p-value µ0 x Hypothesis testing approach – power • Standard tests are designed to minimize probability of Type II error = β • Power of the test = 1 – β • Standard test designed to have maximum possible power (depending on effect size) • Recommended power ∼ 0.8 • Fixed n - increase power by relaxing α? • 10 ...
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• Spring '09
• ValerieOzaki
• Null hypothesis, Hypothesis testing, Statistical hypothesis testing, Statistical significance, Type I and type II errors, Statistical power
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Ask a homework question - tutors are online | 761 | 3,046 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.8125 | 3 | CC-MAIN-2017-51 | latest | en | 0.767657 |
https://www.convert-measurement-units.com/convert+Picovolt+to+Microvolt.php | 1,723,272,419,000,000,000 | text/html | crawl-data/CC-MAIN-2024-33/segments/1722640790444.57/warc/CC-MAIN-20240810061945-20240810091945-00385.warc.gz | 564,597,573 | 13,678 | Convert pV to µV (Picovolt to Microvolt)
## Picovolt into Microvolt
numbers in scientific notation
https://www.convert-measurement-units.com/convert+Picovolt+to+Microvolt.php
# Convert pV to µV (Picovolt to Microvolt):
1. Choose the right category from the selection list, in this case 'Voltage'.
2. Next enter the value you want to convert. The basic operations of arithmetic: addition (+), subtraction (-), multiplication (*, x), division (/, :, ÷), exponent (^), square root (√), brackets and π (pi) are all permitted at this point.
3. From the selection list, choose the unit that corresponds to the value you want to convert, in this case 'Picovolt [pV]'.
4. Finally choose the unit you want the value to be converted to, in this case 'Microvolt [µV]'.
5. Then, when the result appears, there is still the possibility of rounding it to a specific number of decimal places, whenever it makes sense to do so.
With this calculator, it is possible to enter the value to be converted together with the original measurement unit; for example, '990 Picovolt'. In so doing, either the full name of the unit or its abbreviation can be usedas an example, either 'Picovolt' or 'pV'. Then, the calculator determines the category of the measurement unit of measure that is to be converted, in this case 'Voltage'. After that, it converts the entered value into all of the appropriate units known to it. In the resulting list, you will be sure also to find the conversion you originally sought. Alternatively, the value to be converted can be entered as follows: '72 pV to µV' or '2 pV into µV' or '58 Picovolt -> Microvolt' or '44 pV = µV' or '30 Picovolt to µV' or '16 pV to Microvolt' or '87 Picovolt into Microvolt'. For this alternative, the calculator also figures out immediately into which unit the original value is specifically to be converted. Regardless which of these possibilities one uses, it saves one the cumbersome search for the appropriate listing in long selection lists with myriad categories and countless supported units. All of that is taken over for us by the calculator and it gets the job done in a fraction of a second.
Furthermore, the calculator makes it possible to use mathematical expressions. As a result, not only can numbers be reckoned with one another, such as, for example, '(17 * 3) pV'. But different units of measurement can also be coupled with one another directly in the conversion. That could, for example, look like this: '45 Picovolt + 31 Microvolt' or '88mm x 74cm x 60dm = ? cm^3'. The units of measure combined in this way naturally have to fit together and make sense in the combination in question.
The mathematical functions sin, cos, tan and sqrt can also be used. Example: sin(π/2), cos(pi/2), tan(90°), sin(90) or sqrt(4).
If a check mark has been placed next to 'Numbers in scientific notation', the answer will appear as an exponential. For example, 1.801 222 205 831 1×1021. For this form of presentation, the number will be segmented into an exponent, here 21, and the actual number, here 1.801 222 205 831 1. For devices on which the possibilities for displaying numbers are limited, such as for example, pocket calculators, one also finds the way of writing numbers as 1.801 222 205 831 1E+21. In particular, this makes very large and very small numbers easier to read. If a check mark has not been placed at this spot, then the result is given in the customary way of writing numbers. For the above example, it would then look like this: 1 801 222 205 831 100 000 000. Independent of the presentation of the results, the maximum precision of this calculator is 14 places. That should be precise enough for most applications. | 892 | 3,691 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.296875 | 3 | CC-MAIN-2024-33 | latest | en | 0.835329 |
http://aslstem.cs.washington.edu/topics/view/7138 | 1,566,048,544,000,000,000 | text/html | crawl-data/CC-MAIN-2019-35/segments/1566027313259.30/warc/CC-MAIN-20190817123129-20190817145129-00515.warc.gz | 23,288,219 | 7,012 | ## Viewing topic: Lebesgue Integrable
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#### Lebesgue Integrable
• Definition: We denote by L(I) the set of all functions f of the form f=u-v, where u is in U(I) and v is in U(I). Each function f in L(I) is said to be Lebesgue-integrable on I, and its integral is defined by the equation int(f)=int(u) - int(v) where each integral is taken over I.
Source: Mathematical Analysis, second edition by Tom M. Apostol
• There are no comments for this topic. | 135 | 508 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.5625 | 3 | CC-MAIN-2019-35 | latest | en | 0.922688 |
https://www.mrexcel.com/board/threads/vlookup-a-table-with-a-value-before-a-comma.1169834/#post-5685524 | 1,638,777,931,000,000,000 | text/html | crawl-data/CC-MAIN-2021-49/segments/1637964363290.59/warc/CC-MAIN-20211206072825-20211206102825-00234.warc.gz | 981,945,224 | 19,529 | # vlookup a table with a value before a comma
#### Dares2
##### New Member
hi, I have a need to vlookup a surname in a list but the lookup range has both surname, comma then first name so eg vlookup(smith in a list that has smith,david and take the forth value along) Im not sure how to write this formula thanks.
### Excel Facts
Square and cube roots
The =SQRT(25) is a square root. For a cube root, use =125^(1/3). For a fourth root, use =625^(1/4).
#### Fluff
##### MrExcel MVP, Moderator
You can do that like
Excel Formula:
``=VLOOKUP(X2&"*",A2:D100,4,0)``
where X2 is the cell with the value to find.
#### Peter_SSs
##### MrExcel MVP, Moderator
I would suggest including the comma in the lookup value (Y2 formula) to avoid a possible incorrect result (Z2 formula)
21 05 03.xlsm
ABCDEWXYZ
1post #3post #2
2jones, alan147smith98
3smithson, anne258
4smith, david369
VLOOKUP
Cell Formulas
RangeFormula
Y2Y2=VLOOKUP(X2&",*",A2:D100,4,0)
Z2Z2=VLOOKUP(X2&"*",A2:D100,4,0)
#### Dares2
##### New Member
I would suggest including the comma in the lookup value (Y2 formula) to avoid a possible incorrect result (Z2 formula)
21 05 03.xlsm
ABCDEWXYZ
1post #3post #2
2jones, alan147smith98
3smithson, anne258
4smith, david369
VLOOKUP
Cell Formulas
RangeFormula
Y2Y2=VLOOKUP(X2&",*",A2:D100,4,0)
Z2Z2=VLOOKUP(X2&"*",A2:D100,4,0)
Thanks although I couldnt get this to work. May because this is the otherway round ie looking up eg smith within a lookup range of smith,peter
Name Gross pf new Fnights Total Baker, Annie \$1,000.00 4.00 =SUM(C5*D5)+VLOOKUP(A5&"*",C9:E11,2,FALSE) Baker Blue 2237 Bawden Red 2273 Smith Green 2283
#### Dares2
##### New Member
You can do that like
Excel Formula:
``=VLOOKUP(X2&"*",A2:D100,4,0)``
where X2 is the cell with the value to find.
Not sure but couldnt get this to work
Name Gross pf new Fnights Total Baker, Annie \$1,000.00 4.00 =SUM(C5*D5)+VLOOKUP(A5&"*",C9:E11,2,FALSE) Baker Blue 2237 Bawden Red 2273 Smith Green 2283
#### Peter_SSs
##### MrExcel MVP, Moderator
May because this is the otherway round
Yes, it appears that you described this the wrong way around in your original post. But it also looks like you have the wrong column referenced in the lookup range in your latest attempt.
Is below what you want?
BTW, I suggest that you
• Update your Account details (click your user name at the top right of the forum) so helpers always know what Excel version(s) & platform(s) you are using as the best solution often varies by version. (Don’t forget to scroll down & ‘Save’)
• Investigate XL2BB for providing sample data to make it easier for helpers.
21 05 03.xlsm
ABCDE
4NameGross pf newFnightsTotal
5Baker, Annie\$1,000.004\$6,237.00
6
7
8
9BakerBlue2237
10BawdenRed2273
11SmithGreen2283
VLOOKUP (2)
Cell Formulas
RangeFormula
E5E5=C5*D5+VLOOKUP(LEFT(A5,FIND(",",A5)-1),C9:E11,3,FALSE)
#### Dares2
##### New Member
Yes, it appears that you described this the wrong way around in your original post. But it also looks like you have the wrong column referenced in the lookup range in your latest attempt.
Is below what you want?
BTW, I suggest that you
• Update your Account details (click your user name at the top right of the forum) so helpers always know what Excel version(s) & platform(s) you are using as the best solution often varies by version. (Don’t forget to scroll down & ‘Save’)
• Investigate XL2BB for providing sample data to make it easier for helpers.
21 05 03.xlsm
ABCDE
4NameGross pf newFnightsTotal
5Baker, Annie\$1,000.004\$6,237.00
6
7
8
9BakerBlue2237
10BawdenRed2273
11SmithGreen2283
VLOOKUP (2)
Cell Formulas
RangeFormula
E5E5=C5*D5+VLOOKUP(LEFT(A5,FIND(",",A5)-1),C9:E11,3,FALSE)
thanks so much..worked a treat
#### Peter_SSs
##### MrExcel MVP, Moderator
thanks so much..worked a treat
You're welcome. Glad we could help. Thanks for the follow-up.
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Go back | 1,503 | 4,902 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.859375 | 3 | CC-MAIN-2021-49 | latest | en | 0.698379 |
https://solveordie.com/riddles-about-oranges/ | 1,726,358,206,000,000,000 | text/html | crawl-data/CC-MAIN-2024-38/segments/1725700651601.85/warc/CC-MAIN-20240914225323-20240915015323-00022.warc.gz | 496,102,804 | 10,213 | funny
## An orange
What does the orange do, when it takes a test?
It concentrates.
73.41 %
## Mislabeled Jar Puzzle
You have 3 jars that are all mislabeled. One jar contains Apples, another contains Oranges and the third jar contains a mixture of both Apples and Oranges. You are allowed to pick as many fruits as you want from each jar to fix the labels on the jars. What is the minimum number of fruits that you have to pick and from which jars to correctly label them?
Let's take a scenario. Suppose you pick from jar labelled as Apples and Oranges and you got Apple from it. That means that jar should be Apples as it is incorrectly labelled. So it has to be Apples jar. Now the jar labelled Oranges has to be Mixed as it cannot be the Oranges jar as they are wrongly labelled and the jar labelled Apples has to be Oranges. Similar scenario applies if it's a Oranges taken out from the jar labelled as Apples and Oranges. So you need to pick just one fruit from the jar labelled as Apples and Oranges to correctly label the jars.
69.55 %
funny
## Apples and oranges
If you had three apples and four oranges in one hand and four apples and three oranges in the other hand, what would you have?
Very large hands.
69.24 %
## Sweet and juicy
I taste sweet and juicy. I smell fruity. I feel rough and wet. I look like a bumpy ball. What am I?
An orange.
67.33 %
## Oranges
How do you divide 20 oranges equally to 11 girls? No one gets more and no one gets less. All 11 girls should receive equal portions. How?
Juice the oranges and serve the equal quantity of orange juice.
66.24 % | 391 | 1,591 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.125 | 4 | CC-MAIN-2024-38 | latest | en | 0.953874 |
https://math.answers.com/Q/What_are_the_least_common_multiples_of_three_and_ten | 1,713,633,900,000,000,000 | text/html | crawl-data/CC-MAIN-2024-18/segments/1712296817670.11/warc/CC-MAIN-20240420153103-20240420183103-00745.warc.gz | 352,165,573 | 46,267 | 0
# What are the least common multiples of three and ten?
Updated: 8/21/2019
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Q: What are the least common multiples of three and ten?
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20, 40, 60
### What is the least common multiple of ten and twelve?
multiples of 10- 10,20,30,40,50,60.... Multiples of 12- 12, 24, 36, 48, 60..... Answer-60
The LCM is: 210
### What is the least common multiple of ten?
You need at least two numbers to find an LCM.
### Least common multiple of 6 and 10?
Multiples of 6: 6, 12, 18, 24, 30, 36, ................ Multiples of 10: 10, 20, 30, 40, ................ 30 is the LCM (least common multiple) of 6 and 10. correct but LCM stands for lowest common multiple...
9 is.
### What is the lowest common multiple of two and three and ten?
The Least Common Multiple (LCM) for 2 3 10 is 30.
### What are the first ten common multiples of 2 and 3?
Get the first common multiple, then multiply that by 0, 1, 2, 3, 4, etc. to get additional common multiples.
### What are the common multiples of ten eight and seven?
7 is a prime number. The answer is 1.
40, 80, 120 | 376 | 1,192 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.59375 | 4 | CC-MAIN-2024-18 | latest | en | 0.899121 |
https://dwellingexpertise.com/how-much-does-a-liter-of-water-weigh-in-grams/ | 1,709,566,824,000,000,000 | text/html | crawl-data/CC-MAIN-2024-10/segments/1707947476452.25/warc/CC-MAIN-20240304133241-20240304163241-00067.warc.gz | 217,173,975 | 25,933 | # How Much Does a Liter of Water Weigh in Grams?
The weight of 1 liter is 1,000 grams. It is also equivalent to 1,000 milliliters and weighs exactly 1 kilogram when measured at its maximum density.
Water is usually stored in containers, and the capacity of a container is commonly mentioned in liters.
## How Much Does Water Weigh?
The weight of water depends on the temperature at which it is stored.
Water expands when it gets warmer. This makes it weigh less when hot.
Cold water weighs more because it is denser.
In fact, water reaches the highest density when it’s near freezing. When it begins to freeze, it expands and becomes less dense again.
At room temperature, the density of water equals 0.99802 g/ml. This simply means that the weight of a gallon of water (US) is around 8.34 at room temperature.
## What is the Weight of 1 Liter in Kilograms?
A liter of water weighs approximately 1 kilogram (35.2 ounces or 2.2 lb.) at room temperature. Its density is around 1g/ml.
## FAQs
### Q: How much does 500ml of water weigh in grams?
500 ml of water weighs approximately 500 grams at room temperature (17.6 ounces or 1.1lb). This is simply because the density of water is 0.998 g/ml.
### Q: What does 1 liter of water weigh?
A liter of water weighs about 1 kilogram or 2.2lb.
### Q: Does 1 liter weigh 1 kg?
Yes, a liter of water weighs 1 kilogram.
### Q: What does one pound of water weigh?
A pound of water weighs 16 ounces.
### Q: What is the weight of a 16.9 oz bottle of water?
A 16.9 oz bottle of water weighs about 1.1lb or 17.595 plus the weight of the bottle.
### Q: How many grams make 1 liter?
1,000 grams make a liter.
### Q: Does temperature affect the weight of water?
Temperature fluctuations affect the weight of water. When cold, water becomes denser and thus weighs more. Hot water weighs less.
### Q: What is the weight of 2 liters of water?
2 liters of water weigh about 2 kilograms or 2,000 grams.
### Q: What is heavier, water or sand?
Sand is denser than water and, therefore, heavier.
### Q: What is the weight of a cup of water?
A cup of water weighs about 236 grams or 0.52028218695 pounds.
### Q: Do water and ice weigh the same?
No, water weighs more than ice. This is simply because water is denser than ice.
references
When you buy something through our affiliate links, we earn a commission without you having to pay extra. | 608 | 2,399 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.203125 | 3 | CC-MAIN-2024-10 | latest | en | 0.936693 |
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UNIT 4
Expressions, Equations, and Relationships Contents 6.7.A 6.7.A 6.7.A
6.7.C 6.7.A 6.7.D
6.9.A 6.10.A 6.10.A
6.9.A 6.10.A 6.10.A 6.9.B
MODULE 10
Generating Equivalent Numerical Expressions
Lesson 10.1 Lesson 10.2 Lesson 10.3
Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269 Prime Factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 Order of Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . 281
MODULE 11
Generating Equivalent Algebraic Expressions
Lesson 11.1 Lesson 11.2 Lesson 11.3
Modeling Equivalent Expressions . . . . . . . . . . . . . . . . . 293 Evaluating Expressions . . . . . . . . . . . . . . . . . . . . . . . . . 301 Generating Equivalent Expressions. . . . . . . . . . . . . . . . 307
MODULE 12
Equations and Relationships
Lesson 12.1 Lesson 12.2 Lesson 12.3
Writing Equations to Represent Situations . . . . . . . . . . 321 Addition and Subtraction Equations . . . . . . . . . . . . . . . 327 Multiplication and Division Equations . . . . . . . . . . . . . 335
MODULE 13
Inequalities and Relationships
Writing Inequalities . . . . . . . . . . . . . . . . . . . . . Addition and Subtraction Inequalities . . . . . . Multiplication and Division Inequalities with Positive Numbers . . . . . . . . . . . . . . . . . . . . . . Lesson 13.4 Multiplication and Division Inequalities with Rational Numbers . . . . . . . . . . . . . . . . . . . . . . Lesson 13.1 Lesson 13.2 Lesson 13.3
MODULE 14
6.11 6.6.A 6.6.B 6.6.C
. . . . . . . 349 . . . . . . . 355 . . . . . . . 361 . . . . . . . 367
Relationships in Two Variables
Graphing on the Coordinate Plane . . . . . . . . . . . Independent and Dependent Variables in Tables and Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lesson 14.3 Writing Equations from Tables . . . . . . . . . . . . . . Lesson 14.4 Representing Algebraic Relationships in Tables and Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lesson 14.1 Lesson 14.2
. . . . . 379 . . . . . 385 . . . . . 393 . . . . . 399
UNIT 4
Unit Pacing Guide 45-Minute Classes Module 10 DAY 1
DAY 2
DAY 3
DAY 4
DAY 5
Lesson 10.1
Lesson 10.1
Lesson 10.2
Lesson 10.2
Lesson 10.3
DAY 6
DAY 7
Lesson 10.3
Ready to Go On? Texas Test Prep
Module 11 DAY 1
DAY 2
DAY 3
DAY 4
DAY 5
Lesson 11.1
Lesson 11.1
Lesson 11.2
Lesson 11.2
Lesson 11.3
DAY 6
DAY 7
Lesson 11.3
Ready to Go On? Texas Test Prep
Module 12 DAY 1
DAY 2
DAY 3
DAY 4
DAY 5
Lesson 12.1
Lesson 12.1
Lesson 12.2
Lesson 12.2
Lesson 12.3
DAY 6
DAY 7
Lesson 12.3
Ready to Go On? Texas Test Prep
Module 13 DAY 1
DAY 2
DAY 3
DAY 4
DAY 5
Lesson 13.1
Lesson 13.1
Lesson 13.2
Lesson 13.2
Lesson 13.3
DAY 6
DAY 7
DAY 8
DAY 9
Lesson 13.3
Lesson 13.4
Lesson 13.4
Ready to Go On? Texas Test Prep
DAY 1
DAY 2
DAY 3
DAY 4
DAY 5
Lesson 14.1
Lesson 14.1
Lesson 14.2
Lesson 14.2
Lesson 14.3
Module 14
DAY 6
DAY 7
DAY 8
DAY 9
Lesson 14.3
Lesson 14.4
Lesson 14.4
Ready to Go On? Texas Test Prep
Expressions, Equations, and Relationships
263B
Program Resources Plan
Engage and Explore
Online Teacher Edition Access a full suite of teaching resources online— plan, present, and manage classes, assignments, and activities.
Real-World Videos Engage students with interesting and relevant applications of the mathematical content of each module.
ePlanner Easily plan your classes, create and view assignments, and access all program resources with your online, customizable planning tool.
Animated Math Online interactive simulations, tools, and games help students actively learn and practice key concepts.
Professional Development Videos Author Juli Dixon models successful teaching practices and strategies in actual classroom settings.
QR Codes Scan with your smart phone to jump directly from your print book to online videos and other resources.
Explore Activities Students interactively explore new concepts using a variety of tools and approaches.
Teacher’s Edition Support students with point-of-use Questioning Strategies, teaching tips, resources for differentiated instruction, additional activities, and more.
LESSON
7.2 Rates ?
ESSENTIAL QUESTION How do you use rates to compare quantities?
6.4.D
EXPLORE ACTIVITY
LESSON LESSON
Rates
The student is expected to: Proportionality—6.4.D Give examples of rates as the comparison by division of two quantities having different attributes, including rates as quotients.
Mathematical Processes
?
Engage
Calculating Unit Rates
6.4.D Give examples of
rates as the comparison by division of two quantities having different attributes, including rates as quotients. Math On the Spot my.hrw.com
ESSENTIAL QUESTION How do you use rates to compare quantities?
ESSENTIAL QUESTION How do you use rates to compare quantities? Sample answer: You use division to compare two quantities with different units.
Yoga Classes This month’s special:
6 classes for \$90
Using Rates to Compare Prices A rate is a comparison of two quantities that have different units.
Motivate the Lesson
6 classes
Engage with the Whiteboard Have students fill in the table for each brand on the whiteboard. Ask students to write a rate for each brand, using the data from the last row of each table. Then have them compare that rate to the original rate in the problem statement. Help students to see that the two rates are equivalent.
B The cost of 2 cartons of milk is \$5.50. What is the unit price?
C An airplane makes a 2,748-mile flight in 6 hours. What is the airplane’s average rate of speed in miles per hour? 458 miles per hour Interactive Whiteboard Interactive example available online my.hrw.com
The unit price is \$2.75 per carton of milk.
3.84
÷2
÷2
8
1.92
4
0.96 0.48 0.24
÷2
÷2
2
÷2
1
B Brand A costs \$
0.24
÷5 ÷5
÷2
Ounces
Price (\$)
25
4.50
5
0.90 0.18
1
The first quantity in a unit rate can be less than 1.
÷5 ÷5
÷2
The ship travels 0.4 mile per minute.
They are equivalent. per ounce.
YOUR TURN 3. There are 156 players on 13 teams. How many players are on each
Analyze Relationships Describe another method to compare the costs.
team? Personal Math Trainer Online Assessment and Intervention
EXAMPLE 1
To compare the costs, Shana must compare prices for equal amounts of juice. How can she do this?
÷50
Analyze Relationships In all of these problems, how is the unit rate related to the rate given in the original problem?
B; it costs less per ounce.
Reflect 1.
1 carton
2 cartons
Reflect
÷2
0.18
\$5.50 \$2.75 ________ = _______
C A cruise ship travels 20 miles in 50 minutes. 20 miles = ________ 0.4 mile __________ How far does the ship travel per minute? 50 minutes 1 minute
2.
per ounce. Brand B costs \$
C Which brand is the better buy? Why?
Shana is at the grocery store comparing two brands of juice. Brand A costs \$3.84 for a 16-ounce bottle. Brand B costs \$4.50 for a 25-ounce bottle.
÷50
Brand B
Price (\$)
16
\$15 The unit rate _____ 1 class is the same as 15 ÷ 1 = \$15 per class.
÷2
Divide the cost of each bottle by the amount of juice; 3.84 4.50 A: ____ = 0.24; B: ____ = 0.18 16 25
Explain B Michael walks 30 meters in 20 seconds. How many meters does he walk per second? 1.5 meters per second
© Houghton Mifflin Harcourt Publishing Company
EXPLORE ACTIVITY
1 class
÷6
Brand A Ounces
107 miles . The rate is _______ 2 hours
÷6
\$15 \$90 ________ = ______
Gerald’s yoga classes cost \$15 per class.
A Complete the tables.
÷2
Chris drove 107 miles in two hours. You are comparing miles and hours.
To find the unit rate, divide both quantities in the rate by the same number so that the second quantity is 1:
107 miles . The rate is _______ 2 hours
Shana is at the grocery store comparing two brands of juice. Brand A costs \$3.84 for a 16-ounce bottle. Brand B costs \$4.50 for a 25-ounce bottle.
Explore
Connect Vocabulary
A rate is a comparison by division of two quantities that have different units.
6.4.D
\$90
Use the information in the problem to write a rate: _______ 6 classes
Chris drove 107 miles in two hours. You are comparing miles and hours.
Ask: Have you ever wanted to find out which was the best buy between two products when shopping or find out how fast you are walking or running? Begin the Explore Activity to find out how to compare quantities with different units.
To compare the costs, Shana must compare prices for equal amounts of juice. How can she do this?
ADDITIONAL EXAMPLE 1 A The cost of 3 candles is \$19.50. What is the unit price? \$6.50 per candle
EXAMPLE 1 A Gerald pays \$90 for 6 yoga classes. What is the cost per class?
6.4.D
EXPLORE ACTIVITY
6.1.G Display, explain, and justify mathematical ideas and arguments using precise mathematical language in written or oral communication.
Using Rates to Compare Prices
A unit rate is a rate in which the second quantity is one unit. When the first quantity in a unit rate is an amount of money, the unit rate is sometimes called a unit price or unit cost.
12
A Complete the tables. Brand A
photograph? \$
0.50
per photograph
my.hrw.com
ELL
Lesson 7.2
187
188
Unit 3
Remind students that a ratio is a comparison of two quantities expressed with the same units of measure, and a rate is a comparison of two quantities with different units of measure. A rate in which the second quantity is one unit is a unit rate.
Questioning Strategies
Mathematical Processes • How do you determine what number to divide by when finding a unit rate? Divide both quantities by the same number so that the second quantity is 1.
• How is finding a unit rate like simplifying a fraction? You find unit rates by dividing both quantities by the same number, just as you would to simplify a fraction.
PROFESSIONAL DEVELOPMENT Integrate Mathematical Processes This lesson provides an opportunity to address Mathematical Process TEKS 6.1.G, which calls for students to “display, explain, and justify mathematical ideas and arguments using precise mathematical language in written or oral communication.” In the Explore Activity, students
Math Background The words ratio and rate come from the Latin word ratus, which means “calculation.” A unit rate is a rate expressed in its simplest form, a to b, where a may or may not be a whole number and b is 1. The terms have only 1 as a common factor. Many real-world situations involve the use of rate
Brand B
Ounces
Price (\$)
÷2
16
3.84
÷2
8
1.92
4
0.96 0.48 0.24
players per team
4. A package of 36 photographs costs \$18. What is the cost per
lin Harcourt Publishing Company
Texas Essential Knowledge and Skills
Proportionality—
7.2 Rates
© Houghton Mifflin Harcourt Publishing Company
7.2
Proportionality— 6.4.D Give examples of rates as the comparison by division of two quantities having different attributes, including rates as quotients.
÷2 ÷2
2 1
B Brand A costs \$
0.24
Ounces ÷2 ÷2
÷5 ÷5
÷2
Price (\$)
25
4.50
5
0.90 0.18
1
÷2
per ounce. Brand B costs \$
0.18
per ounce.
B; it costs less per ounce.
÷5 ÷5
3 2 1
Teach
Assessment and Intervention
Math on the Spot video tutorials, featuring program authors Dr. Edward Burger and Martha Sandoval-Martinez, accompany every example in the textbook and give students step-by-step instructions and explanations of key math concepts.
The Personal Math Trainer provides online practice, homework, assessments, and intervention. Monitor student progress through reports and alerts. Create and customize assignments aligned to specific lessons or TEKS. • Practice – With dynamic items and assignments, students get unlimited practice on key concepts supported by guided examples, step-by-step solutions, and video tutorials. • Assessments – Choose from course assignments or customize your own based on course content, TEKS, difficulty levels, and more. • Homework – Students can complete online homework with a wide variety of problem types, including the ability to enter expressions, equations, and graphs. Let the system automatically grade homework, so you can focus where your students need help the most! • Intervention – Let the Personal Math Trainer automatically prescribe a targeted, personalized intervention path for your students.
Present engaging content on a multitude of devices, including tablets and interactive whiteboards. Continually monitor and assess student progress with integrated formative assessment.
Math Talk
Differentiated Instruction Print Resources Support all learners with Differentiated Instruction Resources, including • Leveled Practice and Problem Solving • Reteach • Reading Strategies • Success for English Learners • Challenge
Problem Solving
Calculating Unit Rates Math On the Spot my.hrw.com
You can solve rate
A unit rate is a rate in which the second quantity is one unit. When the first quantity in a unit rate is an amount of money, the unit rate is sometimes called a unit price or unit cost.
EXAMPLE 1
6.4.D
This month’s special:
6 classes for \$90
To find the unit rate, divide both quantities in the rate by the same number so that the second quantity is 1:
1 class
÷2
The unit price is \$2.75 per carton of milk.
112 camp _____ ers __ _____ _____ 8 campers per cabin = 14 cabins Method 2 Use equiv alent rates.
\$5.50 \$2.75 ________ = _______ 1 carton
2 cartons
The ship travels 0.4 mile per minute.
Divide to find the unit
rate.
÷2
Divide to find the numbe r of cabins.
16 camp ers _____ _____
112 campers _____ 2 cabins = 14 ______ cabins ×7
The camp needs 14
÷50
Animated Math
cabins.
my.hrw.com
Reflect Analyze Relationships In all of these problems, how is the unit rate related to the rate given in the original problem?
3. There are 156 players on 13 teams. How many players are on each players per team
4. A package of 36 photographs costs \$18. What is the cost per
in Harcourt Publishing Company
They are equivalent.
© Houghton Mifflin Harcourt Publishing Company
Reflect
Assessment Resources Tailor assessments to meet the needs of all your classes and students, including • Leveled Module Quizzes • Leveled Unit Tests • Unit Performance Tasks • Placement, Diagnostic, and Quarterly Benchmark Tests
×7
÷50
12
my.hrw.com
ers per cabin.
C A cruise ship travels 20 miles in 50 minutes. 20 miles = ________ 0.4 mile __________ How far does the ship travel per minute? 50 minutes 1 minute
team?
Math On the Spot
There are 8 camp
÷2
Personal
rate or by using equiv alent rates.
÷2
8 camp ers _____ ____ 2 cabins = 1 cabin
\$15 The unit rate _____ 1 class is the same as 15 ÷ 1 = \$15 per class.
Gerald’s yoga classes cost \$15 per class.
B The cost of 2 cartons of milk is \$5.50. What is the unit price?
2.
Prepare students with practice similar to the Texas assessment program at every module and unit.
16 camp ers _____ _____
÷6
\$90 \$15 ________ = ______ ÷6
The first quantity in a unit rate can be less than 1.
Texas Test Prep
6.4.D At a summer camp , the campers are divided into grou 16 campers and ps. Each group has 2 cabins. How many cabins are needed for 112 campers? Method 1 Find the unit rate. How many campers per cabin?
\$90
Use the information in the problem to write a rate: _______ 6 classes
6 classes
Raise the bar with homework and practice that incorporates higher-order thinking and mathematical processes in every lesson.
with Unit Rates
problems by using a unit
EXAMPLE 2
A Gerald pays \$90 for 6 yoga classes. What is the cost per class?
Yoga Classes
Response to Intervention
5.
What If? Suppose each group has 12 campers and 3 canoe unit rate of camp ers to canoes. s. Find the 12
÷ 3 = 4; there are
4 camper per cano
4 camp ers ____ e or ____ . 1 canoe
Petra jogs 3 miles in 27 minutes. At this rate, how long to jog 5 would
miles? 27 minu ____ tes____ ____ ÷3
9 minute __
it take her
Expressions, Equations, and Relationships
263D
Math Background Algebraic Expressions
TEKS 6.7.B, 6.7.C
LESSONS 11.1 to 11.3 An algebraic expression is a mathematical statement constructed from at least one variable. It may have one or more operation symbols and one or more numbers. The table shows examples and nonexamples of algebraic expressions. Algebraic Expressions Examples
x, 3y - __25 , -2xy, z 2 + 1
Nonexamples
7, 20 ÷ (4 + 1), 4x = 16
Note that an algebraic expression does not contain an equal sign. A mathematical statement that contains an equal sign is an equation. Some students may have trouble understanding that algebraic expressions can be represented in multiple equivalent forms, just like numbers and numerical expressions. Students should recognize that the following 5x . algebraic expressions are equivalent: x + 0, x, 1x, __ 5
Writing One-Variable Equations TEKS 6.7.B, 6.9.A
LESSON 12.1 Students must be able to translate English phrases and sentences into algebraic symbols. To avoid misunderstandings, there are conventions we all use when translating from words to math. The phrase “the difference of 3 and 7” translates to 3 - 7. An equation is a mathematical statement that two quantities are equal. An equation may involve only numbers, as in 6 + 5 = 11, or may have algebraic expressions, as in 2x = 6. When an equation contains one variable, the solution of the equation is a value of the variable that makes the equation true. For instance, x = 3 is the solution of 2x = 6 since 2(3) = 6 is a true statement. An equation such as 2x = 6 is sometimes called an open sentence. That is, it is neither true nor false until additional information (i.e., a value of x) is given. When x is replaced by
263E
a value that is a solution, the open sentence becomes a true statement. If the given value of x is not a solution, the open sentence becomes a false statement. Students should know that an equation may have no solutions, one solution, more than one solution, or infinitely many solutions. For example, the equation x + 5 = 2 + x + 3 has infinitely many solutions. Every real number is a solution of this equation. Such an equation is called an identity.
Solving One-Variable Equations TEKS 6.9.A, 6.9.B, 6.9.C, 6.10.A, 6.10.B
LESSONS 12.2 to 12.3 An equation is like a scale that is perfectly balanced. The quantities on both sides have exactly the same weight. When two quantities a and b cause a scale to balance, the same quantity c can be added to both sides of the scale while preserving the balance. Applying this idea to equations yields the Addition Property of Equality: If a = b, then a + c = b + c. Similarly, it is possible to subtract the same quantity c from both sides of the scale and preserve the balance. Applying this idea to equations gives the Subtraction Property of Equality: If a = b, then a – c = b – c. The Multiplication Property of Equality states that multiplying each side of an equation by the same nonzero number produces a new equation that has the same solutions as the original. In other words, if a = b and c ≠ 0, then ac = bc. (Strictly speaking, multiplying both sides by a constant c = 0 results in a true equation, 0 = 0, but this is not useful because we lose whatever information the original equation contained.) The Division Property of Equality states that dividing each side of an equation by the same nonzero number produces a new equation that has the same solutions as the original. That is, if a = b and c ≠ 0, then __ac = __bc .
Writing One-Variable Inequalities TEKS 6.9.A, 6.9.B, 6.9.C
LESSON 13.1 One of the essential skills of algebra is translating words into mathematics. This can be challenging for many students, especially those for whom English is a second language. Perhaps the most common error is the attempt to make a direct word-to-symbol translation that preserves the order of the words. Although this method works in many cases, it can cause problems. Consider the statement “5 less than a number n is greater than 2.” A student making the “word-order” error might translate this incorrectly as 5 - n > 2. The correct translation is n - 5 > 2. Students should realize that they can check their mathematical translations in much the same way that they can check a solution. In the above example, it is helpful to choose a specific value for n, such as 20, and ask whether “5 less than 20” is represented by 5 - 20.
Solving One-Variable Inequalities TEKS 6.9.B, 6.10.A, 6.10.B
LESSONS 13.2 to 13.4
The Addition and Subtraction Properties of Inequality state that the same quantity may be added to or subtracted from both sides of an inequality without changing the solution set. That is, if a > b, then a + c > b + c and a - c > b - c. Multiplying or dividing both sides of an inequality by a positive number also produces an inequality with the same solution set as the original inequality. In general terms, if a > b and c > 0, then ac > bc and __ac = __bc . When multiplying or dividing both sides of an inequality by a negative number, however, the inequality symbol must be reversed. Thus, if a > b and c < 0, then ac < bc and __ac = __bc . It is often helpful for students to check the solution of an inequality by substituting specific values for the variable. To check that n ≥ -10 is the solution of -18n ≤ 180, choose a value of n that is greater than or equal to -10, such as -2. In the original inequality, this gives -18(-2) ≤ 180 or 36 ≤ 180, which is a true inequality. Checking a single value can never guarantee that a solution to an inequality is correct, but it can help students catch some errors.
Solving an inequality in one variable is similar to solving an equation in one variable. The goal is to isolate the variable on one side of the inequality by writing a series of inequalities that have the same solution set.
Expressions, Equations, and Relationships
263F
UNIT 4
Expressions, Equations, and Relationships
MODULE MODULE
10 10
Generating Equivalent Numerical Expressions 6.7.A MODULE MODULE
11 11
Generating Equivalent Algebraic Expressions 6.7.A, 6.7.C, 6.7.D MODULE MODULE
12 12
Equations and Relationships 6.7.B, 6.9, 6.10
13 13 Inequalities and
MODULE MODULE
Relationships 6.9, 6.10
14 14 Relationships in Two
© Houghton Mifflin Harcourt Publishing Company • Image Credits: Andy Sotiriou/Photodisc/Getty Images
MODULE MODULE
Variables
6.6.A, 6.6.B, 6.6.C, 6.11
CAREERS IN MATH
Unit 4 Performance Task At the end of the unit, check out how botanists use math.
Botanist A botanist is a biologist who studies plants. Botanists use math to analyze data and create models of biological organisms and systems. They use these models to make predictions. They also use statistics to determine correlations. If you are interested in a career in botany, you should study these mathematical subjects: • Algebra • Trigonometry • Probability and Statistics • Calculus Research other careers that require the analysis of data and use of mathematical models.
Unit 4
263
UNIT 4
Careers in Math
Vocabulary
Botanist A botanist uses math to find correlations and to predict future results. You will learn more about using math to make predictions in the Performance Tasks at the end of the unit. For more information about careers in mathematics as well as various mathematics appreciation topics, visit the American Mathematical Society at www.ams.org
Preview
Use the puzzle to preview key vocabulary from this unit. Unscramble the circled letters within found words to answer the riddle at the bottom of the page.
T N E I C I F F E O C Z U S L
Vocabulary Preview Integrating the ELPS Use the puzzle to give students a preview of important concepts in this unit. Students may work individually, in pairs, or in groups. c.4.D Use prereading supports such as graphic organizers, illustrations, and pretaught topic-related vocabulary to enhance comprehension of written text.
O E K O F B S D E O R H B B F
Y S O F P N O X X W U X X F W
H S P J Z U X V P S T R F Q W
S V X S O A A X O S L H U O Z
D L U F E M F L N H O O B E T
F E C P E T U D E Z O P Y K V
P B D M U T A L N W P L H Y F
P C R Y I Z D N T T R Y J K O
O E A O N O C V I Y S C F P P
T R N Q V Q P J O D S U K H U
H V I H Z C Z S Q Q R X P C H
T N U G W C H A X E S O U S U
• A number that is multiplied by a variable in an algebraic expression. (Lesson 11-1) • A value of the variable that makes the equation true. (Lesson 12-1)
J N P C I J L V W B J Q O N U
J F N I C N P K B T A C E C B
coefficient
solution
• The point where the axes intersect to form the coordinate plane. (Lesson 14-1) origin • The part of an expression that is added or subtracted. (Lesson 11-1) term • The two number lines that intersect at right angles to form a coordinate plane. (Lesson 14-1) axes • Tells how many times the base is used in the product. (Lesson 10-1) exponent
Q: Unit Resources
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Before Students understand: • operations with whole numbers, decimals, and fractions • order of operations • properties of operations: inverse, identity, commutative, associative, and distributive properties • graphs in the first quadrant
Go online to access all your unit resources.
In this Unit Students will learn about: • exponents • prime factorizations • numerical and algebraic expressions • equations and inequalities • the coordinate plane
Why did the paper rip when the student tried to stretch out the horizontal axis of his graph?
A: 264
© Houghton Mifflin Harcourt Publishing Company
• The numbers in an ordered pair. (Lesson 14-1) coordinates
Too much X – T
E N S
I
O N!
Vocabulary Preview
After Students will learn how to: • evaluate algebraic expressions with more than one variable • write two-step equations and inequalities to represent real-world problems and write a real-world problem to represent an equation or inequality • solve two-step equations and inequalities • graph linear equations in the form y = mx + b on the coordinate plane
Expressions, Equations, and Relationships
264
Generating Equivalent Numerical Expressions How can you generate equivalent numerical expressions and use them to solve real-world problems?
© Houghton Mifflin Harcourt Publishing Company • Image Credits: Vladimir Ivanovich Danilov / Shutterstock.com
?
ESSENTIAL QUESTION
Module 10
10
LESSON 10.1
Exponents 6.7.A
You can represent real-world problems with numerical expressions and simplify the expressions by applying rules relating to exponents, prime factorization, and order of operations.
LESSON 10.2
Prime Factorization 6.7.A
LESSON 10.3
Order of Operations 6.7.A
Real-World Video
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MODULE
Assume that you post a video on the internet. Two of your friends view it, then two friends of each of those view it, and so on. The number of views is growing exponentially. Sometimes we say the video went viral.
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Math On the Spot
Animated Math
Personal Math Trainer
Go digital with your write-in student edition, accessible on any device.
Scan with your smart phone to jump directly to the online edition, video tutor, and more.
Interactively explore key concepts to see how math works.
Get immediate feedback and help as you work through practice sets.
265
DO NOT EDIT--Changes must be made through "File info" CorrectionKey=B
Complete these exercises to review skills you will need for this chapter.
Use the assessment on this page to determine if students need intensive or strategic intervention for the module’s prerequisite skills.
Whole Number Operations
2 1
Response to Intervention
1. 992 × 16
Enrichment
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2. 578 × 27
3 × 270 80 × 270 (3 × 270) + (80 × 270)
3. 839 × 65
15,606
15,872
Access Are You Ready? assessment online, and receive instant scoring, feedback, and customized intervention or enrichment.
Online Assessment and Intervention
← ← ←
Find the product.
Intervention
Personal Math Trainer
270 × 83 810 + 21,600 22,410
Online Assessment and Intervention
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4. 367 × 23
54,535
8,441
Use Repeated Multiplication EXAMPLE
5×5× 5× 5
↓
25 × 5
Online and Print Resources
Multiply the first two factors.
Multiply the result by the next factor.
Multiply that result by the next factor.
125 × 5
Skills Intervention worksheets
Differentiated Instruction
• Skill 34 Whole Number Operations
• Challenge worksheets
• Skill 35 Use Repeated Multiplication
Extend the Math PRE-AP Lesson Activities in TE
Continue until there are no more factors to multiply.
625 Find the product.
PRE-AP
5. 7×7×7
6. 3×3×3×3
343
7. 6×6×6×6×6
7,776
81
8. 2×2×2×2×2×2
64
© Houghton Mifflin Harcourt Publishing Company
3
270 × 83
EXAMPLE
Personal Math Trainer
Division Facts
• Skill 38 Division Facts
EXAMPLE
54 ÷ 9 =
Think:
54 ÷ 9 = 6
So, 54 ÷ 9 = 6.
9 times what number equals 54? 9 × 6 = 54
Divide. 9. 20 ÷ 4
5
266
10. 21 ÷ 7
3
11. 42 ÷ 7
6
12. 56 ÷ 8
7
Unit 4
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PROFESSIONAL DEVELOPMENT VIDEO my.hrw.com
Author Juli Dixon models successful teaching practices as she explores equivalent numerical expressions in an actual sixth-grade classroom.
Online Teacher Edition Access a full suite of teaching resources online—plan, present, and manage classes and assignments.
Professional Development
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Interactive Answers and Solutions Customize answer keys to print or display in the classroom. Choose to include answers only or full solutions to all lesson exercises.
Interactive Whiteboards Engage students with interactive whiteboard-ready lessons and activities. Personal Math Trainer: Online Assessment and Intervention Assign automatically graded homework, quizzes, tests, and intervention activities. Prepare your students with updated, TEKS-aligned practice tests.
Generating Equivalent Numerical Expressions
266
Have students complete the activities on this page by working alone or with others.
Vocabulary Review Words ✔ factor (factor) factor tree (árbol de factores) ✔ integers (entero) ✔ numerical expression (expresión numérica) ✔ operations (operaciones) ✔ prime factorization (factorización prima) repeated multiplication (multiplicación repetida) simplified expression (expresión simplificada)
Visualize Vocabulary Use the ✔ words to complete the graphic. You may put more than one word in each box.
Visualize Vocabulary
Reviewing Factorization
The sequence diagram helps students review vocabulary associated with factorization and prepares them to work with exponents. After students complete the diagram, discuss the vocabulary as a class.
Factor tree
24
factor, prime factorization
integer
Understand Vocabulary Use the following explanation to help students learn the preview words. You may hear the terms power and exponent used in place of each other. However, they do not mean the same thing. An exponent is the number that is written beside and slightly above the base. It tells you how many times to use the base as a factor. A power is a number that is formed by repeated multiplication by the same factor (the base) and can be represented as the base with an exponent.
Integrating the ELPS Students can use these reading and note-taking strategies to help them organize and understand new concepts and vocabulary. c.4.D Use prereading supports such as graphic organizers, illustrations, and
pretaught topic-related vocabulary to enhance comprehension of written text.
8×3
integers, factors, operations, numerical expression
Preview Words base (base) exponent (exponente) order of operations (orden de las operaciones) power (potencia)
Understand Vocabulary Complete the sentences using the preview words.
1. A number that is formed by repeated multiplication by the same
power
factor is a
2. A rule for simplifying expressions is © Houghton Mifflin Harcourt Publishing Company
2×2×2×3
integers, factors, operations, numerical expression, prime factorization
3. The
base
.
order of operations
.
is a number that is multiplied. The number that
indicates how many times this number is used as a factor is the
exponent
.
Active Reading Three-Panel Flip Chart Before beginning the module, create a three-panel flip chart to help you organize what you learn. Label each flap with one of the lesson titles from this module. As you study each lesson, write important ideas like vocabulary, properties, and formulas under the appropriate flap.
Differentiated Instruction • Reading Strategies ELL Module 10
Grades 6–8 TEKS Before Students understand: • operations with whole numbers, decimals, and fractions • prime numbers • order of operations
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Module 10
In this module Students will learn to: • generate equivalent numerical expressions using exponents • generate equivalent numerical expressions using prime factorization • simplify numerical expressions using the order of operations
After Students will connect: • order of operations and numerical expressions • numerical and algebraic expressions
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MODULE 10
Unpacking the TEKS
Unpacking the TEKS Understanding the TEKS and the vocabulary terms in the TEKS will help you know exactly what you are expected to learn in this module.
6.7.A Generate equivalent numerical expressions using order of operations, including whole number exponents and prime factorization.
Texas Essential Knowledge and Skills Content Focal Areas
Key Vocabulary exponent (exponente) The number that indicates how many times the base is used as a factor. order of operations (orden de las operaciones) A rule for evaluating expressions: first perform the operations in parentheses, then compute powers and roots, then perform all multiplication and division from left to right, and then perform all addition and subtraction from left to right.
Expressions, equations, and relationships—6.7 The student applies mathematical process standards to develop concepts of expressions and equations.
Integrating the ELPS c.4.F Use visual and contextual support … to read grade-appropriate content area text … and develop vocabulary … to comprehend increasingly challenging language.
What It Means to You You will simplify numerical expressions using the order of operations. UNPACKING EXAMPLE 6.7.A
Ellen is playing a video game in which she captures frogs. There were 3 frogs onscreen, but the number of frogs doubled every minute when she went to get a snack. She returned after 4 minutes and captured 7 frogs. Write an expression for the number of frogs remaining. Simplify the expression. 3×2
number of frogs after 1 minute
3×2×2
number of frogs after 2 minutes
3×2×2×2
number of frogs after 3 minutes
3×2×2×2×2
number of frogs after 4 minutes
Since 3 and 2 are prime numbers, 3 × 2 × 2 × 2 × 2 is the prime factorization of the number of frogs remaining. 3 × 2 × 2 × 2 × 2 can be written with exponents as 3 × 24. The expression 3 × 24 – 7 is the number of frogs remaining after Ellen captured the 7 frogs. Use the order of operations to simplify 3 × 24 – 7.
Go online to see a complete unpacking of the .
3 × 24 – 7 = 3 × 16 – 7 = 48 – 7 = 41 41 frogs remain.
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© Houghton Mifflin Harcourt Publishing Company • Image Credits: Patrik Giardino/ Photodisc/Getty Images
Use the examples on this page to help students know exactly what they are expected to learn in this module.
Visit my.hrw.com to see all the unpacked. my.hrw.com
268
Lesson 10.1
Lesson 10.2
Unit 4
Lesson 10.3
6.7.A Generate equivalent numerical expressions using order of operations, including whole number exponents and prime factorization.
Generating Equivalent Numerical Expressions
268
LESSON
10.1 Exponents Texas Essential Knowledge and Skills The student is expected to: Expressions, equations, and relationships—6.7.A Generate equivalent numerical expressions using order of operations, including whole number exponents and prime factorization.
Mathematical Processes
Engage ESSENTIAL QUESTION How do you use exponents to represent numbers? Sample answer: You can use exponents to represent repeated multiplication. For example, in 5 × 5 × 5 × 5, the number 5 is multiplied 4 times, so you can represent it as 5 4.
Motivate the Lesson Ask: Have you ever heard the terms squared or cubed? Both of those expressions are used to describe exponents. Do you know what 3 squared means? Take a guess. Begin the Explore Activity to find out.
6.1.D Communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate.
Explore EXPLORE ACTIVITY Engage with the Whiteboard Have students fill in the table on the whiteboard. Extend the table to include 8 hours, and have students fill in the table for 5, 6, 7, and 8 hours. Ask students if they could predict the total number of bacteria for hour 12 and hour 20. Discuss a rule that students could use to make those kinds of predictions.
Explain ADDITIONAL EXAMPLE 1 Use an exponent to write each expression. A 7 × 7 × 7 × 7 × 7 × 7 76 5 B __23 × __23 × __23 × __23 × __23 ( __23 )
Interactive Whiteboard Interactive example available online my.hrw.com
EXAMPLE 1 c.1.F ELL Students may know other definitions of the words raised and base. Point out that in math, raised means “to multiply by itself,” and base can mean “the foundation,” or that on which something is built. In 24, you build on the base 2 by multiplying it by itself 4 times.
Connect Vocabulary
Questioning Strategies
Mathematical Processes
4 • In B, why is the base __4 in parentheses in the power ( __4 ) ? The base is in parentheses to
5
5
show that the entire fraction is used for repeated multiplication, not just the numerator.
• Can a power have a base and an exponent that are the same number? Justify your answer. Yes; for example, 6 × 6 × 6 × 6 × 6 × 6 = 6 6.
YOUR TURN Avoid Common Errors Students may want to find the product for each expression. Review the direction line. They are not asked to simplify the expression, but to write it in exponential form.
Talk About It Check for Understanding Ask: What is the difference between the two numbers in a power? The first number is the base, which is the number that is multiplied. The second number is the exponent, which tells how many times the base is multiplied by itself.
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Lesson 10.1
LESSON
Expressions, equations, and relationships—6.7.A Generate equivalent numerical expressions using… exponents.
10.1 Exponents
A number that is formed by repeated multiplication by the same factor is called a power. You can use an exponent and a base to write a power. For example, 73 means the product of three 7s:
Math On the Spot
73 = 7 × 7 × 7
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ESSENTIAL QUESTION How do you use exponents to represent numbers?
The base is the number that is multiplied. 6.7.A
EXPLORE ACTIVITY
Power
Identifying Repeated Multiplication A real-world problem may involve repeatedly multiplying a factor by itself. A scientist observed the hourly growth of bacteria and recorded his observations in a table.
© Houghton Mifflin Harcourt Publishing Company • Image Credits: D. Hurst/Alamy
Time (h) 0 1
6 squared, 6 to the power of 2, 6 raised to the 2nd power
73
7 cubed, 7 to the power of 3, 7 raised to the 3rd power
94
9 to the power of 4, 9 raised to 4th power
6.7.A
Use an exponent to write each expression. After 2 hours, there are 2 · 2 = ? bacteria.
2
2× 2=
3
2× 2× 2=
4
2× 2× 2× 2=
4
A 3×3×3×3×3
Math Talk
Find the base, or the number being multiplied. The base is 3.
Mathematical Processes
What is the value of a number raised to the power of 1?
8 16
When a number is raised to the power of 1, the value of the power is equal to the base.
A Complete the table. What pattern(s) do you see in the Total bacteria column?
Sample answer: Each number is 2 times the previous number. B Complete each statement. At 2 hours, the total is equal to the product of two 2s. At 3 hours, the total is equal to the product of
three
2s.
At 4 hours, the total is equal to the product of
four
2s.
Find the exponent by counting the number of 3s being multiplied. The exponent is 5. 3 × 3 × 3 × 3 × 3 = 35 5 factors of 3
B
4 _ _ × 45 × _45 × _45 5
Find the base, or the number being multiplied. The base is _45. Find the exponent by counting the number of times _45 appears in the expression. The exponent is 4.
( )
4 4 _ _ × 45 × _45 × _45 = _45 5 4 __
4 factors of
5
Reflect 1.
62
EXAMPLE 1
Total bacteria 1 2
The exponent tells how many times the base appears in the expression.
© Houghton Mifflin Harcourt Publishing Company
?
Using Exponents
Use exponents to write each expression.
Communicate Mathematical Ideas How is the time, in hours, related to the number of times 2 is used as a factor?
Personal Math Trainer
The number of hours is the number of times the factor
Online Assessment and Intervention
2 is repeated.
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270
2. 4 × 4 × 4 4. _18 × _18
43
( )
1 2 _ 8
3. 6
61
5. 5 × 5 × 5 × 5 × 5 × 5
56
Unit 4
PROFESSIONAL DEVELOPMENT Integrate Mathematical Processes This lesson provides an opportunity to address Mathematical Process TEKS 6.1.D, which calls for students to “communicate mathematical ideas, …using multiple representations, including symbols, diagrams, graphs, and language as appropriate.” Students first use tables to identify patterns involving repeated multiplication. They then use exponents to rewrite expressions that involve repeated multiplication. Finally, students find the value of expressions that are written with exponents. This process helps students understand multiple ways to represent and use exponents.
Math Background The use of the terms squared and cubed is directly related to the measurements of area and volume. The area of a square with sides 5 units long is found by multiplying, 5 × 5, or 52, or 5 squared. The volume of a cube with sides 5 units long is found by multiplying, 5 × 5 × 5, or 53, or 5 cubed.
Exponents
270
EXAMPLE 2
Find the value of each power.
Avoid Common Errors
A 35 243
Since powers relate to multiplication, students may confuse powers with simple multiplication. After students evaluate each power, have them compare it to a simple multiplication problem to show that the two are not equal. For example, they find that 104 = 10,000, ask them to find 10 × 4 = 40. Since the two expressions have different answers, it should be clear that 104 and 10 × 4 are not equivalent.
0
B 17 1 C (-4)3 -64 D 0.62 0.36 Interactive Whiteboard Interactive example available online my.hrw.com
Questioning Strategies
Mathematical Processes • If a > b, which is greater: 2a or 2b ? Explain. 2a; If the exponent a is a bigger number than the exponent b, students find then the base 2 is used as a factor more times.
• If c > d, which is greater: c3 or d3? Explain. c3; The exponent with the greater base has to be greater if the exponent is the same. For example: 53 = 5 × 5 × 5 = 125, while 43 = 4 × 4 × 4 = 64. c.4.E ELL Encourage English learners to use the active reading strategies and the illustrated, bilingual glossary as they encounter new terms and concepts.
Integrating the ELPS
YOUR TURN Engage with the Whiteboard Have students rewrite each power as repeated multiplication. Seeing the power expressed as repeated multiplication can make it easier for students to find the correct value.
Elaborate Talk About It Summarize the Lesson Ask: How can you use an exponent to represent repeated multiplication? How can you find the value of a power? You can write a repeated multiplication, such as 23 = 2 × 2 × 2. To find the value of a power, rewrite the expression without using exponents by multiplying the base the number of times shown in the exponent—for example, 54 = 5 × 5 × 5 × 5 = 625.
GUIDED PRACTICE Engage with the Whiteboard For Exercise 1, have students complete the table on the whiteboard. Discuss different methods students may have for finding the value of each power, such as using parentheses to group the repeated multiplication or multiplying the value of the previous power by 5.
Avoid Common Errors Exercises 2–5 Remind students that their answers should be expressed as a power, not as the product of a repeated multiplication. Exercises 6–20 Some students may multiply a base by its exponent instead of using the base as a factor the number of times indicated by the exponent. Remind them that 43 means that 4 is used as a factor 3 times (4 × 4 × 4). Exercise 15 If students get the answer 8, remind them that the Property of Zero as an Exponent states that the value of any nonzero number raised to the power of 0 is 1.
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DO NOT EDIT--Changes must be made through “File info” CorrectionKey=A
DO NOT EDIT--Changes must be made through “File info” CorrectionKey=B
Guided Practice
Finding the Value of a Power
1. Complete the table. (Explore Activity)
To find the value of a power, remember that the exponent indicates how many times to use the base as a factor.
Exponential form Math On the Spot
Property of Zero as an Exponent The value of any nonzero number raised to the power of 0 is 1.
52
5×5
25
Example: 50 = 1
53
5×5×5
125
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6.7.A
Find the value of each power.
My Notes
B 0.43
625 3,125
Use an exponent to write each expression. (Example 1)
63
107
3. 10 × 10 × 10 × 10 × 10 × 10 × 10
( _34 )5
( _79 )8
5. _79 × _79 × _79 × _79 × _79 × _79 × _79 × _79
Find the value of each power. (Example 2)
Evaluate: 0.4 = 0.4 × 0.4 × 0.4 = 0.064
2
1 __ 16
12. 0.82
0.64
9.
( )
3 0 _ 5
Identify the base and the exponent. The base is _35, and the exponent is 0.
Math Talk
Mathematical Processes
Evaluate.
3
=1
Is the value of 2 the same as the value of 32? Explain.
Any number raised to the power of 0 is 1.
D -112
23 = 2 · 2 · 2 = 8 and 32 = 3 · 3 = 9, so the values are not equivalent.
Identify the base and the exponent. The base is 11, and the exponent is 2. Evaluate. -112 = -(11 × 11) = -121
512
6. 83
3
The opposite of a positive number squared is a negative number.
( _14 )
15. 80 18.
? ?
( -2 )3
1 -8
2,401
7. 74 10.
( _13 )
1 __ 27
3
13. 0.53
0.125
16. 121
12
19.
( - _25 )
2
4 __ 25
8. 103 11.
( _67 )
2
14. 1.12 17.
( _12 )
0
20. -92
1,000 36 __ 49
1.21 1 -81
ESSENTIAL QUESTION CHECK-IN
21. How do you use an exponent to represent a number such as 16?
You use an exponent to write a number that can be written as a product of equal factors. 16 = 4 × 4 (or 2 × 2 × 2 × 2), so it can be written as 42 (or 24).
© Houghton Mifflin Harcourt Publishing Company
Identify the base and the exponent. The base is 0.4, and the exponent is 3.
0
5×5×5×5×5
4. _34 × _34 × _34 × _34 × _34
Evaluate: (-10)4 = -10 × (-10) × (-10) × (-10) = 10,000
( _35 )
5×5×5×5
55
3 factors of 6
Identify the base and the exponent. The base is -10, and the exponent is 4.
5
54
2. 6 × 6 × 6
A (-10)4
© Houghton Mifflin Harcourt Publishing Company
Simplified product
5
EXAMPL 2 EXAMPLE
C
Product
51
YOUR TURN Find the value of each power. 6. 34
81
7. (-1)9
-1
8.
( _25 )
3
8 ___ 125
Personal Math Trainer
9. -122-144
Online Assessment and Intervention
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Lesson 10.1
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22/10/12 10:51 PM
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Unit 4
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DIFFERENTIATE INSTRUCTION Kinesthetic Experience
Critical Thinking
To help students remember the meaning of the base and the exponent in a power, have them use graph paper or square tiles to construct models of the squares of whole numbers 1–10. Label the models as shown below. A visual representation of a square number can help students remember that exponents represent repeated multiplication of the same factor.
Have students explore multiplication of numbers written as powers. Partners can work together to find the values of pairs of expressions such as these:
Differentiated Instruction includes: • Reading Strategies • Success for English Learners ELL • Reteach • Challenge PRE-AP
32 = 3 × 3
32 · 33 and 31 · 34 22 · 24 and 23 · 23 43 · 43 and 41 · 45
243 and 243; 64 and 64; 4,096 and 4,096; The values of both expressions in each pair are the same. The base is used the same number of times in each pair. The exponents in each pair have equal sums. You can add the exponents to multiply powers with the same base, for example, 2 2 · 24 = 26. Exponents
272
Personal Math Trainer Online Assessment and Intervention
Online homework assignment available
Evaluate GUIDED AND INDEPENDENT PRACTICE 6.7.A
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10.1 LESSON QUIZ 6.7.A Use exponents to write each expression. 1. __37 × __37 × __37
Concepts & Skills
Practice
Explore Activity Identifying Repeated Multiplication
Exercises 1, 38–44
Example 1 Using Exponents
Exercises 2–5, 22–37
Example 2 Finding the Value of a Power
Exercises 6–20, 38–44
2. 0.9 × 0.9 × 0.9 × 0.9 Find the value of each power.
Exercise
3. 74
Depth of Knowledge (D.O.K.)
Mathematical Processes
3 4. ( -__34 )
22–37
2 Skills/Concepts
1.C Select tools
Lesson Quiz available online
38–39
2 Skills/Concepts
1.A Everyday life
40
2 Skills/Concepts
1.D Multiple representations
41–42
2 Skills/Concepts
1.A Everyday life
43
3 Strategic Thinking
1.F Analyze relationships
44
2 Skills/Concepts
1.D Multiple representations
45
3 Strategic Thinking
1.G Explain and justify arguments
46–47
3 Strategic Thinking
1.F Analyze relationships
48
3 Strategic Thinking
1.G Explain and justify arguments
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Answers 3 1. ( __37 )
2. 0.94 3. 2,401 27 4. -__ 64
Additional Resources Differentiated Instruction includes: • Leveled Practice Worksheets
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Lesson 10.1
Name
Class
Date
10.1 Independent Practice
Personal Math Trainer
6.7.A
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Online Assessment and Intervention
Write the missing exponent.
2
22. 100 = 10
( )
1 = ___ 1 26. ____ 169 13
23. 8 = 2
2
3
24. 25 = 5
1
27. 14 = 14
28. 32 = 2
2
25. 27 = 3
5
1= 34. __ 9
(
1 _ 3
3
10
)
31. 256 =
4
4
()
64 = __ 8 29. ___ 81 9
2
35. 64 =
2
9 = 36. ___ 16
8
4
32. 16 =
33. 9 =
2
(
3 _ 4
)
Sample answer: 0.32 = 0.09; 0.3 > 0.09 44. Which power can you write to represent the volume of the cube shown? Write the power as an expression with a base and an exponent, and then find the volume of the cube.
1 in.3 ( _13 )3 = _13 × _13 × _13 = __ 27
3
FOCUS ON HIGHER ORDER THINKING
2
2
Work Area
The value of 1 raised to any power is 1. 1 multiplied by
3
2
37. 729 =
1 in. 3
45. Communicate Mathematical Ideas What is the value of 1 raised to the power of any exponent? What is the value of 0 raised to the power of any nonzero exponent? Explain.
Write the missing base. 30. 1,000 =
43. Write a power represented with a positive base and a positive exponent whose value is less than the base.
itself any number of times is 1. The value of 0 raised
3
9
to any power is 0. 0 multiplied by itself any number of times is still 0.
38. Hadley’s softball team has a phone tree in case a game is canceled. The coach calls 3 players. Then each of those players calls 3 players, and so on. How many players will be notified during the third round of calls?
46. Look for a Pattern Find the values of the powers in the following pattern: 101, 102, 103, 104… . Describe the pattern, and use it to evaluate 106 without using multiplication.
27 players
Sample answer: 10; 100; 1,000; 10,000… . Each term in
39. Tim is reading a book. On Monday he reads 3 pages. On each day after that, he reads triple the number of pages as the previous day. How many pages does he read on Thursday?
the pattern is a 1 followed by the same number of zeros as the exponent. 106 = 1,000,000
4
3 pages, or 81 pages 40. Which power can you write to represent the area of the square shown? Write the power as an expression with a base and an exponent, and then find the area of the square.
8.5 = 8.5 × 8.5 = 72.25 mm 2
47. Critical Thinking Some numbers can be written as powers of different bases. For example, 81 = 92 and 81 = 34. Write the number 64 using three different bases.
2
26, 43, and 82
41. Antonia is saving for a video game. On the first day, she saves two dollars in her piggy bank. Each day after that, she doubles the number of dollars she saved on the previous day. How many dollars does she save on the sixth day?
48. Justify Reasoning Oman said that it is impossible to raise a number to the power of 2 and get a negative value. Do you agree with Oman? Why or why not?
26 dollars, or \$64
Sample answer: Agree; because the product of two numbers with the same sign is always positive, and 0
42. A certain colony of bacteria triples in length every 10 minutes. Its length is now 1 millimeter. How long will it be in 40 minutes?
raised to the power of 2 is 0.
34 mm, or 81 mm
Lesson 10.1
EXTEND THE MATH
© Houghton Mifflin Harcourt Publishing Company
© Houghton Mifflin Harcourt Publishing Company
8.5 mm
PRE-AP
Activity Every integer can be written as the sum of square numbers. For example: Sum of 2 squares: 20 = 42 + 22 Sum of 3 squares: 24 = 42 + 22 + 22 Some integers, such as 22, can be written as the sum of square numbers more than one way. Sum of 3 squares: 22 = 32 + 32 + 22 Sum of 4 squares: 22 = 42 + 22 + 12 + 12
273
Activity available online
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Unit 4
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• Can you write the number 36 as the sum of squares in more than one way? Yes; 36 = 32 + 32 + 32 + 32 = 9 + 9 + 9 + 9; 36 = 42 + 42 + 22 = 16 + 16 + 4 • Write a number on one side of an index card and on the reverse write the number as a sum of squares. Challenge a classmate to write the number. For example: Write 62 as the sum of 3 squares.
12 + 52 + 62
• Write the integers 8, 13, and 18 as the sum of 2 squares. 8 = 22 + 22; 13 = 32 + 22; 18 = 32 + 32
Exponents
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LESSON
10.2 Prime Factorization Texas Essential Knowledge and Skills The student is expected to: Expressions, equations, and relationships—6.7.A Generate equivalent numerical expressions using order of operations, including whole number exponents and prime factorization.
Mathematical Processes
Engage
ESSENTIAL QUESTION How do you write the prime factorization of a number? Sample answer: Use a factor tree or a ladder diagram to find the prime factorization of the number, then write the prime factorization using exponents.
Explore Motivate the Lesson
6.1.E Create and use representations to organize, record, and communicate mathematical ideas.
Ask students to name two numbers that can be multiplied to get a specific product. For example, you might ask them to name two numbers that can be multiplied to get 28 (1 and 28; 2 and 14; 4 and 7). Repeat the process using 36.
Explain ADDITIONAL EXAMPLE 1 Rayshawn is designing a mural. The mural must have an area of 42 square yards. What are the possible whole number lengths and widths for the mural? The possible lengths and widths are listed: Length (yd)
42
21
14
7
Width (yd)
1
2
3
6
EXAMPLE 1 Focus on Reasoning
Mathematical Processes Point out to students that you can tell when you have found all the factors of a number when the factor pairs start to repeat.
Questioning Strategies
Mathematical Processes • For any number, which numbers are always factors? 1 and the number itself. • Is it possible for a number to have all even factors? No; 1 is a factor for all numbers.
YOUR TURN Interactive Whiteboard Interactive example available online my.hrw.com
Avoid Common Errors Remind students that when they list the factors of a number they should always begin with 1 and end with the number itself.
EXPLORE ACTIVITY 1 Animated Math Prime Factorization Students use an interactive factor tree to find prime factors of composite numbers. my.hrw.com
Connect Vocabulary
ELL
Remind students that a prime number is a number with exactly 2 factors, 1 and itself, and a composite number is a number that has more than 2 factors.
Engage with the Whiteboard Have students make alternate factor trees for 240 on the whiteboard, next to the given factor tree. Have students start with the following pairs: 8 and 30; 24 and 10; and 12 and 20. Point out to students that while the order of the factors in a factor tree may differ, the prime factors of a number are always the same.
Questioning Strategies
Mathematical Processes • When choosing the first factor pair for the branches of the factor tree for 240, does one of the factors have to be a prime number? Explain. No. A factor tree can start with any factor pair.
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Lesson 10.2
LESSON
Expressions, equations, and relationships— 6.7.A Generate equivalent numerical expressions using prime factorization.
10.2 Prime Factorization ESSENTIAL QUESTION
The prime factorization of 12 is 2 · 3 · 2 or 22 · 3.
Finding Factors of a Number
8 · 30, 10 · 24, 12 · 20, 15 · 16
Recall that area = length · width. For Ana’s garden, 24 ft2 = length · width.
3
2 · 3
2·2·2·2·3·5 Then write the prime factorization using exponents.
24 · 3 · 5
4
6
Math Talk
8 12 24
Mathematical Processes
Give an example of a whole number that has exactly two factors? What type of number has exactly two factors?
The factors of 24 are 1, 2, 3, 4, 6, 8, 12, 24. STEP 3
2 · 6
D Write the prime factorization of 240.
4 · 6 = 6 · 4, so you only list 4 · 6.
You can also use a diagram to show the factor pairs.
2
2 · 12
C Continue adding branches until the factors at the ends of the branches are prime numbers.
List the factors of 24 in pairs. List each pair only once.
1
240
5 · 48 2 · 24
B Choose any factor pair to begin the tree. If a number in this pair is prime, circle it. If a number in the pair can be written as a product of two factors, draw additional branches and write the factors.
Ana wants to build a rectangular garden with an area of 24 square feet. What are the possible whole number lengths and widths of the garden?
© Houghton Mifflin Harcourt Publishing Company • Image Credits: Brand X Pictures/ Getty Images
1 · 240, 2 · 120, 3 · 80, 4 · 60, 5 · 48, 6 · 40,
Math On the Spot
6.7.A
24 = 1 · 24 24 = 2 · 12 24 = 3 · 8 24 = 4 · 6
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Use exponents to show repeated factors.
A List the factor pairs of 240. my.hrw.com
EXAMPL 1 EXAMPLE
Animated Math
Use a factor tree to find the prime factorization of 240.
Whole numbers that are multiplied to find a product are called factors of that product. A number is divisible by its factors. For example, 4 and 2 are factors of 8 because 4 · 2 = 8, and 8 is divisible by 4 and by 2.
STEP 2
Finding the Prime Factorization of a Number The prime factorization of a number is the number written as the product of its prime factors. For example, the prime factors of 12 are 3, 2, and 2.
How do you write the prime factorization of a number?
STEP 1
6.7.A
The possible lengths and widths are: Length (ft)
24
12
8
6
Width (ft)
1
2
3
4
Reflect 5.
What If? What will the factor tree for 240 look like if you start the tree with a different factor pair? Check your prediction by creating another factor tree for 240 that starts with a different factor pair.
Sample answer: The intermediate steps on Sample answer: 13; its factors are 1 and 13; prime number
the factor tree will be different but the final prime factorization will be the same.
240
2 · 120 2 · 60 3 · 20 4 · 5 2 · 2
© Houghton Mifflin Harcourt Publishing Company
?
