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Key to... Workbooks
Key to… workbooks are accessible, using a non-intimidating step-by-step approach, and simple language and clear visual models that help all students grasp concepts quickly.
Key to… workbooks are effective, with fully worked-out examples that help students overcome obstacles and build upon their skills. And a large number of problems ensures completely mastery.
Key to… workbooks are flexible. Depending on your students' needs, you can use Key to... workbooks for basic skills development, remediation, review, or standardized test preparation. Clear and direct instructions enable your students to work on their own and at their own pace.
Key to… workbooks are time saving, with complete lessons so you don't need to prepare worksheets or copy pages.
Note: Key to... workbooks are now published by McGraw-Hill Education and available for purchase at MHEonline.com. If you're a customer in a U.S. territory or outside the United States, find ordering information here.
"I'm a Resource Specialist at the middle school level and have experience teaching high school students in juvenile hall. I've used the Key to… materials in both environments, and they are absolutely wonderful!" —Leyla Momeny, James Lick Middle School, San Francisco, CA | 677.169 | 1 |
CORE MATHEMATICS VERSUS MATHEMATICAL LITERACY
WHY CORE MATHS?
Core Mathematics is an intellectual discipline, an art form and a challenging
game. Core Mathematics is an abstract course, and develops thinking and
problem solving skills which are in high demand in the workplace.
Careers in Mathematics include, all types of Engineering, Mathematical
Sciences, IT, Medicine, Maths Education, Statistics, Finance and Actuarial
Fields, Biomathematics and Biostatics, Computer Science, Operations research.
If a learner is coping well with the course, we recommend that it is taken so as
not to limit career choices later on.
HOWEVER, YOU CAN STILL GO ON TO TERTIARY STUDY WITH
MATHEMATICAL LITERACY!!!!!
WHAT IS MATHEMATICAL LITERACY?
Mathematical Literacy is a course, driven by life-related applications of
Mathematics. It develops the ability and confidence of the learners to think
numerically and spatially, which leads to critical analysis and interpretation of
everyday situations. Mathematical Literacy is a more concrete subject than core
Mathematics and suited to many careers.
WHAT IS THE PURPOSE OF MATHEMATICAL LITERACY?
The purpose of Mathematical Literacy is to provide the learner with the ability and
skills to understand mathematical terminology, and make sense of numerical and
spatial information encountered in every day life, (e.g. tables, statistical trends,
quotations, areas, volumes, percentages, graphs, diagrams, text, finance, bond
rates, interest rates, budgets, ratio and proportion)
Mathematical Literacy focuses on developing a self-managing individual and a
contributing and participating employee when exposed to mathematical issues.
Mathematical literacy contributes to entrepreneurial success.
A BRIEF COMPARISON BETWEEN MATHS AND MATHS LITERACY (NEW
FET CURRICULUM
MATHEMATICS MATHEMATICAL LITERACY
Mathematics focuses on the discipline Mathematical literacy focuses on the
of Mathematics, incorporating abstract role of mathematics in the real world
and hypothetical thinking. using relevant examples in day to day
life.
Applications are most important, not The contexts chosen are employment
necessarily in real life contexts. based being current and relevant.
Content is also emphasized.
Content is expanded on as the learners The contexts become more advanced
progress annually. as the learners progress annually.
Mathematics is designed for those wish Maths Literacy is designed for learners
to pursue careers in the natural wanting to pursue tertiary qualifications
sciences or engineering. in the social and life sciences, e.g. law,
marketing, advertising etc, or
entrepreneurs who wish to start their
own businesses | 677.169 | 1 |
Avondale, AZ AlgebraHasina F.
...When I was a Preschool Teacher, I worked with children with autism. These children had different types of autism. I understand how autism affects a child's social interaction and their learning.
Linda S.eenan M.Jonathan H.
...The basics is the understanding of matrices and the Gauss-Jordan Method. Later you get into inverses, proofs of a vector space(zero, scalar, addition), eigen values, dot product, and much more. Differential equations and Mathematical structures are good prerequisites to take before starting Linear Algebra. | 677.169 | 1 |
Numerous examples and indexed definitions make this detailed guide to polynomial theory in error-correcting codes a highly accessible resource. New codes and a unitary approach to block and convolutional codes will enhance readers? understanding of the topic. more...
Technology, Applications, and Computation
Ancient Roots
Analog or Digital?
Where Are We Now?
Arithmetic and DSP
Discrete Fourier Transform (DFT)
Arithmetic Considerations
Convolution Filtering with Exact Arithmetic
The Double-Base Number System (DBNS)
Motivation
The Double-Base Number System
The Greedy Algorithm
Reduction... more...
Digital arithmetic plays an important role in the design of general-purpose digital processors and of embedded systems for signal processing, graphics, and communications. In spite of a mature body of knowledge in digital arithmetic, each new generation of processors or digital systems creates new arithmetic design problems. Designers, researchers,... more...
The present book deals with the theory of computer arithmetic, its implementation on digital computers and applications in applied mathematics to compute highly accurate and mathematically verified results. The aim is to improve the accuracy of numerical computing (by implementing advanced computer arithmetic) and to control the quality of... more...
Mi Lu received her MS and PhD in electrical engineering from Rice University, Houston. She joined the Department of Electrical Engineering at Texas A&M University in 1987 and is currently a professor. Her research interests include computer arithmetic, parallel computing, parallel computer architectures, VLSI algorithms, and computer networks.... more...
This is the first book promoting Mathematical Philosophy as an interdisciplinary field. It is a collection of articles applying methods of logic and math to solve problems, some from logic itself, others from other sciences. more...
This handbook aims to provide a complete overview of modern floating-point arithmetic. This includes a detailed treatment of the current (IEEE-754) and next (preliminarily called IEEE-754R) standards for floating-point arithmetic. more... | 677.169 | 1 |
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Quantity & Units in Mathematica 9
Nick Lariviere
In this talk from the Wolfram Technology Conference, Nick Lariviere gives a summary of the features and functions of Mathematica's unit system and presents examples from various disciplines, including statistics, fluid dynamics, and general equation solving.
Mathematica provides several convenient ways to find information about functions. In addition to searching the documentation or navigating the guide pages, you can access documentation on functions directly ...
The Virtual Book is a browsable electronic collection of all the Mathematica tutorials grouped according to functionality. It is an excellent place for users of all experience levels to gain more detailed knowledge of Mathematica. Learn more in this "How ... | 677.169 | 1 |
Description
This Smart Worksheet is on complex numbers and takes you through the standard properties and operations on complex numbers. For detailed syllabus content, please see the video listed below.
I've included are a host of tools that allow you to enter problems of your own and check your answers and there's also an email? if you have any questions you'd like to submit.
Hope you enjoy the worksheet and any feedback is always appreciated so give us a buzz and send us an email.
Cheers,
Mark | 677.169 | 1 |
Mathematics Program Goals and Objectives
Goal #1: Problem-Solving.
To develop student's ability to apply both conventional and creative techniques to the solution of mathematical problems.Student Learning Objectives:Students graduating with a major in mathematics will | 677.169 | 1 |
Re: More Algebra Pages
MathsIsFun, Here is my opinion:-
BASICS :- Introduction to Algebra, Balance when adding and subtracting, Introduction to Multiplication : Very neatly presented. The School kids would love them. The presentation is such that even one who has no exposure whatsoever to Basic Mathematics would be able to fully understand the fundamentals of Algebra! Order of Operations :- BODMAS/PEMDAS : Very useful acronyms, a high school student to know that! Equations and formulas :- The concept of variable and contant has been very neatly explained. Nice Images. Basic Algebra Definitions : Neat explanations. Degree of a polynomial is missing! But for that, I think the list of definitions is almost exhaustive! Good work!
EXPONENETS:- Exponents, Negative Exponents well expalined. I don't think the topic can be explained any better! Neat work! quare Roots, Cube roots, and nth rots; Surds :- All well explained. Fractional exponents : Everything what can be said about it isin the page. The graph is a bonus. Laws of Exponents, Operations : Well expalined.
Algebra Expanding, removing the parantheses : Very good work, Lovely Animation, Extremely useful. Kids would love it. Teachers too would find this very helpful. If I were a teacher, I would use this page to explain the concepts. I personally loved the interactive operations.
Multiplying Negatives : The Rules explained with Buttons Neatly. It is good thinking to use a number ray. good work.
Re: More Algebra Pages
algebra section is good menu. what if there are the menu(section) with different levels (categorized with different levels) so that all users with different levels can easily post their problems and view and reply the sections.
Re: More Algebra Pages
gyanashreshta, Please don't feel sorry that your English is not good, or that you are posting in this section.
Your suggestions are expected to be posted here. I don't see any mistakes in your English language or spellings in this post.
Your suggestions are indeed, good. But, it is not possible to categorise age-wise in the indes page. This is becase all over the world, here are different levels of mathematics taught for different age groups. It is not uniform. What is taught in UK may not be exactly taught in US for the same age, same level. Similarly, what is taught in US may not be waht is taught in India or Japan for the same age. There lies the difficulty. However, your suggestion appears good. You can post different levels for your countries, naming that in the topic, in Exercies. Please mark the country name as it would help the students of that age of that particular country. Thanks. | 677.169 | 1 |
Search Results
These roughly 100 lesson plans are related to mathematics technology and are provided by the non-profit Education Development Center. The lesson plans are designed to help students develop both deep conceptual...
These roughly 200 lesson plans are related to mathematical problem solving and are provided by the non-profit Education Development Center. The lesson plans are designed to help students develop both deep conceptual...
This page, from Cut the Not, presents a proof using Euler's formula. Following the proof, users have added their own comments and questions. There is also a list of further resources on complex numbers online.
These applets accompany the textbook, Discrete Mathematics: Mathematical Reasoning and Proof with Puzzles, Patterns and Games, by Doug Ensley and Winston Crawley, published by John Wiley and Sons. The development of...
This is an introduction to Gaussian primes, complex numbers with integers for real and imaginary parts that are divisible by themselves and 1, but no other complex numbers with integer coefficients. This shows... | 677.169 | 1 |
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). | 677.169 | 1 |
Enter your mobile number or email address below and we'll send you a link to download the free Kindle Reading App. Then you can start reading Kindle books on your smartphone, tablet, or computer - no Kindle device required.
While many think of algorithms as specific to computer science, at its core algorithmic thinking is defined by the use of analytical logic to solve problems. This logic extends far beyond the realm of computer science and into the wide and entertaining world of puzzles. In Algorithmic Puzzles, Anany and Maria Levitin use many classic brainteasers as well as newer ex&les from job interviews with major corporations to show readers how to apply analytical thinking to solve puzzles requiring well-defined procedures. The book's unique collection of puzzles is supplemented with carefully developed tutorials on algorithm design strategies and analysis techniques intended to walk the reader step-by-step through the various approaches to algorithmic problem solving. Mastery of these strategies-exhaustive search, backtracking, and divide-and-conquer, among others-will aid the reader in solving not only the puzzles contained in this book, but also others encountered in interviews, puzzle collections, and throughout everyday life. Each of the 150 puzzles contains hints and solutions, along with commentary on the puzzle's origins and solution methods. The only book of its kind, Algorithmic Puzzles houses puzzles for all skill levels. Readers with only middle school mathematics will develop their algorithmic problem-solving skills through puzzles at the elementary level, while seasoned puzzle solvers will enjoy the challenge of thinking through more difficult puzzles.
{"currencyCode":"USD","itemData":[{"priceBreaksMAP":null,"buyingPrice":20.41,"ASIN":"0199740445","moqNum":1,"isPreorder":0},{"priceBreaksMAP":null,"buyingPrice":3.99,"ASIN":"04862707850199740445::kzT9qk6Ub1QOmQs%2FVNDh%2Fl28WSvMhvBggnm2wjRXK4HRPZXM3g1r0kY99bbPxVRn%2F0hPOMdiK%2BNcmAqMNTr7vx6EDNZpHGegP0lPwUoKPgvUyTuv1RHtOg%3D%3D,0486270785::A1J%2BLycEUcNKvM8mQHjCqz%2BWq7LxnNVafD3v68amPMvNiahBM0CJ0%2Fa2INBjPRS5yA7wBw%2FiH6ADDdXRUIS8K64Mx0HLb9vqSvYa%2ByVF7rw%3D,0486281523::obu5mExkfoIwTg%2FHe3eQgzILHDQULbsM7vZMW9nf0zMUsS3KnpY%2BIHJVrQ42CoCoRfsVDOA1pBDeTapBVLgql5qpFotV1erjtwUWfmsNrAlgorithmic Puzzles by Anany Levitin and Maria Levitin is an interesting and novel style of puzzle book. The emphasis lies in training the reader to think algorithmically and develop new puzzle-solving skills: the majority of puzzles are problems where we are asked to find the shortest distance or the fewest moves to get from A to B, or construct a proof that a puzzle has no solution. the book provides plenty of puzzles to keep even the most avid problem-solvers busy for a long time, all with varying levels of difficulty and different styles/contexts. The solutions provided are comprehensive and explain themselves in a friendly, constructive manner, complete with illustrations. In addition to questions and answers, a section of brief hints is also provided to assist the reader in their puzzle-solving endeavours." -- Graham Wheeler, Significance
About the Author
Anany Levitin is a professor of Computing Sciences at Villanova University. He is the author of a popular textbook on design and analysis of algorithms, which has been translated into Chinese, Greek, Korean, and Russian. He has also published papers on mathematical optimization theory, software engineering, data management, algorithm design, and computer science education. Maria Levitin is an independent consultant. After some years working for leading software companies and developing business applications for large corporations, she now specializes in web-based applications and wireless computing.
I bought this book on sight. It's possibly my favorite book of any and all books I own. The puzzles are not only ubiquitous and exciting... they're educational and provide many "Aha!" moments. I've been looking for a book like this for years, and I recommend it to those looking for fun and challenging puzzles of varying difficulty levels. Being a computer science major, many of the puzzles are also fun to implement and solve using programming, emphasizing the "algorithmic" component in the title.
My motivation for reading this book: preparing for coding interview puzzles. This is the perspective from which I evaluate the book.
Conclusion: if you are in the coding interview game do yourself a favor and read at least the first 30 pages. Best return on investment, unless you already have specific training on solving puzzles.
I have at least 5 of the most popular titles specifically dedicated to coding interviews but I find that just the first 30 pages of this book are more valuable than all the other books combined. (This statement is limited to the general problem solving topics and not to technical trivia or dedicated computer science topics like tricky manipulations of a linked list.)
The first pages review very clearly the major strategies for solving "puzzles," which is just another word for problems. If you have a computing or mathematics background then likely nothing on those pages will be new. Regardless, in my case the systematic review of these strategies makes the difference between "I know that I should be able to solve this in minutes but still I run around in loops and getting frustrated" to actually being able to convert the problem space to a graphical representation that helps getting the solution in minutes even with the most simplistic approach of exhaustive search. Of course those 30 pages are about describing the progression of the more sophisticated strategies that, even though I have been aware of them all, are extremely helpful to see in a system.
Contrast to other books I've seen: other books usually start with the problem, then either provide the solution up front, or demonstrate one example of how you could start looking for the solution.Read more ›
This is the perfect book for a math nerd or someone interested in understanding algorithms. While its not a novel, its the perfect thing to take with you for an evening in the park or an afternoon stroll. You can sit and read a puzzle and think about it for a second, or you can read up on some of the different algorithmic techniques used in solving these puzzles. Its one of my go-to's when I feel like taking a break from text books or novels.
I agreed with some reviewers' comment about the Kindle formatting of this book. You are better off with a paper version.
Fortunately I have no need of hints and solutions. I bought this book only to entertain my puzzle-loving daughter, in lieu of a bed time story. So I skipped the tutorial and went all the way to main section. | 677.169 | 1 |
Bezier and B-spline Techniques with Matlab. Edition No. 1
The aim of this book is to teach students the essential of Bezier and B-spline techniques with the aid of examples. Computer codes, which give an easy interface of Bezier and B-spline techniques to the users, are implemented as Matlab programs. The reason to choose Matlab is that it is easy to use and has a good graphical user interface. This book focuses on curves and surfaces using Bezier and B-spline techniques. It is based on the theory "Bezier and B-spline Techniques" which are known in mathematics. Interpolation and approximation methods have been illustrated intensively. Some of algorithms are represented using practical cases for example Casteljau algorithm. Students and researchers can use this book to succeed good understanding of Bezier and B-spline techniques for reliable and efficient studies in accordance within scientific applications | 677.169 | 1 |
La Marque ExcelThe Socratic Method is a "guided question" methodology that allows students to see how a problem is solved rather then just mindlessly applying a formula without a conceptual understanding of the problem. A lot of students have difficulties with math and the math found in other subjects such as physics. Math is an interesting subject with a myriad of techniques for finding an answer.
Geane S.
...Microsoft Word is a powerful word processor that can even do simple spreadsheet calculations, and allows the user to quickly set up documents, letters and envelopes, text with pictures, tables, columns and so much more. As a certified Microsoft instructor I can teach you how to get the most out ...
Ahmed A | 677.169 | 1 |
An Introduction to Differential Equations and Their Applications (Dover Books on Mathematics)
9780486445953
ISBN:
048644595X
Pub Date: 2006 Publisher: Dover Publications
Summary: Starting with an introduction to differential equations, this insightful text then explores 1st- and 2nd-order differential equations, series solutions, the Laplace transform, systems of differential equations, difference equations, nonlinear differential equations and chaos, and partial differential equations. Numerous figures, problems with solutions, and notes. 1994 edition. Includes 268 figures and 23 tables. ...> Stanley J. Farlow is the author of An Introduction to Differential Equations and Their Applications (Dover Books on Mathematics), published 2006 under ISBN 9780486445953 and 048644595X. Five hundred seventeen An Introduction to Differential Equations and Their Applications (Dover Books on Mathematics) textbooks are available for sale on ValoreBooks.com, sixty four used from the cheapest price of $20.97, or buy new starting at $30.30 | 677.169 | 1 |
results for "Mth 221 Discrete Math For Information Technology"
MTH221 Entire Course (DiscreteMath For Information) Complete Course
Visit Website For More Tutorials :
Email Us for Any Question or More Final Exams at : Uopguides@gmail.com
...
1) Suppose we are given the following expression: x + ((xy + x)/y). Represent this
expression as a binary tree.
ANS: on last page.
2) Use the rooted tree below to perform a post-order traversal of the expression.
[pic]
ANS: x y + 2 ^ x 4 -3...
1) Suppose we are given the following expression: x + ((xy + x)/y). Represent this
expression as a binary tree.
ANS: on last...
Caring for Populations through Community Outreach
Chamberlain College of Nursing
NR 443: Community Health Nursing
Caring for Populations through Community Outreach
I selected my work setting as the Health Department, functioning as a Health Promotion Nurse. The identified problem in Atlanta, Georgia...
Caring for Populations through Community Outreach
Chamberlain College of Nursing
NR 443: Community Health Nursing
Caring for Populations through...
Mathematical Database
The principle of mathematical induction can be used to prove a wide range of statements
involving variables that take discrete values. Some typical examples are shown below.
Example 2.2.
Prove that 23n − 1 is divisible by 11 for all positive integers n.
Solution.
Clearly...
Mathematical Database
The principle of mathematical induction can be used to prove a wide range of statements
involving variables that take...
WHY
THEY ARE IMPORTANT?
introduction
Many students seem to have trouble with the notion of a mathematical proof. People that come to a course like Math 216, who certainly
know a great deal of mathematics - Calculus, Trigonometry, Geometry
and Algebra, all of the sudden come to meet a new kind of mathematics...
WHY
THEY ARE IMPORTANT?
introduction
Many students seem to have trouble with the notion of a mathematical proof. People that come to a course like...
COMBANITARICS
* A branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size (enumerative combinatorics), deciding when certain criteria can be met, and constructing and analyzing objects...
COMBANITARICS
* A branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics...
Formulas
Read the following instructions in order to complete this discussion, and review the example of how to complete the math required for this assignment:
• Read about Cowling's Rule for child sized doses of medication (number 92 on page 119 of Elementary and Intermediate Algebra).
...
Formulas
Read the following instructions in order to complete this discussion, and review the example of how to complete the math requiredindicates that to find the treasure, one must get to Castle Rock, walk x paces to the north, and then walk 2x + 4 paces to the east. If they share their information, then they can find x and save a lot of digging. What is x?
So, if you walk x paces north, then 2x+4 paces east, you have moved roughly east...
indicates that to find the treasure, one must get to Castle Rock, walk x paces to the north, and then walk 2x + 4 paces to the east. If they share their...
Chapter 7 Exercises:
7.1.5a) For each of the following relations, determine whether the relation is reflexive, symmetric, antisymmetric, or transitive.
R ⊆ Z+ x Z+ where a R b if a|b (read "a divides b," as defined in Section 4.3)
The relation is reflexive, antisymmetric, and transitive
7.1.6) Which...
Chapter 7 Exercises:
7.1.5a) For each of the following relations, determine whether the relation is reflexive, symmetric, antisymmetric, or transitive.
R ⊆...
digit)that gave me a range between 62.3-72.3". All of the females surveyed fell into the 95%. I used the descriptive statistics under minitab besides simple math to compare the ranges. So to arrive at 95% or 2 SD's I multiplied 2.5 by 2 indicating 5" from the mean was where the percentile fell. |
11....
digit)that gave me a range between 62.3-72.3". All of the females surveyed fell into the 95%. I used the descriptive statistics under minitab besides simple...
page 151 of the text "Elementary and intermediate algebra" By Dugopolski, M. (2012). We are looking to the Body Mass Index "BMI" chart to gather information on the formula that's given in pounds and inches and weight. These below are the four intervals provided…
17 < BMI <= 22 These numbers state that...
page 151 of the text "Elementary and intermediate algebra" By Dugopolski, M. (2012). We are looking to the Body Mass Index "BMI" chart to gatherCHAPTER 2
LOGIC
Introduction:
Logic is the discipline that deals with the methods of reasoning. On an elementary level, logic provides rules and techniques for determining whether a given argument is valid. Logical reasoning...
CHAPTER 2
LOGIC
Introduction:
Logic is the discipline that deals with the methods of... | 677.169 | 1 |
4.
3
Introduction
This document replaces The Ontario Curriculum, Grades 9 and 10: Mathematics, 1999.
Beginning in September 2005, all Grade 9 and 10 mathematics courses will be based on the
expectations outlined in this document.
The Place of Mathematics in the Curriculum
The unprecedented changes that are taking place in today's world will profoundly affect the
future of today's students. To meet the demands of the world in which they will live, students
will need to adapt to changing conditions and to learn independently. They will require the
ability to use technology effectively and the skills for processing large amounts of quantitative
information. Today's mathematics curriculum must prepare students for their future roles in
society. It must equip them with essential mathematical knowledge and skills; with skills of
reasoning, problem solving, and communication; and, most importantly, with the ability and
the incentive to continue learning on their own. This curriculum provides a framework for
accomplishing these goals.
The choice of specific concepts and skills to be taught must take into consideration new appli-
cations and new ways of doing mathematics. The development of sophisticated yet easy-to-use
calculators and computers is changing the role of procedure and technique in mathematics.
Operations that were an essential part of a procedures-focused curriculum for decades can
now be accomplished quickly and effectively using technology, so that students can now solve
problems that were previously too time-consuming to attempt, and can focus on underlying
concepts. "In an effective mathematics program, students learn in the presence of technology.
Technology should influence the mathematics content taught and how it is taught. Powerful
assistive and enabling computer and handheld technologies should be used seamlessly in teach-
ing, learning, and assessment."1 This curriculum integrates appropriate technologies into the
learning and doing of mathematics, while recognizing the continuing importance of students'
mastering essential numeric and algebraic skills.
Mathematical knowledge becomes meaningful and powerful in application. This curriculum
embeds the learning of mathematics in the solving of problems based on real-life situations.
Other disciplines are a ready source of effective contexts for the study of mathematics. Rich
problem-solving situations can be drawn from closely related disciplines, such as computer
science, business, recreation, tourism, biology, physics, or technology, as well as from subjects
historically thought of as distant from mathematics, such as geography or art. It is important
that these links between disciplines be carefully explored, analysed, and discussed to emphasize
for students the pervasiveness of mathematical knowledge and mathematical thinking in all
subject areas.
1. Expert Panel on Student Success in Ontario, Leading Math Success: Mathematical Literacy, Grades 7–12 – The Report of
the Expert Panel on Student Success in Ontario, 2004 (Toronto: Ontario Ministry of Education, 2004), p. 47. (Referred to
hereafter as Leading Math Success.)
5.
4 T H E O N T A R I O C U R R I C U L U M , G R A D E S 9 A N D 1 0 : M A T H E M A T I C S
The development of mathematical knowledge is a gradual process. A coherent and continuous
program is necessary to help students see the "big pictures", or underlying principles, of math-
ematics. The fundamentals of important skills, concepts, processes, and attitudes are initiated in
the primary grades and fostered through elementary school. The links between Grade 8 and
Grade 9 and the transition from elementary school mathematics to secondary school math-
ematics are very important in the student's development of confidence and competence.
The Grade 9 courses in this curriculum build on the knowledge of concepts and skills that
students are expected to have by the end of Grade 8. The strands used are similar to those of
the elementary program, with adjustments made to reflect the new directions mathematics
takes in secondary school. The Grade 9 courses are based on principles that are consistent with
those that underpin the elementary program, facilitating the transition from elementary
school. These courses reflect the belief that students learn mathematics effectively when they
are initially given opportunities to investigate ideas and concepts and are then guided carefully
into an understanding of the abstract mathematics involved. Skill acquisition is an important
part of the program; skills are embedded in the contexts offered by various topics in the math-
ematics program and should be introduced as they are needed.
The Grade 9 and 10 mathematics curriculum is designed to foster the development of the
knowledge and skills students need to succeed in their subsequent mathematics courses, which
will prepare them for the postsecondary destinations of their choosing.
Roles and Responsibilities in Mathematics Programs
Students. Students have many responsibilities with regard to their learning in school. Students
who make the effort required and who apply themselves will soon discover that there is a
direct relationship between this effort and their achievement, and will therefore be more moti-
vated to work. There will be some students, however, who will find it more difficult to take
responsibility for their learning because of special challenges they face. For these students, the
attention, patience, and encouragement of teachers and family can be extremely important
factors for success. However, taking responsibility for one's progress and learning is an impor-
tant part of education for all students, regardless of their circumstances.
Successful mastery of concepts and skills in mathematics requires a sincere commitment to
work and study. Students are expected to develop strategies and processes that facilitate learn-
ing and understanding in mathematics. Students should also be encouraged to actively pursue
opportunities to apply their problem-solving skills outside the classroom and to extend and
enrich their understanding of mathematics.
Parents. Parents have an important role to play in supporting student learning. Studies show
that students perform better in school if their parents or guardians are involved in their educa-
tion. By becoming familiar with the curriculum, parents can find out what is being taught in
the courses their children are taking and what their children are expected to learn. This aware-
ness will enhance parents' ability to discuss their children's work with them, to communicate
with teachers, and to ask relevant questions about their children's progress. Knowledge of the
expectations in the various courses also helps parents to interpret teachers' comments on stu-
dent progress and to work with them to improve student learning.
6.
5I N T R O D U C T I O N
The mathematics curriculum promotes lifelong learning not only for students but also for
their parents and all those with an interest in education. In addition to supporting regular
school activities, parents can encourage their sons and daughters to apply their problem-
solving skills to other disciplines or to real-world situations. Attending parent-teacher interviews,
participating in parent workshops, becoming involved in school council activities (including
becoming a school council member), and encouraging students to complete their assignments
at home are just a few examples of effective ways to support student learning.
Teachers. Teachers and students have complementary responsibilities. Teachers are responsible
for developing appropriate instructional strategies to help students achieve the curriculum
expectations for their courses, as well as for developing appropriate methods for assessing and
evaluating student learning. Teachers also support students in developing the reading, writing,
and oral communication skills needed for success in their mathematics courses. Teachers bring
enthusiasm and varied teaching and assessment approaches to the classroom, addressing differ-
ent student needs and ensuring sound learning opportunities for every student.
Recognizing that students need a solid conceptual foundation in mathematics in order to fur-
ther develop and apply their knowledge effectively, teachers endeavour to create a classroom
environment that engages students' interest and helps them arrive at the understanding of
mathematics that is critical to further learning.
Using a variety of instructional, assessment, and evaluation strategies, teachers provide numer-
ous opportunities for students to develop skills of inquiry, problem solving, and communica-
tion as they investigate and learn fundamental concepts. The activities offered should enable
students not only to make connections among these concepts throughout the course but also
to relate and apply them to relevant societal, environmental, and economic contexts. Oppor-
tunities to relate knowledge and skills to these wider contexts – to the goals and concerns
of the world in which they live – will motivate students to learn and to become lifelong
learners.
Principals. The principal works in partnership with teachers and parents to ensure that each
student has access to the best possible educational experience. To support student learning,
principals ensure that the Ontario curriculum is being properly implemented in all classrooms
using a variety of instructional approaches. They also ensure that appropriate resources are
made available for teachers and students. To enhance teaching and learning in all subjects,
including mathematics, principals promote learning teams and work with teachers to facilitate
participation in professional development. Principals are also responsible for ensuring that
every student who has in Individual Education Plan (IEP) is receiving the modifications
and/or accommodations described in his or her plan – in other words, for ensuring that the
IEP is properly developed, implemented, and monitored.
7.
6
Overview
The Grade 9 and 10 mathematics program builds on the elementary program, relying on the
same fundamental principles on which that program was based. Both are founded on the
premise that students learn mathematics most effectively when they have a thorough under-
standing of mathematical concepts and procedures, and when they build that understanding
through an investigative approach, as reflected in the inquiry model of learning. This curricu-
lum is designed to help students build a solid conceptual foundation in mathematics that will
enable them to apply their knowledge and skills and further their learning successfully.
Like the elementary curriculum, the secondary curriculum adopts a strong focus on the
processes that best enable students to understand mathematical concepts and learn related
skills. Attention to the mathematical processes is considered to be essential to a balanced math-
ematics program. The seven mathematical processes identified in this curriculum are problem
solving, reasoning and proving, reflecting, selecting tools and computational strategies, connecting, represent-
ing, and communicating. Each of the Grade 9 and 10 mathematics courses includes a set of
expectations – referred to in this document as the "mathematical process expectations" – that
outline the knowledge and skills involved in these essential processes. The mathematical
processes apply to student learning in all areas of a mathematics course.
A balanced mathematics program at the secondary level includes the development of algebraic
skills. This curriculum has been designed to equip students with the algebraic skills they need
to understand other aspects of mathematics that they are learning, to solve meaningful prob-
lems, and to continue to meet with success as they study mathematics in the future. The alge-
braic skills required in each course have been carefully chosen to support the other topics
included in the course. Calculators and other appropriate technology will be used when the
primary purpose of a given activity is the development of concepts or the solving of problems,
or when situations arise in which computation or symbolic manipulation is of secondary
importance.
Courses in Grades 9 and 10. The mathematics courses in the Grade 9 and 10 curriculum
are offered in two types, academic and applied, which are defined as follows:
Academic courses develop students' knowledge and skills through the study of theory and abstract
problems.These courses focus on the essential concepts of a subject and explore related concepts as well.
They incorporate practical applications as appropriate.
Applied courses focus on the essential concepts of a subject, and develop students' knowledge and skills
through practical applications and concrete examples. Familiar situations are used to illustrate ideas, and
students are given more opportunities to experience hands-on applications of the concepts and theories
they study.
Students who successfully complete the Grade 9 academic course may proceed to either the
Grade 10 academic or the Grade 10 applied course. Those who successfully complete the
Grade 9 applied course may proceed to the Grade 10 applied course, but must successfully
complete a transfer course if they wish to proceed to the Grade 10 academic course. The
The Program in Mathematics
8.
7T H E P R O G R A M I N M A T H E M A T I C S
Grade 10 academic and applied courses prepare students for particular destination-related
courses in Grade 11. The Grade 11 and 12 mathematics curriculum offers university prepara-
tion, university/college preparation, college preparation, and workplace preparation courses.
When choosing courses in Grades 9 and 10, students, parents, and educators should carefully
consider students' strengths, interests, and needs, as well as their postsecondary goals and the
course pathways that will enable them to reach those goals.
School boards may develop locally and offer two mathematics courses – a Grade 9 course
and a Grade 10 course – that can be counted as two of the three compulsory credits in math-
ematics that a student is required to earn in order to obtain the Ontario Secondary School
Diploma (see Program/Policy Memorandum No. 134, which outlines a revision to section
7.1.2,"Locally Developed Courses", of Ontario Secondary Schools, Grades 9 to 12: Program and
Diploma Requirements, 1999 [OSS]). The locally developed Grade 10 course may be designed
to prepare students for success in the Grade 11 workplace preparation course. Ministry
approval of the locally developed Grade 10 course would authorize the school board to use
it as the prerequisite for that course.
Courses in Mathematics, Grades 9 and 10*
Course Course Credit
Grade Course Name Type Code Value Prerequisite**
9 Principles of Mathematics Academic MPM1D 1
9 Foundations of Mathematics Applied MFM1P 1
10 Principles of Mathematics Academic MPM2D 1 Grade 9 Mathematics,
Academic
10 Foundations of Mathematics Applied MFM2P 1 Grade 9 Mathematics,
Academic or Applied
* See preceding text for information about locally developed Grade 9 and 10 mathematics courses.
** Prerequisites are required only for Grade 10, 11, and 12 courses.
Half-Credit Courses. The courses outlined in this document are designed to be offered as
full-credit courses. However, they may also be delivered as half-credit courses.
Half-credit courses, which require a minimum of fifty-five hours of scheduled instructional
time, must adhere to the following conditions:
• The two half-credit courses created from a full course must together contain all of the
expectations of the full course. The expectations for each half-credit course must be divided
in a manner that best enables students to achieve the required knowledge and skills in the
allotted time.
• A course that is a prerequisite for another course in the secondary curriculum may be
offered as two half-credit courses, but students must successfully complete both parts of the
course to fulfil the prerequisite. (Students are not required to complete both parts unless the
course is a prerequisite for another course they wish to take.)
• The title of each half-credit course must include the designation Part 1 or Part 2. A half
credit (0.5) will be recorded in the credit-value column of both the report card and the
Ontario Student Transcript.
Boards will ensure that all half-credit courses comply with the conditions described above, and
will report all half-credit courses to the ministry annually in the School October Report.
9.
8 T H E O N T A R I O C U R R I C U L U M , G R A D E S 9 A N D 1 0 : M A T H E M A T I C S
Curriculum Expectations
The expectations identified for each course describe the knowledge and skills that students are
expected to acquire, demonstrate, and apply in their class work, on tests, and in various other
activities on which their achievement is assessed and evaluated.
Two sets of expectations are listed for each strand, or broad curriculum area, of each course.
• The overall expectations describe in general terms the knowledge and skills that students are
expected to demonstrate by the end of each course.
• The specific expectations describe the expected knowledge and skills in greater detail. The
specific expectations are arranged under subheadings that reflect particular aspects of the
required knowledge and skills and that may serve as a guide for teachers as they plan learn-
ing activities for their students. The organization of expectations in subgroupings is not
meant to imply that the expectations in any subgroup are achieved independently of the
expectations in the other subgroups. The subheadings are used merely to help teachers focus
on particular aspects of knowledge and skills as they develop and present various lessons and
learning activities for their students.
In addition to the expectations outlined within each strand, a list of seven "mathematical
process expectations" precedes the strands in all mathematics courses. These specific expecta-
tions describe the knowledge and skills that constitute processes essential to the effective study
of mathematics. These processes apply to all areas of course content, and students' proficiency
in applying them must be developed in all strands of a mathematics course. Teachers should
ensure that students develop their ability to apply these processes in appropriate ways as they
work towards meeting the expectations outlined in the strands.
When developing detailed courses of study from this document, teachers are expected to
weave together related expectations from different strands, as well as the relevant process
expectations, in order to create an overall program that integrates and balances concept devel-
opment, skill acquisition, the use of processes, and applications.
Many of the expectations are accompanied by examples and/or sample problems, given in
parentheses. These examples and sample problems are meant to illustrate the kind of skill, the
specific area of learning, the depth of learning, and/or the level of complexity that the expec-
tation entails. They are intended as a guide for teachers rather than as an exhaustive or manda-
tory list. Teachers do not have to address the full list of examples or use the sample problems
supplied. They might select two or three areas of focus suggested by the examples in the list or
they might choose areas of focus that are not represented in the list at all. Similarly, they may
incorporate the sample problems into their lessons, or they may use other problems that are
relevant to the expectation.
