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$87 Computer Arithmetic focuses on arbitrary-precision algorithms for efficiently performing arithmetic operations such as addition, multiplication and division, and their connections to topics such as modular arithmetic, greatest common divisors, the Fast Fourier Transform (FFT), and the computation of elementary and special functions. Brent and Zimmermann present algorithms that are ready to implement in your favorite language, while keeping a high-level description and avoiding too low-level or machine-dependent details. The book is intended for anyone interested in the design and implementation of efficient high-precision algorithms for computer arithmetic, and more generally efficient multiple-precision numerical algorithms. It may also be used in a graduate course in mathematics or computer science, for which exercises are included. These vary considerably in difficulty, from easy to small research projects, and expand on topics discussed in the text. Solutions are available from the authors.
Contains all the important algorithms relevant to integers, modular arithmetic, floating-point numbers and special functions
Reviews & endorsements
"Very few books do justice to material that is suitable for both professional software engineers and graduate students. This book does just that, without losing its focus or stressing one audience over the other."
Marlin Thomas, Computing Reviews
"This is a concise, well-written and beautiful book in modern computer arithmetic. I found the book very pleasant to read."
Song Yan for SIGACT News P. Brent, Australian National University, Canberra Richard Brent is a Professor of Mathematics and Computer Science at the Australian National University, Canberra |
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Derivatives of Exponential Function Teacher Resources
Title
Resource Type
Views
Grade
RatingConstruct tables of values to understand constant percent growth rate in an exponential function. This lesson contains a handout to different problems that help provide insight into exponential functions.
Money, money, money. A complete lesson that makes use of different representations of simple and compound interest, including written scenarios, tables, graphs, and equations to highlight similarities and differences between linear and exponential functions.
Introduce the concept of exponential functions with an activity that extends the definition of exponents to include rational values. Start with a doubling function at integer values of time, then expand table to include frational time units. Lesson includes a detailed commentary on how to work each problem.
Ninth graders investigate the functional relationship of different environmental phenomena. In this math lesson, 9th graders create models of various natural disasters. They use logarithmic and exponential functions to interpret population growth.
Twelfth graders investigate the derivative of a function. In this calculus lesson, 12th graders explore the derivatives of sine, cosine, natural log, and natural exponential functions. The lesson promotes the idea of the derivative as a function and uses numerical and graphical investigations to form conjectures about common derivative formulas.
Students study exponential decay and its application to radiocarbon dating. In this exponential decay lesson, students use candy to model the time it takes for something to decay. Students also graph the data they collect and describe using an algebraic formula that gives the age of an object as a function.
High schoolers explore the calculator function of the TI-Nspire. In this secondary mathematics lesson, students investigate many of the features of the calculator function of the TI-Nspire. High schoolers review basic computation, square roots, absolute values, exponential functions, logarithmic functions, trigonometric functions, summations and matrices as they explore the TI-Npsire.
Ninth graders investigate exponential regression. For this Algebra I lesson, 9th graders explore the rebound heights of a racquetball bouncing and develop an exponential data model. The lesson is intended to be an introduction to exponential regression.
Students explore the concept of piecewise functions. In this piecewise functions lesson, students discuss how to make a piecewise function continuous and differentiable. Students use their Ti-89 to find the limit of the function as it approaches a given x value. Students find the derivative of piecewise functions.
Twelfth graders investigate derivatives. In this calculus lesson, 12th graders use technology to explore the basic derivatives and how to choose the proper formula to use them. The lesson requires the use of the TI-89 or Voyage and the appropriate application.
Sal continues his discussion of decay by showing students the math involved in determining how much a substance is left after one half-life, two half-lives, and even three half-lives have gone by. He sets up a general function of time that can be used to determine the remaining amount of a substance after 10 minutes, or three billion years have elapsed!
In this complex exponential activity, students identify an entire function and explore how to show a function is analytic. This two-page activity contains four problems, as well as explanations and examples.
In this function worksheet, students use various methods to solve functions. They explore the logarithm function, the derivative of an exponential function, and compose a function with a linear equation. This four-page worksheet contains explanations, examples, and four problems.
Students will solve exponential function problems, graph exponential functions, find the value of logarithms, determine future value, and compound interest. In this Pre-calculus lesson, students will use the properties of logs to solve test problems.
In this radiation and temperature worksheet, students use 2 methods to derive the Wein Displacement law that shows the relationship between the temperature of a body to the frequency where the Planck curve's value is at its maximum. Students are given data of the temperatures and wavelengths for 12 different bodies and they find a formula that fits the data.
The major goal of this lesson is to collect data from a variety of experiments, and then determine what type of model best fits the data, and explain why. Students will explore a variety of relationships using pennies, pressure, temperature, light, and pe |
The mission of the Mathematics Department is to build an analytical academic foundation for students to be able to transfer to four-year institutions, enter a vocational program, or seek employment where they would utilize their analytical training.
DEGREE PROGRAMS
Degree: AS - Mathematics - Plan A
Degree Learning Outcomes
The student will:
1.
Graph functions using the first and second derivatives and use integration to find areas and volumes bounded by functions.
2.
Identify and graph the appropriate (logarithmic or exponential) function to model the situation.
3.
Set up a triple integral to find the volume of a given solid bounded by the graphs of equations of surfaces, then evaluate the integral using multivariate change of variables.
4.
Find the four fundamental subspaces of the coefficient matrix of an over-determined system of equations and relate these subspaces to the least squares solution. The student will find the least squares solution to the system of equations .
5.
Given a higher order, linear differential equation, the student will solve it by three methods: Laplace Transform, method of eigenvalues, and solution by series substitution. |
This is one of only a few classic introductory combinatorics textbooks for undergraduates. The author of such a textbook faces an unusally high number of difficult choices of what topics to include, and in what order, so the reviewer's task is to discuss the choices the author decided to make. Enumeration is the topic favored by the author, in that nine of his fourteen chapters have this topic at their center. This is an appropriate choice of focus since in most branches of combinatorics, counting is, if not the goal, then at least a tool.
The order of in which the topics are discussed also favors enumeration. Advanced enumeration techniques are treated relatively early in the book. Möbius inversion is explained in Chapter 6, while generating functions are introduced in Chapter 7, so these advanced topics are part of the first half of the text. There is a separate chapter on some famous counting sequences, such as the Catalan numbers and the Schröder numbers, and even that comes before the parts of the book that are not devoted to enumeration.
The next part of the book has gone through a major change since the last edition. Chapter 9 is about systems of distinct representatives, Chapter 10 is on Combinatorial Designs, and the following three chapters are on graph theory. This probably sets the book apart from most similar textbooks: it is easy to view both systems of distinct representatives and Combinatorial Designs as generalizations of concepts in Graph Theory, and therefore, most authors treat graphs first. The order chosen in this book probably has the effect that if you teach a two-semester course from the book, and cover all topics in it in the order in which the book covers them, then all of Graph Theory will be discussed in the second semester.
The book ends with a chapter on Pólya theory (enumeration under group action).
There are about 40 exercises at the end of each chapter, some of which are, commendably, challenging enough to be interesting even for the instructors. Roughly half of them have a numerical answer or a hint at the end of the book, but none come with full solutions. |
Algebra 1
It is imperative that students do not fall behind in math. Each day builds on itself and it is easy to get overwhelmed. I do have a policy of NO WORK = NO CREDIT. I am strict about this because I need to know a student understands the concept so that it can be applied to multiple types of problems.
Notebooks:
Students will keep an organized notebook/folder for my class. It should contain only math in it. I will give a Notebook Test at the end of each six weeks. It will consist of 10 questions that pertain to the order of the student's notebook. If a student keeps up with all math papers and has them organized, it should be an easy test grade.
Tutorials:
I am available for tutorials Monday - Thursday from 3:30 - 4:00. I can be here before school if it is scheduled in advance. Please make sure questions are asked as they come up and not at the end of the unit. As I said before, it can be ease to fall behind in math.
Bulletins
1st Day Handout
Please be sure to return your signed 1st day handout by the first Friday of school. |
10.) (+) Understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative operation, but still satisfies the associative and distributive properties. [N-VM9]
11.) [N-VM10]
ALGEBRA
Seeing Structure in Expressions
Interpret the structure of expressions. (Polynomial and rational.)
12.) Interpret expressions that represent a quantity in terms of its context.* [A-SSE1]
a. Interpret parts of an expression such as terms, factors, and coefficients. [A-SSE1a]
b. Interpret complicated expressions by viewing one or more of their parts as a single entity. [A-SSE1b]
Example: Interpret P(1+r)n as the product of P and a factor not depending on P.
13.) Use the structure of an expression to identify ways to rewrite it. [A-SSE2]
13.) (+) Know and apply the Binomial Theorem for the expansion of (x + y)n in powers of x and y for a positive integer n, where x and y are any numbers, with coefficients determined, for example, by Pascal's Triangle. (The Binomial Theorem can be proved by mathematical induction or by a combinatorial argument.) [A-APR5]
Example: See x4 - y4 as (x2)2 - (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 - y2)(x2 + y2).
Write expressions in equivalent forms to solve problems.
14.) Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems.* [A-SSE4]
Example: Calculate mortgage payments.
Arithmetic With Polynomials and Rational Expressions
Perform arithmetic operations on polynomials. (Beyond quadratic.)
15.)15.) (+) Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions. [A-APR7]
Understand the relationship between zeros and factors of polynomials.
16.) Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x - a is p(a), so p(a) = 0 if and only if (x - a) is a factor of p(x). [A-APR2]
17.) Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial. [A-APR3]
Use polynomial identities to solve problems.
18.) Prove polynomial identities and use them to describe numerical relationships. [A-APR4]
19.) Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or for the more complicated examples, a computer algebra system. [A-APR6]
Creating Equations*
Create equations that describe numbers or relationships. (Equations using all available types of expressions, including simple root functions.)
20.) Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. [A-CED1]
21.) Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. [A-CED2]
22.) Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. [A-CED3]
Example: Represent inequalities describing nutritional and cost constraints on combinations of different foods.
23.) Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. [A-CED4]
Example: Rearrange Ohm's law V = IR to highlight resistance R.
Reasoning With Equations and Inequalities
Understand solving equations as a process of reasoning and explain the reasoning. (Simple rational and radical.)
24.) Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. [A-REI2]
Solve equations and inequalities in one variable.
25.) Recognize when the quadratic formula gives complex solutions, and write them as a ± bi for real numbers a and b. [A-REI4b] (Alabama)
Solve systems of equations.
26.) (+) Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3 x 3 or greater). [A-REI9] (Alabama)
27.)FUNCTIONS
Conic Sections
Understand the graphs and equations of conic sections. (Emphasize understanding graphs and equations of circles and parabolas.) (Alabama)
Example: Build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.
Build new functions from existing functions. (Include simple radical, rational, and exponential functions; emphasize common effect of each transformation across function types.)
34.)
35.) Find inverse functions. [F-BF4]
Solve an equation of the form f(x) = c for a simple function f that has an inverse, and write an expression for the inverse. [F-BF4a]
Example f(x) = 2x3 or f(x) = (x+1)/(x-1) for x ≠ 1.
Linear, Quadratic, and Exponential Models*
Construct and compare linear, quadratic, and exponential models and solve problems. (Logarithms as solutions for exponentials.)
36.) For exponential models, express as a logarithm the solution to abct = d where a, c, and d are numbers, and the base b is 2, 10, or e; evaluate the logarithm using technology. [F-LE4]
Trigonometric Functions
Extend the domain of trigonometric functions using the unit circle.
37.) Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle. [F-TF1]
37.)
38.) Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle. [F-TF2]
39.) Define the six trigonometric functions using ratios of the sides of a right triangle, coordinates on the unit circle, and the reciprocal of other functions. (Alabama)
Use probability to evaluate outcomes of decisions. (Include more complex situations.)
41.) (+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator). [S-MD6]
42.) (+) Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game). [S-MD7]
Conditional Probability and the Rules of Probability
Understand independence and conditional probability and use them to interpret data. (Link to data from simulations or experiments.)
43.) Describe events as subsets of a sample space (the set of outcomes), using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events ("or," "and," "not"). [S-CP1]
44.) Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B. [S-CP3]
45.) Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities. [S-CP4]
Example: Collect data from a random sample of students in your school on their favorite subject among mathematics, science, and English. Estimate the probability that a randomly selected student from your school will favor science given that the student is in tenth grade. Do the same for other subjects and compare the results.
46.) Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. [S-CP5]
Example: Compare the chance of having lung cancer if you are a smoker with the chance of being a smoker if you have lung cancer.
Use the rules of probability to compute probabilities of compound events in a uniform probability model.
47.) Find the conditional probability of A given B as the fraction of B's outcomes that also belong to A, and interpret the answer in terms of the model. [S-CP6]
48.) Apply the Addition Rule, P(A or B) = P(A) + P(B) - P(A and B), and interpret the answer in terms of the model. [S-CP7]
49.) (+) Apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A)P(B|A) = P(B)P(A|B), and interpret the answer in terms of the model. [S-CP8]
50.) (+) Use permutations and combinations to compute probabilities of compound events and solve problems. [S-CP9] |
EducationPc Calculator is a clever note and formula editor combined with an advanced and strong scientific calculator. Being an editor it is extremely user-friendly allowing all possible typing and other errors to be easily corrected and fast recalculated |
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I remember I faced similar difficulties with converting fractions, long division and syntehtic division. This Algebrator is truly a great piece of math software program. This would simply give step by step solution to any math problem that I copied from homework copy on clicking on Solve. I have been able to use the program through several Intermediate algebra, Algebra 2 and Intermediate algebra. I seriously recommend the program.
Algebrator is a user friendly product and is certainly worth a try. You will also find several interesting stuff there. I use it as reference software for my math problems and can say that it has made learning math much more fun. |
Basic College Math - With Early Integers - 2nd edition
Elayn Martin-Gay firmly believes that every student can succeed, and her developmental math textbooks and video resources are motivated by this belief.Basic College Mathematics with Early Integers, Second Editionwas written to help students effectively make the transition from arithmetic to algebra. The new edition offers new resources like theStudent Organizerand now includesStudent Resourcesin the back of the book to help students on th...show moreeir quest for success37.73 +$3.99 s/h
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Building on rudimentary knowledge of real analysis, point-set topology, and basic algebra, Basic Algebraic Topology provides plenty of material for a two-semester course in algebraic topology.
The book first introduces the necessary fundamental concepts, such as relative homotopy, fibrations and …
This book is a comprehensive collection of known results about the Lozi map, a piecewise-affine version of the Henon map. Henon map is one of the most studied examples in dynamical systems and it attracts a lot of attention from researchers, however it is difficult to analyze analytically. Simpler …
Ever since Lorensen and Cline published their paper on the Marching Cubes algorithm, isosurfaces have been a standard technique for the visualization of 3D volumetric data. Yet there is no book exclusively devoted to isosurfaces. Isosurfaces: Geometry, Topology, and Algorithms represents the first …
Topology is a large subject with many branches broadly categorized as algebraic topology, point-set topology, and geometric topology. Point-set topology is the main language for a broad variety of mathematical disciplines. Algebraic topology serves as a powerful tool for studying the problems in … …
The Geometry of Special Relativity provides an introduction to special relativity that encourages readers to see beyond the formulas to the deeper geometric structure. The text treats the geometry of hyperbolas as the key to understanding special relativity. This approach replaces the ubiquitous ? …
Now with an extensive introduction to fractal geometry
Revised and updated, Encounters with Chaos and Fractals, Second Edition provides an accessible introduction to chaotic dynamics and fractal geometry for readers with a calculus background. It incorporates important mathematical concepts …
This video is about the problem of turning a sphere inside out, by passing the surface through itself, without making any holes or creases. Mathematicians believed the problem to be unsolvable until 1958, when Stephen Smale proved otherwise. The motion of turning a sphere inside out, called a …
Designed for mathematics majors and other students who intend to teach mathematics at the secondary school level, College Geometry: A Unified Development unifies the three classical geometries within an axiomatic framework. The author develops the axioms to include Euclidean, elliptic, and …
Origami5 continues in the excellent tradition of its four previous incarnations, documenting work presented at an extraordinary series of meetings that explored the connections between origami, mathematics, science, technology, education, and other academic fields.
The fifth such meeting, 5OSME ( |
Advanced Mathematics and Mechanics Applications Using MATLAB, 3e
This book is intended for use as a reference or supplementary text in mathematics and mechanics courses that feature computer-based exercises and emphasize physical examples. Topics covered include beam flexure, electrostatic stress analysis, vibration response of linear systems, heat conduction, eigenvalues, truss dynamics, and nonlinear cable dynamics. MATLAB is used to analyze various applications throughout the book. |
TexasInstruments CBR for datalogging. The easy, affordable way to connect math and science to the real world. Designed for teachers who want to start collecting and analysing real world motion data - distance, velocity, and acceleration. All the programs needed are sent to your calculator at the push of a single button. Start sampling right away with the handy default settings, or change your setup - it's easy! Pivoting sensor head: Point the sensor where the action is. Measures distance...
* Different representations can share dynamically-linked data using the TI-Nspire™ applications: Calculator, Graphs, Geometry, Lists & Spreadsheet, Data & Statistics, Notes, Questions and new Vernier DataQuest™. Users can also split the screen, enabling up to four applications to be viewed simultaneously. * Import colour/black and white images from online sources or from your own digital archive and overlay graphs and other elements. * Explore surfaces by rotating them, investigate intersections...
Modern and ergonomic design. Intuitive icon desktop for easy navigation and organization of Handheld Software Applications (Apps). Built-in clock to keep track of time and date and to use for timing experiments. Full QWERTY keyboard, numeric keypad. 128 x 240 pixel display. About 188K bytes of user-available RAM. About 2.7 MB of user-available FLASH ROM (3x the FLASH ROM memory of the TI-92 Plus).*. Electronic upgradability of software including maintenance and feature upgrades. Advanced...
A strategic assessment of TexasInstruments, one of the world's leading semiconductor companies, provides significant competitor information, analysis, and insight critical to the development and implementation of effective marketing and R&D programs.
This is the new technology from TexasInstruments.The touchpad easy navigation through different screens, menus and mathematical objects. The Homescreen with the scratchpad displays menus and clickable icons like a computer and the scratchpad features enables calculations to be performed quickly and easily. Letter Keys-These quick touch alpha keys are positioned at the bottom of the keypad hence it is easier to enter text. Students can take notes on homework and educational professionals can comment...
This TN-Nspire Maths Platform has easy to use computer features. You can use multiple representations of a problem, individually or together on a single screen. The TI-Nspire Maths ICT Platform includes: Dynamic geometry, Data and statistics, Lists and Spreadsheets, and East transfer of files to software on PC. It is ideal to use in Secondary Education (KS 3 upwards) and beyond. Every student learns differently: Some quickly understand equations, other tables, some others graphics. TI-Nspire™ CA
A robust graphing handheld for secondary schools, preloaded with 4 software applications (Apps). Further Apps (up to a maximum of 10) can easily be downloaded from the TI website to suit each individual user's needs. The TI-83 Plus with its advanced scientific, statistical and financial functions is a very useful tool for secondary school teachers and students from 11 to 18 years old. |
Synopses & Reviews
Publisher Comments:
The idea of this book is to give an extensive description of the classical complex analysis, here classical means roughly that sheaf theoretical and cohomological methods are omitted. The first four chapters cover the essential core of complex analysis presenting their fundamental results. After this standard material, the authors step forward to elliptic functions and to elliptic modular functions including a taste of all most beautiful results of this field. The book is rounded by applications to analytic number theory including distinguished pearls of this fascinating subject as for instance the Prime Number Theorem. Great importance is attached to completeness, all needed notions are developed, only minimal prerequisites (elementary facts of calculus and algebra) are required. More than 400 exercises including hints for solutions and many figures make this an attractive, indispensable book for students who would like to have a sound introduction to classical complex analysis. For the second edition the authors have revised the text carefully.
Synopsis:
This book offers an extensive description of the classical complex analysis, roughly meaning that sheaf theoretical and cohomological methods are omitted. Over 400 exercises are included, and the text has been heavily revised for this new edition.
Synopsis:
"Synopsis"
by Springer,
This book offers an extensive description of the classical complex analysis, roughly meaning that sheaf theoretical and cohomological methods are omitted. Over 400 exercises are included, and the text has been heavily revised for this new edition.
"Synopsis"
by Springer, |
Stack Exchange: Mathematics
MathType works with Stack Exchange: Mathematics, allowing you to add equations to a question or answer response. Others can later open these equations in MathType to edit or to use elsewhere in Stack Exchange: Mathematics, or in other applications such as in Word or one of hundreds of other apps & sites that MathType works with.
Stack Exchange: Mathematics is a collaboratively edited question and answer site for people studying math at any level and professionals in related fields. Stack Exchange: Mathematics uses MathJax, a solution for displaying math on the web that works in all modern browsers. MathJax works behind the scenes on a website to display MathML and LaTeX equations.
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Add an equation to Stack Exchange: Mathematics: You can add a MathType equation into a Stack Exchange: Mathematics question or answer response. How-to
Copy equations from Stack Exchange: Mathematics: You can copy and paste equations from Stack Exchange: Mathematics into MathType to edit or use elsewhere. How-to |
13 terms
· textbook: Everyday Math
Algebraic Expression
Predict
Rate
A comparison by division of two quantities with different units. For example, traveling 1oo miles in 2 hours can be expressed as 100 mi/2 hr or 50 miles per hour. In this case, the rate compares distance (miles) to time (hours).
Line Graph
Ordered Number Pairs
A pair of numbers used to locate a point on a coordinate grid. The first number corresponds to position along the horizontal axis, and the second number corresponds to position along the vertical axis.
Circumference
Diameter
A line segment that passes through the center of a circle or sphere and has endpoints on the circle or sphere: also, the length of such a line segment. The diameter of a circle or sphere is twice the length of the radius. |
Pre-Algebra is a common name for a course in middle school mathematics. In the United States, pre-algebra is generally taught between the fifth and eighth grades, although sometimes taught as early as the third grade for gifted students. The objective of pre-algebra is to prepare the student for the study of algebra. |
7th Grade Algebra PDF
THE ALGEBRA AND FUNCTIONS STRAND In Grade 7, there are two reporting clusters within the Algebra and Functions strand: 1) Quantitative Relationships and Evaluating Expressions and 2) Multi-step Problems, Graphing, and Functions. This booklet
7thGrade Pre-Algebra Analyze proportional relationships and use them to solve real-world and mathematical problems. 7.RP.A.1 Compute unit rates associated with ratios of fractions, including ratios of lengths,
6. Chris starts with xdollars in his bank account. He withdraws half of the amount and then deposits 3 times the original amount back into his account, finally he withdraws half of the money in the account.
for fifth grade, Math 7/6 for sixth grade, Math 8/7 for seventh grade, and Algebra 1__ for eighth grade). 2 Students who have missed math concepts in their previous study may be better served beginning one textbook level lower.
Grade 7 Introduction: Summary of Goals i. GRADE SEVEN By the end of grade seven, students are adept at manipulating numbers and ... Grade 7 Algebra & Functions a. Write as a mathematical expression: 1. 5 less than R 2. One fourth as large as the area, where the area = A
7thGrade Math Measurement 4.1, 4.2 Multiple Choice Identify the choice that best completes the statement or answers the question. ____ 1. Beth needs to find the length of fencing needed to enclose a garden. Which of the following measures best
7thGrade Math Placement Many parents are interested in the process of how incoming 7th graders are placed into Algebra I. More than 400 students enroll in Granite Oaks Middle School from the five feeder elementary |
activity, from Shirley Fall, contains ten linked problems which are intended to lead to students developing a rule for integrating. Problem 1 begins by exploring the area enclosed under a straight line by finding the area of a trapezium. Students are required to generalise the area between the limits x=0 and x=a for different…
This Durham Maths Mystery is designed to extend high achieving GCSE students and for use with students studying mathematics Post 16.
Task 1A: students are presented with a blank three by three grid and twenty fact cards. Students have to use the clues on the fact cards to help them sketch the correct graph in the correct place…
This activity, from Shirley Fall, requires students to use and interpret function notation, sketch graphs using key points, explore the common transformations of translate parallel to the y axis, translate parallel to the x axis, stretch parallel to the y axis from the x axis and stretch parallel to the x axis from the y axis.
Ideally…
In this resource from Shirley Fall, students are presented with a general straight line y=mx+c and are shown that the gradient of the line between any two points that lie on the line is always equal to m. Students should be encouraged to explain the proof to check understanding.
Students then move on to curves and need to be clear…
During 1986 a group of teachers met to consider alternative styles of teaching A- level students. The Cockcroft Report is influencing work in lower years, but often post 16 teaching has remained closest to lecturing.
These offerings are to indicate the themes explored and give a flavour of possibilities.
Section 1: Ways of relating |
23 STATE NORMAL SCHOOL.
SUBJECTS OF INSTRUCTION.
MATHEMATICS.
Mr. Webb.
Mr. Hansen.
Mr. Smith.
Algebra. a. This course affords a thorough and complete treatment of elementary algebra; including quadratic equations, equations in quadratic form; simultaneous quadratic equations, theory of quadratic equations; ratio, proportion, and variation; arithmetical, geometrical and harmonical progressions; Logarithms.
Wells's The Esssentials of Algebra is the text book used.
Five hours per week throughout the year.
Algebra. b. Review of algebra a, and a brief course in advanced algebra.
Two hours per week throughout the year.
Plane Geometry. This course covers the five books in plane geometry. It aims to familiarize the student with the forms of rigid deductive reasoning, and to develop accuracy of statement and the power of logical proof. Considerable time is devoted to the demonstration of original theorems and to the solution of practical problems.
Two hours per week throughout the year.
Solid Geometry. Wentworth's Solid Geometry.
Two hours per week throughout the year.
ENGLISH.
Mr. Marshman.
English .a.-This course consists of a thorough study of English Grammar, special attention being paid to the practical |
Physics : Algebra and Trigonometry / With CD-ROM - 3rd edition
Summary: In Hecht's groundbreaking book, you'll find real-life applications, an unparalleled art and photography program, a presentation that anticipates students' questions, and an approach that emphasizes contemporary physics while interweaving historical perspectives. Hecht's coverage of classical physics is clear and insightful. He shows students how 21st-century physics illuminates the classical topics of each chapter, adding excitement to the subject matter. Over 1,300 ...show moreillustrations make it possible for students to visualize a diversity of physical phenomena. Many of these are multi-frame, sequential drawings allowing students to comprehend the temporal unfolding of complex events. A selection of sketch art teaches students how to create problem-solving diagrams. This new edition of the text was designed to aggressively address the issue of problem solving for students (guided by contemporary physics education research). To this end Hecht has provided not only his approach to the five-step problem-solving framework but also a wide range of new problems and solutions specifically designed to build student capability and confidence.