EXPLORE ACTIVITY 1
YOUR TURN List all the factors of each number. 1. 21 3. 42
1, 3, 7, 21 1, 2, 3, 6, 7, 14, 21, 42
Personal Math Trainer
2. 37
1, 37
4. 30
1, 2, 3, 5, 6, 10, 15, 30
Online Assessment and Intervention
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Lesson 10.2
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PROFESSIONAL DEVELOPMENT Integrate Mathematical Processes This lesson provides an opportunity to address Mathematical Process TEKS 6.1.E, which calls for students to “create and use representations to organize, record, and communicate mathematical ideas.” Students use diagrams and factor trees to organize factor pairs of a number to find prime factorizations. They also use ladder diagrams to find prime factorizations, with the “ladder” as the means of recording and communicating the prime factorization.
Math Background Ancient Greeks started to study prime numbers circa 300 B.C.E. They observed that there were an infinite number of prime numbers and that there were irregular gaps between successive prime numbers. In 1984, Samuel Yates coined the term titanic prime. He used this term to refer to any prime number with 1,000 digits or more. When he first defined a titanic prime, only 110 of them were known. Today more than 110,000 titanic primes have been identified.
Prime Factorization
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EXPLORE ACTIVITY 2 Engage with the Whiteboard Have students complete the ladder diagram starting with different combinations of prime factors. Point out to students that while the order in which they used the prime factors may differ, the prime factors of a number are always the same.
Focus on Modeling Mathematical Processes Students who find mental math difficult may find ladder diagrams to be challenging. Model other methods for dividing by 2, such as using long division or a calculator, to show that the ladder diagram is a useful organizational tool. Emphasize that they can use a combination of division methods when using ladder diagrams. Questioning Strategies
Mathematical Processes • How do you know when 2 is a factor of a number? How do you know when 2 is not a factor of a number? Even numbers have 2 as a factor. Odd numbers do not.
• Why do you have to divide by prime numbers when using the ladder diagram? The divisors on the left show the prime factorization, so all of them must be prime numbers.
Mathematical Processes Discuss with students ways to check that 2 · 3 · 3 · 3 is the prime factorization of 54. Students should understand that they can check their work two ways: by making sure that every number in the prime factorization is prime and by multiplying the expression 2 · 3 · 3 · 3 to verify that the product is 54.
Elaborate Talk About It Summarize the Lesson Ask: What is the prime factorization of a number, and how can you find the prime factorization of a number? The prime factorization of a number is an expression that shows the number as the product of its prime factors. You can use a factor tree or a ladder diagram to find the prime factorization of a number.
GUIDED PRACTICE Engage with the Whiteboard For Exercises 1–2, have students draw a diagram to list the factors of each number on the whiteboard. For Exercise 5, have students make several different factor trees on the whiteboard.
Avoid Common Errors Exercise 3 Remind students that you can tell when you have found all the factors of a number when the factor pairs start to repeat and that writing factor pairs in order makes it easier to check that all the factor pairs are listed. Exercises 4–7 Remind students that they can check their work by multiplying their answer for the prime factorization to make sure the product is the original number.
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Lesson 10.2
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Guided Practice
6.7.A
Use a diagram to list the factor pairs of each number. (Example 1)
1. 18
A ladder diagram is another way to find the prime factorization of a number.
1
2
Use a ladder diagram to find the prime factorization of 132.
A Write 132 in the top “step” of the ladder. Choose a prime factor of 132 to write next to the step with 132. Choose 2. Divide 132 by 2 and write the quotient 66 in the next step of the ladder. B Now choose a prime factor of 66. Write the prime factor next to the step with 66. Divide 66 by that prime factor and write the quotient in the next step of the ladder.
2
4
13 26 52
1, 2, 4, 13, 26, 52
Length (ft)
72
36
24
18
12
9
Width (ft)
1
2
3
4
6
8
402
4. 402
201
E Write the prime factorization of 132 using exponents.
36
5. 36
3 · 12
· 2
4 · 3
3 · 67
22 · 3 · 11
2 · 2
2 · 3 · 67
Reflect 6. Complete a factor tree and a ladder diagram to find the prime factorization of 54.
© Houghton Mifflin Harcourt Publishing Company
1
Use a factor tree to find the prime factorization of each number. (Explore Activity 1)
to the left of the steps of the ladder.
22 · 32
Use a ladder diagram to find the prime factorization of each number. (Explore Activity 2)
54
6. 32
2 · 27
9 · 3
3 · 3
prime factorization: 2 · 33
2 32 2 16 2 8 2 4 2 2 1
7. 27
3 27 3 9 3 3 1 3 · 3 · 3 or 33
25
? ?
Communicate Mathematical Ideas If one person uses a ladder diagram and another uses a factor tree to write a prime factorization, will they get the same result? Explain.
ESSENTIAL QUESTION CHECK-IN
8. Tell how you know when you have found the prime factorization of a number.
Yes; there is only one unique prime factorization for
Sample answer: when all the factors are prime and their
every integer greater than 1.
product is the original number Lesson 10.2
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1. This is a list
2. 52
Complete the table with possible measurements of the stage.
The prime factors are 2, 2, 3, and 11. They are written
InCopy Notes
9 18
3. Karl needs to build a stage that has an area of 72 square feet. The length of the stage should be longer than the width. What are the possible whole number measurements for the length and width of the stage? (Example 1)
3 33 11 11 1
D What are the prime factors of 132? How can you tell from the ladder diagram?
7.
6
1, 2, 3, 6, 9, 18
2 132 2 66
C Keep choosing prime factors, dividing, and adding to the ladder until you get a quotient of 1.
2 54 3 27 3 9 3 3 1
3
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EXPLORE ACTIVITY 2
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DIFFERENTIATE INSTRUCTION InDesign Notes
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Kinesthetic Experience
Critical Thinking
Students can use a method called the sieve of Eratosthenes to help identify prime numbers. Start with a 10 × 10 grid showing the numbers 1 to 100. Cross out 1, because 1 is not a prime number. Circle 2 because it is prime. Then cross out all multiples of 2, because multiples of 2 are not prime. Circle the next prime number, 3, and cross out all multiples of 3. Repeat the process until all numbers are circled or crossed out. Students can refer to this chart when deciding whether a number is prime or composite.
Discuss with students how the prime factorization of a number can be used to find all the factors of a number, by using the Associative and Commutative properties. For example, the prime factorization of 30 is 2 · 3 · 5, which can be expressed as 2 · (3 · 5), (2 · 3) · 5, or (2 · 5) · 3.
Differentiated Instruction includes: • Reading Strategies • Success for English Learners ELL • Reteach • Challenge PRE-AP
When you multiply the numbers inside the parentheses, the expressions simplify to 2 · (15), (6) · 5, and (10) · 3.
These numbers along with the factor pair 1 · 30, give all the factors for 30: 1, 2, 3, 5, 6, 10, 15, 30.
Prime Factorization
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Personal Math Trainer Online Assessment and Intervention
Online homework assignment available
Evaluate GUIDED AND INDEPENDENT PRACTICE 6.7.A
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10.2 LESSON QUIZ 6.7.A 1. Find all the factors of 54. 2. Find the prime factorization of 54, and then write it using exponents.
Concepts & Skills
Practice
Example 1 Finding Factors of a Number
Exercises 1–3, 9–10
Explore Activity 1 Finding the Prime Factorization of a Number
Exercises 4–5, 12–15, 17–19
Explore Activity 2 Using a Ladder Diagram
Exercises 6–7, 16
3. Find all the factors of 60. 4. Find the prime factorization of 60, and then write it using exponents. 5. Chanasia has 30 beads. She wants to put them in boxes, so that each box will contain the same whole number of beads. Use factors to list all the different ways she can put the beads into boxes.
Exercise
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Answers 1. 1, 2, 3, 6, 9, 18, 27, 54 2. 2 × 3 × 3 × 3 = 2 · 3
3
3. 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 4. 2 × 2 × 3 × 5 = 2 · 3 · 5
2 Skills/Concepts
1.D Multiple representations
10
2 Skills/Concepts
1.A Everyday life
11
3 Strategic Thinking
1.G Explain and justify arguments
2 Skills/Concepts
1.C Select tools
16
3 Strategic Thinking
1.G Explain and justify arguments
17
3 Strategic Thinking
1.C Select tools
18
3 Strategic Thinking
1.G Explain and justify arguments
19
3 Strategic Thinking
1.F Analyze relationships
20
3 Strategic Thinking
1.G Explain and justify arguments
21–22
3 Strategic Thinking
1.F Analyze relationships
2
5. 1 box with 30 beads 2 boxes with 15 beads each 3 boxes with 10 beads each 5 boxes with 6 beads each 6 boxes with 5 beads each 10 boxes with 3 beads each 15 boxes with 2 beads each 30 boxes with 1 bead each
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Lesson 10.2
Mathematical Processes
9
12–15
Lesson Quiz available online
Depth of Knowledge (D.O.K.)
Additional Resources Differentiated Instruction includes: • Leveled Practice Worksheets
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DO NOT EDIT--Changes must be made through “File info” CorrectionKey=B
Name
Class
Date
10.2 Independent Practice
18. In a game, you draw a card with three consecutive numbers on it. You can choose one of the numbers and find the sum of its prime factors. Then you can move that many spaces. You draw a card with the numbers 25, 26, 27. Which number should you choose if you want to move as many spaces as possible? Explain.
Personal Math Trainer
6.7.A
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Online Assessment and Intervention
26; the prime factors of 25 are 5 and 5, the prime factors
9. Multiple Representations Use the grid to draw three different rectangles so that each has an area of 12 square units and they all have different widths. What are the dimensions of the rectangles?
of 26 are 2 and 13, and the prime factors of 27 are 3, 3, and 3. The sums are 10, 15, and 9. The greatest sum is 15,
1 × 12; 2 × 6; 3 × 4
so choose 26 to move 15 spaces. 19. Explain the Error When asked to write the prime factorization of the number 27, a student wrote 9 · 3. Explain the error and write the correct answer.
10. Brandon has 32 stamps. He wants to display the stamps in rows, with the same number of stamps in each row. How many different ways can he display the stamps? Explain.
9 is not a prime number; prime factorization of 27 = 33.
6 different ways; 1 row of 32 stamps; 2 rows of 16; 4 rows of 8; 32 rows of 1; 16 rows of 2; 8 rows of 4
FOCUS ON HIGHER ORDER THINKING
20. Communicate Mathematical Ideas Explain why it is possible to draw more than two different rectangles with an area of 36 square units, but it is not possible to draw more than two different rectangles with an area of 15 square units. The sides of the rectangles are whole numbers.
11. Communicate Mathematical Ideas How is finding the factors of a number different from finding the prime factorization of a number?
When you find the factors of a number, you find all
36 has five factor pairs, so five different rectangles can
factors, some of which are prime; when you find the
be drawn. 15 has only two factor pairs, so only two
prime factorization, you find only the prime factors.
14. 23
34 · 11
13. 504
23
15. 230
different rectangles can be drawn. 21. Critique Reasoning Alice wants to find all the prime factors of the number you get when you multiply 17 · 11 · 13 · 7. She thinks she has to use a calculator to perform all the multiplications and then find the prime factorization of the resulting number. Do you agree? Why or why not?
2 3 · 32 · 7 2 · 5 · 23
Disagree; the factors that are being multiplied are all prime numbers, so the prime factorization of the
16. The number 2 is chosen to begin a ladder diagram to find the prime factorization of 66. What other numbers could have been used to start the ladder diagram for 66? How does starting with a different number change the diagram?
number is 17 · 13 · 11 · 7. 22. Look for a Pattern Ryan wrote the prime factorizations shown below. If he continues this pattern, what prime factorization will he show for the number one million? What prime factorization will he show for one billion?
3 and 11 can be chosen because they are prime factors. The intermediate steps would be different, but the
10 = 5 · 2
prime factorization is the same.
100 = 52 · 22
17. Critical Thinking List five numbers that have 3, 5, and 7 as prime factors.
1,000 = 53 · 23
one million: 56 · 26; one billion: 59 · 29
Sample answer: 105, 315, 525, 735, 945 Lesson 10.2
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InDesign Notes
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Find the prime factorization of each number. 12. 891
Work Area
Unit 4
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InDesign Notes 1. This is a list
Activity In mathematics, a perfect number is a number that is equal to the sum of all its factors (excluding the number itself ). The number 6 is an example of a perfect number. The factors of 6 are 1, 2, 3, and 6. The sum of the factors excluding 6 is 1 + 2 + 3 = 6. • Find the next largest perfect number, and show why it is perfect. 28; the factors of 28 are 1, 2, 4, 7, 14, and 28, and 1 + 2 + 4 + 7 + 14 = 28. • A student claims that 128 is a perfect number. Prove or disprove the student’s claim. False; The factors of 128 are 1, 2, 4, 8, 16, 32, 64, and 128. Their sum, excluding the number itself, is 127. • Another student says that 496 is a perfect number. Prove or disprove the student’s claim. True; The factors of 496 are 1, 2, 4, 8, 16, 31, 62, 124, 248, and 496. Their sum, excluding the number itself, is 496.
Prime Factorization
280
LESSON
10.3 Order of Operations Texas Essential Knowledge and Skills The student is expected to: Expressions, equations, and relationships—6.7.A Generate equivalent numerical expressions using order of operations, including whole number exponents and prime factorization.
Mathematical Processes
Engage
ESSENTIAL QUESTION How do you use the order of operations to simplify expressions with exponents? Sample answer: Find the value of any expressions within parentheses first. Then evaluate all powers. Then multiply or divide in order from left to right, and finally, add or subtract in order from left to right.
Motivate the Lesson
Ask: Have you ever tried to simplify an expression such as 35 + 20(122 ÷ 9)? Try it. Need help? Begin the Explore Activity to find out how to use the order of operations.
6.1.C Select tools, including real objects, manipulatives, paper and pencil, and technology as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems.
Explore EXPLORE ACTIVITY Engage with the Whiteboard
Write the expression 2 + 3 × 4 on the whiteboard. Point out to students that this expression could have two different results (14 or 20) without guidelines to show which operation should be performed first. Show students that the correct solution is 14 based on the fact that 3 × 4 = 4 + 4 + 4. Thus the expression 2 + 3 × 4 can be written as 2 + 4 + 4 + 4, which equals 14. Now write the expressions 36 - 18 ÷ 6 and 7 + 24 ÷ 6 × 2 on the whiteboard and ask the students to solve them. After a few minutes, have students come up to the whiteboard and solve the equations, showing all the steps. Discuss the solutions with the class. Answers: 33 and 15
Explain ADDITIONAL EXAMPLE 1 Simplify each expression. A 30 - 3 × 23 6 B 128 ÷ (4 × 2)
2
32 C 40 - _____ (7 - 5)3
EXAMPLE 1 Avoid Common Errors Students may find it easier to perform operations from left to right as they appear in an expression, rather than use the order of operations. Remind students that using the order of operations correctly ensures that everyone who simplifies the same expression will get the same answer.
2
36
Interactive Whiteboard Interactive example available online my.hrw.com
Questioning Strategies
Mathematical Processes • Can you use the order of operations with expressions that have no parentheses? Explain. Yes. The order of operations tells the order in which operations should be performed but does not require that an expression include parentheses, exponents, or all the operations. • Are the expressions 3 + 5 · 2 and 3 + (5 · 2) equivalent? Explain. Yes. In both expressions 5 · 2 should be evaluated first, and then 3 should be added to the product.
281
Lesson 10.3
Check for Understanding Ask: In the expression 36 - 18 ÷ 2 + 6 × 1, which operation should you perform first? Explain. Division. This expression has no parentheses or exponents, so the first operation to perform is either multiplication or division from left to right.
LESSON
10.3 Order of Operations ?
ESSENTIAL QUESTION
EXPLORE ACTIVITY (cont’d)
Expressions, equations, and relationships— 6.7.A Generate equivalent numerical expressions using order of operations …
Reflect 1.
In C , why does it makes sense to write the values as powers? What is the pattern for the number of e-mails in each wave for Amy?
Sample answer: By writing the values as powers, you How do you use the order of operations to simplify expressions with exponents?
can see the exponent is equal to the wave number. The pattern would be 31, 32, 33, 34, and so on.
6.7.A
EXPLORE ACTIVITY
Exploring the Order of Operations
Simplifying Numerical Expressions
Order of Operations
A numerical expression is an expression involving numbers and operations. You can use the order of operations to simplify numerical expressions.
1. Perform operations in parentheses. Math On the Spot
2. Find the value of numbers with exponents.
my.hrw.com
EXAMPLE 1
6.7.A
3. Multiply or divide from left to right.
My Notes
4. Add or subtract from left to right.
A 5 + 18 ÷ 32 5 + 18 ÷ 32 = 5 + 18 ÷ 9
Amy 1st wave
C Amy is just one of four friends initiating the first wave of e-mails. Write an expression for the total number of e-mails sent in the 2nd wave.
=5+2
Divide.
=7
B 4 × (9 ÷ 3)2
2nd wave
4 × (9 ÷ 3)2 = 4 × 32
A Use a diagram to model the situation for Amy. Each dot represents one e-mail. Complete the diagram to show the second wave. B Complete the table to show how many e-mails are sent in each wave of Amy’s diagram.
Wave
Number of emails
Power of 3
1st
3
31
2nd
9
32
4
3
Personal Math Trainer
Multiply 4 and 3/Find the value of 32
my.hrw.com
36
Multiply.
2
Online Assessment and Intervention
=
= 36
(12 - 8) 42 = 8 + __ 8 + _______ 2 2
D Identify the computation that should be done first to simplify the expression in C . Then simplify the expression.
9
Evaluate 32.
(12 - 8)2
2
The value of the expression is 4 ×
Perform operations inside parentheses.
=4×9
C 8 + _______ 2
number of people × number of emails in 2nd wave written as a power ×
Evaluate 32.
Perform operations inside parentheses.
16 = 8 + __ 2
Evaluate 42.
= 8+ 8
Divide.
= 16
© Houghton Mifflin Harcourt Publishing Company
Amy and three friends launch a new website. Each friend emails the web address to three new friends. These new friends forward the web address to three more friends. If no one receives the e-mail more than once, how many people will receive the web address in the second wave of e-mails?
© Houghton Mifflin Harcourt Publishing Company
Simplify each expression.
YOUR TURN Simplify each expression using the order of operations. 2. (3 - 1)4 + 3
19
3.
24 ÷ (3 × 22)
2
. Lesson 10.3
281
282
Unit 4
PROFESSIONAL DEVELOPMENT Integrate Mathematical Processes This lesson provides an opportunity to address Mathematical Process TEKS 6.1.C, which calls for students to “select tools, including…paper and pencil…and techniques, including mental math…and number sense…to solve problems.” Students work with pencil and paper using the order of operations to simplify expressions. Since use of the order of operations can involve many steps, students use mental math and number sense in situations such as finding the sum of two numbers with unlike signs and raising a negative number to a positive power.
Math Background A History of Mathematical Notations by Florian Cajori describes the history of various mathematical symbols. According to Cajori, parentheses and brackets have been used as grouping symbols since the sixteenth century. A work published in 1556, General trattato di numeri e misure by Niccolò Tartaglia, is one of the first works in which parentheses are used. Brackets have been found in a manuscript edition of Algebra, by Rafael Bombelli which dates back to 1550.
Order of Operations
282
ADDITIONAL EXAMPLE 2 Simplify each expression using the order of operations. A (-3)3 + 8(15 - 7) 32 B ____ +3×3 (-4)2
11
C 18 - (-3) + 3 × 2 3
37 51
EXAMPLE 2 Avoid Common Errors Point out to students that, now that they are working with integers, not every minus sign will indicate subtraction. Some minus signs are used to indicate negative numbers. Remind students to read equations carefully to distinguish between the two uses of the minus sign.
Engage with the Whiteboard
Interactive Whiteboard Interactive example available online
Cover up the blue text in each part and have students circle the operation to be performed for each step on the whiteboard. Ask students to explain their choices. Then discuss the choices with the class.
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Questioning Strategies
Mathematical Processes • In A, the order of operations says to perform operations in parentheses first. So why isn’t (-2)2 evaluated until the second step? When you are working with negative numbers, parentheses are used to separate addition, subtraction, multiplication, and division signs from negative numbers to avoid confusion. Since there is no operation in (-2)2, the first step is to subtract within the expression (3 - 9). (-3)2
• In B, can you evaluate the expression ____ the same as (-1)2? Explain. No. You cannot 3 cancel the 3s before you have evaluated (-3)2.
YOUR TURN Focus on Communication It may be helpful to review the rules for adding numbers with different signs before students try to work on their own. Remind them to subtract the absolute values of the numbers and then use the sign of the number with the greater absolute value in the difference.
Elaborate Talk About It Summarize the Lesson Ask: Why is it important to use the order of operations? The order of operations is important because correctly using the order or operations ensures that everyone who simplifies the same expression will get the same answer.
GUIDED PRACTICE Engage with the Whiteboard For Exercise 1, have students complete the diagram on the whiteboard. Then have them number the branches at each level to show the numbers of each type of fish that can be formed.
Avoid Common Errors Exercises 2–3 Remind students that when they work with an expression that has both multiplication and division, they should perform the operation that occurs first in the equation from left to right. Exercises 4–5 Remind students that when an expression inside parentheses has more than one operation, they need to perform those operations according to the order of operations.
283
Lesson 10.3
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DO NOT EDIT--Changes must be made through “File info” CorrectionKey=B
Guided Practice
Using the Order of Operations with Integers
1. In a video game, a guppy that escapes a net turns into three goldfish. Each goldfish can turn into two betta fish. Each betta fish can turn into two angelfish. Complete the diagram and write the number of fish at each stage. Write and evaluate an expression for the number of angelfish that can be formed from one guppy. (Explore Activity)
You can use the order of operations to simplify expressions involving integers. Math On the Spot
EXAMPL 2 EXAMPLE
6.7.A
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1 guppy
Simplify each expression using the order of operations.
3 goldfish
A -4(3 - 9) + (-2)2 Perform operations inside parentheses.
B
= -4(-6) + 4
Evaluate (-2)2.
= 24 + 4
Multiply.
= 28
Divide.
= -18
=4+ =4+
© Houghton Mifflin Harcourt Publishing Company
2
3
)2 ÷ 3
9
=
12
angelfish
3
=2+
= -48 ÷ 3 + 1
Multiply.
= -16 + 1
=
Divide.
= -15
=
? ?
= = =
4. 2 + (-24 ÷ 23) -9 = 2 + (-24 ÷
Evaluate (-2)3.
3. 36 ÷ 22 - 4 × 2 = 36 ÷
÷3
7
=
C 6 × (-2)3 ÷ 3 + 1 6 × (-2)3 ÷ 3 + 1 = 6 × (-8) ÷ 3 + 1
×2
Complete to simplify each expression. (Examples 1 and 2)
Evaluate (-3)2.
= -21 + 3
3 3 × 2 = 12 angelfish 2. 4 + (10 - 7)2 ÷ 3 = 4 + (
2
× 2 betta fish
2
(-3)2 -21 + _____ 3 (-3) -21 + _____ = -21 + _93 3
3
-1
-3
-10
8
)-9
9
9 1
4
-4 × 2
-4 × 2 -8
5. -42 × (-3 × 2 + 8) = -42 × ( -6 + 8)
2
= -42 ×
-9
=
-9
=
-16
-32
×
2
ESSENTIAL QUESTION CHECK-IN
6. How do you use the order of operations to simplify expressions with exponents?
Find the value of any expressions within parentheses first. Then evaluate all powers. Then multiply or divide from
left to right, and finally add or subtract from left to right.
Simplify each expression using the order of operations. 4.
-7 × (-4) ÷ 14 - 22
5.
-2
© Houghton Mifflin Harcourt Publishing Company
-4(3 - 9) + (-2)2 = -4(-6) + (-2)2
-5 (-3 + 1)3 - 3
37
Personal Math Trainer Online Assessment and Intervention
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Lesson 10.3
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DIFFERENTIATE INSTRUCTION InDesign Notes 1. This is a list
Unit 4
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InDesign Notes 1. This is a list
Cooperative Learning
Cognitive Strategies
Have students work in groups to decide which operation signs to use to make the number sentences true. They may need to use operations more than once in each number sentence.
Students may be familiar with the abbreviation PEMDAS (or Please Excuse My Dear Aunt Sally) even before being introduced to the order of operations. But its abbreviation may give the impression that multiplication is always done before division and that addition is always done before subtraction. You may wish to present the mnemonic as P E M/D A/S
Differentiated Instruction includes: • Reading Strategies • Success for English Learners ELL • Reteach • Challenge PRE-AP
1. Operation signs: +, -, · 12
4
6
3
7 = 37
12 · 4 - 6 · 3 + 7 = 37
2. Operation signs: +, -, ÷ 18
2
24
12
4 = 22
18 + 2 - 24 ÷ 12 + 4 = 22
The slashes between the M and the D, and the A and the S, can help students remember that from left to right either operation can be performed first.
Order of Operations
284
Personal Math Trainer Online Assessment and Intervention
Online homework assignment available
Evaluate GUIDED AND INDEPENDENT PRACTICE 6.7.A
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10.3 LESSON QUIZ 6.7.A Simplify each expression using the order of operations. 1. 3 + (7 - 5)2 × 6
Concepts & Skills
Practice
Explore Activity Exploring the Order of Operations
Exercise 1
Example 1 Simplifying Numerical Expressions
Exercises 2–3, 7–12, 14–16
Example 2 Using the Order of Operations with Integers
Exercises 4–5, 13
25 2. ______ ×2 (-4 + 8)
3. 8 - 6 ÷ 2 + 3 × 5 4. 7 × 3 - 15 ÷ 5
Exercise
5. 8 + 2(-3 + 12 ÷ 2)2
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1.C Select tools
13
3 Strategic Thinking
1.F Analyze relationships
14
3 Strategic Thinking
1.G Explain and justify arguments
15–16
2 Skills/Concepts
1.A Everyday life
17–19
3 Strategic Thinking
1.F Analyze relationships
2. 16 3. 20
4. 18
Differentiated Instruction includes: • Leveled Practice Worksheets
5. 26
285
Lesson 10.3
Mathematical Processes
2 Skills/Concepts
7–12
Lesson Quiz available online
Depth of Knowledge (D.O.K.)
Class
Date
10.3 Independent Practice
Personal Math Trainer
6.7.A
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15. Ellen is playing a video game in which she captures butterflies. There are 3 butterflies onscreen, but the number of butterflies doubles every minute. After 4 minutes, she was able to capture 7 of the butterflies. a. Look for a Pattern Write an expression for the number of butterflies after 4 minutes. Use a power of 2 in your answer.
Online Assessment and Intervention
3 × 2 × 2 × 2 × 2 = 3 × 24
Simplify each expression using the order of operations. 7. 5 × 2 + 32
19
9. (11 - 8)2 - 2 × 6 2
9 11. 12 + __ 3
9
8. 15 - 7 × 2 + 23
-3
8 + 62 12. _____ +7×2 11
39
b. Write an expression for the number of butterflies remaining after Ellen captured the 7 butterflies. Simplify the expression.
14
10. 6 + 3(13 - 2) - 52
3 × 24 - 7 = 3 × 16 - 7 = 48 - 7 = 41; 41
18
butterflies remain.
13. Explain the Error Jay simplified the expression -3 × (3 + 12 ÷ 3) - 4. For his first step, he added 3 + 12 to get 15. What was Jay’s error? Find the correct answer.
16. Show how to write, evaluate and simplify an expression to represent and solve this problem: Jeff and his friend each text four classmates about a concert. Each classmate then texts four students from another school about the concert. If no one receives the message more than once, how many students from the other school receive a text about the concert?
Jay worked inside the parentheses first, but he should have performed the division 12 ÷ 3 = 4 first; -25.
2 × 42 = 32; 32 students receive a text.
14. Multistep A clothing store has the sign shown in the shop window. Pani sees the sign and wants to buy 3 shirts and 2 pairs of jeans. The cost of each shirt before the discount is \$12, and the cost of each pair of jeans is \$19 before the discount.
SALE
Today ONLY
\$3 off every purchase!
a. Write and simplify an expression to find the amount Pani pays if a \$3 discount is applied to her total.
3 × 12 + 2 × 19 - 3; \$71
17. Geometry The figure shown is a rectangle. The green shape in the figure is a square. The blue and white shapes are rectangles, and the area of the blue rectangle is 24 square inches. a. Write an expression for the area of the entire figure that includes an exponent. Then find the area.
b. Pani says she should get a \$3 discount on the price of each shirt and a \$3 discount on the price of each pair of jeans. Write and simplify an expression to find the amount she would pay if this is true. © Houghton Mifflin Harcourt Publishing Company
Work Area
FOCUS ON HIGHER ORDER THINKING
2 in. 6 in.
62 + 2 × 6 + 24 = 72 square inches
3 × (12 - 3) + 2 × (19 - 3); \$59
b. Find the dimensions of the entire figure.
c. Analyze Relationships Why are the amounts Pani pays in a and b different?
8 in. by 9 in.
In part a, the \$3 discount is applied 1 time; in b it is
18. Analyze Relationships Roberto’s teacher writes the following statement on the board: The cube of a number plus one more than the square of the number is equal to the opposite of the number. Show that the number is -1.
applied 5 times. d. If you were the shop owner, how would you change the sign? Explain.
Sample answer: If the shop owner wants to make more
(-1)3 + ((-1)2 + 1) = -1 + (1 + 1) = -1 + 2 = 1; 1 is
money, the sign should say “\$3 off your entire purchase.”
the opposite of -1.
If customers can take the discount off every item, a lot
19. Persevere in Problem Solving Use parentheses to make this statement true: 8 × 4 - 2 × 3 + 8 ÷ 2 = 25
more money is discounted from each purchase.
8 × 4 - (2 × 3 + 8) ÷ 2 Lesson 10.3
EXTEND THE MATH
PRE-AP
© Houghton Mifflin Harcourt Publishing Company • Image Credits: imagebroker/ Alamy
Name
285
286
Activity available online
Unit 4
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Activity The expression (4 × 4 - 4) × 4 uses exactly 4 fours. When simplified, its value is 48. • Write 10 expressions that use exactly 4 fours and that equal one of the numbers 0 to 9. Use what you know about the order of operations to write the expressions. You can use addition, subtraction, multiplication, division, parentheses, and exponents in the expressions. • Justify the expressions you have written by showing how to simplify them. Sample answers: 4+4-4-4=0
(4 + 4) ÷ (4 + 4) = 1
4÷4+4÷4=2
(4 + 4 + 4) ÷ 4 = 3
4 - (4 - 4) × 4 = 4
( __44 )
(4 + 4) 4 + _____ =6 4 (4 × 4) _____ +4=8 4
4
+4=5
4 + 4 - ( __44 ) = 7
4 + 4 + ( __44 ) = 9 Order of Operations
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MODULE QUIZ
Assess Mastery
10.1 Exponents
Use the assessment on this page to determine if students have mastered the concepts and standards covered in this module.
Find the value of each power.
5. 3
Response to Intervention
2 1
343
1. 73
( ) 2 _ 3
3
Enrichment
10. 120
Access Ready to Go On? assessment online, and receive instant scoring, feedback, and customized intervention or enrichment. Online and Print Resources Differentiated Instruction
Differentiated Instruction
• Reteach worksheets
• Challenge worksheets
• Reading Strategies • Success for English Learners ELL
ELL
6. ( -3
)5
-243
3.
( )
7. ( -2
)4
16
4.
( ) 1 _ 2
6
8. 1.42
1 __ 64
1.96
Additional Resources Assessment Resources includes: • Leveled Module Quizzes
1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120
11. 58
2 × 29
12. 212
22 × 53
13. 2,800
2 4 × 52 × 7
14. 900
22 × 3 2 × 5 2
10.3 Order of Operations Simplify each expression using the order of operations. 15. ( 21 - 3 ) ÷ 32
PRE-AP
Extend the Math PRE-AP Lesson activities in TE
1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 96
Find the prime factorization of each number.
2
17. 17 + 15 ÷ 3 - 24 © Houghton Mifflin Harcourt Publishing Company
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81
2. 92
8 __ 27
49 __ 81
2
Find the factors of each number. 9. 96
Online Assessment and Intervention
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7 _ 9
10.2 Prime Factorization
Intervention
Personal Math Trainer
Personal Math Trainer Online Assessment and Intervention
16. 72 × ( 6 ÷ 3 )
6
98
18. ( 8 + 56 ) ÷ 4 - 32
19. The nature park has a pride of 7 adult lions and 4 cubs. The adults eat 6 pounds of meat each day and the cubs eat 3 pounds. Simplify 7 × 6 + 4 × 3 to find the amount of meat consumed each day by the lions.
7
54 pounds
ESSENTIAL QUESTION 20. How do you use numerical expressions to solve real-world problems?
Write an expression to model the situation. Simplify the expression using the order of operations. First perform operations in parentheses, then find the value of each power, multiply or divide from left to right, and finally add or subtract from left to right.
Module 10
Texas Essential Knowledge and Skills Lesson
Exercises
10.1
1–8
6.7.A
10.2
9–14
6.7.A
10.3
15–19
6.7.A
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TEKS
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MODULE 10 MIXED REVIEW
Texas Test Prep
Texas Testing Tip Students can use logic to eliminate some or all of the answer choices. Item 5 Students can eliminate choices B, C, and D because they all have numbers that are not prime in the product expression. This leaves choice A as the correct answer.
Selected Response 1. Which expression has a value that is less than the base of that expression? A 2 5 B _ 6 C 32 3
( )
Item 8 Students can eliminate choices A and B because they have numbers that are not prime in the product expression.
Avoid Common Errors Item 3 Students sometimes will answer an order of operations question simply by completing the operations from left to right. Remind the students to perform operations in parentheses first. Item 7 Some students may see that 3.6 appears four times and then choose A. Remind them that multiplication is repeated addition and that exponents are needed to represent repeated multiplication.
2
2. After the game the coach bought 9 chicken meals for \$5 each and 15 burger meals for \$6 each. What percent of the total amount the coach spent was used for the chicken meals? 1 A 33 _% 3 B 45% 2 C 66 _% 3 D 90%
5. Which expression shows the prime factorization of 100? A 22 × 52
C
B 10 × 10
D 2 × 5 × 10
1010
A 21
C
B 23
D 27
25
7. Which expression is equivalent to 3.6 × 3.6 × 3.6 × 3.6? 34 × 64
A 3.6 × 4
C
B 36
D 3.64
3
8. Which expression gives the prime factorization of 80?
3. Which operation should you perform first when you simplify 75 - ( 8 + 45 ÷ 3 ) × 7? A addition
A 24 × 10
C
B 2×5×8
D 24 × 5
23 × 5
Gridded Response 9. Alison raised 10 to the 5th power. Then she divided this value by 100. What was the quotient?
B division
multiplication
D subtraction
4. At Tanika’s school, three people are chosen in the first round. Each of those people chooses 3 people in the second round, and so on. How many people are chosen in the sixth round? A 18
.
1
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
2
2
2
2
2
2
3
3
3
3
3
3
4
4
4
4
4
4
5
5
5
5
5
5
6
6
6
6
6
6
B 216
7
7
7
7
7
7
C
8
8
8
8
8
8
9
9
9
9
9
9
243
D 729
288
Online Assessment and Intervention
6. Which number has only two factors?
D 44
C
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© Houghton Mifflin Harcourt Publishing Company
Texas Test Prep
Unit 4
Texas Essential Knowledge and Skills Items
Mathematical Process TEKS
1
6.7.A
6.1.F
2
6.7.A
6.1.A
3*
6.4.E, 6.7.A
6.1.C
4
6.7.A
6.1.A, 6.1.F
5
6.7.A
6.1.E
6
6.7.A
6.1.F
7
6.7.A
6.1.E
8
6.7.A
6.1.E
9
6.7.A
6.1.C
* Item integrates mixed review concepts from previous modules or a previous course.
Generating Equivalent Numerical Expressions
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Generating Equivalent Algebraic Expressions ?
MODULE
11
LESSON 11.1
Modeling Equivalent Expressions
You can model real-world problems with variable expressions, then use algebraic rules to solve the problems.
6.7.C
LESSON 11.2
ESSENTIAL QUESTION
Evaluating Expressions
How can you generate equivalent algebraic expressions and use them to solve real-world problems?
6.7.A
LESSON 11.3
Generating Equivalent Expressions 6.7.C, 6.7.D
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Lloyd Sutton/ Alamy
Real-World Video
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Module 11
Carpenters use formulas to calculate a project’s materials supply. Sometimes formulas can be written in different forms. The perimeter of a rectangle can be written as P = 2(l + w) or P = 2l + 2w.
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Math On the Spot
Animated Math
Personal Math Trainer
Go digital with your write-in student edition, accessible on any device.
Scan with your smart phone to jump directly to the online edition, video tutor, and more.
Interactively explore key concepts to see how math works.
Get immediate feedback and help as you work through practice sets.
289
DO NOT EDIT--Changes must be made through "File info" CorrectionKey=B
Complete these exercises to review skills you will need for this chapter.
Use the assessment on this page to determine if students need intensive or strategic intervention for the module’s prerequisite skills.
Use of Parentheses
2 1
(6 + 4) × (3 + 8 + 1) = 10 × 12
Evaluate.
Intervention
Enrichment
1. 11 + (20 - 13)
2. (10 - 7) - (14 - 12)
3. (4 + 17) - (16 - 9)
4. (23 - 15) - (18 - 13)
5. 8 × (4 + 5 + 7)
6. (2 + 3) × (11 - 5)
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Differentiated Instruction
• Skill 50 Use of Parentheses
• Challenge worksheets
• Skill 53 Words for Operations • Skill 54 Evaluate Expressions
14
30
128
Words for Operations EXAMPLE
Online and Print Resources Skills Intervention worksheets
1
3
Access Are You Ready? assessment online, and receive instant scoring, feedback, and customized intervention or enrichment.
Online Assessment and Intervention
Online Assessment and Intervention
Do the operations inside parentheses first. Multiply.
= 120
Response to Intervention
18
Personal Math Trainer
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Write a numerical expression for the quotient of 20 and 5.
Think: Quotient means to divide.
20 ÷ 5
Write 20 divided by 5.
Write a numerical expression for the word expression. 7. the difference between 42 and 19
PRE-AP
Extend the Math PRE-AP Lesson Activities in TE
42 - 19
20 + 30
9. 30 more than 20
8. the product of 7 and 12 10. 100 decreased by 77
7 × 12
100 - 77
© Houghton Mifflin Harcourt Publishing Company
3
EXAMPLE
Personal Math Trainer
Evaluate Expressions EXAMPLE
Evaluate 2(5) – 32. 2(5) – 32 = 2(5) - 9 = 10 - 9 =1
Evaluate exponents. Multiply. Subtract.
Evaluate the expression. 11. 3(8) - 15
9 8
14. 4(2 + 3) - 12
290
59
12. 4(12) + 11 15. 9(14 - 5) - 42
39
13. 3(7) - 4(2)
13
16. 7(8) - 5(8)
16
Unit 4
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InCopy Notes
PROFESSIONAL DEVELOPMENT VIDEO
InDesign Notes
1. This is a list Bold, Italic, Strickthrough.
1. This is a list
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Author Juli Dixon models successful teaching practices as she explores equivalent algebraic expressions in an actual sixth-grade classroom.
Online Teacher Edition Access a full suite of teaching resources online—plan, present, and manage classes and assignments.
Professional Development
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Interactive Answers and Solutions Customize answer keys to print or display in the classroom. Choose to include answers only or full solutions to all lesson exercises.
Interactive Whiteboards Engage students with interactive whiteboard-ready lessons and activities. Personal Math Trainer: Online Assessment and Intervention Assign automatically graded homework, quizzes, tests, and intervention activities. Prepare your students with updated, TEKS-aligned practice tests.
Generating Equivalent Algebraic Expressions
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Simplifying Expressions
operations, order of operations
Understand Vocabulary
Students can use these reading and note-taking strategies to help them organize and understand new concepts and vocabulary. c.4.D Use prereading supports such as graphic organizers, illustrations, and pretaught topic-related vocabulary to enhance comprehension of written text.
Preview Words
base, exponent
numerical expression
×, ÷, +, -
algebraic expression (expresión algebraica) coefficient (coeficiente) constant (constante) equivalent expression (expresión equivalente) evaluating (evaluar) like terms (términos semejantes) term (término, en una expresión) variable (variable)
23 2+1+3
Understand Vocabulary Complete the sentences using the preview words.
1. An expression that contains at least one variable is an
algebraic expression
.
2. A part of an expression that is added or subtracted is a © Houghton Mifflin Harcourt Publishing Company
Integrating the ELPS
base (base) exponent (exponente) numerical expression (expresión numérica) operations (operaciones) order of operations (orden de las operaciones)
Use the review words to complete the graphic. You may put more than one word in each oval.
The graphic organizer will help students to review concepts related to simplifying expressions. If time allows, discuss as a class the mnemonic Please Excuse My Dear Aunt Sally for order of operations (Parentheses, Exponent, Multiplication, Division, Addition, Subtraction).
Review Words
Visualize Vocabulary
Visualize Vocabulary
Use the following explanation to help students learn the preview words. Variable is an antonym of constant. Variable means “changeable”; something that is constant does not change. In math, a variable is a letter or symbol that represents a number. A letter is used because the number is unknown, and it may vary. A constant is a numeral, not a letter. For an expression to be an algebraic expression, it must contain at least one variable.
Vocabulary
3. A
constant
term
.
is a specific number whose value does not change.
Key-Term Fold Before beginning the module, create a key-term fold to help you learn the vocabulary in this module. Write the highlighted vocabulary words on one side of the flap. Write the definition for each word on the other side of the flap. Use the key-term fold to quiz yourself on the definitions used in this module.
Differentiated Instruction • Reading Strategies ELL
Module 11
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Grades 6–8 TEKS Before Students understand: • operations with whole numbers, decimals, and fractions • order of operations • properties of operations: inverse, identity, commutative, associative, and distributive properties
291
Module 11
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InCopy Notes
InDesign Notes
1. This is a list
1. This is a list
In this module Students will learn to: • determine if two expressions are equivalent using concrete models, pictorial models, and algebraic representations • evaluate algebraic expressions for the given value of a variable • generate equivalent expressions using the properties of operations: inverse, identity, commutative, associative, and distributive properties
291
After Students will connect: • numerical and algebraic expressions • variables and symbols to translate words into math
MODULE 11
Unpacking the TEKS
Unpacking the TEKS Understanding the TEKS and the vocabulary terms in the TEKS will help you know exactly what you are expected to learn in this module.
Use the examples on this page to help students know exactly what they are expected to learn in this module.
6.7.C
What It Means to You
Determine if two expressions are equivalent using concrete models, pictorial models, and algebraic representations.
Texas Essential Knowledge and Skills
You will use models to compare expressions. UNPACKING EXAMPLE 6.7.C
On a math quiz, Tina scored 3 points more than Yolanda. Juan scored 2 points more than Yolanda and earned 2 points as extra credit. Draw models for Tina's and Juan's scores. Use your models to decide whether they made the same score.
Key Vocabulary
Content Focal Areas
equivalent expressions (expresión equivalente) Expressions that have the same value for all values of the variables.
Expressions, equations, and relationships—6.7 The student applies mathematical process standards to develop concepts of expressions and equations.
y+3 Tina
y
3 y+2+2
Integrating the ELPS Juan
y
2
2
Tina and Juan did not make the same score because the models do not show equivalent expressions.
6.7.D Generate equivalent expressions using the properties of operations: inverse, identity, commutative, associative, and distributive properties.
Go online to see a complete unpacking of the .
What It Means to You You will use the properties of operations to find an equivalent expression. UNPACKING EXAMPLE 6.7.D
William earns \$13 an hour working at a movie theater. He worked h hours in concessions and three times as many hours at the ticket counter. Write and simplify an expression for the amount of money William earned.
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\$13 · hours at concessions + \$13 · hours at ticket counter Visit my.hrw.com to see all the unpacked.
© Houghton Mifflin Harcourt Publishing Company • Image Credits: Erik Dreyer/Getty Images
c.4.F Use visual and contextual support … to read grade-appropriate content area text … and develop vocabulary … to comprehend increasingly challenging language.
13h + 13(3h) 13h + 39h
Multiply 13 · 3h.
h(13 + 39)
Distributive Property
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Lesson 11.1
Lesson 11.2
Unit 4
Lesson 11.3
6.7.A Generate equivalent numerical expressions using order of operations, including whole number exponents and prime factorization. 6.7.C Determine if two expressions are equivalent using concrete models, pictorial models, and algebraic representations. 6.7.D Generate equivalent expressions using the properties of operations: inverse, identity, commutative, associative, and distributive properties.
Generating Equivalent Algebraic Expressions
292
LESSON
11.1 Modeling Equivalent Expressions Texas Essential Knowledge and Skills The student is expected to: Expressions, equations, and relationships—6.7.C Determine if two expressions are equivalent using concrete models, pictorial models, and algebraic expressions.
Engage
ESSENTIAL QUESTION How can you write algebraic expressions and use models to decide if expressions are equivalent? Sample answer: Write or model the variables, constants, and operations to represent each expression. Then compare the expressions or models.
Motivate the Lesson Ask: Is a quarter the same as 5 nickels? as 25 pennies? How can you describe something in different ways but not change its value? Begin the Explore Activity to find out.
Mathematical Processes 6.1.D Communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate.
Explore EXPLORE ACTIVITY Focus on Modeling Mathematical Processes Point out to students that a balance scale can represent how two numbers or expressions compare in value. When the numbers or expressions on each side of the scale are equal in value, the scale is in balance. If the numbers or expressions are unequal, one side of the scale is higher than the other side.
Explain ADDITIONAL EXAMPLE 1 A Write each phrase as an algebraic expression. x less than 5 5 - x the product of z and 8 8z B Write a phrase for each algebraic expression. __z the quotient of z and 5 5
9 + y 9 more than y
EXAMPLE 1 Engage with the Whiteboard Write a mathematical operation on the whiteboard. Next to it, have students write an expression that uses the operation. Then ask them to describe their expression with words in several different ways and to list these on the whiteboard. For example, 6x can be described as 6 times x, the product of 6 and x, and 6 multiplied by x. Discuss with students the ways to describe each operation by looking at the list provided above Example 1.
Questioning Strategies
Interactive Whiteboard Interactive example available online my.hrw.com
Mathematical Processes • How do you identify the variable in an algebraic expression? Find a letter or symbol that represents an unknown. • Does it matter which letter you choose when writing an algebraic expression? No. You can choose any letter, but mathematicians often choose the last letters of the alphabet (e.g., x, y, and z) to represent variables. • How can you find the constant in an algebraic expression? Look for a specific number whose value does not change.
Connect Vocabulary
ELL
Explain to students that in the expression 8x + 15, the number 8 is a coefficient. A coefficient is the number multiplied by the variable, x. The number 15 in this expression is a constant.
293
Lesson 11.1
11.1 ?
Modeling Equivalent Expressions
ESSENTIAL QUESTION
Expressions, equations, and relationships— 6.7.C Determine if two expressions are equivalent using concrete models, pictorial models, and algebraic representations.
Writing Algebraic Expressions An algebraic expression is an expression that contains one or more variables and may also contain operation symbols, such as + or -. Math On the Spot my.hrw.com
How can you write algebraic expressions and use models to decide if expressions are equivalent?
A variable is a letter or symbol used to represent an unknown or unspecified number. The value of a variable may change. A constant is a specific number whose value does not change. constant
EXPLORE ACTIVITY
variable
6.7.C
Algebraic Expressions
Modeling Equivalent Expressions
150 + y
w+n
x
15
12 - 7
9 __ 16
Not Algebraic Expressions
Equivalent expressions are expressions that have the same value.
In algebraic expressions, multiplication and division are usually written without the symbols × and ÷. • Write 3 × n as 3n, 3 · n, or n · 3. • Write 3 ÷ n as _n3 . There are several different ways to describe expressions with words.
The scale shown to the right is balanced.
A Write an expression to represent the circles on the left side of the balance.
2+3
B The value of the expression on the left side is
5
.
Operation
C Write an expression to represent the circles on the right side of the balance.
1+4
D The value of the expression on the right side is
5
Words
.
added to plus sum more than
E Since the expressions have the same value, the expressions are
equivalent
subtracted from minus difference less than take away taken from
• • • •
Multiplication
Division
times multiplied by product groups of
• divided by • divided into • quotient
.
F What will happen if you remove a circle from the right side of the balance? © Houghton Mifflin Harcourt Publishing Company
Subtraction • • • • • •
EXAMPLE 1
The scale will no longer be balanced.
6.7.C
A Write each phrase as an algebraic expression.
G If you add a circle to the left side of the balance, what can you do to the right side to keep the scale in balance?
The sum of 7 and x
The algebraic expression is 7 + x.
Add a circle to the right side.
The quotient of z and 3
The operation is division.
The algebraic expression is _3z .
Reflect 1. What If? Suppose there were 2 + 5 circles on the right side of the balance and 3 on the left side of the balance. What can you do to balance the scale? Explain how the scale models equivalent expressions.
B Write a phrase for each expression. 11x
© Houghton Mifflin Harcourt Publishing Company
LESSON
The operation is multiplication.
Sample answer: Add 4 circles to the left side. 4 + 3 is equivalent
The product of 11 and x
to 2 + 5 because the value of both expressions is 7.
8-y
The operation is subtraction.
y less than 8 Lesson 11.1
293
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Unit 4
PROFESSIONAL DEVELOPMENT Integrate Mathematical Processes This lesson provides an opportunity to address Mathematical Process TEKS 6.1.D, which calls for students to “communicate mathematical ideas … using multiple representations, including symbols, diagrams…and language as appropriate.” In this lesson, students use symbols, a model of a scale, and bar models to represent equivalent expressions. They employ these multiple representations to compare algebraic expressions and solve problems in real-world situations.
Math Background François Viète (1540–1603) was a lawyer in France who devoted his spare time to mathematics. In his book In artem analyticam isagoge, he introduced the idea of using vowels for variables and using consonants for constants. This was an important step toward modern algebra. Although Viète also used + and -, he had no symbol for equality. To write “equals,” he would use the Latin word aequatur. Viète is sometimes called the Father of Algebra.
Modeling Equivalent Expressions
294
YOUR TURN Avoid Common Errors ADDITIONAL EXAMPLE 2 Use a bar model to represent each expression. A 5+y
Talk About It Check for Understanding Ask: How can you tell if an expression is an algebraic expression? Look for a variable, a letter or symbol that represents an unknown. If the expression has a variable, it is an algebraic expression.
5+y
y
5
B __n4
Exercise 2 When students multiply a number by a variable, be sure that they write the number first: 4x, not x4. It’s easier to read and understand.
EXAMPLE 2
n
Questioning Strategies
Mathematical Processes • On the bar model for A, why is 7 + x written above the model, while 7 and x are written below? The total that the bar represents is 7 plus an unknown amount; the labels below show what part of the model is 7 and what part is x.
n 4
Interactive Whiteboard Interactive example available online my.hrw.com
• On the bar model for B, how do you know how many pieces to divide the bar into? The expression __z3 means “7 divided into 3 parts,” so you know you need to divide the bar into 3 equal parts.
Engage with the Whiteboard Have students change the constant in Example 2A and/or 2B. Then have students draw a model to represent the new expression. Ask volunteers to explain their models to the class, and then ask the class if the students are correct.
ADDITIONAL EXAMPLE 3 Amanda and Stuart began the week with the same amount of money. Amanda paid \$7 to go to the movies. Stuart spent \$4 on snacks and \$3 on a pen. Write algebraic expressions and draw bar models to represent the money each has left at the end of the week. Do Amanda and Stuart have the same amount of money left? n
n- 7
7 n
Exercise 7 Some students may try to model t - 2 as an addition equation. Remind students that for subtraction expressions, they must “take away” the 2 from the whole, not add to it.
EXAMPLE 3 Questioning Strategies
Mathematical Processes • How can you know which operation to use to solve the problem? Read the problem carefully. Katriana and Andrew “spent” or “took away” money from the money they started the day with. These words describe subtraction. • In Step 1, what do the labels on the model represent? The variable x represents the amount of money Katriana started with, the 5 represents the money she spent, and x - 5 represents the money she has left.
Focus on Reasoning 4
3
n- 4- 3
The models are equivalent, so Amanda and Stuart have the same amount of money left. Interactive Whiteboard Interactive example available online my.hrw.com
295
Lesson 11.1
Mathematical Processes Have students compare and contrast the models presented in the Explore Activity and Example 3. Encourage students to debate the advantages and disadvantages of each model, supporting their arguments with examples.
YOUR TURN Avoid Common Errors If students have difficulty with drawing the models, encourage them to circle the information for Tina in one color and for Juan in a different color. Then remind them to look at the list of ways to describe math operations to determine the operation they should use in the models.
Comparing Expressions Using Models
YOUR TURN Write each phrase as an algebraic expression. 2. n times 7
7n
3. 4 minus y
12
4. 13 added to x x
+ 13
Write a phrase for each expression. x 5. __
4-y
6. 10y 10
the quotient of x and 12
Personal Math Trainer Online Assessment and Intervention
Algebraic expressions are equivalent if they are equal for all values of the variable. For example, x + 2 and x + 1 + 1 are equivalent. Math On the Spot my.hrw.com
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EXAMPLE 3
6.7.C
Katriana and Andrew started the day with the same amount of money. Katriana spent 5 dollars on lunch. Andrew spent 3 dollars on lunch and 2 dollars on an afterschool snack.
multiplied by y
Do Katriana and Andrew have the same amount of money left?
EXAMPL 2 EXAMPLE
6.7.C
Math On the Spot
The variable represents the amount of money both Katriana and Andrew have at the beginning of the day.
Use a bar model to represent each expression.
A 7+x
7+x
x
7 B _3z
Divide z into 3 equal parts.
© Houghton Mifflin Harcourt Publishing Company
z
z 3
2
8. 4y
t
t- 2
y
x
3
2
x- 3- 2
Compare the models.
STEP 3
The models are equivalent, so the expressions are equivalent. Andrew and Katriana have the same amount of money left.
Mathematical Processes
9. On a math quiz, Tina scored 3 points more than Julia. Juan scored 2 points more than Julia and earned 2 points in extra credit. Write an expression and draw a bar model to represent Tina’s score and Juan’s score. Did Tina and Juan make the same grade on the quiz? Explain.
What two phrases can you use to describe the expression _4x ? What is different about the two phrases?
Draw a bar model to represent each expression. 7. t - 2
x−3−2
Math Talk
Write an algebraic expression to represent the money Andrew has left. Represent the expression with a model.
STEP 2
Sample answer: 4 divided by x; x divided into 4. The numerator, or dividend, comes first when using the term divided by. The denominator, or divisor, comes first when using the term divided into.
Combine 7 and x.
x-5
5
x−5
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© Houghton Mifflin Harcourt Publishing Company
Algebraic expressions can also be represented with models.
x
Write an algebraic expression to represent the money Katriana has left. Represent the expression with a model.
STEP 1
Modeling Algebraic Expressions
Juan: y + 2 + 2
Tina: y + 3
4y Personal Math Trainer
Personal Math Trainer
Online Assessment and Intervention
Online Assessment and Intervention
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Lesson 11.1
295
296
y
3
y
2
2
No; the expressions are not equivalent.
Unit 4
Modeling Equivalent Expressions
296
Elaborate Talk About It Summarize the Lesson Ask: How can you find out whether algebraic expressions are equivalent? Draw models of each expression and then compare them. If the models are equivalent, the expressions are equivalent.
GUIDED PRACTICE Engage with the Whiteboard For Exercise 1, have students write expressions that will keep the scale balanced next to the scale on the whiteboard. Emphasize to students that all the expressions are equivalent. For Exercise 4, have students circle the variable and the constant in y + 12. Then have them write three different algebraic expressions that are equivalent to y + 12 on the whiteboard. Sample answer: y + 3 + 9; y + 4 + 8; y + 5 + 7 For Exercise 7, have students complete the bar models for each city on the whiteboard and then explain their reasoning.
Avoid Common Errors Exercise 2 Remind students that the order of the variable and the constant is important in subtraction expressions, y - 5 is not the same as 5 - y. Exercise 3 When students multiply a number by a variable, be sure that they write the number first: 4x, not x4. It’s easier to read and understand. Exercise 6 Some students may try to model m ÷ 4 as an addition equation. Remind students that for division expressions, they must “divide” the whole into parts, not add to it.
297
Lesson 11.1
Guided Practice
Name
5+ 5
2p
3. The product of 2 and p
p 5. __ 10
11. Write an algebraic expression with two variables and one constant.
answers are given. 10 divided into p m
t
Tucson
© Houghton Mifflin Harcourt Publishing Company
2
3
4
14. n divided by 8
n __ 8
15. p multiplied by 4
4p b + 14
Tuesday
5
3
Wednesday
8
No; the models show that the temperature in Phoenix is 1 degree less than the temperature in Tucson.
18. a take away 16 19. k less than 24
24 - k
20. 3 groups of w
3w 1+q
21. the sum of 1 and q
13 __ z
22. the quotient of 13 and z
ESSENTIAL QUESTION CHECK-IN
45 + c
9. How can you use expressions and models to determine if expressions are equivalent?
Write a phrase in words for each algebraic expression. Sample answers given.
Write or model the variables, constants, and operations to represent each
24. m + 83
expression. Then compare the expressions or models.
25. 42s Lesson 11.1
297
298
32. Write an expression that represents Sarah’s total pay last week. Represent her hourly wage with w.
5w + 3w
90x a - 16
17. 90 times x
8. Are the expressions that represent the temperatures in the two cities equivalent? Justify your answer.
? ?
Monday
Noah
t
t- 4
Sarah
Write each phrase as an algebraic expression.
16. b plus 14
t- 2- 3
k less than 5
Read On Bookstore Work Schedule (hours)
x
Variable(s)
the quotient of h and 12
Sarah and Noah work at Read On Bookstore and get paid the same hourly wage. The table shows their work schedule for last week.
15
Constant(s)
the product of 11 and x
12
13. Identify the parts of the algebraic expression x + 15.
7. Represent the temperature in each city with an algebraic expression and a bar model.
28. 2 + g
31. 5 − k
12. What are the variables in the expression x + 8 − y?
m 4
t minus 29 g more than 2
h 30. __
x and y
At 6 p.m., the temperature in Phoenix, AZ, t, is the same as the temperature in Tucson, AZ. By 9 p.m., the temperature in Phoenix has dropped 2 degrees and in Tucson it has dropped 4 degrees. By 11 p.m., the temperature in Phoenix has dropped another 3 degrees. (Example 3)
27. t − 29
29. 11x
Sample answer: x + 24 − y
6. Draw a bar model to represent the expression m ÷ 4. (Example 2)
Phoenix
d
Online Assessment and Intervention
9 divided by d
26. __9
10. Write an algebraic expression with the constant 7 and the variable y.
Write a phrase for each algebraic expression. (Example 1) Sample 4. y + 12
my.hrw.com
33. Write an expression that represents Noah’s total pay last week. Represent his hourly wage with w.
8w 34. Are the expressions equivalent? Did Sarah and Noah earn the same amount last week? Use models to justify your answer.
Yes, Sarah and Noah got paid the same amount last week. Check
© Houghton Mifflin Harcourt Publishing Company
7+ 3
y−3
Personal Math Trainer
6.7.C
Write each phrase as an algebraic expression. (Example 1) 2. 3 less than y
Date
11.1 Independent Practice
1. Write an expression in the right side of the scale that will keep the scale balanced. (Explore Activity)
Class
students’ models.
83 added to m 42 times s
Unit 4
DIFFERENTIATE INSTRUCTION Cognitive Strategies
Cooperative Learning
Ask students for the meanings of the words variable and constant in a context such as the following: The air temperature in the desert was quite variable yesterday; it was cold overnight and warm during the day. The temperature at the equator was constant for 24 hours. Explain that the words have the same meaning in mathematics. A constant is a value that does not change, such as the number 5, and a variable is a symbol for a quantity that is not fixed, such as x.
Have students work in pairs to draw models of balance scales as shown in the Explore Activity. Instruct pairs to take turns writing simple expressions and drawing circles to represent them on the balance pans on each side of the scale. Then ask each student to write and illustrate two sets of equivalent expressions that balance when arranged on the scales. Invite pairs to explain how they chose their arrangements of circles.
Differentiated Instruction includes: • Reading Strategies • Success for English Learners ELL • Reteach • Challenge PRE-AP
Modeling Equivalent Expressions
298
Personal Math Trainer Online Assessment and Intervention
Evaluate GUIDED AND INDEPENDENT PRACTICE
Online homework assignment available
6.7.C
my.hrw.com
11.1 LESSON QUIZ 6.7.C 1. Write each phrase as an algebraic expression: z times 5 6 plus n 2. Write a phrase for each algebraic expression: 6 8y __ m
Concepts & Skills
Practice
Explore Activity Modeling Equivalent Expressions
Exercise 1
Example 1 Writing Algebraic Expressions
Exercises 2–5, 14–33
Example 2 Modeling Algebraic Expressions
Exercise 6
Example 3 Comparing Expressions Using Models
Exercises 7–8, 34–37
3. Use a bar model to represent 4 + m. 4. Jan and Jackie check out the same number of library books. Jan turns in 4 books after 3 weeks. Jackie returns 2 books that week and 4 books later. Write algebraic expressions and draw bar models to represent the books Jan and Jackie have left. Do they have the same number of books left? Justify your answer. Lesson Quiz available online my.hrw.com
1. 5z, 6 + n 2. 8 multiplied by y, 6 divided by m 4+m
3.
Exercise
Depth of Knowledge (D.O.K.)
10–13
2 Skills/Concepts
1.F Analyze relationships
14–33
2 Skills/Concepts
1.C Select tools
34–36
3 Strategic Thinking
1.F Analyze relationships
37–40
2 Skills/Concepts
1.A Everyday life
41
3 Strategic Thinking
1.G Explain and justify arguments
42
3 Strategic Thinking
1.G Explain and justify arguments
43
3 Strategic Thinking
1.E Create and use representations
44
3 Strategic Thinking
1.A Everyday life
45
3 Strategic Thinking
1.F Analyze relationships
Additional Resources Differentiated Instruction includes: • Leveled Practice Worksheets m
4
4. No. Sample answer: The expressions are not equivalent. If b = the number of books each checked out, Jan’s books = b - 4 and Jackie’s books = b - 2 - 4 b
b-4
b-2-4
299
Lesson 11.1
4 b
2
4
Mathematical Processes
DO NOT EDIT--Changes must be made through “File info” CorrectionKey=A
DO NOT EDIT--Changes must be made through “File info” CorrectionKey=A
35. Critique Reasoning Lisa concluded that 3 · 2 and 32 are equivalent expressions. Is Lisa correct? Explain.
39. Abby baked 48 cookies and divided them evenly into bags. Let b represent the number of bags. Write an algebraic expression to represent the number of cookies in each bag.
No; 3 · 2 = 6 and 32 = 9, 6 ≠ 9; The expressions are not
40. Eli is driving at a speed of 55 miles per hour. Let h represent the number of hours that Eli drives at this speed. Write an algebraic expression to represent the number of miles that Eli travels during this time.
equal. 36. Multiple Representations How could you represent the expressions x - 5 and x - 3 - 3 on a scale like the one you used in the Explore Activity? Would the scale balance?
41. Represent Real-World Problems If the number of shoes in a closet is s, then how many pairs of shoes are in the closet? Explain.
x - 3 - 3 on the right side; no; you are removing 5 from
x on the left side and you are removing 6 from x on
_s ; there are half as many pairs of shoes as there are total 2
the right.
shoes. 42. Communicate Mathematical Ideas Is 12x an algebraic expression? Explain why or why not.
37. Multistep Will, Hector, and Lydia volunteered at the animal shelter in March and April. The table shows the number of hours Will and Hector volunteered in March. Let x represent the number of hours Lydia volunteered in March.
Sample answer: 12x is an algebraic expression because it contains a variable.
March Volunteering Will
3 hours
Hector
5 hours
43. Problem Solving Write an expression that has three terms, two different variables, and one constant.
Sample answer: 2x - 8y + 7.
a. Will’s volunteer hours in April were equal to his March volunteer hours plus Lydia’s March volunteer hours. Write an expression to represent Will’s volunteer hours in April.
44. Represent Real-World Problems Describe a situation that can be modeled by the expression x − 8.
Sample answer: Sam started the day with a pack of gum. During the day he gave out 8 pieces of gum.
b. Hector’s volunteer hours in April were equal to 2 hours less than his March volunteer hours plus Lydia’s March volunteer hours. Write an expression to represent Hector’s volunteer hours in April.
5-2+x
45. Critique Reasoning Ricardo says that the expression y + 4 is equivalent to the expression 1y + 4. Is he correct? Explain.
c. Did Will and Hector volunteer the same number of hours in April? Explain.
Sample answer: Yes; 1y is the product of 1 and y. Since 1 times any number is equal to the number, 1 · y = y. The
Yes; The expressions are equivalent.
expression y + 4 is equivalent to 1y + 4.
38. The town of Rayburn received 6 more inches of snow than the town of Greenville. Let g represent the amount of snow in Greenville. Write an algebraic expression to represent the amount of snow in Rayburn.
g+6
Lesson 11.1
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1. This is a list
300
299
24/12/12 12:08 PM
EXTEND THE MATH
Unit 4
6_MTXESE051676_U4M11L1.indd 300
InCopy Notes
InDesign Notes
PRE-AP 1. This is a list
© Houghton Mifflin Harcourt Publishing Company
© Houghton Mifflin Harcourt Publishing Company
3+x
InCopy Notes
55h Work Area
FOCUS ON HIGHER ORDER THINKING
Represent x - 5 on the left side of the scale and
48 __ b
1. This is aonline list Bold, Italic, Strickthrough. my.hrw.com Activity available
10/25/12 3:05 PM
InDesign Notes 1. This is a list
Activity Each season the Ravens and the Hawks baseball teams play the same number of games. So far this year, the Ravens have played 3 games at home and 4 games on the road. The Hawks have played 5 games at home and 3 games on the road. Roberto drew these bar models and says that the Hawks have more games left to play in the season than the Ravens do. Is he correct? If not, what did he do incorrectly when he represented the given information? Ravens:
g
g–5–3
5
Hawks:
3
g
g–3–4
3
4
No, Roberto is not correct. He mixed up the data and mislabeled both models. 1) The tops of both bars should be labeled as g, the variable that stands for games left to play. 2) The Ravens model should have the labels g - 3 - 4, 3, and 4. 3) The Hawks model should have the labels g - 5 - 3, 5 and 3. Comparing the corrected models shows that the expressions are not equivalent: 7 < 8. So, the Ravens have one more game to play than the Hawks do. Modeling Equivalent Expressions
300
LESSON
11.2 Evaluating Expressions Texas Essential Knowledge and Skills The student is expected to:
Engage
ESSENTIAL QUESTION How can you use the order of operations to evaluate algebraic expressions? Sample answer: Substitute the given value for the variable in the expression and then use the order of operations to find the value of the resulting numerical expression.
Expressions, equations, and relationships—6.7.A Generate equivalent numerical expressions using order of operations, including whole number exponents and prime factorization.
Motivate the Lesson Ask: How can you evaluate an expression that includes an unknown value? Begin Example 1 to find out.
Mathematical Processes 6.1.G Display, explain, and justify mathematical ideas and arguments using precise mathematical language in written or oral communication.
Explore Engage with the Whiteboard
Write the expression 2(4 + x) - 5 on the whiteboard. Ask a student to evaluate the expression for x = 2. Then ask the student to explain his/her reasoning. Ask the class if the student’s work and reasoning are correct. If students have difficulty understanding which operation to perform first, review the order of operations.
ADDITIONAL EXAMPLE 1 Evaluate each expression for the given value of the variable. A b - 7; b = 16 28 B __ m; m = 4
9
EXAMPLE 1
7
C 0.2t; t = 1.6
Explain Focus on Math Connections
Mathematical Processes C and D involve an expression with a coefficient, a number that is multiplied by the variable. In the expression 0.5y, the coefficient is 0.5.
0.32
D 8s: s = __12 4 Interactive Whiteboard Interactive example available online my.hrw.com
Questioning Strategies
Mathematical Processes • How might substituting a negative value into one of the expressions affect its value? Substituting a negative value could change the sign of the answer and make it larger or smaller. In A, for example, if x = -15, the answer would be -24.
YOUR TURN ADDITIONAL EXAMPLE 2 Evaluate each expression for the given value of the variable. A 6(y - 6); y = 9 B 6y - 6; y = 9
18 48
C n - y + x; n = 5; y = 3; x = 4 D y - 2y; y = 7 2
E 4y - 3; y = -7
6
35
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Lesson 11.2
Check for Understanding Ask: How do you evaluate an expression for a given variable? Substitute the given value for the variable in the expression. Then perform the operations, using the order of operations to find the value of the expression.
EXAMPLE 2 Focus on Math Connections
-31
Interactive Whiteboard Interactive example available online
301
Mathematical Processes Remind students of the correct order of operations (parentheses, exponents, multiplication/ division, addition/subtraction) and the mnemonic device PEMDAS.
Questioning Strategies
Mathematical Processes • The answers to A and B are not the same, even though the expressions are very similar. Why? The parentheses in 4(x - 4) mean that you subtract first. There are no parentheses in 4x - 4, so you multiply first.
11.2 ?
Evaluating Expressions
ESSENTIAL QUESTION
Expressions, equations, and relationships—6.7.A Generate equivalent numerical expressions using order of operations, including whole number exponents and prime factorization.
Evaluate each expression for the given value of the variable.
Online Assessment and Intervention
Math On the Spot my.hrw.com
Math On the Spot my.hrw.com
A x - 9; x = 15
6
Subtract.
No; in w - x + y, you subtract x from w first. Then you add y to that result. In w - y + x, you subtract y from w first. Then you add x to that result.
When x = 15, x - 9 = 6. 16 B __ n ;n=8 16 __ 8
Substitute 8 for n.
2
Divide.
© Houghton Mifflin Harcourt Publishing Company
16 When n = 8, __ n = 2.
C 0.5y; y = 1.4 0.5(1.4)
Substitute 1.4 for y.
0.7
Multiply.
Math Talk
When y = 1.4, 0.5y = 0.7.
Mathematical Processes
Is w - x + y equivalent to w - y + x? Explain any difference in the order the math operations are performed.
D 6k; k = _13 6 . HINT: Think of 6 as __ 1
()
6 _13
1 for k. Substitute __ 3
2
Multiply.
m ; m = 18 3. __ 6
3
Expressions may have more than one operation or more than one variable. To evaluate these expressions, substitute the given value for each variable and then use the order of operations.
EXAMPLE 2
6.7.A
A 4(x - 4); x = 7
Evaluate each expression for the given value of the variable.
Substitute 15 for x.
4.7
Evaluate each expression for the given value of the variable.
6.7.A
15 - 9
2. 6.5 - n; n = 1.8
Using the Order of Operations
Evaluating Expressions
EXAMPL 1 EXAMPLE
32
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How can you use the order of operations to evaluate algebraic expressions?
Recall that an algebraic expression contains one or more variables. You can substitute a number for that variable and then find the value of the expression. This is called evaluating the expression.
1. 4x; x = 8
When k = _13, 6k = 2.
4(7 - 4)
Substitute 7 for x.
4(3)
Subtract inside the parentheses.
12
Multiply.
When x = 7, 4(x - 4) = 12.
