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I'm looking for some good books including mathematics articles which are appropriate for talented high school students. I'm NOT looking for puzzle or Olympiad problem books. Here are some of my findings, which may serve as examples:
2 Answers
Try the AMS's Mathematical World Series - they aim to "brings the beauty and wonder of mathematics to the advanced high school student". Specifically, Prasolov's Essays on Numbers and Figures is exactly the kind of book you want.
This book is devoted to the geometry and arithmetic of elliptic curves and to elliptic functions with applications to algebra and number theory. It includes modern interpretations of some famous classical algebraic theorems such as Abel's theorem on the lemniscate and Hermite's solution of the fifth degree equation by means of theta functions. Suitable as a text, the book is self-contained and assumes as prerequisites only the standard one-year courses of algebra and analysis.
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books.google.com - MATHEMATICS: A DISCRETE INTRODUCTION teaches students the fundamental concepts in discrete mathematics and proof-writing skills. With its clear presentation, the text shows students how to present cases logically beyond this course. All of the material is directly applicable to computer science and engineering,... A Discrete Introduction
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N-Q.1. Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.
290
Content Area: Mathematics
● Subject: Grade 8
● Category: Functions
Skill: F.8.2 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a linear function represented by a table of values and a linear function repre
Skill: A-CED.1. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.
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Beginner's Guide to Finite Mathematics For Business, Management, and the Social Sciences
Description: This second edition of A Beginner's Guide to Finite Mathematics takes a distinctly applied approach to finite mathematics at the freshman and sophomore level. Topics are presented sequentially: the book opens with a brief review of sets and numbers,More...
This second edition of A Beginner's Guide to Finite Mathematics takes a distinctly applied approach to finite mathematics at the freshman and sophomore level. Topics are presented sequentially: the book opens with a brief review of sets and numbers, followed by an introduction to data sets, histograms, means and medians. Counting techniques and the Binomial Theorem are covered, which provides the foundation for elementary probability theory; this, in turn, leads to basic statistics. This new edition includes chapters on game theory and financial mathematics. Requiring little mathematical background beyond high school algebra, the text will be especially useful for business and liberal arts majors
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Course in Combinatorics
9780521422604
ISBN:
0521422604
Publisher: Cambridge University Press
Summary: This major textbook, a product of many years' teaching, will appeal to all teachers of combinatorics who appreciate the breadth and depth of the subject. The authors exploit the fact that combinatorics requires comparatively little technical background to provide not only a standard introduction but also a view of some contemporary problems. All of the 36 chapters are in bite-size portions; they cover a given topic i...n reasonable depth and are supplemented by exercises, some with solutions, and references. To avoid an ad hoc appearance, the authors have concentrated on the central themes of designs, graphs and codes.
van Lint, Jacobus H. is the author of Course in Combinatorics, published under ISBN 9780521422604 and 0521422604. Nineteen Course in Combinatorics textbooks are available for sale on ValoreBooks.com, fourteen used from the cheapest price of $6.65, or buy new starting at $96.16
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Elements of Mathematics and R.D. Sharma PHYSICS 1. Prepare for class test to be taken in mid of July, 2012. 2. ... (working or demo) alongwith project report to be submitted at the time of final XII Class Exam. 4. Do all NCERT numericals (upto magnetic effects of current) in your note books ...
MATHEMATICS A Textbook for class IX (NCERT) RECOMMENDED BOOKS: MATHEMATICS for class IX- R.S.Aggarwal / R.D.Sharma Month Course Content April CH-1: Number System CH 3: Coordinate Geometry Activity 1: To represent an irrational number on a number line
... students shall revise the lessons and poems done in the class:- a) My mother at sixty six. b) An ... MATHEMATICS-XII 1. Solve the assignments of chapter ... For differentiation and continuity solve for R.D. Sharma with example and exercise. 4. Solve the question paper of first U.T ...
MCB Units: Children, Sports and Games, Mystery, The Class IX SUGGESTIONS TO PARENTS Parents must encourage their children to converse in English. ... Together with MathematicsMathematics by R. D. Sharma RECOMMENDED BOOKS Mathematics: NCERT MONTH NO. OF WORKING DAYS COURSE CONTENT
DIAGNOSIS AND REMEDIATION OF LEARNING PROBLEMS IN MATHEMATICS JULY 13 - 17, 2009 ... Mathematics," and Editor of "The Mathematics Notebook." Professor Sharma provides direct services of ... on the first day of class. Materials: Required materials are included.
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The purpose of this course is to allow the student to gain mastery in working with and evaluating mathematical expressions, equations, graphs, and other topics in a year long algebra course. Topics included are real numbers, simplifying real number expressions with and without variables, solving linear equations and inequalities, solving quadratic equations, graphing linear and quadratic equations, polynomials, factoring, linear patterns, linear systems of equality and inequality, simple matrices, sequences, and radicals. Assessments within the course include multiple-choice, short-answer, or extended response questions. Also included in this course are self-check quizzes, audio tutorials, and interactive games.
Prerequisite: Successful completion of Pre-Algebra
Length: One Semester, 1/2 Credit
Course Outline:
Semester 2
Unit 6: Solving Systems
Section 1: Systems of Equations
Section 2: Solving Systems
Section 3: Systems of Inequalities
Section 4: The Matrix
Section 5: Statistics
Unit 7: Polynomials
Section 1: Scientific Notation
Section 2: Add and Subtract Polynomials
Section 3: Multiply Polynomials
Section 4: Factors and GCF
Section 5: Factoring Trinomials
Section 6: Special Factors
Unit 8: Quadratics and Radicals
Section 1: Quadratic Functions
Section 2: Solving Quadratic Equations
Section 3: Radicals
Section 4: Operations on Radicals
Section 5: Radical Equations
Unit 9: Rational Expressions
Section 1: Inverse Variation
Section 2: Multiplying and Dividing Rational Expressions
Section 3: Adding and Subtracting Rational Expressions
Section 4: Solving Rational Equations
Section 5: Probability
Unit 10: Exponentials
Section 1: Exponential Functions
Section 2: Growth and Decay
Section 3: Geometric Sequences
Course Objectives
Students will
Read, write, evaluate, and understand the properties of mathematical expressions including real numbers, radicals, and polynomials
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Friday, 15 February 2013
MATHEMATICS SYLLABUS FOR CLASS 12[CBSE]
ii. Questions number 1 to 12 one of 3 marks each. Questions number 13 to 22 one of 4 marks each. Questions number 23 to 26 one of 6 marks each.
iii. There will be no over all choice. There will be internal choices in any two questions of 3 marks each, any two questions of 4 marks and any two questions of 6 marks each (Total of six internal choices).
iv. Use of calculator is not permitted. However each your may ask for logarithmic and statistical tables, if required.
Concept notation, order, equality, types of matrices, zero matrix, transpose of a matrix, symmetric and skew symmetric matrices. Addition, multiplication and scalar multiplication of matrices, simple properties of addition, multiplication of matrices and existence of non-zero matrices whose product is the/zero :
Determinant of a square matrix (up to 3 (X) 3 matrices), properties of determinants, minors
geometrically and second derivative test given as a provable tool). Simple problems (that illustrate basic principles and understanding of the subject as well as real-life situations).
3. Integrals :
(Periods 20)
Integration as inverse process of differentiation. Intergration of a variaty of functions by subsitution, by partial fractions and by parts, only simple integrals of the type
to be evaluated.
Definite intergrals as a limit of a sum. Fundamental Theorem of Calculus (without proof). Basic properties of definite intergrals and evaluation of definte integrals.
4. Applications of the Integrals:
Applications in finding the area under simple curves, especially lines, areas of circles/ parabolas/ellipses (in standard form only), area between the two above said curves (the region should be clearly identifiable).
5. Differential Equations :
Definationdy = p(x) y = q(x), where p(x) and q(x) are functions of x. dx
Unit -IV : Vectors and Three-Dimensional Geometry 1. Vectors:
Vectors and scalars, maguitude and direction of a vector. Direction cosines/ratios of vectors. Types of vectors (equal,unit
.
2. Three - dimensional Geometry : (ii) two planes. (iii) a line and a plane. Distance of a point from a plane.
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Learning Basic Math Online: Quick "Brushups" or Certificate Courses
There's a huge array
math classes online available, designed for everyone from grammar
school kids to college students, businesspeople in accounting or
statistics right on down to folks who just want to do
"everyday math" to keep better track of their
finances.
Purely Practical
"Brushup" online mathematics classes can improve
your basic life skills with an overview of practical arithmetic.
Subject will include basic addition and subtraction to fractions,
decimals, computing with integers and application of these skills to
word problems.
At this level, it's
not necessarily bad if the school is unheard of or has no
accreditation. A great many small companies offer these courses,
sometimes for as little as $40. The course may run anywhere from a few
weeks to six months. Many will actually offer refunds if
you're not satisfied with the course.
Of course, if you're
a savvy web searcher and you're willing to spend time
searching around via Google or Yahoo, you'll also find some
free online math courses, though they may simply offer a series of
documents for you to study by, with no actual teacher involvement.
Your Own Pace
Math courses at all levels tend to be
"asynchronous," meaning there's little
formal class time when you and the professor are online together.
That's because so much of the learning in math comes from
simply practicing equations on your own.
Basic online mathematics
classes can help a student at any age who needs to pass a placement
test or qualify for a specific job promotion. Some basic math classes
online will provide you with a certificate of completion, though
it's not universal.
Capella University
Capella is a large, well-known online school with over 35,000 students and solid regional accreditation. It focuses mainly on adult education, with a wide variety of bachelor's and master's degrees in:
- Business
- Education
- Nursing
- Criminal Justice
- Technology
- Health Care
- Human Resources Get info on Capella University
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wide-ranging introduction to the basic techniques in business mathematics and statistics is broken down into ten separate parts, each of which covers standard examination testing areas. Each chapter concludes with a summary, review notes and student exercises which concentrate on the more practical numerical aspects covered in the chapter. The book is intended for ACCA Level 1 - Business Mathematics, CIMA Stage 1 - Quantitative Methods, and ICSA Part 1 Module 2 - Quantitative Studies, and is also suitable for students on any course requiring an understanding of mathematical and statistical techniques.
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Mathematics and the Physical World (Dover Books on Mathematics)
Book Description: Stimulating account of development of mathematics from arithmetic, algebra, geometry and trigonometry, to calculus, differential equations and non-Euclidean geometries. Also describes how math is used in optics, astronomy, motion under the law of gravitation, acoustics, electromagnetism, other phenomena. 147
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Details: This course explores topics in calculus that complement and embellish Math 167: Calculus II. Topics will include applications of calculus to science and social science, calculus topics of hisotrical interst, and tehcnologies for exploring calculus. Co-requisite: Math 167.
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Extending Frontiers of Mathematics - 06 edition
ISBN13:978-0470412220 ISBN10: 0470412224 This edition has also been released as: ISBN13: 978-1597570428 ISBN10: 1597570427
Summary: In the real world of research mathematics, mathematicians do not know in advance if their assertions are true or false. Extending the Frontiers of Mathematics: Inquiries into proof and argumentation requires students to develop a mature process that will serve them throughout their professional careers, either inside or outside of mathematics. Its inquiry-based approach to the foundations of mathematics promotes exploring proofs and other advanced mathematical ideas through these fe...show moreatures: - Puzzles and patterns introduce the pedagogy. These precursors to proofs generate creativity and imagination that the author builds on later - Prove and extend or disprove and salvage, a consistent format of the text, provides a framework for approaching problems and creating mathematical proofs - Mathematical challenges are presented which build upon each other, motivate analytical skills, and foster interesting discussion
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My Insight about Calculus
In the first term of my Calculus, I had a hard time to review all my previous Math subjects, like Trigonometry and Algebra. I don't have any idea in Calculus; I just know that it's harder and more complicated than my previous Math subjects. After the midterm of this Semester I somehow know what Calculus for is, I learned that Differential Calculus can be use in finding the change in ratio. I also know that Calculus can be use to find the maximum and minimum rate of objects. It can also apply in other situation that involve changing and math. Using the concept of function derivatives, it studies the behavior and rate on how different quantities change. Using the process of differentiation, the graph of a function can actually be computed, analyzed, and predicted.
Though it is complicated to use well, calculus does have a lot of practical uses - uses that you probably won't comprehend at first. The most common practical use of calculus is when plotting graphs of certain formulae or functions. Using methods such as the first derivative and the second derivative, a graph and its dimensions can be accurately estimated. These 2 derivatives are used to predict how a graph may look like, the direction that it is taking on a specific point, the shape of the graph at a specific point (if concave or convex), just to name a few.
When do you use calculus in the real world? In fact, you can use calculus in a lot of ways and applications. Among the disciplines that utilize calculus include physics, engineering, economics, statistics, and medicine. It is used to create mathematical models in order to arrive into an optimal solution. For example, in physics, calculus is used in a lot of its concepts. Among the physical concepts that use concepts of calculus include motion, electricity, heat, light, harmonics, acoustics, astronomy, and dynamics. In fact, even advanced physics concepts including electromagnetism and Einstein's theory of...
[continues]
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many think of algorithms as specific to computer science, at its core algorithmic thinking is defined by the use of analytical logic to solve problems. This logic extends far beyond the realm of computer science and into the wide and entertaining world of puzzles. In Algorithmic Puzzles, Anany and Maria Levitin use many classic brainteasers as well as newer ex&les from job interviews with major corporations to show readers how to apply analytical thinking to solve puzzles requiring well-defined procedures. The book's unique collection of puzzles is supplemented with carefully developed tutorials on algorithm design strategies and analysis techniques intended to walk the reader step-by-step through the various approaches to algorithmic problem solving. Mastery of these strategies-exhaustive search, backtracking, and divide-and-conquer, among others-will aid the reader in solving not only the puzzles contained in this book, but also others encountered in interviews, puzzle collections, and throughout everyday life. Each of the 150 puzzles contains hints and solutions, along with commentary on the puzzle's origins and solution methods. The only book of its kind, Algorithmic Puzzles houses puzzles for all skill levels. Readers with only middle school mathematics will develop their algorithmic problem-solving skills through puzzles at the elementary level, while seasoned puzzle solvers will enjoy the challenge of thinking through more difficult puzzles.
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About the Author
Anany Levitin is a professor of Computing Sciences at Villanova University. He is the author of a popular textbook on design and analysis of algorithms, which has been translated into Chinese, Greek, Korean, and Russian. He has also published papers on mathematical optimization theory, software engineering, data management, algorithm design, and computer science education. Maria Levitin is an independent consultant. After some years working for leading software companies and developing business applications for large corporations, she now specializes in web-based applications and wireless computing.
I bought this book on sight. It's possibly my favorite book of any and all books I own. The puzzles are not only ubiquitous and exciting... they're educational and provide many "Aha!" moments. I've been looking for a book like this for years, and I recommend it to those looking for fun and challenging puzzles of varying difficulty levels. Being a computer science major, many of the puzzles are also fun to implement and solve using programming, emphasizing the "algorithmic" component in the title.
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To introduce the basic mathematical tools for generating and solving the governing equations of fluid dynamics. Vector calculus and partial differential equations are the primary topics covered.
Materials
The main text is the second volume of the book I wrote with the title Advanced Engineering Mathematics, Addison-Wesley-Longman, 1998. The text is augmented with notes that I have placed on my homepage at
Description
The course, which is primarily taught to junior level students majoring in oceanography, mechanical engineering and mathematics, covers the basic materials of vector calculus and partial differential equations in the context of fluid flows. After a thorough review of the vector operations (grad, div and curl), conservation laws of mass and linear momentum are introduced. Numerous examples of flows that one typically encounters in a basic fluid dynamics and geophysical fluid dynamics setting are introduced and visualized using Mathematica's symbolic and numerical capabilities. The course ends with the derivation of the Navier-Stokes equation in a rotating frame with special emphasis on the Coriolis force and its impact on the so-called Ekman tranport solution.
The primary goal of the course is to demonstrate the natural relationship between several topics in mathematics and fluid dynamics and oceanography. Mathematica is the primary tool used throughout the course as a symbolic manipulator as well as a numerical workhorse. Visualizing flows, especially through animations, is one of the main strengths of this course. Another strength of the course is the set of computer projects that the students carry out as part of the grade requirement. Examples of past projects include a) Flow past cylinder, b) Oseen vortex, c) Rayleigh-Benard flow, d) Lorenz and Veronis models of convection, and e) Serrin's tornado model.
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Product Details
The book includes two appendices: one listing the problems with their prerequisites, and a second which groups problems by subject matter. These make the book useful for teachers looking for extra challenges for their students.
_CHOICE
Here is a collection of 208 challenging, original problems, with carefully worked, detailed solutions. In addition to problems from The Wohascum County Problem Book, there are about 80 new problems, many of which involve experimentation and pattern finding.
The problems are intended for undergraduates; although some knowledge of linear or abstract algebra is needed for a few of the problems, most require nothing beyond calculus. In fact, many of the problems should be accessible to high school students. On the other hand, some of the problems require considerable mathematical maturity, and most students will find few of the problems routine.
Over four-fifths of the book is devoted to presenting instructive, clear, and often elegant solutions. For many problems, multiple solutions are given. Appendices list the prerequisites for individual problems and arrange them by topic. This should be helpful to classes on problem solving and to individuals or teams preparing for contests such as the Putnam. The index can help, as well, in finding problems with a specific theme, or in recovering a half-remembered problem.
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If you are having difficulty with algebra, these pages may be of some
help because they offer different sorts of explanations, in some sense
more "psychologically complete", than are usually found in algebra texts.
It is my belief that the way algebra is typically
presented to students leaves out some ideas and explanations that are helpful,
even sometimes necessary, for them to be able to do algebra well and to
have a "feel" for it.