Benefits:
NEW! One of the greatest difficulties faced by physics educators is the teaching of problem solving. Hecht formalizes a five-step procedure that is widely used throughout the physics community, and this new edition has taken that approach to the next level. Throughout the text and CD-ROM, each example problem is solved using the five-step problem-solving framework, which includes Translation, Given and Find, Problem Type, Procedure, and Calculation. This provides the student with a consistent and reliable method for approaching and analyzing any physics problem.
Each copy of the text is available with a free tutorial CD-ROM, YOUR PERSONAL TUTOR, written by the author specifically to correlate with the textbook and enhance the physics experience. Hecht has written hundreds and hundreds of problems designed to tutor a student as he or she progresses through the course. These problems consist of the following: "Warm-Ups," "Walk-Throughs," "Interactive Explorations," and "Further Discussions."
"Warm-Ups" are interactive multiple-choice questions that establish a basic level of understanding.
"Walk-Throughs" expertly guide students through typical homework problems that are broken down into pieces and analyzed step-by-step in an interactive manner.
"Interactive Explorations," state-of-the-art physics simulations, allow students to do everything from fire a ball out of a virtual cannon to learn about projectile motion to varying the parameters in a Carno Cycle.
"Further Discussions," enhance the book discussion with additional textual material available only on the CD-ROM.
Problem-solving strategies, approximation techniques, and ways to anticipate common errors and pitfalls are discussed in every chapter in a section called Suggestions on Problem Solving.
A Quick Check is included at the end of every example problem, encouraging students to review their work.
Approximately 25 multiple-choice questions, similar to those found on national medical (MCAT) and optometry (OATP) entrance exams, are included in every chapter.
Answers to all odd-numbered multiple-choice questions and all odd-numbered problems appear in the back of the book.
The complete solutions to roughly one-third of the problems in the text appear at the back of the book. The problems that have a solution listed are designated with a bolded number.
NEW! Marginal notes, called Study Guide, are a guiding aside to give the student perspective and help her focus on the most important aspects of what she is learning.
To guide the students through using the solutions manual, each problem available in the solutions manual is designated with an italicized number.
In addition to detailed summaries, the Core Material and Study Guide sections at the end of each chapter point out what is important and makes specific suggestions for effective problem solving.
Many conceptual discussions are set off from the main text so students can negotiate the primary material more easily.
Hecht focuses on the historical perspective so that students learn not only the ideas behind the study of physics, but where the ideas came from, making the material in the text more accessible and less intimidating.
Available with WebAssign, the online homework service at North Carolina State.
NEW! An element called Calculator Tips appears in the margin at appropriate points in the mathematical development throughout the chapters. These hints are geared to keystrokes on varieties of scientific calculators.
NEW! Internet references appear in the margin, carefully designed to reference stable government and educational sites on the Web.
NEW! In addition, the author has designed a "problem-solving program" to support students as they build confidence and capability in problem-solving. This program consists of four pedagogical components: "Problems," "Coordinated Problems," "Progressive Problems," and "Solutions."
NEW! Graduated in difficulty, "Problems" are based, whenever possible, on real-life data. The goal of this pedagogical component is to eventually bring students up to a level where he can successfully solve these problems.
NEW! To support that effort, "Coordinated Problems," (in magenta) are designed to strengthen technique and build confidence.
NEW! Approximately ten "Progressive Problems" (introduced in blue) per chapter unfold step-by-step to guide the student through the analytic process and avoid common pitfalls. The students are lead to solve these problems in a carefully developed sequence of guided steps.
NEW! Throughout the text, students are encouraged to study worked solutions in the interest of building intuition through experience. The final component, "Solutions," provides an additional selection of five to six model problems and solutions per chapter.
3. KINEMATICS: ACCELERATION. Average Acceleration. Instantaneous Acceleration. Constant Acceleration. The Mean Speed. The Equations of Constant Acceleration. Air Drag. Acceleration Due to Gravity. Straight Up and Down. Two-Dimensional Motion: Projectiles.
4. NEWTON'S THREE LAWS. The Law of Inertia. Force. The Second Law. Interaction: The Third Law. The Effects of Force: Newton's Laws. Weight: Gravitational Force. Coupled Motions. Friction. Equilibrium: Statics.
16. ELECTROSTATICS: ENERGY. Electrical-PE and Potential. Potential of a Point-Charge. The Potential of Several Charges. Conservation of Charge. The Capacitor. Capacitors in Combination. Energy in Capacitors.
Textbook may contain underlining, highlighting or writing. Infotrac or untested CD may not be included.
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0534377297 WE HAVE NUMEROUS COPIES. HARDCOVER with unopened CD light pink liquid damage to bottom edges of last pages, one page from index pulled loose but still there, cover has a few tears which hav...show moree been taped, cover is separating from page binding but page binding appears ok, -heavy wear to cover,edges, and corners with cardboard showing on most corners/edges, most pages clean with minimal markings ...show less
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Resources
Below are some of the resources that students have at their disposal.
The navigation panel on the left also lists more specific resources that students can take advantage of.
Department Office: The Mathematics Department Office, wonderfully staffed by Ms. Melanie Chamberlin, is located in SCI 361. Her extension is x3148. The office contains information about summer work and research opportunities, actuarial opportunities, and information about graduate studies.
Common room (SCI 362): The department common room is located in SCI 362. Whenever it is not being used for a department event, the common room is available to students for study or discussion. If you need to consult someone on a mathematical idea, the common room is one place to try. This room is also used in the evenings as the Math Help Room.
The Mathematics Computer Laboratory (SCI 257): Outfitted with Macintosh computers, the computer lab is open all day and evening most days, and the computers are available for students to use whenever the lab is not being used for a class. Each of the computers is equipped with Mathematica and Joy of Mathematica, as well as a variety of other software. Both Mathematica and Joy of Mathematica are also available at various sites around campus, including the Science Center Minifocus.
Science Library: The Science Library is a wonderful resource. Explore it for suggested alternative texts, for popular mathematics books, or biographies of famous mathematicians. Check out on their webpage of interesting mathematics sources |
Key to Coding: Unit 1: Algebra and Integers This unit builds a foundation of basic understandings of numbers, Lesson 1 Place Value of Whole Numbers and Decimals pgs. models and the Properties of Equality and justify the solution in writing. 153-157 (writing two-step equations), pgs. Pizzazz for applications
Encribd is NOT affiliated with the author of any documents mentioned in this site. All sponsored products, company names, brand names, trademarks and logos found on this document are the property of its respective owners. |
In this book, Pierre de la Harpe provides a concise and engaging introduction to geometric group theory, a new method for studying infinite groups via their intrinsic geometry that has played a major ...
This substantially revised and expanded new edition of the bestselling textbook, addresses the difficulties that can arise with the mathematics that underpins the study of symmetry, and acknowledges ...
An introductory text book for graduates and advanced undergraduates on group representation theory. It emphasizes group theory's role as the mathematical framework for describing symmetry properties ... |
books.google.com - As the title indicates, this text is intended for courses aimed at bridging the gap between lower level mathematics and advanced mathematics. The transition to advanced mathematics presented is discrete since continuous functions are not studied. The text provides a careful introduction to techniques... Discrete Transition to Advanced Mathematics |
...
Show More incorporated additional features in this 9th edition, plus exercises for each section to help students sharpen their advanced math skills and to easily incorporate and apply what they have learned. Quizzes have been revised extensively based on the analysis of previous data usage of other students. Calculus is written mainly for entry-level students, but it is also favored by math majors and engineers. It offers an additional section for differential equations and provides access to an e-book and online media where students can practice their math skills on their own. The interactive internet component also links to appendices such as a precalculus review materials, complex numbers, and case studies showing how this type of math can solve economics-, engineering-, and business-related problems that cannot be solved by algebra alone. Teachers will appreciate the supplemental Maple Software, which offers tests and homework samples online. Files can then be transferred to Excel, but an online gradebook is likewise available to make it even easier to grade and compile output from students. Ron Larson and Bruce H. Edwards employ a gradual approach so as not to overwhelm beginners, starting with limits to differentiation then moving on to integration and exponential and logarithmic functions. Calculus also covers polar coordination, infinite series, conics, and vectors |
Homework
For Friday, September 2:
Turn inHomework #1.
Come to class with any questions that you have about functions or Maple.
The Homework passed out in class said to use the 2-up option when printing.
This means that you should print 2 pages per sheet of paper.
When the Print dialogue box appears, select Properties and go to the Layout tab. Go down to Pages per sheet, and select 2.
If some of the exercises gave you difficulties, review Chapter 3 and Section 4.7 form the Just-In-Time book.
Monday, September 5: We will begin class with the
Gateway Exam.
For Wednesday, September 7:
Work on the Parametric Equations problems.
Copies of the problems are available on ANGEL.
For Friday, September 9:
Do exercises from the Just-In-Time book:
Section 7.1 #1;
Section 7.2 #1,3,6;
Section 7.3 #3,7;
and from The Calculus book (by Thomas):
Section 1.3 #4,5,13,28;
Section 1.5 #1,12,16,20.
Turn in these exercises on Monday. Turn in The Parametric Exercises homework.
For Monday, September 12:
Turn in the homework assigned for Friday.
Do the exercises
Sec. 1.3 #47,51,57,62;
Sec. 1.5 #5,9,21,26,29,31,33.
For Tuesday, September 13:
Section 1.6 #13,15,19,24,35,39,43,55,61,63,67,76,80,82. Wednesday, September 14:
Quiz covering the material disucussed thus far in class.
For Thursday, September 15:
Section 2.1 #1,3,11,15,18,19,21.
turn in these exercises on Friday.
For Friday, September 16:
Turn in the Section 2.1 Homework.
Section 2.2 #2,5,6,11,21,25.
For Monday, September 19:
Quiz about limits.
Section 2.2 #13,22,33,35,54,61,64,73,76.
For Tuesday, September 20:
Section 2.3 #7,9,169,31,37,39.
For Wednesday, September 21:
Section 2.4 #1,5,6,11,14,15,18,36.
,,
Thursday, September 22:
Exam #1.
The average score on the exam was 85.6/140. Equivalent grades are
A 105-136 B 88-104 C 71-87 D 54-70 F <45
Friday, September 23.
We are on a convo schedule.
Period 8 meets 1:20PM-2:00PM.
Period 9 meets 2:05PM-2:45PM.
Section 2.4 #21,25,30,40,52.
For Monday, September 26>I:
Section 2.5 #1,5,13,23,31,39,49,55,66.
Turn in these exercises Tuesday.
For Tuesday, September 27:
Section 2.6 #3,5,17,19,37,40,49,54,69,74
For Wednesday, September 28:
Section 2.6 #8,25,32,39,44,67; Section 3.1 #1.
For Thursday, September 29:
Section 3.1 #5,11,23,28,33,37,49;
Section 3.2 #1,5,10,13,27-30,32.
For Friday, September 30:
Section 3.2 #6,22,37,45,47,49,53,62.
For Monday, October 3:
Section 3.3 #1,6,10,13,27-30,32,37,47,53,62.
For Tuesday, October 4:
We will have a Quiz about the definition of the derivative.
Section 3.4 #1,8,10,14,17,19,27.
For Wednesday, October 5:
Section 3.5 #1,5,24,30,53,55,59,61.
Come with questions relevant to Thursday's exam.
Course Policy
Material added since the beginning of class will be in purple
This class will have four in-class exams - most likely during the 3rd, 5th, 7th and 9th weeks. Note: I don't set grades using a 93/86/77/70 "straight scale" system. I plan to challenge you to excel. This means that the exam averages will generally be lower than what you are used to.
The final exam will be given during finals week. If you are making arrangements for travel home, you should make sure that they will not conflict with the final exam schedule. If your parents will be purchasing airline tickets for you, you should contact them and remind them not to schedule you on a flight that might cause you fail a class.
There will be two parts to the final exam.
For the first part, you will be allowed the use of only a writing utensil.
For the second part, you may use clean Maple worksheets.
The weights for the grade are given below.
You will be expected to pass some "basic skills" quizzes during the term.
These will count as part of your quiz grades.
There may be two other types of quizzes given in class: announced and unannounced.
Exams and quizzes will cover everything up to recent material. The focus will usually beon recent sections, but all earlier material, from pre-school to the material immediately before those sections of concentration, is considered known, and you will be responsible for it.
Homework should have your name, campus mailbox number, and class at the top of the page.
A stapler is probably a good investment for most of you. Multi-page homeworks should be stapled together, not mutilated.
Homework will typically be due at the beginning of class on the due date.
Homework may be turned in later but will be penalized based on just how late it is - typically 10% off per day. i.e. 10 days later, it's too late to get a makeup homework turned in.
Homework turned in late during class will be subject to a 1 point penalty. Homework turned in late the day that it is due, a 5% penalty.
When writing up homework, you should circle (or otherwise clearly indicate) your answers.
If the homework is on a worksheet that is passed out to the class, you will generally be expected to write your answers on a separate sheet of paper, in a well-organized fashion. Answers should be written on the worksheet only if answer blanks have been specifically provided.
If you have any questions while I'm not around, you may e-mail me at
rickert@rose-hulman.edu and I will reply as soon as I can.
You should come to class prepared. This means that I expect you to have done the homework, brought your book to class and launched Maple at the beginning of class.
Silence cell phones, pagers, and similar devices during class. Do not text or hold phone conversations during class - they can leave a message.
If you don't understand something, ASK
If I'm going too fast, STOP ME.
I enjoy mathematics. When I get on a roll, I tend to keep going.
SHOW YOUR WORK. The correct answer will only be worth 1 point. I want to verify that you understand the process.
If you are having problems understanding the material, see me or go to the learning center.
I will assign some `group' projects in this class.
Groups will consist of either three or four members. Write-ups from smaller or larger groups will not be accepted unless prior approval has been given.
Write-ups should be neatly presented. Write-ups returned on the information sheet handed out to the groups will not be accepted. Write-ups handed in at the end of class may be hand-written. Write-ups for work outside of class should be typed. Maple code may be included as part of an appendix or in figures, but should not be considered as a formal write-up. Similarly, scratch-work is unacceptable. Neatly written partial results may be turned in, but scribbles will reduce your grade.
A summary of the grade weights
There will be four in-class exams worth 15% each
The final exam will be worth 30%
Quizzes and homework will be worth 10%
Extra credit that is earned will be added on to your grade after the curve has been determined.
Go to |
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A gateway to the archives of the Math Forum's online electronic community comprising eight geometry discussion groups for all those interested in geometry. Access current news or follow a link to threaded past discussions to browse or search the archive
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A nonprofit educational organization devoted to informing people about Logo and supporting them in their use of Logo-based software and learning environments. The site includes links for workshops and seminars; books and videos; Logo software (at discount
... Help - Graeme McRae
Tips and techniques for math students from sixth grade through high school, and one-on-one help for math students in a variety of formats. Also includes math puzzles and jokes. Tools - Math Forum
A community digital library that supports the use and development of mathematics education software for computers, calculators, PDAs, and other handheld devices, with relevant lesson plans, activities, support materials, stories, discussions, and technology
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PBS TeacherLine - PBS
A professional development resource delivering courses online for PreK-12 teachers. Find course dates, graduate credit opportunities, and other information. Download the course tour for a preview of the experience of taking a course. See also the Virtual
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ApplicationsSystemsof linear equations provide an excellent opportunity to discuss mathematical modeling. The process of using mathematics to solve real-world problems can be broken down into three steps: Step 1.
All other proofs were geometric in nature. REALWORLDAPPLICATIONS Quadratic Equations, the basis of algebra, can be used in an incredible number of ways. Finding the curve on a cartesian grid, ... and quadratic equations are used in the creation of sound systems in homes, movie theaters, ...
Solving SystemsofEquations Using the Elimination Method. ... You wrote a system ofequations to model a real-world situation. ... Building Understand for Solving a System ofEquations through Applications Last modified by: Patti Company:
The expectation is that students will be able to solve all types of linear and quadratic equations over the set ofreal numbers. Solving systemsofequations ... Again, the use of these concepts to solve realworldapplications ... and solving real-world problems involving equations and ...
Therefore the conclusion can be drawn that algebra lessons taught with realworldapplications is very helpful for student understanding. ... The students we worked with in the Algebra 141 class did not know much about how to solve a word problem involving systemsof linear equations.
Sometimes linearity can be bad systemsof linear equations can be efficiently solved "non-linearity" is a measure of how ... Counter mode proves to be very useful in applications where external synchronisation is forced ... RealWorldApplicationsof Cryptography Author: PC Image ...
... and by systemsofequations and/or inequalities, and interpret ... Standard: Score 4.0 In addition to Score 3.0, in-depth inferences and applications that go beyond what was taught ... It is the only point that is a solution to both equations. * * RealWorld Application Peer Talk What prior ...
STANDARD AII.12 The student will compute and distinguish between permutations and combinations and use technology for applications. ... to solve real-world ... Analytical Geometry SystemsofEquations and Inequalities New Reporting Categories Expressions and ...
Mathematical models can help in the understanding of practical systems, ... methods of solving. Introduce vectors and their applications to the realworld. Introduce linear equations: no emphasis on ... reinforces mathematical concepts through their connection to real-worldapplications.
... AII.2 The student will investigate and apply the properties of arithmetic and geometric sequences and series to solve real-world problems, including ... AII.5 The student will solve nonlinear systemsofequations, ... and combinations and use technology for applications.
Heavy emphasis is placed on the real-life applicationsofsystemsofequations. ... Can students solve real-world problems using systemsofequations? 5. Can students graph systemsof inequalities and recognize the solution set?
... that occurs widely in dynamical systems Considered to be complex and no simple analysis Study of chaos can be used in real-worldapplications: secure communication ... Vol. 40, No 10 (1993) pp. 657-674. [7] Lawrence Perko, Differential Equations and Dynamical Systems. Springer ...
Motivation Graphical models are widely used in real-worldapplications, ... on dynamic systems Poisson tracking Signal detection for wireless ... the complexity of the posterior increases with the number of measurements I.e. BP equations are not "closed" Beliefs need not stay ...
It covers linear equations, graphing lines, quadratic equations, function notation, rational ... students to explore connections between the concepts they have just learned and more advanced mathematical concepts or real-worldapplications. ... (and viruses), and the human body systems.
Motivation Graphical models are widely used in real-worldapplications, ... Extending EP on Bayesian networks for dynamic systems Poisson tracking Signal detection ... the complexity of the posterior increases with the number of measurements I.e. BP equations are not "closed" Beliefs ...
Represent constraints by equations or inequalities, and by systemsofequations and/or inequalities and interpret solutions as viable or non-viable options in a modeling context. ... students should work with real-worldapplicationsof absolute value, step, ...
Outline What are Fuzzy Systems? Fuzzy logic applications Fuzzy set and fuzzy operations Origin of ... The ambiguity ofrealworld definitions John is OLD David is TALL ... A quantified framework to deal with the IMPRECISE NATURE OF THE REALWORLD where conventional mathematical equations become ...
... Systems Overview Math Overview Lab Activity Real-Life ... The bulk of the year will be spent solving three variable equations where one variable is unknown while the ... and the library are all good sources. You, the students, are not a source of examples ofrealworldapplicationsof ...
... where a is a constant, non-negative real number and aF is the birthrate of the fish. It is negatively ... where a/b is the stable point for the fish population in a shark-free world, and k ... One of the most interesting applicationsofsystemsof differential equations is the predator ...
Director UTA Energy Systems Research Center University of Texas at Arlington [email protected] ... Students will see the relevancy of linear equations in practical applications. ... Teachers will be able to connect class theory with applications in the realworld. |
Pre-calculus Demystified 2/ solution to mastering pre-calculus! Fully updated throughout, Pre-calculus DeMYSTiFieD, Second Edition features all-new quizzes and test questions, detailed explanations of the exercises, and a completely refreshed design. Author Rhonda Huettenmueller helps you understand the material by organizing the information from simple to complex, and presenting it clearly and concisely. This practical guide covers lines, functions, exponents, logarithms, systems of equations, matrices, conic sections, and more. Step-by-step examples make it easy to... MORE understand the topics, and end-of-chapter quizzes and a final exam reinforce key concepts. Pre-calculus DeMYSTiFieD, Second Edition Contains examples, graphs, questions, and answers Helps you prepare for the CLEP (College-Level Examination Program) Pre-calculus test, SATs, or other college placement exams Provides chapter-opening objectives that describe what you'll learn in each step Includes questions at the end of every chapter to reinforce learning and pinpoint weaknesses Offers "Still Struggling?" elements throughout to help you with difficult subtopics Concludes with a final exam for overall self-assessment The second edition this bestseller is updated with all-new quizzes and test questions, clearer explanations of the exercises, and a completely refreshed design.
Your step-by-step solution to mastering precalculus
Understanding precalculus often opens the door to learning more advanced and practical math subjects, and can also help satisfy college requisites. Precalculus Demystified, Second Edition, is your key to mastering this sometimes tricky subject.
This self-teaching guide presents general precalculus concepts first, so you'll ease into the basics. You'll gradually master functions, graphs of functions, logarithms, exponents, and more. As you progress, you'll also conquer topics such as absolute value, nonlinear inequalities, inverses, trigonometric functions, and conic sections. Clear, detailed examples make it easy to understand the material, and end-of-chapter quizzes and a final exam help reinforce key ideas.
It's a no-brainer! You'll learn about:
Linear questions
Functions
Polynomial division
The rational zero theorem
Logarithms
Matrix arithmetic
Basic trigonometry
Simple enough for a beginner but challenging enough for an advanced student, Precalculus Demystified, Second Edition, Second Edition, helps you master this essential subject. |
Select a content and you will automatically be redirected to the list of performance expectations.
A2.1.
Core Content:
Solving Problems
The first core content area highlights the type of problems students will be able to solve by the end of Algebra 2, as they extend their ability to solve problems with additional functions and equations. When presented with a word problem, students are able to determine which function or equation models the problem and use that information to solve the problem. They build on what they learned in Algebra 1 about linear and quadratic functions and are able to solve more complex problems. Additionally, students learn to solve problems modeled by exponential and logarithmic functions, systems of equations and inequalities, inverse variations, and combinations and permutations. Turning word problems into equations that can be solved is a skill students hone throughout Algebra 2 and subsequent mathematics courses.
A2.2.
Core Content:
Numbers, expressions, and operations
(Numbers, Operations, Algebra)
Students extend their understanding of number systems to include complex numbers, which they will see as solutions for quadratic equations. They grow more proficient in their use of algebraic techniques as they continue to use variables and expressions to solve problems. As problems become more sophisticated and the level of mathematics increases, so does the complexity of the symbolic manipulations and computations necessary to solve the problems. Students refine the foundational algebraic skills they need to be successful in subsequent mathematics courses.
A2.3.
Core Content:
Quadratic functions and equations
(Algebra)
As students continue to solve quadratic equations and inequalities in Algebra 2, they encounter complex roots for the first time. They learn to translate between forms of quadratic equations, applying the vertex form to evaluate maximum and minimum values and find symmetry of the graph, and they learn to identify which form should be used in a particular situation. This opens up a whole range of new problems students can solve using quadratics. These algebraic skills are applied in subsequent high school mathematics and statistics courses.
Students extend their understanding of exponential functions from Algebra 1 with an emphasis on inverse functions. This leads to a natural introduction of logarithms and logarithmic functions. They learn to use the basic properties of exponential and logarithmic functions, graphing both types of function to analyze relationships, represent and model problems, and answer questions. Students employ these functions in many practical situations, such as applying exponential functions to determine compound interest and applying logarithmic functions to determine the pH of a liquid.
A2.5.
Core Content:
Additional functions and equations
(Algebra)
Students learn about additional classes of functions including square root, cubic, logarithmic, and those involving inverse variation. Students plot points and sketch graphs to represent these functions and use algebraic techniques to solve related equations. In addition to studying the defining characteristics of each of these classes of functions, students gain the ability to construct new functions algebraically and using transformations. These extended skills and techniques serve as the foundation for further study and analysis of functions in subsequent mathematics courses.
A2.6.
Core Content:
Probability, data, and distributions
(Data/Statistics/Probability)
Students formalize their study of probability, computing both combinations and permutations to calculate the likelihood of an outcome in uncertain circumstances and applying the binominal theorem to solve problems. They extend their use of statistics to graph bivariate data and analyze its shape to make predictions. They calculate and interpret measures of variability, confidence intervals, and margins of error for population proportions. Dual goals underlie the content in the section: students prepare for the further study of statistics and become thoughtful consumers of data.
A2.7.
Additional Key Content:
(Algebra)
Students study two important topics here. First, they extend their ability to solve systems of two equations in two variables to solving systems of three equations in three variables, which leads to the full development of matrices in Precalculus. Second, they formalize their work with series as they learn to find the terms and partial sums of arithmetic series and the terms and partial and infinite sums of geometric series. This conceptual understanding of series lays an important foundation for understanding calculus.
A2.8.
Core Processes:
Reasoning, problem solving, and communication
Students formalize the development of reasoning at high school as they use algebra and the properties of number systems to develop valid mathematical arguments, make and prove conjectures, and find counterexamples to refute false statements |
Midterm Study Guide NCTM StandardsPresentation Transcript
Number and Operations
Understand numbers, ways of
representing numbers, relationships
among numbers, and number systems.
Understand meanings of operations
and how they relate to one another.
Compute fluently and make reasonable
estimates.
Algebra
Understand patterns, relations, and
functions
Represent and analyze mathematical
situations and structures using algebraic
symbols
Use mathematical models to represent
and understand quantitative relationships
Analyze change in various contexts
Geometry
Analyze characteristics and properties of two- and
three-dimensional geometric shapes and develop
mathematical arguments about geometric
relationships
Specify locations and describe spatial relationships
using coordinate geometry and other
representational systems
Apply transformations and use symmetry to
analyze mathematical situations
Use visualization, spatial reasoning, and geometric
modeling to solve problems
Understand patterns,
relations, and functions
Algebra
Understand patterns,
relations, and functions
Use visualization, spatial
reasoning, and geometric
modeling to solve
problems |
048644595X
9780486445953
An Introduction to Differential Equations and Their Applications:Intended for use in a beginning one-semester course in differential equations, this text is designed for students of pure and applied mathematics with a working knowledge of algebra, trigonometry, and elementary calculus. Its mathematical rigor is balanced by complete but simple explanations that appeal to readers' physical and geometric intuition. Starting with an introduction to differential equations, the text proceeds to examinations of first- and second-order differential equations, series solutions, the Laplace transform, systems of differential equations, difference equations, nonlinear differential equations and chaos, and partial differential equations. Numerous figures, problems with solutions, and historical notes clarify the text.