B 4x - 4; x = 7 4(7) - 4
Substitute 7 for x.
28 - 4
Multiply.
24
Subtract.
When x = 7, 4x - 4 = 24.
C w - x + y; w = 6, x = 5, y = 3 (6) - (5) + (3)
Substitute 6 for w, 5 for x, and 3 for y.
1+3
Subtract.
4
When w = 6, x = 5, y = 3, w - x + y = 4.
D x2 - x; x = 9 (9)2 - (9)
Substitute 9 for each x.
81 - 9
Evaluate exponents.
72
© Houghton Mifflin Harcourt Publishing Company
LESSON
Subtract.
When x = 9, x - x = 72. 2
Lesson 11.2
301
302
Unit 4
PROFESSIONAL DEVELOPMENT Integrate Mathematical Processes This lesson provides an opportunity to address Mathematical Process TEKS 6.1.G, which calls for students to “display, explain, and justify mathematical ideas … using precise mathematical language in written … communication. ” In each Example and Exercise, students use mathematical ideas and language, including the order of operations, to evaluate algebraic expressions by substituting given values for variables. Students then evaluate real-world expressions such as formulas for finding surface area and volume and converting Celsius temperatures to Fahrenheit.
Math Background We translate (or write) words into algebraic expressions by using a consistent, universally understood system. This system has evolved over thousands of years. Archaeological records indicate that Babylonian mathematicians had developed prose-based algebra by 2000 B.C. The adoption of symbols to represent operations was also part of this evolution. The symbols + and - can be traced to Johann Widman (1498); the symbol · can be traced to Gottfried Leibniz (1698); and the symbol ÷ can be traced to Johann Heinrich Rahn (1659).
Evaluating Expressions
302
YOUR TURN Avoid Common Errors Exercises 7–9 Watch for students who substitute the wrong value for the variable. Caution students to be sure that they are substituting the correct value for each variable in expressions with more than one variable.
ADDITIONAL EXAMPLE 3 The expression 3.3m gives the number of feet in m (meters). Use the expression to find the number of feet that is equivalent to 400 meters. 1,320 feet Interactive Whiteboard Interactive example available online my.hrw.com
EXAMPLE 3 Focus on Math Connections
Mathematical Processes Remind students that when there is a coefficient in front of a variable, multiplication is indicated. Therefore, when they replace the variable with a value, they need to insert parentheses. In Step 2, for example, the expression 1.8c + 32 is written as 1.8(30) + 32 when substituting 30 for c.
Questioning Strategies
Mathematical Processes • How do you find the value of the variable c? Read the text of the problem carefully. The value of the variable is given in the last sentence.
• If the expression were written as 32 + 1.8c, would you still perform the multiplication first? Explain. Addition is associative, so changing the order of the terms does not change the expression in any way. You still need to do the multiplication before the addition. c.3.B ELL Be sure English learners understand the references to Celsius and Fahrenheit in Example 3. You may want to point out that both scales measure temperature but the Celsius Scale is part of the Metric System.
Integrating the ELPS
YOUR TURN Avoid Common Errors Exercise 10 Remind students that an exponent tells how many times to use the base as a factor, so x3 means x · x · x, not 3x.
Elaborate Talk About It Summarize the Lesson Ask: How can the order of operations help you evaluate algebraic expressions? When an expression contains more than one operation, the order of operations tells which operation to perform first.
GUIDED PRACTICE Engage with the Whiteboard For Exercises 7–8, have students circle all the important information provided, including key words that indicate operations, on the whiteboard. Then have them write an expression to represent each problem. Finally, have them complete the steps to evaluate each problem.
Avoid Common Errors Exercise 2 Remind students that when there is a coefficient in front of a variable, multiplication is indicated. Therefore, when they replace the variable with a value, they need to insert parentheses. Exercises 3, 5 Remind students that the fraction bar is another way to represent division.
303
Lesson 11.2
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Guided Practice
YOUR TURN Evaluate each expression for n = 5. 5. 4 (n - 4) + 14
18
6. 6n + n2
18
8. bc + 5a
-9
9. a2 - (b + c)
5. _12 w + 2; w = _19
6.7.A
The expression 1.8c + 32 gives the temperature in degrees Fahrenheit for a given temperature in degrees Celsius c. Find the temperature in degrees Fahrenheit that is equivalent to 30 °C.
my.hrw.com
Find the value of c.
12(
Substitute the value into the expression.
1.8(30) + 32
Substitute 30 for c.
54 + 32
Multiply.
86
3
)+5=
36
+5=
The family spent \$41
\$6
Nonstudent tickets
\$12
Parking
\$5
41
to attend the game.
a. Write an expression that represents the perimeter of the rectangular tablecloth. Let l represent the length of the tablecloth and w represent its width. The expression would be
2w + 2l
.
b. Evaluate the expression P = 2w + 2l for l = 5 and w = 7.
2(
10. The expression 6x2 gives the surface area of a cube, and the expression x3 gives the volume of a cube, where x is the length of one side of the cube. Find the surface area and the volume of a cube with a side length of 2 m.
24
m2 ; V =
8
seconds
7
) + 2(
Stan bought
5
) = 14 +
24 feet
10
=
24
of trim to sew onto the tablecloth.
9. Essential Question Follow Up How do you know the correct order in which to evaluate algebraic expressions?
m3
11. The expression 60m gives the number of seconds in m minutes. How many seconds are there in 7 minutes?
420
Women’s Soccer Game Prices Student tickets
8. Stan wants to add trim all around the edge of a rectangular tablecloth that measures 5 feet long by 7 feet wide. The perimeter of the rectangular tablecloth is twice the length added to twice the width. How much trim does Stan need to buy? (Example 3)
86 °F is equivalent to 30 °C.
S=
50
b. Since there are three attendees, evaluate the expression 12x + 5 for x = 3.
1.8c + 32
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6. 5(6.2 + z); z = 3.8
12x + 5 is an expression that represents the cost of one carful of nonstudent soccer fans.
c = 30 °C STEP 2
10.5
4. 9 + m; m = 1.5
1 2__ 18
a. Write an expression that represents the cost of one carful of nonstudent soccer fans. Use x as the number of people who rode in the car and attended the game.
Math On the Spot
6
2. 3a - b; a = 4, b = 6
7. The table shows the prices for games in Bella’s soccer league. Her parents and grandmother attended a soccer game. How much did they spend if they all went together in one car? (Example 3)
You can evaluate expressions to solve real-world problems.
STEP 1
2
3. _8t ; t = 4
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11
Evaluating Real-World Expressions EXAMPL 3 EXAMPLE
16
1. x - 7; x = 23
Online Assessment and Intervention
Evaluate each expression for a = 3, b = 4, and c = -6. 7. ab - c
Evaluate each expression for the given value(s) of the variable(s). (Examples 1 and 2)
Personal Math Trainer
55
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18
4. 3(n + 1)
Substitute for the variables and follow the order Personal Math Trainer
of operations that you would use for a numerical
Online Assessment and Intervention
expression.
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1. This is a list
Home Connection
Cooperative Learning
Have students record real-world math situations they experience at home, using both words and mathematical symbols.
Have students work in groups to solve a magic square. A magic square is an array of numbers in which each row, column, and diagonal has the same sum. Ask students if the array below is a magic square if x = 4; if x = 6; or if x = 0.
Differentiated Instruction includes: • Reading Strategies • Success for English Learners ELL • Reteach • Challenge PRE-AP
Sample answer: Mom works out for the same length of time each day. How long does she work out in a week? 7t, where t represents the length of time she works out each day.
x+7
x
x+2
0.5x + 6
3x - 5
3x
2x + 1 x+6
x+1
The array is a magic square if x = 4, but not if x = 6 or if x = 0.
Evaluating Expressions
304
Personal Math Trainer Online Assessment and Intervention
Online homework assignment available
Evaluate GUIDED AND INDEPENDENT PRACTICE 6.7.A
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11.2 LESSON QUIZ 6.7.A Evaluate each expression for the given value(s) of the variable(s). 1. a + 6; a = 2
Concepts & Skills
Practice
Example 1 Evaluating Expressions
Exercises 1–6
Example 2 Using the Order of Operations
Exercises 1–6, 13
Example 3 Evaluating Real-World Expressions
Exercises 7–8, 10–12, 14, 16
16 2. __ g ;g = 4
3. 7(m - 6); m = 8 4. 7m - 6; m = 8
Exercise
5. s - k + x; s = 7, k = 4, x = 6 6. The expression 4g gives the number of quarts in g gallons. How many quarts are there in 4 gallons? Lesson Quiz available online my.hrw.com
Answers 1. 8 3. 14 4. 50 5. 9 6. 16 quarts
Lesson 11.2
Mathematical Processes
2 Skills/Concepts
1.A Everyday life
13
3 Strategic Thinking
1.G Explain and justify arguments
14–16
3 Strategic Thinking
1.A Everyday life
17
3 Strategic Thinking
1.G Explain and justify arguments
18
3 Strategic Thinking
1.F Analyze relationships
10–12
Additional Resources Differentiated Instruction includes: • Leveled Practice Worksheets
2. 4
305
Depth of Knowledge (D.O.K.)
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Class
Date
11.2 Independent Practice 6.7.A
Movie 16 Ticket Prices
x
0
1
2
3
4
5
6
\$8.75
6x - x2
0
5
8
9
8
5
0
Children
\$6.50
Seniors
\$6.50
The value of 6x - x2 increases
8.75a + 6.5c + 6.5s or 8.75a +
6.5(c + s)
b. The Andrews family bought 2 adult tickets, 3 children’s tickets, and 1 senior ticket. Evaluate your expression in part a to find the total cost of the tickets.
8.75(2) + 6.5(3) + 6.5(1) =
© Houghton Mifflin Harcourt Publishing Company
8.75(2) + 6.5(3 + 1) = \$43.50 c. The Spencer family bought 4 adult tickets and 2 children’s tickets. Did they spend the same as the Andrews family? Explain.
No; 8.75(4) + 6.5(2) = \$48.00
11. The area of a triangular sail is given by the expression _21 bh, where b is the length of the base and h is the height. What is the area of a triangular sail in a model sailboat when b = 12 inches and h = 7 inches?
42
in.2
V = 5,760 ft3
13. Look for a Pattern Evaluate the expression 6x - x2 for x = 0, 1, 2, 3, 4, 5, and 6. Use your results to fill in the table and describe any pattern that you see.
a. Write an expression for the total cost of tickets.
A=
Online Assessment and Intervention
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10. The table shows ticket prices at the Movie 16 theater. Let a represent the number of adult tickets, c the number of children’s tickets, and s the number of senior citizen tickets.
16. The volume of a pyramid with a square base is given by the expression _13s2h, where s is the length of a side of the base and h is the height. Find the volume of a pyramid with a square base of side length 24 feet and a height of 30 feet.
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17. Draw Conclusions Consider the expressions 3x(x - 2) + 2 and 2x2 + 3x - 12. a. Evaluate each expression for x = 2 and for x = 7. Based on your results, do you know whether the two expressions are equivalent? Explain.
from 0 at x = 0 to 9 at x = 3, then
decreases back to 0 at x = 6. Also,
For x = 2, each expression has a value of 2. For x = 7,
and 6, for x = 1 and 5, and for
suggest that the expressions may be equivalent but
the values are the same for x = 0
each expression has a value of 107. These results
x = 2 and 4.
do not prove that the expressions are equivalent.
14. The kinetic energy (in joules) of a moving object can be calculated from the expression _12mv2, where m is the mass of the object in kilograms and v is its speed in meters per second. Find the kinetic energy of a 0.145-kg baseball that is thrown at a speed of 40 meters per second. E=
116
b. Evaluate each expression for x = 1. Based on your results, do you know whether the two expressions are equivalent? Explain.
For x = 1, the 1st expression has a value of -1 and the 2nd expression has a value of -7. Because
the values are different, the expressions are not
joules
equivalent.
15. The area of a square is given by x2, where x is the length of one side. Mary’s original garden was in the shape of a square. She has decided to double the area of her garden. Write an expression that represents the area of Mary’s new garden. Evaluate the expression if the side length of Mary’s original garden was 8 feet.
18. Critique Reasoning Marjorie evaluated the expression 3x + 2 for x = 5 as shown: 3x + 2 = 35 + 2 = 37 What was Marjorie’s mistake? What is the correct value of 3x + 2 for x = 5?
2(x2); 2(64) = 128 square feet
3x means that 3 should be multiplied by the value
12. Ramon wants to balance his checking account. He has \$2,340 in the account. He writes a check for \$140. He deposits a check for \$268. How much does Ramon have left in his checking account?
of x; 17
\$2,468 Lesson 11.2
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Work Area
FOCUS ON HIGHER ORDER THINKING
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Name
EXTEND THE MATH
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Activity Harold has a globe that has a diameter of 10 inches. He builds a globe with a radius twice as long as his original globe. Use the formulas below to find the surface area and the volume of the new globe. Use π = 3.14. Explain your process and show your work. Surface Area (SA) of a sphere = 4πr2, where r is the radius of the sphere. Volume (V) of a sphere = __43 πr3, where r is the radius of the sphere. Sample answer: First, find the radius of the new globe. The diameter of the original globe is 10 inches, so its radius is 10 ÷ 2 = 5 inches. The radius of the new globe is twice as long: 5 inches × 2 = 10 inches. Then use r = 10 inches as the value to substitute for r in each formula. SA of new globe is 4πr2 ≈ 4(3.14)(10)2 ≈ 4(3.14)(100) ≈ 4(314) ≈ 1,256 in2; V of new globe is __43 πr3 ≈ __43 (3.14)(10)3 ≈ __43 (3.14)(1,000) ≈ __43 (3,140) ≈ 4,186 __23 in3
Evaluating Expressions
306
LESSON
11.3 Generating Equivalent Expressions Texas Essential Knowledge and Skills The student is expected to: Expressions, equations, and relationships—6.7.C Determine if two expressions are equivalent using concrete models, pictorial models, and algebraic representations.
Engage
ESSENTIAL QUESTION How can you identify and write equivalent expressions? Sample answer: Substitute the same value into each expression and compare the results, or simplify each expression to see if they are equivalent.
Motivate the Lesson
Ask: Are the expressions 3x + 8 and 2x + 14 equivalent expressions for x = 6? Begin the Explore Activity to find out.
Expressions, equations, and relationships—6.7.D Generate equivalent expressions using the properties of operations: inverse, identity, commutative, associative, and distributive properties.
Explore
Mathematical Processes
EXPLORE ACTIVITY 1
6.1.D Communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate.
Mathematical Processes Focus on Critical Thinking Mathematical Processes Point out to students that even though expressions share an element, such as 5x, and include the same operation, they are not necessarily equivalent expressions. For example, x 2 and x 3 have the same base and look similar but they are not equivalent; x 2 = x · x and x 3 = x · x · x.
Explain EXPLORE ACTIVITY 2 Focus on Modeling Mathematical Processes Point out that the number and arrangement of the algebra tiles mirrors each expression. The first model shows three groups of 1 variable plus 2 ones, and the second model shows 3 variables plus 6 ones. Discuss with students why the two algebraic expressions are equivalent. Questioning Strategies
Mathematical Processes • Earlier, you used counters to make models. How are algebra tiles similar to counters? How are they different? Both counters and algebra tiles are used the same way, one tile or counter for each number being added. The difference is that algebra tiles have an x (variable) tile used to represent an unknown quantity, while counters represent +1 or -1 only. • Which two characteristics do you look for in the models to decide whether the two expressions are equivalent? The models of both expressions should have 1) an equal number of x tiles and 2) an equal number of +1-tiles or -1-tiles.
YOUR TURN Avoid Common Errors Remind students to arrange the tiles in the same groups and order as shown in the expressions. Then compare the number of x tiles and the number of +1-tiles or –1-tiles in each model to check whether the expressions are equivalent.
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Generating Equivalent Expressions
LESSON
11.3 ?
EXPLORE ACTIVITY 1 (cont’d)
Expressions, equations, and relationships— 6.7.D Generate equivalent expressions using the properties of operations: inverse, identity, commutative, associative, and distributive properties. Also 6.7.C
Reflect 1.
Evaluate the expressions for a different value of x; for
ESSENTIAL QUESTION
example, when x = 1, 2x = 2 and x2 = 1.
How can you identify and write equivalent expressions?
EXPLORE ACTIVITY 1
Error Analysis Lisa evaluated the expressions 2x and x2 for x = 2 and found that both expressions were equal to 4. Lisa concluded that 2x and x2 are equivalent expressions. How could you show Lisa that she is incorrect?
6.7.C
EXPLORE ACTIVITY 2
Identifying Equivalent Expressions One way to test whether two expressions might be equivalent is to evaluate them for the same value of the variable.
Modeling Equivalent Expressions
List B
5x + 65
5x + 1
5(x + 1)
5x + 5
1 + 5x
5(13 + x)
Algebra Tiles
You can also use models to determine if two expressions are equivalent. Algebra tiles are one way to model expressions.
Match the expressions in List A with their equivalent expressions in List B. List A
6.7.C
=1 = -1 =x
Determine if the expression 3(x + 2) is equivalent to 3x + 6.
A Model each expression using algebra tiles. 3(x + 2)
3x + 6
A Evaluate each of the expressions in the lists for x = 3.
© Houghton Mifflin Harcourt Publishing Company
5(3) + 65 = 5(3 + 1) = 1 + 5(3) =
List B 5(3) + 1 =
16
20
5(3) + 5 =
20
16
5(13 + 3) =
80
3
B The model for 3(x + 2) has The model for 3x + 6 has
80
3
6
x tiles and x tiles and
6
1 tiles. 1 tiles.
C Is the expression 3(x + 2) equivalent to 3x + 6? Explain.
Yes; each expression is represented by 3 x tiles and
B Which pair(s) of expressions have the same value for x = 3?
5(x + 1) and 5x + 5; 1 + 5x and 5x + 1; 5x + 65 and
6 1 tiles.
5(13 + x)
Reflect
C How could you further test whether the expressions in each pair are equivalent?
2.
Use algebra tiles to determine if 2(x - 3) is equivalent to 2x - 3. Explain your answer.
© Houghton Mifflin Harcourt Publishing Company
List A
No; 2(x - 3) is represented by 2 x tiles and 6 -1 tiles.
Sample answer: Evaluate for several other values of x.
2x - 3 is represented by 2 x tiles and 3 -1 tiles.
D Do you think the expressions in each pair are equivalent? Why or why not?
Sample answer: Yes; it appears that they will always have the same value. Lesson 11.3
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Unit 4
PROFESSIONAL DEVELOPMENT Integrate Mathematical Processes This lesson provides an opportunity to address Mathematical Process TEKS 6.1.D, which calls for students to “communicate mathematical ideas… using multiple representations, including symbols, diagrams…and language as appropriate.” In this lesson’s Explore Activities and Examples, students use words and operational symbols as well as algebra tiles to identify, represent, and compare algebraic expressions and to generate equivalent expressions.
Math Background Algebraic expressions are equivalent if they simplify to the same value. The Commutative, Associative, Distributive, and Identity properties give rules about how to rewrite an expression without changing its value. x + (3x + 2) + (2x + 3) x + 3x + (2 + 2x) + 3 x + 3x + (2x + 2) + 3 (x + 3x + 2x) + (2 + 3) x(1 + 3 + 2) + (2 + 3) x(6) + (5) 6x + 5
Associative Commutative Associative Distributive Addition Commutative
Generating Equivalent Expressions
308
ADDITIONAL EXAMPLE 1 Use a property to write an expression that is equivalent to n × 5. Tell which property you used. 5 × n; Commutative Property of Multiplication Interactive Whiteboard Interactive example available online my.hrw.com
EXAMPLE 1 Questioning Strategies
Mathematical Processes • How does the Commutative Property of Addition allow you to rewrite an expression without changing its value? You can change the order of the terms in an addition expression. • How do you know which properties of operations may help you identify equivalent expressions? Look at the operational symbols that appear in the given expressions. Apply properties of operations that relate to those symbols.
Engage with the Whiteboard Have students circle the operational symbol in Example 1 and write an equivalent expression on the whiteboard. Then have students draw a model of each expression, using algebra tiles to show that the expressions are equivalent.
Focus on Communication
Mathematical Processes Make sure students understand that the properties of operations are rules about how to rewrite expressions by rearranging and combining terms without changing the value of the expression.
YOUR TURN Avoid Common Errors If students have difficulty determining which property to use, remind them to begin by identifying the operation used in the given expression. Then they should look at the list of properties to see which properties apply to that operation. Point out that a given expression may have more than one equivalent expression, as more than one property can be applied.
ADDITIONAL EXAMPLE 2 Use the properties of operations to determine if the expressions are equivalent. A 6 + y; __12 (12 + y) B 2(y - 3); 2y - 6
not equivalent equivalent
Interactive Whiteboard Interactive example available online my.hrw.com
Animated Math Equivalent Expressions Students explore equivalent expressions using an interactive model. my.hrw.com
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Lesson 11.3
EXAMPLE 2 Questioning Strategies
Mathematical Processes • In A, how does the Distributive Property enable you to determine that the expressions are equivalent without evaluating them? The Distributive Property states that multiplying a number by a difference, as in 3(x - 2), is the same as multiplying each number in the difference and subtracting the products: (3)(x) - (3)(2) = 3x - 6.
YOUR TURN Engage with the Whiteboard For Exercises 5–6, have students use algebra tiles to draw a model of each expression on the whiteboard. Then have students explain whether the expressions are equivalent or not.
DO NOT EDIT--Changes must be made through “File info” CorrectionKey=B
DO NOT EDIT--Changes must be made through “File info” CorrectionKey=A
Identifying Equivalent Expressions Using Properties
Writing Equivalent Expressions Using Properties Properties of operations can be used to identify equivalent expressions. Examples 3+4=4+3
Commutative Property of Multiplication: When multiplying, changing the order of the numbers does not change the product.
2×4=4×2
Associative Property of Addition: When adding more than two numbers, the grouping of the numbers does not change the sum.
(3 + 4) + 5 = 3 + (4 + 5)
Associative Property of Multiplication: When multiplying more than two numbers, the grouping of the numbers does not change the product.
(2 × 4) × 3 = 2 × (4 × 3)
Identity Property of Multiplication: Multiplying a number by one does not change its value.
1×7=7
Inverse Property of Addition: The sum of a number and its opposite, or additive inverse, is zero.
- 3 + 3 = 0
© Houghton Mifflin Harcourt Publishing Company
EXAMPL 1 EXAMPLE
Math Talk
What property can you use to write an expression that is equivalent to 0 + c? What is the equivalent expression?
Sample answer: Represent 3 (x - 2) as three groups of x - 2 with each group showing one x tile and two - 1 tiles. To represent the Distributive Property, regroup the tiles to represent 3x - 6 by showing a group of three x tiles and a group of six - 1 tiles.
You can use the Commutative Property of Addition to write an equivalent expression: x + 3 = 3 + x
YOUR TURN For each expression, use a property to write an equivalent expression. Tell which property you used. Sample answers given.
a(bc); Associative Property of Multiplication
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Online Assessment and Intervention
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Distributive Property Commutative Property
YOUR TURN Use the properties of operations to determine if the expressions are equivalent. 5.
6x - 8; 2(3x - 5)
2(3x - 5) = 6x - 10;
6.
7.
2 - 2 + 5x; 5x
2 - 2 + 5x = 5x; equivalent
not equivalent
Jamal bought 2 packs of stickers and 8 individual stickers. Use x to represent the number of stickers in a pack of stickers and write an expression to represent the number of stickers Jamal bought. Is the expression equivalent to 2(4 + x)? Check your answer with algebra tile models.
Jamal bought 2x + 8 stickers. 2(4 + x) = 8 + 2x = 2x + 8; yes
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DIFFERENTIATE INSTRUCTION
Distributive Property
1 (4 + x) B 2 + x; __ 2 1x+ 2 1 (x + 4) = __ __ lk Ta th Ma 2 2 Mathematical Processes 1x = 2 + __ 2 Explain how you could use algebra tiles to 1 x. 2 + x does not equal 2 + __ represent the Distributive 2 Property in A. They are not equivalent expressions.
Mathematical Processes
6.7.D
(3 + 4)y; Distributive Property
3(x - 2) = 3x - 6
3(x 3 - 2) and 3x - 6 are equivalent expressions.
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The operation in the expression is addition.
4. 3y + 4y =
6.7.C
A 3(x - 2); 3x - 6
Use a property to write an expression that is equivalent to x + 3.
3. (ab)c =
EXAMPLE 2
1. This is a list Bold, Italic, Strickthrough.
10/25/12 8:35 PM
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Visual Cues
Cognitive Strategies
Point out to students that it is often helpful to use colored pencils to identify like terms before combining them.
A fun way for students to remember how to combine like terms is to name the variable part of like terms. For example, for the expression 2a + 5b + 4a, students can name variable a apples and variable b bananas. Thus, 2 apples + 5 bananas + 4 apples = 6 apples + 5 bananas = 6a + 5b.
Differentiated Instruction includes: • Reading Strategies • Success for English Learners ELL • Reteach • Challenge PRE-AP
Have students identify the like terms in the following expressions. 1. a + 2b + 2a + b + 2c 2b and b
Like terms: a and 2a,
© Houghton Mifflin Harcourt Publishing Company
9+0=9
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Animated Math
6(2 + 4) = 6(2) + 6(4) 8(5 - 3) = 8(5) - 8(3)
Identity Property of Addition: Adding zero to a number does not change its value.
Math On the Spot
my.hrw.com
Use the properties of operations to determine if the expressions are equivalent.
Properties of Operations Commutative Property of Addition: When adding, changing the order of the numbers does not change the sum.
Distributive Property: Multiplying a number by a sum or difference is the same as multiplying by each number in the sum or difference and then adding or subtracting.
Math On the Spot
2. 18 + 2d 3 + 5d + 3d 3 - 2d 2 Like terms: 2d 3 and 3d 3 3. 5x 3 + 3y + 7x 3 - 2y - 4x 2 Like terms: 5x 3 and 7x 3, 3y and 2y
Generating Equivalent Expressions
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ADDITIONAL EXAMPLE 3 Combine like terms.
EXAMPLE 3 Connect Vocabulary
A 8y2 - 3y2 5y2 B 4m + 3(n + 7m)
25m + 3n
C x + 8y - 5y + 5x
6x + 3y
Interactive Whiteboard Interactive example available online my.hrw.com
ELL
Stress the use of correct mathematical terminology. The parts of an expression that are separated by + or - signs, such as 3x and 5x in the expression 3x + 5x, are called terms. Terms that have identical variable parts are like terms. In the expression 3x + 5x, 3x and 5x are like terms. The properties of operations allow you to rearrange and combine like terms.
Questioning Strategies
Mathematical Processes • In B, why do you have to apply the Distributive Property before adding like terms? Because of the order of operations, you need to multiply before you can add. • In C, why does y + 7y equal 8y? Because y is 1y, so 1y + 7y = (1 + 7)y = 8y.
YOUR TURN Avoid Common Errors Exercise 10 Students may neglect to add single variables. Remind them that b is 1b, so adding b to another b-term increases the coefficient by 1.
Elaborate Talk About It Summarize the Lesson Ask: How do properties of operations help you to write equivalent expressions? Properties of operations allow me to write an expression in different ways without changing its value. I can use the properties to form equivalent expressions by regrouping, reordering, and combining terms.
GUIDED PRACTICE Engage with the Whiteboard For Exercise 1, have students rewrite each expression on the whiteboard by substituting 5 for y and then simplifying the expressions. For Exercise 2, ask a student to circle the part of the model that shows that the two expressions are not equivalent on the whiteboard.
Avoid Common Errors Exercises 3–4 If students have difficulty determining which property to use, remind them to begin by identifying the operation used in the given expression. Then they should look at the list of properties to see which properties apply to that operation. Exercises 5–6 Remind students that when they apply the Distributive Property they must distribute the constant or variable that is outside the parentheses to each term that is inside the parentheses. Exercise 8 Students may neglect to add single variables. Remind them that -x is -1x, so adding -x to another x-term decreases the coefficient by 1.
311
Lesson 11.3
Generating Equivalent Expressions
Parts of an algebraic expression
coefficients like terms
12 + 3y2 + 4x + 2y2 + 4 12 + 3y2 + 4x + 2y2 + 4
Math On the Spot
Online Assessment and Intervention
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EXAMPL 3 EXAMPLE
4 + 4y = 4(y − 1) =
2
6x and 4x are like terms.
Subtract inside the parentheses.
= 2x2
Commutative Property of Multiplication
B 3a + 2(b + 5a)
© Houghton Mifflin Harcourt Publishing Company
4y + 1 =
Distributive Property
= x2(2)
= 3a + 2b + (2 · 5)a
Distributive Property
= 3a + 2b + 10a
Associative Property of Multiplication Multiply 2 and 5.
= 3a + 10a + 2b
= (3 + 10)a + 2b
Distributive Property
= 13a + 2b
Math Talk
Mathematical Processes
Write 2 terms that can be combined with 7y4.
y + 11x - 7x + 7y = y + 7y + 11x - 7x
List B
24 16 21
For each expression, use a property to write an equivalent expression. Tell which property you used. (Example 1) Sample 3. ab =
Distributive Property Commutative Property
ba
+
x-4
2(x - 2)
+
+
answers are given. 5(3x) - 5(2)
4. 5(3x − 2) =
Distributive Prop.
Commutative Prop. of Mult.
not equivalent
6. _12(6x − 2); 3 − x
not equivalent
Combine like terms. (Example 3) 7. 32y + 12y =
= 8y + 4x
4(y + 1) = 1 + 4y =
16 24 21
2. Determine if the expressions are equivalent by comparing the models. (Explore Activity 2) not equivalent
5. _12(4 − 2x); 2 - 2x
y and 7y are like terms; 11x and 7x are like terms. Commutative Property
= y(1 + 7) + x(11 - 7)
4y − 4 =
Use the properties of operations to determine if each pair of expressions is equivalent. (Example 2)
3a + 2(b + 5a) = 13a + 2b
C y + 11x - 7x + 7y
8m + 2 + 4n
2a5 + 5b
List A
A 6x2 - 4x2
3a + 2(b + 5a) = 3a + 2b + 2(5a)
11. 8m + 14 - 12 + 4n =
10. 4a5 - 2a5 + 4b + b =
1. Evaluate each of the expressions in the list for y = 5. Then, draw lines to match the expressions in List A with their equivalent expressions in List B. (Explore Activity 1)
Combine like terms.
6x2 - 4x2 = 2x2
10x2 - 4
Guided Practice
6.7.D
6x2 - 4x2 = x2 (6 - 4)
9. 6x2 + 4(x2 - 1) =
12 + 3y2 + 4x + 2y2 + 4
When an expression contains like terms, you can use properties to combine the like terms into a single term. This results in an expression that is equivalent to the original expression.
2
5y
8. 8y - 3y =
? ?
44y
8. 12 + 3x − x − 12 =
2x
ESSENTIAL QUESTION CHECK-IN
© Houghton Mifflin Harcourt Publishing Company
The parts of the expression that are separated by + or - signs Numbers that are multiplied by at least one variable Terms with the same variable(s) raised to the same power(s)
terms
Combine like terms.
Personal Math Trainer
9. Describe two ways to write equivalent algebraic expressions.
Use properties of operations or combine like terms
y + 11x - 7x + 7y = 8y + 4x
Lesson 11.3
311
312
Unit 4
Generating Equivalent Expressions
312
Personal Math Trainer Online Assessment and Intervention
Online homework assignment available
Evaluate GUIDED AND INDEPENDENT PRACTICE 6.7.C, 6.7.D
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Concepts & Skills
Practice
Explore Activity 1 Identifying Equivalent Expressions
Exercise 1
Explore Activity 2 Modeling Equivalent Expressions
Exercises 2, 14
Example 1 Writing Equivalent Expressions Using Properties
Exercises 3–4, 10–13, 27–29
Use the properties of operations to determine if the expressions are equivalent.
Example 2 Identifying Equivalent Expressions Using Properties
Exercises 5–6, 25
2. 3 + y; __13 (6 + y)
Example 3 Generating Equivalent Expressions
Exercises 7–8, 15–24, 26
11.3 LESSON QUIZ 6.7.D 1. Use one of the properties of operations to write an expression that is equivalent to 4 + m. Tell which property you used.
3. 4(y - 3); 4y - 12 Combine like terms. 4. 9y2 - 6y2
Exercise
5. 5c + 4(d+ 6c)
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1. m + 4; Commutative Property of Addition 2. not equivalent 3. equivalent 4. 3y2 5. 29c + 4d 6. 5x + 6y
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Lesson 11.3
1.C Select tools
3 Strategic Thinking
1.G Explain and justify arguments
26–27
2 Skills/Concepts
1.A Everyday life
28–29
2 Skills/Concepts
1.D Multiple representations
30
3 Strategic Thinking
1.F Analyze relationships
31
3 Strategic Thinking
1.G Explain and justify arguments
32
3 Strategic Thinking
1.F Analyze relationships
25
Lesson Quiz available online
Mathematical Processes
2 Skills/Concepts
10–24
6. x + 10y - 4y + 4x
Depth of Knowledge (D.O.K.)
Additional Resources Differentiated Instruction includes: • Leveled Practice Worksheets
Class
Date
11.3 Independent Practice
Personal Math Trainer
6.7.D, 6.7.C
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27. Multiple Representations Use the information in the table to write and simplify an expression to find the total weight of the medals won by the top medal-winning nations in the 2012 London Olympic Games. The three types of medals have different weights.
Online Assessment and Intervention
2012 Summer Olympics
For each expression, use a property to write an equivalent expression. Tell which property you used. Sample answers given.
dc
10. cd =
12. 4(2x − 3) =
13 + x
11. x + 13 =
Commutative Prop. of Mult.
4(2x) - 4(3)
Distributive Prop.
Silver 29 27 17
113g + 73s + 71b
Write an expression for the perimeters of each given figure. Simplify the expressions.
14. Draw algebra tile models to prove that 4 + 8x and 4(2x + 1) are equivalent.
6x + 10 mm
28.
36.4 + 4x in.
29.
10.2 in.
3x - 1 mm 6 mm
4(2x + 1)
4 + 8x
x + 4 in. x + 4 in.
© Houghton Mifflin Harcourt Publishing Company
17. 6b + 7b − 10 =
13b - 10
16. 32y + 5y =
37y
18. 2x + 3x + 4 =
5x + 4 6a2 + 16
19. y + 4 + 3(y + 2) =
4y + 10
20. 7a2 − a2 + 16 =
21. 3y2 + 3(4y2 - 2) =
15y2 - 6
22. z2 + z + 4z3 + 4z2 =
23. 0.5(x4 - 3) + 12 =
0.5x + 10.5 4
FOCUS ON HIGHER ORDER THINKING
a. Write two equivalent expressions for
4z3 + 5z2 + z
the model.
The expressions become 6 - 9x and 3(2 - 3x).
Yes; Applying the Associative Property of Addition to 3x + 12 - 2x
31. Communicate Mathematical Ideas Write an example of an expression that cannot be simplified, and explain how you know that it cannot be simplified.
you get 3x - 2x + 12 which is equivalent to 1x + 12. Then apply
3x2 - 4x + 7; It does not have any like terms.
the Identity Property of Multiplication to get x + 12.
32. Problem Solving Write an expression that is equivalent to 8(2y + 4) that can be simplified.
26. William earns \$13 an hour working at a movie theater. Last week he worked h hours at the concession stand and three times as many hours at the ticket counter. Write and simplify an expression for the amount of money William earned last week.
13h + 13(3h) = 13h + 39h = 52h Lesson 11.3
PRE-AP
4 - 6x; 2(2 - 3x)
b. What If? Suppose a third row of tiles identical to the ones above is added to the model. How does that change the two expressions?
25. Justify Reasoning Is 3x + 12 - 2x equivalent to x + 12? Use two properties of operations to justify your answer.
EXTEND THE MATH
Work Area
30. Problem Solving Examine the algebra tile model.
4+p
24. _14 (16 + 4p) =
x + 4 in. 10.2 in.
Combine like terms.
2x4
x + 4 in.
6 mm 3x - 1 mm
15. 7x4 − 5x4 =
Bronze 29 23 19
(46 + 38 + 29)g + (29 + 27 + 17)s + (29 + 23 + 19)b;
(2 + a) + b
13. 2 + (a + b) =
Gold 46 38 29
United States China Great Britain
)PVHIUPO.JGGMJO)BSDPVSU1VCMJTIJOH\$PNQBOZt*NBHF\$SFEJUTª\$PNTUPDL +VQJUFSJNBHFT(FUUZ*NBHFT
Name
313
Activity available online
314
Unit 4
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Activity During a basketball game, Joelle scored 8 points on free throws. She also scored 2 points for each inside shot and 3 points for each outside shot she made. Joelle made n inside shots and s outside shots during the game. Write six equivalent expressions for the total number of points Joelle scored. Which properties of operations did you use to identify equivalent expressions? Sample answer: Commutative Property of Addition: 8 + 2n + 3s, 8 + 3s + 2n, 2n + 3s + 8, 2n + 8 + 3s; Distributive Property and Commutative Property of Addition: 2(4 + n) + 3s, 3s + 2(4 + n)
Generating Equivalent Expressions
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DO NOT EDIT--Changes must be made through “File info” CorrectionKey=B
Module Quiz
Assess Mastery
11.1 Modeling Equivalent Expressions
Personal Math Trainer
Write each phrase as an algebraic expression.
Use the assessment on this page to determine if students have mastered the concepts and standards covered in this module.
p __ 6
1. p divided by 6
185 + h
3. the sum of 185 and h 3
Response to Intervention
2 1
4 seasons.
4. the product of 16 and g
6. 8p; p = 9
4x
72
8. 4(d + 7); d = -2
20
7. 11 + r; r = 7 -60 9. ____ m ;m=5
of a triangle with a base of 6 and a height of 8?
Differentiated Instruction
Differentiated Instruction
11.3 Generating Equivalent Expressions
• Reteach worksheets
• Challenge worksheets
11. Draw lines to match the expressions in List A with their equivalent expressions in List B.
• Success for English Learners ELL
ELL
PRE-AP
Extend the Math PRE-AP Lesson Activities in TE
Additional Resources Assessment Resources includes: • Leveled Module Quizzes
18 -12
10. To find the area of a triangle, you can use the expression b × h ÷ 2, where b is the base of the triangle and h is its height. What is the area
Online and Print Resources
16g
Evaluate each expression for the given value of the variable.
© Houghton Mifflin Harcourt Publishing Company
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2. 65 less than j
11.2 Evaluating Expressions
Enrichment
Access Ready to Go On? assessment online, and receive instant scoring, feedback, and customized intervention or enrichment.
Online Assessment and Intervention
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j - 65
5. Let x represent the number of television show episodes that are taped in a season. Write an expression for the number of episodes taped in
Intervention
Personal Math Trainer
Online Assessment and Intervention
ESSENTIAL QUESTION
24 square units
List A 7x + 14 7 + 7x 7( x - 1 )
List B 7( 1 + x ) 7x - 7 7( x + 2 )
12. How can you determine if two algebraic expressions are equivalent?
Sample answer: Model the expressions with bar models or algebra tiles to determine if the expressions are equivalent; write equivalent expressions using properties of operations, order of operations, and combining like terms.
Module 11
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Texas Essential Knowledge and Skills Lesson
Exercises
11.1
1–5
6.7.C
11.2
6–10
6.7.C
11.3
11
6.7.D
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Module 11
TEKS
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Texas Test Prep
Texas Testing Tip Students can circle or underline key words and phrases to identify important information. Item 1 Students should underline the word product. Product means the operation used is multiplication. Thus students can quickly see that choice C is the correct answer. Item 3 Students should underline the phrase divided them evenly between p pages. Since the operation used is division, students can eliminate choices A and B, and then check the order of division in the two remaining answer choices to reveal that choice D is the correct answer.
Selected Response 1. Which expression represents the product of 83 and x? A 83 + x
Item 5 Some students may forget that 7w means the product of 7 and the variable and mistakenly substitute 9 for w to make the number 79. Remind students that the variable in a term is multiplied by the coefficient. Item 6 Caution students to read the question carefully. Some students may not realize that this question is asking for how many pages Katie has left to read and may indicate that choice D is correct.
83x
A the product of r and 9 B the quotient of r and 9
9 less than r
3. Rhonda was organizing photos in a photo album. She took 60 photos and divided them evenly among p pages. Which algebraic expression represents the number of photos on each page? p
A p - 60
C
B 60 - p
60 D __ p
__ 60
4. Using the algebraic expression 4n + 6, what is the greatest whole-number value of n that will give you a result less than 100? A 22
C
B 23
D 25
C
49
D 77
316
7. The expression 12(x + 4) represents the total cost of CDs Mei bought in April and May at \$12 each. Which property is applied to write the equivalent expression 12x + 48? A Associative Property of Addition
Commutative Property of Multiplication
C
D Distributive Property
Gridded Response 8. When traveling in Europe, Bailey converts the temperature given in degrees Celsius to a Fahrenheit temperature by using the expression 9x ÷ 5 + 32, where x is the Celsius temperature. Find the temperature in degrees Fahrenheit when it is 15 °C.
.
5
9
0
0
0
0
0
0
1
1
1
1
1
1
2
2
2
2
2
2
3
3
3
3
3
3
4
4
4
4
4
4
5
5
5
5
5
5
6
6
6
6
6
6
7
7
7
7
7
7
8
8
8
8
8
8
9
9
9
9
9
9
24
5. Evaluate 7w - 14 for w = 9. B 18
200
B Associative Property of Multiplication
D r more than 9
A 2
A 40
D 250
D 83 - x
Avoid Common Errors
6. Katie has read 32% of a book. If she has read 80 pages, how many more pages does Katie have left to read?
C
2. Which phrase describes the algebraic expression _9r ?
C
Online Assessment and Intervention
B 170
B 83 ÷ x
C
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Texas Test Prep
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Texas Essential Knowledge and Skills Items
Mathematical Process TEKS
1
6.7.C
6.1.D
2
6.7.C
6.1.D
3
6.7.C
6.1.A, 6.1.D
4
6.7.A
6.1.F
5
6.7.D
6*
6.5.A
6.1.A
7
6.7.D
6.1.A
8
6.7.D
6.1.A, 6.1.D
* Item integrates mixed review concepts from previous modules or a previous course.
Generating Equivalent Algebraic Expressions
316
Equations and Relationships ?
ESSENTIAL QUESTION How can you use equations and relationships to solve real-world problems?
MODULE
You can model real-world problems with equations, then use algebraic rules to solve the equations.
12
LESSON 12.1
Writing Equations to Represent Situations 6.7.B, 6.9.A, 6.10.B
LESSON 12.2
Addition and Subtraction Equations 6.9.B, 6.9.C, 6.10.A
LESSON 12.3
Multiplication and Division Equations
© Houghton Mifflin Harcourt Publishing Company • Image Credits: JoeFox/Alamy
6.9.B, 6.9.C, 6.10.A
Real-World Video
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Module 12
People often attempt to break World Records. To model how many seconds faster f an athlete's time a seconds must be to match a record time r seconds, write the equation f = a - r.
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Math On the Spot
Animated Math
Personal Math Trainer
Go digital with your write-in student edition, accessible on any device.
Scan with your smart phone to jump directly to the online edition, video tutor, and more.
Interactively explore key concepts to see how math works.
Get immediate feedback and help as you work through practice sets.
317
Are You Ready? Use the assessment on this page to determine if students need intensive or strategic intervention for the module’s prerequisite skills. 3 2 1
Personal Math Trainer
Complete these exercises to review skills you will need for this chapter.
Evaluate Expressions EXAMPLE
Response to Intervention
Online Assessment and Intervention
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Evaluate 8(3+2) - 52 8(3+2) - 52 = 8(5) - 52 = 8(5) - 25 = 40 - 25 = 15
Perform operations inside parentheses first. Evaluate exponents. Multiply. Subtract.
Evaluate the expression.
Online Assessment and Intervention
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Online and Print Resources Skills Intervention worksheets
Differentiated Instruction
• Skill 54 Evaluate Expressions
• Challenge worksheets
7. 2(8 + 3) + 42
38
9. 8(2 +1)2 - 42
56
39
4. 6(8 - 3) + 3(7 - 4)
1
5. 10(6 - 5) - 3(9 - 6)
64
2. 8(2 + 4) + 16
5
3. 3(14 - 7) - 16
Access Are You Ready? assessment online, and receive instant scoring, feedback, and customized intervention or enrichment.
Personal Math Trainer
29
1. 4(5 + 6) - 15
Enrichment
29
6. 7(4 + 5 + 2) - 6(3 + 5) 8. 7(14 - 8) - 62
6
Connect Words and Equations EXAMPLE
PRE-AP
• Skill 56 Connect Words and Extend the Math PRE-AP Equations Lesson Activities in TE
The product of a number and 4 is 32. The product of x and 4 is 32. Represent the unknown with a variable. Determine the operation. 4 × x is 32. 4 × x = 32. Determine the placement of the equal sign.
© Houghton Mifflin Harcourt Publishing Company
Intervention
Write an algebraic equation for the word sentence.
10. A number increased by 7.9 is 8.3.
x + 7.9 = 8.3
12. The quotient of a number and 8 is 4.
x÷8=4
14. The difference between 31 and a number is 7.
318
31 - x = 7
11. 17 is the sum of a number and 6.
17 = x + 6
13. 81 is three times a number.
81 = 3x 15. Eight less than a number is 19.
x - 8 = 19
Unit 4
PROFESSIONAL DEVELOPMENT VIDEO my.hrw.com
Author Juli Dixon models successful teaching practices as she explores equation concepts in an actual sixth-grade classroom.
Online Teacher Edition Access a full suite of teaching resources online—plan, present, and manage classes and assignments.
Professional Development
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Interactive Answers and Solutions Customize answer keys to print or display in the classroom. Choose to include answers only or full solutions to all lesson exercises.
Interactive Whiteboards Engage students with interactive whiteboard-ready lessons and activities. Personal Math Trainer: Online Assessment and Intervention Assign automatically graded homework, quizzes, tests, and intervention activities. Prepare your students with updated, TEKS-aligned practice tests.
Equations and Relationships
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DO NOT EDIT--Changes must be made through "File info" CorrectionKey=B
Visualize Vocabulary Use the ✔ words to complete the graphic.
Visualize Vocabulary The main idea web will help students review vocabulary and concepts related to algebraic expressions. Ask students to think of other algebraic expressions and their components and share them with the class, making sure to identify the variable, coefficient, terms, and constant in each expression.
4x + 5
Use the following explanations to help students learn the preview words. The words expression and equation are not synonyms. An algebraic expression is a mathematical statement that contains one or more variables. An equation is a mathematical statement stating that two expressions are equal. You evaluate an expression and solve an equation.
Students can use these reading and note-taking strategies to help them organize and understand new concepts and vocabulary. c.4.D Use prereading supports such as graphic organizers, illustrations, and pretaught topic-related vocabulary to enhance comprehension of written text.
Review Words ✔ algebraic expression (expresión algebraica) ✔ coefficient (coeficiente) ✔ constant (constante) evaluating (evaluar) like terms (términos semejantes) ✔ term (término, en una expresión) ✔ variable (variable)
Preview Words
5
4x and 5
constant
terms
Understand Vocabulary
equation (ecuación) equivalent expression (expresión equivalente) properties of operations (propiedades de las operaciones) solution (solución)
Match the term on the left to the correct expression on the right.
© Houghton Mifflin Harcourt Publishing Company
Integrating the ELPS
x
variable
algebraic expression
Understand Vocabulary
4
coefficient
Vocabulary
1. algebraic expression
A. A mathematical statement that two expressions are equal.
2. equation
B. A value of the variable that makes the statement true.
3. solution
C. A mathematical statement that includes one or more variables.
Booklet Before beginning the module, create a booklet to help you learn the concepts in this module. Write the main idea of each lesson on each page of the booklet. As you study each lesson, write important details that support the main idea, such as vocabulary and formulas. Refer to your finished booklet as you work on assignments and study for tests.
Differentiated Instruction • Reading Strategies ELL Module 12
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Grades 6–8 TEKS Before Students understand: • operations with rational numbers • properties of operations: inverse, identity, commutative, associative, and distributive properties
319
Module 12
In this module Students will learn to: • write one-variable, one-step equations to represent constraints or conditions within problems • model and solve one-variable, one-step equations that represent problems • write corresponding real-world problems given one-variable, one-step equations
319
After Students will learn how to: • write one-variable, two-step equations to represent real-world problems • write a real-world problem to represent a one-variable, two-step equation • solve one-variable, two-step equations
MODULE 12
Unpacking the TEKS
Unpacking the TEKS Understanding the TEKS and the vocabulary terms in the TEKS will help you know exactly what you are expected to learn in this module.
6.9.A Write one-variable, one-step equations and inequalities to represent constraints or conditions within problems.
Texas Essential Knowledge and Skills Content Focal Areas Expressions, equations, and relationships—6.7 The student applies mathematical process standards to develop concepts of expressions and equations.
Expressions, equations, and relationships—6.10 The student applies mathematical process standards to use equations and inequalities to solve problems.
You will learn to write an equation or inequality to represent a situation.
Key Vocabulary
UNPACKING EXAMPLE 6.9.A
equation (ecuación) A mathematical sentence that shows that two expressions are equivalent.
The Falcons won their football game with a score of 30 to 19. Kevin scored 12 points for the Falcons. Write an equation to determine how many points Kevin’s teammates scored.
inequality (desigualdad) A mathematical sentence that shows the relationship between quantities that are not equal.
Expressions, equations, and relationships—6.9 The student applies mathematical process standards to use equations and inequalities to represent situations.
What It Means to You
6.10.B Determine if the given value(s) make(s) one-variable, one-step equations or inequalities true.
Integrating the ELPS
Kevin’s points
+
Teammates’ points
=
Total points
12
+
t
=
30
What It Means to You You can substitute a given value for the variable in an equation or inequality to check if that value makes the equation or inequality true. UNPACKING EXAMPLE 6.10.B
Melanie bought 6 tickets to a play. She paid a total of \$156. Write an equation to determine whether each ticket cost \$26 or \$28.
content area text … and develop vocabulary … to comprehend increasingly challenging language.
Number of tickets bought 6
· ·
Price per ticket p
=
Total cost
=
156
Substitute 26 and 28 for p to see which equation is true.
Go online to see a complete unpacking of the .
Visit my.hrw.com to see all the unpacked.
6p = 156
6p = 156
? 156 6 · 26 =
? 156 6 · 28 =
? 156 ✓ 156 =
© Houghton Mifflin Harcourt Publishing Company • Image Credits: Robert Llewellyn/Corbis Super RF/Alamy Limited
Use the examples on this page to help students know exactly what they are expected to learn in this module.
? 156 ✗ 168 =
The cost of a ticket to the play was \$26. my.hrw.com
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Lesson 12.1
Lesson 12.2
Unit 4
Lesson 12.3
6.7.B Distinguish between expressions and equations verbally, numerically, and algebraically. 6.9.A Write one-variable, one-step equations and inequalities to represent constraints or conditions within problems. 6.9.B Represent solutions for one-variable, one-step equations and inequalities on number lines. 6.9.C Write corresponding real-world problems given one-variable, one-step equations or inequalities. 6.10.A Model and solve one-variable, one-step equations and inequalities that represent problems, including geometric concepts. 6.10.B Determine if the given value(s) make(s) one-variable, one-step equations or inequalities true.
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LESSON
12.1 Writing Equations to Represent Situations Texas Essential Knowledge and Skills The student is expected to: Expressions, equations, and relationships—6.9.A Write one-variable, one-step equations and inequalities to represent constraints or conditions within problems. Expressions, equations, and relationships—6.7.B Distinguish between expressions and equations verbally, numerically, and algebraically. Expressions, equations, and relationships—6.10.B Determine if the given value(s) make(s) one-variable, one-step equations or inequalities true.
Mathematical Processes 6.1.D Communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate.
ADDITIONAL EXAMPLE 1 Determine whether the given value is a solution of the equation. A x + 15 = 10; x = -5 m B __ = -11; m = 33 3
C 5n = 42; x = 7
yes no
no
Interactive Whiteboard Interactive example available online my.hrw.com
ADDITIONAL EXAMPLE 2 Eli is y years old. His 9-year-old cousin Jen is 4 years younger than he is. Write an equation to represent this situation. Sample answer: y - 4 = 9 Interactive Whiteboard Interactive example available online my.hrw.com
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Engage
ESSENTIAL QUESTION How do you write equations and determine whether a number is a solution of an equation? Write a statement that links two expressions with an equals sign. Substitute a number for the variable and simplify. If the final statement is true, the number is a solution.
Motivate the Lesson Ask: Ty wants to buy a video game. He has \$57, which is \$38 less than he needs. Does the game cost \$90 or \$95? Begin Example 1 to find out how to solve this problem.
Explore Engage with the Whiteboard
Write x + 4 and x + 4 = 9 on the whiteboard. Ask students how they differ. Explain the difference between an expression and an equation. Then have students model both sides of the equation, using algebra tiles on the whiteboard. Ask them what x must represent for the two sides to be equal. Tell them that x = 5 is the solution of the equation.
Explain EXAMPLE 1 Focus on Modeling Mathematical Processes Explain to students that an equation is like a balanced scale. Just as the weights on both sides of a balanced scale are exactly the same, the expressions on both sides of an equation represent exactly the same value. Show examples of equations on a balance scale. Questioning Strategies
Mathematical Processes • In A, why is x = 6 a solution of the equation? because the resulting statement, 15 = 15, is a true statement In B, why is y = -8 not a solution of the equation? because the resulting statement, -2 = -32, is a false statement
YOUR TURN Avoid Common Errors Watch for students who substitute the given value incorrectly. Caution students to doublecheck their work for accuracy.
EXAMPLE 2 Questioning Strategies
Mathematical Processes • Could you write another equation to represent the situation? Yes. I can write the equation p = Total points - Mark’s points or p = 46 - 17 because subtraction is the opposite of addition.
YOUR TURN Avoid Common Errors If students have difficulty writing equations to represent the word problems, encourage them to make a model, like the one in Example 2, to organize the information.
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12.1 ?
Writing Equations to Represent Situations
ESSENTIAL QUESTION
Expressions, equations, and relationships—6.9.A Write one-variable, one-step equations … to represent constraints or conditions within problems. Also 6.7.B, 6.10.B
How do you write equations and determine whether a number is a solution of an equation?
Determine whether the given value is a solution of the equation. 1. 11 = n + 6; n = 5
Online Assessment and Intervention
yes
5+4
5+4=9 a number plus 4 is 9.
n+4
Algebraic
Math On the Spot my.hrw.com
n+4=9
EXAMPL 1 EXAMPLE
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EXAMPLE 2
An equation relates two expressions using symbols for is or equals.
© Houghton Mifflin Harcourt Publishing Company
Mark’s points
17
Math Talk
6.10.B
Mathematical Processes
Use words to describe two expressions that represent the total points scored. What does the equation say about the expressions you wrote?
A x + 9 = 15; x = 6 ? 6 + 9 = 15 Substitute 6 for x. ? 15 = 15 Add. y __ = -32; y = -8 4 -8 ? ___ = -32 Substitute -8 for y. 4
=
Total points
+
p
=
46
HOME
PERIOD
GUEST
Math Talk: The sum of Mark’s points and his teammates’ points; the total points scored; the equation uses the symbol = to show that the two expressions are equal.
Write an equation to represent each situation.
Divide.
4. Marilyn has a fish tank that contains 38 fish. There are 9 goldfish and f other fish.
5. Juanita has 102 beads to make n necklaces. Each necklace will have 17 beads.
6. Craig is c years old. His 12-year-old sister Becky is 3 years younger than Craig.
7. Sonia rented ice skates for h hours. The rental fee was \$2 per hour and she paid a total of \$8.
f + 9 = 38
y
-8 is not a solution of the equation _4 = -32.
C 8x = 72; x = 9 ? 8(9) = 72 ? 72 = 72
+
Teammates’ points
6 is a solution of x + 9 = 15.
? -2 = -32
6.9.A
Mark scored 17 points for the home team in a basketball game. His teammates as a group scored p points. Write an equation to represent this situation.
Determine whether the given value is a solution of the equation.
B
You can represent some real-world situations with an equation. Making a model first can help you organize the information.
Math On the Spot
Equation
a number plus 4
Numerical
yes
Writing Equations to Represent Situations
An equation is a mathematical statement that two expressions are equal. An equation may or may not contain variables. For an equation that has a variable, a solution of the equation is a value of the variable that makes the equation true. An equation represents a relationship between two An expression represents values. a single value.
Words
no
36 3. __ x = 9; x = 4
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Determining Whether Values Are Solutions
Expression
2. y - 6 = 24; y = 18
Substitute 9 for x.
Personal Math Trainer
Multiply.
Online Assessment and Intervention
9 is a solution of 8x = 72.
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c - 3 = 12
17n = 102
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LESSON
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PROFESSIONAL DEVELOPMENT Integrate Mathematical Processes This lesson provides an opportunity to address Mathematical Process TEKS 6.1.D, which calls for students to “Communicate mathematical ideas, …using multiple representations, including symbols…and language as appropriate.” Students begin by distinguishing equations from expressions. Then they determine whether a given number is a solution of the equation. Next, students write equations that represent real-world situations expressed in words. Finally, students use substitution to check whether a value for a variable makes an equation true.
Math Background The equals sign, which first appeared about 450 years ago, is a relatively new math concept. The term equation comes from a Latin word meaning “to set equal.” There are, essentially, four types of equations: • True equation: 2 + 3 = 5 • False equation: 3 + 4 = 8 • Conditional equation: y - 6 = 7 (not true for all values) • Identity: 3n + 5n = 8n (true for all values)
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ADDITIONAL EXAMPLE 3 Suki paid \$132 for 6 DVDs. Write an equation to determine whether each DVD cost \$17 or \$22. Sample equation: 6n = \$132; \$22 Interactive Whiteboard Interactive example available online
EXAMPLE 3 Questioning Strategies
Mathematical Processes • Is the equation x - 47 = 18 true? Explain. The equation is true only when x = 65. An equation is a balanced mathematical statement, and only the correct solution will maintain the balance. • How can you check your answer by using a different mathematical operation? Add \$18 and \$47 to find the original amount on the card. You should get \$65.
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Connect Multiple Representations Mathematical Processes Have students explain why adding \$18 to \$47 to check that \$65 is the answer will yield the same result as subtracting \$47 from \$65 to see that \$18 is the money left over. Students should recognize that addition and subtraction are inverse operations. Animated Math Modeling Equations Students model equations using interactive algebra tiles. my.hrw.com
Focus on Math Connections Point out that an equals sign with a question mark above (≟) is used immediately after a variable has been substituted by a number. This symbol indicates that it is not yet known whether the equation is true or false.
YOUR TURN Avoid Common Errors If students have difficulty writing equations to represent the word problems, encourage them to make a model, like the one in Example 3, to organize the information. Then remind students to check that their solution makes the original equation true.
Elaborate Talk About It Summarize the Lesson Ask: How do you know when a number is a solution to a equation? When the number is substituted for the variable, it makes the equation true. An equation is true when the values of the expressions on opposite sides of the equals sign are the same.
GUIDED PRACTICE Engage with the Whiteboard For Exercises 1–2, have student fill in the boxes and determine whether the given values are solutions to the equations. For Exercise 13, have a student circle the important information in the problem statement on the whiteboard. Then have another student complete the model for the word equation and write the equation next to the model.
Avoid Common Errors Exercises 3–12 Watch for students who substitute the given value incorrectly. Caution students to double-check their work for accuracy. Exercises 14–16 If students have difficulty writing equations to represent the word problems, encourage them to make a model, like the one in Example 3, to organize the information. Then remind students to check that their solution makes the original equation true.
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DO NOT EDIT--Changes must be made through “File info” CorrectionKey=B
Guided Practice
Writing an Equation and Checking Solutions
Determine whether the given value is a solution of the equation. (Example 1) Math On the Spot my.hrw.com
EXAMPL 3 EXAMPLE
6.10.B
Sarah used a gift card to buy \$47 worth of groceries. Now she has \$18 left on her gift card. Write an equation to determine whether Sarah had \$65 or \$59 on the gift card before buying groceries. STEP 1
Amount on card STEP 2
-
Amount spent
=
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STEP 3
=
Amount left on card
x
-
47
=
18
Substitute 65 and 59 for x to see which equation is true.
The amount spent and the amount left on the card are the known quantities. Substitute those values in the equation.
© Houghton Mifflin Harcourt Publishing Company
13
4 yes
7. 21 = m + 9; m = 11 9. d - 4 = 19; d = 15
no no
6. 2.5n = 45; n = 18
yes
8. 21 - h = 15; h = 6
yes
10. 5 + x = 47; x = 52
yes
yes
12. 5q = 31; q = 13
no no
Number of rooms
×
on each floor
=
Total number of rooms
8r = 256
14. In the school band, there are 5 trumpet players and f flute players. There are twice as many flute players as there are trumpet players. Write an equation to represent this situation. (Example 3)
15. Pedro bought 8 tickets to a basketball game. He paid a total of \$208. Write an equation to determine whether each ticket cost \$26 or \$28. (Example 3)
Sample equation: 8x = 208; \$26
8. What expressions are represented in the equation x - 47 = 18? How does the relationship represented in the equation help you determine if the equation is true?
16. The high temperature was 92°F. This was 24°F higher than the overnight low temperature. Write an equation to determine whether the low temperature was 62°F or 68°F. (Example 3)
x - 47 and 18; Subtract 47 from the given number and
Sample equation: x + 24 = 92; 68°F
if the difference is 18 the equation is true.
? ?
YOUR TURN 9. On Saturday morning, Owen earned \$24 raking leaves. By the end of the afternoon he had earned a total of \$62. Write an equation to determine whether Owen earned \$38 or \$31 on Saturday afternoon.
? =4
4. 17y = 85; y = 5
no
Number
Reflect
Personal Math Trainer
Substitute the number for the variable and simplify. If the final statement is true, the number is a solution.
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ESSENTIAL QUESTION CHECK-IN
17. Tell how you can determine whether a number is a solution of an equation.
Online Assessment and Intervention
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52
? _____ = 4
of floors
x - 47 = 18 x - 47 = 18 ? ? 65 - 47 = 18 59 - 47 = 18 ? ? 18 = 18 12 = 18 The amount on Sarah’s gift card before she bought groceries was \$65.
Sample equation: x + 24 = 62; \$38
5
yes
13. Each floor of a hotel has r rooms. On 8 floors, there are a total of 256 rooms. Write an equation to represent this situation. (Example 2)
Let x be the amount on the card. Amount spent
? 23 =
n 2. __ = 4; n = 52 13
-9
11. w - 9 = 0; w = 9
Rewrite the equation using a variable for the unknown quantity and the given values for the known quantities.
-
14
3. 14 + x = 46; x = 32 Animated Math
Amount left on card
Amount on card
? 23 =
5. 25 = _5k ; k = 5
Identify the three quantities given in the problem.
no
1. 23 = x - 9; x = 14
© Houghton Mifflin Harcourt Publishing Company
You can substitute a given value for the variable in an equation to check if that value makes the equation true.
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DIFFERENTIATE INSTRUCTION Visual Cues
Critical Thinking
Display a set of pan balance scales. For each of the following equations, write the left side over the left scale and the right side over the right scale. Ask what the value of the variable must be for the scales to remain balanced.
Give students the following problem to solve.
1. x + 3 = 7 x = 4
Remind students that to compare quantities, they need to use the same units.
Differentiated Instruction includes: • Reading Strategies • Success for English Learners ELL • Reteach • Challenge PRE-AP
3. 4y = 28 y = 7
17 ≠ 350 ÷ 20; No, they do not have the same amount of money.
2. t - 5 = 3 t = 8 w 4. __ = 6 w = 12 2
Rebecca has 17 one-dollar bills. Courtney has 350 nickels. Do the two girls have the same amount of money?
Writing Equations to Represent Situations
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Personal Math Trainer Online Assessment and Intervention
Online homework assignment available
Evaluate GUIDED AND INDEPENDENT PRACTICE 6.9.A, 6.10.B
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12.1 LESSON QUIZ 6.9.A, 6.10.B Determine whether the given value is a solution of the equation. Write yes or no. 1. w – 7 = 20; w = 13
Concepts & Skills
Practice
Example 1 Determining Whether Values Are Solutions
Exercises 1–12
Example 2 Writing Equations to Represent Situations
Exercises 13, 18–21, 24–25
Example 3 Writing an Equation and Checking Solutions
Exercises 14–16, 26
2. 15t = -120; t = -8 y 3. __ = -2; y = 24 12
Exercise
4. -5 = x + 5; x = -10 5. In a choir there are 16 altos and s sopranos. There are twice as many sopranos as altos. Write an equation to represent this situation. 6. Jerome is one-third the age of his aunt, who is 51 years old. Write an equation to determine whether Jerome is 14 or 17.
Depth of Knowledge (D.O.K.) 2 Skills/Concepts
1.A Everyday life
22
3 Strategic Thinking
1.G Explain and justify arguments
23
2 Skills/Concepts
1.F Analyze relationships
24
3 Strategic Thinking
1.G Explain and justify arguments
25–26
3 Strategic Thinking
1.D Multiple representations
27–29
3 Strategic Thinking
1.G Explain and justify arguments
18–21
Lesson Quiz available online my.hrw.com
Answers 1. no 2. yes 3. no 4. yes 5. __2s = 16 6. 3x = 51; Jerome is 17 years old.
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Lesson 12.1
Mathematical Processes
Additional Resources Differentiated Instruction includes: • Leveled Practice Worksheets
Class
Date Personal Math Trainer
12.1 Independent Practice 6.7.B, 6.9.A, 6.10.B
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18. Andy is one-fourth as old as his grandfather, who is 76 years old. Write an equation to determine whether Andy is 19 or 22 years old.
4n = 76; 19 years old 19. A sleeping bag weighs 8 pounds. Your backpack and sleeping bag together weigh 31 pounds. Write an equation to determine whether the backpack without the sleeping bag weighs 25 or 23 pounds.
x + 8 = 31; 23 pounds
22. Write an equation that involves multiplication, contains a variable, and has a solution of 5. Can you write another equation that has the same solution and includes the same variable and numbers but uses division? If not, explain. If possible, write the equation.
© Houghton Mifflin Harcourt Publishing Company
29 32
a. Write an equation that relates Cindy’s age to her dad’s age when Cindy is 18.
Cindy’s Age 2 years old
36 years old
10 years old
?
18 years old
b. Determine if 42 is a solution to the equation. Show your work.
No; 42 - 26 = 16; 16 is not equal to 18.
represents one value such as
Cindy’s father will not be 42 when she is 18.
2 + 5 and an equation represents FOCUS ON HIGHER ORDER THINKING
the relationship between two equation is a statement that the
Yes, because 4 + f = 12 means there are 8 flute players,
equivalent. Each expression is
and 8 is twice 4.
24. Explain the Error The problem states that Ursula earns \$9 per hour. To write an expression that tells how much money Ursula earns for h hours, Joshua wrote _h9 . Sarah wrote 9h. Whose expression is correct and why?
Sarah is correct because the
x is the distance between
28. Problem Solving Ronald paid \$162 for 6 tickets to a basketball game. During the game he noticed that his friend paid \$130 for 5 tickets. The price of each ticket was \$26. Was Ronald overcharged? Justify your answer.
Yes, Ronald was overcharged because 6(26) = 156 and 156 < 162. 29. Communicate Mathematical Ideas Tariq said you can write an equation by setting an expression equal to itself. Would an equation like this be true? Explain.
earnings are the number of
b. Describe what your variable represents.
Work Area
27. Critical Thinking In the school band, there are 4 trumpet players and f flute players. The total number of trumpet and flute players is 12. Are there twice as many flute players as trumpet players? Explain.
two expressions, 2 + 5 and 7, are just part of that statement.
29 - 13 = x; 13 + x = 29
hours times the amount per
Yes, setting an expression equal to itself forms an
hour, or 9h.
equation that will always be true regardless of the value of the variable(s) or numbers in the expression.
Artaville and Greenville Lesson 12.1
EXTEND THE MATH
x - 26 = 18
values such as 2 + 5 = 7. An
a. Write two equations that state the relationship of the distances between Greenville, Artaville, and Jonesborough.
26. Multiple Representations The table shows ages of Cindy and her dad.
23. How are expressions and equations different? Explain using a numerical example.
x - 23 = 48; 71 students
Maybern
is 44 - 7 = x.
20 ÷ x = 4 or x = 20 ÷ 4
21. The table shows the distance between Greenville and nearby towns. The distance between Artaville and Greenville is 13 miles less than the distance between Greenville and Jonesborough. Distance between Greenville and Nearby Towns (miles)
Both equations are correct. Another correct equation
Sample answer: 4x = 20; Yes:
20. Halfway through a bus route, 23 students have been dropped off and 48 students remain on the bus. Write an equation to determine whether there are 61 or 71 students on the bus at the beginning of the route.
Jonesborough
Online Assessment and Intervention
25. Communicate Mathematical Ideas A dog weighs 44 pounds and the veterinarian thinks it needs to lose 7 pounds. Mikala wrote the equation x + 7 = 44 to represent the situation. Kirk wrote the equation 44 - x = 7. Which equation is correct? Can you write another equation that represents the situation?
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Name
PRE-AP
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Activity available online
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Activity Following the release of the hit movie Attack of the Giant Muffins, muffin fever spread across the country. This resulted in a contest to choose the plot of the sequel Attack of the Giant Muffins, Part 2. The table shows the number of contest entries from some towns in one state.
Town Entries Hillville .....................................40 Dos Rios ..................................30 High Corn ..............................105 Moose .......................................n Ho-Hum ..................................25
Write an equation for each statement. 1. The sum of the number of entries from Hillville and Moose was 90. n + 40 = 90 2. The difference between the number of Moose entries and Ho-Hum entries was 25. n - 25 = 25 3. The total number of entries is 5 times the number of Moose entries. 5n = 250 Writing Equations to Represent Situations
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LESSON
12.2 Addition and Subtraction Equations Texas Essential Knowledge and Skills The student is expected to: Expressions, equations, and relationships—6.10.A Model and solve one-variable, one-step equations and inequalities that represent problems, including geometric concepts. Expressions, equations, and relationships—6.9.B Represent solutions for one-variable, one-step equations and inequalities on number lines. Expressions, equations, and relationships—6.9.C Write corresponding real-world problems given one-variable, one-step equations or inequalities.
Mathematical Processes 6.1.E Create and use representations to organize, record, and communicate mathematical ideas.
ADDITIONAL EXAMPLE 1 Solve the equation y + 8 = 22. Graph the solution on a number line. y = 14 10
15
20
Interactive Whiteboard Interactive example available online my.hrw.com
Engage
ESSENTIAL QUESTION How do you solve equations that contain addition and subtraction? Sample answer: Apply the inverse operation—subtraction for addition equations and addition for subtraction equations—to both sides of the equation.
Motivate the Lesson Ask: Does anyone have a puppy? Have you noticed how fast they grow? Begin the Explore Activity to find out how to model one puppy’s weight gain, using an equation and algebra tiles.
Explore EXPLORE ACTIVITY Focus on Modeling Mathematical Processes Remind students that an equation is like a balanced scale. If you increase or decrease the weights by the same amount on both sides, the scale will remain in balance. Emphasize that students must remove the same number of tiles from both sides of the mat.