10.
9T H E P R O G R A M I N M A T H E M A T I C S
Strands
Grade 9 Courses
Strands and Subgroups in the Grade 9 Courses
Principles of Mathematics Foundations of Mathematics
(Academic) (Applied)
The strands in the Grade 9 courses are designed to build on those in Grade 8, while at the
same time providing for growth in new directions in high school.
The strand Number Sense and Algebra builds on the Grade 8 Number Sense and Numeration
strand and parts of the Patterning and Algebra strand. It includes expectations describing
numeric skills that students are expected to consolidate and apply, along with estimation and
mental computation skills, as they solve problems and learn new material throughout the
course. The strand includes the algebraic knowledge and skills necessary for the study and
application of relations. In the Principles course, the strand covers the basic exponent rules,
manipulation of polynomials with up to two variables, and the solving of first-degree equa-
tions. In the Foundations course, it covers operations with polynomials involving one variable
and the solving of first-degree equations with non-fractional coefficients. The strand in the
Foundations course also includes expectations that follow from the Grade 8 Proportional
Reasoning strand, providing an opportunity for students to deepen their understanding of
proportional reasoning through investigation of a variety of topics, and providing them with
skills that will help them meet the expectations in the Linear Relations strand.
Number Sense and Algebra
• Solving Problems Involving Proportional
Reasoning
• Simplifying Expressions and Solving Equations
Linear Relations
• Using Data Management to Investigate
Relationships
• Determining Characteristics of Linear Relations
• Investigating Constant Rate of Change
• Connecting Various Representations of Linear
Relations and Solving Problems Using the
Representations
Measurement and Geometry
• Investigating the Optimal Values of
Measurements of Rectangles
• Solving Problems Involving Perimeter, Area, and
Volume
• Investigating and Applying Geometric
Relationships
Number Sense and Algebra
• Operating with Exponents
• Manipulating Expressions and Solving Equations
Linear Relations
• Using Data Management to Investigate
Relationships
• Understanding Characteristics of Linear
Relations
• Connecting Various Representations of Linear
Relations
Analytic Geometry
• Investigating the Relationship Between the
Equation of a Relation and the Shape of Its
Graph
• Investigating the Properties of Slope
• Using the Properties of Linear Relations to Solve
Problems
Measurement and Geometry
• Investigating the Optimal Values of
Measurements
• Solving Problems Involving Perimeter, Area,
Surface Area, and Volume
• Investigating and Applying Geometric
Relationships
11.
10 T H E O N T A R I O C U R R I C U L U M , G R A D E S 9 A N D 1 0 : M A T H E M A T I C S
The focus of study in the Grade 9 courses is linear relations, with some attention given to the
study of non-linear relations. In the Linear Relations strand, students develop initial under-
standings of the properties of linear relations as they collect, organize, and interpret data drawn
from a variety of real-life situations (applying knowledge gained in the Data Management
strand of the elementary school program) and create models for the data. Students then
develop, make connections among, and apply various representations of linear relations and
solve related problems. In the Analytic Geometry strand of the Principles course, students will
extend the initial experiences of linear relations into the abstract realm of equations in the
form y = mx + b, formulas, and problems.
The strand Measurement and Geometry extends students' understandings from Grade 8 to
include the measurement of composite two-dimensional shapes and the development of for-
mulas for, and applications of, additional three-dimensional figures. Furthermore, in measure-
ment, students investigate the effect of varying dimensions (length and width) on a measure
such as area. Students in the Principles course conduct similar investigations in connection
with volume and surface area. Examination of such relationships leads students to make con-
clusions about the optimal size of shapes (in the Foundations course) or of shapes and figures
(in the Principles course). In geometry, the knowledge students acquired in Grade 8 about the
properties of two-dimensional shapes is extended through investigations that broaden their
understanding of the relationships among the properties.
Grade 10 Courses
Strands and Subgroups in the Grade 10 Courses
Principles of Mathematics Foundations of Mathematics
(Academic) (Applied)
Measurement and Trigonometry
• Solving Problems Involving Similar Triangles
• Solving Problems Involving the Trigonometry of
Right Triangles
• Solving Problems Involving Surface Area and
Volume, Using Imperial and Metric Systems of
Measurement
Modelling Linear Relations
• Manipulating and Solving Algebraic Equations
• Graphing and Writing Equations of Lines
• Solving and Interpreting Systems of Linear
Equations
Quadratic Relations of the Form y = ax2 + bx + c
• Manipulating Quadratic Expressions
• Identifying Characteristics of Quadratic Relations
• Solving Problems by Interpreting Graphs of
Quadratic Relations
Quadratic Relations of the Form y = ax2 + bx + c
• Investigating the Basic Properties of Quadratic
Relations
• Relating the Graph of y = x2 and Its
Transformations
• Solving Quadratic Equations
• Solving Problems Involving Quadratic Relations
Analytic Geometry
• Using Linear Systems to Solve Problems
• Solving Problems Involving Properties of Line
Segments
• Using Analytic Geometry to Verify Geometric
Properties
Trigonometry
• Investigating Similarity and Solving Problems
Involving Similar Triangles
• Solving Problems Involving the Trigonometry of
Right Triangles
• Solving Problems Involving the Trigonometry of
Acute Triangles
12.
11T H E P R O G R A M I N M A T H E M A T I C S
The strands in the two Grade 10 courses have similarities, but there are significant differences
between them in terms of level of abstraction and degree of complexity. Both courses contain
the strand Quadratic Relations in the Form y = ax2 + bx + c. The difference between the
strand in the Principles course and its counterpart in the Foundations course lies in the greater
degree of algebraic treatment required in the Principles course. Both strands involve concrete
experiences upon which students build their understanding of the abstract treatment of qua-
dratic relations. In the Foundations course, problem solving relates to the interpretation of
graphs that are supplied to students or generated by them using technology. In the Principles
course, problem solving involves algebraic manipulation as well as the interpretation of sup-
plied or technologically generated graphs, and students also learn the techniques involved in
sketching and graphing quadratics effectively using pencil and paper.
Both Grade 10 courses extend students' understanding of linear relations through applications
(in the Analytic Geometry strand of the Principles course and in the Modelling Linear
Relations strand of the Foundations course). Students in the Foundations course begin by
extending their knowledge into the abstract realm of equations in the form y = mx + b, for-
mulas, and problems. While students in both courses study and apply linear systems, students
in the Principles course solve multi-step problems involving the verification of properties of
two-dimensional shapes on the xy-plane. The topic of circles on the xy-plane is introduced in
the Principles course as an application of the formula for the length of a line segment.
In both the Trigonometry strand of the Principles course and the Measurement and
Trigonometry strand of the Foundations course, students apply trigonometry and the proper-
ties of similar triangles to solve problems involving right triangles. Students in the Principles
course also solve problems involving acute triangles. Students in the Foundations course begin
to study the imperial system of measurement, and apply units of measurement appropriately to
problems involving the surface area and volume of three-dimensional figures.
13.
12
The Mathematical Processes
Presented at the start of every course in this curriculum document is a set of seven expecta-
tions that describe the mathematical processes students need to learn and apply as they work
to achieve the expectations outlined within the strands of the course. In the 1999 mathematics
curriculum, expectations relating to the mathematical processes were embedded within indi-
vidual strands. The need to highlight these process expectations arose from the recognition
that students should be actively engaged in applying these processes throughout the course,
rather than in connection with particular strands.
The mathematical processes that support effective learning in mathematics are as follows:
• problem solving
• reasoning and proving
• reflecting
• selecting tools and computational strategies
• connecting
• representing
• communicating
The mathematical processes are interconnected. Problem solving and communicating have
strong links to all the other processes. A problem-solving approach encourages students to
reason their way to a solution or a new understanding. As students engage in reasoning, teachers
further encourage them to make conjectures and justify solutions, orally and in writing.The
communication and reflection that occur during and after the process of problem solving help
students not only to articulate and refine their thinking but also to see the problem they are
solving from different perspectives. This opens the door to recognizing the range of strategies
that can be used to arrive at a solution. By seeing how others solve a problem, students can
begin to think about their own thinking (metacognition) and the thinking of others, and to
consciously adjust their own strategies in order to make their solutions as efficient and accu-
rate as possible.
The mathematical processes cannot be separated from the knowledge and skills that students
acquire throughout the course. Students must problem solve, communicate, reason, reflect, and
so on, as they develop the knowledge, the understanding of concepts, and the skills required in
the course.
Problem Solving
Problem solving is central to learning mathematics. It forms the basis of effective mathematics
programs and should be the mainstay of mathematical instruction. It is considered an essential
process through which students are able to achieve the expectations in mathematics, and is an
14.
13T H E M A T H E M A T I C A L P R O C E S S E S
integral part of the mathematics curriculum in Ontario, for the following reasons. Problem
solving:
• is the primary focus and goal of mathematics in the real world;
• helps students become more confident mathematicians;
• allows students to use the knowledge they bring to school and helps them connect math-
ematics with situations outside the classroom;
• helps students develop mathematical understanding and gives meaning to skills and concepts
in all strands;
• allows students to reason, communicate ideas, make connections, and apply knowledge and
skills;
• offers excellent opportunities for assessing students' understanding of concepts, ability to
solve problems, ability to apply concepts and procedures, and ability to communicate ideas;
• promotes the collaborative sharing of ideas and strategies, and promotes talking about math-
ematics;
• helps students find enjoyment in mathematics;
• increases opportunities for the use of critical-thinking skills (e.g., estimating, classifying,
assuming, recognizing relationships, hypothesizing, offering opinions with reasons, evaluating
results, and making judgements).
Not all mathematics instruction, however, can take place in a problem-solving context.
Certain aspects of mathematics must be explicitly taught. Conventions, including the use of
mathematical symbols and terms, are one such aspect, and they should be introduced to stu-
dents as needed, to enable them to use the symbolic language of mathematics.
Selecting Problem-Solving Strategies. Problem-solving strategies are methods that can be
used successfully to solve problems of various types. Teachers who use relevant and meaningful
problem-solving experiences as the focus of their mathematics class help students to develop
and extend a repertoire of strategies and methods that they can apply when solving various
kinds of problems – instructional problems, routine problems, and non-routine problems.
Students develop this repertoire over time, as they become more mature in their problem-
solving skills. By secondary school, students will have learned many problem-solving strategies
that they can flexibly use and integrate when faced with new problem-solving situations, or to
learn or reinforce mathematical concepts. Common problem-solving strategies include the
following: making a model, picture, or diagram; looking for a pattern; guessing and checking;
making assumptions; making an organized list; making a table or chart; making a simpler
problem; working backwards; using logical reasoning.
Reasoning and Proving
An emphasis on reasoning helps students make sense of mathematics. Classroom instruction in
mathematics should always foster critical thinking – that is, an organized, analytical, well-
reasoned approach to learning mathematical concepts and processes and to solving problems.
As students investigate and make conjectures about mathematical concepts and relationships,
they learn to employ inductive reasoning, making generalizations based on specific findings from
their investigations. Students also learn to use counter-examples to disprove conjectures.
Students can use deductive reasoning to assess the validity of conjectures and to formulate proofs.
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14 T H E O N T A R I O C U R R I C U L U M , G R A D E S 9 A N D 1 0 : M A T H E M A T I C S
Reflecting
Good problem solvers regularly and consciously reflect on and monitor their own thought
processes. By doing so, they are able to recognize when the technique they are using is not
fruitful, and to make a conscious decision to switch to a different strategy, rethink the problem,
search for related content knowledge that may be helpful, and so forth. Students' problem-
solving skills are enhanced when they reflect on alternative ways to perform a task even if they
have successfully completed it. Reflecting on the reasonableness of an answer by considering
the original question or problem is another way in which students can improve their ability to
make sense of problems.
Selecting Tools and Computational Strategies
Students need to develop the ability to select the appropriate electronic tools, manipulatives,
and computational strategies to perform particular mathematical tasks, to investigate math-
ematical ideas, and to solve problems.
Calculators, Computers, Communications Technology.Various types of technology are useful
in learning and doing mathematics. Students can use calculators and computers to extend
their capacity to investigate and analyse mathematical concepts and to reduce the time they
might otherwise spend on purely mechanical activities.
Students can use calculators and computers to perform operations, make graphs, manipulate
algebraic expressions, and organize and display data that are lengthier or more complex than
those addressed in curriculum expectations suited to a paper-and-pencil approach. Students
can also use calculators and computers in various ways to investigate number and graphing
patterns, geometric relationships, and different representations; to simulate situations; and to
extend problem solving. When students use calculators and computers in mathematics, they
need to know when it is appropriate to apply their mental computation, reasoning, and esti-
mation skills to predict results and check answers.
The computer and the calculator must be seen as important problem-solving tools to be used
for many purposes. Computers and calculators are tools of mathematicians, and students
should be given opportunities to select and use the particular applications that may be helpful
to them as they search for their own solutions to problems.
Students may not be familiar with the use of some of the technologies suggested in the cur-
riculum. When this is the case, it is important that teachers introduce their use in ways that
build students' confidence and contribute to their understanding of the concepts being investi-
gated. Students also need to understand the situations in which the new technology would be
an appropriate choice of tool. Students' use of the tools should not be laborious or restricted
to inputting and learning algorithmic steps. For example, when using spreadsheets and statisti-
cal software (e.g., Fathom), teachers could supply students with prepared data sets, and when
using dynamic geometry software (e.g.,The Geometer's Sketchpad), they could use pre-made
sketches so that students' work with the software would be focused on manipulation of the
data or the sketch, not on the inputting of data or the designing of the sketch.
Computer programs can help students to collect, organize, and sort the data they gather, and to
write, edit, and present reports on their findings. Whenever appropriate, students should be
encouraged to select and use the communications technology that would best support and
communicate their learning. Students, working individually or in groups, can use computers,
16.
15T H E M A T H E M A T I C A L P R O C E S S E S
CD-ROM technology, and/or Internet websites to gain access to Statistics Canada, mathemat-
ics organizations, and other valuable sources of mathematical information around the world.
Manipulatives.2 Students should be encouraged to select and use concrete learning tools to
make models of mathematical ideas. Students need to understand that making their own mod-
els is a powerful means of building understanding and explaining their thinking to others.
Using manipulatives to construct representations helps students to:
• see patterns and relationships;
• make connections between the concrete and the abstract;
• test, revise, and confirm their reasoning;
• remember how they solved a problem;
• communicate their reasoning to others.
Computational Strategies. Problem solving often requires students to select an appropriate
computational strategy. They may need to apply the standard algorithm or to use technology
for computation. They may also need to select strategies related to mental computation and
estimation. Developing the ability to perform mental computation and to estimate is conse-
quently an important aspect of student learning in mathematics.
Mental computation involves calculations done in the mind, with little or no use of paper and
pencil. Students who have developed the ability to calculate mentally can select from and use
a variety of procedures that take advantage of their knowledge and understanding of numbers,
the operations, and their properties. Using their knowledge of the distributive property, for
example, students can mentally compute 70% of 22 by first considering 70% of 20 and then
adding 70% of 2. Used effectively, mental computation can encourage students to think more
deeply about numbers and number relationships.
Knowing how to estimate, and knowing when it is useful to estimate and when it is necessary
to have an exact answer, are important mathematical skills. Estimation is a useful tool for judg-
ing the reasonableness of a solution and for guiding students in their use of calculators. The
ability to estimate depends on a well-developed sense of number and an understanding of
place value. It can be a complex skill that requires decomposing numbers, compensating for
errors, and perhaps even restructuring the problem. Estimation should not be taught as an iso-
lated skill or a set of isolated rules and techniques. Knowing about calculations that are easy to
perform and developing fluency in performing basic operations contribute to successful
estimation.
Connecting
Experiences that allow students to make connections – to see, for example, how concepts and
skills from one strand of mathematics are related to those from another – will help them to
grasp general mathematical principles. As they continue to make such connections, students
begin to see that mathematics is more than a series of isolated skills and concepts and that they
can use their learning in one area of mathematics to understand another. Seeing the relation-
ships among procedures and concepts also helps deepen students' mathematical understanding.
2. See the Teaching Approaches section, on page 23 of this document, for additional information about the use of
manipulatives in mathematics instruction.
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16 T H E O N T A R I O C U R R I C U L U M , G R A D E S 9 A N D 1 0 : M A T H E M A T I C S
In addition, making connections between the mathematics they study and its applications in
their everyday lives helps students see the usefulness and relevance of mathematics beyond the
classroom.
Representing
In secondary school mathematics, representing mathematical ideas and modelling situations
generally takes the form of numeric, geometric, graphical, algebraic, pictorial, and concrete
representation, as well as representation using dynamic software. Students should be able to go
from one representation to another, recognize the connections between representations, and
use the different representations appropriately and as needed to solve problems. Learning the
various forms of representation helps students to understand mathematical concepts and rela-
tionships; communicate their thinking, arguments, and understandings; recognize connections
among related mathematical concepts; and use mathematics to model and interpret mathemat-
ical, physical, and social phenomena. When students are able to represent concepts in various
ways, they develop flexibility in their thinking about those concepts. They are not inclined to
perceive any single representation as "the math"; rather, they understand that it is just one of
many representations that help them understand a concept.
Communicating
Communication is the process of expressing mathematical ideas and understandings orally,
visually, and in writing, using numbers, symbols, pictures, graphs, diagrams, and words.
Students communicate for various purposes and for different audiences, such as the teacher, a
peer, a group of students, or the whole class. Communication is an essential process in learning
mathematics. Through communication, students are able to reflect upon and to clarify ideas,
relationships, and mathematical arguments.
The development of mathematical language and symbolism fosters students' communication
skills. Teachers need to be aware of the various opportunities that exist in the classroom for
helping students to communicate. For example, teachers can:
• model proper use of symbols, vocabulary, and notations in oral and written form;
• expect correct use of mathematical symbols and conventions in student work;
• ensure that students are exposed to and use new mathematical vocabulary as it is introduced
(e.g., by means of a word wall; by providing opportunities to read, question, and discuss);
• provide feedback to students on their use of terminology and conventions;
• ask clarifying and extending questions and encourage students to ask themselves similar
kinds of questions;
• ask students open-ended questions relating to specific topics or information;
• model ways in which various kinds of questions can be answered.
Effective classroom communication requires a supportive and respectful environment that
makes all members of the class comfortable when they speak and when they question, react to,
and elaborate on the statements of their classmates and the teacher.
The ability to provide effective explanations, and the understanding and application of correct
mathematical notation in the development and presentation of mathematical ideas and solu-
tions, are key aspects of effective communication in mathematics.
18.
17
Basic Considerations
The primary purpose of assessment and evaluation is to improve student learning. Information
gathered through assessment helps teachers to determine students' strengths and weaknesses in
their achievement of the curriculum expectations in each course. This information also serves
to guide teachers in adapting curriculum and instructional approaches to students' needs and
in assessing the overall effectiveness of programs and classroom practices.
Assessment is the process of gathering information from a variety of sources (including assign-
ments, demonstrations, projects, performances, and tests) that accurately reflects how well a
student is achieving the curriculum expectations in a course. As part of assessment, teachers
provide students with descriptive feedback that guides their efforts towards improvement.
Evaluation refers to the process of judging the quality of student work on the basis of estab-
lished criteria, and assigning a value to represent that quality.
Assessment and evaluation will be based on the provincial curriculum expectations and the
achievement levels outlined in this document.
In order to ensure that assessment and evaluation are valid and reliable, and that they lead to the
improvement of student learning, teachers must use assessment and evaluation strategies that:
• address both what students learn and how well they learn;
• are based both on the categories of knowledge and skills and on the achievement level
descriptions given in the achievement chart on pages 20–21;
• are varied in nature, administered over a period of time, and designed to provide opportuni-
ties for students to demonstrate the full range of their learning;
• are appropriate for the learning activities used, the purposes of instruction, and the needs
and experiences of the students;
• are fair to all students;
• accommodate the needs of exceptional students, consistent with the strategies outlined in
their Individual Education Plan;
• accommodate the needs of students who are learning the language of instruction (English
or French);
• ensure that each student is given clear directions for improvement;
• promote students' ability to assess their own learning and to set specific goals;
• include the use of samples of students' work that provide evidence of their achievement;
• are communicated clearly to students and parents at the beginning of the school year or
semester and at other appropriate points throughout the year.
All curriculum expectations must be accounted for in instruction, but evaluation focuses on
students' achievement of the overall expectations. A student's achievement of the overall
expectations is evaluated on the basis of his or her achievement of related specific expectations
(including the process expectations). The overall expectations are broad in nature, and the
Assessment and Evaluation of Student
Achievement
19.
18 T H E O N T A R I O C U R R I C U L U M , G R A D E S 9 A N D 1 0 : M A T H E M A T I C S
specific expectations define the particular content or scope of the knowledge and skills
referred to in the overall expectations. Teachers will use their professional judgement to deter-
mine which specific expectations should be used to evaluate achievement of the overall expec-
tations, and which ones will be covered in instruction and assessment (e.g., through direct
observation) but not necessarily evaluated.
The characteristics given in the achievement chart (pages 20–21) for level 3 represent the
"provincial standard" for achievement of the expectations in a course. A complete picture of
overall achievement at level 3 in a course in mathematics can be constructed by reading from
top to bottom in the shaded column of the achievement chart, headed "70–79% (Level 3)".
Parents of students achieving at level 3 can be confident that their children will be prepared
for work in subsequent courses.
Level 1 identifies achievement that falls much below the provincial standard, while still reflect-
ing a passing grade. Level 2 identifies achievement that approaches the standard. Level 4 iden-
tifies achievement that surpasses the standard. It should be noted that achievement at level 4
does not mean that the student has achieved expectations beyond those specified for a particu-
lar course. It indicates that the student has achieved all or almost all of the expectations for
that course, and that he or she demonstrates the ability to use the specified knowledge and
skills in more sophisticated ways than a student achieving at level 3.
The Ministry of Education provides teachers with materials that will assist them in improving
their assessment methods and strategies and, hence, their assessment of student achievement.
These materials include samples of student work (exemplars) that illustrate achievement at
each of the four levels.
The Achievement Chart for Mathematics
The achievement chart that follows identifies four categories of knowledge and skills in math-
ematics. The achievement chart is a standard province-wide guide to be used by teachers. It
enables teachers to make judgements about student work that are based on clear performance
standards and on a body of evidence collected over time.
The purpose of the achievement chart is to:
• provide a common framework that encompasses the curriculum expectations for all courses
outlined in this document;
• guide the development of quality assessment tasks and tools (including rubrics);
• help teachers to plan instruction for learning;
• assist teachers in providing meaningful feedback to students;
• provide various categories and criteria with which to assess and evaluate student learning.
Categories of knowledge and skills. The categories, defined by clear criteria, represent four
broad areas of knowledge and skills within which the expectations for any given mathematics
course are organized. The four categories should be considered as interrelated, reflecting the
wholeness and interconnectedness of learning.
The categories of knowledge and skills are described as follows:
Knowledge and Understanding. Subject-specific content acquired in each course (knowledge),
and the comprehension of its meaning and significance (understanding).
20.
19A S S E S S M E N T A N D E V A L U A T I O N O F S T U D E N T A C H I E V E M E N T
Thinking. The use of critical and creative thinking skills and/or processes,3 as follows:
• planning skills (e.g., understanding the problem, making a plan for solving the problem)
• processing skills (e.g., carrying out a plan, looking back at the solution)
• critical/creative thinking processes (e.g., inquiry, problem solving)
Communication. The conveying of meaning through various oral, written, and visual forms
(e.g., providing explanations of reasoning or justification of results orally or in writing; com-
municating mathematical ideas and solutions in writing, using numbers and algebraic symbols,
and visually, using pictures, diagrams, charts, tables, graphs, and concrete materials).
Application. The use of knowledge and skills to make connections within and between various
contexts.
Teachers will ensure that student work is assessed and/or evaluated in a balanced manner with
respect to the four categories, and that achievement of particular expectations is considered
within the appropriate categories.
Criteria.Within each category in the achievement chart, criteria are provided, which are
subsets of the knowledge and skills that define each category. For example, in Knowledge and
Understanding, the criteria are "knowledge of content (e.g., facts, terms, procedural skills, use of
tools)" and "understanding of mathematical concepts".The criteria identify the aspects of stu-
dent performance that are assessed and/or evaluated, and serve as guides to what to look for.
Descriptors. A "descriptor" indicates the characteristic of the student's performance, with
respect to a particular criterion, on which assessment or evaluation is focused. In the achieve-
ment chart, effectiveness is the descriptor used for each criterion in the Thinking, Communica-
tion, and Application categories. What constitutes effectiveness in any given performance task
will vary with the particular criterion being considered. Assessment of effectiveness may there-
fore focus on a quality such as appropriateness, clarity, accuracy, precision, logic, relevance,
significance, fluency, flexibility, depth, or breadth, as appropriate for the particular criterion.
For example, in the Thinking category, assessment of effectiveness might focus on the degree
of relevance or depth apparent in an analysis; in the Communication category, on clarity of
expression or logical organization of information and ideas; or in the Application category, on
appropriateness or breadth in the making of connections. Similarly, in the Knowledge and
Understanding category, assessment of knowledge might focus on accuracy, and assessment of
understanding might focus on the depth of an explanation. Descriptors help teachers to focus
their assessment and evaluation on specific knowledge and skills for each category and criterion,
and help students to better understand exactly what is being assessed and evaluated.
Qualifiers. A specific "qualifier" is used to define each of the four levels of achievement – that
is, limited for level 1, some for level 2, considerable for level 3, and a high degree or thorough for
level 4. A qualifier is used along with a descriptor to produce a description of performance at a
particular level. For example, the description of a student's performance at level 3 with respect
to the first criterion in the Thinking category would be: "the student uses planning skills with
considerable effectiveness".
The descriptions of the levels of achievement given in the chart should be used to identify the
level at which the student has achieved the expectations. In all of their courses, students should
be provided with numerous and varied opportunities to demonstrate the full extent of their
achievement of the curriculum expectations, across all four categories of knowledge and skills.
3. See the footnote on page 20, pertaining to the mathematical processes.
22.
21T H E A C H I E V E M E N T C H A R T F O R M A T H E M A T I C S
Expression and organiza-
tion of ideas and mathe-
matical thinking (e.g.,
clarity of expression, logi-
cal organization), using
oral, visual, and written
forms (e.g., pictorial,
graphic, dynamic,
numeric, algebraic
forms; concrete
materials)
Communication for dif-
ferent audiences (e.g.,
peers, teachers) and pur-
poses (e.g., to present
data, justify a solution,
express a mathematical
argument) in oral, visual,
and written forms
Use of conventions,
vocabulary, and terminol-
ogy of the discipline (e.g.,
terms, symbols) in oral,
visual, and written forms
– expresses and orga-
nizes mathematical
thinking with limited
effectiveness
– communicates for
different audiences and
purposes with
limited effectiveness
– uses conventions,
vocabulary, and
terminology of the
discipline with limited
effectiveness
– expresses and orga-
nizes mathematical
thinking with some
effectiveness
– communicates for
different audiences and
purposes with some
effectiveness
– uses conventions,
vocabulary, and
terminology of the
discipline with some
effectiveness
– expresses and orga-
nizes mathematical
thinking with consider-
able effectiveness
– communicates for
different audiences and
purposes with
considerable
effectiveness
– uses conventions,
vocabulary, and
terminology of the
discipline with
considerable
effectiveness
– expresses and orga-
nizes mathematical
thinking with a high
degree of effectiveness
– communicates for
different audiences and
purposes with a high
degree of effectiveness
– uses conventions,
vocabulary, and
terminology of the
discipline with a high
degree of effectiveness
Note: A student whose achievement is below 50% at the end of a course will not obtain a credit for the course.
Application The use of knowledge and skills to make connections within and between various contexts
The student:
Application of knowledge
and skills in familiar
contexts
Transfer of knowledge and
skills to new contexts
Making connections within
and between various con-
texts (e.g., connections
between concepts, repre-
sentations, and forms
within mathematics; con-
nections involving use of
prior knowledge and experi-
ence; connections between
mathematics, other disci-
plines, and the real world)
– applies knowledge
and skills in familiar
contexts with limited
effectiveness
– transfers knowledge
and skills to new
contexts with limited
effectiveness
– makes connections
within and between
various contexts with
limited effectiveness
– applies knowledge
and skills in familiar
contexts with some
effectiveness
– transfers knowledge
and skills to new
contexts with some
effectiveness
– makes connections
within and between
various contexts with
some effectiveness
– applies knowledge
and skills in familiar
contexts with
considerable
effectiveness
– transfers knowledge
and skills to new
contexts with
considerable
effectiveness
– makes connections
within and between
various contexts
with considerable
effectiveness
– applies knowledge
and skills in familiar
contexts with a high
degree of effectiveness
– transfers knowledge
and skills to new
contexts with a high
degree of effectiveness
– makes connections
within and between
various contexts
with a high degree
of effectiveness
Communication The conveying of meaning through various forms
The student:
50–59%
(Level 1)
60–69%
(Level 2)
70–79%
(Level 3)
80–100%
(Level 4)
Categories
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22 T H E O N T A R I O C U R R I C U L U M , G R A D E S 9 A N D 1 0 : M A T H E M A T I C S
Evaluation and Reporting of Student Achievement
Student achievement must be communicated formally to students and parents by means of the
Provincial Report Card, Grades 9–12. The report card provides a record of the student's
achievement of the curriculum expectations in every course, at particular points in the school
year or semester, in the form of a percentage grade. The percentage grade represents the qual-
ity of the student's overall achievement of the expectations for the course and reflects the cor-
responding level of achievement as described in the achievement chart for the discipline.
A final grade is recorded for every course, and a credit is granted and recorded for every
course in which the student's grade is 50% or higher. The final grade for each course in
Grades 9–12 will be determined as follows:
• Seventy per cent of the grade will be based on evaluations conducted throughout the
course. This portion of the grade should reflect the student's most consistent level of
achievement throughout the course, although special consideration should be given to more
recent evidence of achievement.
• Thirty per cent of the grade will be based on a final evaluation in the form of an examina-
tion, performance, essay, and/or other method of evaluation suitable to the course content
and administered towards the end of the course.
24.
23
Some Considerations for Program Planning
in Mathematics
Teachers who are planning a program in mathematics must take into account considerations in
a number of important areas, including those discussed below.
Teaching Approaches
To make new learning more accessible to students, teachers draw upon the knowledge and
skills students have acquired in previous years – in other words, they help activate prior
knowledge. It is important to assess where students are in their mathematical growth and to
bring them forward in their learning.
In order to apply their knowledge effectively and to continue to learn, students must have a
solid conceptual foundation in mathematics. Successful classroom practices involve students in
activities that require higher-order thinking, with an emphasis on problem solving. Students
who have completed the elementary program should have a good grounding in the investiga-
tive approach to learning new concepts, including the inquiry model of problem solving,4 and
this approach is still fundamental in the Grade 9 and 10 program.
Students in a mathematics class typically demonstrate diversity in the ways they learn best. It is
important, therefore, that students have opportunities to learn in a variety of ways – individu-
ally, cooperatively, independently, with teacher direction, through hands-on experience,
through examples followed by practice. In mathematics, students are required to learn con-
cepts, procedures, and processes and to acquire skills, and they become competent in these var-
ious areas with the aid of the instructional and learning strategies best suited to the particular
type of learning. The approaches and strategies used in the classroom to help students meet
the expectations of this curriculum will vary according to the object of the learning and the
needs of the students.
Even at the secondary level, manipulatives are necessary tools for supporting the effective
learning of mathematics. These concrete learning tools invite students to explore and represent
abstract mathematical ideas in varied, concrete, tactile, and visually rich ways. Manipulatives are
also a valuable aid to teachers. By analysing students' concrete representations of mathematical
concepts and listening carefully to their reasoning, teachers can gain useful insights into stu-
dents' thinking and provide supports to help enhance their thinking.5
All learning, especially new learning, should be embedded in well-chosen contexts for learn-
ing – that is, contexts that are broad enough to allow students to investigate initial understand-
ings, identify and develop relevant supporting skills, and gain experience with varied and
interesting applications of the new knowledge. Such rich contexts for learning open the door
for students to see the "big ideas" of mathematics – that is, the major underlying principles,
such as pattern or relationship. This understanding of key principles will enable and encourage
students to use mathematical reasoning throughout their lives.
4. See the resource document Targeted Implementation & Planning Supports (TIPS): Grade 7, 8, and 9 Applied Mathematics
(Toronto: Queen's Printer for Ontario, 2003) for helpful information about the inquiry method of problem solving.
5. A list of manipulatives appropriate for use in intermediate and senior mathematics classrooms is provided in Leading
Math Success, pages 48–49.
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24 T H E O N T A R I O C U R R I C U L U M , G R A D E S 9 A N D 1 0 : M A T H E M A T I C S
Promoting Attitudes Conducive to Learning Mathematics. Students' attitudes have a signifi-
cant effect on how they approach problem solving and how well they succeed in mathematics.
Teachers can help students develop the confidence they need by demonstrating a positive dis-
position towards mathematics.6 Students need to understand that, for some mathematics prob-
lems, there may be several ways to arrive at the correct answer. They also need to believe that
they are capable of finding solutions. It is common for people to think that if they cannot
solve problems quickly and easily, they must be inadequate. Teachers can help students under-
stand that problem solving of almost any kind often requires a considerable expenditure of
time and energy and a good deal of perseverance. Once students have this understanding,
teachers can encourage them to develop the willingness to persist, to investigate, to reason and
explore alternative solutions, and to take the risks necessary to become successful problem
solvers.
Collaborative learning enhances students' understanding of mathematics. Working coopera-
tively in groups reduces isolation and provides students with opportunities to share ideas and
communicate their thinking in a supportive environment as they work together towards a
common goal. Communication and the connections among ideas that emerge as students
interact with one another enhance the quality of student learning.7
Planning Mathematics Programs for Exceptional Students
In planning mathematics courses for exceptional students, teachers should begin by examining
both the curriculum expectations for the course and the needs of the individual student to
determine which of the following options is appropriate for the student:
• no accommodations8 or modifications; or
• accommodations only; or
• modified expectations, with the possibility of accommodations.
If the student requires either accommodations or modified expectations, or both, the relevant
information, as described in the following paragraphs, must be recorded in his or her Individual
Education Plan (IEP). For a detailed discussion of the ministry's requirements for IEPs, see
Individual Education Plans: Standards for Development, Program Planning, and Implementation, 2000
(referred to hereafter as IEP Standards, 2000). More detailed information about planning pro-
grams for exceptional students can be found in the Individual Education Plan (IEP): A Resource
Guide, 2004. (Both documents are available at
Students Requiring Accommodations Only. With the aid of accommodations alone, some
exceptional students are able to participate in the regular course curriculum and to demon-
strate learning independently. (Accommodations do not alter the provincial curriculum
expectations for the course.) The accommodations required to facilitate the student's learning
must be identified in his or her IEP (see IEP Standards, 2000, page 11). A student's IEP is
likely to reflect the same accommodations for many, or all, courses.
There are three types of accommodations. Instructional accommodations are changes in teaching
strategies, including styles of presentation, methods of organization, or use of technology and
multimedia. Environmental accommodations are changes that the student may require in the
6. Leading Math Success, p. 42.
7. Leading Math Success, p. 42.
8. "Accommodations" refers to individualized teaching and assessment strategies, human supports, and/or individualized
equipment.
26.
25classroom and/or school environment, such as preferential seating or special lighting.
Assessment accommodations are changes in assessment procedures that enable the student to
demonstrate his or her learning, such as allowing additional time to complete tests or assign-
ments or permitting oral responses to test questions (see page14 of IEP Standards, 2000, for
more examples).
If a student requires "accommodations only" in mathematics courses, assessment and evalua-
tion of his or her achievement will be based on the appropriate course curriculum expecta-
tions and the achievement levels outlined in this document.
Students Requiring Modified Expectations. Some exceptional students will require modified
expectations, which differ from the regular course expectations. For most of these students,
modified expectations will be based on the regular course curriculum, with changes in the
number and/or complexity of the expectations. It is important to monitor, and to reflect
clearly in the student's IEP, the extent to which expectations have been modified. As noted in
Section 7.12 of the ministry's policy document Ontario Secondary Schools, Grades 9 to 12:
Program and Diploma Requirements, 1999, the principal will determine whether achievement of
the modified expectations constitutes successful completion of the course, and will decide
whether the student is eligible to receive a credit for the course. This decision must be com-
municated to the parents and the student.