There are at least three different kinds of things
taught in algebra courses: (1) language conventions, (2) logical numerical
manipulations using those conventions, and (3) deducing answers to problems
by using the conventions and the logical manipulations of them. I
will explain as I go. But it is important for students to keep in mind
whether in a given lesson they are supposed to be learning a convention,
a manipulation, or a way of solving problems by using conventions and manipulations.
It is also my belief that school "culture"
is such that, even contrary to good teachers' warnings, students will often
think they are simply supposed to memorize formulas and recipes in math,
rather than (also) understand them. Such memorization becomes a problem
in courses, such as algebra, where understanding is at least as important
as specific knowledge.
This is a two-fold problem. (1) Teachers need
to try to give useful and helpful explanations -- and they need to be aware
of as many typical student misunderstandings and confusions as they can;
and teachers need to constantly try to monitor students for confusion and
misunderstandings about what has been presented; waiting until there are
test results is often too late. But also, (2) students need to know that
THEY are the ones who will have to ultimately make the material make sense
to them, and that they need to keep trying until it does. They may have
to consult others, find a different book, or just sit down and think about
the material, if they cannot understand their teacher's explanation about
some aspect of algebra or other. There is simply no guaranty that any particular
explanation will provide automatic understanding. Understanding requires
reflective thinking of one's own. Explanations often are only a help to
such thinking; and what serves as a great explanation for one student may
not be helpful at all to another.
(My own first difficulty in "pre"-algebra was not understanding what
letters such as "x" had to do with anything, and why letters were chosen
to represent quantities at all, or how you worked with them when you had
them. I vividly remember when the light dawned on me about this
particular lack of understanding, in part because I still do not know why
or how the teacher's particular explanation "worked" on me. She was
saying that doing algebra was like unwrapping a package in the reverse
way it was wrapped to begin with. It may be that I figured out what I needed
to know while she was talking instead of because of what
she specifically said, or it may be that what she said had some sort of
meaning to me subconsciously somehow. I don't know, since "unwrapping"
is not the way I see (or even then saw) algebra. But what follows are explanations
of the sort that seem to me the most meaningful about some aspects of algebra
many students typically have trouble with. Further explanations of
other aspects of algebra can be found at A
supplemental introduction to the first chapter of an algebra book and
at Rate, Time Problems.)
The following question was asked on the Math-Help forum. It is typical
of the kinds of problems had by students who don't really understand in
general "what is going on" in algebra -- why you do certain manipulations
of formulas, or how you choose which manipulations to do. Particular algebraic
manipulations do not make sense to them because they don't have a general
sense of what algebra is about, or what the point of the manipulations
is. After I give the response to this question, a response which will include
both general and specific problem-solving ideas, I will make come comments
about how a typical algebra course is structured, and why it is structured
that way.
I have a big exam Monday in algebra and I have
no idea how to do linear equations! Can someone please help quick? Her
are examples of a few that I am having problems with. 3(3 - 4x) + 30=5x - 2(6x-7) and 5x²-[2(2x²+3)]-3=x²-9
I am also having a few problems with this:
x + 3 + 2x = 5 + x + 8 (5) I am supposed to figure out if 5 is the answer,
and tell how I got the answer.
My response: It looks to me from this last problem that you perhaps don't have an
UNDERSTANDING of what doing algebra with equations is all about, which
makes doing any problems a bit difficult. But let's see what we can do
for you here. The following may be too much for you to absorb before your
test, though I hope not; but I think it is stuff you will need to know
for future tests as well, so maybe it will help you for them even if it
is too much for tomorrow's test.
Take the last problem first. Do you understand that if "five is the
answer," that simply means that IF YOU WERE TO SUBSTITUTE 5 AS THE VALUE
EVERYWHERE THERE IS AN X, THE STATEMENT THAT THE LEFT SIDE OF THE EQUATION
EQUALS THE RIGHT SIDE WOULD BE TRUE? If 5 is NOT the solution, then when
you make the substitution, the statement will not be true that the left
side of the equation is equal to the right side.
Take a simple case first: X = 27 - 4 There is only one number X can be for this statement to be true; 23,
right? So 19 would NOT be a solution to this equation; that is, X cannot
equal 19 and the statement still be true that the left side equals the
right side.
Now make it just a bit harder: X + 3 = 27 -
4 There is still only one number this can work for but it is a different
number, because now we know that it is not X that equals 23, but some number
that, when you add 3 to it, gives you 23. So what number gives you 23 when
you add 3 to it? 20, right? That means X must be 20 in that equation.
It gets a little harder when you start putting X's on both sides of
the equations and add or subtract some multiplications and divisions, etc.,
but the idea of what is going on is the same.
So if we look at the equation you gave last: if X does equal 5 in the
equation x + 3 + 2x = 5 + x + 8, then
that would mean 5 + 3 + 10 should equal 5 + 5 + 8. Does it? If it does,
then X does equal 5; if it doesn't, then X cannot equal five.
If we thought X might be some other number, such as 2, then we would
have to replace X with 2 to check. Would 2 + 3 + 4 = 5 + 2 + 8? If not,
then X cannot equal 2 and the above equation still be true. All the "X"
does is to tell you which number it is you don't know, and the problem
is something like: I have a bag with some number of cookies in it; and
if we add three more cookies to the bag and then add twice as many cookies
as we had in the bag to begin with we get the same number of cookies as
if we had taken the bag and added 5 cookies and then 8 more. There is only
one number that could be in the bag to begin with that would make this
statement be true; and that number is what X represents.
Now, of course, this sounds like a silly way to tell someone you had
a bag of five cookies, but they make up silly
problems like this so you can have practice learning the things you need
to in order to learn how to solve real problems where you don't know a
missing number that you need to figure out. Like in my business,
I need to enter in my books how much people paid me for photographs and
how much they paid for the sales tax that I have to turn in to the government.
Sometimes all I have is a check to go by that has the total amount, so
I need to figure out how much of it was tax and how much of it was for
the photography itself. I work where I must charge 8% sales tax. If I get
a check for $37.80 I have to compute how much the photography was and how
much the tax was. If I let X represent the price of the photography, then
I know that:
X + .08X = 37.80, since that means
that the price of the photograph plus 8% of the price of the photograph
is equal to the amount on the check. And I thus have an equation I can
solve to let me know which part of the $37.80 is for tax and which part
is for me.
Back to your equations. The idea is that you sometimes need to
"multiply out" factors, and sometimes need
to factor products into components in order
to be able to get the X's and the numbers to such a point that you can
see what the X is. If I told you that 3X
= 9, you would know right off that X = 3, right? Well, there
are strategies for getting the X's where you can tell what they are supposed
to be, when you can't see it right off or any other
way.
Let's go back to your last equation: x + 3
+ 2x = 5 + x + 8
Normally, what you need to do is to find out how many X's you have altogether
and what the total quantity is that those X's altogether give you. And
to do that, you normally want to try to get all the X's on one side of
the equal sign and all the quantities that don't have an X in them on the
other, so you can find out that, say, 3X = 9 Well, in your equation, you
have 3X's and a quantity 3 on the left side, and on the right side you
have 1X and 13. So we need to try to get the X's to one side and the numbers
that don't have X's to the other.
You have to understand that (when you see no other way to solve the
problem) the most important concept, or principle, or tool is:
Whenever you have any equation, where you
are saying
the left side = the right side
you can always change either side of the equation
by any amount you want to AS LONG AS YOU CHANGE THE OTHER SIDE BY THE EXACT
SAME AMOUNT. That way, both sides will still be equal to each other even
though both will be different from what you started with. There
is a purpose for doing this with equations.
(That is, if someone gives you and me the same amount of money, we can
know that no matter how much money that is -- even if we don't know how
much
it is -- that if we each double our money, we will still have the same
amount as each other. And we know that if we each triple our money and
then lose half of it and then add $5 to that, we still will have the same
amount as each other -- whatever that will be. No matter what you do with
your money, you can know that IF I do the same thing with mine, we will
still end up with the same amount of money as each other, no matter how
much or how little that will be.)
And, unless we see some other way to figure out the unknown quantities,
such as "X", we use this principle over and over in almost every problem
in order to get the X's on one side and the nonX quantities on the other.
And hopefully, then, we can get to a point where we can tell what X must
be equal to, since if you can get any equation into the form of
so-many X's = a kabillion,
you can figure out what ONE X is by dividing a kabillion by however
many "so-many" is.
The Purpose
The difficult part is to figure out which changes
to make to both sides that will be useful and helpful for you to figure
out what the variables (or "unknown quantities" or "unknowns") are. Some
changes will be more useful than others. Many won't be helpful at all.
Normally, but not always, you want to try to do whatever manipulations
will help you get down to a quantity that is equal to ONE variable, that
is, one X or one Y or whatever letter(s) you chose to represent the quantity
you are trying to figure out. But it is not always easy to see which manipulations
might help you do even that. Understanding, combined with practice, help;
but solving any particular (new) problem may also take some trial and error,
a flash of insight, or some luck.
So, back to your equation: x
+ 3 + 2x = 5 + x + 8 we added all the X's up and all the numbers up on each side of the
equation, and we found out that
3X +3
= X + 13. Well, we can subtract the X on the right side in order
to "get rid" of it AS LONG AS WE ALSO subtract X from the left side. Hence,
we get: 2X + 3 = 13 (Or, in intuitive
terms, you might see that if we DON'T add the X to the 13 on the right
side, that is the same as not having added one of the X's in on the left
side to begin with, which would have just given us 2 X's on that side.)
If you still can't see how much X must be for this to be true, we can
use the same principle, this time subtracting 3 from each side of the equation,
so that we are left only with 2X = 10 (which just says that some number
multiplied by 2 equals 10). If these were big numbers and you still couldn't
see it, then we would divide both sides by what we are multiplying the
X by (in this case "2") in order to see how much 1 X is; 5. If we had 33X
= 9999, you would divide both sides by 33 in order to get what one X is.
Do you get the idea here?
Now we go back to the first problems: Take 3(3
- 4x) + 30=5x - 2(6x-7) first.
They have added an extra wrinkle to this one, because instead of telling
you how many X's you have on each side, they have told you that you have
some X's that get manipulated by subtractions and multiplications, etc.
So the first thing to do, usually, but not always (I'll give a counter-example
at the end of all this) is "multiply out" the quantities, giving:
9 - 12X + 30 = 5X - 12X + 14
(Time out, in case you need it: do you see why
it is PLUS 14 here, instead of minus 14? If not, click HERE.)
Back to working out the above equation. We had
it to:
9 - 12X + 30 = 5X - 12X + 14
(There are at least two ways we could proceed from here. I am going
to go through the "standard" way in the text and put the non-standard way
HERE.)
Now it is just like the one we did before where
we had some X's and some non-X quantities mixed together on each side of
the equal sign. In this case we have:
39 - 12X = -7X +14
At this point we start trying to do things equally to BOTH sides of
the equation in order to try to end up with all the X's on one side and
all the non-X's on the other side. So we can either try to get the non-X's
on the right and the X's on the left, or we can try to get the non-X's
on the left and the numbers on the right. It doesn't really matter from
a technical standpoint which way you choose, but from a psychological standpoint,
it is usually easier to work with positive numbers than negative numbers,
so instead of subtracting 39 from both sides and ending up with -25 on
the right side, it is psychologically easier to subtract 14 from both sides
and end up with 25 on the left: 25 - 12X = -7X
Then, if you add 12X to both sides, you end up with 25 = 5X, and since
that just means five times some quantity is equal to 25, you know the quantity
must be 5. So X = 5.
(If we had gone "the other way" and started out
by subtracting 39 from both sides of the above equation, we would have
got -12X = - 7X - 25, and then to get the X's all on the left, we would
have had to add 7X to both sides, giving us -5X = -25, which will still
mean that X = 5, but it is usually psychologically more difficult to work
with and to see that way.)
Now, it is important to make sure you did it right and got the right
answer. To make sure you did it right, you go back to your original equation
and put 5 back in wherever there is an X, and see if it comes out true:
This one is something of a trick AT THE LEVEL YOUR ARE STUDYING because
when you have Xsquared = something, you get TWO answers: a plus and a minus
answer; e.g., if X2 = 9, X can be either plus 3 or negative
3, since (-3)(-3) = 9 just as (3)(3) does. It will turn out that when you
graph equations that have squares (in their most simplified version), you
don't get straight lines, so they are not straight-line or "linear" equations.
So the trick here must be that either the X2 will all disappear,
leaving you with a linear equation or something else will happen where
you end up with something like X2= 36 and you know X will be
either 6 or -6, or something else weird will happen (and it is this last
thing that actually happens here -- your book or your teacher must have
a sense of humor, perhaps a slightly sadistic one?). So, let's see what
happens when we do all the operations in order to get to a simplified statement
of the equation: 5x²-[2(2x²+3)]-3=x²-9
Multiply it all out and you get:
5X2 - [4X2 + 6] - 3 =
X2- 9
X2 - 6 - 3 = X2 - 9
which is just to say that X2 - 9
= X2 - 9
or
9=9
In other words, this equation will be true for every value of X since
the equation is something like: X + 12 = X + 12,
which is always true, no matter what X is. Weird problem to give you.
Finally, above I said that you USUALLY need to multiply out all the
factors they give you, BUT you don't have to ALWAYS multiply factors out,
if you happen to see that a factor is duplicated somewhere, or duplicated
on each side of the equation in a way you can get rid of it.
Suppose we had: 3(2x
- 5) + x = 4 + 3(2x - 5)
Since there is a 3(2x - 5) on both sides of the equation, we can subtract
that quantity from both sides without having to figure out WHAT it is.
That would give us x = 4 without having to do all the work of multiplying
and regrouping and everything.
Or, if you had 6(2x - 5) divided by 3(2x - 5), you can know that is
equal to 2, no matter what (2x - 5) equals.
The Structure of Algebra Courses
Typically
Typically students are taught a number of principles and manipulations
that seem to many of them to have nothing to do with anything. Then they
are given practice in using those principles or performing those manipulations
-- principles and manipulations which may or may not make any sense to
them. For example, they are taught about association
and commutation, and given practice "simplifying" or "multiplying out"
expressions such as a(b + c) or (a +b)(3c - 5d). They are taught the rules
for "order of operations" and then given practice calculating expressions
that are written without parentheses. Or they are taught to factor expressions
like ax + bx into x(a+b). Or they are taught you can do the same things
to both sides of an equation, as long as you do the same thing to each
side.Then they are given equations to solve, which seem
to use a bunch of those manipulations or principles. And finally they
are given problems in words which seem to have something to do with
equations and manipulations.
I write the above the way I did because much of algebra seems rather
arbitrary to students who do not understand the individual manipulations
and principles (in algebra overall, or in a particular chapter or unit)
or who do not understand their point. Some of these students will not be
able to remember the manipulations well enough to do them very well; others
will be able to use them rather mechanically to solve problems of a type
they have been trained to solve, even though they don't really understand
why one goes about those particular mechanics other than that they give
you the answer the teacher likes. For both of these types of students,
new problems will be particularly difficult.
What is actually occurring is that it turns out there are certain (kinds
of) logical, sensible, reasonable principles and manipulations that
tend to be useful in solving certain kinds of problems that are the typical
algebra problems, or, to put it perhaps better -- there are certain (kinds
of) logical principles and manipulations that tend to be useful in solving
the sorts of problems that algebra lends itself to solving. That is why
students are taught these particular manipulations and ideas. It
is important for students to see the logic and the sense in the principles
they are taught, not just to learn the rules as some sort of arbitrary
recipes. The principles themselves are logical, not arbitrary, although
the way they are stated, the order in which they are introduced, and the
particular ones chosen for a particular book or chapter may have been quite
different.
One of the important aspects taught in algebra is precision of expression,
so that one can learn to express in numerical, usable form ideas or problems
that normally occur first in words. The example I gave above is one such
case, where I was entering checks into a ledger and needed to figure out
how much of the amount on the check was for sales tax and how much was
the price of the object. Another case might occur in something like baseball
where a batter might want to know how many hits he may need in his next
10 at bats to raise his average to a certain level. Or you might want to
know how much money you need to take on a certain car trip in order to
pay cash for gasoline. Or you might want to compute how much interest you
paid on your mortgage last year versus how much of what you paid went for
principle. Solving these problems generally requires your being able to
express the problem in numbers related to each other in some way, and then
knowing how to go about manipulating the numbers.
Some of the concepts apply not just to math, but to language
in words. I was driving home tonight listening to a news program on the
radio in which they were talking about celebrating Dr. Seuss's birthday
in many schools, and in one school, a chef had prepared "green eggs and
ham" in honor of the book by that name. Well, I was familiar with the book,
and had read it many times to my own children. However, I was surprised
when the news reporter asked the chef, after he had explained how he made
the eggs green, how he had made the ham green. I was surprised because
it never occurred to me that the ham was supposed to be green. I thought
Green Eggs and Ham meant Ham and Green Eggs; I thought the "Green" went
just with the "Eggs", not with the ham too, or not with the Eggs-And-The-Ham.
To put it in math-like terms, I thought the title and story were about
(Green Eggs) and Ham, not about Green (Eggs and Ham) or Green Eggs and
Green Ham. When I mentioned it to my younger daughter, she put another
interpretation to it -- Green Ham-and-Eggs, that is a greenish mixture
of ham and eggs, not just Green Ham with Green Eggs. So now I am not sure
at all what Dr. Seuss really meant, because he didn't use parentheses in
the story or the book title and he didn't write about the individual components....
