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booklet containing 31 problem sets that involve a variety of math skills, including scientific notation, simple algebra, and calculus. Each set of problems is contained on one page. Learners will use mathematics to explore varied space...(View More) science topics including black holes, ice on Mercury, a mathematical model of the Sun's interior, sunspots, the heliopause, and coronal mass ejections, among many others
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9780471669609186.66
More New and Used from Private Sellers
Starting at $4.31Elementary Linear Algebra, 10th Edition
Summary
This classic treatment of linear algebra presents the fundamentals in the clearest possible way, examining basic ideas by means of computational examples and geometrical interpretation. It proceeds from familiar concepts to the unfamiliar, from the concrete to the abstract. Readers consistently praise this outstanding text for its expository style and clarity of presentation. Clear, accessible, step-by-step explanations make the material crystal clear. The authors spotlight the relationships between concepts to give a unified and complete picture. Established the intricate thread of relationships between systems of equations, matrices, determinants, vectors, linear transformations and eigenvalues.
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03879864 Geometry of Spacetime: An Introduction to Special and General Relativity (Undergraduate Texts in Mathematics)
Hermann Minkowski recast special relativity as essentially a new geometric structure for spacetime. This book looks at the ideas of both Einstein and Minkowski, and then introduces the theory of frames, surfaces and intrinsic geometry, developing the main implications of Einstein's general relativity
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Synopses & Reviews
Please note that used books may not include additional media (study guides, CDs, DVDs, solutions manuals, etc.) as described in the publisher comments.
Publisher Comments:
Everyone studying math needs this book, and it is an essential review guide for examination candidates. Over 500 definitions of all the key terms and concepts. More than 300 useful illustrations and diagrams. Over 100 worked examples. Comprehensive cross-referencing and detailed index. Internet links to recommended websites
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Excursions in Modern Mathematics undergraduate courses in Liberal Arts Mathematics, Quantitative Literacy, and General Education.This very successful liberal arts mathematics textbook is a collection of " excursions" into the real-world applications of modern mathematics. The excursions are organized into four independent parts: 1) The Mathematics of Social Choice, 2) Management Science, 3) Growth and Symmetry, and 4) Statistics. Each part consists of four chapters plus a mini-excursion (new feature in 6/e). The book is written in an informal, very readable style, with pedagogical features that make the material both interesting and clear. The presentation is centered on an assortment of real-world examples and applications specifically chosen to illustrate the usefulness, relevance, and beauty of liberal arts mathematics.
(Note: Each chapter includes a Chapter Opener that includes an outline of the chapter, a Conclusion, a list of Key Terms, , Exercises, a Biographical Note, and References and Further Readings.)
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Live Love AlgorithmsMathematics, data and computing is your passion. Great holiday, birthday or back to school gift for the math student, math teacher, math professor, math tutor, math geek or mathematician.
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A site for answering common algebra questions automatically: users enter their e-mail address and a mathematical expression, and decide whether they wish to expand, factorize, or simplify that expression. The answer is computed
automatically and returned to them within a couple of minutes by e-mail. The service is free.
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I've always enjoyed reading mathematics problem-solving books. So it comes as no surprise that I enjoyed this book as well. Not just because of the collection of problems, but also because of their sheer scope and depth. This is a great collection which is extremely well-organized!
The book is a compilation of advanced problems (from the Putnam exams and various Mathematical Olympiads, etc), arranged by topic. After a brief chapter on the different methods of mathematical proof, the topics covered are Algebra, Real Analysis, Geometry and Trigonometry, Number Theory, and Combinatorics and Probability. These chapters cover just over 300 pages, with well over 1000 problems. The remaining 450 or so pages contain complete solutions to all of the problems.
This extraordinary book can be read for fun. However, it can also serve as a textbook for preparation for the Putnam (or other advanced mathematical competitions), for an advanced problem-solving course, or even as an overview of undergraduate mathematics. Due to the level of the mathematics, it would be difficult to use in a course in which the students had not already been exposed to the topic. I would not advise using the Real Analysis chapter to introduce an undergraduate to that topic. But it could certainly serve as a great review for senior-level students. If you enjoy these sorts of collections of mathematics problems, then this book really is a must-have.
I leave you with an example from the chapter on geometry: Prove that the plane cannot be covered by the interior of finitely many parabolas. Want to see the proof? Get the book!
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History of Mathematics An Introduction
9780073051895
ISBN:
0073051896
Edition: 6 Pub Date: 2005 Publisher: McGraw-Hill College
Summary: David Burton covers the history behind the topics typically covered in an undergraduate maths curriculum or in elementary or high schools. He illuminates the people, stories, and social context behind mathematics' greatest historical advances, while maintaining appropriate focus on the mathematical concepts themselves.
Burton, David M. is the author of History of Mathematics An Introduction, published 2005 u...nder ISBN 9780073051895 and 0073051896. Thirty four History of Mathematics An Introduction textbooks are available for sale on ValoreBooks.com, twenty four used from the cheapest price of $30.60, or buy new starting at $92.22.[read more
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Find a River Grove PrecalculusFinite math is an introductory course in discrete math. A
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Scientists are always trying to get better and more reliable data. One way of getting a more precise measurement might be to switch to an instrument with a more finely divided scale. Figure 4 shows parts of two thermometers placed side by side to record the air temperature in a room UK, two systems of units are in common use. We still use old imperial measures for some things: milk is sold in pints and signposts indicate distances in miles. But for many other everyday measurements metric units have been adopted: we buy petrol in litres and sugar in kilogram bags. A great advantage of metric units is that we no longer have to convert laboriously from imperial units, such as gallons, feet and inches, in order to trade with continental Europe. Also, calculations are complex numbers This unit looks at complex numbers. You will learn how they are defined, examine their geometric representation and then move on to looking at the methods for finding the nth roots of complex numbers and the solutions to simple polynominal equations. First published on Mon, 13 JunComplex numbers You may have met complex numbers before, but not had experience in manipulating them. This unit gives an accessible introduction to complex numbers, which are very important in science and technology, as well as mathematics. The unit includes definitions, concepts and techniques which will be very helpful and interesting to a wide variety of people with a reasonable background in algebra and trigonometry.AuthorExploring data: Graphs and numerical summaries This Unit will introduce you to a number of ways of representing data graphically and of summarising data numerically. You will learn the uses for pie charts, bar charts, histograms and scatterplots. You will also be introduced to various ways of summarising data and methods for assessing location and dispersion.Interpreting data: Boxplots and tables This unit is concerned with two main topics. In Section 1, you will learn about another kind of graphical display, the boxplot. A boxplot is a fairly simple graphic, which displays certain summary statistics of a set of data. Boxplots are particularly useful for assessing quickly the location, dispersion, and symmetry or skewness of a set of data, and for making comparisons of these features in two or more data sets. Boxplots can also be useful for drawing attention to possible outliers in a datSurfaces Surfaces are a special class of topological spaces that crop up in many places in the world of mathematics. In this unit, you will learn to classify surfaces and will be introduced to such concepts as homeomorphism, orientability, the Euler characteristic and the classification theorem of compact surfaces. First published on Thu, 18 Aug 2011 as < Author(s): Creator not set
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and asked whether it is possible to add and multiply these numbers to obtain another real number. We now explain how this can be done using the Least Upper Bound Property of examples just given, it was straightforward to guess the values of sup E and inf E. Sometimes, however, this is not the case. For example, if
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Lin McMullin
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Get ready to master basic arithmetic subjects, principles, and formulas! Master Math: Basic Math and Pre-Algebra is a comprehensive reference guide that explains and clarifies mathematic principles in a simple, easy-to-follow style and format. Beginning with the most basic fundamental topics and progressing through to the more advanced, Master Math:... more...
Get ready to master the concepts and principles of geometry! Master Math: Geometry is a comprehensive reference guide that explains and clarifies the principles of geometry in a simple, easy-to-follow style and format. You'll begin with the language of geometry, deductive reasoning and proofs, and key axioms and postulates. And as you understand... more...... more...
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The reader is introduced to higher mathematics in an experimental way. He works with numerous interactive Java- simulations treating mathematical topics from number theory to infinitesimal calculus and partial differential equations. On the way he playfully learns the EJS simulation technique. Beyond the mathematics simulations the data pool contains... more...
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"Proven and Effective Methods for Passing the End-of-Course Exam"
This 3-day course consists of tried-and-true philosophies, classroom procedures, incremental
approaches to teaching a new topic, questioning techniques, assignments, REVIEW and
RETAIN materials, preparing for each quiz and each test, and helping students gain
confidence and security in taking a test and the final exams.
This course, along with one other course, may be used by an INTERVENTION SPECIALIST
to become a HIGHLY QUALIFIED TEACHER (HQT) in mathematics CONTENT.
Content (Some possibilities) See description above
Algebra
Properties, Sets, and Subsets of REAL NUMBERS
Relations, Functions, Domain, Range; Special Quadratics
Factoring; Completing the Square; Quadratic Formula
Many approaches to GRAPHING (FUNCTIONS and others)
Special Connection and Interaction of GRAPH, FUNCTION, TABLE
Logarithms; Complex Numbers; Conic Sections; Trigonometry Geometry
Importance of TERMINOLOGY, DEFINITIONS, POSTULATES, and THEOREMS in PROOFS
Polygons, Similarity, Special Right Triangles, Pythagorean Theorem
Circles; Solid Figures
Perimeter, Area, Volume
Prove that √2 is irrational; MUCH, MUCH MORE
Course Expectations
Bring a TI-83 or -84, or tell Duane in advance if you need to borrow one Attendance at all sessions Participation in all activities Sharing teaching ideas that have worked for you
Problems to Ponder for the Workshop
A line in the front wall of the classroom is A-S-N parallel to a line in the ceiling
of the classroom. A line in the front wall of the classroom is A-S-N perpendicular to a line in the
ceiling of the classroom. A line in the front wall of the classroom is A-S-N perpendicular to a line in the
back wall of the classroom. A line in the front wall of the classroom is A-S-N skew to a line in the ceiling of
the classroom. Suppose the symbol p(n) means the sum of all the prime numbers that are factors of
n; e.g., p(18) = 2 + 3 = 5; p(7) = 7 Find: p(8); p(9); p(10); p(24); p(28); p(70); p(154); p(1001) FACTOR COMPLETELY: x5 – y5 ; x5 + y5 ; x6 – y6; x6+ y6
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Math Word Problems Demystified 2/E solution to MATH word PROBLEMS! Find yourself stuck on the tracks when two trains are traveling at different speeds? Help has arrived! Math Word Problems Demystified, Second Edition is your ticket to problem-solving success. Based on mathematician George Polya's proven four-step process, this practical guide helps you master the basic procedures and develop a plan of action you can use to solve many different types of word problems. Tips for using systems of equations and quadratic equations are included. Detailed examples and concise explanations make it easy to understand the material, and end-of-chapter quizzes and a final exam help reinforce learning. It's a no-brainer! You'll learn to solve: Decimal, fraction, and percent problems Proportion and formula problems Number and digit problems Distance and mixture problems Finance, lever, and work problems Geometry, probability, and statistics problems Simple enough for a beginner, but challenging enough for an advanced student, Math Word Problems Demystified, Second Edition helps you master this essential mathematics skill.
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More About
This Textbook
Overview
This text gives first-year algebra students the confidence they need to problem solve and apply algebra to real-world situations. The Aufmann Interactive Approach presents students with at least one matched-pair example per objective. The Example is worked out and the Problem is left for the student to solve, with references to worked-out solutions at the back of the text. Beginning Algebra with Applications also features a complete integrated learning system, in which all lessons, exercises, review tests, and ancillaries are organized around objectives, enhancing course organization for instructors and students.
Product Details
Meet the Author
RichardVernon Barker has retired from Palomar College where he was Professor of Mathematics. He is a co-author on the majority of Aufmann texts, including the best-selling developmental paperback series.
Joanne Lockwood is co-author with Dick Aufmann and Vernon Barker on the hardback developmental series, Business Mathematics, Algebra with Trigonometry for College Students, and numerous software ancillaries that accompany Aufmann titles. She is also the co-author of Mathematical Excursions with Aufmann
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We started teaching Mathematica-based mathematics courses in 1989. We made every mistake in the book—mistakes I often see in other attempts at using Mathematica or other math processors at many universities and high schools. But we were fortunate enough to learn from our mistakes and progress to the point at which we have an efficient, effective set of successful courses including precalculus, calculus, vector calculus, differential equations, matrix theory, and probability theory. These courses are in service on campuses and internet distance education at Ohio State and Illinois.
The courses are highly visual, but with a twist articulated by former teacher Scott Mills Gray: "While visualization is an extremely important aspect to a learning system, if the teacher or author is creating all the visualizations then it is of no help to the student. What is more important is that the student have an unbounded set of tools to create their own visualizations." Our students have this set of tools.
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Linear Algebra investigates the linear relationship between multiple variables. On one side it is for many students a first encounter with mathematical abstraction and on the other side it is a topic that occurs in many scientific applications like in numerical or economical models. At different universities I taught my own courses that built on linear algebra.
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Get a quick overview of Mathematica
STEP 2:
Watch a tutorial screencast
Hands-on Start to Mathematica: Follow along in Mathematica as you watch this multi-part screencast that teaches you the basics—how to create your first notebook, calculations, visualizations, interactive examples, and more.
STEP 3:
Explore the Learning Center
Learning Center: Browse through this extensive collection of Mathematica tutorials, examples, and other resources to learn more about using Mathematica.
For Research
Rather than requiring different toolkits for different jobs, Mathematica integrates the world's largest collection of algorithms, high-performance computing capabilities, and a powerful visualization engine in one coherent system, making it ideal for academic research in just about any discipline.
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Number Systems Help
Introduction to Number Systems
Calculus is one of the most important parts of mathematics. It is fundamental to all of modern science. How could one part of mathematics be of such central importance? It is because calculus gives us the tools to study rates of change and motion. All analytical subjects, from biology to physics to chemistry to engineering to mathematics, involve studying quantities that are growing or shrinking or moving—in other words, they are changing . Astronomers study the motions of the planets, chemists study the interaction of substances, physicists study the interactions of physical objects. All of these involve change and motion.
In order to study calculus effectively, you must be familiar with cartesian geometry, with trigonometry, and with functions. We will spend this article reviewing the essential ideas.
Number System
The number systems that we use in calculus are the natural numbers, the integers, the rational numbers, and the real numbers . Let us describe each of these:
The natural numbers are the system of positive counting numbers 1, 2, 3, .... We denote the set of all natural numbers by .
The integers are the positive and negative whole numbers and zero: ..., −3, −2, −1, 0, 1, 2, 3, .... We denote the set of all integers by .
The rational numbers are quotients of integers. Any number of the form p/q , with p, q and q ≠ 0, is a rational number. We say that p/q and r/s represent the same rational number precisely when ps = qr . Of course you know that in displayed mathematics we write fractions in this way:
The real numbers are the set of all decimals, both terminating and non-terminating. This set is rather sophisticated, and bears a little discussion. A decimal number of the form x = 3.16792 is actually a rational number, for it represents
A decimal number of the form
m = 4.27519191919 ...,
with a group of digits that repeats itself interminably, is also a rational number. To see this, notice that
100 · m = 427.519191919 ...
and therefore we may subtract:
100 m = 427.519191919 ...
m = 4.275191919 ...
Subtracting, we see that
99 m = 423.244
or
So, as we asserted, m is a rational number or quotient of integers.
The third kind of decimal number is one which has a non-terminating decimal expansion that does not keep repeating . An example is 3.14159265 .... This is the decimal expansion for the number that we ordinarily call . Such a number is irrational , that is, it cannot be expressed as the quotient of two integers.
In summary: There are three types of real numbers: (i) terminating decimals, (ii) non-terminating decimals that repeat, (iii) non-terminating decimals that do not repeat. Types (i) and (ii) are rational numbers. Type (iii) are irrational numbers.
You Try It : What type of real number is 3.41287548754875 ... ? Can you express this number in more compact form?
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Mathematics for Elementary Teachers - 8th edition
Summary: Now in its eighth edition, this book masterfully integrates skills, concepts, and activities to motivate learning. It emphasizes the relevance of mathematics to help readers learn the importance of the information being covered. This approach ensures that they develop a sold mathematics foundation and discover how to apply the content in the real world.
Gary L. Musser is currently Professor Emeritus from Oregon State University. He earned both his B.S. in Mathematics Education in 1961 and his M.S. in Mathematics in 1963 at the University of Miami in Florida. He taught at the junior and senior high, junior college, college, and university levels for more than 30 years. He served his last 24 years teaching prospective teachers in the Department of Mathematics at Oregon State University. While at OSU, Dr. Musser developed the mathematics component of the elementary teacher program. Soon after Professor William F. Burger joined the OSU Department of Mathematics in a similar capacity, the two of them began to write the first edition of this book. Professor Burger passed away during the preparation of the second edition, and later Professor Blake E. Peterson was hired at OSU. Professor Peterson joined Professor Musser as a coauthor beginning with the fifth edition. Professor Musser has published 40 papers in many journals, including the Pacific Journal of Mathematics, Canadian Journal of Mathematics, The Mathematics Association of America Monthly, the NCTM's The Mathematics Teacher, the NCTM's The Arithmetic Teacher, School Science and Mathematics, The Oregon Mathematics Teacher, and the Computing Teacher. In addition, he is a coauthor of two other college mathematics Books: College Geometry- A Problem-Solving Approach with Applications and Mathematics in Life, Society, and the World. He also coauthored the K-8 series Mathematics in Action. He has given more than 64 invited lectures/workshops at a variety of conferences, including NCTM and MAA conferences, and was awarded 15 federal, state, and local grants to improvethe teaching of mathematics.
While Professor Musser was at OSU, he was awarded the university's prestigious College of Science Carter Award for Teaching. he is currently living in sunny Las Vegas, where he continues to write, ponder the mysteries of the stock market, entertain his faithful yellow lab, Zoey, and enjoy watching his granddaughter blossom into a young lady.
Blake E. Peterson is currently an Associate Professor in the Department of Mathematics Education at Brigham Young University. He was born and raised in Logan, Utah, where he graduated from Logan High School. Before completing his BA in secondary mathematics education at Utah State University, he spent two years in Japan as a missionary for the church of Jesus Christ of Latter Day Saints. After graduation, he took his new wife, Shauna, to southern California, where he taught and coached at Chino High School for two years. In 1988, he began graduate school at Washington State University, where he later completed an M.S. and Ph.D. in pure mathematics.
After completing his Ph.D., Dr. Peterson was hired as a mathematics educator in the Department of Mathematics at Oregon State University in Corvallis, Oregon, where he taught for three years. It was at OSU that he met Gary Musser. He has since moved his wife and four children to Provo, Utah, to assume his position at Brigham Young University. As a professor, his first love is teaching, for which he has received a College Teaching he Award in the College of Science. He has also designed the "Mathematics Teaching with Technology" and "Mathematics Methods" courses at Brigham Young University.
Dr. Peterson has published papers in Rocky Mountain Mathematics Journal, The American Mathematical Monthly, The Mathematical Gazette, and Mathematics magazine, as well as NCTM's mathematics Teacher and Mathematics Teaching in the Middle School. His current research interests are the mathematical dialogue that occurs during teacher collaborations. This research recently took him back to Japan where he studied mathematics student teachers at a Japanese junior high school. In addition to teaching, research, and writing, Dr. Peterson has done consulting for the College Board, founded the Utah Association of mathematics Teacher Educators, is an associate chair of the department of mathematics education at BYU.
Aside from his academic interests, Dr. Peterson enjoys spending time with his family, playing basketball, mountain biking, water skiing, and working in the yard
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Suitable for the GCSE Modular Mathematics, this book covers different concepts through artwork and diagrams.
Synopsis:
This book is revised in-line with the 2007 GCSE Modular Mathematics specification. This Student Book is delivered in colour giving clarity to different concepts through artwork and diagrams. Worked examples, practice exercises and examiners tips ensure students are fully prepared for their exams. It is written by an experienced author team, including Senior Examiners, which means you can trust that the 2007 specification is covered to ensure exam success
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irsch and Goodman offer a mathematically sound, rigorous text to those instructors who believe students should be challenged. The text prepares students for future study in higher-level courses by gradually building students' confidence without sacrificing rigor. To help students move beyond the "how" of algebra (computational proficiency) to the "why" (conceptual understanding), the authors introduce topics at an elementary level and return to them at increasing levels of complexity. Their gradual introduction of concepts, rules, and definitions through a wealth of illustrative examples -- both numerical and algebraic--helps students compare and contrast related ideas and understand the sometimes-subtle distinctions among a variety of situations. This author team carefully prepares students to succeed in higher-level mathematics.
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"Great problem sets! Rigorous and flexible enough to work with various teaching styles."
About the Author
Lewis Hirsch (Ph.D., Pennsylvania State University) currently teaches in the mathematics department at Rutgers University. Dr. Hirsch teaches both developmental mathematics and higher level courses such as college algebra and pre-calculus. His experiences in the classroom make him committed to properly preparing students in lower-level courses so they can succeed in for credit courses, and this is reflected in the way he writes his textbooks.
Dr. Arthur Goodman (Ph.D., Yeshiva University) currently teaches in the mathematics department at Queens College of the City University of New York. Dr. Goodman takes great pride in the mathematical accuracy and in depth explanation in all of his textbooks.
Most Helpful Customer Reviews
This book does a fanstastic job explaining the principles of basic algebra and incorporates really helpful word problems with each section. Each section also has review problems from previous sections so that you don't forget the principles you just learned. There are lots of exercise problems so that you can practice as much as necessary.
I studied more advanced math over 15 years ago, and this was the perfect book to help me get back into math mode. I highly recommend it for anyone who wants to self-study basic algebra.
This text was easy to use and had good examples. There were some mistakes in the answers in the book but overall the text is good. Excellent for the beginning algebra student. There is a student solution manual available for purchase but be aware that like the text, there are only answers to the odd numbered problems. The solution manual does show step by step how to solve the problems.
I used this textbook for a college math class and I did not understand anything it was trying to say. My prof explained how to do the math way better and I only referenced this book to look at the homework questions. If you are not going to be using the homework questions, i would not recommend buying the book because it wont help you a all.
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Eventually a problem or concept surfaces where that missing link creates a huge obstacle (this usually happens in pre-calculus or trigonometry, or if you are lucky your first calculus class). However, these sorts of problems are easily resolved if identified before your math grade is beyond repai...
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Pelham, NY PrecalculusRight triangles are explored through a study of the Pythagorean Theorem as it relates to square roots, irrational numbers and special right triangles. An overview of data analysis and probability teaches students how to represent data and determine probabilities. The course concludes with a study of nonlinear functions and polynomials
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This class presents the fundamental probability and statistical concepts used in elementary data analysis. It will be taught at an introductory level for students with junior or senior college-level mathematical training including a working knowledge of calculus.
This course will introduce you to the fundamentals of probability theory and random processes. The theory of probability was originally developed in the 17th century by two great French mathematicians, Blaise Pascal and Pierre de Fermat, to understand gambling.
A degree in General Mathematics is designed to equip you with the skills necessary to be a professional problem-solver, as a mathematician is not defined by his or her knowledge of laws and theorems, but by his or her critical thinking and reasoning skills.
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Clear, rigorous definitions of mathematical terms are crucial to good scientific and technical writing-and to understanding the writings of others. Scientists, engineers, mathematicians, economists, technical writers, computer programmers, along with teachers, professors, and students, all have the need for comprehensible, working definitions of mathematical... more...
You know mathematics. You know how to write mathematics. But do you know how to produce clean, clear, well-formatted manuscripts for publication? Do you speak the language of publishers, typesetters, graphics designers, and copy editors? Your page design-the style and format of theorems and equations, running heads and section headings, page breaks,... text provides a masterful and systematic treatment of all the basic analytic and geometric aspects of Bergman's classic theory of the kernel and its invariance properties. These include calculation, invariance properties, boundary asymptotics, and asymptotic expansion of the Bergman kernel and metric. Moreover, it presents a unique compendium... more...