When a student is expected to achieve most of the curriculum expectations for the course, the
modified expectations should identify how they differ from the course expectations. When
modifications are so extensive that achievement of the learning expectations is not likely to
result in a credit, the expectations should specify the precise requirements or tasks on which
the student's performance will be evaluated and which will be used to generate the course
mark recorded on the Provincial Report Card. Modified expectations indicate the knowledge
and/or skills the student is expected to demonstrate and have assessed in each reporting period
(IEP Standards, 2000, pages 10 and 11). Modified expectations represent specific, realistic,
observable, and measurable achievements and describe specific knowledge and/or skills that the
student can demonstrate independently, given the appropriate assessment accommodations.The
student's learning expectations must be reviewed in relation to the student's progress at least
once every reporting period, and must be updated as necessary (IEP Standards, 2000, page 11).
If a student requires modified expectations in mathematics courses, assessment and evaluation
of his or her achievement will be based on the learning expectations identified in the IEP and
on the achievement levels outlined in this document. If some of the student's learning expec-
tations for a course are modified but the student is working towards a credit for the course, it
is sufficient simply to check the IEP box. If, however, the student's learning expectations are
modified to such an extent that the principal deems that a credit will not be granted for the
course, the IEP box must be checked and the appropriate statement from Guide to the
Provincial Report Card, Grades 9–12, 1999 (page 8) must be inserted. The teacher's comments
should include relevant information on the student's demonstrated learning of the modified
expectations, as well as next steps for the student's learning in the course.
English As a Second Language and English Literacy Development (ESL/ELD)
Young people whose first language is not English enter Ontario secondary schools with
diverse linguistic and cultural backgrounds. Some may have experience of highly sophisticated
educational systems while others may have had limited formal schooling. All of these students
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26 T H E O N T A R I O C U R R I C U L U M , G R A D E S 9 A N D 1 0 : M A T H E M A T I C S
bring a rich array of background knowledge and experience to the classroom, and all teachers
must share in the responsibility for their English-language development.
Teachers of mathematics must incorporate appropriate strategies for instruction and assessment
to facilitate the success of the ESL and ELD students in their classrooms. These strategies
include:
• modification of some or all of the course expectations, based on the student's level of
English proficiency;
• use of a variety of instructional strategies (e.g., extensive use of visual cues, manipulatives,
pictures, diagrams, graphic organizers; attention to clarity of instructions; modelling of
preferred ways of working in mathematics; previewing of textbooks; pre-teaching of key
specialized vocabulary; encouragement of peer tutoring and class discussion; strategic use
of students' first languages);
• use of a variety of learning resources (e.g., visual material, simplified text, bilingual dictio-
naries, culturally diverse materials);
• use of assessment accommodations (e.g., granting of extra time; use of alternative forms of
assessment, such as oral interviews, learning logs, or portfolios; simplification of language
used in problems and instructions).
Students who are no longer taking ESL or ELD courses may still need program adaptations to
be successful. If a student requires modified expectations or accommodations in a mathematics
course, a checkmark must be placed in the ESL/ELD box on the student's report card (see
Guide to the Provincial Report Card, Grades 9–12, 1999).
For further information on supporting ESL/ELD students, refer to The Ontario Curriculum,
Grades 9 to 12: English As a Second Language and English Literacy Development, 1999.
Antidiscrimination Education in Mathematics
To ensure that all students in the province have an equal opportunity to achieve their full
potential, the curriculum must be free from bias and all students must be provided with a safe
and secure environment, characterized by respect for others, that allows them to participate
fully and responsibly in the educational experience.
Learning activities and resources used to implement the curriculum should be inclusive in
nature, reflecting the range of experiences of students with varying backgrounds, abilities,
interests, and learning styles. They should enable students to become more sensitive to the
diverse cultures and perceptions of others, including Aboriginal peoples. For example, activities
can be designed to relate concepts in geometry or patterning to the arches and tile work often
found in Asian architecture or to the patterns used in Aboriginal basketry design. By discussing
aspects of the history of mathematics, teachers can help make students aware of the various cul-
tural groups that have contributed to the evolution of mathematics over the centuries. Finally,
students need to recognize that ordinary people use mathematics in a variety of everyday con-
texts, both at work and in their daily lives.
Connecting mathematical ideas to real-world situations through learning activities can
enhance students' appreciation of the role of mathematics in human affairs, in areas including
health, science, and the environment. Students can be made aware of the use of mathematics
28.
27in contexts such as sampling and surveying and the use of statistics to analyse trends. Recog-
nizing the importance of mathematics in such areas helps motivate students to learn and also
provides a foundation for informed, responsible citizenship.
Teachers should have high expectations for all students. To achieve their mathematical poten-
tial, however, different students may need different kinds of support. Some boys, for example,
may need additional support in developing their literacy skills in order to complete mathemat-
ical tasks effectively. For some girls, additional encouragement to envision themselves in
careers involving mathematics may be beneficial. For example, teachers might consider pro-
viding strong role models in the form of female guest speakers who are mathematicians or
who use mathematics in their careers.
Literacy and Inquiry/Research Skills
Literacy skills can play an important role in student success in mathematics courses. Many of
the activities and tasks students undertake in math courses involve the use of written, oral, and
visual communication skills. For example, students use language to record their observations,
to explain their reasoning when solving problems, to describe their inquiries in both informal
and formal contexts, and to justify their results in small-group conversations, oral presentations,
and written reports. The language of mathematics includes special terminology. The study of
mathematics consequently encourages students to use language with greater care and precision
and enhances their ability to communicate effectively. The Ministry of Education has facili-
tated the development of materials to support literacy instruction across the curriculum.
Helpful advice for integrating literacy instruction in mathematics courses may be found in the
following resource documents:
• Think Literacy: Cross-Curricular Approaches, Grades 7–12, 2003
• Think Literacy: Cross-Curricular Approaches, Grades 7–12 – Mathematics: Subject-Specific
Examples, Grades 7–9, 2004
In all courses in mathematics, students will develop their ability to ask questions and to plan
investigations to answer those questions and to solve related problems. Students need to learn
a variety of research methods and inquiry approaches in order to carry out these investigations
and to solve problems, and they need to be able to select the methods that are most appropri-
ate for a particular inquiry. Students learn how to locate relevant information from a variety of
sources, such as statistical databases, newspapers, and reports. As they advance through the grades,
students will be expected to use such sources with increasing sophistication. They will also be
expected to distinguish between primary and secondary sources, to determine their validity
and relevance, and to use them in appropriate ways.
The Role of Technology in Mathematics
Information and communication technology (ICT) provides a range of tools that can signifi-
cantly extend and enrich teachers' instructional strategies and support students' learning in
mathematics.Teachers can use ICT tools and resources both for whole-class instruction and to
design programs that meet diverse student needs.Technology can help to reduce the time spent
on routine mathematical tasks and to allow students to devote more of their efforts to thinking
and concept development. Useful ICT tools include simulations, multimedia resources, data-
bases, sites that gave access to large amounts of statistical data, and computer-assisted learning
modules.
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28 T H E O N T A R I O C U R R I C U L U M , G R A D E S 9 A N D 1 0 : M A T H E M A T I C S
Applications such as databases, spreadsheets, dynamic geometry software, dynamic statistical
software, graphing software, computer algebra systems (CAS), word-processing software, and
presentation software can be used to support various methods of inquiry in mathematics. The
technology also makes possible simulations of complex systems that can be useful for problem-
solving purposes or when field studies on a particular topic are not feasible.
Information and communications technology can also be used in the classroom to connect
students to other schools, at home and abroad, and to bring the global community into the
local classroom.
Career Education in Mathematics
Teachers can promote students' awareness of careers involving mathematics by exploring appli-
cations of concepts and providing opportunities for career-related project work. Such activities
allow students the opportunity to investigate mathematics-related careers compatible with
their interests, aspirations, and abilities.
Students should be made aware that mathematical literacy and problem solving are valuable
assets in an ever-widening range of jobs and careers in today's society. The knowledge and
skills students acquire in mathematics courses are useful in fields such as science, business,
engineering, and computer studies; in the hospitality, recreation, and tourism industries; and
in the technical trades.
Health and Safety in Mathematics
Although health and safety issues are not normally associated with mathematics, they may be
important when the learning involves fieldwork or investigations based on experimentation.
Out-of-school fieldwork can provide an exciting and authentic dimension to students' learning
experiences. It also takes the teacher and students out of the predictable classroom environment
and into unfamiliar settings.Teachers must preview and plan activities and expeditions carefully
to protect students' health and safety.
30.
29
Principles of Mathematics, Grade 9, Academic (MPM1D)
This course enables students to develop an understanding of mathematical concepts related to
algebra, analytic geometry, and measurement and geometry through investigation, the effective
use of technology, and abstract reasoning. Students will investigate relationships, which they
will then generalize as equations of lines, and will determine the connections between different
representations of a linear relation. They will also explore relationships that emerge from the
measurement of three-dimensional figures and two-dimensional shapes. Students will reason
mathematically and communicate their thinking as they solve multi-step problems.
PROBLEM SOLVING
REASONING AND
PROVING
SELECTING TOOLS AND
COMPUTATIONAL
STRATEGIES
REFLECTING
CONNECTING
REPRESENTING
COMMUNICATING
• develop, select, apply, and compare a variety of problem-solving strategies as they pose and
solve problems and conduct investigations, to help deepen their mathematical under-
standing;
• develop and apply reasoning skills (e.g., recognition of relationships, generalization
through inductive reasoning, use of counter-examples) to make mathematical conjectures,
assess conjectures, and justify conclusions, and plan and construct organized mathematical
arguments;
• demonstrate that they are reflecting on and monitoring their thinking to help clarify their
understanding as they complete an investigation or solve a problem (e.g., by assessing the
effectiveness of strategies and processes used, by proposing alternative approaches, by
judging the reasonableness of results, by verifying solutions);
• select and use a variety of concrete, visual, and electronic learning tools and appropriate
computational strategies to investigate mathematical ideas and to solve problems;
• make connections among mathematical concepts and procedures, and relate mathematical
ideas to situations or phenomena drawn from other contexts (e.g., other curriculum areas,
daily life, current events, art and culture, sports);
• create a variety of representations of mathematical ideas (e.g., numeric, geometric, alge-
braic, graphical, pictorial representations; onscreen dynamic representations), connect and
compare them, and select and apply the appropriate representations to solve problems;
• communicate mathematical thinking orally, visually, and in writing, using mathematical
vocabulary and a variety of appropriate representations, and observing mathematical
conventions.
Mathematical process expectations. The mathematical processes are to be integrated into
student learning in all areas of this course.
Throughout this course, students will:
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30 T H E O N T A R I O C U R R I C U L U M , G R A D E S 9 A N D 1 0 : M A T H E M A T I C S
Number Sense and Algebra
Overall Expectations
By the end of this course, students will:
• demonstrate an understanding of the exponent rules of multiplication and division, and
apply them to simplify expressions;
• manipulate numerical and polynomial expressions, and solve first-degree equations.
Specific Expectations
Operating with Exponents
By the end of this course, students will:
– substitute into and evaluate algebraic expres-
sions involving exponents (i.e., evaluate
expressions involving natural-number
exponents with rational-number bases
[e.g., evaluate
3
by hand and 9.83 by
using a calculator]) (Sample problem: A
movie theatre wants to compare the
volumes of popcorn in two containers, a
cube with edge length 8.1 cm and a cylin-
der with radius 4.5 cm and height 8.0 cm.
Which container holds more popcorn?);
– describe the relationship between the
algebraic and geometric representations of
a single-variable term up to degree three
[i.e., length, which is one dimensional, can
be represented by x; area, which is two
dimensional, can be represented by (x)(x)
or x2; volume, which is three dimensional,
can be represented by (x)(x)(x), (x2)(x),
or x3];
– derive, through the investigation and exam-
ination of patterns, the exponent rules for
multiplying and dividing monomials, and
apply these rules in expressions involving
one and two variables with positive
exponents;
– extend the multiplication rule to derive and
understand the power of a power rule, and
apply it to simplify expressions involving
one and two variables with positive
exponents.
3
2
Manipulating Expressions and Solving
Equations
By the end of this course, students will:
– simplify numerical expressions involving
integers and rational numbers, with and
without the use of technology;*
– solve problems requiring the manipulation
of expressions arising from applications of
percent, ratio, rate, and proportion;*
– relate their understanding of inverse
operations to squaring and taking the
square root, and apply inverse operations
to simplify expressions and solve
equations;
– add and subtract polynomials with up to
two variables [e.g., (2x – 5) + (3x + 1),
(3x2y + 2xy2) + (4x2y – 6xy2)], using a
variety of tools (e.g., algebra tiles, com-
puter algebra systems, paper and pencil);
– multiply a polynomial by a monomial
involving the same variable [e.g., 2x(x + 4),
2x2(3x2 – 2x + 1)], using a variety of tools
(e.g., algebra tiles, diagrams, computer
algebra systems, paper and pencil);
– expand and simplify polynomial
expressions involving one variable
[e.g., 2x(4x + 1) – 3x(x + 2)], using a
variety of tools (e.g., algebra tiles,
computer algebra systems, paper and
pencil);
*The knowledge and skills described in this expectation
are to be introduced as needed and applied and consoli-
dated throughout the course.
32.
31P R I N C I P L E S O F M A T H E M A T I C S , G R A D E 9 , A C A D E M I C ( M P M 1 D )
– solve first-degree equations, including
equations with fractional coefficients,
using a variety of tools (e.g., computer
algebra systems, paper and pencil) and
strategies (e.g., the balance analogy,
algebraic strategies);
– rearrange formulas involving variables in
the first degree, with and without substi-
tution (e.g., in analytic geometry, in mea-
surement) (Sample problem: A circular
garden has a circumference of 30 m. What
is the length of a straight path that goes
through the centre of this garden?);
– solve problems that can be modelled with
first-degree equations, and compare alge-
braic methods to other solution methods
(Sample problem: Solve the following
problem in more than one way: Jonah is
involved in a walkathon. His goal is to
walk 25 km. He begins at 9:00 a.m. and
walks at a steady rate of 4 km/h. How
many kilometres does he still have left to
walk at 1:15 p.m. if he is to achieve his
goal?).
33.
32 T H E O N T A R I O C U R R I C U L U M , G R A D E S 9 A N D 1 0 : M A T H E M A T I C S
Overall Expectations
By the end of this course, students will:
• apply data-management techniques to investigate relationships between two variables;
• demonstrate an understanding of the characteristics of a linear relation;
• connect various representations of a linear relation.
Specific Expectations
Using Data Management to Investigate
Relationships
By the end of this course, students will:
– interpret the meanings of points on scatter
plots or graphs that represent linear rela-
tions, including scatter plots or graphs in
more than one quadrant [e.g., on a scatter
plot of height versus age, interpret the
point (13, 150) as representing a student
who is 13 years old and 150 cm tall; iden-
tify points on the graph that represent stu-
dents who are taller and younger than this
student] (Sample problem: Given a graph
that represents the relationship of the
Celsius scale and the Fahrenheit scale,
determine the Celsius equivalent of –5°F.);
– pose problems, identify variables, and
formulate hypotheses associated with rela-
tionships between two variables (Sample
problem: Does the rebound height of a
ball depend on the height from which it
was dropped?);
– design and carry out an investigation or
experiment involving relationships
between two variables, including the
collection and organization of data, using
appropriate methods, equipment, and/or
technology (e.g., surveying; using measur-
ing tools, scientific probes, the Internet)
and techniques (e.g., making tables, draw-
ing graphs) (Sample problem: Design and
perform an experiment to measure and
record the temperature of ice water in a
plastic cup and ice water in a thermal mug
over a 30 min period, for the purpose of
comparison. What factors might affect the
outcome of this experiment? How could
you design the experiment to account for
them?);
– describe trends and relationships observed
in data, make inferences from data, com-
pare 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?) (Sample problem: Hypo-
thesize the effect of the length of a
pendulum on the time required for the
pendulum to make five full swings. Use
data to make an inference. Compare the
inference with the hypothesis. Are there
other relationships you might investigate
involving pendulums?).
Understanding Characteristics of Linear
Relations
By the end of this course, students will:
– construct tables of values, graphs, and
equations, using a variety of tools (e.g.,
graphing calculators, spreadsheets, graphing
software, paper and pencil), to represent
linear relations derived from descriptions
of realistic situations (Sample problem:
Construct a table of values, a graph, and
Linear Relations
34.
33P R I N C I P L E S O F M A T H E M A T I C S , G R A D E 9 , A C A D E M I C ( M P M 1 D )
an equation to represent a monthly
cellphone plan that costs $25, plus $0.10
per minute of airtime.);
– construct tables of values, scatter plots, and
lines or curves of best fit as appropriate,
using a variety of tools (e.g., spreadsheets,
graphing software, graphing calculators,
paper and pencil), for linearly related and
non-linearly related data collected from a
variety of sources (e.g., experiments, elec-
tronic secondary sources, patterning with
concrete materials) (Sample problem:
Collect data, using concrete materials or
dynamic geometry software, and construct
a table of values, a scatter plot, and a line
or curve of best fit to represent the fol-
lowing relationships: the volume and the
height for a square-based prism with a
fixed base; the volume and the side length
of the base for a square-based prism with a
fixed height.);
– identify, through investigation, some pro-
perties of linear relations (i.e., numerically,
the first difference is a constant, which rep-
resents a constant rate of change; graphi-
cally, a straight line represents the relation),
and apply these properties to determine
whether a relation is linear or non-linear;
– compare the properties of direct variation
and partial variation in applications, and
identify the initial value (e.g., for a relation
described in words, or represented as a
graph or an equation) (Sample problem:
Yoga costs $20 for registration, plus $8 per
class. Tai chi costs $12 per class. Which
situation represents a direct variation, and
which represents a partial variation? For
each relation, what is the initial value?
Explain your answers.);
– determine the equation of a line of best fit
for a scatter plot, using an informal process
(e.g., using a movable line in dynamic
statistical software; using a process of trial
and error on a graphing calculator; deter-
mining the equation of the line joining
two carefully chosen points on the scatter
plot).
Connecting Various Representations of
Linear Relations
By the end of this course, students will:
– determine values of a linear relation by
using a table of values, by using the equa-
tion of the relation, and by interpolating
or extrapolating from the graph of the
relation (Sample problem: The equation
H = 300 – 60t represents the height of a
hot air balloon that is initially at 300 m
and is descending at a constant rate of
60 m/min. Determine algebraically and
graphically how long the balloon will take
to reach a height of 160 m.);
– describe a situation that would explain the
events illustrated by a given graph of a
relationship between two variables
(Sample problem: The walk of an individ-
ual is illustrated in the given graph, pro-
duced by a motion detector and a graph-
ing calculator. Describe the walk [e.g., the
initial distance from the motion detector,
the rate of walk].);
– determine other representations of a linear
relation, given one representation (e.g.,
given a numeric model, determine a graph-
ical model and an algebraic model; given a
graph, determine some points on the graph
and determine an algebraic model);
– describe the effects on a linear graph and
make the corresponding changes to the
linear equation when the conditions of
the situation they represent are varied
(e.g., given a partial variation graph and
an equation representing the cost of pro-
ducing a yearbook, describe how the
graph changes if the cost per book is
altered, describe how the graph changes if
the fixed costs are altered, and make the
corresponding changes to the equation). | 677.169 | 1 |
A Google User
Works very well Nice scientific calculator with graphing, conversions, physics constants etc. Can do symbolic derivatives. It has just been published and looks very promising! I for one would welcome more symbolic stuff some time in future.Works very well Nice scientific calculator with graphing, conversions, physics constants etc. Can do symbolic derivatives. It has just been published and looks very promising! I for one would welcome more symbolic stuff some time in future. | 677.169 | 1 |
This new volume introduces readers to the current topics of industrial and applied mathematics in China, with applications to material science, information science, mathematical finance and engineering. The authors utilize mathematics for the solution of problems. The purposes of the volume are to promote research in applied mathematics and computational... more...
This volume comprises the Lecture Notes of the CIMPA/TUBITAK Summer School Arrangements, Local systems and Singularities held at Galatasaray University, Istanbul during June 2007. The volume is intended for a large audience in pure mathematics, including researchers and graduate students working in algebraic geometry, singularity theory, topology and... more...
Preempt your anxiety about PRE-ALGEBRA!
Ready to learn math fundamentals but can't seem to get your brain to function? No problem! Add Pre-Algebra Demystified , Second Edition, to the equation and you'll solve your dilemma in no time.
Written in a step-by-step format, this practical guide begins by covering whole numbers, integers, fractions,... more...
Lesson study is a professional development process that teachers engage in to systematically examine their practice. This book examines how it effectively works in different contexts and models of teacher learning, while advancing the knowledge base. more...
Build student success in math with the only comprehensive guide for developing math talent among advanced learners. The authors, nationally recognized math education experts, offer a focused look at educating gifted and talented students for success in math. More than just a guidebook for educators, this book offers a comprehensive approach to mathematics... more... | 677.169 | 1 |
Precalculus Functions And Graphs
9780495108375
ISBN:
0495108375
Edition: 11 Pub Date: 2007 Publisher: Thomson Learning
Summary: Clear explanations, an uncluttered and appealing layout, and examples and exercises featuring a variety of real-life applications have made this text popular among students year after year. This latest edition of Swokowski and Cole's PRECALCULUS: FUNCTIONS AND GRAPHS retains these features. The problems have been consistently praised for being at just the right level for precalculus students like you. The book also p...rovides calculator examples, including specific keystrokes that show you how to use various graphing calculators to solve problems more quickly. Perhaps most important-this book effectively prepares you for further courses in mathematics.
Swokowski, Earl W. is the author of Precalculus Functions And Graphs, published 2007 under ISBN 9780495108375 and 0495108375. Nine Precalculus Functions And Graphs textbooks are available for sale on ValoreBooks.com, six used from the cheapest price of $32.92, or buy new starting at $114.75.[read more | 677.169 | 1 |
Edexcel GCSEMathematics A (2010)
Here, you'll find everything you need to study for or to teach Edexcel
GCSE Mathematics A, including key documents and the latest news.
Students
of our Edexcel GCSE Maths A specification will develop knowledge,
skills and understanding of mathematical methods and concepts, including
working with numbers, algebra, geometry, measures, statistics and
probability.
Published resources
Support for
Why choose this specification?
Our Edexcel GCSE in Mathematics A gives you the flexibility to teach the course content in the order that's right for your students, helping them to make connections across the different areas of mathematics.
Key features of our Edexcel GCSE in Mathematics A:
Flexible delivery approach
More time for cross-curricular work, for example to support work done in subjects that require mathematics (such as science or geography)
Synoptic assessment at the end of the course, allowing for longer questions linking different areas of mathematics | 677.169 | 1 |
Mathematics: Algebra 1 with: Prof. Eric Smith
Whether you are learning Algebra for the first time or brushing up on your math skills, Professor Eric Smith's comprehensive course on Algebra 1 will help you become an expert in no time. Professor Smith begins each lecture with common terminology and methods pertinent to the lesson. He explains concepts clearly, illustrates them with colorful slides, and reinforces them with plenty of sample problems and solutions. This fun, straightforward course in Algebra 1 will keep you engaged and covers everything from Order of Operations to Graphing Functions, Inequalities, Quadratics, and Complex Numbers. Professor Eric Smith has been teaching math at the high school and college level for 10+ years. | 677.169 | 1 |
In the 21st century, technology plays an important role to users, businesses and higher education to raise performance, productivity and endorse gratification. Technology usage increased collaboration, cooperation, and communication among users as easily as to recapture information, entertainment, marketing, political, health information and online... more...
Symposia Mathematica, Volume I focuses on research in the field of mathematics and its applications. This book discusses the definition of S-semigroup, extensions of R modules, structure of H, laws of conservation and equations of motion, and measures of strain. The basic equations for continua with internal rotations, general concepts of the discrete... more...
Group Theory and Its Applications focuses on the applications of group theory in physics and chemistry. The selection first offers information on the algebras of lie groups and their representations and induced and subduced representations. Discussions focus on the functions of positive type and compact groups; orthogonality relations for square-integrable... more...
Currently there is substantial exchange and communication between academic communities around the world as researchers endeavour to discover why so many children 'fail' at a subject that society deems crucial for future economic survival. This book charts current thinking and trends in teacher education around the world, and looks critically at the... more...
Explores the psychology of thinking about post-secondary level mathematics, suggesting that the way it is taught does not correspond to the way it is learned. Addressed to mathematicians and educators in mathematics, considers the nature and cognitive theory of advanced mathematical thinking, and re more...
A collection of annotated texts presenting the principles and key concepts of the theory of didactical situations developed by Guy Brousseau. Concepts such as didactical contract, didactical variable and epistemological obstacle are presented, along with theoretical and experimental research. more... | 677.169 | 1 |
Essentials of College Algebra with Modeling and Visualization with MathXL
Mathxl 12Mo Stu Cpn Business Mathematics
College Algebra with Modeling & Visualization
College Algebra with Modeling and Visualization
College Algebra through Modeling and Visualization
Summary
Today's algebra students want to know thewhybehind what they are learning and it is this that motivates them to succeed in the course. By focusing on algebra in a real-world context, Gary Rockswold gracefully and succinctly answers this need. As many topics taught in today's college algebra course aren't as crucial to students as they once were, Gary has developed this streamlined text, covering linear, quadratic, nonlinear, exponential, and logarithmic functions and systems of equations and inequalities, to get to the heart of what students need from this course. By answering thewhyand streamlining thehow, Rockswold has created a text to serve today's students and help them to truly succeed. | 677.169 | 1 |
For those who hate numbers; and those who like them. Are numbers weird, annoying things that don't mean much? And where did they come from anyway? We don't learn the history of numbers in Math class. This gives them an unreal image that confuses a lot of people. But they came from somewhere for a reason and it helps to know where. This is a short article about the invention of 1, long, long ago.
This International Bestselling book on Vedic Mathematics which will help you do calculations and solve complicated math questions in a matter of a few seconds. Extremely helpful for students giving GMAT, GRE, CAT, SAT, CET and other entrance exams. Over 150,000 copies sold in 14 languages worldwide
One of the best ways to succeed in Geometry is to practice taking real test questions. This volume contains 133 problems on Three-Dimensional Figures divided into four chapters: Definitions and Shapes; Rectangular Solids; Cylinders, Cones, Spheres; and Prisms and Pyramids. Try the problems. With a little Practice, Practice, Practice, you'll be Perfect, Perfect, Perfect. Good Luck!!More people struggle with math than with any other academic discipline. So much so, that it is commonly excusable to not do well in math but to merely survive. There are reasons why math is so hard to learn. This little book identifies and provides insights into the top 10 reasons why math is so hard to learn. If you know what the bumps are, you can slow down and make it over them in one piece!
Practice solving linear equations with these fifty problems in elementary algebra. The student selects a single variable linear equation, solves for the variable, and checks the answer by viewing the step-by-step solution. Problems start with low difficulty and gradually increase to challenging. Most appropriate for 6th to 8th grade students.
Practice solving linear equations with these fifty basic problems in elementary algebra. The student selects a single variable linear equation, solves for the variable, and checks the answer by viewing the step-by-step solution. Problems start with low difficulty and gradually increase to challenging. Most appropriate for 4th and 5th grade studentsMathematics Principles V10 – now comes with an interactive Tablet and Smartphone App.
See Additional Notes at the back of the book for instructions to download the accompanying interactive App which brings the 250+ topics to life by allowing you to insert your own values. Visually on a phone or tablet it looks almost identical to the eBook.
This book teaches the mechanics and methodology of long division, a procedure for dividing numbers without the need for an electronic calculator. Starting with basic concepts, the book explains the method step by step, and then reinforces these concepts using extensive examples and problems with complete solutions. A Tarrington Math Series Book. Most appropriate for grades 5 to 8.
Math is a different kind of subject from all the rest and requires that you understand the differences. If you don't, you will give up too soon and never master math. If you know the secret strategy and apply it from the beginning, you stand a better chance of developing good habits and a good approach to learning math right from the beginning.
2,230 words.
Practice and hone important math skills with twenty division problems, each with an instructive solution. Choose a problem to solve from the problem list, and then confirm your answer by easily navigating the link to view the solution. The book also includes four bonus word problems with complete explanations and answers. Most appropriate for 4th and 5th grade students.
Practice division with | 677.169 | 1 |
A Manual of Geometry and PostScript
0521547881
9780521547888
Details about Mathematical Illustrations:
This practical introduction to the techniques needed to produce high-quality mathematical illustrations is suitable for anyone with basic knowledge of coordinate geometry. Bill Casselman combines a completely self-contained step-by-step introduction to the graphics programming language PostScript with an analysis of the requirements of good mathematical illustrations. The many small simple graphics projects can also be used in courses in geometry, graphics, or general mathematics. Code for many of the illustrations is included, and can be downloaded from the book's web site: scientists, engineers, and even graphic designers seeking help in creating technical illustrations need look no further.
Back to top
Rent Mathematical Illustrations 1st edition today, or search our site for Bill textbooks. Every textbook comes with a 21-day "Any Reason" guarantee. Published by Cambridge University Press. | 677.169 | 1 |
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). | 677.169 | 1 |
books.google.com - This title in the Homework Helpers series tackles the most advanced mathematical course in most high schools: Calculus. The concepts are explained in everyday language before the examples are worked. Good habits, such as checking your answers after every problem, are reinforced. There are practice problems... | 677.169 | 1 |
books.google.com - Pat... Algebra: A Text/Workbook
Beginning Algebra: A Text/Workbook
Pat fresh new technology options. Throughout this text, you'll find hundreds of new and relevant applications, with timely references to topics like gas prices and to companies like Google and Yahoo. Real data is used throughout, and wherever possible, has been updated to reflect recent changes. By showing how mathematics is used every day, the author makes students feel more at ease with the topics at hand. In a course where many students are entering with math anxiety, McKeague almost immediately calms students with chapter and section introductions, provides just in time study tips, and then helps students prepare with Getting Ready for Class boxes found throughout the text. This new edition comes complete with Enhanced WebAssign (EWA), the easy to use homework management system. With EWA, you can assign, collect, grade, and record homework assignments via the web. This complete learning system for students includes text-specific exercises, as well as tutorials, videos, and links to online tutoring, and eBook sections of the text. Important Notice: Media content referenced within the product description or the product text may not be available in the ebook version.
About the author (2009)
Charles | 677.169 | 1 |
This document from SpaceTEC National Aerospace Technical Education Center presents a core readiness course which will serve to prepare individuals entering the aerospace field. The document is 55 pages and contains...
"In the Classroom" highlights how some schools and organizations use Mathematica extensively in their curricula. The section on "Collaborative Initiatives" illustrates how businesses have teamed up with Wolfram Research...
Would Lewis Carroll have approved of using "Alice in Wonderland" to teach algebra? We may never know, but that exact possibility turns up in episode two of the valuable "Teaching Math" series created by staffers at WGBH...
This course, designed for Miami Dade Community College, integrates arithmetic and beginning algebra for the undergraduate student. By applying math to real-life situations most students experience during college, the... | 677.169 | 1 |
Mathematical Applications and Modelling is the second in the series of the yearbooks of the Association of Mathematics Educators in Singapore. The book is unique as it addresses a focused theme on mathematics education. The objective is to illustrate the diversity within the theme and present research that translates into classroom pedagogies.The book,...
"? the games also provide an extremely well-suited platform for the introduction of a unified method for determining complexity using constraint logic ? considers not only mathematically oriented games, but also games that may well be suitable for non-mathematicians ? The book also contains a comprehensive overview of known results on the complexity... more...
The Math in Your Life Health, Safety, and Mathematics Found in Translation The Essentials of Conversion
Making Sense of Your World with Statistics Summarizing Data with a Few Good Numbers Estimating Unknowns Leading You Down the Garden Path with Statistics
Visualizing with Mathematics Seeing Data A Graph Is Worth a Thousand... more...
The new edition of this classic book describes and provides a myriad of examples of the relationships between problem posing and problem solving, and explores the educational potential of integrating these two activities in classrooms at all levels. The Art of Problem Posing, Third Edition encourages readers to shift their thinking about problem... more...
Convex and Discrete Geometry is an area of mathematics situated between analysis, geometry and discrete mathematics with numerous relations to other areas. The book gives an overview of major results, methods and ideas of convex and discrete geometry and its applications. Besides being a graduate-level introduction to the field, it is a practical... more...
Covers percentages, probability, proportions, and more Get a grip on all types of word problems by applying them to real life Are you mystified by math word problems? This easy-to-understand guide shows you how to conquer these tricky questions with a step-by-step plan for finding the right solution each and every time, no matter the kind or... more... | 677.169 | 1 |
RELATIONS and FUNCTIONS
This
Integrating
Technology eBook was designed and developed
for
Mr. Gannaway's
Mathematics II Class at Tift County High
School
Standard
MM2A5a:
Discuss the characteristics of functions and their inverses, including
one-to-oneness, domain and range.
PLO:
Students
will
identify
the difference between a relation and a function, recognize
the domain and range of a function,
locate the slope and y-intercept of a linear function and
recognize the graphs of a linear equations in the slope-intercept
form. | 677.169 | 1 |
The Department of Mathematics at Las Positas College Offers a wide range
of courses in a variety of formats designed
to meet the needs of a diverse student population. Whatever your educational
goals, the Mathematics Department has the courses you need to succeed.
We offer a full lower division curriculum which prepares sudents for
transfer to 4-year institutions, as well as basic skills and college
prepatory math courses. To find out how to select the right first math
course for you, read about Assessment and Placement
below.
Las Positas College
offers four ways to take mathematics:
Non-Traditional
Math X Lab. The Math X Lab features a workbook delivery
system in a self-paced mastery learning approach. The instructor and
instructional assistant are resources and do not lecture. Each course
is divided into two halves (called modules) to allow students more time
to cover the material. Math X Lab courses are numbered Math 36X, 36Y,
55X, 55Y, 65X, 65Y, 71X, 71Y, 107X and 107Y.
On-Line.
An On-Line course is a distance education course delivered over the
internet using web-based tutorial software. To take an on-line class,
a student must have experience using an Internet browser, such as Netscape
Navigator or Internet Explorer and must have daily access to a computer
with an Internet connection and a CD-ROM drive. The instructor maintains
a web-site containing course information uses e-mail, on-line chat rooms
and threaded discussions to assist students taking the course. The only
on-line courses currently available are Math 65 Elementary Algebra and
Math 55 Intermediate Algebra.
ASSESSMENT
AND PLACEMENT Before registering for a math course at LPC
every student must take a Math Assessment test. These exams are given by appointment
in Building 1000. Contact the staff in the Assessment Testing Office, 925.424.1475, for an appointment.
At LPC we use the ACCUPlacer computerized testing system. The system will guide you to the appropriate testing choice to determine your placement into the appropriate math class.
Your
assessment results will tell you the math courses in which you are eligible
to enroll. Please note that students may enroll in Basic Mathematics
(Math 106 or 107X) without taking an assessment test. It has been our
experience that the assessment process is generally quite accurate.
If you feel strongly that you have been placed in a course incorrectly,
see a Counselor about the prerequisite challenge process.
PREREQUISITES Many
students' program may include a sequence of math courses that must be
taken to successfully complete the program. Often these courses must
be taken in a prescribed order. To help you in planning your program,
you may want to refer to the Math Prerequisite
Flow Chart. This chart may be found in the Schedule
of Courses
and is available here. In reading the flow chart, please note that Math 65 Elementary
Algebra (*) can be replaced with the sequence Math 65A or Math
65X followed by Math 65B or Math 65Y. Similarly, Math
55 Intermediate Algebra (**) can be replaced with the sequence Math
55A or Math 55X followed by Math 55B or Math 55Y. | 677.169 | 1 |
I to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material Algebra I
Select this link to open drop down to add material AlgebraIntermediate Algebra carefully builds on the basics learned in Elementary Algebra and introduces the more advanced topics,...