So part of algebra is learning the language
of how to write down consistently precisely what you mean when you are
trying to express or represent something numerically; it is about learning
a precise language. As you learn to do this, you should be thinking
about what a given numerical expression might mean if you write it one
way versus another. If you do that, then numerical expressions will begin
to take on meaning for you and not just be a bunch of symbols you are trying
to manipulate. And if you understand their meaning, you will be able to
figure out ways to manipulate them when you cannot remember by rote how
to do something. For example, if you go to add 1/a to 1/b, you might be
tempted to think it will come out to 2/(a+b), but if you understand that
1/a is a fraction, and that 1/b is a fraction with a different denominator,
you can check to see whether you can add fractions that way by looking
at fractions you know, say adding 1/2 to 1/4. You know that will be 3/4,
so the question is whether your inclined way to do it will also give you
three fourths. However, you will see that it gives you 2/6, which is 1/3
and is not anything near 3/4. That way you know your inclined way of adding
1/a and 1/b is not right. If you weren't thinking about what 1/a and 1/b
meant, you might have gone ahead and just made the erroneous combination.
At a more complex level, suppose you are doing one of those problems where
it takes one guy 3 days to do a job by himself, another guy 2 days, and
a third guy a day and a half, and they want to know how long it will take
them all working together to do the job. If you get an answer of more than
a day and a half, you know something is wrong, because that means that
when the fastest guy has help, it takes him longer to do the job than it
does when he is working alone. (Now, of course, some people DO slow you
down when they try to help you, but that is not the intent of problems
like these.) So you not only want to think about your problems and your
answers in numerical terms, but what those numerical terms mean in your
own language.
Another tool that tends to crop up repeatedly in working problems in
an algebraic way is multiplication across or through parentheses -- expressions
of the sort (a+b)(c+d), or even more complicated by having more components
or more multipliers. You need to be able to do these sorts of multiplications
quite often in order to be able to get to the specific quantities of a
certain variable (or unknown) and the quantities of non-variables. E.g.,
if you have some problem that starts out being expressed as
8(x + 3) - 4(x - 21)=5(x-1) + 3(4x - 2),
you will likely need to multiply all that out in order to figure out
how many X's equal how much.
But, on the other hand, there will be times you will want to be able
to factor expressions into components in order to work with them. For example,
if you have (X2 - 49)/(X + 7) = 14,
you can figure out pretty easily that X = 21 because any expression of
the form (A2 - B2) can be factored, as you normally
would be taught in an algebra class, into (A + B)(A - B), which means that
the above expression will factor into (X + 7)(X
- 7)/(X + 7) = 14 And that means that you can then divide the (X + 7) in the numerator
by the (X + 7) in the denominator and end up with just X - 7 = 14, which
means X must be 21. And if you work out (212 - 49)/(21 + 7),
you will get (441 - 49)/(28), which is 392/28, which is 14, as the original
equation stated. So by factoring you made it easy to calculate something
that would otherwise be difficult to see or figure out.
Algebra courses also often teach about the relationship between graphs,
or certain kinds of lines and curves and shapes (that appear on graphs),
and numerical expressions or representations of those things.
I am not sure whether it is true or not, but it may be helpful to
think of many branches of math as being potentially able to represent or
express certain important characteristics of phenomena of all kinds in
numerical and/or logical terms.
Basically what you want to do when studying algebra
is to make sure you understand 1) what expressions, equations, formulas,
and manipulations really represent and why they are able to represent what
they do, and to make sure you understand 2) how to do the sorts of manipulations
they give you, AND how and why those manipulations work. And you want to
ask your teacher how those representations and manipulations will likely
be useful later when you get to actually working real problems, or at least
the sorts of problems expressed in ordinary language
in the algebra class. Then you also
want to have practiced sufficient representations and manipulations sufficiently
to be able, not only to understand them but, to do them fairly quickly
and automatically.
Otherwise what will tend to happen to you in an algebra class is that
you will just try to memorize sequences of equations and their manipulations
that don't really make any sense to you, but which you can memorize --
UNTIL you get too many to be able to retain or UNTIL you have to figure
out WHICH MANIPULATION is the one you need at a particular time. It is
one thing, for example, to be able to work out problems involving (A2
- B2) when they come at the end of a chapter teaching about
(A2 - B2); it is quite another thing to be able to
recognize something in that form and realize the form can be useful to
you three chapters later in the book when you haven't been working specifically
with that form and aren't looking just for it.
Part of the reason students tend to memorize aspects of algebra that
they instead ought to be trying to understand logically is that they wasted
time at the beginning of algebra trying to understand "conventions" that
did just need to be memorized instead of being understood logically.
How mathematicians understand the use of parentheses (or their absence)
is a matter of convention that just has to be learned by memory, but the
consequences of that choice have to be understood logically as you practice
doing different manipulations, such as adding fractions with different
denominators. For example, you cannot logically add 2/3(a + b) to 3/3a
+ b to get 5/3a + b, because the denominators are not the same even though
they look similar. Suppose, for example, that a = 4 and b = 6. The
first fraction then is 2/30 and the second fraction is 3/18. And
their sum is neither 5/18 nor 5/30.
So, if you have a problem such as
2/4(a + 4) + 3/(3a + 2) = 7/10
and you are trying to solve to find out what "a" is, you have to know how
to add the two fractions together first. That is why algebra books
give you many exercises (sometimes far too many) to practice calculating
or "simplifyng" manipulations such as adding fractions with unlike denominators.
The manipulations may or may not be useful for solving any given problem.
It will sometimes seem as though the manipulations or practice calculations
have no point other than to torture students with homework. But they
have a point in that they tend to be the general kinds of calculations
one will need to be able to do to (easily) solve different kinds of typical
problems (on tests and in real life) that require algebra to solve.
So what students need to do in general in algebra courses is:
1) Learn the symbols and conventions of the language by memory and practice.
This is not a matter of logic. However, understanding the need and
use for the symbols and conventions sometimes is a matter of logic -- seeing
what sorts of distinctions and differences necessitate the symbols and
conventions. 2) Learn to do various kinds of calculations using the conventions.
These are a matter of logic and understanding, prior to practicing them
to help you use them better. 3) Learn to solve equations, normally by using the logical manipulations
and calculations in a logical and creative manner to isolate the unknown
variable on one side of the equation and its eqivalent on the other.
This is a matter of insight, logic, creativity, and sometimes luck.
But insight, creativity, and luck can be improved many times by practice
and understanding.
There is an analogy to much of this in learning computer languages,
or even in building web pages with HTML codes. It is one thing to learn
HTML codes or functions you can make the computer do in a given programming
language, but it is quite another to see how you can build elaborate web
pages or create very complex functions and utilities by simply utilizing
a few simple codes or functions in ingenious combinations.
Rick Garlikov ([email protected])
There are two ways to get this -- the short way
of just following a rule (either the rule that when you multiply a negative
by a negative, you get a positive; or the rule that when you subtract a
negative number, that is the same thing as adding the number), and the
longer way where you understand what you are doing -- which is to realize
that in the above case you are subtracting (double) a-number-that-is- going-to-be-made-smaller-by
7-first. Let's look at an example in numbers first:
Suppose you have 20 things and someone thinks
he wants to buy 9 of them. If he does, you would have 11 left, since 20
- 9 would be 11. Now suppose someone else comes along and wants to buy
some of those things but for some reason he says "I want 7 less than the
last guy bought". Well, he wants 9 - 7 then, or 2. There are at least two
ways you could figure out how many you will have left after this purchase:
1) you could just say, he is buying 2, and since
I have 11, I will have 11 - 2 left, or 9. Or
2) you COULD say (but you probably wouldn't) that
what you will have left will be 11 - (9 - 7), since (9 - 7) is in numbers what the guy told you in
words that he wanted -- seven less than the last guy bought. Hence, you
are subtracting 9 from the 11, BUT you are adding back the seven the guy
did not want to take from you. In essence he is subtracting seven less
than the first guy, so if you subtract the same amount the first guy took,
you will be subtracting 7 too many.
Either way it should come out the same, because
either way you figure it out, it will still be that you will have 9 things
left.
And since, in the equation they gave you they
double the quantity after they subtract something from it, it would be
like the second guy's saying he wanted to have twice as many things as
seven less than the other guy, which, of course is a stupid way to talk.
But suppose he and his brother each have the
same number of kids and that he and his brother are trying to keep up with
the Joneses who have seven more kids than each of them has. So he wants
to buy the same things for his kids and for his brother's kids that the
Joneses buy. So he goes everywhere Jones does, and any time Jones buys
something for each of his kids, he wants to buy the same number of things
for his own kids and for his brother's kids. So he always tells the sales
person, "whatever that guy bought, I want to buy seven fewer -- but
after you figure out how much that is, I need you to double it, since I
am buying the same amount for my brother as I am buying for myself." This
is represented by 2(Jones' purchase - 7), and it will come out to be twice
what Jones purchased, minus 14. So, if you were keeping a running inventory
of what you had left, and you started with, say, 100 of these things, Jones
would leave you with an inventory of (100 - Jones'purchase) and the next
guy's purchase would leave you with that amount minus the following amount:
2(Jones'purchase - 7)
and if you play around with some numbers, you
will see that always ends up where the second purchase is 14 less than
the first purchase, and therefore leaves you with 14 more items left in
your inventory than if the second guy had simply bought twice what Jones
did. (Return to the main text.)
The equation we have arrived at is 9 - 12X
+ 30 = 5X - 12X + 14
Notice that 12X is subtracted on both sides of the equal sign. This
means that the same amount is being taken away from both sides and therefore
the rest of what is left on either side will still be equal. So instead
of adding 12X to both sides, we could actually just eliminate subtracting
12X from both sides, and either way, arrive at: 9 + 30 = 5X + 14, which
is to say that 39 = 5X + 14. So if we didn't add the 14 to the 5X, we would
have 14 less than the 39, which would be 25; meaning that 25 = 5X, which
is just to say that some quantity multiplied by 5 will give 25. And, of
course, that quantity is 5. So X = 5.
(return to the text)
Factoring & "Multiplying
Out"
By "multiplying out" an expression, I simply mean taking an expression
such as a(x + y) and doing the multiplication
which is indicated so that you get ax + ay.
Or, if we were to take an expression such as 5(15x
- 4y), you would get 75x - 20y.
Or if we had (3x + 2)(4x - 5), that
would give us, when we "multiply it out" 12x2
- 15x + 8x - 10 or (combining the "x's")12x2
- 7x - 10.
"Factoring" just means going in the opposite direction; that is, dividing
a quantity into factors or components which, when multiplied together,
would give you that quantity. The factors of 10 then would be 5 and 2.
Factors of ax + ay would be a
and (x + y) so that when you "factor"
ax + ay, you get
a(x + y). From the paragraph above,
if you factor 12x2 - 7x - 10,
you get (3x + 2)(4x - 5).
Multiplying out expressions is pretty much just a mechanical procedure
where you multiply each of the terms times the other terms and then combine
the "like" or similar terms -- numerical quantities, quantities with the
same variables to the same power, etc.; but factoring takes some insight,
and often a great deal of luck in seeing combinations. For example, it
is not easy to see that the above expression 12x2
- 7x - 10 even factors, let alone see what the factors are.
If you have been doing "factoring problems" in a text book chapter, you
start to see patterns that the author has begun to use, or that are stereotypical
in textbooks, but those are not usually in your mind when you are working
real problems later and get into an expression that is factorable but not
obviously so. (return to text)
Sometimes you can do a problem without having to go through lots of
manipulations because you can see the answer right off. Physicist Richard
Feynman, when he was in high school, answered a difficult algebra problem
immediately during a math tournament, without doing ANY algebra or math
at all. It was something of the sort where a rowing team during practice
is rowing upstream against a current that is moving 4 mph relative to the
shore. They are making progress relative to the shore at the rate
of 1.5 mph. The hat of the guy in the back of the boat falls off into the
river without his realizing it, and it floats downstream with the current.
After 15 minutes they realize they have lost the hat, and immediately begin
rowing back downstream to retrieve the hat. If they row with the same strength
or power they were rowing upstream, now that they have the current going
with them, how long will it take them to retrieve the hat? Feynman saw
immediately that it was 15 minutes because he realized that all the rates
relative to the shore were irrelevant to the result, just as when you drive
west on an open stretch of level road for an hour and then drive back at
the same rate, it doesn't take you longer to go one direction rather than
the other, even though the earth is turning eastward at the rate of approximately
1000 mph (near the equator).
I found out, through a different, counter-intuitive, "trick" problem
one time that not only is the circumference of any circle six and a quarter
times its radius, but that when you increase the radius of any circle any
amount, you thereby increase the size of its circumference by roughly 6.25
times that amount. Therefore, if you tell me that you added 10 inches to
the radius of a dime, or 10 inches to the radius of the universe (assuming
it is round, which may not be true, of course), I know that you have increased
the circumference of each by the same amount -- roughly 62.5 inches.
Of course, in an algebra course, teachers won't tend to accept
such reasoning, and want to see a "mathematical proof" -- which can be
given for both of these cases; but I am simply saying that in real life,
one does not necessarily need to solve what look like algebra problems
in an "algebraic way". The idea in real life is to solve problems however
you can. Algebraic manipulations -- particularly standard cookbook manipulations
-- are just one way to solve certain kinds of problems. Feynman himself
seemed to hold that the algorithmic or cookbook types of rules one learns
in algebra class are really just rules for people to follow who don't really
understand mathematical thinking. I wouldn't go quite that far, but I would
say that following such rules is only one way to solve many (apparently)
mathematical problems. (Return to text.)
These are what are usually called "word problems"
because they are logical/numerical problems expressed in ordinary language.
They are attempts at giving you some practice in solving problems the way
they supposedly appear in real life, since real life problems don't appear
already set up in formulas or equations.
However, unfortunately, real life problems
also don't appear formulated as structured as word problems do; and they
don't appear at the end of a unit so that you get an arbitrary, circumstantial,
clue to what is expected. (return to
text)
Languages are "conventions" because what words
or phrases or marks on paper mean is what people decide they mean or what
they happen to grow to mean. There are private conventions, such as secret
codes, and public conventions such as ordinary languages. There are
also private conventions for individuals. For example, if you are
measuring and sawing wood, you might mark the length of the wood to be
cut by putting your mark even with the end of your measuring tape, or you
might put the mark just outside the measuring tape. Either way is
fine, as long as you know when you go to cut the wood, which way
you chose to make the mark. If you made your mark even with the end
of the measuring tape but cut the wood inside of that mark because you
thought you had made the mark on the outer edge of the tape, your wood
will be a fraction shorter than you intended. So it is important
to remember what your mark really means. You can do this by remembering
how you do things each time you do them, even if you do them differently
each time. Or, usually easier, you choose a way to do it every time
and then do it that way each time, and there is less to remember.
You have developed a convention for yourself.
In math and science, ambiguities of ordinary verbal language (such as
in the "Green Ham and Eggs" case) are tried to be eliminated by developing
and improving as necessary, precise symbols that have very specific meanings.
This is intended to prevent confusion in communicating to others what you
are doing. But it also helps you keep straight yourself what you
are doing, by not having to remember each time what you meant by using
a specific symbol.
Conventions have something of an arbitrary nature, however, though once
developed they might have some sort of logic or psychological sense of
appropriateness or seeming reasonableness of their own. They
are arbitrary in that we could have used any symbol to designate what we
want to designate. And in some cases how we use them is arbitrary.
For example, in algebra, "3x + 1" is understood by convention to mean the
quantity that is one more than the product of 3 times the variable "x",
rather than the product of three times the number which is one larger than
x. If "x" were 8, "3x + 1" is 25 in the way the convention or language
of algebra is used because it is 3 times 8, which is 24 and then 1 is added
to give 25. But if there were no such convention, someone might take
it to be equivalent to 27, because they thought it meant 3 times the sum
of eight and one, which would be 3 times 9.
In algebraic notation, that is to say that "3x + 1" just is designated
to mean the same thing as "(3x) + 1" rather than "3(x + 1)".
Without that convention "3x + 1" would be ambiguous and possibly mean different
things to different people, or you might use it one way when setting up
a problem and then mistakenly think it meant something else when working
the problem -- as when you accidentally forget to keep the parentheses
when going from one step to the next while solving a problem and then get
the wrong answer.
In algebra, sometimes they state conventions in ways that make it sound
like they are a conclusion to some sort of logic rather than merely an
arbitrary convention or choice that was once made. This confuses students
who are trying to understand "why" something is the way it is when, in
fact, there is no logical reason, because it is simply a convention.
For example, using the "x" below to mean simply the multiplication symbol
(not an unknown variable) some teachers at the beginning of algebra
will ask students what the answer is to something like
3 + 5 x 8 - 2 x 3
Many students might say it is 186 because they just did the steps in the
order they were given: 3 + 5 is 8, then 8x8 is 64, and subtracting 2 gives
62, and then multiplying that by 3 is 186. That is certainly logical.
But the teacher will often say, "No, the answer is 37" and give as the
reason that "Because in algebra, you always do the multiplications
and divisions first and then do the additions and subtractions."
So the problem becomes 3 + 40 - 6, which is 37. When stated that
way, it leaves many students asking "Why?" in the sense of how that works.
But it doesn't "work" in the sense of being a matter of logic. It
is only true because that is the convention that was established.
It could have been established that you do calculations in order unless
there are parenthesis grouping quantities together, but it just wasn't.
Neither way is necessarily better than the other, but once one way is picked,
if you want to communicate with others, you need to use the conventions
they understand. It is like driving on the right or left side of
the road. It is not that there is some reason for one side's being better
than the other -- if we are going to choose, as a society, from the very
beginning; but once a choice is made, it is better to abide by the choice
so you don't run into others. Sometimes conventions are not as clear
as they ought to be. Crosswalks, for example, in some places mean
that pedestrians already walking in them have the right of way and cars
need to stop for them. But in some places, they mean that cars have to
stop for pedestrians on the curb at a crosswalk who have not yet even stepped
into the street. That is a very dangerous ambiguity for pedestrians
who think the crosswalk means they can step off the curb without even stopping
or looking if approaching drivers think it means there is no reason to
slow down because the pedestrian will surely stop walking when he gets
to the curb and wait till there are no cars approaching before he begins
to cross the street.