Convexity is an ancient idea going back to Archimedes. Used sporadically in the mathematical literature over the centuries, today it is a flourishing area of research and a mathematical subject in its own right. Convexity is used in optimization theory, functional analysis, complex analysis, and other parts of mathematics.
Convex Analysis... more...
Foundations of Analysis covers the basics of real analysis for a one- or two-semester course. In a straightforward and concise way, it helps students understand the key ideas and apply the theorems. The book?s accessible approach will appeal to a wide range of students and instructors.
Each section begins with a boxed introduction that familiarizes... more...
Here's the perfect self-teaching guide to help anyone master differential equations--a common stumbling block for students looking to progress to advanced topics in both science and math. Covers First Order Equations, Second Order Equations and Higher, Properties, Solutions, Series Solutions, Fourier Series and Orthogonal Systems, Partial Differential... more...
This text explores the many transformations that the mathematical proof has undergone from its inception to its versatile, present-day use, considering the advent of high-speed computing machines. Though there are many truths to be discovered in this book, by the end it is clear that there is no formalized approach or standard method of discovery to... more...
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Pre-Geometry Guide
Over 300 rules, definitions and clear examples summarized into one easy-to-use app! Every day we use principles of geometry to help guide decision making and now the keys to this important subject can be at your fingertips! Geometry sharpens our reasoning, logic and problem solving skills and is one of those subjects we need to […]
Punctuation Guide [HD]
Over 400 examples! Mistakes in grammar or punctuation can be annoying to a reader and quickly draws attention away from what is being written. These types of mistakes cause the reader to focus on the grammar instead of what is being communicated, leads them to question on how well educated the person is, and significantly […]
Pre-Algebra Guide [HD]
A Top Seller! Well paying careers demand skills like problem solving, reasoning, decision making, and applying solid strategies etc. and Algebra provides you with a wonderful grounding in those skills – not to mention that it can prepare you for a wide range of opportunities. This is a COMPLETE Pre-Algebra guide to well over 325 […]
Science Terms
A mobile reference of over 275 definitions and terms! Science is a key focus area in school and needed for many of today's top-paying careers. Mastering the many areas, let alone understanding the basics of each area, can be very difficult, confusing and even mind boggling! The good news is this app will help make it easier […]
Pre-Calculus Guide
Calculus may not seem very important to you but the lessons and skills you learn will be with for your whole lifetime! Calculus is the mathematical study of continuous change. It helps you practice and develop your logic/reasoning skills. It throws challenging problems your way which make you think. Although you may never use calculus […]
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Hallandale Cal
...It develops advanced algebra skills such as systems of equations, advanced polynomials, imaginary and complex numbers, quadratics, and concepts and includes the study of trigonometric functions. It also introduces matrices and their properties. The content of this course are important for students? success on both the ACT and college mathematics entrance exams
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Overview
Editorial Reviews
Howard W. Smith
"The authors have provided a uniquely strategy-focused resource supported by a wealth of engaging examples that mathematics teachers can readily use to help students develop a more purposeful, systematic, and successful approach to problem solving."
Daniel Jaye
"This terrific resource helps bothnew and veteran teachers better understand the nature of problem solving as a critical mathematics process. In very simple terms, the authors present and illuminate the strategies that are the backbone of mathematics instruction. This indispensable material is useful at all levels, from basic stages to advanced student work to the development of top problem solvers."
MAA Reviews
"Each chapter covers a different strategy and includes problems that could be solved using that strategy. This would be a great resource for any secondary mathematics teacher and a wonderful resource for the mathematics teacher educator."
Product Details
Meet the Author
Alfred S. Posamentier is professor of mathematics education and dean of the School of Education at the City College of the City University of New York. He has authored and co-authored several resource books in mathematics education for Corwin Press.
Stephen Krulik is professor of mathematics education at Temple University in Philadelphia, where he is responsible for the undergraduate and graduate preparation of mathematics teachers for Grades K-12, as well as in the inservice training of mathematics teachers at the graduate level. He teaches a wide variety of courses, among them the History of Mathematics, Methods of Teaching Mathematics, and the Teaching of Problem Solving. Before coming to Temple University, he taught mathematics in the New York City public schools for 15 years, where he created and implemented several courses designed to prepare students for the SAT examination. Nationally, Krulik has served as a member of the committee responsible for preparing the Professional Standards for Teaching Mathematics of the National Council of Teacher of Mathematics (NCTM). He was also the editor of the NCTM's 1980 yearbook Problem Solving in School Mathematics. He is the author or co-author of more than 20 books for teachers of mathematics, including Assessing Reasoning and Problem Solving: A Sourcebook for Elementary School Teachers. He has served as a consultant to and has conducted many workshops for school district throughout the United States and Canada, as well as delivering major presentations in Austria, Hungary, Australia, and international professional meetings, where his major focus is on preparing all students to reason and problem-solve in their mathematics classroom, as well as in their lives. Krulik received his BA degree in mathematics from Brooklyn College of the City University of New York, and his MA and Ed D in mathematics education from Columbia University's Teachers College.
Table of Contents
Preface
About the Authors
1. Introduction to Problem-Solving Strategies
2. Working Backwards
The Working Backwards Strategy in Everyday Life Problem-Solving Situations
Applying the Working Backwards Strategy to Solve Mathematics Problems
Problems Using the Working Backwards Strategy
3. Finding a Pattern
The Finding a Pattern Strategy in Everyday Life Problem-Solving Situations
Applying the Finding a Pattern Strategy to Solve Mathematics Problems
Problems Using the Finding a Pattern Strategy
4. Adopting a Different Point of View
The Adopting a Different Point of View Strategy in Everyday Life Problem-Solving Situations
Applying the Adopting a Different Point of View Strategy to Solve Mathematics Problems
Problems Using the Adopting a Different Point of View Strategy
5. Solving a Simpler Analogous Problem
The Solving a Simpler Analogous Problem Strategy in Everyday Life Problem-Solving Situations
Applying the Solving a Simpler Analogous Problem Strategy to Solve Mathematics Problems
Problems Using the Solving a Simpler Analogous Problem Strategy
6. Considering Extreme Cases
The Considering Extreme Cases Strategy in Everyday Life Problem-Solving Situations
Applying the Considering Extreme Cases Strategy to Solve Mathematics Problems
Problems Using the Considering Extreme Cases Strategy
7. Making a Drawing (Visual Representation)
The Making a Drawing (Visual Representation) Strategy in Everyday Life Problem-Solving Situations
Applying the Making a Drawing (Visual Representation) Strategy to Solve Mathematics Problems
Problems Using the Making a Drawing (Visual Representation) Strategy
8. Intelligent Guessing and Testing (Including Approximation)
The Intelligent Guessing and Testing (Including Approximation) Strategy in Everyday Life Problem-Solving Situations
Applying the Intelligent Guessing and Testing (Including Approximation) Strategy to Solve Mathematics Problems
Problems Using the Intelligent Guessing and Testing (Including Approximation) Strategy
9. Accounting for All Possibilities
The Accounting for All Possibilities Strategy in Everyday Life Problem-Solving Situations
Applying the Accounting for All Possibilities Strategy to Solve Mathematics Problems
Problems Using the Accounting for All Possibilities Strategy
10. Organizing Data
The Organizing Data Strategy in Everyday Life Problem-Solving Situations
Applying the Organizing Data Strategy to Solve Mathematics Problems
Problems Using the Organizing Data Strategy
11. Logical Reasoning
The Logical Reasoning Strategy in Everyday Life Problem-Solving Situations
Applying the Logical Reasoning Strategy to Solve Mathematics Problems
Problems Using the Logical Reasoning Strategy
Afterword by Herbert A. Hauptman
Sources for Problems
Readings on Problem Solving
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Glencoe Pre-Algebra
"Glencoe Pre-Algebra" is focused, organized, and easy to follow. The program shows your students how to read, write, and understand the unique ...Show synopsis"Glencoe Pre-Algebra" is focused, organized, and easy to follow. The program shows your students how to read, write, and understand the unique language of mathematics, so that they are prepared for every type of problem-solving and assessment situation.Hide synopsis
Description:Fair. 2005. MULTIPE COPIES AVAILABLE. HEAVY COVER WEAR. A used...Fair. 2005. MULTIPE COPIES AVAILABLE. HEAVY COVER WEAR. A used copy with heavy cover wear. Cardboard showing at the corners. Usual school stamps and labels which may include a bookDescription:Book is in good condition. Pages are clean and the binding is...Book is in good condition. Pages are clean and the binding is tight. Multiple copies available. Note: This is the Ohio edition with additional pages for Ohio students
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Mathematical Reasoning: Writing and Proof
Mathematical Reasoning: Writing and Proof...
More
To help students learn how to read and understand mathematical definitions and proofs;
To help students learn how to construct mathematical proofs;
To help students learn how to write mathematical proofs according to ac-cepted guidelines so that their work and reasoning can be understood by others; and
To provide students with material that will be needed for their further study of mathematics.'
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Barron's Mathematics Study Dictionary - 98 edition
Summary: Focused especially for use by students on the middle-to-high school level, this quick-reference source is helpful to anybody who needs to know the meaning of math terms in clear, simple language. An opening alphabetized Wordfinder index contains more than 1,000 words, and directs readers to the page where the word is defined. Where needed, the definition is accompanied by examples. The book also features helpful illustrative diagrams--or instance, a full page demonst...show morerating the geometry of the circle, another page showing quadrilateral geometric shapes, and still others showing ways of charting statistics, measuring vectors, and more. Here is an imaginative new approach to mathematics, a great classroom supplement, a useful homework helper for middle school and high school students, and a reference book that belongs in every school library. ...show less
"I recently completed several years' commitment to elementary school and science glossaries...I therefore can view with appreciation, indeed with some amazement, how Tapson and his editors have handled the nearly insurmountable difficulties inherent in such a project...Just about everything in secondary-level mainstream mathematics, pure and applied, is tidily handled in a classroom-friendly way...I find the book easy to use, clear in presentation, adequate for the basics, and splendid in its efforts to support and extend student inquiry."WoodyBooks Huntington Beach, CA
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There is a newer edition of this item:
Test with success using the Spectrum Math workbook! This book helps students in grade 8 apply essential math skills to everyday life. The lessons focus on ratio and proportion, fractions, percents, calculating interest, perimeter, volume, and statistics, and the activities help extend problem-solving and analytical abilities. The book
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Whether you are a home schooler or just getting the most from your student this math workbook will do the trick. My wife and I have trusted Spectrum workbooks for years, this book also does keep advancing the students skills and abilities. Need one, buy one. Enjoy.
This attractive book has all the lessons needed to practise Grade 8 Maths. I tutor children in Maths in Australia, and choose these American Spectrum books because I think they are an ideal supplement to the work done at school. The children I work with love these books too.
I like the book because it has enforced my 11-year-old daughter's confidence in math through repeated exercises and problems. It's easy to understand and user friendly. Though my daughter is going to the 7th grade this September, she is enrolled in the gifted math program; therefore, 8th grade math preparation is appropriate, I believe. In addition, I told my daughter to do all the lessons in each chapter first and then the pre-test and post-test. I don't want her to feel bad in case she hasn't learned something before and can't do the problems in the pretest if she does the pretest first.
The answer keys in the back contain occasional errors; therefore, parents should pay attention when checking their chidlren's work.
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books.google.com - Many students have trouble the first time they take a mathematics course in which proofs play a significant role. This new edition of Velleman's successful text will prepare students to make the transition from solving problems to proving theorems by teaching them the techniques needed to read and write... to Prove It
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Code No. : BCA 153* L T CPaper: General Proficiency – I 15 w.e.f. session 2005-2006
Code No.: BCA 102 L T CPaper: Mathematics II 3 1 4INSTRUCTIONS TO PAPER SETTERS:1. Question No. 1 should be compulsory and cover the entire syllabus. This question should have objective or short answer type questions. It should be of 25 marks.2. Apart from Question No. 1, rest of the paper shall consist of four units as per the syllabus. Every unit should have two questions. However, student may be asked to attempt only 1 question from each unit. Each question should be 12.5 marks.UNIT-ISETS: Sets, Subsets, Equal Sets Universal Sets, Finite and Infinite Sets, Operation on Sets,Union, Intersection and Complements of Sets, Cartesian Product, Cardinality of Set, SimpleApplications.RELATIONS AND FUNCTIONS: Properties of Relations, Equivalence Relation, PartialOrder Relation Function: Domain and Range, Onto, Into and One to One Functions,Composite and Inverse Functions, Introduction of Trignometric, Logarithmic andExponential Functions. [No. of Hrs: 11]UNIT-IIPARTIAL ORDER RELATIONS AND LATTICES: Partial Order Sets, Representation ofPOSETS using Hasse diagram, Chains, Maximal and Minimal Point, Glb, lub, Lattices &Algebric Systems, Principle of Duality, Basic Properties, Sublattices, Distributed &Complemented Lattics. [No. of Hrs: 11]UNIT-IIIFUNCTIONS OF SEVERAL VARIABLES: Partial Differentiation, Change of Variables,Chain Rule, Extrema of Functions of 2 Variables, Euler's Theorem.3D COORDINATE GEOMETRY: Review of 2D Coordinate Geometry: Equations ofStraight Lines, Circle, Ellipse, Parabola, Hyprbola. 3D Coordinate Geometry: Coordinates inSpace, Direction Cosines, Angle Between Two Lines, Projection of Join of Two Points on aPlane, Equations of Plane, Straight Lines, Conditions for a line to lie on a plane, Conditionsfor Two Lines to be Coplanar, Shortest Distance Between Two Lines, Equations of Sphere,Tangent plane at a point on the sphere. Equations of Ellipsoid, Paraboloid, Hyperbolid andCylinder in Cartesian coordinate. [No. of Hrs: 10]UNIT-IVMULTIPLE INTEGRATION: Double Integral in Cartesian and Polar Coordinates to findArea, Change of Order of Integration, Triple Integral to Find Volume of Simple Shapes inCartesian Coordinates. [No. of Hrs: 12]TEXT BOOKS:1. Kolman, Busby and Ross, "Discrete Mathematical Structure", PHI, 1996.REFERENCE BOOKS:1. H.K. Dass, "Advanced Engineering Mathematics"; S.Chand & Co., 9th Revised Ed., 2001.2. S.K. Sarkar, "Discrete Maths"; S. Chand & Co., 2000 16 w.e.f. session 2005-2006
Code No. : BCA 154* L T CPaper: General Proficiency – II 22 w.e.f. session 2005-2006
Code No. : BCA 253* L T CPaper: General Proficiency – III 29 w.e.f. session 2005-2006
Code No. : BCA 254* L P CPaper: General Proficiency – IV 36 w.e.f. session 2005-2006
Code No. : BCA 354* L P CPaper: Major Project / Seminar 0 10 5Evaluation will be based on Summer Training held after fourth semester and will beconducted by the college committee only. 55 w.e.f. session 2005-2006
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Description of Saxon Math 7/6 7/6 builds on core concepts to prepare students for upper-level algebra and geometry courses.
I have used Saxon with all three of my children. I like it because each lesson is clearly defined with instructions and sample problems to reinforce each lesson. There are tests available after every five lessons with clear guidelines on when to administer them.
I also like that the child can read through the lesson independently and proceed with it if they have a good understanding. Thus it did not require teacher participation for every lesson.
No, Saxon has included supplemental lessons in case a child needs more help in a specific area.
Pre-assessment tests available
Yes
Tools available
Not applicable
Liked most about curriculum
Ease of use, Supplemental materials, sequential math
Liked least about curriculum
thin pages, difficult to copy
Other books used
Assessment types
Practice tests available
Resources also used
Content reusable
Yes
Content consumable
No
Other curriculum considered
A Beka
Reviewed By
Chalice H
Parent Rating
Comments
This is good in that it is the same as what the schools here have been using, and made for an easy transition form school to homeschool. I have found after homeschooling for a few years though that it is not my favorite. Math is a hard subject for me , so when my child had a question I wasn't much help, and there was not a teacher to ask to help explain it. I have found teaching textbooks to be my favorite math cirriculum now.
Every child and the way they learn can be different. Saxon is highly praised and I liked many aspects of the way the material is covered. But, my daughter struggled and felt lost all the time. I switched to Teaching Textbooks math program and she began to understand math better. She went from a struggling C/D student to a self-confident A student in a very short time. She had to repeat the year we 'lost' using Saxon because I don't just 'pass' my kids unless they can do the work. Now I use Teaching Textbooks for math with all my kids and they are all doing very well in math and understand the higher concepts (algebra).
Story of the World seems like a great concept to teach my children. But I felt it talked down to them. They didn't like the way the author wrote the material and after a while stopped reading it all together.
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Marinwood, CA StatisticsMath concepts build on each other so that future topics depend on understanding previous material. Therefore, misunderstanding one topic can cause continuous problems down the road. If this is left unaddressed, knowledge gaps compound over time and the student gets further behind
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0130412244
9780130412249
Trigonometry:A proven motivator for readers of diverse mathematical backgrounds, this book explores mathematics within the context of real life using understandable, realistic applications consistent with the abilities of any reader. Graphing techniques are emphasized, including a thorough discussion of polynomial, rational, exponential, and logarithmic functions and conics. Includes Case Studies; New design that utilizes multiple colors to enhance accessibility; Multiple source applications; Numerous graduated examples and exercises; Discussion, writing, and research problems; Important formulas, theorems, definitions, and objectives; and more. For anyone interested in trigonometry.
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Rent Trigonometry 6th edition today, or search our site for Michael textbooks. Every textbook comes with a 21-day "Any Reason" guarantee. Published by Prentice Hall.
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29,687algebra 1
| 8 other subjects
Language Arts (K-Adult); Math (K-8); ESL
...So it's literally impossible to learn the new concepts.
This situation also requires concentration, and it will take more time, so you can relax, make a plan, and start making steady progress on your plan.
Don't panic, don't rush. You will be tackling your stack...
read more
algebra 1
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This book provides a fun, hands-on approach to learning how mathematics and computing relate to the world around us and help us to better understand it. How can reposting on Twitter kill a movie?s opening weekend? How can you use mathematics to find your celebrity look-alike? What is Homer Simpson?s method for disproving Fermat?s Last Theorem? EachDo you have a handle on basic physics terms and concepts, but your problem-solving skills could use some static friction? Physics Workbook for Dummies helps you build upon what you already know to learn how to solve the most common physics problems with confidence and ease. Physics Workbook for Dummies gets the ball rolling with a brief overview
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Entry Level Mathematics Test Review "Your Best Preparation for the ELM" Cliffs ELM Review is designed specifically to Review, Refresh, Reintroduce, Diagnose and, in effect, give you a "Fighting Chance" by focusing squarely on an ELM-oriented math review. This guide combines insights and strategies for problem types, while reviewing the most needed basic skills: • Arithmetic • Basic and Intermediate Algebra • Geometry • Word Problems • Graphs Each review section includes:a diagnostic testrules and concepts with examplespractice problemscomplete (understandable) explanationsa review testand a glossaryA special section provides the key strategies for common question types. If you're planning to take the ELM, this book is designed for you!!!
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Trigonometry With Infotrac
9780534403928
ISBN:
0534403921
Edition: 5 Pub Date: 2003 Publisher: Thomson Learning
Summary: This text provides students with a solid understanding of the definitions and principles of trigonometry and their application to problem solving. Identities are introduced early in Chapter 1. They are reviewed often and are then covered in more detail in Chapter 5. Also, exact values of the trigonometric functions are emphasized throughout the textbook. There are numerous calculator notes placed throughout the text.
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BOOK ONLY, CD/CODE NOT INCLUDED!! Book is in overall good condition!! Cover shows some edge wear and corners are lightly worn. Pages have a minimal to moderate amount of markings. ...FAST SHIPPING W/USPS TRACKING!!! Read moreShow Less
Has some markings. Has minor shelf and corner wear. Has minor corner curl and creasing to covers. Has water stain at back of book, does not affect text. Binding is in good ...condition 2nd Edition. Read moreShow Less
Pacific Grove, CA 2006 Softcover 2nd Edition Good Condition Has some markings. Has minor shelf and corner wear. Has minor corner curl and creasing to covers. Has water stain at ...back of book, does not affect text. Binding is in good conditionRead moreShow Less
More About
This Textbook
Overview
Pat McKeague's passion and dedication to teaching mathematics and his ongoing participation in mathematical organizations provides the most current and reliable textbook series for both instructors and students. When writing a textbook, Pat McKeague's main goal is to write a textbook that is user-friendly. Students develop a thorough understanding of the concepts essential to their success in mathematics with his attention to detail, exceptional writing style, and organization of mathematical concepts. BASIC COLLEGE MATHEMATICS: A TEXT/WORKBOOK, Second Edition offers a unique and effortless way to teach your course, whether it is a traditional lecture course or a self-paced format. In a lecture-course format, each section can be taught in 45-to-50-minute class sessions, affording instructors a straightforward way to prepare and teach their course. In a self-paced format, Pat's proven EPAS approach (Example, Practice Problem, Answer and Solution) moves students through each new concept with ease and assists students in breaking up their problem-solving into manageable steps. The Second Edition of BASIC COLLEGE MATHEMATICS has new features that will further enhance your students' learning, including boxed features entitled Improving Your Quantitative Literacy, Getting Ready for Chapter Problems, Section Objectives, and Enhanced and Expanded Review Problems. These features are designed so your can students to practice and reinforce conceptual learning. Furthermore, iLrn/MathematicsNow™ for Developmental Math, a new Brooks/Cole technology product, is an assignable assessment and homework system that consists of pre-tests, Personalized Learning Plans, and post-tests to gauge concept mastery.
Related Subjects
Meet the Author
Charles P. "Pat" McKeague earned his B.A. in Mathematics from California State University, Northridge, and his M.S. in Mathematics from Brigham Young University. A well-known author and respected educator, he is a full-time writer and a part-time instructor at Cuesta College. He has published twelve textbooks in mathematics covering a range of topics from basic mathematics to trigonometry. An active member of the mathematics community, Professor McKeague is a popular speaker at regional conferences, includ
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MP: Beginning Algebra with SMART CDMiller/O'Neill Beginning Algebra is an insightful text written by instructors who have first-hand experience with students of developmental mathematics. The authors have placed an emphasis on graphing, by including special sections called, "Connections to Graphing" at the end of Chapters 1-5, before the formal presentation of Graphing appears in Chapter 6. The "Connections to Graphing" sections may be considered optional for those instructors who do not prefer an early introduction to graphing. For those who do prefer graphing early, instructors can use the "Connections to Graphing" sections together where they prefer to introduce graphing.A section on geometry appears in "Chapter R" for instructors who look for such content in Beginning Algebra. Applications that incorporate geometric concepts may also be found throughout the text.Chapter R also contains a section on study skills. This section provides easy to digest tips (in list format) for course success.The authors have crafted the exercise sets with the idea of infusing review. In each set of practice exe
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2006 Hardcover New Book New and in stock. 12/21 problems, and presents examples and exercises from past exams to illustrate the concepts. Anyone preparing for the Mathematical Olympiads will find many useful ideas here, but people generally interested in logical problem solving should also find the problems and their solutions stimulating. The book can be used either for self-study or as topic-oriented material and samples of problems for practice exams. Useful reading for anyone who enjoys solving mathematical problems, and equally valuable for educators or parents who have children with mathematical interest and ability.
What People Are Saying
David Wells
"Within each chapter, three well-chosen examples illustrate a variety of problem-solving strategies and applications of concepts... The thoughtful choice of examples and exercises is one of the book's strengths, providing a wealth of opportunity for students to become experienced problem solvers within a remarkably small number of pages."
—Penn State University
Preface
A Brief History of the American Mathematics Competitions
In the last year of the second millennium, the American High School Mathematics Examination, commonly known as the AHSME, celebrated its fiftieth year. It began in 1950 as a local exam in the New York City area, but within its first decade had spread to most of the states and provinces in North America, and was being administered to over 150,000 students. A third generation of students is now taking the competitions.