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Intermediate Algebra carefully builds on the basics learned in Elementary Algebra and introduces the more advanced topics, enhancing it all with with the modern amenities that only a free online text can deliver.It is essential to lay a solid foundation in mathematics if a student is to be competitive in today's global market. The importance of algebra, in particular, cannot be overstated, as it is the basis of all mathematical modeling used in applications found in all disciplines. Traditionally, the study of algebra is separated into a two parts, Elementary and Intermediate Algebra. This textbook by John Redden, Intermediate Algebra, is the second part. Written in a clear and concise manner, it carefully builds on the basics learned in Elementary Algebra and introduces the more advanced topics required for further study in applications found in most disciplines. Used as a standalone textbook, Intermediate Algebra offers plenty of review as well as something new to engage the student in each chapter. Written as a blend of the traditional and graphical approaches to the subject, this textbook introduces functions early and stresses the geometry behind the algebra. While CAS independent, a standard scientific calculator will be required and further research using technology is encouraged. Intermediate Algebra is written from the ground up in an open and modular format, allowing the instructor to modify it and leverage their individual expertise as a means to maximize the student experience and success. A more modernized element, embedded video examples, are present, but the importance of practice with pencil and paper is consistently stressed. Therefore, this text respects the traditional approaches to algebra pedagogy while enhancing it with the technology available today. The importance of Algebra cannot be overstated; it is the basis for all mathematical modeling used in all disciplines. After completing a course sequence based on Elementary and Intermediate Algebra, students will be on firm footing for success in higher-level studies at the college Intermediate Algebra to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material Intermediate Algebra
Select this link to open drop down to add material Intermediate Algebra to your Bookmark Collection or Course ePortfolio
* An introduction that appeals to the reader's reason rather than to her/his ability to memorize. * A complete tool for...
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* An introduction that appeals to the reader's reason rather than to her/his ability to memorize. * A complete tool for teaching "developmental" students twice a week for 15 weeks. * A way for adults to learn some mathematics—more or less in the same spirit as mathematicians do. * A text, with a story-line, written to be read and reread. * A presentation that pays pedantic attention to the linguistic difficulties the reader is likely to have in mathematics. * A political act to "enable people to get on better terms with reason—to learn to live with the truth." [Colin McGinn] * An anti "Show a Template Example, Drill and Test" manifesto. * An open source package written in LaTeX with lots of vector graphics. * A standalone version of part of From Arithmetic to Differential Calculus. (In Preparation.) * An instance of a model-theoretic approach to mathematical exposition. * A treatment that, while not rigorous in the usual mathematical sense, sins only by venial omission. * A work by a mathematician who, almost fifty years ago, got interested in reconciling "just plain folks" with mathematicsasonable Basic Algebra (RBA) to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material Reasonable Basic Algebra (RBA)
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A Problem Course in Mathematical Logic is intended to serve as the text for an introduction to mathematical logic for...
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A Problem Course in Mathematical Logic is intended to serve as the text for an introduction to mathematical logic for undergraduates with some mathematical sophistication. It supplies definitions, statements of results, and problems, along with some explanations, examples, and hints. The idea is for the students, individually or in groups, to learn the material by solving the problems and proving the results for themselves. The book should do as the text for a course taught using the modified Moore-method.The material and its presentation are pretty stripped-down and it will probably be desirable for the instructor to supply further hints from time to time or to let the students consult other sources. Various concepts and and topics that are often covered in introductory mathematical logic or computability courses are given very short shrift or omitted entirely, among them normal forms, definability, and model Course in Mathematical Logic to your Bookmark Collection or Course ePortfolio
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Select this link to open drop down to add material A Problem Course in Mathematical virtual edition of a developmental algebra textbook. It follows the Suffolk County Community College Mathematics...
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This is a virtual edition of a developmental algebra textbook. It follows the Suffolk County Community College Mathematics syllabus for our developmental algebra courses, with a few additions.This book has a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License and may be used and shared for any non-profit educational purpose is Vital to your Bookmark Collection or Course ePortfolio
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A top-selling teacher resource line, The 100+ Series(TM) features over 100 reproducible activities in each book! --This revised edition of Pre-Algebra links all the activities to the NCTM Standards. The activities were designed to provide students with practice in the skill areas necessary to master the concepts introduced in a course of pre-algebra. Reinforcing operations skills with both decimals and fractions plus activities involving ratios, integers, proportions, percents, rational numbers, simple equations, plotting coordinates, and graphing linear equations are all part of this new edition. Examples of solution methods are presented at the top of each page. New puzzles and riddles have been added to gauge the success of skills learned. It also contains a complete answer key.
Book Description Instructional Fair. PAPERBACK. Book Condition: New. 0742417875 Brand New in Mint condition. Guaranteed delivery in 2-4 days when you order with Expedited Shipping! No Expedited shipping to PO Boxes. Bookseller Inventory # 0742417875 | 677.169 | 1 |
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4434A First Course in Linear Algebra
A First Course in Linear Algebra is an introductory textbook aimed at college-levelPDF versions are available to download for printing or on-screen viewing, an online version is available, and physical copies may be purchased from the print-on-demand service at Lulu.com. GNU Free Documentation LicenseThu, 13 Mar 2008 12:39:13 -0700Linear Algebra
Linear Algebra is free for downloading, It covers the material of an undergraduate first linear algebra course.Sat, 12 Apr 2008 23:32:27 -0700Linear Algebra, Theory and Applications
From the preface: "This is a book on linear algebra and matrix theory. While it is self-contained, it will work best for those who have already had some exposure to linear algebra. It is also assumed that the reader has had calculus. Some optional topics require more analysis than this, however." A solutions manual to the exercises in the textbook is included.Wed, 23 May 2012 14:13:20 -0700A First Course in Linear AlgebraTue, 26 Feb 2013 09:33:26 -0800A First Course in Linear Algebra: Study Guide for the Undergraduate Linear Algebra CourseWed, 20 May 2015 02:25:04 -0700Computational and Algorithmic Linear Algebra and n-Dimensional Geometry
״This is a sophomore level webbook on linear algebra and n-dimensional geometry with the aim of developing in college entering undergraduates skills in algorithms, computational methods, and mathematical modeling. It is written in a simple style with lots of examples so that students can read most of it on their own.״ Each chapter is downloaded separately as a pdf file. Fri, 5 Mar 2010 13:00:31 -0800Elementary Algebra Exercise Book I
This is a free textbook from BookBoon.'Algebra is one of the main branches in mathematics. The book series of elementary algebra exercises includes useful problems in most topics in basic algebra. The problems have a wide variation in difficulty, which is indicated by the number of stars.'Tue, 29 Jan 2013 06:58:43 -0800Elementary Linear Algebra
״This book is an introduction to linear algebra, based on lectures given by me over 17 years, in the (now defunct) first year course MP103 at the University of Queensland. The section on subspaces is meant to be a gentle introduction to the second course, where abstract vector spaces are met in detail. Things of substance are met here, including the rank of a matrix. The section on three dimensional geometry makes use of the earlier sections on linear equations, matrices and determinants and some of the proofs are more algebraic (even pedantic) than some readers would like.״Wed, 10 Mar 2010 13:49:52 -0800Elementary Linear Algebra
This is a free, online textbook offered by BookBoon. "This is an introduction to linear algebra. The main part of the book features row operations and everything is done in terms of the row reduced echelon form and specific algorithms. At the end, the more abstract notions of vector spaces and linear transformations on vector spaces are presented. This is intended to be a first course in linear algebra for students who are sophomores or juniors who have had a course in one variable calculus and a reasonable background in college algebra.״ Wed, 28 Mar 2012 11:45:06 -0700Fundamentals of Matrix Algebra
his text provides the reader with a solid foundation of the fundamental operations and concepts of matrix algebra. The topics include systems of linear equations, matrix arithmetic, transpose, trace, determinant, eigenvalues/vectors, and linear transformations, focusing largely on transformations of the Cartesian plane.The text is designed to be easily read, written in a casual style. Key concepts are explained, but rigorous proofs are omitted. Numerous examples are provided to illuminate new ideas and provide practice. Each section ends with exercises (with answers to odd questions appearing in the back).The text is currently in use by Cadets of the Virginia Military Institute. It is appropriate for undergraduates in mathematics and the sciences, as well as advanced high school students.Tue, 19 Jan 2010 11:35:58 -0800 | 677.169 | 1 |
Introductory Algebra: Equations and Graphs
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Read More of functions in intermediate algebra. This clearly differentiates Yoshiwara from standard introductory algebra texts. The text emphasizes the study of tables and graphs, and the concept of the variable is developed from that platform. Graphs are used extensively throughout the book to illustrate algebraic technique and to help students visualize relationships between variables. The numerous labeled grids paired with exercises throughout the text reinforce the need to draw graphs by hand while helping students to focus on the properties of the graphs by eliminating the time-consuming task for beginning students of choosing the appropriate scales for the axes. Suggestions for calculator activities are included on the text web site by the authors believe that this skill must be learned through practice with pencil and paper CD MISSING.
The book has normal wear and tear. The Cover and Pages are Very Good! Fast shipping. Most orders shipped by the next business day! The CD to accompany the book is Included! Brand New Access Code is included & Unopened! Book has little or no writing or highlighting. All pages included...nothing torn out or missing | 677.169 | 1 |
Synopses & Reviews
Publisher Comments:
This book is a concise introduction to number theory and some related algebra, with an emphasis on solving equations in integers. Finding integer solutions led to two fundamental ideas of number theory in ancient times - the Euclidean algorithm and unique prime factorization - and in modern times to two fundamental ideas of algebra - rings and ideals. The development of these ideas, and the transition from ancient to modern, is the main theme of the book. The historical development has been followed where it helps to motivate the introduction of new concepts, but modern proofs have been used where they are simpler, more natural, or more interesting. These include some that have not yet appeared in textbooks, such as a treatment of the Pell equation using Conway's theory of quadratic forms. Also, this is the only elementary number theory book that includes significant applications of ideal theory. It is clearly written, well illustrated, and supplied with carefully designed exercises, making it a pleasure to use as an undergraduate textbook or for independent study. John Stillwell is Professor of Mathematics at the University of San Francisco. He is the author of several highly regarded books published by Springer-Verlag, including Mathematics and Its History (Second Edition 2001), Numbers and Geometry (1997) and Elements of Algebra (1994).
Synopsis:
Springer, ,
Includes bibliographical references (p. 239-244 | 677.169 | 1 |
"We really can't assume that calculators are helping students," said King. "The goal is to understand the core concepts during the lecture. What we found is that use of calculators isn't necessarily helping in that regard." | 677.169 | 1 |
Elementary Numerical Analysis
9780471433378
ISBN:
0471433373
Edition: 3 Pub Date: 2003 Publisher: Wiley
Summary: Offering a clear, precise, and accessible presentation, complete with MATLAB programs, this new Third Edition of Elementary Numerical Analysis gives students the support they need to master basic numerical analysis and scientific computing. Now updated and revised, this significant revision features reorganized and rewritten content, as well as some new additional examples and problems. The text introduces core areas... of numerical analysis and scientific computing along with basic themes of numerical analysis such as the approximation of problems by simpler methods, the construction of algorithms, iteration methods, error analysis, stability, asymptotic error formulas, and the effects of machine arithmetic.
Kendall Atkinson is the author of Elementary Numerical Analysis, published 2003 under ISBN 9780471433378 and 0471433373. Two hundred ninety one Elementary Numerical Analysis textbooks are available for sale on ValoreBooks.com, twenty used from the cheapest price of $107.94, or buy new starting at $137 576 pages, Printed in Black and White , Same content a... [more]ALTERNATE EDITION: Brand New Softcover International Edition, 576 pages, Printed in Black and White , Same | 677.169 | 1 |
A new ANGLE to learning GEOMETRY
Trying to understand geometry but feel like you're stuck in another dimension? Here's your solution. Geometry Demystified , Second Edition helps you grasp the essential concepts with ease.
Written in a step-by-step format, this practical guide begins with two dimensions, reviewing points, lines, angles, and... more...
This edited collection of chapters, authored by leading experts, provides a complete and essentially self-contained construction of 3-fold and 4-fold klt flips. A large part of the text is a digest of Shokurov's work in the field and a concise, complete and pedagogical proof of the existence of 3-fold flips is presented. The text includes a ten... more...
The word barycentric is derived from the Greek word barys (heavy), and refers to center of gravity. Barycentric calculus is a method of treating geometry by considering a point as the center of gravity of certain other points to which weights are ascribed. Hence, in particular, barycentric calculus provides excellent insight into triangle centers.... more...
The tensorial nature of a quantity permits us to formulate transformation rules for its components under a change of basis. These rules are relatively simple and easily grasped by any engineering student familiar with matrix operators in linear algebra. More complex problems arise when one considers the tensor fields that describe continuum bodies.... more... | 677.169 | 1 |
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College Algebra, 11th Edition
Description
College Algebra, Eleventh Edition, by Lial, Hornsby, Schneider, and Daniels, engages and supports students in the learning process by developing both the conceptual understanding and the analytical skills necessary for success in mathematics. With the Eleventh Edition, the authors adapt to the new ways in which students are learning, as well as the ever-changing classroom environment.
Table of Contents
R. Review of Basic Concepts.
R.1 Sets
R.2 Real Numbers and Their Properties
R.3 Polynomials
R.4 Factoring Polynomials
R.5 Rational Expressions
R.6 Rational Exponents
R.7 Radical Expressions
1. Equations and Inequalities
1.1 Linear Equations
1.2 Applications and Modeling with Linear Equations
1.3 Complex Numbers
1.4 Quadratic Equations
1.5 Applications and Modeling with Quadratic Equations
1.6 Other Types of Equations and Applications
1.7 Inequalities
1.8 Absolute Value Equations and Inequalities
2. Graphs and Functions
2.1 Rectangular Coordinates and Graphs
2.2 Circles
2.3 Functions
2.4 Linear Functions
2.5 Equations of Lines and Linear Models
2.6 Graphs of Basic Functions
2.7 Graphing Techniques
2.8 Function Operations and Composition
3. Polynomial and Rational Functions.
3.1 Quadratic Functions and Models
3.2 Synthetic Division
3.3 Zeros of Polynomial Functions
3.4 Polynomial Functions: Graphs, Applications, and Models
3.5 Rational Functions: Graphs, Applications, and Models
3.6 Variation
4. Inverse, Exponential, and Logarithmic Functions.
4.1 Inverse Functions
4.2 Exponential Functions
4.3 Logarithmic Functions
4.4 Evaluating Logarithms and the Change-of-Base Theorem
4.5 Exponential and Logarithmic Equations
4.6 Applications and Models of Exponential Growth and Decay
5. Systems and Matrices
5.1 Systems of Linear Equations
5.2 Matrix Solution of Linear Systems
5.3 Determinant Solution of Linear Systems
5.4 Partial Fractions
5.5 Nonlinear Systems of Equations
5.6 Systems of Inequalities and Linear Programming
5.7 Properties of Matrices
5.8 Matrix Inverses
6. Analytic Geometry
6.1 Parabolas
6.2 Ellipses
6.3 Hyperbolas
6.4 Summary of the Conic Sections
7. Further Topics in Algebra
7.1 Sequences and Series
7.2 Arithmetic Sequences and Series
7.3 Geometric Sequences and Series
7.4 The Binomial Theorem
7.5 Mathematical Induction
7.6 Counting Theory
7.7 Basics of Probability
Glossary
Solutions to Selected Exercises
Answers to Selected Exercises
Index of Applications
Index
This title is also sold in the various packages listed below. Before purchasing one of these packages, speak with your professor about which one will help you be successful in your course. | 677.169 | 1 |
* The basic approach and application of algebra to problem solving * The number system (in a much broader way than you have known it from arithmetic) * Monomials and polynomials; factoring algebraic expressions; how to handle algebraic fractions; exponents, roots, and radicals; linear and fractional equations * Functions and graphs; quadratic equations; inequalities; ratio, proportion, and variation; how to solve word problems, and more Authors Peter Selby and Steve Slavin emphasize practical algebra throughout by providing you with techniques for solving problems in a wide range of disciplines--{"currencyCode":"USD","itemData":[{"priceBreaksMAP":null,"buyingPrice":11.26,"ASIN":"0471530123","moqNum":1,"isPreorder":0},{"priceBreaksMAP":null,"buyingPrice":8.45,"ASIN":"0965911373","moqNum":1,"isPreorder":0},{"priceBreaksMAP":null,"buyingPrice":14.67,"ASIN":"0965911381","moqNum":1,"isPreorder":0}],"shippingId":"0471530123::qqgAKwt3HECc4cckzegzFmfF0uLzNWkZ54V0zjQHwiQ9rZPoRCTnbL%2BPJ1bL5OMgucew04zNaajD2RVFc3sI2T%2FwLbO1CZllokkDVMzarGU%3D,0965911373::tLPd1aYP5h2zsbYBM0N8Qg1GjX6KnMljWRvg4XdaeGoBkX5ozFdXduXltiH05K6Y3njr1R6nL7IrRf9wps6lzRnhypx2Zvkr1Kuz3HEDbpU%3D,0965911381::9nDjwmbTdKpFfk3D6dKOI%2Byox0ztQbJz6WaDZ9lqhzUXWAolg3%2Fw5WyoVttm9G%2Bz60YcTez8YzkMDVDCcv5NUpo6RvTBfTTBOSyICG05W ``learn-by-doing'' approach, it reviews and teaches elementary and some intermediate algebra. While rigorous enough to be used as a college or high school text, the format is reader friendly, particularly in this Second Edition, and clear enough to be used for self-study in a non-classroom environment. ``Pre-test'' material enables readers to target problem areas quickly and skip areas that are already well understood. Some new material has been added to the Second Edition and redundant or confusing material omitted. The first chapter has undergone major revision. Chapters feature ``post-tests'' for self-evaluation. Thousands of practice problems, questions and answers make this algebra review a unique and practical text.
From the Back Cover
The basic approach and application of algebra to problem solving
The number system (in a much broader way than you have known it from arithmetic)
Functions and graphs; quadratic equations; inequalities; ratio, proportion, and variation; how to solve word problems, and more
Authors Peter Selby and Steve Slavin emphasize practical algebra throughout by providing you with techniques for solving problems in a wide range of disciplines—Most Helpful Customer Reviews
I'd certainly award them to Mr. Selby and Dr. Slavin's book. I'm not finished with the text, but I'm learning so well that I KNOW I will have mastered the material. In fact, I just ordered the book which they recommend as a follow-up, "Geometry and Trigonometry for Calculus." They start out with baby steps -- literally simple addition and subtraction, and work up to fairly sophisticated algebraic expressions and problems. The student finds himself enjoying the material almost like it were a novel. Selby and Slavin give immediate answers for ALL the problems, so there's no frustration wondering if the problem is solved correctly. If the student misses a certain number of problems in a set, the authors recommend review of specific "frames" which cover the material. They treat the student with respect, making it clear that NO ONE (least of all someone who is motivated to study through their text) is a dummy. This is so much easier than trying to keep up with a math teacher who scribbles and then erases the material on a blackboard about as fast as he/she can write. I now know that self-paced learning is the way to go, at least for me. Many years ago, I failed calculus in college, and barely squeaked through college algebra and trigonometry. I fully intend to use this framework as a basis to repeat the course (even if via correspondence) and get this only blotch on an otherwise fine college career off my record. How sad that Mr. Selby has passed on, but how great it is that his books live still.
If you have been away from Algebra for a while or even if you have just started learning Algebra this book is for you. The authors have done a truly masterful job of explaining the basic concepts of Algebra. But more than that, the book has frequent tests and exercises which will help you to see if you are truly learning the stuff. Another great feature is that things are presented in a succinct and cogent manner. There is no patronizing prose or cute stories to bore you. One thing I recommend strongly. Get well versed with chapters 2 through 4. These are the "nail and hammer" chapters in the book. Techniques like factoring, exponent manipulations, and simplification are explained. If you do not get these techniques nailed down you will be hounded with difficulties in later chapters. I re-read the chapters 3 times as well as worked the problems. I also recommend that as you go along begin solving pertinent problems from "REA'S Algebra & Trigonometry Problem Solver". This may take some time but your efforts will be well rewarded. I highly recommend this text for anyone learning Algebra. But specially if you are like myself, just "doing" Algebra after a long hiatus, grab this book. A must have !
This book is the greatest thing since sliced bread. I am getting ready to start college, and I wanted to major in theoretical physics but all I had ever taken was basic math through school. This book helped me learn more in one week than a high school teacher could in a month. If you want to learn algebra, and are good in arithmitic buy this book and see for yourself.
I recently had to get back into Math. I was not anxious since it had been about 10 years since I had to practice and test myself on Math. I asked my College Counselor (College of Marin) who recommended your book "Practical Algebra: A Self Teaching Guide". I was afraid!! But once I got into the book, from Page 1, I felt comfortable. I felt as if the instructor was there with me, supporting me and teaching me all at once. I felt completely at ease with the structure, the pace and the questions. Thank you so much for allowing my return to Math be so painless.
This book, together with Algebra Unplugged, has bolstered my confidence, as I return to school after many years, for my MBA. Not having taken a math class in 30 years, my skills were beyond rusty. Algebra Unplugged is a light-hearted look at algebra concepts and was a great start, but I needed a book that would give me sample problems to work on and this book fit the bill. I'm nearly done with it and feel just about ready to tackle the math challenges ahead of me. (But first, I need to re-learn more advanced material, like Trig, before I hit the Calculus for Managers class.)
I am 46 years old and have not cracked open an Algebra textbook in close to thirty years. The reviews I read about this book helped me to decide to purchase it, since an upcoming Intermediate Algebra course is looming on the horizon.
I have only gone through the first three chapters but I am very pleased with what I have learned so far. The lessons are very methodical which makes retaining the information possible. I do wish there were more practice problems but overall this is one of the best Amazon purchases I've made to date (and I've purchased plenty).
I purchased this book to help me get through a college Algebra class. It was much more informative that the class textbook. The author practically saved my life in this class. There are plenty of explanations of the whys, hows and rules of Algebra. There are plenty of examples and practices. This book is very well laid out with great order of the topics covered. I will probably keep this book and pitch my $100 college textbook when the class is over. | 677.169 | 1 |
Description:
Created by artist Melissa Tomlinson Newell and mathematician Deann Leoni, this page presents lecture-studio courses in both 2-dimensional and 3-dimensional design. These courses allow students to explore elements and principles of design with a focus on critical thinking and problem solving. Syllabi for both courses are available, as well as some sample projects, student work, and instructor reflections. This is an excellent resource for educators looking to explore mathematics with students in new ways, or for design teachers looking to incorporate math skills into the classroom. | 677.169 | 1 |
> Dick Askey says FOIL is "a mindless phrase which almost everyone >who teaches college math knows causes trouble for many students". > I disagee about FOIL. I think it is very helpful and useful, NOT >"mindless." > I currently teach developmental math at the college level. I teach >Math 10, a non-credit course that is taught in two tracks. Track A is >roughly equivalent to high school Algebra I and Track B is roughly >equivalent to high school Algebra II. > I use the term FOIL when teaching my current students for several >reasons: > 1) Since I learned the term, I found it helpful in the past for >teaching alegbra at the high school level and in my present position. > 2) My college students of the past three years are familiar >with the term and use it very well as an aid in multiplying binomials. >Note: A lot of things in math cause trouble for many of my Math 10 >students; FOIL seems to be one concept that most of my students >understand and use well before it is introduced in class. That is not >true for much of other things that my Math 10 students encounter in >Math 10. > 3) The textbook used in Math 10 explains and uses FOIL. That >text is Intermediate Algebra, Second Edition by Mark Dugopolski, >Addison-Wesley, Reading, Massachusetts, 1996. This text was not >selected by me but by the Math 10 coordinator. > > What do others think? I'm interested. >------------------------------------------------------------------------
First, I have to confess that I taught our college high school Algebra I class once, and was so frustrated I never did it again. But I do teach a lot of math for elementary teachers, and I like the method below.
To multiply a polynomial with 2 terms by a polynomial with 3 terms, for example (x-2)(2x^2 - x + 5), do this:
Make a 2 x 3 grid of boxes. Turn the polynomial into a sum by adding the opposite: (x + (-2))(2x^2 + (-x) + 5). Write each term of each sum across the top and down the side, as appropriate. Fill in the boxes with products like a multiplication table. For this example, you will get 2x^2 -x 5 --------------------------------------------- x | 2x^3 | -x^2 | 5x | --------------------------------------------- -2 | -4x^2 | 2x | -10 | ---------------------------------------------
Then add all the products.
Reasons I like this method: *It works for any polynomials, not just binomials (one of Dick's objections) *It's a more visual/geometric way than algebraic/symbolic, which feels more comfortable and understandable to some people. *The reason it works is deeply related to what addition and multiplication are all about, and is connected to length and area, multiplication of whole numbers, multiplication tables, etc. If your students can get these connections, they will be way ahead.
Here's a quick summary of the connections.
For any two positive (real) numbers a and c, you can represent their product by the area of a rectangle with sides a and c. For students who don't get this connection, use whole numbers and graph paper.
To represent the distributive property (a+b)c = ac + bc in this picture, make an (a+b) by c rectangle, and cut it into two rectangles by splitting a+b into a and b. The left side of the equation is the total area, the right side is the areas of the two subrectangles.
The way we multiply multi-digit whole numbers can be shown in this same kind of diagram. Divide the length and width according to place values. For instance, 347 x 26 = (300 + 40 + 7) x (20 + 6) Draw the diagram, which will have 6 boxes. Fill the boxes with the products 300x200, 300x6, etc. (It might be worthwhile for those who haven't caught on to the idea to do this on graph paper with actual lengths, though a number in the 100s is a little mean. In general, use a not-to-scale diagram.) Then add all the products.
The so-called "standard algorithm" for multiplication in columns does something more compressed than this: if the 347 is on top, then the partial products (6940 and 2082) that you add to get the product are not immediately visible in the box diagram. But they are there: add the boxes across or down as appropriate: 6000+800+140 = 6940, 1800+240+42 = 2082.
Almost the same picture would multiply 3x^2 + 4x + 7 by 2x + 6. Then if x=10, this is the multiplication you just did. If x represents some negative number, the diagram is no longer geometrically meaningful, but the distributive property works all the same. (The number system is designed so that all the properties you're used to work even with new and strange numbers.)
Variation: Fill in 0 coefficients for any missing terms in the polynomial and give them their own rows and/or columns. Then like terms in the product will end up in diagonal rows, providing an extra check. This is almost the same as the lattice multiplication method for whole numbers. | 677.169 | 1 |
The new edition of Thomas is a return to what Thomas has always been: the book with the best exercises. For the 11th edition, the authors have added exercises cut in the 10th edition, as well as, going back to the classic 5th and 6th editions for additional exercises and examples. The book's theme is that Calculus is about thinking; one cannot memorize it all. The exercises develop this theme as a pivot point between the lecture in class, and the understanding that comes with applying the ideas of Calculus. In addition, the table of contents has been refined to match the standard syllabus. Many of the examples have been trimmed of distractions and rewritten with a clear focus on the main ideas. The authors have also excised extraneous information in general and have made the technology much more transparent. The ambition of Thomas 11e is to teach the ideas of Calculus so that students will be able to apply them in new and novel ways, first in the exercises but ultimately in their careers. Every effort has been made to insure that all content in the new edition reinforces thinking and encourages deep understanding of the material stl110
Book Description Addison Wesley. Book Condition: New. 03211855871185587т
Book Description Lebanon, Indiana, U.S.A.: Addison-Wesley, 2004. Soft cover. Book Condition: New. 5th or later Edition. Bookseller Inventory # ABE-10208420020
Book Description Book Condition: Brand New. 111349 | 677.169 | 1 |
This course, along with MTH 1122, is designed to meet the requirements of the state certification of elementary teachers. Students are strongly encouraged to complete both courses in sequence at the same institution and should check the specific requirements at the senior institution. The sequence fulfills the general education requirement only for students with a declared major in elementary and/or special education. This course focuses on mathematical reasoning and problem solving. Topics will be selected from the following list: integers, irrational numbers and the real number system, number theory, probability, rational numbers, sets, function, logic, whole numbers, and statistics. The use of calculators and other technology is strongly encouraged. PREREQUISITE: PRE 0420 Intermediate Algebra and PRE 0415 Elementary Geometry with a grade of C or better or two years of college preparatory algebra and one year geometry or placement test score, or consent of instructor. | 677.169 | 1 |
This is a free online course offered by the Saylor Foundation.'This course is an introduction to complex analysis, or the...
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This is a free online course offered by the Saylor Foundation.'This course is an introduction to complex analysis, or the theory of the analytic functions of a complex variable. Put differently, complex analysis is the theory of the differentiation and integration of functions that depend on one complex variable. Such functions, beautiful on their own, are immediately useful in Physics, Engineering, and Signal Processing. Because of the algebraic properties of the complex numbers and the inherently geometric flavor of complex analysis, this course will feel quite different from Real Analysis, although many of the same concepts, such as open sets, metrics, and limits will reappear. Simply put, you will be working with lines and sets and very specific functions on the complex plane—drawing pictures of them and teasing out all of their idiosyncrasies. You will again find yourself calculating line integrals, just as in multivariable calculus. However, the techniques you learn in this course will help you get past many of the seeming dead-ends you ran up against in calculus. Indeed, most of the definite integrals you will learn to evaluate in Unit 7 come directly from problems in physics and cannot be solved except through techniques from complex variables.We will begin by studying the minimal algebraically closed extension of real numbers: the complex numbers. The Fundamental Theorem of Algebra states that any non-constant polynomial with complex coefficients has a zero in the complex numbers. This makes life in the complex plane very interesting. We will also review a bit of the geometry of the complex plane and relevant topological concepts, such as connectedness.In Unit 2, we will study differential calculus in the complex domain. The concept of analytic or holomorphic function will be introduced as complex differentiability in an open subset of the complex numbers. The Cauchy-Riemann equations will establish a connection between analytic functions and differentiable functions depending on two real variables. In Unit 3, we will review power series, which will be the link between holomorphic and analytic functions. In Unit 4, we will introduce certain special functions, including exponentials and trigonometric and logarithmic functions. We will consider the Möbius Transformation in some detail.In Units 5, 6, and 7 we will study Cauchy Theory, as well as its most important applications, including the Residue Theorem. We will compute Laurent series, and we will use the Residue Theorem to evaluate certain integrals on the real line which cannot be dealt with through methods from real variables alone. Our final unit, Unit 8, will discuss harmonic functions of two real variables, which are functions with continuous second partial derivatives that satisfy the Laplace equation, conformal mappings, and the Open Mapping Theorem Complex Analysis to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material Complex Analysis
Select this link to open drop down to add material Complex Analysis to your Bookmark Collection or Course ePortfolio
This is a free, online textbook/course that provides introductory information for math students. "This unit has two aims:...
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This is a free, online textbook/course that provides introductory information for math students. "This unit has two aims: firstly, to help you read and interpret information in the form of diagrams, charts and graphs, and secondly, to give you practice in producing such diagrams yourself. To start you will deal with interpreting and drawing diagrams to a particular scale. You will then learn to extract information from tables and charts. Finally you will learn to draw graphs using coordinate axes, which is a very important mathematical techniqueagrams, Charts and Graphs to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material Diagrams, Charts and Graphs
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This is a free, online textbook/course that provides information on conducting empirical research. The site includes written...
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This is a free, online textbook/course that provides information on conducting empirical research. The site includes written material as well as videos. According to the authors, "This course includes self-guiding materials and activities, and is ideal for independent learners, or instructors trying out this course package is a free online course offered by the Saylor Foundation.'The courses included in this program are designed for the high...
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This is a free online course offered by the Saylor Foundation.'The courses included in this program are designed for the high school student preparing for college or the adult learner who needs a refresher course or two in mathematics.Each of the courses in this series includes instructional videos and practice problems from Khan Academy™ (Khan Academy™ is a library of over 3,000 videos covering a range of topics, including math and physics) that will help you master the foundational knowledge necessary for success in College Algebra (MA001: Beginning Algebra) and beyond.These courses focus on the ways in which math relates to common "real world" situations, transactions, and phenomena, such as personal finance, business, and the sciences. This "real world" focus will help you grasp the importance of the mathematical concepts you encounter in these courses and understand why you need quantitative and algebraic skills in order to be successful both in college and in your day-to-day-life Foundations of Real World Math to your Bookmark Collection or Course ePortfolio
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This free online textbook/course "looks at various aspects of shape and space. It uses a lot of mathematical vocabulary, so...
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This free online textbook/course "looks at various aspects of shape and space. It uses a lot of mathematical vocabulary, so you should make sure that you are clear about the precise meaning of words such as circumference, parallel, similar and cross-section. You may find it helpful to note down the meaning of each new word in your Learning Journal, perhaps illustrating it with a diagram. This module contains some interactive geometry activities which use the Java based software, Geogebra to your Bookmark Collection or Course ePortfolio
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2007 Summer Semester "the World of Mathematics״Mathematics has the history of more than 2,000 years. It is the field still...
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2007 Summer Semester "the World of Mathematics״Mathematics has the history of more than 2,000 years. It is the field still studied actively today, and developments are made rapidly in. Profound nature of numbers and figures, structures of functions and spaces are discovered one after another, and now, Fermat's last theorem, which had been the mystery of 350 years, or Poincare conjecture left pending for 100 years are solved. Recent proceedings of mathematics is quite spectacular. Also, since it allows free thinking and has a wide versatility, mathematics is applied to fundamental science, engineering, economics, sociology, and other disciplines as the common language of science. Sometimes, mathematics has the power to change all over the sociey, as it did when the principle of computer was discovered. Mathematics cannot be divided artificially. However, we are accustomed to, for the sake of convenience, separate it into four fields; algebra, geometry, analysis and applied mathematics. Plus, mathematical science includes mathematics for practical applications to analyze natural and social phenomena In this lecture series, themes characteristic to each fields are discussed by the experts working internationally. They navigate you for your further studies, by focusing mathematics from a wide perspective. Mathematics students learn in the first and second grade would be the basis of further studies. These lectures are expected to provide you with an outlook 2007 / the World of Mathematics to your Bookmark Collection or Course ePortfolio
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This course introduces the basic techniques of demographic analysis. Students will become familiar with the sources of data...
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This course introduces the basic techniques of demographic analysis. Students will become familiar with the sources of data available for demographic research. Population composition and change measures will be presented. Measures of mortality, fertility, marriage and migration levels and patterns will be defined. Life table, standardization and population projection techniques will also be explored. This course was developed by JHSPH faculty with the generous support of the Andrew W. Mellon Foundation and the Bill and Melinda Gates Foundation. Qualified educators may order this course and others on CD-ROM from the Bill and Melinda Gates Instiute for Population and Reproductive Health Demographic Methods to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material Introduction to Demographic Methods
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This is a free online course offered by the Saylor Foundation.'The main purpose of this course is to bridge the gap between...
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This is a free online course offered by the Saylor Foundation.'The main purpose of this course is to bridge the gap between introductory mathematics courses in algebra, linear algebra, and calculus on one hand and advanced courses like mathematical analysis and abstract algebra, on the other hand, which typically require students to provide proofs of propositions and theorems. Another purpose is to pose interesting problems that require you to learn how to manipulate the fundamental objects of mathematics: sets, functions, sequences, and relations. The topics discussed in this course are the following: mathematical puzzles, propositional logic, predicate logic, elementary set theory, elementary number theory, and principles of counting. The most important aspect of this course is that you will learn what it means to prove a mathematical proposition. We accomplish this by putting you in an environment with mathematical objects whose structure is rich enough to have interesting propositions. The environments we use are propositions and predicates, finite sets and relations, integers, fractions and rational numbers, and infinite sets. Each topic in this course is standard except the first one, puzzles. There are several reasons for including puzzles. First and foremost, a challenging puzzle can be a microcosm of mathematical development. A great puzzle is like a laboratory for proving propositions. The puzzler initially feels the tension that comes from not knowing how to start just as the mathematician feels when first investigating a topic or trying to solve a problem. The mathematician"plays" with the topic or problem, developing conjectures which he/she then tests in some special cases. Similarly, the puzzler "plays" with the puzzle. Sometimes the conjectures turn out to be provable, but often they do not, and the mathematician goes back to playing. At some stage, the puzzler (mathematician) develops sufficient sense of the structure and only then can he begin to build the solution (prove the theorem). This multi-step process is perfectly mirrored in solving the KenKen problems this course presents. Some aspects of the solutions motivate ideas you will encounter later in the course. For example, modular congruence is a standard topic in number theory, and it is also useful in solving some KenKen problems. Another reason for including puzzles is to foster creativity Mathematical Reasoning to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material Introduction to Mathematical Reasoning
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This is a free, online textbook/course that "provides a rigorous presentation of the syntax and semantics of sentential...
