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The Academic Language of Math: Making it Work in the Classroom Following decades of research, educators recognize that without sufficient knowledge of the academic ...
2 2004, Enablearning, Inc. The Elements of Algebra Enablearning, Inc. A Letter to Students Why is math so hard for me to learn? Reason #1 - You have to do your homework!
Copyright by The McGraw-Hill Companies, Inc. All rights reserved. Except as permitted under the United States Copyright Act, no part of this publication may be ...
CWS-1 Understanding Multicultural Teaching and Learning PE1 Pedagogy In this rationale and entry the candidate will demonstrate depth and breadth of pedagogical knowledge. |
for a one-semester undergraduate course in topology. The fundamental concepts of general topology are covered rigorously but at a gentle pace and an elementary level. It is accessible to students with only an elementary calculus background. In particular, abstract algebra is not a prerequisite. The first chapter develops the elementary concepts of sets and functions, and in Chapter 2 the general topological space is introduced. Subspaces, continuity, and homeomorphisms are covered in Chapter 3. The remaining chapters cover product spaces, connected spaces, separation properties, and metric spaces. |
Algebra with Applications and Visualization
The Rockswold/Krieger algebra series fosters conceptual understanding by using relevant applications and visualization to show students why math ...Show synopsisThe Rockswold/Krieger algebra series fosters conceptual understanding by using relevant applications and visualization to show students why math matters. It answers the common question â When will I ever use this?â Rockswold teaches students the math in context, rather than including the applications at the end of the presentation. By seamlessly integrating meaningful applications that include real data and supporting visuals (graphs, tables, charts, colors, and diagrams), students are able to see how math impacts their lives as they learn the concepts. The authors believe this approach deepens conceptual understanding and better prepares students for future math courses and life.Hide synopsis
Description:Very good in very good dust jacket. ANNOTATED EDITION, WITH...Very good in very good dust jacket. ANNOTATED EDITION, WITH EXTRA NOTES AND ANSWERS, NO CDS OR ACCESS CODES, EXCELLENT CUSTOMER SERVICE!
Description:New. ATTENTION! New ANNOTATED INSTRUCTOR'S EDITION. Same as...New. ATTENTION! New ANNOTATED INSTRUCTOR'S EDITION. Same as Student Edition with Solutions. Never Used! Ships Next Business Day |
Mathematics
Mathematics has presented difficulty for many of our students. A small, though increasing, number are formally diagnosed with learning disabilities in Math. A much larger group struggle in Math because their weaknesses in studentship skills seem to have a proportionately greater impact in that subject. Difficulties with homework, study skills, and memorization cause students to fall behind in Math, and the cumulative nature of the subject often means that a student who falls behind is in danger of being lost. Many of our students have decided early in their school careers that based on these problems, they "can't do Math," a perception that can easily become self-fulfilling. In the past, our experience has been that some students desire to drop Math entirely once they have met the minimum requirement for graduation. Over the years, we have made several modifications to the Math program resulting in a larger number of students persisting in Math. We believe it is a reasonable expectation that students with the ability to do so will continue their study of Math until graduation.
The initial goal of the Mathematics program is to help students change their attitudes about both their ability and their interest in Mathematics. A significant portion of the first quarter of every academic year is spent on review and on providing an introduction to basic topics of the course. As students see themselves succeed, their attitudes begin to change. Students are encouraged to persist and to acknowledge that what they accept and easily express as inability is instead confusion or frustration and, in most cases, difficulty identifying and expressing the source of trouble. Teachers work with students to help them develop the language and skills they need to be self-advocates in every Math class.
As students begin changing their attitudes about Mathematics, we seek to continually challenge them. Stevenson has always offered the traditional sequence of Pre-Algebra, Elementary Algebra, Geometry, Intermediate Algebra/Trigonometry, Pre-Calculus and Calculus. It is our goal to engage our students in a study of Mathematics that intrigues them and fosters in them a desire to continue their study after leaving Stevenson. For those students who we identify as having severe difficulties or major gaps in mathematics, we have developed skills classes in Algebra and Algebra II. The goal of these skills classes is to fill gaps and strengthen skills so that students can rejoin the traditional sequence of courses. Regular use of scientific calculators, graphing calculators and online interactive math resources are integrated into all coursework. |
commo... read more
Concepts of Mathematical Modeling by Walter J. Meyer This text features examinations of classic models and a variety of applications. Each section is preceded by an abstract and statement of prerequisites. Includes exercises. 1984 edition.
A Concept of Limits by Donald W. Hight An exploration of conceptual foundations and the practical applications of limits in mathematics, this text offers a concise introduction to the theoretical study of calculus. Many exercises with solutions. 1966 edition.
Stochastic Modeling: Analysis and Simulation by Barry L. Nelson Coherent introduction to techniques also offers a guide to the mathematical, numerical, and simulation tools of systems analysis. Includes formulation of models, analysis, and interpretation of results. 1995 edition.
Elements of Pure and Applied Mathematics by Harry Lass This completely self-contained survey explores important topics in pure and applied mathematics. Each chapter can be read independently, and all are unified by cross-references to the complete work. 1957 edition.
A Short Course in Discrete Mathematics by Edward A. Bender, S. Gill Williamson Explores Boolean functions and computer arithmetic; logic; number theory and cryptography; sets and functions; equivalence and order; and induction, sequences, and series. Assumes some familiarity with calculus. Original 2005100 Great Problems of Elementary Mathematics by Heinrich Dörrie Problems that beset Archimedes, Newton, Euler, Cauchy, Gauss, etc. Features squaring the circle, pi, similar problems. No advanced math is required. Includes 100 problems with proofs.
Product Description:
common workplace procedures. Each chapter begins with a brief review of relevant mathematics, followed by an examination of the material's typical industrial applications. The author demonstrates the problem-solving power of interweaving analytic and computing methods and integrates MATLAB code into the narrative flow. Topics include the Monte Carlo method, the discrete Fourier transform, linear programming, regression, microeconomics, ordinary and partial differential equations, and frequency domain methods. A concluding chapter on technical writing explains how to present mathematical data in a variety of situations and offers helpful suggestions for assembling formal technical reports, progress reports, executive summaries, and other statements |
Course MAT201
Calculus III (Multivariable Calculus)
A continuation of MAT103/104, the third semester in the calculus sequence gives a thorough introduction to multivariable calculus and mathematical methods needed to understand real world questions involving quantities changing over time in 3-space. Topics include limits, continuity and differentiability in several variables, extrema, Lagrange multipliers, Taylor's theorem, multiple integrals, integration on curves and surfaces, Green's theorem, Stokes' s theorem, divergence theorem. Emphasizes concrete computations over more theoretical considerations. However, this course demands that students move beyond thinking of mathematics as a set of rules and algorithms to memorize, and begin to approach problems with greater independence and maturity. Exams test for thorough conceptual understanding and computational fluency in standard cases. Although demanding, the exams do not typically involve any proofs, and there is little emphasis on exceptional cases. Offered both Fall and Spring. Prerequisite: MAT104 or equivalent.
The first part of the course introduces basic objects in space: lines, planes, curves, (quadric) surfaces, and basic properties like arc length, surface area and volume.
The second part moves on to quantities that change with position in 3-space, like temperature or population density, which can be described by functions of several variables. We study their graphs, level sets, rates of change (partial derivatives) and their extreme points (maxima/minima) using Lagrange multipliers and Taylor's theorem.
The third part is devoted to vector fields, quantities that change with both position in space and with time. Think of how storms move, how radio signals make cell phones work or how blood flows. Although we do not explicitly study such applications, we lay the mathematical foundation for more advanced courses in science, engineering, and economics. The course ends with the theorems of Green, Gauss, and Stokes.
Description of classes
Classes meet 3 times per week, for 50 minutes. Sections are generally offered MWF at 10, 11 and 12:30 in both semesters.
The course is organized into small sections of 20 to 30 students. There is one course head who coordinates with all the instructors to write the exams. All students have the same homework assignments and take the same midterm and final exam. The midterm and final count for the bulk of the course grade, typically about 70%. These exams are graded by all the instructors and graduate student AI's together to ensure uniformity across all sections. Typically there are two to four take-home quizzes, the same for all sections. Homework and quizzes together usually account for about 30% of the course grade.
In order to do well in the course, we anticipate that most students will need to spend approximately ten hours per week reading the text, reviewing class notes, solving homework problems and working through lots of extra practice problems to prepare for quizzes and exams. The course will be quite fast-paced and it is essential to work steadily throughout the semester. Frequent feedback will be given to help students keep up and monitor progress.
The first half of MAT201 generalizes MAT103 to higher dimensions, and the second half generalizes MAT104.
Students who have already taken MAT175 should not sign up for MAT201 because there is too much overlap between these two courses. MAT175 is intended for students who will not take futher mathematics courses at Princeton; in rare cases it may be possible for a highly motivated student who received a grade of at least B+ to attempt MAT202 afterwards, but he/she should expect to work extremely hard in order to succeed.
Who Takes This Course
Most students in this course are incoming freshmen or sophomores who consider majoring in one of the sciences or engineering. More mathematically inclined economics majors will take this course along with MAT202 (instead of MAT175). It gives a solid introduction to multivariable calculus suitable for most students who want to use mathematics as an analytic tool in later studies in other fields. Although it is not a prerequisite, many students in the course will have had a more basic multivariable calculus course in high school.
Most students in the course are freshmen; in the fall they are students who got a 5 on the BC exam in high school (or its equivalent). In the spring, they are mostly continuing from MAT104.
Students who took AB calculus only should take MAT104 instead. There is one possible exception to this rule: students with a 5 on the AB exam and a very strong interest in math as a major along with a math SAT score of at least 750 can consider taking MAT215 or MAT214 instead. This is rather rare, and such students should consult the math placement officer.
Students who consider a major in physics or applied math should consider MAT203 or MAT215 instead if they have a 5 on the BC exam (or equivalent) and a math SAT score of at least 750.
Future math majors usually learn multivariable analysis (calculus) in MAT218 instead (after MAT215 and MAT217). Some, especially those who are more interested in applied math opt for MAT203.
Some economics majors take MAT175 instead for a much more basic treatment of some of the ideas in MAT104, MAT201 and MAT202 in one semester.
A very solid knowledge of single-variable calculus and precalculus is needed: how to analyze and graph functions, how to compute and interpret derivatives, how to interpret, set up, and calculate definite integrals with speed and accuracy. An interest in thinking rigorously about problems involving space and time is also needed.
If your background is weak or rusty consider MAT104 to get an excellent review of the knowledge assumed in this course.
Keep in mind that a score of 5 on the BC calculus exam is minimally equivalent to MAT104, probably equivalent to a grade of C. As a result, many students who scored a 5 on the BC calculus exam or took a similar course in high school (and did well) opt to start in MAT104 and find it to be quite challenging. Although many of the topics in MAT104 are somewhat familiar after a BC calculus course, the depth of coverage and mastery of the subject required here is much greater, with more emphasis on independent thinking. Additional topics that are not covered in most BC calculus courses are also included.
Note that a score of 7 on the IB MathHL exam (not SL!) or an A on the British A-levels exam is treated as equivalent to a 5 on the BC exam here and at many other universities. These scores indicate that MAT201 is a reasonable starting point for you.
If in doubt, sign up for MAT201 and be prepared to re-evaluate during the two weeks of classes.
You should probably take MAT201 and switch down after a couple of weeks, if necessary. Taking MAT175 limits your options and not every program accepts it as a substitute for MAT201; you should not take it if you may later need further math courses at Princeton. Also, keep in mind that these courses are not generally offered at the same time, so plan your schedule carefully to leave room for MAT175 if you think you may want to switch.
Working problems from these sample quizzes and exams can give you a good idea of the expectations and content in this course as you think about which course is right for you. Just reading the questions or the solutions can be very misleading however. Try the problems yourself!
I already took multivariable in high school, do I have to take this course?
Most students in MAT201 have had some multivariable calculus and/or linear algebra before, but rarely with the same depth and thoroughness. If you need the course for upper division courses in your major, then you are probably better off to take MAT201 even though some material will be review.
Not convinced? Take the sample final. Can you do any of the problems? For most students, the answer will be no. Review your old notes and try again. Can you do at least 60% of the exam?
In rare cases, the placement officer will decide that your prior work is indeed equivalent to MAT201 at Princeton. It will be helpful if you can bring your graded exams from the course you took to show the placement officer. He/she may also require you to take an exam to demonstrate your knowledge.
You might consider MAT203 or MAT218 instead (if you take MAT215 and MAT217 first) but these courses are not for everyone -- they require an intense commitment and interest in math for its own sake, not just fulfilling a requirement.
Can I take MAT201 and MAT202 in the same semester?
It is not impossible, but we do not recommend it. It makes midterm week particularly unpleasant, but if you have a very good reason for it and you are a very strong student, it can be done. It will likely mean that you will get a lower grade in one of them that you would otherwise have done.
How much work is this course?
Most math courses require a steady time commitment. We expect that the weekly problem sets will take at least three hours to complete, although this can vary quite a lot depending on your background and goals. To do well on math exams, you need to work through a lot of extra problems. All in all, you should be ready to spend up to ten hours per week working outside of class.
If I think MAT201 is too hard, what should I do?
You may have a couple of options, depending on which courses you have already taken here at Princeton and depending on your major. You may consider switching into MAT175 but only if you are completely sure that you don't need to take any further math courses and that your program will allow you to substitute MAT175 for MAT201. If you want to major in engineering, then your only option is to consider dropping back to MAT104 to get a thorough review of all the material assumed in MAT201. As a future BSE major, starting in MAT104 will not throw you off-track, and taking the time to strengthen your foundations can really pay off in the long run.
Try an old final exam in MAT104. Can you do at least half of the problems correctly? (Try the problems -- don't just read the questions!) If you do need to switch down to MAT104 because your knowledge of one-variable calculus is insufficient, then you should decide this as quickly as possible because there is very little overlap between the first half of MAT201 and MAT104. If you wait too long, it will be very difficult to catch up and do well in MAT104.
Peer tutoring can be arranged through your residence college, and there is also help available at the McGraw Study Halls. Talk to your instructor!
If I think MAT201 is too easy, what should I do?
Have you had a quiz yet? You may be in for a surprise. Homework and the first couple of weeks of class can be misleading since the first few topics are not too difficult and the homework problems are quite routine compared to the exam questions. Try a sample quiz or midterm for this course. Remember -- don't just read the question. See if you can produce correct solutions to most of the problems in the allotted time.
If you are also taking PHY103, you might consider just enjoying the fact that MAT201 is too easy. The combination of a demanding physics class and a demanding math class has been responsible for quite a few academic distress stories at Princeton.
I need both MAT104 and MAT201 for my major. After checking the math placement information, I think MAT201 is probably the right course for me, but I don't qualify for AP credit for MAT104. Do I have to take MAT104 or can I sign up for MAT201 instead?
Be cautious. Students often underestimate the difficulty of MAT104 and of MAT201 because many have seen some of the techniques taught in the first few weeks of these courses. Consider the information in the previous two questions as you think about your decision.
If you are really sure that you belong in MAT201, you can sign up for it. If you pass MAT201 in your freshman year, you will automatically receive AP credit for MAT104. Just be ready to re-consider and switch down to MAT104 early on if necessary since there is very little overlap with the first half of MAT201 and the material in MAT104.
If I want to switch courses, what should I do?
Details of departmental drop/add/swap procedures and information about who to contact for advice about courses can be found on the undergraduate home page. But don't delay -- If you are going to switch, do it soon!
I would like to switch sections within MAT201, what should I do?
If you have a time conflict, there is usually no problem. If you want to switch to another section at the same time, be prepared for a possible refusal --- the instructor may not have room for another student. Details of departmental drop/add procedures and information about who to contact for advice about courses can be found on the general Math FAQ page. But don't delay -- if you are going to switch, do it soon!
I can't fit this course into my schedule. Can I take this course for Princeton credit at another university?
Yes, but it may be difficult to find an equivalent course. Many multivariable courses at other universities cover only about half of 201. Check out our summer course approval procedures.
I have more questions that are not answered here. What should I do?
First, check the undergraduate home page for more information about how our courses work in general and about who to contact if you need to discuss your situation with someone from the math department. Also: representatives from the math department will be available at freshman registration. |
Core-Plus Mathematics Project (CPMP)
Summary
CPMP is a text-based curriculum featuring integrated development of fundamental concepts and skills in algebra and functions, geometry and trigonometry, statistics and probability, and discrete mathematics. It utilizes student‑centered investigations of math concepts in the context of realistic problems and applications, leading to an understanding of concepts, principles, techniques, and productive habits of mind. New technology tools enhance learning, teaching, and problem solving across the curriculum. The first three courses focus on broad mathematics concepts to prepare students for college and/or the workforce. The fourth course is flexible and is designed to help students prepare for college mathematics.
Target Population
Developed for grades 9-12 in traditional high school classrooms.
Goal(s)
Focus on college-preparatory mathematics through the use of an alternative four-year math curriculum.
Program Highlights
Core-Plus Mathematics covers four years of high school math with a coordinated program.
Core-Plus Mathematics boosts students' scores on standardized math tests, including college entrance examinations, as well as their quantitative thinking skills.
Core-Plus was identified as an "exemplary" program by the U.S. Department of Education.
The Facts
Year introduced:1995
Number of schools:The program is implemented in over 500 high schools in multiple districts nationwide.
Costs:Student editions of each "course" (the textbook/student materials for one grade level) cost approximately $60 per student.
Region:National
Partners
Higher Education
Western Michigan University
Business
Glencoe/McGraw Hill
Non-Profit / Philanthropy
Government
National Science Foundation
Evidence of Impact
Research shows that the CPMP curriculum has boosted students' mathematical achievement on standardized tests.
Study 1:
A Matched Study of Washington State 10th Grade Assessment Scores of Students in Schools Using the CPMP Program: One study, conducted by a consultant, found that the pass rates for 10th grade students on the 2004-2005 WASL mathematics test were significantly higher in schools using the Core-Plus Mathematics Program. The higher scores were consistent across the board for students from various economic backgrounds.
Study 2:
CPMP Perspectives and Student Achievement: A study conducted by independent researchers found an overall boost in achievement/quantitative thinking skills among students in the CPMP curriculum compared to students in traditional math courses
Extent of evidence:Four independent studies.
Evidence of replicability:The Core-Plus Mathematics Program was originally funded by NSF in 1992 and developed as a three-year course but has expanded to serve all four high school grades. The program has been implemented in over 500 schools and in at least 40 states across the country. |
Buy ePub
Are you having trouble with algebra? Do you wish someone could explain algebra concepts to you in a clear, simple way? From the most basic algebraic expressions to more challenging polynomial functions, this book takes a step-by-step approach to teaching algebraic concepts. ALGEBRA I AND ALGEBRA II SMARTS! is designed for students to use alone or with a tutor or parent, provides clear lessons with easy-to-learn techniques and plenty of examples. Whether you are looking to learn this information for the first time, on your own or with a tutor, or you would like to review some algebra skills, this book will be a great choice. |
These authors understand what it takes to be successful in mathematics, the skills that students bring to this course, and the way that technology can be used to enhance learning without sacrificing math skills. As a result, they have created a textbook with an overall learning system involving preparation, practice, and review to help students get the most out of the time they put into studying. In sum, Sullivan and Sullivan's Precalculus: Enhanced with Graphing Utilitiesgives students a model for success in mathematics. This is just the standalone book.
decent book need for class so not much i can review for it comes with a lot
decent book need for class so not much i can review for it comes with a lot of cds though!!!
decent book need for class so not much i can review for it comes with a lot of cds though!!! decent book need for class so not much i can review for it comes with a lot of cds though!!! decent book need for class so not much i can review for it comes with a lot of cds though!!!
Actually a really nice book
i get this book for a summer course i am taking. i didn't want to spend $200 on a book i was only going to use for 6 weeks so i found it on half.com. the book has a lot of examples that help you learn. what i also liked about it that you can register with mathxl and you can do all homework on it and it grades it for you. and i'll give you lots of sources to get help. and you can do the problems as many times as you want until you get the grade you want or until you learn it. it's a great tool. i strongly suggest it. the only reason i'm giving it a good rating is because it was so expensive. |
Symbolic Representation
This article describes how students generalize and formalize patterns using student developed schemas including subtracting out and building up. The article includes samples of student work including students explanations for each schema. This article could supplement readings from section II and IV.
This article is a brief overview of the results of a research project focusing on solution strategies algebra students used to solve non-linear function problems. The article focuses on the variety of strategies, the relationship to achievement and using multiple representations. Examples of the constructed response items and student solutions are given. This reading could be used to supplement readings from section II and IV.
Bibliographic Citation:
Senk, Sharon and Thompson, Denisse (2006). Strategies Used by Second-Year Algebra Students to Solve Problems. Journal for Research in Mathematics Education 37(2), pgs.116-128.
This article describes the results of a multi-year research project on algebraic reasoning in middle school students. The article describes middle school studentsí understandings and/or misunderstandings of two core algebraic ideas ñ equivalence and variables. The article gives student response examples and discusses implications for instruction. This article could be used to supplement the readings from sections II and IV.
This article describes the implementation of an activity designed to confront a common misconception about variable. The article describes common misconceptions held by middle level students and describes a problem situation designed to uncover studentsí misconceptions. This article could be used to supplement the readings from sections II and IV.
This article illustrates the misconceptions that students have when using the equals sign and describes an activity used with students to develop the foundation for an accurate conception of equivalency. The author states that developing an accurate understanding of the equal sign is the basis for comprehending equations and inequalities. This article could be used to supplement the readings from sections II and IV.
Video instructional series consisting of 26 programs designed to be used by students or as a review of content matter for teachers. Illustrations and examples are used to solve real-world problems. This series could be used to supplement the readings from section I.
Bibliographic Citation:
Consortium for Mathematics and Its Applications and Chedd-Angier (1991). Annenberg Media. To view these video clips:
Curriculum Topic Study is supported with funding from the National Science Foundation (NSF) Award #0353315. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation. |
Algebra SVMI MARS Tasks
Year Task Strand Description Standard Text
2008 Expressions Representation This task asks students find algebraic expressions for area
and perimeter of parallelograms and trapezoids. Successful
students could show how the formula for area of a trapezoid
is derived from the area of the two triangles made by
decomposing the shape.
2008 Buying Chips Algebra This task asks students to form and solve a pair of linear
and Candy equations in a practical situation. Successful students could
use substitution or systems of equations to find their
solutions.
2008 Sorting Representations This task asks students to find relationships between graphs,
Functions equations, tables and rules. Successful students could
describe how to look at an equation and predict the shape of
the graph.
2008 Sidewalk Functions This task asks students to work with patterns and find the nth
Patterns term of a sequence. Successful students could write an
equation to finding the nth term.
2008 Functions Functions and This task asks students to work with graphs and equations of
Representations linear and non-linear functions. Students need to identify
points on a graph, write a linear equation. Successful
students knew the difference between quadratic and
exponential equations and could give the equation of a
parabola.
2007 Graphs Functions This task asks students to match linear and quadratic
equations with their graphs. Interpret the meaning of the
intersections of the two lines, graph the equation y=3x, and
read points off the graph. Successful students could use
algebra to find the intersecting points by writing and solving
an equation.
2007 House Prices Data Analysis This task asks students to work with scatterplots in the
context of wages and house prices. Students were asked to
make a general statement about the correlation of the
variables in each scatterplot, read points from the graph, and
identify outliers. Successful students could give an equation
for the graph with a positive correlation and show the
location on the graph where house payments exceeded
monthly income.
Algebra SVMI MARS Tasks
2007 Ash's Puzzle Mathematical This task asks students to investigate and find numbers that
Reasoning fit a given set of rules and write rules to describe how to find
numbers with certain characteristics. Successful students
could consider all or most possibilities.
2007 How Old Are Algebraic This task asks students to form algebraic expressions to
They? Properties and describe relationships between the ages of some children,
Representations use these expressions to write and solve equations to find
their ages, and solve for the time when one child will be
twice as old as the other child.
2007 Two Solutions Algebra This task asks students to find two possible solutions to a
variety of types of equations, such as 121=x2and x2< x3.
Students are then asked to sort equations into those with only
2 solutions, more than 2 solutions, and an infinite number of
solutions. Successful students could solve the equations, use
substitution, and had other strategies to help them find the
two solutions.
2006 Swimming Geometry and This task asks students to work with trapezoids, volume,
Pool Measurement rates and time graphs in the context of a swimming pool.
Successful students reason about rates per second to find the
total time to fill a swimming pool and choose a time/depth
graph to match the geometric situation of filling the pool.
Students working at a high level could develop a formula
and calculate the volume of water in a swimming pool with
two trapezoidal sides.
2006 Odd Sums Mathematical This task asks students to work with odd, even and
Reasoning consecutive numbers. Make and justify conjectures about
consecutive numbers. Successful students could give
examples of two consecutive numbers to make a given odd
number or 3 consecutive numbers to make an even number.
Students were able to give a rule to determine if an even
number could be written as the sum of 3 consecutive
numbers. Students working at a high level could write a
justification for why any odd number can be written as the
sum of two consecutive numbers.
2006 Patchwork Functions and This task asks students to recognize and extend a number
Quilt Relations pattern for a geometric pattern. Students expressed the rule
using algebra and used inverse operations to solve a
problem. Successful students could identify and extend a
pattern and write an equation to show the pattern. Students
Algebra SVMI MARS Tasks
could use their equations to solve the pattern extensions of
either variable.
2006 Printing Algebraic This task asks students to compare price plans using graphs
Tickets Properties and and formulae. Use inequalities in a practical context of
Representations buying tickets. Successful students were able to write an
equation to find the cost of buying tickets with an initial set
up cost and graph that equation. Students could look at a
graph and use inequalities to determine when to use different
printing companies. Students working at a high level could
use two equations to solve for the break-even cost, when
both printers would charge the same.
2006 Graphs Algebraic This task asks students to relate line graphs to their
Properties and equations. Successful students could match key parts of
Representations graphs with their equations and write an equation of a line
that would pass through a given point.
2005 Magic Squares Algebraic Use symbolic algebraic notation to calculate values in
Properties and "magic" squares where each row, column and diagonal adds
Representations to the same number.