Explain EXAMPLE 1 Focus on Math Connections
Mathematical Processes Remind students that subtraction is the inverse, or opposite, of addition. If an equation contains addition, solve it by subtracting from both sides to “undo” the addition.
Questioning Strategies
Mathematical Processes • Would you solve 15 + a = 26 differently from a + 15 = 26? no because addition is associative and the order of the variable and the added number does not change the process of subtracting 15 from both sides
• How can you use substitution to check an answer to an addition equation? Substitute the value for the variable into the original equation and simplify. If the result is a true statement, the value is the solution. • Why is the graph only a single point? because there is only one answer for the equation
YOUR TURN Avoid Common Errors Watch for students who perform the inverse operation of subtraction only on the side with the variable. Stress that to keep the equation “balanced” the same amount must be taken away from each side.
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Lesson 12.2
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12.2 ?
ESSENTIAL QUESTION
Expressions, equations, and relationships—6.10.A Model and solve one-variable, one-step equations ... that represent problems, including geometric concepts. Also 6.9.B, 6.9.C
How do you solve equations that contain addition or subtraction?
EXPLORE ACTIVITY
Using Subtraction to Solve Equations Removing the same number of tiles from each side of an equation mat models subtracting the same number from both sides of an equation. Math On the Spot my.hrw.com
You can subtract the same number from both sides of an equation, and the two sides will remain equal. When an equation contains addition, solve by subtracting the same number from both sides.
6.10.A
Modeling Equations
EXAMPLE 1
A puppy weighed 6 ounces at birth. After two weeks, the puppy weighed 14 ounces. How much weight did the puppy gain?
6
+
Weight gained
=
Weight after 2 weeks
+
x
=
14
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Petra Wegner / Alamy
a + 15 = 26
Notice that the number 15 is added to a.
a + 15 = 26 - 15 -15 ___ a = 11
Subtract 15 from both sides of the equation.
Check: a + 15 = 26
To answer this question, you can solve the equation 6 + x = 14. Algebra tiles can model some equations. An equation mat represents the two sides of an equation. To solve the equation, remove the same number of tiles from both sides of the mat until the x tile is by itself on one side.
B How many 1 tiles must you remove on the left 6 side so that the x tile is by itself? Cross out these tiles on the equation mat.
1
1
1
1
1
1
x
1
1
1
1
1
1
1
1
1
1
1
1
1
1
Substitute 11 for a.
26 = 26
6+ x
5 6 7 8 9 10 11 12 13 14 15
14
Reflect 2.
C Whenever you remove tiles from one side of the mat, you must remove the same number of tiles from the other side of the mat. Cross out the tiles that should be removed on the right side of the mat. D How many tiles remain on the right side of the mat? This is the solution of the equation.
8
8
ounces.
variable is isolated, or alone, on one side of the equation.
Mathematical Processes
Why did you remove tiles from each side of your model?
You want to remove tiles equally from both The x tile is alone on one side of the mat. The solution is sides to keep the equation balanced. the number of small tiles on the other side. Communicate Mathematical Ideas How do you know when the model shows the final solution? How do you read the solution?
Lesson 12.2
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Communicate Mathematical Ideas How do you decide which number to subtract from both sides?
Subtract the number added to the variable so that the
Math Talk
Reflect 1.
? 11 + 15 = 26
Graph the solution on a number line.
A Model 6 + x = 14.
The puppy gained
6.10.A, 6.9.B
Solve the equation a + 15 = 26. Graph the solution on a number line.
Let x represent the number of ounces gained. Weight at birth
Subtraction Property of Equality
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Personal Math Trainer Online Assessment and Intervention
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3. Solve the equation 5 = w + 1.5. Graph the solution on a number line. w=
-5 -4 -3 -2 -1
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LESSON
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0 1 2 3 4 5
3.5
Unit 4
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PROFESSIONAL DEVELOPMENT Integrate Mathematical Processes This lesson provides an opportunity to address Mathematical Process TEKS 6.1.E, which calls for students to “create and use representations to organize, record, and communicate mathematical ideas.” Students use algebra tiles and number lines to model the solutions to one-step equations. They proceed to solve algebraic equations symbolically by using inverse operations to isolate the variable. They then apply the algebraic method to solve equations representing real-world situations.
Math Background In Elements, Book I, Euclid listed five axioms that he called “common notions.” 1. Things which are equal to the same thing are also equal to one another. 2. If equals be added to equals, the wholes are equal. 3. If equals be subtracted from equals, the remainders are equal. 4. Things which coincide with one another are equal to one another. 5. The whole is greater than the part.
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Solve the equation y −13 = –12. Graph the solution on a number line. y=1 -5
0
5
Interactive Whiteboard Interactive example available online my.hrw.com
EXAMPLE 2 Focus on Math Connections
Mathematical Processes Remind students that addition and subtraction are inverse operations. If an equation contains subtraction, solve it by adding to both sides to “undo” the subtraction.
Questioning Strategies
Mathematical Processes • How is solving an equation containing subtraction similar to solving an equation containing addition? Solve both types of equations by using the inverse operation to get the variable by itself on one side of the equation. • Does it make sense that the solution is greater than 18? Explain. Yes. The equation indicates that subtracting 21 from some numbers gives an answer of 18, so the solution must be greater than 18.
Engage with the Whiteboard Cover up the sentences in blue next to each step of the solution and have students write a description of what is happening in each step. Then have the students graph the solution on the number line.
Focus on Modeling
Mathematical Processes Guide students to see that it makes sense to draw only the needed portion of the number line when graphing a solution, especially when the numbers are large.
YOUR TURN Focus on Math Connections
ADDITIONAL EXAMPLE 3 Write and solve an equation to find the measure of the unknown angle. z + 48° = 90°; z = 42°
z
Mathematical Processes Remind students that to add or subtract fractions with different denominators, it is necessary to rewrite the fractions with a common denominator. A fraction greater than 1 should be written as a mixed number for ease of graphing.
EXAMPLE 3 Questioning Strategies
Mathematical Processes • How do you know that x + 60 = 180 is the correct equation to use to find the measure of x? The two angles shown in the drawing are supplementary. • Does the sketch of the unknown angle x appear to be twice 60°? Explain. Yes. If I divide x into two halves, the two angles appear to have measures that are similar to the 60° angle.
48°
Interactive Whiteboard Interactive example available online my.hrw.com
Focus on Math Connections
Mathematical Processes Have students explain what will be true about the unknown measure of any angle if its supplement is less than 90°. Elicit that it will be an obtuse angle. Then ask the same question about angles whose supplements are greater than 90°.
Solve the equation n + 6.5 = 20. Then write a real-world problem that involves adding these two quantities. n = 13.5; Sample answer: Ivan needed 6.5 more points to take the lead with 20 points. How many points had he already scored? Interactive Whiteboard Interactive example available online my.hrw.com
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Lesson 12.2
Focus on Math Connections
Mathematical Processes Point out to students that the angle represented here is a right angle and that a right angle measures exactly 90°. Remind them that if the sum of the measures of two angles is 90°, the two angles are complementary angles. This information should help students to write and solve an equation to find the measure of the unknown angle.
EXAMPLE 4 Connect to Daily Life
Mathematical Processes Point out to students that the equation has decimals containing hundredths and that money is commonly represented as decimals containing hundredths. Thus, a good real-world application for this equation would be a situation involving money. Encourage students to brainstorm situations from everyday life that could make sense.
DO NOT EDIT--Changes must be made through “File info” CorrectionKey=B
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Solving Equations that Represent Geometric Concepts
When an equation contains subtraction, solve by adding the same number to both sides.
You can write equations to represent geometric relationships.
Math On the Spot
Math On the Spot
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Recall that a straight line has an angle measure of 180°. Two angles whose measures have a sum of 180° are called supplementary angles. Two angles whose measures have a sum of 90° are called complementary angles.
You can add the same number to both sides of an equation, and the two sides will remain equal.
EXAMPLE 3 EXAMPL 2 EXAMPLE
Find the measure of the unknown angle.
6.10.A, 6.9.B
STEP 1
Solve the equation y - 21 = 18. Graph the solution on a number line. y - 21 = 18
Notice that the number 21 is subtracted from y.
y - 21 = 18 + 21 +21 ___ y = 39
Add 21 to both sides of the equation. STEP 2
Subtract.
=
180°
Write a description to represent the model. Include a question for the unknown angle.
STEP 3
Write an equation.
STEP 4
Solve the equation.
x + 60 = 180
35 36 37 38 39 40 41 42 43 44 45
x+
Reflect © Houghton Mifflin Harcourt Publishing Company
60°
The sum of an unknown angle and a 60° angle is 180°. What is the measure of the unknown angle?
Substitute 39 for y.
Graph the solution on a number line.
4.
+
60°
x
60 = 180 -60 -60 _ _
Subtract 60 from each side.
Communicate Mathematical Ideas How do you know whether to add on both sides or subtract on both sides when solving an equation?
x
If the equation contains addition, subtract. If the
The unknown angle measures 120°.
=
120
The final answer includes units of degrees.
equation contains subtraction, add. YOUR TURN 6. Write and solve an equation to find the measure of the unknown angle.
YOUR TURN Solve the equation h - _12 = _43 .
5.
x + 65 = 90; x = 25°
-2
Graph the solution on a number line. h=
-1
0
1
5 _ , or 1_14 4
2 Personal Math Trainer
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Online Assessment and Intervention
Online Assessment and Intervention
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x
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18 = 18
Write the information in the boxes. Unknown angle
Check: y - 21 = 18 ? 39 - 21 = 18
6.10.A
65°
7. Write and solve an equation to find the measure of a complement of an angle that measures 42°.
x + 42 = 90; x = 48°
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DIFFERENTIATE INSTRUCTION Critical Thinking
Cognitive Strategies
Present the magic square shown below. Explain that the sum of any three numbers added across, down, or diagonally should have the same sum.
Have students work with a partner to create “mind-reading” puzzles that depend upon inverse operations to return to the original number. The following puzzle is an example. Pick a number.
Differentiated Instruction includes: • Reading Strategies • Success for English Learners ELL • Reteach • Challenge PRE-AP
Have students write and solve equations to fill in the table. The magic sum for this square is 465. 248
31
186
93
155
217
124
279
62
5. Subtract 4.
2. Subtract 1.
7. Subtract 7.
4. Subtract 5.
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Questioning Strategies
Mathematical Processes • Using number sense, what can you determine about the value of x, given that the sum of two numbers is 25? The value of x will be less than 25.
• How else could you write the equation to solve for x? Explain. Sample answer: You could write x + 21.79 = 25; addition is commutative, so x + 21.79 = 21.79 + x. • How is the question in a real-world problem related to its equation? The question asks about the value of the variable.
Mathematical Processes Guide students to write problems that use sensible data, and have them explain, step by step, how to solve their equations. Challenge each student to create original word problems based on unique situations.
Elaborate Talk About It Summarize the Lesson Ask: How do you solve and check an equation containing only addition or only subtraction? You apply the inverse operation—subtraction for addition equations and addition for subtraction equations—to both sides of the equation.
GUIDED PRACTICE Engage with the Whiteboard For Exercise 1, have a student complete the verbal model on the whiteboard and then draw a model with algebra tiles below the verbal model. Ask students to provide an explanation of the solution.
Avoid Common Errors Exercises 2–6 Remind students to use the inverse operation of subtraction to undo addition or addition to undo subtraction. Exercise 7 Remind students to begin by identifying the angle they are using, as right, straight, or obtuse so that they will know what angle measure to use in their equation.
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Guided Practice
Writing Real-World Problems for a Given Equation Math On the Spot
1. A total of 14 guests attended a birthday party. Three guests stayed after the party to help clean up. How many guests left when the party ended? (Explore Activity) a. Let x represent the number
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EXAMPL 4 EXAMPLE
b.
6.9.C
+
left at end of party
Write a real-world problem for the equation 21.79 + x = 25. Then solve the equation.
x
21.79 + x = 25 STEP 1
Number that
of guests who left when the party ended. Number that
stayed to clean 3
+
c. Draw algebra tiles to model the equation.
Examine each part of the equation.
11
Total at party
=
14
+ + +
+
friends left when the party ended.
=
x is the unknown or quantity we are looking for. 21.79 is added to x. = 25 means that after adding 21.79 and x, the result is 25. STEP 2
Write a real-world situation that involves adding two quantities. Joshua wants to buy his mother flowers and a card for Mother’s Day. Joshua has \$25 to spend and selects roses for \$21.79. How much can he spend on a card?
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Brand X Pictures/Getty Images
STEP 3
Math Talk
Mathematical Processes
How is the question in a real-world problem related to its equation?
Solve the equation. 21.79 + x = 25 -21.79 -21.79 __ __ x=
3.21
The final answer includes units of money in dollars.
Solve each equation. Graph the solution on a number line. (Examples 1 and 2) 2. 2 = x - 3
x=
-5 -4 -3 -2 -1
5
3. s + 12.5 = 14 -5 -4 -3 -2 -1
0 1 2 3 4 5
s=
1.5
0 1 2 3 4 5
Solve each equation. (Examples 1 and 2) 4. h + 6.9 = 11.4 h=
4.5
6. n + _12 = _74
5. 82 + p = 122
40
p=
n=
5 _ 4
7. Write and solve an equation to find the measure of the unknown angle. (Example 3)
Joshua can spend \$3.21 on a Mother’s Day card.
x + 45 = 180; x = 135°
Reflect 8.
+ + + + + + + + + + + + + +
45°
x
8. Write a real-world problem for the equation x - 75 = 200. Then solve the equation. (Example 4)
What If? How might the real-world problem change if the equation were x - 21.79 = 25 and Joshua still spent \$21.79 on roses?
Check students’ answers; x = 275
21.79 is subtracted from the unknown value, and the result would remain 25. The variable would be the amount Joshua started with and
? ?
25 would the amount of money available after buying roses.
ESSENTIAL QUESTION CHECK-IN
9. How do you solve equations that contain addition or subtraction?
You apply the inverse operation—subtraction for
YOUR TURN 9. Write a real-world problem for the equation x - 100 = 40. Then solve the equation.
Check students’ problems; x = 140
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equations—to both sides of the equation.
Online Assessment and Intervention
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You can write a real-world problem for a given equation. Examine each number and mathematical operation in the equation.
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Online homework assignment available
Evaluate GUIDED AND INDEPENDENT PRACTICE 6.9.B, 6.9.C, 6.10.A
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12.2 LESSON QUIZ 6.10.A, 6.9.C Solve each equation. 1. y + 8.3 = 12.7 2. w − 14 = 23 3. 18 + x = 6 4. t – 1.9 = 15.7 5. __12 = a - __45 6. Ellie spent \$88.79 at the computer store. She then had \$44.50 left to buy a cool hat. How much money did she originally have? Write and solve an equation to answer the question. 7. Write a real-world problem for the equation x + 12 = 35. Then solve the equation. Lesson Quiz available online my.hrw.com
2. w = 37 3. x = –12 4. t = 17.6 5. a = __74 6. x − \$88.79 = \$44.50; x = \$133.29 7. See students’ answers; x = 23
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Concepts & Skills
Practice
Explore Activity Modeling Equations
Exercise 1
Example 1 Using Subtraction to Solve Equations
Exercises 2–6, 10, 12–14
Example 2 Using Addition to Solve Equations
Exercises 2–6, 11, 15–16
Example 3 Solving Equations that Represent Geometric Concepts
Exercise 7
Example 4 Writing Real-World Problems for a Given Equation
Exercises 8, 17
Exercise
Depth of Knowledge (D.O.K.)
Mathematical Processes
10–16
2 Skills/Concepts
1.A Everyday life
17–18
3 Strategic Thinking
1.F Analyze relationships
19
3 Strategic Thinking
1.A Everyday life
20
3 Strategic Thinking
1.G Explain and justify arguments
21
3 Strategic Thinking
1.F Analyze relationships
Additional Resources Differentiated Instruction includes: • Leveled Practice Worksheets
Name
Class
Date
12.2 Independent Practice 6.9.B, 6.9.C, 6.10.A
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Write and solve an equation to answer each question.
Handy Dandy Grocery Regular price
Sample answer: m - 123.45 = 36.55; \$160
Sample answer: e + 8 = 31; 23
11. My sister is 14 years old. My brother says that his age minus twelve is equal to my sister’s age. How old is my brother?
Both a and r are equal to \$1.50, so the discount is the same. 20. Critical Thinking An orchestra has twice as many woodwind instruments as brass instruments. There are a total of 150 brass and woodwind instruments.
17. Represent Real-World Problems Write a real-world situation that can be represented by 15 + c = 17.50. Then solve the equation and describe what your answer represents for the problem situation.
Sample answer: x + 8.95 = 21.35; \$12.40
a. Write two different addition equations that describe this situation. Use w for woodwinds and b for brass.
w + b = 150; w = b + b = 2b
Check students’ answers. c = 2.50
13. The Acme Car Company sold 37 vehicles in June. How many compact cars were sold in June?
w = 100; b = 50
Number sold
SUV
8
Compact
?
18. Critique Reasoning Paula solved the equation 7 + x = 10 and got 17, but she is not certain if she got the correct answer. How could you explain Paula’s mistake to her?
Sample answer: x + 8 = 37;
21. Look for a Pattern Assume the following: a + 1 = 2, b + 10 = 20, c + 100 = 200, d + 1,000 = 2,000, ... a. Solve each equation for each variable.
a = 1; b = 10, c = 100, d = 1,000, ...
29 compact cars
b. What pattern do you notice between the variables?
of subtracting. She should have
14. Sandra wants to buy a new MP3 player that is on sale for \$95. She has saved \$73. How much more money does she need?
c. What would be the value of g if the pattern continues?
g = 1,000,000
the equation and found that
Sample answer: 73 + b = 95 or
x = 3.
95 - b = 73; \$22
Lesson 12.2
EXTEND THE MATH
Every variable is ten times the one before it.
subtracted 7 from both sides of
© Houghton Mifflin Harcourt Publishing Company
b. How many woodwinds and how many brass instruments satisfy the given information?
Acme Car Company - June Sales Type of car
\$3.99
b. Which fruit has a greater discount? Explain.
t = \$773
12. Kim bought a poster that cost \$8.95 and some colored pencils. The total cost was \$21.35. How much did the colored pencils cost?
\$2.99
5-pound bag of oranges
1.49 + a = 2.99; 2.49 + r = 3.99
Sample answer: 548 = t - 225;
Sample answer: 14 = b - 12; b = 26
5-pound bag of apples
a. Write an equation to find the discount for each situation using a for apples and r for oranges.
16. Brita withdrew \$225 from her bank account. After her withdrawal, there was \$548 left in Brita’s account. How much money did Brita have in her account before the withdrawal?
elephants
Work Area
19. Multistep Handy Dandy Grocery is having a sale this week. If you buy a 5-pound bag of apples for the regular price, you can get another bag for \$1.49. If you buy a 5-pound bag of oranges at the regular price, you can get another bag for \$2.49.
Online Assessment and Intervention
15. Ronald spent \$123.45 on school clothes. He counted his money and discovered that he had \$36.55 left. How much money did he originally have?
10. A wildlife reserve had 8 elephant calves born during the summer and now has 31 total elephants. How many elephants were in the reserve before summer began?
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FOCUS ON HIGHER ORDER THINKING
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Activity available online
PRE-AP
Unit 4
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Activity Use the code shown in the table below to send messages. A
B
C
D
E
F
G
H
I
J
K
L
M
1
2
3
4
5
6
7
8
9
10
11
12
13
N
O
P
Q
R
S
T
U
V
W
X
Y
Z
14
15
16
17
18
19
20
21
22
23
24
25
26
• Write a short secret message on a piece of paper. • Use the chart to find the number that matches each letter of your message. Write the number above each letter in your message. • To send your message in code, write a series of equations on a separate piece of paper. Use n as your variable and write the equations in the same order as the letters appear in your message. For example, for the letter G, n = 7. You might write n + 5 = 12. • Trade equations and decode classmates’ messages. Addition and Subtraction Equations
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LESSON
12.3 Multiplication and Division Equations Texas Essential Knowledge and Skills The student is expected to: Expressions, equations, and relationships—6.10.A Model and solve one-variable, one-step equations and inequalities that represent problems, including geometric concepts.
Engage
ESSENTIAL QUESTION How do you solve equations that contain multiplication or division? Sample answer: You apply the inverse operation—division for multiplication equations and multiplication for division equations—to both sides of the equation.
Motivate the Lesson Ask: How many of you like to bake? Can you imagine writing an equation to calculate the ingredients you need? Begin the Explore Activity to see an example for a cookie recipe.
Expressions, equations, and relationships—6.9.B Represent solutions for one-variable, one-step equations and inequalities on number lines. Expressions, equations, and relationships—6.9.C Write corresponding real-world problems given one-variable, one-step equations or inequalities.
Mathematical Processes
Explore EXPLORE ACTIVITY Engage with the Whiteboard For B, have a student circle the remaining two groups of the model on the whiteboard. Then have students fill in the answer for C and the final answer. Ask how the model helped them understand the problem.
6.1.C Select tools, including real objects, manipulatives, paper and pencil, and technology as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems.
Explain EXAMPLE 1 Focus on Math Connections
ADDITIONAL EXAMPLE 1 Solve each equation. Graph the solution on a number line.
-5
0
B -27 = -9z z = 3 -5
0
Questioning Strategies
Mathematical Processes • How will you know what number to divide both sides of an equation by in order to solve it? Divide both sides by the number that the variable is multiplied by.
A 6y = -24 y = -4 -10
Mathematical Processes Remind students that division is the inverse, or opposite, of multiplication. To solve an equation that contains multiplication, use division to “undo” the multiplication.
5
Interactive Whiteboard Interactive example available online my.hrw.com
• Why must you divide both sides of the equation by the same number? You do so to maintain the equality, or the balance, of the equation.
Focus on Critical Thinking Mathematical Processes Challenge students to describe how solving a multiplication equation is similar to solving an addition or subtraction equation. Students should indicate that the same inverse operation must be applied to both sides of the equation, leaving the variable isolated on one side of the equation.
YOUR TURN Avoid Common Errors Watch for students who multiply both sides of the equation when they should divide. Remind students to be certain they are using the inverse operation and to check their answers.
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12.3 ?
Multiplication and Division Equations
ESSENTIAL QUESTION
Expressions, equations, and relationships—6.10.A Model and solve one-variable, one-step equations . . . that represent problems, including geometric concepts. Also 6.9.B, 6.9.C
How do you solve equations that contain multiplication or division?
EXPLORE ACTIVITY
Using Division to Solve Equations Separating the tiles on both sides of an equation mat into an equal number of groups models dividing both sides of an equation by the same number. Math On the Spot my.hrw.com
You can divide both sides of an equation by the same nonzero number, and the two sides will remain equal. When an equation contains multiplication, solve by dividing both sides of the equation by the same nonzero number.
6.9.B, 6.9.C, 6.10.A
Modeling Equations
EXAMPLE 1
Deanna has a cookie recipe that requires 12 eggs to make 3 batches of cookies. How many eggs are needed per batch of cookies?
Number of eggs per batch
·
=
A 9a = 54 Total eggs
x
1 1 1 1
x
1 1 1 1
x
1 1 1 1
© Houghton Mifflin Harcourt Publishing Company
3x
4
Divide both sides of the equation by 9.
Check: 9a = 54 ? 9(6) = 54 54 = 54
12
18 = -3d
x
1 1 1 1
x
1 1 1 1
x
1 1 1 1
4
eggs are needed per batch of cookies.
Reflect 1.
Notice that 9 is multiplied by a.
9a __ __ = 54 9 9
0 1 2 3 4 5 6 7 8 9 10
Substitute 6 for a. Multiply on the left side.
B 18 = -3d
B There are 3 x-tiles, so draw circles to separate the tiles into 3 equal groups. One group has been circled for you.
C How many +1-tiles are in each group? This is the solution of the equation.
9a = 54 a=6
3 x 12 · = To answer this question, you can use algebra tiles to solve 3x = 12.
A Model 3x = 12.
6.10.A, 6.9.B
Solve each equation. Graph the solution on a number line.
Let x represent the number of eggs needed per batch. Number of batches
Division Property of Equality
Look for a Pattern Why does it make sense to arrange the 12 tiles in 3 rows of 4 instead of any other arrangement of 12 + 1-tiles, such as 2 rows of 6?
Notice that -3 is multiplied by d.
18 -3d ___ = ____ -3 -3
Math Talk
Mathematical Processes
Check: 18 = -3d ? 18 = -3(-6)
Why is the solution to the equation the number of tiles in each group?
18 = 18
The number of +1-tiles in each group equals one x-tile.
-10-9 -8 -7 -6 -5 -4 -3 -2 -1 0
Substitute -6 for d. Multiply on the right side.
3 rows are evenly divisible by 3.
Divide both sides of the equation by -3.
-6 = d
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LESSON
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Online Assessment and Intervention
Solve the equation 3x = -21. Graph the solution on a number line. 2. x =
-7
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0
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PROFESSIONAL DEVELOPMENT Integrate Mathematical Processes This lesson provides an opportunity to address Mathematical Process TEKS 6.1.C, which calls for students to “select tools…and techniques, including mental math…and number sense to solve problems.” Students first use number sense to identify which inverse operation they should use to isolate the variable on one side of an equation. They then use pencil and paper to solve equations, to check their solutions by using substitution, and to graph the solutions on a number line. In Example 3, students use number sense to translate words to algebraic equations. Students should be encouraged to use mental math to check their answers.
Math Background Division is defined as multiplication by the reciprocal. To solve 3x = 15, for example, we could multiply both sides by __13 , which would be equivalent to dividing both sides by 3. While most students would rather divide by 3 than multiply by __13 , this alternate interpretation is useful when trying to solve equations such as __2 x = 4. Dividing by __2 is equivalent to multiplying 3 3 by __32 .
Multiplication and Division Equations
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ADDITIONAL EXAMPLE 2 Solve each equation. Graph the solution on a number line. y A __7 = -2 y = -14
-20
-15
-10
15
20
y
B 5 = __3 y = 15 10
Interactive Whiteboard Interactive example available online my.hrw.com
EXAMPLE 2
Focus on Math Connections
Mathematical Processes Remind students that multiplication and division are inverse operations. To solve an equation that contains division, use multiplication to “undo” the division.
Questioning Strategies
Mathematical Processes • How is solving an equation containing division similar to solving an equation containing multiplication? You solve both by using the inverse operation on both sides to get the variable by itself. • Does it make sense that the solution is greater than 15? Yes. Since some number divided by 2 is 15, the unknown number will be greater than (double) the unknown number.
Engage with the Whiteboard Cover up the sentences in blue next to each step of the solution in B and have students write a description of what is happening in each step on the whiteboard. Then have the students graph the solution on the number line.
Avoid Common Errors Watch for students who divide both sides of the equation when they should multiply. Remind students to be certain that they are using the inverse operation and to check their answers.
ADDITIONAL EXAMPLE 3 The area of Danielle’s garden is one-twelfth the area of her entire yard. The area of the garden is 10 square feet. What is the area of the yard? Write and solve an equation to solve the y problem. 10 = __ ; 120 square feet 12 Interactive Whiteboard Interactive example available online my.hrw.com
EXAMPLE 3
Questioning Strategies
Mathematical Processes • Why does it make sense to express Juanita’s total scrapbooking time as the decimal 2.5? How else might you express the time? You can compute easily with the number 2.5. Another way is to express the time as the mixed number 2__12 or as the fraction __52 . • Is Julia’s scrapbooking speed best described as a ratio, a rate, or a unit rate? Explain. It is a unit rate because it not only compares quantities in different units, but it also has a denominator of 1. • Suppose Juanita works at her usual rate for 6 hours one weekend. How many pages can she expect to complete? about 54 pages
Focus on Math Connections
Mathematical Processes In solving the problem about Juanita’s scrapbooking, students need to subtract after they solve the division equation, to compare her rate last week to her usual rate. This Example not only reinforces the usefulness of the four-step problem-solving process but also prepares students for solving two-step equations—those containing more than one operation.
Integrating the ELPS
c.4.G ELL Encourage English learners to take notes on new terms or concepts and to write them in familiar language.
Focus on Reasoning
Mathematical Processes In this multistep problem, students first need to find the total number of cards Roberto started with. Have students describe why __5x = 9 is the correct equation for finding that number.
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Using Multiplication to Solve Equations
Using Equations to Solve Problems You can use equations to solve real-world problems. Math On the Spot
Math On the Spot
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Multiplication Property of Equality
My Notes
You can multiply both sides of an equation by the same number, and the two sides will remain equal.
EXAMPLE 3
Problem Solving
Juanita is scrapbooking. She usually completes about 9 pages per hour. One night last week she completed pages 23 through 47 in 2.5 hours. Did she work at her average rate?
Solve each equation. Graph the solution on a number line.
© Houghton Mifflin Harcourt Publishing Company
Math Talk
0
10
20
30
40
50
Divide on the left side.
Notice that r is divided by the number 2. Multiply both sides of the equation by 2. 0 5 10 15 20 25 30 35 40 45 50
Check: 15 = __r 2 ? 30 15 = ___ 2 15 = 15
• Compare the number of pages Juanita can expect to complete with the number of pages she actually completed. Justify and Evaluate Solve
Both equations are solved by applying an inverse operation to both sides of the equation. The inverse operations applied are different.
Substitute 50 for x.
B 15 = __r 2 15 = __r 2 2 · 15 = 2 · __r 2 30 = r
• Solve an equation to find the number of pages Juanita can expect to complete.
How is solving a multiplication equation similar to solving a division equation? How are they different?
Multiply both sides of the equation by 5. -20 -10
Formulate a Plan
Mathematical Processes
Notice that x is divided by the number 5.
Substitute 30 for r.
TICKET TICKET
Identify the important information. • Worked for 2.5 hours • Starting page: 23 Ending page: 47 • Scrapbooking rate: 9 pages per hour
6.10.A, 6.9.B
x = 10 A __ 5 x = 10 __ 5 x = 5 · 10 5 · __ 5 x = 50 x = 10 Check: __ 5 ? 50 = ___ 10 5 10 = 10
Analyze Information
EXAMPL 2 EXAMPLE
6.9.C 6.9.C
Let n represent the number of pages Juanita can expect to complete in 2.5 hours if she works at her average rate of 9 pages per hour. Write an equation. n =9 ___ 2.5
n = 2.5 · 9 2.5 · ___ 2.5
Write the equation. Multiply both sides by 2.5.
n = 22.5 Juanita can expect to complete 22.5 pages in 2.5 hours. Juanita completed pages 23 through 47, a total of 25 pages. Because 25 > 22.5, she worked faster than her expected rate.
Divide on the right side.
Justify and Evaluate
You used an equation to find the number of pages Juanita could expect to complete in 2.5 hours if she worked at her average rate. You found that she could complete 22.5 pages.
Solve the equation __ = -1. Graph the solution on a number line. 9 y 3. _ = -1 9
y=
-9
Online Assessment and Intervention
The answer makes sense, because Juanita completed 25 pages in 2.5 hours, which is equivalent to a rate of 10 pages in 1 hour. Since 10 > 9, you know that she worked faster than her average rate.
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Since 22.5 pages is less than the 25 pages Juanita completed, she worked faster than her average rate.
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-10 -9 -8 -7 -6 -5-4 -3 -2 -1 0
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When an equation contains division, solve by multiplying both sides of the equation by the same number.
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DIFFERENTIATE INSTRUCTION Number Sense
Have students use a fraction bar to indicate division of both sides of the equation. This provides an easy visual check to compare the coefficient of the variable and the number chosen for dividing both sides of the equation.
Kinesthetic Experience
Give each group of students a set of nine cards numbered 1–9. On signal, each group picks a card at random and passes it to a different group. The number on this card becomes the divisor in a division equation. On a second signal, each group passes a different card to the same group, and this card becomes the quotient. Each group solves their equation, using the following form: (x ÷ first card) = (second card). The first group to solve their equation wins.
Differentiated Instruction includes: ••Reading Strategies ••Success for English Learners ELL ••Reteach ••Challenge PRE-AP
Multiplication and Division Equations
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EXAMPLE 4 Focus on Reasoning
Mathematical Processes Encourage students to begin by analyzing the equation and using number sense to find situations from everyday life that could make sense. For example, if an equation contained decimals, a situation involving money would most likely be a good real-world situation.
Questioning Strategies
Mathematical Processes • Using number sense, what can you determine about the value of x given that the product of two numbers is 72? That the value of x will be less than 72.
• How is the question in a real-world problem related to its equation? The question asks about the value of the variable.
Mathematical Processes Guide students to write problems that use sensible data, and have them explain step by step how to solve their equations. Challenge each student to create original word problems based on unique situations.
Elaborate Talk About It Summarize the Lesson Ask: How do you solve equations that contain multiplication or division? You apply the inverse operation—division for multiplication equations and multiplication for division equations—to both sides of the equation.
GUIDED PRACTICE Engage with the Whiteboard For Exercise 1, have students complete the verbal model on the whiteboard and then draw a model with algebra tiles below the verbal model. Ask students to explain their reasoning.
Avoid Common Errors Exercises 2–3 Remind students to use the inverse operation of division to undo multiplication or multiplication to undo division. Exercise 4 Remind students that the formula for area of a rectangle is A = l · w.
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Lesson 12.3
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DO NOT EDIT--Changes must be made through “File info” CorrectionKey=B
Guided Practice YOUR TURN 4. Roberto is dividing his baseball cards equally among himself, his brother, and his 3 friends. Roberto was left with 9 cards. How many cards did Roberto give away? Write and solve an equation to solve the problem.
Sample answer: _5x = 9; x = 45; Roberto gave away
Personal Math Trainer
1. Caroline ran 15 miles in 5 days. She ran the same distance each day. Write and solve an equation to determine the number of miles she ran each day. (Explore Activity)
b.
Number of
days
Math On the Spot my.hrw.com
8x = 72 Caroline ran
Examine each part of the equation. x is the unknown or quantity we are looking for.
2. x ÷ 3 = 3
= 72 means that after multiplying 8 and x, the result is 72.
x=
Write a real-world situation that involves multiplying two quantities. A hot air balloon flew at 8 miles per hour. How many hours did it take this balloon to travel 72 miles?
© Houghton Mifflin Harcourt Publishing Company
+ + + + +
Total number of miles
=
15
+ + + + +
+ + + + +
miles each day.
Solve each equation. Graph the solution on a number line. (Examples 1 and 2)
8 is multiplied by x.
3. 4x = -32
9
x=
0 1 2 3 4 5 6 7 8 9 10
Solve the equation.
-8
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0
4. The area of the rectangle shown is 24 square inches. How much longer is its length than its width? (Example 3)
8x = 72 8x __ __ = 72 8 8
3
+ + + + +
=
6 in.
24 = 6w; w = 4; 6 - 4 = 2 inches longer
Divide both sides by 8.
? ?
x=9 The balloon traveled for 9 hours.
ESSENTIAL QUESTION CHECK-IN
5. How do you solve equations that contain multiplication or division?
You apply the inverse operation—division for multiplication equations and multiplication for division YOUR TURN 5. Write a real-world problem for the equation 11x = 385. Then solve the equation.
Check students’ problems; x = 35
equations—to both sides of the equation.
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Lesson 12.3
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w
© Houghton Mifflin Harcourt Publishing Company
6.9.C
Write a real-world problem for the equation 8x = 72. Then solve the equation.
STEP 3
x
·
c. Draw algebra tiles to model the equation.
You can write a real-world problem for a given equation.
STEP 2
· Number of miles
run each day
5
Writing Real-World Problems
STEP 1
.
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45 - 9 = 36 cards.
EXAMPL 4 EXAMPLE
number of miles run each day
a. Let x represent the
Online Assessment and Intervention
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Unit 4
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Multiplication and Division Equations
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Personal Math Trainer Online Assessment and Intervention
Online homework assignment available
Evaluate GUIDED AND INDEPENDENT PRACTICE 6.9.B, 6.10.A
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Concepts & Skills
Practice
Explore Activity Modeling Equations
Exercise 1
Example 1 Using Division to Solve Equations
Exercise 3
Example 2 Using Multiplication to Solve Equations
Exercise 2
Example 3 Using Equations to Solve Problems
Exercises 4, 6–12
5. The area of a rectangle is 48 square inches. The length is 8 inches. What is the measure of its width? Write and solve an equation.
Example 4 Writing Real-Word Problems
Exercises 13–14
Lesson Quiz available online
Exercise
12.3 LESSON QUIZ 6.10.A Solve each equation. 1. __4x = 12.5 2. 8w = -120 z 3. 6 = ___ 4.5
4. 20 = 2.5m
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Depth of Knowledge (D.O.K.)
Mathematical Processes
2 Skills/Concepts
1.A Everyday life
13
3 Strategic Thinking
1.C Select tools
1. x = 50
14
3 Strategic Thinking
1.F Analyze relationships
2. w = -15
15
3 Strategic Thinking
1.G Explain and justify arguments
16–17
3 Strategic Thinking
1.F Analyze relationships
3. z = 27
6–12
4. m = 8 5. Sample answer: 8x = 48 inches; x = 6 in.
Additional Resources Differentiated Instruction includes: • Leveled Practice Worksheets
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Lesson 12.3
DO NOT EDIT--Changes must be made through “File info” CorrectionKey=A
DO NOT EDIT--Changes must be made through “File info” CorrectionKey=B
Name
Class
Date
12.3 Independent Practice 6.9.B, 6.9.C, 6.10.A
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Write and solve an equation to answer each question. Sample answers are given. 6. Jorge baked cookies for his math class’s end-of-year party. There are 28 people in Jorge’s math class including Jorge and his teacher. Jorge baked enough cookies for everyone to get 3 cookies each. How many cookies did Jorge bake?
14. Representing Real-World Problems Write and solve a problem involving money that can be solved with a division equation and has a solution of 1,350.
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Sample answer: Marcy split her income from last week
Online Assessment and Intervention
equally between paying her student loans, rent, and
11. Dharmesh has a square garden with a perimeter of 132 feet. Is the area of the garden greater than 1,000 square feet?
savings. She put \$450 in savings. How much did Marcy earn last week? _3x = 450, x = 1,350; Marcy earned
S
\$1,350 last week. S
c __ = 3; 84 cookies 28
FOCUS ON HIGHER ORDER THINKING
4s = 132; s = 33 ft;
15. Communicate Mathematical Ideas Explain why 7 · _7x = x. How does this relate to solving division equations?
33 × 33 = 1,089 square feet
7. Sam divided a rectangle into 8 congruent rectangles that each have the area shown. What is the area of the rectangle before Sam divided it?
7x 7 · _7x = __ = 1x = x; 7 divided by 7 is 1. 1x is the same as x. 7
Yes, the area of the garden is greater than 1,000 square feet.
When you solve division equations, you multiply both
Area = 5 cm2
sides of the equation by the number that will give you 1x, or x.
12. Ingrid walked her dog and washed her car. The time she spent walking her dog was one-fourth the time it took her to wash her car. It took Ingrid 14 minutes to walk the dog. How long did it take Ingrid to wash her car?
8. Carmen participated in a read-a-thon. Mr. Cole pledged \$4.00 per book and gave Carmen \$44. How many books did Carmen read?
4k = 44; 11 books
16. Critical Thinking A number tripled and tripled again is 729. What is the number?
w __ = 14; 56 minutes 4
3(3x) = 729;
13. Representing Real-World Problems Write and solve a problem involving money that can be solved with a multiplication equation.
9. Lee drove 420 miles and used 15 gallons of gasoline. How many miles did Lee’s car travel per gallon of gasoline?
15m = 420; 28 mi/gal
10. On some days, Melvin commutes 3.5 hours per day to the city for business meetings. Last week he commuted for a total of 14 hours. How many days did he commute to the city?
9x = 729;
x = 81;
the number is 81.
17. Multistep Andre has 4 times as many model cars as Peter, and Peter has one-third as many model cars as Jade. Andre has 36 model cars. a. Write and solve an equation to find how many model cars Peter has.
4p = 36; p = 9; Peter has 9 model cars.
\$168 for babysitting over 6 weeks. If she earned the same
b. Using your answer from part a, write and solve an equation to find how many model cars Jade has.
amount each week, how much
1 _ j = 9; j = 27; Jade has 27 model cars. 3
did she earn for one week? 6x = 168; x = \$28.
© Houghton Mifflin Harcourt Publishing Company
a __ = 5; 40 square centimeters 8
© Houghton Mifflin Harcourt Publishing Company
Work Area
3.5d = 14; 4 days
Lesson 12.3
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EXTEND THE MATH
PRE-AP
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Unit 4
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Activity available online
28/01/14 9:52 PM
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Activity Here is one way you can name 14, using four 7s and common arithmetic operations: 7×7 ____ +7 7
Use four 7s and the signs for common operations to name each given number. 1. 2 = __________ 2. 3 = __________ 3. 4 = __________ 4. 5 = __________
7+7+7
7+7
77 Sample solutions: 1. _77 + _77 2. _______ 3. __ - 7 4. 7 - _____ 7 7 7
Multiplication and Division Equations
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MODULE QUIZ
Assess Mastery
12.1 Writing Equations to Represent Situations
Use the assessment on this page to determine if students have mastered the concepts and standards covered in this module.
Determine whether the given value is a solution of the equation.
3 1
no
1. p - 6 = 19; p = 13
yes
b 3. __ = 5; b = 60 12
Response to Intervention
2
Personal Math Trainer
5. 18 - h = 13; h = -5
no
Online Assessment and Intervention
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2. 62 + j = 74; j = 12
yes
4. 7w = 87; w = 12
no
6. 6g = -86; g = -16
no
Write an equation to represent the situation. 7. The number of eggs in the refrigerator e decreased by 5 equals 18.
Intervention
e - 5 = 18
Enrichment
Access Ready to Go On? assessment online, and receive instant scoring, feedback, and customized intervention or enrichment.
Personal Math Trainer
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p + 17 = 29
12.2 Addition and Subtraction Equations Solve each equation.
Online and Print Resources
9. r - 38 = 9
Differentiated Instruction
Differentiated Instruction
• Reteach worksheets
• Challenge worksheets
• Reading Strategies • Success for English Learners ELL
ELL
11. n + 75 = 155
PRE-AP
Extend the Math PRE-AP Lesson Activities in TE
Additional Resources Assessment Resources includes: • Leveled Module Quizzes
r = 47 n = 80
10. h + 17 = 40
h = 23
12. q - 17 = 18
q = 35
12.3 Multiplication and Division Equations Solve each equation. © Houghton Mifflin Harcourt Publishing Company
Online Assessment and Intervention
8. The number of new photos p added to the 17 old photos equals 29.
13. 8z = 112 f 15. __ = 24 28
z = 14 f = 672
d 14. __ =7 14
16. 3a = 57
d = 98 a = 19
ESSENTIAL QUESTION 17. How can you solve problems involving equations that contain addition, subtraction, multiplication, or division?
Write an equation for the situation. Then apply the inverse operation to get the variable alone on one side of the equation.
Module 12
Texas Essential Knowledge and Skills Lesson
Exercises
12.1
1–8
6.7.B, 6.9.A, 6.10.B
12.2
9–12
6.9.B, 6.9.C, 6.10.A, 6.10.B
12.3
13–16
6.9.B, 6.9.C, 6.10.A, 6.10.B
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Module 12
TEKS
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Personal Math Trainer
MODULE 12 MIXED REVIEW
Texas Test Prep
Texas Test Prep
Item 8 Students can write the equation 17x = 680 to represent the situation and then start substituting answers in the equation to find which answer makes the sentence true. Choices A and B result in false equations, but choice C results in 680 = 680. Thus, choice C is the correct answer.
Selected Response 1. Kate has gone up to the chalkboard to do math problems 5 more times than Andre. Kate has gone up 11 times. Which equation represents this situation? A a - 11 = 5 B 5a = 11 C
Item 1 Some students will see the word times, automatically think of multiplication, and quickly pick choice B. Encourage them to read carefully to identify all key words or phrases such as more times, which indicates addition, not multiplication. Item 4 Some students may select choice A because they looked only at the endpoints under the arrow and didn’t pay attention to the direction in which the arrow was pointing. Remind them that they need to analyze art or diagrams carefully to fully understand them.
a - 5 = 11
2. For which equation is y = 7 a solution?
A 6x = 42
6 C _ x = 42
B 42 - x = 6
D 6 + x = 42
A t = 19
C
B t = 108
D t = 684
t = 120
8. The area of a rectangular deck is 680 square feet. The deck’s width is 17 feet. What is its length?
A 7y = 1 B y - 26 = -19 C
6. Jeordie spreads out a rectangular picnic blanket with an area of 42 square feet. Its width is 6 feet. Which equation could you use to find its length?
7. What is a solution to the equation 6t = 114?
D a + 5 = 11
Avoid Common Errors
Online Assessment and Intervention
y+7=0
y D _ = 14 2
A 17 feet
C
B 20 feet
D 51 feet
40 feet
3. Which is an equation? A 17 + x
C
20x = 200
B 45 ÷ x
D 90 - x
4. The number line below represents which equation?
9. Sylvia earns \$7 per hour at her afterschool job. One week she worked several hours and received a paycheck for \$91. Write and solve an equation to find the number of hours in which Sylvia would earn \$91.
.
1
3
0
0
0
0
0
0
1
1
1
1
1
1
2
2
2
2
2
2
3
3
3
3
3
3
3 + 7 = -4
4
4
4
4
4
4
D 3 - 7 = -4
5
5
5
5
5
5
6
6
6
6
6
6
7
7
7
7
7
7
8
8
8
8
8
8
9
9
9
9
9
9
-5 - 4 - 3 - 2 - 1
0 1 2 3 4 5
A -4 + 7 = 3 B -4 - 7 = 3 C
5. Becca hit 7 more home runs than Beverly. Becca hit 21 home runs. How many home runs did Beverly hit?
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Gridded Response
A 3
C
B 14
D 28
© Houghton Mifflin Harcourt Publishing Company
Texas Testing Tip Instead of solving equations directly, students can work backward by substituting answers into given equations to see if they are correct. Item 7 Students can try substituting the answers in the equation in order. When they try choice A, they should find that the result is a true equation, indicating that this is the correct solution.
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21
Unit 4
Texas Essential Knowledge and Skills Items
Mathematical Process TEKS
1
6.9.A
6.1.A
2
6.10.B
6.1.C
3
6.7.B
6.1.F
4*
6.3.D, 6.9.A, 6.9.B
6.1.D
5
6.10.A
6.1.A, 6.1.F
6
6.10.A
6.1.A, 6.1.F
7
6.10.B
6.1.C
8
6.10.A
6.1.A, 6.1.F
9
6.10.A, 6.10.B
6.1.A, 6.1.F
* Item integrates mixed review concepts from previous modules or a previous course.
Equations and Relationships
344
Inequalities and Relationships ?
ESSENTIAL QUESTION How can you use inequalities and relationships to solve real-world problems?
MODULE
You can model real-world problems with inequalities, then use algebraic rules to solve the inequalities.
13
LESSON 13.1
Writing Inequalities 6.9.A, 6.9.B, 6.10.B
LESSON 13.2
Addition and Subtraction Inequalities 6.9.B, 6.9.C, 6.10
LESSON 13.3
Multiplication and Division Inequalities with Positive Numbers 6.9.B, 6.9.C, 6.10
LESSON 13.4
Multiplication and Division Inequalities with Rational Numbers
© Houghton Mifflin Harcourt Publishing Company
6.9.B, 6.10.A, 6.10.B
Real-World Video
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Module 13
Some rides at amusement parks indicate a minimum height required for riders. You can model all the heights that are allowed to get on the ride with an inequality.
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Math On the Spot
Animated Math
Personal Math Trainer
Go digital with your write-in student edition, accessible on any device.
Scan with your smart phone to jump directly to the online edition, video tutor, and more.
Interactively explore key concepts to see how math works.
Get immediate feedback and help as you work through practice sets.
345
Are You Ready? Use the assessment on this page to determine if students need intensive or strategic intervention for the module’s prerequisite skills. 2 1
Understand Integers EXAMPLE
Response to Intervention
-735 Write an integer to represent each situation.
Intervention
–75
Enrichment
Access Are You Ready? assessment online, and receive instant scoring, feedback, and customized intervention or enrichment.
Online Assessment and Intervention
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Skills Intervention worksheets
Differentiated Instruction
• Skill 33 Understand Integers
• Challenge worksheets
• Skill 59 Solve Multiplication Equations
2. a football player’s 3. spending \$1,200 4. a climb of 2,400 gain of 9 yards on a flat screen feet 2,400 9 TV -1,200
Integer Operations EXAMPLE
Online and Print Resources
• Skill 47 Integer Operations
Online Assessment and Intervention
Decide whether the integer is positive or negative: into the ground → negative Write the integer.
A water well was drilled 735 feet into the ground.
1. a loss of \$75
Personal Math Trainer
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3 × 8 = 24 -30 ÷ (-5) = 6
The product or quotient of two integers is positive if the signs of the integers are the same.
7 × (-4) = -28 -72 ÷ 9 = -8
The product or quotient of two integers is negative if the signs of the integers are different.
Find the product or quotient.
PRE-AP PRE-AP
Extend the Math Lesson Activities in TE
5. 6 × 9
54
9. 3 × (-7)
-21 10.
6. 15 ÷ (-5) -64 ÷ 8
-3 -8
7. -8 × 6
-48
11. -8 × (-2)
16
8. -100 ÷ (10) -10 12. 32 ÷ 2
16 © Houghton Mifflin Harcourt Publishing Company
3
Personal Math Trainer
Complete these exercises to review skills you will need for this chapter.
Solve Multiplication Equations 3 _ h = 15 4
EXAMPLE
4 _ _ · 3 h = 15 · _43 3 4 ·4 _____ h = 15 3
Write the equation. 3 . Multiply both sides by the h is multiplied by __ 4 4 , to isolate reciprocal, __ the variable. 3
h = 20
Simplify.
14. _35 n = 21
35
Solve. 13. 9p = 108
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12
15. _47 k = 84
147
3 16. __ e = 24 20
160
Unit 4
PROFESSIONAL DEVELOPMENT VIDEO my.hrw.com
Author Juli Dixon models successful teaching practices as she explores inequality concepts in an actual sixth-grade classroom.
Online Teacher Edition Access a full suite of teaching resources online—plan, present, and manage classes and assignments.
Professional Development
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Interactive Answers and Solutions Customize answer keys to print or display in the classroom. Choose to include answers only or full solutions to all lesson exercises.
Interactive Whiteboards Engage students with interactive whiteboard-ready lessons and activities. Personal Math Trainer: Online Assessment and Intervention Assign automatically graded homework, quizzes, tests, and intervention activities. Prepare your students with updated, TEKS-aligned practice tests.
Inequalities and Relationships
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DO NOT EDIT--Changes must be made through "File info" CorrectionKey=B
Visualize Vocabulary Use the ✔ words to complete the graphic.
Visualize Vocabulary The graphic organizer helps students review vocabulary associated with evaluating expressions. If time allows, brainstorm other vocabulary that can be added to the chart.
>, <
4x + 4 = 12; x = 2
greater than, less than
solution
3x - 5
Understand Vocabulary
algebraic expression
Use the following explanation to help students learn the preview words. Inequalities are similar to equations in that they represent a statement relating two expressions with a symbol. Equations use an equal sign, while inequalities use an inequality symbol.
Students can use these reading and note-taking strategies to help them organize and understand new concepts and vocabulary. c.4.D Use prereading supports such as graphic organizers, illustrations, and
pretaught topic-related vocabulary to enhance comprehension of written text.
✔ algebraic expression (expresión algebraica) evaluating (evaluar) ✔ greater than (mayor que) ✔ less than (menor que) like terms (términos semejantes) ✔ numerical expression (expresión numérica) properties of operations (propiedades de las operaciones) ✔ solution (solución) term (término, en una expresión)
Preview Words Match the term on the left to the correct expression on the right.
© Houghton Mifflin Harcourt Publishing Company
Integrating the ELPS
6×4
numerical expression
Review Words
Understand Vocabulary
The solution of an inequality is a value or values that make the inequality true.
Evaluating Expressions
Vocabulary
1. solution of an inequality
A. A value or values that make the inequality true.
2. coefficient
B. A specific number whose value does not change.
3. constant
C. The number that is multiplied by the variable in an algebraic expression.
coefficient (coeficiente) constant (constante) solution of an inequality (solución de una desigualdad) variable (variable)
Two-Panel Flip Chart Create a two-panel flip chart to help you understand the concepts in this module. Label one flap “Adding and Subtracting Inequalities.” Label the other flap “Multiplying and Dividing Inequalities.” As you study each lesson, write important ideas under the appropriate flap.
Module 13
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Grades 6–8 TEKS Before Students understand: • operations with rational numbers • properties of operations: inverse, identity, commutative, associative, and distributive properties
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Module 13
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InCopy Notes
InDesign Notes
1. This is a list
1. This is a list
In this module Students will learn how to: • write one-variable, one-step inequalities to represent constraints or conditions within problems • model and solve one-variable, one-step inequalities that represent problems • write corresponding real-world problems given one-variable, one-step inequalities
347
After Students will learn how to: • write one-variable, two-step inequalities to represent real-world problems • write a real-world problem to represent a one-variable, two-step inequality • solve one-variable, two-step inequalities
MODULE 13
Unpacking the TEKS
Unpacking the TEKS Understanding the TEKS and the vocabulary terms in the TEKS will help you know exactly what you are expected to learn in this module.
Use the examples on this page to help students know exactly what they are expected to learn in this module.
6.9.B Represent solutions for onevariable, one-step equations and inequalities on number lines.
Texas Essential Knowledge and Skills Content Focal Areas Expressions, equations, and relationships—6.9 The student applies mathematical process standards to use equations and inequalities to represent situations. Expressions, equations, and relationships—6.10 The student applies mathematical process standards to use equations and inequalities to solve problems.
What It Means to You You will learn to graph the solution of an inequality on a number line.
Key Vocabulary
UNPACKING EXAMPLE 6.9.B
equation (ecuación) A mathematical sentence that shows that two expressions are equivalent.
The temperature in a walk-in freezer must stay under 5 °C. Write and graph an inequality to represent this situation.
inequality (desigualdad) A mathematical sentence that shows the relationship between quantities that are not equal. solution of an inequality (solución de una desigualdad) A value or values that make the inequality true.
Write the inequality. Let t represent the temperature in the freezer. The temperature must be less than 5 °C. t<5 Graph the inequality. 0
5
10
6.10.A
c.4.F Use visual and contextual support … to read grade-appropriate content area text … and develop vocabulary … to comprehend increasingly challenging language.
Model and solve one-variable, one-step equations and inequalities that represent problems, including geometric concepts.
You can model and solve a one-variable, one-step inequality. UNPACKING EXAMPLE 6.10.A
Donny buys 3 binders and spends more than \$9. How much did he spend on each binder? Let x represent the cost of one binder.
Go online to see a complete unpacking of the .
Number of binders · Cost of a binder > Total cost of binders
x
>
Use algebra tiles to model 3x > 9 and solve the inequality. x>3 Donny spent more than \$3 on each binder.
+ + +
3 Visit my.hrw.com to see all the unpacked.
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What It Means to You
Lesson 13.1
Lesson 13.2
·
9
>
© Houghton Mifflin Harcourt Publishing Company • Image Credits: Image Source/Corbis
Integrating the ELPS
+ + + + + + + + +
Unit 4
Lesson 13.3
Lesson 13.4
6.9.A Write one-variable, one-step equations and inequalities to represent constraints or conditions within problems. 6.9.B Represent solutions for one-variable, one-step equations and inequalities on number lines. 6.9.C Write corresponding real-world problems given one-variable, one-step equations or inequalities. 6.10.A Model and solve one-variable, one-step equations and inequalities that represent problems, including geometric concepts. 6.10.B Determine if the given value(s) make(s) one-variable, one-step equations or inequalities true.
Inequalities and Relationships
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LESSON
13.1 Writing Inequalities Texas Essential Knowledge and Skills The student is expected to: Expressions, equations, and relationships—6.9.A Write one-variable, one-step equations and inequalities to represent constraints or conditions within problems. Expressions, equations, and relationships—6.9.B
Engage
ESSENTIAL QUESTION How can you use inequalities to represent real-world constraints or conditions? Sample answer: You can choose a letter to represent the variable value in the situation and then use one of the inequality symbols to describe its range of values.
Motivate the Lesson Ask: If you know the lowest and highest temperatures recorded for yesterday, how could you describe yesterday’s temperature at any given time of day? For example, what could you say about the temperature at noon? Begin the Explore Activity to find out.
Represent solutions for one-variable, one-step equations and inequalities on number lines. Expressions, equations, and relationships—6.10.B Determine if the given value(s) make(s) one-variable, one-step equations or inequalities true.
Mathematical Processes 6.1.E Create and use representations to organize, record, and communicate mathematical ideas.
Explore EXPLORE ACTIVITY Engage with the Whiteboard
Have students graph -2 °F on the number line on the whiteboard and then graph -1 °F, 0 °F, 3 °F, 5 °F, and 6 °F in a different color on the same number line. Then have them write inequalities comparing each of the temperatures from B to -2 °F on the whiteboard. Students will see that all five temperatures are greater than -2 °F. Ask students to compare some of the numbers to the left of -2 as well, so that they use both the > and < symbols.
Explain ADDITIONAL EXAMPLE 1 Graph the solutions of each inequality. Check the solutions. A b ≥ -4 -5
0
5
5
Sample check: -3 > -5 Interactive Whiteboard Interactive example available online my.hrw.com
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Lesson 13.1
Mathematical Processes Remind students that when a variable is less than a given number, all the values to the left of the given number on the number line make that inequality true. They are all solutions of the inequality. For example, if x < 4, then every number less than 4 is a solution. Mathematical Processes • What is the difference between the graph of y ≤ 4 and the graph of y < 4? In the first graph, 4 is included in the solution set. In the second graph, 4 is not included in the solution set.
B -3 > s 0
Focus on Communication
Questioning Strategies
Sample check: -1 ≥ -4
-5
EXAMPLE 1
• What is the difference between using a solid circle and an open circle? A solid circle includes the number at that point; an open circle does not. • How do you know when to use a solid circle or an open circle? A solid circle is used to represent ≤ or ≥ on a graph; an open circle is used to represent < or > on a graph. • How can you check that the graph of an inequality is correct? Pick a point on the shaded portion of the graph. Any point selected should make the inequality a true inequality.
LESSON
Expressions, equations, and relationships— 6.9.A Write … inequalities to represent constraints or conditions within problems. Also 6.9.B, 6.10.B.
13.1 Writing Inequalities ESSENTIAL QUESTION
Math On the Spot my.hrw.com
How can you use inequalities to represent real-world constraints or conditions?
EXAMPLE 1
Yes; yes; both numbers make the inequality true.
6.9.A
EXPLORE ACTIVITY
Using Inequalities to Describe Quantities You can use inequality symbols with variables to describe quantities that can have many values. Symbol
Meaning
Word Phrases
<
Is less than
Fewer than, below
>
Is greater than
More than, above
Is less than or equal to
At most, no more than
Is greater than or equal to
At least, no less than
A y ≤ -3
© Houghton Mifflin Harcourt Publishing Company
STEP 1
Draw a solid circle at -3 to show that -3 is a solution.
STEP 2
Shade the number line to the left of -3 to show that numbers less than -3 are solutions.
Mathematical Processes
Is -4 _14 a solution of y ≤ -3? Is -5.6?
-5 -4 -3 -2 -1
STEP 3
Draw an empty circle at 1 to show that 1 is not a solution.
STEP 2
Shade the number line to the right of 1 to show that numbers greater than 1 are solutions. -5 -4 -3 -2 -1
are located to the right of -2.
STEP 3
D How many other numbers have the same relationship to -2 as the temperatures in B ? Give some examples.
D
Use an open circle for an inequality that uses > or <.
0 1 2 3 4 5
Check your answer. Substitute 2 for m. 1<2 1 is less than 2, so 2 is a solution.
Reflect
infinitely many; any number greater than -2; sample answer: 1, 2, 8.5, 10
1.
on a number
How is x < 5 different from x ≤ 5?
For x < 5, 5 is not a solution and is not included in the graph. For x ≤ 5, 5 is a solution and is included
a ray extending to the right with its open endpoint at -2 >
-4 is less than -3, so -4 is a solution.
STEP 1
They are all greater than -2. All of the temperatures
D
-4 ≤ -3
C How do the temperatures in B compare to -2? How can you see this relationship on the number line?
Complete this inequality: x
0 1 2 3 4 5
B 1
0 1 2 3 4 5 6 7 8
F Let x represent all the possible answers to
Use a solid circle for an inequality that uses ≥ or ≤.
Choose a number that is on the shaded section of the number line, such as -4. Substitute -4 for y.
B The temperatures 0 °F, 3 °F, 6 °F, 5 °F, and -1 °F have also been recorded in Florida. Graph these temperatures on the number line.
E Suppose you could graph all of the possible answers to line. What would the graph look like?
6.9.B
Graph the solutions of each inequality. Check the solutions.
Math Talk
A The lowest temperature ever recorded in Florida was -2 °F. Graph this temperature on the number line. -8 -7 -6 -5 -4 -3 -2 -1
A solution of an inequality that contains a variable is any value of the variable that makes the inequality true. For example, 7 is a solution of x > -2, since 7 > -2 is a true statement.
© Houghton Mifflin Harcourt Publishing Company
?
Graphing the Solutions of an Inequality
in the graph.
.
-2 Lesson 13.1
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Unit 4
PROFESSIONAL DEVELOPMENT Integrate Mathematical Processes This lesson provides an opportunity to address Mathematical Process TEKS 6.1.E, which calls for students to “create and use representations to… communicate mathematical ideas.” In the Explore Activity and the Examples, students use number lines, word expressions, and mathematical symbols to express inequalities. They use graphs to represent inequalities and to determine if a given number is a solution.
Math Background Inequalities have a number of properties, including the Transitive Property of Inequality. This property states that for any real numbers a, b, c, if a > b and b > c, then a > c. if a < b and b < c, then a < c. if a > b and b = c, then a > c. if a < b and b = c, then a < c. This may seem like common sense, but the Transitive Property is not necessarily true in daily life. If Team A defeats Team B, and Team B defeats Team C, you can’t assume that Team A will defeat Team C. Writing Inequalities
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ADDITIONAL EXAMPLE 2 A Write an inequality that represents the phrase “y minus 3 is less than or equal to -5.” Then graph the inequality. y - 3 ≤ -5 -5
0
5
B The temperature of the river is greater than -1 °C. Write and graph an inequality to represent this situation. t > -1 -5
0
5
Interactive Whiteboard Interactive example available online my.hrw.com
YOUR TURN Avoid Common Errors If students have trouble determining which side of the number line to shade, remind them that the inequality sign always points to the lesser of two numbers. Since t ≤ -4, they should shade the number line to the left of -4, because the numbers decrease to the left on a number line.
EXAMPLE 2 Focus on Reasoning
Mathematical Processes Point out to students that sometimes a problem may provide clues and facts that you must use to find a solution. Encourage them to use logical reasoning to solve this kind of problem. For example, in A, tell students that the first step is to identify key words or phrases that indicate operations or relationships. Then they can proceed to write an equation.
Questioning Strategies
Mathematical Processes • In A, how do you know which operation to use to write the inequality? The word sum indicates addition. • In A, how do you know which inequality symbol to use? The phrase greater than indicates the symbol >. • In B, how do you know which inequality symbol to use? The phrase keeps the temperature below 5 °C tells me that the temperature is less than 5, which indicates that the symbol < should be used.
Animated Math Modeling Inequalities Students model inequalities using interactive algebra tiles.
Exercise 3 Some students may read the problem quickly and use > instead of ≥. Encourage them to begin by underlining the key words or phrases before trying to graph the inequality.
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Elaborate Talk About It Summarize the Lesson Ask: How can you make sure you have graphed an inequality correctly? Test one of the values on the number line. If it makes the inequality true, the number line is correct.
GUIDED PRACTICE Engage with Whiteboard For Exercise 3, have students begin by underlining the key words or phrases on the whiteboard and then write an equation, using the same method shown in Example 2. Finally, ask students to graph the inequality on the number line.
Avoid Common Errors Exercises 1 and 2 Some students may shade in the wrong direction when they attempt to graph the solution set of an inequality, such as 1 ≤ x. Reading 1 ≤ x as “x is greater than or equal to 1” serves as a reminder to shade to the right. Exercise 4 Some students may have difficulty determining which inequality symbol to use. Encourage them to begin by underlining key words or phrases before graphing the inequality.
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Lesson 13.1
Graph the solution of the inequality t ≤ -4. - 10 - 9 - 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1
0 1 2 3 4 5 6 7 8 9 10
Personal Math Trainer
Personal Math Trainer
Online Assessment and Intervention
3.
1 + y ≥ 3; y = 1 is not a solution because 1 + 1 is not
Online Assessment and Intervention
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greater than or equal to 3.
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Writing Inequalities
Write an inequality that represents the phrase the sum of 1 and y is greater than or equal to 3 . Check to see if y = 1 is a solution.
Write and graph an inequality to represent each situation.
You can write an inequality to model the relationship between an algebraic expression and a number. You can also write inequalities to represent certain real-world situations.
The highest temperature in February was 6 °F.
5.
Each package must weigh more than 2 ounces.
Math On the Spot
6.9.A, 6.10.B
A Write an inequality that represents the phrase the sum of y and 2 is greater than 5. Draw a graph to represent the inequality. STEP 1
y+2
>
Animated Math
© Houghton Mifflin Harcourt Publishing Company
Guided Practice 1. Graph 1 ≤ x. Use the graph to determine which of these numbers are solutions of the inequality: -1, 3, 0, 1 (Explore Activity and Example 1)
Use an open circle at 3 and shade to the right of 3.
3, 1
0 1 2 3 4 5
Check your solution by substituting a number greater than 3, such as 4, into the original inequality. 4+2>5
0 1 2 3 4 5
-5 -4 -3 -2 -1
0 1 2 3 4 5
3. Write an inequality that represents the phrase “the sum of 4 and x is less than 6.” Draw a graph that represents the inequality, and check your solution. (Example 2)
-5 -4 -3 -2 -1
0 1 2 3 4 5
-5 -4 -3 -2 -1
0 1 2 3 4 5
4+x<6 4. During hibernation, a garter snake’s body temperature never goes below 3 °C. Write and graph an inequality that represents this situation. (Example 2)
6 is greater than 5, so 4 is a solution.
B To test the temperature rating of a coat, a scientist keeps the temperature below 5 °C. Write and graph an inequality to represent this situation.
Let t be temperature in °C; t ≥ 3
Write the inequality. Let t represent the temperature in the lab. t<5
STEP 2
-5 -4 -3 -2 -1
2. Graph -3 > z. Check the graph using substitution. (Example 1)
Substitute 4 for y.
6>5
STEP 1
0 1 2 3 4 5 6 7 8 9 10 11 12
my.hrw.com
5
Graph the solution. For y + 2 to have a value greater than 5, y must be a number greater than 3.
-5 -4 -3 -2 -1
STEP 3
-2 -1
w>2
Write the inequality. The sum of y and 2 is greater than 5.
STEP 2
t≤6
0 1 2 3 4 5 6 7 8 9 10 11 12
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EXAMPL 2 EXAMPLE
4.
? ?
The temperature must be less than 5 °C.
Graph the inequality.
ESSENTIAL QUESTION CHECK-IN
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2.
5. Write an inequality to represent this situation: Nina wants to take at least \$15 to the movies. How did you decide which inequality symbol to use?
d ≥ 15, where d represents dollars. “At least” means
0 1 2 3 4 5 6 7 8 9 10
she wants to take \$15 or more than \$15. Lesson 13.1
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Unit 4
DIFFERENTIATE INSTRUCTION Critical Thinking
Cooperative Learning
Have students work together to consider absolute-value inequalities, such as | x | < 2.
Have students work in groups of 4. Have each group make a set of inequality symbol cards, a variable card, and 6 number cards (3 negative numbers and 3 positive numbers). Have the students take turns using the cards to create an inequality, such as this one:
Differentiated Instruction includes: • Reading Strategies • Success for English Learners ELL • Reteach • Challenge PRE-AP
First have students find numbers that make the inequality true. Then have them use the numbers to sketch a graph of what they think the solution should be. -5
-2
0
2
5
x
-4.5
Then have the groups record the inequalities and graph them on a number line.
Writing Inequalities
352
Personal Math Trainer Online Assessment and Intervention
Online homework assignment available
Evaluate GUIDED AND INDEPENDENT PRACTICE 6.9.A, 6.9.B, 6.10.B
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13.1 LESSON QUIZ 6.9.A Graph each inequality. 1. a ≤ -2 2. n < 4
Concepts & Skills
Practice
Explore Activity Using Inequalities to Describe Quantities
Exercise 1
Example 1 Graphing the Solutions of an Inequality
Exercises 1–2, 6–11, 16–19
Example 2 Writing Inequalities
Exercises 3–4, 12–15, 16–19
3. h > -1.5 4. t ≤ 3 Write an inequality that matches the number line model. Use x for the variable. 5. 6. 7. 8.
-10
-5
0
-5
0
5
-5
0
5
-5
0
5
Lesson Quiz available online my.hrw.com
2. 3. 4.
-5
0
5
-5
0
5
-5
0
5
-5
0
5
5. x ≤ -5 6. x > -1 7. x ≥ -0.5 8. x < 2 9. x ≥ 2.5
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Depth of Knowledge (D.O.K.)
Lesson 13.1
Mathematical Processes
6
2 Skills/Concepts
1.F Analyze relationships
7–15
2 Skills/Concepts
1.D Multiple representations
16
2 Skills/Concepts
1.F Analyze relationships
17–19
2 Skills/Concepts
1.A Everyday life
20
3 Strategic Thinking
1.G Explain and justify arguments
21–22
3 Strategic Thinking
1.F Analyze relationships
9. The weight of a package is at least 2.5 pounds. Write an inequality to represent this situation.
1.
Exercise
Differentiated Instruction includes: • Leveled Practice Worksheets
DO NOT EDIT--Changes must be made through “File info” CorrectionKey=A
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Class
Name
Date
Write and graph an inequality to represent each situation.
13.1 Independent Practice
Personal Math Trainer
6.9.A, 6.9.B, 6.10.B
10
Online Assessment and Intervention
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0.03, 0, 1.5, _12
-5
9. x ≥ -9 10. n > 2.5 11. -4 _12 >x
© Houghton Mifflin Harcourt Publishing Company
x>6
13.
x ≤ -3
14.
x < 1.5
15.
x ≥ -3.5
14
15
16
17
18
0
1
2
3
t < 3.5
-3
-2
-1
19
20
4
5
g > 150
50 100 150 200 250 300
0 1 2 3 4 5 6 7 8 9 10 11 12
- 10 - 9 - 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1
0
1
2
3
4
- 12 - 11 - 10 - 9 - 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1
0
1
2
20. Communicate Mathematical Ideas Explain how to graph the inequality 8 ≥ y.
Sample answer: Make a solid circle at 8 because of the
-5
-4
-3
-2
-1
0
1
2
2.5 3
4
5
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
-4 12
- 10 - 9 - 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1
inequality symbol, greater than or equal to. Then shade in the numbers to the left of 8, which are the numbers that make y less than 8.
0 1 2 3 4 5 6 7 8 9 10
21. Represent Real-World Problems The number line shows an inequality. Describe a real-world situation that the inequality could represent.
0 1 2 3 4 5 6 7 8 9 10
Sample answer: Steve has more than \$2.75 in his wallet.