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This is a free, online textbook/course that "provides a rigorous presentation of the syntax and semantics of sentential and predicate logic. The distinctive emphasis is on strategic argumentation." It "includes self-guiding materials and activities, and is ideal for independent learners, or instructors trying out this course packageCourse HighlightsMath I is a mandatory course for students specializing in natural sciences in Junior Divisions (1st & 2nd...
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Course HighlightsMath I is a mandatory course for students specializing in natural sciences in Junior Divisions (1st & 2nd years) of the College of Arts and Sciences. Math I is offered in "Math IA" and "Math IB". Both classes cover almost the same content although Math IA focuses on theoretical aspects while Math IB emphasizes in practical computation.In Math I, students learn differential and integral calculus to acquire basic knowledge on differential and integral required for university-level study. Thus, differential and integral of high-school mathematics are prerequisite for taking IB ( Differential and Integral Calculus) to your Bookmark Collection or Course ePortfolio
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Mathematical calculator Percentage calculator Online and offline handbook (NEC 2011, PUE Ukraine, PUE Russia ,and more..). Save the result of some calculation in HTML format, and then view and send. For each calculation is given a separate help. Dark and light themes application.
If you like the application, | 677.169 | 1 |
Wednesday, July 15, 2015
In the fall of 2014, we
switched my high school son's math program. As we were finishing up the school
year, he told me that he was struggling with math. My son who had previously
received A type grades in math had not only dropped his grade by two levels, but
his confidence in his math skills as well.
Since I had experience using
Volume I of the Learn Math Fast System curriculum with my youngest (you can
read my review about it here), and having been really impressed with it, I decided to try it with my
oldest. Because my son hopes to get into the mechanical engineering field
someday, I wanted a math program that would help him understand math, in this case Algebra.
This summer I had him begin the
Learn Math Fast System Volume III, Pre-Algebra book as a refresher and to get him used to J.K. Mergens' (the book's author) method of teaching. In addition, I
wanted to make sure that we covered any gaps he may have had from using his previous
math programs.
Volume III includes 25
lessons covering topics such as, "solving for x, ratios, proportions,
exponents, square roots, terms, expressions, like terms, order of operations,
distributive property of multiplication, graphs, linear equations, the
y-intercept, and the slope formula." All of the solutions to the problems are found at the end of the book.
My son had difficulty understanding how to do the slope formula problems with his old math program, but
after using the Learn Math Fast System book he was able to do these types of
problems without any difficulties. Also, since the lessons were short, he was able
to finish the book within about a four week period (doing two lessons a day).
Next week my son will be completing
Volume V of the Learn Math Fast System book which covers "Algebra I with a special focus on
quadratic equations and formulas." It includes 36 lessons, 7
chapter review tests, and a final test. Topics in this book include: "terminology, absolute value, terms with
exponents, simplifying expressions, multiplying and dividing terms with
exponents, square roots, cube roots, Laws of Algebra, multiplying polynomials,
factoring polynomials, factoring out a common factor, prime factorization,
quadratic equations, a perfect square, completing the square, quadratic
formula, and intro to functions."
Most
of the lessons could be completed in less than an hour, which allowed him to go
through two lessons a day. If the lesson was long, he would only do one lesson.
Some of the review tests I would have him complete in one day, others I would
allow him to take two days to complete.
According
to the information on the curriculum's website, this book can be completed within a three to
four month period. I would estimate that it has taken my son about three months to
finish this book. (His last few lessons will be completed next week.)
I
am happy to report that not only did his grade increase to an A, but his
confidence in math is back again!
·It provides
step-by-step instructions on how to do the problems and includes a good amount
of examples.
·It is written in
an easy-to-understand manner.
·There aren't
hundreds of problems needing to be completed.
·I can email (or
call) J.K., the author, with any questions and receive a response within 24
hours.
·The subject can
be learned in half the time it would take with other math programs.
·The high school student can complete this curriculum independently.
My son will be moving on to
Volume VI, Algebra 2, as I prefer to have him continue with Algebra while it's
still fresh in his mind and then he will move on to Pre-Geometry (Volume IV)
and Geometry (Volume VII).
If you are looking for a
math curriculum for your student written in an easy-to-understand manner, then
I highly recommend you try any of the Learn Math Fast System books.
By the way, if you'd like to
try this curriculum use the code SLICE to get $5.00 off your order (expires
12/31/15).
I know your time is precious, so any comment you share below will be greatly valued.
If you enjoyed this post, please Pin or Share it now by clicking on the icons below. Don't miss any of my posts. Become a subscriber by clicking on the Bloglovin' button or the RSS button below.
Monday, May 04, 2015
This fall I will be offering a Spanish Level I course for students
in Grades 8 – 12 (Students in 7th grade may also participate if they
have taken a prior Spanish course.)
This course
will be taught online in a group setting using Google+ Hangouts. In order to
participate, students must have a Google+
account and a scanner. Students will be required to scan their homework
assignments and email them to me. All tests and quizzes will be emailed to the students. Students will receive a Certificate of Completion
award at the completion of the full course.
Wednesday, April 08, 2015
I have been a homeschool parent for the past 10 years. In May my oldest son will finish his first year of high school lessons, and this past winter I added a new title to my homeschool resume. This new title is Guidance Counselor. Having absolutely no past experience in this regard, I find this new position challenging.
For those of you who have not yet started down this road called the high school years, I thought I'd share my journey with you in a new series of posts which will read more like a diary. My hope is that by reading my "entries," it will assist you in preparing for this phase of your homeschooling career, or if you're already driving down this path, that we will be able to come alongside each other in order to help steer one another in the right direction.
Thursday, October 16, 2014
I believe that most people who know me probably think that I have it all together as a homeschool parent. My house is pretty organized (on most occasions), I am a strong cheerleader for the homeschooling movement, I lead a homeschool support group, and my children are performing academically well. However, what people may not know is that I have a deep, dark secret, one that I feel lead to share.
Monday, September 29, 2014
I first learned about Excelerate Spanish through an article I read in the 2014 annual print book of The Old Schoolhouse Magazine. What captured my attention was that Excelerate Spanish used a method of teaching foreign language that was new to me, Teaching Proficiency Through Reading and Storytelling (TPRS).
Friday, September 05, 2014
I am very eager to share this review with you! If you and your family enjoy audio dramas as much as my family does, then you will be excited to learn about Heirloom Audio Productions' Christian audio drama, Under Drake's Flag. This story was written by author G.A. Henty, who is well known for his historical adventure stories. My children and I were delighted to add this to our collection. Under Drake's Flag is an enthralling story about a boy named Ned Hawkshaw who embarks on an exciting adventure which offers him the opportunity to set sail with one of the most admired sea captains of his time, Sir Francis Drake. This sixteenth-century teenager experiences first-hand the struggle between Great Britain and Spain for preeminence over the high seas.
You will be kept in suspense as you listen to Ned as he fights off a shark attack. You will also find yourself asking questions such as, "Will he survive the shipwreck? Is he going to get through these battles? How is he going to handle the Spanish Inquisition?"
Thursday, September 04, 2014
Even though communication skills are vital to their success, many students (and most parents!) are still terrified of public speaking
Let us help. Join us for the
Communicators for Christ Conference
Through this two day integrated communication and leadership training conference hosted by the Institute for Cultural Communicators (ICC), we'll help students overcome their hesitation to effectively engage audiences of all sizes. Our activities, workshops, and exercises provide a safe and encouraging environment where everyone has a chance to succeed, regardless of their public speaking experience. This allows them to see past their fears to a greater mission of impacting those around them - one person at a time.
Hi! I'm Clara If you're looking for a place to discuss the topic of raising boys, homeschooling, and other fun subjects related to everyday life, then you've come to the right place. To learn more about me, click here. | 677.169 | 1 |
Are you taking calculus right now and it's kicking your butt? You're not alone; when I was teaching calculus, I realized that textbooks suck!
I wrote the Practically Cheating Calculus Handbook so that you don't have to struggle any more. This handbook contains hundreds of step-by-step explanations for calculus problems from differentiation to differential equations -- in plain English!
Practice division withEnhance essential elementary algebra skills with these twenty-one practice problems. Each problem has an expression with mixed operators to reinforce the concept of operator precedence and use of parentheses. Choose a problem from the problem list, and then confirm your answer by easily navigating the link to the complete instructive solution. Most appropriate for 4th and 5th grade students second firstPractice and hone important addition skills with this second first | 677.169 | 1 |
First course calculus texts have traditionally been either ?engineering/science-oriented? with too little rigor, or have thrown students in the deep end with a rigorous analysis text. The How and Why of One Variable Calculus closes this gap in providing a rigorous treatment that takes an original and valuable approach between calculus and analysis.... more...
Now considered a classic text on the topic, Measure and Integral: An Introduction to Real Analysis provides an introduction to real analysis by first developing the theory of measure and integration in the simple setting of Euclidean space, and then presenting a more general treatment based on abstract notions characterized by axioms and with less... more... | 677.169 | 1 |
Beginning and Intermediate Algebra
9780073229713
0073229717
Summary: Intended for schools that want a single text covering the standard topics from Beginning and Intermediate Algebra. Topics are organized by using the principles of the AMATYC standards as a guide, giving strong support to teachers using the text. The book's organization and pedagogy are designed to work "for students" with a variety of learning styles and for teachers with varied experiences and backgrounds. The inclu...sion of multiple perspectives -- verbal, numerical, algebraic, and graphical -- has proven popular with a broad cross section of students. Use of a graphing calculator is assumed. BEGINNING AND INTERMEDIATE ALGEBRA: THE LANGUAGE AND SYMBOLISM OF MATHEMATICS is a reform-oriented book.
James Hall is the author of Beginning and Intermediate Algebra, published 2007 under ISBN 9780073229713 and 0073229717. Twenty six Beginning and Intermediate Algebra textbooks are available for sale on ValoreBooks.com, twelve used from the cheapest price of $0.18, or buy new starting at $96 | 677.169 | 1 |
0028253264Glencoe Algebra 1: Integration, Applications, Connections
Glencoe's Algebra 1 and Algebra 2 balance sound skill and concept development with applications, connections, problem solving, critical thinking, and technology. Whether your students are getting ready for college or the workplace, this program gives them the skills they need for success. Integration occurs within and across lessons and exercises at the point of instruction. Continual real-life, practical applications keep students excited about learning. Interdisciplinary connections show your students how mathematics relates to science, history, geography, health, careers and life. Long-term Investigation projects engage students in interesting and relevant applications of mathematics. Modeling Mathematics empowers students to think mathematically as they use charts, graphs, concrete models, hands-on manipulatives, open-ended questions, and chapter projects to discover mathematical concepts. Graphing Technology lessons makes the focus on modeling and functions attractive and accessible to all | 677.169 | 1 |
Book Review: The Princeton Companion to Mathematics, Timothy Gowers, ed
Review by Tom Siegfried
Math is everywhere, from the gas station and grocery store
to the stock market and science magazines. And it shows up, of course, in
schools at all levels. But the educational system doesn't provide enough math
for most people to appreciate its scope, or understand its intrinsic powers or
practical applications.
For those with a deep interest in understanding such things,
this book provides a reasonably accessible, technically precise and thorough
account of all of math's major aspects—from the basics of algebra, geometry,
algorithms and proofs to the essential features of Hilbert spaces and
Hamiltonians. Much is understandable to anyone with a good high school math
background; sometimes more advanced education is better.
Added attractions include biographical sketches of close to
100 famous mathematicians, a comprehensive chronology of mathematical events
throughout history and engaging discussions of math's influence. This book
covers such diverse areas as communications, chemistry, biology, economics,
image compression, the flow of "traffic" in all sorts of networks (including
transportation), music and medical statistics.
Students of math will find this book a helpful reference for
understanding their classes; students of everything else will find helpful
guides to understanding how math describes it all. | 677.169 | 1 |
College Algebra
Browse related Subjects ...
Read More Edition introduces each function--linear, power, quadratic, exponential, polynomial--and presents a study of the basic form of expressions for that function. Readers are encouraged to examine the basic forms, see how they are constructed, and consider the role of each component. Throughout the text, there are Tools sections placed at the ends of chapters to help readers acquire the skills they need to perform basic algebraic manipulations.
Read Less
Very good. Looseleaf. Has minor wear and/or markings. SKU: 9780470556641-3-0-3 Orders ship the same or next business day. Expedited shipping within U.S. will arrive in 3-5 days. Hassle free 14 day return policy. Contact Customer Service for questions. ISBN: 9780470556641.
Fine. 0470556641New. 0470556641 | 677.169 | 1 |
Group theory is the branch of mathematics that studies symmetry, found in crystals, art, architecture, music and many other contexts. But its beauty is lost on students when it is taught in a technical style that is difficult to understand. Visual Group Theory assumes only a high school mathematics background and covers a typical undergraduate course in group theory from a thoroughly visual perspective.
The more than 300 illustrations in Visual Group Theory bring groups, subgroups, homomorphisms, products, and quotients into clear view. Every topic and theorem is accompanied with a visual demonstration of its meaning and import, from the basics of groups and subgroups through advanced structural concepts such as semidirect products and Sylow theory.
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If you teach abstract algebra, then this book should be a part of the resources you use. While the phrase "visual abstract algebra" may seem to be a contradiction, the diagrams in this book are an existence proof to the contrary. They are clear, colorful and concise very easy to understand and sure to aid the students that have difficulty in internalizing the abstract nature of the subject matter. Especially appealing are the colorized tables of groups and their operations.
The approach is a very slow one in the sense that a foundation of common operations and rearrangements that are groups that are first examined with text and images. A large number of exercises are included at the end of each chapter and detailed solutions with colored images found in an appendix.
this book could also serve as a text in a first course in abstract algebra provided that the course is limited to groups only or you used supplementary material for rings and fields. If your course is restricted to groups only, then this is the best book available. --Charles Ashbacher, Journal of Recreational Mathematics
Book Description
This text approaches the learning of group theory visually. It allows the student to see groups, experiment with groups and understand their significance. It is ideal as a supplement for a first course in group theory or alternatively as recreational reading.
Most Helpful Customer Reviews
...I only like it. Given that I am not a fan of the formal turgid theorem/proof type of math text, I waited with much anticipation to get a copy of this text. Without a doubt, this is a very original and fun text that presents a novel approach to teaching group theory. However, if I had come to group theory with this text as my first introduction, I do not think it would have been effective and I would have quickly become frustrated. Simply put, the author goes to far to build up intuition about groups before he actually defines a group mathematically. I think it would have been better to define a group early and then start to build up intuition after the definition is secure. Sort of "Here is the abstract definition, lets now start to understand why anyone would want to define a group in the first place and lets see some examples involving symmetry and other physical situations where groups arise".
A better choice for the absolute novice in group theory is the book Groups by Jordan and Jordan. This book is not well known in the USA but is simple, intuitive, and well-written. For the person interested in exploring the entire landscape of abstract algebra (groups, rings, fields, Galois theory) "A Book of Abstract Algebra" by Pinter is unmatched for its clarity. I also recommend the older book by Keesee "Elementary Abstract Algebra".
In summary, I cannot recommend Visual Group Theory as a textbook for a course in group theory although some might find it a refreshing supplement to an overly formal abstract algebra text/course. I think its target audience falls outside workers in the mathematical sciences and is geared toward chemists, molecular biologists, future/current high school math teachers, and weekend math warriors who are looking for intellectual stimulation.
I am reading this book half way through, it is amazing how 'readable' this book is compared to other Abstract Algebra books. Although this book inherits much ideas and notations from the other older book: "Groups and Their Graphs" by Grossman and Magnus, it elaborates with more Cayley Diagrams (over 300 of them) and more detailed explanations.
The book is well written and well illustrated. I took one night to read through the first 5 chapters, that shows how easy it is to understand the concepts. (It helps, may be, because I had read the Grossman and Magnus book, however the later is not a pre-requisite reading).
I am also very pleased that the author uses Cayley diagrams to show how Subgroups, Cosets and Normal Subgroups could be visualized. This is a real break-through in teaching abstract stuff like Group Theory - a real tough nut to crack for most Math students. Don't forget the inventor of Group - Evariste Galois - had hard time making himself understood by even the greatest mathematicians of his time - Cauchy, Fourier, Poisson, etc.
This is the book to read before anyone attempting to study Group Theory in a formal textbook way.
I was a physics B.S. who is now pursuing a Ph.D. in applied physics. Was very intimidated by the very math-y and formal books on group theory. Have been working through all the problems through the first 5/6 chapters so far, the ones whose answers are in the back of the book. Already feel much more confident about the fundamentals of group theory. The referenced software the author created is also very helpful and creative. Plan to continue to work problems all the way to the end, after which I'm confident I'll feel like I've got a solid grasp on group theory.
This is a great book for anyone interested in mathematics. I bought it just to read after reading about it in another text but find myself returning to it again and again to sharpen my understanding with the examples. | 677.169 | 1 |
Course Flow Chart
Information for students taking an algebra or calculus course
Page Image
Page Content
Let's face it. Taking an algebra or calculus course requires that you understand thoroughly the prerequisite material. We have put in place several mechanisms to determine which math course you are mathematically prepared to take, and several more ways in which you can review to become better prepared. The more effort you put into reviewing material on your own, the higher the math class you can start taking.
The flow chart above provides the sequence of courses which, when followed, best-prepares students for mathematical success when started at the appropriate level. The first step is to get assessed to see which math class you are prepared to take. From this assessment, a student should know whether they are qualified to take MATH 1070, MATH 1110, MATH 1130, or MATH 1401.
*Note that the course title for MATH 1070 has changed from "Algebra for Social Sciences and Business" to "College Algebra for Business." (As of July 2015)
Have your Mathematics Ability Assessed:
Upon admission to UCD, your goal should be to take the highest level math class which you are comfortable and have the skill set needed to succeed. UCD provides several options for students to determine which math class you are prepared to take. Some of the exams which can be used include the SAT, ACT, MyMathTest, and ALEKS. Information about assessing readiness for other UCD math classes such as MATH 1130 and MATH 1401 are also available.
Tab Content Two
UC Denver Math Course Prerequisites:
It should be noted that if you want to take Calculus I, MATH 1401, you must meet one of the prerequisite requirements before registering for the course, so if you have not already met one of the prerequisite requirements, it is particularly important for you to take one of these exams early so that you will have plenty of time to review material and if necessary, re-take a prerequisite exam before registering for classes. For MATH 1110 and MATH 1070 no exam prerequisite are enforced, so the scores are only recommended. For other MATH courses, students must have the prerequisite course on their transcript before being able to register.
To summarize:
MATH 1401: Prerequisite exam is required before registering. No exceptions to the prerequisite requirements are made.
MATH 1070 and MATH 1110: Prerequisite exam scores are recommended only -- they are not strictly enforced. It is strongly recommended that students take one of these exams to determine whether they are sufficiently prepared, and if they are not, we recommend they spend some time reviewing the prerequisite material. Options for reviewing material are described below.
Other MATH classes: Students must have the prerequisite math course listed on their transcript with a passing grade of C- or better before they can register online. Exceptions to this, such as a course taken at another institution that may have a title not consistent with UCD courses, may seek approval from the Department of Mathematical and Statistical Sciences. Please bring course materials (place taken course, title, description, prerequisite course, text used) to the CU Building, sixth floor, and leave this information along with your contact information, and we will evaluate the course as soon as possible (usually within 2 working days).
For MATH 1070 and MATH 1110, it is your responsibility to be sure you are sufficiently prepared to be successful. The percentage of students who pass these classes who have not met the prerequisite requirement is less than 40%. Students who meet the prerequisite requirements are considered to have a sufficient mathematical background to pass their math class. Of the students who meet the appropriate prerequisites, between 60% and 80% successfully complete the course with a grade of C or better. Students who meet the suggested prerequisites and who do not pass the class often have poor study habits or have non-academic pressures or issues which keep them from focusing on their academics.
Tab Content Three
If you need to review prerequisite materials before taking MATH 1070, 1110, 1130, or 1401, and would like more specific information about which topics you should review, we strongly recommend taking MyMathTest, an online exam, for free at the UCD Math Education Resource Center (MERC) lab, located in the North Classroom, room 4015 any time the lab is open and available. See the MERC website for available lab hours. You should be prepared to spend about 1.5 hours learning how to input mathematical solutions into MyMathTest before taking a detailed assessment. MyMathTest also has tutorials available so that as you practice for the exam, you can click on tutorial links that allow students to get more information about particular topics.
What You Can Do To Be Better-Prepared: If results from MyMathTest or other tests indicate you are not prepared to take the desired course, we strongly recommend that you take one of the following actions:
Use an online tutorial program such as ALEKS. We recommend ALEKS because it is relatively inexpensive, $30 for 6 weeks of access, and because it gives detailed feedback as far as which mathematical topics you are strong and weak. It is not meant to take the place of a math class, but to help students who are deficient in only a couple of topics or for whom it has been more than 6 months since they have seen the mathematical material and hence need to review. Tutorials are positively reviewed by students, and once students get to 80% correct in each topical math area they are considered prepared.
Take or re-take the prerequisite course (see the flow chart above).
Hire a tutor. It is well-known that the quickest way to learn new mathematical topics or correct mathematical misconceptions is to get immediate feedback directly. We recommend students meet with a tutor at minimum one hour per week.
International Students: UCD currently does not require international students to submit an SAT or ACT math score with their application prior to being admitted. We strongly urge students to consider taking a self-guided math test such as ALEKS prior to their program and then test at the UCD MERC lab (NC 4015). It is our hope that with some recent math experience, students will not feel as anxious when taking a UCD math placement exam. We also strongly encourage students to receive math-specific English as a Foreign Language (EFL) tutoring. What we are learning is that many students may have the math skills but lack proficiency in English to be successful in Calculus I or Algebra.
For international students, ALEKS is incorporated into their intensive English language learning curriculum at Spring International. It is hoped that these international students will be able to review the math they have taken and be prepared to take and successfully complete the highest possible level math class appropriate for the desired degree. | 677.169 | 1 |
How to Solve Word Problems in Mathematics (Paperback) - Wayne, David S.
Product Overview
Most 9th grade math, or "Algebra 1," textbooks are structured in such a way that students find it extremely difficult to apply pertinent mathematical concepts and skills to the solving of word problems. This book soothes math students'' fears with numerous solved practice problems, step-by-step problem-solving procedures, and crystal-clear explanations of important mathematical concepts. Designed to be used independently or in conjunction with standard textbooks.
Specifications
Physical
Dimensions
(in Inches) 8.25H x 5.25L x 0.5T
From the Publisher
Editors Note 1 2Product Attributes
Book Format
Paperback
Number of Pages
0164
Publisher
McGraw-Hill
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This is a stand alone instructional resource created in PowerPoint and designed to teach students how to use the order of...
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This is a stand alone instructional resource created in PowerPoint and designed to teach students how to use the order of operations to simplify numerical and algebraic expressions. This lesson is intended for 8th or 9th grade Algebra 1 students, but could also be used with younger students to introduce order of operations or with older students to review the order of operations. Students should complete this lesson individually and at their own pace Order of Operations StAIR to your Bookmark Collection or Course ePortfolio
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This is a lesson plan for solving systems of linear equations to Algebra 1 students. Methods to be taught include...
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This is a lesson plan for solving systems of linear equations to Algebra 1 students. Methods to be taught include substitution, elimination, and graphing. The lesson also includes the use of graphing calculators and spreadsheets to solve systems of equations. The lesson involves practice with real world application problems, as well as creation and presentation of original problems by Systems of Equations Lesson Plan to your Bookmark Collection or Course ePortfolio
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Synopses & Reviews
Publisher Comments:
Without math, how would romance survive? Where would we be without pluses and minuses? Without public opinion and box office tallies? Without measuring cups, calorie counters, or computer technology? More than just a class you tried to avoid, math defines modern life. Around the home, business applications, even theoretical physics…. Everyone uses it, just do the math. Math is too complicated. Math is too hard. Not anymore. The Handy Math Answer Book eliminates the instant headache and helps the many math- challenged among us better understand and enjoy the magic of numbers. You can count on it. From modern-day challenges such as balancing a checkbook, following the stock market, buying a home, and figuring out credit card finance charges to appreciating historical developments like the use of algebra by Mesopotamian mathematicians, The Handy Math Answer Book addresses more than 1,000 questions relating to mathematics. Handy Math provides a complete overview, beginning with the early historyPythagoras and "the music of the spheres," Archimedes and his "Eureka!" moment in the bathtub, and how some of the first calendars were invented. Catch yourself falling for the gravity of Newton and winding your way around modern-day string theory. Refresh yourself on the basics and fundamentals of algebra, calculus, geometry (including why the word derived from "earth measuring"), and trigonometry. Organized in sixteen chapters that cluster similar topics in an easily accessible format, Handy Math Handy Maths straightforward language is supported by more than 200 charts, graphs, illustrations, and photographs.
Synopsis:About the Author
Patricia Barnes-Svarney has been a nonfiction science and science fiction writer for more than 15 years. She has a master's degree in geography/geomorphology and has worked professionally as a geomorphologist and oceanographer. She has written or coauthored more than 300 articles for science magazines and journals. Thomas E. Svarney is a scientist, naturalist, and artist. They are the coauthors of numerous science books, including The Handy Geology Answer Book. | 677.169 | 1 |
Math Assignment Help
Mathematics is the abstract science of number, quantity, and space, either as abstract concepts or as applied to other disciplines such as physics and engineering.
Mathematics evolved as an ever-expanding series of abstractions or subject matter. Mathematics emerged with the abstraction of numbers. It was a realization that a pair of legs and a pair of hands has something in common and that is the quantity of them, the number which was coined to be called as "2."
Areas of Mathematics:
Calculus: Calculus is a major branch of mathematics which includes various topics like Relations and functions, Limits and Continuity, Differentiation and Differential equations, Indefinite integrals and definite integrals, Application of derivatives and various series.
Trigonometry: Trigonometry is the study of triangles. It's about analyzing the relationship between the sides of triangles and their relation with the angles.
Geometry: Geometry is the study of points, lines, surfaces, solids, and higher dimensional analogues, and their properties.
Linear Algebra: Linear algebra deals with vector spaces and the linear mappings between them along with lines, planes, and subspaces.
Discrete Mathematics: Discrete mathematics deals with discrete mathematical structures rather than continuous objects. Discrete objects are those which are separated from each other and have no neighbors whatsoever.
Engineering Mathematics: Engineering Mathematics comes under applied Mathematics which studies the application of Mathematics in the engineering industry.
Topology: Topology is a branch of mathematics that deals with geometrical properties and spatial relations unaffected by the continuous change of shape or size of figures.
Boolean Algebra: Boolean algebra is the logical calculus of truth values and is a part of abstract algebra.
Topics covered under Mathematics:
Circle: A circle is the locus of a point which moves in a plane such that its distance from a fixed point is always constant. The fixed point is called the center of the circle and the constant distance, the radius of the circle.
Parabola: The locus of a point which moves in such a way that the distance from a fixed point called focus equals perpendicular distance from a fixed straight line called the directrix.
Probability: Probability denotes the possibility of occurrence or non-occurrence of an event. It represents how likely we can expect an event to happen.
Ellipse: An ellipse is the locus of a point which moves in a plane such that the ratio of its distances from a fixed point (called focus) and from a fixed straight line (called directrix) is always constant and less than 1. And this constant ratio is called the eccentricity of the ellipse.
Matrices: A matrix is an array of numbers, which are usually real numbers, arranged into a fixed number of rows and columns.
Hyperbola: A hyperbola is the locus of a point which moves such that, ratio of its distance from a fixed point (focus) and its distance from a fixed straight line (directrix), is a constant (eccentricity). This constant (eccentricity) is greater than unity.
Binomial Theorem: Any formula by which any power of a binomial expression can be expanded in the form of a series is known as Binomial Theorem | 677.169 | 1 |
6Foundations of Geometry
Summary
Foundations of Geometry, Second Editionimplements the latest national standards and recommendations regarding geometry for the preparation of high school mathematics teachers ;and encourages students to make connections between their college courses and classes they will later teach. This text's coverage begins with Euclid's Elements, lays out a system of axioms for geometry, and then moves on to neutral geometry, Euclidian and hyperbolic geometries from an axiomatic point of view, and then non-Euclidean geometry. Good proof-writing skills are emphasized, along with a historical development of geometry. The Second Editionstreamlines and reorganizes material in order to reach coverage of neutral geometry as early as possible, adds more exercises throughout, and facilitates use of the open-source software Geogebra. | 677.169 | 1 |
A collection of almost 200 single concept lessons. The lessons are divided into the following categories: Geometry and Measurement, Patterns and Logic, Probability and Statistics, Recreational and Creative Math,...
This unit consists of two computer programs. The first teaches X,Y plotting; the second is a demonstration of coordinate transformations, matrices, vector equations of lines and perspective and will draw a picture of...
Located at the University of Illinois at Urbana-Champaign, The Office for Mathematics, Science, and Technology Education is primarily interested in creating resources for educators working on these topics. First-time...
High school or college students taking an introductory trigonometry course may find this site useful. Three modules comprise the site, and each provides an overview of basic concepts. Some of the most common... | 677.169 | 1 |
reading many revews I am under the impression that the best book for someone who hasn't practiced any sort of math in over ten years would be the EZ Gmat-math strategies book. Any input would be greatly appreciated. Thanks!
Depending on your starting level you might need more or less study material; even before that, thou, you might start exploring the Gmat Club and the quantitative section. If you are following the OG12 you might look up answers explanations here and then figure out you weaknesses and strenghts. Basically everything you might need you can find it here, including theory: | 677.169 | 1 |
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This lesson from Illuminations asks students to solve a system of linear equations using a practical math problem. The lesson involves question for students; participants are asked to give a short presentation to the...
Gilbert Strang, of the Massachusetts Institute of Technology, highlights calculus in a series of short videos that introduces the basic ideas of calculus ? how it works and why it is important. The intended audience i...
This lesson from Illuminations asks students to use a geometry applet to analyze the characteristics of a square. Graphs are created to show relationships between characteristics (side length, diagonal length, perimeter...
This math unit from Illuminations includes 4 lessons which use iteration, recursion and algebra to model and analyze a population of fish. Graphs, equations, tables and technological tools are used in each lesson. Each...
This learning object from Wisc-Online covers solving systems of linear equations using the substitution method. The unit looks at the common solution to two or more linear equations in two variables. Practice questions... | 677.169 | 1 |
Concise Introduction to MATLAB
9780073385839
ISBN:
0073385832
Pub Date: 2007 Publisher: McGraw-Hill Companies, The
Summary: 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 textbook.
Palm, William J., III is the author of Concise Introduction to MATLAB, published 2007 under ISBN 9780073385839 and 0073385832. Six hundred sixty three Concise Introduction to MATLAB textbooks are available for sale on ValoreBooks.com, seventy eight used from the cheapest price of $19.12, or buy new starting at $67Writing programs to do tedious computations quickly and making a graphical representation of that data. One example is the differential equation solver. Very versatile built in functions to do that task. | 677.169 | 1 |
This course is an elementary introduction to number theory with no algebraic prerequisites. Topics covered include primes,...
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This course is an elementary introduction to number theory with no algebraic prerequisites. Topics covered include primes, congruences, quadratic reciprocity, diophantine equations, irrational numbers, continued fractions, and partitions781 Theory of Numbers (MIT) to your Bookmark Collection or Course ePortfolio
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The goal of this class is to prove that category theory is a powerful language for understanding and formalizing common...
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The goal of this class is to prove that category theory is a powerful language for understanding and formalizing common scientific models. The power of the language will be tested by its ability to penetrate into taken-for-granted ideas, either by exposing existing weaknesses or flaws in our understanding, or by highlighting hidden commonalities across scientific996 Category Theory for Scientists (MIT) to your Bookmark Collection or Course ePortfolio
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Name the Book Chapter Worksheets and Practice tests to your Bookmark Collection or Course ePortfolio
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Contest Problem Book VI
Compiled and with solutions by Leo J. Schneider
Contest Problem Book VI chronicles the high school competitions sponsored by the Mathematical Association of America. It contains 180 challenging problems from the six years of the American High School Mathematics Examination [AHSME], 1989 through 1994, as well as a selection of other problems. Many problem solving techniques for problems in this book show alternative approaches that appear in print for the first time.
Some aspects of mathematical problem solving unique to competitions is discussed. Useful tools are selected from important areas of high school mathematics.
A Problems Index classified the 180 problems in the book into subject areas: Algebra (with 65 subclasses), Complex Numbers (with 11 subclasses), Discrete Mathematics (with 20 subclasses), Geometry (with 43 subclasses), Number Theory (with 24 subclasses), Statistics (with 5 subclasses), and Trigonometry (with 12 subclasses). Many subclasses have sub-subclasses, some with over a dozen. The Pigeon Hole Principle proves that some problems must appear in more than one class. That, in fact, is the case! Outstanding problems combine elementary techniques from diverse areas of mathematics, occasionally three or more. You will find many of them here. | 677.169 | 1 |
This richly illustrated and clearly written undergraduate textbook captures the excitement and beauty of geometry. The approach is that of Klein in his Erlangen programme: a geometry is a space together with a set of transformations of the space. The authors explore various geometries: affine, projective, inversive, hyperbolic and elliptic. In each case they carefully explain the key results and discuss the relationships between the geometries. New features in this second edition include concise end-of-chapter summaries to aid student revision, a list of further reading and a list of special symbols. The authors have also revised many of the end-of-chapter exercises to make them more challenging and to include some interesting new results. Full solutions to the 200 problems are included in the text, while complete solutions to all of the end-of-chapter exercises are available in a new Instructors' Manual, which can be downloaded from
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'… this is a remarkable and nicely written introduction to classical geometry.' Zentralblatt MATH
'… could form the basis of courses in geometry for mathematics undergraduates. It will also appeal to the general mathematical reader.' John Stone, The Times Higher Education Supplement
'It conveys the beauty and excitement of the subject, avoiding the dryness of many geometry texts.' J. I. Hall, Mathematical Association of America
'To my mind, this is the best introductory book ever written on introductory university geometry … readers are introduced to the notions of Euclidean congruence, affine congruence, projective congruence and certain versions of non-Euclidean geometry (hyperbolic, spherical and inversive). Not only are students introduced to a wide range of algebraic methods, but they will encounter a most pleasing combination of process and product.' P. N. Ruane, MAA Focus
'… an excellent and precisely written textbook that should be studied in depth by all would-be mathematicians.' Hans Sachs, American Mathematical Society
Book Description
Popular with students and instructors alike, this accessible and highly readable undergraduate textbook has now been revised to include end-of-chapter summaries, more challenging exercises, new results and a list of further reading. Complete solutions to all of the exercises are also provided in a new Instructors' Manual available online.
The thing that attracts me about this volume is the beautiful paper its printed on. The book is weighty, and VERY well bound considering its in paperback. Also it is printed in font size that is readable even to those who need spectacles.
* Is it introductory, intermediate, or higher undergraduate level?
No book can cover everything in a subject. So deciding the level its pitched at is important. Overall the balance of its explanations are starting just in A- level territory, and largely between first-year / second- year crossover studies in Geometry. The earlier parts of the book is Euclidean / Affine geometry for the first 5 chapters, and the sixth and above are Spherical geometries topics.
* The authors traits throughout the book
Within each of the sections, that has a steady well - designed incline in difficulty. Throughout it has a belting way of using many, many diagrams and graphs to get the ideas over to the willing to stick with it reader.
* The balance of the book before chapter 6
Its not an insult to say it starts in a- Level arena in the first chapters, the stuff in Conics and ellipse, parabola and hyperbola topics is expressively explained and thoroughly described graphically and in texts. I loved the ways set theory and theorems is intertwined with the affine transformations and projected geometries. Its just (i.m.h.o) Its hard to find a more descriptive manner to explain these concepts and still be useful!