2005 Vacations Functions and Match graphic displays to the written descriptions of how
Relations some students are paying for their summer vacations. Write
a formula that describes each of the matched relationships
Algebraic and then write a possible description for a new vacation
Properties and saving formula.
Representations
2005 Multiples of Algebraic Given a statement regarding multiples of three, test it to see
Three Properties and if it is true, find examples that match the statement and
Representations explain and justify conclusions.
2005 Scatter Data Analysis Explain the information presented in a scatter plot of
Diagram students' scores on two tests. Evaluate statements made
about the relationships found from the data and revise the
statements if necessary.
2005 Fractions Functions and Extend a sequence of fractions and compare the values.
Sequences Relations Make conjectures about the patterns in the values of the
terms as well as their equivalent decimal values.
2004 Square Functions and Find and extend number patterns in a geometric context.
Patterns Relations Find and use rules or formulas to solve problems.
2004 Population Data Analysis Analyze a scatterplot for trend, graph a line represent
average density, graph specific point for a given piece of
data, locate points on a graph to meet criteria for largest
Algebra SVMI MARS Tasks
population or lowest density, and calculate density
relationships.
2004 From 2 to 3 Geometry and Reason about a net and how it would fold into a 3-
Dimensions Measurement dimensional prism. Find the number of faces, edges, and
vertices. Calculate perimeter and area of net and volume of
prism. Understand how features in the net relate to features
in the 3-dimensional object, deciding which will remain and
which will combine when folded.
2004 Graphs Functions and Convert description of a function from a context to equation
Relations and graph. Match function descriptions and equations to
their graphical representation.
2004 Fibonacci Functions and Extend a pattern, work a pattern backwards, and generate a
Sequences Relations sequence using a given pattern. Add and divide algebraic
terms with two variables, solve simultaneous equations and
use substitution to find missing expressions.
2003 Vacuum Geometry and Given a radius, make an arc to show area covered by a
Cleaning Measurement vacuum cleaner. Use information about furniture to draw
areas accessible to vacuum cleaner and uncleaned area on a
scaled diagram.
2003 Snakes Data Analysis Read and interpret scatter plots. Locate points on a scatter
plots to identify which scatter plot best fits the coordinates or
values given.
2003 Crisscross Algebraic Investigate number patterns on a hundreds chart. Describe
Numbers Properties and rules or patterns in words or symbols. Use algebra to prove
Representations why the rules hold true for all cases.
2003 Conference Functions and Find and extend patterns in a geometric context. Use inverse
Tables Relations relationships to solve problems. Describe a rule or write a
formula to explain how to find any number in the pattern.
2003 Number Algebraic Combine numbers and variables using addition or
Towers Properties and multiplication to fill in blanks in a number tower. Use
Representations symbol manipulation to prove why expressions from the
number tower are equivalent to given expressions. Find
values of unknowns in equations |
A FREE, Powerful Algebra
End-of-Course Prep Tool
To help teachers and students succeed on the Algebra 1 End-of-Course exam
(EOC), the University of Florida and Study Edge have created Algebra Nation – a FREE, online, easy-to-use,
EOC preparation resource aligned with the latest state standards.
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Synopses & Reviews
Publisher Comments:
Taking an innovative approach to both content and methods, this book explores the foundations, basic concepts, and fundamental results of probability theory. Geared toward those unfamiliar with probability theory, it offers a firm basis for the study of topics related to the probability of mathematical statistics and to information theory.
The effective construction of probability spaces receives particular attention. Author Alfred Rényi—former Director of the Mathematical Institute of the Hungarian Academy of Sciences and an expert in the fields of probability theory, mathematical statistics, and number theory—considered effective construction of probability spaces particularly important to applying methods and results of probability theory to other branches of mathematics. Professor Rényi discusses basic theorems of probability theory in terms specific to the theorem in question, rather than in the most general form. His rigorous treatment also covers the mathematical notions of experiments and independence, the laws of chance for independent random variables, and the effects of dependence. Two brief appendixes offer helpful background in measure theory and functional analysis.
Synopsis:
Taking an innovative approach to both content and methods, this book explores the foundations, basic concepts, and fundamental results of probability theory, plus mathematical notions of experiments and independence. 1970 edition.
Synopsis:Taking an innovative approach to both content and methods, this book explores the foundations, basic concepts, and fundamental results of probability theory, plus mathematical notions of experiments and independence. 1970 edition.
"Synopsis"
by Firebrand, |
The author does a pretty good job of presenting the basic mathematical techniques for economics students. The explanations are, for the most part, clear and understandable. The mathematics covered in this book ranges from the very basic to the more advanced topics.
Although the author starts off with the most basic math, there are, nevertheless, more advanced concepts introduced within these basic chapters. For example, the chapter on arithmetic not only covers the basic elements of arithmetic but also has sections on the elasticity of demand and logarithms. However, it is unfortunate that the author combines the most basic math with the more advanced topics within the same chapter.
At the beginning of each chapter, the learning objectives are presented. The book contains many examples and exercises, and with few exceptions the author provides the answers to all the exercises unlike other books that give the answers only to selected problems.
In reading this book certain assumptions are made. For example, the author uses Excel in some of the chapters and it is assumed that the student has some prior knowledge of using this software package. Also, it is assumed that the student has some knowledge of economics.
There are certain areas in this book that can be a matter of confusion for readers. First, the author uses two different symbols for the square root, i.e., a check mark and the more familiar traditional square root symbol. Second, the author does not clearly explain when the antilog in division is taken or not taken when using logarithms. Third, an example on matrix algebra shows calculated numbers to two decimal places when, in fact, these numbers are actually carried out to more than two decimal places for purposes of getting the final answer. This can be frustrating to students trying to understand the example if they are not aware that the author has done this. Fourth, there are some errors in the book. It is important that readers be aware of these types of problems.
However, overall the mathematics is explained clearly. In addition, the author provides remedial math topics for those students needing such a review. Therefore, I would recommend this book.
I would recommend this book to anyone who wants to have some understanding of macroeconomics. This book can be used as a supplemental text or for independent study or as review/reference material. The author, for the most part, presents clear and understandable concepts in a concise manner.
As far as prerequisites are concerned, some basic knowledge of economics and money and banking will make reading this book easier and more understandable. Although calculus is not required, some knowledge of basic or college algebra is helpful.
Since this book covers intermediate macroeconomics primarily intended for those at the undergraduate, upper division level, college students, especially, should find this book useful as a supplement to their textbooks. It may help to clarify problems that students may encounter in their course work. In the words of the author: "The book can be used by undergraduates or graduate business students as a supplement to current standard texts or by instructors as an independent text supplemented by empirical and/or policy readings. The book may also be useful to graduate economics students as a review of the analytical core of macroeconomic theory."
However, the benefits derived from this book need not be limited only to college students. Those wishing to use this book for independent self-study can, with some extra effort, gain valuable information since the author does a fairly good job of presenting in a clear manner the fundamental concepts of macroeconomics.
At the beginning of each chapter, the author gives a chapter summary and outline. The chapter summaries briefly present the concepts that will be further explained in the main text. The outlines are a repetition of the same sections that are listed in the table of contents. Furthermore, in order to reinforce the concepts presented in the main text, the author not only provides the usual solved problems but also includes multiple choice and true or false questions which give the student a wide range of practice.
However, there are some weaknesses of this book which readers should be aware. First, due to the brevity of some of the concepts presented, it may be advantageous to refer to other economics books in order to get more in-depth information, and thereby, a better understanding. This is particularly true for those using this book for independent self-study. Second, some terms and concepts are defined and presented for the first time in the "Solved Problems" section at the end of the chapter rather than in the main text. Third, this book does not contain a glossary.
This is an excellent book. I would recommend it for anyone wishing to learn mathematical economics. It is equally beneficial to college students, who can use this book as a supplement to their textbook, and to those who want to use this book for self-study. For anyone who is weak in mathematical knowledge, the author presents the necessary mathematical skills needed to understand how to work the problems. As the author states: "No mathematical proficiency beyond the high school level is assumed at the start." All the required concepts are clearly explained.
The problems range from the easier to the more challenging ones. For example, some of the chapters include doing proofs, although these are limited to a minimum. In fact, only the chapter on "Simultaneous Differential and Difference Equations," out of all the chapters, contains a greater proportion of problems dealing with proofs - a little less than one-half of the total problems at the end of this chapter involve proofs.
This 3rd edition excludes three chapters on linear programming which were included in the 2nd edition. However, three other chapters were added instead: comparative statics and concave programming, simultaneous differential and difference equations, and optimal control theory.
I believe the author has done a very good job in presenting a difficult subject in an understandable manner.
This book is an intermediate microeconomics book intended primarily for undergraduate, upper division students. However, those not attending college can also benefit from this book by using it for independent study or reference material. There are no major prerequisites other than some knowledge of basic economics. For those who have a calculus background, the author does include one to three solved problems at the end of the majority of the chapters which are clearly marked.
Of the 14 chapters that make up this book, one is an added chapter and three are optional chapters. The added chapter introduces such topics as the Lerner and Herfindahl indexes which deal with measuring monopoly power, peak-load and cost-plus pricing, and game theory. Some of the concepts presented in the three optional chapters are: the Hicksian and Slutsky substitution effects of a price change, deriving indifference curves from inferred consumers' preferences, price index numbers, utility theory under uncertainity, the Cobb-Douglas production function, linear programming, and general equilibrium and welfare economics.
A weakness of this book concerns how some of the concepts are presented. In other words, the author introduces some new ideas at the end of some of the chapters in the "problems" section rather than in the main text. However, overall, I think this book provides valuable information that can benefit many people, even those who have a minimal economics background. The author presents a reasonably clear explanation of the subject matter covered which includes providing a glossary at the end of each chapter. |
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An Introduction to Calculus (For 8th and 9th Graders!)
The Problem
An Introduction to Calculus is a course modeled after Duke TiP (Talent Identification Program) Summer Studies Program. This class selects highly talented 8th and 9th graders in my county, with only a background in Algebra I to Geometry, and presents a Polynomial Based-Calculus Course. The intention is that this course will provide a novel challenge to the classes participants without over burdening them. As about 40% of individuals enrolled in college calculus courses either make a D or fail, the material covered in this class, coupled with its delivery, which emphasises independent learning and communication of mathematics will support these students as they enter a full calculus course.
The course follows an independently constructed curriculm, ranging from a basic introduction of limits to the derivative, it's applications, the Mean-Value Theorem and Rolle's Theorem, optimization and related rates, and integrals. The course culminates with a guided independent project in an area of interest, and each student is expected to present a lesson to the class on this interest.
Students who complete the class sucessfully will exit with a plan for advancement in science and mathematics. Because of limited mathematics and science opportunities within the high school and a community college and university only miles away, this class hopes to utilize both of these resources to present opportunities which may best serve these students, as they have demonstrated their abilities through their participation. |
Linear Algebra
9780201526752
ISBN:
0201526751
Pub Date: 1995 Publisher: Addison-Wesley
Summary: Fraleigh and Beauregard's text is known for its clear presentation and writing style, mathematical appropriateness, and overall student usability. Its inclusion of calculus-related examples, true/false problems, section summaries, integrated applications, and coverage of Cn make it a superb text for the sophomore or junior-level linear algebra course. This Third Edition retains the features that have made it successf...ul over the years, while addressing recent developments of how linear algebra is taught and learned. Key concepts are presented early on, with an emphasis on geometry.
Fraleigh, John B. is the author of Linear Algebra, published 1995 under ISBN 9780201526752 and 0201526751. Five hundred fifteen Linear Algebra textbooks are available for sale on ValoreBooks.com, one hundred twenty one used from the cheapest price of $23.27, or buy new starting at $95Fraleigh and Beauregard's text is known for its clear presentation and writing style, mathematical appropriateness, and overall usability. Its inclusion of calculus-related e [more]
Fraleigh and Beauregard's text is known for its clear presentation and writing style, mathematical appropriateness, and overall usability. Its inclusion of calculus-related examples, true/false problems, section s |
Did You Know?
Students with learning disabilities in mathematics generally have problems in one or more of three areas:
Doing the actual calculations.
Understanding abstract mathematical concepts.
Doing the calculations and understanding the mathematical concepts.
Students with learning disabilities often have average calculation skills, however, they have difficulty understanding algebraic concepts. Other characteristics of students with learning disabilities in mathematics include difficulties in following a series of mathematical steps to solve a problem, in applying math concepts to word problems, in solving oral problems or interpreting graphs. Students with learning disabilities in mathematics may also have difficulty understanding the instructor in class and taking notes. They may also have poor study skills and/or test anxiety. |
Physics Cheat Sheet DEMO is an interactive physics package that helps students solve and visualize numerous physics equations. By checking their homework problems with Physics Cheat Sheet DEMO, students will better develop the mathematical thinking skills needed to succeed in physics. Physics Cheat Sheet DEMO was designed for use in high school and college physics courses |
Offers an introduction to Lie theory, representation theory, invariant theory, and algebraic groups. This book presents the principal theorems of representation theory for the classical matrix groups as motivation for the general theory of reductive groups. It is suitable for physicists as well as mathematicians. more...
Writings by early mathematicians feature language and notations that are quite different from what we're familiar with today. Sourcebooks on the history of mathematics provide some guidance, but what has been lacking is a guide tailored to the needs of readers approaching these writings for the first time. How to Read Historical Mathematics fills... more...
Every discipline has it own unique history. Markedly different from others, the history of mathematics is primarily concerned with investigations into the origin of discoveries in mathematics and, to a lesser extent, an investigation into methods and notations of the past. Again, the history of mathematics covers the whole of mankind?s civilization.... more... |
Erudite and entertaining overview follows development of mathematics from ancient Greeks, through Middle Ages and Renaissance to the present. Chapters focus on Logic and Mathematics, the Number, the Fundamental Concept, Differential Calculus, the Theory of Probability and much more. Exercises and problems. DOWNLOAD LINKS |
The following is a summary of main duties for some occupations in this unit group:
Mathematicians conduct research to extend mathematical knowledge in traditional areas of mathematics such as algebra, geometry, probability and logic and apply mathematical techniques to the solution of problems in scientific fields such as physical science, engineering, computer science or other fields such as operations research, business or management.
Statisticians conduct research into the mathematical basis of the science of statistics, develop statistical methodology and advise on the practical application of statistical methodology. They also apply statistical theory and methods to provide information in scientific and other fields such as biological and agricultural science, business and economics, physical sciences and engineering, and the social sciences.
Actuaries apply mathematical models to forecast and calculate the probable future costs of insurance and pension benefits. They design life, health, and property insurance policies, and calculate premiums, contributions and benefits for insurance policies, and pension and superannuation plans. They may assist investment fund managers in portfolio asset allocation decisions and risk management. They also use these techniques to provide legal evidence on the value of future earnings.
Jobs for Mathematicians, Statisticians and Actuaries in Saguenay--Lac-Saint-Jean Region |
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The Millennium Prize Problems In order to celebrate mathematics in the new millennium, The Clay Mathematics Institute of Cambridge, Massachusetts (CMI) established seven Prize Problems. The Prizes were conceived to record some of the most difficult problems with which mathematicians were grappling at the turn of the second millennium; to elevate in the consciousness of the […]
English | 2013 | ISBN: 0262019353 | ISBN-13: 9780262019354 | 424 pages | EPUB | 5,3 MB Many books explain what is known about the universe. This book investigates what cannot be known. Rather than exploring the amazing facts that science, mathematics, and reason have revealed to us, this work studies what science, mathematics, and […]
English | 2013 | ISBN: 1461485282 | 292 pages | PDF | 4,3 MB Mathematics in the Real World is a self-contained, accessible introduction to the world of mathematics for non-technical majors. With a focus on everyday applications and context, the topics in this textbook build in difficulty and are presented sequentially, starting with a […] |
, algebra 2 and calculus algebra 1, algebra 2 and prealgebra algebra 1, algebra 2 and algebra 1, algebra 2 and calculus |
Synopses & Reviews
Publisher Comments:
The easiest way to solve the hardest problems! Geometry's extensive use of figures and visual calculations make its word problems especially difficult to solve. This book picks up where most textbooks leave off, making techniques for solving problems easy to grasp and offering many illustrative examples to make learning easy. Each yearmore than two million students take high school or remedial geometry courses. Geometry word problems are abstract and especially hard to solve--this guide offers detailed, easy-to-follow solution procedures. Emphasizes the mechanics of problem-solving. Includes worked-out problems and a 50-question self-test with answers.
Synopsis:
About the Author
Dawn Sova, Ph.D., is an Adjunct Professor at Montclair State University. She holds undergraduate degrees in mathematics and English, as well as a doctorate in English Literature. She serves as faculty advisor for courses offered through the Thomas Edison State College Distance Independent Adult Learning Program, and has worked as a consultant to several school districts in creating math and verbal SAT review curricula.
"Synopsis"
by McGraw, |
The implicit function theorem is part of the bedrock of mathematical analysis and geometry. Finding its genesis in eighteenth century studies of real analytic functions and mechanics, the implicit and inverse function theorems have now blossomed into powerful tools in the theories of partial differential equations, differential geometry, and geometric... more...
This volume contains definitions of terms associated with arithmetic, algebra and trigonometry, with natural overlap into geometry, topology, and other related areas. There are more than 2500 definitions, from Abelian cohomology to zero ring and from the very basic to the highly advanced. more...
The "Handbook of Typography for the Mathematical Sciences" explains how to use TeX, LaTeX and AMS TeX during the typesetting process so that readers can take a more active role in ensuring that their work is properly represented in print. more...
Here's the perfect self-teaching guide to help anyone master differential equations--a common stumbling block for students looking to progress to advanced topics in both science and math. Covers First Order Equations, Second Order Equations and Higher, Properties, Solutions, Series Solutions, Fourier Series and Orthogonal Systems, Partial Differential... more...
This text explores the many transformations that the mathematical proof has undergone from its inception to its versatile, present-day use, considering the advent of high-speed computing machines. Though there are many truths to be discovered in this book, by the end it is clear that there is no formalized approach or standard method of discovery to... more...
This book treats all of the most commonly used theories of the integral. After motivating the idea of integral, we devote a full chapter to the Riemann integral and the next to the Lebesgue integral. Another chapter compares and contrasts the two theories. The concluding chapter offers brief introductions to the Henstock integral, the Daniell integral,... more...
This text provides a masterful and systematic treatment of all the basic analytic and geometric aspects of Bergman's classic theory of the kernel and its invariance properties. These include calculation, invariance properties, boundary asymptotics, and asymptotic expansion of the Bergman kernel and metric. Moreover, it presents a unique compendium... more...
Introduces geometric measure theory through the notion of currents. This book provides background for the student and discusses techniques that are applicable to complex geometry, partial differential equations, harmonic analysis, differential geometry, and many other parts of mathematics. more...
This work examines a rich tapestry of themes and concepts and provides a comprehensive treatment of an important area of mathematics, while simultaneously covering a broader area of the geometry of domains in complex space. At once authoritative and accessible, this text touches upon many important parts of modern mathematics: complex geometry, equivalent... more... |
Advantages: A lot of information, explained clearly, covers all units Disadvantages: Could have more colour
...When I was studying ALevel Maths it required me to have a lot of textbooks available for studying and most of the time to prepare for exams with. For my exams in Mathematics, I was given the textbook Edexcel AS and A Level Modular Mathematics by Keith Pledger, however if you have to purchase it, they are around as expensive as most textbooks are at £15. I was quite glad that it combined both...
...Being a student I probably use this book more frequently than teachers do and so I believe I am more able to write a 'real' review.
This 370-page book aims to cover all aspects of the A-level IT course. The book is presented well with short paragraphs and bullet point used to outline key points.
The book consists of five sections, in total 71 chapters. The aim of using so many chapters is to... |
MAZ505: AlgModel (2013-2014)
Major Concepts/Content: Algebraic Modeling will help students understand the connection between math and their daily lives. Students will explore Algebra 1 topics such as linear, quadratic, exponential and piecewise functions by modeling real world situations. Students will identify key characteristics, represent problems algebraically and graphically, determine lines/curves of best fit and make predictions. Concepts and solutions are presented in non-threatening, easy-to-understand language with numerous examples to illustrate ideas. Whether the student will go on to study early childhood education, graphic arts, automotive technologies, criminal justice or something else, the student will discover that the practical applications of mathematical modeling will continue to be useful well after they have finished this course.
Major Instructional Activities: This course is designed to help students make connections between algebra and real world applications through activities, modeling and extensive conversations. Students will be expected to explore real world data, make conjectures about different situations and communicate their thoughts in a variety of ways. Students will become proficient in a variety of technologies including: graphing calculators and software for graphing /modeling, word processing, spreadsheet processing and presentation software.
Major Evaluative Techniques: Students will demonstrate their knowledge through tests, hands-on demonstrations, technical reports, projects, case studies, and reflections.
Course Objectives: Throughout this course, students will create and use mathematical models employing algebraic modeling techniques with the following mathematical concepts |
Synopses & Reviews
Publisher Comments:
Crossing the River with Dogs: Problem Solving for College Students, 2nd edition promotes the philosophy that students learn best by working in groups and the skills required for real workplace problem solving are those skills of collaboration. The text aims to improve students writing, oral communication, and collaboration skills while teaching mathematical problem-solving strategies.
Focusing entirely on problem solving and using issues relevant to college students for examples, the authors continue their approach of explaining classic as well as non-traditional strategies through dialogs among fictitious students. This text is appropriate for a problem solving, quantitative reasoning, liberal arts mathematics, mathematics for elementary teachers, or developmental mathematics course.
Synopsis:
"Synopsis"
by Wiley, |
Mental Math for Pilots (Professional Aviation series)
Book Description: For pilots looking to improve their math skills in the cockpit and easily perform math calculations in their heads, this book offers numerous tips and invaluable tricks to help in all areas of cockpit calculations. Pilots are guided through basic and more advanced formulas with explanations on how to perform them without needing paper or electronic calculators, step-by-step instructions, practice exercises, and personal advice from experienced pilots. Easy and quick methods for calculating airborne math problems, enroute descents, and visual descent points are covered. Numerous references, math memorization tables, lists of formulas, and definitions for terms and abbreviations are provided. This book will be useful for pilots gearing up for airline interviews, preparing for checkrides or proficiency checks, or wanting to improve their in-flight calculations performance |
MAZ501: Discrete Math (2013-2014)
Major Concepts/Content: This discrete mathematics course by design shows a different view of mathematics than as seen in traditional mathematics courses. It is an applications driven course that is based upon the study of events that occur in small, or discrete, chunks. Discrete concepts are used extensively in business, industry, government, and the digital world. The major areas of study are counting and probability, graph theory, the mathematics of social choice (voting and fair division), and coding and encryption. Some of the questions investigated in discrete math are: What does a bar code mean? What is the most efficient way a delivery truck can visit ten destinations? Should you buy a lottery ticket?
Major Instructional Activities: Probability applications include predicting outcomes using combinations, permutations, and counting principles. Mathematics of social choice investigates election theory and fair division. The graph theory component is comprised of the following: (1) graphs and directed graphs- shortest paths and graph coloring; (2) various trees; and (3) circuits and networks. The role of coding and encryption in the digital world will be investigated. The concept of recursion is a strategy embedded throughout discrete studies. Concepts will be explored and simulated using a variety of technology tools.
Major Evaluative Techniques: The assessments, formal and informal, will be used to describe and identify student progression toward the discrete mathematics expectations. Students will be asked to demonstrate the depth of their knowledge through tasks which mirror realistic situations. Students will be required to develop verbal, written and technological skills in the process of solving the problem as well as use critical thinking in working towards a solution.
Course Objectives: Upon completion of the course, students will be able to use:
graphs of vertices and edges to model a problem situation.
critical path analysis to solve scheduling problems.
graph coloring techniques.
minimal spanning trees to solve problems.
codes, including error-correcting codes, and decoding techniques.
bin-packing techniques to solve problems.
tree diagrams, Venn diagrams and other pictorial representations to find the number of outcomes in a problem situation |
Introductory Algebra-Text - 8th edition
Summary: Lial/Hornsby/McGinnis's Introductory Algebra, 8e, gives students the necessary tools to succeed in developmental math courses and prepares them for future math courses and the rest of their lives. The Lial developmental team creates a pattern for success by emphasizing problem-solving skills, vocabulary comprehension, real-world applications, and strong exercise sets. In keeping with its proven track record, this revision includesan effective new design, many new exe...show morercises and applications, and increased Summary Exercises to enhance comprehension and challenge students' knowledge of the subject matter. ...show less
0321279212 Has heavy shelf wear, but still a good reading copy. Name written on inside of cover A portion of your purchase of this book will be donated to non-profit organizations. We are a tested a...show morend some shelf wear, highlighting, underlining and/or writing. Great used condition. Has some water damage. A portion of your purchase of this book will be donated to non-profit organizat...show moreions. We are a tested and15.75 +$3.99 s/h
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2005-01 |
This lecture set explains not only the key ideas of mathematics but also its impact on the history of philosophical ideas, focusing on the two areas of mathematics - probability and statics, and geometry.
This is a collection of video lectures for Math 110 - College Algebra taught by Professor Richard Delaware, which consists of 40 video lectures emphasizing the concepts of college algebra. It covers a variety of topics on algebra: numbers, graphs, equations and inequalities, polynomial and rational functions, systems of linear equations, and sequences.
The MathHistory series consists of 12 lectures on the History of Mathematics, which look at important aspects in mathematics from the ancient Greek geometry and number theory to non-Euclidean geometry.
Calculus Revisited is a series of videos and related resources that covers the materials normally found in a freshman-level introductory calculus course. The lectures taught by Prof. Herbert Gross come from MIT OCW.
History of Mathematics. Starting with Greek mathematics, Professor NJ Wildberger discusses Hindu, Chinese and Arabic influences on algebra, then the development of coordinate geometry, calculus and mechanics, and the course of geometry from projective to non-Euclidean in the 19th century. |
Algebraic Geometry Teacher Resources
Find Algebraic Geometry educational ideas and activities
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In this Algebra I/Geometry/Algebra II worksheet, students solve problems that reflect the types of assessment items found on the Regents Examination for the state of New York. The seventy-one page document contains a combination of thirty-four multiple choice and free response test items along with samples of student work, rubrics, and subject matter alignment. Answers are not included.