But conventions, once established, also have a resulting logic that
is not arbitrary. For example, because the English drive on the left
side of the road, and Americans drive on the right side of the road, the
way they each have to make right or left turns from two-way streets onto
two-way streets in their own countries is different. When Americans
turn right, they stay near the curb. When they turn left, they have to
"swing wide" to get to the right side of the street onto which they are
turning. The English have to do just the opposite, hugging the curb
lane to make left turns and swinging wide to the outside lane to make right
turns. That is a logical result of choosing which side of the road
to drive on, even though the choice itself had no logic but was simply
a matter of convention. Moreover, there is another logical result
of the convention. Although we are taught to look both ways before
crossing a street, we tend to look one way, toward the traffic approaching
on the curb side, as we step off the curb; and we look the other way as
we get to the center of the street. In America, one generally then
looks left before stepping off the curb, and right once one gets to the
middle of the street. In England, that can put you in front of a
bus, because the traffic next to the curb is coming from your right, not
your left; and the traffic on the far side of the center line is coming
from your left not your right. The English have the same problem in reverse
when they are in America. That is a very difficult habit to overcome when
visiting the other country and walking across streets. But the reason
you have to look the way you do as you cross a street is a logical result
of the side of the road the country chose to have drivers use. So
conventions which are arbitrary, nevertheless can have consequences that
logically depend on them. (Return to text.)
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Introduction to Fitting Things In Spaces:
If you aren't the greatest at maximizing closet space, you may have some direct experience being frustrated by this one. It will often come in handy to know how much room you are working with, and how much of a certain object or material can possibly go there.
Sample Problem
Shin is building a shelf to put over his desk and hold his books. Each book is of an inch thick. Okay, they're issues of Vogue, but same difference. The board Shin is using is 2 ft long. How many "books" can Shin fit on the shelf?
This problem can be turned into an inequality. We're turning it into an inequality rather than an equation, because we might have a little space left over on the shelf. Let x be the number of "books" on the shelf. We'll stop putting quotes around "books" now, since it's exhausting. Paying attention to units, we see the shelf is 24 inches long. Taking this information and writing it in symbols, we find that
Now that we have this inequality, we can forget about books for the time being and solve the inequality. To do this, we multiply both sides by to find that
and simplify to find that
x ≤ 32
Now we need to think about books again. Sorry. It'll all be over soon. This answer means Shin can fit up to 32 books on his shelf.
Okay, you can stop thinking about books now. We told you it would fly by.
Algebra could also be used to figure out how many cars you can fit in a parking lot, how many boxes of cereal you can fit on a shelf, how many airplane runways you could fit on a piece of land, or how many full-sized marshmallows you can fit in your mouth without choking. Don't try this at home.
In the next section we will be dealing a lot with specific practical applications of algebra. In the meantime, rest assured that algebra is useful.
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This course targets students majoring in both computational and biological sciences,
broadly defined to include mathematical, computer science, bio-medical and environmental majors.
The goal of this course is to give students an understanding of the biological-mathematical interface,
and how mathematics contributes to the study of biological phenomena. Biological systems
have a very high level of complexity and practically every phenomenon is the result of
complex interactions between various levels of organization. To apply modeling
(both mathematical and experimental models), we always simplify the natural system by
making both implicit and explicit assumptions, and this course teaches students to see
the hidden assumptions and understand their role in the results of model applications.
The course introduces general mathematical methods in biology, such as scaling,
approximations of stochastic and individual-based biological models by differential equations,
and linearization and stability analysis, using both classic and recent examples. The course covers
fundamental and applied models operating at different organization levels, from processes inside individual
cells to those that form ecosystems. Specific examples include: dynamics of infectious diseases
(flu epidemics and AIDS), natural recourse management (fisheries), forest dynamics, interacting
species (resource competition, predator-prey, and host-parasite models), spatial models,
enzyme kinetics, chemostat theory, and bioremediation. In this course biology students
learn to formulate their specific questions in a mathematical way, while mathematics
students learn what constitutes biologically relevant questions, and how to accept
the high level of uncertainty that exists in biological research. A substantial part
of the course will use analytical methods in concert with computer simulations, using the Mathematica software.
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This courses uses the 2011 edition of the Jurgensen, Brown, and Jurgensen textbook, ... Brown, and Jurgensen, Geometry. Houghton Mifflin, 2011. Teachers use other texts for supplementary ideas, such as Discovering Geometry by Michael Serra, and also current mathematical
This course use the 2000 edition of the Jurgensen, Brown, and Jurgensen textbook, Geometry, published by Houghton Mifflin. ... This text matches both the 2000 edition of the National Council of Teachers of mathematics curriculum standards and the 2000 edition of the Massachusetts State
Ray Jurgensen, Richard Brown, and John Jurgensen Students and Grade Levels ... Geometry, Pupil's Edition ... provide service to teachers. This service is available from 8 a.m. to 5 p.m. CST, Monday through Friday. Page 4
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Follow Us
Applied Mathematics
Mathematics is an essential part of scientific development. By itself, mathematics is a subject of great depth and beauty. But mathematics is also crucial in the development of natural sciences, engineering and social sciences. At UC Merced, several professors focus on interdisciplinary, applied mathematics. Their focus is in solving real-world problems using modeling, analysis and scientific computing.
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The first new American research university in the 21st century, with a mission of research, teaching and service.
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4. R Bhatia, Pinching,trimming,truncating and averaging of matrices, paper available on the course website.
------------------------------------------------------------
Useful preparation for the course "Numerical ranges (classical and higher-rank) with applications to quantum information theory" , to be given by John Holbrook during the second week, might include some familiarity with:
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Saul Stahl
When reviewing a book on modern algebra the issue is not only how good the book good is, but also for whom it is good. This book is an excellent book for an upper-level, undergraduate, one or two semester course, in modern algebra, for a typical University student population that is not especially strong in proofs.
What I particularly like about the book is the following.
Doable Exercises: The strongest point of the book is the richness and diverse flavor of over 1000 exercises. There are proof exercises but most exercises are non-routine computations or verifications. The issue with general proof exercises is that weak students can attempt them and get nowhere, thus wasting time and encouraging them to give up. An exercise that has thinking aspects but is based on non-routine computation or verification can be done with enough work even by a weak student. This stimulates and motivates.
An illustration of the computational-verification flavor of the exercises is afforded by the first section in the chapter on group theory which has 37 exercises.
10 of them are of the form "compute the group of symmetries of x1/x2 + x3/x4"
12 of them are of the form "compute the product and find the axes and angles of the rotations of the following two rotations of the tetrahedron in Figure 9.3: A = (1 2 3) and B = (2 4 3)"
3 of them ask about the dihedral group Dn (e.g. how many elements does the group have; how many have order 2, etc.)
4 of them are of the form "Describe the vertex symmetries of the cube in Figure 9.5"
6 of them are "verification proofs", for example, "show that all even permutations form a group"
2 of them are more serious proofs, for example, "Prove that for every positive integer k there is a polygon whose group of vertex symmetries contains k elements."
Exercise Richness: The text excels in both quantity and quality of exercises. The exercises have a rich diversity of color as the following examples illustrate.
The book has a standard appendix on mathematical induction. There are 18 exercises; a) there are routine exercises such as proof of the sum of squares or cubes of the first n integers; b) there are also inequality proofs such as "prove (by induction!) that 2n > n2 for n > 4" ; c) there are number theory exercises such as "prove (by induction!) that 11n+2+122n+1 is divisible by 133" ; and there are d) geometric and e) integral exercises.
The exercises on polynomials are enriched by many exercises requesting factorizations over finite fields.
History: The author wrote this book from the historical point of view. This can indeed be exciting to a student interested in what mathematics is like. I myself found it interesting to see original excerpts from the masters such as al-Khwarizmi (solution of the quadratic equation), Cardano (solution of the cubic), Abel (unsolvability of the quintic), Galois (foundations of Galois theory) and of course Cayley (enumeration of groups by looking at permutation groups). I believe the real strength in using a historical approach is the wealth of computational examples it invites. This is felt throughout the book where exercises challenge students to apply the theory to solve equations of degree 3 or 4 over the complex and finite fields as well as factorizations over rings over the integers.
Modern Look: The book has all the characteristics of many modern textbooks: a) accompanying diagrams, b) adequate illustrative examples in each section, c) chapter summaries, d) a list of new terms at the end of each chapter, e) chapter review exercises, f) supplementary chapter exercises, g) solutions to odd number exercises, h) appendices covering induction and logic in adequate depth, i) a modest bibliography and a j) neat collection of one-paragraph biographies of about two dozen mathematicians.
Semester Coverage: The book has 14 chapters and 60 sections (each with several dozen exercises) making it usable for either a one or two semester course. The section lengths are just right for coverage in one day. The book uses an example-abstract approach vs. an abstract axiom-example approach. This means, for example, that the definition of group is delayed a few weeks into the semester. Personally, I prefer such an approach and I think the students, exposed to an axiomatic approach for the first time, find it easier.
Non-standard Applications: Every author tries to include non-standard applications, that is, applications of modern algebra not found in almost all other text books. This book emphasizes a) the 15 puzzle, b) the RSA algorithm, c) Dedekind ideal theory, and (as already mentioned) the historically motivated d) solvability of equations and e) geometric constructibility.
I have never seen another modern algebra book with a presentation of the quadratic reciprocity law. True to the book's spirit, both the historical (mathematical theorem with the second most proofs) and aesthetic (the golden theorem) aspects of quadratic reciprocity are mentioned. The law is presented with accompanying diagrams and computational exercises showing the theorem's power. Of course, instructors who wish to can comfortably omit teaching the "Number Theory" chapter.
I found two topics lacking in the book.
The Beautiful Cayley Counting Theorem: At this I was surprised, since the Cayley theory lends itself to many clever and combinatorial computational exercises that would be consistent with the book's goals. This is not a serious omission, however, unless the student target population consists of Chemistry majors. I would recommend to the author to include an additional chapter in edition 3 or 4.
The Sylow Theorems: I and many colleagues prefer to omit this topic anyway.
Russell Jay Hendel, [email protected], holds a Ph.D. in theoretical mathematics and an Associateship from the Society of Actuaries. He teaches at Towson University. His interests include discrete number theory, applications of technology to education, problem writing, actuarial science and the interaction between mathematics, art and poetry.
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On selecting a constituent part of MU the "Overview of publishing activities" page will be displayed with information relevant to the selected constituent part. The "Overview of publishing activities" page is not available for non-activated items.
This book presents methods of solving problems in three areas of classical elementary mathematics: Equations and systems of equations of various kinds, algebraic inequalities, and elementary number theory, in particular divisibility and diophantine equations. In each topic, brief theoretical discussions are immediately followed by carefully worked out examples of increasing degrees of difficulty, and by exercises which range from routine to rather challenging problems. While this book emphasizes some methods that are not usually covered in beginning university courses, it nevertheless teaches techniques and skills which are useful not only in the specific topics covered here. There are approximately 330 examples and 760 exercises.
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Calculus: Single Variable (Coursera)
This course provides a brisk, entertaining treatment of differential and integral calculus, with an emphasis on conceptual understanding and applications to the engineering, physical, and social sciences.
Calculus is one of the grandest achievements of human thought, explaining everything from planetary orbits to the optimal size of a city to the periodicity of a heartbeat. This brisk course covers the core ideas of single-variable Calculus with emphases on conceptual understanding and applications. The course is ideal for students beginning in the engineering, physical, and social sciences.
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...
More hosted by Wolfram Research, Inc., and is offered as a free service to the mathematics community.
Technical Requirements: Basic browser. A Java-enabled browser provides enhancements but site is fully viewable without Java. Much of the material is powered by "Live Graphics 3D" which produces images similar to those in Mathematica.
Discussion for Eric Weisstein's World of Mathematics
Jessica Lynn
(Other)
Like a great interactive online textbook for math, one could easily get lost clicking from link to link. I bookmarked this page for further use. The visuals are great, the content is crisp and the information is easily accessible. This would be a great resource for teachers or for anyone looking for help with a math project.
9 years ago
jason miller
(Student)
Although I am new to Merlot, this isn't the first time I've been on mathworld. I have used this site before for research on a mathematics paper. Just for fun I checked out how diverse this site is and was even more impressed with it. I will bookmark it for future use. A good tool for instructors and students alike. Easy to use, and well organized.
10 years ago
Barbra Bied Sperling
(Staff)
The giant of Mathematics reference tools on the net. It was such an amazingly exhaustive catalogue of mathematics concepts, with superior illustrations, that it was also published in print. An absolutely phenomenal contribution by one person, comparable in scale to a mathematician's life's work.
Used in course
11 years ago
Christopher Taylor
(Student)
The sheer volume of material here is staggering. This site certainly outlines pretty much anything the average mathematics student could want to know. I spent a good twenty minutes just looking around, making sure that everything I could want to know was there. Like those hard to remember formulae from high school geometry, for example, because who can remember those when they need them...in their college math classes. Anyway, this site was very dense, from a materials stand-point, but never difficult to navigate. I would highly recommend it to anyone without a textbook handy. It has pretty much everything you could be looking for.
12 years ago
Ben Flores
(Student)
The most outstanding math site I have ever visit, so easy to use an so much to see, I will have to get back to it with more time, I learn that usually Physicis use the term sphere to mean the solid ball, but mathemathicians give a total different meaning, and that is the outer surface of a bubble.
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Description
This well-respected text gives an introduction to the theory and application of modern numerical approximation techniques for students taking a one- or two-semester course in numerical analysis. With an accessible treatment that only requires a calculus prerequisite, Burden and Faires explain how, why, and when approximation techniques can be expected to work, and why, in some situations, they fail. A wealth of examples and exercises develop students' intuition, and demonstrate the subject's practical applications to important everyday problems in math, computing, engineering, and physical science disciplines. The first book of its kind built from the ground up to serve a diverse undergraduate audience, three decades later Burden and Faires remains the definitive introduction to a vital and practical subject. Contents
Mathematical Preliminaries and Error Analysis | Solutions of Equations in One Variable | Interpolation and Polynomial Approximation | Initial-Value Problems for Ordinary Differential Equations | Direct Methods for Solving Linear Systems | Iterative Techniques in Matrix Algebra | Approximation Theory | Approximating Eigenvalues | Numerical Solutions of Nonlinear Systems of Equations | Boundary-Value Problems for Ordinary Differential Equations | Numerical Solutions to Partial Differential Equations Additional ResourcesCompanion WebsiteRelated TopicsCalculus and Analysis
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The irresistibly engaging book that "enlarges one's wonder at Tammet's mind and his all-embracing vision of the world as grounded in numbers." --Oliver Sacks, MD THINKING IN NUMBERS is the book that Daniel Tammet, mathematical savant and bestselling author, was born to write. In Tammet's world, numbers are beautiful and mathematics illuminates... more...
Maths is everywhere, often where we least expect it. Award-winning professor Steven Strogatz acts as our guide as he takes us on a tour of numbers that - unbeknownst to the most of us - form a fascinating and integral part of our everyday lives. In The Joy of X , Strogatz explains the great ideas of maths - from negative numbers to calculus, fat... more...
All the math basics you'll ever need! It's not too late to learn practical math skills! You may not need to use quadratic equations very often, but math does play a large part in everyday life. On any given day, you'll need to know how long a drive will take, what to tip a waiter, how large a rug to buy, and how to calculate a discount. With The... more...
This text embodies at advanced and postgraduate level the professional and technical experience of two experienced mathematicians. It covers a wide range of applications relevant in many areas, including actuarial science, communications, engineering, finance, gambling, house purchase, lotteries, management, operational research, pursuit and search.... more...
Towards the end of the nineteenth century, Frege gave us the abstraction principles and the general notion of functions. Self-application of functions was at the heart of Russell's paradox. This led Russell to introduce type theory in order to avoid the paradox. Since, the twentieth century has seen an amazing number of theories concerned with types... more...
Rod structures are widely used in modern engineering. These are bars, beams, frames and trusses of structures, gridwork, network, framework and other constructions. Numerous applications of rod structures in civil engineering, aircraft and spacecraft confirm the importance of the topic. On the other hand the majority of books on structural mechanics... more...
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Engineering Mathematics by K.A. Stroud
... This is one of the reasons Engineering Maths is so popular as incomplete explanations often hinder learning more than help.
The one thing I would stress is that this book really can only be considered as the first half of a two part set. I don't know about other Engineering courses but ... Read review
category books comics magazines about speedy hen ltd by continuing with this checkout and ordering from speedy hen you are accepting our current terms and conditions details of which can be found by clicking here content note tables graphs country of publication united states date of publication 10 03 2011 edited by a k haghi format hardback genre level 1 adult non fiction specialist genre level 2 engineeri
Great Value, Very clear, easy to understand, You can pass exams, earn more money, buy more beer
You might not find it useful but everyone I know does
"As someone studying for an engineering degree this book has proved invaluable. it starts at the basic components of A-Level Mathematics and works it way through a large portion of the maths needed for a first year engineering course.
The book is divided into programmes so it works similarly to a lectured course. it directs the reader to answer certain questions and achieve the correct answers before progressing. it seems like an overly-simplistic ..."
Read review
"...form of Maths, Physics or Engineering in your degree then I highly recommend that you purchase this book! Maths comes into so many things and covers a very wide range of topics but for the basic things and a wide range of what you will study in your first few years at university this book has it well covered.
Unlike other books that I have looked at, this book does not assume that you have gained an A or A* at A-level Further mathematics or mathematics. ..."
Read review
"1236 PAGES
As an Engineering student I own several maths text books but this beauty is by far the best. This book differs from most maths textbooks in that it is easy to read and understand. It is designed for people who use maths.. but not "true mathematicians" .. which is why its so great!! If you're really awesome at maths (550+ / 600 in Further Maths without working) you could probably cope with something harder and poncier! If you see maths ..."