Also it has the useful graphical diagrams of whats going in with finding solutions to equations in the second and third dimensions using intersections between graphical planes. Its beautifully described and yet able to stand up to examination.Read more ›
Most Helpful Customer Reviews on Amazon.com (beta)
Amazon.com:
10 reviews
53 of 53 people found the following review helpful
Good and enjoyable for a wide range of readers7 May 2003
By
William A. Huber
- Published on Amazon.com
Format: Paperback
A quarter century ago I noticed that some of the graduate physics students in my university were carrying around copies of Scientific American. Armed with that clue, I dug out every article on the newly discovered fundamental particles. Within the space of a week of fairly easy reading I was able to acquire a good sense of what this subject was all about. These articles explained the basic stuff our professors assumed we must know (but most of us surely didn't). Brannan, Esplen, and Gray's Geometry accomplish for math what those Scientific American articles did for physics: speaking at a level accessible to anyone with a good high school education, they bring the interested reader up to speed in affine, projective, hyperbolic, inversive, and spherical geometry. They provide the simple explanations, diagrams, and computational details you are assumed to know-but probably don't-when you take advanced courses in topology, differential geometry, algebraic geometry, Lie groups, and more. I wish I had had a book like this when I learned those subjects. Individual chapters of about 50 pages focus on distinct geometries. Each one is written to be studied in the course of five evenings: a week or two of work apiece. Although they build sequentially, just about any of them can be read after mastering the basic ideas of projective geometry (chapter 3) and inversive geometry (chapter 5). This makes the latter part of the book relatively accessible even to the less-committed reader and an effective handbook for someone looking for just an overview and basic formulas. The approach is surprisingly sophisticated. The authors do not shy away from introducing and using a little bit of group theory, even at the outset. (Scientific American, even in its heyday, never dared do that.) They present all geometries from a relatively modern point of view, as the study of the invariants of a transitive group of transformations on a set. Many explanations and proofs are based on exploiting properties of these transformations. This brings a welcome current of rigor and elegance to a somewhat static subject long relegated to out of date or sloppy authors (with the exception of a few standouts, such as Lang & Murrow's "Geometry"). One nice aspect is the authors' evident awareness of and appreciation for the history of mathematics. Marginal notes begin at Plato and wind up with Felix Klein's Erlangen program some 2300 years later. Although the text does not necessarily follow the historical development of geometry, its references to that development provide a nice context for the ideas. This is an approach that would improve the exposition of many math texts at all levels. The authors are British and evidently write for students with slightly different backgrounds than American undergraduates. Obvious prerequisites are a mastery of algebra and a good high school course in Euclidean geometry. Synopses of the limited amounts of group theory and linear algebra needed appear in two brief appendices. However, readers had better be intuitively comfortable with matrix operations, including diagonalization and finding eigenspaces, because matrices and complex numbers are used throughout the book for performing computations and developing proofs. A knowledge of calculus is not needed. Indeed, calculus is not used in the first two-thirds of the book, appearing only briefly to derive a distance formula for hyperbolic geometry (a differential equation for the exponential map is derived and solved). During the last third of the book (the chapters on hyperbolic and spherical geometry), some basic familiarity with trigonometric functions and hyperbolic functions is assumed (cosh, sinh, tanh, and their inverses). Definitions of these functions are not routinely provided, but algebraic identities appear in marginal notes where they are needed. Now for the quibbles. The book has lots of diagrams, but not enough of them. The problems are usually trivial, tending to ask for basic calculations to reinforce points in the text. The text itself does not go very deeply into any one geometry, being generally content with a few illustrative theorems. An opportunity exists here to create a set of gradually more challenging problems that would engage smarter or more sophisticated readers, as well as show the casual reader where the theories are headed. This book is the work of three authors and it shows, to ill effect, in Chapter 6 ("non-euclidean geometry"). Until then, the text is remarkably clean and free of typographical and notational errors. This chapter contains some glaring errors. For example, a function s(z) is defined at the beginning of a proof on page 296, but the proof confusingly proceeds to refer to "s(0,c)", "s(a,b)", and so on. The written-by-committee syndrome appears in subtler ways. There are few direct cross-references among the chapters on inversive, hyperbolic, and spherical geometry, despite the ample opportunities presented by the material. Techniques used in one chapter that would apply without change to similar situations in another are abandoned and replaced with entirely different techniques. Within the aberrant Chapter 6, some complex derivations could be replaced by much simpler proofs based on material earlier in the chapter. The last chapter attempts to unify the preceding ones by exhibiting various geometries as sub-geometries of others. It would have been better to make the connections evident as the material was being developed. It is disappointing, too, that nothing in this book really hints at the truly interesting developments in geometry: differentiable manifolds, Lie groups, Cartan connections, complex variable theory, quaternion actions, and much more. Indeed, any possible hint seems willfully suppressed: the matrix groups in evidence, such as SL(2, R), SU(1,1, C), PSL(3, R), O(3), and so on, are always given unconventional names, for instance. Even where a connection is screaming out, it is not made: the function abstractly named "g" on pages 296-97 is the exponential map of differential geometry, for instance. Despite these limitations, Brannan et al. is a good and enjoyable book for anyone from high school through first-year graduate level in mathematics.
21 of 21 people found the following review helpful
A lovely Introduction to all kinds of Plane Geometries16 Feb. 2003
By
J. J. K. Swart
- Published on Amazon.com
Format: Paperback
This book gives a beautiful overview of geometry of 2 dimensions. All of the book is about many plane geometries I have heard of, but didn't really know. This book changed that. The first chapter treats some basics about conics. The second chapter is on affine geometry. The third and fourth chapters are about projective geometry. In the fifth chapter you will be led through Inversive geometry which functions as a base for the sixth and seventh chapter. The sixth chapter has as itst title Non-Euclidean geometry, but it is in fact the Hyperbolic geometry of Boljay in a formulation of Henry Poincaré. The seventh chapter is about Spherical Geometry. In the eighth chapter all of these geometries are demonstrated to be special cases of the Kleinian vieuw of geometry: that is, every geometry can be seen as consisting of the invariants of a specific group of transformations of the 2 dimensional plane into itself. It is clearly demonstrated that this is less trivial than you would expect. I learned two things from this book. The first is, that you can, in principle, prove every theorem of geometry by just using Euclidean geometry. But if you do this, the amount of work it takes can be very huge indeed. It is a far better strategy to try to determine what geometry is best suited for the problem at hand, and solve it within that geometry. Since the book gives a very clear picture not only of the particular geometries, but also to how the geometries relate to each other, you have, as an extra bonus, insight in the level of abstraction and the scope of your theorem. The second thing I learned is how you can use geometry to make concepts as simple as 'triangle' precise. What I mean is this: a right angle triangle is not the same as an equilateral triangle. But both are the same in the sense that they are both triangles. The question is this: how can two 'things' be the same and at the same time not 'the same'? The book gives an answer to this 'question about the meaning of abstractions'. It gives the following solution. Take a triangle, ANY triangle. Consider the group of all affine transformations A (which consists of an uncountably infinite set of transformations.) If you subject this one triangle Tr to every affine transformation in this group A, you will have created a set consisting of exactly ALL triangles. In other words, the abstract idea of 'triangle' consists of ONE triangle Tr together with the set of ALL affine transformations. You can denote this as the pair (Tr, A). In the same way you can express the abstract idea of ellipse by the pair (El, A), and the abstract idea of parabola by the pair (Par, A). And, by passing to the more abstract Projective geometry, you can express the abstract idea of 'conic' by giving just one quadratic curve, be it a parabola, ellipse or hyperbola, by the pair (Qu, P), whereby P is the group of all projective transformations. The book presupposes some group theory and some knowledge of linear algebra. Furthermore you have to know a little calculus. I have very little knowledge of group theory, and I have just about enough knowledge and skill about linear algebra to know the difference between an orthogonal and unitary matrix, and to know what eigenvectors are. I have studied the first 5 chapters of CALCULUS from Tom M. Apostol, which does not go too deep into linear algebra. This proved to be enough. I have only one point of critique. Virtually all problems in the book are of the 'plug in type', even those at the end of every chapter (from which, by the way, you cannot find the solutions at the end of the book, while the solutions of those in the text can be found in an appendix). If you have understood the text, you have no difficulties whatsoever to solve them. The problems are not challenging enough to give you a real skill in all of these geometries, although they do become more challenging in later chapters. They are only intended to help you to understand the basic principles of all of these geometries, no more, no less. So if you want to have a tool to help you in obtaining a greater skill in, say, the special theory of relativity by studying hyperbolic geometry, this is not a suitable book. That is why I have given it 4 stars, and not the full 5 stars. I also have a piece of advise. Although the problems are, from a conceptual point of view, not challenging, a mistake is easily made. Therefore it is best to solve the problems by making use of a mathematical program like Maple or Mathematica. If you then have made a mistake, you can backtrack exactly where you have made it, and let the program take care of all of the tedious calculations. This has also stimulated to try to calculate some outcomes by following a different approach, and then to compare the results. I have enjoyed studying this book immensely.
18 of 20 people found the following review helpful
A Nice Introduction to Geometrys - Precise and Accurate!2 Jan. 2005
By
Shankar N. Swamy
- Published on Amazon.com
Format: Paperback
This book is at the level of a freshman mathematics course.
Mainly deals with Affine, Projective, Inversive, Spherical and Non-Euclidean geometrys. The beauty of the book is in its accuracy. Someone has done a good job of technical editing! There is always a risk of getting things wrong when attempting to make mathematics accessible at a lower level. The authors seem to have avoided that pitfall with significant success. The subject matter is focused and to the point. At each point, it precisely explains what is intended and moves on without digressions.
I have had significant interests in geometries, and work in a area that uses some elementary projective geometry. At times I get asked some relatively simple questions such as "why do we need 4x4 matrices in Computer Graphics?" Often I just answer such questions to the minimum (" ... it makes applying translations easier ..."). I never proffer a deeper answer because most people I run into either have no background to understand a more technical explanation in terms of the algebra of projective planes or they don't care - they don't need to, for most of their work!. (Many of the computer graphics folks I have met think that the homogeneous coordinates is an ad-hok concept that was invented as a "trick"!)
Occasionally, I do run into some who are interested in knowing the analytical reasoning behind some of the transformations used everyday in computer graphics. This book demonstrated to me how to talk to some of those without having to use very abstract concepts of geometry. I read it first in 1999. I have revisited it since, many times for the nice figures they provide. First time, it took me about three "after work" months to study through the book - not bad at all for a 350+ pages mathematics book!
By looking at the diagrams in the book, I learned how to draw simple diagrams instead of abstract symbols to explain the concepts, theorems and problems. For a book that is as simple, the technical content is remarkably precise and accurate. The book assumes minimal background in mathematics.
Recommended for people interested in computer graphics and want to understand the transformations in there deeper (for whatever reason!), under-graduate students interested in geometry, and for anyone with a casual interest in geometry.
6 of 6 people found the following review helpful
My Favorite Geometry Book.26 Jan. 2009
By
J. Wrenholt
- Published on Amazon.com
Format: Hardcover
I have purchased and worked through several geometry books in the past year. This is, by far, the best one I've come across. It is modern, it is fun, and it is enlightening.
I love the clear worked out numerical examples. As I write geometric computer programs it gives me a way to check each function as I go along. It has an abundance of wonderful illustrations that help you understand the theorems and concepts.
I found this Geometry to be clearly and fully presented for an independent reader, and not just a supplement to some lecture course. There is a minimum amount of mathematical history included, but it is well-chosen and well-written.
Plus, I learned how to draw the tessellations of the Poincare disc, ala M.C. Escher. So that alone made it worth the price and the effort.
Definitely 5 stars.
5 of 5 people found the following review helpful
Excellent Undergraduste intro to geometry1 Jan. 2014
By
Barbara Herrington
- Published on Amazon.com
Format: Paperback
Verified Purchase
I am a physicist and wish I had had a good undergraduate class in geometry to lay the visual foundation for much of the rest of the undergraduate math I took. This book has been a great book for establishing that foundation as well as clearly explaining the underlying language, concepts and relationships between the different geometries.
This is a good place to start to really understand linear algebra, tensor analysis, transforms and differential geometry.
I recently learned that before 1900, much more emphasis was placed on geometry study in undergraduate math curriculums than we get today. This book has helped to fill in what physicists have lost in not having more geometry in their undergraduate math curriculums.
I used GeoGebra (an excellent, free, virus/malware free graphics application available online) in helping me visualize both the problems and the examples which was helpful in finding mistakes in calculations for the problems. | 677.169 | 1 |
Teaching-Learning Tool for Integral Calculus
Sakda Noinang
g4737169@student.mahidol.ac.th
Institute for Innovation and Development of Learning Process
Mahidol University, Bangkok
Thailand
Benchawan Wiwatanapataphee
scbww@mahidol.ac.th
Department of Mathematics, Faculty of Science
Mahidol University, Bangkok
Thailand
Yong Hong Wu
y.wu@curtin.edu.au
Department of Mathematics and Statistics
Curtin University of Technology, Perth
Australia
Abstract: This paper presents an efficient Mathematics teaching-learning tool for integral calculus courses. The tool,
consisting of a set of PowerPoint slides with Maple animation and interactive Maplets with Maple worksheets, is
developed to help instructors to teach in class and to provide students with best opportunity for self-planned learning
and self-assessment. More specifically, the PowerPoint slides with Maple animation help instructors to explain certain
concepts and methods more effectively and clearly; while the interactive Maplets and Maple worksheets reinforce
students' conceptual understanding of integral calculus.
1. Introduction
Computer algebra systems (CAS) have been used widely as tools in Mathematics education.
A number of symbolic packages for mathematics courses have been developed using
MATHEMATICA, MATLAB or MAPLE over the last two decades. It is found that these symbolic
packages enable students to achieve high level of logical-analytical reasoning by visually
supporting the concepts and the proofs with graphics presented through the packages [4, 5, 15].
Many researchers have measured and evaluated student learning of mathematical concepts via
computer generated dynamic visualizations [6, 7, 9, 11, 14, 17]. A CAS Maple has been used to
teach double integration [13], number theory [2], graph sketching [10], mathematical analysis of
electronic signals and circuits [16]. Recently, Cook developed a set of MAPLE graphing tools-
calcIIIplots for the teaching of 3-dimensional calculus [3]. Man introduced CAS Maple and its
applications in mathematics education at school level [12]. Students were provided with
opportunities in using MAPLE to perform mathematics exploration or problem solving. Symbolic
packages also have been used in teaching other subjects such as physics and engineering [1, 8, 19]
and computation [18]. Tonkes et al. developed a learning model and designed a workbook for
teaching computation using MATLAB to the first year students at university of Queensland.
Throughout the learning model, students work through the workbook to cover all topics required
for improving the proficiency in MATLAB. They found that their learning model together with
workbook improved learning outcomes over historical experiences. Hence, it is noted that the use
of symbolic packages is essential in developing logical/analytical reasoning as well as for
implementing habits for justifying the results.
Based on a powerful scientific programming environments and libraries packages of special
routines, Maple is one of the most popular systems because it is well-suited to aid college students
to learn mathematics through verifying calculation and plotting complicated graphs, and also
combines mathematical capabilities with a text editor. In this paper, we develop an efficient
Mathematic teaching-learning tool to aid in teaching and learning Mathematics concepts of
multivariate integral calculus for science and engineering students. The Mathematics tool consists
of PowerPoint slides integrated with Maple animations, interactive Maplets and Maple worksheets.
It provides students with the best opportunity to have self-motivation, self-planned learning and
self-assessment. Interactive Maplets with Maple worksheets were designed to reinforce students'
conceptual understanding of integral calculus.
2. Microsoft PowerPoint with Maple Animation
The PowerPoint presentation consists of a number of PowerPoint slides integrated with
Maple animations. It has been designed for instructors to introduce Mathematics concepts and
theories to students. Figures 2.1-2.2 show some PowerPoint slides we produced for our integral
calculus course.
Figure 2.1 demonstrates the concept of z-cap region and shows how to evaluate a triple
integral over a z-cap region.
Figure 2.2 presents a PowerPoint slide for evaluating a volume integral. To clearly identify
the problem, double clicking the hyperlink figure opens a 3-dimension plot object of the solid
laying above the cone z = x 2 + y 2 and below the sphere x 2 + y 2 + z 2 = z
Figure 2.1 Diagram showing how to evaluate triple integral over a z-cap region: (a) A PowerPoint
slide showing the method of evaluation; (b) Maple animation showing the concept of a z-cap region
Figure 2.2 PowerPoint slides (a) and (b) presenting an application example of volume integral: find
the volume of the solid that lie above the cone z = x 2 + y 2 and below the sphere x 2 + y 2 + z 2 = z
3. Maplets of Line/Surface/Volume Integrals
A symbolic package for integral calculus was developed by Maple. The symbolic package
consists of three Maplets which are used to introduce the concepts of line integrals, surface
integrals, and volume integrals. The package enables students to achieve high level of
logicalanalytical reasoning by visually supporting the Mathematics concepts with graphics.
Students learn Mathematics concepts step by step through these Maplets. In each Maplet, there are
three main functions including a few input functions with help sections for defining a problem, a
graphic visualization function, and an output function for showing the symbolic results. Included
here are examples of using Maplets to check solutions of problems in integral calculus.
Example 1 A wire takes the shape of the semicircle x 2 + y 2 = 1 , y ≥ 0 , and is thicker near its base
than near the top. Find (a) the mass of the wire and (b) the center of mass of the wire if the linear
density at any point is proportional to its distance from the line y = 1.
Solution The semi circle can be parameterized by the parametric equations
x = cos t , y = sin t , 0 ≤ t ≤ π
(a)The mass of the wire.
As the linear density is
ρ ( x, y ) = k (1 − y ) ,
the mass of the wire can be determined by
m = ∫ k (1 − y )ds . (3.1)
C
Figure 3.1 shows how to evaluate the above line integral with respect to arc length. Firstly,
we select the integral type: line integral with respect to arc length. Then we enter the integrand
which is the linear density function of the wire, and then we enter the parametric equations of the
curve C. Finally, by clicking on the "Evaluate Integral" button, the mass of the wire is calculated
step by step as shown on the result area yielding the result m = kπ − 2k .
(b)The center of mass of the wire.
Let ( x, y ) be the center of mass of the wire, then
1 1
x = ∫ x ρ ( x, y )ds , y = ∫ y ρ ( x, y )ds (3.2)
m C m C
By symmetry we see that x = 0 , so we only need to find y .
1 1
y = ∫ y ρ ( x, y )ds =
kπ − 2k ∫C
yk (1 − y )ds . (3.3)
m C
By performing the same steps as above, y can be determined as shown in the result area of Figure
⎛ −4 + π ⎞
3.2. Hence the center of mass is at the point with coordinates ⎜ 0, − ⎟ ≈ (0, 0.38) .
⎝ 2(π − 2) ⎠
Figure 3.1 Maplet window for evaluation of two-dimension line integrals
Figure 3.2 Maplet window for calculating the center of mass of the wire
Example 2 Find the rate of flow through the unit sphere, where F ( x, y, z ) = zi + y j + xk is a
velocity field describing the flow of a fluid with density 1.
Solution The unit sphere x 2 + y 2 + z 2 = 1 can be parameterized by
x = sin u cos v , y = sin u sin v , z = cos u ,
where the parameter domain is
D = {(u, v) 0 ≤ u ≤ π , 0 ≤ v ≤ 2π } .
The velocity field describing the flow of a fluid with density 1 is
F ( x, y, z ) = zi + y j + xk .
So, the rate of flow through the unit sphere is
∫∫ S
F id S = ∫∫ F in dS ,
S
(3.4)
ru × rv
where the unit normal vector n = .
ru × rv
Figure 3.3 shows how to use the Maplet to evaluate the surface integrals. Firstly, define the
parametric equations of the unit sphere, then the vector field F ( x, y, z ) , and then click on the
"Evaluate Integral" button. The rate of flow through the unit sphere will then be calculated step by
step as shown on the result area and the visualization of the rate of flow through the unit sphere is
4
shown on the plot area. The result obtained is π .
3
Figure 3.3 Maplet window for evaluating the surface integral representing the rate of flow through
the unit sphere
4. Assessment Process
Pre and post testing are used to measure students' achievement. All equations in pre and
post testing including basic concepts of each topic and its applications are designed in parallel. To
measure students' knowledge in the basic concepts, multiple choice tests and short answer tests are
used. Writing test is used for the application part.
Instructors describe the objectives of learning for each topic and assign students to complete
workbook which is designed to enhance students' learning outcomes. There are 3 main topics,
line/surface/volume integrals, in this workbook. Figure 6 shows an example in the line integral
topic which was produced to help instructor in measuring students' learning progress and help
students in self-assessment.
Figure 4.1 An example of workbook for line integral problems
5. Conclusion
An efficient Mathematics teaching-learning tool for integral calculus courses has been
developed to enhance students' interest in mathematics. It enables students to have self-motivation,
self-planned learning and self-assessment. Teacher and students have more time to cover a wider
range of problems in class. Interactive Maplets of line/surface/volume integrals enable students to
analyze mathematics concepts step by step. Students were encouraged to use Maplets with Maple
worksheets on homework to check their answers. Maplets provide fast solutions with good
visualization of applied mathematics problems. These can help students to identify patterns, see
connections and allow students to deepen their knowledge of multivariable calculus.
References
[1] Beltzer, A. I. and Shenkman, A. W. (1995). Use of symbolic computation in engineering
education. IEEE Transactions on Education, 38(2):177–184.
[2] Cheung, Y. L. (1996). Learning unmber theory with a computer algebra system. International
Journal of Mathematical Education in Science and Technology, 27(3):379–385.
[3] Cook, D. (2006). Maple graphing tools for calculus iii. Mathematics and Computer Education,
40(1):36–41.
[4] Drijvers, P. (2002). Learning mathematics in a computer algebra environment: obstacles are
opportunities. ZDM The International Journal on Mathematics Education, 34(5):221–228.
[5] Fuchs, K. J. (2001). Computer algebra systems in mathematics education. ZDM The
International Journal on Mathematics Education, 35(1):20–23.
[6] Hayden, M. B. and Lamagna, E. A. (1998). Newton: An interactive environment for exploring
mathematics. Journal of Symbolic Computation, 25:195–212.
[7] Heid, M. (1988). Resequencing skills and concepts in applied calculus using the computer as a
tool. Journal of Research in Mathematics, 19(1):3–25.
[8] Johnson, D. and Buege, J. (1995). Rethinking the way we teach undergraduate physics and
engineering with mathematica. Mathematics with Vision: Proceedings of the First International
Mathematica Symposium, pages 233–242.
[9] Kendal, M. and Stacey, K. (2002). Teacher in transition: Moving towards CAS-supported
classroom. ZDM The International Journal on Mathematics Education, 34(5):196–201.
[10] Kong and Kwok (1999). An interactive teaching and learning environment for graph
sketching. Computers and education, 32(1):1–17.
[11] Kramarski, B. and Hirsch, C. (2003). Using computer algebra systems in mathematical
classrooms. Journal of Computer Assisted Learning, 19:35–45.
[12] Man, Y. K. (2007). Introducing comprter algebra to school teachers of mathematics. Teaching
mathematics and its applications, 20(1):23–26.
[13] Mathews, J. H. (1990). Using a computer algebra system to teach double integration.
International Journal of Mathematical Education in Science and Technology, 21(5):723–732.
[14] Perjési, I. H. (2003). Application of cas for teaching of integral-transforming theorems. ZDM
The International Journal on Mathematics Education, 35(2):43–47.
[15] Peschek, W. and Schneider, E. (2002). Cas in general mathematics education. ZDM The
International Journal on Mathematics Education, 34(5):189–195.
[16] Røyrvik, O. (2002). Teaching electrical engineering using maple. Internaltional Journal of
Electrical Engineering, 39(4):297–300.
[17] Tall, D. (1991). Recent developments in the use of the computer to visualize and symbolize
calculus concepts. MAA Notes, 20:15–25.
[18] Tonkes, E. J., Loch, B. I., and Stace, A. W. (2005). An innovation learning model for
computation in first year mathematics. International Journal of Mathematical Education in
Science and Technology, 36(7):751–758.
[19] Ward, J. P. (2003). Modern mathematics for engineers and scientists. Teaching mathematics
and its applications, 22(1):37–44. | 677.169 | 1 |
Mathematics
The Queen of the Sciences
Essays on Mathematical Topics
Real, rational, irrational, imaginary. An explanation of numbers of different kinds, a little about infinite series and an surprising relationship. Types of integers: odd, even, squares; perfect, amicable and prime numbers.
An introductory look at the strange world of logarithms, how they are used for calculations, how they are evaluated with series, and using logarithms to solve algebraic equations. Definitions of index and base.
Graphs are a way of visualising algebraic functions. The Cartesian coordinate system is introduced along with a description of graph drawing from first principles. There are examples of different types of graphs. | 677.169 | 1 |
To log in and use all the features of Khan Academy, please enable JavaScript in your browser.
Linear algebra
Have you ever wondered what the difference is between speed and velocity? Ever try to visualize in four dimensions or six or seven? Linear algebra describes things in two dimensions, but many of the concepts can be extended into three, four or more. Linear algebra implies two dimensional reasoning, however, the concepts covered in linear algebra provide the basis for multi-dimensional representations of mathematical reasoning. Matrices, vectors, vector spaces, transformations, eigenvectors/values all help us to visualize and understand multi dimensional concepts. This is an advanced course normally taken by science or engineering majors after taking at least two semesters of calculus (although calculus really isn't a prereq) so don't confuse this with regular high school algebra.
Vectors and spaces
We will begin our journey through linear algebra by defining and conceptualizing what a vector is (rather than starting with matrices and matrix operations like in a more basic algebra course) and defining some basic operations (like addition, subtraction and scalar multiplication).
Given a set of vectors, what other vectors can you create by adding and/or subtracting scalar multiples of those vectors. The set of vectors that you can create through these linear combinations of the original set is called the "span" of the set.
If no vector in a set can be created from a linear combination of the other vectors in the set, then we say that the set is linearly independent. Linearly independent sets are great because there aren't any extra, unnecessary vectors lying around in the set. :)
In this tutorial, we'll define what a "subspace" is --essentially a subset of vectors that has some special properties. We'll then think of a set of vectors that can most efficiently be use to construct a subspace which we will call a "basis".
In this tutorial, we define two ways to "multiply" vectors-- the dot product and the cross product. As we progress, we'll get an intuitive feel for their meaning, how they can used and how the two vector products relate to each other.
This tutorial is a bit of an excursion back to you Algebra II days when you first solved systems of equations (and possibly used matrices to do so). In this tutorial, we did a bit deeper than you may have then, with emphasis on valid row operations and getting a matrix into reduced row echelon form.
We will define matrix-vector multiplication and think about the set of vectors that satisfy Ax=0 for a given matrix A (this is the null space of A). We then proceed to think about the linear combinations of the columns of a matrix (column space). Both of these ideas help us think the possible solutions to the Matrix-vector equation Ax=b. | 677.169 | 1 |
Precalculus Mathematics for Calculus
9780534492779
ISBN:
0534492770
Edition: 5 Pub Date: 2005 Publisher: Thomson Learning
Summary: This best selling author team explains concepts simply and clearly, without glossing over difficult points. Problem solving and mathematical modeling are introduced early and reinforced throughout, so that when students finish the course, they have a solid foundation in the principles of mathematical thinking. This comprehensive, evenly paced book provides complete coverage of the function concept and integrates subs...tantial graphing calculator materials that help students develop insight into mathematical ideas. The authors' attention to detail and clarity, as in James Stewart's market-leading Calculus text, is what makes this text the market leader.
Redlin, Lothar is the author of Precalculus Mathematics for Calculus, published 2005 under ISBN 9780534492779 and 0534492770. One hundred ninety eight Precalculus Mathematics for Calculus textbooks are available for sale on ValoreBooks.com, fifty eight used from the cheapest price of $31.79, or buy new starting at $201 | 677.169 | 1 |
Math.NET aims to provide a self contained clean framework for symbolic mathematical (Computer Algebra System) and numerical/scientific computations, including a parser and support for linear algebra, complex differential analysis, system solving and more | 677.169 | 1 |
Discrete Mathematics and Its Applications
9780073312712
0073312711
Summary: Discrete Mathematics and its Applications, Sixth Edition, is intended for one- or two-term introductory discrete mathematics courses taken by students from a wide variety of majors, including computer science, mathematics, and engineering. This renowned best-selling text, which has been used at over 500 institutions around the world, gives a focused introduction to the primary themes in a discrete mathematics course ...and demonstrates the relevance and practicality of discrete mathematics to a wide a wide variety of real-world applications...from computer science to data networking, to psychology, to chemistry, to engineering, to linguistics, to biology, to business, and to many other important fields.
Rosen, Kenneth H. is the author of Discrete Mathematics and Its Applications, published 2006 under ISBN 9780073312712 and 0073312711. Four Discrete Mathematics and Its Applications textbooks are available for sale on ValoreBooks.com, three used from the cheapest price of $20.41, or buy new starting at $77.31.[read more | 677.169 | 1 |
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Opening with a cartoon showing the weights of three combinations of fish, this activity challenges students to determine the weight of each fish. This activity is part of the Figure This! collection of challenges emphasizing real-world uses of mathematics. The introduction discusses algebraic reasoning and notes its importance to scientists, engineers, and psychologists. Students are encouraged to begin by adding the weights on all three scales. The answer page describes three strategies for solving the problem. Related questions invite students to use the strategies to solve similar problems. Answers to all questions and links to resources are included.
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This lesson is designed for students to gather and analyze data about baseball figures. The student will use the Internet or other resources to collect statistical data on the top five home run hitters for the current season as well as their career home run totals. The students will graph the data and determine if it is linear or non-linear.
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Microsoft Mathematics 4.0 download
Microsoft Mathematics provides a set of mathematical tools that help students get school work done quickly and easily. Students can learn to solve equations step-by-step while gaining a better understanding of fundamental concepts in pre-algebra.
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Create professional-quality mathematics worksheets to provide students in grades K to 10 with the skills development and practice they need as part of a complete numeracy program. Over 70 mathematics worksheet activities can be produced to advance and reinforce skills in number operations, number concepts, fractions, numeration, time, measurement, money, problem .... Free download of Math Resource Studio 4.4.2
CurveFitter program Fit thousands .... Free download of Regression Analysis - CurveFitter 4.5.10
EqPlot plots 2D graphs from complex equations. The application comprises algebraic, trigonometric, hyperbolic and transcendental functions. EqPlot can be used to verify the results of nonlinear regression analysis program. Graphically Review Equations: EqPlot gives engineers and researchers the power to graphically review equations, by putting a large number of equations at .... Free download of EqPlot 1.3.10 | 677.169 | 1 |
Paperback
Item is available through our marketplace sellers and in stores.
Overview
Create your own path to GED success with help from McGraw-Hill's GED test series
The newly revised McGraw-Hill's GED test series helps you develop the skills you need to pass all five areas of the GED test.
Presented in a clear, appealing format, these books offer many opportunities for test practice and explain the essential concepts of each subject so you can succeed on every portion of the GED exam. The series covers: Language Arts, Reading • Language Arts, Writing • Mathematics • Science • Social Studies
McGraw-Hill's GED Mathematics guides you through the GED preparation process step-by-step. A Pretest helps you find out your strengths and weaknesses so you can create a study plan to fit your needs. The following chapters introduce you to math concepts on which hundreds of GED questions are based. Then check your understanding of these ideas with the Posttest, presented in the GED format. You can then see how ready you are for the big exam by taking the full-length Practice Test.
McGraw-Hill's GED Mathematics includes:
Clear instructions to show you how to use number grids and coordinate plane grids | 677.169 | 1 |
Mathematical Methods in Artificial Intelligence
Mathematical Methods in Artificial Intelligence introduces the
student to the important mathematical foundations and tools in AI
and describes their applications to the design of AI algorithms.
This useful text presents an introductory AI course based on the
most important mathematics and its applications. It focuses on
important topics that are proven useful in AI and involve the most
broadly applicable mathematics.
The book explores AI from three different viewpoints: goals,
methods or tools, and achievements and failures. Its goals of
reasoning, planning, learning, or language understanding and use
are centered around the expert system idea. The tools of AI are
presented in terms of what can be incorporated in the data
structures. The book looks into the concepts and tools of limited
structure, mathematical logic, logic-like representation, numerical
information, and nonsymbolic structures.
The text emphasizes the main mathematical tools for representing
and manipulating knowledge symbolically. These are various forms of
logic for qualitative knowledge, and probability and related
concepts for quantitative knowledge. The main tools for
manipulating knowledge nonsymbolically, as neural nets, are
optimization methods and statistics. This material is covered in
the text by topics such as trees and search, classical mathematical
logic, and uncertainty and reasoning. A solutions diskette is
available, please call for more information | 677.169 | 1 |
books.google.fr - This classic text has entered and held the field as the standard book on the applications of analysis to the transcendental functions. The authors explain the methods of modern analysis in the first part of the book and then proceed to a detailed discussion of the transcendental function, unhampered... Course of Modern Analysis | 677.169 | 1 |
The Guide was designed to support educators by providing an overview of the new test design. Information about how the assessment shifts informed test development and how the CCLS will be measured on the new Regents Exam in Algebra I (CC) is specified.
The three webcasts provide: (1) background on the Common Core State Standards along with teaching principles and tools that are crucial for implementing changes in pedagogy and a mathematics curriculum (2) information for teachers on the test design of the Regents Exam in Algebra I (Common Core) and how it measures the Common Core Learning Standards (CCLS) and (3) background information on the sample questions for Algebra I (Common Core).
As NY guidance documents have indicated we must simplify radicals as we solve quadratics using the quadratic formula. Anyone using the modules will have to add this supplement to module 4 as they care covering the quadratic formula. You do not need to do extensive work around this, but enough so that students can utilize it as they solve quadratics.
New modules are posted for most grade levels. The full Module 1 is up for 6-10 and most module 2's at the k-5 level. I have a busy week, but will work on getting everything up next week. Until then here is the link to engageny. | 677.169 | 1 |
1
00:00:08 --> 00:00:13.13
OK, this is the lecture on
linear transformations.
2
00:00:13.13 --> 00:00:18
Actually, linear algebra
courses used to begin with this
3
00:00:18 --> 00:00:24
lecture, so you could say I'm
beginning this course again by
4
00:00:24 --> 00:00:27
talking about linear
transformations.
5
00:00:27 --> 00:00:31
In a lot of courses,
those come first before
6
00:00:31 --> 00:00:33
matrices.
7
00:00:33 --> 00:00:37
The idea of a linear
transformation makes sense
8
00:00:37 --> 00:00:42
without a matrix,
and physicists and other --
9
00:00:42 --> 00:00:45
some people like it better that
way.
10
00:00:45 --> 00:00:48
They don't like coordinates.
11
00:00:48 --> 00:00:51
They don't want those numbers.
12
00:00:51 --> 00:00:57
They want to see what's going
on with the whole space.
13
00:00:57 --> 00:01:01
But, for most of us,
in the end, if we're going to
14
00:01:01 --> 00:01:04.35
compute anything,
we introduce coordinates,
15
00:01:04.35 --> 00:01:08
and then every linear
transformation will lead us to a
16
00:01:08 --> 00:01:08
matrix.
17
00:01:08 --> 00:01:13
And then, to all the things
that we've done about null space
18
00:01:13 --> 00:01:15
and row space,
and determinant,
19
00:01:15 --> 00:01:19
and eigenvalues -- all will
come from the matrix.
20
00:01:19 --> 00:01:28
But, behind it -- in other
words, behind this is the idea
21
00:01:28 --> 00:01:32
of a linear transformation.
22
00:01:32 --> 00:01:40
Let me give an example of a
linear transformation.
23
00:01:40 --> 00:01:42
So, example.
24
00:01:42 --> 00:01:43
Example one.
25
00:01:43 --> 00:01:47.19
A projection.
26
00:01:47.19 --> 00:01:53
I can describe a projection
without telling you any matrix,
27
00:01:53 --> 00:01:55
anything about any matrix.
28
00:01:55 --> 00:02:01
I can describe a projection,
say, this will be a linear
29
00:02:01 --> 00:02:05
transformation that takes,
say, all of R^2,
30
00:02:05 --> 00:02:11
every vector in the plane,
into a vector in the plane.
31
00:02:11 --> 00:02:17.03
And this is the way people
describe, a mapping.
32
00:02:17.03 --> 00:02:21
It takes every vector,
and so, by what rule?
33
00:02:21 --> 00:02:26
So, what's the rule,
is, I take a -- so here's the
34
00:02:26 --> 00:02:32.5
plane, this is going to be my
line, my line through my line,
35
00:02:32.5 --> 00:02:37
and I'm going to project every
vector onto that line.
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00:02:37 --> 00:02:45.06
So if I take a vector like b --
or let me call the vector v for
37
00:02:45.06 --> 00:02:51
the moment -- the projection --
the linear transformation is
38
00:02:51 --> 00:02:54
going to produce this vector as
T(v).