Students explore careers that require math knowledge and solve real life math problems. As a class, they compare and contrast classroom math to real life math and explore the links between algebra, geometry and math skills used in school and in a career. In pairs, students complete worksheets. They play a quick response game where they list five ways a given job uses math.
Students assess how algebra, telescopes, space exploration and optics are so important in astronomy. They encounter studies on the Hubble Space Telescope, Hubble Deep Field and how algebra helps to determine the effects of contamination on Hubble's optics. Students are introduced to the development of the Next Generation Space Telescope.
Help 9th graders identify lines, points, rays, and planes in geometry. They practice identifying, measuring, and drawing angles of different degrees. This is a fundamental lesson to help students learn the building blocks of geometry.
Study geometry through the concept of quilt design. High schoolers examine the geometric shapes in various quilts and then create their own quilts using geometric shapes that fit together. In the end, they write a paragraph to describe their quilt pattern.
Students explore properties of triangles. In the geometry lesson, students construct the centroid, circumcenter, and the orthocenter of a triangle. The dynamic nature of Geometer's Sketchpad allows students to discover important properties regarding Euler's line. Additionally the lesson considers a proof of the Pythagorean Theorem and investigation into Fermat's point.
Students explore the concept of iteration, recursion, and algebra to analyze a changing fish population. In this iteration, recursion, and algebra lesson plan, students explore the effects a change in a parameter has on a graph. Students use an applet to change parameters of a fish population and see the effect on the graph.
Students relate miniature golf to reflection of an image. In this algebra instructional activity, students collect and graph data as they study linear equations. They apply properties of graphing to solve real life scenarios.
Young scholars identify he proportion of a cereal box. In this algebra lesson, students define the relationship between patterns, functions and relations. They model their understanding using the cereal box.
Students investigate linear equations through models. In this algebra instructional activity, students investigate solving one and two step equations. Review solving a basic equation before starting this assignment.
Students discuss what good presentation looks like. In this geometry lesson, students discuss the 7 important steps required to be a good presenter. They start with their names and what they will be discussing and end with a thank you to all who helped with the creation of their work. They may work in groups. |
Systems and their mathematical description play an important role in all branches of science. This book offers an introduction to mathematical modeling techniques. It is intended for undergrad students in applied natural science, in particular earth and environmental science, environmental engineering, as well as ecology, environmental chemistry, chemical engineering, agronomy, and forestry. The focus is on developing the basic methods of modeling. Students will learn how to build mathematical models of their own, but also how to analyze the properties of existing models. The book neither derives mathematical formulae, nor does it describe modeling software, instead focusing on the fundamental concepts behind mathematical models. A formulary in the appendix summarizes the necessary mathematical knowledge. To support independent learners, numerous examples and problems from various scientific disciplines are provided throughout the book. Thanks in no small part to the cartoons by Nikolas Sturchler, this introduction to the colorful world of modeling is both entertaining and rich in content. less |
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...The Calculator is small and handy. It supports compound expressions,...to the
number value and indicates that the calculator must multiply this
value by the specific coefficient....the cosine of 20 degrees? (By default, The Calculator works with radians) Type cos(20g) and get the...Some frequently used values, such as physical and
mathematical constants, may be written in a special file,...string memorizing
* Hot key call.
...Developed by the Dovada research team, this student calculator is ideal for use in the school, home,...or any engineering or research areas, anywhere scientific calculator or graphic calculator is regularly used or required....that science or maths homework help.
The student calculator graphically displays many of the standard mathematical formulae...the basic structure of mathematic formulae. |
Learning trigonometry concepts can be a difficult and frustrating process. The tenth edition of this successful book helps readers gain a strong understanding of these concepts by discovering how trigonometry is relevant in their lives through rich applications. It follows a right triangle-first approach and is graphing optional. Readers will find new and updated applications as well as additional exercises and solutions. Greater emphasis is also placed on relevant applications more than other books in the field. All of this will help readers c... MOREomprehend and retain the material. |
Staffing
Other requisites
This course is equivalent to MAT1102. Students cannot enrol in MAT1502 if they have successfully completed MAT1102 or are currently enrolled in MAT1502.
Rationale
Mathematical concepts and processes provide powerful descriptive and investigative tools for engineering and surveying. In particular, the techniques of calculus, matrices, vectors and complex numbers are fundamental for modelling and analysis in a wide range of applications. This course provides opportunity for students to advance their mastery of these topics, and explore their applications to engineering and surveying.
Synopsis
It is assumed that students entering this course already have well-established algebra, function, graphing and trigonometry competencies, and have already developed introductory level skills in matrices, vectors and calculus. This course advances conceptual and technical competencies in these fields by investigating limits, continuity, inverse functions, compositions, rational functions and implicit functions. Differentiation and integration are advanced and used in engineering applications and problem-solving. Vector algebra is extended and applied to the description of lines and planes in space. Matrix algebra is extended to determinants, and used for modelling and to solve systems of linear equations in a range of settings. Euler notation is used to represent complex numbers and functions.
Objectives
On completion of this course students will be able to:
demonstrate advances in understanding of mathematical concepts that are essential for tertiary studies in engineering and surveying;
demonstrate proficiency in the skills and competencies covered in this course;
interpret and solve a range of authentic problems involving mathematical concepts relevant to this course and to engineering and surveying;
effectively communicate the mathematical concepts, reasoning and technical skills contained in this course;
use computing aids for computation, graphing, matrix manipulation, concept development and problem solving in algebra and calculus.
Topics
Description
Weighting(%)
1.
Mathematical writing and communication.
10.00
2.
The library of common types of functions of one variable and their graphs; function concepts, and their applications.
20.00
3.
Calculus: higher derivatives and their applications; definite, indefinite and improper integrals, integration techniques; applications of calculus to engineering and science.
30.00
4.
Vector algebra and cross product, and application to describing lines and planes in space.
15.00
5.
Matrix algebra, inverse and determinants; applications to solving systems of linear equations in engineering and other contexts.
15.00
6.
Complex number applications, Euler form and complex functions.
10.00
Text and materials required to be purchased or accessed
ALL textbooks and materials available to be purchased can be sourced from USQ's Online Bookshop (unless otherwise stated). (
Assessment details
USQ will make the examination dates available during the semester. Students must refer to the official USQ examination timetable when it is published.
Important assessment information
Attendance requirements:
It is students' responsibility to study all the course materials, including those posted on the course website, in order to maximise their chance of meeting the course objectives. On-campus students (Semester 1 only) should participate responsibly in all scheduled activities: lectures, tutorials, computer laboratories.
Requirements for students to complete each assessment item satisfactorily:
To complete each of the assessment items satisfactorily, students must obtain at least 50% of the marks available for each assessment item final grades for students will be assigned on the basis of the weighted aggregate of the marks obtained for each of the summative assessment items in the course.
Examination information:
An open examination is one in which candidates may have access to any printed or written material and a calculator during the examination.
Examination period when Deferred/Supplementary examinations will be held:
Any Deferred or Supplementary examinations for this course will be held during the next examination period. |
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Complex number or complex algebra is very confusing topic. JEE syllabus is however not so much, and an example from jee papers is solved here with help of simple concepts and method of complex algebra. This free IIT JEE and CBSE online preparation lecture discusses a Complex Numbers question that appeared in IIT JEE 2009 papers. Algebra is a pretty vast unit, and Complex numbers is considered one of the most important and confusing topics from the unit. The reason why it is confusing is that after studying real numbers for so long, the concept of a number with a real and an imaginary part is very new. Topics like Arguments of a complex number are important and must be prepared properly. This IIT JEE video lecture would help you understand the techniques of solving complex numbers questions to some extent. This JEE and CBSE class 11 Maths lecture has been provided by More such questions are available at , and You can connect with Exponent Education on as well. IIT JEE 2012 and AIEEE 2012 are the last engineering entrance exams hosted separately. From 2013, a common entrance exam called JEE (initially called ISEET) would enter the picture which would give combined weightage to board exam scores and performance in a PCM advanced level test and an aptitude test. For the preparation of this entrance test it is important to score well in class 12 board exams. Use NCERT books strictly for the preparation of Physics and Chemistry. For Math, start with NCERT and then later you can move on to more advanced level IIT JEE books like TMH Maths, Hall and Knight, M.L Khanna etc. Make sure that you pay attention to your class curriculum right from class 11th, so that your fundamental concepts are strong. CBSE curriculum uses NCERT books; students from other boards can refer to their course books or NCERT books only. Practice a lot of questions to strengthen your problem solving skills using model test papers, CBSE question banks, CBSE sample papers, CBSE papers of previous years, IIT JEE practice papers etc. IIT JEE question banks, CBSE solved question papers, IIT JEE question papers of previous years etc. would give you a good idea of what kind of questions to expect in the exam. You can take help of online lectures and online portals as well to study. There are various online CBSE and IIT JEE portals that can help you manage exam-related stress and keep calm, in addition to helping you prepare. Your CBSE results or ISC results are more important than ever now, therefore avoid spending unnecessarily on expensive IIT JEE coaching institutes and fancy residential IIT coaching centres like the ones in Kota, and rather pay attention to your school curriculum. You can use online coaching sites or IIT JEE video coaching tools instead to save time and energy, and to be well-prepared for your class XI and class XII papers as well as the common entrance test, i.e. JEE or ISEET. |
books.google.com - A... analysis
Complex analysis: an introduction to the theory of analytic functions of one complex variable
A for accuracy, and notations and terminology have been modernized. Chapter 2, Complex Functions, features a brief section on the change of length and area under conformal mapping, and much of Chapter 8, Global-Analytic Functions, has been rewritten in order to introduce readers to the terminology of germs and sheaves while still emphasizing that classical concepts are the backbone of the theory. Chapter 4, Complex Integration, now includes a new and simpler proof of the general form of Cauchy's theorem. There is a short section on the Riemann zeta function, showing the use of residues in a more exciting situation than in the computation of definite integrals.
From inside the book
Review: Complex Analysis: An Introduction to the Theory of Analytic Functions of Complex Variable
User ReviewRead full review
Review: Complex Analysis: An Introduction to the Theory of Analytic Functions of Complex Variable
User Review - Ronald Lett - Goodreads
A beautiful exposition of complex analysis. One warning, though: you should have a good understanding of complex algebra and calculus before reading this text, as it is dense.Read full review |
Dr. Eva Nosal
University of Calgary, Mount Royal College,
and Calgary Christian High School
Crossing into University Math
Students entering post secondary math courses for the first time
are often faced with unfamiliar settings, methodology, and
expectations. How do they adjust and cope? Are they sufficiently
prepared? Is there a gap between the philosophies of learning
at high school and at university? How does our secondary school
system differ from European and Asian systems? Who does well and
why? This talk will address such issues and give suggestions that
may help students successfully transfer from high school to
university mathematics.
Biographical Note: Eva Nosal graduated from the Pedagogical
University of Prague in 1960 with a M.Sc. in education. After
one year of teaching high school, she joined the Technical
University of Prague as an Assistant Professor of mathematics
and descriptive geometry. In 1968, Eva moved to Canada where
she obtained her Canadian teaching certificate. She further
earned a M.Sc. and Ph.D., both in mathematics, from the
University of Calgary. She has been teaching mathematics and
physics in Calgary at high school, college, and university
levels for the past twenty years. |
itioning between two worlds, students in the middle grades (4-9) are no longer elementary students but are not quite ready for the challenges of secondary school. The state departments of education are beginning to recognize that the preparation of teachers for these students must change. Teaching and Learning Middle Grades Mathematics is the ideal text for future teachers who are completing their pre-service instruction. Through readings, lessons, sample middle grades exercises, and more, future teachers learn to address the teaching and ... MORElearning of algebraic and geometric thinking at the level appropriate for middle grades students. The lessons in this text follow a popular collaborative teaching method used in middle schools called Launch, Explore, Share and Summarize that involves very little lecturing, a lot of group work, and class discussions. Teaching and Learning Middle Grades Mathematics will serve as a life long resources to students, as each lesson is filled with student pages which are worksheets that can be modified for use in actual middle grades classrooms. The text comes packaged with a CD-ROM that will be a valuable resource, containing professional readings that correlate directly to the lessons in the text. |
Linear Algebra with thorough and accessible book from one of the leading figures in the field of linear algebra provides readers with both a challenging and broad understanding of linear algebra. The author infuses key concepts with their modern practical applications to offer readers examples of how mathematics is used in the real world. Topics such as linear systems theory, matrix theory, and vector space theory are integrated with real world applications to give a clear understanding of the material and the application of the concepts to solve real world p... MOREroblems. Each chapter contains integrated worked examples and chapter tests. The book stresses the important role geometry and visualization play in understanding linear algebra.For anyone interested in the application of linear algebra theories to solve real world problems. For sophomore-level or junior/senior-level first courses in Linear Algebra; assumes a Calculus prerequisite. This thorough and accessible text from one of the leading figures in the use of technology in linear algebra gives students a challenging and broad understanding of the subject. the author infuses key concepts with their modern practical applications to offer students examples of how mathematics is used in the real world. Each chapter contains integrated worked examples and chapter tests. the book stresses the important role geometry and visualization play in understanding linear algebra. This edition will continue to be packaged with the ancillary ATLAST computer exercise guide, as well as new MATLAB and Maple guides, which are also free in the package. |
Synopses & Reviews
Publisher Comments:
A plain-English guide to the basics of trig
From sines and cosines to logarithms, conic sections, and polynomials, this friendly guide takes the torture out of trigonometry, explaining basic concepts in plain English, offering lots of easy-to-grasp example problems, and adding a dash of humor and fun. It also explains the "why" of trigonometry, using real-world examples that illustrate the value of trigonometry in a variety of careers.
Mary Jane Sterling (Peoria, IL) has taught mathematics at Bradley University in Peoria for more than 20 years. She is also the author of the highly successful Algebra For Dummies (0-7645-5325-9).
Synopsis:
Get up to speed quickly with worked-out problems
Understand the how and the why of trigonometrySynopsis:
"Synopsis"
by Ingram, |
Linear Algebra
56 Lectures
The Span of a Set of Vectors. In this video, I look at the notion of a span of a vector set. I work in R2 just to keep things simple, but the results can be generalized! I show how to justify that two vectors do in fact span all of R2.
Row Reducing a Matrix - Systems of Linear Equations - Part 1. Basic notation and procedure as well as a full example are shown. The last part of the second part got cut off, but is finished in another video!!!
An Introduction to the Dot Product. In this video, I give the formula for the dot product of two vectors, discuss the geometric meaning of the dot product, and find the dot product between some vectors.
Word Problems Involving Velocity or Other Forces (Vectors), Ex 2. In this problem we are given the bearing and velocity of a plane and the bearing and velocity of the wind; we want to find out the actual velocity of the plane after taking the wind into consideration. (a nice little problem!)
Vector Addition and Scalar Multiplication, Example 1. In this video, we look at vector addition and scalar multiplication algebraically using the component form of the vector. I do not graph the vectors in this video (but do in others).
Vector Basics - Drawing Vectors/ Vector Addition. In this video, I discuss the basic notion of a vector, and how to add vectors together graphically as well as what it means graphically to multiply a vector by a scalar.
Linear Independence and Linear Dependence, Ex 1. In this video, I explore the idea of what it means for a set of vectors to be linearly independent or dependent. I then work an example showing that a set of vectors is linearly dependent.
Homogeneous Systems of Linear Equations - Trivial and Nontrivial Solutions, Part 1. In this video, I show what a homogeneous system of linear equations is, and show what it means to have only trivial solutions. In the next video, I work out an example that has nontrivial solutions.
Homogeneous Systems of Linear Equations - Trivial and Nontrivial Solutions, Part 2. In this video, I show how to find solutions to a homogeneous system of linear equations that has nontrivial solutions.
Basis for a Set of Vectors. In this video, I give the definition for a apos; basis apos; of a set of vectors. I think proceed to work an example that shows three vectors that I picked form a basis for R_3.
Linear Transformations , Example 1, Part 1 of 2. In this video, I introduce the idea of a linear transformation of vectors from one space to another. I then proceed to show an example of whether or not a particular transformation is linear or not.
Linear Transformations , Example 1, Part 2 of 2. In this video, I introduce the idea of a linear transformation of vectors from one space to another. I then proceed to finish an example of whether or not a particular transformation is linear or not.
Is our political ideology simply the result of a genetic coin toss? Mounting evidence suggests that biology may be a factor. In this video, Academic Earth explores some of the key research into the biology of politics. |
Optimization---Theory and Practice offers a modern and well-balanced presentation of various optimization techniques and their applications. The book's clear structure, sound theoretical basics complemented by insightful illustrations and instructive examples, makes it an ideal introductory textbook and provides the reader with a comprehensive foundation in one of the most fascinating and useful branches of mathematics. |
Algebra: A Complete Course
An alternative approach to Pre-Algebra, Algebra 1, or Algebra 2!
The reason that we named our program "Algebra: A Complete Course," is that we believe the best way to learn Algebra is to start at the beginning and end at the end! In this program you will find a complete study of the essential material covered in a traditional Algebra 1 and Algebra 2 course.
However, we need to continue a little further with this answer because Algebra 1 and Algebra 2 are terms that refer mostly to the traditional way that Algebra has been taught. Traditional Algebra 1 classes attempt to cover most of Algebra in the first year, but the methods that are used, and the speed with which the material is covered, hinders student understanding of the material. Instead, the student is just exposed to memorizing rules, formulas, tricks, and shortcuts. By the time they get to what is called an Algebra 2 course, (sometimes after they take a Geometry course), they have forgotten almost all of the Algebra that they memorized. So, that Algebra 2 course (which is by definition, a rehash of whatever has been called "Algebra 1"), must repeat practically all of the Algebra 1 course. In fact, it usually repeats a lot of the Pre-Algebra material as well. This is usually referred to as the "spiral method" of learning, and it is not very effective in helping students to excel, especially at this level of mathematics.
We think that this huge overlap is generally unproductive, and largely unnecessary if the concepts are taught analytically. Therefore we call our program "Algebra: A Complete Course," because we employ a mastery-learning approach, sometimes moving at a slower pace, but without the overlap. As a result, students often complete the course even more quickly.
PROGRAM DESCRIPTION:
There are 176 video lessons contained in 10 unit directories. The program covers Pre-Algebra, Algebra I and Algebra II, and is a firm foundation for students advancing to VideoText's Geometry: A Complete Course, covering Geometry and Trigonometry.
Materials in the complete course include: 176 Video Lessons – Each of the 5-10 minute lessons explore Algebra concepts in a detailed logical order. Because no shortcuts or tricks are used, the methods are easy to follow and promote clear understanding.
360 pages of Course Notes – These notes allow students to review the logical development of a concept. Each page chronologically follows the video lesson, repeating exactly what was shown on the screen.
590 pages of Student WorkText – These pages review the concept developed in each lesson. More examples are given and exercises are provided for students. The explanations are virtually free of complicated language, making it easy for students to follow the logic of each concept.
Solutions Manuals – These manuals provide detailed, step-by-step solutions for every problem in the student WorkText. This resource is a powerful tool when used by students to complete an error-analysis of their work, and to check their thought processes.
Progress Tests – These tests, with the answer keys included, are designed to have students demonstrate understanding, lesson-by-lesson, and unit-by-unit. There are two versions of each test, allowing for retesting or review, to make sure students have mastered concepts.
CLAIMED CREDITS:
When a student completes Algebra: A Complete Course, the student can claim credits for all of the following:
Pre-Algebra
Algebra I
Algebra II
With the completion of Geometry: A Complete Course, the student can also claim credits for:
Geometry
Trigonometry
Pre-Calculus
ALGEBRA PROGRAM OVERVIEW:
The VideoText Algebra program addresses two of the most important aspects of mathematics instruction. First, the inquiry-based video format contributes to the engaging of students more personally in the concept development process. Through the frequent use of the pause button, you, as the instructor, can virtually require interaction and dialogue on the part of your student. As well, students who work on their own, can "simulate" having an instructor present by pausing the tape every time a question is asked and trying to answer it correctly before continuing. Second, each incremental concept is explored in detail, using no shortcuts, tricks, rules, or formulas, and no step in the process in ignored. As such, the logic and the continuity of the development assures students that they understand completely. Subsequently, learning is more efficient, and all of the required concepts (topics) of the subject can be covered with mastery. Of course, the benefits of these efforts can be seen even more clearly in a description of a typical session as follows:
After a brief 2 or 3 sentence introduction of the concept to be considered, usually by examining the description and the objective given at the beginning of the video lesson, you and your student can begin. You should pause the DVD frequently, usually every 15-20 seconds (or more often if appropriate), to engage your student in discussion. This means that for a 5- 10 minute VideoText lesson, it may take 15-20 minutes to finish developing the concept. Dialogue is a cornerstone. In addition, during this time, your student should probably not be allowed to take notes. He or she must not have their attention divided, or they risk missing important links. Neither should you be dividing your attention by looking at notes or writing on a note pad or an overhead projector. Everyone is concentrating on concept development and understanding. Please understand that a student who is accustomed to working alone, or can be motivated to study independently, has, with the VideoText, a powerful resource to explore and master mathematical concepts by simulating the dialogue normally encountered with a "live" instructor. And, because of the extensive detail of the explanations along with the computer generated graphics and animation, students are never shortchanged when it comes to the insight necessary to fully comprehend.
Once the concept is developed and the VideoText lesson is completed, you can then employ the course notes to review, reinforce, or to check on your student's comprehension. These course notes are exact replications of the content that was viewed in the VideoText lesson, illustrating the same terms, problems, numbers, and. logical sequences. In fact, at this time, if your student needs a little more help, he or she can use these course notes while viewing the lesson again, using them as a guide through the reexamination of the concept. .The key here is that students concentrate on understanding first, and take care of documentation later.
Please understand that it is not the intent of the program to let the VideoText lesson take the place of instruction. Actually, the video should never tell your student anything that hasn't been considered or discussed (while the DVD is paused), and it should never answer questions that have not already been resolved. As such, it becomes a "new breed of overhead projector whereby you, as the teacher, or your student working alone, can "write" on this overhead simply by pressing the "play" button. This is a critical point to be understood and should serve to help you examine all of the materials and strategies from the proper perspective.
Finally, your student can begin to do some work independently, either by your introduction of additional examples from the WorkText, or by the student immediately going to the WorkText on his or her own. The primary feature of the WorkText, beside providing problem banks with which students can work on mastery, is that objectives are restated, important terms are reviewed, and additional example are considered, in noticeable detail, taking students, once again, through the logic of the concept development process. The premise here is simple. When students work with an instructor, whether doing exercises on their own or working through them with other students, they are concentrating more on "how to do" the problems. Then, when they leave the instructor, they simply don't take the discussion of the concept with them. The goal of the program is to provide a resource which will help students "relive" the concept development on their own, whether for review or for additional help. That is the focus of the Student WorkText.
In addition, there are detailed Solutions Manuals, which afford the opportunity to check work and engage in error analysis. Of course, there are also quizzes, unit tests, cumulative reviews, and final examinations to help you to further assess your student's progress. In fact, the assessment package often utilizes open-response questions which require the learners to state, in writing, their understanding of the concept. This often reveals much more about a student's understanding of a concept than just checking to see if an answer on a test is correct.
As you can see, the highly interactive quality of this program, at a personal level, affords students a much greater opportunity than usual to grow mathematically and develop confidence in their ability. As well, they can review the video lessons as often as they wish to further ground that understanding.
ALGEBRA SCOPE AND SEQUENCE RATIONAL:
There are two basic premises which drive concept development in Algebra, and these two essentials shape the logical scope and sequence of algebraic content. First, it is generally understood that the study of Algebra is the study of relations. In the same way that Geometry focuses on spatial concepts, and Calculus is concerned with rates of change, Algebra is a comprehensive exploration of mathematical relationships, including both equations and inequalities. As such, no treatment of Algebra should ever separate equations from inequalities, especially when it utilizes a format which addresses them in different "chapters." In fact, a true adherence to the National Council of Teachers of mathematics (NCTM) standards, requires us to deal specifically with functions, and we know that the set of functions is a subset of the set of relations, without regard initially to the differences between equations and inequalities. Therefore, in this course, equations and inequalities are studied together, and distinctions are made only when necessary, to clarify functional differences. As an aside, documentation exists to show that students generally have little or no trouble working with all types of relations at the same time and, in fact, understand the logic of studying them together.
The second premise is that the concepts of Algebra develop by degrees. This means, of course, that relations of first-degree should be mastered first. In fact, as instructors, we all understand that relations of any degree other than one must be "reduced" to relations of first-degree, or "factored" into linear or first-degree factors, before they can be resolved. The impact of this understanding on the scope and sequence of Algebra content, is to organize the various types of relations, by degree. In this course, first-degree relations are examined exhaustively before higher-order relations are encountered. Unit II deals with first-degree relations with one variable. Unit III then addresses first-degree relations with two variables. Unit IV considers first-degree relations with three or more variables. The idea here is to help students master first-degree relations, before moving on to relations of other degrees (or orders). This is not only more mathematically correct than the traditional treatment, but it allows students to reinforce more efficiently, one-variable concepts by immediately moving to two-variable concepts, and then to concepts involving three or more variables. And we all know that a system of relations with three variables is resolved using the same approach as a system with only two variables.
Moving on to Unit V, students quickly review exponent notation, including the various properties of powers and operations with powers, and investigate relations with integral degrees of 2 or higher. Unit VI continues this exploration with a focus on algebraic fractions, in which negative, integral exponents make a prominent appearance. In Unit VII, fractional exponents are introduced, which obviously pave the way for a study of radicals and roots. This, of course, is the seed from which rational-degree relations develop, or, as they are more commonly called, relations with radicals in them.
Then, after a review of second-degree relations with one variable (Unit VIII – The Quadratic Relations) and two variables (Unit IX – The Conic Sections), the study of Algebra is completed by examining the only type of exponent not yet investigated – the variable, or placeholder. This is the start of a study of literal-degree relations, and is the basis for the development of exponential and logarithmic functions. It is only after considering all possible degrees, that we can say we have studied a complete course in Algebra. In that context, it is quite artificial to define, for everyone, what "Algebra 1" is, or "Algebra 2", or even "Pre-Algebra".