0 - 10 - 9 - 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1
-5
-4
-3
-2
-1
0
-5
-4
-3
-2
-1
0
1
2
3
4
Lesson 13.1
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D x is less than or equal to -2 or x is greater than or equal to 3; -2 ≥ x or x ≥ 3
- 10 - 9 - 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1
0 1 2 3 4 5 6 7 8 9 10
- 10 - 9 - 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1
0 1 2 3 4 5 6 7 8 9 10
C
D
Unit 4
6_MTXESE051676_U4M13L1.indd 354
Activity available online
Activity The four graphs at right show constraints on A both ends of the graph. Challenge students to describe each graph in words and with an inequality statement. Tell them that graphs C and D describe the solutions for a single variable and that they should use the word “or” B to describe these situations.
C x is less than -2 or x is greater than 3; -2 > x or x>3
5
or fractional parts of students.
No; 46 is not greater than or equal to 48.
B x is greater than -2 and is less than or equal to 3; -2 < x ≤ 3
4
not make sense to have negative numbers of students
b. Can a child who is 46 inches tall ride the roller coaster? Explain.
A x is greater than -2 and less than 3: -2 < x < 3
3
3, and 4 as solutions is correct. In this example, it does
c ≥ 48
PRE-AP
2
Sample answer: The number line that shows only 0, 1, 2,
38 40 42 44 46 48 50 52 54 56 58
EXTEND THE MATH
1
22. Critique Reasoning Natasha is trying to represent the following situation with a number line model: There are fewer than 5 students in the cafeteria. She has come up with two possible representations, shown below. Which is the better representation, and why?
5
16. A child must be at least 48 inches tall to ride a roller coaster. a. Write and graph an inequality to represent this situation.
Work Area
FOCUS ON HIGHER ORDER THINKING
Write an inequality that matches the number line model. 12.
13
© Houghton Mifflin Harcourt Publishing Company
8. -7 < h
-4
0 -2 -1
12
19. The goal of the fundraiser is to make more than \$150.
Graph each inequality. 7. t ≤ 8
11
18. The temperature is less than 3.5 °F.
6. Which of the following numbers are solutions to x ≥ 0? -5, 0.03, -1, 0, 1.5, -6, _12
s ≥ 14.5
17. The stock is worth at least \$14.50.
28/01/14 10:12 PM
my.hrw.com
-5
0
5
-5
0
5
-5
0
5
-5
0
5
Writing Inequalities
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LESSON
13.2 Addition and Subtraction Inequalities Texas Essential Knowledge and Skills The student is expected to: Expressions, equations, and relationships—6.10.A Model and solve one-variable, one-step inequalities that represent problems, including geometric concepts. Expressions, equations, and relationships—6.9.B Represent solutions for one-variable, one-step equations and inequalities on number lines. Expressions, equations, and relationships—6.9.C Write corresponding real-world problems given one-variable, one-step equations or inequalities. Expressions, equations, and relationships—6.10.B Determine if the given value(s) make(s) one-variable, one-step equations or inequalities true.
Engage
ESSENTIAL QUESTION How can you solve an inequality involving addition or subtraction? Sample answer: You can use the Addition and Subtraction Properties of Inequality: add or subtract the same amount from both sides to isolate the variable.
Motivate the Lesson Ask: You’ve used algebra tiles to model an equation. Do you think you can use algebra tiles to model an inequality? Take a guess. Begin the Explore Activity to find out how.
Explore EXPLORE ACTIVITY Engage with the Whiteboard
Have a student use algebra tiles to make a model of the equation x + 5 = 8 on the whiteboard. Then have another student make a model of the inequality x + 5 ≥ 8 on the whiteboard. Then have all students graph the results on number lines. Discuss with the students how the two models are similar and how they are different. Emphasize the differences.
Mathematical Processes 6.1.D Communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate.
Explain EXAMPLE 1 Focus on Math Connection
ADDITIONAL EXAMPLE 1 Solve each inequality. Graph and check the solution. A x + 3 ≥ -2 -5
x ≥ -5 0
B -6 < y -5
5
Questioning Strategies
Mathematical Processes • In A, Step 1, why do you subtract 5 from both sides? The inequality has + 5 in it. To isolate the variable, you need to add the inverse, which is the same as subtracting 5.
y > -1
-5
0
Mathematical Processes Solving inequalities is very similar to solving equations. Remind students that they used the properties of equality and inverse operations to solve equations in an earlier module. Similarly, you can use the Addition and Subtraction Properties of Inequality and inverse operations to solve inequalities. Show how solving x + 5 = -12 is similar to solving x + 5 < -12.
5
Interactive Whiteboard Interactive example available online my.hrw.com
• In B, Step 2, why do you shade to the right when the inequality is ≤? Because the variable is on the right-hand side of the inequality, you need to read the inequality starting with the variable so that you will know how to shade the graph correctly. If you read the inequality 11 ≤ y as “y is greater than or equal to 11,” you will see that you need to shade to the right. • How can you check that the graph of an inequality is correct? Pick a point on the shaded portion of the graph. Any point selected should make the inequality a true inequality.
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Lesson 13.2
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LESSON
13.2 ?
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ESSENTIAL QUESTION
Expressions, equations, and relationships—6.10.A Model and solve one-variable, one-step… inequalities that represent problems. Also 6.9.B, 6.9.C, 6.10.B.
Math On the Spot my.hrw.com
How can you solve an inequality involving addition or subtraction?
EXPLORE ACTIVITY
Using Properties of Inequalities
6.10.A
EXAMPLE 1
Modeling One-Step Inequalities
A x + 5 < -12
On a day in January in Watertown, NY, the temperature was 5 °F at dawn. By noon it was at least 8 °F. By how many degrees did the temperature increase?
STEP 1
Increase in temperature
Solve the inequality. x + 5 < -12
A Let x represent the increase in temperature. Write an inequality. +
6.9.B, 6.10.B
Solve each inequality. Graph and check the solution.
You can use algebra tiles to model an inequality involving addition.
Temperature at dawn
Subtraction Property of Inequality
You can add the same number to You can subtract the same number both sides of an inequality and the from both sides of an inequality inequality will remain true. and the inequality will remain true.
8
5 ____ -5 ____ x < -17
Use the Subtraction Property of Inequality. Subtract 5 from both sides.
STEP 2
Graph the solution.
STEP 3
Check the solution. Substitute a solution from the shaded part of your number line into the original inequality. ? -18 + 5 < -12 Substitute -18 for x into x + 5 < -12
5
x
+
B The model shows 5 + x ≥ 8. How many tiles must you remove from each side to isolate x on one side of the inequality? Circle these tiles.
5
8
+ + + + + 5
+ +
x
≥ ≥
C What values of x make this inequality true? Graph the solution of the inequality on the number line. x≥
3
-5 -4 -3 -2 -1
8
Mathematical Processes
What would it tell you if the inequality is false when you check the solution?
No; Yes; It can only increase by an amount greater than or equal to 3.
0 1 2 3 4 5
Reflect 1.
Math Talk
+ + + + + + + +
Analyze Relationships How is solving the inequality 5 + x ≥ 8 like solving the equation 5 + x = 8? How is it different?
To solve both, you subtract 5 from both sides. But there is only one solution for the equation and
Math Talk
The number line is shaded in the wrong direction, and the number you chose is not a solution.
-13 < -12 STEP 1
8 ≤y - 3 + 3 + 3 _ _ 11 ≤ y
Lesson 13.2
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356
Use the Addition Property of Inequality. Add 3 to both sides. You can rewrite 11 ≤ y as y ≥ 11.
STEP 2
Graph the solution.
STEP 3
Check the solution. Substitute a solution from the shaded part of your number line into the original inequality.
infinitely many solutions for the inequality.
6_MTXESE051676_U4M13L2.indd 355
Solve the inequality.
Mathematical Processes
Could the temperature have increased by 2 degrees by noon? Could it have increased by 5 degrees? Explain.
The inequality is true.
B 8≤ y-3
5 6 7 8 9 10 11 12 13 14 15
? 8 ≤ 12 - 3
Substitute 12 for y in 8 ≤ y - 3
8≤ 9
The inequality is true.
© Houghton Mifflin Harcourt Publishing Company
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Janusz Wrobel/ Alamy
-20 -19 -18 -17 -16 -15 -14 -13 -12 -11 -10
Unit 4
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PROFESSIONAL DEVELOPMENT Integrate Mathematical Processes This lesson provides an opportunity to address Mathematical Process TEKS 6.1.D, which calls for students to “communicate mathematical ideas… using multiple representations…as appropriate.” In the Explore Activity, students use algebra tiles and a number line to model solving a one-step inequality. In the Examples, students represent inequalities symbolically, graph them on a number line, and write word problems involving real-world situations that correspond to specific inequalities.
Math Background The Commutative and Associative properties hold for addition but not for subtraction, as shown below. Caution students not to use either property when simplifying inequalities that involve subtraction. Commutative Property: Subtraction: 5 - 3 ≠ 3 - 5 Associative Property: Subtraction: 5 - (3 - 1) ≟ (5 - 3) - 1 5-2≟2-1 3≠1
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Mathematical Processes Exercise 3 Some students may shade in the wrong direction when they attempt to graph the solution set of an inequality with the variable on the right-hand side of the inequality symbol. Remind students that reading the inequality beginning with the variable will help them understand which side of the number line to shade.
ADDITIONAL EXAMPLE 2 Write a real-world problem that can be described by 14 < x + 7. Sample: A cook has 7 pounds of potatoes. He needs more than 14 pounds for dinner. What inequality describes the amount of potatoes the cook needs to buy? Interactive Whiteboard Interactive example available online my.hrw.com
EXAMPLE 2 Questioning Strategies
Mathematical Processes • What is the first step in writing a real-world problem to describe an inequality? First, analyze the numbers and relationships given in the inequality. Then think of real-world situations for which those numbers would make sense. • Are there any situations where the negative numbers on the number line would make sense? Yes. For example, it would make sense to use negative numbers for a situation involving changes in temperature.
Connect Vocabulary ELL
Mathematical Processes To help students understand the suggested comparison, relate the words “no more than 60 pounds” to the inequality symbol and the number 60 in the inequality. You may want to draw a simple sketch of a dog in a box on a weight scale to help students visualize the problem.
YOUR TURN Connect to Daily Life
Mathematical Processes Some students may need help conceiving inequality situations. You may want to make a list of various types of situations that students can use for inspiration. Invite the class to help you make the list.
Elaborate Talk About It Summarize the Lesson Ask: How does solving an addition inequality compare with solving an addition equation? The process is essentially the same: subtracting the same amount from each side to isolate the variable.
GUIDED PRACTICE Engage with the Whiteboard For Exercise 1, have a student circle the tiles that need to be removed from each side of the inequality on the whiteboard. Then ask the student to write the inequality and show how to “remove” the three tiles from the left-hand side of the inequality. Finally, have the student show how the solution matches the model. For Exercises 2–3, have students make models to represent the inequality on the whiteboard and explain how to solve it, using the same method that was used for Exercise 1.
Avoid Common Errors Exercises 2–5 Watch for students who incorrectly apply one of the Properties of Inequality. When this is the case, have the student circle the variable and the operation sign associated with it. Remind the student to apply the opposite operation to both sides of the inequality.
357
Lesson 13.2
DO NOT EDIT--Changes must be made through “File info” CorrectionKey=B
DO NOT EDIT--Changes must be made through “File info” CorrectionKey=B
Solve each inequality. Graph and check the solution. 2. y - 5 ≥ -7
y≥ - 2
-5 -4 -3 -2 -1
0 1 2
Personal Math Trainer
3. 21 > 12 + x
Online Assessment and Intervention
x< 9
3 4 5
0 1 2 3 4 5 6 7
Personal Math Trainer
5. Write a real-world problem that can be modeled by x - 13 > 20. Solve your problem and tell what values make sense for the situation.
Check students’ problems.
Online Assessment and Intervention
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8 9 10
Guided Practice
Interpreting Inequalities as Comparisons
1. Write the inequality shown on the model. Circle the tiles you would remove from each side and give the solution. (Explore Activity)
You can write a real-world problem for a given inequality. Examine each number and mathematical operation in the inequality.
EXAMPL 2 EXAMPLE
Inequality: Math On the Spot 6.9.C
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Write a real-world problem for the inequality 60 ≥ w + 5. Then solve the inequality. STEP 1
Solution:
3+x≤5
+ + +
x≤2
+
+ + + + +
Solve each inequality. Graph and check the solution. (Example 1)
Examine each part of the inequality.
2. x + 4 ≥ 9
x≥5
3. 5 > z - 3
z<8
w is the unknown quantity.
STEP 2
STEP 3
8 9 10
t>7
0 1 2 3 4 5 6 7
60 is greater than or equal to a number added to 5.
4. t + 5 > 12
Write a comparison that the inequality could describe. June’s dog will travel to a dog show in a pet carrier. The pet carrier weighs 5 pounds. The total weight of the pet carrier and the dog must be no more than 60 pounds. What inequality describes the weight of June’s dog?
6. Write a real-world problem that can be represented by the inequality y - 4 < 2. Solve the inequality and tell whether all values in the solution make sense for the situation. (Example 2)
0 1 2 3 4 5 6 7 8 9 10
0 1 2 3 4 5 6 7 8 9 10
Solve the inequality.
he had less than \$2 left on the card. How much money was on the
60 ≥ w + 5 -5 ____ 55 ≥ w
-5 ____
card before the purchase?; y < 6. Only values between 4 and 6 make sense, since these values result in positive dollar values.
June’s dog currently weighs ≤ 55 pounds.
? ?
Reflect
4. If you were to graph the solution, would all points on the graph make sense for the situation?
ESSENTIAL QUESTION CHECK-IN
7. Explain how to solve 7 + x ≥ 12. Tell what property of inequality you would use.
No; the dog’s weight cannot be 0 or a negative value,
Use the Subtraction Property of Inequality to subtract 7 from both sides of
so only positive numbers make sense.
the inequality. The answer is any number greater than or equal to 5. Lesson 13.2
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5. y - 4 < 2
8 9 10
y<6
© Houghton Mifflin Harcourt Publishing Company
0 1 2 3 4 5 6 7
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DIFFERENTIATE INSTRUCTION Cooperative Learning
Number Sense
Let students work in small groups to describe a situation from science that suggests inequalities. Topics may include comparing speeds, temperatures, weights, and so on. Have each group write an inequality for their situation and share the result with the class. Sample answers: The weight of sample A < 2.5 grams; the weight of sample A > the weight of sample B.
Start with the solution to an inequality, x > -3 Discuss with students that many different addition and subtraction inequalities have this same solution. Demonstrate that by “working backward” they can create an inequality that has this same solution. x + 4 > -3 + 4 x+4>1 Ask students to write three additional inequalities that have the solution x > -3. Then record them in a list for the class.
Differentiated Instruction includes: • Reading Strategies • Success for English Learners ELL • Reteach • Challenge PRE-AP
358
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Online homework assignment available
Evaluate GUIDED AND INDEPENDENT PRACTICE 6.10.A, 6.9.B, 6.9.C, 6.10.B
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13.2 LESSON QUIZ 6.10.A Solve each inequality. Graph and check the solution. 1. x + 20 ≥ -7
Concepts & Skills
Practice
Explore Activity Modeling One-Step Inequalities
Exercise 1
Example 1 Using Properties of Inequalities
Exercises 2–5, 8–11, 17
Example 2 Interpreting Inequalities as Comparisons
Exercises 6, 12–17
2. y - 30 ≤ 32 3. -16 < z -14 4. -2.5 > t + 3.5
Exercise
Write and solve an inequality for each situation. 5. Jeremy’s goal is to earn at least \$80 toward a week at soccer camp. He has earned \$32 so far. How much more does he need to earn? 6. The maximum weight an airline allows for a suitcase is 50 pounds. Ella’s suitcase weighs 8 pounds empty. a. If Ella meets the requirements, what could be the weight of the contents of her suitcase?
Depth of Knowledge (D.O.K.)
8–11
2 Skills/Concepts
1.D Multiple representations
12–16
2 Skills/Concepts
1.A Everyday life
17
3 Strategic Thinking
1.G Explain and justify arguments
18
3 Strategic Thinking
1.F Analyze relationships
19
3 Strategic Thinking
1.G Explain and justify arguments
20
3 Strategic Thinking
1.F Analyze relationships
Additional Resources Differentiated Instruction includes: • Leveled Practice Worksheets
b. Do all values in the solution make sense for the situation? Explain. Lesson Quiz available online my.hrw.com
-30
-25
-20
60
65
0
5
x ≥ -27 2.
55
y ≤ 62 3.
-5
z > -2 (or -2 < z) 4.
-10
-5
t < -6 (or -6 > t) 5. m + 32 ≥ 80; m ≥ 48
359
Lesson 13.2
0
Mathematical Processes
6. a. w + 8 ≤ 50; w ≤ 42; b. No. A negative value would make no sense, and 0 would mean that the suitcase is empty.
DO NOT EDIT--Changes must be made through “File info” CorrectionKey=A
DO NOT EDIT--Changes must be made through “File info” CorrectionKey=B
Name
Class
Date
13.2 Independent Practice
17. Multistep The table shows Marco’s checking account activity for the first week of June.
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6.9.B, 6.9.C, 6.10.A, 6.10.B
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a. Marco wants his total deposits for the month of June to exceed \$1,500. Write and solve an inequality to find how much more he needs to deposit to meet this goal.
Online Assessment and Intervention
x > 50
y ≥ -10
-8
-7
-6
-5
-3
z ≤ -11
11. 15 ≥ z + 26 - 15 - 14 - 13 - 12 - 11 - 10 - 9
-4
-8
-7
-6
Yes; he has to spend less than \$356.68. He spent \$93.32 in the first week. \$93.32(3) = \$279.96, which is less than \$356.68.
-2
FOCUS ON HIGHER ORDER THINKING
18. Critique Reasoning Kim solved y - 8 ≤ 10 and got y ≤ 2. What might Kim have done wrong?
-5
Write an inequality to solve each problem.
Kim added 8 to the left side of the inequality and subtracted
12. The water level in the aquarium’s shark tank is always greater than 25 feet. If the water level decreased by 6 feet during cleaning, what was the water level before the cleaners took out any water?
8 from the right side of the inequality to isolate y. She should have added 8 to both sides of the inequality.
© Houghton Mifflin Harcourt Publishing Company
w - 6 > 25; w > 31 feet
19. Critical Thinking José solved the inequality 3 > x + 4 and got x < 1. Then, to check his solution, he substituted -2 into the original inequality to check his solution. Since his check worked, he believes that his answer is correct. Describe another check José could perform that will show his solution is not correct. Then explain how to solve the inequality.
13. Danny has at least \$15 more than his big brother. Danny’s big brother has \$72. How much money does Danny have?
72 ≤ d - 15; d ≥ 87 dollars
Substitute 0 for x in the original inequality. 0 is less than
14. The vet says that Ray’s puppy will grow to be at most 28 inches tall. Ray’s puppy is currently 1 foot tall. How much more will the puppy grow?
1 but is not a solution. To solve the inequality, subtract 4
12 + x ≤ 28; x ≤ 16 inches
from both sides to get x <-1.
15. Pierre’s parents ordered some pizzas for a party. 4.5 pizzas were eaten at the party. There were at least 5_12 whole pizzas left over. How many pizzas did Pierre’s parents order?
20. Look for a Pattern Solve x + 1 > 10, x + 11 > 20, and x + 21 > 30. Describe a pattern. Then use the pattern to predict the solution of x + 9,991 > 10,000.
p - 4.5 ≥ 5.5; p ≥ 10 pizzas
x > 9 for each inequality; in each case the number
16. To get a free meal at his favorite restaurant, Tom needs to spend \$50 or more at the restaurant. He has already spent \$30.25. How much more does Tom need to spent to get his free meal?
x + 30.25 ≥ 50
added to x is 9 less than the number on the right side of each inequality, so x > 9 is the solution.
x ≥ \$19.75
Lesson 13.2
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EXTEND THE MATH
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You may wish to present the model shown at the right for x - 2 ≤ 4. Remind students that x - 2 and x + (-2) are equivalent. Also remind them of the concept of zero pairs. Have students write an explanation of the way to use algebra tiles to solve subtraction inequalities and be prepared to demonstrate the method to other students.
Unit 4
6_MTXESE051676_U4M13L2.indd 360
Activity available online
PRE-AP
Activity Challenge pairs of students to use what they know about adding and subtracting integers to model subtraction inequalities with algebra tiles.
Work Area
© Houghton Mifflin Harcourt Publishing Company
- 12 - 11 - 10 - 9
\$22.82
c. There are three weeks left in June. If Marco spends the same amount in each of these weeks that he spent during the first week, will he meet his goal of spending less than \$450 for the entire month? Justify your answer.
8 9 10
10. y - 5 ≥ -15
\$24.00
Purchase – Water bill
x + 93.32 < 450; x < 356.68; less than \$356.68
y≥8
0 1 2 3 4 5 6 7
\$46.50
Purchase – Movie Theatre
b. Marco wants his total purchases for the month to be less than \$450. Write and solve an inequality to find how much more he can spend and still meet this goal.
0 10 20 30 40 50 60 70 80 90 100
9. 193 + y ≥ 201
\$520.45
Purchase – Grocery Store
x + 520.45 > 1500; x > 979.55; more than \$979.55
Solve each inequality. Graph and check the solution. 8. x - 35 > 15
Deposit – Paycheck
28/01/14 11:00 PM
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x + (-2) ≤ 4
+
− −
+ +
+ +
x-2+2≤4+2
+
− −
+ +
+ +
+ +
+ +
+ +
+ +
+ +
x≤6
+
360
LESSON
13.3
Multiplication and Division Inequalities with Positive Numbers
Texas Essential Knowledge and Skills The student is expected to: Expressions, equations, and relationships—6.10.A Model and solve one-variable, one-step inequalities that represent problems, including geometric concepts. Expressions, equations, and relationships—6.9.B
Engage ESSENTIAL QUESTION How can you solve an inequality involving multiplication or division with positive numbers? Sample answer: You can use the Multiplication and Division Properties of Inequality: multiply or divide each side by the same positive amount to isolate the variable.
Motivate the Lesson
Ask: If you get a monthly allowance of \$36, how could you use the inequality 4x ≤ 36 to plan a weekly budget? Begin the Explore Activity to find out how to solve multiplication inequalities.
Represent solutions for one-variable, one-step equations and inequalities on number lines. Expressions, equations, and relationships—6.9.C Write corresponding real-world problems given one-variable, one-step equations or inequalities. Expressions, equations, and relationships—6.10.B Determine if the given value(s) make(s) one-variable, one-step equations or inequalities true.
Explore EXPLORE ACTIVITY Connect Multiple Representations Mathematical Processes In the activity, the real-world problem is represented first by an inequality in words, then as an inequality in symbols. The inequality is modeled and solved with algebra tiles. Finally, the solution is shown on a number line. Discuss with students how each representation leads to the next, and discuss the advantages and disadvantages of each type of representation.
Mathematical Processes 6.1.B Use a problem-solving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem-solving process and the reasonableness of the solution.
ADDITIONAL EXAMPLE 1 Solve each inequality. Graph and check the solution. A 7x > 28
x>4
0 y B __5 ≤ 30
5
10
150
155
Interactive Whiteboard Interactive example available online my.hrw.com
361
Lesson 13.3
EXAMPLE 1 Focus on Math Connections
Mathematical Processes Present the equation 12x = 24. Encourage students to solve for x, x = 2. The solution to the equation is one point on the number line. The solution to the inequality, x < 2, is a section of the number line, in this case, everything to the left of 2.
Questioning Strategies
Mathematical Processes • In A, Step 1, why do you divide by 12? To isolate x, it is necessary to “remove” the 12. Since x is multiplied by 12, using the inverse operation, division, will “undo” the multiplication. • In A, Step 2, 24 ÷ 12 is 2. Why isn’t 2 a solution? Since the inequality sign is the less than sign, the solution is less than 2. • When graphing inequalities, how do you know when to use a solid circle or an open circle? A solid circle is used to represent ≤ or ≥ on a graph. An open circle is used to represent < or > on a graph.
y ≤ 150
145
Explain
Engage with the Whiteboard For A, have a volunteer make a model with algebra tiles on the whiteboard. Have the student solve the inequality, using the same method that was used in B of the Explore Activity. Then have students compare and contrast the model with the number line provided in the example.
DO NOT EDIT--Changes must be made through “File info” CorrectionKey=B
LESSON
13.3
DO NOT EDIT--Changes must be made through “File info” CorrectionKey=A
Multiplication and Division Inequalities with Positive Numbers
Expressions, equations, and relationships—6.10.A Model and solve one-variable, one-step inequalities that represent problems. Also 6.9.B, 6.9.C, 6.10.B.
Solving Inequalities Involving Multiplication and Division Math On the Spot
You can use properties of inequality to solve inequalities involving multiplication and division with positive integers.
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ESSENTIAL QUESTION
Multiplication and Division Properties of Inequality How can you solve an inequality involving multiplication or division with positive numbers?
• You can multiply both sides of an inequality by the same positive number and the inequality will remain true. • You can divide both sides of an inequality by the same positive number and the inequality will remain true.
6.10.A
EXPLORE ACTIVITY
Modeling One-Step Inequalities
EXAMPLE 1
You can use algebra tiles to solve inequalities that involve multiplying positive numbers.
Solve each inequality. Graph and check the solution.
A 12x < 24
Dominic is buying school supplies. He buys 3 binders and spends more than \$9. How much did he spend on each binder?
STEP 1
A Let x represent the cost of one binder. Write an inequality. Number of binders
3
Cost of a binder
B The model shows the inequality from There are
3
A
3
Mathematical Processes
9
Are all negative numbers solutions to 12x < 24? Explain.
.
+ + +
>
+ + + + + + + + +
Yes; because 12 times a negative number is a negative number, and all negative numbers are less than 24.
3
C What values make the inequality you wrote in true? Graph the solution of the inequality.
A
x>3
-5 -4 - 3 - 2 - 1
0 1 2 3 4 5
Reflect 1. Analyze Relationships Is 3.25 a solution of the inequality you wrote in A ? If so, does that solution make sense for the situation?
Yes, 3.25 is a solution. Dominic may have paid \$3.25 for each binder.
Divide both sides by 12.
x<2
Math Talk
equal groups.
How many units are in each group?
Solve the inequality. 12x __ ___ < 24 12 12
9
x-tiles, so draw circles
to separate the tiles into © Houghton Mifflin Harcourt Publishing Company
>
x
>
6.9.B, 6.10.B
Use an open circle to show that 2 is not a solution.
STEP 2
Graph the solution.
STEP 3
Check the solution by substituting a solution from the shaded part of the graph into the original inequality. ? Substitute 0 for x in the original inequality. 12(0) < 24
-5 -4 - 3 - 2 - 1
0 < 24
y B _3 ≥ 5
STEP 1
0 1 2 3 4 5
The inequality is true.
Solve the inequality. 3 ( _3 ) ≥ 3(5) y
Multiply both sides by 3.
y ≥ 15
Use a closed circle to show that 15 is a solution.
STEP 2
Graph the solution.
STEP 3
Check the solution by substituting a solution from the shaded part of the graph into the original inequality. 18 ? __ ≥5 Substitute 18 for x in the original inequality. 3
© Houghton Mifflin Harcourt Publishing Company
?
5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
2. Represent Real-World Problems Rewrite the situation in represent the inequality 3x < 9.
A
to
Dominic is buying school supplies. He buys 3 binders and spends less than \$9. How much did he spend on each binder?
6≥5 Lesson 13.3
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InCopy Notes 1. This is a list
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InDesign Notes 1. This is a list
PROFESSIONAL DEVELOPMENT Integrate Mathematical Processes This lesson provides an opportunity to address Mathematical Process TEKS 6.1.B, which calls for students to “use a problem-solving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem-solving process and the reasonableness of the solution.” Students directly apply each of these steps of the problem-solving model in Example 2, to determine the possible side lengths of a square flag.
The inequality is true.
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InCopy Notes
10/29/12 7:05 PM
InDesign Notes
1. This is a list Bold, Italic, Strickthrough.
1. This is a list
Math Background As with addition and subtraction, not all properties that hold for multiplication hold for division. For example, the Commutative and Associative properties hold for multiplication but not for division. Caution students to avoid rearranging or regrouping numbers that are being divided. Commutative Property: Division: 15 ÷ 3 ≠ 3 ÷ 15 Associative Property: Division: 50 ÷ (10 ÷ 5) ≟ (50 ÷ 10) ÷ 5 50 ÷ 2 ≟ 5 ÷ 5 25 ≠ 1
Multiplication and Division Inequalities with Positive Numbers
362
YOUR TURN Avoid Common Errors Exercise 4 Some students may divide both sides by 4 instead of multiplying. Rewriting the equation as z ÷ 4 = 11 may help these students see which operation they should use.
ADDITIONAL EXAMPLE 2 Kayla earned more than \$50 babysitting. Her mother paid her \$4 an hour to babysit her little brother. Write and solve an inequality to find the possible number of hours Kayla babysat. 4h > 50; h > 12.5; Kayla babysat more than 12.5 hours. Interactive Whiteboard Interactive example available online my.hrw.com
EXAMPLE 2 Focus on Reasoning
Mathematical Processes Point out to students that sometimes a problem may provide clues and facts that you must use to find a solution. Encourage students to begin by identifying the important information. They can underline or circle the information in the problem statement.
Questioning Strategies
Mathematical Processes • What information is given in the problem? Because the flag being made is a square, all the sides have the same length, and the perimeter needs to be 22 inches or longer. • How do you find the perimeter of a figure? Perimeter = sum of the lengths of the sides
Focus on Modeling Mathematical Processes Have students draw a square to represent the square flag. Then have students label each of the sides of the square with an x. Guide them to see that the four sides (x + x + x + x) are the same as the 4x shown in the inequality in the Solve step. c.2.I.4 ELL Encourage class discussion to develop the scenario in Reflect. English learners will benefit from hearing and participating in classroom discussions.
Integrating the ELPS
YOUR TURN Focus on Critical Thinking Mathematical Processes Point out to students that the unknown is the total weight of sand needed for 6 paperweights. Help students to see that the total weight divided by 6 is equivalent to the weight of a single paperweight.
Elaborate Talk About It Summarize the Lesson Ask: How do you solve a division inequality? Multiply both sides of the inequality by the same number to isolate the variable.
GUIDED PRACTICE Engage with the Whiteboard For Exercise 1, have students circle the groups of tiles on the whiteboard and then write the inequality and the solution below the model. Have students make a number line to show the solution. Ask the class to compare and contrast the model with the number line.
Avoid Common Errors Exercises 2–3 Watch for students who incorrectly apply one of the Properties of Inequality. Remind them that they should apply the opposite operation to both sides of the inequality. Exercise 4 If students have difficulty writing the inequality, encourage them to make a model to help them understand the situation better.
363
Lesson 13.3
DO NOT EDIT--Changes must be made through “File info” CorrectionKey=B
DO NOT EDIT--Changes must be made through “File info” CorrectionKey=A
YOUR TURN Solve each inequality. Graph and check the solution. 3. 5x ≥ 100 4. _4z < 11
Reflect
Personal Math Trainer
x ≥ 20
15 16 17 18 19 20 21 22 23 24 25
z < 44
40 41 42 43 44 45 46 47 48 49 50
5. Represent Real-World Problems Write and solve a real-world problem for the inequality 4x ≤ 60.
Online Assessment and Intervention
Sample problem: Cy is making a square flag. He wants
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the perimeter to be no more than 60 inches. What are the possible side lengths? x ≤ 15 inches
Solving Real-World Problems
You can use multiplication and division inequalities to model and solve real-world problems.
EXAMPL 2 EXAMPLE
Personal Math Trainer Math On the Spot
Problem Solving
6.10.A
Online Assessment and Intervention
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6. A paperweight must weigh less than 4 ounces. Brittany wants to make 6 paperweights using sand. Write and solve an inequality to find the possible weight of the sand she needs.
w __ < 4; w < 24; Brittany must use less than 24 oz of sand. 6
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Cy is making a square flag. He wants the perimeter to be at least 22 inches. Write and solve an inequality to find the possible side lengths.
Guided Practice 1. Write the inequality shown on the model. Circle groups of tiles to show the solution. Then write the solution. (Explore Activity)
Formulate a Plan
Inequality:
Write and solve a multiplication inequality. Use the fact that the perimeter of a square is 4 times its side length.
Solution:
© Houghton Mifflin Harcourt Publishing Company
Let x represent a side length.
4x __ __ ≥ 22 4 4
Divide both sides by 4.
x ≥ 5.5
2. 8y < 320
35 36 37 38 39 40 41 42 43 44 45
The side lengths must be greater than or equal to 5.5 in.
? ?
Check the solution by substituting a value in the solution set in the original inequality. Try x = 6.
30 31 32 33 34 35 36 37 38 39 40
ESSENTIAL QUESTION CHECK-IN
Use the Division Property of Inequality to divide both sides by 5 to get x < 8.
The statement is true.
Cy’s flag could have a side length of 6 inches.
To check, substitute a number from the solution into the original inequality. Lesson 13.3
364
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1. This is a list
r ≥ 33
5. Explain how to solve and check the solution to 5x < 40 using properties of inequalities.
Substitute 6 for x.
InCopy Notes
3. _3r ≥ 11
+ + + + + + + +
b __ ≥ 14; b ≥ 84; Karen had at least 84 books. 6
Justify and Evaluate
24 ≥ 22
y < 40
<
4. Karen divided her books and put them on 6 shelves. There were at least 14 books on each shelf. How many books did she have? Write and solve an inequality to represent this situation. (Example 2)
Cy’s flag should have a side length of 5.5 inches or more.
? 4(6) ≥ 22
+ +
Solve each inequality. Graph and check the solution. (Example 1)
Justify and Evaluate Solve
4x ≥ 22
2x < 8 x<4
© Houghton Mifflin Harcourt Publishing Company
Analyze Information
Find the possible lengths of 1 side of a square that has a perimeter of at least 22 inches.
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InCopy Notes
InDesign Notes 1. This is a list
1. This is a list Bold, Italic, Strickthrough.
DIFFERENTIATE INSTRUCTION
28/01/14 11:13 PM
InDesign Notes 1. This is a list
Cooperative Learning
Critical Thinking
Have students work in pairs. Give each pair an index card. Have students write an inequality word problem on the card, then exchange problems with another pair and work together to solve the problems.
Let students work together to decide which of the following statements, a + c > b + c or ac > bc, are always true if a, b, and c are real numbers and a > b. If the statement is not always true, have students give an example to show that the statement is false.
Differentiated Instruction includes: • Reading Strategies • Success for English Learners ELL • Reteach • Challenge PRE-AP
If students have trouble writing word problems, suggest that they use the word problems in the Independent Practice as a template. They can change the names, numbers, and/or the direction of the inequality to create new problems.
a + c > b + c is always true; ac > bc is not always true. a = 2, b = 1, c = 3 2‧3<1‧3
6 < 3, false
Multiplication and Division Inequalities with Positive Numbers
364
Personal Math Trainer Online Assessment and Intervention
Online homework assignment available
Evaluate GUIDED AND INDEPENDENT PRACTICE 6.10.A, 6.9.B, 6.9.C, 6.10.B
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13.3 LESSON QUIZ 6.10.A Solve each inequality. Graph and check the solution. 1. 4x ≥ 12 y 2. __3 ≤ 6
Concepts & Skills
Practice
Explore Activity Modeling One-Step Inequalities
Exercise 1
Example 1 Solving Inequalities Involving Multiplication and Division with Positive Integers
Exercises 2–3, 10–11, 15–18
Example 2 Solving Real-World Problems
Exercises 4, 6–9, 12–14
3. 2p > 24 4. __8t < 20 Write and solve an inequality for each problem. 5. Kellin pays more than \$40 a day to rent a canoe. What amount might he pay to rent the canoe for a 7-day trip? 6. Amanda earns \$15 an hour. She needs at least \$90 to buy a computer desk. How many hours does she need to work to buy the desk?
Exercise
Depth of Knowledge (D.O.K.)
6–9
2 Skills/Concepts
1.B Problem-solving model
10–11
2 Skills/Concepts
1.D Multiple representations
12–14
2 Skills/Concepts
1.B Problem-solving model
15–18
2 Skills/Concepts
1.D Multiple representations
19–22
2 Skills/Concepts
1.A Everyday life
23–24
3 Strategic Thinking
1.F Analyze relationships
25
3 Strategic Thinking
1.B Problem-solving model
Lesson Quiz available online my.hrw.com
5
10
15
20
15
20
160
165
x≥3 2. 10
y ≤ 18 3. 10
p > 12 4. 155
t < 160 w 5. __ > 40; w > 280; more than \$280 7
6. 15h ≥ 90; h ≥ 6
365
Lesson 13.3
Mathematical Processes
Additional Resources Differentiated Instruction includes: • Leveled Practice Worksheets
DO NOT EDIT--Changes must be made through “File info” CorrectionKey=B
DO NOT EDIT--Changes must be made through “File info” CorrectionKey=B
Class
Date
The sign shows some prices at a produce stand.
13.3 Independent Practice 6.9.B, 6.9.C, 6.10.A, 6.10.B
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Write and solve an inequality for each problem.
22. The produce buyer for a local restaurant wants to buy more than 30 lb of onions. The produce buyer at a local hotel buys exactly 12 pounds of spinach. Who spends more at the produce stand? Explain.
numbers don’t make sense.
The restaurant produce buyer; he will spend more than
14. Multistep Lina bought 4 smoothies at a health food store. The bill was less than \$16.
\$37.50; the hotel’s produce buyer will spend \$36.
a. Write and solve an inequality to represent the cost of each smoothie.
8. In a litter of 7 kittens, each kitten weighs more than 3.5 ounces. Find the possible total weight of the litter. w __ 7 > 3.5; w > 24.5; more than
4s < 16; s < 4; the cost of each
_r 5
24. Represent Real-World Problems Write and solve a word problem that can be represented with 240 ≤ 2x.
Sample answer: Jake and his brother want to earn at
less than \$4.
least \$240. If each brother earns the same amount, how
c. Graph the values that make sense for this situation on the number line.
Solve each inequality. Graph and check the solution.
-1
x≤6
p 15. __ ≤ 30 13
0 1 2 3 4 5 6 7 8 9 10
t>0
16. 2t > 324 17. 12y ≥ 1
0 1 2 3 4 5
x 18. ___ < 11 9.5
much money does each brother have to earn?; x ≥ 120;
each brother must earn at least \$120.
0 1 2 3 4 5
Solve each inequality.
25. Persevere in Problem Solving A rectangular prism has a length of 13 inches and a width of _12 inch. The volume of the prism is at most 65 cubic inches. Find all possible heights of the prism. Show your work.
p ≤ 390
()
13 ∙ _12 ∙ h ≤ 65; 6.5h ≤ 65; h < 10; all heights greater
t > 162 1 y ≥ __ 12
than 0 inches but no more than 10 inches
x < 104.5 Lesson 13.3
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InCopy Notes
InDesign Notes
EXTEND THE MATH
Unit 4
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InCopy Notes
1. This is a list
Activity available online A Mystery Rectangle
There are only 3 possible rectangles.
Clue 2: Each dimension must be a whole number.
Perimeter
21
24
90
22
25
94
23
26
98
1. This is a list
my.hrw.com
Activity Challenge students to translate Clues 1, 3, and 4 into inequalities or equations. Direct them to use the clues together to find all the possible dimensions of the rectangle. List them in the table below.
Length
28/01/14 11:19 PM
InDesign Notes
1. This is a list Bold, Italic, Strickthrough.
PRE-AP
Width
is the
side and divided by 5 on the right side.
amount greater than \$0 and
of Will’s backyard is at least 11 feet.
2 __ . What 25
r ≤ 2; the student may have multiplied by 5 on the left
Each smoothie can cost any
15.5w ≥ 170.5; w ≥ 11; the width
_2 and 5
23. Critique Reasoning A student solves ≤ gets r ≤ correct solution? What mistake might the student have made?
b. What values make sense for this situation? Explain.
9. To cover his rectangular backyard, Will needs at least 170.5 square feet of sod. The length of Will’s yard is 15.5 feet. What are the possible widths of Will’s yard?
Work Area
FOCUS ON HIGHER ORDER THINKING
smoothie was less than \$4.
24.5 ounces.
© Houghton Mifflin Harcourt Publishing Company
No; 1.25x ≤ 3; x ≤ 2.4 so 2.4 pounds of onions are the most
Florence can buy. 2.4 < 2.5, so she cannot buy 2.5 pounds.
negative amount so negative
14x ≥ 84; x ≥ 6; at least 6 hours
1. This is a list
21. Florence wants to spend no more than \$3 on onions. Will she be able to buy 2.5 pounds of onions? Explain.
No; Steve would not pay a
7. Tamar needs to make at least \$84 at work on Tuesday to afford dinner and a movie on Wednesday night. She makes \$14 an hour at her job. How many hours does she need to work on Tuesday?
-5 -4 - 3 - 2 - 1
at most \$2.75
13. If you were to graph the solution for exercise 12, would all points on the graph make sense for the situation? Explain.
to 7 inches
11. _2t > 0
20. Gary has enough money to buy at most 5.5 pounds of potatoes. How much money does Gary have?
r __ < 32; r < 992; less than \$992 31
6s ≤ 42; s ≤ 7; less than or equal
10. 10x ≤ 60
3_13 pounds
Online Assessment and Intervention
12. Steve pays less than \$32 per day to rent his apartment. August has 31 days. What are the possible amounts Steve could pay for rent in August?
6. Geometry The perimeter of a regular hexagon is at most 42 inches. Find the possible side lengths of the hexagon.
Price per Pound Produce \$1.25 Onions \$0.99 Yellow Squash \$3.00 Spinach \$0.50 Potatoes
19. Tom has \$10. What is the greatest amount of spinach he can buy?
Personal Math Trainer
© Houghton Mifflin Harcourt Publishing Company
Name
Clue 1: A rectangle has a perimeter that is less than 100 units. 2w + 2l < 100 Clue 3: The width must be less than 25, but greater than 20. w > 20; w < 25 Clue 4: The length must be 3 units longer than the width. l=w+3
Multiplication and Division Inequalities with Positive Numbers
366
LESSON
13.4
Multiplication and Division Inequalities with Rational Numbers
Texas Essential Knowledge and Skills The student is expected to: Expressions, equations, and relationships—6.9.B Represent solutions for one-variable, one-step equations and inequalities on number lines. Expressions, equations, and relationships—6.10.A Model and solve one-variable, one-step equations and inequalities that represent problems, including geometric concepts. Expressions, equations, and relationships—6.10.B Determine if the given value(s) make(s) one-variable, one-step equations or inequalities true.
Mathematical Processes 6.1.F Analyze mathematical relationships to connect and communicate mathematical ideas.
Engage ESSENTIAL QUESTION How do you solve inequalities that involve multiplication and division of integers? Sample answer: You multiply or divide each side of the inequality to isolate the variable. If the integer is negative, you must reverse the direction of the inequality symbol.
Motivate the Lesson Ask: Have you ever had a friend ask you a question using double-negative language, such as “You don’t want to not go, do you?” Well, double negatives can change things around. Begin the Explore Activity and find out how multiplying or dividing both sides of an inequality by a negative number can change things.
Explore EXPLORE ACTIVITY Engage with the Whiteboard Have students draw number lines on the whiteboard to graph each example in A. First, have them graph the given inequality. Then have them graph the inequality after multiplying each side by the same number in a different color. For each example, ask if the original symbol still makes the inequality true after multiplying. If it doesn’t, ask what symbol would make the inequality true.
Explain ADDITIONAL EXAMPLE 1 Solve each inequality. Graph and check the solution. A -3x < 18 -10
B -__6 > -7 y
35
x > -6 -5
0
45
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Lesson 13.4
Questioning Strategies
Mathematical Processes
number you need to use to multiply both sides of the inequality
Interactive Whiteboard Interactive example available online
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ELL c.1.F Remind students that the word reverse means to turn backward in position or direction. Connect this meaning to reversing the inequality, where the point of the symbol changes direction.
Connect Vocabulary
y • In B, why might you rewrite -__3 as ( -__13 )y before multiplying? to make it easier to see what
y < 42 40
EXAMPLE 1
• How can you check that the graph of an inequality is correct? Pick a point on the shaded portion of the graph. Any point selected should make the inequality a true inequality.
DO NOT EDIT--Changes must be made through "File info" CorrectionKey=A
LESSON
13.4
DO NOT EDIT--Changes must be made through "File info" CorrectionKey=B
Multiplication and Division Inequalities with Rational Numbers
Expressions, equations, and relationships—6.9.B Represent solutions for one-step inequalities on number lines. Also 6.10.A, 6.10.B
Multiplication and Division Properties of Inequality Math On the Spot
Recall that you can multiply or divide both sides of an inequality by the same positive number, and the statement will still be true.
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ESSENTIAL QUESTION
Multiplication and Division Properties of Inequality How do you solve inequalities that involve multiplication and division of integers?
EXPLORE ACTIVITY
• If you multiply or divide both sides of an inequality by the same negative number, you must reverse the inequality symbol for the statement to still be true.
6.10.A
Investigating Inequality Symbols
EXAMPLE 1
You have seen that multiplying or dividing both sides of an inequality by the same positive number results in an equivalent inequality. How does multiplying or dividing both sides by the same negative number affect an inequality?
Solve each inequality. Graph and check the solution.
My Notes
A Complete the tables. New inequality
New inequality is true or false?
3<4
2
true
2 ≥ -3
6< 8
3
© Houghton Mifflin Harcourt Publishing Company
5>2
-1
-8 > -10
-8
6 ≥ -9
-5 > -2
false
x < -13
Divide each side by:
New inequality
New inequality is true or false?
4<8
4
1< 2
true
4 ≤ -3
false
4 ≥ -5
3
-16 ≤ 12
-4
15 > 5
-5
-3 > -1
Divide both sides by -4. Reverse the inequality symbol.
-4x 52 ____ < ___ -4 -4
false
64 > 80
Solve the inequality. -4x > 52
true
Inequality
12 ≥ -15
A -4x > 52 STEP 1
Multiply each side by:
Inequality
6.9.B, 6.10.B
STEP 2
Graph the solution.
STEP 3
Check your answer using substitution. ? Substitute -15 for x in -4x > 52. -4(-15) > 52 60 > 52
y
-15 -14 -13 -12 -11 -10 -9
-8
The statement is true.
B - _3 < -5
true
STEP 1
Solve the inequality. y
- _3 < -5
false
( )
y -3 - _3 > -3(-5)
Multiply both sides by -3. Reverse the inequality symbol.
y > 15
B What do you notice when you multiply or divide both sides of an inequality by the same negative number?
The inequality is no longer true. C How could you make each of the multiplication and division inequalities that were not true into true statements?
STEP 2
Graph the solution.
STEP 3
Check your answer using substitution. ? 18 < - ___ -5 3 -6 < -5
Reverse the inequality symbol.
Lesson 13.4
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10 11 12 13 14 15 16 17 18 19 20
© Houghton Mifflin Harcourt Publishing Company
?
y Substitute 18 for y in - __ < -5. 3 The inequality is true.
Unit 4
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PROFESSIONAL DEVELOPMENT Integrate Mathematical Processes This lesson provides an opportunity to address Mathematical Process TEKS 6.1.F, which calls for students to “analyze mathematical relationships to connect and communicate mathematical ideas.” In the Explore Activity, students use tables and numerical inequalities to explore the effects of multiplying or dividing an inequality by a negative number. Students examine examples and counter-examples, leading to the conclusion that the inequality symbol must be reversed when multiplying or dividing both sides of the inequality by a negative number.
Math Background The solution set also can be described by using interval notation. In this notation, parentheses indicate that the endpoints are not included in the solution set. Brackets indicate that the endpoints are included in the solution set. Suppose a and b are any real numbers: Inequality
Interval notation
a
(a, b)
a
(a, b] [a, b) [a, b]
Multiplication and Division Inequalities with Rational Numbers
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YOUR TURN Avoid Common Errors Exercise 2 Some students may divide both sides by -6 instead of multiplying. Rewriting the equation as 7 ≥ t ÷ (-6) may help these students see which operation they should use.
ADDITIONAL EXAMPLE 2 During the month, Jenna uses the dining commons at the college for dinner. Each dinner costs \$12. Each time she uses her dining commons card, her balance changes by -12. Last month the balance change was an amount greater than or equal to -324. How many times did she use her dining commons card? -12n ≥ -324; n ≤ 27; Jenna used the dining commons card 27 or fewer times. Interactive Whiteboard Interactive example available online my.hrw.com
EXAMPLE 2 Focus on Reasoning
Mathematical Processes Point out to students that sometimes a problem may provide clues and facts that must be used to find a solution. Encourage students to begin by identifying the important information. They can underline or circle the information in the problem statement.
Questioning Strategies
Mathematical Processes • Why is “40 feet below sea level” written as a negative number? Because the words “below sea level” indicate a negative direction, down. • What is the unknown in this situation? The unknown is the time it took the submersible to reach its final elevation of more than 40 feet below sea level.
Focus on Modeling Mathematical Processes It may help students to understand the problem better if they draw a vertical number line to represent this situation. They can label the descent in intervals of -5 feet for each second.
YOUR TURN Focus on Modeling Mathematical Processes To help students visualize the \$35 being deducted each month, draw a vertical number line from 0 to -315. Show the decreases in multiples of -35 for each month. Help students understand that what they need to find is the number of months it takes to go no farther than -315.
Elaborate Talk About It Summarize the Lesson Ask: How do you know when to reverse the inequality symbol when solving an inequality and when not to reverse it? If you multiply or divide both sides by a negative number, you must reverse the inequality symbol. If you multiply or divide by a positive number, the inequality symbol stays the same.
GUIDED PRACTICE Engage with the Whiteboard For Exercises 1–2, have students solve and graph each inequality on the whiteboard, showing all their work. Then have students explain how they knew which operation to use to solve each inequality.
Avoid Common Errors Exercises 3–5 Watch for students who may forget to reverse the inequality symbol when multiplying or dividing by a negative number. Remind them that they should always check their solution. Exercise 5 Remind students that “not colder than -80 °C” means that the temperature is at least -80 °C or warmer. Warmer temperatures are above -80° on a thermometer or to the right of -80 on a number line.
369
Lesson 13.4
DO NOT EDIT--Changes must be made through "File info" CorrectionKey=B
DO NOT EDIT--Changes must be made through "File info" CorrectionKey=B
Solve each inequality. Graph and check the solution. 1. -10y < 60 2. 7 ≥ - __t 6
y > -6
t ≥ -42
- 10 - 9 - 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1
0 1
-47 -46 -45 -44 -43 -42 -41 -40
Personal Math Trainer Online Assessment and Intervention
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3. Every month, \$35 is withdrawn from Tom’s savings account to pay for his gym membership. He has enough savings to withdraw no more than \$315. For how many months can Tony pay for his gym membership?
Personal Math Trainer Online Assessment and Intervention
m ≤ 9; Tom can pay for no more than 9 months of his
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gym membership using this account.
Solving a Real-World Problem
EXAMPL 2 EXAMPLE
Problem Solving
Guided Practice Math On the Spot my.hrw.com
6.10.A
z ≤ -3
1. -7z ≥ 21
A marine submersible descends more than 40 feet below sea level. As it descends from sea level, the change in elevation is -5 feet per second. For how many seconds does it descend?
2. -__t > 5 4
t < -20
t >5 4. -___
Rewrite the question as a statement. • Find the number of seconds that the submersible decends below sea level.
10
-10 -9 - 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1 -50 -40 -30 -20 -10
x < -6
3. 11x < -66
Analyze Information
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Jeffrey L. Rotman/Peter Arnold Inc/Getty Images
Solve each inequality. Graph and check the solution. (Explore Activity and Example 1)
t < -50
0
10
20
30
40
-8 -7 - 6 - 5 - 4 - 3 - 2 - 1
0
50
0 1 2
-100 -90 -80 -70 -60 -50 -40 -30 -20 -10
0
5. For a scientific experiment, a physicist must make sure that the temperature of a metal does not get colder than -80 °C. The metal begins the experiment at 0 °C and is cooled at a steady rate of -4 °C per hour. How long can the experiment run? (Example 2)
List the important information: • The final elevation is greater than 40 feet below sea level or < -40 feet. • The rate of descent is -5 feet per second.
a. Let t represent time in hours. Write an inequality. Use the fact that the rate of change in temperature times the number of seconds equals the final temperature.
Formulate a Plan
Write and solve an inequality. Use this fact:
-4 · t ≥ -80
Rate of change in elevation × Time in seconds = Total change in elevation
b. Solve the inequality in part a. How long will it take the physicist to change the temperature of the metal?
20 or fewer hours
Justify and Evaluate Solve
c. The physicist has to repeat the experiment if the metal gets cooler than -80 °C. How many hours would the physicist have to cool the metal for this to happen?
more than 20 hours
-5t < -40 -5t > ____ -40 ____ -5 -5 t> 8
Rate of change × Time < Maximum elevation Divide both sides by -5. Reverse the inequality symbol.
? ?
The submersible descends for more than 8 seconds.
6. Suppose you are solving an inequality. Under what circumstances do you reverse the inequality symbol?
Justify and Evaluate
when you divide or multiply both sides by a negative number
Check your answer by substituting a value greater than 8 seconds in the original inequality. ? Substitute 9 for t in the inequality -5t < -40. -5(9) < -40 -45 < -40
ESSENTIAL QUESTION CHECK-IN
© Houghton Mifflin Harcourt Publishing Company
Although elevations below sea level are represented by negative numbers, we often use absolute value to describe these elevations. For example, -50 feet relative to sea level might be described as 50 feet below sea level.
The statement is true. Lesson 13.4
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DIFFERENTIATE INSTRUCTION Cooperative Learning
Modeling
Have students work in pairs to explain to each other how to solve multiplication and division inequalities. Have one student explain how to solve an inequality that involves multiplying or dividing by a positive number, and have the other student explain how to solve an inequality that involves multiplying or dividing by a negative number. Then have each pair work with specific inequalities. Invite several pairs of students to share their explanations with the class.
Draw a number line with 1 and 3 graphed.
Differentiated Instruction includes: • Reading Strategies • Success for English Learners ELL • Reteach • Challenge PRE-AP
Sample inequalities for a pair of students:
Use similar steps to show that dividing both sides of a numerical inequality by -1 will change the sign as well.
-3x ≥ 60 and __6x < -4
Ask if it is correct to write 1 < 3. Yes
Multiply both numbers by -1 and graph the products. Ask if it is correct to write -1 < -3. No
Why? -1 is farther to the right than -3. Ask if it is correct to write -1 > -3. Yes
What changed? the direction of the inequality sign
Multiplication and Division Inequalities with Rational Numbers
370
Personal Math Trainer Online Assessment and Intervention
Online homework assignment available
Evaluate GUIDED AND INDEPENDENT PRACTICE 6.9.B, 6.10.A, 6.10.B
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13.4 LESSON QUIZ 6.9.B Solve each inequality. Graph and check the solution. 1. -__3x > -7 2. 10z ≤ -20 3. -6t ≥ 54
Concepts & Skills
Practice
Explore Activity Investigating Inequalities Involving Multiplication and Division of Integers
Exercises 1–4, 7–12, 17
Example 1 Multiplication and Division Properties of Inequality
Exercises 1–4, 7–12, 18–24
Example 2 Solving a Real-World Problem
Exercises 5, 13–16
4. -__2r < 7 5. Melissa’s rent check is lost in the mail. There is a -\$9 late fee charged to Melissa’s account for every day the rent is late. Her account started out at \$0, and after the late fees showed a balance that was less than -\$36. How many days late was the check? Lesson Quiz available online my.hrw.com
Answers 1. x < 21 15
20
25
0
5
-10
-5
-15
-10
3. t ≤ -9 -15
4. r > -14 -20
5. -9d < -36; d > 4; The check was more than 4 days late.
371
Depth of Knowledge (D.O.K.)
Lesson 13.4
Mathematical Processes
2 Skills/Concepts
1.D Multiple representations
3 Strategic Thinking
1.G Explain and justify arguments
14–16
2 Skills/Concepts
1.A Everyday life
17
2 Skills/Concepts
1.F Analyze relationships
18–23
2 Skills/Concepts
1.C Select tools
24
2 Skills/Concepts
1.C Select tools
25
3 Strategic Thinking
1.F Analyze relationships
26
3 Strategic Thinking
1.E Create and use representations
7–12 13
Additional Resources Differentiated Instruction includes: • Leveled Practice Worksheets
2. z ≤ -2 -5
Exercise
Name
Class
Date
Solve each inequality.
13.4 Independent Practice 6.9.B, 6.10.A, 6.10.B
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Solve each inequality. Graph and check your solution. q q≤7 7. - __ ≥ -1 7 0 1 2 3 4 5 6 7 8 9 10
x > -5
-10 -9 - 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1
y 9. 0.5 ≤ __ 8
11. -12 > 2x
© Houghton Mifflin Harcourt Publishing Company
0
2
x>4 1 x < __ 20
21. 0.4 < -x
x ≥ 161 x < -0.4
x ≤ -30 23. - ___ 0.8
x ≥ 24
24. Use the order of operations to simplify the left side of the inequality below. What values of x make the inequality a true statement? - _12 (32 + 7)x > 32
- _12 (32 + 7)x > 32; - _12 (9 + 7)x > 32; - _12(16)x > 32; -8x > 32; x < -4. The solution would be all values less than -4.
FOCUS ON HIGHER ORDER THINKING
Work Area
25. Counterexamples John says that if one side of an inequality is 0, you don’t have to reverse the inequality symbol when you multiply or divide both sides by a negative number. Find an inequality that you can use to disprove John’s statement. Explain your thinking.
10 seconds or more
x < -6
If you divide both sides of -7z ≥ 0 by -7, you get z ≥ 0.
0 1 2
x≥3
x ≤ -0.5 12. - __ 6 -5 -4 - 3 - 2 - 1
14. A veterinarian tells Max that his cat should lose no more than 30 ounces. The veterinarian suggests that the cat should lose 7 ounces or less per week. What is the shortest time in weeks and days it would take Max’s cat to lose the 30 ounces?
15. The elevation of an underwater cave is -120 feet relative to sea level. A submarine descends to the cave. The submarine’s rate of change in elevation is no greater than -12 feet per second. How long will it take to reach the cave?
r < -6
-8 -7 - 6 - 5 - 4 - 3 - 2 - 1
8
1 22. 4x < __ 5
0
y≥4
-10 -9 - 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1
Online Assessment and Intervention
4 weeks and 2 days
0 1 2 3 4 5 6 7 8 9 10
10. 36 < -6r
x < - __ 1 20. - __
x ≤ -23 19. - __ 7
0 1 2 3 4 5
13. Multistep Parav is playing a game in which he flips a counter that can land on either a -6 or a 6. He adds the point values of all the flips to find his total score. To win, he needs to get a score less than -48. a. Assuming Parav only gets -6s when he flips the counter, how many times does he have to flip the counter?
-6x < -48; x > 8
16. The temperature of a freezer is never greater than -2 °C. Yesterday the temperature was -10 °C, but it increased at a steady rate of 1.5 °C per hour. How long in hours and minutes did the temperature increase inside the freezer?
This is incorrect because if you choose a value from the possible solutions, such as z = 1, and substitute it into the original equation, you get -7 ≥ 0, which is not true.
less than 5 hours and 20 minutes 26. Communicate Mathematical Thinking Van thinks that the answer to -3x < 12 is x < -4. How would you convince him that his answer is incorrect?
17. Explain the Error A student's solution to the inequality -6x > 42 was x > -7. What error did the student make in the solution? What is the correct answer?
b. Suppose Parav flips the counter and gets five 6s and twelve -6s when he plays the game. Does he win? Explain.
No; his score is 5(6) +
I would remind him of the properties of inequality that we learned. I would show him that if you substitute -5 into the original inequality, the statement is not true.
The student did not reverse the inequality sign. The answer
Therefore his answer is not correct. Then I would show
should be x < -7.
him that the several solutions from x > -4 do hold true.
© Houghton Mifflin Harcourt Publishing Company
8. -12x < 60
x ≤ -9
18. 18 ≤ -2x
Personal Math Trainer
12(-6) = -42, which is not less than -48. Lesson 13.4
EXTEND THE MATH
PRE-AP
Activity Present the statements at the right. Challenge students to make the statements true by filling in the blanks with <, >, or =. For each choice, students should give an example to demonstrate that the statement is true. 1. > example: a = 3, b = 4, c = -2 (3)(-2) > (4)(-2); -6 > -8 2. < example: x = -5, y = -3, z = 2 (-5)(2) < (-3)(2); -10 < -6 3. = example: n = -3, m = 2, p = 0 (-3)(0) = (2)(0); 0 = 0
371
372
Activity available online
Unit 4
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Algebraically Speaking 1. If a < b and c < 0, then ac
bc.
2. If x < y and z > 0, then xz
yz.
3. If n < m and p = 0, then np
mp.
4. If f < 0 and g < 0, then fg
0.
4. > example: f = -6, g = -2 (-6)(-2) > 0; 12 > 0
Multiplication and Division Inequalities with Rational Numbers
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DO NOT EDIT--Changes must be made through “File info” CorrectionKey=B
MODULE QUIZ
Assess Mastery
13.1 Writing Inequalities
Personal Math Trainer Online Assessment and Intervention
Write an inequality to represent each situation, then graph the solutions.
Use the assessment on this page to determine if students have mastered the concepts and standards covered in this module.
1. There are fewer than 8 gallons of gas in the tank. 0 1 2 3 4 5 6 7 8 9 10
3
Response to Intervention
2 1
0 1 2 3 4 5 6 7 8 9 10
Enrichment
- 10 - 9 - 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1
Access Ready to Go On? assessment online, and receive instant scoring, feedback, and customized intervention or enrichment.
Personal Math Trainer
v ≤ -4
my.hrw.com
Solve each inequality. Graph the solution. 4. c - 28 > -32
Online and Print Resources Differentiated Instruction
Differentiated Instruction
• Reteach worksheets
• Challenge worksheets
• Reading Strategies • Success for English Learners ELL
ELL
0 1 2 3 4 5 6 7 8 9 10
-10 -9 - 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1
5. 0
v + 17 ≤ 20
0 1 2 3 4 5 6 7 8 9 10
6. Today’s high temperature of 80 °F is at least 16 ° warmer than yesterday’s high temperature. What was yesterday’s high temperature? 80
PRE-AP
Extend the Math PRE-AP Lesson activities in TE
Additional Resources Assessment Resources includes: • Leveled Module Quizzes
≥ 16 + T; T ≤ 64 °F
13.3, 13.4 Multiplication and Division Inequalities © Houghton Mifflin Harcourt Publishing Company
Online Assessment and Intervention
p≥3
2. There are at least 3 pieces of gum left in the pack.
3. The valley was at least 4 feet below sea level.
Intervention
my.hrw.com
f<8
Solve each inequality. Graph the solution. 8. __a2 < 4
7. 7f ≤ 35 0 1 2 3 4 5 6 7 8 9 10
-10 -9 - 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1
0 1 2 3 4 5 6 7 8 9 10 k <3 10. ___ -3
9. -25g ≥ 150 0
-10 -9 - 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1
0
Module 13
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Texas Essential Knowledge and Skills Lesson
Exercises
13.1
1–3
6.9.A, 6.9.B, 6.10.B
13.2
4–6
6.9.B, 6.9.C, 6.10.A, 6.10.B
13.3
7–10
6.9.B, 6.9.C, 6.10.A, 6.10.B
13.4
7–10
6.9.B, 6.10.A, 6.10.B
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Module 13
TEKS
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MODULE 13 MIXED REVIEW
Texas Test Prep
Texas Test Prep
Item 6 Students can look at each graph and test the ending point to see if it makes the inequality true. Then, using the ≤ symbol, students can identify that choice B is the correct answer.
Selected Response 1. Em saves at least 20% of what she earns each week. If she earns \$140 each week for 4 weeks, which inequality describes the total amount she saves? A t > 112 B t ≥ 112 C
t < 28
D t ≤ 28
2. Which number line represents the inequality r > 6?
Avoid Common Errors
A
Item 1 If students do not read the item carefully, they may find the amount saved per week rather than the total amount saved. Remind them to read all items fully and carefully. Item 7 When students divide \$150 by \$12, they will get 12.5. Many students might insert this answer, not realizing that the question asks for full weeks, which requires the answer to be rounded up to 13. Remind students to read the questions carefully and be sure to understand what is being asked.
0 1 2 3 4 5 6 7 8 9 10
5. The number line below represents the solution to which inequality? 0 1 2 3 4 5 6 7 8 9 10 m A __ > 2.2 4
C
m __ > 2.5 3
B 2m < 17.6
D 5m > 40
6. Which number line shows the solution to w - 2 ≤ 8? A
0 1 2 3 4 5 6 7 8 9 10
B
0 1 2 3 4 5 6 7 8 9 10
C
0 1 2 3 4 5 6 7 8 9 10
B
0 1 2 3 4 5 6 7 8 9 10
Online Assessment and Intervention
D
0 1 2 3 4 5 6 7 8 9 10
C
0 1 2 3 4 5 6 7 8 9 10
Gridded Response
D
0 1 2 3 4 5 6 7 8 9 10
3. For which inequality below is z = 3 a solution? A z+5≥9
7. Hank needs to save at least \$150 to ride the bus to his grandparent’s home. If he saves \$12 a week, what is the least number of weeks he needs to save?
B z+5>9
3
0
0
0
0
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D z+5<8
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4. What is the solution to the inequality −6x < −18? A x>3 B x<3 C
x≥3
D x≤3
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.
1
z+5≤8
C
© Houghton Mifflin Harcourt Publishing Company
Texas Testing Tip Students can use the given solutions to work backwards and find a solution. Item 3 Students are given the solution of 3 in the main problem. Instead of solving every problem to see which would give 3 as a true solution, the students can substitute 3 into each inequality. The choice with the true statement is the correct answer.
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Unit 4
Texas Essential Knowledge and Skills Items
Mathematical Process TEKS
1
6.10.A
6.1.A, 6.1.F
2
6.9.B
6.1.D, 6.1.E
3
6.10.B
6.1.D
4
6.10.A
6.1.D
5
6.9.B, 6.10.A
6.1.D
6
6.9.B, 6.10.A
6.1.D, 6.1.E
7*
6.3.D, 6.9.A, 6.10.A
6.1.A
* Item integrates mixed review concepts from previous modules or a previous course.
Inequalities and Relationships
374
Relationships in Two Variables ?
ESSENTIAL QUESTION How can you use relationships in two variables to solve real-world problems?
MODULE
You can use tables, graphs, and equations in two variables to model real-world problems, then use algebraic methods to solve the problems.
14
LESSON 14.1
Graphing on the Coordinate Plane 6.11
LESSON 14.2
Independent and Dependent Variables in Tables and Graphs 6.6.A, 6.6.C
LESSON 14.3
Writing Equations from Tables 6.6.B, 6.6.C
LESSON 14.4
© Houghton Mifflin Harcourt Publishing Company • Image Credits: Blickwinkel/Alamy
Representing Algebraic Relationships in Tables and Graphs 6.6.A, 6.6.B, 6.6.C
Real-World Video
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Module 14
A two-variable equation can represent an animal’s distance over time. A graph can display the relationship between the variables. You can graph two or more animals’ data to visually compare them.
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Math On the Spot
Animated Math
Personal Math Trainer
Go digital with your write-in student edition, accessible on any device.
Scan with your smart phone to jump directly to the online edition, video tutor, and more.
Interactively explore key concepts to see how math works.
Get immediate feedback and help as you work through practice sets.
375
Are You Ready? Use the assessment on this page to determine if students need intensive or strategic intervention for the module’s prerequisite skills. 2 1
Multiplication Facts EXAMPLE
Response to Intervention
1. 7 × 6
Enrichment
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5.
Skills Intervention worksheets
Differentiated Instruction
• Skill 36 Multiplication Facts
• Challenge worksheets
42
2. 10 × 9
90
3. 13 × 12
6.
x
1
2
3
4
y
7
14
21
28
156
4. 8 × 9
72
x
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9
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y is 7 times x.
Online and Print Resources
• Skill 69 Graph Ordered Pairs (First Quadrant)
Use a related fact you know. 7 × 7 = 49 Think: 8 × 7 = (7 × 7) + 7 = 49 + 7 = 56
8×7=
Write the rule for each table.
Access Are You Ready? assessment online, and receive instant scoring, feedback, and customized intervention or enrichment.
Online Assessment and Intervention
Online Assessment and Intervention
Multiply.
Intervention
Personal Math Trainer
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7.
y is 6 more than x.
x
1
2
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4
y
-5
-10
-15
-20
8.
x
0
4
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12
y
0
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y is -5 times x.
PRE-AP
y is one-half x.
Extend the Math PRE-AP Lesson Activities in TE
EXAMPLE
10 8
B
D
6 4
A
Start at the origin. Move 9 units right. Then move 5 units up. Graph point A(9, 5).
C
2 O
© Houghton Mifflin Harcourt Publishing Company
3
Personal Math Trainer
Complete these exercises to review skills you will need for this chapter.
E 2
4
6
8 10
Graph each point on the coordinate grid above.
9. B (0, 8)
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10. C (2, 3)
11. D (6, 7)
12. E (5, 0)
Unit 4
PROFESSIONAL DEVELOPMENT VIDEO my.hrw.com
Author Juli Dixon models successful teaching practices as she explores graphing in the coordinate plan in an actual sixth-grade classroom.
Online Teacher Edition Access a full suite of teaching resources online—plan, present, and manage classes and assignments.
Professional Development
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Interactive Answers and Solutions Customize answer keys to print or display in the classroom. Choose to include answers only or full solutions to all lesson exercises.
Interactive Whiteboards Engage students with interactive whiteboard-ready lessons and activities. Personal Math Trainer: Online Assessment and Intervention Assign automatically graded homework, quizzes, tests, and intervention activities. Prepare your students with updated, TEKS-aligned practice tests.
Relationships in Two Variables
376
DO NOT EDIT--Changes must be made through "File info" CorrectionKey=B
Parts of the Algebraic Expression 14 + 3x Definition
Understand Vocabulary Use the following explanations to help students learn the preview words. A coordinate plane is formed by two number lines that intersect at right angles. Coordinate planes are used in geographic maps and for locating images on computer screens.
Students can use these reading and note-taking strategies to help them organize and understand new concepts and vocabulary.
Mathematical Representation
Review Word
A specific number whose value does not change
14
constant
A number that is multiplied by a variable in an algebraic expression
3
coefficient
A letter or symbol used to represent an unknown
x
variable
Preview Words
Understand Vocabulary Complete the sentences using the preview words.
1. The numbers in an ordered pair are © Houghton Mifflin Harcourt Publishing Company
Integrating the ELPS
✔ coefficient (coeficiente) ✔ constant (constante) equation (ecuación) negative number (número negativo) positive number (número positivo) scale (escala) ✔ variable (variable)
Use the ✔ words to complete the chart.
The chart helps students review vocabulary associated with algebraic expressions. Write additional expressions on the board and have students identify the parts of each expression.
Review Words
Visualize Vocabulary
Visualize Vocabulary
The two lines that make a coordinate plane are called the axes. The x-axis is the horizontal number line that runs left to right on the coordinate plane. The y-axis is the vertical line that runs up and down on the coordinate plane.