39
00:02:54 --> 00:02:57
So T -- it's like a function.
40
00:02:57 --> 00:03:00
Exactly like a function.
41
00:03:00 --> 00:03:05
You give me an input,
the transformation produces the
42
00:03:05 --> 00:03:07
output.
43
00:03:07 --> 00:03:10
So transformation,
sometimes the word map,
44
00:03:10 --> 00:03:11.98
or mapping is used.
45
00:03:11.98 --> 00:03:14
A map between inputs and
outputs.
46
00:03:14 --> 00:03:18
So this is one particular map,
this is one example,
47
00:03:18 --> 00:03:22
a projection that takes every
vector -- here,
48
00:03:22 --> 00:03:26
let me do another vector v,
or let me do this vector w,
49
00:03:26 --> 00:03:27
what is T(w)?
50
00:03:27 --> 00:03:28
You see?
51
00:03:28 --> 00:03:32
There are no coordinates here.
52
00:03:32 --> 00:03:36
I've drawn those axes,
but I'm sorry I drew them,
53
00:03:36 --> 00:03:41
I'm going to remove them,
that's the whole point,
54
00:03:41 --> 00:03:46
is that we don't need axes,
we just need -- so guts -- get
55
00:03:46 --> 00:03:49
it out of there,
I'm not a physicist,
56
00:03:49 --> 00:03:52
so I draw those axes.
57
00:03:52 --> 00:03:55
So the input is w,
the output of the projection
58
00:03:55 --> 00:03:58
is, project on that line,
T(w).
59
00:03:58 --> 00:03:58
OK.
60
00:03:58 --> 00:04:02
Now, I could think of a lot of
transformations T.
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00:04:02 --> 00:04:06
But, in this linear algebra
course, I want it to be a linear
62
00:04:06 --> 00:04:07
transformation.
63
00:04:07 --> 00:04:12
So here are the rules for a
linear transformation.
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00:04:12 --> 00:04:16
Here, see, exactly,
the two operations that we can
65
00:04:16 --> 00:04:19
do on vectors,
adding and multiplying by
66
00:04:19 --> 00:04:25
scalars, the transformation does
something special with respect
67
00:04:25 --> 00:04:27
to those operations.
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00:04:27 --> 00:04:30
So, for example,
the projection is a linear
69
00:04:30 --> 00:04:34
transformation because --
for example,
70
00:04:34 --> 00:04:40
if I wanted to check that one,
if I took v to be twice as
71
00:04:40 --> 00:04:44
long, the projection would be
twice as long.
72
00:04:44 --> 00:04:50
If I took v to be minus -- if I
changed from v to minus v,
73
00:04:50 --> 00:04:53
the projection would change to
a minus.
74
00:04:53 --> 00:04:57
So c equal to two,
c equal minus one,
75
00:04:57 --> 00:04:59
any c is OK.
76
00:04:59 --> 00:05:04
So you see that actually,
those combine,
77
00:05:04 --> 00:05:08
I can combine those into one
statement.
78
00:05:08 --> 00:05:14
What the transformation does to
any linear combination,
79
00:05:14 --> 00:05:20
it must produce the same
combination of T(v) and T(w).
80
00:05:20 --> 00:05:26
Let's think about some --
I mean, it's like,
81
00:05:26 --> 00:05:33
not hard to decide,
is a transformation linear or
82
00:05:33 --> 00:05:35
is it not.
83
00:05:35 --> 00:05:43.25
Let me give you an example so
you can tell me the answer.
84
00:05:43.25 --> 00:05:52
Suppose my transformation is --
here's another example two.
85
00:05:52 --> 00:05:55
Shift the whole plane.
86
00:05:55 --> 00:06:03
So here are all my vectors,
my plane, and every vector v in
87
00:06:03 --> 00:06:11
the plane, I shift it over by,
let's say, three by some vector
88
00:06:11 --> 00:06:12
v0.
89
00:06:12 --> 00:06:15
Shift whole plane by v0.
90
00:06:15 --> 00:06:24
So every vector in the plane --
this was v, T(v) will be v+v0.
91
00:06:24 --> 00:06:26
There's T(v).
92
00:06:26 --> 00:06:27
Here's v0.
93
00:06:27 --> 00:06:29
There's the typical v.
94
00:06:29 --> 00:06:31
And there's T(v).
95
00:06:31 --> 00:06:34
You see what this
transformation does?
96
00:06:34 --> 00:06:37
Takes this vector and adds to
it.
97
00:06:37 --> 00:06:40
Adds a fixed vector to it.
98
00:06:40 --> 00:06:45
Well, that seems like a pretty
reasonable, simple
99
00:06:45 --> 00:06:49
transformation,
but is it linear?
100
00:06:49 --> 00:06:53
The answer is no,
it's not linear.
101
00:06:53 --> 00:06:56.54
Which law is broken?
102
00:06:56.54 --> 00:06:59
Maybe both laws are broken.
103
00:06:59 --> 00:07:01
Let's see.
104
00:07:01 --> 00:07:08.82
If I double the length of v,
does the transformation produce
105
00:07:08.82 --> 00:07:13
something double -- do I double
T(v)?
106
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No.
107
00:07:15 --> 00:07:18
If I double the length of v,
in this transformation,
108
00:07:18 --> 00:07:22
I'm just adding on the same one
-- same v0, not two v0s,
109
00:07:22 --> 00:07:26
but only one v0 for every
vector, so I don't get two times
110
00:07:26 --> 00:07:27
the transform.
111
00:07:27 --> 00:07:29.23
Do you see what I'm saying?
112
00:07:29.23 --> 00:07:32
That if I double this,
then the transformation starts
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00:07:32 --> 00:07:37.41
there and only goes one v0 out
and doesn't double T(v).
114
00:07:37.41 --> 00:07:43
In fact, a linear
transformation -- what is T of
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00:07:43 --> 00:07:43
zero?
116
00:07:43 --> 00:07:50
That's just like a special
case, but really worth noticing.
117
00:07:50 --> 00:07:56
The zero vector in a linear
transformation must get
118
00:07:56 --> 00:07:59
transformed to zero.
119
00:07:59 --> 00:08:04
It can't move,
because, take any vector V here
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00:08:04 --> 00:08:10
--
well, so you can see why T of
121
00:08:10 --> 00:08:12
zero is zero.
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00:08:12 --> 00:08:19
Take v to be the zero vector,
take c to be three.
123
00:08:19 --> 00:08:27
Then we'd have T of zero vector
equaling three T of zero vector,
124
00:08:27 --> 00:08:32
the T of zero has to be zero.
125
00:08:32 --> 00:08:32
OK.
126
00:08:32 --> 00:08:39
So, this example is really a
non-example.
127
00:08:39 --> 00:08:45
Shifting the whole plane is not
a linear transformation.
128
00:08:45 --> 00:08:51
Or if I cooked up some formula
that involved squaring,
129
00:08:51 --> 00:08:56
or the transformation that,
also non-example,
130
00:08:56 --> 00:09:03
how about the transformation
that, takes any vector and
131
00:09:03 --> 00:09:06
produces its length?
132
00:09:06 --> 00:09:10
So there's a transformation
that takes any vector,
133
00:09:10 --> 00:09:15
say, any vector in R^3,
let me just -- I'll just get a
134
00:09:15 --> 00:09:18
chance to use this notation
again.
135
00:09:18 --> 00:09:23
Suppose I think of the
transformation that takes any
136
00:09:23 --> 00:09:27
vector in R^3 and produces this
number.
137
00:09:27 --> 00:09:30
So that, I could say,
is a member of R^1,
138
00:09:30 --> 00:09:34
for example,
if I wanted.
139
00:09:34 --> 00:09:36
Or just real numbers.
140
00:09:36 --> 00:09:39
That's certainly not linear.
141
00:09:39 --> 00:09:44
It's true that the zero vector
goes to zero.
142
00:09:44 --> 00:09:49
But if I double a vector,
it does double the length,
143
00:09:49 --> 00:09:51
that's true.
144
00:09:51 --> 00:09:56
But suppose I multiply a vector
by minus two.
145
00:09:56 --> 00:10:00
What happens to its length?
146
00:10:00 --> 00:10:01
It just doubles.
147
00:10:01 --> 00:10:04.59
It doesn't get multiplied by
minus two.
148
00:10:04.59 --> 00:10:09
So when c is minus two in my
requirement, I'm not satisfying
149
00:10:09 --> 00:10:10
that requirement.
150
00:10:10 --> 00:10:15
So T of minus v is not minus v
-- minus, the length,
151
00:10:15 --> 00:10:16
it's just the length.
152
00:10:16 --> 00:10:19
OK, so that's another
non-example.
153
00:10:19 --> 00:10:23
Projection was an example,
let me give you another
154
00:10:23 --> 00:10:25
example.
155
00:10:25 --> 00:10:36
I can stay here and have a --
this will be an example that is
156
00:10:36 --> 00:10:43
a linear transformation,
a rotation.
157
00:10:43 --> 00:10:48
Rotation by -- what shall we
say?
158
00:10:48 --> 00:10:51
By 45 degrees.
159
00:10:51 --> 00:10:53
OK?
160
00:10:53 --> 00:10:57.32
So again, let me choose this,
this will be a mapping,
161
00:10:57.32 --> 00:11:01
from the whole plane of
vectors, into the whole plane of
162
00:11:01 --> 00:11:05
vectors, and it just -- here is
the input vector v,
163
00:11:05 --> 00:11:09
and the output vector foam this
45 degree rotation is just
164
00:11:09 --> 00:11:12
rotate that thing by 45 degrees,
T(v).
165
00:11:12 --> 00:11:14
So every vector got rotated.
166
00:11:14 --> 00:11:19
You see that I can describe
this without any coordinates.
167
00:11:19 --> 00:11:22
And see that it's linear.
168
00:11:22 --> 00:11:28
If I doubled v,
the rotation would just be
169
00:11:28 --> 00:11:30
twice as far out.
170
00:11:30 --> 00:11:35
If I had v+w,
and if I rotated each of them
171
00:11:35 --> 00:11:44
and added, the answer's the same
as if I add and then rotate.
172
00:11:44 --> 00:11:49
That's what the linear
transformation is.
173
00:11:49 --> 00:11:52
OK, so those are two examples.
174
00:11:52 --> 00:11:57
Two examples,
projection and rotation,
175
00:11:57 --> 00:12:04
and I could invent more that
are linear transformations where
176
00:12:04 --> 00:12:08
I haven't told you a matrix yet.
177
00:12:08 --> 00:12:13.25
Actually, the book has a
picture of the action of linear
178
00:12:13.25 --> 00:12:17
transformations -- actually,
the cover of the book has it.
179
00:12:17 --> 00:12:22.1
So, in this section seven point
one, we can think of a --
180
00:12:22.1 --> 00:12:26
actually, here let's take this
linear transformation,
181
00:12:26 --> 00:12:30
rotation, suppose I have,
as the cover of the book has,
182
00:12:30 --> 00:12:32
a house in R^2.
183
00:12:32 --> 00:12:37
So instead of this,
let me take a small house in
184
00:12:37 --> 00:12:37
R^2.
185
00:12:37 --> 00:12:41
So that's a whole lot of
points.
186
00:12:41 --> 00:12:44.4
The idea is,
with this linear
187
00:12:44.4 --> 00:12:49
transformation,
that I can see what it does to
188
00:12:49 --> 00:12:52
everything at once.
189
00:12:52 --> 00:12:58
I don't have to just take one
vector at a time and see what T
190
00:12:58 --> 00:13:03
of V is, I can take all the
vectors on the outline of the
191
00:13:03 --> 00:13:06
house, and see where they all
go.
192
00:13:06 --> 00:13:12
In fact, that will show me
where the whole house goes.
193
00:13:12 --> 00:13:18
So what will happen with this
particular linear
194
00:13:18 --> 00:13:19
transformation?
195
00:13:19 --> 00:13:24
The whole house will rotate,
so the result,
196
00:13:24 --> 00:13:29
if I can draw it,
will be, the house will be
197
00:13:29 --> 00:13:30
sitting there.
198
00:13:30 --> 00:13:31
OK.
199
00:13:31 --> 00:13:37
And, but suppose I give some
other examples.
200
00:13:37 --> 00:13:44
Oh, let me give some examples
that involve a matrix.
201
00:13:44 --> 00:13:51
Example three -- and this is
important -- coming from a
202
00:13:51 --> 00:13:55
matrix at -- we always call A.
203
00:13:55 --> 00:14:00
So the transformation will be,
multiply by A.
204
00:14:00 --> 00:14:06.22
There is a linear
transformation.
205
00:14:06.22 --> 00:14:11
And a whole family of them,
because every matrix produces a
206
00:14:11 --> 00:14:16
transformation by this simple
rule, just multiply every vector
207
00:14:16 --> 00:14:19
by that matrix,
and it's linear,
208
00:14:19 --> 00:14:19
right?
209
00:14:19 --> 00:14:24
Linear, I have to check that
A(v) -- A times v plus w equals
210
00:14:24 --> 00:14:29
Av plus A w, which is fine,
and I have to check that A
211
00:14:29 --> 00:14:32
times vc equals c A(v).
212
00:14:32 --> 00:14:32.81
Check.
213
00:14:32.81 --> 00:14:34
Those are fine.
214
00:14:34 --> 00:14:38.29
So there is a linear
transformation.
215
00:14:38.29 --> 00:14:44
And if I take my favorite
matrix A, and I apply it to all
216
00:14:44 --> 00:14:49
vectors in the plane,
it will produce a bunch of
217
00:14:49 --> 00:14:50
outputs.
218
00:14:50 --> 00:14:55
See, the idea is now worth
thinking of, like,
219
00:14:55 --> 00:14:58
the big picture.
220
00:14:58 --> 00:15:01
The whole plane is transformed
by matrix multiplication.
221
00:15:01 --> 00:15:04
Every vector in the plane gets
multiplied by A.
222
00:15:04 --> 00:15:07
Let's take an example,
and see what happens to the
223
00:15:07 --> 00:15:08
vectors of the house.
224
00:15:08 --> 00:15:11.03
So this is still a
transformation from plane to
225
00:15:11.03 --> 00:15:14.07
plane, and let me take a
particular matrix A -- well,
226
00:15:14.07 --> 00:15:17
if I cooked up a rotation
matrix, this would be the right
227
00:15:17 --> 00:15:18
picture.
228
00:15:18 --> 00:15:24
If I cooked up a projection
matrix, the projection would be
229
00:15:24 --> 00:15:25.26
the picture.
230
00:15:25.26 --> 00:15:28.48
Let me just take some other
matrix.
231
00:15:28.48 --> 00:15:32
Let me take the matrix one zero
zero minus one.
232
00:15:32 --> 00:15:36
What happens to the house,
to all vectors,
233
00:15:36 --> 00:15:41
and in particular,
we can sort of visualize it if
234
00:15:41 --> 00:15:48
we look at the house --
so the house is not rotated any
235
00:15:48 --> 00:15:51
more, what do I get?
236
00:15:51 --> 00:15:59
What happens to all the vectors
if I do this transformation?
237
00:15:59 --> 00:16:03
I multiply by this matrix.
238
00:16:03 --> 00:16:08
Well, of course,
it's an easy matrix,
239
00:16:08 --> 00:16:11
it's diagonal.
240
00:16:11 --> 00:16:16
The x component stays the same,
the y component reverses sign,
241
00:16:16 --> 00:16:19
so that like the roof of that
house, the point,
242
00:16:19 --> 00:16:23
the tip of the roof,
has an x component which stays
243
00:16:23 --> 00:16:27
the same, but its y component
reverses, and it's down here.
244
00:16:27 --> 00:16:30.17
And, of course,
what we get is,
245
00:16:30.17 --> 00:16:33
the house is,
like, upside down.
246
00:16:33 --> 00:16:37
Now, I have to put -- where
does the door go?
247
00:16:37 --> 00:16:42
I guess the door goes upside
down there, right?
248
00:16:42 --> 00:16:46
So here's the input,
here's the input house,
249
00:16:46 --> 00:16:48
and this is the output.
250
00:16:48 --> 00:16:49
OK.
251
00:16:49 --> 00:16:54
This idea of a linear
transformation is like kind of
252
00:16:54 --> 00:17:00
the abstract description of
matrix multiplication.
253
00:17:00 --> 00:17:03
And what's our goal here?
254
00:17:03 --> 00:17:09
Our goal is to understand
linear transformations,
255
00:17:09 --> 00:17:16
and the way to understand them
is to find the matrix that lies
256
00:17:16 --> 00:17:17
behind them.
257
00:17:17 --> 00:17:20
That's really the idea.
258
00:17:20 --> 00:17:26
Find the matrix that lies
behind them.
259
00:17:26 --> 00:17:30
Um, and to do that,
we have to bring in
260
00:17:30 --> 00:17:31
coordinates.
261
00:17:31 --> 00:17:34
We have to choose a basis.
262
00:17:34 --> 00:17:40
So let me point out what's the
story -- if we have a linear
263
00:17:40 --> 00:17:44
transformation -- so start with
-- start.
264
00:17:44 --> 00:17:49
Suppose we have a linear
transformation.
265
00:17:49 --> 00:17:53.03
Let -- from now on,
let T stand for linear
266
00:17:53.03 --> 00:17:54
transformations.
267
00:17:54 --> 00:17:58
I won't be interested in the
nonlinear ones.
268
00:17:58 --> 00:18:02
Only linear transformations I'm
interested in.
269
00:18:02 --> 00:18:02.97
OK.
270
00:18:02.97 --> 00:18:06
I start with a linear
transformation T.
271
00:18:06 --> 00:18:11
Let's suppose its inputs are
vectors in R^3.
272
00:18:11 --> 00:18:11
OK?
273
00:18:11 --> 00:18:16
And suppose its outputs are
vectors in R^2,
274
00:18:16 --> 00:18:17
for example.
275
00:18:17 --> 00:18:17
OK.
276
00:18:17 --> 00:18:22
What's an example of such a
transformation,
277
00:18:22 --> 00:18:25
just before I go any further?
278
00:18:25 --> 00:18:29
Any matrix of the right size
will do this.
279
00:18:29 --> 00:18:35
So what would be the right
shape of a matrix?
280
00:18:35 --> 00:18:42
So, for example -- I'm wanting
to give you an example,
281
00:18:42 --> 00:18:46
just because,
here, I'm thinking of
282
00:18:46 --> 00:18:53
transformations that take
three-dimensional space to
283
00:18:53 --> 00:18:57.74
two-dimensional space.
284
00:18:57.74 --> 00:19:03
And I want them to be linear,
and the easy way to invent them
285
00:19:03 --> 00:19:06
is a matrix multiplication.
286
00:19:06 --> 00:19:10
So example, T of v should be
any A v.
287
00:19:10 --> 00:19:15
Those transformations are
linear, that's what 18.06 is
288
00:19:15 --> 00:19:16.18
about.
289
00:19:16.18 --> 00:19:21.55
And A should be what size,
what shape of matrix should
290
00:19:21.55 --> 00:19:23
that be?
291
00:19:23 --> 00:19:30
I want V to have three
components, because this is what
292
00:19:30 --> 00:19:36
the inputs have -- so here's the
input in R^3,
293
00:19:36 --> 00:19:40
and here's the output in R^2.
294
00:19:40 --> 00:19:43
So what shape of matrix?
295
00:19:43 --> 00:19:49
So this should be,
I guess, a two by three matrix?
296
00:19:49 --> 00:19:50
Right?
297
00:19:50 --> 00:19:54
A two by three matrix.
298
00:19:54 --> 00:19:59
A two by three matrix,
we'll multiply a vector in R^3
299
00:19:59 --> 00:20:04
-- you see I'm moving to
coordinates so quickly,
300
00:20:04 --> 00:20:07.26
I'm not a true physicist here.
301
00:20:07.26 --> 00:20:12
A two by three matrix,
we'll multiply a vector in R^3
302
00:20:12 --> 00:20:17
an produce an output in R^2,
and it will be a linear
303
00:20:17 --> 00:20:20
transformation,
and OK.
304
00:20:20 --> 00:20:24
So there's a whole lot of
examples, every two by three
305
00:20:24 --> 00:20:27
matrix give me an example,
and basically,
306
00:20:27 --> 00:20:31
I want to show you that there
are no other examples.
307
00:20:31 --> 00:20:35
Every linear transformation is
associated with a matrix.
308
00:20:35 --> 00:20:40
Now, let me come back to the
idea of linear transformation.
309
00:20:40 --> 00:20:47
Suppose I've got this linear
transformation in my mind,
310
00:20:47 --> 00:20:51
and I want to tell you what it
is.
311
00:20:51 --> 00:20:58
Suppose I tell you what the
transformation does to one
312
00:20:58 --> 00:21:00
vector.
313
00:21:00 --> 00:21:00
OK.
314
00:21:00 --> 00:21:03.81
You know one thing,
then.
315
00:21:03.81 --> 00:21:05
All right.
316
00:21:05 --> 00:21:12
So this is like the -- what I'm
speaking about now is,
317
00:21:12 --> 00:21:20.59
how much information is needed
to know the transformation?
318
00:21:20.59 --> 00:21:26
By knowing T,
I -- to know T of v for all v.
319
00:21:26 --> 00:21:29
All inputs.
320
00:21:29 --> 00:21:35
How much information do I have
to give you so that you know
321
00:21:35 --> 00:21:39
what the transformation does to
every vector?
322
00:21:39 --> 00:21:44
OK, I could tell you what the
transformation -- so I could
323
00:21:44 --> 00:21:48
take a vector v1,
one particular vector,
324
00:21:48 --> 00:21:52
tell you what the
transformation does to it --
325
00:21:52 --> 00:21:54
fine.
326
00:21:54 --> 00:21:59
But now you only know what the
transformation does to one
327
00:21:59 --> 00:21:59
vector.
328
00:21:59 --> 00:22:02
So you say, OK,
that's not enough,
329
00:22:02 --> 00:22:06
tell me what it does to another
vector.
330
00:22:06 --> 00:22:08
So I say, OK,
give me a vector,
331
00:22:08 --> 00:22:13
you give me a vector v2,
and we see, what does the
332
00:22:13 --> 00:22:16
transformation do to v2?
333
00:22:16 --> 00:22:20
Now, you only know -- or do you
only know what the
334
00:22:20 --> 00:22:22
transformation does to two
vectors?
335
00:22:22 --> 00:22:27
Have I got to ask you -- answer
you about every vector in the
336
00:22:27 --> 00:22:30
whole input space,
or can you, knowing what it
337
00:22:30 --> 00:22:34
does to v1 and v2,
how much do you now know about
338
00:22:34 --> 00:22:36.98
the transformation?
339
00:22:36.98 --> 00:22:42
You know what the
transformation does to a larger
340
00:22:42 --> 00:22:49
bunch of vectors than just these
two, because you know what it
341
00:22:49 --> 00:22:53.57
does to every linear
combination.
342
00:22:53.57 --> 00:22:59
You know what it does,
now, to the whole plane of
343
00:22:59 --> 00:23:03
vectors, with bases v1 and v2.
344
00:23:03 --> 00:23:07
I'm assuming v1 and v2 were
independent.
345
00:23:07 --> 00:23:11
If they were dependent,
if v2 was six times v1,
346
00:23:11 --> 00:23:16
then I didn't give you any new
information in T of v2,
347
00:23:16 --> 00:23:20.48
you already knew it would be
six times T of v1.
348
00:23:20.48 --> 00:23:24
So you can see what I'd headed
for.
349
00:23:24 --> 00:23:30
If I know what the
transformation does to every
350
00:23:30 --> 00:23:34
vector in a basis,
then I know everything.
351
00:23:34 --> 00:23:42
So the information needed to
know T of v for all inputs is T
352
00:23:42 --> 00:23:45
of v1, T of v2,
up to T of vm,
353
00:23:45 --> 00:23:51
let's say, or vn,
for any basis -- for a basis v1
354
00:23:51 --> 00:23:53.73
up to vn.
355
00:23:53.73 --> 00:24:05
This is a base for any -- can I
call it an input basis?
356
00:24:05 --> 00:24:12
It's a basis for the space of
inputs.
357
00:24:12 --> 00:24:20
The things that T is acting on.
358
00:24:20 --> 00:24:25
You see this point,
that if I have a basis for the
359
00:24:25 --> 00:24:32
input space, and I tell you what
the transformation does to every
360
00:24:32 --> 00:24:38
one of those basis vectors,
that is all I'm allowed to tell
361
00:24:38 --> 00:24:43
you, and it's enough to know T
of v for all v-s,
362
00:24:43 --> 00:24:46
because why?
363
00:24:46 --> 00:24:51.53
Because every v is some
combination of these basis
364
00:24:51.53 --> 00:24:56.65
vectors, c1v1+...+cnvn,
that's what a basis is,
365
00:24:56.65 --> 00:24:57
right?
366
00:24:57 --> 00:24:59
It spans the space.
367
00:24:59 --> 00:25:05
And if I know what T does to
this, and what T does to v2,
368
00:25:05 --> 00:25:11
and what T does to vn,
then I know what T does to V.
369
00:25:11 --> 00:25:16
By this linearity,
it has to be c1 T of v1 plus O
370
00:25:16 --> 00:25:20
one plus cn T of vn.
371
00:25:20 --> 00:25:22
There's no choice.
372
00:25:22 --> 00:25:29
So, the point of this comment
is that if I know what T does to
373
00:25:29 --> 00:25:36
a basis, to each vector in a
basis, then I know the linear
374
00:25:36 --> 00:25:37
transformation.
375
00:25:37 --> 00:25:45
The property of linearity tells
me all the other vectors.
376
00:25:45 --> 00:25:47
All the other outputs.
377
00:25:47 --> 00:25:47
OK.
378
00:25:47 --> 00:25:52
So now, we got -- so that light
we now see, what do we really
379
00:25:52 --> 00:25:57
need in a linear transformation,
and we're ready to go to a
380
00:25:57 --> 00:25:57
matrix.
381
00:25:57 --> 00:25:58
OK.
382
00:25:58 --> 00:26:03
What's the step now that takes
us from a linear transformation
383
00:26:03 --> 00:26:08
that's free of coordinates to a
matrix that's been created with
384
00:26:08 --> 00:26:11
respect to coordinates?
385
00:26:11 --> 00:26:17
The matrix is going to come
from the coordinate system.
386
00:26:17 --> 00:26:20
These are the coordinates.
387
00:26:20 --> 00:26:23
Coordinates mean a basis is
decided.
388
00:26:23 --> 00:26:30
Once you decide on a basis --
this is where coordinates come
389
00:26:30 --> 00:26:30
from.
390
00:26:30 --> 00:26:35
You decide on a basis,
then every vector,
391
00:26:35 --> 00:26:40.5
these are the coordinates in
that basis.
392
00:26:40.5 --> 00:26:47
There is one and only one way
to express v as a combination of
393
00:26:47 --> 00:26:53
the basis vectors,
and the numbers you need in
394
00:26:53 --> 00:26:57
that combination are the
coordinates.
395
00:26:57 --> 00:27:00
Let me write that down.
396
00:27:00 --> 00:27:03
So what are coordinates?
397
00:27:03 --> 00:27:06
Coordinates come from a basis.
398
00:27:06 --> 00:27:11
Coordinates come from a basis.
399
00:27:11 --> 00:27:18
The coordinates of v,
the coordinates of v are these
400
00:27:18 --> 00:27:25
numbers that tell you how much
of each basis vector is in v.
401
00:27:25 --> 00:27:31
If I change the basis,
I change the coordinates,
402
00:27:31 --> 00:27:32
right?
403
00:27:32 --> 00:27:39
Now, we have always been
assuming that were working with
404
00:27:39 --> 00:27:43
a standard basis,
right?
405
00:27:43 --> 00:27:46
The basis we don't even think
about this stuff,
406
00:27:46 --> 00:27:51
because if I give you the
vector v equals three two four,
407
00:27:51 --> 00:27:55
you have been assuming
completely -- and probably
408
00:27:55 --> 00:27:59
rightly -- that I had in mind
the standard basis,
409
00:27:59 --> 00:28:03
that this vector was three
times the first coordinate
410
00:28:03 --> 00:28:07
vector, and two times the
second, and four times the
411
00:28:07 --> 00:28:08
third.
412
00:28:08 --> 00:28:17
But you're not entitled -- I
might have had some other basis
413
00:28:17 --> 00:28:19
in mind.
414
00:28:19 --> 00:28:23.83
This is like the standard
basis.
415
00:28:23.83 --> 00:28:32
And then the coordinates are
sitting right there in the
416
00:28:32 --> 00:28:34
vector.
417
00:28:34 --> 00:28:37
But I could have chosen a
different basis,
418
00:28:37 --> 00:28:41
like I might have had
eigenvectors of a matrix,
419
00:28:41 --> 00:28:45.42
and I might have said,
OK, that's a great basis,
420
00:28:45.42 --> 00:28:50
I'll use the eigenvectors of
this matrix as my basis vectors.
421
00:28:50 --> 00:28:56
Which are not necessarily these
three, but some other basis.
422
00:28:56 --> 00:29:00
So that was an example,
this is the real thing,
423
00:29:00 --> 00:29:05
the coordinates are these
numbers, I'll circle them again,
424
00:29:05 --> 00:29:07
the amounts of each basis.
425
00:29:07 --> 00:29:08
OK.
426
00:29:08 --> 00:29:13
So, if I want to create a
matrix that describes a linear
427
00:29:13 --> 00:29:17.91
transformation,
now I'm ready to do that.
428
00:29:17.91 --> 00:29:19
OK, OK.
429
00:29:19 --> 00:29:28
So now what I plan to do is
construct the matrix A that
430
00:29:28 --> 00:29:37
represents, or tells me about,
a linear transformation,
431
00:29:37 --> 00:29:42
linear transformation T.
432
00:29:42 --> 00:29:42
OK.
433
00:29:42 --> 00:29:51
So I really start with the
transformation --
434
00:29:51 --> 00:29:57.97
whether it's a projection or a
rotation, or some strange
435
00:29:57.97 --> 00:30:04
movement of this house in the
plane, or some transformation
436
00:30:04 --> 00:30:10.85
from n-dimensional space to --
or m-dimensional space to
437
00:30:10.85 --> 00:30:15
n-dimensional space.
n to m, I guess.
438
00:30:15 --> 00:30:19
Usually, we'll have T,
we'll somehow transform
439
00:30:19 --> 00:30:22
n-dimensional space to
m-dimensional space,
440
00:30:22 --> 00:30:26
and the whole point is that if
I have a basis for n-dimensional
441
00:30:26 --> 00:30:29.48
space -- I guess I need two
bases, really.
442
00:30:29.48 --> 00:30:32
I need an input basis to
describe the inputs,
443
00:30:32 --> 00:30:36
and I need an output basis to
give me coordinates -- to give
444
00:30:36 --> 00:30:40
me some numbers for the output.
445
00:30:40 --> 00:30:43
So I've got to choose two
bases.
446
00:30:43 --> 00:30:50
Choose a basis v1 up to vn for
the inputs, for the inputs in --
447
00:30:50 --> 00:30:53.17
they came from R^n.
448
00:30:53.17 --> 00:31:00
So the transformation is taking
every n-dimensional vector into
449
00:31:00 --> 00:31:03
some m-dimensional vector.
450
00:31:03 --> 00:31:10
And I have to choose a basis,
and I'll call them w1 up to wn,
451
00:31:10 --> 00:31:13
for the outputs.
452
00:31:13 --> 00:31:16
Those are guys in R^m.
453
00:31:16 --> 00:31:23
Once I've chosen the basis,
that settles the matrix -- I
454
00:31:23 --> 00:31:27
now working with coordinates.
455
00:31:27 --> 00:31:33
Every vector in R^n,
every input vector has some
456
00:31:33 --> 00:31:34
coordinates.
457
00:31:34 --> 00:31:39
So here's what I do,
here's what I do.
458
00:31:39 --> 00:31:43
Can I say it in words?
459
00:31:43 --> 00:31:45
I take a vector v.
460
00:31:45 --> 00:31:48
I express it in its basis,
in the basis,
461
00:31:48 --> 00:31:50
so I get its coordinates.
462
00:31:50 --> 00:31:55
Then I'm going to multiply
those coordinates by the right
463
00:31:55 --> 00:32:00
matrix A, and that will give me
the coordinates of the output in
464
00:32:00 --> 00:32:02
the output basis.
465
00:32:02 --> 00:32:07
I'd better write that down,
that was a mouthful.
466
00:32:07 --> 00:32:15
What I want -- I want a matrix
A that does what the linear
467
00:32:15 --> 00:32:18
transformation does.
468
00:32:18 --> 00:32:25
And it does it with respecting
these bases.
469
00:32:25 --> 00:32:33
So I want the matrix to be --
well, let's suppose -- look,
470
00:32:33 --> 00:32:38.5
let me take an example.
471
00:32:38.5 --> 00:32:41
Let me take the projection
example.
472
00:32:41 --> 00:32:44
The projection example.
473
00:32:44 --> 00:32:49
Suppose I take -- because we've
got that -- we've got that
474
00:32:49 --> 00:32:53
projection in mind -- I can fit
in here.
475
00:32:53 --> 00:32:56
Here's the projection example.
476
00:32:56 --> 00:33:02
So the projection example,
I'm thinking of n and m as two.
477
00:33:02 --> 00:33:08
The transformation takes the
plane, takes every vector in the
478
00:33:08 --> 00:33:13
plane, and, let me draw the
plane, just so we remember it's
479
00:33:13 --> 00:33:18
a plane -- and there's the thing
that I'm projecting onto,
480
00:33:18 --> 00:33:23
that's the line I'm projecting
onto -- so the transformation
481
00:33:23 --> 00:33:30.16
takes every vector in the plane
and projects it onto that line.
482
00:33:30.16 --> 00:33:34
So this is projection,
so I'm going to do projection.
483
00:33:34 --> 00:33:34
OK.
484
00:33:34 --> 00:33:39
But, I'm going to choose a
basis that I like better than
485
00:33:39 --> 00:33:41.22
the standard basis.
486
00:33:41.22 --> 00:33:45
My basis -- in fact,
I'll choose the same basis for
487
00:33:45 --> 00:33:49
inputs and for outputs,
and the basis will be -- my
488
00:33:49 --> 00:33:53
first basis vector will be right
on the line.
489
00:33:53 --> 00:33:57
There's my first basis vector.
490
00:33:57 --> 00:33:59
Say, a unit vector,
on the line.
491
00:33:59 --> 00:34:03
And my second basis vector will
be a unit vector perpendicular
492
00:34:03 --> 00:34:04
to that line.
493
00:34:04 --> 00:34:08
And I'm going to choose that as
the output basis,
494
00:34:08 --> 00:34:08
also.
495
00:34:08 --> 00:34:11
And I'm going to ask you,
what's the matrix?
496
00:34:11 --> 00:34:13
What's the matrix?
497
00:34:13 --> 00:34:16
How do I describe this
transformation of projection
498
00:34:16 --> 00:34:19
with respect to this basis?
499
00:34:19 --> 00:34:20
OK?
500
00:34:20 --> 00:34:22
So what's the rule?
501
00:34:22 --> 00:34:28.53
I take any vector v,
it's some combination of the
502
00:34:28.53 --> 00:34:34
first basis ve- vector,
and the second basis vector.
503
00:34:34 --> 00:34:37
Now, what is T of v?
504
00:34:37 --> 00:34:43
Suppose the input is -- well,
suppose the input is v1.
505
00:34:43 --> 00:34:48
What's the output?
v1, right?
506
00:34:48 --> 00:34:51
The projection leaves this one
alone.
507
00:34:51 --> 00:34:57
So we know what the projection
does to this first basis vector,
508
00:34:57 --> 00:34:59
this guy, it leaves it.
509
00:34:59 --> 00:35:04
What does the projection do to
the second basis vector?
510
00:35:04 --> 00:35:07
It kills it,
sends it to zero.
511
00:35:07 --> 00:35:11
So what does the projection do
to a combination?
512
00:35:11 --> 00:35:14
It kills this part,
and this part,
513
00:35:14 --> 00:35:17
it leaves alone.
514
00:35:17 --> 00:35:23
Now, all I want to do is find
the matrix.
515
00:35:23 --> 00:35:30
I now want to find the matrix
that takes an input,
516
00:35:30 --> 00:35:37
c1 c2, the coordinates,
and gives me the output,
517
00:35:37 --> 00:35:39
c1 0.