The examination has expanded and developed over the years in a number of ways. Initially it was a 50-question test in three parts. Part I consisted of 15 relatively routine computational problems; Part II contained 20 problems that required a thorough knowledge of high school mathematics, and perhaps some ingenuity; those in Part III were the most difficult, although some of these seem, based on the latter problems on the modern examination, relatively straightforward. The points awarded for success increased with the parts, and totaled 150. The exam was reduced to 40 questions in 1960 by deleting some of the more routine problems. The number of questions was reduced again, to 35, in 1968, but the number of parts was increased to four. The number of problems on the exam was finally reduced to 30, in 1974, and the division of the exam into parts with differing weights on each part was eliminated. After this time, each problem would be weighted equally. It continued in this form until the end of the century, by which time the exam was being given to over 240,000 students at over 5000 schools. One might get the impression that with a reduction in the number of problems the examination was becoming easier over the years, but a brief look at the earlier exams (which can be found The Contest Problem Book, Volumes I through V) will dissuade one from this view. The number of problems has been reduced, but the average level of difficulty has increased. There are no longer many routine problems on the exams, and the middle-range problems are more difficult than those in the early years. Since 1974, students from the United States have competed in the International Mathematical Olympiad (IMO), and beginning in 1972 students with very high scores on the AHSME were invited to take the United States of America Mathematical Olympiad (USAMO). The USAMO is a very difficult essay-type exam that is designed to select the premier problem-solving students in the country. There is a vast difference between the AHSME, a multiple-choice test designed for students with a wide range of abilities, and the USAMO, a test for the most capable in the nation. As a consequence, in 1983 an intermediate exam, the American Invitational Mathematics Examination was instituted, which the students scoring in approximately the top 5% on the AHSME were invited to take. Qualifying for the AIME, and solving even a modest number of these problems, quickly became a goal of many bright high school students, and was seen as a way to increase the chance of acceptance at some of the select colleges and universities. The plan of the top high school problem solvers was to do well enough on the AHSME to be invited to take the AIME, solve enough of the AIME problems to be invited to take the USAMO, and then solve enough USAMO problems to be chosen to represent the United States in the International Mathematical Olympiad. Also, of course, to do well in the IMO, that is, to win a Gold Medal! But I digress, back to the history of the basic exams. The success of the AHSME led in 1985 to the development of a parallel exam for middle school students, called the American Junior High School Mathematics Examination (AJHSME). The AJHSME was designed to help students begin their problem-solving training at an earlier age. By the end of the 20th century nearly 450,000 students were taking these exams, with representatives in each state and province in North America. In 2000 a major change was made to the AHSME-AJHSME system. Over the years there had been a reduction in the number of problems on the AHSME with a decrease in the number of relatively elementary problems. This reduction was dictated in large part by the demands of the school systems. Schools have had a dramatic increase in the number of both curricular and extra-curricular activities, and time schedules are not as flexible as in earlier years. It was decided in 2000 to reduce the AHSME examination to 25 questions so that the exams could be given in a 75 minute period. However, this put students in the lower high school grades at an additional disadvantage, since it resulted in a further reduction of the more elementary problems. The Committee on the American Mathematics Competitions (CAMC) was particularly concerned that a capable student who had a bad experience with the exam in grades 9 or 10 might be discouraged from competing in later years. The solution was to revise the examination system by adding a competition specifically designed for students in grades 9 and 10. This resulted in three competitions, which were renamed AMC 8, AMC 10 and AMC 12. The digits following AMC indicate the highest grade level at which students are eligible to take the exam. There was no change in the AJHSME except for being renamed AMC 8, nor, except for the reduction in problems, was there a change in AHSME. The new AMC 10 was to consist of problems that could be worked with the mathematics generally taught to students in grades 9 and lower and there would be overlap, but not more than 50\%, between the AMC 10 and AMC 12 examinations. Excluded from the AMC 10 would be problems involving topics generally seen only by students in grades 11 and 12, including trigonometry, logarithms, complex numbers, functions, and some of the more advanced algebra and geometry techniques. The AMC 10 was designed so that students taking this competition are able to qualify for the AIME, however only approximately the top 1\% do so. The reason for making the qualifying score for AMC 10 students much higher than for AMC 12 students was three-fold. First, there are students in grades 9 and 10 who have the mathematical knowledge required for the AMC 12, and these students should take the AMC 12 to demonstrate their superior ability. Having to score at the 1% level on the AMC 10 is likely to be seen to be riskier for these students than having to score at the 5% level on the AMC 12. Second, the committee wanted to be reasonably sure that a student who qualified for the AIME in grades 9 or 10 would also qualify when taking the AMC 12 in grades 11 and 12. Not to do so could discourage a sensitive student. Third, the AIME can be very intimidating to students who have not prepared for this type of examination. Although there has been a concerted effort recently to make the first group of problems on the AIME more elementary, there have been years when the median score on this 15-question test was 0. It is quite possible for a clever 9th or 10th grader without additional training to do well on the AMC 10, but not be able to begin to solve an AIME problem. This, again, could discourage a sensitive student from competing in later years. The primary goal of the AMC is to promote interest in mathematics by providing a positive problem-solving experience for all students taking the exams. The AMC exam is also the first step in determining the top problem-solving high school students in the country, but that goal is decidedly secondary. My Experience with the American Mathematics Competitions My first formal involvement with the AMC began in 1996 when I was appointed to the CAMC as a representative from Pi Mu Epsilon, the National Honorary Mathematics Society. Simultaneously, I began writing problems for the AHSME and the AJHSME. In 1997 I joined the committee that constructs the examination for the AJHSME, based on problems submitted from a wide range of people in the United States and Canada. At the same time, I had been helping some local students in middle school prepare for the AJHSME and for the MathCounts competition, and had discovered how excited these students were even when they didn't do as well in the competitions as they had expected. The next year, when they were in 9th grade, I encouraged them to take the AHSME, since that was the only mathematical competition that was available to them. The level of difficulty on this AHSME was so much higher than the exams they were accustomed to taking that most of them were devastated by the experience. I believe that for all but two of these students this was their last competitive problem-solving experience. At the next meeting of the CAMC I brought my experience to the attention of the members and showed figures that demonstrated that only about 20% of the 9th grade students and less than 40% of the 10th grade students who had taken the AJHSME in grade 8 were taking the AHSME. Clearly, the majority of the 9th and 10th grade teachers had learned the lesson much earlier than I had, and were not encouraging their students to take the AHSME. At this meeting I proposed that we construct an intermediate exam for students in grades 9 and 10, one that would provide them with a better experience than the AHSME and encourage them to continue improving their problem-solving skills. As any experienced committee member knows, the person who proposes the task usually gets assigned the job. In 1999 Harold Reiter, the Director of the AHSME, and I became joint directors of the first AMC 10, which was first given on February 15, 2000.Since 2001 I have been the director of AMC 10. I work jointly with the AMC 12 director, Dave Wells, to construct the AMC 10 and AMC 12 exams. In 2002 we began to construct two sets of exams per year, the AMC 10A and AMC 12A, to be given near the beginning of February, and the AMC 10B and AMC 12B, which are given about two weeks later. This gives a student who has a conflict or unexpected difficulty on the day that the A version of the AMC exams are given a second chance to qualify for the AIME. For the exam committee, it means, however, that instead of constructing and refining 30 problems per year, as was done in 1999 for the AHSME, we need approximately 80 problems per year, 25 for each version of the AMC 10 and AMC 12, with an overlap of approximately ten problems. There are a number of conflicting goals associated with constructing the A and B versions of the exams. We want the versions of the exams to be comparable, but not similar, since similarity would give an advantage to the students taking the later exam. Both versions should also contain the same relative types of problems, but be different, so as not to be predictable. Additionally, the level of difficulty of the two versions should be comparable, which is what we have found most difficult to predict. We are still in the process of grappling with these problems but progress, while slow, seems to be steady. The Basis and Reason for this Book When I became a member of the Committee on American Competitions, I found that students in the state of Ohio had generally done well on the exams, but students in my local area were significantly less successful. By that time I had over 25 years experience working with undergraduate students at Youngstown State University and, although we had not done much with problem-solving competitions, our students had done outstanding work in undergraduate research presentations and were very competitive on the international mathematical modeling competition sponsored by COMAP. Since most of the Youngstown State students went to high school in the local area, it appeared that their performance on the AHSME was not due to lack of ability, but rather lack of training. The mathematics and strategies required for successful problem solving is not necessarily the same as that required in general mathematical applications. In 1997 we began to offer a series of training sessions at Youngstown State University for high school students interested in taking the AHSME, meeting each Saturday morning from 10:00 until 11:30. The sessions began at the end of October and lasted until February, when the AHSME was given. The sessions were attended by between 30 and 70 high school students. Each Saturday about three YSU faculty, a couple of very good local high school teachers, and between five and ten YSU undergraduate students presented some topics in mathematics, and then helped the high school students with a collection of exercises. The first year we concentrated each week on a specific past examination, but this was not a successful strategy. We soon found that the variability in the material needed to solve the problems was such that we could not come close to covering a complete exam in the time we had available. Beginning with the 1998--1999 academic year, the sessions were organized by mathematical topic. We used only past AHSME problems and found a selection in each topic area that would fairly represent the type of mathematical techniques needed to solve a wide range of problems. The AHSME was at that time a 30-question exam and we concentrated on the problem range from 6 to 25. Our logic was that a student who could solve half the problems in this range could likely do all the first five problems and thus easily qualify for the AIME. Also, the last few problems on the AHSME are generally too difficult to be accessible to the large group we were working with in the time we had available. This book is based on the philosophy of sessions that were run at Youngstown State University. All the problems are from the past AMC (or AHSME, I will not subsequently distinguish between them) exams. However, the problems have been edited to conform with the modern mathematical practice that is used on current AMC examinations. So, the ideas and objectives of the problems are the same as those on past exams, but the phrasing, and occasionally the answer choices, have been modified. In addition, all solutions given to the Examples and the Exercises have been rewritten to conform to the material that is presented in the chapter. Sometimes this solution agrees with the official examination solution, sometimes not. Multiple solutions have occasionally been included to show students that there is generally more than one way to approach the solution to a problem. The goal of the book is simple. To promote interest in mathematics by providing students with the tools to attack problems that occur on mathematical problem-solving exams, and specifically to level the playing field for those who do not have access to the enrichment programs that are common at the top academic high schools. The material is written with the assumption that the topic material is not completely new to the student, but that the classroom emphasis might have been different. The book can be used either for self study or to give people who would want to help students prepare for mathematics exams easy access to topic-oriented material and samples of problems based on that material. This should be useful for teachers who want to hold special sessions for students, but it should be equally valuable for parents who have children with mathematical interest and ability. One thing that we found when running our sessions at Youngstown State was that the regularly participating students not only improved their scores on the AMC exams, but did very well on the mathematical portion of the standardized college admissions tests. (No claim is made concerning the verbal portion, I hasten to add.) I would like to particularly emphasize that this material is not a substitute for The Contest Problem Book, Volumes I through VIII. Those books contain multiple approaches to solutions to the problems as well as helpful hints for why particular ``foils'' for the problems were constructed. My goal is different, I want to show students how a few basic mathematical topics can be used to solve a wide range of problems. I am using the AMC problems for this purpose because I find them to be the best and most accessible resource to illustrate and motivate the mathematical topics that students will find useful in many problem-solving situations. Finally, let me make clear that the student audience for this book is perhaps the top 10--15\% of an average high school class. The book is not designed to meet the needs of elite problem solvers, although it might give them an introduction that they might otherwise not be able to find. References are included in the Epilogue for more advanced material that should provide a challenge to those who are interested in pursuing problem solving at the highest level. Structure of the Book Each chapter begins with a discussion of the mathematical topics needed for problem solving, followed by three Examples chosen to illustrate the range of topics and difficulty. Then there are ten Exercises, generally arranged in increasing order of difficulty, all of which have been on past AMC examinations. These Exercises contain problems ranging from relatively easy to quite difficult. The Examples have detailed solutions accompanying them. The Exercises also have solutions, of course, but these are placed in a separate Solutions chapter near the end of the book. This permits a student to read the material concerning a topic, look at the Examples and their solutions, and then attempt the Exercises before looking at the solutions that I have provided.Within the constraints of wide topic coverage, problems on the most recent examinations have been chosen. It is, I feel, important to keep in mind that a problem on an exam as recent as 1990 was written before many of our current competitors were born! The first four chapters contain rather elementary material and the problems are not difficult. This material is intended to be accessible to students in grade 9. By the fifth chapter on triangle geometry there are some more advanced problems. However, triangle geometry is such an important subject on the examinations, that there are additional problems involving these concepts in the circle geometry and polygon chapters. Chapters 8 and 9 concern counting techniques and probability problems. There is no advanced material in these chapters, but some of the probability problems can be difficult. More counting and probability problems are considered in later chapters. For example, there are trigonometry and three-dimensional geometry problems that require these notions. Chapters 10 and 11 concern problems with integer solutions. Since these problems frequently occur on the AMC, Chapter 10 is restricted to those problems that essentially deal with the Fundamental Theorem of Arithmetic, whereas Chapter 11 considers the more advanced topics of modular arithmetic and number bases. All of this material should be accessible to an interested younger student. Chapter 12 deals with sequences and series, with an emphasis on the arithmetic and geometric sequences that often occur on the AMC. Sequences whose terms are recursive and repeat are also considered, since the AMC sequence problems that are not arithmetic or geometric are frequently of this type. This material and that in Chapter 13 that deals with statistics may not be completely familiar to younger students, but there are only a few concepts to master, and some of these problems appear on the AMC 10. The final four chapters contain material that is not likely to be included on an AMC 10. Definitions for the basic trigonometric and logarithm functions are given in Chapters 14 and 17, respectively, but these may not be sufficient for a student who has not previously seen this material before. Chapter 15 considers problems that have a three-dimensional slant, and Chapter 16 looks at functions in a somewhat abstract setting. The final chapter on complex numbers illustrates that the knowledge of just a few concepts concerning this topic is all that is generally required, even for the AMC 12. One of the goals of the book is to permit a student to progress through the material in sequence. As problem-solving abilities improve, more difficult notions can be included, and problems presented that require greater ingenuity. When reviewing this material I hope that you will keep in mind that the intended student audience for this book is perhaps the top 10--15\% of an average high school class. The more mature (think parental) audience is probably the working engineer or scientist who has not done problems of this type for many years, if ever, but enjoys a logical challenge and/or wants to help students develop problem solving
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For those interested in using Mathematica in teaching
Mathematical modeling at Texas A&M University has been taught
by having the students work through Mathematica notebooks which
assign readings in the textbook "A First Course in Mathematical
Modeling" by Frank R. Giordano and Maurice D. Weir, contain
interactive examples and explanations, and assign exercises in which
students create and analyze a mathematical model. Students complete
an individual and group project during the class.
The objective of the course is
1. Illustrate the broad range of problems which can be modeled
mathematically.
2. Synthesize mathematical models from non-mathematical descriptions
of problems.
3. Interpret the results of models and evaluate their implications.
4. Show the necessity of simplification and approximation in models
and identify their effects.
5. Work cooperatively in groups.
Currently there are 11 Mathematica notebooks. Eight Mathematica
notebooks cover the first 9 topics of the syllabus below. The
remaining 3 additional Mathematica notebooks create new Mathematica
functions and illustrate their use with examples.
1. Simplex: Creates a tableau for a standard linear programming
maximization problem. These tableaus are useful for solving linear
programming problems by the standard maximization method, dual
method, Big M method, or for achieving a sensitivity analysis.
2. Natural Spline: Determines the equations for the natural cubic
spline functions which fit the data and includes examples of how to
graph these functions automatically.
3. Clamped Spline: Determines the equations for the clamped cubic
spline functions which fit the data for specified derivatives at the
endpoints and includes examples of how to graph these functions
automatically.
Handouts:
Theory of the Simplex Method
Theory of the Dual Simplex Method
Dimensional Analysis
Syllabus:
1. Nuclear Arms Race
Develop a simple probabilistic model
Observe an equilibrium point
Changing assumptions affects parameter values
Sensitivity of the equilibrium point to change in parameters
2. The Modeling Process-identifying a problem
Vehicular Stopping Distance
Automobile Gas Mileage
Elevator Service during the morning rush hour
3. Using Geometric Similarity in the Modeling Process
Overall winner in weight lifting across weight classes
Volume of lumber from the diameter of a tree at waist level
Predicting Pulse rate from body weight
Vehicular Stopping distance
4. Model Fitting--identifying the optimal parameters for a model
Error criteria--Least square and Chebyshev
Identifying Kepler's third law from observational data
5. Models requiring optimization--Linear programming/Critical points
Linear programming economic models
Inventory problem--minimize delivery and storage cost
6. Experimental Modeling
Problems with using high order polynomials to interpolate
data
Splines
7. Project 1: Find the flow rate and water use of a small town from
.5in height measurements of the water tank
Numerical differentiation of data
Fitting curves to transformed data
Error analysis
8. Dimensional Analysis and Similitude
Range of a cannonball
Damped pendulum
Terminal speed of a raindrop
Turkey cooking times
Model design for determining the drag force on a submarine
9. Simulation Modeling
Monte Carlo simulation of area and volume
Simulation of gas station delivery and storage cost for a
stochastic demand
Simulation of harbor waiting times for stochastic arrival and
unloading times
10. Project 2: The projects for Spring 93 were
A: Identifying a near optimal ratio of slurry, greens, and paper
for composting.
B: Identifying a near optimal work schedule for a coal tipple.
11. Differential equation models
Population models
Drug dosage
Managing the fishing industry
The notebooks on the Simplex and the splines are available from
Mathsource. From Mosaic you could access it by
Simplex #0207-447 and Natural
and Clamped Cubic spline coefficients #0207-436. If you are
interested in additional information, discussion of the advantages or
disadvantages of teaching in a format where students work in groups
or independently at a computer while the teacher acts a coach and
answers questions and stimulates inquiry, or any of the other
materials, please contact me.
Joseph Herrmann
Joseph M. Herrmann
Department of Mathematics, Texas A&M University
College Station, Texas 77843-3368
(409) 845-1474
herrmann at math.tamu.edu
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Algebra With Pizzazz Pdf
PDF Group - Lawton Chiles Middle School notes from the authors middle school math with pizza is a series of five books designed to provide practice with skills and concepts taught PDF Group - Lawton Chiles Middle School
PDF The Authors - Lawton Chiles Middle School not from the authors middle school matn with pizzazz is a series of five books designed to provide practice with skills and concepts taught PDF The Authors - Lawton Chiles Middle School
PDF ALEKS Math Assessment aleks math assessment by david l johnson a ll math teachers face the chal-lenge of responding to students individually in the classroom assessing each students PDF ALEKS Math Assessment
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Linear Algebra Grading Plan
I described my structure for my linear algebra class last week. This week, I will describe my grading system.
First, recall that this is typically a sophomore-level class (although 9 of my 16 students are first-years). It is typically the first mathematics course that a mathematics student takes after calculus II. The purpose is to learn about linear algebra—including abstract vector spaces—and gently introduce the students to proofs.
The grading system hinges on what I want the grades of , , and to mean. My linear algebra class does not vary from my usual interpretation: To get a , you must demonstrate conceptual understanding of the material. To get a , you must demonstrate both conceptual understanding and computational proficiency. To get an , you must additionally demonstrate an ability to learn independently.
Below, I describe the graded portions of the class, followed by a set of rules that determines the semester grade for each student. I reserve the right to raise a student's grade from what this rubric says, but I will never lower it. The rest of this post is taken directly from my syllabus (modulo some formatting and possible editorial comments); I would appreciate any thoughts that you have on how to improve this.
Here we go; it may be helpful to see my structuring if you have questions about anything.
Presentations
Each time you present a Presentation Problem, I will make a note of it. It is better to correctly do a problem than to incorrectly do a problem, and it is better to correctly do a difficult problem than it is to do a relatively easy problem. Both the quality and quantity of the Presentations you give will be considered in determining your semester grade.
Part of the goal of this class is to help you hone your mathematical judgment. Because of this, I will limit the number of presentations that I confirm are correct (I will confirm 2 per day for the first three classes, 1 per day for the enxt three classes, and then 5 for the entirety of the remaining part of the semester). Please note that I will still do something during the semester to correct any misconceptions the class has—I just won't necessarily do it immediately following the presentation unless the majority of the class wants to spend a Confirmation.
Giving a Presentation can never hurt your grade; the worst it can do is to fail to help your grade.
Daily Homework
You will be expected to submit you work for the Routine Problems and Presentation Problems at the end of each Monday and Friday during the first half of the semester. These will be graded on completion: as long as you make an honest effort at solving every Routine and Presentation Problem due that day, you will receive full credit. You are not expected to use for Daily Homework.
Since these are due at the end of the class, you may write on the homework with a marker (I will supply them, although I would appreciate it if any student brought his/her own). Your grade for Daily Homework will be based on the work you came to class with—the work in marker will not be graded.
Submitting Daily Homework can never hurt your grade; the worst it can do is to fail to help your grade.
Portfolio Homework
Each Monday and Friday during the first half of the semester, I will denote 1—2 of the Presentation Problems to be Portfolio Homework. This is homework that you are to write up nice solutions on .
Each problem will receive one of two grades: Complete or Resubmit. I will read your solution until I find an error. Once I find an error—it could be mathematical, grammatical, etc—I will mark it with Resubmit (or some suitable shorthand) and stop reading it. You can resubmit the same problem multiple times without penalty; if you eventually get a grade of Complete, you have been 100% successful on the problem, whether it was on your first attempt or your twenty-first.
You can only submit up to two Portfolio Homework problems per class day. This means that you should start working on them immediately—you do not want to have a lot of problems left to do during the last week of the semester, since you will not be able to submit all of them.
Submitting Portfolio Homework can never hurt your grade; the worst it can do is to fail to help your grade.
Mind Maps
You will be required to keep a "mind map" of all of the ideas described by the Chapter titles and Sub-Chapter titles in the table of contents of the course notes. Do this online using coggle.it, name it YOURNAME239MindMap, and share your work with me by using the "Share" button. You are to update this mind map weekly; I will check to make sure that it is up-to-date periodically, although I may not announce when I do so. (This is not in the syllabus: the whole point of the mind map is that I want students to start to intentionally make connections among the different topics..)
Quizzes
We will be having quizzes during the second half of the semester. Each question will be linked to a Learning Goal (and clearly labelled indicating which Goal it is attached to), and will be graded either as Acceptable (if there are no errors of any sort) or Not Acceptable (if an error exists).
You will need to get five problems correct for every Learning Goal to be considered to have successfully learned a particular Learning Goal. There will be a few Learning Goals (This is not in the syllabus: the topics are and factorization; the only time I am going to mention them at all in class is when I put them on the quiz. I will grade those quizzes, and perhaps answer some specific questions of students who come to office hours. But students really are expected to truly learn these topics on their own if they want an .) that are not covered in class; these are only for students who are aiming to get an in the class, and they are meant to be learned on your own.
Calculators will not be allowed on quizzes. You should also not need them.
Submitting a Quiz can never hurt your grade; the worst it can do is to fail to help your grade. Because of this, the course policy is that make-up quizzes will not be given; you should plan on "making up" the quiz by doing well on later quizzes.
Examinations
The quizzes largely take the place of most examinations. We will, however, have a final exam. There will be two components: We will have an in-class final exam on Thursday, May 15th at 8:00 am. The location is our usual classroom. Also, there will be a take-home portion of the final exam will be assigned in the last week of classes to be handed in at the final exam.
For students who wish to get an for the semester, there will be a brief oral examination. You will receive the topic prior to the oral examination. This oral examination can only be done once you have at least 4 correct answers for each of the quiz Learning Goals (so this does not need to be done during Finals Week).
Project
You are encouraged, but not required to do a project for this course. These projects will be mini-research projects. Your job is to find a problem (I will provide some possible problems), try to solve the problem, create a poster for it, and be prepared to answer questions about the topic and poster. Toward the end of the semester, we will have a poster presentation session for the class. The poster presentation will likely occur on Scholarship and Creativity Day.
Department Colloquia
Part of being a mathematician is to listen to mathematics. Because of this, you will be expected to attend some number of the Mathematics Colloquium, which occurs most every other Thursday at 2:40 pm (if you have a scheduling conflict, please let me know).