Vocabulary
coordinate plane 2. A lines that intersect at right angles.
coordinates
.
is formed by two number
axes (ejes) coordinate plane (plano cartesiano) coordinates (coordenadas) dependent variable (variable dependiente) independent variable (variable independiente) ordered pair (par ordenado) origin (origen) quadrants (cuadrantes) x-axis (eje x) x-coordinate (coordenada x) y-axis (eje y) y-coordinate (coordenada y)
c.4.D Use prereading supports such as graphic organizers, illustrations, and pretaught topic-related vocabulary to enhance comprehension of written text.
Layered Book Before beginning the module, create a layered book to help you learn the concepts in this module. Label each flap with lesson titles from this module. As you study each lesson, write important ideas such as vocabulary and formulas under the appropriate flap. Refer to your finished layered book as you work on exercises from this module.
Differentiated Instruction • Reading Strategies ELL
Module 14
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Grades 6–8 TEKS Before Students understand: • how to recognize the difference between additive and multiplicative numerical patterns given in a table or graph • how to graph a relationship on a number line • how to identify and locate ordered pairs of whole numbers in the first quadrant
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Module 14
In this module Students will learn to: • identify independent and dependent quantities from tables and graphs • write an equation that represents the relationship between independent and dependent quantities from a table • represent a given situation using verbal descriptions, tables, graphs, and equations in the form y = kx or y=x+b • graph points in all four quadrants using ordered pairs of rational numbers
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After Students will connect: • tables and verbal descriptions with a linear relationship • graphs and equations with a linear relationship • ordered pairs with an equation
MODULE 14
Unpacking the TEKS
Unpacking the TEKS Understanding the TEKS and the vocabulary terms in the TEKS will help you know exactly what you are expected to learn in this module.
Use the examples on this page to help students know exactly what they are expected to learn in this module.
6.6.B Write an equation that represents the relationship between independent and dependent quantities from a table.
Texas Essential Knowledge and Skills Content Focal Areas
Key Vocabulary equation (ecuación) A mathematical sentence that shows that two expressions are equivalent.
Expressions, equations, and relationships—6.6 The student applies mathematical process standards to use multiple representations to describe algebraic relationships.
What It Means to You You will learn to write an equation that represents the relationship in a table. UNPACKING EXAMPLE 6.6.B
Emily has a dog-walking service. She charges a daily fee of \$7 to walk a dog twice a day. Create a table that shows how much Emily earns for walking 1, 6, 10, and 15 dogs. Write an equation that represents the situation. Dogs walked
1
6
10
15
Earnings (\$)
7
42
70
105
Earnings is 7 times the number of dogs walked. Let the variable e represent earnings and the variable d represent the number of dogs walked.
Measurement and data—6.11 The student applies mathematical process standards to use coordinate geometry to identify locations on a plane.
e=7×d
6.6.C Represent a given situation using verbal descriptions, tables, graphs, and equations in the form y = kx or y = x + b.
Go online to see a complete unpacking of the .
What It Means to You You can use words, a table, a graph, or an equation to model the same mathematical relationship.
Key Vocabulary
UNPACKING EXAMPLE 6.6.C
coordinate plane (plano cartesiano) A plane formed by the intersection of a horizontal number line called the x-axis and a vertical number line called the y-axis.
The equation y = 4x represents the total cost y for x games of miniature golf. Make a table of values and a graph for this situation.
Visit my.hrw.com to see all the unpacked.
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Number of games, x Total cost (\$), y
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y
Total cost (\$)
c.4.F Use visual and contextual support … to read grade-appropriate content area text … and develop vocabulary … to comprehend increasingly challenging language.
© Houghton Mifflin Harcourt Publishing Company • Image Credits: PhotoDisc/Getty Images
Integrating the ELPS
20 16 12 8 4 x O
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6
8
Number of games my.hrw.com
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Lesson 14.1
Lesson 14.2
Unit 4
Lesson 14.3
Lesson 14.4
6.6.A Identify independent and dependent quantities from tables and graphs. 6.6.B Write an equation that represents the relationship between independent and dependent quantities from a table. 6.6.C Represent a given situation using verbal descriptions, tables, graphs, and equations in the form y = kx or y = x + b. 6.11 Graph points in all four quadrants using ordered pairs of rational numbers.
Relationships in Two Variables
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LESSON
14.1 Graphing on the Coordinate Plane Engage
Texas Essential Knowledge and Skills
ESSENTIAL QUESTION
The student is expected to:
How do you locate and name points in the coordinate plane? Sample answer: Points in the coordinate plane are located and named by their locations from the origin along the x-axis first, followed by the y-axis. The order of the coordinates is important.
Measurement and data—6.11 Graph points in all four quadrants using ordered pairs of rational numbers.
Motivate the Lesson Ask: Have you ever tried to find a city or town by its location on a map grid? Maps are like a coordinate plane. Begin Example 1 to find out how to locate a point on a coordinate plane.
Mathematical Processes 6.1.E Create and use representations to organize, record, and communicate mathematical ideas.
Explore
Engage with the Whiteboard
Identify the coordinates of each point. Name the quadrant where each point is located.
To introduce students to the four-quadrant coordinate plane, sketch a simple “treasure map” with a coordinate plane on the whiteboard. Mark a point for “Start” at the origin and a point for “Treasure” in Quadrant I. Ask students to draw a path to the treasure using the grid lines and then to describe the path in words, such as, “Walk east 3 steps. Then walk north 5 steps.” Repeat several times with new coordinates for the “Treasure.”
y 2
B
x -4
O
-2
2
4
A
-2
EXAMPLE 1 Focus on Communication
Mathematical Processes Point out to students that coordinates describe a location in relation to the origin, so it is important to always start at the origin when identifying the coordinates of a point.
Point A: (3, -2), Quadrant IV; Point B: (-4, 1), Quadrant II Interactive Whiteboard Interactive example available online my.hrw.com
Students may give incorrect coordinates for a point because they transposed the x- and y-coordinates. Remind students that the x-coordinate is the first number in an ordered pair.
y
EXAMPLE 2
2
-2
R 2
x
4
P
Interactive Whiteboard Interactive example available online my.hrw.com
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Lesson 14.1
ELL c.2.C Some students may have difficulty remembering what the x- and y-coordinates mean in an ordered pair. Encourage students to think of plotting points as physical movements, run and jump. The x-coordinate tells how far to run to the right or left, and the y-coordinate tells how far to jump up or down. So, when plotting points, students should always “run before they jump.”
Connect Vocabulary
O
-2
Mathematical Processes • Is point (2, 3) the same as point (3, 2)? Explain. No, they are not the same point. Point (2, 3) lies 2 units to the right of the origin and 3 units up, while point (3, 2) lies 3 units to the right and 2 units up.
Avoid Common Errors
P(0, -2), Q(-4, 1.5), R(3.5, 0)
-4
Questioning Strategies
ADDITIONAL EXAMPLE 2 Graph and label each point on the coordinate plane.
Q
Explain
Questioning Strategies
Mathematical Processes • Describe how graphing the point (0, -3) is similar to graphing the point (-3, 0). How is it different? Sample answer: They are similar because you start at the origin and move three units to graph each point. They are different because in (0, -3), you move down from the origin. In (-3, 0), you move left from the origin.
14.1 ?
Graphing on the Coordinate Plane
Measurement and data—6.11 Graph points in all four quadrants using ordered pairs of rational numbers.
Reflect 1.
If both coordinates of a point are negative, in which quadrant is the
2.
Describe the coordinates of all points in Quadrant I.
point located?
ESSENTIAL QUESTION
Both coordinates are positive.
How do you locate and name points in the coordinate plane? 3.
Naming Points in the Coordinate Plane
The horizontal axis is called the x-axis.
The vertical axis is called the y-axis.
The point where the axes intersect is called the origin.
The two axes divide the coordinate plane into four quadrants.
left of the origin. The x-coordinate in (3, 5) is 3 which
y
The two number lines are called the axes.
Math On the Spot
2
x-axis x
-6
-4
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O
2 4 Origin
will be to the right of the origin.
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4
y-axis
Personal Math Trainer
6
Online Assessment and Intervention
-4
my.hrw.com
4.
© Houghton Mifflin Harcourt Publishing Company
EXAMPL 1 EXAMPLE
Point A is 1 unit left of the origin, and 5 units down. It has x-coordinate -1 and y-coordinate -5, written (-1, -5). It is located in Quadrant III. Point B is 2 units right of the origin, and 3 units up. It has x-coordinate 2 and y-coordinate 3, written (2, 3). It is located in Quadrant I.
Math On the Spot my.hrw.com
2
(-1, -3); III
x -4
-2
O
2
4
-2 H -4
G
EXAMPLE 2
6.11 y 4
Point A is 5 units left and 2 units up from the origin.
x O
H
F
2
Graph and label each point on the coordinate plane. A(-5, 2), B(3, 1.5), C(0, -3)
B
2 -2
F
(-2, 4); II (3, 2); I
4
Points that are located on the axes are not located in any quadrant. Points on the x-axis have a y-coordinate of 0, and points on the y-axis have an x-coordinate of 0.
y
-4
(4, -4); IV
E
Graphing Points in the Coordinate Plane
6.11
4
G E
5.
The numbers in an ordered pair are called coordinates. The first number is the x-coordinate and the second number is the y-coordinate.
y
Identify the coordinates of each point. Name the quadrant where each point is located.
-6
An ordered pair is a pair of numbers that gives the location of a point on a coordinate plane. The first number tells how far to the right (positive) or left (negative) the point is located from the origin. The second number tells how far up (positive) or down (negative) the point is located from the origin.
Identify the coordinates of each point. Name the quadrant where each point is located.
Communicate Mathematical Ideas Explain why (-3, 5) represents a different location than (3, 5).
The x-coordinate in (-3, 5) is -3 which will be to the
A coordinate plane is formed by two number lines that intersect at right angles. The point of intersection is 0 on each number line. •
III
© Houghton Mifflin Harcourt Publishing Company
LESSON
4
Point B is 3 units right and 1.5 units up from the origin. Graph the point halfway between (3, 1) and (3, 2).
-2 -4 A
Point C is 3 units down from the origin. Graph the point on the y-axis. Lesson 14.1
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A
2
B x
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Unit 4
PROFESSIONAL DEVELOPMENT Integrate Mathematical Processes This lesson provides an opportunity to address Mathematical Process TEKS 6.1.E, which calls for students to “create and use representations to organize, record, and communicate mathematical ideas.” Students use coordinate planes to locate points. Then students solve a real-world problem on a coordinate plane in which the scale on each axis represents a real-world situation. In this way, students are able to connect a coordinate plane to the real world.
Math Background The concept of the rectangular coordinate system is generally credited to French mathematician and philosopher René Descartes and, therefore, is sometimes referred to as the Cartesian plane. Every point on the plane can be located because all real numbers, not just integers, are used. The points represented by integer coordinates are sometimes called lattice points.
Graphing on the Coordinate Plane
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YOUR TURN Avoid Common Errors Students may graph the points incorrectly by using the x- and y-coordinates in the wrong order. Remind students to run before they jump.
EXAMPLE 3
The graph shows the location of a fountain, a slide, and a sandbox in a park. The scale on each axis represents yards. Give the coordinates for the sandbox and use them to describe the location of the sandbox relative to the fountain.
Focus on Math Connections
Mathematical Processes Point out to students that the coordinate plane also indicates directions. The x-axis points east (to the right) and west (to the left) and the y-axis points north (up) and south (down). For Example 3, have students add the correct north, south, east, and west labels to the axes of the coordinate plane.
Questioning Strategies
Mathematical Processes • Describe the direction you would go from Gary’s house to Jen’s house. I would travel east to go from Gary’s house to Jen’s house.
y
• What would the coordinates of Gary’s house be if he lived 30 miles directly east of Jen? Explain. (35, 15); Jen lives at (5, 15), so 30 miles directly east would be (35, 15).
20
10
x
Fountain -20
Slide
O
-10
-10
10
20
Sandbox
-20
The sandbox is at (10, -15). This is 10 yards east and 15 yards south of the fountain. Interactive Whiteboard Interactive example available online my.hrw.com
YOUR TURN Focus on Math Connections
Mathematical Processes Show students how to translate directions to movements by using the axes. Remind students that “20 miles south” is on the y-axis below the origin and that “20 miles west” is on the x-axis to the left of the origin. Both movements are in a negative direction, so the coordinates of Ted’s home are (-20, -20). Since Ned lives “50 miles directly north of Ted’s house,” only the y-coordinate changes. Because north is a positive direction, -20 + 50 = 30. So, Ned’s home is located at (-20, 30).
Elaborate Talk About It Summarize the Lesson Have students complete a graphic organizer that shows the number of each quadrant and the signs of the coordinates in each quadrant. y
Q __ II Q __ I - __) + (__, + __) + (__,
x
O
III IV Q __ Q __ - __) - (__, + __) (__,
GUIDED PRACTICE Engage with the Whiteboard For Exercises 1–2, have students fill in the blanks by identifying the coordinates of each point on the whiteboard. Then have them name the quadrant where each point is located. Also ask students to describe how they would graph each point.
Avoid Common Errors Exercises 3–4, 6–7 Students may graph the points incorrectly by using the x- and y-coordinates in the wrong order. Remind students to run before they jump.
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Lesson 14.1
DO NOT EDIT--Changes must be made through “File info” CorrectionKey=B
DO NOT EDIT--Changes must be made through “File info” CorrectionKey=A
6.
P(-4, 2)
7.
Q(3, 2.5)
8.
R(-4.5, -5)
9. 10.
S(4, -5)
4
P -4
2
T
Online Assessment and Intervention
x
-2
O
2
left
1. Point A is 5 units
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1 unit
-2
up
S
2
2. Point B is
T(-2.5, 0)
and
3
.
2
A -4
(2, -3)
IV
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O
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x
4
-2 B
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Each grid square is _12 unit on a side.
y
Gary
Jen
-20 -10
x
City O 10
Each grid square is 5 miles on a side.
-10
Gary’s house is at (-25, 15), which is 25 miles west and 15 miles north of the city.
-20
20
N E
W
( (
S
)
-1
O -1 -2
1
2
B
The first number, the x-coordinate, tells how many units to the right or left the point is located from the origin. The second number, the y-coordinate, tells how many units up or down the point is located from the origin.
? ?
How are north, south, east, and west represented on the graph in Example 3?
ESSENTIAL QUESTION CHECK-IN
9. Give the coordinates of a point that could be in each of the four quadrants, a point on the x-axis, and an point on the y-axis.
Use the graph in the Example. Ted lives 20 miles south and 20 miles west of the city represented on the graph in Example 3. His brother Ned lives 50 miles north of Ted’s house. Give the coordinates of each brother’s house.
Personal Math Trainer
(5, -2). x-axis: (-3, 0); y-axis: (0, 5).
Online Assessment and Intervention
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Lesson 14.1
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x -2
Mathematical Processes
Jen’s house is located 6 grid squares to the right of Gary’s house. Since each grid square is 5 miles on a side, her house is 6 · 5 = 30 miles east of Gary’s.
Ted (-20, -20), Ned (-20, 30)
2 1
8. Vocabulary Describe how an ordered pair represents a point on a coordinate plane. Include the terms x-coordinate, y-coordinate, and origin in your answer.
Math Talk
B Describe the location of Jen’s house relative to Gary’s house.
)
6. Plot point A at -_12 , 2 . 7. Plot point B at 2 _12 , -2 .
North and south are the positive and negative directions along the y-axis; east and west are the positive and negative directions on the x-axis.
10
y
A
5. Describe the scale of the graph.
6.11
20
4. Point D at (5, 0)
For 5–7, use the coordinate plane shown. (Example 3)
Math On the Spot my.hrw.com
EXAMPL 3 EXAMPLE
© Houghton Mifflin Harcourt Publishing Company
II
units right of the origin
3. Point C at (-3.5, 3)
The scale of an axis is the number of units that each grid line represents. So far, the graphs in this lesson have a scale of 1 unit, but graphs frequently use other units.
11.
Graph and label each point on the coordinate plane above. (Example 2)
A Use the scale to describe Gary’s location relative to the city.
(-5, 1)
4
C
units down from the origin.
Its coordinates are
The graph shows the location of a city. It also shows the location of Gary’s and Jen’s houses. The scale on each axis represents miles.
y
of the origin and
from the origin.
Its coordinates are
-4
R
Identify the coordinates of each point in the coordinate plane. Name the quadrant where each point is located. (Example 1)
Personal Math Trainer
Q
© Houghton Mifflin Harcourt Publishing Company
Graph and label each point on the coordinate plane.
Guided Practice
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Unit 4
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DIFFERENTIATE INSTRUCTION Curriculum Integration
Cooperative Learning
Have students draw coordinate grid lines on maps of Texas. Instruct students to draw the x- and y-axes through the state capital and the other lines at __12 -inch increments above, below, to the left, and to the right of the axes. Have the students label the grid lines, beginning with the axes, with the appropriate numbers and give coordinates for various cities and towns on the map.
Have students work in three teams to play coordinate tic-tac-toe. Use a coordinate plane that is 5 units from the origin in all directions. One player on each team alternates calling out the coordinates of a point. Another player on each team locates the point and marks it on the coordinate plane. The first team to place three marks in an uninterrupted row horizontally, vertically, or diagonally wins the round.
Differentiated Instruction includes: • Reading Strategies • Success for English Learners ELL • Reteach • Challenge PRE-AP
Graphing on the Coordinate Plane
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Personal Math Trainer Online Assessment and Intervention
Evaluate GUIDED AND INDEPENDENT PRACTICE
Online homework assignment available
6.11
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14.1 LESSON QUIZ 6.11 Use the coordinate plane shown. Each unit represents 1 city block. y Henry
Concepts & Skills
Practice
Example 1 Naming Points in the Coordinate Plane
Exercises 1–2, 10–13
Example 2 Graphing Points in the Coordinate Plane
Exercises 3–4, 10–13
Example 3 Reading Scales on Axes
Exercises 5–7, 14–15
4 2
x -4
-2
O
Emma -2 Ice cream -4 shop
2
4
Library
1. Write the ordered pairs that represent Henry and the library. 2. Describe Henry’s location relative to the library. 3. Henry wants to meet his friend Emma at an ice cream shop before they go to the library. The ice cream shop is 7 blocks west of the library. Plot and label a point representing the ice cream shop. What are the coordinates of the point? 4. Emma describes her current location: “I’m directly west of the library, halfway to the ice cream shop.” Plot and label a point representing Emma’s location. What are the coordinates of the point? Lesson Quiz available online my.hrw.com
1. Henry (-3, 4), Library (2, -3) 2. Henry is 5 blocks west and 7 blocks north of the library. 3. (-5, -3) 4. (-1.5, -3)
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Lesson 14.1
Exercise
Depth of Knowledge (D.O.K.)
Mathematical Processes
10
2 Skills/Concepts
1.A Everyday life
11
3 Strategic Thinking
1.F Analyze relationships
12–13
2 Skills/Concepts
1.A Everyday life
14–15
2 Skills/Concepts
1.D Multiple representations
16
3 Strategic Thinking
1.F Analyze relationships
17
3 Strategic Thinking
1.G Explain and justify arguments
18–19
3 Strategic Thinking
1.F Analyze relationships
Additional Resources Differentiated Instruction includes: • Leveled Practice Worksheets
Class
Date
14.1 Independent Practice 6.11
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17. Critical Thinking Choose scales for the coordinate plane shown so that you can graph the points J(2, 40), K(3, 10), L(3, -40), M(-4, 50), and N(-5, -50). Explain why you chose the scale for each axis.
Online Assessment and Intervention
For 10–13, use the coordinate plane shown. Each unit represents 1 kilometer. 10. Write the ordered pairs that represent the location of Sam and the theater.
FOCUS ON HIGHER ORDER THINKING
Personal Math Trainer
Theater
Sam: (4, 2); Theater: (-3, 5)
4
11. Describe Sam’s location relative to the theater.
Sam is 3 km south and 7 km east of the
Beth -4
-2
theater.
Sam
2 2
S
(-3, -4)
-40
L
down (in a negative direction). 19. Represent Real-World Problems Zach graphs some ordered pairs in the coordinate plane. The x-values of the ordered pairs represent the number of hours since noon, and the y-values represent the temperature at that time.
y V
© Houghton Mifflin Harcourt Publishing Company
N
4
direction) along the x-axis. Then count 12 grid squares
For 14–15, use the coordinate plane shown. 14. Find the coordinates of points T, U, and V.
U 1.0
T (0.75, -1.0); U (0.75, 1.25); V (-0.75, 1.25)
0.5
Quadrants I and IV; time is always positive, but
x -1.0 -0.5 O
0.5
temperatures can be positive or negative. In
1.0
-0.5
W
16. Explain the Error Janine tells her friend that ordered pairs that have an x-coordinate of 0 lie on the x-axis. She uses the origin as an example. Describe Janine’s error. Use a counterexample to explain why Janine’s statement is false.
each grid square. On the
2
Count 18 grid squares to the right (in a positive
(-3, 0.5)
W(-0.75, -1.0)
K
18. Communicate Mathematical Ideas Edgar wants to plot the ordered pair (1.8, -1.2) on a coordinate plane. On each axis, one grid square equals 0.1. Starting at the origin, how can Edgar find (1.8, -1.2)?
13. Beth describes her current location: “I’m directly south of the theater, halfway to the restaurant.” Plot and label a point representing Beth’s location. What are the coordinates of the point?
15. Points T, U, and V are the vertices of a rectangle. Point W is the fourth vertex. Plot point W and give its coordinates.
I used a scale of 1 unit for
-4 -2 -20
O
y-coordinates ranged from -50 to 50.
E
W
Restaurant
20
The x-coordinates ranged from -5 to 3, and the
N
-4
J
units for each grid square.
4
-2
12. Sam wants to meet his friend Beth at a restaurant before they go to the theater. The restaurant is 9 km south of the theater. Plot and label a point representing the restaurant. What are the coordinates of the point?
40
y-axis I used a scale of 10
x O
M
x
y
Work Area y
quadrants I and IV, the x-coordinate is always positive,
T
-1.0
but the y-coordinate can be positive or negative.
b. In what part of the world and at what time of year might Zach collect data so that the points he plots are in Quadrant IV?
Janine is describing points that lie on the y-axis. Ordered pairs that lie on the x-axis have a y-coordinate of 0. The origin lies on the x- and y-axis. Any
Sample answer: in a region with a cold climate during
other point with an x-coordinate of 0 lies on the y-axis such as (0, 3).
the winter
Lesson 14.1
EXTEND THE MATH
PRE-AP
383
Activity available online
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Name
384
Unit 4
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Activity Plot the points for each set of ordered pairs below. Then connect the points in the order shown to reveal a figure. Name the figure and find its area. Set 1: (2, 5), (2, -1), (-3, -1), (-3, 5) Set 2: (-4, -3), (6, -3), (6, 4) Set 3: (1, 3), (-4, 3), (-4, -2), (1, -2) Write the coordinates for another set of points that form a figure. Find its area. Then challenge a classmate to draw the figure and find its area. Set 1: rectangle, A = 30 square units Set 2: triangle, A = 35 square units Set 3: square, A = 25 square units
Graphing on the Coordinate Plane
384
LESSON
14.2
Independent and Dependent Variables in Tables and Graphs
Texas Essential Knowledge and Skills The student is expected to: Expressions, equations, and relationships—6.6.A Identify independent and dependent quantities from tables and graphs. Expressions, equations, and relationships—6.6.C Represent a given situation using verbal descriptions, tables, graphs, and equations in the form y = kx or y = x + b.
Mathematical Processes 6.1.D Communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate.
Engage ESSENTIAL QUESTION How can you identify independent and dependent quantities from tables and graphs? Sample answer: The dependent variable is the quantity that depends on the other variable. On a graph, the independent variable is shown on the horizontal axis and the dependent variable is shown on the vertical axis.
Motivate the Lesson Ask: What is the relationship between the amount of time a person works and the amount of money that person earns? Begin the Explore Activity to find out what independent and dependent quantities are and how to recognize them.
Explore EXPLORE ACTIVITY 1 Connect to Vocabulary
ELL
Have students describe the meaning of the following phrases: Sample answers are given. • independently wealthy doesn’t need to work for money • Independence Day day of freedom • working independently doesn’t need help to do a job • dependent child needs a parent • insulin-dependent needs insulin daily • dependent clause cannot stand alone in a sentence Then ask students to define independent and dependent variables. An independent variable stands alone and isn’t changed by the other variables. A dependent variable depends on or is changed by another variable.
Explain EXPLORE ACTIVITY 2 Connect Multiple Representations Mathematical Processes Have students complete this table, which represents the situation about the art teacher and clay, to reinforce that both a table and a graph can represent this relationship. Clay bought by teacher (lb)
0
10
20
30
Clay available for classes (lb)
20
30
40
50
Engage with the Whiteboard Ask a student volunteer to locate the point on the graph that shows the 50 pounds of clay that is available for the art class. Have the volunteer draw a line from the y-axis to the point and a line from the point to the x-axis. Have students repeat this process for several more values.
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Lesson 14.2
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LESSON
14.2 ?
Independent and Dependent Variables in Tables and Graphs
ESSENTIAL QUESTION
EXPLORE ACTIVITY (cont’d)
Expressions, equations, and relationships—6.6.A Identify independent and dependent quantities from tables and graphs. Also 6.6.C.
Reflect
1. Analyze Relationships Describe how the value of the independent variable is related to the value of the dependent variable. Is the relationship additive or multiplicative?
The value of y is always 50 times the value of x; multiplicative. 2. What are the units of the independent variable and of the dependent variable?
How can you identify independent and dependent quantities from tables and graphs?
independent variable: hours; dependent variable: miles.
6.6.A
3. A rate is used in the equation. What is the rate?
Identifying Independent and Dependent Quantities from a Table
50 miles per hour
Many real-world situations involve two variable quantities in which one quantity depends on the other. The quantity that depends on the other quantity is called the dependent variable, and the quantity it depends on is called the independent variable.
EXPLORE ACTIVITY 2
A freight train moves at a constant speed. The distance y in miles that the train has traveled after x hours is shown in the table. 0
1
2
3
Distance y (mi)
0
50
100
150
Identifying Independent and Dependent Variables from a Graph In Explore Activity 1, you used a table to represent a relationship between an independent variable (time) and a dependent variable (distance). You can also use a graph to show a relationship of this sort.
A What are the two quantities in this situation?
time and distance
An art teacher has 20 pounds of clay but wants to buy more clay for her class. The amount of clay x purchased by the teacher and the amount of clay y available for the class are shown on the graph.
© Houghton Mifflin Harcourt Publishing Company
Which of these quantities depends on the other?
Distance depends on time. time, x
What is the independent variable?
A If the teacher buys 10 more pounds of clay, how many
distance, y
What is the dependent variable?
B How far does the train travel each hour?
pounds will be available for the art class?
50 miles
=
Distance traveled per hour
=
50
↓ y
·
30 lb; find the point on the graph with a
Time (hours)
y-coordinate of 50. Then find the x-coordinate of this point, which is 30.
x
Lesson 14.2
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Clay Used in Art Class
lb
How can you use the graph to find this information?
↓ ·
30
If the art class has a total of 50 pounds of clay available, how many pounds of clay did the teacher buy?
The relationship between the distance traveled by the train and the time in hours can be represented by an equation in two variables. Distance traveled (miles)
© Houghton Mifflin Harcourt Publishing Company • Image Credits: © Hill Street Studios/Corbis
Time x (h)
6.6.A
y
Clay available for classes (Ib)
EXPLORE ACTIVITY 1
80 60 40 20 O
x 20 40 60 80
Clay bought by teacher (Ib)
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PROFESSIONAL DEVELOPMENT Integrate Mathematical Processes This lesson provides an opportunity to address Mathematical Process TEKS 6.1.D, which calls for students to “communicate mathematical ideas… using multiple representations…as appropriate.” Students use tables, graphs, equations, and language to describe and model relationships between independent and dependent variables. In this way, students use multiple representations to model real-world situations involving independent and dependent variables.
Math Background Although the term function is not mentioned in this lesson, the tables in the lesson represent functions. A function is a rule that relates two quantities so that each input value corresponds to one output value exactly. When y is a function of x, x is called the independent variable and y is called the dependent variable. Whenever a value is assigned to x, a value is automatically assigned to y by an applicable rule or correspondence.
Independent and Dependent Variables in Tables and Graphs
386
EXPLORE ACTIVITY 2 CONTINUED Questioning Strategies
Mathematical Processes • Why does the graph show only Quadrant I? Negative amounts do not make sense in this situation, so the values and the graph are limited to positive x- and y-values. • Why does the graph start at (0, 20)? The art teacher had 20 pounds of clay to start with.
• As the x-value is increasing, what is happening to the y-value? The y-value is also increasing.
ADDITIONAL EXAMPLE 1 A The table below shows a relationship between two variables, x and y. Describe a possible situation the table could represent. Describe the independent and dependent variables in this situation. Independent variable, x
1
2
Dependent variable, y
8
16 24 32
3
4
Sample answer: The table could represent the amount a person earns at a rate of \$8 per hour. The independent variable, x, is the number of hours the person works. The dependent variable, y, is the total earnings. B The graph below shows a relationship between two variables, x and y. Describe a possible situation the graph could represent. Describe the independent and dependent variables. y 6
2
x 2
4
6
Sample answer: The graph could represent the progress of a rock climber, starting at a 2-foot height and continuing at a pace of 1 foot every second. The independent variable is the number of seconds, and the dependent variable is the total number of feet climbed after x seconds. Interactive Whiteboard Interactive example available online my.hrw.com
387
Engage with the Whiteboard Have a volunteer sketch a graph of the relationship shown in the table in A. Have a second volunteer make a table of the relationship shown in the graph in B. This will help students to see that both a table and a graph can represent the same relationship.
Connect Multiple Representations Mathematical Processes Point out to students that each of these situations can be represented by a verbal description, a table, a graph, or an equation. Questioning Strategies • How can the relationship in A be represented by an equation? The table begins with a y-value of 10, so the y-value always will be 10 units greater than the x-value. Then, as x increases by 1, y also increases by 1, resulting in the equation y = x + 10. • How would you describe the relationship in A? Explain. The relationship is an additive relationship because the value of y is always 10 units greater than the value of x. • How can the relationship in B be represented by an equation? The graph begins at the origin, so both variables begin at 0. Then, as x increases by 1, y increases by 12, resulting in the equation y = 12x. • How could you check that the equation is correct for either A or B? Pick a point from either the table or the graph and substitute it into the equation. The result should be a true equation.
Focus on Reasoning
Mathematical Processes Ask students to identify the independent and dependent quantities in the following situations. • A veterinarian must weigh an animal before determining the amount of medication it needs. independent quantity, weight of animal, dependent quantity: amount of medication
4
O
EXAMPLE 1
Lesson 14.2
• A company charges \$10 per hour to rent a jackhammer. independent quantity: time, dependent quantity: cost c.4.D ELL Encourage English learners to use the active reading strategies presented at the beginning of the module.
Integrating the ELPS
DO NOT EDIT--Changes must be made through “File info” CorrectionKey=B
DO NOT EDIT--Changes must be made through “File info” CorrectionKey=A
Describing Relationships Between Independent and Dependent Variables
B What are the two quantities in this situation?
the clay bought by the teacher and the amount of clay available to the class Math On the Spot
Which of these quantities depends on the other?
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The amount of clay available to the class depends on the amount of clay bought by the teacher.
Thinking about how one quantity depends on another helps you identify which quantity is the independent variable and which quantity is the dependent variable. In a graph, the independent variable is usually shown on the horizontal axis and the dependent variable on the vertical axis.
EXAMPLE 1
clay bought by teacher
A The table shows a relationship between two variables, x and y. Describe a possible situation the table could represent. Describe the independent and dependent variables in the situation.
clay available for the class
What is the dependent variable?
C The relationship between the amount of clay purchased by the teacher and the amount of clay available to the class can be represented by an equation in two variables. Amount of clay available (pounds) ↓ y
=
Amount of clay Current amount + purchased (pounds) of clay (pounds) ↓
=
+
20
x
2
3
11
12
13
B The graph shows a relationship between two variables. Describe a possible situation that the graph could represent. Describe the independent and dependent variables.
Reflect
© Houghton Mifflin Harcourt Publishing Company
1
10
The independent variable, x, is the number of days she has been adding money to her savings. The dependent variable, y, is her savings after x days.
value of x.
y 36 24 12
As x increases by 1, y increases by 12. The relationship O is multiplicative. The value of y is always 12 times the value of x.
4. In this situation, the same units are used for the independent and dependent variables. How is this different from the situation involving the train in the first Explore?
x 2
4
6
The graph could represent the number of eggs in cartons that each hold 12 eggs.
The other situation involves two different units
The independent variable, x, is the number of cartons. The dependent variable, y, is the total number of eggs.
(miles and hours). 5. Analyze Relationships Tell whether the relationship between the independent variable and the dependent variable is a multiplicative or an additive relationship.
Reflect 7. What are other possible situations that the table and graph in Example 1 could represent?
Sample answer: Table: Paul has 10 DVDs and buys more. Independent: number of DVDs he buys; dependent:
6. What are the units of the independent variable, and what are the units of the dependent variable? ; dependent variable:
0
Dependent variable, y
The table could represent Jina’s savings if she starts with \$10 and adds \$1 to her savings every day.
The value of y is always 20 units greater than the
pounds
Independent variable, x
As x increases by 1, y increases by 1. The relationship is additive. The value of y is always 10 units greater than the value of x.
D Describe in words how the value of the independent variable is related to the value of the dependent variable.
independent variable:
6.6.A
© Houghton Mifflin Harcourt Publishing Company
What is the independent variable?
number he has after he buys x DVDs. Graph: 12 photos
pounds
fit on each page of a yearbook. Independent: number of pages; dependent: total number of photos on x pages. Lesson 14.2
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DIFFERENTIATE INSTRUCTION Curriculum Integration
Cooperative Learning
Music: The notes you hear played by a musical instrument are an example of a dependent relationship. For example, a clarinet’s pitch at a particular moment depends on the number of holes covered by the musician. A harp’s pitch depends on the length of the string being plucked.
One way to remember which is the independent variable and which is the dependent variable is to use the names of the two variables in a sentence that makes sense. For example:
Differentiated Instruction includes: • Reading Strategies • Success for English Learners ELL • Reteach • Challenge PRE-AP
Dollars Earned depends on Hours Worked, but Hours Worked does not depend on Dollars Earned. So, Dollars Earned must be the dependent variable and Hours Worked must be the independent variable.
Independent and Dependent Variables in Tables and Graphs
388
YOUR TURN Avoid Common Errors If students have difficulty distinguishing between independent and dependent variables, remind them that the independent variable causes a change in the dependent variable, while the dependent variable could not cause a change in the independent variable.
Elaborate Talk About It Summarize the Lesson Ask: How do you know which is the dependent variable and which is the independent variable in a table or graph? In a table, the independent variable usually is represented by the variable x. The dependent variable usually is represented by the variable y. On a graph, the independent variable usually is shown on the horizontal axis and the dependent variable on the vertical axis.
GUIDED PRACTICE Engage with the Whiteboard For Exercises 1–2, have a student sketch the graph to represent the table on the whiteboard. Have the student explain how to know which quantity should be represented by the x-axis and which by the y-axis. For Exercise 3, have a student volunteer label the axes in the graph to represent the real-world situation suggested by the student.
Avoid Common Errors Exercise 3 If students have difficulty determining whether a relationship is additive or multiplicative, remind them that in a multiplicative relationship the graph will pass through the origin, but in an additive relationship the graph will not pass through the origin. Exercise 4 Remind students that if the independent variable is on the horizontal axis of a graph, the dependent variable is on the vertical axis of the graph.
389
Lesson 14.2
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DO NOT EDIT--Changes must be made through “File info” CorrectionKey=B
Guided Practice
8.
x
0
1
2
3
y
15
16
17
18
Personal Math Trainer Online Assessment and Intervention
1. A boat rental shop rents paddleboats for a fee plus an additional cost per hour. The cost of renting for different numbers of hours is shown in the table.
0
1
2
3
Cost (\$)
10
11
12
13
What is the independent variable, and what is the dependent variable? How do you know? (Explore Activity 1)
my.hrw.com
Sample answer: Bridget’s grandmother gave her a
Time is the independent variable and cost is the
collection of 15 perfume bottles. Bridget adds one
dependent variable, because cost depends on the
bottle per week to the collection. The independent
number of hours rented.
variable is the number of weeks. The dependent variable
2. A car travels at a constant rate of 60 miles per hour. (Explore Activity 1)
is the number of perfume bottles in her collection. The
a. Complete the table.
value of y is always 15 units greater than the value of x. 9.
x
0
1
2
3
4
y
0
16
32
48
64
1
2
3
0
60
120
180
dependent variable. c. Describe how the value of the dependent variable is related to the value of the independent variable.
The value of y is always 60 times the value of x.
profit per T-shirt. The independent variable is the
Use the graph to answer the questions.
number of T-shirts he sells, and the dependent variable
3. Describe in words how the value of the dependent variable is related to the value of the independent variable. (Explore Activity 2)
is his profit in dollars. The value of y is always 16 times
The dependent variable is 5 times the value of the
the value of x.
independent variable.
y 18
4. Describe a real-world situation that the graph could represent. (Example 1)
12 6
O
0
Distance y (mi)
Time is the independent variable and distance is the
that are printed with funny slogans. He makes a \$16
10.
Time x (h)
b. What is the independent variable, and what is the dependent variable?
Sample answer: Colin created a website to sell T-shirts
© Houghton Mifflin Harcourt Publishing Company
Time (hours)
4
20 10
O
x 2
4
6
Sample answer: The graph could represent the total
x 2
y 30
cost y of buying x carnival tickets for \$5 each.
6
Sample answer: Tickets to the school musical cost \$3
? ?
each. The independent variable is the number of tickets
ESSENTIAL QUESTION CHECK-IN
5. How can you identify the dependent and independent variables in a real-world situation modeled by a graph?
purchased, and the dependent variable is the total cost. The value of y is always 3 times the value of x.
© Houghton Mifflin Harcourt Publishing Company
Describe a real-world situation that the variables could represent. Describe the relationship between the independent and dependent variables.
Sample answer: The dependent variable is the quantity that depends on the other variable. On a graph, the independent variable is usually shown on the horizontal axis and the dependent variable on the vertical axis. Lesson 14.2
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Independent and Dependent Variables in Tables and Graphs
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Personal Math Trainer Online Assessment and Intervention
Online homework assignment available
Evaluate GUIDED AND INDEPENDENT PRACTICE 6.6.A, 6.6.C
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14.2 LESSON QUIZ 6.6.A The graph below shows the relationship between the number of tickets Lisa is ordering for a raffle and the cost. Use the graph to answer questions 1–3.
Total Cost (including postage)
Lisa’s Ticket Order
Concepts & Skills
Practice
Explore Activity 1 Identifying Independent and Dependent Quantities from a Table
Exercises 1–2, 7
Explore Activity 2 Identifying Independent and Dependent Variables from a Graph
Exercises 3, 6, 8
Example 1 Describing Relationships Between Independent and Dependent Variables
Exercises 4, 6
y 12
Exercise
8
Depth of Knowledge (D.O.K.)
Mathematical Processes
6
2 Skills/Concepts
1.A Everyday life
7
3 Strategic Thinking
1.F Analyze relationships
Number of Tickets
8–9
3 Strategic Thinking
1.G Explain and justify arguments
1. What are the dependent and independent variables?
10
3 Strategic Thinking
1.F Analyze relationships
4
x O
2
4
6
2. Is the relationship between the two variables additive or multiplicative? 3. Describe the relationship between the two quantities in words.
Additional Resources Differentiated Instruction includes: • Leveled Practice Worksheets
Use the table for question 4. x
0
1
2
3
4
y
0
7
14
21
28
4. Describe a possible situation that can be represented by the table. Identify the dependent and independent variables in this situation. Lesson Quiz available online my.hrw.com
Answers 1. Independent variable, x, is the number of tickets bought; dependent variable, y, is the total cost. 2. It is an additive relationship. 3. The total cost is the number of tickets bought plus \$5 for postage for the tickets.
391
Lesson 14.2
4. Sample answer: Parking at the airport costs \$7 per day. Independent variable, x, is the number of days a vehicle is parked; dependent variable, y, is the total cost for parking.
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Class
Date
14.2 Independent Practice
8. Ty borrowed \$500 from his parents. The graph shows how much he owes them each month if he pays back a certain amount each month.
Personal Math Trainer
6.6.A, 6.6.C
a. How many hours did the soccer team practice before the season began?
6 hours b. What are the two quantities in this situation?
hours practiced during the season and total
Total practice time for year (hours)
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6. The graph shows the relationship between the hours a soccer team practiced after the season started and their total practice time for the year.
a. Describe the relationship between the number of months and the amount Ty owes. Identify an independent and dependent variable and explain your thinking.
Online Assessment and Intervention
y 10
Ty starts out owing \$500 and every
8 6 4 2
400 300 200 100
O
2
8 10
b. How long will it take Ty to pay back his parents?
10 months FOCUS ON HIGHER ORDER THINKING
d. Analyze Relationships Describe the relationship between the quantities in words.
Work Area
9. Error Analysis A discount store has a special: 8 cans of juice for a dollar. A shopper decides that since the number of cans purchased is 8 times the number of dollars spent, the cost is the independent variable and the number of cans is the dependent variable. Do you agree? Explain.
Total practice time for the year is 6 hours more than practice time during the season.
Sample answer: I disagree because the amount
e. Is the relationship between the variables additive or multiplicative? Explain.
a shopper pays depends on the number of cans
Additive; the total practice increases by 1 hour as the
purchased. So, the number of cans is the independent
practice time during the season increases by 1 hour.
variable, and cost is the dependent variable.
b. What is the independent variable?
6
owes.
dependent: total practice time for year
a. What is the dependent variable?
4
Months
months; dependent variable: amount he
4
independent: hours practiced during season;
7. Multistep Teresa is buying glitter markers to put in gift bags. The table shows the relationship between the number of gift bags and the number of glitter markers she needs to buy.
2
by \$50; independent variable: number
x
O
c. What are the dependent and independent variables?
© Houghton Mifflin Harcourt Publishing Company
500
month the amount he owes decreases
Practice time during the season (hours)
practice time for year
Ty’s Loan Payments
0
1
2
3
Number of markers, y
0
5
10
15
10. Analyze Relationships Provide an example of a real-world relationship where there is no clear independent or dependent variable. Explain.
Sample answer: Andrea is 4 years older than Lisa. You
number of markers number of gift bags
could say that Andrea’s age depends on Lisa’s because you can add 4 to Lisa’s age. You can also say that Lisa’s
c. Describe the relationship between the quantities in words.
© Houghton Mifflin Harcourt Publishing Company
Name
Amount Ty owes (dollars)
DO NOT EDIT--Changes must be made through “File info” CorrectionKey=A
age depends on Andrea’s age because you can subtract
The number of glitter markers is 5 times the number of gift bags.
4 from Andrea’s age.
d. Is the relationship additive or multiplicative? Explain.
The relationship is multiplicative because y increases by a factor of 5 as x increases by 1. Lesson 14.2
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PRE-AP
Introduce students to independent and dependent variables in situations that involve decimals or fractions. For example: Gina is charged \$0.15 for each text message that she sends. 1. What is the independent variable? a a. number of texts sent b. charge per text c. total amount charged for texting
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c
a. number of texts sent b. charge per text c. total amount charged for texting 3. Write an equation that expresses the situation. Let b be the amount of Gina’s bill. Let s be the number of texts sent. b = 0.15s
Independent and Dependent Variables in Tables and Graphs
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LESSON
14.3 Writing Equations from Tables Texas Essential Knowledge and Skills The student is expected to: Expressions, equations, and relationships—6.6.B Write an equation that represents the relationship between independent and dependent quantities from a table. Also 6.6.C
Engage
ESSENTIAL QUESTION How can you use an equation to show a relationship between two variables? Use a table to find the relationship between the two variables. Use that relationship to write an equation.
Motivate the Lesson Ask students to imagine a hot dog stand that charges \$3 per hot dog. How much would 4 hot dogs cost? 30 hot dogs? Begin the Explore Activity to find out how to write an equation that will help you predict the total cost for any number of hot dogs.
Mathematical Processes 6.1.B Use a problem-solving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem-solving process and the reasonableness of the solution.
Explore EXPLORE ACTIVITY Focus on Patterns
Mathematical Processes Point out to students that to write an equation from the data in the table, they need to look for a pattern in the data. First, they should look for changes in both the input values and the output values. Then they need to see how the changes are related. For example:
8=8·1 16 = 8 · 2 24 = 8 · 3
Write an equation that expresses y in terms of x. A x
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y
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x
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y
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So, the pattern is y = 8 · x, where x is the number of dogs walked and y is the amount of money earned.
y = 3x B
y=x+5
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Animated Math Writing Equations from Tables Students generate patterns with an interactive model, record the data in a table, and write equations to represent the pattern. my.hrw.com
Lesson 14.3
EXAMPLE 1 Focus on Reasoning
Interactive Whiteboard Interactive example available online
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Explain Mathematical Processes In A, point out to students that the y-value is always less than the x-value. Therefore, the operation in the equation must be subtraction, division, or multiplication by a factor that is less than 1. In B, point out to students that the y-value is always more than the x-value. Therefore, the operation in the equation must be addition or multiplication by a factor that is greater than 1.
Questioning Strategies
Mathematical Processes • For A, how can you write an equation that expresses x in terms of y? I compared the x- and y-values and found that each x-value is twice the corresponding y-value, which gives me the equation x = 2y.
YOUR TURN Engage with the Whiteboard For Exercises 2–5, have students write a pattern on the whiteboard for each table. Then have them use the pattern to write an equation to represent each table. Ask students to explain their reasoning.
Writing Equations from Tables
14.3 ?
ESSENTIAL QUESTION
Expressions, equations, and relationships—6.6.B Write an equation that represents the relationship between independent and dependent quantities from a table. Also 6.6.C.
Writing an Equation Based on a Table The relationship between two variables where one variable depends on the other can be represented in a table or by an equation. An equation expresses the dependent variable in terms of the independent variable.
Math On the Spot
When there is no real-world situation to consider, we usually say x is the independent variable and y is the dependent variable. The value of y depends on the value of x.
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How can you use an equation to show a relationship between two variables?
EXAMPLE 1
6.6.B, 6.6.C
EXPLORE ACTIVITY
Write an equation that expresses y in terms of x.
Writing an Equation to Represent a Real-World Relationship
Animated Math
A
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Many real-world situations involve two variable quantities in which one quantity depends on the other. This type of relationship can be represented by a table. You can also use an equation to model the relationship.
1
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5
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Earnings
\$8
\$16
\$24
\$40
\$80
\$160
For 1 dog, Amanda earns 1 · 8 = \$8. For 2 dogs, she earns 2 · 8 = \$16.
© Houghton Mifflin Harcourt Publishing Company
3
4
5
1
1.5
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2.5
Compare the x- and y-values to find a pattern.
Use the pattern to write an equation expressing y in terms of x. y = 0.5x
B
x
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8
10
y
5
7
9
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13
Compare the x- and y-values to find a pattern.
STEP 1
Each y-value is 3 more than the corresponding x-value.
Use the pattern to write an equation expressing y in terms of x.
STEP 2
Math Talk
Mathematical Processes
number of dogs walked.
y=x+3
How can you check that your equations are correct?
for each dog she walks.
C Write an equation that relates the number of dogs Amanda walks to the amount she earns. Let e represent earnings and d represent dogs.
For each table, write an equation that expresses y in terms of x. 2.
e=8·d D Use your equation to complete the table for 5, 10, and 20 walked dogs. E Amanda’s earnings depend on the number of dogs walked
x
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10
y
10
9
8
3.
x
10
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14
y
25
30
35
y=x-2
y = 2.5x
. 4.
Reflect
Personal Math Trainer
1. What If? If Amanda changed the amount earned per dog to \$11, what equation could you write to model the relationship between number of dogs walked and earnings?
2
STEP 2
Substitute each value of x in the equation. If the equation is correct, the result is the corresponding y-value.
Each earnings amount is 8 times the corresponding 8
1 0.5
1 Each y-value is __ , or 0.5 times, the corresponding x-value. 2
A For each column, compare the number of dogs walked and earnings. What is the pattern?
B Based on the pattern, Amanda earns \$
x y STEP 1
The table shows how much Amanda earns for walking 1, 2, or 3 dogs. Use the table to determine how much Amanda earns per dog. Then write an equation that models the relationship between number of dogs walked and earnings. Use your equation to complete the table. Dogs walked
6.6.B, 6.6.C
Online Assessment and Intervention
e = 11 · d
x
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y
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8
y=x+5
5.
x
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© Houghton Mifflin Harcourt Publishing Company
LESSON
y = 2x
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Lesson 14.3
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PROFESSIONAL DEVELOPMENT Integrate Mathematical Processes This lesson provides an opportunity to address Mathematical Process TEKS 6.1.B, which calls for students to “use a problem-solving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem-solving process and the reasonableness of the solution.” Each of these steps is explicitly used in solving a real-world problem in this lesson.
Math Background A rule that relates the x- and y-values in a table also can be called a relation. A relation can describe a function if, for each x-value (input), there is only one y-value (output). There are several different ways to describe the variables of a function: Independent Variable
Dependent Variable
x-value
y-value
Domain
Range
Input
Output
x
f(x)
Writing Equations from Tables
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ADDITIONAL EXAMPLE 2 Meredith is playing a video game. She earns the same number of points for each alien she captures. She earned 750 points for capturing 5 aliens and 1,350 points for capturing 9 aliens. Write an equation to represent the relationship. Then solve the equation to find how many points Meredith will earn if she captures 27 aliens. p = 150a, where p represents the number of points and a represents the number of aliens captured; 4,050 points Interactive Whiteboard Interactive example available online my.hrw.com
EXAMPLE 2 Focus on Reasoning
Mathematical Processes Point out to students that sometimes a problem may provide clues and facts that you must use to find a solution. Encourage students to begin by identifying the important information. They can underline or circle the information in the problem statement.
Engage with the Whiteboard Have students extend the table on the whiteboard, continuing with sale prices of \$400 through \$1,200 in increments of \$100. Then have students find the donation amount for these sale prices. After they have completed the table, ask students to identify any patterns in the table.
Questioning Strategies • What is the independent variable in this situation? the dependent variable? The independent variable is the sale price of a painting. The dependent variable is the amount donated to charity. • How else could you solve this problem? Write and solve a proportion to find the amount 50 75 x x of the donation if a painting sells for \$1,200: ___ = ____ or ___ = ____ . 200 1,200 300 1,200
YOUR TURN Avoid Common Errors If students have difficulty identifying the independent and dependent variables, remind them to begin by using the given information to make a table and then look for a pattern.
Elaborate Talk About It Summarize the Lesson Ask: How can you use a table to write an equation that represents the relationship in the table? In the table, find the relationship between the independent and dependent variables. Then write the equation that represents the relationship.
GUIDED PRACTICE Engage with the Whiteboard For Exercises 1–2, have students write a pattern on the whiteboard for each table. Then have students use the pattern to write an equation to represent each table. Ask students to explain their reasoning. For Exercise 5, have students fill in the missing information in the table on the whiteboard. Then have them identify the pattern and write an equation.
Avoid Common Errors Exercises 1–4 Some students may write an equation that expresses x in terms of y instead of y in terms of x. Remind them that the form of the equation should be y = kx or y = x + b. Exercise 5 If students have difficulty identifying the independent and dependent variables, remind them to begin by using the given information to make a table and then look for a pattern.
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DO NOT EDIT--Changes must be made through “File info” CorrectionKey=B
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Using Tables and Equations to Solve Problems Problem Solving
EXAMPL 2 EXAMPLE
6. When Ryan is 10, his brother Kyle is 15. When Ryan is 16, Kyle will be 21. When Ryan is 21, Kyle will be 26. Complete the table for Ryan and Kyle. Write and solve an equation to find Kyle’s age when Ryan is 52.
Personal Math Trainer
You can use tables and equations to solve real-world problems. 6.6.B, 6.6.C
Math On the Spot
Online Assessment and Intervention
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A certain percent of the sale price of paintings at a gallery will be donated to charity. The donation will be \$50 if a painting sells for \$200. The donation will be \$75 if a painting sells for \$300. Find the amount of the donation if a painting sells for \$1,200.
Ryan
10
16
21
Kyle
15
21
26
k = r + 5; 57 years old
Analyze Information
Guided Practice
You know the donation amount when the sale price of a painting is \$200 and \$300. You need to find the donation amount if a painting sells for \$1,200.
Write an equation to express y in terms of x. (Explore Activity, Example 1) 1.
Formulate a Plan
You can make a table to help you determine the relationship between sale price and donation amount. Then you can write an equation that models the relationship. Use the equation to find the unknown donation amount. 3. 200
300
Donation amount (\$)
50
75
50 ÷ 2 50 25 ___ = ______ = ___ = 25% 200 200 ÷ 2 100
75 ÷ 3 75 25 ___ = _______ = ___ = 25% 300 300 ÷ 3 100
Write an equation. Let p represent the sale price of the painting. Let d represent the donation amount to charity. © Houghton Mifflin Harcourt Publishing Company
The donation amount is equal to 25% of the sale price. d = 0.25 · p Find the donation amount when the sale price is \$1,200.
One way to determine the relationship between sale price and donation amount is to find the percent.
d = 0.25 · 1,200
p is the independent variable; its value does not depend on any other value. d is the dependent variable; its value depends on the price of the painting.
26
36
2.
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12
y = 4x
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4.
1
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1.35
2.70
6.75
Number of songs = n; Cost =
? ?
x
12
24
36
48
y
2
4
6
8
y = _6x
1.35n \$33.75
.
ESSENTIAL QUESTION CHECK-IN
6. Explain how to use a table to write an equation that represents the relationship in the table.
Justify and Evaluate
16
The total cost of 25 songs is
Substitute values from the table for p and d to check that they are solutions of the equation d = 0.25 · p. Then check your answer of \$300 by substituting for d and solving for p.
6
Total cost (\$)
Simplify to find the donation amount.
d = 0.25 · p 300 = 0.25 · p p = 1,200
y
Substitute \$1,200 for the sale price of the painting.
d = 0.25 · p d = 0.25 · 300 d = 75
40
y=x+3
When the sale price is \$1,200, the donation to charity is \$300.
d = 0.25 · p d = 0.25 · 200 d = 50
30
5. Jameson downloaded one digital song for \$1.35, two digital songs for \$2.70, and 5 digital songs for \$6.75. Complete the table. Write and solve an equation to find the cost to download 25 digital songs. (Example 2)
d = 0.25 · p
d = 300
20
© Houghton Mifflin Harcourt Publishing Company
Sale price (\$)
10
y=x-4
Justify and Evaluate Solve
Make a table.
x
Compare the x- and y-values to find a pattern. Use the pattern to write an equation expressing y in terms of x.
✓ Lesson 14.3
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DIFFERENTIATE INSTRUCTION Curriculum Integration
Cognitive Strategies
Discuss the relationship between Celsius temperature and Kelvin temperature. Show students the following table and ask them to write an equation to convert from degrees Celsius to degrees Kelvin.
Some students may find it helpful to include a “Process” column in a table to help them identify patterns. Have students complete the table below.
Differentiated Instruction includes: • Reading Strategies • Success for English Learners ELL • Reteach • Challenge PRE-AP
Celsius (°C)
Kelvin (°K)
-100
173
K = C + 273
x
Process
y
-3
-2 = -3 + 1
-2
0 = -1 + 1
-2
-50
223
-1
0
273
0
50
323
1
100
373
2
-1 = -2 + 1 1=0+1 2=1+1 3=2+1
-1 0 1 2 3
Each value of y is one more than the value of x. Writing Equations from Tables
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Personal Math Trainer Online Assessment and Intervention
Online homework assignment available my.hrw.com
Evaluate GUIDED AND INDEPENDENT PRACTICE 6.6.B, 6.6.C
Concepts & Skills
Practice
Explore Activity Writing an Equation to Represent a Real-World Relationship
Exercises 1–4, 8, 11
Write an equation that expresses y in terms of x.
Example 1 Writing an Equation Based on a Table
Exercises 1–4, 9–10
1.
Example 2 Using Tables and Equations to Solve Problems
Exercises 5, 11
14.3 LESSON QUIZ 6.6.B
x
1
2
3
4
5
y
5
10 15 20 25
x
10 20 30 40 50
y
7
2. 17 27 37 47
3. Jaime bought 2 puzzles for \$5.00 and 3 puzzles for \$7.50. Write and solve an equation to find the cost of 15 puzzles. 4. A submarine descends to -100 feet in 2 minutes and -250 feet in 5 minutes. Write and solve an equation to find the depth of the submarine in 8 minutes. Lesson Quiz available online my.hrw.com
2. y = x - 3
3. c = 2.50p; \$37.50
4. d = -50m; -400 feet
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Exercise
Depth of Knowledge (D.O.K.)
Mathematical Processes
7
3 Strategic Thinking
1.F Analyze relationships
8
2 Skills/Concepts
1.A Everyday life
9–10
3 Strategic Thinking
1.F Analyze relationships
11
3 Strategic Thinking
1.A Everyday life
12
3 Strategic Thinking
1.G Explain and justify arguments
13–14
3 Strategic Thinking
1.F Analyze relationships
15
3 Strategic Thinking
1.G Explain and justify arguments
Additional Resources Differentiated Instruction includes: • Leveled Practice Worksheets
DO NOT EDIT--Changes must be made through “File info” CorrectionKey=B
DO NOT EDIT--Changes must be made through “File info” CorrectionKey=A
Name
Class
Date
14.3 Independent Practice
12. Communicate Mathematical Ideas For every hour that Noah studies, his test score goes up 3 points. Explain which is the independent variable and which is the dependent variable. Write an equation modeling the relationship between hours studied h and the increase in Noah’s test score s.
Personal Math Trainer
6.6.B, 6.6.C
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Online Assessment and Intervention
Independent: hours studied; its value does not depend on
7. Vocabulary What does it mean for an equation to express y in terms of x?
another variable; dependent: test score; its value depends on the number of hours that Noah studies; s = 3h
The variable y is on one side of the equation. The expression on the other side of the equation shows the relationship
FOCUS ON HIGHER ORDER THINKING
between x and y.
13. Make a Conjecture Compare the y-values in the table to the corresponding x-values. Determine whether there is an additive relationship or a multiplicative relationship between x and y. If possible, write an equation modeling the relationship. If not, explain why.
8. The length of a rectangle is 2 inches more than twice its width. Write an equation relating the length l of the rectangle to its width w.
l = 2w + 2
9. Look for a Pattern Compare the y-values in the table to the corresponding x-values. What pattern do you see? How is this pattern used to write an equation that represents the relationship between the x- and y-values? 20
24
28
32
y
5
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8
© Houghton Mifflin Harcourt Publishing Company
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y-values and corresponding x-values. 14. Represent Real-World Problems Describe a real-world situation in which there is an additive or multiplicative relationship between two quantities. Make a table that includes at least three pairs of values. Then write an equation that models the relationship between the quantities.
10. Explain the Error A student modeled the relationship in the table with the equation x = 4y. Explain the student’s error. Write an equation that correctly models the relationship. x
1
Not possible; there is no consistent pattern between the
The y-value is _14 of the x-value. Write an equation that relates y to _14 of x.
y
x y
Sample answer: The distance Yasmine traveled in miles is equal to 50 times the number of hours she drove; d = 50 × t (multiplicative relationship). Time (h)
The student switched the variables; y = 4x
2
Distance (mi) 100
3
4
5
150
200
250
15. Critical Thinking Georgia knows that there is either an additive or multiplicative relationship between x and y. She only knows a single pair of data values. Explain whether Georgia has enough information to write an equation that models the relationship between x and y.
11. Multistep Marvin earns \$8.25 per hour at his summer job. He wants to buy a video game system that costs \$206.25. a. Write an equation to model the relationship between number of hours worked h and amount earned e.
No; with only one pair of values, Georgia cannot tell
e = 8.25h
© Houghton Mifflin Harcourt Publishing Company
x
Work Area
whether the relationship is additive or multiplicative, so
b. Solve your equation to find the number of hours Marvin needs to work in order to afford the video game system.
she cannot write an equation for the relationship.
206.25 = 8.25h; 25 = h; 25 hours
Lesson 14.3
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InCopy Notes 1. This is a list
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InDesign Notes
EXTEND THE MATH
1. This is a list
PRE-AP
Activity available online
my.hrw.com
Activity To introduce the idea of relationships that are not purely additive or multiplicative, have students find the y-values in the following tables. Remind them to use the order of operations. Have them compare and contrast these relationships with additive and multiplicative relationships. 1. y = 2x + 1 3. y = __12 x + 3 x
0
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4. y = __13 x -1
2. y = 3x - 2 x
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Writing Equations from Tables
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LESSON
14.4
Representing Algebraic Relationships in Tables and Graphs
Texas Essential Knowledge and Skills The student is expected to: Expressions, equations, and relationships—6.6.C Represent a given situation using verbal descriptions, tables, graphs, and equations in the form y = kx or y = x + b. Expressions, equations, and relationships—6.6.A
Engage ESSENTIAL QUESTION How can you use verbal descriptions, tables, and graphs to represent algebraic relationships? Sample answer: You can make a table from the verbal description and then make a graph from the ordered pairs in the table. From a graph, you can make a table and write an equation.
Motivate the Lesson Ask: Have you ever thought about running in a marathon? Do you know how many kilometers you could run in an hour? in two hours? Begin Explore Activity 1 to find out how to make a table and a graph to estimate how far you could run in a given period of time.
Identify independent and dependent quantities from tables and graphs. Expressions, equations, and relationships—6.6.B Write an equation that represents the relationship between independent and dependent quantities from a table.
Mathematical Processes 6.1.D Communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate.
Explore EXPLORE ACTIVITY 1 Connect Multiple Representations
Mathematical Processes Point out to students that the ordered pairs from each table are used to make the graphs. However, after the lines are drawn on the graphs, they represent a more complete picture of the relationship than the tables do because all positive real numbers, not just integers, are included in the graph.
Explain EXPLORE ACTIVITY 2 Focus on Math Connections
Mathematical Processes Point out to students that when finding an equation from a graph, it is easier to first make a table of values from the graph. Then they can look for a pattern for the equation.
Questioning Strategies
Mathematical Processes • Is the relationship additive or multiplicative? Explain how you know. The relationship is additive, because the line drawn through the points does not go through the origin. • Explain how you can find the entrance fee for the museum from the graph. The starting point of the graph is (0, 5). This ordered pair represents the cost of Cherise’s expenses at the museum without any purchases at the gift shop, so it represents the entrance fee, \$5.
c.4.C ELL Be sure English learners understand the context in Explore Activity 2. You may want to discuss the terms “museum”, “souvenir”, and “entrance fee” before starting the activity.
Integrating the ELPS
Engage with the Whiteboard Ask a student volunteer to complete the table and identify the pattern. Then have the student write the equation to represent the total amount spent at the museum gift shop. Finally, discuss with the class what the independent and dependent variables are.
399
Lesson 14.4
Representing Algebraic Relationships in Tables and Graphs
?
ESSENTIAL QUESTION
How can you use verbal descriptions, tables, and graphs to represent algebraic relationships?
Writing an Equation from a Graph Cherise pays the entrance fee to visit a museum, then buys souvenirs at the gift shop. The graph shows the relationship between the total amount she spends at the museum and the amount she spends at the gift shop. Write an equation to represent the relationship.
A Read the ordered pairs from the graph. Use them to complete a table comparing total spent y to amount spent at the gift shop x.
6.6.C
EXPLORE ACTIVITY 1
Representing Algebraic Relationships Angie’s walking speed is 5 kilometers per hour, and May’s is 4 kilometers per hour. Use tables and graphs to show how the distance each girl walks is related to time.
A For each girl, make a table comparing time and distance. 0
1
2
3
4
Angie’s distance (km)
0
5
10
15
20
Time (h)
0
1
2
3
4
May’s distance (km)
0
4
8
12
16
For every hour May walks, she travels 4 km.
Distance (km)
Distance (km)
© Houghton Mifflin Harcourt Publishing Company
20 16 12 8 4 1
2
3
4
Time (h)
5
10
15
20
Total amount (\$)
5
10
15
20
25
y 32
shop amount.
C Write an equation that expresses the total amount y in terms of the gift shop amount x.
28 24 20 16 12 8 4 x
O
4
8 12 16 20 24
Reflect Math Talk
20
Mathematical Processes
16
2. Identify the dependent and independent quantities in this situation.
Dependent: total amount spent; independent: amount
Why does it make sense to connect the points in each graph?
12 8
4 x
x O
5
y= x+5
May
y
0
The total amount is 5 more than the gift
B For each girl, make a graph showing her distance y as it depends on time x. Plot points from the table and connect them with a line. Angie
B What is the pattern in the table? For every hour Angie walks, she travels 5 km.
Time (h)
y
6.6.C, 6.6.B
EXPLORE ACTIVITY 2
O
1
2
3
4
5
Time (h)
Reflect 1. Analyze Relationships How can you use the tables to determine which girl is walking faster? How can you use the graphs?
The girls can walk for fractional parts of an hour and can travel fractional parts of a kilometer.
3. Draw a line through the points in the graph. Find the point that represents Cherise spending \$25 at the gift shop. Use this point to find the total she would spend if she spent \$25 at the gift shop. Then use your equation from C to verify your answer.
© Houghton Mifflin Harcourt Publishing Company • Image Credits: © Thinkstock/ Corbis
14.4
Expressions, equations, and relationships—6.6.C Represent a given situation using tables, graphs, and equations…. Also 6.6.A, 6.6.B
Total amount (\$)
LESSON
(25, 30); \$30; 30 = 25 + 5; 30 = 30
Compare the distances walked after the same amount of time; compare the steepness of the lines. Lesson 14.4
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Unit 4
PROFESSIONAL DEVELOPMENT Integrate Mathematical Processes This lesson provides an opportunity to address Mathematical Process TEKS 6.1.D, which calls for students to “communicate mathematical ideas...using multiple representations…as appropriate.” Students use verbal descriptions to make tables and draw graphs that represent real-life situations. They also represent information from graphs by using tables and equations; and represent equations by using tables and graphs.
Math Background The equations in this lesson are linear equations. A linear equation is an equation whose solutions fall on a line on a coordinate plane. All solutions of a particular linear equation fall on the line, and all the points on the line are solutions of the equation. Linear equations have constant slope. In the slope-intercept form, y = mx + b, m is the slope and b is the y-intercept. Linear equations fit into the general category of polynomial equations. Linear equations are called first-degree equations, because the greatest power of x is 1.
Representing Algebraic Relationships in Tables and Graphs
400
ADDITIONAL EXAMPLE 1 Graph each equation.
A y=x+2
Check for Understanding Ask: Can the points on the graphs only have whole number coordinates? Explain your answer. No, the coordinates can be any pair of rational numbers that satisfies the equation. For example, (0.5, 1.5) is on the graph of y = x + 1.
y 6 4 2
x
O
EXAMPLE 1
2
4
6
B y = 3x
Questioning Strategies
Mathematical Processes • By looking at the graph, how can you tell if the relationship in A is additive or multiplicative? The relationship is additive because the line drawn through the points does not go through the origin. • By looking at the graph, how can you tell if the relationship in B is additive or multiplicative? The relationship is multiplicative because the line drawn through the points goes through the origin.
y 6 4 2 O
x 2
4
6
Interactive Whiteboard Interactive example available online
Students may plot the points in the table but forget to draw a line connecting the points. Remind them that they must connect the points with a line for the graph to be correctly drawn.
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Elaborate Talk About It Summarize the Lesson Ask: How can you use tables and graphs to represent algebraic relationships? You can make a table from the verbal description and then make a graph from the ordered pairs in the table. From a graph, you can make a table and write an equation.
GUIDED PRACTICE Engage with the Whiteboard For Exercises 1–2, have students complete the table and graph the points on the coordinate grid on the whiteboard. Then have another student identify the pattern and write the equation.
Avoid Common Errors Exercises 1–2 Students may plot the points in the table but forget to draw a line connecting the points. Remind them that they must connect the points with a line for the graph to be drawn correctly.
401
Lesson 14.4
Graphing an Equation
An ordered pair (x, y) that makes an equation like y = x + 1 true is called a solution of the equation. The graph of an equation represents all the ordered pairs that are solutions.
EXAMPL 1 EXAMPLE
4. Graph y = x + 2.5.
Personal Math Trainer Math On the Spot
Online Assessment and Intervention
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6.6.C
Graph each equation.
y
x
x + 2.5 = y
(x, y)
0 1 2 3
0 + 2.5 = 2.5 1 + 2.5 = 3.5 2 + 2.5 = 4.5 3 + 2.5 = 5.5
(0, 2.5) (1, 3.5) (2, 4.5) (3, 5.5)
10 8 6 4 2 x
A y=x+1
O
STEP 1
Make a table of values. Choose some values for x and use the equation to find the corresponding values for y.
STEP 2
Plot the ordered pairs from the table. Mathematical Processes
Is the ordered pair (3.5, 4.5) a solution of the equation y = x + 1? Explain.
y
x
x+1=y
(x, y)
1+1=2
(1, 2)
8
10
2
2+1=3
(2, 3)
6
3
3+1=4
(3, 4)
4
4
4+1=5
(4, 5)
2
5
5+1=6
(5, 6)
O
x 1
2
3
4
5
B y = 2x
© Houghton Mifflin Harcourt Publishing Company
3
4
5
STEP 1
Make a table of values. Choose some values for x and use the equation to find the corresponding values for y.
STEP 2
Plot the ordered pairs from the table.
STEP 3
Draw a line through the plotted points to represent all of the ordered pair solutions of the equation.
Frank mows lawns in the summer to earn extra money. He can mow 3 lawns every hour he works. (Explore Activity 1 and Explore Activity 2) 1. Make a table to show the relationship between the number of hours Frank works, x, and the number of lawns he mows, y. Graph the relationship and write an equation.
Yes; (3.5, 4.5) is on the graph. You can also substitute for the variables in the equation to check.
Hours worked
Lawns mowed
0
0 3 6 9
1
2 3
y
10 8 6 4 2
y = 3x
x O
1
2
3
4
5
Hours worked
Graph y = 1.5x. (Example 1)
y 5
2. Make a table to show the relationship. x y
y
x
2x = y
(x, y)
1
2×1=2
(1, 2)
2
2×2=4
(2, 4)
6
3
2×3=6
(3, 6)
4
4
2×4=8
(4, 8)
2
5
2 × 5 = 10
(5, 10)
O
2 1
10
3. Plot the points and draw a line through them.
8
? ?
x 1
2
3
4
4 3
0 1 2 3 0 1.5 3 4.5
x O
1
2
3
4
5
ESSENTIAL QUESTION CHECK-IN
© Houghton Mifflin Harcourt Publishing Company
Draw a line through the plotted points to represent all of the ordered pair solutions of the equation.
1
2
Guided Practice Math Talk
Lawns mowed
STEP 3
1
4. How can a table represent an algebraic relationship between two variables?
5
It shows pairs of values that satisfy the relationship.
Lesson 14.4
401
402
Unit 4
DIFFERENTIATE INSTRUCTION Cooperative Learning
Modeling
Have students work in pairs to write an equation with two variables. Each equation should be an additive equation. Collect students’ equations and randomly redistribute them. Have the students make tables for the equations and write solutions of the equations as ordered pairs. Then have the students graph the equations.