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Rent An Introduction to Differential Equations and Their Applications 1st edition today, or search our site for Stanley J. textbooks. Every textbook comes with a 21-day "Any Reason" guarantee. Published by Dover Publications, Incorporated. |
A lot of learning Math is learning problem solving skills, or even more importantly it's about learning how to learn new problem solving skills. Sure there are many advanced Math skills that can be useful in industry, but learning how to think about problems in multiple ways can be helpful for developing solutions to problems. Additionally, Math can be useful for determining complexity and optimizing programs.
It sounds to me like you're more interested in software development than in computer science. These two fields are often confused for each other, but are certainly not the same thing.
The only thing I'm likely to find torrents of would be pirated copies of textbooks. A digital textbook and dead tree textbook are going to be pretty comparable. I was referring to non-textbook materials, as that's what the parent and GP post were about.
When I've looked I've come across a complete lack of material for most topics beyond say third semester calculus and second semester differential equations that one can read to actually learn a subject. While there are certainly some example problems out there, a cohesive narration of how to go about solving more advanced problems especially with a consistent notation seems to be lacking. Sure there's resources like Pauls Online Math Notes, but that drops off before then. Wikipedia has some formulas and descriptions, but often doesn't have example problems. For many topics, you can find small pieces of information all over the web, but if you want to actually read up on a specific subject I haven't seen anything on the internet that rivals a good old fashioned text book.
Now, I don't see any reason for there to be new editions as often as there are. Many of the textbooks I read in my spare time are actually pretty old, but outside of some of the topics that rely on technology there isn't a whole lot of reason to have new editions. Even something such as numerical analysis (which should probably have a technology based theme for CAS) doesn't really need to be updated very often as the algorithms don't change, just the languages that may be used.
Perhaps I'm just biased, but a well written text book seems much more useful than gleaning bits of information from a variety of sources that all use different notations and symbols for learning about a topic in math.As someone who is currently in the diagnosis process of celiac I concur with it being terrible. A lot of people think it's like lactose intolerance where you're ill for an hour or two then fine. If I have any gluten at all then I'm not able to leave the house for 24-36 hours and it takes 4 or 5 days for me to actually feel okay again.
That said, I haven't had as hard of a time finding gluten free foods as I thought I would. Sure I had to radically alter my diet, and I can't get a straight answer out of a lot of manufacturers, but there are a lot of things that will actually say gluten free on them (and I reward companies that come out and say it with my money, so that I don't have to search through a list of ingredients).
I was recently helping my parents pick out a new hub at bestbuy since they needed one solely to add an ethernet port to their network for their tivo to plug into. Speed didn't really matter since it was just for getting subscription data. When we went and found the wired networking stuff they had exactly 2 to choose from. An $80 gigabit switch and a $30 10Base-T switch. Not a 10/100 hub or switch, just a 10Base-T one. I had no idea they still made these lol.
It took two or three tries for it to actually show up in my mailbox at all. If you're logged into an apps account, going to plus.google.com doesn't work at all, you can't see anything beyond the error message about it not working because your organization doesn't support profiles.
When you click an invite to google+ and you're logged into a google apps account you get a message that says:
Oops... you need a Google profile to use this feature.
Google Profiles is not available for your organization.
From the reading I have done on the web it says that google profiles are not yet available for google apps, though they are working on that in the future. I can't, in fact, find any other references besides your own saying anything except that google profiles are not yet available for google apps users. |
Fractions - Part 1: Fundamentals, covers types of fractions, mixed numbers and their meaning. How to find equivalent fractions and visualize relative values.
MathDBase.com will be a math database site containing comprehensive course information and thousands of solved problems from: Arithmetic, Algebra, Geometry, Trigonometry, Analytic Geometry, Pre-Calculus, Calculus, Ordinary Differential Equations and Linear Algebra. In addition, there will be extensive tables of formulas and lots of other resources.
If you have a math topic or series request, make your request in the comments section. If you would like to just view all of our videos, please subscribe to our You Tube channel (MathDBase) and leave a comment or send an e-mail. Your request will be developed and uploaded as soon as possible. |
calculus
Offering
10+ subjects
including calculus |
McAuliffe Middle School
Introduction to Algebra
Curriculum Letter
2012-2013
Room 13 Mr. Lynes (714)816-3320 ext. 77213
Dear Students and Parents,
Welcome to Math 7/8! I am looking forward to working with you and
helping to make your year in math a very positive learning experience. Please
keep this information packet in your notebook for future reference.
Learning Environment
In order to maintain our classroom as a place of learning, students are
expected to be respectful to me and to one another. They should be in their
seats ready to begin when the bell rings. In addition, students are expected to
follow directions and be cooperative at all times. Incentives for positive behavior
include positive comments, display of student work, positive phone calls, Eagle
Express Cards and Student of the Month. Warnings for less than acceptable
classwork or behavior will be followed by "Failure to do Assignment" notices,
progress reports, detentions, or phone calls home. A good citizen will participate
in class, follow directions, be cooperative, and will bring materials to class.
Students who try their best and make a good effort on all assignments will
receive a positive effort grade. Class warnings regarding behavior, failure to
bring materials or to complete assignments will result in the lowering of a
student's effort and/or citizenship grade.
Major Academic Goals
Students will:
1. Be fluent in the required California state standards for math.
2. Recognize and appreciate the relevance of math beyond the classroom.
3. Demonstrate their understanding of math concepts through open-ended
questions, investigation and research, problem solving situations, and
projects.
Homework/Classwork
Students should expect to have homework at least four nights a week. To
receive full credit, students must show all work and have a complete heading.
Homework and classwork will also include warm-ups, problem solving, note
taking, and group work. Each of these will be explained as we do them in class.
Homework represents 15% of a student's total grade.
Grading Policy
Grades are based on individual assignments, quizzes, tests, projects, and
class participation. Point values vary for each assignment.
GRADES ARE WEIGHTED AS FOLLOWS: Access grades/assignments online:
Homework and Classwork 15%
Weekly Quizzes 15%
Tests 70%
All assignments are to be done neatly, completely, on time, and with outstanding
effort. NO UNEXCUSED LATE WORK WILL BE ACCEPTED.
Citizenship and Effort grades are based on the following class expectations
1. Arrive to class on time and with proper supplies.
2. Ask appropriate questions.
3. Listen to and follow directions carefully.
4. Be kind, polite, and cooperative.
Tutoring
I will be available to all students at least two days a week before school
from 8:15 until 8:45. Students may make an appointment for additional time. I
encourage all students to take advantage of this time if needed. I am here to
help you succeed.
Absences
Students are responsible for making up both classwork and homework on
the days that are missed. Students will have one day to make up work for each
day they are absent from class.
Materials
Two spiral (graph) notebooks 2 pencils with erasers
Folder (to hold loose papers) 1 red correcting pen (ball-point)
If you have any questions or concerns, please contact me at school:
(714) 816-3320 ext. 77213. My conference period is 1:35 – 2:20 pm.
Sincerely,
Mr. Lynes
Please sign and return this portion indicating receipt of Mr. Lynes'
Introduction to Algebra curriculum letter.
Student Name Parent Name Parent |
More About
This Textbook
Editorial Reviews
Booknews
Noting that spreadsheets and mathematics go together like Euclid and geometry, the authors offer 100-plus activities (including optional Internet-based ones) to teach new computer and math skills: 39 to build basic skills, problems to hopefully apply these skills, and higher-level challenges in "crunching, charting, and cruising the numbers." TechTips and TechLessons for using the most widely used spreadsheets in schools accompany each step-by-step lesson. Spiral wire binding |
Mathematical Methods in Linquistics is far more about mathematical methods than about linguistics, although in many places linquistics is used as a source of examples.
Instead it covers such mathematical topics as sets (including infinite sets), relations, a good deal of mathematical logic, automata (up to turing machines), the lambda calculus, lattices and more.
This would be an excellent book for an advanced undergraduate or graduate student in either mathematics or computer science to use either as a review text, or as a study guide for further investigation.
8 of 10 people found the following review helpful
5.0 out of 5 starsComprehensiveMarch 22 2007
By JDCA - Published on Amazon.com
Format:Paperback|Amazon Verified Purchase
This book is well written and detailed. I found it particularly useful for my semantics course. It covers the necessary logic one need to do semantics. It also discuses type theory and the lambda calculus. This book is a great complement to any semantics text.
5 of 6 people found the following review helpful
5.0 out of 5 starsExtremely good.Jan. 27 2010
By E. Ebrahim - Published on Amazon.com
Format:Paperback|Amazon Verified Purchase
This book is incredible. It's a great introduction to axiomatic set theory at the beginning, and it does a great introduction to model theory. I haven't gotten to the linguistics part of it yet, but it covers math and logic in an extremely effective way. |
Applied Trigonometry
Topics including right triangles, trigonometric functions, graphing; basic identities,
triangle solutions, and complex numbers; emphasis on applications. Not open to students
with credit in 145.
Objectives
1. To introduce students to trigonometric functions. 2. To practice solving applied problems using trigonometry. 3. To introduce students to the role of trigonometry in the analysis and solution
of mechanical problems. 4. To develop analytical skills using new tools provided by a trigonometric approach
to problem solving. |
Spring 2014 quarter
college algebra, introductory computer science and programming, and problem solving.
Description
Computers are a driving force of our modern world and increasingly influence our lives. Mathematics and mathematical models lay at the foundation of modern computers; furthermore, we increasingly rely on mathematics as a language for understanding the natural world, such as complex climate models that predict major changes in weather patterns world wide over the next 50 years. Mathematics and computational thinking enable people as citizens to make good decisions on a wide range of issues from interpreting the evidence for climate change to understanding the potential impacts of technology; as such, they are an integral part of a liberal arts education. In this program, we will explore connections between mathematics, computer science, the natural sciences and graphic arts.
We will develop mathematical abstractions and the skills to express, analyze and solve simple problems in the sciences and the arts and explore how to program interesting visual shapes using simple geometry. Class sessions include seminars, lectures, problem-solving workshops, programming labs, problem sets and seminars with writing assignments. The emphasis will be on fluency in mathematical and statistical thinking and expression along with reflections on mathematics and society. Topics will include concepts of algebra, algorithms, programming and problem solving, with seminar readings about the role of mathematics in education, the sciences and society.
This program is intended for students who want to gain a fundamental understanding of mathematics and computing before leaving college or before pursuing further work in the sciences or the arts. |
Publisher's Description
Microsoft Mathematics provides a set of mathematical tools that help students get school work done quickly and easily. With Microsoft Mathematics, students can learn to solve equations step-by-step while gaining a better understanding of fundamental concepts in pre-algebra, algebra, trigonometry, physics, chemistry, and calculus.
Microsoft Mathematics includes a full-featured graphing calculator that's designed to work just like a handheld calculator. Additional math tools help you evaluate triangles, convert from one system of units to another, and solve systems of equations.
This is an outstanding program (especially for the price) for students in middle school and above. I haven't found anything else like it in terms of completeness of capabilities. My 14-year old uses it constantly for homework. Microsoft is to be commended for providing such a wonderful tool.
This is not a MATLAB replacement. It gets confused easily if you made the functions too complicated. Cannot export graph data to the likes of excel.
Summary
I am a physicist. I thought this program would be for school children only. Essentially it is a graphics calculator. It is much more advanced than I initially gave it credit for. It is useful for quick visualisation or a quick check of integrals, units |
The family in this book is moving to a new neighborhood. They have a lot of work to do! They need to unload the moving truck, unpack boxes, and put everything away. The kids make new friends and discover all the fun they can have with the empty boxes. While building forts from the empty packing boxes, the kids discover many new shapes and their dimensions.... more...
egghead's Guide to Geometry will help students improve their understanding of the fundamental concepts of geometry. With the help of Peterson's new character, egghead, students can strengthen their math skills with narrative cartoons and graphics. Along the way there are plenty of study tips and exercises, making this the perfect guide for students... more...
The focus of this book and its geometric notions is on real vector spaces X that are finite or infinite inner product spaces of arbitrary dimension greater than or equal to 2. It characterizes both euclidean and hyperbolic geometry with respect to natural properties of (general) translations and general distances of X. Also for these spaces X, it studies... more...
This book mainly deals with the Bochner–Riesz means of multiple Fourier integral and series on Euclidean spaces. It aims to give a systematical introduction to the fundamental theories of the Bochner–Riesz means and important achievements attained in the last 50 years. For the Bochner–Riesz means of multiple Fourier integral, it... more...
If you've ever thought that mathematics and art don't mix, this stunning visual history of geometry will change your mind. As much a work of art as a book about mathematics, Beautiful Geometry presents more than sixty exquisite color plates illustrating a wide range of geometric patterns and theorems, accompanied by brief accounts of the fascinating... more...
The Yang-Mills theory of gauge interactions is a prime example of interdisciplinary mathematics and advanced physics. Its historical development is a fascinating window into the ongoing struggle of mankind to understand nature. The discovery of gauge fields and their properties is the most formidable landmark of modern physics. The expression of the... more...
Maximize student use of the TI-Nspire while processing and learning algebraic concepts with this resource. Lessons provided delve into the five environments of the TI-Nspire including calculator, graphs and geometry, lists and spreadsheets, notes, and data analysis. |
Blog
One of the common challenges for many Algebra students is not having a solid understanding of Pre-Algebra. So many students complain that they forgot or never fully learned fractions, decimals, percentages, ratios and word problems. Even memorization of
multiplication & division tables can fade over time. These concepts are critical for middle and high school math. A better understanding can...
read more
Most students that struggle in math find Algebra is when things "start to get difficult". This is very common because Algebra brings all the basic rules of arithmetic together. If you were relieved when the chapter on decimals or fractions was over, it turns
out it is never over. The good news is that math concepts are all linked together and there is usually more than one way to solve a problem...
read more |
Geared specifically toward the homeschool classroom, Saxon Algebra 1 is a college-prep course designed to build the mathematical foundation necessary for students to transition successfully into higher-level math courses.
This helped my first two children to be ready for calculus in college--they were able to test out of college algebra. My youngest child needed to go at a slower rate and had to take college Math in college. So it is great for those who have a bent for math and science. |
MATH-I-NESS
Improve your reasoning skills in math by making sense of today's mathematical
theory and application. This course is constructed to exceed the expectations
and outcomes set for the Michigan Department of Education for this age group.
There are no limitations for exploring Calculus, Technical Math and the Art of
Geometry. |
In this lesson you will learn how Algebra is used in everyday life and how to solve basic problems using multiplication and division along with addition and subtraction from Algebra 101. This application includes a detailed description of basic algebra functions, an unlimited number of practice problems and a step by step solution to each problem |
Beginning Algebra - 4th edition
Summary: This text reflects the compassion and insight of its experienced author team with features developed to address the specific needs of developmental level students. Throughout the text, the authors communicate to students the very points their instructors are likely to make during lecture, and this helps to reinforce the concepts and provide instruction that leads students to mastery and success lightly used instructor's edition. Inside may contain answers/notes in margins. Ships same day or next business day. Free USPS Tracking Number. Excellent Customer Service. Ships from TN
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INSTRUCTOR EDITION!!!I ALL ANSWERS INCLUDED.dentical to student edition!!SHIPS FAST!! SAME DAY OR W/N 24 HOURS.EXPEDITED SHIPPING AVAILABLE TOO!!
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New Hard cover New. No dust jacket. 100% BRAND NEW ORIGINAL US STUDENT Edition / Mint condition / Never been read / Shipped out in one business day with free tracking.
$295.85 |
algebra 1, algebra 2 and algebra 1, algebra 2 and calculus algebra 1, algebra 2 and geometry |
Word problems lose some of their mystery with this helpful series geared toward algebra students. Algebra Word Problem Tutor: Problems Involving Speed, Distance & Time movie This step-by-step instructional focuses on speed, distance, and time, demonstrating how to glean pertinent information from word problems to form the correct equations. |
What is Algebra? Why Take Algebra?
Simply put, Algebra is about finding the unknown or it is about putting real life problems into equations and then solving them. Unfortunately many textbooks go straight to the rules, procedures and formulas, forgetting that these are real life problems being solved.
A branch of mathematics that substitutes letters for numbers. An algebraic equation represents a scale, what is done on one side of the scale with a number is also done to the other side of the scale. The numbers are the constants. Algebra can include real numbers, complex numbers, matrices, vectors etc.
In the simplest of form, I could say: A clown was carrying a handful of balloons. Along came the wind and blew 8 away, leaving him only with 9. How many did he start with? In algebra, this problem would then be converted to:
x - 8 = 9
The x replaces the unknown that we are trying to find out, we know the wind blew 8 away and we know that the man was left with 9 balloons. Remember, in Algebra x seems to be teh favorite letter to substitute for the unknown.
The goal in algebra is to find out the unknown. Therefore, we often end up with x = something. In this problem, when you apply agebra, you'll learn that it's like using a scale, you want to isolate x and in so doing you'll do the same thing on each side of the = sign. To isolate x in x-8=9, I will need to add 8 to the left side of the = sign and add 8 to the right side of the = sign. I am then left with x = 8 + 9 therefore solving my real life problem that x = 17, meaning that the balloon man started with 17 balloons.
Why Do I
Need Algebra? Only you can answer this question. I've always said math is an opportunity gateway and you can't get to higher maths without taking algebra. Algebra develops your thinking, specifically logic, patterns, problem solving, deductive and inductive reasoning. The more math you have, the greater the opportunity for jobs in engineering, actuary, physics, programming etc. Higher math is often an important requirement for entrance to college or universities.
Ultimately, you need to do your own homework to determine if your goals mean sticking it out in math, however, you can never go wrong if you do. |
Art Within Math Watch this short video and try to identify the location of the sculptures. Then see how art and these sculptures relate to the science of mathematics. Author(s): No creator set
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Mathematics Framework for the 2009 National Assessment of Educational Progress This is an assessment framework, not a curriculum framework. In broad terms, this framework attempts to answer the question: What mathematics should be assessed in 2009 on NAEP at grades 4, 8 and 12? The answer to this question must necessarily take into account the constraints of a large-scale assessment such as NAEP, with its limitations on time and resources. Of critical importance is the fact that this document does not attempt to answer the question: What mathematics should be taught (or ho Author(s): No creator set
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Basic Concepts of Mathematics This book helps the student complete the transition from purely manipulative to rigorous mathematics. The clear exposition covers many topics that are assumed by later courses but are often not covered with any depth or organization: basic set theory, induction, quantifiers, functions and relations, equivalence relations, properties of the real numbers (including consequences of the completeness axiom), fields, and basic properties of n-dimensional Euclidean spaces. Author(s): No creator set
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Calculus on the Web COW is an internet utility for learning and practicing calculus. The principal purpose of COW is to provide you, the student or interested user, with the opportunity to learn and practice problems in calculus (and in the future other topics in mathematics) in a friendly environment via the internet. The most important feature of the COW is that you get to know whether your answer is correct almost immediately. It is as if you had a tutor looking over your shoulder and helping you along as you wo Author(s): No creator set
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Related contentReports from the Curriculum Foundations Project This page holds the archived reports from the Curriculum Foundations Project. This project held workshops in several different subject areas in order to initiate a dialogue among the representatives from each partner discipline, with mathematicians present to listen and serve as a resource when questions about the mathematics curriculum arose. Users can access Microsoft Word reports as well as a compressed full report. Author(s): No creator set
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John Napier Scotsman John Napier is best known to for his treatise on Protestant religion. However, it was his interest in a completely different subject that radically altered the course of mathematics. After forty years of dabbling in maths, he revealed his table oHighlights of Calculus, Spring 2010" " Sc Author(s): Strang, Gilbert
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Connections: linking mathematics to social studies, art, and science This publication offers online resources that connect mathematics to three subject areas: social studies, art, and science. Each section contains lesson plans, problems to solve, and examples of mathematics at work within contexts not usually associated with school mathematics. Author(s): No creator set
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Ratios for all occasions A central theme in the middle school mathematics curriculum, proportional reasoning is based on making sense of ratios in a variety of contexts. The resources chosen for this unit provide practice in solving problems, often informally, in the format of ga Author(s): No creator set
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Applied Finite Mathematics This module contains all 10 chapters of the Applied Finite Mathematics open textbook by Rupinder Sekhon. NOTE: This book is a work in progress and has not yet been marked up in CNXML. You can download individual chapter files from their respective modules. Author(s): Rupinder Sekhon
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RelatedLugosi teaches math - convergence of series 3 Béla Lugosi teaches advanced mathematical concepts in this video. How one uses power series to solve varying kinds of problems is discussed, and Lugosi explains how this one application shows the power of power series. It also show that power series have a very important and powerful and analytical technique. A great series for any advanced mathematics student who is working in Calculus and Calculus II. Author(s): No creator set |
A study of the real and complex number systems, algebraic expressions and equations, polynomial and rational functions and their graphs, inequalities and their graphs, exponential and logarithmic functions and their graphs, systems of equations, and conic sections. Mat 1050 is not a help in preparing a student to take this course. Please be aware that this course is intended to be a preparation for calculus (MAT 2150 and MAT 2210) and will be taught in such a manner.
This is a 3 credit hour course and at least 6 hours of study outside of class are expected.
Goals
*Initiate student-faculty contact
*Cooperate with other students
*Be an active learner
*Spend time on task
*Exhibit high expectations
*Respect diverse talents
Objectives
Each student will demonstrate an understanding of the fundamental properties of the following:
*The fundamental properties of real numbers
*The common notations of algebra
*Methods and strategies for solving inequalities and linear and quadratic equations
The lowest 100-point chapter test score will be dropped. A practice of giving make-up tests is not made. If a student knows that he or she will be absent the day of a test, he or she may take the test before the given day. If a make-up test is given, it will be much harder than the original test and must be made up before the following chapter test is given. No make-up tests will be given after the last class day. Each student is only allowed one make up test per course per semester. If every student in the class is present the day of the test, each student will receive 2 bonus points on the given test.
There will be eleven 10-point pop quizzes given. The one with the lowest score will be dropped. They may only be made up if the instructor knows in advance that you will be absent from class.
There will be a 3-point online homework assignment given over each section. These homework assignments may be taken up to 5 times each, and only the top grade for each section will be recorded. Be sure to watch the deadline for these quizzes, because they cannot be made up for any reason.
There will be a 3-point online quiz given over each section. These quizzes may be taken up to 5 times each, and only the top grade for each sections will be recorded. Be sure to watch the deadline for these quizzes, because after the expiration date, they cannot be made up for any reason.
The 100-point comprehensive multiple choice final exam will be given for Section 001: Wednesday, December 4, from 8:00 - 10:30; for Section 002: Monday, December 2, from 10:45 - 1:15; for Section 013, Tuesday, December 3, from 1:30 - 4:00.
The last day to drop this class with a "W" grade is Wednesday October 16.
A Blackboard website will be utilized in this class. You will find the "class notes" posted under "Course Documents."
Supplemental Instruction: A Supplemental Instruction (SI) component is provided for all students who want to improve their understanding of the material taught in this course. SI sessions are led by a student who has already mastered the course material and has been trained to facilitate group sessions where students can meet to compare class notes, review and discuss important concepts, develop strategies for studying, and prepare for exams. Attendance at SI sessions is free and voluntary. Students may attend as many times as they choose. SI sessions begin the second week of class and continue throughout the semester. A session schedule will be announced in class. For information about the program and session schedule/updates, visit
Grade Components
Name
Weight
Subject
Test 1
100 points
Chapter 1
Test 2
100 points
Chapter 2
Test 3
100 points
Chapter 3
Test 4
100 points
Chapter 4
Test 5
100 points
Chapters 5, 6
Final Exam
100 points
comprehensive multiple choice
Pop Quizzes
100 points *
random pop quizzes
Online Homework
100 points
to be taken online in MyMathLab by the appropriate due dates
Online Quizzes
100 points
to be taken online in MyMathLab by the appropriate due dates
Final Grades
A:
90-100
B+:
88
C+:
78
D+:
68
F:
0-54
A-:
89
B:
80-87
C:
70-77
D:
60-67
B-:
79
C-:
69
D-:
55-59
Attendance Policy
Each students should arrive on time and prepared for class.
Do not attend class if you are going to sleep through any part of it.
Students, who submit written notification to their instructors within two weeks of the beginning of the semester, shall be excused from class or other scheduled academic activity to observe a religious holy day of their faith. Excused absences are limited to two class sessions (days) per semester. Students shall be permitted a reasonable amount of time to make up tests or other work missed due to an excused absence for a religious observance. Students should not be penalized due to absence from class or other scheduled academic activity because of religious observances.
Student Conduct & Honor Code
Students have the responsibility to know and observe the UNCP Academic Honor Code. This code forbids cheating, plagiarism, abuse of academic materials, fabrication or falsification of information, and complicity in academic dishonesty. Any special requirements or permission regarding academic honesty in this course will be provided to students in writing at the beginning of the course, and are binding of the students. Academic evaluations in this course include a judgment that the student's work is free from academic dishonesty of any type and grades in this course therefore should be and will be adversely affected by academic dishonesty. Students who violate this code can be dismissed from the University. The normal penalty for a first offense is an F in the course. Standards of academic honor will be enforced in this course. Students are expected to report cases of academic dishonesty to the instructor. In general, faculty members should, and will, take preventative measures to avoid cases of academic dishonesty (for example, examinations should be carefully proctored). However, a faculty member's failure to take such measures is no excuse for academic dishonesty. Academic honesty and integrity, in the final analysis, are matters of personal honesty and individual integrity on the part of every student. More information on the academic honor code can be found at
The UNCP Academic Honor Code will be strictly observed in this class.
Any behaviors that disrupt the teaching and/or educational process will not be tolerated. If a student displays such behavior, the student will be asked to leave the classroom for the remaining class period. If the disruptive behavior continues, the Office for Academic Affairs will be asked to administratively withdraw the student from the class.