The logical scope of Algebra covers relations of all degrees, including numeric and literal, while the sequence of concepts begins with a mastery of first-degree relations and grows systematically to include increasingly more sophisticated degrees. One more organizational quality is noteworthy here. The normal flow of each unit is based on the logical introduction of any new mathematical symbolism. First, the new "thing" is defined and described in detail. Then, operations involving the new "thing" are explored. Finally, relations involving this new "thing" are examined, and strategies are developed to resolve them. This cycle is introduced and explained in Unit I, and is evidenced in each successive unit. For example, in Unit V, polynomials are introduced. This is new mathematical symbolism for the student, and it must be defined carefully. Then, operations with polynomials must be examined. All of this culminates, of course, in learning to solve relations with polynomials. This logical cycle of exploration in mathematics is helpful to students, providing them with some anticipation of the levels of exploration necessary to develop algebraic concepts. Please understand that the organizational argument presented here is not meant to stifle the creativity of the instructor. Neither should it prohibit the instructor from utilizing a modular approach to concept development. It does, however, serve to remedy the fragmented, isolated topic, "chapter" approach, to a subject which has been traditionally presented to us in "textbooks", without that element of developmental continuity. To that end, it speaks loudly to the curricular issues which all instructors face, and the attitudinal issues students deal with when they are presented with the fact that "everyone must pass Algebra". |
Math Department - Profiles
Faculty Profiles
Ft. Steilacoom
- For all of my classes, students are most successful if they do some homework for the class every day, before the next class meeting. The next day's class will make sense when you've already practiced the previous topics.
- The homework is not due the next day – there are several days to do the work even though it is best to do it soon.
- In my courses in Statistics, Calculus, and Precalculus, the required homework is primarily or totally online, in WAMAP. There may also be suggested problems from the book, but they are not generally required.
- In my Math 60, 98, 171, 172, and 173 courses, the homework is primarily from the textbook. Typically all of the homework for a chapter is saved and turned in at the end of the chapter. In addition, for Math 60 and 98 there are some problems to do online in WAMAP.
Attendance
There is no grade for attendance. Students must attend class regularly to succeed. If a student is absent on the day of a quiz, usually no make-up is possible.
Group Work & Projects
- During class there is often informal work done in groups (not turned in).
- Occasionally there may be work done with a partner and turned in for a grade.
- In some courses, there is a project to do outside of class in a group. Definitely in Math& 146 and Math& 171 and 172. Sometimes in other courses. These types of projects are done over several weeks, while we proceed with regular course topics in class.
Math 98 Homework is due weekly. It is mostly online homework using WAMAP, but some homework done on paper will also be required.
Math 107 Homework is primarily done on paper (problems from the book). It is due on the day of the exam for that chapter.
Math 146 Homework is primarily done on paper (problems from the book). It is due on the day of the exam for that chapter.
Late homework is not accepted since I view homework as the student's effort at learning the material prior to taking the test.
Attendance
I do not take attendance every day, however when I have in-class activities, attendance is part of the grade for that activity. It usually constitutes about 20-25% of the activity grade.
Group Work & Projects
How often do you do group work and projects in your courses or require students to work in groups?
Math 98 – Weekly
Math 107 – There is one major group project for the quarter
Math 146 – There is one major group project for the quarter
Teaching Philosophy / Approach
I'd call my approach the Google-map teaching approach. If you'd ever used Google Maps to get directions for driving somewhere new, then you've probably seen the options for printing out only the directions or printing out the map and the directions. My choice is to print out the maps and the directions because with the map I have the big picture. If I get lost on the detail (e.g. I miss a turn), then by having the map, I have a sense of how to correct my mistake. Without the map, I would be lost.
The same goes for learning. Learning a bunch of skills in sequence without understanding how these skills fit into the big picture ultimately leaves a student confused about when to use specific skills. I think it is the reason students forget math so quickly. With the big mathematical picture, a student has a better sense of their objective and can find the skill needed to achieve that objective. Consequently, I try to regularly remind student about the big picture and how the skill being taught fits into it.
Beyond that approach, I assume students are interested in learning so they can improve the quality of their lives. I won't nag them about assignments so they can assume responsibility for themselves. I also don't give extra credit since I prefer the student put their effort into being successful the first time rather than as a way of making up for insufficient effort.
Other Comments
I can't learn for you. In 50 minutes a day, I can't demonstrate every possible problem you may ever encounter. For you to learn requires plenty of "struggle time", multiple sources of information, time for reflection, a search for patterns, and plenty of practice.
I typically assign homework daily. In most classes I assign online homework using WAMAP.org, which is typically due 3 days after assigned. The homework is graded, but students can retry new versions of problems they miss to earn full credit. Occasionally I will assign extra problems from the book or as worksheets for topics that the online homework is not good for.
Attendance
Students are responsible for announcements made and material taught during class, but I do not take roll or give a grade for attendance. There are regular unannounced quizzes given, but the lowest few are dropped, so it is not a big deal if a student misses a day or two.
Group Work & Projects
I encourage students to work on classwork in groups regularly. I typically only give one or two assignments each quarter than require students to work in groups outside of class. In most cases, I am willing to allow students to work individually if they choose.
Teaching Philosophy / Approach
I believe that learning only comes through doing, so while my class does include lecture, whenever possible I try to give students a chance in-class to practice the material. I provide time everyday for homework questions. To address a variety of learning styles I try to explain concepts in multiple ways, including using graphical and numerical approaches in addition to algebraic ones. I use technology when appropriate to simplify calculations or illuminate a concept. As much as possible, I try to provide examples of how the material being learned is applicable either in general life, or in later classes the student might take.
In pre-college classes, I tend to focus on skill development. In college-level classes, my focus tends to be on conceptual understanding, moving beyond calculation and manipulation skills. The further a student progresses through our math sequence, the more essential it becomes that they be able to adapt their knowledge to new situations, and not simply replicate examples.
In addition to homework, I try to provide a number of graded quizzes prior to exams to give students an opportunity to practice in a testing environment before exams. My exams are typically 70% demonstration of skills. The remaining 30% is typically application of those skills to new situations.
I assign homework in all pre-college math classes and Math 156 (Finite Math) and Math 148 (Elements of Calculus) – this homework is due at the beginning of class on the day the class takes the exam. Late homework is not accepted. In Math 146, homework is assigned and some problems/practice tests are handed out in class, but are not turned in for credit.
Attendance
I do not have an attendance policy.
Group Work & Projects
The only course where I require a project is Math 146 (Statistics). Students can work in groups or alone for the statistics project. In all my other classes, I encourage students to work in groups, but do not require a project or group work.
Teaching Philosophy / Approach
At the beginning of every class session, I answer questions on material we have already covered. After that, I cover new material, mostly on the white boards. Students have some input regarding when the test will be once we know when we will finish a chapter/unit.
Other Comments
I use numerous handouts and formula sheets, depending on the class I am teaching. |
5.0 out of 5 starsEssential reference for any 3D graphics work.30 Jun 2004
By Francis J. Kane - Published on Amazon.com
Format:Hardcover
Finally, no more searching through all my college math textbooks for the reference I need for real-time 3D software development. The basics of vectors and matrices are of course included, but in much more depth than you got in school, more than likely - and with emphasis on how they are useful in 3D game programming. So many game developers lack an intuitive feel for such basics as transformation matrices, dot products, and cross products and are hobbled by this; just read up to chapter three and the lights will go on, so to speak. The chapter on lighting is particularly, well, enlightening - not only are the various lighting models explained in detail (including some I was unfamiliar with before), but the author provides means for accomplishing them in real-time using texture and vertex shaders.
The notation used in the book is modern and consistent, and the code samples clearly written. I believe this is the first volume to combine complete mathematical explanations of essential 3D computer graphics operations with practical advice on how to implement the sometimes complex math efficiently in real-time systems.
The chapters on picking and collision detection are also complete and include practical advice on implementation in addition to the theory behind it.
This is not a book for most high school math students - the author assumes you've at least been through some higher level math and can talk the basic language of mathematics. However, it does not presuppose that you are familiar with anything but basic calculus, and more importantly, it doesn't assume that you're familiar with some quirky notational system specific to the author. I haven't been in a math class for ten years, but I had no trouble understanding any concepts introduced in this book upon the first read.
I don't forsee this volume leaving my desk anytime soon!
39 of 43 people found the following review helpful
3.0 out of 5 starsMath majors rejoice1 Mar 2007
By GameMaker - Published on Amazon.com
Format:Hardcover|Amazon Verified Purchase
To be honest, while I find this book to be a decent reference, I find it to be pretty inaccessible in terms of sitting down and reading through it in an attempt to learn the concepts. As a non-math major (I'm actually an engineer and software developer) these math concepts are by no means beyond me. But rather than simply being presented with equation after equation, proof after proof, what I find a lot more valuable is more discussion on the usage of these equations. Specifically I'd like to see examples, diagrams, and code, and there is precious little of any of that in this book.
In other words, this book is very much like what you expect to find in a very dry upper devision college math text for the consumption of math majors who are used to such things. But for a non math major just trying to make use of these concepts in order to get the job done and make games? eh, not so much.
Still, I do think this book is useful as a reference when I want to look up an equation as there are a ton of them crammed into this book, but for me, I just don't find this book to be very good as a learning tool.
16 of 17 people found the following review helpful
5.0 out of 5 starsThis book is fantastic3 Aug 2004
By Waylon - Published on Amazon.com
Format:Hardcover
This book is great. Its material is well explained, the topics covered are complete (for the most part), and the examples make sense. It is a fantastic reference that should be on the shelf of any professional game programmer or aspiring game programmer. However, this book isn't a hand holding guide to making "cool" games, as some reviewers expected it to be. There is no single book for that. There are so many topics to cover, it would be impossible to put them all into one text. Please don't be fooled by reviews from non-professionals, as this book is a must have. For a list of beginner books to give yourself an introduction to game programming, feel free to send me an email. |
TRIGONOMETRY CHALLENGE is designed to supplement your classroom and textbook instruction. The topics included in the program are: The Pythagorean Theorem, Degree and Radian Conversion, Using the Sine Function, Using the Cosine Function, Using the Tangent Function, The Inverse Functions, The Law of Sines, The Law of Cosines. The activities are constructed from ramdom variables. Student work is graded instantly. Grades can be printed on "certificates" or stored on diskettes. The program is simple to install and use. An unlimited use site license is $159.00. Licensed sites can optionally allow students and teachers to use a copy of the program on their home computers.AC Circuits Challenge - This interactive computer program consists of several circuit analysis activities.This interactive computer program consists of several circuit analysis activities. Realistic troubleshooting activities are also included involving resistors,...
Basic Circuits Challenge - Basic Circuits Challenge consists of a set of fifteen activities to help you teach basic electrical concepts.Basic Circuits Challenge consists of a set of fifteen activities to help you teach basic electrical concepts. Exercise Titles: Electric...
DC Circuits Challenge - This interactive computer program consists of several circuit analysis activities.This interactive computer program consists of several circuit analysis activities. Realistic troubleshooting activities are also included involving resistors,...
Ohmmeter Challenge - Ohmmeter Challenge is designed to help you teach students to analyze wiring and troubleshoot circuits using digital ohmmeters.Ohmmeter Challenge is designed to help you teach students to analyze wiring and troubleshoot circuits using digital... |
Math 450 - introduction to modelling
This will be an awesome but tough class where
we learn to model problems and systems using
mathematics and computers. We'll be using the
python computer language (which I will teach
everybody at the start of the course). We'll
cover statistical models, automata models,
and classical applied-math models.
We'll also discuss the nature of modelling,
based on readings from Nate Silvers Signal and the Noise.
At the end of the course, you'll have a the start of
a very valuable skill set. |
Math Essentials
Includes operations and problem solving with proper fractions, improper fractions, and mixed numbers without the use of a calculator. Emphasizes applications and includes U. S. customary units of measure. Credit is not applicable toward graduation. Lecture 1 hour per week.
Includes basic operations with algebraic expressions and solving simple algebraic equations using signed numbers with emphasis on applications. Credit is not applicable toward graduation. Lecture 1 hour per week.
Includes solving first degree equations and inequalities containing one variable, and using them to solve application problems. Emphasizes applications and problem solving. Credit is not applicable toward graduation. Lecture 1 hour per week.
Includes finding the equation of a line, graphing linear equations and inequalities in two variables and solving systems of two linear equations. Emphasizes writing and graphing equations using the slope of the line and points on the line, and applications. Credit is not applicable toward graduation. Lecture 1 hour per week.
MTE 6 - Exponents, Factoring and Polynomial Equations (1 credit)
Prerequisite(s): MTE 5 or qualifying placement score.
The student will learn to perform operations on exponential expressions and polynomials. Students will also learn techniques to factor polynomials and use these techniques to solve polynomial equations. Emphasis should be on learning all the different factoring methods, and solving application problems using polynomial equations. Credit is not applicable toward graduation. Lecture 1 hour per week.
Includes an introduction to functions in ordered pair, graph, and equation form. Also introduces quadratic functions, their properties and their graphs. Credit is not applicable toward graduation. Lecture 1 hour per week. |
Precise Calculator has arbitrary precision and can calculate with complex numbers, fractions, vectors and matrices. Has more than 150 mathematical functions and statistical functions and is programmable (if, goto, print, return, for). |
Driving Directions
Plain text directions to get you to the Department of Mathematics and Computer Science offices are as follows:
From the Pennsylvania Turnpike (Interstate 76)
1. Leave the turnpike at Exit 226, the "Carlisle" exit; stay in the right lane to follow US Route 11 south.
2. Follow US Route 11 approximately 2.5 miles into Carlisle where it becomes North Hanover Street.
3. Continue south on Hanover to its intersection with Louther Street.
4. Turn right (west) onto Louther Street and follow it three and a half blocks.
5. The Math/CS department is in Tome Hall in the Rector Science Complex on the right side of Louther Street.
6. The Math/CS department is on the second floor of Tome hall.
From Interstate 81:
1. Take Exit 47 if on I-81N and make a left at the end of the ramp OR take Exit 47B if on I-81S and make a right at the end of the ramp. You are now on Hanover Street (PA Route 34) heading into Carlisle.
2. Follow Hanover Street approximately 1.5 miles to the sixth traffic light at Louther Street.
3. Turn left (west) onto Louther Street and follow it three and a half blocks.
4. The Math/CS department is in Tome Hall in the Rector Science Complex on the right side of Louther Street.
5. The Math/CS department is on the second floor of Tome hall. |
It's really complicated for me to understand this alone so I think I need someone to give an advice. I require help on the subject of answers for math books. It's giving me problems every time I attempt to understand it because I can't seem to figure out how to solve it. I read some books about it but it's really confusing. Can I ask assistance from anyone of you guys here? I need someone who can explain how to solve some problems concerning answers for math books.
I really don't know why God made algebra, but you will be happy to know that someone also came up with Algebrator! Yes, Algebrator is a program that can help you solve math problems which you never thought you would be able to. Not only does it provide a solution the problem, but it also gives a detailed description of how it got to that solution. All the Best!
I myself have been using this program since a year now, and it has never let me down. It won't just solve a problem for you, but it'll also explain every step that was taken to arrive at a particular solution. And that's the best feature in my opinion.I used to face a lot of problems tackling questions based on answers for math books but ever since I started using software, math has beenreally easy for me.
Algebrator is a user friendly product and is definitely worth a try. You will find lot of exciting stuff there. I use it as reference software for my math problems and can swear that it has made learning math much more fun. |
Algebra Connections
Description
For one-semester undergraduate courses in algebra for the middle grades.
Strong mathematics performance in the middle grades is more important than ever–and teachers entering the field need to prepare for this endeavor in new and innovative ways. This new approach introduces some basic concepts of number theory and modern algebra that underlie middle grade arithmetic and algebra, with a focus on collaborative learning combined with extensive in-class and out-of-class assignments. The primary goal is to help future teachers (both in-service and pre-service) gain a fundamental understanding of the key mathematical ideas that they will be teaching so that, in turn, they can help their students learn important mathematics.
This text presents is designed to equip future middle grade mathematics teachers with the skills needed for teaching NCTM (National Council of Teachers of Mathematics) Standards-based curricula. Throughout the text, the reader will find a number of Classroom Connections, Classroom Discussions, and Classroom Problems. These instructional components are designed to deepen the connections between the college-level abstract algebra and number theory the students are studying now and the algebra they will teach.
Other titles in the Prentice Hall Connections in Mathematics Courses for Teachers include:
Geometry Connections: Mathematics for Middle School Teachers
Algebra Connections: Mathematics for Middle School Teachers
Data and Probability Connections: Mathematics for Middle School Teachers
Calculus Connections: Mathematics for Middle School Teachers
Features
Classroom Connections – Directly connect college-level abstract algebra and number theory to standards-based middle grade mathematics curricula. – Students benefit by learning about the mathematics that underlies algebraic concepts in the middle school curriculum. – Specific examples from middle-grade curricular materials show students the direct connections between the mathematics they are learning and the mathematics they will be teaching.
Focus on the mathematics in new reform materials, including Connected Mathematics, Mathematics in Context, MathThematics, MathScape. – Students are shown how to effectively utilize these newly developed, innovative materials. – Provides many in-depth examples.
Classroom Problems and Classroom Discussions – Focus on discovery and collaborative learning. – Teaches students to make sense of mathematics and learn many habits of discovery. – Encourages group work, prompting students to share ideas and understand the ideas of others. – Encourages writing solutions with complete justifications. |
Professional Commentary: This task challenges a student to use knowledge of functions to match verbal descriptions of a context with equations and graphs. A student must be able to construct an argument based on properties of graphs and equations to justify matches....
Professional Commentary: The task challenges a student to demonstrate understanding of the concepts representing and analyzing mathematical situations and structures using algebra. A student must be able to: use symbolic algebra to represent and explain mathematical relationships in a cost-analysis problem judge the meaning, utility, and reasonableness of results of symbolic manipulations in the given real-world...
Professional Commentary: This online historical map collection includes more than 34,000 maps and images, including rare examples of 18th and 19th century North and South American. There are also maps of the World, Europe, Asia and Africa....
Professional Commentary: This task challenges a student to represent a context by constructing two equations from a table. A student must be able to solve two equations with two unknowns and interpret the results in terms of the context of the situation including identifying the correct monetary units....
Professional Commentary: This task challenges a student to use knowledge of graphs and their significant features to identify the linear equations for various lines. A student must be able to quantify the solution for intersecting lines and be able to write a linear equation going through a given point....
Professional Commentary: The task challenges a student to demonstrate an understanding of the graphical properties of a quadratic function given by its formula. A student must: make sense of and be able to identify and perform transformations on functions understand properties of functions identify, formulate, and confirm conjectures The task was developed by the Mathematics...
Professional Commentary: The task challenges a student to demonstrate understanding of the concepts representing and analyzing mathematical situations and structures with algebraic symbols. A student must: understand the meaning of equivalent forms of expressions, equations, inequalities, or relations determine equivalent forms of equations, inequalities, and systems of equations and solve them employ forms of mathematical...
Professional Commentary: The task challenges a student to demonstrate understanding of the concepts of relations and functions. A student must: analyze mathematical situations, including growing geometric patterns, and use models to solve problems involving quantity and change represent, analyze, and generalize a variety of functions including linear relationships from growing geometric patterns determine and express mathematical...
Professional Commentary: The task challenges a student to demonstrate understanding of the concepts of algebraic properties and representations. A student must: represent and analyze mathematical situations and structures using algebraic symbols make sense of areas and perimeters of parallelograms and trapezoids using algebraic expressions understand how to use symbolic algebra to represent and explain mathematical relationships...
Professional Commentary: The task challenges a student to demonstrate understanding of the concepts of relations and functions. A student must: understand functions and select, convert flexibly among, and use various representations for them identify linear points on a coordinate grid and name them determine the equation for a linear function from a graph or from coordinates... |
Introductory Algebra for College Students
9780132356794
ISBN:
0132356791
Edition: 5 Pub Date: 2008 Publisher: Prentice Hall
Summary: The Blitzer Algebra Series combines mathematical accuracy with an engaging, friendly, and often fun presentation for maximum student appeal. Blitzer's personality shows in his writing, as he draws students into the material through relevant and thought-provoking applications.
Blitzer, Robert F. is the author of Introductory Algebra for College Students, published 2008 under ISBN 9780132356794 and 0132356791.... Five hundred twenty three Introductory Algebra for College Students textbooks are available for sale on ValoreBooks.com, two hundred eleven used from the cheapest price of $0.01, or buy new starting at $485th edition. Book is in overall good condition. Some water damage causing pages to stain and lightly ripple. Does not affect the text. Cover shows some edge wear and corners a [more]
5 5th HC teacher ed,just like the student version cover to cover,will have answers and or marginal notes,promo labels are covered with black book tape,inc [more]
ALTERNATE EDITION: A brand new never used 5th HC teacher ed,just like the student version cover to cover,will have answers and or marginal notes,promo labels are covered with black book tape,includes a testprep video,smoke free env[less] |
Algebra Workbook
Printed
eBook
Help students in grades 6–9 master the skills necessary to succeed in algebra using Algebra. This 128-page book allows for differentiated instruction so that each student can learn at his or her own pace. It is perfect for extra practice at home or school and includes more than 100 pages of exciting activities! The activities cover skills such as operations with real numbers, variables and equations, factoring, rational expressions, ratios and proportions, graphing, and radicals. The book includes 96 durable flash cards and an award certificate |
Kip Maths Expert Yrs 11-12 Only
Retail Product
Purchase this item today in our secure online shop
Kip Maths Expert (KipME) @ Home (11-12) maths development CD program is designed to extend students' knowledge and skills in Maths. This program provides maths content for Years11 - 12 with specially designed work programs for each state in Australia.
The CD program features:
Over 20,000 maths questions and answers
Self directed learning
Guided, program paced learning
Hint, Tips and topic introductions for Maths topics at all levels
Fully work solutions for senior maths subjects
A high function graphics calculator
Quick checks on your own progress levels
Customised curriculums for each state and grade in Australia
Search function helps you quickly locate topics of interest
Self-motivated students who really want to extend themselves will love this program.
Parents who want to help their child focus on particular areas of maths difficulty will find an answer in this program.
Students who are looking for that extra edge in senior secondary maths will find a comprehensive help base in this program
Purchase price includes 12 months of product updates and customer support - visit our Customer Service Page for more information.
Each high school age student in your family can setup with their own login and individualised maths program based on School year and stream.
This item provides content access to curriculum for Years 11 - 12 only. If you wish to purchase content for Years 7 - 10 only, or for all secondary years 7 - 12 please select the appropriate product by clicking on the links.
Click on the Add to Cart button below to purchase KipME@Home (11-12) and proceed to the Shopping Cart for checkout. |
Product Description
Stuck on a problem, and not sure what you're doing wrong? Turn to Bagatrix's Linear Algebra Solved! CD-ROM to find the answer, as well as what you're doing wrong. Enter in your own homework problem or choose from a list of unlimited example problems, and get step-by-step explanations. This program creates math-formatted documents, detailed graphs, interactive tests, and even helps you track your progress |
ISBN: 0073385832
Description
A Concise Introduction to Matlab is a simple, concise book designed to cover all the major capabilities of MATLAB that are useful for beginning students. Thorough coverage of Function handles, Anonymous functions, and Subfunctions. In addition, key applications including plotting, programming, statistics and model building are also all covered.
MATLAB is presently a globally available standard computational tool for engineers and scientists. The terminology, syntax, and the use of the programming language are well defined and the organization of the material makes it easy to locate information and navigate through the textbookA Concise Introduction to Matlab |
Synopses & Reviews
Publisher Comments:
Grasp the principles and concepts you need to score high in pre-calculus
Getting ready for calculus but feel confused? Have no fear! This un-intimidating guide walks you through all the essential topics, from absolute value and quadratic equations to logarithms and exponential functions to trig identities and matrix operations. You'll understand the concepts — not just the number crunching — and see how to perform all the tasks you need to score high at exam time.
Pre-calculus 101 — get a review of the Algebra II you need to know, real numbers, and how to graph, solve, and perform operations with functions
What's your angle? — take a tour of the essentials of trigonometry, from angles, right triangles, and trig ratios to graphing the parent graph of the six basic trig functions
Keep it simple — discover how to simplify trig expressions and solve for an unknown variable using formulas and identities (and solve triangles that aren't right triangles using Law of Sines and Law of Cosines)
Not just plane thinking — delve into analytic geometry and system solving with the understanding of complex numbers, polar-coordinate graphing, conics, systems of equations, sequences, and more
Open the book and find:
An overview of pre-calculus
Tips on graphing trig functions like a pro
How to apply the major theorems and formulas
The lowdown on analytic geometry
Guidance on identifying limits and continuity
The 4-1-1 on working with a graphing calculator
The laws for solving oblique triangles
Everything you need to prepare for calculus
Learn to:
Grasp essential pre-calculus topics
Apply the major theorems and formulas
Tackle proofs with confidence and ease
Synopsis:
The fun and easy way to learn pre-calculusAbout the Author
Yang Kuang, PhD, is a professor of mathematics at Arizona State University. He currently serves on the calculus committee where he and other members discuss what and how to teach calculus to students majoring in math and physical sciences. Elleyne Kase is a professional writer. |
Algebra for College Students
9780136129080
ISBN:
0136129080
Edition: 3 Pub Date: 2007 Publisher: Prentice Hall
Summary: "Algebra for College Students "is typically used in a very comprehensive 1-semester Intermediate Algebra course serving as a This Alge...bra for College students text may also be used in a 1-semester, lower-level College Algebra course as a prerequisite to Precalculus.
Angel, Allen R. is the author of Algebra for College Students, published 2007 under ISBN 9780136129080 and 0136129080. Four hundred sixty nine Algebra for College Students textbooks are available for sale on ValoreBooks.com, one hundred four used from the cheapest price of $133.30, or buy new starting at $149Algebra for College Students is typically used in a very comprehensive 1-semester Intermediate Algebra course serving as a gateway course to other college-level mathematics c [more]
Algebra for College Students is typically used in a very comprehensive 1-semester Intermediate Algebra course serving as a gateway course to other college-level mathematics courses. The goal of the Intermediate Algebra |
Math in 10...
1. Be more than just a number here
Our department is very proactive in mentoring our students. Through classroom discussions, out-of-class meetings, departmental social events, phone calls, and e-mails, we work hard to help our students make the most of their education. We connect our students interested in particular fields with alumni working in those areas. Beginning in their first year, we inform and advise students on valuable enrichment opportunities, such as research programs, conferences, and study abroad.