Draw an equilateral triangle and a square, each with a side length of 6 inches, on the chalkboard. Have students find the perimeter of each. Ask students to come up with a formula for the perimeter of an equilateral triangle and a square. Then have students make a table and a graph for each formula.
Differentiated Instruction includes: • Reading Strategies • Success for English Learners ELL • Reteach • Challenge PRE-AP
Representing Algebraic Relationships in Tables and Graphs
402
Personal Math Trainer Online Assessment and Intervention
Evaluate GUIDED AND INDEPENDENT PRACTICE
Online homework assignment available
6.6.A, 6.6.C
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14.4 LESSON QUIZ 6.6.C The graph shows the number of bracelets Olivia can make in an hour. Number of Bracelets
y
Concepts & Skills
Practice
Explore Activity 1 Representing Algebraic Relationships
Exercise 1
Explore Activity 2 Writing an Equation from a Graph
Exercises 1, 5–7
Example 1 Graphing an Equation
Exercises 2–3, 8–9, 11
8 6
Exercise
4
5–6
2
7 2
4
1.D Multiple representations
3 Strategic Thinking
1.F Analyze relationships
8–9
2 Skills/Concepts
1.D Multiple representations
10
3 Strategic Thinking
1.G Explain and justify arguments
11
3 Strategic Thinking
1.F Analyze relationships
12
3 Strategic Thinking
1.G Explain and justify arguments
13–14
3 Strategic Thinking
1.F Analyze relationships
6
Hours
1. Read the ordered pairs from the graph to make a table. 2. Write an equation to model the relationship. The equation y = x + 2 represents the total cost of doing x loads of laundry at a laundromat, including buying a box of detergent.
Additional Resources Differentiated Instruction includes: • Leveled Practice Worksheets
3. Make a table that represents the relationship between number of loads and total cost. 4. Make a graph showing the relationship. Lesson Quiz available online my.hrw.com
0
1
2
Number of bracelets
0
4
8
2. y = 4x 3.
0
1
2
3
Total cost (\$)
2
3
4
5
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Lesson 14.4
y
4. Total Cost
1.
Mathematical Processes
2 Skills/Concepts
x O
Depth of Knowledge (D.O.K.)
6 4 2
x O
2
4
6
DO NOT EDIT--Changes must be made through “File info” CorrectionKey=B
Class
Date
14.4 Independent Practice
Personal Math Trainer
6.6.A, 6.6.B, 6.6.C
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11. Multistep The equation y = 9x represents the total cost y for x movie tickets. a. Make a table and a graph to represent the relationship between x and y.
Online Assessment and Intervention
Students at Mills Middle School are required to work a certain number of community service hours. Students may work additional hours beyond the requirement.
5
10
15
20
Total hours
20
25
30
35
40
50
Total (h)
0
30
3
4
5
Total cost (\$), y
9
18
27
36
45
y 50 40 30 20 10
of tickets; the total cost depends on how many
20
O
2
Dependent: total cost; independent: number
40
10
6. Write an equation that expresses the total hours in terms of the additional hours.
1
b. Critical Thinking In this situation, which quantity is dependent and which is independent? Justify your answer.
y
5. Read the ordered pairs from the graph to make a table.
Number of tickets, x
2
3
4
5
Number of tickets
c. Multiple Representations Eight friends want to go see a movie. Would you prefer to use an equation, a table, or a graph to find the cost of 8 movie tickets? Explain how you would use your chosen method to find the cost.
7. Analyze Relationships How many community service hours are students required to work? Explain.
Sample answer: an equation; substitute 8 for x in
20 hours; when 0 additional hours are worked,
y = 9x to get y = 9(8) = \$72.
the total is 20 hours.
FOCUS ON HIGHER ORDER THINKING
Work Area
12. Critical Thinking Think about graphing the equations y = 5x and y = x + 500. Which line would be steeper? Why?
Beth is using a map. Let x represent a distance in centimeters on the map. To find an actual distance y in kilometers, Beth uses the equation y = 8x.
The graph of y = 5x would be steeper because y
8. Make a table comparing a distance on the map to the actual distance.
increases more rapidly for each value of x.
Map distance (cm)
1
2
3
4
5
Actual distance (km)
8
16
24
32
40
9. Make a graph that compares the map distance to the actual distance. 10. Critical Thinking The actual distance between Town A and Town B is 64 kilometers. What is the distance on Beth’s map? Did you use the graph or the equation to find the answer? Why?
8 cm; sample answer: I used the equation because the scales on the graph don’t extend far enough.
Actual distance (km)
© Houghton Mifflin Harcourt Publishing Company
x 1
tickets were purchased.
x 10 20 30 40 50
y = x + 20
O
13. Persevere in Problem Solving Marcus plotted the points (0, 0), (6, 2), (18, 6), and (21, 7) on a graph. He wrote an equation for the relationship. Find another ordered pair that could be a solution of Marcus’s equation. Justify your answer.
Sample answer: (30, 10); every y-value is _13 of the x-value.
y 50
So, 10 = _13 (30).
40 30 20 10 O
x 1
2
3
4
5
14. Error Analysis The cost of a personal pizza is \$4. A drink costs \$1. Anna wrote the equation y = 4x + 1 to represent the relationship between total cost y of buying x meals that include one personal pizza and one drink. Describe Anna’s error and write the correct equation.
Map distance (cm)
© Houghton Mifflin Harcourt Publishing Company
Name
Total cost (\$)
DO NOT EDIT--Changes must be made through “File info” CorrectionKey=B
Anna’s equation does not show that every meal includes both a pizza and a drink; the correct equation is y = 5x.
Lesson 14.4
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EXTEND THE MATH
PRE-AP
404
Unit 4
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Activity available online
29/01/14 2:18 AM
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Activity Ask students if they know that you can find the approximate temperature by listening to crickets chirp? The chirp rate of a cricket varies by temperature. The hotter it is, the more chirps per minute. The temperature in °F can be found by multiplying the number of chirps in one minute by __14 and adding 40. Write the equation that represents this situation. Then make a table of values to find the temperature for 20, 40, 60, 80, and 100 chirps per minute. t = __14 c + 40, where t is the Fahrenheit temperature and c is the number of chirps in a minute c
20
40
60
80
100
t
45
50
55
60
65
Representing Algebraic Relationships in Tables and Graphs
404
MODULE QUIZ
Assess Mastery
14.1 Graphing on the Coordinate Plane
Use the assessment on this page to determine if students have mastered the concepts and standards covered in this module.
Graph each point on the coordinate plane.
3
Response to Intervention
2 1
Differentiated Instruction
Differentiated Instruction
• Reteach worksheets
• Challenge worksheets
• Success for English Learners ELL
ELL
Additional Resources Assessment Resources includes: • Leveled Module Quizzes
E
2 -6
6. F(-4, 6)
-2O
D
x 2
6
C
-6
14.3 Writing Equations from Tables Write an equation that represents the data in the table. 8.
x
3
5
8
10
y
21
35
56
70
9.
x
5
10
15
20
y
17
22
27
32
y = x + 12
y = 7x
14.4 Representing Algebraic Relationships in Tables and Graphs
PRE-AP
Extend the Math PRE-AP Lesson activities in TE
A
independent: number of packages; dependent: total cost
Online and Print Resources
2. B(3, 5) 4. D(-3, -5)
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B
6
7. Jon buys packages of pens for \$5 each. Identify the independent and dependent variables in the situation.
Enrichment
Graph each equation. © Houghton Mifflin Harcourt Publishing Company
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1. A(-2, 4)
14.2 Independent and Dependent Variables in Tables and Graphs
Access Ready to Go On? assessment online, and receive instant scoring, feedback, and customized intervention or enrichment.
Online Assessment and Intervention
y
F
3. C(6, -4) 5. E(7, 2)
Intervention
Personal Math Trainer
Personal Math Trainer Online Assessment and Intervention
10. y = x + 3
11. y = 5x
8
40
6
30
4
20
2
O
10 2
4
6
8
O
2
4
6
8
ESSENTIAL QUESTION 12. How can you write an equation in two variables to solve a problem?
Decide which variable depends on the other. Use a table to find the relationship between the variables and write an equation.
Module 14
Texas Essential Knowledge and Skills Lesson
Exercises
14.1
1–6
6.11
14.2
7
6.6.A, 6.6.C
14.3
8–9
6.6.B, 6.6.C
14.4
10–11
6.6.A, 6.6.B, 6.6.C
405
Module 14
TEKS
405
Personal Math Trainer
MODULE 14 MIXED REVIEW
Texas Test Prep
Texas Testing Tip Some items are called context-based items, which means the student has to examine each answer choice in order to determine the correct answer. Item 2 If students don’t remember that for every point in quadrant II the x-coordinate is a negative number and the y-coordinate is a positive number, they may need to plot each point to see that choice C is the correct answer.
Selected Response 1. What are the coordinates of point G on the coordinate grid below? y 4 2 x -4
Item 5 To find the point that the graph of y = 10 + x does not pass through, students may need to graph each point on a coordinate grid to see that choice C is the correct answer.
Item 4 Students often will get the independent and dependent quantities backward in problems, thereby choosing A for the answer. Remind students that the dependent quantity depends on the independent quantity. Therefore, the number of points earned depends on the number of prizes captured.
O
2
4
-2
G
-4
Avoid Common Errors Item 1 Students may forget what the first and second numbers in an ordered pair mean. Remind students that the first number is the x-coordinate and the second is the y-coordinate.
-2
A (4, 3)
C
B (4, -3)
D (-4, -3)
(-4, 3)
2. A point is located in quadrant II of a coordinate plane. Which of the following could be the coordinates of that point? A (-5, -7)
C
B (5, 7)
D (5, -7)
(-5, 7)
3. Matt had 5 library books. He checked 1 additional book out every week without returning any books. Which equation describes the number of books he has, y, after x weeks? A y = 5x
C
y = 1 + 5x
B y= 5-x
D y= 5+x
4. Stewart is playing a video game. He earns the same number of points for each prize he captures. He earned 1,200 points for 6 prizes, 2,000 points for 10 prizes, and 2,600 points for 13 prizes. Which is the dependent variable in the situation? A the number of prizes captured B the number of points earned C
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Online Assessment and Intervention
5. Dwayne graphed the equation y = 10 + x. Which point does the graph not pass through? A (0, 10)
C
B (3, 13)
D (5, 15)
(8, 2)
6. Amy gets paid by the hour. Her little sister helps. As shown below, Amy gives her sister part of her earnings. Which equation represents Amy’s pay when her sister’s pay is \$13? Amy’s pay in dollars
10
20
30
40
Sister’s pay in dollars
2
4
6
8
13 A y = __ 5 x B 13 = __ 5
C
5y = 13
D 13 = 5x
Gridded Response 7. Betty earns \$7.50 per hour at a part-time job. Let x be the number of hours and y be the amount she earns. Betty makes a graph to show how x and y are related. If she earns \$60, how many hours did she work?
8
.
0
0
0
0
0
0
1
1
1
1
1
1
2
2
2
2
2
2
3
3
3
3
3
3
4
4
4
4
4
4
5
5
5
5
5
5
6
6
6
6
6
6
7
7
7
7
7
7
8
8
8
8
8
8
9
9
9
9
9
9
© Houghton Mifflin Harcourt Publishing Company
Texas Test Prep
the number of hours
D the number of prizes available
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Unit 4
Texas Essential Knowledge and Skills Items
Mathematical Process TEKS
1
6.11
6.1.E
2
6.11
6.1.F
3
6.6.C
6.1.A
4
6.6.A
6.1.A
5
6.11
6.1.F
6
6.6.B
6.1.A, 6.1.E
7*
6.2.E, 6.6.C
6.1.A, 6.1.F
* Item integrates mixed review concepts from previous modules or a previous course.
Relationships in Two Variables
406
UNIT 4
Expressions, Equations, and Relationships
Additional Resources Personal Math Trainer my.hrw.com
Online Assessment and Intervention
Assessment Resources • Leveled Unit Tests: A, B, C, D • Performance Assessment
Study Guide Review Vocabulary Development Integrating the ELPS Encourage English learners to refer to their notes and the illustrated, bilingual glossary as they review the unit content. c.4.E Read linguistically accommodated content area material with a decreasing need for linguistic accommodations as more English is learned.
MODULE 10 Generating Equivalent Numerical Expressions 6.7.A
Key Concepts • A power is a number that is formed by repeated multiplication by the same factor. An exponent and a base can be used to write a power. (Lesson 10.1) • Factors are whole numbers that are multiplied to find a product. (Lesson 10.2) • To simplify an expression with more than one operation, there is a specific order in which to apply the operations. (Lesson 10.3)
MODULE 11 Generating Equivalent Algebraic Expressions 6.7.C, 6.7.D
Key Concepts • An algebraic expression is an expression that contains one or more variables and may also contain operation symbols, such as + or -. A variable is a letter or symbol used to represent an unknown number. (Lesson 11.1) • To evaluate an expression, substitute a number for the variables and find the value of the expression. (Lesson 11.2) • To generate equivalent expressions, use the properties of operations to combine like terms. (Lesson 11.3)
407
Unit 4
DO NOT EDIT--Changes must be made through "File info" CorrectionKey=A
Study Guide MODULE MODULE
?
10
Review
Find the value of each power. (Lesson 10.1)
Generating Equivalent Numerical Expressions
base (base (en numeración)) exponent (exponente)
7. 75
power (potencia)
EXAMPLE 1 Find the value of each power.
0.9 = 0.9 × 0.9 = 0.81
MODULE MODULE
4 C. (_41 )
( ) ( )( )( )( )
Any number raised to the power of 0 is 1.
2
1 4 1 _ 1 _ 1 _ 1 1 _ _ ___ 4 = 4 4 4 4 = 256
?
180 = 1
EXAMPLE 2 Find the prime factorization of 60.
© Houghton Mifflin Harcourt Publishing Company
B. 27 ÷ 32 × 6
= 4 × 13 = 52
= 27 ÷ 9 × 6
32 = 9
=3×6
Divide.
Multiply.
= 18
Multiply.
3.62
2. 9 × 9 × 9 × 9
8 ___ 343
(_27 )
3
11
23 × 3 × 7
9. 168
22 - (3 + 4) 12. __________ 12 ÷ 4 2
Generating Equivalent Algebraic Expressions
How can you generate equivalent algebraic expressions and use them to solve real-world problems?
21; 6, 14; 7, 12 3 Key Vocabulary algebraic expression (expresión algebraica) coefficients (coeficiente) constant (constante) equivalent expressions (expresiónes equivalente) evaluating (evaluar) term (término (en una expresión))
B. w - y 2 + 3w; w = 2, y = 6
2(52 - 9)
52 = 25
2 - 62 + 3(2)
62 = 36
= 2(16)
Subtract.
= 2 - 36 + 6
Multiply.
= 32
Multiply.
= -28
Add and subtract from left to right.
When w = 2 and y = 6, w - y 2 + 3w = -28.
EXAMPLE 2 Determine whether the algebraic expressions are equivalent: 5(x + 2) and 10 + 5x. 5(x + 2) = 5x + 10
= 10 + 5x
Distributive Property Commutative Property
5(x + 2) is equal to 10 + 5x. They are equivalent expressions.
EXERCISES Use exponents to write each expression. (Lesson 10.1) 1. 3.6 × 3.6
45
When x = 5, 2(x2 – 9) = 32.
EXAMPLE 3 Simplify each expression.
23 = 8
29
ESSENTIAL QUESTION
The prime factorization of 60 is 22 × 3 × 5.
= 4 × (8 + 5)
8. 29
A. 2(x2 - 9); x = 5
60 = 22 × 3 × 5
A. 4 × (23 + 5)
6.
EXAMPLE 1 Evaluate each expression for the given value of the variable.
60 = 2 × 2 × 3 × 5
2 60 2 30 3 15 5 5 1
52 × 3
11. 2 × 52 – (4 + 1)
B. 180
169
10. Eduardo is building a sandbox that has an area of 84 square feet. What are the possible whole number measurements for the length and width of the sandbox? (Lesson 10.2) 1, 84; 2, 42; 3, 28; 4,
order of operations (orden de las operaciones)
How can you generate equivalent numerical expressions and use them to solve real-world problems?
5. 132
Write the prime factorization of each number. (Lesson 10.2)
Key Vocabulary
ESSENTIAL QUESTION
A. 0.92
1
4. 120
© Houghton Mifflin Harcourt Publishing Company
UNIT 4
DO NOT EDIT--Changes must be made through “File info” CorrectionKey=B
(_45 )
3
94
3. _45 × _45 × _54
EXERCISES Write each phrase as an algebraic expression. (Lesson 11.1) 1. x subtracted from 15
Unit 4
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InCopy Notes
InDesign Notes
1. This is a list
1. This is a list
408
15 - x
2. 12 divided by t
12 __ t
Unit 4
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InCopy Notes 1. This is a list Bold, Italic, Strickthrough.
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InDesign Notes 1. This is a list
Expressions, Equations, and Relationships
408
MODULE 12 Equations and Relationships 6.7.B, 6.9.A, 6.9.B, 6.9.C, 6.10.A, 6.10.B
Key Concepts • An equation is a mathematical statement that two expressions are equal, which, if it includes a variable, has a solution. (Lesson 12.1) • Both sides of an equation remain equal after adding, or subtracting, the same number from both sides. (Lesson 12.2) • Both sides of an equation remain equal after multiplying, or dividing, both sides by the same number. (Lesson 12.3)
409
Unit 4
DO NOT EDIT--Changes must be made through “File info” CorrectionKey=B
DO NOT EDIT--Changes must be made through "File info" CorrectionKey=A
EXAMPLE 2
Write a phrase for each algebraic expression. (Lesson 11.1)
4. s + 7
67
7. s - 5t + s2; s = 4, t = -1
Add 12 to both sides. ? Check: 22 - 12 = 10 Substitute.
33
(Lesson 12.1)
40 in2
1. 7x = 14; x = 3
13. 7x + 4(2x - 6)
© Houghton Mifflin Harcourt Publishing Company
MODULE MODULE
?
12
yes
2. y + 13 = -4; y = -17
3. Don has three times as much money as his brother,
equivalent
d __ = 25 3
who has \$25.
4. There are s students enrolled in Mr. Rodriguez’s class. There are 6 students absent and 18 students present
m2 - 2m - 5
today.
15x - 24
s - 6 = 18
Equations and Relationships
Key Vocabulary equation (ecuación) solution (solución)
ESSENTIAL QUESTION
12 is not a solution of r - 5 = 17.
-42 is a solution of _6x = -7.
InDesign Notes
1. This is a list
1. This is a list
q = 2.1
8. 3.5 + x = 7
x = 49
10. _27 = 2x
x = 3.5
x = _17
x – 12.50 = 34.25; \$46.75
equation to solve the problem. (Lesson 12.3)
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InCopy Notes
7. 9q = 18.9
t = -48
12. Tom read 132 pages in 4 days. He read the same number of pages each day. How many pages did he read each day? Write and solve an
Unit 4
6_MTXESE051676_U4EM.indd 409
6. _4t = -12
the problem. (Lesson 12.2)
B. _6x = -7; x = -42 -42 ? ____ = -7 Substitute. 6 -7 = -7
p = 23
11. Sonia used \$12.50 to buy a new journal. She has \$34.25 left in her savings account. How much money did Sonia have before she bought the journal? Write and solve an equation to solve
EXAMPLE 1 Determine if the given value is a solution of the equation.
7 ≠ 17
5. p - 5 = 18
9. 18 = x - 31
How can you use equations and relationships to solve real-world problems?
A. r - 5 = 17; r = 12 ? 12 - 5 = 17 Substitute.
no
Write an equation to represent the situation. (Lesson 12.1)
not equivalent
Combine like terms. (Lesson 11.3) 12. 3m - 6 + m2 - 5m + 1
-30 = -30
EXERCISES Determine whether the given value is a solution of the equation.
Determine if the expressions are equivalent. (Lesson 11.3)
11. 2.5(3 + x); 2.5x + 7.5
Divide both sides by 5. ? Check: 5(-6) = -30 Substitute.
10 = 10
-34
8. x - y3; x = -7, y = 3
9. The expression _12 (h)(b1 + b2) gives the area of a trapezoid, with b1 and b2 representing the two base lengths of a trapezoid and h representing the height. Find the area of a trapezoid with base lengths 4 in. and 6 in. and a height of 8 in. (Lesson 11.2)
10. 7 + 7x; 7(x + _17 )
p = -6
y = 22
6. 3(7 + x2); x = 2
25
5p -30 __ = ____ 5 5
+12 = +12
Evaluate each expression for the given value of the variable. (Lesson 11.2) 5. 8z + 3; z = 8
B. 5p = -30
A. y - 12 = 10
the sum of s and 7
410
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3. 8p
the product of 8 and p
4p = 132; 33 pages
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MODULE 13 Inequalities and Relationships 6.9.A, 6.9.B, 6.9.C, 6.10.A, 6.10.B
Key Concepts • An inequality is a mathematical statement that uses one of the following inequality symbols: greater than, >, less than, <, greater than or equal to, ≥, or less than or equal to, ≤. (Lesson 13.1) • All inequalities have many solutions. (Lesson 13.2) • Reverse the inequality symbol when multiplying or dividing both sides of an inequality by a negative number. (Lesson 13.4)
MODULE 14 Relationships in Two Variables 6.6.A, 6.6.B, 6.6.C, 6.11
Key Concepts • An ordered pair is a pair of numbers in the form (x, y) that gives the location of a point on a coordinate plane. (Lesson 14.1) • The quantity that depends on the other quantity is called the dependent variable, and the quantity it depends on is called the independent variable. (Lesson 14.2) • Tables and graphs can be used to represent the relationship between an independent and dependent variable. (Lesson 14.4)
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13
Inequalities and Relationships
9.
Key Vocabulary solution of an inequality (solución de una desigualdad)
ESSENTIAL QUESTION
Sample answer: Juan’s dog lost 3 pounds and still weighs at least 11 pounds.
How can you use inequalities and relationships to solve real-world problems?
10. Omar wants a rectangular vegetable garden. He only has enough space to make the garden 5 feet wide, and he wants the area of the garden to be more than 80 square feet. Write and solve an inequality to find the possible lengths of the garden. (Lesson 13.3)
EXAMPLE 1 Write and graph an inequality to represent each situation. A. There are at least 5 gallons of water in an aquarium.
5ℓ > 80, ℓ > 16
B. The temperature today will be less than 35 °F.
g≥5
MODULE
t < 35
?
30 31 32 33 34 35 36 37 38 39 40
0 1 2 3 4 5 6 7 8 9 10
Solve each inequality. Graph and check your solutions. A. x - 7 ≤ 2
B. -5y < -15 y>3
Divide by -5. Reverse the symbol.
© Houghton Mifflin Harcourt Publishing Company
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
s < 2.5
2. Tina got a haircut, and her hair is still at least 15 inches long.
h ≥ 15
5. 9q > 10.8 7. -_45 x < 8
q > 15
4. _4t ≤ -1
t ≤ -4
q > 1.2
6. 87 ≤ 25 + x
x ≥ 62
x > -10
8. -4 ≥ -0.5x
x≥8
(4, -2) is in quadrant IV.
EXAMPLE 2 Tim is paid \$8 more than the number of bags of peanuts he sells at the baseball stadium. The table shows the relationship between the money Tim earns and the number of bags of peanuts Tim sells. Identify the independent and dependent variables, and write an equation that represents the relationship.
10 11 12 13 14 15 16 17 18 19 20
Solve each inequality. Graph and check your solutions. (Lessons 13.2, 13.3, 13.4) 3. q - 12 > 3
̵5 ̵ 4 ̵3 ̵2 ̵1 O 1 2 3 4 5 ̵1 ̵2 (4, ̵2) ̵3 Quadrant III Quadrant IV ̵4 ̵5
(Lesson 13.1)
\$2.50 per share.
axes (ejes)
(4, -2) is located 4 units to the right of the origin and 2 units down from the origin.
y 5 4 Quadrant II 3 2 1
Key Vocabulary coordinate plane (plano cartesiano)
EXAMPLE 1 Graph the point (4, -2) and identify the quadrant where it is located.
EXERCISES Write and graph an inequality to represent each situation.
1. Orange Tech’s stock is worth less than
Relationships in Two Variables
How can you use relationships in two variables to solve real-world problems?
0 1 2 3 4 5 6 7 8 9 10
5 6 7 8 9 10 11 12 13 14 15
14
ESSENTIAL QUESTION
EXAMPLE 2
x≤9
Write a real-world comparison that can be described by x - 3 ≥ 11. (Lesson 13.2)
# of bags of peanuts, x
0
1
2
Money earned, y
8
9
10 11
3
© Houghton Mifflin Harcourt Publishing Company
MODULE
The number of bags is the independent variable, and the money Tim earns is the dependent variable. The equation y = x + 8 expresses the relationship between the number of bags Tim sells and the amount he earns.
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Unit 4 Performance Tasks The Performance Tasks provide students with the opportunity to apply concepts from this unit in real-world problem situations.
CAREERS IN MATH For more information about careers in mathematics as well as various mathematics appreciation topics, visit the American Mathematical Society at www.ams.org
CAREERS IN MATH Botanist In Performance Task Item 1, students can see how a botanist uses mathematics on the job.
Possible Points (Total: 6)
a
1 point for correctly writing the expression 205 + 2d.
b
1 point for correctly setting the expression 205 + 2d = 235. 1 point for correct answer: 15 days
c
1 point for correct expression for Suntracker: 195 +2.5d. 1 point for determining the heights of each sunflower variety after 22 days: Suntracker: h = 195 + 2.5(22) = 250 Sunny Yellow: h = 205 + 2(22) = 249 1 point for stating that Suntracker is taller.
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6.1.A, 6.1.F
6.1.A
Possible Points (Total: 6)
a
1 point for correctly defining a variable: Let w = the number of hours Vernon practiced soccer over the weekend. 1 point for the correct equation: 4__13 + w = 5__34 .
b
1 point for correctly finding the LCM, 12, and showing how to find it. Students can use a number line, a list of multiples, or prime factorization to find the LCM. 4 = 2 × 2; 3 = 3; LCM = 2 × 2 × 3 = 12
c
5 1 point for correctly solving the equation: w = 1__ . 12 1 point for showing stepped-out solution to equation. 1 point for correctly interpreting the equation in terms of the problem: Vernon 5 practiced 1__ hours over the weekend. 12
DO NOT EDIT--Changes must be made through "File info" CorrectionKey=A
DO NOT EDIT--Changes must be made through “File info” CorrectionKey=B
EXERCISES Graph and label each point on the coordinate plane. (Lesson 14.1)
1.
y
1. (4, 4) 2. (-3, -1)
(-1, 4)
4
(4, 4)
2
3. (-1, 4)
x ̵4
̵2
(-3, -1)
O
2
CAREERS IN MATH Botanist Dr. Adama is a botanist. She measures the daily height of a particular variety of sunflower, Sunny Yellow, beginning when the sunflower is 60 days old. At 60 days, the height of the sunflower is 205 centimeters. Dr. Adama finds that the growth rate of this sunflower is 2 centimeters per day after the first 60 days.
a. Write an expression to represent the sunflower’s height d days after the 60th day.
4
̵2
205 + 2d
̵4
b. How many days after the 60th day does it take for the sunflower to reach 235 centimeters? Show your work.
235 = 205 + 2d; 30 = 2d; 15 = d
Use the graph to answer the questions. (Lesson 14.2)
It takes 15 days for the sunflower to reach 235 centimeters. c. Dr. Adama is studying a different variety of sunflower, Suntracker, which grows at a rate of 2.5 centimeters per day after the first 60 days. If this sunflower is 195 centimeters tall when it is 60 days old, write an expression to represent Suntracker’s height d days after the 60th day. Which sunflower will be taller 22 days after the 60th day? Explain how you found your answer.
8 6 4
Suntracker: 195 + 2.5d; after 22 days: Suntracker: h = 195 + 2.5(22)
2 O
2
4
6
8
Time (h)
= 250; Sunny Yellow: h = 205 + 2(22) = 249. Suntracker is taller.
10
2. Vernon practiced soccer 5_43 hours this week. He practiced 4_13 hours on weekdays and the rest over the weekend.
time
4. What is the independent variable?
a. Write an equation that represents the situation. Define your variable.
distance
5. What is the dependent variable?
Let w = the number of hours Vernon practiced soccer over the weekend; 4_13 + w = 5_34
© Houghton Mifflin Harcourt Publishing Company
6. Describe the relationship between the independent variable and the dependent variable.
b. What is the least common multiple of the denominators of 5_34 and 4_13 ? Show your work.
The dependent variable is 3 times the independent variable.
Using prime factorization: 4 = 2 × 2; 3 = 3; LCM = (2)(2)(3) = 12
7. Use the data on the table to write an equation to express y in terms of x. Then graph the equation. (Lessons 14.3, 14.4) x y
0
1
-2 -1
2
3
0
1
y= x-2
c. Solve the equation and interpret the solution. Show your work.
5 9 9 4 4 4_13 + w = 5_34; 4__ + w = 5__ ; w = 5__ - 4__ = 1__ 12 12 12 12 12
y 4 2
(2, 0)
̵4
̵2
O ̵2 ̵4
2
5 hours over the weekend. Vernon practiced 1__ 12
(3, 1)
© Houghton Mifflin Harcourt Publishing Company
Distance (km)
10
x
4
(1, -1) (0, -2) Unit 4
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Additional Resources Personal Math Trainer my.hrw.com
Online Assessment and Intervention
Assessment Resources • Leveled Unit Tests: A, B, C, D • Performance Assessment
MIXED REVIEW
Texas Test Prep Texas Testing Tip Students should always read each question carefully to identify key words or phrases, such as no more than, to help identify what the question is really asking. Item 5 Students should underline the phrase no more than; it will help them to realize that n cannot be larger than 7, but it can be equal to 7. This will help them choose the correct inequality. Item 7 Students who do not read the problem carefully may choose B because they see the word times. But reading carefully they will see the phrase “3 more times than,” which will reveal choice D as the correct answer.
Avoid Common Errors Item 8 Some students may fail to pay attention to the direction the graph is pointing. As a result, they may choose answer choice A or D, because they both include the number 4. 4 in the solution. Remind the students that the direction of the inequality is as important as its endpoint. Item 13 Some students may give 2 as their answer because they transposed the x- and y-coordinates. Remind students that the y-coordinate is the second number in an ordered pair.
Texas Essential Knowledge and Skills Items
Mathematical Process TEKS
1
6.7.A
6.1.D
2
6.7.A
6.1.F
3
6.7.C
6.1.A
4
6.9.B, 6.10.A
6.1.D
5
6.9.A
6.1.A, 6.1.F
6
6.10.B
6.1.F
7
6.9.A
6.1.A, 6.1.F
8
6.9.B, 6.10.A
6.1.D
9
6.6.A
6.1.A
10*
6.4.C, 6.4.E
6.1.F
11*
6.3.E
6.1.F
12
6.10.A
6.1.A
13
6.11
6.1.E
14
6.7.A, 6.7.C
6.1.A
* Item integrates mixed review concepts from previous modules or a previous course.
415
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Personal Math Trainer
Texas Test Prep Selected Response 1. Which expression is equivalent to 2.3 × 2.3 × 2.3 × 2.3 × 2.3? A 2.3 × 5
2. Which operation should you perform first when you simplify 63 – (2 + 54 × 6) ÷ 5? A addition
3. Sheena was organizing items in a scrapbook. She took 25 photos and divided them evenly among p pages. Which algebraic expression represents the number of photos on each page? A p – 25
© Houghton Mifflin Harcourt Publishing Company
B 25 – p p __ C 25 25 D __ p
D 4 – 6 = –2
6. For which of the inequalities below is v = 4 a solution? A v+5≥9
̵4
5
6 C __ 15 18 __ D 45
A 1.8 centimeters B 11.4 centimeters C 13.7 centimeters D 114 centimeters
A j–9=3
Gridded Response
C j–3=9
12. The area of a rectangular mural is 84 square feet. The mural’s width is 7 feet. What is its length in feet?
B 3j = 9
5
10
m A __ > 1.1 4 m B __ < 1.2 3 C 2m < 8.8 D 5m > 22
Hot ! Tip
When possible, use logic to eliminate at least two answer choices.
Unit 4
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O ̵2
11. One inch is 2.54 centimeters. About how many centimeters is 4.5 inches?
7. Sarah has read aloud in class 3 more times than Joel. Sarah has read 9 times. Which equation represents this situation?
0
̵2
D the number of stars available
2 _ A 5 __ B 12 25
D j+3=9
0
x ̵4
C the number of hours played
8. The number line below represents the solution to which inequality?
4. The number line below represents which equation?
G 2
10. Which ratio is not equivalent to the other three?
D v+5<8
D subtraction
C 4 + 6 = –2
A n<7
C v+5≤8
C multiplication
y 4
B the number of points earned
B v+5>9
B division
13. What is the y-coordinate of point G on the coordinate grid below?
A the number of stars picked up
D n≥7
D 2.35
B –2 – 6 = 4
5. No more than 7 copies of a newspaper are left in the newspaper rack. Which inequality represents this situation?
C n>7
C 25 × 35
A –2 + 6 = 4
9. Brian is playing a video game. He earns the same number of points for each star he picks up. He earned 2,400 points for 6 stars, 4,000 points for 10 stars, and 5,200 points for 13 stars. Which is the independent variable in the situation?
Online Assessment and Intervention
B n≤7
B 235
-5
my.hrw.com
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Gridded responses cannot be negative numbers. If you get a negative value, you likely made an error. Check your work!
Hot ! Tip
14. When traveling in Canada, Patricia converts the temperature given in degrees Celsius to a Fahrenheit temperature by using the expression 9x ÷ 5 + 32, where x is the Celsius temperature. Find the temperature in degrees Fahrenheit when it is 25 °C.
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UNIT 4 Expressions, Equations, and Relationships Contents 6.7.A 6.7.A 6.7.A 6.7.C 6.7.A 6.7.D 6.9.A 6.10.A 6.10.A 6.9.A 6.10.A 6.10.A 6.9.B MODUL...
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I have mixed feelings about the recent changes to this article. It looks to me that the text is unnecessarily complicated, and the part contained in the ==Definition== section is no definition at all, rather some generalities, which I am having trouble following. Then, the way the article is now, one starts with the euclidean space, only to go to the topological space, then followed by a metric space, then back to the real numbers.
Is there any way to integrate the recently inserted material? Oleg Alexandrov 02:44, 28 July 2005 (UTC)
## Punctured Neighbourhood
I would like to add Puncture Neighbourhood, but I just fail. It is the same as a normal neighbourhood, only it does not actually contain p.
## "wiggle" or "move"
eek! Sorry that's just too much badly defined for my tast. I believe that a simple sketch would explain this much butter, something like (--p--). (I hope you understand this) Possibly have a group of sketches showing what is and what is not a neighbourhood.
## 'wiggle' or 'move' 2
I second that.
This "wiggle" or "move" definition is pretty shit.
It's necessary a great deal of competence in making abstract mathematical concepts intuitive. In some cases it can rather make it even harder to comprehend. ?The preceding unsigned comment was added by 200.164.220.194 (talk) 23:33, 6 March 2007 (UTC).
I changed "wiggle" to "perturb". It sounds more formal, but I am not sure it is better... Best, Sam nead 17:31, 29 May 2007 (UTC)
## Which comes first: neighborhood of a point or of a set?
Oleg -- I looked through some books. The order is given below is the order with which I looked at them. "Counterexamples in topology" defines the neighborhood of a set and takes a point neighborhood as a special case. "General topology" by Kelly defines the neighborhood of a point and doesn't seem to mention the neighborhood of a set. "Topology, a first course" by Munkres defines neighborhoods of points and insists that neighborhoods be open. Spanier doesn't seem to mention neighborhoods at all. "Topology and geometry" by Bredon defines the neighborhood of a point. "Set theory and metric spaces" by Kaplansky defines the neighborhood of a point. Those are all of the reference that spring off the shelves.
So, if we have a bibliocracy, then we should remove the definition of the neighborhood of a set. Hmmm. I have to admit my preference for the way you suggested -- that we define the neighborhood of a set first and then take the neighborhood of a point as a special case. This way, we still present the "neighborhood of a point" idea, and retain the generality. I prefer this because it fits correctly with the notions of tubular neighborhood, regular neighborhood, and absolute neighborhood retract (ANR) in geometry and topology. I'll make the edits in a few minutes. best, Sam nead 16:55, 29 May 2007 (UTC)
My preference is actually for neighborhood of a point to be first, as that's the most fundamental definition, and your references seem to agree. The neighborhood of a set would then be an extension of this concept. Any strong feelings about this? :) Oleg Alexandrov (talk) 01:50, 30 May 2007 (UTC)
I think I have a definite preference for the way it is written, as of today. Let me try to justify this preference. 1) This way is slick. We get two definitions for the price of one. Now, "bogus generality" is something to be avoided, but this is not bogus generality because 2) the idea of a tubular or regular neighborhood of a submanifold is as important, if not more important, as the idea of the neighborhood of a point. At least, that is the case in geometry and topology: for example cut-and-paste arguments rely on tubular neighborhoods. Also, if you want to glue two manifolds-with-boundary in the smooth category, you have to build collar neighborhoods of the boundaries. (Hmm. Even then, the way you perform the gluing effects the smooth structure you get -- this is a step in Milnor's construction of exotic seven-spheres.)
Back on topic: 3) The references I gave above which didn't define the neighborhood of a set were textbooks. The reference which did define the neighborhood of a set was Counterexamples. That is not a textbook but rather is a reference book. As an encyclopedia, I think we are bound to resemble the latter not the former. Best, Sam nead 05:34, 30 May 2007 (UTC)
I find the "textbooks" vs "references books" argument unconvincing. Wikipedia articles should be as elementary as possible, so that people can learn things, see Wikipedia:Make technical articles accessible. So we should imitate textbooks, not reference books.
Also, defining the neighborhood of a point first does not interfere at all with tubular neighbourhoods and all that stuff, since we immediately define the neighbourhood of a set too.
Lastly, in analysis the neighbourhood of a point is used all the time, while neighbourhood of a set are used much more rarely. Oleg Alexandrov (talk) 15:14, 30 May 2007 (UTC)
I agree that articles should be accessible, but I do not agree that Wikipedia is more like a text book. An encyclopedia is a reference, almost by definition! Also, I won't be convinced by your last point without examples. Finally, may I suggest that you look through the book "Counterexamples in topology"? I think you will find it interesting. best, Sam nead 23:30, 31 May 2007 (UTC)
I strongly agree with Oleg here. Wikipedia is aimed at a general audience. Introductions should be intelligible to someone who does not already know the subject, to the extent possible. This means avoiding unnecessary generality. Also, for what its worth, the Encyclopedic Dictionary of Mathematics (MIT Press) defines neighborhoods in terms of a point.--agr 02:41, 1 June 2007 (UTC)
Agree with Oleg. Define it for points first, then sets (possibly even separately from the main definition). I think you'll run into problems, for instance, axiomatizing the space if all of your neighborhoods are set-based. (I'm not saying it's impossible, but I just don't think you can carry it off quite as smoothly.) Notions such as a neighborhood basis (etc.) are conventionally thought of in terms of points. Also, for what it's worth, even Bourbaki defines neighborhoods in terms of points. Cheers, Silly rabbit 02:54, 1 June 2007 (UTC)
I'll add another voice of agreement with Oleg. The neighbourhood of a point is far and away the more common usage in my experience, and it is easy enough to generalise this up to neighbourhoods of sets as needed. Given the nature of Wikipedia it seems sensible to start with the more common usage first. -- Leland McInnes 03:05, 1 June 2007 (UTC)
Gah! Where did all of you people come from? Ok, the point I find persusive here is the invocation of the MIT Encyclopedia and of Bourbaki. However, I don't buy this lowest common denominator business: an encyclopedia is a reference book, not a textbook. If a page of Wikipedia is too hard to read, then there should be obvious links to the necessary background. It seems impossible for every page to be free of prerequisites. Although I suppose that neighborhoods are pretty basic!
Next, that's two people who have said that point neighborhoods are important in analysis. I'd like some examples, please. My preference would be for examples not taken from first year calculus, by the way. Here -- I'll start: if f(z) is a function in the plane, analytic in a neighborhood of the point x, then f(z) agrees with its power series expansion (computed at x) inside of a small disk about x. Furthermore the power series converges absolutely inside the disk. All bets are off if f is not analytic in a neighborhood about x. (Ok, so I didn't manage to get out of the undergraduate syllabus. But at least that is a pretty result which mentions the concept.) Best, Sam nead 05:50, 1 June 2007 (UTC)
I agree with Oleg, an encyclopedia is indeed a reference, but it is generally accepted that the articles should build from the most basic and generally used case up to the more abstract and difficult concepts, I have heard the analogy of a pyramid, and I think that works pretty well. We are trying to inform the general reader first. This doesn't mean making the entire article LCD, just the beginning so it is accessible. --Cronholm144 06:39, 1 June 2007 (UTC)
Yes Wikipedia is a reference, but we are also admonished to make technical articles accessible, and in this case that means it will be more helpful to start with the more common and more easily intuitive case first. I'm not suggesting neighbourhoods of sets shouldn't be mentioned, merely that discussion of that can come afterwards. -- Leland McInnes 13:17, 1 June 2007 (UTC)
One more voice of agreement with Oleg. The neighborhood of a point is the first approach taken in every topology class I've ever been a part of. I will concede that there are textbooks which do not do this, but these are the exception, not the rule. Keep in mind, as well, that when a textbook chooses to discuss neighborhoods first—before the axioms for a topology are given—then an open set can be defined as a set that is a neighborhood for each of its points. While Wikipedia is not a textbook, I think it is proper to take the best pedagogy from textbooks to apply to our pages here for the benefit of our general readership. VectorPosse 07:22, 1 June 2007 (UTC)
Agreed completely. Wikipedia should not confuse unnecessarily. There is no such thing as neighborhood of a set in a Calculus course. Jmath666 20:27, 5 June 2007 (UTC)
## Pictures
Oleg -- Those pictures are very cool! Thank you for adding them -- they really help the article! Question: how did you draw them? Request: could you add a similar picture to the tubular neighborhood page? This could be something as simple as a graph of a function in the plane, with a (green) shaded region around it. (I like green!) Best, Sam nead 05:39, 30 May 2007 (UTC)
I used Inkscape, it is free on both Windows and Linux. I could draw the picture you want, but perhaps you want to give it a try to take the picture I made as a base and modify it to do what you want, that way you may learn something new. :) Oleg Alexandrov (talk) 05:42, 30 May 2007 (UTC)
Fair enough. I'll take a look. Best, Sam nead 15:04, 30 May 2007 (UTC)
I think that the section "Significance of neighbourhoods in analysis of real functions", especially the sentence "Other notions of distance will (as they ought to) lead to the same results in analysis, if they are properly formulated." is flatly wrong. The example I am most familiar with is the quasi-conformal distortion of a continuous map. This, by definition, depends on the metrics on both the domain and range.
Perhaps the writer was thinking of the continuity of a map -- it is certainly true that metrics giving the same topology give the same notion of continuity. But this fails for more refined analytical concepts. (Other possible examples: BMO, exotic differentiable structures). If nobody objects I will trim the section, by quite a bit. Hmm. Advise me: am I expected to find out the identity of the writer and inform them of my intentions? Or should I just "Be bold"? Best, Sam nead 17:31, 29 May 2007 (UTC)
Ok, I've been bold. I thought about it, and I think I know what the writer of the "Significance" section was after. The point is that different metrics on a set may (or may not) define the same topology. When they do define the same topology then the metrics are called equivalent. Obviously, equivalent metrics give identical notions of continuity. However, as noted above, they need not have identical analytic properties.
I've deleted the entire discussion and I will check and make sure that the notion of equivalent metrics appears someplace more appropriate. Best, Sam nead 17:52, 29 May 2007 (UTC)
## Regular neighborhood
The page Dehn twist refers to a 'regular neighborhood', which is also in the 'see also' sec'n of this article, but I cannot find a reference. Is it the same as 'Uniform neighborhood'? If not, can we add a section to this article explaining what 'regular neighborhood' is and delete the 'see also' (unless reg. neighborhood is a vast and interesting topic deserving of its own article). Thanks. Zero sharp 15:07, 19 July 2007 (UTC)
I think in this case it means symmetric around c.--Cronholm144 15:36, 19 July 2007 (UTC)
Regular neighborhood is indeed a big topic needing its own article. There are even "relative regular neighborhoods", which should be considered a "power steering" version of the vanilla regular neighborhood. To explain regular neighborhood, one would first need to specify a category (such as PL or DIFF) and then explain something like collapsing or whatnot. On the other hand, all that is meant in the Dehn twist article is taking a closed annulus containing the curve as its core. --C S 16:43, 13 September 2007 (UTC) | 3,156 | 13,639 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.609375 | 3 | CC-MAIN-2013-48 | latest | en | 0.928585 |
https://wikimili.com/en/Orthogonal_group | 1,627,725,016,000,000,000 | text/html | crawl-data/CC-MAIN-2021-31/segments/1627046154085.58/warc/CC-MAIN-20210731074335-20210731104335-00428.warc.gz | 607,364,271 | 58,360 | # Orthogonal group
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In mathematics, the orthogonal group in dimension n, denoted O(n), is the group of distance-preserving transformations of a Euclidean space of dimension n that preserve a fixed point, where the group operation is given by composing transformations. The orthogonal group is sometimes called the general orthogonal group, by analogy with the general linear group. Equivalently, it is the group of n×n orthogonal matrices, where the group operation is given by matrix multiplication (an orthogonal matrix is a real matrix whose inverse equals its transpose). The orthogonal group is an algebraic group and a Lie group. It is compact.
## Contents
The orthogonal group in dimension n has two connected components. The one that contains the identity element is a subgroup, called the special orthogonal group, and denoted SO(n). It consists of all orthogonal matrices of determinant 1. This group is also called the rotation group, generalizing the fact that in dimensions 2 and 3, its elements are the usual rotations around a point (in dimension 2) or a line (in dimension 3). In low dimension, these groups have been widely studied, see , and . In the other connected component all orthogonal matrices have –1 as a determinant.
By extension, for any field F, a n×n matrix with entries in F such that its inverse equals its transpose is called an orthogonal matrix overF. The n×n orthogonal matrices form a subgroup, denoted O(n, F), of the general linear group GL(n, F); that is
${\displaystyle \operatorname {O} (n,F)=\left\{Q\in \operatorname {GL} (n,F)\mid Q^{\mathsf {T}}Q=QQ^{\mathsf {T}}=I\right\}.}$
More generally, given a non-degenerate symmetric bilinear form or quadratic form [1] on a vector space over a field, the orthogonal group of the form is the group of invertible linear maps that preserve the form. The preceding orthogonal groups are the special case where, on some basis, the bilinear form is the dot product, or, equivalently, the quadratic form is the sum of the square of the coordinates.
All orthogonal groups are algebraic groups, since the condition of preserving a form can be expressed as an equality of matrices.
## Name
The name of "orthogonal group" originates from the following characterization of its elements. Given a Euclidean vector space E of dimension n, the elements of the orthogonal group O(n) are, up to a uniform scaling (homothecy), the linear maps from E to E that map orthogonal vectors to orthogonal vectors.
## In Euclidean geometry
The orthogonal group O(n) is the subgroup of the general linear group GL(n, R), consisting of all endomorphisms that preserve the Euclidean norm, that is endomorphisms g such that ${\displaystyle \|g(x)\|=\|x\|.}$
Let E(n) be the group of the Euclidean isometries of a Euclidean space S of dimension n. This group does not depend on the choice of a particular space, since all Euclidean spaces of the same dimension are isomorphic. The stabilizer subgroup of a point xS is the subgroup of the elements g ∈ E(n) such that g(x) = x. This stabilizer is (or, more exactly, is isomorphic to) O(n), since the choice of a point as an origin induces an isomorphism between the Euclidean space and its associated Euclidean vector space.
There is a natural group homomorphism p from E(n) to O(n), which is defined by
${\displaystyle p(g)\cdot (y-x)=g(y)-g(x),}$
where, as usual, the subtraction of two points denotes the translation vector that maps the second point to the first one. This is a well defined homomorphism, since a straightforward verification shows that, if two pairs of points have the same difference, the same is true for their images by g (for details, see Affine space § Subtraction and Weyl's axioms).
The kernel of p is the vector space of the translations. So, the translation form a normal subgroup of E(n), the stabilizers of two points are conjugate under the action of the translations, and all stabilizers are isomorphic to O(n).
Moreover, the Euclidean group is a semidirect product of O(n) and the group of translations. It follows that the study of the Euclidean group is essentially reduced to the study of O(n).
### SO(n)
By choosing an orthonormal basis of a Euclidean vector space, the orthogonal group can be identified with the group (under matrix multiplication) of orthogonal matrices, which are the matrices such that
${\displaystyle QQ^{\mathsf {T}}=I.}$
It follows from this equation that the square of the determinant of Q equals 1, and thus the determinant of Q is either 1 or –1. The orthogonal matrices with determinant 1 form a subgroup called the special orthogonal group, denoted SO(n), consisting of all direct isometries of O(n), which are those that preserve the orientation of the space.
SO(n) is a normal subgroup of O(n), as being the kernel of the determinant, which is a group homomorphism whose image is the multiplicative group {–1, +1}. Moreover, the orthogonal group is a semidirect product of SO(n) and the group with two elements, since, given any reflection r, one has O(n) \ SO(n) = r SO(n).
The group with two elements I} (where I is the identity matrix) is a normal subgroup and even a characteristic subgroup of O(n), and, if n is even, also of SO(n). If n is odd, O(n) is the internal direct product of SO(n) and I}. For every positive integer k the cyclic group Ck of k-fold rotations is a normal subgroup of O(2) and SO(2).
### Canonical form
For any element of O(n) there is an orthogonal basis, where its matrix has the form
${\displaystyle {\begin{bmatrix}{\begin{matrix}R_{1}&&\\&\ddots &\\&&R_{k}\end{matrix}}&0\\0&{\begin{matrix}\pm 1&&\\&\ddots &\\&&\pm 1\end{matrix}}\\\end{bmatrix}},}$
where the matrices R1, ..., Rk are 2-by-2 rotation matrices, that is matrices of the form
${\displaystyle {\begin{bmatrix}a&b\\-b&a\end{bmatrix}},}$
with ${\displaystyle a^{2}+b^{2}=1.}$
This results from the spectral theorem by regrouping eigenvalues that are complex conjugate, and taking into account that the absolute values of the eigenvalues of an orthogonal matrix are all equal to 1.
The element belongs to SO(n) if and only if there are an even number of –1 on the diagonal.
The special case of n = 3 is known as Euler's rotation theorem, which asserts that every (non-identity) element of SO(3) is a rotation about a unique axis-angle pair.
### Reflections
Reflections are the elements of O(n) whose canonical form is
${\displaystyle {\begin{bmatrix}-1&0\\0&I\end{bmatrix}},}$
where I is the (n–1)×(n–1) identity matrix, and the zeros denote row or column zero matrices. In other words, a reflection is a transformation that transforms the space in its mirror image with respect to a hyperplane.
In dimension two, every rotation is the product of two reflections. More precisely, a rotation of angle 𝜃 is the product of two reflections whose axes have an angle of 𝜃 / 2.
Every element of O(n) is the product of at most n reflections. This results immediately from the above canonical form and the case of dimension two.
The Cartan–Dieudonné theorem is the generalization of this result to the orthogonal group of a nondegenerate quadratic form over a field of characteristic different from two.
The reflection through the origin (the map v ↦ −v) is an example of an element of O(n) that is not the product of fewer than n reflections.
### Symmetry group of spheres
The orthogonal group O(n) is the symmetry group of the (n − 1)-sphere (for n = 3, this is just the sphere) and all objects with spherical symmetry, if the origin is chosen at the center.
The symmetry group of a circle is O(2). The orientation-preserving subgroup SO(2) is isomorphic (as a real Lie group) to the circle group, also known as U(1), the multiplicative group of the complex numbers of absolute value equal to one. This isomorphism sends the complex number exp(φi) = cos(φ) + i sin(φ) of absolute value 1 to the special orthogonal matrix
${\displaystyle {\begin{bmatrix}\cos(\varphi )&-\sin(\varphi )\\\sin(\varphi )&\cos(\varphi )\end{bmatrix}}.}$
In higher dimension, O(n) has a more complicated structure (in particular, it is no longer commutative). The topological structures of the n-sphere and O(n) are strongly correlated, and this correlation is widely used for studying both topological spaces.
## Group structure
The groups O(n) and SO(n) are real compact Lie groups of dimension n(n − 1)/2. The group O(n) has two connected components, with SO(n) being the identity component, that is, the connected component containing the identity matrix.
### As algebraic groups
The orthogonal group O(n) can be identified with the group of the matrices A such that ${\displaystyle A^{\mathsf {T}}A=I.}$ Since both members of this equation are symmetric matrices, this provides ${\displaystyle \textstyle {\frac {n(n+1)}{2}}}$ equations that the entries of an orthogonal matrix must satisfy, and which are not all satisfied by the entries of any non-orthogonal matrix.
This proves that O(n) is an algebraic set. Moreover, it can be proved that its dimension is
${\displaystyle {\frac {n(n-1)}{2}}=n^{2}-{\frac {n(n+1)}{2}},}$
which implies that O(n) is a complete intersection. This implies that all its irreducible components have the same dimension, and that it has no embedded component. In fact, O(n) has two irreducible components, that are distinguished by the sign of the determinant (that is det(A) = 1 or det(A) = –1). Both are nonsingular algebraic varieties of the same dimension n(n – 1) / 2. The component with det(A) = 1 is SO(n).
### Maximal tori and Weyl groups
A maximal torus in a compact Lie group G is a maximal subgroup among those that are isomorphic to Tk for some k, where T = SO(2) is the standard one-dimensional torus. [2]
In O(2n) and SO(2n), for every maximal torus, there is a basis on which the torus consists of the block-diagonal matrices of the form
${\displaystyle {\begin{bmatrix}R_{1}&&0\\&\ddots &\\0&&R_{n}\end{bmatrix}},}$
where each Rj belongs to SO(2). In O(2n + 1) and SO(2n + 1), the maximal tori have the same form, bordered by a row and a column of zeros, and 1 on the diagonal.
The Weyl group of SO(2n + 1) is the semidirect product ${\displaystyle \{\pm 1\}^{n}\rtimes S_{n}}$ of a normal elementary abelian 2-subgroup and a symmetric group, where the nontrivial element of each {±1} factor of {±1}n acts on the corresponding circle factor of T × {1} by inversion, and the symmetric group Sn acts on both {±1}n and T × {1} by permuting factors. The elements of the Weyl group are represented by matrices in O(2n) × {±1}. The Sn factor is represented by block permutation matrices with 2-by-2 blocks, and a final 1 on the diagonal. The {±1}n component is represented by block-diagonal matrices with 2-by-2 blocks either
${\displaystyle {\begin{bmatrix}1&0\\0&1\end{bmatrix}}\quad {\text{or}}\quad {\begin{bmatrix}0&1\\1&0\end{bmatrix}},}$
with the last component ±1 chosen to make the determinant 1.
The Weyl group of SO(2n) is the subgroup ${\displaystyle H_{n-1}\rtimes S_{n}<\{\pm 1\}^{n}\rtimes S_{n}}$ of that of SO(2n + 1), where Hn−1 < {±1}n is the kernel of the product homomorphism {±1}n → {±1} given by ${\displaystyle \left(\epsilon _{1},\ldots ,\epsilon _{n}\right)\mapsto \epsilon _{1}\cdots \epsilon _{n}}$; that is, Hn−1 < {±1}n is the subgroup with an even number of minus signs. The Weyl group of SO(2n) is represented in SO(2n) by the preimages under the standard injection SO(2n) → SO(2n + 1) of the representatives for the Weyl group of SO(2n + 1). Those matrices with an odd number of ${\displaystyle {\begin{bmatrix}0&1\\1&0\end{bmatrix}}}$ blocks have no remaining final −1 coordinate to make their determinants positive, and hence cannot be represented in SO(2n).
## Topology
### Low-dimensional topology
The low-dimensional (real) orthogonal groups are familiar spaces:
### Fundamental group
In terms of algebraic topology, for n > 2 the fundamental group of SO(n, R) is cyclic of order 2, [4] and the spin group Spin(n) is its universal cover. For n = 2 the fundamental group is infinite cyclic and the universal cover corresponds to the real line (the group Spin(2) is the unique connected 2-fold cover).
### Homotopy groups
Generally, the homotopy groups πk(O) of the real orthogonal group are related to homotopy groups of spheres, and thus are in general hard to compute. However, one can compute the homotopy groups of the stable orthogonal group (aka the infinite orthogonal group), defined as the direct limit of the sequence of inclusions:
${\displaystyle \operatorname {O} (0)\subset \operatorname {O} (1)\subset \operatorname {O} (2)\subset \cdots \subset O=\bigcup _{k=0}^{\infty }\operatorname {O} (k)}$
Since the inclusions are all closed, hence cofibrations, this can also be interpreted as a union. On the other hand, is a homogeneous space for O(n + 1), and one has the following fiber bundle:
${\displaystyle \operatorname {O} (n)\to \operatorname {O} (n+1)\to S^{n},}$
which can be understood as "The orthogonal group O(n + 1) acts transitively on the unit sphere Sn, and the stabilizer of a point (thought of as a unit vector) is the orthogonal group of the perpendicular complement, which is an orthogonal group one dimension lower. Thus the natural inclusion O(n) → O(n + 1) is (n − 1)-connected, so the homotopy groups stabilize, and πk(O(n + 1)) = πk(O(n)) for n > k + 1: thus the homotopy groups of the stable space equal the lower homotopy groups of the unstable spaces.
From Bott periodicity we obtain Ω8OO, therefore the homotopy groups of O are 8-fold periodic, meaning πk + 8(O) = πk(O), and one need only to list the lower 8 homotopy groups:
{\displaystyle {\begin{aligned}\pi _{0}(O)&=\mathbf {Z} /2\mathbf {Z} \\\pi _{1}(O)&=\mathbf {Z} /2\mathbf {Z} \\\pi _{2}(O)&=0\\\pi _{3}(O)&=\mathbf {Z} \\\pi _{4}(O)&=0\\\pi _{5}(O)&=0\\\pi _{6}(O)&=0\\\pi _{7}(O)&=\mathbf {Z} \end{aligned}}}
#### Relation to KO-theory
Via the clutching construction, homotopy groups of the stable space O are identified with stable vector bundles on spheres (up to isomorphism), with a dimension shift of 1: πk(O) = πk + 1(BO). Setting KO = BO × Z = Ω−1O × Z (to make π0 fit into the periodicity), one obtains:
{\displaystyle {\begin{aligned}\pi _{0}(KO)&=\mathbf {Z} \\\pi _{1}(KO)&=\mathbf {Z} /2\mathbf {Z} \\\pi _{2}(KO)&=\mathbf {Z} /2\mathbf {Z} \\\pi _{3}(KO)&=0\\\pi _{4}(KO)&=\mathbf {Z} \\\pi _{5}(KO)&=0\\\pi _{6}(KO)&=0\\\pi _{7}(KO)&=0\end{aligned}}}
#### Computation and interpretation of homotopy groups
##### Low-dimensional groups
The first few homotopy groups can be calculated by using the concrete descriptions of low-dimensional groups.
• π0(O) = π0(O(1)) = Z/2Z, from orientation-preserving/reversing (this class survives to O(2) and hence stably)
• π1(O) = π1(SO(3)) = Z/2Z, which is spin comes from SO(3) = RP3 = S3/(Z/2Z).
• π2(O) = π2(SO(3)) = 0, which surjects onto π2(SO(4)); this latter thus vanishes.
##### Lie groups
From general facts about Lie groups, π2(G) always vanishes, and π3(G) is free (free abelian).
##### Vector bundles
From the vector bundle point of view, π0(KO) is vector bundles over S0, which is two points. Thus over each point, the bundle is trivial, and the non-triviality of the bundle is the difference between the dimensions of the vector spaces over the two points, so π0(KO) = Z is dimension.
##### Loop spaces
Using concrete descriptions of the loop spaces in Bott periodicity, one can interpret the higher homotopies of O in terms of simpler-to-analyze homotopies of lower order. Using π0, O and O/U have two components, KO = BO × Z and KSp = BSp × Z have countably many components, and the rest are connected.
#### Interpretation of homotopy groups
In a nutshell: [5]
Let R be any of the four division algebras R, C, H , O , and let LR be the tautological line bundle over the projective line RP1, and [LR] its class in K-theory. Noting that RP1 = S1, CP1 = S2, HP1 = S4, OP1 = S8, these yield vector bundles over the corresponding spheres, and
• π1(KO) is generated by [LR]
• π2(KO) is generated by [LC]
• π4(KO) is generated by [LH]
• π8(KO) is generated by [LO]
From the point of view of symplectic geometry, π0(KO) ≅ π8(KO) = Z can be interpreted as the Maslov index, thinking of it as the fundamental group π1(U/O) of the stable Lagrangian Grassmannian as U/O ≅ Ω7(KO), so π1(U/O) = π1+7(KO).
The orthogonal group anchors a Whitehead tower:
${\displaystyle \ldots \rightarrow \operatorname {Fivebrane} (n)\rightarrow \operatorname {String} (n)\rightarrow \operatorname {Spin} (n)\rightarrow \operatorname {SO} (n)\rightarrow \operatorname {O} (n)}$
which is obtained by successively removing (killing) homotopy groups of increasing order. This is done by constructing short exact sequences starting with an Eilenberg–MacLane space for the homotopy group to be removed. The first few entries in the tower are the spin group and the string group, and are preceded by the fivebrane group. The homotopy groups that are killed are in turn π0(O) to obtain SO from O, π1(O) to obtain Spin from SO, π3(O) to obtain String from Spin, and then π7(O) and so on to obtain the higher order branes.
## Of indefinite quadratic form over the reals
Over the real numbers, nondegenerate quadratic forms are classified by Sylvester's law of inertia, which asserts that, on a vector space of dimension n, such a form can be written as the difference of a sum of p squares and a sum of q squares, with p + q = n. In other words, there is a basis on which the matrix of the quadratic form is a diagonal matrix, with p entries equal to 1, and q entries equal to –1. The pair (p, q) called the inertia, is an invariant of the quadratic form, in the sense that it does not depend on the way of computing the diagonal matrix.
The orthogonal group of a quadratic form depends only on the inertia, and is thus generally denoted O(p, q). Moreover, as a quadratic form and its opposite have the same orthogonal group, one has O(p, q) = O(q, p).
The standard orthogonal group is O(n) = O(n, 0) = O(0, n). So, in the remainder of this section, it is supposed that neither p nor q is zero.
The subgroup of the matrices of determinant 1 in O(p, q) is denoted SO(p, q). The group O(p, q) has four connected components, depending on whether an element preserves orientation on either of the two maximal subspaces where the quadratic form is positive definite or negative definite. The component of the identity, whose elements preserve orientation on both subspaces, is denoted SO+(p, q).
The group O(3, 1) is the Lorentz group that is fundamental in relativity theory. Here the 3 corresponds to space coordinates, and 1 corresponds to the time coordinate.
Over the field C of complex numbers, every non-degenerate quadratic form in n variables is equivalent to ${\displaystyle x_{1}^{2}+\cdots +x_{n}^{2}}$. Thus, up to isomorphism, there is only one non-degenerate complex quadratic space of dimension n, and one associated orthogonal group, usually denoted O(n, C). It is the group of complex orthogonal matrices, complex matrices whose product with their transpose is the identity matrix.
As in the real case, O(n, C) has two connected components. The component of the identity consists of all matrices of determinant 1 in O(n, C); it is denoted SO(n, C).
The groups O(n, C) and SO(n, C) are complex Lie groups of dimension n(n − 1)/2 over C (the dimension over R is twice that). For n ≥ 2, these groups are noncompact. As in the real case, SO(n, C) is not simply connected: For n> 2, the fundamental group of SO(n, C) is cyclic of order 2, whereas the fundamental group of SO(2, C) is Z.
## Over finite fields
### Characteristic different from two
Over a field of characteristic different from two, two quadratic forms are equivalent if their matrices are congruent, that is if a change of basis transforms the matrix of the first form into the matrix of the second form. Two equivalent quadratic forms have clearly the same orthogonal group.
The non-degenerate quadratic forms over a finite field of characteristic different from two are completely classified into congruence classes, and it results from this classification that there is only one orthogonal group in odd dimension and two in even dimension.
More precisely, Witt's decomposition theorem asserts that (in characteristic different from two) every vector space equipped with a non-degenerate quadratic form Q can be decomposed as a direct sum of pairwise orthogonal subspaces
${\displaystyle V=L_{1}\oplus L_{2}\oplus \cdots \oplus L_{m}\oplus W,}$
where each Li is a hyperbolic plane (that is there is a basis such that the matrix of the restriction of Q to Li has the form ${\displaystyle \textstyle {\begin{bmatrix}0&1\\1&0\end{bmatrix}}}$), and the restriction of Q to W is anisotropic (that is, Q(w) ≠ 0 for every nonzero w in W).
Chevalley–Warning theorem asserts that over a finite field the dimension of W is at most two.
If the dimension of V is odd, the dimension of W is thus equal to one, and its matrix is congruent either to ${\displaystyle \textstyle {\begin{bmatrix}1\end{bmatrix}}}$ or to ${\displaystyle \textstyle {\begin{bmatrix}\varphi \end{bmatrix}},}$ where 𝜙 is a non-square scalar. It results that there is only one orthogonal group that is denoted O(2n + 1, q), where q is the number of elements of the finite field (a power of an odd prime). [6]
If the dimension of W is two and –1 is not a square in the ground field (that is, if its number of elements q is congruent to 3 modulo 4), the matrix of the restriction of Q to W is congruent to either I or I, where I is the 2×2 identity matrix. If the dimension of W is two and –1 is a square in the ground field (that is, if q is congruent to 1, modulo 4) the matrix of the restriction of Q to W is congruent to ${\displaystyle \textstyle {\begin{bmatrix}1&0\\0&\phi \end{bmatrix}},}$𝜙 is any non-square scalar.
This implies that if the dimension of V is even, there are only two orthogonal groups, depending whether the dimension of W zero or two. They are denoted respectively O+(2n, q) and O(2n, q). [6]
The orthogonal group Oϵ(2, q) is a dihedral group of order 2(qϵ), where ϵ = ±.
Proof
For studying the orthogonal group of Oϵ(2, q), one can suppose that the matrix of the quadratic form is ${\displaystyle Q={\begin{bmatrix}1&0\\0&-\omega \end{bmatrix}},}$ because, given a quadratic form, there is a basis where its matrix is diagonalizable. A matrix ${\displaystyle A={\begin{bmatrix}a&b\\c&d\end{bmatrix}}}$ belongs to the orthogonal group if ${\displaystyle AQA^{\text{T}}=Q,}$ that is, a2ωb2 = 1, acωbd = 0, and c2ωd2 = –ω. As a and b cannot be both zero (because of the first equation), the second equation implies the existence of ϵ in Fq, such that c = ϵωb and d = ϵa. Reporting these values in the third equation, and using the first equation, one gets that ϵ2 = 1, and thus the orthogonal group consists of the matrices
${\displaystyle {\begin{bmatrix}a&b\\\epsilon \omega b&\epsilon a\end{bmatrix}},}$
where a2ωb2 = 1 and ϵ = ±1. Moreover, the determinant of the matrix is ϵ.
For further studying the orthogonal group, it is convenient to introduce a square root α of ω. This square root belongs to Fq if the orthogonal group is O+(2, q), and to Fq2 otherwise. Setting x = a + αb, and y = aαb, one has
${\displaystyle xy=1,\qquad a={\frac {x+y}{2}}\qquad b={\frac {x-y}{2\alpha }}.}$
If ${\displaystyle A_{1}={\begin{bmatrix}a_{1}&b_{1}\\\omega b_{1}&a_{1}\end{bmatrix}}}$ and ${\displaystyle A_{2}={\begin{bmatrix}a_{2}&b_{2}\\\omega b_{2}&a_{2}\end{bmatrix}}}$ are two matrices of determinant one in the orthogonal group then
${\displaystyle A_{1}A_{2}={\begin{bmatrix}a_{1}a_{2}+\omega b_{1}b_{2}&a_{1}b_{2}+b_{1}a_{2}\\\omega b_{1}a_{2}+\omega a_{1}b_{2}&\omega b_{1}b_{2}+a_{1}a_{1}\end{bmatrix}}.}$
This is an orthogonal matrix ${\displaystyle {\begin{bmatrix}a&b\\\omega b&a\end{bmatrix}},}$ with a = a1a2 + ωb1b2, and b = a1b2 + b1a2. Thus
${\displaystyle a+\alpha b=(a_{1}+\alpha b_{1})(a_{2}+\alpha b_{2}).}$
It follows that the map ${\displaystyle (a,b)\mapsto a+\alpha b}$ is a homomorphism of the group of orthogonal matrices of determinant one into the multiplicative group of Fq2.
In the case of O+(2n, q), the image is the multiplicative group of Fq, which is a cyclic group of order q.
In the case of O(2n, q), the above x and y are conjugate, and are therefore the image of each other by the Frobenius automorphism. This meand that ${\displaystyle y=x^{-1}=x^{q},}$ and thus ${\displaystyle x^{q+1}=1.}$ For every such x one can reconstruct a corresponding orthogonal matrix. It follows that the map ${\displaystyle (a,b)\mapsto a+\alpha b}$ is a group isomorphism from the orthogonal matrices of determinant 1 to the group of the (q + 1)-roots of unity. This group is a cyclic group of order q + 1 which consists of the powers of ${\displaystyle g^{q-1},}$ where g is a primitive element of Fq2,
For finishing the proof, it suffices to verify that the group all orthogonal matrices is not abelian, and is the semidirect product of the group {1, –1} and the group of orthogonal matrices of determinant one.
The comparison of this proof with the real case may be illuminating.
Here two group isomorphisms are involved:
{\displaystyle {\begin{aligned}\mathbb {Z} /(q+1)\mathbb {Z} &\to T\\k&\mapsto g^{(q-1)k},\end{aligned}}}
where g is a primitive element of Fq2 and T is the multiplicative group of the element of norm one in Fq2 ;
{\displaystyle {\begin{aligned}\mathbb {T} &\to \operatorname {SO} ^{+}(2,\mathbf {F} _{q})\\x&\mapsto {\begin{bmatrix}a&b\\\omega b&a\end{bmatrix}},\end{aligned}}}
with ${\displaystyle a={\frac {x+x^{-1}}{2}}}$ and ${\displaystyle b={\frac {x-x^{-1}}{2\alpha }}.}$
In the real case, the corresponding isomorphisms are:
{\displaystyle {\begin{aligned}\mathbb {R} /2\pi \mathbb {R} &\to C\\\theta &\mapsto e^{i\theta },\end{aligned}}}
where C is the circle of the complex numbers of norm one;
{\displaystyle {\begin{aligned}\mathbb {C} &\to \operatorname {SO} (2,\mathbb {R} )\\x&\mapsto {\begin{bmatrix}\cos \theta &\sin \theta \\-\sin \theta &\cos \theta \end{bmatrix}},\end{aligned}}}
with ${\displaystyle \cos \theta ={\frac {e^{i\theta }+e^{-i\theta }}{2}}}$ and ${\displaystyle \sin \theta ={\frac {e^{i\theta }-e^{-i\theta }}{2i}}.}$
When the characteristic is not two, the order of the orthogonal groups are [7]
${\displaystyle \left|\operatorname {O} (2n+1,q)\right|=2q^{n^{2}}\prod _{i=1}^{n}\left(q^{2i}-1\right),}$
${\displaystyle \left|\operatorname {O} ^{+}(2n,q)\right|=2q^{n(n-1)}\left(q^{n}-1\right)\prod _{i=1}^{n-1}\left(q^{2i}-1\right),}$
${\displaystyle \left|\operatorname {O} ^{-}(2n,q)\right|=2q^{n(n-1)}\left(q^{n}+1\right)\prod _{i=1}^{n-1}\left(q^{2i}-1\right).}$
In characteristic two, the formulas are the same, except that the factor 2 of ${\displaystyle \left|\operatorname {O} (2n+1,q)\right|}$ must be removed.