Disruptive behavior is defined as repeated, continuous, and/or other multiple student behaviors that hamper the ability of instructors to teach and students to learn. Examples include, but are not limited to:
* Eating in class
* Failing to respect the rights of other students
* Talking when the instructor is speaking
* Constant questions or interruptions
* Overt inattentiveness
* Creating excessive noise
* Entering class late or leaving early
* Use of pagers or cell phones in the classroom
**There will be absolutely no electronic communication devices allowed in the classroom to include laptop computers, ipads, or cell phones. Texting during class - sending or receiving - is totally prohibited. All cell phones must be put away in a backpack or purse (not just in a pocket) before entering the classroom.
Extreme examples include, but are not limited to:
* Use of profanity or pejorative language
* Intoxication
* Verbal abuse
* Harassment
* Threats to harm oneself or others
* Physical violence
Conditions attributed to physical or psychological disabilities are not considered a legitimate excuse for disruptive behavior.
Office Location and Phone Number
1228 Oxendine Science Building
910-521-6256
Course Calendar and Office Hours
Class will meet Mondays, Wednesdays, and Fridays from August 14 through November 25, with the exception of the following holidays: September 2 and October 10, 11.
Office Hours: 1:15 - 3:00 MW; 1:30 - 3:00 F; by appointment T
Email Address
linda.hafer@uncp.edu
Other Information
I want to see you succeed in this course. I hope you will feel comfortable coming to see me whenever you have questions or need help with course material. UNCP also has a group of professors known as the Go-to Faculty, who are available to answer your questions, help you resolve problems, and locate helpful resources on campus. For a list of Go-to Faculty or more details, see their Web site ( or contact Mark Canada (mark.canada@uncp.edu, 521-6431, Old Main 209.
Federal laws require UNCP to accommodate students with documented learning, physical, chronic health, psychological, visual or hearing disabilities. In post-secondary school settings, academic accommodations are not automatic; to receive accommodations, students must make a formal request and must supply documentation from a qualified professional to support that request. Students who believe they qualify must contact Disability Support Services (DSS) in DF Lowry Building, Room 107 or call 910-521-6695 to begin the accommodation process. All discussions remain confidential. Accommodations cannot be provided retroactively. More information for students about the services provided by DSS and the accommodation process may be found at the following links:
Tutoring is available for most subjects in the Center for Academic Excellence to groups of up to five students per session, with peer tutors who show proficiency in courses and have been trained in effective tutoring strategies. To get the most effective results, students should sign up for tutoring as soon as possible after the beginning of the semester, come to tutoring sessions with specific questions prepared regarding course material, and attend the sessions consistently. Sign up for tutoring by contacting Courtney Walters at 910-775-4408 or courtney.walters@uncp.edu.
Student Support Services provides eligible students with one-on-one and group tutoring, personal counseling, and assistance with applying for financial aid. Contact the TRIO office in the A and B suites in Jacobs Hall.
The Resource Learning Lab in the Center for Academic Excellence offers computer based, self-paced tutoring in basic writing skills from composing sentences, paragraphs, and essays, to addressing common writing problems, basic reading comprehension, and word problem dissection. These programs are 4 – 8 weeks long and offer non-credit, collectable test performance data on each student's progress. The Resource Learning Lab also offers self-help DVDs for academic study skills such as Values and Goals, Time Management, Critical Thinking and Problem Solving, Active Listening and Note Taking, Researching, Reading and Writing, and Studying and Test Taking. The Resource Learning Lab is available to all students, whether right out of high school or non-traditional students needing a refresher, by contacting Mark Hunt at 910-775-4393 or mark.hunt@uncp.edu.
Academic Resource Mentors (ARMs) are available in the Center for Academic Excellence for all students regardless of academic standing or class standing. Participants are matched up with a peer mentor to meet for one 30 minute session each week to discuss progress in current classes and help students further develop their academic skill sets such as time management, test preparation, utilization of textbooks, note taking, and other valuable skills. Mentors also help students navigate the university's policies and procedures such as grade replacements and academic appeals. ARMs host regular skill enhancement workshops that are available to all students, including those not regularly seeing a mentor. Workshops address specific topics such as understanding your educational expenses, preparing for your future career, the importance of being involved on campus, and what to do in order to recover from academic struggles. Sign up for services by contacting Jennifer Bruner at 910-775-4391 or jennifer.bruner@uncp.edu.
The University Writing Center staff works one-to-one with UNCP students at any stage in the writing process, from brainstorming topics to drafting, revising, editing, and formatting. UNCP students from any course or department are welcome to use the Center. Tutors work with students on all types of writing assignments, including application essays and personal statements. The University Writing Center is located in D.F. Lowry room 308. For more information, and to make an appointment, students should visit
For the Emergency Information Hotline, phone 521-6888.
The instructor reserves the right to change or modify any of the above conditions. |
Course Number and Title
Number of Credits
Minimum Number of Instructional Minutes Per Semester
Prerequisites
MATH Placement Test score of 9, or MATH122 (C or better), or MATH120 (C or better) and High School Trigonometry
Corequisites
None
Other Pertinent Information
A comprehensive departmental final examination is included in this course.
Catalog Course Description
This course introduces the foundations of analysis designed to precede the calculus sequence with emphasis on functions and graphs. Topics include properties of absolute value, polynomial, rational, exponential, logarithmic and trigonometric functions; techniques for solving equations and inequalities, and an introduction to the concept of limits and the difference quotient.
Required Course Content and Direction
Learning Goals:
Course Specific:
The student will be able to:
apply and extend algebraic skills.
analyze calculus-related concepts, such as limits, tangent to a curve, and the difference quotient.
apply the properties of functions, such as finding the domain and range and developing the skill to graph them.
accurately use a graphing calculator to find transformations of graphs.
Category III: CRITICAL THINKING/PROBLEM SOLVING
The students will be able to:
Demonstrate an understanding of solving problems by:
recognizing the problem
reviewing information about the problem
developing plausible solutions
evaluating the results
These skills are developed in VII.B.2.i, VII.3.b, VII.3.c, and VII.B.5.h.
Planned Sequence of Topics and/or Learning Activities:
The following is a list of the minimum amount of course material to be covered by the instructor. Accompanying each topic is an approximate number of lessons required to study the topic.
Review Topics (10 lessons)
exponents & scientific notation
radicals & rational exponents
polynomials
factoring polynomials
rational expressions
linear equations
quadratic equations
linear inequalities
Graphs, Functions, and Models (9 lessons)
graphs& graphing utilities
lines and slopes
distance & midpoint formulas, circles
basic functions
graphs of functions
transformations of functions
combinations of functions, composite functions
inverse functions
modeling with functions
Polynomial & Rational Functions (9 lessons)
complex numbers
quadratic functions & applications
polynomials functions, application & graphs
dividing polynomials; remainder and factor theorem
zeros of polynomial functions
rational functions and graphs polynomial & rational inequalities
Exponential & Logarithmic Functions (5 lessons)
exponential functions
logarithmic functions
properties of logarithms
exponential & logarithmic equations
optional; modeling with exponential and logarithmic functions
Trigonometric Functions & Analytical Trigonometry (10 topics)
angles and their measure
trigonometric functions; the unit circle
right triangle trigonometry
trigonometric functions of any angle
graphs of sine and cosine functions
graphs of other trigonometric functions
inverse trigonometric functions
applications of trigonometric functions
verifying trigonometric identities
sum and difference formulas
double angle and half angle identities
Additional Topics in Trigonometry (5 topics)
optional: product to sum and sum to product formulas
trigonometric equations
the law of sines
the law of cosines
Assessment Methods for Core Learning Goals:
All Core Critical Thinking and Problem Solving, College Level Mathematics or Science, and Discipline-Specific Course Objectives will be assessed as follows:
The student will apply mathematical concepts and principles to identify and solve problems presented through informal assessment, such as oral communication among students and between teacher and students and, for the core, formal assessment using open-ended questions reflecting theoretical and applied situations.
Reference, Resource, or Learning Materials to be used by Students:
Departmentally selected textbook and graphing calculator is required. Details provided by the instructor or each course section. See Course Format. |
This student-friendly textbook for the Statistics 1 Module of A-Level Maths comprehensively covers the Edexcel exam specification. It contains straightforward, accessible notes explaining all the theory, backed up with useful step-by-step examples. There are practice questions throughout the book to test understanding, with recap and exam-style questions at the end of each section (detailed answers to all the questions are included at the back). Finally, there's a CD-ROM containing two complete Statistics 1 practice exam papers - ideal to print out for realistic practice before the final tests |
books.google.com - Leading experts in the rapidly evolving field of Clifford (geometric) algebras have contributed to these comprehensive volumes. They consist of thematically organized chapters that present a broad overview of cutting-edge topics in mathematical physics and the physical applications of Clifford algebras.Volume... Algebras and Their Applications in Mathematical Physics |
Welcome to the NHI Math Module Start Page
The NHI Math Module is a self directed Learning Module that will be of use to students taking classes offered through the Minnesota Department of Transportation. This is a valuable tool to access and improve the math skills you will need to use on the job. To get started:
Click the Presentation 1- Introduction link below to open the Power Point presentation which will demonstrate how to use the NHI Math Module.
After you have familiarized yourself with the navigation of the module, click on the NHI Math Module link below. This is a self directed learning module. You can start or stop the module at any time. Be sure to take the quizzes to determine how you are doing.
After you finish the module and you feel you have mastered the learning materials, return to this page and click on the Presentation 2-Conclusion link to finish the NHI Math Module. |
This classroom activity presents Calculus II students with some Flash tutorials involving work and pumping liquids along with...
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This classroom activity presents Calculus II students with some Flash tutorials involving work and pumping liquids along with some simple questions concerning the amount of work involved in pumping water out of two full containers having the same shape and size but different spatial orientations.Students are given opportunities to address this question by means of a ConcepTest and a write-pair-share activity. The results are quite revealing and show that while students may have learned how to perform the necessary calculations, their conceptual understanding concerning work may remain faulty.
This classroom activity presents Calculus II students with some Flash tutorials involving work and pumping liquids along with...
see more
This classroom activity presents Calculus II students with some Flash tutorials involving work and pumping liquids along with some questions concerning the amount of work involved in pumping water out of two full containers having the same shape and size but different spatial orientations. Students are given opportunities to address this question by means of a write-pair-share activity in which they construct an integral equation and solve for an upper limit of integrationA web hosted courseware of undergraduate single and many-variable calculus for physics and engineering students, with...
see more
A web hosted courseware of undergraduate single and many-variable calculus for physics and engineering students, with animated and interactive graphics. It is based on a course "Mathematical Introduction for Physicists" of the Tel-Aviv University.In addition to text, examples and exercises, the courseware takes advantage of modern technology with interactive and animated graphics (over 110) that can be projected in class, and accessed at any time by the students.Math Animated is technically based on SVG and MathML - open standards, developed by the Web Consortium, without the need of proprietary software. It runs on the most popular platforms: Windows, Mac and Linux.About 10% of the material, including text and graphics is open and does not require any registration. |
Enduring Understandings: ... physical spaces and that the geometric figure in
questions can actually be ... Construct suitable mathematical models to represent
a geometric situation. 2. ... How is visualization essential to the study of geometry
?
Students use conceptual understanding of mathematics when they identify and
apply principles, know and .... Enduring Understandings: Mathematics can be ...
Essential Questions: Is your plan working ... |
Netaji Subash Chandra Bose Bhavan, HPU, Summerhill, Shimla 171005.
Syllabus ( for Entrance)
b) MATHEMATICS
ALGEBRA :
Algebra of complex numbers, modulus and argument, triangle
inequality, nth roots of unity. Theory of quadratic equations
and quadratic expressions, relationship between the roots
and coefficients, sign of a quadratic expression, greatest
and least values of quadratic expression. Arithmetic geometric
and harmonic progressions, sums of arithmetic, geometric
and harmonic progressions, infinite geometric series, sums
of the squares and cubes of the first n natural numbers.
Mathematical induction, permutations and combinations, Binomial
theorem for a positive integral index. Determinants of order
two and three, solutions of simultaneous linear equations
in two and three variables.
TRIGONOMETRY :
Trigonometric functions and their graphs, addition and subtraction
formulae, formula involving multiple and sub multiple angles,
general solution of trigonometric equations, relations between
the sides and angles of triangle, properties of a triangle,
solutions of triangles, heights and distances, trigonometric
functions.
ANALYTICAL GEOMETRY OF TWO DIMENSIONS :
Equation of straight line in various forms, angle between
tow lines, distance of a point from a line, line through
the point of intersection of two given lines, concurrency
of lines. Equation of a circle in various forms, equations
of tangent and normal, intersection of a circle with a straight
line, equation of a circle through the points of intersection
of two circles and that of a circle and a straight, line.
Equations of the conic sections in the standard form, focus,
directrix, eccentricity of the conic section, parametric
equations, equations of tangent and normal
CALCULUS :
Into, onto and one-to-one functions, Sum, difference, product
and quotient of two functions, composite function; absolute
value, greatest integer, polynomial, rational, trigonometric,
exponential and logarithmic functions, even and odd functions,
inverse of a function. Limit and continuity of a function,
limit and continuity of the sum, difference, product and
quotient of two functions, continuity of composite function.
Derivative of a function, derivative of composite and implicit
functions, derivatives of polynomial, rational, trigonometric,
inverse trigonometric, exponential and logarithmic functions.
Geometrical interpretation of derivative, tangents and normal.
Monotonicity, maximum and minimum values of a function.
Derivatives upto order three.
INTEGRATION, DIFFERENTIAL EQUATIONS :
Integration as the inverse proves of differentiation, integration
by parts, integration by the methods of substitution and
partial fraction, Definite integral and its application
for the determination of areas. Properties of definite integrals.
Formational of differential equations. First order equation,
variables separable and homogeneous equations. |
Get everything you need for a successful and pain-free year of learning math! This kit includes Saxon's 3rd Edition Algebra 1 textbook and tests/worksheets book & answer key. as well as the DIVE Algebra 1 CD-ROM. A balanced, integrated mathematics program that has proven itself a leader in the math teaching field, Algebra 1 covers signed numbers, exponents, and roots; absolute value; equations and inequalities; scientific notations; unit conversions; polynomials; graphs; factoring; quadratic equations; direct and inverse variations; exponential growth; statistics; and probability.
The DIVE software teaches each Saxon lesson concept step-by-step on a digital whiteboard, averaging about 10-15System Requirements:
Mac OS 10.3.9-10.4.x
Windows 98, 2000, ME, XP, Vista
Quicktime Download Required.
This Kit Includes:
Saxon Math Textbook, 564 pages, hardcover
Saxon Tests Book
Saxon Answer Key
DIVE CD-ROM Saxon Algebra 1 Kit & DIVE CD-ROM, 3rd Edition
Review 1 for Saxon Algebra 1 Kit & DIVE CD-ROM, 3rd Edition
Overall Rating:
5out of5
Date:February 7, 2011
dlcs
Age:35-44
Gender:female
Quality:
5out of5
Value:
5out of5
Meets Expectations:
5out of5
We returned all of our Teaching Textbooks curriculum. They were lacking in content. We are thrilled to be using Saxon. This is our fourth math curriculum. Wish we had started with Saxon!!
Share this review:
+3points
3of3voted this as helpful.
Review 2 for Saxon Algebra 1 Kit & DIVE CD-ROM, 3rd Edition
Overall Rating:
5out of5
Date:June 18, 2008
Cha
My family has loved the Saxon methods ever since level 4/5 when we started using it. My son loves it's easy to use format and has rapidly moved along in his math skills, with very little to no supervision to complete the lessons. He even starts his days with Saxon math by his own choice. He really loves this math and he's developmentaly delayed. I've introduced others to Saxon math and found much of the same success as my son.
Share this review:
+4points
4of4voted this as helpful.
Review 3 for Saxon Algebra 1 Kit & DIVE CD-ROM, 3rd Edition
Overall Rating:
1out of5
Date:May 8, 2008
SuperMommy5
If you home school and if you are leaning toward another math program like Teaching Textbooks or something that doesn't require constant supervision then SAXON IS NOT FOR YOU.Last year I said I'd never use Saxon again, but I balked at the price of Teaching Textbooks (about 3+ times the price of Saxon). Now I wish we had eaten rice and beans for a few months to pay for it.Saxon in general is not friendly to children who are weak in Math, nor their parents. They offer very few practice problems for each new concept. Their story problems are designed to confuse you with the language, so that the math concept is doubly confusing.I have used Saxon for five years, and I can't believe we have wasted so much time and expended so much anxiety when math was as easily presentable as Math U See or Teaching Textbooks. |
More About
This Textbook
Overview
Clearly presented elements of one of the most penetrating concepts in modern mathematics include discussions of fields, vector spaces, homogeneous linear equations, extension fields, polynomials, algebraic elements, as well as sections on solvable groups, permutation groups, solution of equations by radicals, and other concepts. 1966 2001
concise and self-contained introduction to galois theory
You don't need any algebra background to read and appreciate this book. Only the knowledge of the definitions of groups and normal subgroups is needed. You can find these in any modern algebra book. I read it as a college sophomore without much prior knowledge in this field. I was able to enjoy it pretty much. It might be a little too dense for beginners, but it is almost entirely self contained. It is written based on lecture notes, so don't expect it to be in a very organized format. The only thing I don't like is that it doesn't have an index, but it's okay since the book is very thin.
Was this review helpful? YesNoThank you for your feedback.Report this reviewThank you, this review has been flagged. |
Calculus is a branch of mathematics that deals with rate of
change, area, and volume. For example, while using algebraic
formulae you can calculate the area of a rectangle, using calculus
formulae allows you to calculate the area under any curve. Almost
all modern physics uses calculus as a foundation, and other
sciences as well as economics also use calculus. These problems
show the theory and applications of calculus. Their difficulty is
indicated by bullets. |
Title: Mathematical Proofs
Description:
In this lesson, from Science NetLinks, students explore the nature of logic, evidence, and proofs in the context of mathematics. This lesson would be appropriate after students are familiar with the Pythagorean Theorem.
Standard(s): [MA2010] DM1 (9-12) 11: Solve application-based logic problems using Venn diagrams, truth tables, and matrices [MA2010] MI1 (9-12) 12: Summarize the history of probability, including the works of Blaise Pascal; Pierre de Fermat; Abraham de Moivre; and Pierre-Simon, marquis de Laplace. (Alabama)
Subject: Mathematics Title: Mathematical Proofs Description: In this lesson, from Science NetLinks, students explore the nature of logic, evidence, and proofs in the context of mathematics. This lesson would be appropriate after students are familiar with the Pythagorean Theorem. Thinkfinity Partner: Science NetLinks Grade Span: 9,10,11,12
Web Resources
Podcasts
Teacher Tools |
Buy Used
$3.98 basic concepts of college algebra and trigonometry are presented in a simple manner and are consistently motivated and reinforced by examples. In this third edition, the author has added and/or changed several examples to better illustrate the concept under discussion. Readers are prompted to organize their work and to decide when a meaningful shortcut might be used. Unifying themes found throughout the book are vital to precalculus students. |
Math Review for Physics
Get math review for physics and study guides here. Learn about geometry, trigonometry, and algebra for physics or brush up on your skills. Thorough explanations and practice examples will help you review math concepts require to understand physics.
Study Guides
Introduction
Let's go from two dimensions to three. Here are some formulas for surface areas and volumes of common geometric solids. The three-space involved is flat ; that is, it obeys the laws of euclidean geometry. These ...
Introduction
A logarithm (sometimes called a log ) is an exponent to which a constant is raised to obtain a given number. Suppose that the following relationship exists among three real numbers a , and x , ...
Introduction
There are six basic trigonometric functions . They operate on angles to yield real numbers and are known as sine, cosine, tangent, cosecant, secant , and cotangent . In formulas and equations, they are ...
Introduction
The following paragraphs depict common trigonometric identities for the circular functions. Unless otherwise specified, these formulas apply to angles θ and ϕ in the standard range as follows:
Basic Mathematics for Physics Practice Test
A good score is at least 37 correct. Answers are given at the end. It is best to have a friend check your score the first time so that you won't memorize the answers if you want to take the ...
Introduction to Prefix Multipliers
Sometimes the use of standard units is inconvenient or unwieldy because a particular unit is very large or small compared with the magnitudes of phenomena commonly encountered in real life. We've already seen some good ... |
Tailored to both the specification and the tier, this Student Book delivers exactly what students and teachers need to cover the unit in exactly the right depth.
Synopsis:
* Supports teachers' understanding of AO2 and AO3 through clearly labelled AO2/3 questions in the exercises. * Packed with graded questions reflect the level of demand required, so students and teachers can see their progression. * Includes worked examples throughout the book to break the maths down into easy chunks. * Uses feedback to highlight common errors . |
In response to the need to students to internalise questions and challenges for them to become meaningful several countries are International Science and Maths Olympiads Supports learning in range of aspects of geometry and algebra "Every student can learn mathematics" mathematics curriculum approach
These include mapping the human genome, advances in medical imaging, and advances in During and after their return from the KTO, a significant proportion of Gulf War. What was the overall exposure of troops to Leishmania tropica? |
More About
This Textbook
Overview
The Blitzer Algebra Series combines mathematical accuracy with an engaging, friendly, and often fun presentation for maximum appeal. Blitzer's personality shows in his writing, as he draws readers into the material through relevant and thought-provoking applications. Every Blitzer page is interesting and relevant, ensuring that students will actually use their textbook to achieve success!
Editorial Reviews
From The Critics
This textbook introduces the basic skills of algebra and shows how algebra can model and solve real-world problems. Some of the applications are bicycle manufacturing costs, hate crime incidents, salaries, and telephone calling plans. The third edition has been rewritten to make it more accessible, and adds more supplemental resourcesPreface
Introduction
Introductory Algebra for College Students, Third Edition, provides comprehensive, in-depth coverage of the topics required in a one-term course in beginning or introductory algebra. The book is written for college students who have no previous experience in algebra and for those who need a review of basic algebraic concepts. The primary goals of the Third Edition are to help students acquire a solid foundation in the basic skills of introductoryof the things I enjoy most about teaching in a large urban community college is I selected all updated real-world data to beReadability and Level. The chapters have been extensively rewritten to make them more accessible. The Third Edition pays close attention to ensuring that the amount of detail and depth of coverage is appropriate for a liberal arts college algebra course. Every section has been rewritten to contain a better range of simple, intermediate, and challenging examples. Voice balloons allow for more specific annotations in examples, further clarifying procedures and concepts for students or section.
Check Point Examples. Each worked example is followed by a similar matched problem for the student to work while reading the material. This actively involves the student in the learning process. Answers to all Check Points are given in the answer section.
Updated Real-World Data. Real-world data is used to bring relevance to examples, discussions, and applications. Real-world data from the previous edition has been replaced to include data that extends as far up to the present as possible. Updated real-world data was selected on the basis of being interesting and intriguing to students Critical Thinking Exercises, Technology Exercises, and Review Exercises. This format makes it easy to create well-rounded homework assignments. Many new exercises have been added, with attention paid to making sure that the practice and application exercises are appropriate for the level and graded in difficulty.
Rewritten Exercise Sets. In order to update applications and take them to a new level, most application problems from the previous edition have been replaced with new exercises. At the same time, applications were carefully chosen to avoid readers becoming overwhelmed by an excessive number of options. Expanded writing exercises offer students the opportunity to write about every objective covered in each section, as well as to discuss, interpret, and give opinions about data. Each review exercises now contains the section number and example number of a similar worked-out example.
Expanded Supplements Package. The Third Edition is supported by a wealth of supplements designed for added effectiveness and efficiency. These items are described on pages xii through xivPreserved and Expanded from the Second Edition. The features described below that helped make the Second Edition so popular continue in the Third Edition.
Graphing. Chapter 1 contains an introduction to graphing, a topic that is integrated throughout the book. Line, bar, circle, and rectangular coordinate graphs that use real data appear in nearly every section and exercise set. Many examples and exercises use graphs to explore relationships between data and to provide ways of visualizing a problem's solution.
Geometric Problem Solving. Chapter 3 on problem solving contains a section that teaches geometric concepts that are important to a student's understanding of algebra. There is frequent emphasis on problem solving in geometric situations, as well as on geometric models that allow students to visualize algebraic formulas o; cumulative review exercises.
Supplements for the Instructor
Printed Resources
Annotated Instructor's Edition (0-13-032841-3)
Answers to exercises on the same text page or in Graphing Answer Section.
To the flanked 2002
Much better
This book is considerably better than the 2nd edition, which was an excellent text to start. As an instructor, I hear feedback from many students, and the response has been most favorable thusfar. I personally appreciate the way the book has been redesigned. It does a wonderful job supporting my lectures, and answering student questions before they ever get to me. Well done!
Was this review helpful? YesNoThank you for your feedback.Report this reviewThank you, this review has been flagged. |
More About
This Textbook
Overview
Stefan Waner and Steven Costenoble's APPLIED CALCULUS, THIRD EDITION is designed to address the considerable challenge of generating enthusiasm and developing mathematical sophistication in an audience that is often ill-prepared for and alienated by the traditional applied mathematics courses offered on many college campuses. The new edition retains its engaging conversational style and focus on real data and real world applications of mathematics—a strategy that has proven to be pedagogically successful. The wealth of applications, the highly effective integrated, yet optional, use of graphing calculators or spreadsheets, and the robust supplemental Web site that has received praise from around the world, are what make Waner/Costenoble's text an outstanding choice.
Editorial Reviews
From the Publisher
"In an age where students are increasingly driven by immediate results, this text is a breath of fresh air. It is very readable, so I can confidently assign reading."
"I really enjoy the exercises in the book. Using real-life data in the exercises...gives me opportunities to talk about difficulties in using real data, and helps the students see that there is a difference between understanding calculus and being able to apply it to real models."