2. Explore all the areas
Our faculty members have a wide range of expertise and interests. You will investigate a range of topics in pure and applied mathematics. Our faculty members have special interests in applied graph theory, computation theory, differential equations and mathematical modeling, mathematics education, number theory, and statistics.
3. Participate in academic year research
Our students have the chance to become involved in research both on and off campus. Several projects enable students to participate in and present their work at academic and professional conferences on campus and around the country.
Student research topics have included origami, Galois groups, computer chip layouts, DNA nanostructures, space robotics, HIV infection models, and professional soccer league possession analysis. Some of our students have been able to research, study, and present at such schools and venues as Brigham Young University, Mathematical Association of America meetings such as the summer MathFest, and the NASA Goddard Space Flight Center.
Our students and faculty have received grants and funding for their research from such organizations as: Vermont Consortium for Language and Academics, The Vermont Genetics Network, The Louisville Institute, Vermont EPSCoR, National Security Agency, NASA, and the National Science Foundation.
4. Present at HRUMC, a great regional undergraduate conference
This one-day conference celebrates undergraduate mathematics research with students and faculty from across the Northeast, and provides students with the experience of attending and presenting at a professional mathematics meeting. There are also exciting plenary talks by nationally renowned mathematicians, and career and graduate school panel discussions. This is a great opportunity to meet a wide range of other students also excited about mathematics. Each year, several of our students present their advanced work in mathematics, computer science, statistics, and more.
5. Take advantage of summer learning opportunities
Many of our students take part in Research Experience for Undergraduates (REU) programs over the summer. These opportunities enable our students to learn the fundamentals of mathematics research and to explore the many applications of math in the real world. Our students have recently participated in programs at Texas A&M, University of Georgia, George Washington University, Northern Arizona University, Claremont Colleges, and Princeton University/the Institute for Advanced Study.
6. Learn from presentations by visiting math experts
We host a variety of talks and lectures by distinguished guests and alumni as a part of our regular Colloquium Series. Some recent titles include: "Counting Self-Intersection Points of Loops on Surfaces," "Symmetry and the Monster: Mathematics as the Ultimate Complex System," "Graphs and Symmetry," "Differences in Global Mathematics Education," "Mathematics of Juggling", "Chaos and the Mathematics of Prediction," and "The Shape of Space."
Our Alumni Panels offer the chance for our math students to hear from and talk with math alumni who are using their degrees in different ways. Panelists share their career paths and/or graduate school experience, and give advice to students.
Saint Michael's also co-hosts a joint Combinatorics Seminar Series with the University of Vermont that brings together academic and industry researchers to discuss the theory and applications of combinatorics.
7. Integrate mathematics with other majors and minors
Mathematics has been a cornerstone of the liberal arts for millennia and remains a critical field in the 21st century.Along with pursuing the math major at Saint Michael's, you can choose to minor or double major in another field. Many of mathematics majors also take Saint Michael's courses in education, economics, computer science, engineering, as well as the lab sciences.
8. Become a member of Pi Mu Epsilon
Saint Michael's is home to the Vermont Alpha Chapter of Pi Mu Epsilon (PME), the National Mathematics Honors Society. PME currently has over 340 chapters at colleges and universities throughout the United States. Math students who meet the requirements can be inducted into the society during their junior or senior year. We have regular events, such as an annual induction ceremony and reception, special speakers, and recognition at graduation.
9. Take seminars and special topics courses in mathematics
Our spring semester math seminar courses offer opportunities to study and present on compelling topics in the math field. Our two most recent seminars focused on environmental issues related to water and the mathematics of games. Our fall semester special topics courses provide the opportunity to study mathematical fields not considered in our normal course sequence. Future topics planned are topology, graph theory, and population dynamics.
10. Prepare for life after Saint Michael's
Our graduates excel in a range of professional fields including:
Business, Finance, and Investment Banking
Accounting and Tax Services
Mathematics Education
Actuarial Analysis
Medical School
Computer Science and Software Development
Data and Statistical Analysis
Operations and Administration
Research and Development
Careers in mathematics intensive fields such as bioinformatics and economics
Our alumni attend graduate school in mathematics, statistics, and other fields at places such as the University of Minnesota, Clarkson University, Colorado State University, North Carolina State University, the University of Notre Dame, the University of Maine, and the University of Vermont. |
Math is a wide subject, ranging from K to 11, college and university. Then there is algebra, trigonometry, geometry, arithmetic, calculus, number theory, ... etc.
Not all teachers answer all math questions (many do). If you would give a little more detail on which branch of math to which your question relates, it will not only help teachers sort out the questions quickly, but also will attract attention of your colleagues who work in the same area to help you. Example subjects could be
"math-percentages"
"math-factorization"
"probability"
"trig"
"math-geometry-volumes"
"calculus-linearization"
"calculus-limits"
etc.
In short, the more precisely your "school subject" is described, the more chance the question will be answered quickly. Try not to put just "Math" unless you really have no idea.
It's up to you!
How to get quick responses to your math questions - MathMate, Saturday, May 25, 2013 at 10:17am
Also,
- DO check that the question you are about to post is complete and accurate before pressing the "post" button.
- Do not omit parentheses. If the numerator or denominator of a fraction has more than one term, you need parentheses around them when you transcribe the expression onto a single line.
- If you have a figure that accompanies the question, either describe it completely in words, or post it at an image server and (try to) post the link.
- Some math symbols (omit spaces shown):
° & d e g ;
π & p i ;
≥ & g e ;
≤ & l e ;
α & a l p h a ;
∫ & i n t ;
Σ & S i g m a ;
√ & r a d i c ;
² & s u p 2 ;
³ & s u p 3 ;
italics < i >italics< / i >
bold < b > bold < / b >
How to get quick responses to your math questions - Jay, Wednesday, June 5, 2013 at 7:30pm
Calc and Physics - Damon and Reiny. Thanks for your quick and thorough responses...
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Physical Edu - Responses to stressors include physical and emotional behavioral ...
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College Algebra - (I am not understanding what the last person means so can ...
Eng. Quick ? - Quick ?. What's the correct spelling? Advisor or when I googled ...
English - I won the school marathon! 1. Congratulations! 2. Good for you! 3. ... |
books.google.com - David Poole's innovative book emphasizes vectors and geometric intuition from the start and better prepares students to make the transition from the computational aspects of the course to the theoretical. Poole covers vectors and vector geometry first to enable students to visualize the mathematics while... Algebra |
Excerpt
This book aims to provide thorough coverage of the main topics of abstract algebra while remaining accessible to students with little or no previous exposure to abstract mathematics. It can be used either for a one-semester introductory course on groups and rings or for a full-year course. More specifics on possible course plans using the book are given in this preface.
Style of Presentation
Over many years of teaching abstract algebra to mixed groups of undergraduates, including mathematics majors, mathematics education majors, and computer science majors, I have become increasingly aware of the difficulties students encounter making their first acquaintance with abstract mathematics through the study of algebra. This book, based on my lecture notes, incorporates the ideas I have developed over years of teaching experience on how best to introduce students to mathematical rigor and abstraction while at the same time teaching them the basic notions and results of modern algebra.
Two features of the teaching style I have found effective are repetition and especially an examples first, definitions later order of presentation. In this book, as in my lecturing, the hard conceptual steps are always prepared for by working out concrete examples first, before taking up rigorous definitions and abstract proofs. Absorption of abstract concepts and arguments is always facilitated by first building up the student's intuition through experience with specific cases.
Another principle that is adhered to consistently throughout the main body of the book (Parts A and B) is that every algebraic theorem mentioned is given either with a complete proof, or with a proof broken up into to steps that the student can easily fill in, without recourse to outside references. The book aims to provide a self-contained treatment of the main topics of algebra, introducing them in such a way that the student can follow the arguments of a proof without needing to turn to other works for help.
Throughout the book all the examples, definitions, and theorems are consecutively numbered in order to make locating any particular item easier for the reader.
Coverage of Topics
In order to accommodate students of varying mathematical, backgrounds, an optional Chapter 0, at the beginning, collects basic material used in the development of the main theories of algebra. Included are, among other topics, equivalence relations, the binomial theorem, De Moivre's formula for complex numbers, and the fundamental theorem of arithmetic. This chapter can be included as part of an introductory course or simply referred to as needed in later chapters.
Special effort is made in Chapter 1 to introduce at the beginning all main types of groups the student will be working with in later chapters. The first section of the chapter emphasizes the fact that concrete examples of groups come from different sources, such as geometry, number theory, and the theory of equations.
Chapter 2 introduces the notion of group homomorphism first and then proceeds to the study of normal subgroups and quotient groups. Studying the properties of the kernel of a homomorphism before introducing the definition of a normal subgroup makes the latter notion less mysterious for the student and easier to absorb and appreciate. A similar order of exposition is adopted in connection with rings. After the basic notion of a ring is introduced in Chapter 6, Chapter 7 begins with ring homomorphisms, after which consideration of the properties of the kernels of such homomorphisms gives rise naturally to the notion of an ideal in a ring.
Each chapter is designed around some central unifying theme. For instance, in Chapter 4 the concept of group action is used to unify such results as Cayley's theorem, Burnside's counting formula, the simplicy of A5, and the Sylow theorems and their applications.
The ring of polynomials over a field is the central topic of Part B, Rings and Fields, and is given a full chapter of its own, Chapter 8. The traditional main topic in algebra, the solution of polynomial equations, is emphasized. The solutions of cubics and quartics are introduced in Chapter 8. In Chapter 9 Euclidean domains and unique factorization domains are studied, with a section devoted to the Gaussian integers. The fundamental theorem of algebra is stated in Chapter 10. In Chapter 11 the connection among solutions of quadratic, cubic, and quartic polynomial equations and geometric constructions is explored.
In Chapter 12, after Galois theory is developed, it is applied to give a deeper understanding of all these topics. For instance, the possible Galois groups of cubic and quartic polynomials are fully worked out, and Artin's Galois-theoretic proof of the fundamental- theorem of algebra, using nothing from analysis but the intermediate value theorem, is presented. The chapter, and with it the main body of the book, culminates in the proof of the insolubility of the general quintic and the construction of specific examples of quintics that are not solvable by radicals.
A brief history of algebra is given in Chapter 13, after Galois theory (which was the main historical source of the group concept) has been treated, thus making a more meaningful discussion of the evolution of the subject possible.
A collection of additional topics, several of them computational, is provided in Part C. In contrast to the main body of the book (Parts A and B), where completeness is the goal, the aim in Part C is to give the student an introduction to—and some taste of—a topic, after which a list of further references is provided for those who wish to learn more. Instructors may include as much or as little of the material on a given topic as time and inclination indicate.
Each chapter in the book is divided into sections, and each section provided with a set of exercises, beginning with the more computational and proceeding to the more theoretical. Some of the theoretical exercises give a first introduction to topics that will be treated in more detail later in the book, while others introduce supplementary topic not otherwise covered, such as Cayley digraphs, formal power series, and the existence of transcendental numbers.
Suggestions for Use
A one-semester introductory course on groups and rings might include Chapter 0 (optional); Chapters 1, 2, and 3 on groups; and Chapters 6, 7, and 8 on rings.
For a full-year course, Parts A and B, Chapters 1 through 12, offer a comprehensive treatment of the subject. Chapter 9, on Euclidean domains, and Chapter 11, on geometric constructions, can be treated as optional supplementary topics, depending on time arid the interest of the students and the instructor.
An instructor's manual, with solutions to all exercises plus further comments and suggestions, is available. Instructors can obtain it by directly contacting the publisher, Prentice Hall.
Acknowledgments
It is a pleasure to acknowledge various contributors to the development of this book. First I should thank the students of The College of New Jersey who have taken courses based on a first draft. I am grateful also to my colleagues Andrew Clifford, Tom Hagedorn, and Dave Reimer for useful suggestions.
Special thanks are due to my colleague Ed Conjura, who taught from a craft of the book and made invaluable suggestions for improvement that have been incorporated into the final version.
I am also most appreciative of the efforts of the anonymous referees engaged by the publisher, who provided many helpful and encouraging comments.
My final word of gratitude goes to my family—to my husband, John Burgess, and to our sons, Alexi and Fokion—for their continuous understanding and support throughout the preparation of the manuscript.
"Synopsis"
by Libri,
For a one-semester course covering groups and rings or a two-semester course in Abstract Algebra.This text provides thorough coverage of the main topics of abstract algebra while offering nearly 100 pages of applications. A repetition and examples first approach introduces students to mathematical rigor and abstraction while teaching them the basic notions and results of modern algebra |
Access Math Tasks, Solutions, Videos and Khan Academy Practice
Welcome Teachers and Students! This MFM1P Grade 9 Applied Math Help Resource has been compiled to assist in closing the gap that exists between academic and applied student achievement in the province of Ontario. All resources include math tasks, solutions, videos, practice links and more to ensure every student can achieve at the highest level in their Ontario grade 9 applied math course.
If you have resources you would be willing to share, please contact me so I can add it to the database. I hope you find these resources useful!
Many of these resources were modified from the TIPS4RM resource by great math teachers and friends, Dave Bracken and Michael Smith with further additions and modifications by Kyle Pearce.
MFM1P Specific Expectations
NA1.04 - Make comparisons using unit rates (e.g., if 500 mL of juice costs $2.29, the unit rate is 0.458¢/mL; this unit rate is less than for 750 mL of juice at $3.59, which has a unit rate of 0.479¢/mL);
MFM1P Specific Expectations
LR2.03 - Identify, through investigation, some properties of linear relations (i.e., numerically, the first difference is a constant, which represents a constant rate of change; graphically, a straight line represents the relation), and apply these properties to determine whether a relation is linear or non-linear.
MFM1P Specific Expectations
LR4.07 - – select a topic involving a two-variable relationship, pose a question on the topic, collect data to answer the question, and present its solution using appropriate representations of the data.
MFM1P Specific Expectations
LR1.04 - Describe trends and relationships observed in data, make inferences from data, compare the inferences with hypotheses about the data, and explain any differences between the inferences and the hypotheses (e.g., describe the trend observed in the data. Does a relationship seem to exist? Of what sort? Is the outcome consistent with your hypothesis? Identify and explain any outlying pieces of data. Suggest a formula that relates the variables. How might you vary this experiment to examine other relationships?).
MFM1P Specific Expectations
MG1.03 - Solve problems that require maximizing the area of a rectangle for a fixed perimeter or minimizing the perimeter of a rectangle for a fixed area (Sample problem:You have 100 m of fence to enclose a rectangular area to be used for a snow sculpture competition. One side of the area is bounded by the school, so the fence is required for only three sides of the rectangle. Determine the dimensions of the maximum area that can be enclosed.).
Course Description - Grade 9 Applied Math
This course enables students to develop an understanding of mathematical concepts related to introductory algebra, proportional reasoning, and measurement and geometry through investigation, the effective use of technology, and hands-on activities. Students will investigate real-life examples to develop various representations of linear relations, and will determine the connections between the representations. They will also explore certain relationships that emerge from the measurement of three-dimensional figures and two-dimensional shapes. Students will consolidate their mathematical skills as they solve problems and communicate their thinking. |
xxTexas Instruments TI-83 Plus Graphing Calculator
The TI-83 Plus is an easy-to-use graphing calculator for math and science that lets students graph and compare functions, as well as perform data plotting and analysis. Its FLASH ROM memory allows students to update and add software applications (apps).
This Texas Instruments calculator is preloaded with the periodic table, organizer, science tools, probability simulations, a puzzle pack, and more. Approved for use with the PSAT, SAT, and ACT college entrance exams as well as IB and AP tests, you can count on the TI-83 during exam time. With several useful features, the Texas Instruments TI-83 Plus is sure to help students get through even the toughest coursework.
additional information
See N Solve Fraction Calculator Show students how
to solve a fraction problem step by step with this cleverly designed calculator. It features instruction in how to add, subtract, multiply, and divide fractions |
Middle School Math Grade: 6
Prerequisite: Placement by district assessment
A grade level course designed to provide students with sufficient mathematical foundation to meet 6th grade state and district standards.
Advanced Middle School Math Grade: 6
Prerequisite: Placement by district assessment Designed for students who have demonstrated an exceptionally high level of understanding and proficiency of 5th grade standards.
This course covers the same concepts as the grade level class but with greater depth, complexity, and pace.
Pre-Algebra Grade: 7
Prerequisite: Completion of Middle School Math
A grade level course designed to provide students with sufficient mathematical foundation to meet 7th grade state and district standards. Successful completion of this course adequately prepares students for Algebra 1.
Algebra Foundations Grade: 8
Prerequisite: This course is for students who have not met the prerequisites for
Algebra 1.
A course designed to introduce key Algebra concepts and skills in depth. This course will not cover all Algebra 1 standards. Successful completion will result in preparedness for, and placement in, Algebra 1.
Algebra 1 Grade: 8
Prerequisite: In addition to completion of Pre-Algebra, prerequisites may include district placement test, state test results, semester grades, and teacher recommendation.
A grade level course designed to provide students with sufficient mathematical foundation to meet Algebra 1 standards. This course meets the UC/CSU criteria for first year Algebra. Successful completion will result in placement in Geometry. |
Mathematical Reasoning : Writing and Proof - 03 edition
Summary: Focusing on the formal development of mathematics, this book demonstrates how to read and understand, write and construct mathematical proofs. It emphasizes active learning, and uses elementary number theory and congruence arithmetic throughout. Chapter content covers an introduction to writing in mathematics, logical reasoning, constructing proofs, set theory, mathematical induction, functions, equivalence relations, topics in number theory, and topics in set theory...show more. For learners making the transition form calculus to more advanced mathematics. ...show less
Mathematical Reasoning: Writing and Proof is designed to be a text for to first course in the college mathematics curriculum that focuses on the development of mathematics. The primary goals of the text are as
To help students learn how to read and understand mathematical definitions and proofs;
To help students learn how to construct mathematical proofs;
To help students learn how to write mathematical proofs according to accepted guidelines so that their work and reasoning can be understood by others; and
To provide students with some mathematical material that will be needed for their further study of mathematics.
This type of course is becoming a standard part of the mathematics major at most colleges and universities. It is often referred to as a "transition course" from the calculus sequence to the upper level courses in the major. The transition is from the problem-solving orientation of calculus to the more abstract and theoretical upper-level courses. This is needed today because the principal goals of most calculus courses are developing students' understanding of the concepts of calculus and improving their problem-solving skills. Consequently, most students complete their study of calculus without seeing a formal proof or having constructed a proof of their own. This is in contrast to many upper-level mathematics courses, where the emphasis is on the formal development of abstract mathematical ideas, and the expectations are that students will be able to read and understand proofs and to construct and write coherent, understandable mathematical proofs.
Important Features of the Book
Mathematical Reasoning was written to assist students with the transition from calculus to upper level mathematics courses. Students should be able to use this text with a background of one semester of calculus. Following are some of the important ways this text will help with this transition.
1. Emphasis on Writing in Mathematics The issue of writing mathematical exposition is addressed throughout the book. Guidelines for writing mathematical proofs are incorporated into the text. These guidelines are introduced as needed and begin in Chapter 1. Appendix A contains a summary of all the guidelines for writing mathematical proofs that are introduced in the text. In addition, every attempt has been made to ensure that each proof presented in this text is written according to these guidelines in order to provide students with examples of well-written proofs.
2. Instruction in the Process of Constructing Proofs One of the primary goals of this book is to develop students' abilities to construct mathematical proofs. Another goal is to develop their abilities to write the proof in a coherent manner that conveys an understanding of the proof to the reader. These are two distinct skills.
Instruction on how to write proofs begins in Section 1.2 and is developed further in Chapter 3. In addition, Chapter 5 is devoted to developing students' abilities to construct proofs using mathematical induction. Students are taught to organize their thought processes when attempting to construct a proof with a so-called know-show table. (See Sections 1.2 and 3.1.) Students use this table to work backward from what it is they are trying to prove while at the same time working forward from the assumptions of the problem.
3. Emphasis on Active Learning One of the underlying premises of this text is that the best way to learn and understand mathematics is to be actively involved in the learning process. However, it is unreasonable to expect students to go out and learn mathematics on their own. Students actively involved in learning mathematics need appropriate materials that will provide guidance and support in their learning of mathematics. This text provides these by incorporating two or three Preview Activities for each section and some activities within each section based on the material in that section. These activities can be done individually or in a collaborative learning setting, where students work in groups to brainstorm, make conjectures, test each others' ideas, reach consensus, and, it is hoped, develop sound mathematical arguments to support their work.
The Preview Activities at the beginning of each section should be completed by the students prior to the classroom discussion of the section. The purpose of the Preview Activities is to prepare students to participate in the classroom discussion of the section. Some Preview Activities will review prior mathematical work that is necessary for the new section. This prior work may contain material from previous mathematical courses or it may contain material covered earlier in this text. Other preview activities will introduce new concepts and definitions that will be used when that section is discussed in class.
In addition to the Preview Activities, each section of the text contains two or three activities related to the material contained in that section. These activities can be used for in-class group work or can be assigned as homework in addition to the exercises at the end of each section.
Content and Organization
Mathematical content is needed as a vehicle for learning how to construct and write proofs. The mathematical content for this text is drawn primarily from elementary number theory, including congruence arithmetic; elementary set theory; functions, including injections, surjections, and the inverse of a function; relations and equivalence relations; further topics in number theory such as greatest common divisors and prime factorizations; and cardinality of sets, including countable and uncountable sets. This material was chosen because it can be used to illustrate a broad range of proof techniques and it is needed as a prerequisite for many upper level mathematics courses.
The chapters in the text can roughly be divided into the following (possibly overlapping) classes:
Constructing and Writing Proofs: Chapters 1, 3, and 5
Content: Chapters 4, 6, 7, 8, and 9
Logic: Chapter 2
The first chapter sets the stage for the rest of the book. It introduces the writing guidelines, discusses conditional statements, and begins instruction in the process of constructing a direct proof of a conditional statement. This is not meant to be a thorough introduction to methods of proof. Before this is done, it is necessary to introduce the students to the parts of logic that are needed to aid in the construction of proofs. This is done in Chapter 2.
Students need to learn some logic and gain experience in the traditional language and proof methods used in mathematics. Since this is a text that deals with constructing and writing mathematical proofs, the logic that, is presented in Chapter 2 is intended to aid in the construction of proofs. The goals are to provide students with a thorough understanding of conditional statements, quantifiers, and logical equivalencies. Emphasis is placed on writing correct and useful negations of statements, especially those involving quantifiers. The logical equivalencies that are presented provide the logical basis for some of the standard proof techniques, such as proof by contrapositive, proof by contradiction, and proof using cases.
The standard methods for mathematical proofs are discussed in detail in Chapter 3. The mathematical content that is introduced to illustrate these proof methods is some elementary number theory, including congruence arithmetic. These concepts are used consistently throughout the text as a way to demonstrate ideas in direct proof, proof by contrapositive, proof by contradiction, proof by cases, and proofs using mathematical induction. This gives students a strong introduction to an important mathematical idea, while providing the instructor a consistent reference point and an example of how mathematical notation can greatly simplify a concept.
In Chapter 4, we take a break from introducing new proof techniques. Concepts of set theory are introduced, and the methods of proof studied in Chapter 3 are used to prove results about sets and operations on sets. The idea of an "element-chasing proof" is introduced in Section 4.2.
The three sections of Chapter 5 are devoted to proofs using mathematical induction. Again, the emphasis is not only on understanding mathematical induction but also on developing the ability to construct and write proofs that use mathematical induction.
The last four chapters are considered "mathematical content" chapters. Chapter 6 provides a thorough study of functions. The idea is to begin with a review of functions from previous courses so that students have a base from which to work. This notion of function is then extended to the general definition of function. Various proof techniques are employed in the study of injections, surjections, composition of functions, and inverses of functions.
Chapter 7 introduces the concepts of relations and equivalence relations. Section 7.4 is included to provide a link between the concept of an equivalence relation and the number theory that has been discussed throughout the text.
Chapter 8 continues the study of number theory. The highlights include problems dealing with greatest common divisors, prime numbers, the Fundamental Theorem of Arithmetic, and linear Diophantine equations.
Finally, Chapter 9 deals with further topics in set theory, focusing on cardinality, finite sets, countable sets, and uncountable sets.
A standard one-semester course in constructing and writing proofs should cover the first six chapters of the text and at least one of Chapter 7, Chapter 8, or Chapter 9. A class consisting of well-prepared and motivated students could cover two of the last three chapters. In addition, there are a few options that an instructor could choose to tailor the course to her or his needs. For example,
Chapter 5 can be covered before Chapter 4 if it is desired to cover all methods of proof before beginning the "content" portion of the course. The only part of Chapter 5 that would need to be skipped is the material in Section 5.2 dealing with the cardinality of the power set. If desired, this material could be included when the power set is discussed in Chapter 4.
Instructors who would like to cover topics in both Chapters 7 and 8 can omit a few selected sections from earlier chapters. Although it is an important and interesting section, Section 5.3 is not used in the remainder of the book. The same is true for Section 6.5. Finally, Section 3.5 can be skipped as long as the concept of a constructive proof is discussed during other parts of the course.
Supplementary Materials for the Instructor
The instructor's manual for this text includes suggestions on how to use the text, how to incorporate writing into the course, and how to use the preview activities and activities. The manual also includes solutions for all of the preview activities, activities, and exercises. In addition, for each section, there is a description of the purpose of each preview activity and how it is used in the corresponding section, and there are suggestions about how to use each activity in that section. The intention is to make it as easy as possible for the instructor to use the text in an active learning environment.
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Guidelines for MAA Short Courses
Overview
An important activity supported by the MAA is the offering of Short Courses. Each MAA Short Course consists of a coherent sequence of presentations on a single theme of mathematical interest. A course ordinarily extends over two days immediately preceding the Joint Mathematics Meetings held in January and/or the summer MathFests, and is designed for non-specialists who are seeking professional development, enrichment, and education.
Selection of Topic and Organizers
The MAA Short Course Subcommittee welcomes suggestions for possible topics, which may be communicated to the chair or any member of the committee. The topic may introduce participants to a new area or new developments in an old area. Topics may come from "pure" areas or may have application in the physical, biological, social, or information sciences, or in technology or business. Topics that include techniques of problem-solving, topics that are of current interest, or topics that could be introduced into undergraduate courses are especially welcome.