Grading
Here is how your semester grade will be determined:
To get a for the semester, you must:
You successfully answered at least one Query
You presented (perhaps unsuccessfully) at least a few times
You received credit on Daily Homework on all but at most three attempts
You have grades of Acceptable on all but at most two Portfolio Homework problems
You were successful on all of the Quiz Learning Goals in the Conceptual Learning Goal section
You maintained a mostly complete and mostly up-to-date mind map for the entire semester
You got at least a on the final exam.
To get a for the semester, you must:
You successfully answered at least two Queries
You many successful presentations
You received credit on Daily Homework on all but at most two attempts
You have grades of Acceptable on all Portfolio Homework problems
You were successful on all of the Quiz Learning Goals in the Conceptual Learning Goal and Computation Learning Goal sections
You maintained a complete and up-to-date mind map for the entire semester
You attend at least 1 Mathematics Colloquium this semester
You got at least a on the final exam.
To get a for the semester, you must:
You successfully answered at least two Queries
You many successful presentations of difficult problems
You received credit on Daily Homework on all but at most two attempts
You have grades of Acceptable on all Portfolio Homework problems
You were successful on all of the Quiz Learning Goals
You maintained a complete and up-to-date mind map for the entire semester
Sorry about not being clear. They students need to think about 16 problems per presentation day, although they only present 6 in class (which leads to a much more reasonable 8 or 9 minutes per presentation).
So, of the 16, 6 are presented (6 of 16 are accounted for); the students watch solutions to another 6 problems on YouTube (12 of the 16 are accounted for); and roughly 4 problems are not discussed at all (16 of the 16 are accounted for).
The 4 problems that are not discussed at all tend to be the easiest problems—usually computation problems that are similar to ones I already did in a video. Students are allowed to request a video on any of these if they are confused.
Does this help clarify? We only discuss a small, carefully chosen subset (roughly 6 of 16) of the problems in class.
Bret
I am envious that you are teaching Linear Algebra. That is one of my favorite courses! I noticed that you have a presentation requirement both in LinAlg and in your Problem Solving class. I have thought about, but not implemented, such a thing in my own courses. I can see a lot of reasons why this is a good thing to do. OTOH, I also remember dropping several courses as an undergrad exactly because of such a requirement — I was extremely shy and did not want to get up in front of people ever! While I'd agree this was probably ridiculous, especially given my current line of work, it does leave me with a good deal of empathy for students who would be uncomfortable in such a situation. Do you have any of those kinds of students? Do you have any alternate assignments for them? Or do you just encourage and coach them into getting over their fears?
I am definitely loving linear algebra. This is my third time teaching it here, fourth overall.
I also would have hated a course built on course presentations. Not because I was shy, but rather because I was just kind of a punk sometimes.
Here is the funny thing: there is a good chance that the only reason I am teaching linear algebra this semester is because students didn't want to present. I was originally slated to teach abstract algebra, but very few students signed up for my section (the other was full). Rumor has it that they did not like that I am a presentation-y teacher (there were issues aside from presentations, but the rumor was that my workload requirement was thought to be higher than the other instructor's).
The only issue I have come across are students who cannot present due to various medical issues. When this is the case, I always offer to make accommodations. My preferred accommodation is to present to just me in the office, although I have sometimes allowed for students to submit written work instead.
It hasn't come up, but—aside from a medical issue—I don't think that I would grant many other exceptions. For one, I think that it is a valuable skills for anyone to be reasonably comfortable talking about technical ideas in front of an audience. I can sell this pretty easily at a liberal arts school. My other reason—which I usually explciitly tell the students—is that I want to teach the students to think critically about what is presented; I find that students do not do this when I lecture, but are better at doubting the results if another student does it.
That said, I am not in love with the way I do presentations yet. I need to improve. If you (or anyone else) has any great ideas, let me know.
Bret
| 677.169 | 1 |
MERLOT Search - materialType=Simulation&category=2515&sort.property=overallRating
A search of MERLOT materialsCopyright 1997-2014 MERLOT. All rights reserved.Sat, 20 Sep 2014 13:31:27 PDTSat, 20 Sep 2014 13:31:27 PDTMERLOT Search - materialType=Simulation&category=2515&sort.property=overallRating
4434Connect the Dots
This site provides a visual approach to exploring finite cyclic groupsVector Addition (Physics, Math)
Graphically adds any two vectors to get a thirdGroup Games
An excellent group theoretic approach to the popular Rubik's cube game.The Vector Cross Product
Visually demonstrate and explore the vector cross product.Math Warehouse
This site has has interactive explanations and simulations of math from alegrbra to trigonometry. Just click the "interactive" tab on the top left menu and you can choose different simulations. It includes, the complete definition of parabolas, reaching beyond the ability to graph into the realm of why the graph appears as it does. It also has vivid descriptions of angles including circle angles for geometry. It also has calculators for principal nth roots, gdc, matrices, and prime factorization. It's definitely worth checking out. Quote from site: "A parabola is actually a locus of a point and a line. The point is called the focus and the line the directrix. That means that all points on a parabola are equidistant from the focus and the directrix. To change the equation and the graph of the interactive parabola below just click and drag either the point A, which is the focus, or point B, which controls the directrix." This is an interactive site that allows people to change the graph to understand why directrix and focus dictate parabolic graphs. GraphGraphical representation of eigenvectors
The Graphical representation of eigenvectors simulation aims to help students make connections between graphical and mathematical representations of eigenvectors and eigenvalues. The simulation depicts the two components of a unit vector in the xy-plane, and the same vector under several different transformations that can be chosen by the user. A slider allows students to change the orientation of the initial vector. The simulation shows whether or not the vector is an eigenvector, and if so displays the associated eigenvalue. The simulation includes a small challenge in asking students to find the elements of one of the transformation matrices 4
| 677.169 | 1 |
The revised second edition of this hands-on workbook presents exercises, problems, and quizzes with solutions and answers as it takes college-bound students through all math and science topics covered on the ACT. Separate math chapters cover:
Pre-algebra, elementary algebra, and intermediate algebra
Plane geometry, coordinate geometry, and trigonometry
The science sections emphasize the scientific method and focus on how to read scientific passages. Science topics covered include:
Data representation passages
Research summaries passages
Conflicting viewpoints passages
Additional features include a glossary of science terms and test-taking strategies for success. The workbook concludes with a full-length math and science practice test.
Table of Contents:
ACT Overview and Strategies PART 1: THE MATH SECTION
1 Overview of the Math Test
2 Pre-Algebra
3 Elementary Algebra
4 Intermediate Algebra
5 Plane and Solid Geometry
6 Coordinate Geometry
7 Trigonometry PART 2: THE SCIENCE SECTION
8 Overview of the Science Test
9 General Strategies for Science
10 Data Representation
11 Research Summaries
12 Conflicting Viewpoints PART 3: MODEL ACT TESTS
Practice Mathematics Test
Practice Science Test APPENDIXES
Appendix A: Some Useful Math Formulas
Appendix B: Some Useful Scientific Units
About The Author: Roselyn Teukolsky, M.S., is a math teacher at Ithaca High School, Ithaca, NY, and has many years' experience helping students prepare for the SAT.
For more information about the author, visit her website:
GREAT NEWS! Barrons has a test prep Website for the ACT and SAT I exams! You can take complete ACT and SAT I practice tests, select questions from either the Math or Verbal sections, or concentrate on particular question types. When you are finished, you'll get immediate feedback, including answers to all ACT and SAT I questions with full explanations. Instant results pinpoint your strengths and weaknesses and let you know where you need to practice most. Head on over to BarronsTestPrep.com to start scoring big on the ACT and SAT I exam today!
| 677.169 | 1 |
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About the book:
This book is intended to introduce students to algebraic geometry; to give them a sense of the basic objects considered, the questions asked about them, and the sort of answers one can expect to obtain. It thus emplasizes the classical roots of the subject. For readers interested in simply seeing what the subject is about, this avoids the more technical details better treated with the most recent methods. For readers interested in pursuing the subject further, this book will provide a basis for understanding the developments of the last half century, which have put the subject on a radically new footing. Based on lectures given at Brown and Harvard Universities, this book retains the informal style of the lectures and stresses examples throughout; the theory is developed as needed. The first part is concerned with introducing basic varieties and constructions; it describes, for example, affine and projective varieties, regular and rational maps, and particular classes of varieties such as determinantal varieties and algebraic groups. The second part discusses attributes of varieties, including dimension, smoothness, tangent spaces and cones, degree, and parameter and moduli spaces.
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Hardcover, ISBN 3540977163 Publisher: Springer-Verlag, 1995 Used - Very Good, Usually ships in 24 hours, There is a name inside the cover. There is some yellowing on the edges of the pages. Otherwise this book is in perfect condition.. Shipped from Amazon. Amazon Prime. Eligible for FREE Super Saver Shipping.
| 677.169 | 1 |
Elementary Linear Algebra - 8th edition
Summary: This classic treatment of linear algebra presents the fundamentals in the clearest possible way, examining basic ideas by means of computational examples and geometrical interpretation. It proceeds from familiar concepts to the unfamiliar, from the concrete to the abstract.
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| 677.169 | 1 |
This classic of the mathematical literature forms a comprehensive study of the inequalities used throughout mathematics. First published in 1934, it presents clearly and exhaustively both the statement and proof of all the standard inequalities of analysis. The authors were well known for their powers of exposition and were able here to make the subject...
| 677.169 | 1 |
More About
This Textbook
Overview
Introductory courses in Linear Algebra can be taught in a variety of ways and the order of topics offered may vary based on the needs of the students. Linear Algebra with Applications, Alternate Eighth Edition provides instructors with an additional presentation of course material. In this edition earlier chapters cover systems of linear equations, matrices, and determinants. The more abstract material on vector spaces starts later, in Chapter 4, with the introduction of the vector space R(n). This leads directly into general vector spaces and linear transformations. This alternate edition is especially appropriate for students preparing to apply linear equations and matrices in their own fields.
Clear, concise, and comprehensive--the Alternate Eighth Edition continues to educate and enlighten students, leading to a mastery of the matehmatics and an understainding of how to apply it.
New and Key Features of the Alternate Eighth Edition:
- Updated and revised throughout with new section material and exercises included in every chapter.
- Provides students with a flexible blend of theory, important numerical techniques and interesting relevant applications.
- Includes discussions of the role of linear algebra in many areas such as the operation of the Google search engine and the global structure of the worldwide air transportation network.
- A MATLAB manual that ties into the regular course material is included as an appendix. These ideas can be implemented on any matrix algebra software package. A graphing calculator manual is also included.
- A Student Solutions Manual that contain solutions to selected exercises is available as a supplement, An Instructor Complete Solutions Manual containing worked solutions to all exercises is also
| 677.169 | 1 |
1
00:00:00 --> 00:00:05
OK, this is linear algebra
lecture nine.
2
00:00:05 --> 00:00:11
And this is a key lecture,
this is where we get these
3
00:00:11 --> 00:00:18
ideas of linear independence,
when a bunch of vectors are
4
00:00:18 --> 00:00:24
independent -- or dependent,
that's the opposite.
5
00:00:24 --> 00:00:27
The space they span.
6
00:00:27 --> 00:00:33.65
A basis for a subspace or a
basis for a vector space,
7
00:00:33.65 --> 00:00:37.85
that's a central idea.
8
00:00:37.85 --> 00:00:42
And then the dimension of that
subspace.
9
00:00:42 --> 00:00:47
So this is the day that those
words get assigned clear
10
00:00:47 --> 00:00:48
meanings.
11
00:00:48 --> 00:00:54.91
And emphasize that we talk
about a bunch of vectors being
12
00:00:54.91 --> 00:00:56
independent.
13
00:00:56 --> 00:01:02
Wouldn't talk about a matrix
being independent.
14
00:01:02 --> 00:01:05
A bunch of vectors being
independent.
15
00:01:05 --> 00:01:08
A bunch of vectors spanning a
space.
16
00:01:08 --> 00:01:11
A bunch of vectors being a
basis.
17
00:01:11 --> 00:01:14
And the dimension is some
number.
18
00:01:14 --> 00:01:17
OK, so what are the
definitions?
19
00:01:17 --> 00:01:21
Can I begin with a fact,
a highly important fact,
20
00:01:21 --> 00:01:27
that, I didn't call directly
attention to earlier.
21
00:01:27 --> 00:01:32
Suppose I have a matrix and I
look at Ax equals zero.
22
00:01:32 --> 00:01:38
Suppose the matrix has a lot of
columns, so that n is bigger
23
00:01:38 --> 00:01:39
than m.
24
00:01:39 --> 00:01:43
So I'm looking at n equations
-- I mean, sorry,
25
00:01:43 --> 00:01:47
m equations,
a small number of equations m,
26
00:01:47 --> 00:01:49
and more unknowns.
27
00:01:49 --> 00:01:53
I have more unknowns than
equations.
28
00:01:53 --> 00:01:56
Let me write that down.
29
00:01:56 --> 00:02:00
More unknowns than equations.
30
00:02:00 --> 00:02:03
More unknown x-s than
equations.
31
00:02:03 --> 00:02:10
Then the conclusion is that
there's something in the null
32
00:02:10 --> 00:02:16
space of A, other than just the
zero vector.
33
00:02:16 --> 00:02:23
The conclusion is there are
some non-zero x-s such that Ax
34
00:02:23 --> 00:02:24
is zero.
35
00:02:24 --> 00:02:28.26
There are some special
solutions.
36
00:02:28.26 --> 00:02:29.25
And why?
37
00:02:29.25 --> 00:02:32.47
We know why.
38
00:02:32.47 --> 00:02:36
I mean, it sort of like seems
like a reasonable thing,
39
00:02:36 --> 00:02:41
more unknowns than equations,
then it seems reasonable that
40
00:02:41 --> 00:02:43
we can solve them.
41
00:02:43 --> 00:02:46
But we have a,
a clear algorithm which starts
42
00:02:46 --> 00:02:51
with a system and does
elimination, gets the thing into
43
00:02:51 --> 00:02:55.62
an echelon form with some pivots
and pivot columns,
44
00:02:55.62 --> 00:03:01
and possibly some free columns
that don't have pivots.
45
00:03:01 --> 00:03:09
And the point is here there
will be some free columns.
46
00:03:09 --> 00:03:17
The reason, so the reason is
there must -- there will be free
47
00:03:17 --> 00:03:21
variables, at least one.
48
00:03:21 --> 00:03:24
That's the reason.
49
00:03:24 --> 00:03:30
That we now have this --
a complete, algorithm,
50
00:03:30 --> 00:03:36
a complete systematic way to
say, OK, we take the system Ax
51
00:03:36 --> 00:03:41
equals zero, we row reduce,
we identify the free variables,
52
00:03:41 --> 00:03:46
and, since there are n
variables and at most m pivots,
53
00:03:46 --> 00:03:50
there will be some free
variables, at least one,
54
00:03:50 --> 00:03:54
at least n-m in fact,
left over.
55
00:03:54 --> 00:03:57
And those variables I can
assign non-zero values to.
56
00:03:57 --> 00:04:00
I don't have to set those to
zero.
57
00:04:00 --> 00:04:03
I can take them to be one or
whatever I like,
58
00:04:03 --> 00:04:06
and then I can solve for the
pivot variables.
59
00:04:06 --> 00:04:09
So then it gives me a solution
to Ax equals zero.
60
00:04:09 --> 00:04:12
And it's a solution that isn't
all zeros.
61
00:04:12 --> 00:04:19
So, that's an important point
that we'll use now in this
62
00:04:19 --> 00:04:20
lecture.
63
00:04:20 --> 00:04:27
So now I want to say what does
it mean for a bunch of vectors
64
00:04:27 --> 00:04:29
to be independent.
65
00:04:29 --> 00:04:29
OK.
66
00:04:29 --> 00:04:35
So this is like the background
that we know.
67
00:04:35 --> 00:04:42
Now I want to speak about
independence.
68
00:04:42 --> 00:04:42
OK.
69
00:04:42 --> 00:04:44
Let's see.
70
00:04:44 --> 00:04:53
I can give you the abstract
definition, and I will,
71
00:04:53 --> 00:05:02
but I would also like to give
you the direct meaning.
72
00:05:02 --> 00:05:08.86
So the question is,
when vectors x1,
73
00:05:08.86 --> 00:05:15
x2 up to --
Suppose I have n vectors are
74
00:05:15 --> 00:05:16
independent if.
75
00:05:16 --> 00:05:22.5
Now I have to give you -- or
linearly independent -- I'll
76
00:05:22.5 --> 00:05:27
often just say and write
independent for short.
77
00:05:27 --> 00:05:27
OK.
78
00:05:27 --> 00:05:30
I'll give you the full
definition.
79
00:05:30 --> 00:05:35
These are just vectors in some
vector space.
80
00:05:35 --> 00:05:39
I can take combinations of
them.
81
00:05:39 --> 00:05:45
The question is,
do any combinations give zero?
82
00:05:45 --> 00:05:52
If some combination of those
vectors gives the zero vector,
83
00:05:52 --> 00:05:58
other than the combination of
all zeros, then they're
84
00:05:58 --> 00:05:59
dependent.
85
00:05:59 --> 00:06:06
They're independent if no
combination gives the zero
86
00:06:06 --> 00:06:14
vector -- and then I have,
I'll have to put in an except
87
00:06:14 --> 00:06:17
the zero combination.
88
00:06:17 --> 00:06:20
So what do I mean by that?
89
00:06:20 --> 00:06:25
No combination gives the zero
vector.
90
00:06:25 --> 00:06:33
Any combination c1 x1+c2 x2
plus, plus cn xn is not zero
91
00:06:33 --> 00:06:37
except for the zero combination.
92
00:06:37 --> 00:06:44
This is when all the c-s,
all the c-s are zero.
93
00:06:44 --> 00:06:46
Then of course.
94
00:06:46 --> 00:06:49
That combination -- I know I'll
get zero.
95
00:06:49 --> 00:06:53
But the question is,
does any other combination give
96
00:06:53 --> 00:06:54
zero?
97
00:06:54 --> 00:06:57.57
If not, then the vectors are
independent.
98
00:06:57.57 --> 00:07:02
If some other combination does
give zero, the vectors are
99
00:07:02 --> 00:07:04
dependent.
100
00:07:04 --> 00:07:04.33
OK.
101
00:07:04.33 --> 00:07:06
Let's just take examples.
102
00:07:06 --> 00:07:11
Suppose I'm in,
say, in two dimensional space.
103
00:07:11 --> 00:07:12
OK.
104
00:07:12 --> 00:07:18
I give you -- I'd like to first
take an example -- let me take
105
00:07:18 --> 00:07:25
an example where I have a vector
and twice that vector.
106
00:07:25 --> 00:07:27
So that's two vectors,
V and 2V.
107
00:07:27 --> 00:07:30
Are those dependent or
independent?
108
00:07:30 --> 00:07:34
Those are dependent for sure,
right, because there's one
109
00:07:34 --> 00:07:36
vector is twice the other.
110
00:07:36 --> 00:07:41.15
One vector is twice as long as
the other, so if the word
111
00:07:41.15 --> 00:07:45
dependent means anything,
these should be dependent.
112
00:07:45 --> 00:07:47
And they are.
113
00:07:47 --> 00:07:51
And in fact,
I would take two of the first
114
00:07:51 --> 00:07:54
-- so here's,
here is a vector V and the
115
00:07:54 --> 00:07:59
other guy is a vector 2V,
that's my -- so there's a
116
00:07:59 --> 00:08:03
vector V1 and my next vector V2
is 2V1.
117
00:08:03 --> 00:08:08.42
Of course those are dependent,
because two of these first
118
00:08:08.42 --> 00:08:13
vectors minus the second vector
is zero.
119
00:08:13 --> 00:08:18
That's a combination of these
two vectors that gives the zero
120
00:08:18 --> 00:08:18
vector.
121
00:08:18 --> 00:08:20.12
OK, that was clear.
122
00:08:20.12 --> 00:08:24
Suppose, suppose I have a
vector -- here's another
123
00:08:24 --> 00:08:24
example.
124
00:08:24 --> 00:08:26
It's easy example.
125
00:08:26 --> 00:08:30
Suppose I have a vector and the
other guy is the zero vector.
126
00:08:30 --> 00:08:36
Suppose I have a vector V1 and
V2 is the zero vector.
127
00:08:36 --> 00:08:40
Then are those vectors
dependent or independent?
128
00:08:40 --> 00:08:42
They're dependent again.
129
00:08:42 --> 00:08:46
You could say,
well, this guy is zero times
130
00:08:46 --> 00:08:47
that one.
131
00:08:47 --> 00:08:51
This one is some combination of
those.
132
00:08:51 --> 00:08:54
But let me write it the other
way.
133
00:08:54 --> 00:08:58
Let me say --
what combination,
134
00:08:58 --> 00:09:03
how many V1s and how many V2s
shall I take to get the zero
135
00:09:03 --> 00:09:03
vector?
136
00:09:03 --> 00:09:08
If, if V1 is like the vector
two one and V2 is the zero
137
00:09:08 --> 00:09:12
vector, zero zero,
then I would like to show that
138
00:09:12 --> 00:09:16
some combination of those gives
the zero vector.
139
00:09:16 --> 00:09:18
What shall I take?
140
00:09:18 --> 00:09:21
How many V1s shall I take?
141
00:09:21 --> 00:09:23
Zero of them.
142
00:09:23 --> 00:09:26
Yeah, no, take no V1s.
143
00:09:26 --> 00:09:28
But how many V2s?
144
00:09:28 --> 00:09:29
Six.
145
00:09:29 --> 00:09:29
OK.
146
00:09:29 --> 00:09:30
Or five.
147
00:09:30 --> 00:09:36
Then -- in other words,
the point is if the zero
148
00:09:36 --> 00:09:42
vector's in there,
if the zero --
149
00:09:42 --> 00:09:46
if one of these vectors is the
zero vector, independence is
150
00:09:46 --> 00:09:47
dead, right?
151
00:09:47 --> 00:09:51
If one of those vectors is the
zero vector then I could always
152
00:09:51 --> 00:09:54
take -- include that one and
none of the others,
153
00:09:54 --> 00:09:57
and I would get the zero
answer, and I would show
154
00:09:57 --> 00:09:58
dependence.
155
00:09:58 --> 00:09:58.32
OK.
156
00:09:58.32 --> 00:10:01
Now, let me,
let me finally draw an example
157
00:10:01 --> 00:10:04
where they will be independent.
158
00:10:04 --> 00:10:07
Suppose that's V1 and that's
V2.
159
00:10:07 --> 00:10:10
Those are surely independent,
right?
160
00:10:10 --> 00:10:15
Any combination of V1 and V2,
will not be zero except,
161
00:10:15 --> 00:10:17.85
the zero combination.
162
00:10:17.85 --> 00:10:20
So those would be independent.
163
00:10:20 --> 00:10:25
But now let me,
let me stick in a third vector,
164
00:10:25 --> 00:10:25
V3.
165
00:10:25 --> 00:10:31
Independent or dependent now,
those three vectors?
166
00:10:31 --> 00:10:33
So now n is three here.
167
00:10:33 --> 00:10:39
I'm in two dimensional space,
whatever, I'm in the plane.
168
00:10:39 --> 00:10:43
I have three vectors that I
didn't draw so carefully.
169
00:10:43 --> 00:10:47
I didn't even tell you what
exactly they were.
170
00:10:47 --> 00:10:53
But what's this answer on
dependent or independent?
171
00:10:53 --> 00:10:54
Dependent.
172
00:10:54 --> 00:10:57
How do I know those are
dependent?
173
00:10:57 --> 00:11:02
How do I know that some
combination of V1,
174
00:11:02 --> 00:11:05
V2, and V3 gives me the zero
vector?
175
00:11:05 --> 00:11:08
I know because of that.
176
00:11:08 --> 00:11:14
That's the key fact that tells
me that three vectors in the
177
00:11:14 --> 00:11:18
plane have to be dependent.
178
00:11:18 --> 00:11:19
Why's that?
179
00:11:19 --> 00:11:24
What's the connection between
the dependence of these three
180
00:11:24 --> 00:11:26
vectors and that fact?
181
00:11:26 --> 00:11:26
OK.