Related Subjects
Meet the Author |
Rangeview High School
Course Syllabus
2012-2013
Course Title: Integrated Math 3 (Core-Plus Mathematics Project)
Instructor's Name: Analyn A. Alquitran
Contact Numbers: 303- 695-6848 Voicemail Box: 303-326-3876
School e-mail address: aaalquitran@aps.k12.co.us
Teacher availability: By Appointment
Teacher availability:
Concepts and Skills/ Essential Questions
1st Quarter: Standard #4: Shape, Dimension, and Geometric Relationships. Standard #2:
Patterns, Functions, and Algebraic Structures. Standard #3: Data Analysis, Statistics, and
Probability
Experiment with transformations in the plane
Understand congruence in terms of rigid motions
Prove geometric theorems
2nd Quarter: Standard #2: Patterns, Functions, and Algebraic Structures. Standard #4:
Shape, Dimension, and Geometric Relationships
Analyze functions using different representations
Experiment with transformations in the plane
Understand congruence in terms of rigid motions
Prove geometric theorems involving similarity
3rd Quarter: Standard #2: Patterns, Functions, and Algebraic Structures. Standard #4: Shape,
Dimension, and Geometric Relationships
Interpret functions that arise in applications in terms of the context
Analyze functions using different representations
Understand the relationship between zeroes and factors of polynomials
Prove geometric theorems and make geometric constructions
4th Quarter: Standard #2: Patterns, Functions, and Algebraic Structures.
Analyze functions using different representations
Build new functions from existing functions
Formulate the concept of a function and use function notation
Construct and compare linear, quadratic, and exponential models and solve
problems.
1
Grading guide: This is a standards-based grading class.
Standards-based guideline:
Conversion of grade book body of evidence to letter Used for
grades. grade reports
and
transcripts.
In a variety of assessments, the student consistently and
independently achieves proficiency in grade level
concepts/skills and demonstrates advanced A
application/analysis when the opportunity exists.
In a variety of assessments, the student achieves proficiency
in grade level concepts/skills. B
With teacher or peer support, the student is able to achieve
proficiency in grade level concepts/ skills. C
The student demonstrates limited understanding/
application of grade level concepts/skills and does not meet D
the identified goals at this time.
The student rarely demonstrates understanding of grade
level concepts/skills or there is insufficient evidence to F
accurately determine the proficiency level.
Progress indicator: used in teacher grade book for individual assignments/assessments.
Advanced
Proficient
Partially Proficient
Unsatisfactory
Unsatisfactory/Missing
Capital letters: summative or "major" assignments/assessments
Lower Case: formative or practice assignments/assessments
(+)= denotes upper range within progress indicators
( -)= denotes lower range within progress indicators
Classroom Behavior Expectations and Policies: Students are expected to adhere
with the Classroom expectations, work cooperatively with peers in every group activity,
2
work independently on a given On Your own tasks and respect every individual in the
classroom.
Policies for Absences and Make-up work: Students who were absent must
complete their make-up work in 5 days following their return to school. They must
secure notes on the work missed from the absence packet or from their group
coordinator prior to scheduling a make-up session.
Other Policies:
Hall passes: Each student is eligible for3 hall pass privilege for emergency
purposes only. Student is expected to stay after class 1 minute thereafter
as a replacement for using 4 or more pass privileges.
Tardy Policy: Please see student handbook.
Student who is caught cheating will automatically receive a rating of
unsatisfactory on the assignment, and a discipline referral.
Substitutes: In case the regular course teacher is out, students are
expected to complete and finish the day's class work. Any report
submitted by the substitute to the regular teacher is final.
Classroom managers selected for specific job assignment in the group are
expected to perform their duty based on job description. The teacher
reserves the right to provide a replacement for non-performance.
Homework: Collection of homework is during the first 15 minutes of
class.
Students are expected to remain seated for the duration of the class
period, unless otherwise instructed.
Material or Supplies Required or Recommended for Course:
TI-83 or TI-84 Calculator (required) Eraser (recommended)
Graphing Notebook (required) 1 box Kleenex (recommended)
12 pencils (required) AAA batteries (recommended)
Graph Paper (cm or inch grid) 1 clear plastic ruler (recommended)
Please sign and return.
Student Name : __________________________________
Subject: Integrated Math Course 4
Teacher: Analyn A. Alquitran
Parent Signature: _________________________________
Date |
Find a Magnolia Park, CA Math
...Below I list them and then briefly indicate their contents.
1) Introduction to logic A-
2) Intermediate logic A
3) Modal reasoning A
4) Form and meaning A
5) Philosophy of mathematics A
Introduction to logic teaches the basics of classical logic, that is, propositional logic & first-order logic... |
Recognizing Quantities and Relationships in Word Problems Help
Introduction to Recognizing Quantities and Relationships in Word Problems
Success in solving word problems depends on the mastery of three skills— "translating" English into mathematics, setting the variable equal to an appropriate unknown quantity, and using knowledge of mathematics to solve the equation or inequality. This book will help you develop the first two skills and some attention will be given to the third. |
Mathematics for Physics
Michael M. Woolfson and Malcolm S. Woolfson
The broad scope provides great flexibility, supporting use on a wide range of courses.
Content spans both first and second year, bringing lecturers the convenience of a single text to support two years of teaching; and students the reassurance of a consistent approach between courses and, as one text for two courses, greater value for money.
The layering of mathematical theory with applications to physics strikes a careful balance between theory and practice. It enables the student to grasp the underlying concepts, and increases the students' motivation by demonstrating the concepts' practical relevance.
Extensive learning support, through exercises, problems, and computer programs, encourages active learning to make the studying experience as effective as possible.
Worked solutions to all exercises and problems help students to check they understand every step in a mathematical task, empowering them to apply familiar skills to new, unfamiliar problems.
Online Resource Centre features figures from the book, and a suite of FORTRAN, C, and MATLAB computer programs, to support both teaching and learning. |
Theory of Probability : Explorations classical foundations to advanced modern theory, this self-contained and comprehensive guide to probability uses a pedagogical focus on discovery and elementary methods of proof, to weave together mathematical proofs, historical context and richly detailed illustrative applications. A fascinating reference and essential resource, for engineers, computer scientists and mathematicians. |
Editorial Reviews
From the PublisherB & W illustrations throughout
From the Inside FlapThis is a helpful math book,it covers a lot of subjects from basic math to some pretty difficult topics.My only problem with it is that there are a bunch of wrong answers in the "answer" section,but it just made me hunt and work that much harder to find the correct answers.I hope they do a revision of this book because I would like to give it as a gift,but not at this time with all the mistakes in it.
Overall, the book seems to be a good, quick overview of math. Great if you've been out of school for a while. But it's tarnished by the unbelievable number of wrong answers in the back of the book. Math is an absolute, either right or wrong. How can these errors be allowed in a MATH book?
Disappointing! At first glance this book does have lots potential in terms of simplifying concepts and formulas, but it is loaded with mistakes - not only in the answer sections, but also regarding some of the methods. I applied their "Method for Finding Square Roots" to numbers other than the given examples, and it did not work. This is not a good buy - even if one merely needs a refresher in math. |
Algebra : Combined Approach - With 2 CDs - 3rd edition
ISBN13:978-0131868465 ISBN10: 0131868462 This edition has also been released as: ISBN13: 978-0131870017 ISBN10: 0131870017
Summary: The engaging Martin-Gay workbook series presents a reader-friendly approach to the concepts of basic math and algebra, giving readers ample opportunity to practice skills and see how those skills relate to both their lives and the real world. The goals of the workbooks are to build confidence, increase motivation, and encourage mastery of basic skills and concepts. Martin-Gay enhances users' perception of math by exposing them to real-life situations through graphs a...show morend applications; and ensures that readers have an organized, integrated learning system at their fingertips. The integrated learning resources program features book-specific supplements including Martin-Gay's acclaimed tutorial videotapes, CD videos, and MathPro 5. This book includes key topics in algebra such as linear equations and inequalities with one and two variables, systems of equations, polynomial functions and equations, quadratic functions and equations, exponential functions and equations, logarithmic functions an equations, rational and radical expressions, and conic sections. For professionals who wish to brush up on their algebra skills. ...show less
2006 Paperback2006 Paperback Good Domestic Media Mail shipping ONLY. This is a former library copy with library stickers and stamps. 100% of this purchase will support literacy programs through a nonprofit organ...show moreization!1868462 |
For more information or to request copies of other placement tests, call: (800) 416-8171 Or e-mail us: ... The math placement tests are only one tool used to place a student who is new to the Saxon Math program. You must also consider the ... 76. If __w x = 3, what does ...
to the Saxon math program. This test includes selected content from Math 54, Math 65, Math 76, Math 87, and Algebra 1/2. Please note that this placement test is not infallible. It is simply one indicator ... tests. We can also be contacted at 2450 John Saxon Blvd., Norman, ...
Saxon Homeschool Placement Guide Saxon books are skill level books, not grade level books. It is essential that each student is placed in the text that meets the skill level of the individual student. ... Algebra A Math 76 Math 65 Math 54
Saxon Math 54 (2nd Edition) Tex T: Saxon Math 54 (2nd Edition) In T roduc TI on ... your child will be taking two tests after the final lesson of the year. For your convenience, the testing schedule is included in this lesson plan. Created Date:
set of blackline masters provides simulated tests using concepts taught in Saxon Math. This set of blackline masters contains every ... Synthesis of research on reviews and tests. Educational Leadership, 48,71–76. Dhaliwal, V. (1987). A study of short-term and long-term memory of serial tasks ...
Saxon program should start in Saxon's Math 54, Math 65, Math 76, Math 87, Algebra 1/2, or Algebra 1 textbook. Please note that this placement test is not a fool-proof placement ... ment tests. We can also be contacted at 1320 W. Lindsey, Norman, OK 73069; or by e-mail at [email protected]
If a student is struggling somewhat with these assignments and scoring less than 80% on his tests ... lesson in your Saxon textbook and do the practice problems in the book. SAMPLE Test Scores ... ____Day 93 Complete DIVE Lesson 76 or read Lesson 76 (pp313 – 314).
The Saxon Mathematics 7/6 Tests and Worksheet booklet is represented by the abbreviation WORK. Each weekly assignment is summarized in the first rows of the week's daily course plan along with the goals and notes for that week.
Saxon math 3 is a course designed to challenge ... Students can be evaluated through tests, daily practice sets ... First Semester: Lesson 1 - 75 Second Semester: Lesson 76 - 140 Course Objectives: At the end of this course students should be able to: 1. Memorize all addition ...
The Unit 4 and 5 tests consist of a series of written assessments which can be located in the Saxon Student Workbook. ... Day 76Saxon Math 3, Lesson 71, "Writing Three-Digit Numbers Using Digits" index card marker scissors
MATH – Sarah has been using the Saxon76 Math Textbook and Test Book. She has completed 30 out of 140 lessons, which is about ¼ of the book and has taken 5 tests. Her average for the quarter is 93%. That's it. Repeat for each subject.
Saxon Everland Road Hungerford, Berkshire RG17 0DX United Kingdom ... 76-22-2 EINECS: 200-945-0 bornan-2-one Xn R20-68/22; ... Due to missing tests no recommendation to the glove material can be given for the product/ the preparation/ the |
Intermediate Algebra Graphs and Models
9780321416162
ISBN:
0321416163
Edition: 3 Pub Date: 2007 Publisher: Prentice Hall
Summary: The Third Edition of the Bittinger Graphs and Models series helps readers succeed in algebra by emphasizing a visual understanding of concepts. This latest edition incorporates a new Visualizing for Success feature that helps readers make intuitive connections between graphs and functions without the aid of a graphing calculator. In addition, readers learn problem-solving skills from the Bittinger hallmark five-step ...problem-solving process coupled with Connecting the Concepts and Aha! Exercises. As you have come to expect with any Bittinger text, we bring you a complete supplements package including MyMathLabtrade; and the New Instructor and Adjunct Support Manual. KEY TOPICS: Basics of Algebra and Graphing; Functions, Linear Equations, and Models; Systems of Linear Equations and Problem Solving; More Equations and Inequalities; Polynomials and Polynomial Functions; Rational Expressions, Equations, and Functions; Exponents and Radicals; Quadratic Functions and Equations; Exponential and Logarithmic Functions; Conic Sections; Sequences, Series, and the Binomial Theorem. MARKET: For all readers interested in Algebra.
Bittinger, Marvin L. is the author of Intermediate Algebra Graphs and Models, published 2007 under ISBN 9780321416162 and 0321416163. One hundred fifty four Intermediate Algebra Graphs and Models textbooks are available for sale on ValoreBooks.com, twenty seven used from the cheapest price of $1.00, or buy new starting at $11 |
Professional Commentary: Students use algebra or geometry software to discover that the height at which guy wires supporting two vertical poles cross does not depend on how far apart the poles are. This counterintuitive result can lead to conjecturing and testing hypotheses as to what factors determine the height at which the wires cross....
Professional Commentary: Students are asked to minimize the labor costs of hiring different numbers of workers for different shifts at different hourly wages in a pizza shop. Students use the graph of a system of linear inequalities to solve this linear programming problem geometrically....
Professional Commentary: Students are asked to maximize profits for an athletic shoe company that produces two kinds of shoes. This is the kind of product-mix problem that occurs whenever a company produces more than one item....
use matrices and technology to solve the Meadows or Malls problem, a linear programming problem with six variables. Students who have not done linear programming problems before are advised to begin with The Busing Problem before attempting Meadows or Malls....
Professional Commentary: This activity focuses on having students create and solve systems of linear equations in real-world settings. By solving a system of two equations in two unknowns, students find the equilibrium point for supply and demand.... |
Diran Basmadjian and Ramin Farnood
A FIRST LOOK AT MODELING
The Physical Laws
The Rate of the Variables: Dependent and Independent Variables
The Role of Balance Space: Differential and Integral Balances
The Role of Time: Unsteady State and Steady State Balances
Information Derived from Model Solutions
Choosing a Model
Kick-Starting the Modeling Process
Solution Analysis
Practice Problems
SOLUTION OF LINEAR SYSTEMS BY SUPERPOSITION METHODS
Superposition by Addition of Simple Flows: Solutions in Search of a Problem
Superposition by Multiplication: Product Solutions
Solution of Source Problems: Superposition by Integration
More Superposition by Integration: Duhamel's Integral and the Superposition of Danckwerts
Practice Problems |
Learn more with other tutorials...
You may think of using symbolics for problems involving undefined variables. While this is a very useful feature, it is also important to note that symbolics can be used to solve problems, which can be calculated numerically.... (Show more)(Show less)
Matrices are challenging, but they are really important in applied mathematics – they are a critical STEM topic. Engineers and scientists use matrices to solve challenging problems in many, many dimensions. Mathcad's matrix and graphing tools offer capabilities that can help students' explore matrices early in their school experience so that they are both prepared to use and aware of the importance of matrices. With currently available technologies matrices can be used, explored, and visualized effectively in Algebra 1 class. Systems of equations in three variables need not be avoided any longer. Matrices can be an efficient and powerful way to solve systems, with increased clarity now that we have tools to graph 3D plots.... (Show more)(Show less) |
dynamic new course combining classbook, CD-ROM and online components to offer flexible, time saving and supportive materials. Cambridge Essentials ...Show synopsisA dynamic new course combining classbook, CD-ROM and online components to offer flexible, time saving and supportive materials. Cambridge Essentials Mathematics Extension 7 Pupil Book is aimed at National Curriculum Levels 4-7. The book gives a map for the pupil and teacher of how to cover all aspects of the topic whilst focussing on delivering exercises with strong progression. The pupil CD-ROM replicates the book page with buttons acting as links to prior knowledge, keywords and explanations. Functional Maths questions are included at National Curriculum Level 6 |
Practical Math Success - 3rd edition
Summary: This book is geared toward anyone wishing to overcome math anxiety. Updated and re-evaluated by math experts to ensure the most current lessons and practice exercises, this resource includes: essential math basics and tips for test-takers. |
Summary
Intended for a first-year-level developmental mathematics course in intermediate algebra,Exploring Intermediate Algebra: A Graphing Approachis designed to assist students in making connections between mathematics and its applications. Its goal is to develop a student's mathematical skills through appropriate use of applications and to use technology to establish links between abstract mathematical concepts and visual or concrete representations. The provenAufmann Interactive Method(AIM) ensures that students try concepts and manipulate real-life data as they progress through the material. Every objective contains one or more sets of matched-pair examples encouraging students to interact with the text. The first example in each set is completely worked out; the second example, called 'You Try It,' prompts students to practice concepts at the time they are presented in the text.Complete worked-out solutionsto these examples in an appendix at the end of the book help students by providing immediate feedback, concept reinforcement, and identification of mistakes to prevent frustration. Technology is integrated throughout the text to assist students in making connections between abstract mathematical concepts and a concrete representation provided by technology. Eduspace, powered by Blackboard, for the Aufmann/Lockwood/BoswellExploring Intermediate Algebracourse features algorithmic exercises and test bank content in question pools. |
About:
Measurement and Geometry: The Metric System of Measurement
Metadata
Name:
Measurement and Geometry: The Metric System of Measurement
ID:
m35019
Language:
English
(en)
Summary:
This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This module discusses the Metric System of measurement. By the end of the module students should be more familiar with some of the advantages of the base ten number system, know the prefixes of the metric measures, be familiar with the metric system of measurement and be able to convert from one unit of measure in the metric system to another unit of measure |
Pedagogy
Research on Learning
Results 1 - 17 of 17 matches
Geologic Puzzles: Morrison Formationpart of Pedagogy in Action:Library
Effect of Proportionality Constant on Exponential Graph (k>0>0 and C is an arbitrarily fixed value in f(x)=Ce^(kx).
Effect of Initial Value on Graph of Exponential Function (C < 0<0 and k is an arbitrarily fixed value in f(x)=Ce^(kx
Effect of Proportionality Constant on Exponential Graph (k < 0<0 and C is an arbitrarily fixed value in f(x)=Ce^(kx).
Effect of Initial Value on Graph of Exponential Function (C>0>0 and k is an arbitrarily fixed value in f(x)=Ce^(kx).
Effect of Coefficient of x^0 on Parabola Vertex0 (i.e., the constant, c) on the vertex of a parabola where a and b are arbitrarily fixed values in f(x)=ax^2+bx+c.
Effect of Coefficient of x^2 on Parabola Shape2 on the shape of a parabola where b and c are arbitrarily fixed values in f(x)=ax^2+bx+c.
Effect of Coefficient of x on Parabola Vertex (b < 0) on the vertex of a parabola where a>0, b<0 and a and c are fixed values in f(x)=ax^2+bx+c.
Volumes of Solids of Revolutionpart of Pedagogy in Action:Library:Interactive Lectures:Examples This write-pair-share activity presents Calculus II students with a worksheet containing several exercises that require them to find the volume of solids of revolution using disk, washer and shell methods and to sketch three-dimensional representations of the resulting solids.
How Much Work is Required: Intuition vs. Mathematical Calculationpart of Pedagogy in Action:Library:Interactive Lectures:Examples This classroom activity presents Calculus II students with some Flash tutorials involving work and pumping liquids and a simple question concerning the amount of work involved in pumping water out of two full containers having the same shape and size but different spatial orientations.
Riemann Sums and Area Approximationspart of Pedagogy in Action:Library:Interactive Lectures:Examples After covering the standard course material on area under a curve, Riemann sums and numerical integration, Calculus I students are given a write-pair-share activity that directs them to predict the best area approximation methods for each of several different functions. Afterwards, the instructor employs a Web-based applet that visually displays each method and provides the corresponding numerical approximations.
U.S. Population Growth: What Does the Future Hold?part of Pedagogy in Action:Library:Interactive Lectures:Examples College Algebra or Liberal Arts math students are presented with a ConcepTest, a Question of the Day and a write-pair-share activity involving U.S. population growth. The results are quite revealing and show that while students may have learned how to perform the necessary calculations, their conceptual understanding concerning exponential growth may remain faulty. Student knowledge (or lack thereof) of the size of our population and its annual growth rate may also be surprising.
The Crusty Loaf of Bread: An Exploration of Area of a Surface of Revolutionpart of Pedagogy in Action:Library:Interactive Lectures:Examples This write-pair-share activity for Calculus II students involves a hypothetical hemispherical loaf of bread with a 12-inch diameter that has been sliced into twelve one-inch-thick slices. The objective is to determine which slice contains the most upper crust (i.e., most area of its surface of revolution).
Partial Derivatives: Geometric Visualizationpart of Pedagogy in Action:Library:Interactive Lectures:Examples This write-pair-share activity presents Calculus III students with a worksheet containing several exercises that require them to find partial derivatives of functions of two variables. Afterwards, a series of Web-based animations are used to illustrate the surface of each function, the path of the indicated partial derivative for a specified value of the variable and the value of the derivative at each point along the path.
Mathematical Curve Conjecturespart of Pedagogy in Action:Library:Interactive Lectures:Examples In this activity, a six-foot length of nylon rope is suspended at both ends to model a mathematical curve known as the hyperbolic cosine. In a write-pair-share activity, students are asked to make a conjecture concerning the nature of the curve and then embark on a guided discovery in which they attempt to determine a precise mathematical description of the curve using function notation. |
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Algebra
Ordinary algebra is a topic almost everyone studies to some extent in high school. Even so, it's easy to forget basic skills, and many people find themselves having difficulty in math classes later in life because of those forgotten skills. In fact, many students who take an algebra course in college find themselves struggling with the subject and realize they could benefit from some additional assistance. If you need to refresh your algebra skills and knowledge, turn to 24HourAnswers.com. We offer Algebra homework help to get you caught up and ready to take this subject by storm.
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Exponential and Logarithmic Functions
Systems of Equations and Matrices
Conic Sections
Sequences, Series, and Probability
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Mastering any difficult challenge in life requires a great deal of practice, and learning algebra is no exception. A truly great website for getting help and extra practice in ordinary algebra at all levels is the Virtual Math Lab of West Texas A&M University. This site offers a variety of online tutorials that enable you to practice and perfect your algebra skills.
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To fulfill our mission of educating students, our college homework help center is standing by 24/7, ready to assist students who need Algebra homework help.Use your Web-enabled computer to submit your homework, or participate in an online tutoring session at a time that is convenient for you. You'll have the opportunity to study at a comfortable pace and practice as often as you wish. Many of our students even discover that they learn more efficiently through online studying than when sitting in a classroom.
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More About
This Textbook
Overview
This book offers an overview of the central ideas in calculus and gives examples of how calculus is used to translate many real-world phenomena into mathematical functions. Beginning with an explanation of the two major parts of calculus - differentiation and integration - Gudmund R Iversen illustrates how calculus is used in statistics: to distinguish between the mean and the median; to derive the least squares formulas for regression co-efficients; to find values of parameters from theoretical distributions; and to find a statistical p-value when using one of the continuous test variables such as the t-variable.
Editorial Reviews
Booknews
Five therapeutic supervisors, representing different orientations such as psychoanalytic self psychology, person-centered therapy, and cognitive-behavior therapy, describe their approaches to supervision of a real-life therapist and client. The therapist and client then comment on the impact of the five supervisors on their work in therapy. Of interest to counselors, therapists, and supervisors, and those in training |
VISIT…
PREFACE
The Army Institute for Professional Development (AIPD) administers the
consolidated Army Correspondence Course Program (ACCP), which provides high
quality, economical training to its users. The AIPD is accredited by the
Accrediting Commission of the Distance Education and Training Council (DETC),
the nationally recognized accrediting agency for correspondence institutions.
Accreditation is a process that gives public recognition to educational
institutions which meet published standards of quality. The DETC has developed
a thorough and careful evaluation system to assure that institutions meet
standards of academic and administrative excellence before it awards
accreditation.
The many TRADOC service schools and DOD agencies that produce the ACCP materials
administered by the AIPD develop them to the DETC standards.
The AIPD is also a charter member of the Interservice Correspondence Exchange
(ICE). The ICE brings together representatives from the Army, Navy, Air Force,
Marine Corps, and Coast Guard to meet and share ideas on improving distance
education.
TABLE OF CONTENTS
INTRODUCTION
Supplementary Requirements
Credit Hours
Administrative Instructions
Grading and Certification Instruction
LESSON 1: ALGEBRA (Tasks. This lesson is common to all missile repairer tasks)
The Arithmetic
Addition
Subtraction
Polynomials
Signs of Grouping
Multiplication
Division
The Mathematics
Solving Equations
Exponents, Radicals, and Complex Numbers
Quadratic Equations
REVIEW EXERCISES
LESSON 2: LOGARITHMS (Tasks. This lesson is common to all missile repairer tasks)
Terminology
Systems
Parts of a Logarithm
Procedures
Finding the Logarithm of a Number
Negative Characteristics
Antilogarithms
Computations with Logarithms
Powers of Ten
Simplification
Rules
Reciprocals
Numerical Prefixes
REVIEW EXERCISES
LESSON 3: TRIGONOMETRY (Tasks. This lesson is common to all missile repairer tasks)
Derivation
Trigonometric Functions
Use
Quadrants
Radian Measure
Graphic Representation
REVIEW EXERCISES
LESSON 4: VECTOR ALGEBRA (Tasks. This lesson is common to all missile repairer tasks)
Vector Quantities
Vector Notation
Resultant Vectors
Vector Representation
Calculations
Addition and Subtraction
Multiplication
Division
Raising a Vector to a Power
Root of a Vector
REVIEW EXERCISES
END- OF-SUBCOURSE EXAMINATION
EXERCISE SOLUTIONS
STUDENT INQUIRY SHEET
INTRODUCTION
This is the first of three subcourses that are an introduction to or refreshers
for your knowledge of basic electricity. This reviews the mathematics you need
to understand the basic operating principles of guided missile systems and
electronic and radar circuits. Covered are algebra, logarithms, trigonometry,
and vector algebra.
Supplementary Requirements
There are no supplementary requirements in material or personnel for this
subcourse. You will need only this book and will work without supervision.
Credit Hours
Five credit hours will be awarded for the successful completion of this
subcoursea score of at least 75 on the endofsubcourse examination.
Administrative Instructions
Change Sheets. If a change sheet has been sent to you with this subcourse, be
sure you post the changes in the book before starting the subcourse.
Errors on TSC Form 59. Before you begin this subcourse, make sure that the
information already typed on your TSC Form 59 (ACCP Examination Response Sheet)
is correct. You will find the correct subcourse number and subcourse edition
number on the front cover of this book. If any of the information on your TSC
Form 59 is incorrect, write to:
The Army Institute for Professional Development (IPD)
US Army Training Support Center
Newport News, VA 236280001
A new, correctly filledout form will be sent to you. Do not correct the form
yourself or send it to IPD.
Questions, Changes, Corrections. If you have questions about enrollment or
other administrative matters, write to IPD. If a change occurs or a correction
needs to be made in your status (name, grade, rank, address, unit of assignment,
etc.) notify IPD as soon as possible. These kinds of changes or corrections can
be sent along on a separate sheet of paper with your completed TSC Form 59.