Awareness of the audience is an important factor in choosing a topic. One of the most common reasons participants attend is intellectual curiosity. Also, many college teachers attend to glean new material for their classes. With regard to background, short course participants can range from undergraduate mathematics majors to research mathematicians. The overlap of expertise of mathematicians occurs generally at the level of advanced undergraduate mathematics. Therefore, the course should be accessible to someone who has studied mathematics at that level. In particular, it cannot be assumed they know the technical terminology and important results of the topic of the Short Course. The topic should therefore be chosen with these considerations in mind.
Development of the Proposal
The proposal selection process begins with a review of informal proposals, which consist of a brief description of the topic and name(s) of the organizer(s). Once the committee has approved a topic and organizer(s), the organizer(s) will develop a full proposal as described below, in consultation with the committee.
Proposal Content
Description of topic
Rationale – why is this topic of particular interest or import? Who will benefit? How will this short course serve the mission of the MAA? (See for a description of the mission and goals of the MAA.) We suggest that the writer think of this portion of the proposal as convincing not only the Committee, but also a potential participant. Indeed, this section could be written so that it can be easily transformed into a suitable Focus advertisement once the proposal is approved. It should, therefore, excite interest in the course, as well as accurately describe goals, content, format, and prerequisites.
Organizer(s), their backgrounds, affiliation, and contact information
Speaker(s), their backgrounds, affiliation, and contact information
Organization of the course. While not required, the Committee encourages opportunities for hands-on experience and participant discussion.
Timetable for the course (see typical timetable later in these guidelines) Be sure to allow time for evaluation (discussed below).
Audio-visual or other equipment needs. All requests for equipment are subject to approval of the Associate Secretary. Please note that computers are not provided for MAA short courses.
Evaluation of the Course
All short course participants are asked to complete an evaluation prepared by the Committee on Short Courses. A member of the committee will distribute the forms to attendees during the short course and collect them at the end of the course. Attendees who do not turn them in will be requested to mail them to the chair of the committee. The information obtained from these evaluations is used by the committee to improve future offerings and to identify future topics, organizers, or presenters. The evaluation information is also made available to the organizers and to the presenters.
Course Organization
Organizers have some flexibility as to the format and schedule of the lectures. Although the timetable, number, and length of the talks vary with each course, a typical MAA Short Course consists of six invited presentations, each sixty to seventy-five minutes long. The organizer should pay particular attention to the need for breaks interspersed throughout the day. Sometimes scheduled breaks (with refreshments) must coordinate with breaks of other events. Short Courses often conclude with a panel discussion by presenters, providing an opportunity for substantial audience participation.
Organizers may obtain more detailed outlines of previous short courses from the MAA Associate Secretary.
Responsibilities of the Committee
Once the Committee Chair has received the full proposal, s/he will distribute it to the full Committee for review. The Committee will evaluate the proposal content for completeness, timeliness of the topic (for example, has a similar topic been offered recently?), and appropriateness for and appeal to the MAA membership. As needed, the Committee will consult with the organizer(s) for clarification and revision. The Committee will inform the organizer(s) and the MAA Associate Secretary of its decision in a timely manner.
The MAA Associate Secretary will schedule the short course for the scientific program for the appropriate meeting and handle the publicity. The MAA Meetings staff handles the on-site needs for the course. The Committee will work with the organizer(s) to develop a suitable advertisement of the short course for Focus and transmit it to the Associate Secretary. The Committee will continue to work with the organizer(s) as needed to ensure a successful course. If possible, a member of the Committee will attend the short course; in that case, the Committee representative will welcome the participants and introduce the speaker if the speaker and organizer are the same person.
Responsibilities of Organizers
Course organizers are responsible for choosing, coordinating, and introducing the speakers whose presentations will comprise the course program. The organizer will receive an honorarium of $300 (to be split among co-organizers) and work with a total budget of $2500 to cover all travel expenses, meals, and lodging for presenters and organizers. Ordinarily, there are no honoraria for the presenters. Financial constraints need to be kept in mind when selecting speakers, who should be encouraged to use other sources of travel support whenever possible, as should the organizers. In general, organizers are asked to aid the MAA staff in identifying all possible ways of minimizing costs. The cost of the refreshment breaks, room rental, any on-site staff, and audio-visual expenses are covered separately by the MAA.
During initial conversations with prospective presenters, the organizer should outline the purpose of the short course and the presenter's responsibilities. The organizer should also discuss the nature and extent of MAA financial support offered to the presenters. Organizers should aim for a balanced group of presenters that will provide an interesting mix of ideas and insights on the theme topic. Throughout, the organizer should keep the committee chair and the Associate Secretary informed of any changes.
After initial conversations and informal agreement by presenters, the organizer contacts each presenter formally by sending a letter that outlines the responsibilities and timetable for all parties. The formal letter should contain:
Dates, time and place of the MAA short course
Tentative schedule of presentations
Guidelines for Presenters (below)
Request for synopsis of presentation and reading lists
Request for course materials
Copy of MAA reimbursement policy
Information on audio-visual equipment and other supplies
The last two items can be obtained from the Associate Secretary.
The inclusion of this information in the letter of invitation is intended to formalize the arrangements and to place in writing the understanding of the commitments involved. It also enables the presenters to know how their topic fits in with those of other presenters.
In order to facilitate contacts, the organizer(s) should maintain an up-to-date contact list with information on each presenter, including address, email, telephone, and fax. This list is provided to each presenter, the Chair of the Short Course Subcommittee, and the MAA Associate Secretary.
Organizers coordinate the schedule of presentations so that the course theme is developed in a coherent fashion which avoids excessive overlap. The organizer typically gives the opening overview of the course, introduces speakers, and is responsible for the concluding session of the course. Speakers are expected to share the outline of their presentations with the organizer and with each other well before the meeting. The organizer may wish to provide connective and introductory material prior to each presentation. There should be as much uniformity of notation and terminology as possible in the presentations.
It is also essential that the Associate Secretary be given information on any unusual expenses that might be warranted for the Short Course so that the Executive Committee will have the information it needs to set the fees for the Short Course.
Titles and brief descriptions of all presentations are needed well in advance (See Timetable and Deadlines below). Organizers will request that speakers prepare written course materials that include an outline of the presentation and references for further study; the organizer sends these to the Associate Secretary to be posted on the MAA website one month in advance of the course.
The MAA welcomes manuscripts based on lecture notes of short courses for possible publication. Organizers may discuss this possibility with the acquisitions editor of the MAA. In addition to course materials, organizers should send a letter of welcome to all registered participants informing them of the time and place of the course, how to obtain the course materials, and any additional items to be supplied by the participant, e.g., laptop computer.
Guidelines for Presenters
Speakers should begin with something everyone can follow, while going into enough technical detail to enable the audience to see the whole topic in perspective. Speakers should keep in mind that the expected audience comes with diverse backgrounds, depth and breadth of knowledge, however most members of the audience may be assumed to have a graduate-level mathematics education. Participants come with a keen interest in collegiate mathematics and a desire to develop professionally with the aim of enriching their knowledge of mathematics and improving their skills in presenting such material to students.
The following suggestions are the result of experience with participants' evaluations of previous Short Courses. Speakers should pay careful attention to them.
Content
Move quickly to the pearls of the topic.
Start at an elementary level, such as typically evidenced in an expository article in the American Mathematical Monthly.
Illustrate main ideas with a single, simple, example or with a well-conceived set of examples given throughout the presentation.
Use real data when possible. Such examples are often the only thing that is remembered by the audience afterwards.
Provide a brief history leading up to the topic.
Define all terms. Use standard notation.
Review all key elementary results to be used.
Avoid audience burnout: do not move too fast or give too many details.
Remember, there has never yet been a talk that was too simple or too elementary.
Coordinate your talk (through the organizer or directly) with those of other presenters to avoid excessive overlap and acknowledge overlap when it occurs.
State some open problems or unresolved issues in the field.
Delivery
Use multicolored, neatly prepared slides in a format appropriate for the projection equipment. Make sure the text is large enough for everyone to read, and avoid using more than 10 lines per slide.
Use well-prepared pictures and diagrams.
Be consistent with material distributed prior to the presentation.
Use gender-neutral wording as appropriate.
Include key bibliographic references.
Timetable and Deadlines
JMM Short Courses
Mathfest Short Courses
Informal proposals due to committee
December 15 of year - 2
July 1 of year - 1
Approval of full proposal by Committee; proposers notified
April 15 of year - 1
October 15 of year - 1
Send formal letter to presenters, including Guidelines for Presenters
April 30 of year - 1
October 30 of year - 1
Send presentation titles, brief descriptions, schedule and Focus ad to Associate Secretary, copies to the committee chair. Also notify of special needs with financial implications
Cancellation Policy
If one month prior to the meetings, there are fewer than 20 persons pre-registered for an MAA short course, it may be cancelled. Before such a cancellation is made, organizers will be notified and alternatives will be discussed, such as offering the MAA short course but reducing the financial reimbursement support for the presenters. If the course must be cancelled, the MAA staff will notify all those registered and reimburse their registration fee.
Contact Information
The Chair of the Committee on Short Courses and the MAA Associate Secretary are the two primary contacts for information about Short Courses and their organization. The Associate Secretary will be able to inform organizers of any other MAA personnel who will be needed for helping with organizational, technical, or audio-visual details of the course. Current contacts are: |
Mathematics
Concentration in Mathematics for Elementary Mathematics
All majors in Elementary Education --Childhood and Early Childhood--must complete a concentration in a liberal arts discipline. Our concentration is a focused sequence of courses (30 credits) in which the elementary education major experiences the core areas of mathematics: calculus, algebra, geometry, and probability and statistics. The concentration prepares the future teacher for further specialization in mathematics education or for deeper study in mathematics.
MATH 140
Mathematical Concepts for Elementary Education I
MATH 141
Mathematical Concepts for Elementary Education II
MATH 221
Calculus I
MATH 222
Calculus II
MATH 223
Calculus III
OR
MATH 233
Linear Algebra I
MATH 239
Introduction to Mathematical Proof
MATH 242
Elements of Probability and Statistics
OR
MATH 262
Applied Statistics
OR
MATH 360
Probability and Statistics I
NOTE: Math 223 is a prerequisite for Math 360.
MATH 335
Foundations of Geometry
INTD 376
Advanced Math Topics for Childhood and Early Childhood Education
Students are welcome to consult with advisors in the mathematics department for how to best plan their mathematics program. |
Icons for Handouts
pages are handed out and used in class, and
you'll be responsible for the material. Make sure to print your own
copy if you miss that class. (Note: Other materials are
required but are not handed out in print because many students prefer
to read them on the Web.)
pages recap the lecture material,
maybe expanding on some parts of it. These can be a great supplement to
your class notes.
marks practice problems.
points you to TI calculator procedures.
tells you that the document contains an Excel
workbook or gives instructions for using Excel.
marks interesting supplements to the topics in
the lecture. While you won't be held responsible for this material,
I recommend it to deepen your understanding. Advanced math
is not required for materials with this mark.
marks challenging material that goes
beyond the course requirements, and uses more math background. Read
these handouts to find out more of the theory behind the operations
you learn in class.
marks an external Web page, one
that is not on the TC3 server.
indicates a video available in the Media Center
on campus.
Please use your browser's Back button to return to the page
you were viewing. |
Mathematics in science
The issue
Mathematics enables students to understand and describe many
scientific phenomena, but there is concern that assessment of
the sciences at A-level do not properly reflect the analytical
nature of the subjects. However, there is little
evidence of the type and amount of mathematics that are
required in order to access the science in the current A-level
specifications.
Policy advice
There should be a review of the current mathematical
requirements for each of the sciences at A-level to ensure the
inclusion of appropriate areas of mathematics to support the
science.
A framework should be developed to regulate the way mathematics
within science A-levels are assessed, covering frequency and extent
with which mathematical requirements are included in
assessments, and the number of marks available for each level of
difficulty.
University admissions should stipulate more clearly the
appropriate mathematical requirements for STEM course entry.
The scientific community, including SCORE organisations, should
work with ACME to
consider appropriate and realistic post-16 options in mathematics
to support the teaching and learning of the sciences at
A-level.
Activities
The Institute of Physics led the 22-month research project
on behalf of SCORE. The overall objective for the research was to
provide evidence on the type, extent and difficulty of mathematics
assessed in science A-levels and to establish whether the current
assessments reflected the mathematical requirements of the
sciences.
The findings show that a large number of mathematical
requirements listed in the biology, chemistry and physics
specifications are assessed in a limited way or not at all within
these papers. The mathematical requirements that are assessed are
covered repeatedly and often at a lower level of difficulty than
required for progression into higher education and employment. Also
highlighted is a disparity between awarding organisations in their
assessment of the use of mathematics within biology, chemistry and
physics A-level. |
08176419ane Networks and their Applications
This concise, fast-paced text introduces the concepts and applications behind plane networks. It presents fundamental material from linear algebra and differential equations, and offers several different applications of the continuous theory. Practical problems, supported by MATLAB files, underscore the theory; additional material can be downloaded from the author's website |
The following is a list of Student Learning Outcomes
for College Algebra:
Make sure you have accomplished all of these objectives
for your final exam.
1. The student will review the rules of real and complex numbers; adding,
subtracting, multiplying and dividing and factoring polynomials; review
the laws of exponents; simplifying radicals; and simplify polynomials,
solve linear, fractional, and quadratic equations; solve equations with
variables under radicals by clearing exponents; use a "u" to help solve
an equation; use factoring with fractional exponents to solve equations;
and graph inequalities using intersections and unions, graph rational and
quadratic inequalities, and be capable of using these concepts in conjunction
with subsequent material.
2. The student will work on learning to solve word problems all
semester.
3. The student will know and be able to find the x and y intercepts
of a relation and graph it by plotting points and using symmetry. The student
will be able to write the equation of a circle in standard form and be
able to determine the center and the radius of the circle from this form.
4. The student will be able find and use the slope of a straight line,
the slopes of parallel or perpendicular lines, and the point-slope, slope-intercept,
and standard forms for the equation of a straight line.
5. Students should be able to determine the domain and the range of
all graphs.
6. The student will be able to determine which relations are functions;
graph functions including the use of translations and reflections; find
the domain and range of all functions; evaluate functions at specified
values, determine if a function is one-to-one.
7. The student will be able to graph a quadratic function, specifying
the vertex, axis of symmetry, maximum and minimum values, and intercepts
and write the equation for the parabola in standard form.
8. The student will be able to graph piecewise or step functions, absolute
value functions and relations, and other graphs by making a "t" chart.
9. Students will be able to graph rational functions, determine asymptotes,
x and y intercepts, domain and range for rational functions. Students will
be able to determine if a rational function has a hole in it and name the
point where that hole occurs.
10. Students will be able to combine functions by addition, subtraction,
multiplication, division, and composition; and find the inverse of a function.
11. The student will be able to graph polynomial functions. The
students will be able to divide a polynomial by a binomial using synthetic
division and use the remainder and factor theorems and the fundamental
theorem of algebra to find the roots of a polynomial or, given the roots,
determine the polynomial.
12. The student will be able to manipulate and solve exponential
and logarithmic functions, including common and natural logarithms, and
related application problems and be able to graph these functions. The
student will be able to use a calculator to evaluate logarithms.
13.. The student will be able to solve a system of equations by the
use of matrices. The student will be able to perform algebraic operations
on matrices. The student will be able to find determinants and use Cramer's
Rule and Gaussian Elimination.
14. Students must be able to use a graphing calculator to graph functions,
determine asympotes, turning points, name relative maxiums and minimums,
zeros and holes in graphs. Students should know how to use the 2nd Calc
menu and the table menu. |
Math Courses Online - Nancy Parham
Online courses offered by Cal State Bakersfield, Fresno, Los Angeles, San Bernardino, San Marcos, and Cal Poly San Luis Obispo: designed for students preparing to take the math exams ELM, GRE, CBEST, or SAT, or adults who are reentering college after
...more>>
The Math Dude - Mike DeGraba
Mike DeGraba is the Math Dude, bringing engaging explanations to Algebra I students in this series of videos. The 5- to 7-minute episodes are available via Flash on the web, podcast, RSS feed, and if you live in Montgomery County, MD, cable TV. Episodeshematics Illuminated - Annenberg Media
A video course for adult learners, high school and college teachers; 13 half-hour video programs, online text, course guide, and Web site; graduate credit available. "Rather than a series of problems to be solved, mathematics is presented as play we engage
...more>>
Math for Morons Like Us - ThinkQuest 1998
Students talk to students about math: a site designed to help you understand math concepts better. Tutorials, sample problems, and quizzes for Pre-Algebra, Algebra, Geometry, Algebra II, and Pre-Calc/Calculus, designed assuming you know some of the basic
...more>>
Math Fundamentals Problem of the Week - Math Forum
Math problems for students working with concepts of number, operation, and measurement, as well as introductory geometry, data, and probability. The goal is to challenge students with non-routine problems and encourage them to put their solutions into
...more>>
Mathie x Pensive - Gregory Taylor
Musings on mathematics, teaching, and more by an Ottawa (Canada) high school math teacher who majored in computer science and graduated from the mathematics program at the University of Waterloo before receiving his B. Ed from Queen's University. Posts,
...more>>
Math in Daily Life - Annenberg Media
Students often question how they will use basic mathematical concepts, algebra, and geometry throughout their lives - but the average person uses math at least three times a day. Read how math affects daily decision-making in this series of short articles
...more>>
mathletics - Wayne Winston
From the author of the book by the same title, which reveals "how professional baseball, football and basketball teams use math to improve their performance" à la Moneyball, and also discusses line-up evaluations, the effectiveness of running andPlus Workbooks for Sale
Almost 60 workbooks with more than 3,100 pages of worksheets in subjects organized by grade level: K-1, 1-2, 2-3, 3-4, 4-5, 5-6, and 6-7. Each worksheet is 8-1/2 by 11 printed in high resolution. View sample worksheets and/or print out an order form.
...more>> |
gets you up to speed on Maya's latest features and expands your skills with advanced instruction on cloth, fur, and fluids.
Influence and persuasion aren't just abstract concepts of interest exclusive to psychologists and sociologists. Rather, they're a fundamental part of your everyday life. Whether you realize it or not, you're constantly surrounded by people and groups trying to influence the way you think, act, and feel.
Statistics is one of the most important areas of Math to understand. It has applications in science, engineering, business, economics, policical science, and more. In this 4 Hour Course, Jason Gibson teaches the fundamental concepts needed to truly Master Statistics with step-by-step video tutorials.
Statistics is one of the most important areas of Math to understand. It has applications in science, engineering, business, economics, policical science, and more. In this 6 Hour Course, Jason Gibson teaches the fundamental concepts needed to truly Master Statistics with step-by-step video tutorials. The lessons begin with fundamental definitions of statistics such as populations, samples, and frequency distributions. Next, the student will learn how to represent data in charts and histograms. Then we'll cover important topics on how to represent data using the mean, median, mode, and standard |
An understanding of the principle elements of algebra is essential to upper-level math and good standardized test scores. Introduce your junior high students to advanced math with this kit's 160 colorful lessons. The colorful student workbook reviews basic math skills before introducing algebra,... |
A diagnostic test with explained answers to help students identify their strengths and weaknesses Advice on using a graphing calculator A detailed review of all test topics, including polynomial, trigonometric, exponential, logarithmic, and rational functions; coordinate and three-dimensional geometry; numbers and operations; and much more
Now fully adapted to reflect Canadian drug information, measurement, legislation, and standards, Henke's Med Math, First Canadian Edition is an excellent choice for giving students an in-depth understanding of dosage calculation and principles of drug administration. It presents step-by-step approaches to solving problems and includes dosage problems that simulate actual clinical experience. Each chapter includes numerous examples, self-tests, and proficiency tests. Multiple dosage calculation methodologiesŚformula, ratio, proportion, and dimensional analysisŚare taught, allowing students to adopt their preferred method. In addition, calculations are included for intravenous pediatric, PCA, and critical care medications.
This course covers the maths behind how your computer stores and manipulates data. You'll learn how to read binary and hexadecimal, how both integers and floating point numbers are stored and the limitations of using them. Advice on best practices and how to work effectively with boolean values and bitwise operators |
Algebra Help
Do you need help with algebra? We have tens of video clips on algebra. Along with our video tutorials, we have published tens of printable algebra worksheets and tens of word problems on algebra for you to practice algebra and to get better grades.
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Algebra
Algebra is a branch of mathematics concerning the study of structure, relation and quantity.
Together with geometry, analysis, combinatorics, and number theory, algebra is one of the main branches of mathematics. Elementary algebra is often part of the curriculum in secondary education and provides an introduction to the basic ideas of algebra, including effects of adding and multiplying numbers, the concept of variables, definition of polynomials, along with factorization and determining their roots.
Algebra is much broader than elementary algebra and can be generalized. In addition to working directly with numbers, algebra covers working with symbols, variables, and set elements. Addition and multiplication are viewed as general operations, and their precise definitions lead to structures such as groups, rings and fields.
Do you teach a algebra class? Or Do you or does your child need algebra help?
Learning algebra is now easier. This page lists algebra topics and activities that can be done to teach algebra. Tulyn makes teaching algebra easy. Tulyn makes learning algebra easy.
You can navigate through these pages to locate algebra video clips,
practice word problems on algebra and algebra worksheets.
What Else Do You Study/Teach In Your Algebra Class?
Please list the activities you do in your algebra class.
How Others Use Our Site
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Math Resources and Portals
CR Algebra Review
Prealgebra Review A review of the concepts in Prealgebra as preparation to enter Elementary Algebra (Math 380). Elementary Algebra Review A review of the concepts in Elementary Algebra as preparation to enter Intermediate Algebra (Math 120). Intermediate Algebra Review A review of the concepts in Intermediate Algebra as preparation to enter a Transfer Level math class (Math 5, 15, 25, or 30). Important Information Procedure: Each review course is broken up into 6 modules. Each module has 2 or 3 skills (labeled A, B, ...) to be reviewed.
Authentic Assessment in Mathematics Home Page
The Geometry Forum Summer '94 Workshop at Swarthmore College, Swarthmore, PA. The goal of this project was to collect and organize available INTERNET resources on Authentic Mathematical Assessment applicable to Secondary Schools. What is it?
Go to my home page Participate in The Most Pleasing Rectangle Web Poll which recently moved to jimloy.com. "He must be a 'practical' man who can see no poetry in mathematics." - W.
Jim Loy's Mathematics Page |
Best mathematical tool for school and college! If you are a student, it will help you to learn algebra!
Note: Polynomials appear in settings ranging from basic chemistry and physics to economics and social science; they are used in calculus and numerical analysis to approximate other functions. In advanced mathematics, polynomials are used to construct polynomial rings, a central concept in algebra and algebraic geometry. |
Solve real‐life and mathematical problems using numerical and algebraic expressions and equations |
Objectives: This is the second of two courses designed for future elementary teachers. Students will continue to review mathematical topics, to recognize the connections between topics, to reflect on how these topics are best taught in the elementary classroom and on how the topics relate to the strands of the Mathematics Content Standards for California Schools. Topics include probability and statistics concepts, geometrical figures, measurement, and motions in Geometry.
This course is designed to prepare prospective elementary teachers for credential and/ or graduate programs in elementary education by examining important concepts in Data analysis, Predictions, Experiments, Probability, Number Theory, Plane figures, Space figures, Tessellations, Symmetry, Area, Perimeter, Volume and Surface Area, Constructions and Mappings with emphasis on problem solving and critical thinking. Only a few of the topics will be "taught", while most of the topics will simply be reviewed. Students in the course should already be familiar with most of the course material. If so, what is the point of the course? We are here to make connections between topics, to think about these topics in new ways, and to think about why mathematics is the way it is. Students will examine their beliefs about mathematics and get some experience in what it means to 'do mathematics". Through activities and discussions students will gain insights into how to teach mathematics in the elementary classroom. The course should also be helpful in studying for the mathematics portions of the CSET.
Material Covered: Chapters 8-15 (see tentative schedule).
Attendance: Students are required to attend all class sessions. In case an absence occurs because of circumstances which are beyond the student's control, the student must make up all missing work within 1 week or as agreed with the instructor. Absences without
excusable reason and/or excessive tardiness will result in lowering of your attendance score 6 testsPapers: This semester you will write 5 papers:
Paper #1: Euclid & his famous work "Elements"
Paper #2: "Amazing Constructions" In this paper you will describe the construction of a
particular geometric figure with an unmarked rule & compass. Each student will get an individual
problem.
Paper #3: "The Pythagorean theorem." Write about the history of this theorem, state the
number of proofs that are known today, and perform at least one of this proofs in your paper.
Paper #5: "The history of pi" Make a research and answer the question how Archimedes and
other mathematicians found pi.
How Papers Will Be Graded
Each writing assignment I give will be in the form of a reaction or reflection paper: at least 2 pages in length, 10 points possible credit. Basically, I want to know that you gave the topic some thought and made the paper easy to read. My grading method is:
Format: 3 points
Your paper should be typed on a computer . Your name, the due date for assignment, and the course number on each page. The title of the assignment should appear at the top of the first page.
Content: 4 points
Here I am looking engagement and/or originality. I want to know that you understood the topic and thought about it. Reciting or reporting facts may be useful, but that is not the point of these assignments. I want you to comment on the facts or ideas that you are writing about.
Flow: 3 points
I expect that you will connect your ideas into a readable whole. Often you will be able to do this without rewriting. Please reread what you wrote and edit it, if necessary.
Projects: This semester you will have two projects: (For the due date see tentative schedule)
Project #1: " Beautiful Tessellations"
This is an art project. You will create your own drawing using tessellations: translations, rotations, and reflections of polygons such as squares, equilateral triangles, and regular hexagons to obtain a non polygonal figure. Though you may not copy Escher's work, you can use his paintings as an inspiration.
Project #2: "The Magic Polyhedra"
In this project every student will make a model of a regular or semi regular polyhedron. You can use any type of paper, cardboard, plastic, wood and so on. The particular type of a polyhedron for each student will be assigned in class.
Presentation: This semester each student will present a 7-10 minutes lecture on the current material as a real teacher. Topics will be assigned during the first two weeks.
Cell phone policy: All cell phones are strictly prohibited on the tests. All other time they must be put on vibration.