182
00:11:26 --> 00:11:29
So here's the connection.
183
00:11:29 --> 00:11:33
I take the matrix A that has V1
in its first column,
184
00:11:33 --> 00:11:39
V2 in its second column,
V3 in its third column.
185
00:11:39 --> 00:11:42
So it's got three columns.
186
00:11:42 --> 00:11:48
And V1 -- I don't know,
that looks like about two one
187
00:11:48 --> 00:11:48
to me.
188
00:11:48 --> 00:11:52
V2 looks like it might be one
two.
189
00:11:52 --> 00:11:59
V3 looks like it might be maybe
two, maybe two and a half,
190
00:11:59 --> 00:12:00
minus one.
191
00:12:00 --> 00:12:00
OK.
192
00:12:00 --> 00:12:07
Those are my three vectors,
and I put them in the columns
193
00:12:07 --> 00:12:09
of A.
194
00:12:09 --> 00:12:13
Now that matrix A is two by
three.
195
00:12:13 --> 00:12:20.26
It fits this pattern,
that where we know we've got
196
00:12:20.26 --> 00:12:25
extra variables,
we know we have some free
197
00:12:25 --> 00:12:33
variables, we know that there's
some combination --
198
00:12:33 --> 00:12:36
and let me instead of x-s,
let me call them c1,
199
00:12:36 --> 00:12:39
c2, and c3 -- that gives the
zero vector.
200
00:12:39 --> 00:12:42
Sorry that my little bit of art
got in the way.
201
00:12:42 --> 00:12:43
Do you see the point?
202
00:12:43 --> 00:12:47
When I have a matrix,
I'm interested in whether its
203
00:12:47 --> 00:12:49
columns are dependent or
independent.
204
00:12:49 --> 00:12:53
The columns are dependent if
there is something in the null
205
00:12:53 --> 00:12:55.23
space.
206
00:12:55.23 --> 00:12:59.31
The columns are dependent
because this,
207
00:12:59.31 --> 00:13:05
this thing in the null space
says that c1 of that plus c2 of
208
00:13:05 --> 00:13:08.76
that plus c3 of this is zero.
209
00:13:08.76 --> 00:13:12
So in other words,
I can go out some V1,
210
00:13:12 --> 00:13:17.78
out some more V2,
back on V3, and end up zero.
211
00:13:17.78 --> 00:13:19
OK.
212
00:13:19 --> 00:13:27
So let -- here I've give the
general, abstract definition,
213
00:13:27 --> 00:13:35
but let me repeat that
definition -- this is like
214
00:13:35 --> 00:13:40
repeat -- let me call them Vs
now.
215
00:13:40 --> 00:13:47
V1 up to Vn are the columns of
a matrix A.
216
00:13:47 --> 00:13:53
In other words,
this is telling me that if I'm
217
00:13:53 --> 00:14:00
in m dimensional space,
like two dimensional space in
218
00:14:00 --> 00:14:07
the example, I can answer the
dependence-independence question
219
00:14:07 --> 00:14:14.7
directly by putting those
vectors in the columns of a
220
00:14:14.7 --> 00:14:16
matrix.
221
00:14:16 --> 00:14:22
They are independent if the
null space of A,
222
00:14:22 --> 00:14:24
of A, is what?
223
00:14:24 --> 00:14:32
If I have a bunch of columns in
a matrix, I'm looking at their
224
00:14:32 --> 00:14:38
combinations,
but that's just A times the
225
00:14:38 --> 00:14:39
vector of c-s.
226
00:14:39 --> 00:14:47
And these columns will be
independent if the null space of
227
00:14:47 --> 00:14:51
A is the zero vector.
228
00:14:51 --> 00:15:02
They are dependent if there's
something else in there.
229
00:15:02 --> 00:15:11
If there's something else in
the null space,
230
00:15:11 --> 00:15:23
if A times c gives the zero
vector for some non-zero vector
231
00:15:23 --> 00:15:29
c in the null space.
232
00:15:29 --> 00:15:32
Then they're dependent,
because that's telling me a
233
00:15:32 --> 00:15:36
combination of the columns gives
the zero column.
234
00:15:36 --> 00:15:38.96
I think you're with be,
because we've seen,
235
00:15:38.96 --> 00:15:42
like, lecture after lecture,
we're looking at the
236
00:15:42 --> 00:15:46
combinations of the columns and
asking, do we get zero or don't
237
00:15:46 --> 00:15:47.62
we?
238
00:15:47.62 --> 00:15:50
And now we're giving the
official name,
239
00:15:50 --> 00:15:54
dependent if we do,
independent if we don't.
240
00:15:54 --> 00:15:57
So I could express this in
other words now.
241
00:15:57 --> 00:16:02
I could say the rank -- what's
the rank in this independent
242
00:16:02 --> 00:16:03
case?
243
00:16:03 --> 00:16:05
The rank r of the,
of the matrix,
244
00:16:05 --> 00:16:10
in the case of independent
columns, is?
245
00:16:10 --> 00:16:12
So the columns are independent.
246
00:16:12 --> 00:16:16
So how many pivot columns have
I got.
247
00:16:16 --> 00:16:16
All n.
248
00:16:16 --> 00:16:21
All the columns would be pivot
columns, because free columns
249
00:16:21 --> 00:16:27
are telling me that they're a
combination of earlier columns.
250
00:16:27 --> 00:16:32.24
So this would be the case where
the rank is n.
251
00:16:32.24 --> 00:16:38
This would be the case where
the rank is smaller than n.
252
00:16:38 --> 00:16:45
So in this case the rank is n
and the null space of A is only
253
00:16:45 --> 00:16:47
the zero vector.
254
00:16:47 --> 00:16:50
And no free variables.
255
00:16:50 --> 00:16:52
No free variables.
256
00:16:52 --> 00:16:57
And this is the case yes free
variables.
257
00:16:57 --> 00:17:05
If you'll allow me to stretch
the English language that far.
258
00:17:05 --> 00:17:11
That's the case where we have,
a combination that gives the
259
00:17:11 --> 00:17:12
zero column.
260
00:17:12 --> 00:17:19
I'm often interested in the
case when my vectors are popped
261
00:17:19 --> 00:17:20
into a matrix.
262
00:17:20 --> 00:17:26
So the, the definition over
there of independence didn't
263
00:17:26 --> 00:17:30
talk about any matrix.
264
00:17:30 --> 00:17:35
The vectors didn't have to be
vectors in N dimensional space.
265
00:17:35 --> 00:17:40
And I want to give you some
examples of vectors that aren't
266
00:17:40 --> 00:17:44
what you think of immediately as
vectors.
267
00:17:44 --> 00:17:48
But most of the time,
this is -- the vectors we think
268
00:17:48 --> 00:17:50
of are columns.
269
00:17:50 --> 00:17:53
And we can put them in a
matrix.
270
00:17:53 --> 00:17:59
And then independence or
dependence comes back to the
271
00:17:59 --> 00:18:00
null space.
272
00:18:00 --> 00:18:00
OK.
273
00:18:00 --> 00:18:04
So that's the idea of
independence.
274
00:18:04 --> 00:18:09.15
Can I just, yeah,
let me go on to spanning a
275
00:18:09.15 --> 00:18:09
space.
276
00:18:09 --> 00:18:15
What does it mean for a bunch
of vectors to span a space?
277
00:18:15 --> 00:18:21
Well, actually,
we've seen it already.
278
00:18:21 --> 00:18:25
You remember,
if we had a columns in a
279
00:18:25 --> 00:18:32
matrix, we took all their
combinations and that gave us
280
00:18:32 --> 00:18:34
the column space.
281
00:18:34 --> 00:18:41
Those vectors that we started
with span that column space.
282
00:18:41 --> 00:18:49
So spanning a space means -- so
let me move that important stuff
283
00:18:49 --> 00:18:51
right up.
284
00:18:51 --> 00:18:52
OK.
285
00:18:52 --> 00:19:02
So vectors -- let me call them,
say, V1 up to -- call you some
286
00:19:02 --> 00:19:10
different letter,
say Vl -- span a space,
287
00:19:10 --> 00:19:18
a subspace, or just a vector
space I could say,
288
00:19:18 --> 00:19:26
span a space means,
means the space consists of all
289
00:19:26 --> 00:19:34
combinations of those vectors.
290
00:19:34 --> 00:19:39
That's exactly what we did with
the column space.
291
00:19:39 --> 00:19:46
So now I could say in shorthand
the columns of a matrix span the
292
00:19:46 --> 00:19:48
column space.
293
00:19:48 --> 00:19:55
So you remember it's a bunch of
vectors that have this property
294
00:19:55 --> 00:20:01
that they span a space,
and actually if I give you a
295
00:20:01 --> 00:20:07.72
bunch of vectors and say --
OK, let S be the space that
296
00:20:07.72 --> 00:20:13
they span, in other words let S
contain all their combinations,
297
00:20:13 --> 00:20:17
that space S will be the
smallest space with those
298
00:20:17 --> 00:20:18
vectors in it,
right?
299
00:20:18 --> 00:20:24
Because any space with those
vectors in it must have all the
300
00:20:24 --> 00:20:27
combinations of those vectors in
it.
301
00:20:27 --> 00:20:32
And if I stop there,
then I've got the smallest
302
00:20:32 --> 00:20:37
space, and that's the space that
they span.
303
00:20:37 --> 00:20:37
OK.
304
00:20:37 --> 00:20:41
So I'm just -- rather than,
needing to say,
305
00:20:41 --> 00:20:47.25
take all linear combinations
and put them in a space,
306
00:20:47.25 --> 00:20:51
I'm compressing that into the
word span.
307
00:20:51 --> 00:20:54.24
Straightforward.
308
00:20:54.24 --> 00:20:54
OK.
309
00:20:54 --> 00:20:58.78
So if I think of a,
of the column space of a
310
00:20:58.78 --> 00:20:59
matrix.
311
00:20:59 --> 00:21:03
I've got their -- so I start
with the columns.
312
00:21:03 --> 00:21:06
I take all their combinations.
313
00:21:06 --> 00:21:09
That gives me the columns
space.
314
00:21:09 --> 00:21:12
They span the column space.
315
00:21:12 --> 00:21:15
Now are they independent?
316
00:21:15 --> 00:21:18
Maybe yes, maybe no.
317
00:21:18 --> 00:21:23
It depends on the particular
columns that went into that
318
00:21:23 --> 00:21:23
matrix.
319
00:21:23 --> 00:21:28
But obviously I'm highly
interested in a set of vectors
320
00:21:28 --> 00:21:31
that spans a space and is
independent.
321
00:21:31 --> 00:21:37.29
That's, that means like I've
got the right number of vectors.
322
00:21:37.29 --> 00:21:42
If I didn't have all of them,
I wouldn't have my whole space.
323
00:21:42 --> 00:21:48
If I had more than that,
they probably wouldn't --
324
00:21:48 --> 00:21:52
they wouldn't be independent.
325
00:21:52 --> 00:22:00
So, like, basis -- and that's
the word that's coming -- is
326
00:22:00 --> 00:22:02
just right.
327
00:22:02 --> 00:22:08
So here let me put what that
word means.
328
00:22:08 --> 00:22:16
A basis for a vector space is,
is a, is a sequence of vectors
329
00:22:16 --> 00:22:20
--
shall I call them V1,
330
00:22:20 --> 00:22:26
V2, up to let me say Vd now,
I'll stop with that letters --
331
00:22:26 --> 00:22:28
that has two properties.
332
00:22:28 --> 00:22:32
I've got enough vectors and not
too many.
333
00:22:32 --> 00:22:36
It's a natural idea of a basis.
334
00:22:36 --> 00:22:41
So a basis is a bunch of
vectors in the space and it's a
335
00:22:41 --> 00:22:46
so it's a sequence of vectors
with two properties,
336
00:22:46 --> 00:22:49
with two properties.
337
00:22:49 --> 00:22:55
One, they are independent.
338
00:22:55 --> 00:23:02
And two -- you know what's
coming?
339
00:23:02 --> 00:23:07
-- they span the space.
340
00:23:07 --> 00:23:07
OK.
341
00:23:07 --> 00:23:17
Let me take -- so time for
examples, of course.
342
00:23:17 --> 00:23:26
So I'm asking you now to put
definition one,
343
00:23:26 --> 00:23:37
the definition of independence,
together with definition two,
344
00:23:37 --> 00:23:41
and let's look at examples,
because this is -- this
345
00:23:41 --> 00:23:45
combination means the set I've
-- of vectors I have is just
346
00:23:45 --> 00:23:50
right, and the -- so that this
idea of a basis will be central.
347
00:23:50 --> 00:23:54
I'll always be asking you now
for a basis.
348
00:23:54 --> 00:23:58
Whenever I look at a subspace,
if I ask you for -- if you give
349
00:23:58 --> 00:24:02
me a basis for that subspace,
you've told me what it is.
350
00:24:02 --> 00:24:05
You've told me everything I
need to know about that
351
00:24:05 --> 00:24:06
subspace.
352
00:24:06 --> 00:24:09
Those -- I take their
combinations and I know that I
353
00:24:09 --> 00:24:11
need all the combinations.
354
00:24:11 --> 00:24:12
OK.
355
00:24:12 --> 00:24:13
Examples.
356
00:24:13 --> 00:24:17
OK, so examples of a basis.
357
00:24:17 --> 00:24:23.02
Let me start with two
dimensional space.
358
00:24:23.02 --> 00:24:27
Suppose the space -- say
example.
359
00:24:27 --> 00:24:32.44
The space is,
oh, let's make it R^3.
360
00:24:32.44 --> 00:24:36
Real three dimensional space.
361
00:24:36 --> 00:24:39
Give me one basis.
362
00:24:39 --> 00:24:42
One basis is?
363
00:24:42 --> 00:24:47
So I want some vectors,
because if I ask you for a
364
00:24:47 --> 00:24:52
basis, I'm asking you for
vectors, a little list of
365
00:24:52 --> 00:24:53
vectors.
366
00:24:53 --> 00:24:56
And it should be just right.
367
00:24:56 --> 00:25:01
So what would be a basis for
three dimensional space?
368
00:25:01 --> 00:25:05
Well, the first basis that
comes to mind,
369
00:25:05 --> 00:25:10
why don't we write that down.
370
00:25:10 --> 00:25:15
The first basis that comes to
mind is this vector,
371
00:25:15 --> 00:25:18
this vector,
and this vector.
372
00:25:18 --> 00:25:18
OK.
373
00:25:18 --> 00:25:20
That's one basis.
374
00:25:20 --> 00:25:25
Not the only basis,
that's going to be my point.
375
00:25:25 --> 00:25:29
But let's just see -- yes,
that's a basis.
376
00:25:29 --> 00:25:34
Are, are those vectors
independent?
377
00:25:34 --> 00:25:38
So that's the like the x,
y, z axes, so if those are not
378
00:25:38 --> 00:25:40
independent, we're in trouble.
379
00:25:40 --> 00:25:42
Certainly, they are.
380
00:25:42 --> 00:25:46
Take a combination c1 of this
vector plus c2 of this vector
381
00:25:46 --> 00:25:50
plus c3 of that vector and try
to make it give the zero vector.
382
00:25:50 --> 00:25:52
What are the c-s?
383
00:25:52 --> 00:25:56
If c1 of that plus c2 of that
plus c3 of that gives me 0 0 0,
384
00:25:56 --> 00:25:59
then the c-s are all -- 0,
right.
385
00:25:59 --> 00:26:03
So that's the test for
independence.
386
00:26:03 --> 00:26:09
In the language of matrices,
which was under that board,
387
00:26:09 --> 00:26:13
I could make those the columns
of a matrix.
388
00:26:13 --> 00:26:17
Well, it would be the identity
matrix.
389
00:26:17 --> 00:26:21
Then I would ask,
what's the null space of the
390
00:26:21 --> 00:26:24
identity matrix?
391
00:26:24 --> 00:26:27
And you would say it's only the
zero vector.
392
00:26:27 --> 00:26:31
And I would say,
fine, then the columns are
393
00:26:31 --> 00:26:32.19
independent.
394
00:26:32.19 --> 00:26:36
The only thing -- the identity
times a vector giving zero,
395
00:26:36 --> 00:26:40
the only vector that does that
is zero.
396
00:26:40 --> 00:26:40
OK.
397
00:26:40 --> 00:26:45
Now that's not the only basis.
398
00:26:45 --> 00:26:47
Far from it.
399
00:26:47 --> 00:26:53
Tell me another basis,
a second basis,
400
00:26:53 --> 00:26:55
another basis.
401
00:26:55 --> 00:27:02
So, give me -- well,
I'll just start it out.
402
00:27:02 --> 00:27:04
One one two.
403
00:27:04 --> 00:27:06
Two two five.
404
00:27:06 --> 00:27:12.69
Suppose I stopped there.
405
00:27:12.69 --> 00:27:18
Has that little bunch of
vectors got the properties that
406
00:27:18 --> 00:27:21
I'm asking for in a basis for
R^3?
407
00:27:21 --> 00:27:25
We're looking for a basis for
R^3.
408
00:27:25 --> 00:27:29
Are they independent,
those two column vectors?
409
00:27:29 --> 00:27:30
Yes.
410
00:27:30 --> 00:27:33
Do they span R^3?
411
00:27:33 --> 00:27:33.35
No.
412
00:27:33.35 --> 00:27:34
Our feeling is no.
413
00:27:34 --> 00:27:36
Our feeling is no.
414
00:27:36 --> 00:27:41
Our feeling is that there're
some vectors in R3 that are not
415
00:27:41 --> 00:27:43
combinations of those.
416
00:27:43 --> 00:27:43.29
OK.
417
00:27:43.29 --> 00:27:47
So suppose I add in -- I need
another vector then,
418
00:27:47 --> 00:27:51
because these two don't span
the space.
419
00:27:51 --> 00:27:51
OK.
420
00:27:51 --> 00:27:56
Now it would be foolish for me
to put in three three seven,
421
00:27:56 --> 00:27:58
right, as the third vector.
422
00:27:58 --> 00:28:00
That would be a goof.
423
00:28:00 --> 00:28:04
Because that,
if I put in three three seven,
424
00:28:04 --> 00:28:08.65
those vectors would be
dependent, right?
425
00:28:08.65 --> 00:28:12
If I put in three three seven,
it would be the sum of those
426
00:28:12 --> 00:28:16
two, it would lie in the same
plane as those.
427
00:28:16 --> 00:28:18
It wouldn't be independent.
428
00:28:18 --> 00:28:21.31
My attempt to create a basis
would be dead.
429
00:28:21.31 --> 00:28:25
But if I take -- so what vector
can I take?
430
00:28:25 --> 00:28:29
I can take any vector that's
not in that plane.
431
00:28:29 --> 00:28:33
Let me try -- I hope that 3 3 8
would do it.
432
00:28:33 --> 00:28:37
At least it's not the sum of
those two vectors.
433
00:28:37 --> 00:28:40
But I believe that's a basis.
434
00:28:40 --> 00:28:45
And what's the test then,
for that to be a basis?
435
00:28:45 --> 00:28:51.57
Because I just picked those
numbers, and if I had picked,
436
00:28:51.57 --> 00:28:57
5 7 -14 how would we know do we
have a basis or don't we?
437
00:28:57 --> 00:29:02
You would put them in the
columns of a matrix,
438
00:29:02 --> 00:29:08
and you would do elimination,
row reduction --
439
00:29:08 --> 00:29:15
and you would see do you get
any free variables or are all
440
00:29:15 --> 00:29:18
the columns pivot columns.
441
00:29:18 --> 00:29:25
Well now actually we have a
square -- the matrix would be
442
00:29:25 --> 00:29:27
three by three.
443
00:29:27 --> 00:29:33
So, what's the test on the
matrix then?
444
00:29:33 --> 00:29:42
The matrix -- so in this case,
when my space is R^3 and I have
445
00:29:42 --> 00:29:48
three vectors,
my matrix is square and what I
446
00:29:48 --> 00:29:57
asking about that matrix in
order for those columns to be a
447
00:29:57 --> 00:29:58.34
basis?
448
00:29:58.34 --> 00:30:05
So in this -- for R^n,
if I have -- n vectors give a
449
00:30:05 --> 00:30:12
basis if the n by n matrix with
those columns,
450
00:30:12 --> 00:30:18
with those columns,
is what?
451
00:30:18 --> 00:30:21
What's the requirement on that
matrix?
452
00:30:21 --> 00:30:24
Invertible, right,
right.
453
00:30:24 --> 00:30:27
The matrix should be
invertible.
454
00:30:27 --> 00:30:32
For a square matrix,
that's the, that's the perfect
455
00:30:32 --> 00:30:33
answer.
456
00:30:33 --> 00:30:34
Is invertible.
457
00:30:34 --> 00:30:39
So that's when,
that's when the space is the
458
00:30:39 --> 00:30:42
whole space R^n.
459
00:30:42 --> 00:30:46
Let me, let me be sure you're
with me here.
460
00:30:46 --> 00:30:48
Let me remove that.
461
00:30:48 --> 00:30:54
Are those two vectors a basis
for any space at all?
462
00:30:54 --> 00:31:00
Is there a vector space that
those really are a basis for,
463
00:31:00 --> 00:31:06
those, that pair of vectors,
this guy and this 1,
464
00:31:06 --> 00:31:08
1 1 2 and 2 2 5?
465
00:31:08 --> 00:31:12
Is there a space for which
that's a basis?
466
00:31:12 --> 00:31:12
Sure.
467
00:31:12 --> 00:31:16
They're independent,
so they satisfy the first
468
00:31:16 --> 00:31:22
requirement, so what space shall
I take for them to be a basis
469
00:31:22 --> 00:31:22
of?
470
00:31:22 --> 00:31:26
What spaces will they be a
basis for?
471
00:31:26 --> 00:31:28
The one they span.
472
00:31:28 --> 00:31:30
Their combinations.
473
00:31:30 --> 00:31:31
It's a plane,
right?
474
00:31:31 --> 00:31:34
It'll be a plane inside R^3.
475
00:31:34 --> 00:31:39
So if I take this vector 1 1 2,
say it goes there,
476
00:31:39 --> 00:31:43
and this vector 2 2 5,
say it goes there,
477
00:31:43 --> 00:31:49
those are a basis for --
because they span a plane.
478
00:31:49 --> 00:31:52
And they're a basis for the
plane, because they're
479
00:31:52 --> 00:31:52
independent.
480
00:31:52 --> 00:31:56
If I stick in some third guy,
like 3 3 7, which is in the
481
00:31:56 --> 00:31:58
plane -- suppose I put in,
try to put in 3 3 7,
482
00:31:58 --> 00:32:01.45
then the three vectors would
still span the plane,
483
00:32:01.45 --> 00:32:04
but they wouldn't be a basis
anymore because they're not
484
00:32:04 --> 00:32:06.32
independent anymore.
485
00:32:06.32 --> 00:32:06
OK.
486
00:32:06 --> 00:32:15
So, we're looking at the
question of -- again,
487
00:32:15 --> 00:32:24.81
the case with independent
columns is the case where the
488
00:32:24.81 --> 00:32:32
column vectors span the column
space.
489
00:32:32 --> 00:32:36.93
They're independent,
so they're a basis for the
490
00:32:36.93 --> 00:32:38
column space.
491
00:32:38 --> 00:32:38
OK.
492
00:32:38 --> 00:32:41
So now there's one bit of
intuition.
493
00:32:41 --> 00:32:44
Let me go back to all of R^n.
494
00:32:44 --> 00:32:46
So I -- where I put 3 3 8.
495
00:32:46 --> 00:32:47.18
OK.
496
00:32:47.18 --> 00:32:51
The first message is that the
basis is not unique,
497
00:32:51 --> 00:32:53.48
right.
498
00:32:53.48 --> 00:32:56
There's zillions of bases.
499
00:32:56 --> 00:33:00
I take any invertible three by
three matrix,
500
00:33:00 --> 00:33:03
its columns are a basis for
R^3.
501
00:33:03 --> 00:33:07
The column space is R^3,
and if those,
502
00:33:07 --> 00:33:13
if that matrix is invertible,
those columns are independent,
503
00:33:13 --> 00:33:16
I've got a basis for R^3.