Read review
"...maths student. If your an Engineering student, Chemist, Physicist, Cybernetician etc etc this book is perfect for you. It contains vast amounts of useful and accessible knowledge. What I mean by accessible is that this book requires no prior research. It takes topics and breaks them down into constituent parts and takes you through very slowly. Asking you little questions throughout and giving you the answers. It then moves on to the more complex ..."
Read review
Those with superior maths/further maths skills may find it less helpful
"As a 3rd year Mechanical Engineering student at UCL I have found this textbook to be the most useful textbook in my study and revision arsenal.
Although engineers are required to cover and learn a great deal of math concepts, they must more importantly be able to apply them in everyday problems with confidence. This is where this particular book is heads and shoulders above the rest.
Instead of assuming that the reader is already a master of ..."
Read review
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TI-84 Plus Silver Step-by-Step Instructions
PDF (Acrobat) Document File
Be sure that you have an application to open this file type before downloading and/or purchasing.
0.1 MB
PRODUCT DESCRIPTION
Basic instructions for the TI84-Plus that include: descriptions of menu keys; how to create lists and then use the list to find measures of central tendency; how to graph lines; how to change the viewing window. Instructions include screen captures at each step so students can make sure they are doing the right thing.
2009, Sashaa Murphy
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{"currencyCode":"USD","itemData":[{"priceBreaksMAP":null,"buyingPrice":22.21,"ASIN":"0817636773","isPreorder":0},{"priceBreaksMAP":null,"buyingPrice":24.14,"ASIN":"0817639144","isPreorder":0},{"priceBreaksMAP":null,"buyingPrice":8.5,"ASIN":"0486425649","isPreorder":0}],"shippingId":"0817636773::aslNSo%2BtXl0k%2FkXCp03FKOeMQWYN2Paaz11UWvXRdpt%2B1Rur132Br2hpUc1%2FOFTuV4v1dnjsshzmSqS3oRsswJcIr7D4NFULtzokNfdJ%2BKQ%3D,0817639144::XH08Ien6rt5hoHD386VYzfajgbsGJdVQKs8Gig%2B0P5%2B0J2WHdlNz15f1UEdjZdLY2n1GteySYi%2BRMi00bOK%2BDQ1vqX%2B0b6niWIF%2FW54oZCs%3D,0486425649::qrFr9P1nCG4Iret4ZuAAEt8DgABQG3bZZnmxIKNEsoNYy5hFHUXYtFFzhec8o7AQELnf4tYGNKzeJUe9ieakba%2B%2BLN8SNyFYqNOVNGTb7 idea behind teaching is to expect students to learn why things are true, rather than have them memorize ways of solving a few problems, as most of our books have done. [This] same philosophy lies behind the current text by Gelfand and Shen. There are specific 'practical' problems but there is much more development of the ideas … [The authors] have shown how to write a serious yet lively look at algebra."—The American Mathematics Monthly
"Were 'Algebra' to be used solely for supplementary reading, it could be wholeheartedly recommended to any high school student of any teacher … In fact, given the long tradition of mistreating algebra as a disjointed collection of techniques in the schools, there should be some urgency in making this book compulsory reading for anyone interested in learning mathematics."—The Mathematical Intelligencer
This text, which is intended to supplement a high school algebra course, is a concise and remarkably clear treatment of algebra that delves into topics not covered in the standard high school curriculum. The numerous exercises are well-chosen and often quite challenging.
The text begins with the laws of arithmetic and algebra. The authors then cover polynomials, the binomial expansion, rational expressions, arithmetic and geometric progressions, sums of terms in arithmetic and geometric progressions, polynomial equations and inequalities, roots and rational exponents, and inequalities relating the arithmetic, geometric, harmonic, and quadratic (root-mean-square) means. The book closes with an elegant proof of the Cauchy-Schwarz inequality.
Topics are chosen with higher mathematics in mind. In addition to gaining facility with algebraic manipulation, the reader will also gain insights that will help her or him in more advanced courses.
The exercises, which are numerous, often involve searching for patterns that will enable the reader to tackle the problem at hand. Many of the exercises are quite challenging because they require some ingenuity. Some of the exercises are followed by complete solutions. These are instructive to read because the authors present alternate solutions that offer additional insights into the problem.
This book inspires even those with minimal interest in mathematics. If you are passionate about math, this is a must for you. The book is simply a refresher for high school algebra. It contains numerous gems that you could hardly find in a standard algebra text. If you are a teacher, you would have learned much to improve your teaching style and knows how to make your math classes more interesting...overall, a key source to keep on your bookshelf
Well, H. Wu on his page and N.F Taussig here have written quite good reviews, so I guess I can't really add anything new. Still, I feel the need to praise this book some more. Could it be used for a main text or should it be just a supplement? I don't know, but there is much more mathematics contained in these 149 pages than in any standard 500 page high school text on the market today. That's the unsurprising result of accomplished mathematicians writing a math book. Sure, some topics are missing. You won't find 3 or 4 chapters devoted to the several "different" ways to graph a line. There aren't fifty problems in a row that start with "suppose Sam rows upstream at 5 miles per hour and it takes her seven times as long as..." Unfortunately, there isn't even a treatment of complex numbers, the only omission that seems wrong.
You will find several interesting and serious topics that would be dangerous to bright students who insist they hate math, or rather what they've been told is math. Imagine their initial embarrassment when they find out that they can enjoy the subject! Maybe more importantly, imagine their relief when they realize that there IS a reason why we "FOIL", there IS a reason why negative times negative is positive, there IS a reason why we say a^(-1)=1/a, and it's not because "the teacher said so" or "that's just the rule" (ok, it is the rule, but now you'll see why). And there's no attempt to sneak anything by the reader. The authors are quick to acknowledge any gaps in their reasoning, and to assure the reader that in the future he or she will fill them.
It's this honesty and attention to rigor without being too formal or dry that give this book some extra charm. It moves smoothly from basic arithmetic (which everyone should still read if only to learn a different way of explaining it to a student/younger sibling/child) all the way to proofs, both algebraic and visual when possible, of some important inequalities. Cauchy's inductive proof, first for powers of two and then filling in the gaps, of the AM-GM inequality is here, as is the standard proof of Cauchy-Schwarz by the discriminant of a polynomial. Go to your local high school and look at its algebra book. I doubt that's in there.
I bought this book for my daughter (10 years old) and we read it together. We went very slow and I supplement it with a work book. She likes it. I was impressed by the beauty of this book. It might be a little too slim for a textbook but every kid who wants to learn algebra should read it. More than teaching algebra it shows what math should be: simple and beautiful.
My daughter's math textbook is 5 pounds and I can't even stand looking at it. I understand that not every is enthusiastic about math and not everyone can feel the beauty of math. But you don't have to make math so ugly.
Learning math with a 5 lb textbook is simply terrifying but if your kid goes to public school you probably have no choice. Let you kid read a good book like this one, as early as possible, before he(she) grows a life time aversion to math.
The material in the book, the knowledge, its great and wonderful. The style with which the information is presented is beautiful because it does so in the form of questions but in the process of answering the questions is how you obtain the information; the book makes you think.
The way with which the book is bound is another story though. My book started coming apart after only a few days, the pages are glued with no string to hold the pages together at all; they easily rip at the binding and from what I notice is that there is no way around it. It just spontaneously happening while I was trying to hold the pages open in a way so that the book is full open.
It is getting 4 stars because of the information and not five because of the binding.
this isn't a primary algebra book; I would most definitely recommend it to use as a supplement along with any other text though because it is great in that role.
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Student Explorations in Mathematics - National Council of Teachers of Mathematics (NCTM)
An official journal of the NCTM, Student Explorations in Mathematics publishes resources for teachers and teacher educators at grades 5-10. Each issue develops a single mathematical theme or concept in such a way that fifth grade students can understand the first one or two pages, and so that high school students will be challenged by the last page. The content and style of the notes are intended to interest students in the power and beauty of mathematics and to introduce teachers to some of the challenging areas of mathematics within the reach of their students. Article downloads are free to individual members who subscribe to SEM. Read submission guidelines and browse back issues.
more>>
Adventures in Statistics - Scavo, Petraroja
A Web unit preprint of a paper by teachers Tom Scavo and Byron Petraroja that describes a mathematics project involving fifth grade students and the area of classrooms, including measurement, graphing, computation, data analysis, and presentation of results;
...more>>
Algebra 4 All
A community of educators sharing resources and supporting one another in the practice of teaching algebra: lesson sharing, applets for students and teachers, discussion forum, blogs, media, and other content related to the functions-based approach to
...more>>
The Algebra Survival Kit - Josh Rappaport
A kit that includes a 520-page handbook covering the main content areas of Algebra 1 in accordance with the NCTM Standards. Sections are tabbed, and pages are written in flash card format with questions on the front and answers on the back. Also, a poster,
...more>>
Algorithmic Vectorial Geometry - Jean Paul Jurzak
In French. The author writes: "This work studies vectorial geometry under a new aspect which allow to solve most of the exercises of vectorial geometry without the traditional support of a drawing. For students, teachers, and informaticians."
...more>>
Alive Maths/Maths à Vivre - Nathalie Sinclair
Hands-on, interactive "microworlds," in which students investigate patterns and relationships, pose questions, play with the variables, and solve problems: Play with Lulu on a grid; practice triangle reflections; play with fractions and decimal patterns
...more>>
Analysis & Knowledge - Sanford Aranoff
Download data analysis software, such as Patterns, which analyzes and charts trends in stock market using Andrews concepts. See also this former Rutgers University physics professor and current high school educator's thoughts on teaching and help file
...more>>
AngryMath - Daniel R. Collins
Blog by an adjunct math lecturer at Kingsborough Community College who believes that "Math is a battle. It is a battle that feels like it must be fought ...," and for whom "math isn't beautiful or fun, but it is powerful, and that's what we need from
...more>>
Avances de Investigación en Educación Matemática
The official publication of the Spanish Society for Research in Mathematics Education (SEIEM, Sociedad Española de Investigación Matemática) welcomes contributions in either Spanish or Portuguese. Freely download PDFs of past articles,b's law - Brian Lawler
Blog begun March, 2011, by a professor of math education at California State University San Marcos. Posts, which "try to disrupt," have included "continued reaction to tracking," "The Problem of the Skateboarder Problem in IMP," "A Deconstruction of Learning
...more>>
CalculusABC.com - Chris Watson
A site for calculus teachers to share questions they've written and discuss
issues relating to the field. Sort multiple choice questions according to calculator use and representation type (verbal, numerical, analytical, or graphical). With web forums
...more>>
Constructing Math Instruction - Chris Robinson
Blog by teacher "averse to binary thinking that refuses to countenance all the complexities of the minefield that is the real classroom." Posts, which date back to December, 2012, have included "Number Sense and My Students"; "Analyzing Student Questions";
...more>>
The Cow in the Classroom - Ivars Peterson (MathLand)
Math Curse by Jon Scieszka and Lane Smith spoofs the types of word problems that educators and textbook writers invent to dress up arithmetic exercises and, supposedly, to demonstrate the relevance of math to everyday life. Canadian economist and humorist
...more>>
Daniel Willingham
From the author of Why Don't Students Like School? and When Can You Trust the Experts? (subtitled: "How to tell good science from bad in education"). See, in particular, Willingham's articles, such as "Why transfer is hard," "Why students remember or
...more>>
Developing Function Sense with SAQs - Judah Schwartz
A work in progress, this is an online book on the philosophy of teaching functions in middle and high school algebra. He has come to believe "that approaching algebra through the study of functions using symbolic and graphical representations simultaneously
...more>>
Double Division - Jeff Wilson
Double Division is a method for doing manual division that reinforces the principles of division and gives students success with a less frustrating alternative. The online calculator shows and explains each step. Feedback from teachers who've used theEASI Street to Science, Engineering, and Math (SEM)
Equal access to software and information: an NSF-sponsored project to collect and disseminate information on tools that make these fields more accessible to professionals with disabilities. Online workshops, Webcasts, links to programs for the visually
...more>>
Educational Aspirations
"Ramblings from a 21st century educator" who specializes in differentiating math and seeks to make the subject "more engaging and relevant to students' lives." Blog posts, which date back to July, 2011, have included "Measuring Student Growth," "Twitter
...more>>
Edward Burger
Journal publications, Thinkwell textbooks, and other professional activities of the Francis Christopher Oakley Third Century Professor of Mathematics, formerly Baylor University's Robert Foster Cherry Professor for Great Teaching. Burger, who co-authoredThe Exponential Curve - Dan Greene
A blog by a math teacher at Downtown College Prep charter high school (San Jose, California), where the students "... are primarily Latino, are far below grade level in their math and reading skills, and will be the first in their families to go to college.
...more>>
Finding Ways to Nguyen Students Over - Fawn Nguyen
Nguyen blogs mostly about her teaching middle school math. Posts, which date back to November, 2011, have included "Teaching 101 -- How I Got Like This," "Puzzles and Brainteasers, Set 11," "Pi Day Activities," "Always Sometimes Never," "Using Excel for
...more>>
garysmathsblog
One maths teacher's "personal musings" about his experiences, as well as maths resources and teaching tips to help teach maths. Blog posts, which date back to May, 2011, have included "Fraction word problem," "Dyscalculia," "Mean, median and mode rhyme,"
...more>>
gealgerobophysiculus - Evan Weinberg
Thoughts on teaching, learning, and "the crazy process of having fun doing both with students." Posts, which date back to September, 2011, have included "How China Keeps Me Learning: Part I," "Lens Ray Tracing in Geogebra," "Take Time to Tech – Perspectives
...more>>
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51
FREE
About the Book
Designed for students who are new to the graphing calculator, or for people who would like to brush up on their skills, this instructional graphing calculator videotape covers basic calculations, the custom menu, graphing, advanced graphing, matrix operations, trigonometry, parametric equations, polar coordinates, calculus, Statistics I and one variable data, and Statistics II with Linear Regression. This wonderful tool is 105 minutes in length and covers all the important functions of a graphing calculator.
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Emphasizing fundamental mathematical ideas rather than proofs, Introduction to Stochastic Processes, Second Edition provides quick access to important foundations of probability theory applicable to problems in many fields. Assuming that you have a reasonable level of computer literacy, the ability to write simple programs, and the access to software for linear algebra computations, the author approaches the problems and theorems with a focus on stochastic processes evolving with time, rather than a particular emphasis on measure theory.
For those lacking in exposure to linear differential and difference equations, the author begins with a brief introduction to these concepts. He proceeds to discuss Markov chains, optimal stopping, martingales, and Brownian motion. The book concludes with a chapter on stochastic integration. The author supplies many basic, general examples and provides exercises at the end of each chapter.
New to the Second Edition: Expanded chapter on stochastic integration that introduces modern mathematical finance Introduction of Girsanov transformation and the Feynman-Kac formula Expanded discussion of Itô's formula and the Black-Scholes formula for pricing options New topics such as Doob's maximal inequality and a discussion on self similarity in the chapter on Brownian motion Applicable to the fields of mathematics, statistics, and engineering as well as computer science, economics, business, biological science, psychology, and engineering, this concise introduction is an excellent resource both for students and professionals
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Almost all adults suffer a little math anxiety, especially when it comes to everyday problems they think they should be able to figure out in their heads. Want to figure the six percent sales tax on a $34.50 item? A 15 percent tip for a $13.75 check? The carpeting needed for a 12˝-by-17-foot room? No one learns how to do these mental calculations... more...
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...
Sales Handle A no-nonsense practical guide to trigonometry, providing concise summaries, clear model examples, and plenty of practice, making this workbook the ideal complement to class study or self-study, preparation for exams or a brush-up on rusty skills. About the Book Established as a successful practical workbook series with over 30... more...
Despite what we may sometimes imagine, popular mathematics writing didn't begin with Martin Gardner. In fact, it has a rich tradition stretching back hundreds of years. This entertaining and enlightening anthology--the first of its kind--gathers nearly one hundred fascinating selections from the past 500 years of popular math writing, bringing to... more...
The easy way to brush up on the math skills you need in real life Not everyone retains the math they learned in school. Like any skill, your ability to speak "math" can deteriorate if left unused. From adding and subtracting money in a bank account to figuring out the number of shingles to put on a roof, math in all of its forms factors into daily... more...
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97801302275gebra: A Combined Approach
This CD allows students the opportunity to practice exercises that correlate to the exercises at the end of the sections in the textbook. They are algorithmically generated so students can retry an exercise as many times as they would like with new values each time. Every exercise is accompanied by an example and an interactive guided solution that gives students helpful feedback if they enter an incorrect answer. Selected exercises also include a video clip so students can have help visualizing concepts
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Summary: CONTEMPORARY MATHEMATICS FOR BUSINESS AND CONSUMERS, BRIEF is a 14-chapter educational adventure into today's business world and its associated mathematical procedures. The book is designed to provide solid mathematical preparation and foundation for students going on to business courses and careers. It begins with a business-oriented review of the basic operations, including whole numbers, fractions, and decimals. Once students have mastered these operations, they a...show morere introduced to the concept of basic equations and how they are used to solve business problems. From that point, each chapter presents a business math topic that utilizes the student's knowledge of these basic operations and equations. In keeping with the philosophy of "practice makes perfect," the text contains over 2,000 realistic business math exercises--many with multiple steps and answers designed to prepare students to use math to make business decisions and develop critical-thinking and problem-solving skills. Many of the exercises in each chapter are written in a "you are the manager" format, to enhance student involvement. The exercises cover a full range of difficulty levels, from those designed for beginners to those requiring moderate to challenge-level skillsBetter World Books Mishawaka, IN
With CD! Shows definite wear, and perhaps considerable marking on inside. Find out why millions of customers rave about Better World Books. Experience the best customer care and a 100% satisfaction gu...show morearantee. ...show less
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Course Description: This standards-based seventh grade course emphasizes the development and understanding of the real number system and algebraic thinking. Students acquire skills in adding, subtracting, multiplying, and dividing signed numbers including integers. Students solve one step equations involving real numbers. Problem solving in the course includes consumer applications of ratio, proportion, and percent. It continues to develop other important mathematics topics including patterns, functions, Math 8 or Algebra I in the eighth grade:
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School Teachers
Future elementary and middle school teachers need a clear, coherent presentation of the mathematical concepts, procedures, and processes they will be ...Show synopsisFuture elementary and middle school teachers need a clear, coherent presentation of the mathematical concepts, procedures, and processes they will be called upon to teach. This text uniquely balances "what" they will teach (concepts and content) with "how" to teach (processes and communication). As a result, students using "Mathematics for Elementary School Teachers" leave the course knowing more than basic math skills; they develop a deep understanding of concepts that enables them to effectively teach others. This Fourth Edition features an increased focus on the 'big ideas' of mathematics, as well as the individual skills upon which those ideas are built
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Student learning outcomes: Mathematics, Secondary develop both increased depth and breadth of mathematical content knowledge.