Final Exam: Final exam will be held on: Wednesday, May 6, 2- 4:30.
About your binder
Your binder should have a cover sheet (index) and then be organized into the following sections: |
Help with finding a beginner math book.
So, I'm going to be blunt here. I suck at math, and I'm starting to realize how much I need a more advance level of math understanding then I have now to program effectively. So does anyone have any good suggestions on where to start? I would label myself as a complete beginner.
Here are a few books I have been looking at buying.
1 - Maran Illustrated Effortless Algebra
2 - Practical Algebra: A Self-Teaching Guide, Second Edition
Should I start with Algebra or start somewhere else? I have very low understanding of algebra (Had to dropout of high school because of certain circumstances). So any advice on good books , or where to start would be a great help. Thanks in advance
Not a book but try Khan Academy they have practice problems and videos for everything from telling time to differential calculus (they have videos for integral calculus but not practice problems).
I don't know what level of math you are compared to me but I am 14(years) and found that algebra self teaching guide looked nice. I actually just bought the book myself, If you want I can give you an update when I have read some of it. Allthough I do consider myself a bit over average in math for my age.
If you're a complete beginner, then unfortunately you kind of have to go get proficient with algebra as it's used in almost every other field of math. Algebra is so painfully boring though -_- I always hated algebra class.
As a single word, "algebra" can mean:
* Use of letters and symbols to represent values and their relations, especially for solving equations. This is also called "Elementary algebra". Historically, this was the meaning in pure mathematics too, like seen in "fundamental theorem of algebra", but not now.
* In modern pure mathematics, a major branch of mathematics which studies relations and operations. It's sometimes called abstract algebra, or "modern algebra" to distinguish it from elementary algebra.
I assume you're all talking about elementary algebra? Because actual algebra is fairly cool. Also actual algebra is mostly irrelevant to programming. Although I've had some use for linear algebra. Anyway, if you're into computer science, you should probably concentrate on discrete maths. When you know your basics, at least.
Then you get in calculus and realize it's 99% algebra, 1% miscellaneous concepts.
Now I wouldn't say that's true at all. Sure it uses algebra, but it's definitely not mostly algebra. You can probably get by in calculus with just a basic understanding of algebra.
@hamsterman,
Yea I guess that's what I was talking about. I didn't know the difference between elementary algebra (which is all I've done so far) and abstract algebra. At some point I'll be taking abstract algebra and linear algebra, I hope those are more exciting than elementary algebra.
Calculus is built off the concept of Algebra. Anything you do involves isolating a variable, finding a zero, maximum/min point, area under a curve, etc. Even using power rules to take the derivative of something is just a shortcut for using algebra to solve a limit. Algebra is everywhere, it's almost everything you do. If you're using a variable for something, you're using algebra. The better understanding you have of it, the better you will be at maths, period. Regardless of the field of study.
But that's like saying being a good typist makes me a programmer. Just because the skill is necessary doesn't mean the skill is the task. Calculus is not algebra, it just uses it. There's a reason it's called calculus and not "algebra, and more!".
But that isn't following the logic i was trying to get at, and isn't really a valid comparison. Programming isn't built off of typing (entirely). It's built off of logic and thought process. Now programming isn't logic and reason, but you use logic and reason in almost every aspect of programming, short of memorizing the syntax.
In exactly the same way, calculus isn't algebra, but algebra is used in almost every aspect of calculus.
Can someone spell out for me what you refer as algebra to? I assume quadratic equations and polynomials in general are involved. Trigonometric and exponential functions too? What else?
@ResidentBiscuit, to be honest, I'm not a huge fan of abstract algebra myself - algebraic structures are very hard to visualize. Also it smells like number theory... Don't take me wrong though, it's still cool. As for linear algebra, the basics are fun. More advanced stuff turns to abstract algebra and polynomials a bit too much. Maybe that's just me. And maybe you won't be going too deep.
@Thumper, it's not good to make absolute statements. Calculus is as much about algebra as about arithmetic (it involves numbers) or geometry (integral is area) or topology (limit is only defined in a topology) or number theory (not in high school) and I might come up with more. All of mathematics is tangled up. You certainly need to manipulate equations to do calculus, but it is not about manipulating them. As far as calculus is concerned, there are no equations, only curves.
@ResidentBiscuit
I take nothing personally :P
We're getting so far into semantics that we're slipping away from what we're both trying to say here, lol.
@hamsterman
Why i can't find words to describe it like that, i don't know. What i was attempting to get across is that you can't have one without the other. And now it seems so simple to describe that i feel like a fool for not being able to a moment ago.
I got that was your point, but I don't think it is a good point. Algebra is an approach to calculus. For number theory there are analytic, algebraic, combinatorial, probabilistic and possibly some more approaches. I wish there was "number theoretic calculus", to illustrate my point.
Now that I look at it, I'm not sure. When you say this, do you mean it as a bad thing? |
This collection is included inLens:Community College Open Textbook Collaborative By: CC Open Textbook Collaborative Supplement
Summary: This module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr.
Operations with algebraic expressions and numerical evaluations are introduced in this chapter. Coefficients are described rather than merely defined. Special binomial products have both literal and symbolic explanations and since they occur so frequently in mathematics, we have been careful to help the student remember them. In each example problem, the student is "talked" through the symbolic form.
This module provides an exercise supplement for the chapter "Algebraic Expressions and Equations". |
The Four Pillars of Geometry
by John Stillwell Publisher Comments
For two millennia the right way to teach geometry was the Euclidean approach, and in many respects, this is still the case. But in the 1950s the cry "Down with triangles!" was heard in France and new geometry books appeared, packed with linear algebra... (read more)
Fractals
by John Briggs Publisher Comments
Fractals are unique patterns left behind by the unpredictable movements -- the chaos -- of the world at work. The branching patterns of trees, the veins in a hand, water twisting out of a running tap -- all of these are fractals. Learn to recognize them... (read more)
Riemannian Geometry
by Manfredo P. Do Carmo Publisher Comments
Riemannian Geometry is an expanded edition of a highly acclaimed and successful textbook (originally published in Portuguese)... (read more)
A Vector Space Approach to Geometry
by Melvin Hausner Publisher Comments
This examination of geometry's correlation with other branches of math and science features a review of systematic geometric motivations in vector space theory and matrix theory; the use of the center of mass in geometry, with an introduction to... (read more)
Complex Spaces in Finsler, Lagrange and Hamilton Geometries
by Gheorghe Munteanu Publisher Comments
This book presents the most recent advances in complex Finsler geometry and related geometries: the geometry of complex Lagrange, Hamilton and Cartan Spaces. The last three spaces were initially introduced to and have been investigated by the author of... (read more)
Modern Geometry / With CD (02 Edition)
by David A. Thomas Publisher Comments
MODERN GEOMETRY was written to provide undergraduate and graduate level mathematics education students with an introduction to both Euclidean and non-Euclidean geometries, appropriate to their needs as future junior and senior high school mathematics... (read more)
Convex Polyhedra (Springer Monographs in Mathematics)
by A. D. Alexandrov Publisher Comments
This classic geometry text explores the theory of 3-dimensional convex polyhedra in a unique fashion, with exceptional detail. Vital and clearly written, the book includes the basics of convex polyhedra and collects the most general existence theorems... (read more)
Analytic Geometry (7TH 92 Edition)
by Gordon B. Fuller Publisher Comments
Tailored for a first course in the study of analytic geometry, the text emphasizes the essential elements of the subject and stresses the concepts needed in calculus. This new edition was revised to present the subject in a modern, updated manner. Color... (read more)
Analytical Conics
by Barry Spain Publisher Comments
This concise text introduces students to the elements of analytical geometry, covering basic ideas and methods. Topics include transformation of axes, the line at infinity, conics and pencils of conics, homographic correspondence, line-coordinates, and... (read more)
Elementary Geometry for College Students (5TH 11 Edition)
by Daniel C. Alexander Publisher Comments
Building on the success of its first four editions, the Fifth Edition of this market-leading text covers the important principles and real-world applications of plane geometry, with a new chapter on locus and concurrence and by adding 150-200 new... (read more)
Three Lectures on Fermat's Last Theorem
by Louis Joel Mordell Publisher Comments
Attempted by the greatest mathematicians including Euler, Legendre, Gauss, Abel, Dirichlet, Cauchy, and Kummer, and here is Mordell. Considered a classic, and unabridged.... (read more)
Geometric Analysis and Applications to Quantum Field Theory
by Peter Bouwknegt Publisher Comments
In the last decade there has been an extraordinary confluence of ideas in mathematics and theoretical physics brought about by pioneering discoveries in geometry and analysis. The various chapters in this volume, treating the interface of geometric |
Mathematics 1
Covers the parts of calculus and algebra which have proved fundamental to all of mathematics and its applications. It is the first of a pair of courses, MATH1110 and MATH1120, designed to cover a range of mathematical topics of importance to students in the Sciences, Engineering or Commerce.
In algebra, students learn concepts and symbolic manipulation when calculating with large numbers of variables. In calculus, they learn concepts used when working with continuously changing variables. Both ways of thinking are essential in the mathematics met by students in the Sciences, Engineering and Commerce.
Not to be counted for credit with MATH1210.
Available in 2014
At the successful completion fo this course students will have 1. gained the necessary background to study further university level mathematics as required in their program of study. 2. gained mathematical knowledge and skills in the areas of calculus, functions, vectors and complex numbers. 3. improved their analytical ability, in particular their skills at problem-solving. |
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PROFESSOR STRANG: Just to give an overview in three lines: the text is the book of that name, Computational Science and Engineering. That was completed just last year, so it really ties pretty well with the course. I don't cover everything in the book, by all means. And I don't, certainly, don't stand here and read the book. That would be no good. But you'll be able, if you miss a class -- well, don't miss a class. But if you miss a class, you'll be able, probably, to see roughly what we did.
OK, so the first part of the semester is applied linear algebra. And I don't know how many of you have had a linear algebra course, and that's why I thought I would start with a quick review. And you'll catch on. I want matrices to come to life, actually. You know, instead of just being a four by four array of numbers, there are four by four, or n by n or m by n array of special numbers. They have a meaning. When they multiply a vector, they do something. And so it's just part of this first step is just, like, getting to recognize, what's that matrix doing? Where does it come from? What are its properties? So that's a theme at the start.
Then differential equations, like Laplace's equation, are beautiful examples. So here we get, especially, to numerical methods; finite differences, finite elements, above all. So I think in this class you'll really see how finite elements work, and other ideas. All sorts of ideas. And then the last part of the course is about Fourier. That's Fourier series, that you may have seen, and Fourier integrals. But also, highly important, Discrete Fourier Transform, DFT. That's a fundamental step for understanding what a signal contains. Yeah, so that's great stuff, Fourier.
OK, what else should I say before I start? I said this was my favorite course, and maybe I elaborate a little. Well, I think what I want to say is that I really feel my life is here to teach you and not to grade you. I'm not going to spend this semester worrying about grades, and please don't. They come out fine. We've got lots to learn. And I'll do my very best to explain it clearly. And I know you'll do your best. I know from experience. This class goes for it and does it right. So that's what makes it so good.
OK. Homeworks, by the way, well, the first homework will simply be a way to get a grade list, a list of everybody taking the course. They won't be graded in great detail. Too large a class. And you're allowed to talk to each other about homework. So homework is not an exam at all. So let me just leave any discussion of exams and grades for the future. I'll tell you, you'll see how informally the first homework will be. And I hope it'll go up on the website. The first homework will be for Monday. So it's a bit early, but it's pretty open-ended. If you could take three problems from 1.1, the first section of the book, any three, and any three problems from 1.2, and print your name on the homework -- because we're going to use that to create the grade list -- I'll be completely happy. Well, especially if you get them right and do them neatly and so on. But actually we won't know. So that's for Monday. OK. And we'll talk more about it. I'll announce the TA on the website and the TA hours, the office hours, and everything. There'll be a Friday afternoon office hour, because homeworks will typically come Monday.
OK. Questions about the course before I just start? OK. Another time for questions, too.
OK, so can we just start with that matrix? So I said about matrices, I'm interested in their properties. Like, I'm going to ask you about that. And then, I'm interested in their meaning. Where do they come from? You know, why that matrix instead of some other? And then, the numerical part is how do we deal with them? How do we solve a linear system with that coefficient matrix? What can we say about the solution? So the purpose. Right.
OK, now help me out. So I guess my plan with the video taping is, whatever you say, I'll repeat. So say it as clearly as possible, and it's fantastic to have discussion, conversation here. So I'll just repeat it so that it safely gets on the tape.
So tell me its properties. Tell me the first property that you notice about that matrix. Symmetric. Symmetric. Right. I could have slowed down a little and everybody probably would have said that at once. So that's a symmetric matrix. Now we might as well pick up some matrix notation. How do I express the fact that this a symmetric matrix? In simple matrix notation, I would say that K is the same as K transpose. The transpose, everybody knows, it comes from -- oh, I shouldn't say this -- flipping it across the diagonal. That's not a very "math" thing to do. But that's the way to visualize it. And let me use a capital T for transpose. So it's symmetric. Very important. Very, very important. That's the most important class of matrices, symmetric matrices. We'll see them all the time, because they come from equilibrium problems. They come from all sorts of -- they come everywhere in applications. And we will be doing applications. The first week or week and a half, you'll see pretty much discussion of matrices and the reasons, what their meaning is. And then we'll get to physical applications; mechanics and more.
OK. All right. Now I'm looking for properties, other properties, of that matrix. Let me write "2" here so that you got a spot to put it. What are you going to tell me next about that matrix? Periodic. Well, okay. Actually, that's a good question. Let me write periodic down here. You're using that word, because somehow that pattern is suggesting something. But you'll see I have a little more to add before I would use the word periodic. So that's great to see that here.
What else? Somebody else was going to say something. Please. Sparse! Oh, very good. Sparse. That's also an obvious property that you see from looking at the matrix. What does sparse mean? Mostly zeros. Well that isn't mostly zeros, I guess. I mean, that's got what, out of sixteen entries, it's got six zeros. That doesn't sound like sparse. But when I grow the matrix -- because this is just a four by four. I would even call this one K_4. When the matrix grows to 100 by 100, then you really see it as sparse. So if that matrix was 100 by 100, how many non-zeros would it have? So if n is 100, then the number of non-zeros -- wow, that's the first MATLAB command I've written. A number of non-zeros of K would be -- anybody know what it would be? I'm just asking to go up to five by five. I'm asking you to keep that pattern alive. Twos on the diagonal, minus ones above and below. So yeah, so 298, would it be? A hundred diagonal entries, 99 and 99, maybe 298? 298 out of 100 by 100 would be what? It's been a long summer. Yeah, a lot of zeros. A lot. Right. Because the matrix has got what 100 x 100, 10,000 entries. Out of 10,000. So that's sparse. But we see those all the time, and fortunately we do. Because, of course, this matrix, or even 100 by 100, we could deal with if it was dense. But 10,000, 100,000, or 1 million, which happens all the time now in scientific computation. A million by million dense matrix is not a nice thing to think about. A million by million matrix like this is a cinch.
OK. So sparse. What else do you want to say? Toeplitz. Holy Moses. Exactly right. But I want to say, before I use that word, so that'll be my second MATLAB command. Thanks. Toeplitz. What's that mean? So this matrix has a property that we see right away, which is? I want to stay with Toeplitz but everybody tell me something more about properties of that matrix.
Tridiagonal. Tridiagonal, so that's almost a special subcase of sparse. It has just three diagonals. Tridiagonal matrices are truly important. They come in all the time, we'll see that they come from second order differential equations, which are, thanks to Newton, the big ones. Ok, now it's more than tridiagonal and what more? So what further, we're getting deeper now. What patterns do you see beyond just tridiagonal, because tridiagonal would allow any numbers there but those are not, there's more of a pattern than just three diagonals, what is it? Those diagonals are constant. If I run down each of those three diagonals, I see the same number. Twos, minus ones, minus ones, and that's what the word Toeplitz means. Toeplitz is constant diagonal. Ok. And that kind of matrix is so important. It corresponds, yeah, if we were in EE, I would use the words time invariant filter, linear time invariant. So it's linear because we're dealing with a matrix. And it's time invariant, shift invariant. I just use all these equivalent words to mean that we're seeing the same thing row by row, except of course, at shall I call that the boundary? That's like, the end of the system and this is like the other end and there it's chopped off. But if it was ten by ten I would see that row eight times. 100 by 100 I'd see it 98 times. So it's constant diagonals and the guy who first studied that was Toeplitz. And we wouldn't need that great historical information except that MATLAB created a command to create that matrix. K, MATLAB is all set to create Toeplitz matrices. Yeah, so I'll have to put what MATLAB would put. I realize I'm already using the word MATLAB.
I think that MATLAB language is really convenient to talk about linear algebra. And how many know MATLAB or have used it? Yeah. You know it better than I. I talk a good line with MATLAB but I, the code never runs. Never! I always forget some stupid semicolon. You may have had that experience. And I just want to say it now that there are other languages, and if you want to do homeworks and want to do your own work in other languages, that makes sense. So the older established alternatives were Mathematica and Maple and those two have symbolic, they can deal with algebra as well as numbers. But there are newer languages. I don't know if you know them. I just know my friends say, Yes they're terrific. Python is one. And R. I've just had a email saying, Tell your class about R. And others. Ok, so but we'll use MATLAB language because that's really a good common language.
Ok, so what is a Toeplitz matrix? A Toeplitz matrix is one with constant diagonals. You could use the word time invariant, linear time invariant filter. And to create K, this is an 18.085 command. It's just set up for us. I can create K by telling the system the first row. Two, minus one, zero, zero. That would, then if it wasn't symmetric I would have to give the first column also. Toeplitz would be constant diagonal, it doesn't have to be symmetric. But if it's symmetric, then the first row and first column are the same vector, so I just have to give that vector. Okay, so that's the quickest way to create K. And of course, if it was bigger then I would, rather than writing 100 zeros, I could put zeros of 98 and one. Wouldn't I have to say that? Or is it one and 98? You see why it doesn't run. Well I guess I'm thinking of that as a row. I don't know. Anyway.
I realize getting this videotaped means I'm supposed to get things right! Usually it's like, we'll get it right later. But anyway, that might work. Okay. So there's a command that you know. Zeros that creates a matrix of this size with all zeros. Okay. That would create the 100 by 100. Good. Ok. Oh, by the way, as long as we're speaking about computation I've gotta say something more. We said that the matrix is sparse. And this 100 by 100 matrix is certainly sparse. But if I create it this way, I've created all those zeros and if I ask MATLAB to work with that matrix, to square it or whatever, it would carry all those zeros and do all those zero computations. In other words, it would treat K like a dense matrix and it would just, it wouldn't know the zeros were there until it looked. So I just want to say that if you have really big systems Sparse MATLAB is the way to go. Because Sparse MATLAB keeps track only of the non-zeros. So it knows-- and their locations, of course. What the numbers are and their location. So I could create a sparse matrix out of that, like KS for K sparse. I think if I just did sparse(K) that would create a sparse matrix. And then if I do stuff to it, MATLAB would automatically know those zeros were there and not spend it's time multiplying by zero But of course, this isn't perfect because I've created the big matrix before sparsifying it. And better to have created it in the first place as a sparse matrix. Ok.
So those were properties that you could see. Now I'm looking for little deeper. What's the first question I would ask about a matrix if I have to solve a system of equations, say KU=F or something. I got a 4 by 4 matrix, four equations, four unknowns. What would I want to know next? Is it invertible? Is the matrix invertible? And that's an important question and how do you recognize an invertible matrix? This one is invertible. So let me say K is invertible. And what does that mean? That means that there's another matrix, K inverse such that K times K inverse is the identity matrix. The identity matrix in MATLAB would be eye(n) and it's the diagonal matrix of one. It's the unit matrix is the matrix that doesn't do anything to a vector. So this K has an inverse. But how do you know? How can you recognize that a matrix is invertible? Because obviously that's a critical question and many, many-- since our matrices are not-- a random matrix would be invertible, for sure, but our matrices have patterns, they're created out of a problem and the question of whether that matrix is invertible is fundamental. I mean finite elements has these, zero energy modes that you have to watch out for because, what are they? They produce non-invertible stiffness matrix. Ok. So how did we know, or how could we know that this K is invertible? Somebody said invertible and I wrote it down. Yeah?
Well ok. Now I get to make a speech about determinants. Don't deal with them! Don't touch determinants. I mean this particular four by four happens to have a nice determinant. I think it's five. But if it was a 100 by 100 how would we show that the matrix was invertible? And what I mean by this is the whole family is invertible. All sizes are invertible. K_ n is invertible for every n, not just this particular guy, whose determinant we could take. But as five by five, six by six, we would be up in the-- but you're completely right. The determinant is a test. Alright. But I guess I'm saying that it's not the test that I would use.
So what I do? I would row reduce. That's the default option in linear algebra. If you don't know what to do with a matrix, if you want to see what's going on, row reduce. What does that mean? That means, shall I try it? So let me just start it just so I'm not using a word that we don't need. Ok. And actually, maybe the third lecture, maybe next Monday we'll come back to row reduce. So I won't make heavy weather of that, certainly not now. So what is row reduce, just so you know. I want to get that minus one to be a zero. I'm aiming for a triangular matrix. I want to clean out below the diagonal because if my matrix is triangular then I can see immediately everything. Right? Ultimately I'll reach a matrix U that'll be upper triangular and that first row won't change but the second row will change. And what does it change to? How do I clean out, get a zero in that where the minus one is right now? Well I want to use the first row, the first equation. I want to add some multiple of the first row to the second row. And what should that multiple be? I want to multiply that row by something. And I'll say "add" today. Later I'll say "subtract." But what shall I do? Just tell me what the heck to do. I've got that row and I want to use it, I want to take a combination of these two rows. This row and some multiple of this one that'll produce a zero. This is called the pivot. That's the first pivot P-I-V-O-T. Pivot. And then that's the pivot row. And what do I do? Tell me what to do. Add half this row to this one. When I add half of that row to that one, what do I get? I get that zero. What do I get here for the second pivot? What is it? 1.5, 3/2. Because half of that is, so 3/2. And the rest won't change. So I'm happy with that zero. Now I've got a couple more entries below that first pivot, but they're already zero. That's where the sparseness pays off. The tridiagonal really pays off. So those zeros say the first column is finished. So I'm ready to go on to the second column. It's like I got to this smaller problem with the 3/2 here. And a zero there. What do I do now? There is the second pivot, 3/2. Below it is a non-zero. I gotta get rid of it. What do I multiply by now? 2/3. 2/3 of that new, second row added to the third row will clean out the third row. This was already cleaned out. This is already a zero. But I want to have 2/3 of this row added to this one so what's my new third row? Starts with zero and what's the third pivot now? You see the pivots appearing? The third pivot will be 4/3 because I've got 2/3 this minus one and two is 6/3 so I have 6/3. I'm taking 2/3 away, I get 4/3 and that minus one is still there. So you see that I'm-- this is fast. This is really fast. And the next step, maybe you can see the beautiful patterns that are coming. Do you want to just guess the fourth pivot? 5/4, good guess, right. 5/4.
Now this is actually how MATLAB would find the determinant. It would do elimination. I call that elimination because it eliminated all those numbers below the diagonal and got zeros. Now what's the determinant? If I asked you for the determinant, and I will very rarely use the word determinant, but I guess I'm into it now, so tell me the determinant. Five. Why's that? I guess I did say five earlier. But how do you know it's five? Whatever the determinant of that matrix is, why is it five? Because it's a triangular matrix. Triangular matrices, you've got all these zeros. You can see what's happening. And the determinant of a triangular matrix is just the product down the diagonal. The product of these pivots. The determinant is the product of the pivots. And that's how MATLAB would compute a determinant. And it would take two times 3/2 times 4/3 times 5/4 and it would give answer five.
My friend Alan Edelman told me something yesterday. MATLAB computes in floating point. So 4/3, that's 1.3333, etc. So MATLAB would not, when it does that multiplication, get a whole number. Right? Because in MATLAB that would be 1.333 and probably it would make that last pivot a decimal, a long decimal. And then when it multiplies that it gets whatever it gets. But it's not exactly five I think. Nevertheless MATLAB will print the answer five. It's cheated actually. It's done that calculation and I don't know if it takes the nearest integer when it knows that the-- I shouldn't tell you this, this isn't even interesting. If the determinant of an integer matrix, whole number is a whole number, so MATLAB says, Better get a whole number. And somehow it gets one. Actually, it doesn't always get the right one. So maybe later I'll know the matrix whose determinant might not come out right. But ours is right, five.
Now where was this going? It got thrown off track by the determinant. What's the real test? Well so I said there are two ways to see that a matrix is invertible. Or not invertible. Here we're talking about the first way. How do I know that this matrix-- I've got an upper triangular matrix. When is it invertible? When is an upper triangular matrix invertible? Upper triangular is great. When you've got it in that form you should be able to see stuff. So this key question of invertible, which is not obvious for a typical matrix is obvious for a triangular matrix. And why? What's the test? Well, we could do the determinant but we can say it without using that long word. The diagonal is non-zero. K as invertible because the diagonal-- no, it's got a full set of pivots. It's got four non-zero pivots. That's what it takes. That's what it's going to take to solve systems. So this is the first step in solving this system. In other words, to decide if a matrix is invertible, you just go ahead and use it. You don't stop first necessarily to check invertibility. You go forward, you get to this point and you see non-zeros there and then you're practically got to the answer here. I'll leave for another day the final back to going back upwards that gives you the answer.