Correspondence with IPD. In any correspondence with IPD, always write your
name, Social Security Number, and the school code of your enrollment on each
page.
Grading and Certification Instructions
When you have completed the subcourse, review any of the material covered that
you are not sure of. Then take the endofsubcourse examination. When you have
completed the examination in the book, you must transfer your answers to TSC
Form 69. The instructions on the form itself tell you how to mark your answers.
Follow the instructions carefully.
Once you have transferred your answers to the TSC Form 59, fold the form as it
was folded when sent to you. Do not staple or mutilate this form! Place the
form in the selfaddressed envelope provided and mail it to IPD. No postage is
needed. TSC Form 59 is the only material that you are required to return to
IPD. If you return it as soon as you have completed this subcourse, you will
get your next subcourse sooner.
Grading. The highest score possible on the endofsubcourse examination is 100.
The grade structure for all ACCP subcourses is given below:
Superior 95100
Excellent 8594
Satisfactory 7584
Unsatisfactory 074
Your TSC Form 59 will be machine graded, and you will be notified of the
results. Your grade on the examination will be your grade for the subcourse.
No credit is given for grades below satisfactory (75).
Certificates. When you have completed the subcourse successfully, IPD will send
you a subcourse completion certificate. Keep it with your other personal copies
of personnel material. Subcourse completion certificates can be used to support
accreditation and other personnel actions.
* * * IMPORTANT NOTICE * * *
THE PASSING SCORE FOR ALL ACCP MATERIAL IS NOW 70%.
PLEASE DISREGARD ALL REFERENCES TO THE 75% REQUIREMENT.
MM0702, Lesson 1
Lesson 1
ALGEBRA
Task. The skills and knowledge taught in this subcourse are common to all missile repairer tasks.
Objectives. When you have completed this lesson you should be able to correctly solve equations using algebraic
principles and involving exponents, radicals, and complex numbers.
Conditions. You will have this subcourse book and work without supervision.
Standard. You must score at least 75 on the end-of-subcourse examination that covers this lesson and lessons 2, 3,
and 4 (answer 23 of the 30 questions correctly).
Algebra extends the scope of arithmetic by introducing the concept of negative values and the use of letters for
numbers. Letters used to represent quantities are called literal numbers. An algebraic expression is any combination
of signs, numerals, and literal numbers. For example, a + b and x/2y are algebraic expressions. Signs are used to
indicate whether numbers are positive (+) or negative (-), or they may indicate operations to be performed, such as,
add (+) or subtract (-).
THE ARITHMETIC
Multiplication of two algebraic quantities need not be indicated by a sign. Just a position of terms can indicate
multiplication. Thus, a X b or a . b may be written ab. In the expression ab, a and b are known as factors of the
product. Each factor of a product is known as the coefficient of the other factor(s).
The absolute value of a number refers to its magnitude, regardless of the sign preceding it. Numbers not preceded by
a sign are assumed to be positive. For example, 8 is the absolute value of both +8 and -8. To designate absolute
value, write 8.
An exponent is a number or letter which indicates the power to which a quantity (called the base) is to be raised. It
means the number of times the quantity is multiplied by itself. Thus the expression ex is read e to the x power,
signifying that e is multiplied by itself x times.
Any arithmetical or literal number, or the product or quotient of the numbers, is called a term. For example, 4, x,
25b, and x/y are terms.
Terms which have identical literal parts are called similar or like terms, while those with unlike literal numbers are
known as unlike terms. Thus 7a and a are like terms, while 7a and 7a2, or 8a, and 8b are unlike terms.
Like terms may be added to or subtracted from each other. For example, 4x may be added to 5x to produce 9x, or
10y2 - 3y2 equals 7y2, since the literal numbers are the same in each case. The sum of numbers such as 4y and 5m2
must be indicated as 4y + 5m2 since y and m 2 are unlike terms.
Addition
To add two numbers with the same sign, add their absolute values and write
the common sign before the sum.
+6 added to +3 equals +9
-6 added to -3 equals -9
1
MM0702, Lesson 1
To add two numbers with opposite signs, take the difference of their absolute values and write the sign of the larger
absolute value.
+6 added to -3 equals +3
-6 added to +3 equals -3
Subtraction
To subtract a quantity from another, change the sign of the quantity to be subtracted, then add the quantities following
the rules of addition.
(-6) - (-3) becomes (-6) + (+3) = -3
(-3) - (-6) becomes (-3) + (+6) = +3
(+6) - (-3) becomes (+6) + (+3) = +9
(+6) - (+3) becomes (+6) + (-3) = +3
Polynomials
An algebraic expression containing two or more terms joined together by a plus (+) or minus (-) sign is called a
polynomial. Thus, the expressions a + b - c and ab - ac are polynomials. A polynomial with only two terms is known
as a binomial; one with three terms is known as a trinomial; and a single term such as a, b, 10a2b is a monomial.
Polynomials are added and subtracted the same way single terms are.
Addition Subtraction
4a + 2b + 3c 3m2 - 6n2
+6a + b + 7c -3m2 - 8n2
10a + 3b + 10c 6m2 + 2n2
Signs of Grouping
Certain symbols such as parentheses ( ), brackets [ ] and braces { } are used to group several quantities which are
affected by the same operation and must be treated as a single quantity. For example, in the expression (4a2 - 3ab) -
(3a2 + 2ab), the entire quantity of (3a2 + 2ab) is subtracted from the first term. In removing a symbol or grouping
that is preceded by a minus sign, change the signs of all terms included by the symbol.
-(3a2 + 4ab - x) becomes - 3a2 - 4ab + x
When one symbol of grouping appears within another, it is best to remove one set of grouping symbols at a time
starting with the innermost symbol first.
-[2am - (2a2 + 5am) + a2]
Remove inside parentheses to obtain
-[2am - 2a2 - 5am + a2]
Remove brackets to obtain
- 2am + 2a2 + 5am - a2
2
MM0702, Lesson 1
Combine terms to obtain
3am + a2
Multiplication
Multiplication of polynomials is similar to the arithmetical multiplication of numbers consisting of several digits. Thus
(2c2 - 3c - 5) (- 4c + 7) may be solved as follows:
When multiplying, the sign, the exponent, and the coefficient must be considered.
Sign. The product of two terms with like signs is positive. The product of two terms with unlike terms is negative.
(+a) X (+b) = ab
(-a) X (-b) = ab
(-a) X (+b) = -ab
Be careful to keep the proper sign when multiplying a long series of terms.
(-a) X (-b) X (-c) = -abc
Exponent. The exponent of any letter in the product is the sum of the exponents of the factors with the same base.
a2 X a3 = a(2 + 3) = a5
2 3
a X b X a4 X b2 = a(2 + 4) b(3 + 2) = a6 b5
Coefficient. The arithmetical coefficient of the product is the product of the absolute values of the coefficients of the
terms being multiplied.
6x2 X 2x5 = 12x7
6X2 X 2y5 = 12X2y5
Division
Division of one polynomial by another is similar to long division in arithmetic. One difference is that the dividend,
divisor, and remainder (if there is one) must be arranged in order of ascending or descending powers of some letter.
30c43−82 c2=5c=11 c3
3c2−42 c
3
MM0702, Lesson 1
Solution:
In the division of polynomials, the sign, exponent, and coefficient must again be taken into consideration.
Sign. The quotient of two positive or two negative quantities is positive; the quotient of a positive and negative
quantity is negative.
a
(+a) ÷ (+b) = +
b
a
(-a) ÷ (-b) = +
b
a
(-a) ÷ (+b) = -
b
Exponent. The exponent of any letter in the quotient is the difference of the exponents of the factors with the same
base.
x
4
= x (4-2) = x2
x
2
In division, there is the possibility of obtaining zero and negative exponents. Any quantity with a zero exponent is
equal to one.
a
5
= a (5 - 5) = a0 = 1
a
5
Any quantity with a negative exponent is equal to the reciprocal of that quantity with the corresponding positive
exponent.
1 1
a a
a-3 = 3 or −3 = a3
Coefficient. The coefficient of the quotient is the absolute arithmetic value of the dividend divided by the divisor.
4
MM0702, Lesson 1
4x
2
= 2x
2x
5
MM0702, Lesson 1
THE MATHEMATICS
Solving Equations
Axioms. An equation is a statement that two quantities are equal. In solving an equation, you have to use axioms
(statements accepted as true without proof). The following are a few of the more commonly used axioms plus
examples that illustrate how they are used to solve problems.
• If the same number is added to or subtracted from each side of an equation, the result is still an equation.
x-5 = 3
Add 5 to each side.
x-5+ 5 = 3+ 5
x = 8
• If both sides of an equation are multiplied or divided by the same quantity (not zero), the result is still an
equation.
5x = 25
Divide by 5.
5x 25
=
5 5
x = 5
• If like roots or powers are taken of both sides of an equation, the result is still an equation.
x = 2
Square both sides.
x = 22
x = 4
Forming and Solving Equations. The solving of general equations cannot be explained by any set of rules, for these
rules would not hold true in every case. The most important single thing to remember is to thoroughly understand
what must be translated into mathematical language from the facts or wording of the problem being considered. The
items below will help you to understand this.
• Carefully read the problem. Be sure you understand all facts and relationships.
• Determine exactly what you are looking for (the unknown quantity) and designate it by a different letter. If more
than one unknown exists, try to represent them in terms of each other.
• Select two expressions based on the facts of the problem that represent the same quantity and place them equal
to each other. The equation, thus formed, can then be solved for the unknown.
6
MM0702, Lesson 1
Example:
The first angle of a given triangle is 40o less than the second angle. The first angle is greater than the third angle by
10o. Since the sum of the three angles in any triangle is 180o, how many degrees does each angle contain?
Solution:
First angle = second angle - 40o
First angle = third angle + 10O
Let x = first angle.
Therefore, second angle = x + 40O and third angle = x - 10O
Select two expressions and place them equal to each other.
x + (x + 40O) + (x - = 180O
10O)
3x + 30O = 180O
3x = 180O - 30O
3x = 150O
x = 50O = first angle
50O + 40O = 90O = second angle
50O + 10O = 40O = third angle
Factoring. Factoring is the process of finding two or more quantities, each called a factor, whose product is equal to a
given quantity. For example, factoring the expression ax + ab - az would produce the expression a(x + b - z) which is
still equal in value to the original expression.
Fractions and Fractional Equations. A fraction is an indicated division in which the numerator is the dividend, and
the denominator is the divisor. The value or ratio of the fraction is unchanged if both numbers are multiplied or
divided by the same number (not zero). Dividing both numerator and denominator by the same number is called
reduction to lower terms.
14 2X7 2 6 ax 2 X 3ax
= = = =
21 3X7 3 9a 3 X 3a
2x
3
Addition and Subtraction. Fractions may be added to, or subtracted from, each other only when they have a common
denominator. If the denominators are not alike, you may have to restate the fractions in terms of equivalent fractions
with each having what is known as a least common denominator (LCD).
x 3x 2x
+ -
4 5 6
Since the least common denominator is 60, the problem resolves into the following equivalent expression.
15 x 36 x 20 x 15 x 36 x − 20 x 51 x − 20 x 31 x
+ - = = =
60 60 60 60 60 60
Multiplication. Multiplication of two fractions involves finding the product of both the numerators and the
denominators. Frequently, it is possible to further reduce the resulting answer by factoring out common terms and
dividing or "cancelling out" these terms.
7
MM0702, Lesson 1
a a 2a c a cc
2 2 3 2
2ac 1 X 2ac X 1 =
X X = =
d 2a
b b X d X 2a 2abd bd
8
MM0702, Lesson 1
Complex fractions. Complex fractions are fractions which contain fractions in both the numerator and denominator.
They may be solved as simple division problems.
Solution:
Simultaneous Linear Equations. Simultaneous linear equations are two or more equations that contain only first
powers of the unknown quantities and no products of unknowns. They are also equations that have only certain
common values of the unknowns. You can solve for the unknowns obtained by graphing, by elimination by addition
or subtraction, by elimination by substitution, or by using determinants. Of these methods, elimination by subtraction
is the most common. Solve for e and i in the following two equations.
2e + 10i = 25 (1)
5e - 8i = 46 (2)
Solution:
Multiply equation (1) by 4, and get
8e + 40i = 100. (3)
Multiply equation (2) by 5, and get
25e - 40i = 230. (4)
Add equation (4) to equation (3).
8e + 40i = 100 (3)
25e - 40i = 230 (4)
33e + 0 = 330 (5)
Divide equation (5) by 33, and get
e = 10. (6)
Substitute the value of e from equation (6) in either equation (1) or (2)
2(10) + 10i = 25 (1)
20 + 10i = 25
10i = 25 - 20
i = 5 1
=
10 2
9
MM0702, Lesson 1
Therefore, for the given conditions:
e = 10.
1
i =
2
Exponents, Radicals, and Complex Numbers
Exponents. Remember that an exponent is a number or letter multiplied by itself some indicated number of times.
Remember, too, that a number raised to the zero power is equal to 1 (xO = 1), and that a negative exponent is the
same as the reciprocal of the quantity to the same positive exponent (x-2 = 1/x 2). Now consider the significance of a
fractional exponent. By squaring the quantity x1/2, you obtain (x1/2)(x1/2). By adding the exponents (x1/2 + 1/2), as in
any other multiplication of exponential numbers, we now obtain x1 or x as the result. By reversing these procedures,
you can state that the square root of x is x1/2. Also, by the same reasoning, (x1/3) is the cube root of x and so on. In
the expression x2/3 (which reads x to two-thirds power), you can say that it is equivalent to the cube root of x2 or it
x
3 2
can be expressed mathematically as
−
Radicals. Sometimes it is necessary to simplify an expression involving radicals (square roots, cube roots, etc.)
without changing its value. It may be possible to divide the quantity under the radical (the radicand) into two factors
and then take the root of one of the factors (see below).
80 = 16 X 5 = 4 5
3 3
16 m = 8 X 2m = 2 3 2m
Should the quantity under the radical be a fraction, multiply both numerator and denominator by a number which will
make it possible to extract the root of the denominator.
2
3
=
2 3
x =
3 3
6 6
=
9 9
=
6 1
=
3 3 6
x x x x
3
40 40 x 40 x 8 X 5 x 2 5 x 2 3
3 = 3 X =3 =3 = = 5 x
x x
x
2 2 3 3 3
Radicals may be multiplied by other radicals provided they are of the same root. To do this, multiply the coefficients
to get the new coefficient and multiply the radicands to get the new radicand of the product.
4x 3x = 7x
3 3 3
5y − 3y = 2y
Radicals may be multiplied by other radicals provided they are of the same root. To do this, multiply the coefficients
to get the new coefficient and multiply the radicands to get the new radicand of the product.
2 a x 3 b = 6 ab
10
MM0702, Lesson 1
Division of radicals is the reverse of multiplication. Coefficient is divided by coefficient, and radicand is divided by
radicand.
6 5
3 6
=2
5
6
It is usually better to eliminate the fraction under the radical. The value 2√5/6 may be further changed as follows.
2
5
6
=2
5 6
x =2
6 6
30
36
2 1
= 30 = 30
6 3
Since fractions involving radicals in the denominator are normally clumsy to manipulate when solving algebraic
expressions, it is often better to rearrange the fraction so that a whole number, a fraction, or mixed number will
appear in the denominator only. To eliminate a radical from the denominator of a binomial expression, use the
process known as rationalization of the denominator.
In this instance, the numerator and denominator are multiplied by the conjugate of the denominator. (A conjugate is
the same expression with the sign reversed between the terms.) To illustrate this procedure, consider the expression
5
.
3 − 2
5
Multiply by
3 − 2
3 2
3 2
where
3 2
is the conjugate of
3 − 2
Therefore,
5 3 2 5 3 2
x =
3 − 2 3 2 3 − 2 3 2
15 5 2
=
9 − 3 2 3 2 − 2
15 5 2
=
9−2
15 5 2
=
7
11
MM0702, Lesson 1
Complex Numbers. Thus far you have dealt only with the roots of positive numbers. Consider now negative
quantities appearing under radicals, such as √-x or √-3. There is no quantity that when squared, will produce -x or -3;
therefore, the square roots of negative numbers are designated as "imaginary numbers." The expressions √-x and √-3
can each be visualized as (√-1) (√x) and (√-1) (√3). The term √-1 has been arbitrarily designated "i," an imaginary
number equal to √-1. (In electrical work and in this subcourse, the symbol "j" is used since the symbol "i" is used to
designate current.) The term √-x can now be further expressed as j√x, and the term √-3 can be expressed as j√3.
The following relationship for j (or i) will prove helpful when solving problems involving imaginary numbers (figure 1-
1).
−1 = j
-1 = j2
−−1 = j3
Figure 1-1. Plotting Real and Imaginary Numbers.
The quantity √-1 is usually referred to as the "j operator" and is frequently used in the solution of alternating current
problems. Real and imaginary quantities can be graphically represented by four positions of a unit vector as in figure
1-1. Positive real numbers are plotted horizontally to the right, and negative real numbers are plotted horizontally to
the left. Positive imaginary numbers are plotted vertically above the horizontal axis, and negative imaginary numbers
are plotted vertically below the horizontal axis.
A complex number is the sum or difference of a real quantity and an imaginary quantity. Thus, 5 + j3 and 2 - j2 are
complex numbers. Complex numbers can be easily combined if you combine the real and imaginary portions
separately.
Add (5 + j3) and (2 -j2).
5 j3
2 − j2
¿
7 j
¿
12
MM0702, Lesson 1
Subtract (5 + j3) from (6-j4).
6 − j4
−5 − j3
¿
1 − j7
¿
Multiplication of complex numbers is identical to the procedure for the multiplication of binomials; however,
whenever the term j2 appears in the final result it is replaced by its actual value of -1.
5 j3 2 − j2
5 j3
2 − j2
10 j6
j6 ¿
2
− j10 −
10 − j4 − j 6
2
¿¿¿
Therefore,
10 - j4 - j26 = 10 - j4 - (-1)6
= 10 - j4 + 6
= 16 - j4
Division of complex numbers entails the rationalization of the denominator and division of the real number you get in
the denominator into the numerator. Consider the expression
(5 + j3) (2 - j2).
You first rationalize the denominator by multiplying both numerator and denominator by the conjugate of the
denominator.
5 j3 2 j2 10 j16 j 2 6
X =
2 − j2 2 j2 4 − j24
Simplify the new expression by replacing j2 with -1 and combining the terms as follows.
10 j16 j 2 6 = 10 j16 −16
4 − j24 4 − −4
= 10 j16 − 6
44
= 4 j16
8
0.5 + j2
Quadratic Equations
13
MM0702, Lesson 1
Equations assuming the general form ax2 + bx + c = 0 are known as second degree or quadratic equations in x.
(The degree is established by the highest exponent value of x, which in this instance is x2.) A quadratic equation may
be solved by several methods, including graphing, completing the square, and factoring, or by using the formula
14
MM0702, Lesson 1
b
2
−b ± − 4ac
x=
2a
where a, b and c are respectively the coefficients of x2, x, and the constant term. Before applying the formula, arrange
the equation into the general form ax2 + bx + c = 0. The following example illustrates the use of the quadratic
formula.
Solve for x in the equation
x + 3x2 = 10.
Rearrange terms.
3x2 + 8x - 10 = 0
The coefficients for use in the formula are
a = 3,
b = 8,
c = -10.
Substituting in the formula and solving, you get
−8 ± 64 120
x =
6
−8 ± 184
=
6
−8 ± 13.56
=
6
−21 .56 5.56
= or
6 6
= - 3.59 or 0.93.
The quantity b2 - 4ac, which appears under the radical in the formula, is called the discriminant. It indicates the type
of roots. If b2 - 4ac is positive, there are two real and unequal roots; if b2 - 4ac is negative, the roots are imaginary
and unequal; if b2 -4ac equals zero, the roots are real and equal.
15
MM0702, Lesson 1
REVIEW EXERCISES
1. Which of the following is the proper notation of the absolute value of a
number?
a. 5.
b. 3 + 6 + 4 =
a. 16.
b. 13.
c. 13.
d. 16.
4. What is the correct answer to the following problem?
Subtract:
3x2 6y2
3X2 8y2
a. 14y2.
b. 6x2 + 2y2.
c. 6x2 2y2.
d. 6x2 14y2.
13
MM0702, Lesson 1
5. Which is the number surrounded by braces?
a. 5.
b. (7).
c. [10].
d. {9}.
negative exponent?
x3 =
a. x3.
b. 1/x3.
c. x3/1.
x
3
d. .
8. Which of the following is the correct answer to the problem?
Solve for x.
x2 = 8 4
a. = 4.
b. = 2.
c. = 8.
d. = 2.
14
MM0702, Lesson 1
9. Which of the following is the correct answer to the problem?
Factor ax2 + bx = cy dy2
a. y(ax + b) = x(c dy).
b. ax3 + b = c dy3.
c. x(ax + b) = y(c dy).
d. ab + x3 = cd y3.
10. Which of the following is the correct expression for i or j?
a.. 1.
b. j2.
c. j3.
d.. −1 1 Review Exercise Solutions
15
MM0702, Lesson 1
REVIEW EXERCISES SOLUTIONS
1. Which of the following is the proper notation of the absolute value of a number?
a. 5.
b. --3+ 6+ 4=
a. 16.
b. 13.
c. -13.
d. -16.
4. What is the correct answer to the following problem?
Subtract:
3x2 - 6y2
-3X2 - 8y2
a. -14y2.
b. 6x2 + 2y2.
c. 6x2 - 2y2.
d. 6x2 - 14y2.
5. Which is the number surrounded by braces?
a. 5.
b. (7).
c. [10].
d. {9}.
13
MM0702, Lesson 1 negative exponent?
x-3 =
a. x3.
b. 1/x 3.
c. x3/1.
x
3
d. .
8. Which of the following is the correct answer to the problem?
Solve for x.
x2 = 8 - 4
a. = 4.
b. = 2.
c. = 8.
d. = -2.
9. Which of the following is the correct answer to the problem?
Factor ax2 + bx = cy - dy2
a. y(ax + b) = x(c - dy).
b. ax3 + b = c - dy3.
c. x(ax + b) = y(c - dy).
d. ab + x3 = cd - y3.
10. Which of the following is the correct expression for i or j?
a.. 1.
b. j2.
c. j3.
d. −1
Lesson 1
14
MM0702, Lesson 2
Lesson 2
LOGARITHMS
Task. Skills and knowledge taught in this lesson are common to all missile repairer tasks.
Objectives. When you have completed this lesson, you should be able to describe the parts of a logarithm and use
logarithmic procedures to solve problems correctly.
Conditions. You will have the subcourse book and work without supervision.
Standard. You must score at least 75 on the end-of-subcourse examination that covers this lesson and lessons 1, 3,
and 4 (answer 23 of the 30 questions correctly).
In lesson 1, you saw that exponential terms with like bases can be multiplied, divided, squared, etc., by keeping their
bases and performing simple addition, subtraction, multiplication, or division of their exponents to get the solution
(see the following four examples).
a2 X a3 = a2+3 = a5
a3 X a-2 = a3-2 = a
(a2)3 = a2+2+2 = a6
a
4 = a4/2 = a2
Sometimes complicated arithmetical computations can be simplified by writing the numbers as powers of some base.
Consider a very simple example in which powers of 2 are used. The first 12 powers of 2 are arranged as shown.
Exponent 1 2 3 4 5 6 7 8 9 10 11 12
Powers of 2 2 4 8 16 32 64 128 256 512 1024 2048 4096
Problem:
Find the product of 16 and 64 using the powers of base 2 chart above.
Solution:
Since 16 x 64 = 24 x 26 = 210, you can find the product of 16 and 64 as follows.
Find the corresponding exponents 4 and 6 above 16 and 64 in the chart. The sum of these exponents is 10, and
the number that corresponds to the exponent 10 in the table, 1,024, is the product of 16 and 64.
Other examples follow.
15
MM0702, Lesson 2
Example 1.
4096÷512 = 212 ÷ 29
= 2(12 -9)
= 23
= 8
Example 2.
2084 X 4096 X 256
2 X2 X2
11 12 8
=
64 X 512 X 32 X 16
2 X2 X2 X2
6 9 5 4
2
11 12 8
=
2
6 9 5 4
2
31
=
2
24
2
= 31 − 24
2
= 7
= 128
As you can see, you are limited in the choice of numbers that can be manipulated by this short method because the
table is small. While it would be possible by advanced mathematics to compute fractional exponents for numbers
between those given in the second row of the table above, it is more practical for computational purposes to use a
table based on powers of 10 instead of powers of 2.
Exponents used for the computation above are called logarithms. Thus, since 8 = 23, the logarithm of 8 to the base 2
is 3. In abbreviated form this is written as
log28 = 3.
When 10 is used as the base, you have the following.
log101 = 0 since 1 = 100
log1010 = 1 since 10 = 101
log10100 = 2 since 100 = 102
log101,000 = 3 since 1,000 = 103
log100.1 = 1
1 since 0.1 = 1 = 10-
10
1
16
MM0702, Lesson 2
log100.01 = 1
2 since 0.01 = 2 =
10
10-2
The log of any number between 0 and 10 will be a fraction between 0 and 1; the log of any number between 10 and
100 will be a value between 1 and 2, and so forth.
17
MM0702, Lesson 2
You are now ready to consider a more formal statement concerning logarithms. The logarithm of a quantity is the
exponent or the power to which a given number, called the base, must be raised to equal that quantity. If N = 10x,
the exponent x is the log of N to the base 10.
TERMINOLOGY
Systems
Two numbers have been selected as bases, resulting in two systems of logarithms. One base, 2.718, usually indicated
by the letter (e), is used in the "natural" logarithm system. The other base is 10 and is used in "common" system of
logarithms. In the common system, the base 10 is usually omitted in the logarithmic expression. Thus, log101,000 = 3
is usually written log 1,000 = 3. In the natural system, the base e is usually written in the logarithmic expression. The
logarithms discussed in this subcourse are "common" logarithms.
Parts of a Logarithm
For numbers not exact powers of 10, the logarithm consists of two parts: a whole number, called the "characteristic",
and the decimal part, called the "mantissa." For example, the logarithm of 595 is 2.7745. In this instance, the
characteristic is found by inspection while the mantissa is obtained from log-arithmic tables.
Characteristic. The characteristic of a logarithm can be determined by the following rules.
• The characteristic of the logarithm of a number greater than 10 is positive and is one less than the number of
digits to the left of the decimal point.