Students will be able to communicate mathematics both orally and in writing.
Students will demonstrate the ability to solve challenging problems from the mathematical sciences and write mathematical proofs.
Students will demonstrate the ability to solve problems in a variety of areas of mathematical sciences including statistics, algebra, calculus and geometry.
Students will experience and learn an assortment of pedagogical tools and practices that can be adapted to a variety of school environments and learners.
Students will demonstrate that they can identify and utilize the appropriate strategies and tools (including the use of technology) to solve mathematics problems.
Students will transition their thinking about learning mathematics from a student's perspective to a teacher's perspective.
Students will demonstrate that they can apply various strategies and tools (including the use of technology) to teaching mathematical concepts, thinking, and content appropriate for secondary students.
Students will produce evidence of insights gained from interactions with in-service teachers, results of working collaboratively with peers to solve mathematics problems, and evidence of reflection on experiences in public school mathematics classrooms.
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3.2" backlit color display
TI-Nspire CAS (computer algebra system) student software for home kit
Enables you to continue or complete assigned work outside of the classroom.
Calculator function
Lets you enter and view expressions, equations and formulas, select syntax, symbols and variables from a template that supports standard mathematical notation and scroll through previous entries to explore your outcomes.
Graphing function
Allows you to utilize images and overlay with graphical elements on the screen. A geometry function makes it easy to create and explore geometric shapes for interactive learning.
Lists and spreadsheets function
Enables you to capture and track the values of a graph and collected data to observe numerical patterns, organize the results of statistical analysis, label columns, insert formulas into cells and more.
Data and statistics function
Helps you summarize statistical data using different graphical methods and perform a variety of descriptive and inferential statistics calculations on real data sets.
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The Learning Guide for Introductory and Intermediate Learning Guide helps students learn how to make the most of their textbook and its companion learning tools, including MyMathLab. Organized by the textbook's learning objectives, this workbook provides additional practice for each section and guidance for test preparation. Published in an unbound, binder-ready format, the Learning Guide can serve as the foundation as the student's course notebook.
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complements and extends the Edexcel GCSE Modular Mathematics Examples and Practice book for Foundation Stage 2.This book complements and extends the Edexcel GCSE Modular Mathematics Examples and Practice book for Foundation Stage
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Pre-Algebra
The Pre-Algebra curriculum consists of over 130 lessons offering more than 45 hours of instructional video. This curriculum will provide a solid foundation in the skills necessary to move on to Algebra. Pre-Algebra also offers basic geometry instruction as well as an introduction to more advanced topics like Probability and Statistics.
Sample Pre-Algebra topics Include:
Numbers and Operations
Mathematical Reasoning
Algebraic Expressions and Equations
Integers, Decimals, Fractions and Percents
Ratios and Proportions
Square Roots
Measurement
Basic Geometry
Graphs and Functions
In addition, printed textbooks are available to supplement the electronic courseware. Along with their Summary and Example sections, these textbooks offer Independent Practice problems that match the courseware for students to work on their own.
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... a brief description of MathStudiestopics, ... The Internal Assessment (IA) portion of the IBMathStudiesSL curriculum is the MathStudies project. ... Course: IBMathStudiesSL Due date: February 2013 Do not turn this in!
... Mathematical Studies (SL), Mathematics (SL) and ... Topics The course consists of seven topics taken over 2 years (150 hours): Algebra Functions and Equations ... The exploration (IA) will also contribute to the final IB grade.
... Mathematical Studies (SL), Mathematics (SL) and ... The exploration (IA) will also contribute to the final IB grade. IB Assessments: ... Written work that involves exploration into a math related topic chosen by the student Cheating:
IBMathStudies is a Survey course of a wide range of topics. ... Mathematical StudiesSL Total 150 hours ... Failure to do a mathstudies project will result in a forfeiture of the IB diploma. 130 hours for Topics and 20 hours for Project equal 150 hours for the course.
MathStudiesSL (Discrete) George Magliaro ... (IA); these are activities assessed by the ... principles, practices and skills using actual case studies. Content Topics explored in IB Business and Management provide a broader understanding of
IBMathStudiesSL no required reading assignments ; Students coming from IBSL or IBHL: 15-20 ... essays taken from the topics studied in the current year that students have one hour and a half to ... IA ; 2 Lab Reports per quarter- IA . Group 4 Project .
Also utilize the International Baccalaureate website ... IB Precalculus SL Mathematics Year 1 To the Parent or Guardian: ... Even if you are unable to help your student with their mathstudies, ...
IBMathStudiesSLIB English II HL AP U.S ... Latin SL, Topics, Chemistry SL, Physics HL and SL, MathStudies, Music SL and Art HL were higher than the World Wide average in 2013. 585 students received a ... Virgin ia Department of Education and Southern Association of Colleges and ...
The International Baccalaureate® (IB) is an educational ... IB Diploma Programme students study six subjects at higher or standard level. ... grade, and if you do not complete your IA tasks you shall not be awarded the diploma.
20th Century TopicsSLIB History of Europe HL IB Psychology SLIB Economics SLIB ... Precalculus Advanced prepares students for the IBMathSL, Calculus AB AP, or Statistics AP. 23 ... This IB Standard Level course is for students who expect to go on to study subjects with a
Mathematical StudiesSL is a two year math course intended for ... demonstrates their understanding of the topics being studied along with two IB external assessment papers. ... IB Mathematical StudiesSL is a 2-year course.
The IA portion will consist of laboratory investigations. ... Mathematical StudiesSL is a two year math course intended for ... demonstrates their understanding of the topics being studied along with two IB external assessment papers.
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Customer Reviews for Weimar Institute Math-It Guide Book on CD-ROM
Students from kindergarten to grade eight can benefit from the math teaching method of Elmer W. Brooks as presented in this CD-ROM. All basic math skills from number recognition to statistics and transversals are covered, and is addressed in clear, concise terms. This CD-ROM be used as a basic math skills tool, explaining each term and concept for all covered grades.
Now in CD-ROM format!
Customer Reviews for Math-It Guide Book on CD-ROM
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Prealgebra - 5th edition
Summary: Prealgebra, 5/e, is a consumable worktext that helps students make the transition from the concrete world of arithmetic to the symbolic world of algebra. The Aufmann team achieves this by introducing variables in Chapter 1 and integrating them throughout the text. This text's strength lies in the Aufmann Interactive Method, which enables students to work with math concepts as they're being introduced. Each set of matched-pair examples is organized around an objective...show more and includes a worked example and a You Try It example for students. In addition, the program emphasizes AMATYC standards, with a special focus on real-sourced data. The Fifth Edition incorporates the hallmarks that make Aufmann developmental texts ideal for students and instructors: an interactive approach in an objective-based framework; a clear writing style; and an emphasis on problem solving strategies, offering guided learning for both lecture-based and self-paced courses. The authors introduce two new exercises designed to foster conceptual understanding: Interactive Exercises and Think About It
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vision, the geometric laws that relate different views of a scene. Geometry, one of the oldest branches of ... multipleviews of a scene from the perspective of various types of geometries. A key feature is that it ... role incomputer communications. Producers and users of images, in particular three-dimensional images, ...
numerous computervision algorithms included in the OpenCV library. You will learn how to read, write, ... a variety of computervision algorithms and be exposed to important concepts in image analysis that will ... mathematical morphology and image filtering. The detection and use of interest points incomputervision is ...
filtering. The detection and use of interest points incomputervision is presented with applications for ... Exploit the image geometryin order to match different views of a pictured scene Calibrate the camera from ... programming. It can be used as a companion book in university-level computervision courses. It constitutes an ...
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Description
This text is appropriate for undergraduate courses on numerical methods and numerical analysis found in engineering, mathematics & computer science departments.
Each chapter uses introductory problems from specific applications. These easy-to-understand problems clarify for the reader the need for a particular mathematical technique. Numerical techniques are explained with an emphasis on why they work. CourseSmart textbooks do not include any media or print supplements that come packaged with the bound book.
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Prealgebra, 5th Edition
Prealgebra, 5/e, is a consumable worktext that helps students make the transition from the concrete world of arithmetic to the symbolic world of algebra. The Aufmann team achieves this by introducing variables in Chapter 1 and integrating them throughout the text. This text's strength lies in the Aufmann Interactive Method, which enables students to work with math concepts as they're being introduced. Each set of matched-pair examples is organized around an objective and includes a worked example and a You Try It example for students. In addition, the program emphasizes AMATYC standards, with a special focus on real-sourced data. The Fifth Edition incorporates the hallmarks that make Aufmann developmental texts ideal for students and instructors: an interactive approach in an objective-based framework; a clear writing style; and an emphasis on problem solving strategies, offering guided learning for both lecture-based and self-paced courses. The authors introduce two new exercises designed to foster conceptual understanding: Interactive Exercises and Think About It exercises242.95
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What are the most useful books an undergraduate in math should read? I found Alock's "How to study as a math major" and "how to study for a math degree" very useful. Are there any other good readings before starting a math degree (or any other material which may come in handy, like shaum's mathematical handbook of formulas and tables?)?
[EDIT: can you suggest some readings to prepare for physics courses that will be included in the degree too?]
[EDIT_2: Thank you for your answers, but?]
3 Answers
Shaum's is better for physics than for mathematics (for physics it is invaluable), at least that is what I found. It really depends on what your focus is, it seems like you would be best served by a good applied math book to go along with physics (I would also suggest getting really good at programming and statistics if you are doing mathematical physics). But another student would be better suited to a book more loaded towards number theory (I do not know of any at an undergrad level), or mathematical history.
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and the physical world.
Stimulating account of development of mathematics from arithmetic, algebra, geometry and trigonometry, to calculus, differential equations, and non ...Show synopsisStimulating account of development of mathematics from arithmetic, algebra, geometry and trigonometry, to calculus, differential equations, and non-Euclidean geometries. Also describes how math is used in optics, astronomy, and other phenomenaA good copy, Unmarked, Solid, Cover shows wear, A good copy...A good copy, Unmarked, Solid, Cover shows wear, A good copy overall. We take great pride in accurately describing the condition of our books, ship within 48 hours and offer a 100% money back guarantee.
Description:Good+ Spine and covers are heavily tanned, however text is only...Good+ Spine and covers are heavily tanned, however text is only lightly and evenly tanned. Edges and spine folds are worn. But, don't judge this book just by the cover-because the binding is tight and the text is clean-no names, inscriptions, underlining or highlighting. It's a decent copy.; "A history of mathematics, man's greatest invention for the investigation of the physical world. Morris Kline, professor of mathematics at New York University, shows the simultaneous growth of mathematics and the physical sciences, and the interaction of one domain with the other from the time the Greeks first recognized the mathematical design of nature up to modern science which, according to Kline, is a collection of mathematical theories adorned with physical facts."; 546
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California Mathematics enables students understand large numbers and addition, subtraction, multiplication, and division of whole numbers. Also describe and compare simple fractions and decimal, understand the properties of, and the relationships between, plane geometric figures and to collect, represent, and analyze data to answer questions.
Prentice Hall Science Explorer, the nation's leading middle school science program, is the perfect fit for today's classroom. The book provides students with more options so they can deliver lessons aligned to their standards and preferences.
This high-interest, relevant program teaches students the reading, writing, listening, and speaking skills they need to achieve success at school and in the workplace. Realistic work scenarios demonstrate the importance of communication skills in the book.
This textbook is organized to support students learning of the California Science Content Standards, understanding this organization can help students master the standards provided in the book. Every chapter begins with a Focus on the Big Idea question that is linked to a California Science Standard. Focus on the Big Idea poses a question for students to think about as they study the chapters. They will discover the answers to the question as they readThe updated Magruder's American Government meets the changing needs of today's high school students and teachers. The program's engaging narrative is enhanced with numerous primary sources, political cartoons, charts, graphs, and photos, making the structure and principles of government accessible and motivating to students of all abilitiesStudents of grades 10 and 11 will take the test called the Texas Assessment of Knowledge and Skills (TAKS) in science. The objectives are based on the learning requirements listed in the Texas Essential Knowledge and Skills (TEKS). Students can take steps to improve their performance and increase the chances of doing well on the test, and this biology textbook will help them take these steps. Each chapter has many features that prepare for the TAKS, followed by an overview of the Biology TEKS.
Oklahoma's Priority Academic Student Skills (PASS) objectives list the inquiry skills and science content you are expected to learn in biology. The goal of the PASS standards is for all students to have scientific literacy.
Prentice Hall Literature Missouri edition is divided into four strands: Reading; Writing; Listening and Speaking; and Information Literacy. The MAP Reading Skills Review in the book provides six short tests with questions like the ones students will find on the MAP.
Prentice Hall Mathematics Course 2: A structured approach to a variety of topics such as ratios, percents, equations, inequalities, geometry, graphing and probability.Test Taking Strategies provide a guide to problem solving strategies that are necessary for success on standardized tests. Checkpoint Quizzes assess student understanding after every few lessons. Daily Guided Problem Solving in the text is supported by the Guided Problem Solving worksheet expanding the problem, guiding the student through the problem solving process and providing extra practice
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Info for
Punch Line Algebra Book A 7 11
What is the answer to page 7.8 in punchline algebra book a, Abigbrasstool answer: for my 7.8 math sheet i got:_a_tuba_glue_ (including the spaces).
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Untitled document [ The punchline algebra set consists of two binders, each containing 192 pages. book a includes topics often taught in the first semester of an algebra 1 course, while.
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Phschool.com - prentice hall bridge page, Take a closer look at some of the leading instructional materials for secondary school classrooms..
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Pearson prentice hall - pearsoncmg.com, Please click on the title of a book in the list below to login or register..
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Intermediate Algebra
9780321233868
ISBN:
0321233867
Edition: 7 Pub Date: 2006 Publisher: Addison-Wesley
Summary: The goal of Intermediate Algebra: Concepts and Applications, 7e is to help today's students learn and retain mathematical concepts by preparing them for the transition from "skills-oriented" intermediate algebra courses to more "concept-oriented" college-level mathematics courses, as well as to make the transition from "skill" to "application." This edition continues to bring your students a best-selling text that in...corporates the five-step problem-solving process
Bittinger, Marvin A. is the author of Intermediate Algebra, published 2006 under ISBN 9780321233868 and 0321233867. One hundred forty three Intermediate Algebra textbooks are available for sale on ValoreBooks.com, one hundred forty two used from the cheapest price of $0.75, or buy new starting at $34
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2003 Paperback New Book New and in stock. 4/11John Bird's approach to mathematics, based on numerous worked examples and interactive problems, is ideal for level 2 and 3 vocational courses including the BTEC National specifications.
Theory is kept to a minimum, with the emphasis firmly placed on problem-solving skills, making this a thoroughly practical introduction to the mathematics that students need to master.
Now in its seventh edition, Engineering Mathematics has helped thousands of students to succeed in their exams. The new edition includes a section at the start of each chapter that explains and how it relates to real life engineering projects.
It is supported by a fully updated companion website with resources for both students and lecturers. It has 1000 worked problems, 238 multiple-choice questions, and full solutions to all 1800 further questions contained in the 237 practice exercises. All 525 illustrations used in the text can also be downloaded for use in the classroom.
Audience: Students following vocational engineering courses / first year undergraduates. Suitable for all Level 3 engineering programmes, and core units at Level 3. Matched to New BTEC National specifications: Mathematics for Technicians; Further Mathematics for Technicians; AVCE: Applied Mathematics for Engineering; Further Mathematics for Engineering.