504
00:33:16 --> 00:33:19
So there're many,
many bases.
505
00:33:19 --> 00:33:24
But there is something in
common for all those bases.
506
00:33:24 --> 00:33:31
There's something that this
basis shares with that basis and
507
00:33:31 --> 00:33:33.87
every other basis for R^3.
508
00:33:33.87 --> 00:33:35
And what's that?
509
00:33:35 --> 00:33:41
Well, you saw it coming,
because when I stopped here and
510
00:33:41 --> 00:33:47
asked if that was a basis for
R^3, you said no.
511
00:33:47 --> 00:33:53
And I know that you said no
because you knew there weren't
512
00:33:53 --> 00:33:55
enough vectors there.
513
00:33:55 --> 00:33:59
And the great fact is that
there're many,
514
00:33:59 --> 00:34:04
many bases, but -- let me put
in somebody else,
515
00:34:04 --> 00:34:06
just for variety.
516
00:34:06 --> 00:34:11
There are many,
many bases, but they all have
517
00:34:11 --> 00:34:15
the same number of vectors.
518
00:34:15 --> 00:34:20
If we're talking about the
space R^3, then that number of
519
00:34:20 --> 00:34:22
vectors is three.
520
00:34:22 --> 00:34:28
If we're talking about the
space R^n, then that number of
521
00:34:28 --> 00:34:29
vectors is n.
522
00:34:29 --> 00:34:35
If we're talking about some
other space, the column space of
523
00:34:35 --> 00:34:40
some matrix, or the null space
of some matrix,
524
00:34:40 --> 00:34:47
or some other space that we
haven't even thought of,
525
00:34:47 --> 00:34:52
then that still is true that
every basis -- that there're
526
00:34:52 --> 00:34:58.51
lots of bases but every basis
has the same number of vectors.
527
00:34:58.51 --> 00:35:01.77
Let me write that great fact
down.
528
00:35:01.77 --> 00:35:05
Every basis -- we're given a
space.
529
00:35:05 --> 00:35:06
Given a space.
530
00:35:06 --> 00:35:12
R^3 or R^n or some other column
space of a matrix or the null
531
00:35:12 --> 00:35:18.07
space of a matrix or some other
vector space.
532
00:35:18.07 --> 00:35:26
Then the great fact is that
every basis for this,
533
00:35:26 --> 00:35:33
for the space has the same
number of vectors.
534
00:35:33 --> 00:35:43
If one basis has six vectors,
then every other basis has six
535
00:35:43 --> 00:35:44
vectors.
536
00:35:44 --> 00:35:54
So that number six is telling
me like it's telling me how big
537
00:35:54 --> 00:35:58
is the space.
538
00:35:58 --> 00:36:02
It's telling me how many
vectors do I have to have to
539
00:36:02 --> 00:36:03
have a basis.
540
00:36:03 --> 00:36:05
And of course we're seeing it
this way.
541
00:36:05 --> 00:36:08
That number six,
if we had seven vectors,
542
00:36:08 --> 00:36:10
then we've got too many.
543
00:36:10 --> 00:36:13
If we have five vectors we
haven't got enough.
544
00:36:13 --> 00:36:18
Sixes are like just right for
whatever space that is.
545
00:36:18 --> 00:36:21.47
And what do we call that
number?
546
00:36:21.47 --> 00:36:27
That number is -- now I'm ready
for the last definition today.
547
00:36:27 --> 00:36:30
It's the dimension of that
space.
548
00:36:30 --> 00:36:36
So every basis for a space has
the same number of vectors in
549
00:36:36 --> 00:36:36
it.
550
00:36:36 --> 00:36:42
Not the same vectors,
all sorts of bases --
551
00:36:42 --> 00:36:47.68
but the same number of vectors
is always the same,
552
00:36:47.68 --> 00:36:51
and that number is the
dimension.
553
00:36:51 --> 00:36:53
This is definitional.
554
00:36:53 --> 00:36:58
This number is the dimension of
the space.
555
00:36:58 --> 00:36:58
OK.
556
00:36:58 --> 00:36:59
OK.
557
00:36:59 --> 00:37:01
Let's do some examples.
558
00:37:01 --> 00:37:05.71
Because now we've got
definitions.
559
00:37:05.71 --> 00:37:12
Let me repeat the four things,
the four words that have now
560
00:37:12 --> 00:37:15
got defined.
561
00:37:15 --> 00:37:19
Independence,
that looks at combinations not
562
00:37:19 --> 00:37:19
being zero.
563
00:37:19 --> 00:37:23
Spanning, that looks at all the
combinations.
564
00:37:23 --> 00:37:27
Basis, that's the one that
combines independence and
565
00:37:27 --> 00:37:28
spanning.
566
00:37:28 --> 00:37:33
And now we've got the idea of
the dimension of a space.
567
00:37:33 --> 00:37:40
It's the number of vectors in
any basis, because all bases
568
00:37:40 --> 00:37:42.79
have the same number.
569
00:37:42.79 --> 00:37:43
OK.
570
00:37:43 --> 00:37:45.47
Let's take examples.
571
00:37:45.47 --> 00:37:50
Suppose I take,
my space is -- examples now --
572
00:37:50 --> 00:37:55
space is the,
say, the column space of this
573
00:37:55 --> 00:37:57
matrix.
574
00:37:57 --> 00:38:01
Let me write down a matrix.
1 1 1, 2 1 2,
575
00:38:01 --> 00:38:07
and I'll -- just to make it
clear, I'll take the sum there,
576
00:38:07 --> 00:38:12
3 2 3, and let me take the sum
of all -- oh,
577
00:38:12 --> 00:38:17
let me put in one -- yeah,
I'll put in one one one again.
578
00:38:17 --> 00:38:19
OK.
579
00:38:19 --> 00:38:21
So that's four vectors.
580
00:38:21 --> 00:38:25.74
OK, do they span the column
space of that matrix?
581
00:38:25.74 --> 00:38:29
Let me repeat,
do they span the column space
582
00:38:29 --> 00:38:31.15
of that matrix?
583
00:38:31.15 --> 00:38:31
Yes.
584
00:38:31 --> 00:38:35.72
By definition,
that's what the column space --
585
00:38:35.72 --> 00:38:38
where it comes from.
586
00:38:38 --> 00:38:41
Are they a basis for the column
space?
587
00:38:41 --> 00:38:42
Are they independent?
588
00:38:42 --> 00:38:44
No, they're not independent.
589
00:38:44 --> 00:38:47
There's something in that null
space.
590
00:38:47 --> 00:38:51
Maybe we can -- so let's look
at the null space of the matrix.
591
00:38:51 --> 00:38:56
Tell me a vector that's in the
null space of that matrix.
592
00:38:56 --> 00:39:03
So I'm looking for some vector
that combines those columns and
593
00:39:03 --> 00:39:05
produces the zero column.
594
00:39:05 --> 00:39:11
Or in other words,
I'm looking for solutions to A
595
00:39:11 --> 00:39:12
X equals zero.
596
00:39:12 --> 00:39:17
So tell me a vector in the null
space.
597
00:39:17 --> 00:39:20
Maybe -- well,
this was, this column was that
598
00:39:20 --> 00:39:24
one plus that one,
so maybe if I have one of those
599
00:39:24 --> 00:39:28
and minus one of those that
would be a vector in the null
600
00:39:28 --> 00:39:28
space.
601
00:39:28 --> 00:39:32
So, you've already told me now,
are those vectors independent,
602
00:39:32 --> 00:39:37
the answer is -- those column
vectors, the answer is -- no.
603
00:39:37 --> 00:39:38
Right?
604
00:39:38 --> 00:39:40
They're not independent.
605
00:39:40 --> 00:39:43.33
Because -- you knew they
weren't independent.
606
00:39:43.33 --> 00:39:47
Anyway, minus one of this minus
one of this plus one of this
607
00:39:47 --> 00:39:50.03
zero of that is the zero vector.
608
00:39:50.03 --> 00:39:50
OK.
609
00:39:50 --> 00:39:52
OK, so they're not independent.
610
00:39:52 --> 00:39:56
They span, but they're not
independent.
611
00:39:56 --> 00:39:59
Tell me a basis for that column
space.
612
00:39:59 --> 00:40:02.73
What's a basis for the column
space?
613
00:40:02.73 --> 00:40:07
These are all the questions
that the homework asks,
614
00:40:07 --> 00:40:10
the quizzes ask,
the final exam will ask.
615
00:40:10 --> 00:40:16
Find a basis for the column
space of this matrix.
616
00:40:16 --> 00:40:16
OK.
617
00:40:16 --> 00:40:21
Now there's many answers,
but give me the most natural
618
00:40:21 --> 00:40:21.67
answer.
619
00:40:21.67 --> 00:40:23
Columns one and two.
620
00:40:23 --> 00:40:25
Columns one and two.
621
00:40:25 --> 00:40:27
That's the natural answer.
622
00:40:27 --> 00:40:31
Those are the pivot columns,
because, I mean,
623
00:40:31 --> 00:40:33
we s- we begin systematically.
624
00:40:33 --> 00:40:37
We look at the first column,
it's OK.
625
00:40:37 --> 00:40:40
We can put that in the basis.
626
00:40:40 --> 00:40:43
We look at the second column,
it's OK.
627
00:40:43 --> 00:40:46
We can put that in the basis.
628
00:40:46 --> 00:40:49
The third column we can't put
in the basis.
629
00:40:49 --> 00:40:52
The fourth column we can't,
again.
630
00:40:52 --> 00:40:58
So the rank of the matrix is --
what's the rank of our matrix?
631
00:40:58 --> 00:40:59
Two.
632
00:40:59 --> 00:41:00.03
Two.
633
00:41:00.03 --> 00:41:08
And, and now that rank is also
-- we also have another word.
634
00:41:08 --> 00:41:12
We, we have a great theorem
here.
635
00:41:12 --> 00:41:15
The rank of A,
that rank r,
636
00:41:15 --> 00:41:22
is the number of pivot columns
and it's also -- well,
637
00:41:22 --> 00:41:27
so now please use my new word.
638
00:41:27 --> 00:41:34
This, it's the number two,
of course, two is the rank of
639
00:41:34 --> 00:41:41.13
my matrix,
it's the number of pivot
640
00:41:41.13 --> 00:41:46.9
columns, those pivot columns
form a basis,
641
00:41:46.9 --> 00:41:50
of course, so what's two?
642
00:41:50 --> 00:41:53
It's the dimension.
643
00:41:53 --> 00:41:59
The rank of A,
the number of pivot columns,
644
00:41:59 --> 00:42:04.07
is the dimension of the column
space.
645
00:42:04.07 --> 00:42:06
Of course, you say.
646
00:42:06 --> 00:42:08
It had to be.
647
00:42:08 --> 00:42:11
Right.
648
00:42:11 --> 00:42:15
But just watch,
look for one moment at the,
649
00:42:15 --> 00:42:19
the language,
the way the English words get
650
00:42:19 --> 00:42:20
involved here.
651
00:42:20 --> 00:42:25
I take the rank of a matrix,
the rank of a matrix.
652
00:42:25 --> 00:42:31
It's a number of columns and
it's the dimension of -- not the
653
00:42:31 --> 00:42:37
dimension of the matrix,
that's what I want to say.
654
00:42:37 --> 00:42:41
It's the dimension of a space,
a subspace, the column space.
655
00:42:41 --> 00:42:45
Do you see, I don't take the
dimension of A.
656
00:42:45 --> 00:42:47
That's not what I want.
657
00:42:47 --> 00:42:51.56
I'm looking for the dimension
of the column space of A.
658
00:42:51.56 --> 00:42:56
If you use those words right,
it shows you've got the idea
659
00:42:56 --> 00:42:56
right.
660
00:42:56 --> 00:42:58
Similarly here.
661
00:42:58 --> 00:43:01
I don't talk about the rank of
a subspace.
662
00:43:01 --> 00:43:03
It's a matrix that has a rank.
663
00:43:03 --> 00:43:05
I talk about the rank of a
matrix.
664
00:43:05 --> 00:43:09
And the beauty is that these
definitions just merge so that
665
00:43:09 --> 00:43:13.76
the rank of a matrix is the
dimension of its column space.
666
00:43:13.76 --> 00:43:15
And in this example it's two.
667
00:43:15 --> 00:43:19
And then the further question
is, what's a basis?
668
00:43:19 --> 00:43:22
And the first two columns are a
basis.
669
00:43:22 --> 00:43:24.14
Tell me another basis.
670
00:43:24.14 --> 00:43:26
Another basis for the columns
space.
671
00:43:26 --> 00:43:29
You see I just keep hammering
away.
672
00:43:29 --> 00:43:32
I apologize,
but it's, I have to be sure you
673
00:43:32 --> 00:43:33
have the idea of basis.
674
00:43:33 --> 00:43:37
Tell me another basis for the
column space.
675
00:43:37 --> 00:43:42
Well, you could take columns
one and three.
676
00:43:42 --> 00:43:46
That would be a basis for the
column space.
677
00:43:46 --> 00:43:50
Or columns two and three would
be a basis.
678
00:43:50 --> 00:43:53
Or columns two and four.
679
00:43:53 --> 00:44:00
Or tell me another basis that's
not made out of those columns at
680
00:44:00 --> 00:44:01
all?
681
00:44:01 --> 00:44:05
So -- I guess I'm giving you
infinitely many possibilities,
682
00:44:05 --> 00:44:07
so I can't expect a unanimous
answer here.
683
00:44:07 --> 00:44:10
I'll tell you -- but let's look
at another basis,
684
00:44:10 --> 00:44:11
though.
685
00:44:11 --> 00:44:14
I'll just -- because it's only
one out of zillions,
686
00:44:14 --> 00:44:18
I'm going to put it down and
I'm going to erase it.
687
00:44:18 --> 00:44:23
Another basis for the column
space would be -- let's see.
688
00:44:23 --> 00:44:27
I'll put in some things that
are not there.
689
00:44:27 --> 00:44:31
Say, oh well,
just to make it -- my life
690
00:44:31 --> 00:44:33
easy, 2 2 2.
691
00:44:33 --> 00:44:35
That's in the column space.
692
00:44:35 --> 00:44:39
And, that was sort of obvious.
693
00:44:39 --> 00:44:43
Let me take the sum of those,
say 6 4 6.
694
00:44:43 --> 00:44:47.45
Or the sum of all of the
columns, 7 5 7,
695
00:44:47.45 --> 00:44:48
why not.
696
00:44:48 --> 00:44:50
That's in the column space.
697
00:44:50 --> 00:44:55
Those are independent and I've
got the number right,
698
00:44:55 --> 00:44:57
I've got two.
699
00:44:57 --> 00:45:00
Actually, this is a key point.
700
00:45:00 --> 00:45:06.04
If you know the dimension of
the space you're working with,
701
00:45:06.04 --> 00:45:13
and we know that this column --
we know that the dimension,
702
00:45:13 --> 00:45:18
DIM, the dimension of the
column space is two.
703
00:45:18 --> 00:45:25.55
If you know the dimension,
then -- and we have a couple of
704
00:45:25.55 --> 00:45:32
vectors that are independent,
they'll automatically be a
705
00:45:32 --> 00:45:32
basis.
706
00:45:32 --> 00:45:38
If we've got the number of
vectors right,
707
00:45:38 --> 00:45:42
two vectors in this case,
then if they're independent,
708
00:45:42 --> 00:45:45
they can't help but span the
space.
709
00:45:45 --> 00:45:50.38
Because if they didn't span the
space, there'd be a third guy to
710
00:45:50.38 --> 00:45:54
help span the space,
but it couldn't be independent.
711
00:45:54 --> 00:45:58.25
So, it just has to be
independent if we've got the
712
00:45:58.25 --> 00:46:00
numbers right.
713
00:46:00 --> 00:46:01
And they span.
714
00:46:01 --> 00:46:01
OK.
715
00:46:01 --> 00:46:02
Very good.
716
00:46:02 --> 00:46:05
So you got the dimension of a
space.
717
00:46:05 --> 00:46:09
So this was another basis that
I just invented.
718
00:46:09 --> 00:46:09
OK.
719
00:46:09 --> 00:46:13
Now, now I get to ask about the
null space.
720
00:46:13 --> 00:46:17.82
What's the dimension of the
null space?
721
00:46:17.82 --> 00:46:22
So we, we got a great fact
there, the dimension of the
722
00:46:22 --> 00:46:25
column space is the rank.
723
00:46:25 --> 00:46:29.34
Now I want to ask you about the
null space.
724
00:46:29.34 --> 00:46:35
That's the other part of the
lecture, and it'll go on to the
725
00:46:35 --> 00:46:36
next lecture.
726
00:46:36 --> 00:46:36.54
OK.
727
00:46:36.54 --> 00:46:41
So we know the dimension of the
column space is two,
728
00:46:41 --> 00:46:43
the rank.
729
00:46:43 --> 00:46:45
What about the null space?
730
00:46:45 --> 00:46:48
This is a vector in the null
space.
731
00:46:48 --> 00:46:51
Are there other vectors in the
null space?
732
00:46:51 --> 00:46:51
Yes or no?
733
00:46:51 --> 00:46:52
Yes.
734
00:46:52 --> 00:46:55
So this isn't a basis because
it's doesn't span,
735
00:46:55 --> 00:46:56
right?
736
00:46:56 --> 00:47:01
There's more in the null space
than we've got so far.
737
00:47:01 --> 00:47:03
I need another vector at least.
738
00:47:03 --> 00:47:06
So tell me another vector in
the null space.
739
00:47:06 --> 00:47:10
Well, the natural choice,
the choice you naturally think
740
00:47:10 --> 00:47:14
of is I'm going on to the fourth
column, I'm letting that free
741
00:47:14 --> 00:47:18
variable be a one,
and that free variable be a
742
00:47:18 --> 00:47:22
zero, and I'm asking is that
fourth column a combination of
743
00:47:22 --> 00:47:24
my pivot columns?
744
00:47:24 --> 00:47:25
Yes, it is.
745
00:47:25 --> 00:47:28
And it's -- that will do.
746
00:47:28 --> 00:47:34
So what I've written there are
actually the two special
747
00:47:34 --> 00:47:36
solutions, right?
748
00:47:36 --> 00:47:42
I took the two free variables,
free and free.
749
00:47:42 --> 1.
I gave them the values 1 0 or 0
750
1. --> 00:47:46
751
00:47:46 --> 00:47:48
I figured out the rest.
752
00:47:48 --> 00:47:52
So do you see,
let me just say it in words.
753
00:47:52 --> 00:47:55
This vector,
these vectors in the null space
754
00:47:55 --> 00:47:58
are telling me,
they're telling me the
755
00:47:58 --> 00:48:01
combinations of the columns that
give zero.
756
00:48:01 --> 00:48:07
They're telling me in what way
the, the columns are dependent.
757
00:48:07 --> 00:48:10
That's what the null space is
doing.
758
00:48:10 --> 00:48:11
Have I got enough now?
759
00:48:11 --> 00:48:13
And what's the null space now?
760
00:48:13 --> 00:48:16.35
We have to think about the null
space.
761
00:48:16.35 --> 00:48:19
These are two vectors in the
null space.
762
00:48:19 --> 00:48:20.46
They're independent.
763
00:48:20.46 --> 00:48:23
Are they a basis for the null
space?
764
00:48:23 --> 00:48:26.35
What's the dimension of the
null space?
765
00:48:26.35 --> 00:48:30
You see that those questions
just keep coming up all the
766
00:48:30 --> 00:48:30
time.
767
00:48:30 --> 00:48:33
Are they a basis for the null
space?
768
00:48:33 --> 00:48:37
You can tell me the answer even
though we haven't written out a
769
00:48:37 --> 00:48:38
proof of that.
770
00:48:38 --> 00:48:39
Can you?
771
00:48:39 --> 00:48:40
Yes or no?
772
00:48:40 --> 00:48:45
Do these two special solutions
form a basis for the null space?
773
00:48:45 --> 00:48:48
In other words,
does the null space consist of
774
00:48:48 --> 00:48:50
all combinations of those two
guys?
775
00:48:50 --> 00:48:51
Yes or no?
776
00:48:51 --> 00:48:51.87
Yes.
777
00:48:51.87 --> 00:48:52
Yes.
778
00:48:52 --> 00:48:54.57
The null space is two
dimensional.
779
00:48:54.57 --> 00:48:57
The null space,
the dimension of the null
780
00:48:57 --> 00:49:01
space, is the number of free
variables.
781
00:49:01 --> 00:49:08
So the dimension of the null
space is the number of free
782
00:49:08 --> 00:49:09
variables.
783
00:49:09 --> 00:49:15
And at the last second,
give me the formula.
784
00:49:15 --> 00:49:20
This is then the key formula
that we know.
785
00:49:20 --> 00:49:26
How many free variables are
there in terms of R,
786
00:49:26 --> 00:49:31
the rank, m --
the number of rows,
787
00:49:31 --> 00:49:34
n, the number of columns?
788
00:49:34 --> 00:49:36.09
What do we get?
789
00:49:36.09 --> 00:49:40
We have n columns,
r of them are pivot columns,
790
00:49:40 --> 00:49:46
so n-r is the number of free
columns, free variables.
791
00:49:46 --> 00:49:50
And now it's the dimension of
the null space.
792
00:49:50 --> 00:49:52
OK.
793
00:49:52 --> 00:49:53.6
That's great.
794
00:49:53.6 --> 00:49:57
That's the key spaces,
their bases,
795
00:49:57 --> 00:49:59
and their dimensions.
796
00:49:59 --> 00:50:02
Thanks.
| 677.169 | 1 |
The following computer-generated description may contain errors and does not represent the quality of the book: This work is intended to furnish complete information concerning the subject of which it treats, and forms one number of a well graded and progressive series of Arithmetics now in preparation. It has been the design of the author, in dealing with the different subjects of this volume, to furnish a scientific treatise relating to numbers and their applications, adapted to the wants and necessities of the commercial school, and of that class of high schools whose boards of education have wisely decided that the true science of numbers cannot be successfully comprehended by the immature minds of pupils belonging to the grades. The aim has been to make the work, from the very beginning, purely analytical, to teach the close relations existing between associated facts, and to aid the thinking mind in the deduction of such additional facts as are the natural result of these relations. A careful study of the Properties and Relations of Numbers, as treated in this volume, will be found invaluable to the student, and will have a tendency to elevate his thinking powers to that degree of eminence, whereby he will discern that reason, instead of memory, is the mental faculty to be developed by mathematical study. The work will be found clear and accurate in definitions and logical in explanations, and we believe that a critical examination by thorough business scholars who are conversant with, and know the demands of the present comfm mercial age, will result in an endorsement of the methods of solution concerning the different subjects herein presented.
| 677.169 | 1 |
Math Course Takes 'Real Life' Approach to Algebra
Educational courseware publisher American Education Corp. is taking a new approach to answering the age-old question, "What does algebra have to do with real life?" The company has announced the release of a new course for its A+nyWhere Learning System program. Algebra I: A Function Approach Part 1 is the first semester segment of a full-year algebra course geared to grades 9 and 10, and, in addition to the fundamental concepts and tools of algebra, the course aims to relate the material to "real life."
Taking the fundamentals and applying them to real-world situations using exercises in relevant scenarios allows students to realize the practical uses of linear and quadratic equations, graphs and coordinates, functions, and other algebraic concepts.
The A+nyWhere program is computer based, so students taking courses like Algebra I can use a number of tools incorporated into the software to aid in their assignments and overall comprehension of the material. These tools include onscreen standard and scientific calculators, pictures and diagrams, video tutorials, exercises, practice exams, and, for upper-level courses, interactive
| 677.169 | 1 |
App Detail » Hands-On Equations 3 - The Fun Way to Learn Algebra
What's New
A Progress Bar in each lesson provides the student with the percentage of exercises completed for that lesson.
App Description
"Great intro to Algebra!"
This Hands-On Equations Level 3 app is intended for students who have already completed Level 1 and 2 of Hands-On Equations and who would like the challenge of more sophisticated equations involving negative numbers.