Number Characteristic
5 0
23 1
567.8 2
8432.29 3
• The characteristic of the logarithm of a number less than one is negative and is equal to one more than the
number of zeros immediately to the right of the decimal point.
Number Characteristic
. -1
532
. -2
034
. -3
00509
Mantissa. The mantissa of a logarithm is found from a table of logarithms. Numbers which have the same sequence
of digits, but differ only in the location of the decimal point, have the same mantissa.
Number Mantissa
595 .7745
.7745
59.5
. .7745
595
18
MM0702, Lesson 2
PROCEDURES
Finding the Logarithm of a Number
Table 2-1 at the end of the lesson is a complete, four place logarithm table. The first column in the table contains the
first two digits of numbers whose mantissas are given in the tables, and the top row contains the third digit. To find
the logarithm of a number such as 58.4, first determine the characteristic by inspection. In this instance it is 1 (one
less than the number of digits to the left of the decimal point). Now look in the table to determine the mantissa.
Remember that the mantissa is the decimal portion of the logarithm and, although it is customary to omit the decimal
point in the construction of tables, it must be put in when writing the complete logarithm. Referring to table 2-1,
look down the left-hand column for the first two digits, find 58; then go across to the column labeled 4, find the
mantissa: .7664.
The logarithm for 58.4 is therefore:
log 58.4 = 1.7664.
To find the logarithm of a number with more than three digits, use the process called interpolation. To illustrate this
process, determine the logarithm of 5,956. The mantissa for 5,956 is not listed in the table; however, the mantissas
for 5,950 and 5,960 are listed. (The mantissa for 5,950 is the same for 595, etc.) Since 5,956 lies between 5,950 and
5,960, its mantissa must lie between the mantissas for these two numbers.
Arrange the numbers in tabular form.
Numbers Characteristic Mantissa
5,960 3. 7,752
5,956 3. ?
5,590 3. 7,745
Since 5,956 lies 6/10 of the way between 5,950 and 5,960, the mantissa must be 6/10 of the way between .7745 and .
7752. Since the difference between the two is .0007, and 6/10 of .0007 is .00042, add .00042 to the mantissa of 5,950,
and get the result, .77492. The complete logarithm of 5,956, therefore, is 3.77492, rounded off to 3.7749.
Negative Characteristics
When the characteristic of a logarithm is negative, do not put the minus sign in front of the logarithm; it applies only
to the characteristic and not the mantissa. Instead, add 10 to the negative characteristic and indicate the subtraction
of 10 at the end of the logarithm. Thus the characteristic, if -2, is written:
8. Mantissa -10.
Another method frequently used is to place the negative sign directly above the characteristic. For example:
2. Mantissa 4. Mantissa
19
MM0702, Lesson 2
Antilogarithms
The number corresponding to a given logarithm is called the antilogarithm of that number. It is written "antilog" or
log-1. To find the antilog, reverse the process for determining the logarithm. To find the antilog of 1.8102, locate the
mantissa, .8102, in table 2-1. It is in line with 64 and column 6. Thus, the number corresponding to this mantissa
has the digits, 646. To determine the correct position of the decimal point, reverse the procedure used to determine
the characteristic. Since the characteristic is 1, there must be two digits to the left of the decimal point. The correct
answer, therefore, is 64.6. In some instances, the exact mantissa cannot be found in the tables. You must then use
the process of interpolation.
Problem:
Find the antilog of 3 .7690.
Solution:
Again referring to table 2-1, you find that .769 lies between the mantissa .7686 (587) and .7694 (588). The difference
between the mantissas .7686 and .7694 is .0008. The difference between .769 (the given mantissa) and .7686 is .0004.
Find the number corresponding to .7690 by taking .0004/.0008 of 1 or .5 and adding it to 587 and get 5875. Since the
given characteristic was 3 , you know that there must be two zeros to the decimal point in front of the first digit.
Therefore, the correct answer is 0.005875.
Computations with Logarithms
To compute using logarithms, the general rules described for the manipulation of exponents apply.
Multiplication. To multiply two numbers, add their logarithms and find the antilog of the result.
6,952 X 437 = ?
Log 6,952 = 3.8421
Log 437 = 2.6405 (sum)
6.4826
antilog 6.4826 = 3,038,000
Actual multiplication of 6,952 x 437 would give the result 3,038,024, or an error of .0008 of one percent. The error is
due to the number of places in the logarithm tables. Greater accuracy would result from using 5 or 7 place tables.
Division. To divide two quantities, subtract the logarithm of the divisor from the logarithm of the dividend and find
the antilog of the result to obtain the quotient.
6,952 ÷ 437 = ?
log 6,952 = 3.8421
log 437 = 2.6405 (subtract)
1.2016
antilog 1.2016 = 15.908
20
MM0702, Lesson 2
Therefore,
6,952 ÷ 437 = 15.908.
Powers. To raise a quantity to a power, multiply the logarithm of the quantity by the exponent or the power and find
the antilog of the result.
(5.2)6 = ?
log (5.2)6 = 6 log 5.2
6 log 5.2 = 0.7160 X 6
= 4.2960
antilog 4.296 = 19,768
Therefore,
(5.2)6 = 19,768.
To raise a quantity to a negative power, proceed as follows.
(45.6)-3 = ?
log (45.6)-3 = 3 log 45.6
log 45.6 = 1.6590 X -3
= -4.9770
Since logarithm tables list only positive values of the mantissa, change the logarithm above by subtracting from
10.0000 - 10).
10.0000 - 10
4.9770
5.0230−10 or
5.0230
The antilog of
5 .0230 = 0.00001054.
Therefore,
(45.6)-3 = 0.00001054.
Roots. To find the root of a quantity, divide the logarithm of the number by the indicated root and find the antilog.
3
1.572 = ?
log = 1 log
3
1.572 3
1.572
Log 1.572 = 0.1965
0. 1965 = 0.0655
3
antilog 0.0655 = 1.163
21
MM0702, Lesson 2
Therefore,
3
1.572 = 1.163.
Sample solution of complex problem. Solve
4
439 X 6,793 X 8. 43
4.6
3
X 14. 9 X 1.02
Next, simplify the problem by arranging the terms in the following manner.
Numerator:
log 439 = 2.6425
log 6,793 = 3.8321
4
log 8. 43 = 8.43 ÷ 4 = 0.2315 (sum)
6.7061
Denominator:
4 .6
3 = 1.9884
log = 3
log 4.6
log 14.9 = 1.1732
log 1.02 = 1.02 ÷ 2 = 0.0043 (sum)
3.1659
Now, subtract the log of the denominator (3.1659) from the log of the numerator (6.7063).
6.7061
−3.1659
3.5402
antilog 3.5402 = 3,470
Therefore,
4
439 X 6,793 X 8. 43
4.6
3 = 3,470
X 14. 9 X 1.02
POWERS OF TEN
Frequently it is desirable to determine quickly and with a reasonable degree of accuracy the result of an expression
which contains large or cumbersome numbers. At first glance, the expression
536 ,000 ,000 X .00625 X 482 (1)
.000006 X 6, 213
22
MM0702, Lesson 2
brings visions of tedious multiplication and division by conventional methods. The use of logarithms would shorten
the calculations, provided logarithm tables were readily available. Suppose, however, our problem could be reduced to
some simple expression such as:
23
MM0702, Lesson 2
5. 4 X . 6. 3 X 4.8 (2)
6 X 6. 2
One could arrive at an approximate but fairly accurate answer by merely performing several simple multiplication and
division steps. The solution to (1), gotten by conventional multiplication, is
4,331,509,201 (3)
or approximately
4,332,000,000. (4)
The approximate solution to (2) is
4.32. (5)
Simplification
Ignoring the number of zeros or the placement of the decimal points for a moment, notice the simplicity of the digits
appearing in (4) and (5). If a simple method were available for determining the position of the decimal point, we
would have a virtual "engineers' shorthand" at our disposal. Such a system, known as the powers of 10, does exist
and is based upon the relatively simple rules of exponents described in the preceding portions of this subcourse.
To begin with, the number 536,000,000 can be written
536 X 1,000,000
One million can be expressed (using the principle of exponents) as 106. The number can now be written
536 X 106.
The value 536, however, can be further reduced to 5.36 x 100, and 100 is equal to 102. The number 536,000,000 may
now be written
5.36 X 102 X 106.
Combining the exponents, you get a further simplification,
5.36 X 10) (2+6)
or
5.36 X 108.
The number .000625 can also be rewritten
6.25 X .0001.
Recalling again the laws of exponents and powers of 10, you can see that .0001 is equivalent to 10-4. The expression
now becomes 6.25 x 10-4.
Every number in the original expression can now be rewritten in the form of some small number multiplied by some
power of 10.
24
MM0702, Lesson 2
536,000,000 X .000625 X 482
.000006 X 6,213
=
10 X 6.25 X 10 4.82 X 10
8 −4 2
5.36 X
6 X 10 X 6.213 X 10
−6 3
Regrouping the terms, you get
10 X 10 10
8 −4 2
5.36 X 6.25 X 4.82
6 X 6.213 10 X 10
−6 3
=
10
6
5.36 X 6.25 X 4.82
=
10
−3
6 X 6.213
10
9
5.36 X 6.25 X 4.82 X
6 X 6.213
You can now perform either an exact or approximate multiplication and division of small numbers. Assuming you are
interested only in an approximate answer, you can reduce the fraction as follows: (≅ means "approximately equal to")
Referring again to the exact answer of 4,331,509,201, you find that you are accurate to within 2/10 of 1 percent,
normally an acceptable degree of accuracy.
Rules
Two basic rules for converting numbers to powers of 10 can be stated as follows:
1. To express a large number as a small number times a power of 10, move the decimal point left to the desired
location and count the number of places to the original decimal point. The number of places moved will be the
proper positive exponent of 10.
45 = 4.5 X 101
25
MM0702, Lesson 2
450 = 4.5 X 102
4,500 = 4.5 X 103
45,632 = 4.5632 X 104
26
MM0702, Lesson 2
2. To express a decimal as a whole number times a power of 10, move the decimal to the right and count the
number of places moved. The number of places moved is the proper negative exponent of 10.
0.6 = 6 X 10-1
0.06 = 6 X 10-2
0.006 = 6 X 10-3
0.0006 = 6 X 10-4
Reciprocals
Frequently in electronics, expressions will be encountered that involve reciprocal quantities (1 over a particular
number), such as
1 1 1
= .
R t R 1 R 2
x c =
1
2 π fc
.
Such problems are easily solved by utilizing the powers of 10 system. For example, the quantity
1
40, 000 X 0.00025 X 0.000125
can be solved as follows:
Convert all quantities in the denominator to their proper power of 10.
1
10 X 2.5 X 10 X 1.25 X 10
4 4 −4
4 X
Multiply the numbers and combine the exponents to obtain one final power of 10.
1
10
−4
12.5 X
Divide 1 by 12.5, move 10-4 to the numerator, and reverse the sign of the exponent (remember 1/x 2 = x-2). You now
get
.08 X 104
Moving the decimal point 4 places to the right, you get the answer, 800.
Numerical Prefixes
Throughout your study of electronics you will frequently encounter quantities which are exceptionally large or
exceedingly small in value. To make it easier to express various units such as volts, cycles, amperes, and ohms in
terms of their absolute values, metric prefixes such as kilo, mega, micro, and milli are added to the word. The use of
these prefixes or their abbreviations reduces the requirements for writing large numbers because they are based on
27
MM0702, Lesson 2
powers of 10. At the end of this lesson is a list of these prefixes and their abbreviations are in table 2-2. Table 2-3 is
a conversion table for multiplying units exponentially. Table 2-4 shows exponential differences between units.
28
MM0702, Lesson 2
Table 2-1a. Four-Place Common Logarithms of Numbers.
29
MM0702, Lesson 2
Table 2-1. Four-Place Common Logarithms of Numbers (cont).
30
MM0702, Lesson 2
Table 2-1. Four-Place Common Logarithms of Numbers (cont).
Table 2-2. Abbreviations for Metric Prefixes Table 2-3. Exponential Multiplication Tables.
31
MM0702, Lesson 2
Table 2-4. Exponential Differences Between Units.
Lesson 2 Review Exercises
32
MM0702, Lesson 2
REVIEW EXERCISES. 48 often 2 Review Exercises Solutions
31
MM0702, Lesson 2
REVIEW EXERCISES ANSWER KEY .48 of tenLesson 2
31
MM0702, Lesson 3
Lesson 3
TRIGONOMETRY
Task. The skills and knowledge taught in this lesson are common to all missile repairer tasks.
Objectives. When you have completed this lesson, you should be able to explain how trigonometric functions are
derived and be able to use those functions to solve trigonometric problems correctly.
Conditions. You will have this subcourse book and work without supervision.
Standard. You must score at least 75 on the end-of-subcourse examination that covers this lesson and lessons 1, 2,
and 4 (answer 23 of the 30 questions correctly).
The development and operation of the guided or ballistic missile are based, to a large degree, upon the basic principles
of trigonometry. The problems of determining the position of the target in relation to the launch point and the
calculations incident to sending a missile to some predetermined point in space or to a point on the surface of the
earth all involve some form of trigonometry. Furthermore, many electrical problems, when reduced to triangular
equivalents, can be easily and quickly solved by use of trigonometry.
Trigonometry is that phase of mathematics which is concerned with the relationships of sides and angles of a triangle
to each other. Several special relationships called trigonometric functions hold true in a right triangle. These
trigonometric functions will be the subject of discussion in this lesson.
In the right triangle (figure 3-1), for a given condition of angles, the sides of the triangle will always increase or
decrease in the same mathematical proportion and the angles will always total 180°. For example, consider the
following triangle.
Figure 3-1. Right Triangle Enlarged.
32
MM0702, Lesson 3
In it, the angle θ = 30°, and the length of the hypotenuse is equal to 1 ft. (θ means "unknown angle," but is often
used as angle θ.) Accurate physical measurement of side BC will show that it is equal to 6 in or 0.5 ft, and side AC
will be 0.866 ft. Double the length of AB to AB', making it exactly 2 ft long. Actual measurement will show that
B'C' is exactly 1.0 ft long, and side AC' is now 1.732 ft long. Stating this relationship in a slightly different manner,
you can say that side BC divided by AB is equal to side B'C' divided by AB' or
BC B'C'
=
AB AB'
0.5 1
=
1 2
0.5 = 0.5,
providing that angle θ did not change from its original value of 30°.
DERIVATION
Bearing in mind the principles established in the previous paragraph, the ratio of any two sides of a right triangle will
always produce the same quotient, regardless of the magnitude of the sides, provided the angles remain the same. If
each of the possible ratios of the sides of a right triangle were listed and given a specific name, you would have the
so-called trigonometric functions. Consider the general right triangle in figure 3-2.
Figure 3-2. Right Triangle.
For the angle θ, there are six possible side ratios which can exist.
Side opposite Hypotenuse
1. 4.
Hypotenuse Side opposite
Side adjacent Hypotenuse
2. 5.
Hypotenuse Side adjacent
Side opposite Side adjacent
3. 6.
Side adjacent Side opposite
33
MM0702, Lesson 3
Note that the last three ratios are merely the reciprocals of the first three. By relabeling the above ratios as the sine,
the cosine, the tangent, the cosecant, the secant, and the cotangent respectively, of the angle é, you can say the
following.
1. sine of angle θ
Side opposite = (written as: sin θ
Hypotenuse
2.
Side adjacent = cosine of angle θ (cos θ)
Hypotenuse
3.
Side opposite = tangent of angle θ (tan θ)
Side adjacent
4.
Hypotenuse = cosecant of angle θ (csc θ
Side opposite
5.
Hypotenuse = secant of angle θ (sec θ)
Side adjacent
6.
Side adjacent = cotangent of angle θ (cot θ)
Side opposite
Since the trigonometric functions (sine, cosine, etc.) are merely numerical quotients of a mathematical ratio, you can
see that by knowing any two conditions within any of the preceding formulas, you could obtain the third factor. Since
the numerical equivalents for the trigonometric functions of any angle from 0 to 90 are listed in tables, it becomes a
simple matter to solve mathematical problems involving these ratios. However, to better understand these tables and
their uses, construct an abbreviated table based upon the angles 0°, 30°, 45° 60°, and 90°. To simplify your
calculations, you need to accept certain axioms from geometry:
• In a 30° - 60° right triangle, the side opposite the 30° angle is equal to one-half of the hypotenuse.
• In a 45°- 45° right triangle, the hypotenuse is equal to the side opposite multiplied by √2.
You also need to simplify your problem by making the hypotenuse a length of one unit. A slight amount of
imagination must be exercised in visualizing a right triangle with one of the acute angles as 0° and the other as 90°.
However, if you imagine the conditions at "very nearly" 0° and 90°, the problem becomes more realistic.
CONDITION I. Find the value of the trigonometric functions when θ = 0° (figure 3-3).
34
MM0702, Lesson 3
Figure 3-3. Condition I Triangle.
35
MM0702, Lesson 3
Note. For 0°, the hypotenuse would coincide with the side adjacent making the opposite side equal to zero.
Opposite 0
sin 0° = = = 0,
Hypotenuse 1
Adjacent 1
cos 0° = = = 1,
Hypotenuse 1
Opposite 0
tan 0° = = = 0,
Adjacent 1
Hypotenuse
csc 0° = = 1 = infinity (∞).
Opposite
Hypotenuse 1
sec 0° = = = 1,
Adjacent 1
Adjacent 1
cot 0° = = = infinity (∞).
Opposite 0
CONDITION II. Find the value of the trigonometric functions when θ = 30° (figure 3-4).
Figure 3-4. Condition II Triangle.
By the Pythagorean theorem,
12 = (.5)2 + (adjacent side)2
1 = .25 + (adjacent side)2
1 = - (.25) = (adjacent side)2
.75 = adjacent side
0.866 = adjacent side
Therefore,
36
MM0702, Lesson 3
Opposite 0.5
sin 30° = = = 0.5,
Hypotenuse 1
Adjacent 0.866
cos 30° = = = 0.866,
Hypotenuse 1
Opposite 0.5
tan 30° = = = 0.577
Adjacent 0.866
Hypotenuse 1
csc 30° = = = 2
Opposite 0.5
Hypotenuse 1
sec 30° = = = 1.155,
Adjacent .866
Adjacent .866
cot 30° = = = 1.732.
Opposite .500
CONDITION III. Find the value of the trigonometric function when θ = 45° (figure 3-5).
Figure 3-5. Condition III Triangle.
x2 + x2 = (1)2
2x2 = 1
x2 = .5
x = .5
x = 0.707
37
MM0702, Lesson 3
Therefore,
Opposite 0.707
sin 45° = = = 0.707,
Hypotenuse 1
Adjacent 0.707
cos 45° = = = 0.707,
Hypotenuse 1
Opposite 0.707
tan 45° = = = 1
Adjacent 0.707
Hypotenuse 1
csc 45° = = = 0.707,
Opposite 0.707
Hypotenuse 1
sec 45° = = = 1.414,
Adjacent 0.707
Adjacent 0.707
cot 45° = = = 1
Opposite 0.707
CONDITION IV. Find the value of the trigonometric functions for θ = 60°
Figure 3-6. Condition IV Triangle.
38
MM0702, Lesson 3
Opposite 866
sin 60° = = = 0.866
Hypotenuse 1
Adjacent .5
cos 60° = = = 0.5
Hypotenuse 1
Opposite .866
tan 60° = = = 1.732
Adjacent .500
Hypotenuse 1
csc 60° = = = 1.155
Opposite .866
Hypotenuse 1
sec 60° = = = 2
Adjacent .5
Adjacent .5
cot 60° = = = 0.577
Opposite .866
CONDITION V. Find the value of the trigonometric functions when θ = 90° (figure 3-7).
Figure 3-7. Condition V Triangle.
In this instance, the hypotenuse coincides with the opposite side, and the adjacent side is reduced to zero.
Therefore,
Opposite 1
sin 90° = = = 1
Hypotenuse 1
Adjacent 0
cos 90° = = = 0
Hypotenuse 1
Opposite 1
tan 90° = = = Infinity (∞)
Adjacent 0
Hypotenuse 1
csc 90° = = = 1
Opposite 1
39
MM0702, Lesson 3
Adjacent 0
sec 90° = = = 0
Opposite 1
The next step is to list the previously calculated data into tabular form (table 3-1).
Table 3-1. Trigonometric Functions.
Except for the intermediate values of angles from 0° to 90°, we have constructed an actual table of trigonometric
functions. Note some characteristics of the table. Notice that as the sine increases in value from 0 to 1 (from 0° to
90°), the cosine decreases from 1 to 0 at the same rate. Similar relationships hold true for the other functions. Refer
to table 3-8 for a complete table of trigonometric functions.
TRIGONOMETRIC FUNCTIONS
Use
To illustrate the use of trigonometric functions and the use of the trigonometric table developed in the preceding
paragraph, follow the solving of the problem illustrated in figure 3-8.
Figure 3-8. Hypothetical Triangle.
40
MM0702, Lesson 3
Problem:
A target is located 173.2 miles north and 100 miles east of your location. Determine the range and azimuth.
Solution:
First determine θ, remembering that the tangent of
Side opposite
θ =
Side adjacent
100
tan θ =
173. 2
tan θ = 0.577
Returning to the table developed earlier, you find that 0.577 in the tangent column corresponds to 30°. Therefore,
= 30°
With θ established, you can now solve for R. Recalling that the sine of an angle is equal to
Side opposite,
Hypotenuse
you can write the following relationship.
Side opposite
sin θ =
Hypotenuse
100
sin 30° =
R
Again referring to the table, you find that the sin of 30ø is 0.5. Substituting θ in the formula, you have
100
0.5 = ,
R
0.5R = 100
R = 200 miles.
For the given conditions, your answer is now
range (R) = 200 miles, and
azimuth (θ) = 30°.
41
MM0702, Lesson 3
Quadrants
The functions of angles greater than 90° can be determined by resolving them into an equivalent angle between 0°
and 90°. Consider the following coordinate system (figure 3-9) consisting of a Y or vertical axis and an X or
horizontal axis. Let all values of X to the right of the origin be positive and all values to the left be negative. Let
values of Y above the X axis be positive and those below be negative. Furthermore, label the four quadrants counter-
clockwise I through IV.
Consider a unit length vector rotating counterclockwise about the point of origin, starting at the positive X axis at the
zero point. At any given instant for the angles 0ø through 360°, the trigonometric functions can be resolved into X
and Y components and the unit vector in relation to the angle B between the unit vector and the X axis (figure 3-10).
Summarizing these relationships can be tabulated as in table 3-2.
Specific examples of determining the trigonometric functions of angles greater than 90ø would be as in table 3-3.
Figure 3-9. Coordinate System.
Figure 3-10. Trigonometric Functions as X and Y Components and Unit Vector Relationships. (cont)
42
MM0702, Lesson 3
Figure 3-10. Trigonometric Functions as X and Y Components and Unit Vector Relationships.
Table 3-2. Signs of Trigonometric Functions.
Table 3-3. Determining Trigonometric Functions, Angles Greater than 90°.
43
MM0702, Lesson 3
Radian Measure
In addition to designing the measurement of angles in terms of degrees, it is often necessary or convenient to utilize
the system of radian measure (figure 3-11). Specifically, a radian can be defined as "the angle at the center of a circle
of radius r that subtends an arc of length r."
Figure 3-11. Radian.
More precisely, you can state that 360° (one complete circle) is equal to 2 π radians (π = approximately 3.141593).
Therefore,
2 π radians
1° = = 0.0174533 radians
360°
or
360 °
1 radian = = 57.29578 = 57° 17' 45".
2π
A tabular listing of the more common radian and degree equivalent is listed in table 3-4.
Table 3-4. Degree-Radian Equivalents.
44
MM0702, Lesson 3
Graphic Representation
Frequently, it is necessary to represent the variations of the trigonometric functions in graphic form (figure 3-12). In
this subcourse, only the method utilized for developing the sine, cosine, and tangent functions will be discussed;
however you would develop the other function curves the same way. Assume that you want to plot the positive or
negative value of the sine of an angle as the vertical (Y) ordinate of a graph and the value of θ (the particular angle
in question) as the horizontal ordinate of the same graph. The following equation would then summarize our
problem.
Y = sin θ
Figure 3-12. Graphic Representation of the Sine Function.
For a moment, consider again the unit vector rotating about the origin of a rectangle coordinate system. As θ
increases from 0° to 360°, the sine of the angle will vary between the extremities of +1 and -1 in value. You will
draw your gragh immediately to the right and plot the various values of Y or the sine of θ throughout one complete
revolution (table 3-5).
Table 3-5. Values of the Sine θ for One Revolution.
45
MM0702, Lesson 3
The plot of the cosine of angle θ would follow the same procedure you used in plotting the sine curve. The general
expression Y = cos θ would define the plotted curve (figure 3-13 and table 3-6).
Figure 3-13. Graphic Representation of the Cosine Function.
Table 3-6. Values of the Cosine θ for One Revolution.
The plot of the tangent curve produces a different curve in that it is discontinuous at several points. See figure 3-14
and table 3-7.
Remember, the term infinity does not refer to a number of specific magnitude. It more accurately refers to the
meaning "approaching an infinite value."
46
MM0702, Lesson 3
Figure 3-14. Graphic Representation of the Tangent Function.
Table 3-7. Values of the Tangent θ for One Revolution.
47
MM0702, Lesson 3
Table 3-8. Values of Trigonometric Functions.
Lesson 3 Review Exercises
48
MM0702, Lesson 3
REVIEW EXERCISESLesson 3 Review Exercises Solutions
48
MM0702, Lesson 3
REVIEW EXERCISES SOLUTIONS48
MM0702, Lesson 4
Lesson 4
VECTOR ALGEBRA
Task. The skills and knowledge taught in this lesson are common to all missile repairer tasks.
Objectives. When you have finished this lesson, you should be able to explain what vector algebra is and be able to
use it to solve problems correctly.
Conditions. You will have the subcourse book and work without supervision.
Standard. You must score at least 75 on the end-of-subcourse examination that covers this lesson and lessons 1, 2,
and 3 (answer 23 of the 30 questions correctly).
In the first three lessons, you studied basic algebra, logarithms, and trigonometry. They are all necessary preparation
for this lesson because vector algebra combines their properties.