Related Subjects
Meet the Author
John Bird is the former Head of Applied Electronics in the Faculty of Technology at Highbury College, Portsmouth, UK. More recently, he has combined freelance lecturing at the University of Portsmouth, with examiner responsibilities for Advanced Mathematics with City and Guilds, and examining for the International Baccalaureate Organisation. He is the author of over 125 textbooks on engineering and mathematical subjects, with worldwide sales of one million copies. He is currently a Senior Training Provider at the Defence School of Marine Engineering in the Defence College of Technical Training at HMS Sultan, Gosport, Hampshire
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"A good textbook." ― Mathematical Gazette. This introduction to Euclidean geometry emphasizes both the theory and the practical application of isometries and similarities to geometric transformations. Each chapter begins with an optional commentary on the history of geometry. Contents in... read more
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Taxicab Geometry: An Adventure in Non-Euclidean Geometry by Eugene F. Krause Fascinating, accessible introduction to unusual mathematical system in which distance is not measured by straight lines. Illustrated topics include applications to urban geography and comparisons to Euclidean geometry. Selected answers to problemsProblems and Solutions in Euclidean Geometry by M. N. Aref, William Wernick Based on classical principles, this book is intended for a second course in Euclidean geometry and can be used as a refresher. More than 200 problems include hints and solutions. 1968 editionProduct Description:
"A good textbook." ― Mathematical Gazette. This introduction to Euclidean geometry emphasizes both the theory and the practical application of isometries and similarities to geometric transformations. Each chapter begins with an optional commentary on the history of geometry. Contents include modern elementary geometry, isometries and similarities in the plane, vectors and complex numbers in geometry, inversion, and isometries in space. Numerous exercises appear throughout the text, many of which have corresponding answers and hints at the back of the book. Prerequisites for this text, which is suitable for undergraduate courses, include high school algebra, geometry, and elementary trigonometry. 1972
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that they understand, and with the expectation that their backgrounds may have some gaps. Smith/Minton provide exceptional, reality-based applications that appeal to students' interests and demonstrate the elegance of math in the world around us. New features include: • Many new exercises and examples (for a total of 7,000 exercises and 1000 examples throughout the book) provide a careful balance of routine, intermediate and challenging exercises • New exploratory exercises in every section that challenge students to make connections to previous introduced material. • New commentaries ("Beyond Formulas") that encourage students to think mathematically beyond the procedures they learn. • New counterpoints to the historical notes, "Today in Mathematics," stress the contemporary dynamism of mathematical research and applications, connecting past contributions to the present. • An enhanced discussion of differential equations and additional applications of vector calculus. • Exceptional Media Resources: Within MathZone, instructors and students have access to a series of unique Conceptual Videos that help students understand key Calculus concepts proven to be most difficult to comprehend, 248 Interactive Applets that help students master concepts and procedures and functions, 1600 algorithms , and 113 e-Professors
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Longmont Math extend your skills to thinking mathematically and further developing organized problem solving skills. Calculus is the pathway to solutions to problems you can solve in no other way. Without it, we wouldn't have many of the modern conveniences and technologies and other achievements we take for granted in today's world
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Calculus is one of the greatest achievements of the human intellect. Inspired by problems in astronomy, Newton and Leibniz developed the ideas of calculus 300 years ago. Since then, each century has demonstrated the power of calculus to illuminate questions in mathematics, the physical sciences, engineering, and the social and biological sciences. Calculus has been so successful both because its central theme—change—is pivotal to an analysis of the natural world and because of its extraordinary power to reduce complicated problems to simple procedures.
This book offers a fresh approach to algebra that focuses on teaching readers how to truly understand the principles, rather than viewing them merely as tools for other forms of mathematics. It relies on a storyline to form the backbone of the chapters and make the material more engaging. Conceptual exercise sets are included to show how the information is applied in the real world. Using symbolic notation as a framework, business professionals will come away with a vastly improved skill set.
In this long-awaited follow-up to the best-selling first edition of How to Draw Cars Like a Pro, renowned car designer Thom Taylor goes back to the drawing board to update his classic with all-new illustrations and to expand on such topics as the use of computers in design today. Taylor begins with advice on selecting the proper tools and equipment, then moves on to perspective and proportion, sketching and cartooning, various media, and light, shadow, reflection, color, and even interiors. Written to help enthusiasts at all artistic levels, his book also features more than 200 examples from many of today's top artists in the automotive field. Updated to include computerized illustration techniques
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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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Introduction to the Mathematics of Money: Saving and Investing - 07 edition
Summary: This is an undergraduate textbook on the basic aspects of personal savings and investing with a balanced mix of mathematical rigor and economic intuition. It uses routine financial calculations as the motivation and basis for tools of elementary real analysis rather than taking the latter as given. Proofs using induction recurrence relations and proofs by contradiction are covered. Inequalities such as the Arithmetic-Geometric Mean Inequality and the Cauchy-Schwarz Inequality are use...show mored. Basic topics in probability and statistics are presented. The student is introduced to elements of saving and investing that are of life-long practical use. These include savings and checking accounts certificates of deposit student loans credit cards mortgages buying and selling bonds and buying and selling stocks. The book is self contained and accessible. The authors follow a systematic pattern for each chapter including a variety of examples and exercises ensuring that the student deals with realities rather than theoretical idealizations. It is suitable for courses in mathematics investing banking financial engineering and related topics. ...show less105.82 +$3.99 s/h
VeryGood
Herb Tandree Philosophy Books Stroud, Glos,
2006
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Calculus is one of the greatest achievements of the human intellect. Inspired by problems in astronomy, Newton and Leibniz developed the ideas of calculus 300 years ago. Since then, each century has demonstrated the power of calculus to illuminate questions in mathematics, the physical sciences, engineering, and the social and biological sciences. Calculus has been so successful both because its central theme-change-is pivotal to an analysis of the natural world and because of its extraordinary power to reduce complicated problems to simple procedures.
The Calculus 3 Tutor Volume 1 is a 10 hour DVD course that picks up right where the very popular Advanced Calculus 2 Tutor ends and teaches students how to do well in Calculus 3 by fully worked example problems. This DVD course is essential for any student taking Calculus 3 at the university level.
The Calculus 3 Tutor Volume 2 is a 11 hour DVD course that picks up right where the very popular Calculus 3 Tutor: Volume 1 ends and continues to teach students how to do well in Calculus 3 by fully worked example problems. This DVD course is essential for any student taking Calculus 3 at the university levelCalculus can be an intimidating subject if you don't have a good teacher. It is important to have a teacher that takes things step-by-step so the students don't get lost. That is exactly what this DVD set provides. I have tutored many many people in Math through Calculus and I have found that if you start off with the basics and take things one step at a time - anyone can learn complex Math topics. This 2-DVD set contains 8 hours of fully worked example problems in Calculus 1 and 2. After viewing this DVD course in Calculus 1 & 2 you'll discover that the material isn't hard at all if it is presented in a clear manner. No knowledge is assumed on the part of the student. Each example builds in complexity so before you know it you'll be working the 'tough' problems with ease!Understanding Calculus: Problems, Solutions, and Tips immerses you in the unrivaled learning adventure of this mathematical field in 36 half-hour lectures that cover all the major topics of a full-year calculus course in high school at the College Board Advanced Placement AB level or a first-semester course in college. With crystal-clear explanations of the beautiful ideas of calculus, frequent study tips, pitfalls to avoid, and-best of all-hundreds of examples and practice problems that are specifically designed to explain and reinforce major concepts, this course will be your sure and steady guide to conquering calculus.
Calculus II is the payoff for mastering Calculus I. This second course in the calculus sequence introduces you to exciting new techniques and applications of one of the most powerful mathematical tools ever invented.
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Curriculum report on mathematics places a renewed emphasis on the practical uses of mathematical skills and ways of thinking. This work reflects this emphasis in its range of entries. Whilst retaining the information required to understand topics such as directed numbers, mean, root and vector, the book now includes entries on, for example, estimate, half life, hypothsis, linear programming and tree diagram. In addition, the layout of the series has been redesigned for the new edition, and the text completely reset. Illustrations are now expanded upon in linked caption text, increased inter-entry spacing is easier on the eye, and running heads at the top of each page improve quick reference. Richard Browne has revised the text of Gem "Basic Facts: Mathematics" for this edition. He has taught mathematics at secondary level for 11 years, and is presently Professional Officer with responsibility for Mathematics at the School Examinations and Assessment Council.
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Math 360 Assignment, math grammatically
Math 360 is "Foundations of Higher Mathematics" and required for Math majors with a pre-req of calc I. It's mainly proofs and logic, set theory and what not. Being the first day, I assume we start off basic.
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Precalculus
This course reviews the concepts from Algebra II that are central to calculus and explores several discrete math topics. Calculus topics focus on the study of functions: polynomial, trigonometric, logarithmic, and exponential. Discrete topics include polar coordinates, sequences and series, permutations and combinations, the Binomial Theorem, and conic sections. Throughout the course, students are expected to use the graphing calculator to solve problems in each topic area.
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Intermediate Algebra: Connecting Concepts through Applications
9780534496364
ISBN:
0534496369
Edition: 1 Pub Date: 2011 Publisher: Brooks Cole
Summary: INTERMEDIATE ALGEBRA: CONNECTING CONCEPTS THROUGH APPLICATIONS shows students how to apply traditional mathematical skills in real-world contexts. The emphasis on skill building and applications engages students as they master concepts, problem solving, and communication skills. It modifies the rule of four, integrating algebraic techniques, graphing, the use of data in tables, and writing sentences to communicate so...lutions to application problems. The authors have developed several key ideas to make concepts real and vivid for students. First, the authors integrate applications, drawing on real-world data to show students why they need to know and how to apply math. The applications help students develop the skills needed to explain the meaning of answers in the context of the application. Second, they emphasize strong algebra skills. These skills support the applications and enhance student comprehension. Third, the authors use an eyeball best-fit approach to modeling. Doing models by hand helps students focus on the characteristics of each function type. Fourth, the text underscores the importance of graphs and graphing. Students learn graphing by hand, while the graphing calculator is used to display real-life data problems. In short, INTERMEDIATE ALGEBRA: CONNECTING CONCEPTS THROUGH APPLICATIONS takes an application-driven approach to algebra, using appropriate calculator technology as students master algebraic concepts and skills.
Clark, Mark is the author of Intermediate Algebra: Connecting Concepts through Applications, published 2011 under ISBN 9780534496364 and 0534496369. Seven hundred fifteen Intermediate Algebra: Connecting Concepts through Applications textbooks are available for sale on ValoreBooks.com, two hundred twenty three used from the cheapest price of $28.36, or buy new starting at $170 INSTRUCTORS EDITION534496371
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Hitchcock, TX CalculusThomas and received an A in the course. Linear Algebra is the study of matrices and their properties. The applications for linear algebra are far reaching whether you want to continue studying advanced algebra or computer science use Linux several hours a day. I currently have a Linux server as my personal home computer, currently Arch Linux x86_64 SMP preempt 3.11.4. I was first exposed to UNIX system administration in 1991 as the custodian of my workgroup's computing workstation, I have continued to maintain my skills.
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34,"ASIN":"0816051240","isPreorder":0},{"priceBreaksMAP":null,"buyingPrice":85.53,"ASIN":"0691118809","isPreorder":0}],"shippingId":"0816051240::G4D%2F7sWDtpeFxLZYcVelLUlsJ578eXAO7X3o3gVoNyqoZAfXC7A7K5j7OxOf0hIffb21y10dedz6G2h0Vg6HzenXpZLcRPmxwyLoh79b7DI%3D,0691118809::HX914JRgBY25z6jlFIdTXiBIvoyZnh5wlaLh3cq12TxQu3RBORrJMmeHtn7bKq7xRdR%2BWUzPi0riKWO2ZecBHvegEt5kViZNSyL0zKeEBearcher, author, and educator Tanton has compiled this encyclopedia to share his enthusiasm for thinking about and doing mathematics. More than 800 alphabetically arranged entries present a wide variety of mathematical definitions, theorems, historical figures, formulas, examples, charts, and pictures. Many cross-references serve to connect concepts or extend a concept further. A mathematical time line listing major accomplishments is available following the entries, along with a list of current mathematics organizations. The bibliography contains print and Web resources, and the index is helpful in locating terms and concepts.
Each entry varies in length depending on the term, concept, or person being described. Six longer essays describe the history of the branches of mathematics. The writing style is straightforward and readable and sometimes contains parenthetical notes that add background or context. If an entry contains a word or words in capital letters, that term or person is also an entry in the encyclopedia.
This remarkable book is not just a collection of facts about mathematics, but is a fairly detailed treatment (within the limits imposed by space considerations) of various mathematical terms and topics. It does not restrict itself to simple mathematics and devotes full attention to several advanced concepts, but is always clearly written. I really commend the author for including proofs for some of the more important theorems and results (e.g. proof of the fundamemtal theorem of algebra, derivation of the least-squares method, and many more).
And yes, you *will* learn tons and tons of things from this excellent book. It is a must read for anybody interested in mathematics!
Given these four, there is hardly a topic from among the current 495 math fields of study that isn't at least explained in enough detail to save LOTS of time on link expeditions. At minimum, these give head starts on alphabetized keywords that will quickly fill holes in any research project, class, or syllabus.
Looking for divisibility rules for numbers that you didn't think had divisibility rules? Looking for names of symbols you didn't think had names? Dr. Tanton provides the facts and the explanations along with the stories behind the topics.
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and Intermediate Algebra
Developmental mathematics is the gateway to success in academics and in life. George Woodbury strives to provide his students with a complete ...Show synopsisDevelopmental mathematics is the gateway to success in academics and in life. George Woodbury strives to provide his students with a complete learning package that empowers them for success in developmental mathematics and beyond. The Woodbury suite consists of a combined text written from the ground up to minimize overlap between elementary and intermediate algebra, a new workbook that helps students make connections between skills and concepts, and a robust set of MyMathLab resources. Note: this item is for the textbook only; supplements are available separately.Hide synopsis
Description:Good. Hardcover. May include moderately worn cover, writing,...Good. Hardcover. May include moderately worn cover, writing, markings or slight discoloration. SKU: 97803216654851665485.
Description:New in new dust jacket. Brand New as listed. ISBN 9780321771940....New in new dust jacket. Brand New as listed. ISBN 97803217719401760203....New in new dust jacket. Brand New as listed. ISBN 9780321760203. Clean! Out of sight Shipping & Customer Service! We process all orders same day
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Numerical Computing in C#
In this lesson I will show how to numerically solve algebraic and ordinary differential equations, and perform numerical integration with Simpson method. I will start with the solution of algebraic equations. The secant method is one of the simplest methods for solving algebraic equations. It is usually used as a part of a larger algorithm to improve convergence. As in any numerical algorithm, we need to check that the method is converging to a given precision in a certain number of steps. This is a precaution to avoid an infinite loop.
Our second example is a Simpson integration algorithm. The Simpson algorithm is more precise the naive integration algorithm I have used there. The basic idea of the Simpson algorithm is to sample the integrand in a number of points to get a better estimate of its variations in a given interval.
Finally, let me show a simple code for solving first order ordinary differential equations. The code uses a Runge-Kutta method. The simplest method to solve ODE is to do a Taylor expansion, which is called Euler's method. Euler's method approximates the solution with the series of consecutive secants. The error in Euler's method is O(h) on every step of size h. The Runge-Kutta method has an error O(h^4) Runge-Kutta methods with a variable step size are often used in practice since they converge faster than fixed size methods
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The Basics of Computer Arithmetic Made Enjoyable and Accessible-with a Special Program Included for Hands-on Learning "The combination of this book and its associated virtual computer is fantastic! Experience over the last fifty years has shown me that there's only one way to truly understand how computers work; and that is to learn one computer and... more...
Presents the research in design and analysis of algorithms, computational optimization, heuristic search and learning, modeling languages, parallel and distribution computing, simulation, computational logic and visualization. This book emphasizes a variety of novel applications in the interface of CS, AI, and OR/MS. more...
Aims to reinforce the interface between physical sciences, theoretical computer science, and discrete mathematics. This book assembles theoretical physicists and specialists of theoretical informatics and discrete mathematics in order to learn about developments in cryptography, algorithmics, and more. more...
Real-world problems and modern optimization techniques to solve them Here, a team of international experts brings together core ideas for solving complex problems in optimization across a wide variety of real-world settings, including computer science, engineering, transportation, telecommunications, and bioinformatics. Part One—covers methodologies... more...
About the Book: The book `Fundamental Approach to Discrete Mathematics` is a required part of pursuing a computer science degree at most universities. It provides in-depth knowledge to the subject for beginners and stimulates further interest in the topic. The salient features of this book include: Strong coverage of key topics involving recurrence... more...
The field of discrete calculus, also known as 'discrete exterior calculus', focuses on finding a proper set of definitions and differential operators that make it possible to operate the machinery of multivariate calculus on a finite, discrete space. In contrast to traditional goals of finding an accurate discretization of conventional multivariate... more...
A Complete Introduction to probability AND its computer Science Applications USING R Probability with R serves as a comprehensive and introductory book on probability with an emphasis on computing-related applications. Real examples show how probability can be used in practical situations, and the freely available and downloadable statistical programming... more...
Discrete mathematics and theoretical computer science are closely linked research areas with strong impacts on applications and various other scientific disciplines. Both fields deeply cross fertilize each other. One of the persons who particularly contributed to building bridges between these and many other areas is Laszlo Lovasz, a scholar whose
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The Lone Star College–Tomball Math Center has videotapes, supplemental textbooks, manipulatives for the hands-on (kinesthetic) learner, graphing calculators, math topic handouts, and computer math programs available for students to use in the Math Center.
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Hygiene AlgebraThen extend your skills to thinking mathematically and further developing organized problem solving skills. Calculus is the pathway to solutions to problems you can solve in no other way. Without it, we wouldn't have many of the modern conveniences and technologies and other achievements we take for granted in today's world.
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providing an original treatment of fractals that is at once accessible to beginners and sufficiently rigorous for serious mathematicians. The workis designed to give young, non-specialist mathematiciansa solid foundation in the theory of fractals, and, in the process, to equip them with exposure to a variety of geometric, analytical, and algebraic tools with applications across other areas.
Product Details
Table of Contents
Introduction.- Part 1. The Sierpiński Gasket.- Definition and General Properties.- The Laplace Operator on the Sierpiński Gasket.-Harmonic Functions on the Sierpiński Gasket.- Part 2. The Apollonian Gasket.- Introduction.- Circles and Disks on Spheres.- Definition of the Apollonian Gasket.- Arithmetic Properties of Apollonian Gaskets.- Geometric and Group-Theoretic Approach.- Many-Dimensional Apollonian Gaskets.-
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Mathematical Background/Clarifying Examples:
The focus should be on the multiple representations of trigonometric expressions. Students should be able to demonstrate, using a table of values or a technologically-produced graph, that two trigonometric expressions are equivalent.