### The Dickson invariant
For orthogonal groups, the Dickson invariant is a homomorphism from the orthogonal group to the quotient group Z/2Z (integers modulo 2), taking the value 0 in case the element is the product of an even number of reflections, and the value of 1 otherwise. [8]
Algebraically, the Dickson invariant can be defined as D(f) = rank(If) modulo 2, where I is the identity ( Taylor 1992 , Theorem 11.43). Over fields that are not of characteristic 2 it is equivalent to the determinant: the determinant is −1 to the power of the Dickson invariant. Over fields of characteristic 2, the determinant is always 1, so the Dickson invariant gives more information than the determinant.
The special orthogonal group is the kernel of the Dickson invariant [8] and usually has index 2 in O(n, F ). [9] When the characteristic of F is not 2, the Dickson Invariant is 0 whenever the determinant is 1. Thus when the characteristic is not 2, SO(n, F ) is commonly defined to be the elements of O(n, F ) with determinant 1. Each element in O(n, F ) has determinant ±1. Thus in characteristic 2, the determinant is always 1.
The Dickson invariant can also be defined for Clifford groups and pin groups in a similar way (in all dimensions).
### Orthogonal groups of characteristic 2
Over fields of characteristic 2 orthogonal groups often exhibit special behaviors, some of which are listed in this section. (Formerly these groups were known as the hypoabelian groups, but this term is no longer used.)
• Any orthogonal group over any field is generated by reflections, except for a unique example where the vector space is 4-dimensional over the field with 2 elements and the Witt index is 2. [10] A reflection in characteristic two has a slightly different definition. In characteristic two, the reflection orthogonal to a vector u takes a vector v to v + B(v, u)/Q(u) · u where B is the bilinear form[ clarification needed ] and Q is the quadratic form associated to the orthogonal geometry. Compare this to the Householder reflection of odd characteristic or characteristic zero, which takes v to v − 2·B(v, u)/Q(u) · u.
• The center of the orthogonal group usually has order 1 in characteristic 2, rather than 2, since I = −I.
• In odd dimensions 2n + 1 in characteristic 2, orthogonal groups over perfect fields are the same as symplectic groups in dimension 2n. In fact the symmetric form is alternating in characteristic 2, and as the dimension is odd it must have a kernel of dimension 1, and the quotient by this kernel is a symplectic space of dimension 2n, acted upon by the orthogonal group.
• In even dimensions in characteristic 2 the orthogonal group is a subgroup of the symplectic group, because the symmetric bilinear form of the quadratic form is also an alternating form.
## The spinor norm
The spinor norm is a homomorphism from an orthogonal group over a field F to the quotient group F×/(F×)2 (the multiplicative group of the field F up to multiplication by square elements), that takes reflection in a vector of norm n to the image of n in F×/(F×)2. [11]
For the usual orthogonal group over the reals, it is trivial, but it is often non-trivial over other fields, or for the orthogonal group of a quadratic form over the reals that is not positive definite.
## Galois cohomology and orthogonal groups
In the theory of Galois cohomology of algebraic groups, some further points of view are introduced. They have explanatory value, in particular in relation with the theory of quadratic forms; but were for the most part post hoc, as far as the discovery of the phenomena is concerned. The first point is that quadratic forms over a field can be identified as a Galois H1, or twisted forms (torsors) of an orthogonal group. As an algebraic group, an orthogonal group is in general neither connected nor simply-connected; the latter point brings in the spin phenomena, while the former is related to the discriminant.
The 'spin' name of the spinor norm can be explained by a connection to the spin group (more accurately a pin group). This may now be explained quickly by Galois cohomology (which however postdates the introduction of the term by more direct use of Clifford algebras). The spin covering of the orthogonal group provides a short exact sequence of algebraic groups.
${\displaystyle 1\rightarrow \mu _{2}\rightarrow \mathrm {Pin} _{V}\rightarrow \mathrm {O_{V}} \rightarrow 1}$
Here μ2 is the algebraic group of square roots of 1; over a field of characteristic not 2 it is roughly the same as a two-element group with trivial Galois action. The connecting homomorphism from H0(OV), which is simply the group OV(F) of F-valued points, to H12) is essentially the spinor norm, because H12) is isomorphic to the multiplicative group of the field modulo squares.
There is also the connecting homomorphism from H1 of the orthogonal group, to the H2 of the kernel of the spin covering. The cohomology is non-abelian so that this is as far as we can go, at least with the conventional definitions.
## Lie algebra
The Lie algebra corresponding to Lie groups O(n, F ) and SO(n, F ) consists of the skew-symmetric n × n matrices, with the Lie bracket [ , ] given by the commutator. One Lie algebra corresponds to both groups. It is often denoted by ${\displaystyle {\mathfrak {o}}(n,F)}$ or ${\displaystyle {\mathfrak {so}}(n,F)}$, and called the orthogonal Lie algebra or special orthogonal Lie algebra. Over real numbers, these Lie algebras for different n are the compact real forms of two of the four families of semisimple Lie algebras: in odd dimension Bk, where n = 2k + 1, while in even dimension Dr, where n = 2r.
Since the group SO(n) is not simply connected, the representation theory of the orthogonal Lie algebras includes both representations corresponding to ordinary representations of the orthogonal groups, and representations corresponding to projective representations of the orthogonal groups. (The projective representations of SO(n) are just linear representations of the universal cover, the spin group Spin(n).) The latter are the so-called spin representation, which are important in physics.
More generally, given a vector space ${\displaystyle V}$ (over a field with characteristic not equal to 2) with a nondegenerate symmetric bilinear form ${\displaystyle (\cdot ,\cdot )}$, the special orthogonal Lie algebra consists of tracefree endomorphisms ${\displaystyle \phi }$ which are skew-symmetric for this form (${\displaystyle (\phi A,B)+(A,\phi B)=0}$). Over a field of characteristic 2 we consider instead the alternating endomorphisms. Concretely we can equate these with the alternating tensors ${\displaystyle \Lambda ^{2}V}$. The correspondence is given by:
${\displaystyle v\wedge w\mapsto (v,\cdot )w-(w,\cdot )v}$
This description applies equally for the indefinite special orthogonal Lie algebras ${\displaystyle {\mathfrak {so}}(p,q)}$ for symmetric bilinear forms with signature ${\displaystyle (p,q)}$.
Over real numbers, this characterization is used in interpreting the curl of a vector field (naturally a 2-vector) as an infinitesimal rotation or "curl", hence the name.
The orthogonal groups and special orthogonal groups have a number of important subgroups, supergroups, quotient groups, and covering groups. These are listed below.
The inclusions O(n) ⊂ U(n) ⊂ USp(2n) and USp(n) ⊂ U(n) ⊂ O(2n) are part of a sequence of 8 inclusions used in a geometric proof of the Bott periodicity theorem, and the corresponding quotient spaces are symmetric spaces of independent interest – for example, U(n)/O(n) is the Lagrangian Grassmannian.
### Lie subgroups
In physics, particularly in the areas of Kaluza–Klein compactification, it is important to find out the subgroups of the orthogonal group. The main ones are:
${\displaystyle \mathrm {O} (n)\supset \mathrm {O} (n-1)}$ – preserve an axis
${\displaystyle \mathrm {O} (2n)\supset \mathrm {U} (n)\supset \mathrm {SU} (n)}$U(n) are those that preserve a compatible complex structure or a compatible symplectic structure – see 2-out-of-3 property; SU(n) also preserves a complex orientation.
${\displaystyle \mathrm {O} (2n)\supset \mathrm {USp} (n)}$
${\displaystyle \mathrm {O} (7)\supset \mathrm {G} _{2}}$
### Lie supergroups
The orthogonal group O(n) is also an important subgroup of various Lie groups:
{\displaystyle {\begin{aligned}\mathrm {U} (n)&\supset \mathrm {O} (n)\\\mathrm {USp} (2n)&\supset \mathrm {O} (n)\\\mathrm {G} _{2}&\supset \mathrm {O} (3)\\\mathrm {F} _{4}&\supset \mathrm {O} (9)\\\mathrm {E} _{6}&\supset \mathrm {O} (10)\\\mathrm {E} _{7}&\supset \mathrm {O} (12)\\\mathrm {E} _{8}&\supset \mathrm {O} (16)\end{aligned}}}
#### Conformal group
Being isometries, real orthogonal transforms preserve angles, and are thus conformal maps, though not all conformal linear transforms are orthogonal. In classical terms this is the difference between congruence and similarity, as exemplified by SSS (side-side-side) congruence of triangles and AAA (angle-angle-angle) similarity of triangles. The group of conformal linear maps of Rn is denoted CO(n) for the conformal orthogonal group, and consists of the product of the orthogonal group with the group of dilations. If n is odd, these two subgroups do not intersect, and they are a direct product: CO(2k + 1) = O(2k + 1) × R, where R = R∖{0} is the real multiplicative group, while if n is even, these subgroups intersect in ±1, so this is not a direct product, but it is a direct product with the subgroup of dilation by a positive scalar: CO(2k) = O(2k) × R+.
Similarly one can define CSO(n); note that this is always: CSO(n) = CO(n) ∩ GL+(n) = SO(n) × R+.
### Discrete subgroups
As the orthogonal group is compact, discrete subgroups are equivalent to finite subgroups. [note 1] These subgroups are known as point groups and can be realized as the symmetry groups of polytopes. A very important class of examples are the finite Coxeter groups, which include the symmetry groups of regular polytopes.
Dimension 3 is particularly studied – see point groups in three dimensions, polyhedral groups, and list of spherical symmetry groups. In 2 dimensions, the finite groups are either cyclic or dihedral – see point groups in two dimensions.
Other finite subgroups include:
### Covering and quotient groups
The orthogonal group is neither simply connected nor centerless, and thus has both a covering group and a quotient group, respectively:
These are all 2-to-1 covers.
For the special orthogonal group, the corresponding groups are:
Spin is a 2-to-1 cover, while in even dimension, PSO(2k) is a 2-to-1 cover, and in odd dimension PSO(2k + 1) is a 1-to-1 cover; i.e., isomorphic to SO(2k + 1). These groups, Spin(n), SO(n), and PSO(n) are Lie group forms of the compact special orthogonal Lie algebra, ${\displaystyle {\mathfrak {so}}(n,\mathbb {R} )}$ – Spin is the simply connected form, while PSO is the centerless form, and SO is in general neither. [note 3]
In dimension 3 and above these are the covers and quotients, while dimension 2 and below are somewhat degenerate; see specific articles for details.
## Principal homogeneous space: Stiefel manifold
The principal homogeneous space for the orthogonal group O(n) is the Stiefel manifold Vn(Rn) of orthonormal bases (orthonormal n-frames).
In other words, the space of orthonormal bases is like the orthogonal group, but without a choice of base point: given an orthogonal space, there is no natural choice of orthonormal basis, but once one is given one, there is a one-to-one correspondence between bases and the orthogonal group. Concretely, a linear map is determined by where it sends a basis: just as an invertible map can take any basis to any other basis, an orthogonal map can take any orthogonal basis to any other orthogonal basis.
The other Stiefel manifolds Vk(Rn) for k < n of incomplete orthonormal bases (orthonormal k-frames) are still homogeneous spaces for the orthogonal group, but not principal homogeneous spaces: any k-frame can be taken to any other k-frame by an orthogonal map, but this map is not uniquely determined.
## Notes
1. Infinite subsets of a compact space have an accumulation point and are not discrete.
2. O(n) ∩ GL(n, Z) equals the signed permutation matrices because an integer vector of norm 1 must have a single non-zero entry, which must be ±1 (if it has two non-zero entries or a larger entry, the norm will be larger than 1), and in an orthogonal matrix these entries must be in different coordinates, which is exactly the signed permutation matrices.
3. In odd dimension, SO(2k + 1) ≅ PSO(2k + 1) is centerless (but not simply connected), while in even dimension SO(2k) is neither centerless nor simply connected.
## Citations
1. For base fields of characteristic not 2, the definition in terms of a symmetric bilinear form is equivalent to that in terms of a quadratic form, but in characteristic 2 these notions differ.
2. Hall 2015 Theorem 11.2
3. Hall 2015 Section 1.3.4
4. Hall 2015 Proposition 13.10
5. Wilson, Robert A. (2009). The finite simple groups. Graduate Texts in Mathematics. 251. London: Springer. pp. 69–75. ISBN 978-1-84800-987-5. Zbl 1203.20012.
6. ( Taylor 1992 , p. 141)
7. Knus, Max-Albert (1991), Quadratic and Hermitian forms over rings, Grundlehren der Mathematischen Wissenschaften, 294, Berlin etc.: Springer-Verlag, p. 224, ISBN 3-540-52117-8, Zbl 0756.11008
8. ( Taylor 1992 , page 160)
9. ( Grove 2002 , Theorem 6.6 and 14.16)
10. Cassels 1978 , p. 178
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In mathematics, the classical groups are defined as the special linear groups over the reals R, the complex numbers C and the quaternions H together with special automorphism groups of symmetric or skew-symmetric bilinear forms and Hermitian or skew-Hermitian sesquilinear forms defined on real, complex and quaternionic finite-dimensional vector spaces. Of these, the complex classical Lie groups are four infinite families of Lie groups that together with the exceptional groups exhaust the classification of simple Lie groups. The compact classical groups are compact real forms of the complex classical groups. The finite analogues of the classical groups are the classical groups of Lie type. The term "classical group" was coined by Hermann Weyl, it being the title of his 1939 monograph The Classical Groups. | 12,524 | 48,291 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 73, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.125 | 4 | CC-MAIN-2021-31 | latest | en | 0.911553 |
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June 5, 2000, 22:22 Parallel simulation of unsteady flows #1 MLJ Guest Posts: n/a If we use SPMD and domain docomposition to calculate unsteady flow on distributed memory parallel computers, do we have to make sure that the calculations in different domains are advanced at the same time with the same time step size? If not, would the flow signals be propagated correctly across the domain interfaces?
June 7, 2000, 06:09 Re: Parallel simulation of unsteady flows #2 Rich E Guest Posts: n/a Yes, you must have a spatially constant time step in order to get a physically correct transient solution. Not only that but your global time step size is set by the stabillity limit of your smallest cell. Hence big cells will use a much smaller timestep than they otherwise could. For a steady state solution, you are not intersted in a physically correct transient so not only each domain, but each cell can use a different time step, set by it's own stability limit.
June 8, 2000, 02:23 Re: Parallel simulation of unsteady flows #3 MLJ Guest Posts: n/a Suppose a certain domain is slightly smaller than the others. Then the calculation in this domain may be faster than others. How can we make sure that this will not cause incorrect propagation of the flow signals?
June 8, 2000, 04:37 Re: Parallel simulation of unsteady flows #4 Steve Amphlett Guest Posts: n/a Isn't this the challenge? Firstly you have to implement a synchronization and message passing system (off-the-shelf or DIY?) and use it to ensure all domains come together at the end of each time increment. Then you have to devise a method of dividing your domain up so that each piece takes approximately the same time to solve (or you'll just waste CPU time). You could have a small, lightweight lead process to do all the inter-process communication, or you could fold this work into one of the domains, which then becomes the lead domain.
June 8, 2000, 08:07 Re: Parallel simulation of unsteady flows #5 Rich E Guest Posts: n/a You should have communication across domain boundaries (at least) once per timestep. If one domain takes less physical time to complete a time step than the others then it will just have to wait until the rest 'catch up'. Of course this time spent waiting is very ineffiecient, fortunatly there are several excellent (and free) applications which will divde your mesh into equallly (computationally) sized domains so that the problem is equally balanced between all processors. See the parallel computing section in this website's resources list
June 14, 2000, 01:24 Re: Parallel simulation of unsteady flows #6 Biju Uthup Guest Posts: n/a Hi, Any of you dealing with three dimensional unsteady viscous flows for internal flows caused by instabilities due to vortex interactions on parallel machines? Could any of you put me in touch with some one working something close to this.
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Say we are given a complex smooth projective toric variety $X$. How can one read off hodge theoretic information from combinatorial data? For example I would like to extract dimensions of the various $H^{pq}(X,\mathbb C)$ from the fan or the $sl_2(\mathbb C)$ action on $H^*(X,\mathbb C)$ from the moment polytope etc.
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Say the toric variety $X$ is defined by the lattice polytope $P$ (dimension = $n$). Let $f_i$ denote the number of $i$-dimensional faces of $P$, and set $h_p = \sum_{i=p}^n (-1)^{i-p} \binom{i}{p} f_i$. Then $h^{p,q}(X) = h_p$ if $p=q$ and $0$ otherwise. | 199 | 613 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.65625 | 3 | CC-MAIN-2016-26 | latest | en | 0.796956 |
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A107030 Number triangle associated with the Riordan arrays (1/(1+x),x/(1+x)^k),k>=0. 2
1, 2, 1, 2, 2, 1, 2, 4, 2, 1, 2, 8, 6, 2, 1, 2, 16, 20, 8, 2, 1, 2, 32, 70, 38, 10, 2, 1, 2, 64, 252, 196, 62, 12, 2, 1, 2, 128, 924, 1062, 426, 92, 14, 2, 1, 2, 256, 3432, 5948, 3112, 792, 128, 16, 2, 1, 2, 512, 12870, 34120, 23686, 7302, 1326, 170, 18, 2, 1, 2, 1024, 48620 (list; table; graph; refs; listen; history; text; internal format)
OFFSET 0,2 LINKS Table of n, a(n) for n=0..68. FORMULA Number triangle T(n, k)=(k-1)*C(k(n-k), n-k)-(k-2)*sum{j=0..n-k, C(k(n-k), j)} EXAMPLE Triangle begins 1; 2,1; 2,2,1; 2,4,2,1; 2,8,6,2,1; 2,16,20,8,2,1; CROSSREFS Reversal of A107027. Row sums are A107028. Diagonal sums are A107031. Columns include A040000, A000079, A000984, A047098. Sequence in context: A263666 A107027 A355395 * A271362 A354555 A263643 Adjacent sequences: A107027 A107028 A107029 * A107031 A107032 A107033 KEYWORD easy,nonn,tabl AUTHOR Paul Barry, May 09 2005 STATUS approved
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Last modified May 29 13:20 EDT 2023. Contains 363042 sequences. (Running on oeis4.) | 610 | 1,459 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.453125 | 3 | CC-MAIN-2023-23 | latest | en | 0.579645 |
http://wiki.stat.ucla.edu/socr/index.php?title=AP_Statistics_Curriculum_2007_Distrib_Multinomial&diff=7310&oldid=7309 | 1,521,325,782,000,000,000 | text/html | crawl-data/CC-MAIN-2018-13/segments/1521257645362.1/warc/CC-MAIN-20180317214032-20180317234032-00713.warc.gz | 335,124,215 | 10,350 | AP Statistics Curriculum 2007 Distrib Multinomial
(Difference between revisions)
Revision as of 00:14, 3 May 2008 (view source)IvoDinov (Talk | contribs) (→Examples of Multinomial experiments: fixed a typo on p_3=p_{blue} = 4/9)← Older edit Revision as of 00:28, 3 May 2008 (view source)IvoDinov (Talk | contribs) (→Examples of Multinomial experiments: Added another example to demonstrate complementation in multinomial probabilities.)Newer edit → Line 25: Line 25: Thus, if we draw 5 marbles with replacement from the urn, the probability of drawing no red, 2 green, and 3 blue marbles is ''0.0975461''. Thus, if we draw 5 marbles with replacement from the urn, the probability of drawing no red, 2 green, and 3 blue marbles is ''0.0975461''. + + * Let's again use the urn containing 9 marbles, where the number of red, green and blue marbles are 2, 3 and 4, respectively. This time we select 5 marbles from the urn, but are interested in the probability (''P(B)'') of the event ''B={selecting 2 green marbles}''! (Note that 2 < 5) + **To solve this problem, we classify balls into '''green''' and '''others'''! Thus the multinomial experiment consists of 5 trials (k = 5), $r_1=r_{green} = 2$, $r_2=r_{other} = 3$. In this case, the probabilities of drawing a '''green''' or '''other''' marble are 3/9, and 6/9, respectively. Notice now the P('''other''') is the sum of the probabilities of the '''other''' colors (complement of green)! Hence, + : $P(B) = {5\choose 2, 3}p_1^{r_1}p_2^{r_2} = {5! \over 2! \times 3! }\times (3/9)^2 \times (6/9)^3=0.329218.$ + + This probability is equivalent to the binomial probability (success=green; failure=other color), ''B(n=5, p=1/3)''. ===Synergies between Binomial and Multinomial processes/probabilities/coefficients=== ===Synergies between Binomial and Multinomial processes/probabilities/coefficients===
General Advance-Placement (AP) Statistics Curriculum - Multinomial Random Variables and Experiments
The multinomial experiments (and multinomial distributions) directly extend the their bi-nomial counterparts.
Multinomial experiments
A multinomial experiment is an experiment that has the following properties:
• The experiment consists of k repeated trials.
• Each trial has a discrete number of possible outcomes.
• On any given trial, the probability that a particular outcome will occur is constant.
• The trials are independent; that is, the outcome on one trial does not affect the outcome on other trials.
Examples of Multinomial experiments
• Suppose we have an urn containing 9 marbles. Two are red, three are green, and four are blue (2+3+4=9). We randomly select 5 marbles from the urn, with replacement. What is the probability (P(A)) of the event A={selecting 2 green marbles and 3 blue marbles}?
• To solve this problem, we apply the multinomial formula. We know the following:
• The experiment consists of 5 trials, so k = 5.
• The 5 trials produce 0 red, 2 green marbles, and 3 blue marbles; so r1 = rred = 0, r2 = rgreen = 2, and r3 = rblue = 3.
• For any particular trial, the probability of drawing a red, green, or blue marble is 2/9, 3/9, and 4/9, respectively. Hence, p1 = pred = 2 / 9, p2 = pgreen = 1 / 3, and p3 = pblue = 4 / 9.
Plugging these values into the multinomial formula we get the probability of the event of interest to be:
$P(A) = {5\choose 0, 2, 3}p_1^{r_1}p_2^{r_2}p_3^{r_3}$
$P(A) = {5! \over 0!\times 2! \times 3! }\times (2/9)^0 \times (1/3)^2\times (4/9)^3=0.0975461.$
Thus, if we draw 5 marbles with replacement from the urn, the probability of drawing no red, 2 green, and 3 blue marbles is 0.0975461.
• Let's again use the urn containing 9 marbles, where the number of red, green and blue marbles are 2, 3 and 4, respectively. This time we select 5 marbles from the urn, but are interested in the probability (P(B)) of the event B={selecting 2 green marbles}! (Note that 2 < 5)
• To solve this problem, we classify balls into green and others! Thus the multinomial experiment consists of 5 trials (k = 5), r1 = rgreen = 2, r2 = rother = 3. In this case, the probabilities of drawing a green or other marble are 3/9, and 6/9, respectively. Notice now the P(other) is the sum of the probabilities of the other colors (complement of green)! Hence,
$P(B) = {5\choose 2, 3}p_1^{r_1}p_2^{r_2} = {5! \over 2! \times 3! }\times (3/9)^2 \times (6/9)^3=0.329218.$
This probability is equivalent to the binomial probability (success=green; failure=other color), B(n=5, p=1/3).
Synergies between Binomial and Multinomial processes/probabilities/coefficients
${n\choose i}=\frac{n!}{k!(n-k)!}$
${n\choose i_1,i_2,\cdots, i_k}= \frac{n!}{i_1! i_2! \cdots i_k!}$
• The Binomial vs. Multinomial Formulas
$(a+b)^n = \sum_{i=1}^n{{n\choose i}a^1 \times b^{n-i}}$
$(a_1+a_2+\cdots +a_k)^n = \sum_{i_1+i_2\cdots +i_k=n}^n{ {n\choose i_1,i_2,\cdots, i_k} a_1^{i_1} \times a_2^{i_2} \times \cdots \times a_k^{i_k}}$
$p=P(X=r)={n\choose r}p^r(1-p)^{n-r}, \forall 0\leq r \leq n$
$p=P(X_1=r_1 \cap X_2=r_2 \cap \cdots \cap X_k=r_k | r_1+r_2+\cdots+r_k=n)={n\choose i_1,i_2,\cdots, i_k}p_1^{r_1}p_2^{r_2}\cdots p_k^{r_k}, \forall r_1+r_2+\cdots+r_k=n$
Example
Suppose we study N independent trials with results falling in one of k possible categories labeled $1,2, \cdots, k$. Let pi be the probability of a trial resulting in the ith category, where $p_1+p_2+ \cdots +p_k = 1$. Let Ni be the number of trials resulting in the ith category, where $N_1+N_2+ \cdots +N_k = N$.
For instance, suppose we have 9 people arriving at a meeting according to the following information:
P(by Air) = 0.4, P(by Bus) = 0.2, P(by Automobile) = 0.3, P(by Train) = 0.1
• Compute the following probabilities
P(3 by Air, 3 by Bus, 1 by Auto, 2 by Train) = ?
P(2 by air) = ?
SOCR Multinomial Examples
Suppose we row 10 loaded hexagonal (6-face) dice 8 times and we are interested in the probability of observing the event A={3 ones, 3 twos, 2 threes, and 2 fours}. Assume the dice are loaded to the small outcomes according to the following probabilities of the 6 outcomes (one is the most likely and six is the least likely outcome).
x 1 2 3 4 5 6 P(X=x) 0.286 0.238 0.19 0.143 0.095 0.048
P(A)=?
Of course, we can compute this number exactly as:
$P(A) = {10! \over 3!\times 3! \times 2! \times 2! } \times 0.286^3 \times 0.238^3\times 0.19^2 \times 0.143^2 = 0.00586690138260962656816896.$
However, we can also find a pretty close empirically-driven estimate using the SOCR Dice Experiment.
For instance, running the SOCR Dice Experiment 1,000 times with number of dice n=10, and the loading probabilities listed above, we get an output like the one shown below.
Now, we can actually count how many of these 1,000 trials generated the event A as an outcome. In one such experiment of 1,000 trials, there were 8 outcomes of the type {3 ones, 3 twos, 2 threes and 2 fours}. Therefore, the relative proportion of these outcomes to 1,000 will give us a fairly accurate estimate of the exact probability we computed above
$P(A) \approx {8 \over 1,000}=0.008$.
Note that that this approximation is close to the exact answer above. By the Law of Large Numbers (LLN), we know that this SOCR empirical approximation to the exact multinomial probability of interest will significantly improve as we increase the number of trials in this experiment to 10,000. | 2,331 | 7,334 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 14, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.90625 | 5 | CC-MAIN-2018-13 | longest | en | 0.871023 |
https://eurekamathanswerkeys.com/area-and-perimeter/ | 1,716,060,452,000,000,000 | text/html | crawl-data/CC-MAIN-2024-22/segments/1715971057494.65/warc/CC-MAIN-20240518183301-20240518213301-00555.warc.gz | 210,639,044 | 11,126 | Area and Perimeter is an important and basic topic in the Mensuration of 2-D or Planar Figures. The area is used to measure the space occupied by the planar figures. The perimeter is used to measure the boundaries of the closed figures. In Mathematics, these are two major formulas to solve the problems in the 2-dimensional shapes.
Each and every shape has two properties that are Area and Perimeter. Students can find the area and perimeter of different shapes like Circle, Rectangle, Square, Parallelogram, Rhombus, Trapezium, Quadrilateral, Pentagon, Hexagon, and Octagon. The properties of the figures will vary based on their structures, angles, and size. Scroll down this page to learn deeply about the area and perimeter of all the two-dimensional shapes.
## Area and Perimeter Definition
Area: Area is defined as the measure of the space enclosed by the planar figure or shape. The Units to measure the area of the closed figure is square centimeters or meters.
Perimeter: Perimeter is defined as the measure of the length of the boundary of the two-dimensional planar figure. The units to measure the perimeter of the closed figures is centimeters or meters.
### Formulas for Area and Perimeter of 2-D Shapes
1. Area and Perimeter of Rectangle:
• Area = l × b
• Perimeter = 2 (l + b)
• Diagnol = √l² + b²
Where, l = length
2. Area and Perimeter of Square:
• Area = s × s
• Perimeter = 4s
Where s = side of the square
3. Area and Perimeter of Parallelogram:
• Area = bh
• Perimeter = 2( b + h)
Where, b = base
h = height
4. Area and Perimeter of Trapezoid:
• Area = 1/2 × h (a + b)
• Perimeter = a + b + c + d
Where, a, b, c, d are the sides of the trapezoid
h is the height of the trapezoid
5. Area and Perimeter of Triangle:
• Area = 1/2 × b × h
• Perimeter = a + b + c
Where, b = base
h = height
a, b, c are the sides of the triangle
6. Area and Perimeter of Pentagon:
• Area = (5/2) s × a
• Perimeter = 5s
Where s is the side of the pentagon
a is the length
7. Area and Perimeter of Hexagon:
• Area = 1/2 × P × a
• Perimeter = s + s + s + s + s + s = 6s
Where s is the side of the hexagon.
8. Area and Perimeter of Rhombus:
• Area = 1/2 (d1 + d2)
• Perimeter = 4a
Where d1 and d2 are the diagonals of the rhombus
a is the side of the rhombus
9. Area and Perimeter of Circle:
• Area = Πr²
• Circumference of the circle = 2Πr
Where r is the radius of the circle
Π = 3.14 or 22/7
10. Area and Perimeter of Octagon:
• Area = 2(1 + √2) s²
• Perimeter = 8s
Where s is the side of the octagon.
### Solved Examples on Area and Perimeter
Here are some of the examples of the area and perimeter of the geometric figures. Students can easily understand the concept of the area and perimeter with the help of these problems.
1. Find the area and perimeter of the rectangle whose length is 8m and breadth is 4m?
Solution:
Given,
l = 8m
b = 4m
Area of the rectangle = l × b
A = 8m × 4m
A = 32 sq. meters
The perimeter of the rectangle = 2(l + b)
P = 2(8m + 4m)
P = 2(12m)
P = 24 meters
Therefore the area and perimeter of the rectangle is 32 sq. m and 24 meters.
2. Calculate the area of the rhombus whose diagonals are 6 cm and 5 cm?
Solution:
Given,
d1 = 6cm
d2 = 5 cm
Area = 1/2 (d1 + d2)
A = 1/2 (6 cm + 5cm)
A = 1/2 × 11 cm
A = 5.5 sq. cm
Thus the area of the rhombus is 5.5 sq. cm
3. Find the area of the triangle whose base and height are 11 cm and 7 cm?
Solution:
Given,
Base = 11 cm
Height = 7 cm
We know that
Area of the triangle = 1/2 × b × h
A = 1/2 × 11 cm × 7 cm
A = 1/2 × 77 sq. cm
A = 38.5 sq. cm
Thus the area of the triangle is 38.5 sq. cm.
4. Find the area of the circle whose radius is 7 cm?
Solution:
Given,
We know that,
Area of the circle = Πr²
Π = 3.14
A = 3.14 × 7 cm × 7 cm
A = 3.14 × 49 sq. cm
A = 153.86 sq. cm
Therefore the area of the circle is 153.86 sq. cm.
5. Find the area of the trapezoid if the length, breadth, and height is 8 cm, 4 cm, and 5 cm?
Solution:
Given,
a = 8 cm
b = 4 cm
h = 5 cm
We know that,
Area of the trapezoid = 1/2 × h(a + b)
A = 1/2 × (8 + 4)5
A = 1/2 × 12 × 5
A = 6 cm× 5 cm
A = 30 sq. cm
Therefore the area of the trapezoid is 30 sq. cm.
6. Find the perimeter of the pentagon whose side is 5 meters?
Solution:
Given that,
Side = 5 m
The perimeter of the pentagon = 5s
P = 5 × 5 m
P = 25 meters
Therefore the perimeter of the pentagon is 25 meters.
### FAQs on Area and Perimeter
1. How does Perimeter relate to Area?
The perimeter is the boundary of the closed figure whereas the area is the space occupied by the planar.
2. How to calculate the perimeter?
The perimeter can be calculated by adding the lengths of all the sides of the figure.
3. What is the formula for perimeter?
The formula for perimeter is the sum of all the sides. | 1,504 | 4,754 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.75 | 5 | CC-MAIN-2024-22 | latest | en | 0.901072 |
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Number 46344: mathematical and symbolic properties | Crazy Numbers
Discover a lot of information on the number 46344: properties, mathematical operations, how to write it, symbolism, numerology, representations and many other interesting things!
## Mathematical properties of 46344
Is 46344 a prime number? No
Is 46344 a perfect number? No
Number of divisors 16
List of dividers
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1, 2, 3, 4, 6, 8, 12, 24, 1931, 3862, 5793, 7724, 11586, 15448, 23172, 46344
Sum of divisors 115920
Prime factorization 23 x 3 x 1931
Prime factors
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2, 3, 1931
## How to write / spell 46344 in letters?
In letters, the number 46344 is written as: Forty-six thousand three hundred and forty-four. And in other languages? how does it spell?
46344 in other languages
Write 46344 in english Forty-six thousand three hundred and forty-four
Write 46344 in french Quarante-six mille trois cent quarante-quatre
Write 46344 in spanish Cuarenta y seis mil trescientos cuarenta y cuatro
Write 46344 in portuguese Quarenta e seis mil trezentos quarenta e quatro
## Decomposition of the number 46344
The number 46344 is composed of:
3 iterations of the number 4 : The number 4 (four) is the symbol of the square. It represents structuring, organization, work and construction.... Find out more about the number 4
1 iteration of the number 6 : The number 6 (six) is the symbol of harmony. It represents balance, understanding, happiness.... Find out more about the number 6
1 iteration of the number 3 : The number 3 (three) is the symbol of the trinity. He also represents the union.... Find out more about the number 3 | 583 | 2,043 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.984375 | 3 | CC-MAIN-2024-22 | latest | en | 0.756256 |
https://slideplayer.com/slide/4798246/ | 1,627,987,970,000,000,000 | text/html | crawl-data/CC-MAIN-2021-31/segments/1627046154457.66/warc/CC-MAIN-20210803092648-20210803122648-00549.warc.gz | 538,194,856 | 23,772 | Slide 15-1Copyright © 2003 Pearson Education, Inc. Exchange rates and the Foreign Exchange Market Money, Interest Rates and Exchange Rates Price Levels.
Presentation on theme: "Slide 15-1Copyright © 2003 Pearson Education, Inc. Exchange rates and the Foreign Exchange Market Money, Interest Rates and Exchange Rates Price Levels."— Presentation transcript:
Slide 15-1Copyright © 2003 Pearson Education, Inc. Exchange rates and the Foreign Exchange Market Money, Interest Rates and Exchange Rates Price Levels and the Exchange Rate in the Long Run Chapter II Exchange Rates
Slide 15-2Copyright © 2003 Pearson Education, Inc. Introduction The model of long-run exchange rate behavior provides the framework that actors in asset markets use to forecast future exchange rates. Predictions about long-run movements in exchange rates are important even in the short run. In the long run, national price levels play a key role in determining both interest rates and the relative prices at which countries’ products are traded. The theory of purchasing power parity (PPP) explains movements in the exchange rate between two countries’ currencies by changes in the countries’ price levels.
Slide 15-3Copyright © 2003 Pearson Education, Inc. The Law of One Price Law of one price Identical goods sold in different countries must sell for the same price when their prices are expressed in terms of the same currency. –This law applies only in competitive markets free of transport costs and official barriers to trade. –Example: If the dollar/pound exchange rate is \$1.50 per pound, a sweater that sells for \$45 in New York must sell for £30 in London.
Slide 15-4Copyright © 2003 Pearson Education, Inc. –It implies that the dollar price of good i is the same wherever it is sold: P i US = (E \$/€ ) x (P i E ) where: P i US is the dollar price of good i when sold in the U.S. P i E is the corresponding euro price in Europe E \$/€ is the dollar/euro exchange rate The Law of One Price
Slide 15-5Copyright © 2003 Pearson Education, Inc. Purchasing Power Parity Theory of Purchasing Power Parity (PPP) The exchange rate between two counties’ currencies equals the ratio of the counties’ price levels. It compares average prices across countries. It predicts a dollar/euro exchange rate of: E PPP \$/€ = P US /P E (15-1) where: P US is the dollar price of a reference commodity basket sold in the United States P E is the euro price of the same basket in Europe
Slide 15-6Copyright © 2003 Pearson Education, Inc. Purchasing Power Parity By rearranging Equation (15-1), one can obtain: P US = (E \$/€ ) x (P E ) PPP asserts that all countries’ price levels are equal when measured in terms of the same currency.
Slide 15-7Copyright © 2003 Pearson Education, Inc. The Relationship Between PPP and the Law of One Price The law of one price applies to individual commodities, while PPP applies to the general price level. If the law of one price holds true for every commodity, PPP must hold automatically for the same reference baskets across countries if each commodity is traded. Purchasing Power Parity
Slide 15-8Copyright © 2003 Pearson Education, Inc. Absolute PPP and Relative PPP Absolute PPP –It states that exchange rates equal relative price levels. Relative PPP –It states that the percentage change in the exchange rate between two currencies over any period equals the difference between the percentage changes in national price levels. –Relative PPP between the United States and Europe would be: (E \$/€,t - E \$/€, t –1 )/E \$/€, t –1 = US, t - E, t (15-2) where: t = inflation rate Purchasing Power Parity
Slide 15-9Copyright © 2003 Pearson Education, Inc. Monetary approach to the exchange rate A theory of how exchange rates and monetary factors interact in the long run. The Fundamental Equation of the Monetary Approach Price levels can be expressed in terms of domestic money demand and supplies: –In the United States: P US = M s US /L (R \$, Y US ) (15-3) –In Europe: P E = M s E /L (R €, Y E ) (15-4) A Long-Run Exchange Rate Model Based on PPP
Slide 15-10Copyright © 2003 Pearson Education, Inc. The monetary approach makes a number of specific predictions about the long-run effects on the exchange rate of changes in: –Money supplies –An increase in the U.S. (European) money supply causes a proportional long-run depreciation (appreciation) of the dollar against the euro. –Interest rates –A rise in the interest rate on dollar (euro) denominated assets causes a depreciation (appreciation) of the dollar against the euro. –Output levels –A rise in U.S. (European) output causes an appreciation (depreciation) of the dollar against the euro. A Long-Run Exchange Rate Model Based on PPP
Slide 15-11Copyright © 2003 Pearson Education, Inc. The empirical support for PPP and the law of one price is weak in recent data. The prices of identical commodity baskets, when converted to a single currency, differ substantially across countries. Relative PPP is sometimes a reasonable approximation to the data, but it performs poorly. Empirical Evidence on PPP and the Law of One Price
Slide 15-12Copyright © 2003 Pearson Education, Inc. Empirical Evidence on PPP and the Law of One Price Figure 15-3: The Dollar/DM Exchange Rate and Relative U.S./German Price Levels, 1964-2000
Slide 15-13Copyright © 2003 Pearson Education, Inc. The failure of the empirical evidence to support the PPP and the law of one price is related to: Trade barriers and nontradables Departures from free competition International differences in price level measurement Explaining the Problems with PPP
Slide 15-14Copyright © 2003 Pearson Education, Inc. Trade Barriers and Nontradables Transport costs and governmental trade restrictions make trade expensive and in some cases create nontradable goods. –The greater the transport costs, the greater the range over which the exchange rate can move. Explaining the Problems with PPP
Slide 15-15Copyright © 2003 Pearson Education, Inc. Departures from Free Competition When trade barriers and imperfectly competitive market structures occur together, linkages between national price levels are weakened further. Pricing to market –When a firm sells the same product for different prices in different markets. –It reflects different demand conditions in different countries. –Example: Countries where demand is more price-inelastic will tend to be charged higher markups over a monopolistic seller’s production cost. Explaining the Problems with PPP
Slide 15-16Copyright © 2003 Pearson Education, Inc. International Differences in Price Level Measurement Government measures of the price level differ from country to country because people living in different counties spend their income in different ways. PPP in the Short Run and in the Long Run Departures from PPP may be even greater in the short- run than in the long run. –Example: An abrupt depreciation of the dollar against foreign currencies causes the price of farm equipment in the U.S. to differ from that of foreign’s until markets adjust to the exchange rate change. Explaining the Problems with PPP
Slide 15-17Copyright © 2003 Pearson Education, Inc. Explaining the Problems with PPP Figure 15-4: Price Levels and Real Incomes, 1992
Slide 15-18Copyright © 2003 Pearson Education, Inc. The Real Exchange Rate It is a broad summary measure of the prices of one country’s goods and services relative to the other's. It is defined in terms of nominal exchange rates and price levels. The real dollar/euro exchange rate is the dollar price of the European basket relative to that of the American: q \$/€ = (E \$/€ x P E )/P US (15-6) –Example: If the European reference commodity basket costs €100, the U.S. basket costs \$120, and the nominal exchange rate is \$1.20 per euro, then the real dollar/euro exchange rate is 1 U.S. basket per European basket. Beyond Purchasing Power Parity: A General Model of Long-Run Exchange Rates
Slide 15-19Copyright © 2003 Pearson Education, Inc. Real depreciation of the dollar against the euro –A rise in the real dollar/euro exchange rate –That is, a fall in the purchasing power of a dollar within Europe’s borders relative to its purchasing power within the United States –Or alternatively, a fall in the purchasing power of America’s products in general over Europe’s. A real appreciation of the dollar against the euro is the opposite of a real depreciation. Beyond Purchasing Power Parity: A General Model of Long-Run Exchange Rates
Slide 15-20Copyright © 2003 Pearson Education, Inc. Demand, Supply, and the Long-Run Real Exchange Rate In a world where PPP does not hold (if some goods are non-traded), the long-run values of real exchange rates depend on demand and supply conditions. Beyond Purchasing Power Parity: A General Model of Long-Run Exchange Rates
Slide 15-21Copyright © 2003 Pearson Education, Inc. There are two specific causes that explain why the long-run values of real exchange rates can change: –A change in world relative demand for American products –An increase (fall) in world relative demand for U.S. output causes a long-run real appreciation (depreciation) of the dollar against the euro. –A change in relative output supply –A relative expansion of U.S (European) output causes a long- run real depreciation (appreciation) of the dollar against the euro. Beyond Purchasing Power Parity: A General Model of Long-Run Exchange Rates
Slide 15-22Copyright © 2003 Pearson Education, Inc. Nominal and Real Exchange Rates in Long-Run Equilibrium Changes in national money supplies and demands give rise to the proportional long-run movements in nominal exchange rates and international price level ratios predicted by the relative PPP theory. From Equation (15-6), one can obtain the nominal dollar/euro exchange rate, which is the real dollar/euro exchange rate times the U.S.-Europe price level ratio: E \$/€ = q \$/€ x (P US /P E ) (15-7) Beyond Purchasing Power Parity: A General Model of Long-Run Exchange Rates
Slide 15-23Copyright © 2003 Pearson Education, Inc. Equation (15-7) implies that for a given real dollar/euro exchange rate, changes in money demand or supply in Europe or the U.S. affect the long-run nominal dollar/euro exchange rate as in the monetary approach. –Changes in the long-run real exchange rate, however, also affect the long-run nominal exchange rate. Beyond Purchasing Power Parity: A General Model of Long-Run Exchange Rates
Slide 15-24Copyright © 2003 Pearson Education, Inc. The most important determinants of long-run swings in nominal exchange rates (assuming that all variables start out at their long-run levels): A shift in relative money supply levels A change in relative output demand A change in relative output supply Beyond Purchasing Power Parity: A General Model of Long-Run Exchange Rates
Slide 15-25Copyright © 2003 Pearson Education, Inc. When all disturbances are monetary in nature, exchange rates obey relative PPP in the long run. In the long run, a monetary disturbance affects only the general purchasing power of a currency. –This change in purchasing power changes equally the currency’s value in terms of domestic and foreign goods. When disturbances occur in output markets, the exchange rate is unlikely to obey relative PPP, even in the long run. Beyond Purchasing Power Parity: A General Model of Long-Run Exchange Rates
Slide 15-26Copyright © 2003 Pearson Education, Inc. Beyond Purchasing Power Parity: A General Model of Long-Run Exchange Rates Table 15-1: Effects of Money Market and Output Market Changes on the Long-Run Nominal Dollar/Euro Exchange Rate, E \$/€ 3.
Slide 15-27Copyright © 2003 Pearson Education, Inc. Beyond Purchasing Power Parity: A General Model of Long-Run Exchange Rates Figure 15-5: The Real Dollar/Yen Exchange Rate, 1950-2000
Slide 15-28Copyright © 2003 Pearson Education, Inc. Figure 15-6: Sectoral Productivity Growth Differences and the Change in the Relative Price of Nontraded Goods, 1970-1985 Beyond Purchasing Power Parity: A General Model of Long-Run Exchange Rates
Slide 15-29Copyright © 2003 Pearson Education, Inc. In general, interest rate differences between countries depend not only on differences in expected inflation, but also on expected changes in the real exchange rate. Relationship between the expected change in the real exchange rate, the expected change in the nominal rate, and expected inflation: (q e \$/€ - q \$/€ )/q \$/€ = [(E e \$/€ - E \$/€ )/E \$/€ ] – ( e US - e E ) (15-8) International Interest Rate Differences and the Real Exchange Rate
Slide 15-30Copyright © 2003 Pearson Education, Inc. Combining Equation (15-8) with the interest parity condition, the international interest gap is equal to: R \$ - R € = [(q e \$/€ - q \$/€ )/q \$/€ ] + ( e US - e E ) (15-9) Thus, the dollar-euro interest difference is the sum of two components: –The expected rate of real dollar depreciation against the euro –The expected inflation difference between the U.S. and Europe When the market expects relative PPP to prevail, the dollar-euro interest difference is just the expected inflation difference between U.S. and Europe. International Interest Rate Differences and the Real Exchange Rate
Slide 15-31Copyright © 2003 Pearson Education, Inc. Real Interest Parity Economics makes an important distinction between two types of interest rates: Nominal interest rates –Measured in monetary terms Real interest rates –Measured in real terms (in terms of a country’s output) –Referred to as expected real interest rates
Slide 15-32Copyright © 2003 Pearson Education, Inc. Real Interest Parity The expected real interest rate (r e ) is the nominal interest rate (R) less the expected inflation rate ( e ). Thus, the difference in expected real interest rates between U.S. and Europe is equal to: r e US – r e E = (R \$ - e US ) - (R € - e E ) By combining this equation with Equation (15-9), one can obtain the desired real interest parity condition: r e US – r e E = (q e \$/€ - q \$/€ )/q \$/€ (15-10)
Slide 15-33Copyright © 2003 Pearson Education, Inc. Real Interest Parity The real interest parity condition explains differences in expected real interest rates between two countries by expected movements in the real exchange rates. Expected real interest rates in different countries need not be equal, even in the long run, if continuing change in output markets is expected.
Slide 15-34Copyright © 2003 Pearson Education, Inc. Exchange Rates Management Textbook, Table 17-1 p. 484 Industrialized countries operate under a hybrid system of managed floating exchange rates. –A system in which governments attempt to moderate exchange rate movements without keeping exchange rates rigidly fixed. A number of developing countries have retained some form of government exchange rate fixing. Regional currencies arrangements See Textbook: Krugman and Obstfeld, Table 17-1 p. 482-84
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https://news.ycombinator.com/item?id=11636002 | 1,550,735,488,000,000,000 | text/html | crawl-data/CC-MAIN-2019-09/segments/1550247503249.58/warc/CC-MAIN-20190221071502-20190221093502-00161.warc.gz | 622,530,455 | 16,463 | A beginner's guide to Big O notation (2009) 312 points by g4k on May 5, 2016 | hide | past | web | favorite | 65 comments
On the topic of efficient algorithms, I recently read a nice note in an Algorithms textbook -> It would appear that Moore’s law provides a disincentive for developing polynomial algorithms. After all, if an algorithm is exponential, why not wait it out until Moore’s law makes it feasible? But in reality the exact opposite happens: Moore’s law is a huge incentive for developing efficient algorithms, because such algorithms are needed in order to take advantage of the exponential increase in computer speed.Here is why. If, for example, an O(2n) algorithm for Boolean satisfiability (SAT) were given an hour to run, it would have solved instances with 25 variables back in 1975, 31 variables on the faster computers available in 1985, 38 variables in 1995, and about 45 variables with today’s machines. Quite a bit of progress—except that each extra variable requires a year and a half’s wait, while the appetite of applications (many of which are, ironically, related to computer design) grows much faster. In contrast, the size of the instances solved by an O(n) or O(n log n) algorithm would be multiplied by a factor of about 100 each decade. In the case of an O(n2) algorithm, the instance size solvable in a fixed time would be multiplied by about 10 each decade. Even an O(n6) algorithm, polynomial yet unappetizing, would more than double the size of the instances solved each decade. When it comes to the growth of the size of problems we can attack with an algorithm, we have a reversal: exponential algorithms make polynomially slow progress, while polynomial algorithms advance exponentially fast! For Moore’s law to be reflected in the world we need efficient algorithms.
In the quote, O(2n) and O(n6) are meant to be O(2^n) and O(n^6), right?
Yup, that's correct. Sadly, I can't update the comment now :(
Looks like someone finally got it to the front page:
Looks like the asymptotic complexity of getting this article to the frontpage is pretty high.
It's all about putting the year on there.
> O(N) describes an algorithm whose performance will grow linearly and in direct proportion to the size of the input data setNitpick, but this is technically wrong. There needs to be a "no faster than" somewhere in there. Merge sort is, for example, included in O(2^n) by the actual definition of big O (vs big theta, which is defined as a tight bound). So there exist algorithms that are O(N) that don't grow linearly.f(x) is O(g(x)) just means there is some constant K such that f(x) <= Kg(x) for all x beyond some value. This has nothing to do with 'worst case performance'.
You'll be a lot happier if you mentally replace O(n) with Theta(n) every time you see it, rather than feeling the need to nitpick when you know that that's what people mean when they say O(n), 99% of the time.
Problem is that Theta(n) is an almost useless notation.Thus, most times people say O(n) on practice[1], they mean O(n). But most times people define O(n), they cite the Theta(n) definition.It isn't hard to put an "up to", "no more than" or "or less" at the definition, and it does not make it threatening or hard to read.[1] There's probably a small corner in Hell reserved for people that mean O(n) on average but refuse both to say that qualifier and care about what data distribution created that average.
There seems to be a misconception here that this has something to do with best/average/worst case input. "f(x) is O(g(x))" is a purely mathematical statement about two functions, and means that f(x) is smaller than or equal to (K times) g(x) for all x beyond some threshold. I.e. O(g(x)) is a set of functions that includes f(x).When we say "quicksort is O(g(n))" for some g we also need to specify, or assume, the distribution of the input (i.e. best/average/worst case), and we are free to assume whatever distribution we like. Then, assuming that distribution, the running time of the algorithm is just a function of the input size, and we can use the above definition to make a statement about that function (i.e. what complexity class it is in).A complete statement looks like "quicksort is O(n^2) in the worst case", which means "for every n, pick the list of length n that takes the longest to sort (i.e. the worst-case input), and the number of steps executed by quicksort will be no more than Kn^2 as long as n is large enough, for some fixed K".The difference between O, Theta and Omega only concerns what functions the set contains - in particular, Theta(g(x)) is a subset of O(g(x)). This is a separate issue to whether we are talking about best/average/worst case input.
I disagree.In my experience, outside of technicality-penis-measuring contests (like this thread, or a few particular minutes during the class I was leading a few hours ago), every time someone says O(foo), they're describing an algorithm that they believe to be Theta(foo). On the other hand, I've never seen someone _define_ big-Oh and accidentally define big-Theta, which is unsurprising, given that the big-Theta definition is literally twice as much work to write out as the big-Oh.Relatedly, Theta is a highly useful concept, except for that fact that it's much easier to type O(n) and also mildly faster to whiteboard it (relative to Theta(n)). Why do you think it's almost useless? Mind you, I don't think it's useful enough to care about the fact that most non-specialists can't keep track of the difference between these complexity classes.Also, I question your claims about Hell, and what I take to be your implication that you disapprove of both saying using big-Oh to refer to average-case without saying so, and of saying "average" without specifying a distribution or an aggregation function.To the former, that's how people use the language, and meaning average-case is 100% as reasonable as assuming that big-Oh always means worst-case. Big-Oh is a system for dividing arbitrary functions into nested equality classes (well, I guess maybe the right name for this is inequality classes?), and <> is just as useful an input function as <>. There's no technical reason, and no pragmatic reason, to insist otherwise.As for distribution, if someone doesn't specify distribution, it means they're implying that it's true across all the reasonable distributions that they can think would apply, just like it always means something like that when a human leaves out a detail. Every single time I've looked into a case like this, the assumed distribution is a uniform random distribution of all possible inputs. There are always things left unsaid, why claim that this particular elision is a sin?
What do you mean? Why's Theta(n) useless?
Do you mean O(n^2)? Because merge sort certainly won't be exponential.
No, I mean O(2^n). O(2^n) includes everything that grows slower than 2^n as well. It is no relation to the worst case performance of merge sort, just a clarification on the definition of Big O.
> Big O specifically describes the worst-case scenario, and can be used to describe the execution time required or the space used (e.g. in memory or on disk) by an algorithm.From higher up in the article. You picked the O(N) section description to pick a nit with the whole article when it covers exactly what you want it to earlier.
That's not what the grandparent is getting at. Big O describes an upper bound (and often, but not necessarily, refers to the worst-case scenario).
I like this page http://bigocheatsheet.com/Espcially the graph which hammers home how quickly things go wrong.
That table is so wrong it makes me mad.
In what way? Having skimmed it, I see a couple of minor mistakes but the bulk of it seems correct.The chart says you can search a Cartesian tree in O(log n) expected time, but I don't see how that's possible given that there's no ordering between a node's children.Also, it's possible to do mergesort in O(1) space; in fact, it's one of the exercises in TAOCP, if I recall correctly. But the algorithm is so complicated that nobody uses it in practice.
The article is mostly good, but I have one nitpick.> An example of an O(2^N) function is the recursive calculation of Fibonacci numbersNo, the naive recursive evaluation of the Fibonacci sequence has complexity O(1.618^N) (or just O(Fib(n)). It is unequal to O(2^N) because the base of the power is different.
To be fair, as Lxr points out below, big O notation is an upper bound on the growth rate of a function, and since 2^N grows faster (asymptotically) than (1.618...)^N, O(Fib(N)) <= O(1.618...^N) <= O(2^N) (with some abuse of notation), and the article is technically correct. You may be thinking of big theta notation, which requires both an upper bound and lower bound.This example is helpful: https://en.wikipedia.org/wiki/Big_O_notation#Use_in_computer...
You are right. I made a mistake, not thinking through big O vs. theta notation. Thanks.
Technically 1.618^N is still O(2^N) but I agree with your point.
I was about to correct you until I realized you used big-O instead of big-theta notation.Informally we tend to use big-O to mean big-theta which only adds to the confusion.
Actually we don't use it informally as big-Theta, because big-Theta assumes that the lower and upper bound are asymptotically the same.For example Quicksort is O(n^2) but Omega(nlogn) it is neither Theta(nlogn) nor Theta(n^2).You probably meant that informally we just assume that the stated bound is as tight as possible.
> For example Quicksort is O(n^2) but Omega(nlogn) it is neither Theta(nlogn) nor Theta(n^2).No, this is flat out wrong. Big O is an upper bound on an asymptotic growth rate. Big Omega is a lower bound on the asymptotic growth rate. Big Theta is a tight bound on the asymptotic growth rate. These are independent to the average case, best case, or worst case run time of a given algorithm.Quicksort has an average run time of Theta(n lg n). Equivalently, its average run time is O(n lg n) and Omega(n lg n). It has a worst case run time of Theta(n^2). Equivalently, its worst case run time is O(n^2) and Omega(n^2).> Actually we don't use it informally as big-Theta,Wrong again.
You can state that quicksort has run time O(n^2). The function you're describing with O(n^2) is f : S -> R, where S is the set of input values and R is the set of real numbers, and f(x) is the running time of running quicksort on input value x, and n : S -> N is the size of input value x. The notation O(g(n)) describes the function f in terms of the function n.That f is in O(g(n)) means there exist constants C and N_0 such that for all x in S, provided that n(x) >= N_0, it is true that |f(x)| <= |C g(n(x))|.And that f is in Omega(g(n)) means there exist C > 0 and N_0 such that for all x in S, provided that n(x) >= N_0, it is true that |f(x)| >= |C g(n(x))|.
Perhaps it is you that need to be educated.They are independent only if you explicitly state that you are computing them for best/average/worst case independently. Is it commonly done? Sure. But there is nothing in the definition that forces us to do that.If the things were as you state would you have any use for Omega and Omicron then? Wouldn't just Theta suffice?
> But there is nothing in the definition that forces us to do that.That's true. You're right.> If the things were as you state would you have any use for Omega and Omicron then? Wouldn't just Theta suffice?Couldn't there be cases where we don't have a known tight asymptotic bound but do have an upper and/or lower bound? And although it's an abuse of notation, you do often see big-O used in place of Theta. From CLRS:"In the literature, we sometimes find O-notation informally describing asymptotically tight bounds, that is, what we have defined using Theta-notation."
My interpretation is that "O(...)" informally means "Theta(...) in the average/amortized case". In the average case, Quicksort is Theta(n log n).
Common mistake. When people say O(2^n) they USUALLY mean something closer to 2^O(n).The reason is that f(n) = O(g(n)) means that for some constant k and integer N, if N < n then |f(n)| < kg(n). In other words "grows no faster than a constant times". However when you've got exponential growth the slightest of things can cause the exponent to be slightly different, or there to be a small polynomial factor.That was the case in his example. (The exponent was different.)
The best layman's example of O(log N) I've heard is finding a name in the phone book. You open it around the middle, select which half gets you closer, and repeat. It's then easy to "get" why it's not a big deal if it's a huge phone book versus a tiny phone book.
That seems more like a layman's explanation of a binary search than O(log N). Binary search just happens to be O(log N)
Binary search is an example of a O(log N) algorithm, as OP said. Having tangible examples is great for explaining algorithmic complexity to people.One I've used is sorting with a deck of cards. Take a single suit, A-K, and have people sort it using bubble or insertion or other sorts. It's only 13 cards, but the difference even between some of the O(n^2) algorithms becomes obvious when swapping is postponed until the latest stage versus earliest.Then you can also demonstrate radix sort. One of the most underrated sorting algorithms, in my opinion. A lot of people focus on the (potentially) large constant. But it can be appropriate, or may be reduced depending on your use-case. And it's a fantastic sort when you can't give a neat numeric quality for sorting and have to sort over multiple dimensions. Again with cards, you can do two passes. One with buckets by value, and one with buckets by suit. After the second, you collect each of the 4 stacks and the deck is sorted.Cards, phone books, recipes, these are things that most people encounter (ok, maybe not phone books anymore), in their lives. They offer great aids to demonstrating CS concepts to either new students or people who have only a passing curiosity (or none but they shouldn't have asked :P).
I'm not sure how drawing that distinction helps a layman.
Clarifying something as an example of a principle instead of the principle itself seems pretty important, regardless of lay status.
Ah, when you phrase it that way I see what you mean, I agree.
A phone book search does O(log N) string comparisons, but its running time is not O(log N) unless you consider all string comparisons to take a constant amount of time. Because for the names in the phone book to be distinct, the strings have to be about log N in size, so each string comparison costs O(log N) in the worst case. So the cost of phone book search is O((log N)^2).I've skipped some details here: string comparisons can finish early, and maybe N should be the size of the phone book rather than the number of names in the phone book -- but even if you consider these factors, the runtime complexity is still more than O(log N).
Thats not a good example for Big O because number of operations computes exactly to logN in worst case.Try with mergesort and see how it ends being nlogn
Isn't O(log n) just an upper bound limit ? So isn't it more correct to say, that in worst case the number of steps are < O(log n) ?
Big-O describes an upper limit in itself. Something that is O(N) is also O(2^N). No less-than needed, it's included in the definition of big-O.
Yes, but context matters; Big O is often used informally. The interviewer's going to look at you weird when you tell him the O(N) algorithm is O(2^N).
We have the same problem with things being "in NP". It's clear that all of P is in NP, but not clear whether all of NP is in P. But people often say "NP" with the implication of "(apparently, as far as we know) not in P", which is not actually correct based on the meanings of those terms.(In this case the problem is probably made worse by the mental interpretation of "NP" as "Not Polynomial", when it really means "Nondeterministic machine can solve in Polynomial time", and if a deterministic machine can solve something in polynomial time, a nondeterministic machine can do so as well!)
A phone book? How is Amazon's Kindle App O(log N)? ;-)
Nice! Very succinct and clear. I wrote an intro myself, as someone who'd recently learned the concept. Wordier, but might be helpful if you think like I do. http://nathanmlong.com/2015/03/understanding-big-o-notation/
Super interesting comparing this to the one we have on Interview Cake: https://www.interviewcake.com/article/big-o-notation-time-an...I like how Rob Bell's piece has headings with the most common time costs--sorta makes it easier to see at a glance what you're going to get, and I imagine gives folks a quick sense of, "Right, I've seen that! Been /wondering/ what that means."
I think the nicest way to learn this if you don't have a formal CS education is to still pick up a Discrete Math text book (like Chapter 3 in Rosen) and then read chapters 3 and 4 in CLRS.
If something puzzles you, don't be afraid to start drawing yourself a diagram. O(n^2) vs. O(n) amortized for naive string concatenation vs. growing a buffer by doubling:`````` x xx xxx xxxx xxxxx xxxxxx xxxxxxx xxxxxxxx x xx xxxx xxxxxxxx `````` This can be a good way of getting an intuitive feel for what's going on. Note in the 2nd case, you can stack all of 1st two rows representing cost into the 3rd row. In fact, you can always stack the execution costs into row (n-1) and never exceed the size of row n.
Looks like you're alluding to the recurrence tree method for solving recurrences mentioned in CLRS. And you're right, it's a great way to visualize and get an intuitive feel of big-oh
It would be sensible to put the terms like "constant" and "logarithmic" in there, IMO.
Is there a beginner's guide to proving different time complexities?
> O(N) describes an algorithm whose performance will grow linearly and in direct proportion to the size of the input data set.Argh, I hate this every time I see Big O notation covered.Big O != performance. If you have an O(N) algorithm that walks the data set in the order that it's laid out in memory(assuming contiguous data) it will beat your O(NlogN) and sometimes even O(logN).meant to omit nlogn, that's what I get for any early morning rant pre-coffee.Radix Sort is the classic example I always bring up. On machine word size keys, with a separate pointer look-up table(to get final value) it will beat QSort, MergeSort and all the other NlogN sorts by 10-50x. This includes having to walk the data 3-4 times depending on how you want to split the radix to line up with cache sizes.Friends don't let Friends use Big O to describe absolute performance.
When talking about theory, I have two points: - First: O(N) < O(N log N), so your example isn't good - Second: Your description of Radix sort is actually O(N), and the standard sorting algorithms are O(N log N), so it again isn't a surprise, theoretically speaking...The thing that Big-O knowingly hides is the constant factor. For example, some algorithms are 2N and some are 1000N. For large enough N, this may not matter, but for any reasonably sized N, it does.In practice, of course that Big-O is not sufficient to describe absolute algorithm performance, as it depends on so many factors such as implementation, computer architecture, relative size of data, etc. But, when looking at an algorithm and knowing its Big-O complexity can often be a useful input to your decision on whether to use it. So I disagree with this Big-O bashing...
> If you have an O(N) algorithm that walks the data set in the order that it's laid out in memory(assuming contiguous data) it will beat your O(NlogN)Not sure what you're trying to say with this particular statement. Of course it will. NLogN grows way faster than N.And Radix sort is of course drastically faster than comparison sorts if the word size is smaller than logN regardless of the particularities of the implementation.People don't and shouldn't use big O to describe absolute performance but it's a great place to start and you can't reach the level of implementation details performance tuning if you can't understand or formalize the basics.But you do have a point overall.In practice, performance may be different. Big O assumes a particular 'virtual machine' with a particular word size and a particular time it takes to execute an instruction.
Yeah, meant to omit NlogN there, I blame it on lack of coffee.My meta point is the constant factors that Big O throws out usually end up dominating real-world performance.
That's why some Data Stuctures courses use tilde notation instead (same as big O, but the constant is preserved).
In big-O senses Radix sort is one of those more persistent misconceptions. At scale Radix sort has to be nlog(n) because you should really consider the possibility all your numbers are unique.This does not even start on the reality that memory access really is O(n^(1/3)), addition is O(n) and a lot of other things we assume are O(1) are not.
The statement you quoted is correct. It doesn't say "O(log(N)) algorithms are faster than O(N) algorithms." It says performance of a particular algorithm degrades linearly as data set grows, if that algorithm is O(N). Which is true. (asymptotically)
I was nitpicking the performance part of the statement. N does scale linearly.The point I was trying(and failed) to drive home is that it matters how you walk your N not just what the N is.If you've got to do a pointer indirection for each N rather than a linear read that can end up dominating the non-constant factors in a lot of cases.
You're not understanding what I'm saying. It's of course true that some O(N) algorithms are slower than other O(N) algorithms. But the statement you quote doesn't imply the contrary. Obviously walking an array can be thousands of times faster than walking a linked list. That doesn't change the fact that walking an array of size 3*N is 3 times slower than walking an array of size N, which is (roughly) what big-O notation means and what the quote you're objecting to correctly claims.> If you've got to do a pointer indirection for each N rather than a linear read that can end up dominating the non-constant factors in a lot of cases.Huh? Pointer indirection and cache miss for each element is already non-constant (you have to do it N times) so I don't understand what it means that this "can end up dominating the non-constant factors".
Not really.Take the linked-list case for example. Let's say the allocator you have is a small-block allocator, of which most good malloc implementations use for linked-list style allocations.Since the SBA allocates in chunks your cache locality(driving your "constant-factor" performace) isn't linear any more. It'll change in jumps & starts as you break chunk boundraries. Also another class allocating in-between your linked list allocations(really common) will break up the chunk locality and now your real-world O(N) performance no longer scales linearly.
You're overlooking the word "grow".There's nothing wrong with that sentence. No absolutes.
Of course an O(N) method will beat an O(NlgN) method, because O(N) < O(NlgN).Radix sort is a linear time sort because the hardware (random access memory) is built to allow constant-time lookup of memory. Radix sort wouldn't be linear time on a sequential access memory system, like tape, but merge sort would still be O(NlgN) on such a system.
It doesn't invalidate what you say, but you're assuming there's a bounded number of different values in the array -- 2^k where k is your word size.Under this assumption, a correctly implemented quicksort also runs in O(N) time.
>By one estimate, the health care costs of obesity are responsible for nearly 21 percent of total health care spending in the U.S.BigO has nothing to do with performance even though that what we most use it for.Its simply a classification of growth characteristics, relationship between two functions.
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Search: | 5,610 | 24,105 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.375 | 3 | CC-MAIN-2019-09 | latest | en | 0.913425 |
http://www.studymode.com/essays/Experiment-1-Application-Of-Statistical-Concepts-1248905.html | 1,524,449,787,000,000,000 | text/html | crawl-data/CC-MAIN-2018-17/segments/1524125945669.54/warc/CC-MAIN-20180423011954-20180423031954-00290.warc.gz | 518,085,171 | 24,879 | # Experiment 1 Application of Statistical Concepts in the Determination of Weight Variation in Samples
Pages: 8 (2315 words) Published: November 22, 2012
Experiment 1: APPLICATION OF STATISTICAL CONCEPTS IN THE DETERMINATION OF WEIGHT VARIATION IN SAMPLES LEE, Hyun Sik
Chem 26.1 WFV/WFQR1
-------------------------------------------------
Nov. 23, 2012
A skillful researcher aims to end his study with a precise and accurate result. Precision refers to the closeness of the values when some quantity is measured several times; while accuracy refers to the closeness of the values to the true value. The tool he utilizes to prevent errors in precision and accuracy is called statistics. In order to become familiar to this tactic, the experiment aims to help the researchers become used to the concepts of statistical analysis by accurately measuring the weights of ten (10) Philippine 25-centavo coins using the analytical balance, via the “weighing by difference” method. Then, the obtained data divided into two groups and are manipulated to give statistical significance, by performing the Dixon’s Q-test, and solving for the mean, standard deviation, relative standard deviation, range, relative range, and confidence limit—all at 95% confidence level. Finally, the results are analyzed between the two data sets in order to determine the reliability and use of each statistical function.
RESULTS AND DISCUSSION
This simple experiment only involved the weighing of ten 25-centavo coins that are circulating at the time of the experiment. In order to practice calculating for and validating accuracy and precision of the results, the coins were chosen randomly and without any restrictions. This would give a random set of data which would be useful, as a statistical data is best given in a case with multiple random samples. Following the directions in the Analytical Chemistry Laboratory Manual, the coins were placed on a watch glass, using forceps to ensure stability. Each was weighed according to the “weighing by difference” method. The weighing by difference method is used when a series of samples of similar size are weighed altogether, and is recommended when the sample needed should be protected from unnecessary atmosphere exposure, such as in the case of hygroscopic materials. Also, it is used to minimize the chance of having a systematic error, which is a constant error applied to the true weight of the object by some problems with the weighing equipment. The technique is performed with a container with the sample, in this experiment a watch glass with the coins, and a tared balance, in this case an analytical balance. The procedure is simple: place the watch glass and the coins inside the analytical balance, press ON TARE to re-zero the display, take the watch glass out, remove a coin, then put the remaining coins back in along with the watch glass. Then, the balance should give a negative reading, which is subtracted from the original 0.0000g (TARED) to give the weight of the last coin. The procedure is repeated until the weights of all the coins are measured and recorded. The weights of the coins are presented in table 1, as these raw data are vital in presenting the results of this experiment. Table 1. Weights of 25-centavo coins measured using the “weighing by difference” method| Sample No.| Weight, g|
1| 3.6072| Data Set 2| Data Set 1|
2| 3.7549| | |
3| 3.6002| | |
4| 3.5881| | |
5| 3.5944| | |
6| 3.5574| | |
7| 3.5669| |
8| 3.5919| |
9| 3.5759| |
10| 3.6485| |
Note that the data are classified into two groups, Data Set 1 which includes samples numbered 1~6 and Data Set 2 which includes samples numbered 1~10. Since the number of samples is limited to 10, the Dixon’s Q-test was performed at 95% confidence level in order to look for outliers in each data set. The decision to use the Q-test despite the fact that there were only a few, limited number of samples and to use the confidence level of 95% was carried out as specified in the Laboratory Manual. Significance of Q-test
The Dixon’s Q-test aims to identify and... | 919 | 4,081 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.171875 | 3 | CC-MAIN-2018-17 | latest | en | 0.922581 |
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