VECTOR QUANTITIES
Some quantities have magnitude only; others have both magnitude and direction. Quantities with magnitude only are
known as scalar quantities, while those with both magnitude and direction are known as vector quantities. Forces,
velocity, and acceleration are examples of vector quantities. Scalar quantities can be added, subtracted, multiplied, and
divided directly. Vector quantities, because of the incorporation of a second dimension (direction), must be treated
differently.
VECTOR NOTATION
A vector quantity (figure 4-1) can be represented by a line segment. The length of the line represents the magnitude
of the quantity, while the angular position of the line segment and an arrow point indicate the quantity's direction.
Figure 4-1. Vector Quantity.
49
MM0702, Lesson 4
Usually, this quantity is known as vector OA; however, it may be written OA or OA. Occasionally, it may be
·
designated by a single letter such as M , or, or
M , or M.
RESULTANT VECTOR
Generally, vectors are combined as indicated in figure 4-2. Since the two vectors OA and BC have the same
direction, they can be combined to produce one single vector OC (figure 4-3), having a magnitude of 21 units and
keeping the original direction.
Figure 4-2. Two Vectors of the Same Direction.
Figure 4-3. Two Vectors of the Same Direction Combined.
Resolving the principle shown in figure 4-3 into a practical illustration, suppose a boat is traveling due east at 10 mph,
and a current is flowing due east at 11 miles per hour. The result will be a speed of 21 miles per hour for the boat in
due east direction. Conversely, if the vector BC were in the opposite direction figure 4-4), the result would be the
new vector OC (figure 4-5). It would have a magnitude of 1 and be in the direction as illustrated.
Figure 4-4. Vectors of Opposite Directions.
Figure 4-5. Vectors of Opposite Directions Combined.
50
MM0702, Lesson 4
Two vectors not in the same direction can be combined using the parallelogram method. By placing the heels of the
vectors together and completing a parallelogram (figure 4-6), the diagonal of this parallelogram represents the resultant
vector. For example, combining vector OA and vector OB and completing the parallelogram, you get OC as
the resultant vector.
Figure 4-6. Parallelogram Method of Combining Vectors.
Three or more vectors can be combined in much the same way. By keeping their same relative directions and placing
them "heel to toe," the resultant vector is the vector joining the heel of the first vector to the toe of the last vector.
For example, the following three vectors, OA , PQ , and MN (figure 4-7) are combined to produce the
resultant vector ON . Combining the vectors, you get the vector in figure 4-8.
Figure 4-7. Three Vectors To Be Combined.
Figure 4-8. Parallelogram Method To Find the Resultant Vector.
51
MM0702, Lesson 4
VECTOR REPRESENTATION
Vectors are usually described in polar or rectangular form. The polar form shows the magnitude of the vector and the
angle that it makes with the horizontal. For example, 10/30° describes a vector 10 units long at an angle of 30°. In
rectangular form, the vector is resolved into its horizontal and vertical components, which are its projections on the
horizontal (X) and vertical (Y) axes, and which have as their origin, the origin of the original vector. Thus, for vector
OA (figure 4-9), the vertical component can be expressed
Y= OA sin θ,
and the horizontal component can be expressed
X= OA cos θ.
Figure 4-9. Polar Form of Representing Vectors.
Conversely, the original vector OA could be determined from the two component vectors X and Y (figure 4-10).
Assume:
X and Y are given (figure 4-10).
θ and magnitude of OA is unknown.
Therefore,
Y
tan θ = -, and
X
the magnitude of OA = √-X2 + .Y2.
Figure 4-10. Determining Vector in Polar Form.
52
MM0702, Lesson 4
When dealing with the electricity, you usually have to express a vector in terms of rectangular coordinates; however,
the vertical axis of the coordinate system is designated as the imaginary axis (j component), while the horizontal axis
is designated the real axis. In rewriting vector 10/30° in terms of its real and imaginary components, consider first the
following illustration (figure 4-11).
Figure 4-11. Rectangular Form of Representing Vectors.
By stating that 10/30° could be represented in terms of vertical and horizontal components, you have essentially stated
that
10/30° = b + ja.
However,
b = Z cos θ,
and
a = Z sin θ.
Therefore,
10/30° = (10 cos θ) + j(10 sin θ)
= (10 cos 30°) + j(10 sin 30°)
= (10 X 0.866) + j(10 X 0.5)
= 8.66 + j5.
CALCULATIONS
Addition and Subtraction
Since the addition and subtraction of vectors by graphic means is not sufficiently accurate without a precise measuring
instrument, the usual practice is to convert vectors to their rectangular form and then to add or subtract them
algebraically.
Problem:
Add 35/40° and 47/55°
53
MM0702, Lesson 4
Solution:
35/40° = 35 cos40° + j35 sin 40°
= 35 (.7660) + j35 (.6428)
= 26.81 + j22.50
47/55° = 47 cos 55° + j47 sin 55°
= 47(.5736) + j47 (.8192)
= 26.96 + j38.50
26.81 + j22.50
26.96 + j38.50
53.77 + j61.00
To convert to polar form, remember that
scalar value of the vertical component
tan θ =
scalar value of the horizontal component
61.00
=
53.77
= 1.134.
Therefore,
θ = 48.6°.
Remembering that the horizontal component (b) was equal to Z to cos θ, we can solve the value of Z.
In polar form, the new vector representing the sum of 35/40° and 47/55° is 81.3/48.6°.
Multiplication
To multiply two vectors in polar form, multiply the magnitudes together and add the angles.
55/40° X 47/55° = (55 X 47) / 40° +
55°
= 2,585/95°
You can get the same result by converting the vectors to rectangular form, performing the multiplication, and
converting the product back to polar form.
Division
In dividing vectors, divide the magnitudes and subtract the angles.
54
MM0702, Lesson 4
60
60/40° ÷ 30/20° = /40° - / 20° = 2/20°
30
RAISING A VECTOR TO A POWER
Since a power of a quantity is essentially a repeated multiplication process, (15/20°)3 is essentially
(15/20°) (15/20°) (15/20°),
which revolves into
(153) /20 + 20 + 20 = 3,375 / 60°.
ROOT OF A VECTOR
To extract the root of a vector in polar form, extract the root of the magnitude and divide the angle by the degree of
the root. For example,
3
8/60°
can be rewritten as
3
8/60° ÷ 3 = 2/20°.
Lesson 4 Review Exercises
55 Review Exercises Solutions
56 |
Mathematics Major Guide for 2013-2014
What is Mathematics?
A person who majors in mathematics is trained to solve problems by ignoring superfluous detail, looking for structure, and designing a logical method of attack that considers a wide range of possible outcomes and tries to eliminate preconceived notions. Mastering the content and techniques used in Mathematics is what makes this desirable outcome possible.
Career Opportunities in Mathematics
Because math majors are trained to think clearly and logically, as well as to quantify, they are sought by business, government and industry. In particular, mathematics majors do many things besides teaching.
Recent graduates have ended up in the following fields: Physician, Actuarial Science (analyzing risks for insurance companies), Banking (designing programs so that computers can do lots of things, e.g., billing and payroll), Education (teaching at all levels, including college), Operations Research (finding the optimal way to schedule alternatively, organize industrial operations, e.g., refineries, assembly lines, and inventory control), Computer Industry (designing hardware and software), Health Professions (data mining and data compression), Cryptologist (discovering how to send messages difficult to decipher, or deciphering messages from hostile nations and groups), Law (Law schools appreciate the logical training math majors receive), Investment and Securities (research departments), and Systems Analyst (helping teams of engineers solve real world problems).
Salary Trends in Mathematics
Mathematics majors typically receive some of the highest salaries among all college graduates. Even better, the kinds of jobs math majors typically fill rank near the top on job-satisfaction surveys because mathematicians usually have considerable autonomy in structuring their jobs.
High School Preparation
Since the tools one uses to quantify problems are algebraic, geometric, and analytic (i.e., function driven) in nature, potential math majors should have had four years of high school mathematics: two courses in algebra, one in geometry, and one on functions (usually called precalculus). It is NOT necessary to have taken AP calculus, or even any calculus, in high school to successfully major in mathematics at the University of Tennessee.
Many math majors liked math in high school, but even if you didn't, you may still want to major in math. In college courses, you will discover that mathematics is the language used by the world's most creative minds to discuss the world's most novel, exciting ideas. Math really can be the key to the cosmos.
Highlights of Mathematics
The mathematics program at UT is designed to serve students with a broad range of interests and inclinations. Talented and highly motivated students may choose to participate in the departmental Honors Program, which features an accelerated curriculum leading to graduate courses as early as the junior year. Several recent graduates of this fast-track curriculum have received prestigious fellowships to some of the top graduate schools in the country.
Math major classes at the upper division level are small (rarely over 20 students per class), so math majors tend to know each other well. The departmental Junior Colloquium offers biweekly talks designed for undergraduates, given by mathematicians from UT and other universities.
The faculty of the Mathematics Department at the University of Tennessee is widely recognized for their internationally respected research programs and scholarly output. Therefore, math majors benefit from a well-informed, up-to-date faculty with multiple contacts throughout the national and international mathematical community.
How to Major in Mathematics
There are only a few fixed requirements for a math major. During the first two years you will complete the calculus sequence and other basic courses that are prerequisites to higher level mathematics. These courses include "Introduction to Abstract Mathematics," which helps in the transition to the more abstract methods of thinking that take place in upper division mathematics courses. This course may be taken as early as the freshman year and will help determine the path that you follow towards completion of the major requirements.
Requirements for Mathematics
Prerequisites to the major are Mathematics 141-142 (or Honors equivalent: 147-148) and 171 or Computer Science 102.
The major consists of 37 hours in twelve courses divided into four categories: (1) core courses, (2) courses for breadth, (3) courses for depth, and (4) additional courses (to reach 37 hours). Note: Courses used for depth (3) may also be used for breadth (2).
1. For the Core, complete all of the following (or honors equivalents):
The requirements to graduate with honors in mathematics are the same as those for the mathematics major except the total requirement is 38 hours and includes:
For Depth (3), complete 2 pairings, one of which must be an honors sequence (MATH 447-MATH 448, MATH 457-MATH 458) or a math graduate sequence.
Graduate with an overall GPA of at least 3.25 and an MGPA of at least 3.4.
Complete at least 4 hours of MATH 497.
Complete at least 3 hours of MATH 498 and submit a completed thesis at least 30 calendar days prior to the end of the final semester of enrollment.
Complete a total of 24 hours of honors courses or mathematics courses numbered 510 or higher (except seminars), which may include courses used to fulfill other requirements to graduate with a mathematics honors concentration.
Please see the undergraduate catalog for specific information on the Honors Concentration.
Special Programs, Co-ops, and Internships
The Mathematics Department has a tutorial center, which is staffed primarily by undergraduate math majors that provides part-time employment and educational experience for students interested in teaching after they graduate.
Students interested in industrial employment should look into our Co-operative Education Program in which, beginning at the Sophomore level, students alternate periods (usually semesters) of full-time jobs with periods of full-time study. This program provides professional training, on-the-job experience, and income for math majors and other applied majors, e.g., computer science, engineering, and statistics. Frequently, successful students end up taking their first job after graduation from a company where they had co-operative experience. If you are interested in this program, take several courses from the Computer Science Department and/or the Statistics Department, along with your math major courses, and contact the Co-op office in 100 Dunford Hall early during your first year here web site ( or the Ready for the World web site ( for more information on upcoming cultural programs and activities. Learn more about UT's Ready for the World initiative to help students gain the international and intercultural knowledge they need to succeed in today's world.
Students are also encouraged to develop a global perspective within their academic program through study abroad. Studying abroad options do exist for mathematics and statistics majors! Possibilities include (but are not limited to) studying abstract algebra in Fiji, Galois theory in Hungary, linear programming in Malaysia, advanced stochastic processes in South Africa, or Bayesian inference in London. In addition to taking major-related courses abroad, many math or statistics majors have elected to fulfill their language requirement and/or general education courses overseas.
Consult an academic advisor early in your academic career about the best time for you to study abroad as well as what courses you may need to take. For more information about program options, the application process, and how to finance study abroad, please visit the Programs Abroad Office website.
Academic Plan and Milestones
Following an academic plan will help students stay on track to graduate in four years. Beginning with first-time, first-year, full-time, degree-seeking students entering in the Fall 2013 semester, UT has implemented Universal Tracking (uTrack), an academic monitoring system designed to help students stay on track for timely graduation. In order to remain on track, students must complete the minimum requirements for each tracking semester, known as milestones. Milestones may include successful completion of specified courses and/or attainment of a minimum GPA.
For More Information
Note
The information on this page should be considered general information only. For more specific information on this and other programs refer to the UT catalog or contact the department and/or college directly. |
Gilbert J. Cuevas
Bibliography:
ISBN:
9780078250835
Publisher:
Glencoe/McGraw-Hill School Pub Co
Publication Date:
2002
Binding:
Hardcover
Synopsis:
From the first day your students begin to learn the vocabulary of algebra until the day they take final exams and standardized tests, these programs strengthen student understand ing and provide the tools students need to succeed.
Gilbert J. Cuevas |
English.. [via]
Basic Multivariable Calculus fills the need for a student-oriented text devoted exclusively to the third-semester course in multivariable calculus. In this text, the basic algebraic, analytic, and geometric concepts of multivariable and vector calculus are carefully explained, with an emphasis on developing the student's intuitive understanding and computational technique. A wealth of figures supports geometrical interpretation, while exercise sets, review sections, practice exams, and historical notes keep the students active in, and involved with, the mathematical ideas. All necessary linear algebra is developed within the text, and the material can be readily coordinated with computer laboratories. Basic Multivariable Calculus is the product of an extensive writing, revising, and class-testing collaboration by the authors of Calculus III (Springer-Verlag) and Vector Calculus (W.H. Freeman & Co.). Incorporating many features from these highly respected texts, it is both a synthesis of the authors' previous work and a new and original textbook. [via]
Designed for courses in advanced calculus and introductory real analysis, Elementary Classical Analysis strikes a careful balance between pure and applied mathematics with an emphasis on specific techniques important to classical analysis without vector calculus or complex analysis. Intended for students of engineering and physical science as well as of pure mathematics.
This volume provides a detailed account of the theory of symplectic reduction by stages, along with numerous illustrations of the theory. It gives special emphasis to group extensions, including a detailed discussion of the Euclidean group, the oscillator group, the Bott-Virasoro group and other groups of matrices. The volume also provides ample background theory on symplectic reduction and cotangent bundle reduction.
[via]
More editions of Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems (Texts in Applied Mathematics):
A monograph on some of the ways geometry and analysis can be used in mathematical problems of physical interest. The roles of symmetry, bifurcation and Hamiltonian systems in diverse applications are explored. [via]
More editions of Lectures on Geometric Methods in Mathematical Physics (CBMS-NSF Regional Conference Series in Applied Mathematics):
The use of geometric methods in classical mechanics has proven fruitful, with wide applications in physics and engineering. In this book, Professor Marsden concentrates on these geometric aspects, especially on symmetry techniques. The main points he covers are: the stability of relative equilibria, which is analyzed using the block diagonalization technique; geometric phases, studied using the reduction and reconstruction technique; and bifurcation of relative equilibria and chaos in mechanical systems. A unifying theme for these points is provided by reduction theory, the associated mechanical connection and techniques from dynamical systems. These methods can be applied to many control and stabilization situations, and this is illustrated using rigid bodies with internal rotors, and the use of geometric phases in mechanical systems. To illustrate the above ideas and the power of geometric arguments, the author studies a variety of specific systems, including the double spherical pendulum and the classical rotating water molecule. [via]
The, fluid mechanics, electromagnetism, plasma dynamics and control theory are given using both invariant and index notation. The prerequisites required are solid undergraduate courses in linear algebra and advanced calculus. [via]
More editions of Manifolds, Tensor Analysis and Applications (Global analysis, pure and applied):
Vector Calculus helps students foster computational skills and intuitive understanding. This edition offers revised coverage in several areas and a new section looking at applications to Differential Geometry, Physics and Forms of Life as well as a large number of new exercises and expansion of the book's signature Historical Notes. [via] |
Welcome to CPM Educational Program,
an educational non-profit organization dedicated to improving grades 6-12 mathematics instruction. CPM offers professional development and curriculum materials. We invite you to learn more about the CPM mathematics program by clicking the "Learn about CPM" link at left. The other sections offer support materials for teachers, parents and students.
Headlines
The statistics and probability lessons from the CPM Core Connections series have been posted here. This series of Core Connections Statistics Supplements replaces the Statistics and Probability Resource that was previously available. These files are available for download free of charge.
For a class not using Common Core State Standards (CCSS)–aligned curriculum materials, these lessons supplement the statistics and probability content to meet the statistics and probability standards of the CCSS. Even if CCSS-aligned curriculum is being used in the classroom, in the transition period to CCSS, it is possible, even likely, that a class will not cover all of the statistics and probability content standards. This content will have to be taught in a subsequent course. These CPM Statistics Supplements provide the materials for any delayed statistics content.
Conference Handouts and Resources
News
New curriculum and technical support available
Three CPM mentor teachers are now available to help parents, students and teachers who have questions about the CPM program. This support is primarily for questions about using the program, its technology and the website. To use CPM's support service, go to to see the available services. Follow the prompts to get help. Note that this service is not a "homework helpline." That support is at
CPM now offers a new series of textbooks to meet the grade 6-8 and high school CCSS content and practice standards: Core Connections, Courses 1 - 3 and Core Connections Algebra 1 & 2 and Geometry. Learn how this series as well as the original Connections series of CPM textbooks are fully aligned with the CCSS Content and Mathematical Practice Standards. CPM can also provide professional development centered around embedding the eight CCSS Mathematical Practices into your current lessons and current textbook from any publisher. Start moving on the path to the CCSS today!
Parent e-book licenses are now available!
Parents may purchase a one-year e-book license of their student's book for $10 by calling CPM and using a credit card. See order form for a complete list of available one-year licenses. Contact CPM at (209) 745-2055.
Sample Problem
Core Connections Geometry : 4.2.5: How much can I expect to win?
Expected Value
Different cultures have developed creative forms of games of chance. For example, native Hawaiians play a game called Konane, which uses markers and a board and is similar to checkers. Native Americans play a game called To-pe-di, in which tossed sticks determine how many points a player receives.
When designing a game of chance, attention must be given to make sure the game is fair. If the game is not fair, or if there is not a reasonable chance that someone can win, no one will play the game. In addition, if the game has prizes involved, care needs to be taken so that prizes will be distributed based on their availability. In other words, if you only want to give away one grand prize, you want to make sure the game is not set up so that 10 people win the grand prize!
Today your team will analyze different games to learn about expected value, which helps to predict the result of a game of chance.
4-101 - TAKE A SPIN
Consider the following game: After you spin the wheel at right, you win the amount spun.
a.) If you play the game 10 times, how much money would you expect to win? What if you played the game 30 times? 100 times? Explain your process. [answer]
b.) What if you played the game n times? Write an equation for how much money someone can expect to win after playing the game n times. [answer]
c.) If you were to play only once, what would you expect to earn according to your equation in part (b)? Is it actually possible to win that amount? Explain why or why not. [answer] |
BCC's mission is to prepare students for graduation, transfer and careers; the Math Department's mission is to help students gain quantitative literacy, understand mathematical ideas, and use them to excel in their future work. We support degree programs of study, and students will find that the rigor and demands of the courses offered here are aligned with many four-year colleges and universities. The Math Department acknowledges the recommendations of professional mathematical societies such as AMATYC, 100% Math, and the NCTM standards.
BCC math courses range from arithmetic through calculus and many are offered in three formats: the traditional teacher-paced lecture format, the self-paced MAT 800 format through pre-calculus, or the online MAT 800 format.
In the MAT 800 series, students advance at their own rates and credits are earned individually. Self-motivated students can move quickly through their math credits, while those students who have not recently had math courses or who are lacking in confidence can move more slowly with the individualized faculty assistance needed to build solid foundations for long term success.
There are no lectures in this setting. Instead, students work with their texts, computers, teachers, and tutors, if desired, to learn the material. They decide when to take tests, and then are allowed to retest until they pass. Students may select MAT 800 for one or two credits, and then may choose to add more once these are completed. Each student works with his or her teacher to plan the pace at which the credits should be completed.
Many students who take the Learning Skills Assessment place into Basic Math or Introductory Algebra. Our mission, as pre-college level math teachers, is to help each student master skills, learn techniques, and gain confidence in order to build a solid foundation for college-level math. Pre-college-level courses may be teacher-paced (MAT 018, MAT 028, MAT 029), the self-paced MAT 800 "modules" (MAT 011 through MAT 029C), or MAT 800 online. Course credits at this level do not transfer.
For information on what is covered in MAT 011 through MAT 029C, click here.
College-level math
Although specific programs may require more or less math, College Algebra, Elementary Statistics, and Math for Art and Nature fulfill the BCC general education graduation requirement. Of these three, College Algebra is the most widely transferable and prepares students for pre-calculus. It is available in the traditional teacher-paced format as well as the self-paced MAT 800 format. The Math Department offers courses that meet the requirements at institutions where the majority of BCC students expect to transfer. Degree and program requirements vary among institutions; the responsibility for a realistic plan belongs to each student. |
Algebra 2
Description
Help your students discover the logic, order, beauty, and practicality of algebra. Throughout the course, students are encouraged to use their reasoning ability as they work with the axioms, rules, and principles of algebra. Concepts are developed and mastered through an abundance of worked examples and exercises, with an emphasis on word problems that relate to the physical world. Reviews at the end of each unit measure student progress, and special sections challenge the mathematically talented student. This text calls for a scientific calculator that has the trigonometric functions, statistics, powers, and roots |
5 Amazing Algebra Apps for Students
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Students have all the luck nowadays with so many useful and amazing apps to help them learn. A long time ago when I was a student, I found algebra a tricky subject. We had a look through some of the apps available for students to give them a helping hand to get their head around some of the main concepts in algebra.
Algebra Tutor allows students to practice and learn Prealgebra and Algebra topics. It has walk through step-by-step solutions to see where you made mistakes and you can see your stats for every problem type. It has 35 different practice concepts from fractions to polynomials.
Algebra is a Algebra tutoring and homework help app that has a huge range of chapters teaching all aspects of algebra like linear equations, simplifying, and factoring. The app includes:
Multiple video example problems
Interactive practice problems with built-in support
A Challenge Problem
Multiple-choice self-tests
Extra problem worksheets
Review notes
Math Ref is an award winning app that includes over 1,400 different formulas, references, tips and examples of equations and concepts such as vector calculus, linear algebra and differential equations. It includes topics like:
Algebra
Geometry
Trigonometry
Linear Algebra
Series & Sequences
Derivatives
Integration
Table of Integrals
Vector Calculus
Differential Equations
Discrete
Probability and Statistics
Physics
Chemistry (includes Periodic Table)
Algorithms
Financial (includes Real Estate)
Prime Numbers
Greek Alphabet
Algebra Calculator is a simple tool for calculating algebra problems directly from your iphone or ipad. It includes a quadratic equation calculator, a sum of squares calculator, and a foil methods calculator.
Free Graphing Calculator is a nifty app that allows you to visualize algebraic equations. You can graph up to four equations at once, and use an in-built scientific calculator to help you with your calculations.
Have you found any more Algebra apps that are useful? Post any of your suggestions below!
About the author : silvia
I must recommend DragonBox to you. I got a bit excited and reviewed it on my blog a while back. If you haven't come across it, it is a game that guides you through the process of solving equations superbly well.
avatargeneration
Thanks for the share! We found it too have a post about it:
Let us know if you find any more interesting tools that we miss. Thanks!
Ian
All kids should start by playing – this would teach them the concepts of algebra through fun and get them excited about taking on the challenges of the complex algebra that the above apps engage them in. |
Rutherford, CA SATNor will a detailed and dry derivation necessarily deepen comprehension. Equations do, however, correspond to literal processes in the real world. I find it much more pragmatic to emphasize the beauty and utility of mathematical concepts and functions, and the power of as it gives students a context and appreciation that makes recall a cinchWhen I took over a college learning center, I had a summer workshop learning how to help ADD/HD students. For 11 years I helped many stay in school. I am familiar with the three major kinds of dyslexia and how to overcome the difficulties. |
Topics In ContemporaryTopics in Contemporary Mathematics," 8/e, is uniquely designed to help students see math at work in the contemporary world by presenting problem solving in purposeful and meaningful contexts.Strong technology focus encourages students to learn and apply their knowledge using the most up-to-date web links maintained by the author on a companion web site. Instructors may also use this site to access PowerPoint slides for convenient class presentations. In addition to these web resources, lecture and practice test videos have been developed to pr... MOREovide extra support and foster confidence outside of the classroom. For those students in Florida, a CLAST Test guide and video are available as well.A variety of pedagogical features reinforce ideas and motivate students to learn. "Getting Started" offers a motivating introduction for the techniques and ideas in each section. Through web references and "Web It" exercises, students utilize the Internet as an educational and creative tool to study mathematical concepts. "Collaborative Learning" encourages student interaction as they work together to solve problems. The "Graph It" feature found in the book margins provides step-by-step directions for solving specific examples using the TI-83 graphing calculator.Problem-solving approach throughout the text helps students learn techniques and methods that will benefit them throughout their lives and careers. These special examples use George Polya's problem-solving strategy (RSTUV--Read, Select, Think of a plan, Use the techniques, Verify) and a unique two-column format for describing the general problem-solving method and demonstrating specific uses. Eduspace, powered by Blackboard, for theBello/Britton "Topics in Contemporary Math" course features algorithmic exercises, test bank content in question pools and video explanations.
Note: Each chapter concludes with a Summary, Research Questions, a Practice Test, and Answers to Practice Test |
Math Camp Activities
The primary focus of Raleigh Math Camp is problem-solving, and students will have many opportunities to learn and practice the art of solving mathematical problems.
Mornings will be spent on in-depth, hands-on type problems, which will require a variety of tools and methods to solve. We will also make use of group discussions, journal writing, mental math exercises, TI83/TI84 graphing calculators, and different online resources, in order to sharpen our problem-solving skills.
Afternoons will spent on more traditional "word problems" from a variety of sources (e.g., textbooks, online, SAT problems, etc.). Students will work not only on problems chosen by the instructor, but also on specific problems which they have chosen and which target specific skills which they wish to focus on.
As part of our problem-solving activities, students at Raleigh Math Camp will have the opportunity to review a number of important Algebra concepts, including: |
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