In Level 3, the student solves equations such as 4x – (-4) = -8 and 3(-x) + 2 = -10 + x, which contain negative constants. The green numbered cube, which represents the opposite of the red numbered cube, is introduced at this level.
Since the red and green number cubes are opposite of each other, when they are together on the same side of the balance, their value is zero and may therefore be removed without affecting the balance.
A video introduction is provided for each lesson. (See the sample YouTube video at Each video introduction is followed by two examples and ten exercises. It is essential for the student to view the lesson video prior to attempting the examples and exercises for that lesson.
Hands-On Equations is the ideal introduction to algebra for elementary and middle school students. Not only will students have fun and be fascinated with the program, their sense of self-esteem will be dramatically enhanced as they experience success with sophisticated algebraic equations.
High school students struggling with algebra will likely experience success and understanding for the first time as they work with this app.
More information on Hands-On Equations can be found at and on YouTube.
| 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. One hundred seventeen Precalculus Functions And Graphs textbooks are available for sale on ValoreBooks.com, sixty six used from the cheapest price of $4.83, or buy new starting at $219.13.[read more
| 677.169 | 1 |
Discrete Mathematics with Proof
Eric Gossett
This is a well-written, comprehensive textbook with a lot of information, good examples and plenty of exercises. This reviewer, an enumerative combinatorialist, finds the organization of chapters rather odd, but it is plausible that some instructors prefer to teach in this way.
The book starts with two long and technical chapters on sets, Boolean logic, and various proof techniques. This is appropriate for those students who never had a class on this, but in programs where a separate course on Transition into Advanced Mathematics is required, these two chapters can safely be omitted.
Then there is a chapter on Algorithms. This contains pattern matching, and the Halting problem. This reviewer thinks that it might be too early in the semester for the advanced ideas related to the Halting problem, but that is a judgement call.
The next four chapters consist of material that is, in many programs, taught in a combinatorics course. The chapters are, Counting, Finite Probability, Recursion, and, by a strange choice of a title, "Combinatorics." The latter chapter discusses design theory, error correcting codes, Ramsey numbers, integer partitions and set partitions. This is certainly combinatorics, but so are the topics discussed in the previous three chapters, and also graphs, trees, partially ordered sets, hypergraphs, generating functions, magic squares, permutations, and countless other areas. Choosing a small subset of them and calling it combinatorics is as though someone wrote a book on Paris and called it Western Europe.
Then there is a chapter on formal models of computer science, stopping at Turing machines, and not mentioning complexity classes like P and NP. Perhaps the P versus NP question could have been included; at least it always fascinates the students of this reviewer.
After this, there are very thorough chapters on Graphs and Trees, and a last chapter on Functions, Relations, Databases and Circuits. Again, it struck me as strange to see functions and relations being discussed in the last chapter of a book of this high level. It would seem more natural to discuss them at the beginning, where sets and logic are discussed.
To summarize, if you like the above organization of topics, then this book is definitely for you. The individual chapters are reader-friendly and well supported by examples and exercises. If you do not like the way the parts are put together, you may still want to use the book as a source of examples and exercises for your course.
| 677.169 | 1 |
Education has introduced ALEKS 360, a mathematics solution that combines an artificial intelligence and personalized learning program with a fully integrated, interactive e-book package. ALEKS 360 delivers assessments of students' math knowledge, guiding them in the selection of appropriate new study material, and recording their progress toward mastery of course goals. Through adaptive questioning, ALEKS accurately assesses a student's knowledge state and delivers targeted instruction on the exact topics a student is most ready to learn.
The e-books featured within ALEKS 360 are interactive versions of their physical counterparts, which offer virtual features such as highlighting and note-taking capabilities, as well as access to multimedia assets such as images, video, and homework exercises. E-books are accessible from ALEKS Student Accounts and the ALEKS Instructor Module for convenient, direct access.
The initial e-books to be offered in ALEKS 360 include: Introductory Algebra, Second Edition, by Julie Miller and Molly O'Neill; Intermediate Algebra, Second Edition, by Miller and O'Neill; College Precalculus, Second Edition, by John W. Coburn; and College Algebra, Second Edition, by Coburn.
| 677.169 | 1 |
Essential Discrete Mathematics for Computer Science
9780130186614
0130186619
Summary: Written for freshman/sophomore, one-semester introductory courses in discrete mathematics designated for computer science students, this text introduces the mathematics of computer science.
Krone, Joan is the author of Essential Discrete Mathematics for Computer Science, published 2002 under ISBN 9780130186614 and 0130186619. Twenty one Essential Discrete Mathematics for Computer Science textbooks are availa...ble for sale on ValoreBooks.com, eleven used from the cheapest price of $28.54, or buy new starting at $67.21
| 677.169 | 1 |
Essentials of College Algebra - 10th edition
Summary: Key Benefit:Essentials of College Algebraby Lial, Hornsby, and Schneider, gives readers a solid foundation in the basic functions of college algebra and their graphs, starting with a strong review of intermediate algebra concepts and ending with an introduction to systems and matrices. This brief version of theCollege Algebra, Tenth Editionhas been specifically designed to provide a more compact and less expensive book for courses that do not include the more advanced topics covered ...show morein the longer book. Focused on helping readers develop both the conceptual understanding and the analytical skills necessary to experience success in mathematics, the authors present each mathematical topic in this text using a carefully developed learning system to actively engage students in the learning process. The book addresses the diverse needs of today's students through a clear design, current figures and graphs, helpful features, careful explanations of topics, and a comprehensive package of supplements and study aids. Key Topics: R. Review of Basic Concepts, Sets, Real Numbers and Their Properties, Polynomials, Factoring Polynomials, Rational Expressions, Rational Exponents, Radical Expressions, Equations and Inequalities, Linear Equations, Applications and Modeling with Linear Equations, Complex Numbers, Quadratic Equations, Applications and Modeling with Quadratic Equations, Other Types of Equations and Applications, Inequalities, Absolute Value Equations and Inequalities, Graphs and Functions, Rectangular Coordinates and Graphs, Circles, Functions, Linear Functions, Equations of Lines; Curve Fitting, Graphs of Basic Functions, Graphing Techniques, Function Operations and Composition, Polynomial and Rational Functions, Quadratic Functions and Models, Synthetic Division, Zeros of Polynomial Functions, Polynomial Functions: Graphs, Applications, and Models, Rational Functions: Graphs, Applications, and Models, Variation, Inverse, Ex6499X Satisfaction Guaranteed. Please contact us with any inquiries. We ship daily.46 +$3.99 s/h
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| 677.169 | 1 |
MATH 575
Development Of Mathematics 3 u
A study of the development of mathematical notation and ideas from prehistoric times to the present, with special emphasis being placed on elementary mathematics through the calculus. The development and historic background of the new math will be included.
Prereq: Consent of instructor.
MATH 580
Patterns Of Problem Solving 3 u
This course will expose students to a variety of techniques useful in solving mathematics problems. The experiences gained from this course can be applied to problems arising in all fields of mathematics. The student will have the chance to see how some general techniques can be used as tools in many areas. Homework for this course will consist mostly of solving a large number of mathematics problems.
Prereq: MATH 280 or consent of instructor. (Consent will be given to students with substantial interest in problem solving, and adequate preparation.)
MATH 615
Modern Algebra And Number Theory For The Elementary Teacher 3 u
An introduction to modern algebra with special emphasis on the number systems and algorithms which underlie the mathematics curriculum of the elementary school. Topics include sets, rings, integral domains, rational numbers, real numbers, complex numbers and polynomials. Students may not receive credit for both MATH 615 and MATH 652.
Prereq: MATH 149 and MATH 152.
MATH 616
Geometry For The Elementary Teacher 3 u
A study of the intuitive, informal geometry of sets of points in space. Topics include elementary constructions, coordinates and graphs, tessellations, transformations, problem solving, and symmetries of polygons and polyhedra.
Prereq: MATH 149 and MATH 152.
MATH 664
Advanced Calculus 3 u
This course presents a rigorous treatment of the differential and integral calculus of single variable functions, convergence theory of numerical sequences and series, uniform convergence theory of sequences and series of functions, metric spaces, function of several real variables, and the inverse function theorem. This course contains a written component.
Prereq: MATH 301.
methods, orthogonalization, gradient methods. Consideration of stability and elementary error analysis. Extensive use of microcomputers and programs using a high level language such as PASCAL.
Prereq: MATH 171 and MATH 355/555
MATH 690
Workshop 1-3 u
MATH 694
Seminar 2 u
MATH 696
Special Studies 1-3 u
Prereq: Consent of instructor.
MATH 790
Workshop 1-6 u
MATH 794
Seminar 1-3 u
MATH 798
Individual Studies 1-3 u
MATH 799
Thesis Research 1-6 u
Students must complete a Thesis Proposal Form in the Graduate Studies Office before registering for this course.
COMPUTER SCIENCE (COMPSCI)
COMPSCI 507
Microcomputer Applications 3 u
This course will treat a variety of applications of microcomputers, as well as their architecture, design and social impact.
Prereq: COMPSCI 171
| 677.169 | 1 |
More About
This Textbook
Overview
David Poole's innovative book prepares students to make the transition from the computational aspects of the course to the theoretical by emphasizing vectors and geometric intuition from the start. Designed for a one- or two-semester introductory course and written in simple, "mathematical English" the book presents interesting examples before abstraction. This immediately follows up theoretical discussion with further examples and a variety of applications drawn from a number of disciplines, which reinforces the practical utility of the math, and helps students from a variety of backgrounds and learning styles stay connected to the concepts they are learning. Poole's approach helps students succeed in this course by learning vectors and vector geometry first in order to visualize and understand the meaning of the calculations that they will encounter and develop mathematical maturity for thinking abstractly.
Meet the Author
David Poole is Professor of Mathematics at Trent University, where he has been a faculty member since 1984. Dr. Poole has won numerous teaching awards: Trent University's Symons Award for Excellence in Teaching (the university's top teaching award), three merit awards for teaching excellence, a 2002 Ontario Confederation of University Faculty Associations Teaching Award (the top university teaching award in the province), a 2003 3M Teaching Fellowship (the top university teaching award in Canada, sponsored by 3M Canada Ltd.), a 2007 Leadership in Faculty Teaching Award from the province of Ontario, and the Canadian Mathematical Society's 2009 Excellence in Teaching Award. From 2002-2007, Dr. Poole was Trent University's Associate Dean (Teaching & Learning). His research interests include discrete mathematics, ring theory, and mathematics education. He received his B.Sc. from Acadia University in 1976 before earning his M.Sc. (1977) and Ph.D. (1984) from McMaster University. When he is not doing mathematics, David Poole enjoys hiking and cooking, and he is an avid film buff
| 677.169 | 1 |
Fibonacci sequence is one of the most rapidly growing area of mathematics which has a wide variety of applications in science and mathematics. The manuscript of proposed book is centered on... More > generalization of Fibonacci pol-ynomials. The book will consists of three chapters. Each chapter will be di-vided into several sections.
In Mathematics, the Polynomials are an important class of simple and smooth functions. Here, simple means they are constructed using only multiplication and addition and smooth means they are infinitely differentiable, i.e., we can say that they have derivatives of all finite orders. Because of their simple structure, the polynomials are very easy to evaluate, and are used extensively in numerical analysis for polynomial interpolation or to numerically integrate more complex functions. In linear algebra, characteristic polynomial of a square matrix encodes several important properties of the matrix.< Less
| 677.169 | 1 |
Applied Basic Math Worksheets - 2nd edition
Summary: Worksheets for Classroom or Lab Practice offer extra practice exercises for every section of the text, with ample space for students to show their work. These lab- and classroom-friendly workbooks also list the learning objectives and key vocabulary terms for every text section, along with vocabulary practice problems.6.63 +$3.99 s/h
Good
HPB-Palatine
| 677.169 | 1 |
Basic College Mathematics - 4th edition
Summary: Elayn Martin-Gay firmly believes that every student can succeed, and her developmental math textbooks and video resources are motivated by this belief. Basic College Mathematics, Fourth Edition was written to help readers effectively make the transition from arithmetic to algebra. The new edition offers new resources like the Student Organizer and now includes Student Resources in the back of the book to help students on their quest for success30.4548.56
| 677.169 | 1 |
Gill, CO AlMy MATLAB usage is primarily restricted to coursework. I most recently used it for a linear programming class, as well as conducting NLP research. Because of my programming background, I am confident in my ability to use MATLAB at a high level.
| 677.169 | 1 |
Algebra and Trigonometry
2nd
Algebra and Trigonometry by James Stewart
Book Description
Choose the algebra textbook know through explanations you can understand, as well as tons of practice and step-by-step problem-solving help. Make ALGEBRA AND TRIGONOMETRY your choice today.
Buy Algebra and Trigonometry book by James Stewart from Australia's Online Bookstore, Boomerang Books.
Books By Author James Stewart
On Lombok you can trek up Gunung Rinjani, one of Indonesia's highest peaks, or escape to the remote, white sandy beaches of the Gili Islands. This title helps to ensure you make the most of these alluring islands, with tips on everything from indulgent spa retreats and fantastic shops, to the best hotels, restaurants and bars to suit every budget.
Whether you're looking for the things not to miss at the Top 10 sights, or want to find the best nightspots of Dubrovnik & the Dalmatian Coast, this title features dozens of Top 10 lists - from the Top 10 museums to the Top 10 events and festivals - there's even a list of the Top 10 things to avoid.
A guide to Scotland. It features in-depth coverage of its burgeoning food scene, artistic innovations and awe-inspiring wild places - from remote, Gaelic-speaking islands to untamed, ruggedly beautiful glens that takes you to the most rewarding spots, with striking colour photography bringing everything to life.
A guide to Dubrovnik & the Dalmatian Coast, one of Europe's most vibrant destinations. Whether it's the Top 10 unspoilt beaches, historic towns, museums and galleries, pristine islands, sailing destinations, churches and cathedrals, liveliest festivals, and restaurants and cafes, it lets you explore different corners of this beautiful region
| 677.169 | 1 |
This is an early, incomplete draft of a linear algebra textbook. Please see for further information, and a hyperlinked downloadable copy cropped for viewing... More > on-screenIt is a first linear algebra course for mathematically advanced students. It is intended for a student who, while not yet very familiar with abstract reasoning, is willing to study more rigorous... More > mathematics that is presented in a "cookbook style" calculus type course. It is also supposed to be a first course introducing a student to rigorous proof and formal definitions.
The target audience explains the specific blend of elementary ideas and concrete examples, which are usually presented in introductory linear algebra texts with more abstract definitions and constructions typical for advanced books.
The book emphasizes topics that are important for analysis, geometry, probability, etc., and does not include some traditional topics. For example, I am only considering vector spaces over the fields of real or complex numbers, since I feel time would be better spent on some more classical topics, which will be required in other disciplines.< Less
(CC BY) This textbook, published by Textbook Equity, is an open education resource released by The Saylor Foundation under a Creative Commons license. The original textbook, written by Kenneth... More > Kuttler, was chosen by the Saylor Foundation, after a thorough peer review, to be the companion textbook to their online course.
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. However, 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.< Less
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... More > has had calculus. Some optional topics require more analysis than this, however. I think that the subject of linear algebra is likely the most significant topic discussed in undergraduate mathematics courses. Part of the reason for this is its usefulness in unifying so many different topics. Linear algebra is essential in analysis, applied math, and even in theoretical mathematics. This is the point of view of this book, more than a presentation of linear algebra for its own sake. This is why there are numerous applications, some fairly unusual.< Less
Linear algebra is a powerful set of tools for modelling vector quantities and vector transformations. Linear algebra is used in machine learning, computer graphics, chemistry, economics, quantum... More > mechanics, signal processing, etc. Vectors are used all over the place! Linear algebra is the vector-upgrade to your modelling skills. This book contains lessons on linear algebra written in a style that is precise and concise. Each lesson covers one concept at the level required for a university course. The focus of the book is on showing the intricate connections between the abstract concepts of linear algebra, their geometrical interpretation, and the practical computational aspects. This book differs from regular textbooks by the conversational tone it is written in and its focus on intuitive explanations of the procedures of linear algebra. Each concept is illustrated through concrete examples and in connection with applications. The computer algebra system SymPy is used to illustrate tedious computational tasks.< Less
This is a paper-bound edition of a graduate algebra text originally published by Harper & Row as Groups, Rings, Modules, and written by M. Auslander and D. Buchsbaum. I changed the title to... More > conform to the one that Auslander and I had originally preferred.< Less
| 677.169 | 1 |
More About
This Textbook
OverviewWhat People Are Saying
Julian Simon
Sylvanus Thompson's Cal.
Editorial Reviews
From the PublisherRelated Subjects
Meet the Author
Table of Contents
Preface to the 1998 Edition
Preliminary Chapters by Martin Gardner
1. What is a Function?
2. What is a Limit?
3. What is a Derivative?
Calculus Made Easy by Silvanus P. Thompson
Publisher's Note on the Third Edition
Prologue
1. To Deliver Your from the Preliminary Terrors
2. On Different Degrees of Smallness
3. On Relative Growings
4. Simplest Cases
5. Next Stage. What to Do with Constants
6. Sums, Differences, Products, and Quotients
7. Successive Differentiation
8. When Time Varies
9. Introducing a Useful Dodge
10. Geometrical Meaning of Differentiation
11. Maxima and Minima
12. Curvature of Curves
13. Partial Fractions and Inverse Functions
14. On True Compound Interest and the Law of Organic Growth
15. How to deal with Sines and Cosines
16. Partial Differentiation
17. Integration
18. Integrating as the Reverse of Differentiating
19. On Finding Areas by Integrating
20. Dodges, Pitfalls, and Triumphs
21. Finding Solutions
22. A Little More about Curvature of Curves
23. How to Find the Length of an Arc on a Curve
Table of Standards
Epilogue and Apologue
Answers to Exercises
Appendix: Some Recreational Problems Relating to Calculus, by Martin Gardner
Index
About 30, 2005
I was lost for 4 semester but now understand
I made it through all my calculus classes as an engineering major, but honestly never understood any of the theory behind what I was doing. Now, as a masters student, I need to understand the concepts of calculus. I am not joking when I say that I learned far more from reading this book than I did in 4 semesters of college calc.
3 out of 4 people found this review helpful.
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ChrisWardNJ
Posted July 13, 2012
Barely usable, and vastly inferior to the paper copy with this cover.
Lots of spelling errors left over from the scanning. Much of the layout is lost. It is a really good book, but I read it in a printed form, and this copy is derived from a "Project Gutenberg" copy. Which may be fair use for an out of copyright book, but this edition uses the cover with Martin Gardner's name on it, and I think it is unfair. The cover would not be out of copyright, and I don't think that he would want his name used on this edition. It does preserve the essential text but needs to be copy edited.
1 out of 1 people found this review helpful.
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Anonymous
Posted August 1, 2003
Gardner's Update is Superb!
This update of a classic is a must for those seeking an authoritative source of introductory calculus. The authors have brought only the critical elements to their discussion and exercises. Readers will find this a wonderful introduction or review to the basic principles of calculus. Well done!
1 out of 2 people found this review helpful.
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Anonymous
Posted April 29, 2014
Rating is only for Nook version of Gardiner revision
Numerous equations are in a tiny font that ruins the Nook version of this wonderful classic. I own the print version of the Gardiner revision. It's wonderful. The Nook version of this revision is a disaster. Shame on the publisher. .
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SeverusSnape2010
Posted February 6, 2012
Very helpful
I have just started reading the book, but so far it provides helpful chapters covering derivatives and so much more for calculus. It also has many examples with the answers so you can check your work.
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KP_in_Michigan
Posted March 21, 2010
An oldy but a goody.
This was very useful to one well equipped in algebra. They say those who have difficulty with calculus have a hard time with algebra. This author made sure to include plenty of examples. I recommend this book.
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| 677.169 | 1 |
Functions, Data, and Models - 10 edition
Summary: This is a college algebra-level textbook written to provide the kind of mathematical knowledge and experiences that students will need for courses in other fields, such as biology, chemistry, business, finance, economics, and other areas that are heavily dependent on data either from laboratory experiments or from other studies. The focus is on the fundamental mathematical concepts and the realistic problem-solving via mathematical modeling rather than the development of algebraic skills that mi...show moreght be needed in calculus. Functions, Data, and Models presents college algebra in a way that differs from almost all college algebra books available today. Rather than going over material covered in high school courses the Gordons teach something new. Students are given an introduction to data analysis and mathematical modeling presented at a level that students with limited algebraic skills can understand. The book contains a rich set of exercises, many of which use real data. Also included are thought experiments or what if questions that are meant to stretch the student s mathematical thinking
| 677.169 | 1 |
MAT140 - College Algebra and Trigonometry
This course is an in-depth survey of algebraic and geometric problem solving techniques, including solutions of polynomial equations and inequalities, curve sketching techniques, and trigonometry from the triangular and functional standpoint. The course will make active use of technology by requiring the use of both a graphing calculator and computer software.
| 677.169 | 1 |
Castro Valley PrecalculusThey are able to do the operations on those numbers in monomials and polynomials. They also learn how to solve and graph quadratic equations by factoring, completing the square, or using the quadratic formula, including in the complex numbers. Student knows how to apply these techniques in solving word problems
| 677.169 | 1 |
MAD2104Taken for Credit:N/AAttendance: Not MandatoryTextbook Use: It's a must haveRater Interest: It's my lifeGrade Received: A
I don't understand why everyone finds discrete math so difficult. I got A's on all of my quizzes and 100% on the midterm. You just need to apply yourself and you'll be fine. Attending class is curcial becasuse she gives extra credit towards the exmas and hints of what to expect. Attend class, do the HW, study your notes, & memorize proofs.
First off...this is math, so it's going to be challenging. She gives 5 pts extra credit on each of her exams if you attended all of her lectures. She's a sweet lady, and her accent isn't too bad. She tries to explain how to do the problems but sometimes gets caught up for a minute before remembering how to finish. Best math prof Ive had though.
Her class is difficult, but manageable. I got an A because I worked hard. Do not slack in this class. Study from the book, and attend class only to get the extra credit she gives for the test. I used chegg to check my hw answers, and only wrote down notes when she went over examples. It does get easier towards the end, if that is any consolation.
She is not a horrible teacher but you really have to pay attention and study for of her quizzes and exams. She tries to give extra credit for attendance which is helpful but her tests are extremely hard. DO NOT EXOECT TO GET AN A UNLESS YOU ARE A MATH GENIUS.
She will expect you to be a mathematician before entering class. Her passing rate isn't that high. When you ask her to explain a question, don't expect to understand the answer. Not even people who have taken AP Calc. have understood her class.
She is a HORRIBLE.She does say what is going tobeon the quiz but it is not what she teaches in class. On the board she puts 2X2 onthe quiz its 2342342.2342Xsqrt2345.3she hates partial creditHer accent is notbad what kills me is that I have to concentrate just to listen to herShe looses herself while doing the examples on the board. rude to students
Hard Class. Unclear and hard to understand. Quizzes and tests are literally minutes so you have to be VERY VERY QUICK if you want to finish them. You've been warned! She tells you what will be on the test. You need to read the book on its entirety because her lecture is confusing. Do all the homework.
Does not make the subject palatable. She expects her students to "discover" calculus themselves. Leads the class using proofs. Tests mostly on previous knowledge of trig/algebra and expects students to have a solid background in calculus before entering. Pins blame on student's algebra skills on why they can't understand her. Steer elsewhere.
Professor Shershin is HORRIBLE at teaching. She's not clear at all. Starts with one problem and moves on to another without finishing the previous one. Failed my first quiz, and even tutoring couldn't help pass. DO NOT TAKE HER.
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Intermediate Algebra - 05 edition
Summary: Intermediate Algebra is a book for the student. The authors' goal is to help build students' confidence, their understanding and appreciation of math, and their basic skills by presenting an extremely user-friendly text that models a framework in which students can succeed. Unfortunately, students who place into developmental math courses often struggle with math anxiety due to bad experiences in past math courses. Developmental students often have never developed no...show morer applied a study system in mathematics. To address these needs, the authors have framed three goals for Intermediate Algebra: 1) reduce math anxiety, 2) teach for understanding, and 3) foster critical thinking and enthusiasm.