So K is invertible. That means full set of pivots. n non-zero pivots. And here they are, two, 3/2, 4/3 and 5/4. Worth knowing because this matrix K is so important. We'll see it over and over again. Part of my purpose today is to give some matrices a name because we'll see them again and you'll know them and you'll recognize them. While I'm on this invertible or not invertible business I want to ask you to change K. To make it not invertible. Change that matrix. How could I change that matrix? Well, of course, many ways. But I'm interested in another matrix and this'll be among my special matrices. And it will start out the same. It'll have these same diagonals. It'll be Toeplitz. I'm going to call it C and I want to say the reason I'm talking about it now is that it's not going to be invertible. And I'm going to tell you a C and see if you can tell me why it is not invertible. So here's the difference; I'm going to put minus one in the corners. Still zeros there. So that matrix C still has that pattern. It's still a Toeplitz matrix, actually. That would still be the matrix Toeplitz of two, minus one, zero, minus one. I claim that matrix is not invertible and I claim that we can see that without computing determinants, we can see it without doing elimination, too. MATLAB would see it by doing elimination. We can see it by just human intelligence. Now why? How do I recognize a matrix that's not invertible? And then, by converse, how a matrix that is invertible. I claim-- and let may say first, let me say why that letter C. That letter C stands for circulant. it's because this word circulant, why circulant, it's because that diagonal which only had three guys circled around to the fourth. This diagonal that only had three entries circled around to the fourth entry. This diagonal with two zeros circled around to the other two zeros. The diagonal are not only constant, they loop around. And you use the word periodic. Now for me, that's the periodic matrix. See, a circulant matrix comes from a periodic problem. Because it loops around. It brings numbers, zero is the same as number four or something. And why is that not invertible?
The thing is can you find a vector? Because matrices multiply vectors, that's their whole point. Can you see a vector that it takes to zero? Can you see a solution to Cu=0? I'm looking for a u with four entries so that I get four zeros. Do you see it? All ones. All ones. That will do it. So that's a nice, natural entry, a constant. And do you see why when I-- we haven't spoken about multiplying matrices times vectors. And most people will do it this way. And let's do this one this way. You take row one times that, you get two, minus one, zero, minus one. You get the zero because of that new number. Here we always got zero from the all ones vector and now over here that minus one, you see it's just right. If all the rows add to zero then this vector of all ones will be, I would use the word in the null space if you wanted a fancy word, a linear algebra word. What does that mean? It solves Cu=0. And why does that show that the matrix isn't invertible? Because that's our point here. I have a solution to Cu=0. I claim that the existence of such a solution has wiped out the possibility that the matrix is invertible because if it was invertible, what would this lead to? If invertible, if C inverse exists what would I do to that equation that would show me that C inverse can't exist? Multiply both sides by C inverse.
So you're seeing, just this first day you're seeing some of the natural steps of linear algebra. Row reduction, multiply when you want to see what's happening, multiply both sides by C inverse. That's the same as in ordinary language, Do the same thing to all the equations. So I multiply both sides by the same matrix. And here I would get (C inverse)(Cu)=(C inverse)(0). So what does that tell me? I made it long, I threw in this extra step. You were going to jump immediately to C inverse C is I is the identity matrix and when the identity matrix multiplies a vector u, you get u. And on the right side, C inverse, whatever it is if it existed, times zero would have to be zero. So this would say that if C inverse exists, then the only solution is u equals u. That's a good way to recognize invertible matrices. If it is invertible then the only solution to Cu=0 u=0. And that wasn't true here. So we conclude C is not invertible. C is therefore not invertible.
Now can I even jump in. I've got two more matrices that I want to tell you about that are also close cousins of K and C. But let me just explain physically a little bit about where these matrices are coming from. So maybe next to K-- so I'm not going to put periodic there. Right? That's the one that I would call periodic. This one is fixed at the ends. Can I draw a little picture that aims to show that? Aims to show where this is coming from. It's coming from I think of this as controlling like four masses. Mass one, mass two, mass three and mass four with springs attached and with endpoints fixed. So if I put some weights on those masses-- we'll do this; masses and springs is going to be the very first application and it will connect to all these matrices. And all I'm doing now is just asking to draw the system. Draw the mechanical system. Actually I'll usually draw it vertically. But anyway, it's got four masses and the fact that this minus one here got chopped off, what would I call that end? I'd call that a fixed end. So this is a fixed, fixed matrix. Both ends or fixed. And it's the matrix that would govern and the springs and masses all the same is what tells me that the thing is Toeplitz.
Now what's the picture that goes with C? What's the picture with C? Do you have an instinct of that? So C is periodic. So again we've got four masses connected by springs. But what's up with those masses to make the problem cyclic, periodic, circular, whatever word you like. They're arranged in a ring. The fourth guy comes back to the first one. So the four masses would be, so in some kind of a ring, the springs would connect them. I don't know if that's suggestive, but I hope so. And what's the point of, can we just speak about mechanics one moment? How does that system differ from this fixed system? Here the whole system can't move, right? If there no force, then nothing can happen. Here the whole system can turn. They can all displace the same amount and just turn without any compression of the springs, without any force having to do anything. And that's why the solution that kills this matrix is one, one, one, one. So one, one, one, one would describe a case where all the displacements were equal. In a way it's like the arbitrary constant in calculus. You're always adding plus C. So here we've got a solution of all ones that produces zero the way the derivative of a constant function is the zero function. So this is just like an indication.
Yes, perfect. I've got two more matrices. Are you okay for two more? Yes okay, what are they? Okay a different blackboard for the last two. So one of them is going to come by freeing up this end. So I'm going to take that support away. And you might imagine like a tower oscillating up and down or you might turn it upside down and like a hanging spring, or rather four springs with four masses hanging onto them. But this end is fixed and this is not fixed anymore, this is now free. And can I tell you the matrix, the free-fixed matrix. Free-fixed. Because it's the top end that I changed, I'm going to call it T. So all the other guys are going to be the same but the top one, the top row, the boundary row, boundary conditions are always the tough part, the tricky part, the key part of a model, and here the natural boundary condition is to have a one there. That two changed to a one. Now if I asked you for the properties of that matrix-- so that's the third. shall I do the fourth one? So you have them all, you'll have the whole picture.
The fourth one, well you can guess. What's the fourth? What am I going to do? Free up the other end. So this guy had one free end and the other guy has B for both ends. B for both ends are going to be free. So this is free-fixed. This'll be free-free. So that means I have this free end, the usual stuff in the middle, no change, and the last row is what? What am I going to put in the last row? Minus one, one. Minus one, one. So I've changed the diagonal. There I put a single one in because I freed up one end. With B I freed both ends and I got two minus ones. Now what do you think? So we've drawn the free-fixed one and what's your guess? They're all symmetric. That's no accident. They're all tridiagonal, no accident again. Why are they tridiagonal? Physically they're tridiagonal because that mass is only connected to it's two neighbors, it's not connected to that mass. That's why we get a zero in the two, four position. Because two is not connected to four. So it's tridiagonal. And it's not Toeplitz anymore, right? Toeplitz says constant diagonals and these are not quite constant. I would create K, I would take T equal K if I was going to create this matrix and then I would say T of one, one equal one. That command would fix up the first entry.
Yeah, that's a serious question. Maybe, can I hang on until Friday, and even maybe next week. Because it's very important. When I said boundary conditions are the key to problems, I'm serious. If I had to think okay, what do people come in my office ask about questions, I say right away, What's the boundary condition? Because I know that's where the problem is. And so here we'll see these guys clearly. Fixed and free, very important. But also let me say two more words, I never can resist. So fixed means the displacement is zero. Something was set to zero. The fifth guy, the fifth over here, that fifth column was knocked out. Free means that in here it could mean that the fifth guy is the same as the fourth. The slope is zero. Fixed is u is zero. Free is slope is zero. So here I have a slope of zero at that end, here I have it at both ends. So maybe that's a sort of part answer.
Now I wanted to get to the difference between these two matrices. And the main properties. So what are we see? Symmetric again, tridiagonal again, not quite Toeplitz, but almost, sort of morally Toeplitz. But then the key question was invertible or not. Key question was invertible or not. Right. And what's your guess on these two? Do you think that one's invertible or not? Make a guess. You're allowed to guess. Yeah it is. Why's that? Because this thing has still got a support. It's not free to shift forever. It's held in there. So that gives you a hint about this guy. Invertible or not for B? No. And now prove that it's not. Physically you were saying, well this free guy with this thing gone now, this is now free-free. Physically we're saying the whole thing can move, there's nothing holding it. But now, for linear algebra, that's not the proper language. You have to say something about that matrix. Maybe tell me something about Bu=0. u What are you going to take for u? Yeah. Same u. We're lucky in this course, u equal is the guilty main vector many times. Because again the rows are all adding to zero and the all ones vector is in the null space.
If I could just close with one more word. Because it's the most important. Two words, two words. Because they're the most important words, they're the words that we're leading to in this chapter. And I'm assuming that for most people they will be new words, but not for all. It's a further property of this matrix. So we've got, how many? Four properties, or five? I'm going to go for one more. And I'm just going to say that name first so you know it's coming.
And then I'll say, I can't resist saying a tiny bit about it. I'll use a whole blackboard for this. So I'm going to say that K and T are, here it comes, take a breath; positive definite matrices. So if you don't know what that means, I'm happy. Right? Because well, I can tell you one way to recognize a positive definite matrix. And while we're at it, let me tell you about C and B. Those are positive semi-definite because they hit zero somehow. Positive means up there, greater than zero. And what is greater than zero that we've already seen? And we'll say more. The pivots were.
So if I have a symmetric matrix and the pivots are all positive then that matrix is not only invertible, because I'm in good shape, the determinant isn't zero, I can go backwards and do everything, those positive numbers are telling me that more than that, the matrix is positive definite. So that's a test.
We'll say more about positive definite, but one way to recognize it is compute the pivots by elimination. Are they positive? We'll see that all the eigenvalues are positive. The word positive definite just brings the whole of linear algebra together. It connects to pivots, it connects to eigenvalues, it connects to least squares, it's all over the place. Determinants too. Questions or discussion. It's a big class and we're just meeting for the first time but there's lots of time to, chance to ask me. I'll always be here after class.
So shall we stop today? I'll see you Friday or this afternoon. If this wasn't familiar, this afternoon would be a good idea. Thank you |
A course of elementary mathematics : affording aid to candidates for admission into either of the military colleges, to applicants for appointments in the Indian civil service, and to students of mathematics generally online
The preparation necessary for the profitable study of the following
course of Mathematics is a knowledge of common x\rithmetic, and some
acquaintance with the principles of Geometry, as taught in Euclid's
Elements. A student ignorant of these initiatory, but most important
departments of elementary science, would scarcely seek his first lessons
therein from a book such as this. The Elements of Euclid is a work
by itself ; universally known and esteemed, and everywhere to be easily
procured : to transfer its pages to the present performance, could be of
no possible advantage to the learner. And the same may be said of
common Arithmetic : both this and Euclid are more conveniently studied
from the ordinary manuals in popular use. We shall therefore commence
the volume now in the hands of the reader, with a treatise on Algebea
the indispensable foundation of the entire fabric of modern analytical
science.
I. ALGEBRA.
1. Preliminary Notions. Algebra may be regarded simply as
an extension of the principles of Arithmetic. In the latter science the
symbols of quantity, to which its rules and operations are applied, are
limited to the nine digits or figures 1, 2, 3, 4, 5, 6, 7, 8, 9, together with
the cypJier or zero, 0. And not only is the notation of Arithmetic limited
to these ten symbols, but each symbol is employed by every computer in
the same sense : the character or symbol 4, for instance, stands for four,
always ; 6 for six; 8 for eight, and so on : the symbols of Arithmetic are
thus fixed in meaning, as well as limited in number.
It is otherwise in Algebra: in this science the symbols of quantity
comprehend not only the figures of arithmetic, but also the letters of the
alphabet : the figures being, as in arithmetic, of invariable signification,
but the letters admitting of arbitrary interpretation. It is this latter
circumstance namely, the possession of a set of symbols which we may
employ to represent anything we please that gives to Algebra its pecu-
liarity and its power. In Arithmetic, known quantities only can be
denoted by symbols : in Algebra a quantity altogether unknown, in value,
at the outset of an inquiry, may be represented an alphabetical letter
serving this purpose, and then the rules of the science, to be hereafter
developed, will enable us ultimately to interpret its meaning, consistently
-84592
2 DEFINITIONS SYMBOLS OF QUANTITY SIGNS OF OPERATION.
with the conditions which connect it, in that inquiry, with the known
quantities concerned.
2. Definitions Symbols of QuantitySigns of Opera-
tion. As noticed above, the symbols by which the quantities operated
upon in algebra are represented, are the figures of ordinary arithmetic,
and the letters of the alphabet : the marks or signs by which these opera-
tions are indicated, are called signs of operation : the principal of these
are the following :
+ , plus, the sign of addition, implying that the quantity to which it is
prefixed is to be added.
, minus, the sign of subtraction, denoting that the quantity to which
it is prefixed is to be subtracted.
Thus 5 + 2, which is read 5 plus 2, signifies that 2 is to be added to
5 ; and 52, which is read 5 minus 2, indicates that 2 is to be subtracted
from 5. In like manner a+b, or a plus 6, implies that b is to be added
to a, that is, that the quantity represented by b is to be added to that
represented by a. And ab, or a minus b, implies that b is to be sub-
tracted from a. Of course we cannot actually perforin the addition and
subtraction operations thus indicated, till we know what numbers or
quantities a and b stand for.
It may be remarked here, that although the letters a, 6, &c. are but
the representatives of quantities or numerical values, yet, for brevity of
expression, we refer to them as the quantities themselves.
The crooked mark ~ placed between two quantities denotes the dif-
ference between those quantities : thus a~6 means the difference between
a and b, whether that difference be the result of subtracting b from a, or
a from b.
X , the sign of multiplication, when placed between two quantities,
implies that those quantities are to be multiplied together : thus 4x6,
or 6 X 4 means that 4 and 6 are to be multiplied together, and axb,or
bxa, implies in like manner the product of a and b.
Instead of the sign x , a dot placed between the factors is often used
for the sign of multiplication: thus 4.6, or 6.4, and a.b, or b.a, each
implies the product of the quantities between which the dot is placed.
It must be observed, however, that the dot should range with the lower
part of the figures or letters, and not with the upper part, to avoid con-
founding it with the decimal point, as, in the case of figures, might
otherwise happen: thus 6.4 means 24, but 6-4 means 6 and 4 tenths.
In the case of letters however, the dot is usually dispensed with alto-
gether, and the factors simply written side by side, without any inter-
vening sign at all : thus, ab, ex, bxy, abxz, &c. mean the same as a x ft.
cxx, bxxxy, axbxxxz; or as a.b, c.x, b.x.y, a.b.x.z, &c. This
suppression of the intervening sign of multiplication between the fac-
tors is not allowable when those factors are numbers, as is obvious : if
6 X 4, or 6.4 were written 64, sixty-four would be implied, and not 24,
as intended. But when a single numerical factor enters with the letters,
then the multiplying sign may be omitted, since no ambiguity can arise :
thus, 6xax6 or Q.a.b, may be more conveniently written Qdb, which
means 6 times the product of a and h, or as it is' more briefly read,
6 times a, b. It is proper, as here, always to place the numerical factor
first, and the literal factors afterwards ; and also to arrange these latter
in the order in which they succeed each other in the alphabet. The
numerical factor, thus placed first, is called the coefficient of the quantity
DEFINITIONS SYMBOLS OF QUANTITY SIGNS OF OPERATION. 8
multiplied by it: thus 6 is the coefiBcient of ah in 6a6, and 15 is the
coefficient of xyz in Ibccyz.
-T-, the sign of division, when placed before a quantity, supplies the
place of the words '< divided by," so that 8-r-2 means 8 divided by 2,
12-r3 means 12 divided by 3, a-^-b means a divided by b, and so on;
but, as in common arithmetic, division is more frequently indicated by
writing the dividend above and the divisor below a horizontal bar of
separation, thus:
is the same as a-r6, and ~ is the same as Zopy-r-2db.
zao
8. The four signs now explained indicating the four fundamental
operations both of arithmetic and algebra are, of course, those of most
frequent occurrence in calculation. Algebraists, however, economize their
signs of operation as much as possible, and never introduce them need-
lessly. This has been already exemplified in the case of multiplication :
the absence of sign between letters, placed side by side, as much implies
the multiplication together of the numbers those letters represent, as if
each were separated from the others by an oblique cross, or a dot. In
like manner, when a row of additive and subtractive quantities are con-
nected together by the proper signs, if the Jlrst of these quantities be
additive, or plus, the sign + is suppressed as superfluous; thus : a 6 +
c-f-rf 4 is the same as -f-a & + c+<? 4, and implies that a, c, and d
are additive ; or, as they are more frequently called, positive quantities,
and that b, and 4, are subtractive, or negative quantities. If the letters
a, b, c, d stood respectively for 2, 4, 3, 8, the interpretation of the
expression just written would be 5.
4. The term coefficient has been already defined : it is the numerical
multiplier of the algebraic quantity to which it is prefixed : when this
numerical multiplier is simply 1, it is not inserted : it is superfluous to
introduce unit-factors; a-\-h,is as well understood to be once a plus once
b, as \a-\-\h', but if the question were asked What is the coefficient
of <* or of 6 in the expression a -ffe? the answer would be, not nothing y
but 1.
6. =, equal to, is the sign of equality: it implies that what is written
on one side of it is equal to what is written on the other, thus :
7-f-4=ll, 7-4=3, 7-4-1-1=4, Bx-\-2x-x=ix, &c.
6. Any quantity of how many letters soever it may consist is called
a simple quantity, or a quantity of but one term, provided it be not sepa-
rated into distinct parts by the interposition of a plus or minus sign ;
thus, each of the following is a simple quantity, or a quantity of but one
term :
^ _ ^ _ 2ax 14 ,
Sahx, labay, ^, , &c.
7. Each of the following, however, is a compound quantity : the first
consists of two terms, the second of three terms, and the third of four
terms :
4
2a-i-86, 6a2b-\-c, 5ah+2cd~-Bm-\ - .
8. We shall now add a few exercises by which the learner may satisfy
himself as to whether he correctly understands what has already been
explained or not.
B ti
4 ADDITION OF ALGEBRA.
Exercises on the Definitions. In the following exercises we shall
suppose a=4:, 6^3, c=5, and d=7.
9. Addition of Algebra. In Arithmetic, addition means the
collection into one sum of a set of quantities, all of which are additive or
positive : in Algebra the term is extended to the finding the aggregate
or balance of a set of quantities, some of which only may be positive, and
the others negative ; the result of such addition being plus or minus
according as the sum of the positive, or the sum of the negative terms,
preponderate. The sum of a set of algebraic quantities in the absence
of all interpretation of the symbols can be exhibited only when the
quantities are all alike; that is, when, so far as the letters are concerned,
they do not differ from one another. That a set of like quantities may be
added together, without our requiring to know what those quantities are,
is plain: thus, it is clear that 2a + 3a + 5a=10(2, whatever a may stand
for, and that 66 + 3a6 4a6=5a6, whatever ah may stand for. If how-
ever the quantities are not all like quantities, then to incorporate the
entire set into a single term as here, would be impossible. For instance,
if we had the set of quantities 4ax + 2a^ 3a^ + 26, all we could do would
be to actually collect the first three into one term, and then to annex to
the sum the fourth term 26 with its proper sign ; we should thus say that
4aa; + 2aic 3aa; + 26=3a2' + 26, an expression which we cannot further
reduce or simplify till we know something about the values of the letters.
10. Addition therefore divides itself into two cases : 1. When the
quantities to be added are all like quantities, and 2. When they are not
all like quantities.
Case I, When the quantities are all like quantities, that is, when
they differ in nothing but in their coefficients.
Rule I. Find the sum of the positive coefficients. 2. Find the sum
of the negative coefficients. 3. Take the difference of these two sums,
prefix to it the sign of the greater sum, and then annex the letters
common to all the quantities : the correct sum will thus be exhibited.
Note. When there are two or more columns to be added up, we
always commence with the first column on the left, and not, as in
arithmetic, with the first on the right, as it is more convenient to write
ADDITION OF ALGEBRA.
the several results, with their signs, from left to right, than from right
to left.
The following four examples are worked : the learner should clearly see
how the results are obtained, and satisfy himself of their accuracy, before
attempting the exercises below.
EuLE. Collect the like quantities from the several expressions, and
add them together : to the sum connect, with their proper signs, those of
the quantities which have no like.
Note. Athough it is of no consequence which set of like quantities
be added first, yet the custom is to commence with the quantity at the
top of the Jirst column on the left, and to put down, under that column,
the sum of all the quantities like it; then to collect the quantities like that
at the top of the second column, and so on, as in the following examples :
(1)
2x7y-{- iz
8z-\-2x y
2y-{-5z-\- a
4x-Zz+ 7y
8a;-3y+142+a
(2)
7ax2by-{-z
Zby -\-9z ax
62 -\-2ax-\-by
5by-Zax-\-Q
(3)
AocyzZocy-\- 2yz
5xy -\-Syz 7xyz
-9ax +xyz+ll
-iyz -\-7xy 8
5ax+7by+4z-^e 2xyz-{-9xy-\- 6yz9ax-\-d
12. ExEECisEs TO BE WORKED. In the following examples the
several expressions may be taken as they are, and placed one under
another as above ; or the arrangement of the terms may be changed, so |
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Discrete Math
Discrete mathematics is a branch of math that deals with structures having specific values, as opposed to continuously varying values. There are analogies in signal processing to digital and analog signals, and in science to quantum and classical modeling of atomic and molecular systems. Discrete math is naturally extended, through computer programming algorithms, to analysis of complex infrastructure systems like telecommunication routing, utility distribution, highway and traffic pattern studies, and many others.
Any course in discrete math will cover the following topics:
logic and proofs
sets, functions, sequences and sums
algorithms, integers, and matrices
induction and recursion
mathematical induction
counting
discrete probability
advanced counting techniques
relations
graphs
trees
boolean algebra
modeling computation
A fabulous resource for discrete mathematics can be found at the online forum Science Direct, where refereed journal articles are available, although not for free. This is the real deal, where the latest research is being done in the field.
To fulfill our mission of educating students, our online tutoring centers are standing by 24/7, ready to assist students who need extra practice in discrete math. |
East Camden, NJ StatisticsFirst, I might provide an example that starts with any combination of the two ideas. Then, I would integrate all three concepts into one problem to see if the student can separate the question into the building blocks from Step 2 to decompose a potentially tricky question. Furthermore, I might introduce problems and potential traps that might come up with derivatives? |
Analytic Trigonometry with Applications - 7th edition
ISBN13:978-0470000960 ISBN10: 0470000961 This edition has also been released as: ISBN13: 978-0534358389 ISBN10: 0534358381
Summary: Best-selling authors Raymond Barnett, Michael Ziegler, and Karl Byleen wrote this accurate, time-tested book for students to enjoy reading. Their use of numerous applications from a variety of fields convinces even the most skeptical students that trigonometry is truly useful. To gain students' interest quickly, the text moves directly into trigonometric concepts and applications, and reviews essential material from prerequisite courses only as needed.
Unusual and interesting exercises from many different fields provide variety and relevance. Both application-oriented exercises as well as standard drill are included.
Coverage begins with a right triangle approach . The authors cover the basics of right triangle applications, properties of circular trigonometric functions, elementary trigonometric identities, and the law of sines and cosines.
The authors organize material in a natural manner, focusing first on classical applications of trigonometric ratios, then upon the more sophisticated concept of these ratios as functions.
Color is used functionally to clarify figures and identify important definitions, theorems, processes, and comments. Many competitors are two color.
Graphing calculators enhance learning where appropriate, although they are not required. Problems for graphing calculators are clearly identified by a calculator icon and can be omitted without loss of continuity. The interplay of numeric, symbolic, graphic, and verbal interpretations substantially deepens student understanding.
Almost every concept is illustrated by an example followed by a matching problem to encourage an active rather than passive involvement in the learning process.
Explore-Discuss Boxes (1-4) appear in each section of the text and offer several problems that can be used for written student responses, class discussions, or as group activities. These problems either motivate or expand upon the section's material.
A comprehensive Group Activity Box has been added to the end of each chapter.
Red numbers indicate exercises that involve the written communication of mathematics.
Added emphasis on concept development, understanding and communication, rather than drill
Mathematical reform elements have been incorporated throughout the book using the rule of four. The interplay of numeric, symbolic, graphing and verbal interpretations substantially deepens the understanding of many concepts.
Key pedagogical elements help student comprehension, such as annotations of examples and developments to help through critical stages. Caution warnings identified by an icon alert students to potential problem areas, chapter reviews, and generic calculator steps, and dashed 'think boxes' are all included to enhance student understanding.
The content satisfies the requirements for many succeeding courses, including calculus, analytic geometry, physics, and technical mathematics courses.
More than 2,000 problems are graded in difficulty from the routine to the challenging. Answers to most of the odd-numbered problems and almost all chapter and cumulative review exercises are included at the end of the book.
More graphing material has been added and interspersed in context throughout the book. New material that requires the use of a graphing calculator is identified by an icon, and can be omitted without loss of continuity |
Synopses & Reviews
Publisher Comments:
Complex analysis is one of the most attractive of all the core topics in an undergraduate mathematics course. Its importance to applications means that it can be studied both from a very pure perspective and a very applied perspective. This book takes account of these varying needs and backgrounds and provides a self-study text for students in mathematics, science and engineering. Beginning with a summary of what the student needs to know at the outset, it covers all the topics likely to feature in a first course in the subject, including: complex numbers, differentiation, integration, Cauchy's theorem, and its consequences, Laurent series and the residue theorem, applications of contour integration, conformal mappings, and harmonic functions. A brief final chapter explains the Riemann hypothesis, the most celebrated of all the unsolved problems in mathematics, and ends with a short descriptive account of iteration, Julia sets and the Mandelbrot set. Clear and careful explanations are backed up with worked examples and more than 100 exercises, for which full solutions are provided.
Synopsis:
Table of Contents
What Do I Need to Know?- Complex Numbers.- Prelude to Complex Analysis.- Differentiation.- Complex Integration.- Cauchy's Theorem.- Some Consequences of Cauchy's Theorem.- Laurent Series and the Residue Theorem.- Applications of Contour Integration.- Further Topics.- Conformal Mappings.- Final Remarks.- Solutions to Exercises.- Bibliography.- Index.
"Synopsis"
by Springer, |
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mediate Algebra: Concepts and Applications
The Bittinger Concepts and Applications Program delivers proven pedagogy, guiding students from skills-based math to the concepts-oriented math ...Show synopsisThe Bittinger Concepts and Applications Program delivers proven pedagogy, guiding students from skills-based math to the concepts-oriented math required for college courses.Hide synopsis
Description:Good. Used-Good Hardcover. Annotated instructor's edition. Book...Good. Used-Good Hardcover. Annotated instructor's edition. Book contains same material as original text, but includes the answers or additional annotations. 9th |
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