518
00:35:39 --> 00:35:44
You see that in this basis,
the coordinates of the input
519
00:35:44 --> 00:35:49.33
were c1, c2, and the coordinates
of the output are c1,
520
00:35:49.33 --> 0.
521
0. --> 00:35:49
522
00:35:49 --> 00:35:53
And of course,
not hard to find a matrix that
523
00:35:53 --> 00:35:54
will do that.
524
00:35:54 --> 00:35:58
The matrix that will do that is
the matrix one,
525
00:35:58 --> 00:36:01
zero, zero, zero.
526
00:36:01 --> 00:36:09
Because if I multiply input by
that matrix A -- this is A times
527
00:36:09 --> 00:36:16
input coordinates -- and I'm
hoping to get the output
528
00:36:16 --> 00:36:18
coordinates.
529
00:36:18 --> 00:36:23
And what do I get from that
multiplication?
530
00:36:23 --> 00:36:28.37
I get the right answer,
c1 and zero.
531
00:36:28.37 --> 00:36:32
So what's the point?
532
00:36:32 --> 00:36:35
So the first point is,
there's a matrix that does the
533
00:36:35 --> 00:36:36
job.
534
00:36:36 --> 00:36:39
If there's a linear
transformation out there,
535
00:36:39 --> 00:36:41
coordinate-free,
no coordinates,
536
00:36:41 --> 00:36:45
and then I choose a basis for
the inputs, and I choose a basis
537
00:36:45 --> 00:36:48.44
for the outputs,
then there's a matrix that does
538
00:36:48.44 --> 00:36:48
the job.
539
00:36:48 --> 00:36:50
And what's the job?
540
00:36:50 --> 00:36:53
It multiplies the input
coordinates and produces the
541
00:36:53 --> 00:36:56
output coordinates.
542
00:36:56 --> 00:37:01
Now, in this example -- let me
repeat, I chose the input basis
543
00:37:01 --> 00:37:04
was the same as the output
basis.
544
00:37:04 --> 00:37:08
The input basis and output
basis were both along the line,
545
00:37:08 --> 00:37:11
and perpendicular to the line.
546
00:37:11 --> 00:37:16
They're actually the
eigenvectors of the projection.
547
00:37:16 --> 00:37:22
And, as a result,
the matrix came out diagonal.
548
00:37:22 --> 00:37:27
In fact, it came out to be
lambda.
549
00:37:27 --> 00:37:31
This is like,
the good basis.
550
00:37:31 --> 00:37:38
So the good -- the eigenvector
basis is the good basis,
551
00:37:38 --> 00:37:45
it leads to the matrix --
the diagonal matrix of
552
00:37:45 --> 00:37:50
eigenvalues lambda,
and just as in this example,
553
00:37:50 --> 00:37:56
the eigenvectors and
eigenvalues of this linear
554
00:37:56 --> 00:38:01
transformation were along the
line, and perpendicular.
555
00:38:01 --> 00:38:08
The eigenvalues were one and
zero, and that's the matrix that
556
00:38:08 --> 00:38:09
we got.
557
00:38:09 --> 00:38:10
OK.
558
00:38:10 --> 00:38:13
So that's a,
like, the great choice of
559
00:38:13 --> 00:38:17
matrix, that's the choice a
physicist would do when he had
560
00:38:17 --> 00:38:21
to finally -- he or she had to
finally bring coordinates in
561
00:38:21 --> 00:38:25
unwillingly, the coordinates to
be chosen, the good coordinates
562
00:38:25 --> 00:38:28
are the eigenvectors,
because, if I did this
563
00:38:28 --> 00:38:32
projection in the standard basis
-- which I could do,
564
00:38:32 --> 00:38:33
right?
565
00:38:33 --> 00:38:40.46
I could do the whole thing in
the standard basis -- I better
566
00:38:40.46 --> 00:38:43
try, if I can do that.
567
00:38:43 --> 00:38:50
What are we calling -- so I'll
have to tell you now which line
568
00:38:50 --> 00:38:53
we're projecting on.
569
00:38:53 --> 00:38:56
Say, the 45 degree line.
570
00:38:56 --> 00:39:03
So say we're projecting onto 45
degree line, and we use not the
571
00:39:03 --> 00:39:08
eigenvector basis,
but the standard basis.
572
00:39:08 --> 00:39:12
The standard basis,
v1, is one, zero,
573
00:39:12 --> 00:39:15
and v2 is zero,
one.
574
00:39:15 --> 00:39:22
And again, I'll use the same
basis for the outputs.
575
00:39:22 --> 00:39:26
Then I have to do this -- I can
find a matrix,
576
00:39:26 --> 00:39:30
it will be the matrix that we
would always think of,
577
00:39:30 --> 00:39:33
it would be the projection
matrix.
578
00:39:33 --> 00:39:37
It will be, actually,
it's the matrix that we learned
579
00:39:37 --> 00:39:41
about in chapter four,
it's what I call the matrix --
580
00:39:41 --> 00:39:45.52
do you remember,
P was A, A transpose over A
581
00:39:45.52 --> 00:39:47
transpose A?
582
00:39:47 --> 00:39:51
And I think,
in this example,
583
00:39:51 --> 00:39:55
it will come out,
one-half, one-half,
584
00:39:55 --> 00:39:57.76
one-half, one-half.
585
00:39:57.76 --> 00:40:04
I believe that's the matrix
that comes from our formula.
586
00:40:04 --> 00:40:10
And that's the matrix that will
do the job.
587
00:40:10 --> 00:40:17
If I give you this input,
one, zero, what's the output?
588
00:40:17 --> 00:40:21
The output is one-half,
one-half.
589
00:40:21 --> 00:40:26
And that should be the right
projection.
590
00:40:26 --> 00:40:33.33
And if I give you the input
zero, one, the output is,
591
00:40:33.33 --> 00:40:40
again, one-half,
one-half, again the projection.
592
00:40:40 --> 00:40:44
So that's the matrix,
but not diagonal of course,
593
00:40:44 --> 00:40:49
because we didn't choose a
great basis, we just chose the
594
00:40:49 --> 00:40:50
handiest basis.
595
00:40:50 --> 00:40:54
Well, so the course has
practically been about the
596
00:40:54 --> 00:40:57
handiest basis,
and just dealing with the
597
00:40:57 --> 00:41:00
matrix that we got.
598
00:41:00 --> 00:41:03
And it's not that bad a matrix,
it's symmetric,
599
00:41:03 --> 00:41:08
and it has this P squared equal
P property, all those things are
600
00:41:08 --> 00:41:09
good.
601
00:41:09 --> 00:41:13
But in the best basis,
it's easy to see that P squared
602
00:41:13 --> 00:41:17
equals P, and it's symmetric,
and it's diagonal.
603
00:41:17 --> 00:41:21
So that's the idea then,
is, do you see now how I'm
604
00:41:21 --> 00:41:25
associating a matrix to the
transformation?
605
00:41:25 --> 00:41:31
I'd better write the rule down,
I'd better write the rule down.
606
00:41:31 --> 00:41:34
The rule to find the matrix A.
607
00:41:34 --> 00:41:36
All right, first column.
608
00:41:36 --> 00:41:40
So, a rule to find A,
we're given the bases.
609
00:41:40 --> 00:41:45
Of course, we don't -- because
there's no way we could
610
00:41:45 --> 00:41:51
construct the matrix until we're
told what the bases are.
611
00:41:51 --> 00:41:58.34
So we're given the input basis,
and the output basis,
612
00:41:58.34 --> 00:42:00
v1 to vn, w1 to wm.
613
00:42:00 --> 00:42:02
Those are given.
614
00:42:02 --> 00:42:10
Now, in the first column of A,
how do I find that column?
615
00:42:10 --> 00:42:15
The first column of the matrix.
616
00:42:15 --> 00:42:21
So that should tell me what
happens to the first basis
617
00:42:21 --> 00:42:22
vector.
618
00:42:22 --> 00:42:27
So the rule is,
apply the linear transformation
619
00:42:27 --> 00:42:28
to v1.
620
00:42:28 --> 00:42:31
To the first basis vector.
621
00:42:31 --> 00:42:37
And then, I'll write it -- so
that's the output,
622
00:42:37 --> 00:42:38
right?
623
00:42:38 --> 00:42:43.46
The input is v1,
what's the output?
624
00:42:43.46 --> 00:42:48.55
The output is in the output
space, it's some combination of
625
00:42:48.55 --> 00:42:53
these guys, and it's that
combination that goes into the
626
00:42:53 --> 00:42:57
first column -- so,
let me -- I'll put this word --
627
00:42:57 --> 00:43:00
right, I'll say it in words
again.
628
00:43:00 --> 00:43:02.77
How to find this matrix.
629
00:43:02.77 --> 00:43:06
Take the first basis vector.
630
00:43:06 --> 00:43:11
Apply the transformation,
then it's in the output space,
631
00:43:11 --> 00:43:16
T of v1, so it's some
combination of these outputs,
632
00:43:16 --> 00:43:17
this output basis.
633
00:43:17 --> 00:43:21
So that combination,
the coefficients in that
634
00:43:21 --> 00:43:26
combination will be the first
column -- so a1,
635
00:43:26 --> 00:43:28
a row 2, column 1,
w2, am1, wm.
636
00:43:28 --> 00:43:35
There are the numbers in the
first column of the matrix.
637
00:43:35 --> 00:43:41
Let me make the point by doing
the second column.
638
00:43:41 --> 00:43:44
Second column of A.
639
00:43:44 --> 00:43:47
What's the idea,
now?
640
00:43:47 --> 00:43:55
I take the second basis vector,
I apply the transformation to
641
00:43:55 --> 00:44:03
it, that's in -- now I get an
output, so it's some combination
642
00:44:03 --> 00:44:11
in the output basis --
and that combination is the
643
00:44:11 --> 00:44:17
bunch of numbers that should go
in the second column of the
644
00:44:17 --> 00:44:18
matrix.
645
00:44:18 --> 00:44:18
OK.
646
00:44:18 --> 00:44:20
And so forth.
647
00:44:20 --> 00:44:25
So I get a matrix,
and the matrix I get does the
648
00:44:25 --> 00:44:26.43
right job.
649
00:44:26.43 --> 00:44:32
Now, the matrix constructed
that way, and following the
650
00:44:32 --> 00:44:37
rules of matrix multiplication.
651
00:44:37 --> 00:44:43
The result will be that if I
give you the input coordinates,
652
00:44:43 --> 00:44:49
and I multiply by the matrix,
so the outcome of all this is A
653
00:44:49 --> 00:44:56
times the input coordinates
correctly reproduces the output
654
00:44:56 --> 00:44:57
coordinates.
655
00:44:57 --> 00:44:59
Why is this right?
656
00:44:59 --> 00:45:02
Let me just check the first
column.
657
00:45:02 --> 00:45:09
Suppose the input coordinates
are one and all zeros.
658
00:45:09 --> 00:45:11
What does that mean?
659
00:45:11 --> 00:45:12
What's the input?
660
00:45:12 --> 00:45:16
If the input coordinates are
one and other -- and the rest
661
00:45:16 --> 00:45:19
zeros, then the input is v1,
right?
662
00:45:19 --> 00:45:23
That's the vector that has
coordinates one and all zeros.
663
00:45:23 --> 00:45:24
OK?
664
00:45:24 --> 00:45:27
When I multiply A by the one
and all zeros,
665
00:45:27 --> 00:45:32
I'll get the first column of A,
I'll get these numbers.
666
00:45:32 --> 00:45:35
And, sure enough,
those are the output
667
00:45:35 --> 00:45:37.19
coordinates for T of v1.
668
00:45:37.19 --> 00:45:40
So we made it right on the
first column,
669
00:45:40 --> 00:45:44
we made it right on the second
column, we made it right on all
670
00:45:44 --> 00:45:48
the basis vectors,
and then it has to be right on
671
00:45:48 --> 00:45:49.42
every vector.
672
00:45:49.42 --> 00:45:49
OK.
673
00:45:49 --> 00:45:53
So there is a picture of the
matrix for a linear
674
00:45:53 --> 00:45:55
transformation.
675
00:45:55 --> 00:46:01.97
Finally, let me give you
another -- a different linear
676
00:46:01.97 --> 00:46:03
transformation.
677
00:46:03 --> 00:46:10
The linear transformation that
takes the derivative.
678
00:46:10 --> 00:46:13
That's a linear transformation.
679
00:46:13 --> 00:46:21
Suppose the input space is all
combination c1 plus c2x plus c3
680
00:46:21 --> 00:46:23
x squared.
681
00:46:23 --> 00:46:27
So the basis is these simple
functions.
682
00:46:27 --> 00:46:30
Then what's the output?
683
00:46:30 --> 00:46:32
Is the derivative.
684
00:46:32 --> 00:46:38
The output is the derivative,
so the output is c2+2c3 x.
685
00:46:38 --> 00:46:44
And let's take as output basis,
the vectors one and x.
686
00:46:44 --> 00:46:49
So we're going from a
three-dimensional space of
687
00:46:49 --> 00:46:54
inputs to a two-dimensional
space of outputs by the
688
00:46:54 --> 00:46:57
derivative.
689
00:46:57 --> 00:47:07.56
And I don't know if you ever
thought that the derivative is
690
00:47:07.56 --> 00:47:08
linear.
691
00:47:08 --> 00:47:18
But if it weren't linear,
taking derivatives would take
692
00:47:18 --> 00:47:21
forever, right?
693
00:47:21 --> 00:47:25
We are able to compute
derivatives of functions exactly
694
00:47:25 --> 00:47:27
because we know it's a linear
transformation,
695
00:47:27 --> 00:47:31
so that if we learn the
derivatives of a few functions,
696
00:47:31 --> 00:47:34
like sine x and cos x and e to
the x, and another little short
697
00:47:34 --> 00:47:38
list, then we can take all their
combinations and we can do all
698
00:47:38 --> 00:47:40
the derivatives.
699
00:47:40 --> 00:47:42
OK, now what's the matrix?
700
00:47:42 --> 00:47:44
What's the matrix?
701
00:47:44 --> 00:47:49
So I want the matrix to
multiply these input vectors --
702
00:47:49 --> 00:47:53
input coordinates,
and give these output
703
00:47:53 --> 00:47:54
coordinates.
704
00:47:54 --> 00:47:58
So I just think,
OK, what's the matrix that does
705
00:47:58 --> 00:47:59
it?
706
00:47:59 --> 00:48:02.5
I can follow my rule of
construction,
707
00:48:02.5 --> 00:48:06
or I can see what the matrix
is.
708
00:48:06 --> 00:48:10
It should be a two by three
matrix, right?
709
00:48:10 --> 00:48:13
And the matrix -- so I'm just
figuring out,
710
00:48:13 --> 00:48:15
what do I want?
711
00:48:15 --> 00:48:17.75
No, I'll -- let me write it
here.
712
00:48:17.75 --> 00:48:20
What do I want from my matrix?
713
00:48:20 --> 00:48:22
What should that matrix do?
714
00:48:22 --> 00:48:26.14
Well, I want to get c2 in the
first output,
715
00:48:26.14 --> 00:48:29
so zero, one,
zero will do it.
716
00:48:29 --> 00:48:32
I want to get two c3,
so zero, zero,
717
00:48:32 --> 00:48:33.21
two will do it.
718
00:48:33.21 --> 00:48:37
That's the matrix for this
linear transformation with those
719
00:48:37 --> 00:48:39
bases and those coordinates.
720
00:48:39 --> 00:48:43
You see, it just clicks,
and the whole point is that the
721
00:48:43 --> 00:48:47
inverse matrix gives the inverse
to the linear transformation,
722
00:48:47 --> 00:48:51
that the product of two
matrices gives the right matrix
723
00:48:51 --> 00:48:54
for the product of two
transformations --
724
00:48:54 --> 00:49:03
matrix multiplication really
came from linear
725
00:49:03 --> 00:49:06
transformations.
726
00:49:06 --> 00:49:17
I'd better pick up on that
theme Monday after Thanksgiving.
727
00:49:17 --> 00:49:24
And I hope you have a great
holiday.
728
00:49:24 --> 00:49:30
I hope Indian summer keeps
going.
729
00:49:30 --> 00:49:33
OK, see you on Monday. | 677.169 | 1 |
Linear Algebra and Its Applications, 4th Edition
9780321385178
ISBN:
0321385179
Edition: 4th Pub Date: 2011 Publisher: Pearson
Summary: Linear algebra is relatively easy for students during the early stages of the course, when the material is presented in a familiar, concrete setting. But when abstract concepts are introduced, students often hit a brick wall. Instructors seem to agree that certain concepts (such as linear independence, spanning, subspace, vector space, and linear transformations), are not easily understood, and require time to assimi...late. Since they are fundamental to the study of linear algebra, students'understanding of these concepts is vital to their mastery of the subject. David Lay introduces these concepts early in a familiar, concrete Rn setting, develops them gradually, and returns to them again and again throughout the text so that when discussed in the abstract, these concepts are more accessible.
David C. Lay is the author of Linear Algebra and Its Applications, 4th Edition, published 2011 under ISBN 9780321385178 and 0321385179. Four hundred forty two Linear Algebra and Its Applications, 4th Edition textbooks are available for sale on ValoreBooks.com, one hundred thirty nine used from the cheapest price of $37.95, or buy new starting at $47.50.[read more]
Ships From:Multiple LocationsShipping:Standard, ExpeditedComments:RENTAL: Supplemental materials are not guaranteed (access codes, DVDs, workbooks). With CD! within U.S. will arrive in 3-5 days. Hassle free 14 day return policy. Contact Customer Service for questions.[less]
The Class was Introduction to Linear Algebra. It was about learning the basics of linear algebra.
The book was very helpful for studying for the tests and trying to get a better understanding of what was being taught in class. The step by step processes outlined in the book definitely helped to make the material easier | 677.169 | 1 |
Product Description
Help students study exactly what they need to know in order to meet Common Core State Standards! This laminated study chart is intended for 8th grade students and covers the number system, exponents and powers, linear equations, congruent & similar figures, angles, functions, and more. Definitions, examples, graphs, sample problems, and tips help students quickly grasp the basic concepts of math. Three laminated pages (double-sided), three-hole-punched. 8.5" x 11". | 677.169 | 1 |
9780292755314 / 0292755317
Shipping prices may be approximate. Please verify cost before checkout.
About the book:
There is no question that native cultures in the New World exhibit many forms of mathematical development. This Native American mathematics can best be described by considering the nature of the concepts found in a variety of individual New World cultures. Unlike modern mathematics in which numbers and concepts are expressed in a universal mathematical notation, the numbers and concepts found in native cultures occur and are expressed in many distinctive ways. Native American Mathematics, edited by Michael P. Closs, is the first book to focus on mathematical development indigenous to the New World.
Spanning time from the prehistoric to the present, the thirteen essays in this volume attest to the variety of mathematical development present in the Americas. The data are drawn from cultures as diverse as the Ojibway, the Inuit (Eskimo), and the Nootka in the north; the Chumash of Southern California; the Aztec and the Maya in Mesoamerica; and the Inca and Jibaro of South America. Among the strengths of this collection are this diversity and the multidisciplinary approaches employed to extract different kinds of information. The distinguished contributors include mathematicians, linguists, psychologists, anthropologists, and archaeologists292755317 Publisher: Univ of Texas Pr, 1986 Usually ships in 1-2 business days
Hardcover, ISBN 0292755317 Publisher: Univ of Texas Pr, 1986 02927553172755317 Publisher: Univ of Texas Pr, 1986 Usually dispatched within 1-2 business days, NEW Book, unused. Sent Airmail from New York. Please allow 7-15 Business days for delivery. Excellent Customer Service.
Hardcover, ISBN 0292755317 Publisher: Univ of Texas Pr University of Texas Press, 1986 Used - Good. Former Library book. Shows some signs of wear, and may have some markings on the inside. Shipped to over one million happy customers. Your purchase benefits world literacy!
Hardcover, ISBN 0292755317 Publisher: Univ of Texas Pr Used - Very Good, Usually ships in 1-2 business days, In near perfect condition. Slightly used former library book. May have markings on outside of book.
Hardcover, ISBN 0292755317 Publisher: Univ of Texas Pr, 1986 Used - Acceptable, Usually ships in 1-2 business days, No guarantee on products that contain supplements and some products may include highlighting and writing. | 677.169 | 1 |
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Deep Dive into Mathematica's Numerics: Applications and Tips
Andrew Moylan
In this course from the Wolfram Mathematica Virtual Conference 2011, you'll learn how to best use Mathematica's numerics functions in advanced settings. Topics include techniques and best practices for using multiple numerics functions together, advanced numeric features, and understanding precision and accuracy | 677.169 | 1 |
This advice I give to my students about solving problems in any science like physics and chemistry.
Avoid the frantic plug-n-chug syndrome- usually a mindless act resorting to Pavlovian responses. Learn to think. Do
not be intimidated by the sound of the problem or that you may not know instantly the solution.
(1) Read the problem in a relaxed manner (don't even have the pen in your hand). This is to get a general flavor, perhaps
to identify the class of problem at hand.
(2) Re-read, clause-by-clause, jot down information as you read and start drawing a sketch of the situation. Draw it
sufficiently large and carefully so that it does not introduce additional question(s).
(3) Identify the precise question(s).
(4) Write down the general concepts, unifying relations, and equations you think may apply.
(5) The approach to the problem and consequently the solution will materialize from this logical progression.
(6) Record the logical steps as you effect the solution. This will be valuable not only in establishing a logical progression
of thought, but also in checking and troubleshooting the answer, if an error is suspected.
(7) Simplify the equations before plugging-in numbers. That is, do the algebra first; re-arrange the equations, eliminate
variables, etc. so that the desired quantity stands alone.
(8) Insert values and their units into the appropriate equation(s). Any necessary unit conversions will be transparent.
(9) Execute the calculation. Do not round off intermediate results.
(10) Check the answer for reality. Does it make sense?
(11) Report the final answer with the proper units and significant figures in a box or underline it. Be sure you answered
all the questions; there may be more than one.
As experience breeds confidence, many of the steps can be consolidated or performed simultaneously. | 677.169 | 1 |
Mathematics
Interested in Science? Mathematics is of the essence.
The Department of Mathematics offers rigorous and insightful instruction. Our courses progress from foundational topics to advanced theories and techniques. Faculty members are active in research and will invite, stimulate and support your curiosity and understanding. | 677.169 | 1 |
Created by David Liao, this site offers a way for scientists, educators and others to investigate biological systems using a physical sciences perspective. On the site, visitors will find video tutorials, classroom fact...
This website considers the placement and advising program for undergraduate mathematics students currently in place at the University of Arizona. Donna Krawczyk and Elias Toubassi describe the methods they use to...
An article that discusses how, in effect, a virus is caught up in a version of the prisoners' dilemma, a widely studied game in which acting for individual advantage clashes with acting for the collective benefit. Game...
This web page provides a preliminary look at the pedagogy behind a vision to improve the teaching of mathematics and to provide math relevant to students studying emerging technologies. Recommendations stress critical...
This site from Northern Illinois University provides online notes for students using the Abstract Algebra textbook (which is also available online). The materials cover the topics of integers, functions, groups,... | 677.169 | 1 |
...
Show More data and encouraging modeling and problem-solving. Algebra and Problem Solving. Functions, Linear Functions, and Inequalities. Systems of Linear Equations and Inequalities. Polynomials, Polynomial Functions, and Factoring. Rational Expressions, Functions, and Equations. Radicals, Radical Functions, and Rational Exponents. Quadratic Equations and Functions. Exponential and Logarithmic Functions. Conic Sections and Nonlinear Systems of Equations. Polynomial and Rational Functions. Sequences, Probability, and Mathematical Induction. For anyone interested in introductory and intermediate algebra and for the combined introductory and intermediate algebra | 677.169 | 1 |
Description: The author gives an introduction to basic features of the PostScript language and shows how to use it for producing mathematical graphics. The book includes the discussion of the mathematics involved in computer graphics, and some comments on good style in mathematical illustration. | 677.169 | 1 |
A Web tutorial on the fundamentals of trigonometry. The lessons in PDF format include an introduction to angles and their measurement, definitions of the trigonometric functions, graphs of trigonometric waves, an introduction to
trigonometric identities, and techniques for solving trigonometric equations. This tutorial is part of a collection of sites that includes tutorials on Algebra and Introductory Calculus developed by Dr. D. P. Story. | 677.169 | 1 |
Abstract Algebra A Geometric Approach
9780133198317
ISBN:
0133198316
Pub Date: 1995 Publisher: Prentice Hall
Summary: Appropriate for a 1 or 2 term course in Abstract Algebra at the Junior level. This book explores the essential theories and techniques of modern algebra, including its problem-solving skills, basic proof techniques, many unusual applications, and the interplay between algebra and geometry. It takes a concrete, example-oriented approach to the subject matter.
Shifrin, Theodore is the author of Abstract Algebr...a A Geometric Approach, published 1995 under ISBN 9780133198317 and 0133198316. Two hundred seventeen Abstract Algebra A Geometric Approach textbooks are available for sale on ValoreBooks.com, fifty nine used from the cheapest price of $43.90, or buy new starting at $101.19 | 677.169 | 1 |
The book provides a unified presentation of new methods, algorithms, and select applications that are the foundations of multidimensional image construction and reconstruction. The self-contained survey chapters present cutting-edge research and results in the field. more...
The new edition of this classic book describes and provides a myriad of examples of the relationships between problem posing and problem solving, and explores the educational potential of integrating these two activities in classrooms at all levels. The Art of Problem Posing, Third Edition encourages readers to shift their thinking about problem... more...
Updated and expanded, this second edition satisfies the same philosophical objective as the first -- to show the importance of problem posing. Although interest in mathematical problem solving increased during the past decade, problem posing remained relatively ignored. The Art of Problem Posing draws attention to this equally important act and is... more...
This heavily-illustrated collection of research articles is the first comprehensive book on concept mapping in mathematics. It flows from a historical development overview into numerous applications and contains case studies and field-tested resources.Convex optimization has an increasing impact on many areas of mathematics, applied sciences, and practical applications. It is now being taught at many universities and being used by researchers of different fields. As convex analysis is the mathematical f more...
"? the games also provide an extremely well-suited platform for the introduction of a unified method for determining complexity using constraint logic ? considers not only mathematically oriented games, but also games that may well be suitable for non-mathematicians ? The book also contains a comprehensive overview of known results on the complexity... more... | 677.169 | 1 |
CGP Study Guide explains everything students need to know for Key Stage Three Maths - all fully up-to-date for the new curriculum from September 2014 onwards. It's ideal for students working at a higher level (it covers what would have been called Levels 5-8 in the pre-2014 curriculum). Every topic is explained with clear, friendly notes and worked examples, and there's a range of practice questions to test the crucial skills. We've also included a digital Online Edition of the whole book to read on a PC, Mac or tablet - just use the unique code printed at the front of the book to access it. For extra practice, a matching KS3 Maths Workbook is also available (9781841460383).
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My son was having problems in maths due to lessons missed after illness. I had very little knowledge of modern secondary level maths so this book was perfect to help us both to tackle some tricky new topics.
I purchased this book roughly 6-7 weeks before the actual mathematics exam, thinking that it was too late to revise the majority of the year 9 curriculm. But I found it simple, fun and easy to understand because of its unique teaching methods. For instance it had useful pictures, strange jokes and questions at the end of each chapter to test your knowledge. All of which proved to be extremely helpful, due to the amount of time I had to revise.
my 13 year old daughter was falling behind in her maths so i approached her maths teacher who recommended this book. we bought it for her and 4 weeks later she scored a 7C and secured her place in the top maths set for year 8. well worth every penny. i highly recommend this book.
A really good book that is worth reading it helps with your maths and covers all the catagories studied through Key Stage Three. With funny jokes and pictures it really makes Maths come to life, a brilliant read!!!
CGP books are absolutely fantastic for learning KS3 maths; I thoroughly recommend them for your first choice of guidebooks. There are 4 chapters, `Numbers Mostly' `Algebra' `Shapes' & `Statistics and Probability'. These are split into sections which explain all of the different parts - in amazing detail. CGP tell you everything you need to know and then test you at the end of the section; about 40 in-depth questions that will test your revision knowledge as far as it will go, and the answers are at the back of the book. You will find this book covers most topics, so this is a great buy.
I have purchased this for my daughter as she struggles with maths a little and after a. Opulent of pages read she understood maths more, this book is written to make you understand maths in a better or easier way as my daughter found out by using this, I would defiantly recommend to someone who struggles with maths. | 677.169 | 1 |
Textbooks
This course is a mathematical content course for prospective elementary and middle school teachers. The course objectives focus on the mathematical content needs of those preparing to teach prekindergarten through grade 8, to include mathematical content which addresses Standards I, II, V, and VI of the TExES Mathematics Generalist EC-6 Standards and the TExES Mathematics 4-8 Standards. These standards are stated below:
Standard I. Number Concepts: The mathematics teacher understands and uses numbers, number systems and their structure, operations and algorithms, quantitative reasoning, and technology appropriate to teach the statewide curriculum (Texas Essential Knowledge and Skills [TEKS]) in order to prepare students to use mathematics.
Standard II. Patterns and Algebra: The mathematics teacher understands and uses patterns, relations, functions, algebraic reasoning, analysis, and technology appropriate to teach the statewide curriculum (Texas Essential Knowledge and Skills [TEKS]) in order to prepare students to use mathematics.
Standard V. Mathematical Processes: The mathematics teacher understands and uses mathematical processes to reason mathematically, to solve mathematical problems, to make mathematical connections within and outside of mathematics, and to communicate mathematically.
Standard VI. Mathematical Perspectives: The mathematics teacher understands the historical development of mathematical ideas, the interrelationship between society and mathematics, the structure of mathematics, and the evolving nature of mathematics and mathematical knowledge.
1A requirement of this course is for you to demonstrate basic arithmetic skills needed to teach elementary/middle school mathematics. To demonstrate these skills you will be given an arithmetic proficiency exam on which you must earn at least an 80% to earn any of the 60 points which make up this component of your grade. A study guide will be provided to help you prepare for this exam. The exam will be given once in-class (Sept. 3) and twice out-of-class. Times for the out-of-class administrations of the arithmetic proficiency exam can be found in the calendar at the end of this syllabus.
The standard percentage grading scale of 90-80-70-60 will be used to assign course grades at the end of the semester
Grading Standards
See course requirements above.
Final Exam
12/6/2010 8 - 10 A.M.
Submission Format Policy
The final exam and all other exams will be taken in classMake-up exams are not generally given; however, such exams may be given (at the instructor's discretion) for an absence that is a result of a documented medical or personal emergency or an approved university activity. Further, timely notification (for emergencies, on or before the scheduled day of the exam and for approved university activities, a week prior to the scheduled day of the exam) is necessary to receive consideration to make up the missed exam. Further, make-up exams must be taken prior to the class period following the missed exam. In addition, no student will be allowed to make up more than one exam.
No late homework or other class work will be accepted unless you have a documented medical emergency. Homework is due at the beginning of class. If you are going to be absent, you may turn your homework in early, at your own risk you can give it to another student in class to turn in for you, you scan it and email it to me prior to the beginning of class, or you can fax it to my attention at 940-397-4442 prior to the beginning of class. Any student who misses class should check WebCT for new assignments. If no new assignment is posted on WebCT, the student should send the instructor an email requesting newly assigned workAny student who accumulates more than 4 absences prior to Oct. 18th (last day to withdraw from the course) is subject to dismissal from class with a grade of "F." Further, any student with more than six absences at the end of the semester is subject to a one letter grade reduction in her/his course average. For example, a student with seven absences and a C average will be assigned a course grade of D. Exceptions to this attendance policy may be made (at the instructor's discretion) for extremely rare extenuating circumstances (such as a lengthy hospitalization).
Other Policies
Homework: Homework will be assigned regularly from the book, as well as occasionally from class handouts.
Calculator Policy: Calculators may not be used on the Arithmetic Proficiency Exam. For most other work calculators will be approved for use. Calculators on cell phones are not approved for use.
Cell Phones and Pagers: Please turn these off (or place on silent mode) during class and do not send or read received text messages during class. Incompliant students will not be allowed to attend class.
Adhering to Scheduled Class Time: Students who need to leave class early should have permission from the instructor or be counted absent for the entire class period. Students should make every effort to arrive to class on time and remain for the entire period. Excessively leaving class early or arriving to class late will result in dismissal from the class with a grade of F.
Student Rights: All students should refer to the MSU Student Handbook for information related to student responsibilities, rights and activities. Topics such as Student Affairs and Student Life, Academic Issues, Financial Issues, University Policies and Procedures, and Code of Student Conduct are included in this handbook.
Evidence of Cheating: Evidence of cheating on any of the assessments from which your course grade will be determined will result in a grade of F on the evaluated work and possibly a grade of F in the course. Further, university policy concerning reporting evidence of cheating to department chairs, college deans, etc. will be followed.
Disability Policy: In accordance with the law, MSU provides academic accommodations to students with documented disabilities. Students with disabilities must be registered with Disability Support Services before classroom accommodations can be provided. The DSS Office is located in Clark Student Center, Room 168, phone 397-4140 | 677.169 | 1 |
Welcome!
Welcome to the Department of Mathematics and Statistics. We offer a wide variety of undergraduate and graduate degree programs designed for students with diverse career or higher educational goals. Our faculty members maintain active research programs in the fields of combinatorics, algebra, analysis, applied mathematics and applied statistics. In a nod toward the unity of mathematics, we offer the following question—whose answer requires several of the above fields, as well as geometry:
A collection of small waves are travelling through shallow water and happen to collide. What happens next?
The first half of the above sentence is governed by the famous KdV equations. (Jerry Bona, UIC, spoke at our colloquium about these waves not long ago.)
The second half of the above sentence is governed by cells in the totally positive part of the Grassmannian and plabic graphs. (Dr. Lauve can tell you more about this aspect of the theory of totally positive matrices.)
See our Faculty Research page for a list of local people to ask for more details, or consult the original sources: | 677.169 | 1 |
eMathInstruction was founded on the simple premise that 21st century technology can and will change the entire landscape of mathematics education. Our eTextbooks, answer keys and printed booklets will help both students and teachers navigate the changing curricula of the modern era. | 677.169 | 1 |
Introduction to Dynamic Programming provides information pertinent to the fundamental aspects of dynamic programming. This book considers problems that can be quantitatively formulated and deals with mathematical models of situations or phenomena that exists in the real world. Organized into 10 chapters, this book begins with an overview of the fundamental... more...
Linear Regression and its Application to Economics presents the economic applications of regression theory. This book discusses the importance of linear regression for multi-dimensional variables. Organized into six chapters, this book begins with an overview of the elementary concepts and the more important definitions and theorems concerning two-dimensional... more...
Mathematical Software deals with software designed for mathematical applications such as Fortran, CADRE, SQUARS, and DESUB. The distribution and sources of mathematical software are discussed, along with number representation and significance monitoring. User-modifiable software and non-standard arithmetic programs are also considered. Comprised of... more...
Presents nearly all the important elementary and analytical methods of statistics, designed for the needs of the geoscientist and completely free from higher mathematics. Translated from the second German edition. more...
Featuring key topics within finance, small business management, and entrepreneurship to develop and maintain prosperous business ventures With a comprehensive and organized approach to fundamental financial theories, tools, and management techniques, Entrepreneurial Finance: Fundamentals of Financial Planning and Management for Small Business... more...
This introductory text is geared toward engineers, physicists, and applied mathematicians at the advanced undergraduate and graduate levels. It applies the mathematics of Cartesian and general tensors to physical field theories and demonstrates them chiefly in terms of the theory of fluid mechanics. Numerous exercises appear throughout the text. 1962... more...
This distinctly nonclassical treatment focuses on developing aspects that differ from the theory of ordinary metric spaces, working directly with probability distribution functions rather than random variables. The two-part treatment begins with an overview that discusses the theory's historical evolution, followed by a development of related mathematical... more...
This book can be used as either a primary text or a supplemental reference for courses in applied mathematics. Its core chapters are devoted to linear algebra, calculus, and ordinary differential equations. Additional topics include partial differential equations and approximation methods. Each chapter features an ample selection of solved problems.... more...
Assuming no further prerequisites than a first undergraduate course in real analysis, this concise introduction covers general elementary theory related to orthogonal polynomials. It includes necessary background material of the type not usually found in the standard mathematics curriculum. Suitable for advanced undergraduate and graduate courses,... more... | 677.169 | 1 |
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