The authors' writing style is extremely student-friendly. They talk to students in their own language and walk them through the concepts, explaining not only how to do the math, but also why it works and where it comes from, rather than using the "monkey-see, monkey-do" approach that some books take90 +$3.99 s/h
VeryGood
BookBuyers Online1 CA San Jose, CA
2004 Hardcover Very good Promo-
$12.50 +$3.99 s/h
VeryGood
A2ZBooks Ky Burgin, KY
2004 Hardcover Very Good Condition. No Dust Jacket Very good condition copy, Text appears to be clean, Wear to over all book from storage, stickers on cover, Bump to book edges, Text body appears t...show moreo be clean, and free from previous owner annotation, under12.50 +$3.99 s/h
VeryGood
a2zbooks Burgin, KY
Very good condition copy, Text appears to be clean, Wear to over all book from storage, stickers on cover, Bump to book edges, Text body appears to be clean, and free from previous owner annotation, ...show moreunder
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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
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Find an Elk Grove Village PrecalculusThis is especially true in the area of economics. Only the simplest differential equations admit solutions given by explicit formulas. Beyond this, numerical methods using computers are called upon to approximately solve differential equations
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4th edition. Book is in overall good condition. Some water damage causing pages to stain and lightly ripple. Does not affect the text. Cover shows some edge wear and corners are ...lightly worn. Pages have a minimal to moderate amount of markings. FAST SHIPPING W/USPS TRACKING!!! general survey of mathematical topics helps a diverse audience, with different backgrounds and career plans, to understand mathematics. Blitzer provides the applications and technology readers need to gain an appreciation of mathematics in everyday life.
Demonstrates how mathematics can be applied to readers' lives in interesting, enjoyable, and meaningful ways. Features abundant, step-by-step, annotated Examplesthat provide a problem-solving approach to reach the solution; annotations are conversational in tone, explaining key steps and ideas as the problem is solved. Begins each section with a compelling vignette highlighting an everyday scenario, posing a question about it, and exploring how the chapter section subject can be applied to answer the question.
A highly readable reference for anyone who needs to brush up their mathematics skills.
Editorial Reviews
From The Critics
The "less-expanded" version, which is available under ISBN 0-13-065601-1, is about 100 pages shorter. This text is intended for a one- or two-term course in liberal arts mathematics, finite mathematics, and mathematics for education majors, as well as for courses specifically designed to meet state-mandated requirements. Blitzer teaches mathematics at Miami-Dade Community College. Annotation c. Book News, Inc., Portland, OR
Related Subjects
Meet the Author
Bob Blitzer is a native of Manhattan and received a Bachelor of Arts degree with dual majors in mathematics and psychology (minor: English Literature) from the City College of New York. His unusual combination of academic interests led him toward a Master of Arts in Mathematics from the University of Miami and a doctorate in behavioral sciences from Nova University.
He is most energized by teaching mathematics, and has taught a variety of mathematics courses at Miami-Dade Community College for nearly 30 years. He has received numerous teaching awards, including Innovator of the Year from the League for Innovations in the Community College, Teacher of the Year (13 years in a row) and was among the first group of recipients at Miami-Dade Community College for an endowed chair based on excellence in the classroom. In addition to Thinking Mathematically, Bob has written Introductory Algebra for College Students, Intermediate Algebra for College Students, Introductory and Intermediate Algebra for College Students, Algebra for College Students, College Algebra, Algebra and Trigonometry, and Precalculus, all published by Prentice Hall.
Read an Excerpt
PREFACE:
Preface
To the Student a clearPreface
PREFACE:
Preface
To the Student aclear
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Main Content
Course Descriptions
110 Basic Concepts of Mathematics A review course for students who wish to develop quantitative skills. Topics covered include: number systems, linear equations and inequalities, exponents, polynomial and rational expressions, polynomial equations, relations and functions. Not open to students with demonstrated quantitative skills.
150 The Nature of Mathematics A study of the nature and development of some of the most important mathematical ideas. Topics may include, but are not limited to: infinity, variation, symmetry, numbers and notation, topology, mathematics and calculating machines, dimension, coordinate systems, dynamical systems, randomness, and probability.
170-180 Calculus with Analytic Geometry I-II A study of the differentiation and integration of algebraic and transcendental functions with applications. Topics in analytic geometry include a study of conics. Four credits, each semester.
Prerequisite: 2 years of high school algebra and a half year of trigonometry.
220 Vector Analysis and Differential Equations A study of vector analysis and ordinary differential equations and their applications. Topics include vector fields, line and surface integrals, first order differential equations, linear differential equations, and systems of differential equations.
Prerequisite: Mathematics 210
260 Problem-Solving Via the solution of interesting problems, this course isolates and draws attention to the most important problem-solving techniques encountered in undergraduate mathematics. The aim is to show how a basic set of simple techniques can be applied in diverse ways to solve a variety of problems.
Prerequisite: Mathematics 180
310 Linear Analysis A study of linear algebra with emphasis on its application to the solution of differential equations. Topics include linear systems, matrices, vector spaces, and linear transformations.
Prerequisite: Mathematics 220
330-340 Mathematical Statistics A study of probability distributions and their application to statistical inference. Topics include probability, probability distributions, and parametric and non-parametric statistics.
Prerequisite: Mathematics 210.
350 Introduction to Complex Variables Topics for discussion include complex numbers and their properties, analytic functions, integration in the complex plane, Cauchy's integral formula, Taylor and Laurent series, and methods of contour integration.
Prerequisite: Mathematics 220
360 Modern Geometry An axiomatic approach to geometry including both Euclidean and non- Euclidean geometries.
380 Operations Research A study of the fundamental ideas of operations research and the application of mathematics to decision problems. Topics include linear optimization models, the simplex method, network models, dynamic optimization of inventory scheduling, integer programming, combinatorial models, and optimization with a non-linear objective function.
410-420 Advanced Calculus I - II Designed to bridge the gap between manipulative elementary calculus and theoretical real analysis. The fundamentals of elementary calculus are treated in a more rigorous manner. Point set topology is introduced and general theorems concerning continuity, differentiation, and integration on the real line and in Euclidean n-space are proved. Sequences and series of constants, and sequences and series of functions are also covered.
Prerequisite: Mathematics 210
430 Introduction to Modern Algebra A study of algebraic systems, including groups, rings, and fields.
Prerequisite: Permission of the instructor.
450 History of Mathematics Introduction to the history and development of mathematics from prehistory to the present.
Prerequisite: Mathematics 220.
480 Topics in Mathematics This course will consist of a detailed investigation of a topic important to contemporary mathematics. The topic will be chosen by the department for its relevance to current mathematical thought and its accessibility to students.
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College Mathematics - 7th edition
Summary: Continuing the tradition of excellence in applied mathematics, Cleaves and Hobbs 7th edition is designed for students in a broad range of career programs that require a solid understanding of basic math, elementary algebra, trigonometry, and geometry. Topics are introduced and reinforced using a step-by-step "spiral learning" approach supported by numerous examples and applications. Throughout the text, examples are presented in both symbolic and narrative...show more form and all concepts are applied directly to careers and professions
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(Prerequisites:
CS 01.104 Introduction to Scientific Programming, Math 01.131 Calculus II, and Math 03.150 Discrete Mathematics with a grade of C- or better in all prerequisites)
This
course will use mathematics-specific technologies to help students discover
mathematics and to develop a better understanding of new content. Throughout
the course students will become aware of the broad range of mathematics-specific
technologies available to mathematicians, become proficient in the use
of these, and pursue the advantages, disadvantages, and limitations of
such technologies. Students will solve problems and advance their understanding
of topics in the areas of
pre-calculus,
calculus, geometry, and statistics.
Essence
of the Course and Outline:
The
emphasis in this course will be on mathematics-specific technologies and
on the discovery of mathematics using such tools. Throughout the course,
students will become aware of the broad range of mathematics-specific technologies
available and become proficient in the use of several of these. Some objectives
of the course follow:
Programming:
Students will develop and use programs for graphing calculators and computers
that illustrate mathematical concepts, simulate mathematical and probabilistic
events, and carry out routine computations.
Data
Collection:
Students will use calculator based probes to gather and analyze data, creating
appropriate mathematical models to fit the data.
Internet:
Students will use geometry software to discover theorems in Geometry.
Using
spreadsheets as a tool for investigation statistics, discovering mathematics,
and problem solving.
Mathematics
word processing (Math Type or Equation Editor)
Graphing
calculators - emphasis on using programming for problem solving and on
the utility of programming in forcing logical thinking and precise mathematical
communication.
CBL
Calculator Based Laboratory) or CBR (Calculator Based Ranger) - for data
collection and discovering the relationship between reality and mathematics
modeling.
Resources
for mathematicians available on the Internet.
Optional
Technologies:
Additional
geometry Software such as Cabri Geometry, Geometry SuperSupposer software,
Logo (programming language) and other software associated with spatial
visualization (Gyrographics, The Right Turn, Building Perspective, Kaleidomania,
and The Super Factory).
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You are here
MATH 1143 - Liberal Arts Math I
This is a one semester course whose basic objective is to develop an interest and appreciation for Mathematics in students with little background in the subject. Included in the course are topics from the following areas: Problem Solving, Inductive Reasoning, Logic, Sets, Probability, Statistics, Consumer Math, and Geometry. It may also include topics from the following areas: History of Math, Number Systems, Metric, Algebra, Linear Programming, Finite Math, Matrices, Computer Applications.
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Prealgebra is just like a follow-up from elementary math course. The of the student
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Essential Discrete Mathematics for Computer Science
9780130186614
0130186619
Summary: Written for freshman/sophomore, one-semester introductory courses in discrete mathematics designated for computer science students, this text introduces the mathematics of computer science.
Krone, Joan is the author of Essential Discrete Mathematics for Computer Science, published 2002 under ISBN 9780130186614 and 0130186619. Twenty one Essential Discrete Mathematics for Computer Science textbooks are availa...ble for sale on ValoreBooks.com, eleven used from the cheapest price of $5.99, or buy new starting at $58.83
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MERLOT Search - category=2513&createdSince=2013-01-03&sort.property=dateCreated
A search of MERLOT materialsCopyright 1997-2014 MERLOT. All rights reserved.Sun, 21 Sep 2014 12:09:44 PDTSun, 21 Sep 2014 12:09:44 PDTMERLOT Search - category=2513&createdSince=2013-01-03&sort.property=dateCreated
4434Beginning and Intermediate AlgebraMCC MAT050 Quantitative Literacy Syllabus-Wilderson
MAT050 syllabus, Develops number sense and critical thinking strategies, introduce algebraic thinking, and connect mathematics to real world applications. Topics in the course include ratios, proportions, percents, measurement, linear relationships, properties of exponents, polynomials, factoring, and math learning strategies. This course prepares students for Math for Liberal Arts, Statistics, Integrated Math, and college level career math courses.Online Introductory Statistics - under construction
Stage 1 of a ePortfolio for Promising Course Redesign. Page is still under contruction. The page describes an online course for introductory statistics.Prealgebra Module 1 Section 1: Order Relations of Whole Numbers
Prealgebra Lesson on Order Relations of Whole Numbers. Objectives: Student will use order relations to compare whole numbers. Student will use rules of estimation and rounding to compare whole numbersMathematics Grade 1Section 4: Order of Operations
The student will use the order of operations to solve equations and evaluate expressions.
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Easy Access Study Guide
This 100% free easy access study guide requires no username or password to access the resources.
Using the Site
Take advantage of the tools and resources by selecting a feature from the lefthand menu or entering your search terms into the search bar.
About the Authors
Ron Larson received his Ph.D. in mathematics from the University of Colorado in 1970. At that time he accepted a position with Penn State University in Erie, Pennsylvania, and currently holds the rank of professor of mathematics at the university. Ron is the lead author of over two dozen mathematics textbooks from 6th grade through calculus. Many of his texts, such as the 10th edition of his calculus text, are leaders in their markets. Ron Larson is one of the pioneers in the use of multimedia to enhance the learning of mathematics. He has authored multimedia programs that range from 1st grade through calculus. To help with the development of his programs, Ron founded Larson Texts, Inc., which with its wholly owned subsidiary, Big Ideas Learning, LLC., employs about 70 people.
Robyn Silbey holds a Masters of Science degree in Mathematics Education from McDaniel College and a Bachelors of Science degree in Elementary Education from the University of Maryland, College Park. Robyn taught for 36 years in Montgomery County, Maryland, a large, urban/suburban school district with a widely diverse population. She was a classroom teacher for 11 years, and a school-based math coach for 25 years. For over 30 years, she has authored and co-authored books, computer software, workbooks and articles. Robyn currently collaborates with school districts and individual public, private and charter schools around the country to raise teacher quality and student achievement.
About Mathematical Practices
The Common Core Standards for Mathematical Practice describe the "Habits of Mind" that nurture and develop critical thinking and problem solving in mathematics. These practices rest on important processes and proficiencies with longstanding importance in mathematics education. The first four are NCTM process standards of problem solving, abstract reasoning, communication, and modeling with mathematics. The second four are the strands of mathematical proficiency specified in the National Research Council's report Adding It Up: using tools strategically, attending to precision, making use of structure, and expressing regularity in repeated reasoning. In short, the Mathematical Practices ask us as teachers to move from a teacher centered classroom to a student centered classroom. You can download a poster of the eight Mathematical Practices.
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horizontal and vertical lines in a matrix are called rows and columns , respectively. The numbers in the matrix are called its entries or its elements . To specify a matrix's size, a matrix with m rows and n columns is called an m -by- n matrix or m × n matrix, while m and n are called its dimensions . The above is a 4-by-3 matrix.
Upper triangular matrix If a square matrix in which all the elements that are below the main diagonal are zeros. the matrix must be square.
Lower triangular matrix If a matrix in which all the elements that are above the main diagonal are zeros. the matrix must be square.
5.
TYPES OF MATRICES
Determinant of a matrix. The determinant of a matrix A (n, n) is a scalar or polynomial, which is to obtain all possible products of a matrix according to a set of constraints, being denoted as [A]. The numerical value is also known as the matrix module.
EXAMPLE:
6.
TYPES OF MATRICES
Band matrix:
In mathematics, particularly in the theory of matrices, a matrix is banded sparse matrix whose nonzero elements are confined or limited to a diagonal band: understanding the main diagonal and zero or more diagonal sides. Formally, an n * n matrix A = a (i, j) is a banded matrix if all elements of the matrix are zero outside the diagonal band whose rank is determined by the constants K1 and K2: Ai, j = 0 if j <i - K1 j> i + K2, K1, K2 ≥ 0.
7.
TYPES OF MATRICES
Transpose Matrix If we have a matrix (A) any order mxn, then its transpose is another array (A) of order nxm where they exchange the rows and columns of the matrix (A). The transpose of a matrix is denoted by the symbol "T" and is, therefore, that the transpose of the matrix A is represented by AT. Clearly, if A is an array of size mxn, At its transpose will nxm size as the number of columns becomes row and vice versa.If the matrix A is square, its transpose is the same size.
EXAMPLE:
8.
TYPES OF MATRICES
Two matrices of order n are reversed if your product is the unit matrix of order n. A matrix has inverse is said to be invertible or scheduled, otherwise called singular. Properties (A ° B) -1 = B-1 to-1 (A-1) -1 = A (K • A) -1 = k-1 to-1 (A t) -1 = (A -1) t
Given the matrices m-by-n, A and B, their sum A + B is the matrix m-by-n calculated by adding the corresponding elements (ie (A + B) [i, j] = A [i, j] + B [i, j]). That is, adding each of the homologous elements of the matrices to add. For example:
10.
BASIC OPERATIONS
SCALAR MULTIPLICATION Given a matrix A and a scalar c, cA your product is calculated by multiplying the scalar by each element of In (ie (cA) [I j] = cA [R, j]).
The product of two matrices can be defined only if the number of columns in the left matrix is the same as the number of rows in the matrix right. If A is an m × n matrix B is a matrix n × p, then their matrix product AB is m × p matrix (m rows, p columns) given by:
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Courses in Mathematics for Educators (MAED)
MAED 205 -
Math as a Second Language
Deep conceptual understanding of the operations of arithmetic and interrelationships among arithmetic, algebra, and geometry; applications to the K-8 classroom. Pre/co-requisite: Admission to the VMI program.
Measurement (length, area and volume), probability, application to problem solving, and the ways in which these concepts develop across the K-12 curriculum. Pre/co-requisites: MAED 205, MAED 201, and MAED 215, or Instructor permission.
Continued study of calculus and its relationship to the K-8 curriculum. Topics include infinite series, calculating area, the definite integral, Fundamental Theorem of Calculus. Pre/co-requisite: MAED 235, or Instructor permission.
Credits: 1.00 or 2.00
MAED 295 -
Advanced Special Topics
See Schedule of Courses for specific title.
Credits: 1.00 to 18.00
MAED 300 -
Statistics & Research I
Introduction to statistics with emphasis on research in K-8 education. Representing and summarizing data, measures of relationship between variables, inference from sample data to population. Pre/co-requisites: MAED 205, MAED 210, and MAED 215, or Instructor permission.
Credits: 3.00
MAED 305 -
Statistics & Research II
Error bars in graphs, margins of error in surveys, and confidence intervals; interpret and critique educational research studies; analysis of school assessment data activities. Pre/co-requisites: MAED 300, or Instructor permission.
This course concludes the VMI's school-based-research component. Teachers synthesize their coursework and field experiences and revisit key mathematical concepts from arithmetic through calculus. Pre/co-requisite: Enrollment in VMI program.
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What does e + π mean and how can we evaluate it? What is the difference in the meaning of the equals sign between x2 −1 = 0, x2 −1 = (x−1)(x+1), (x2 −1)/(x−1) = x+1 and √x2 = x? What does it mean for a line to be straight? Are there lines that are not straight? In Math 499 we will be addressing these questions and more!
In this class we will explore the foundations of mathematics and how we acquire and process mathematical knowledge. We will revisit K-12 mathematics from the point of view of a mathematician. We will explore the roles of metaphors, models, and definitions. We will discuss the use of symbols and see that even in mathematics their meanings are often contextual. We will compare and contrast proofs and convincing arguments and think about the roles they play in developing and understanding mathematics. We will discuss the relationship between mathematics and our physical world and how we use mathematics to understand the physical world. We will consider various algorithms common in K- 12 mathematics and discuss why and how they work. We also will read and discuss the literature on how K-12 mathematics is taught and how we learn and process that knowledge. Throughout the semester, you will also the opportunity to observe and participate in classes at AUGUSTUS HAWKINS High School. This is a new school with a modern curriculum implementing an initiative called the Algebra Project.
This class has no prerequisites. In particular, it is not necessary to have taken any college level math classes; you are only expected to know how to count (albeit fairly well!). However, students must be willing to engage with the material at a mathematically sophisticated level. There will be very little lecturing. There will be a lot of discussion, group work, and both oral and written presentations. This class will be valuable for math majors, anyone with an interest in teaching mathematics, and sociology and psychology majors interested in the science of learning.The Fall 2013 Career Fair is here! Each year, more than 400 organizations seeking to employ USC students from diverse disciplines attend our career fairs. All students are encouraged to attend and explore the wide variety of internship and full-time employment opportunities that are represented:
Thursday, September 19, 2013
10:00 am – 2:30 pm
Trousdale Parkway
Part-time, Full-Time, and Internship positions available from over 150 companies. Log-in to connect SC for more information.
The USC Dornsife Environmental Studies Program Catalina Sustainability Semester is a situated learning experience for students that are majoring and/or minoring in either environmental studies or biology. Recommended preparation includes completion of ENST 100 or BISC 120L. Students will learn about coastal ecology and management through scientific diving, laboratory and field studies, and personal interaction with marine managers and scientists, while simultaneously gaining a better understanding and appreciation for the Southern California coastal environment.
Students enrolled in the Catalina Sustainability Semester will live and study at the USC Wrigley Marine Science Center for the entire semester (weekend transportation to the mainland is generally available). Rates for room and board are comparable to those on the University Park Campus. Courses are offered in a block format in which a single class meets on a daily basis (i.e., Monday through Friday, although participation in some weekend activities may be required). Each course will run for approximately four weeks,after which another class will take its place (four courses total; see below for details).
Course participants are expected to become scientific diver certified in accordance with the standards of the American Academy of Underwater Sciences (AAUS). As part of this training, the USC Dive Safety Officer or his designee will require each student to submita completed medical history and dive physical examination. Students will be assessed for water safety and ability to perform ecosystem measurements underwater
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CSI: Algebra -- STEM Project -- Unit 1 -- Variables & Expressions
PDF (Acrobat) Document File
Be sure that you have an application to open this file type before downloading and/or purchasing.
2.27 MB | 12 pages
PRODUCT DESCRIPTION
Nothing like a good criminal investigation to liven up number sense! In this project, students will work in teams to investigate the culprit of six fictional thefts. The criminal has left six messages, layered with algebra. Teams will work to build a case and present their findings to the court. Hopefully they are convincing enough to win the verdict.
This is more of an Algebra 1 unit, but I have used this with much success with students still working to pass their state graduation exam. The puzzle solving hook causes many different students to engage in solving traditionally mundane problems. Be the cool teacher :-)
In this 12 page document you will be given a mapping to the Content Standards, an outline for how to implement the project, and six number theory criminal scene puzzles.
This puzzle is mapped to Chapter 1 of Glencoe: Algebra and includes problems featuring the following skills: Order of Operations, Translating Verbal and Algebraic Expressions, Open Sentences, Distributing, Combining Like Terms
An answer key has been added to the end of the file for your convenience
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Anaheim Hills, CA StatisticsKnowing the basic properties of common will save you a lot of time in your calculus studies. Basic functions include trigonometry functions, exponential function, polynomials, and many more. Each set of functions has unique properties that make them useful in different ways.
3
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Downers Grove Algebra 2The two branches are connected by the Fundamental Theorem of Calculus discovered independently by Isaac Newton and Gottfried Leibnitz. My first exposure to calculus was in high school. I was fortunate enough to have had a teacher who was a true scholar
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This simple example shows how algebra can be useful in the real world by exploring the question: Should Grandpa start receiving his Social Security benefits at age 62 or should he wait until age 65? This resource is from PUMAS - Practical Uses of...(View More) Math and Science - a collection of brief examples created by scientists and engineers showing how math and science topics taught in K-12 classes have real world applications.(View Less)
In this problem set, students are led through a series of calculations to determine the best launch site for a TV satellite. This resource is from PUMAS - Practical Uses of Math and Science - a collection of brief examples created by scientists and...(View More) engineers showing how math and science topics taught in K-12 classes have real world applications.(View Less)
In this activity, students develop reasonable calendar designs for an imaginary planet using factoring as a problem solving strategy. This resource is from PUMAS - Practical Uses of Math and Science - a collection of brief examples created by...(View More) scientists and engineers showing how math and science topics taught in K-12 classes have real world applications.(View Less)
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Let's have a dialogue! I've studied Discrete Mathematics in College when I took Discrete Math I-II in preparation for graduate school. I'm finding a lot of computer science majors are required to take this course
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GeoGebra is a free software package
combining dynamic geometry, algebra and calculus. You can use it in a
lab environment, or for teacher-led classroom demonstrations. It
can be used as a standalone system, or you can create interactive web
pages for student exploration outside of class. Try out its
interactive features in the Riemann sum mini-applet at the right, or by clicking on the following links:
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The text books on Algebra 1 and 2 I can use and tutor well include "Elementary Linear Algebra with Applications" by Kolman and Hill, and others. The text books on Calculus I can use and tutor well include "Calculus" by Larson. The text books on Geometry I can use and tutor well include Essentia...
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This ebook is available for the following devices:
iPad
Windows
Mac
Sony Reader
Cool-er Reader
Nook
Kobo Reader
iRiver Story
Computational Geometry is an area that provides solutions to geometric problems which arise in applications including Geographic Information Systems, Robotics and Computer Graphics. This Handbook provides an overview of key concepts and results in Computational Geometry. It may serve as a reference and study guide to the field. Not only the most advanced methods or solutions are described, but also many alternate ways of looking at problems and how to solve them.
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