Website Links:
1. Sum and Difference Formulas: This website provides a list of the Sum and Difference Formulas. Sum and Difference Formulas
d. Use the double-angle and half-angle identities.
Mathematical Background/Clarifying Examples: Half-angle Identities and Double-Angle Identities should be used to find exact values. Double-Angle Identities, but not Half-Angle Identities, should also be used to verify other identities. Half-Angle Identities may be used in proofs as a possible extension.
Mathematical Background/Clarifying Examples:
Students should be able to use previously learned identities in order to solve trigonometric equations. It is essential that students are comfortable with the identities so that they are able to determine reasonable identities to use for given equations.
Mathematical Background/Clarifying Examples:
Students should be able to manipulate trigonometric equations using various algebraic methods such as factoring in order to arrive at solutions. It is important to use graphing and tables of values as means to verify solutions to trigonometric equations.
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Extended Mathematics for Cambridge IGCSE
About this title: This book meets the needs of all students following the Cambridge International Examinations (CIE) syllabus for IGCSE Extended Mathematics. Updated for the most recent syllabus it provides complete content coverage with thousands of practice questions in an attractive and engaging format for both native and non-native speakers of English. The book is easy-to-use with an accessible format of worked examples and practice questions. Each book is accompanied by a free CD which provides a wealth of support for students, such
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Algebra for Students DVD Series
In Algebra for Students, students will learn about the power of algebra as a tool for representing, analyzing and generalizing situations, and will explore several functions, including linear, quadratic and exponential. Real-world applications of algebra are shown in multiple forms through tables, graphs and equations, and common errors and misconceptions are highlighted. Students will also learn how to translate verbal expressions to algebraic expressions while considering the reasonableness of solutions within the context of the situation. Teacher's guides are included and available online
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Problems With a Point is a site developed for mathematics students and teachers in grades 6-12. The site contains practice problems on various topics that designed to help students understand mathematical concepts and...
This website provides a basic overview of Data Mining and some applications for the process. The site lists some typical tasks addressed by data mining, such as identifying cross-sell opportunities and predicting aA short article designed to provide an introduction to functional equations, those in which a function is sought which is to satisfy certain relations among its values at all points. For example, we may look for...
Written by Leo Moser and presented by the Trillia Group, this virtual text introduces visitors to the theory of numbers. After agreeing to the terms and conditions of use, users will be able to download the full...
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Overview
Calculus III is not really a continuation of Calculus I
and II. It takes both of them to a whole new dimension - the third
dimension. We will learn calculus that can be applied to the three
dimensional world in which we live (but which we frequently ignore because
it cannot be completely reproduced on paper or on screens).
Reading
I have intentionally chosen a very readable text.
In addition to planning time to do homework, please take time to carefully
read the sections in the book. Notice use of the words "time" and
"carefully". Read the sections slowly and actively. If you
do not understand some statement reread it, think of some potential meanings
and see if they are consistent, and if all else fails, ask me. If you
do not believe a statement, check it with your own examples. Finally,
if you understand and believe the statements, consider how you would
convince someone else that they are true, in other words, how would you
prove them?
Because the text is exceptionally accessible, we will
structure classtime more as an interactive discussion of the reading than
lecture. For each class day there is an assigned
reading. Read the section before coming to class. After
completing questions from the reading we will discuss problems not a part of
the problem sets during the class discussion.
Learning Outcomes
Upon successful completion of Math 223 - Calculus III, a
student will be able to:
Represent vectors analytically and geometrically, and compute dot and
cross products for presentations of lines and planes,
Use technological tools such as computer algebra systems or graphing
calculators for visualization and calculation of multivariable calculus
concepts.
Grading
Your grade in this course will be based upon your
performance on various aspects. The weight assigned to each is
designated below: Exams:
Assignments: (5% each*, complete 10)
Exam 1
13%
Problem Sets (5) 25%
Exam 2
13%
More (2)
10%
Final Exam 25%
Lab Writeups
(3) 15%
More may include extra problem sets, papers, or lab
writeups.
*Problem set 3 is rather long and therefore worth 7%.
Problem set 5 is rather short and therefore worth 3%.
Problem Sets
There will be five pairs of problem sets distributed
throughout the semester. You must complete one of each
pair. Problem sets are due on the scheduled dates. You are
encouraged to consult with me outside of class on any questions toward
completing the homework. You are also encouraged to work together on
homework assignments, but each must write up their own well-written
solutions. A good rule for this is it is encouraged to speak to each
other about the problem, but you should not read each other's solutions.
A violation of this policy will result in a zero for the entire
assignment and reporting to the Dean of Students for a violation of academic
integrity. I strongly recommend reading the suggestions on working
such problems before beginning the first set. Each question will be
counted in the following manner:
0 – missing question or plagiarised work
1 – question copied
2 – partial question
3 – completed question (with some solution)
4 – completed question correctly and well-written
Each entire problem set will then be graded on a 90-80-70-60% (decile)
scale. Late items will not be accepted. Problem sets will be
returned on the following class day along with solutions to the
problems. Because solutions will be provided, comments will be
somewhat limited on individual papers. Please feel free to discuss any
homework with me outside of class or during review.
**New: Points lost on problem sets may be reearned (or preearned) by
finding errors in the textbook (there are a few - both mathematical
and writing) as follows: The first student who notifies me via email
of an error will receive one problem set point. I
will keep the errors listed here for you to check.
Solutions and Plagiarism
There are plenty of places that one can find all kinds of
solutions to problems in this class. Reading them and not referencing
them in your work is plagiarism, and will be reported as an academic
integrity violation. Reading them and referencing them is not quite
plagiarism, but does undermine the intent of the problems. Therefore,
if you reference solutions you will receive 0 points, but you will *not* be
reported for an academic integrity. Simply - please do not read any
solutions for problems in this class.
Laboratory Activities and Writeups
We will regularly be spending parts of classes on maple
activities. Activity files are in my outbox in a folder called
"MultiMaple". You may access them via a browser here
(after logging in with your Geneseo account). Please come to class
prepared for the activity (i.e. with a maple-installed computer and the file
loaded), but without having completed it before. We will not use class
time to prepare. I strongly recommend reading the suggestions
on writing lab writeups before submitting one. Follow-up
questions are posted here and will be updated so as to include
questions for each lab. Lab writeups may be turned in no more than
three class days after the lab activity.
Reports
After attending a mathematics department colloquium (or
other approved mathematics presentation) you may write a report. In
your report, please explain the main point of the presentation and include a
discussion of how this presentation affected your views on mathematics.
A – Well written, answers the
questions, and is interesting and insightful
B – Well written and answers the
questions
C – Well written or answers the
questions (convinces the reader that you were there)
D – attempted
Papers are due within a classweek of the colloquium presentation. I
will gladly look at papers before they are due to provide comments.
Exams
There will be two exams during the semester and a final
exam during finals week. If you must miss an exam, it is necessary
that you contact me before the exam begins. Exams require that you
show ability to solve unfamiliar problems and to understand and explain
mathematical concepts clearly. The bulk of the exam questions will
involve problem solving and written explanations of mathematical
ideas. The first two exams will be an hour worth of material that I
will two evening hours to complete. The final exam will be half an
exam focused on the final third of the course, and half a cumulative
exam. Exams will be graded on a scale approximately (to be
precisely determined by the content of each individual exam) given by
100 – 80% A
79 – 60% B
59 – 40% C
39 – 20% D
below 20% E
For your interpretive convenience, I will also give you an exam grade
converted into the decile scale. The exams will be challenging and
will require thought and creativity (like the problems). They will not
include filler questions (like the exercises) hence the full usage of the
grading scale.
Feedback
Occasionally you will be given anonymous feedback
forms. Please use them to share any thoughts or concerns for how the
course is running. Remember, the sooner you tell me your concerns, the
more I can do about them. I have also created a web-site
which
accepts anonymous comments. If we have not yet discussed this in
class, please encourage me to create a class code. This site may also
be accessed via our course page on a link
entitled anonymous
feedback. Of course, you are always welcome to approach me
outside of class to discuss these issues as well.
Social Psychology
Wrong answers are important. We as individuals
learn from mistakes, and as a class we learn from mistakes. You may
not enjoy being wrong, but it is valuable to the class as a whole - and to
you personally. We frequently will build correct answers through a
sequence of mistakes. I am more impressed with wrong answers in class
than with correct answers on paper. I may not say this often, but it
is essential and true. Think at all times - do things for
reasons. Your reasons are usually more interesting than your
choices. Be prepared to share your thoughts and ideas. Perhaps
most importantly "No, that's wrong." does not mean that your comment is not
valuable or that you need to censor yourself. Learn from the
experience, and always try again. Don't give up.
Math Learning
Center
This center is located in South Hall 332 and is open
during the day and some evenings. Hours for the center will be announced in
class. The Math Learning Center provides free tutoring on a walk-in basis.
Academic Dishonesty
While working on assignments with one another is
encouraged, all write-ups of solutions must be your own. You are expected to
be able to explain any solution you give me if asked. Exams will be done
individually unless otherwise directed. The Student Academic Dishonesty
Policy and Procedures will be followed should incidents of academic
dishonesty occur.
Disability Accommodations
SUNY Geneseo will make reasonable accommodations for
persons with documented physical, emotional or learning disabilities.
Students should consult with the Director in the Office of Disability
Services (Tabitha Buggie-Hunt, 105D Erwin, [email protected]) and their
individual faculty regarding any needed accommodations as early as possible
in the semester.
Religious Holidays
It is my policy to give students who miss class because
of observance of religious holidays the opportunity to make up missed
work. You are responsible for notifying me by September 10 of plans to
observe a holiday.
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First Course in Abstract Algebra
Considered a classic by many, John Fraleigh's A First Course in Abstract Algebra is an in-depth introductory text for the Abstract Algebra course. ...Show synopsisConsidered a classic by many, John Fraleigh's A First Course in Abstract Algebra is an in-depth introductory text for the Abstract Algebra course. Focused on groups, rings and fields, this text gives students a firm foundation for more specialized work by emphasizing an understanding of the nature of algebraic structures. It is geared towards high level courses at schools with strong math programs. *New exercises have been written, while past exercises have been revised and modifed. *Classical approach to abstract algebra. *Focus on applications of abstract algebra. *Classic text for the high end of the market - known and loved in this discipline. It is geared towards high level courses at schools with strong maths programs *Accessible pedagogy includes historical notes written by Victor Katz (author of another AWL book The History of Mathematics), an authority in this area *By opening with a study of group theory, this text provides students with an easy transition to axiomatic mathematics 8178089971 Book is lightly used with little or no...Very Good. 8178089971
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Replies to: Best calculator??
Well if you're talking about a non graphical one I would strongly recommend a Casio. First I bought a Texas Instruments one (called TI-30X) which I personally did not like at all. My math teacher took one look at it and told me to throw it away and never use it again. He told me to buy a Casio FX 991MS and I did. I love it it has everything a non-graphical calculator would possibly have, it solves quadratics, matricies, vectors, eq'n with multiple variables and more. Seriously I'm so into this calculator I'm going to buy another one just in case my first one got lost and I couldn't find one near my university!
On a side note I haven't gone to university this calculator has only been used for Calculus 1 but I'm pretty sure its one of the best.
Enjoy!
If you get into higher level mathematics, you may find TI-89's ability to do symbolic evalution helpful. Every engineering student I knew had one. I resisted for a very long time, since I had graphing calc software on my palm pilot that did most of what I needed. It wasn't until I took Complex Analysis (a senior level math course) I finally broke down and bought a used TI-89 because I couldn't do symbolic evaluation otherwise.
I've used a TI-83+ for 5 years now and it's been pretty good to me. It is VERY helpful to have a graphing calculator rather than a scientific calculator for calculus. For physics, it won't matter as much. At some point in your engineering courses, you'll probably have to solve a large systems of equations, and the matrix feature of the TI-83+ will become very helpful.
I have a TI-84 Silver Plus, and I love every square millimeter of it. I've had no problems with it. Granted, I've only used it for high school physics and calculus, but I love it.
Of course, I think Texas Instruments rules the world, but I'm probably a little biased...
It reallly depends on the instructors, because some don't allow graphing calculators. I would recommend a ti-89 and some regular non graphing calculator that can do sin,cos, tan for the those classes that have calculator restrictions.
The best calculators for number-crunching are those with an RPN mode, like the HP-33S (scientific) or HP-50G (graphing). The 33S is the most powerful scientific on the market: it can be used in either RPN or algebraic mode, and it has programming capability (which Casio and TI scientifics lack). The 33S is not as powerful as a graphing calculator, but it can be used in many situations where graphing calcs are banned, such as engineering licensing exams.
The best calculators for symbolic math are graphing calculators, particularly the TI-89 or HP-50G. The 89 is noted for its ease of use, and is very popular with students. The 50G is more powerful and flexible, but it has a significantly steeper learning curve, and is more commonly used by professionals.
The 89 and 50G also support calculations with units, which is handy for engineers. The 50G has the broadest units support, but the 89 probably has all the units you would need as an undergraduate.
Real engineers use Hp's. New interns and students use Ti. you can tell whos the new hire, by who is carrying a Ti.
I use the Hp 50G and the 33S.
I wont debate this subject here as I know im right, but HP makes the best calculators, the only reason Ti is popular is due to them buying out the market with bribes. There calculators actually do suck.
If you are looking for the best graphing calculator its either the HP 48GX or the Hp 50G.
For a scientific it is byfar the 33S.
If you plan to take the FE and PE then buy the 33S as the Ti 36 is awful.
HP 50g all the way, it is far more powerfull than a Ti-89. Someone who knows what they are doing can do complex mathmatics with an HP so much faster than with Ti. The only reason the school system in the U.S. uses TI calculators is because when they got new math curriculum in the 70s TI payed off people to only use TI for the text, even though they are no good. TI is only a player in the calculator market because of backdoor bribery.
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Soalan Matematik Tingkatan 4
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Basic Mathematics - 7th edition
Summary: Patient and clear in his explanations and problems, Pat McKeague helps students develop a thorough understanding of the concepts essential to their success in mathematics. Each chapter opens with a real-world application. McKeague builds from the chapter-opening applications, such as the average amount of caffeine in different beverages, and uses the application as a common thread to introduce new concepts, making the material more accessible and engaging for student...show mores. Diagrams, charts, and graphs are emphasized to help students understand the material covered in visual form. McKeague's unique and successful EPAS system of Example, Practice, Answer, and Solution actively involves students with the material and thoroughly prepares them for working the Problem Sets. The Sixth Edition of BASIC MATHEMATICS also features a robust suite of online course management, testing, and tutorial resources for instructors and students. This includes iLrn Testing and Tutorial, vMentor live online tutoring, the Digital Video Companion CD-ROM with MathCue, a Book Companion Web Site featuring online graphing calculator resources, and The Learning Equation (TLE), powered by iLrn. TLE provides a complete courseware package, featuring a diagnostic tool that gives instructors the capability to create individualized study plans. With TLE, a cohesive, focused study plan can be put together to help each student succeed in math
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Also Available in:
Vedic mathematics Made Easy
(Paperback)
Vedic mathematics Made Easy Book Description
About the Book :
A Simplified Approach For Beginners Can you multiply 231072 by 110649 and get the answer in just a single line? Can you find the cube root of 262144 or 704969 in two seconds? Can you predict the birth-date of a person without him telling you? Can you predict how much money a person has without him telling you? Can you check the final answer without solving the question? Or, in a special case, get the final answer without looking at the question? Can you solve squares, square roots, cube-roots and other problems mentally?All this and a lot more is possible with the techniques of Vedic Mathematics described in this book. The techniques are useful for students, professionals and businessmen. The techniques of Vedic Mathematics have helped millions of students all over the world get rid of their fear of numbers and improve their scores in quantitative subjects. Primary and secondary school students have found the Vedic mathematics approach very exciting. Those giving competitive exams like MBA, MCA, CET, UPSC, GRE, GMAT etc. have asserted that Vedic Mathematics has helped them crack the entrance tests of these exams.
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The book Vedic mathematics Made Easy by Dhaval Bathia
(author) is published or distributed by Jaico Publishing House [8179924076, 9788179924075].
This particular edition was published on or around 2005-1-1 date.
Vedic mathematics Made Easy has Paperback binding and this format has 256 number of pages of content for use.
The printed edition number of this book is 1.
This book by Dhaval Bathia
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Precalculus With Limits
9780073365800
ISBN:
0073365807
Pub Date: 2007 Publisher: McGraw-Hill College
Summary: The Barnett, Ziegler, Byleen College Algebra series is designed to be user friendly and to maximize student comprehension, emphasizing computational skills, ideas, and problem solving as opposed to mathematical theory. Suitable for a one or two semester college algebra with trigonometry or precalculus course, Precalculus with Limits introduces a unit circle approach to trigonometry and includes a chapter on limits to... provide students with a solid foundation for calculus concepts.The large number of pedagogical devices employed in this text will guide a student through the course. Integrated throughout the text, students and instructors will find Explore-Discuss boxes which encourage students to think critically about mathematical concepts. In each section, the worked examples are followed by matched problems that reinforce the concept being taught. In addition, the text contains an abundance of exercises and applications that will convince students that math is useful. A MathZone site featuring algorithmic exercises, videos, and other resources accompanies the text.
Barnett, Raymond A. is the author of Precalculus With Limits, published 2007 under ISBN 9780073365800 and 0073365807. Thirty five Precalculus With Limits textbooks are available for sale on ValoreBooks.com, thirty three used from the cheapest price of $19.89, or buy new starting at $83.35
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Student Solutions Manual for Larson/Hostetler/Edwards' Algebra and Trigonometry: A Graphing Approach and Precalculus: A Graphing Approach
Summary
This manual offers step-by-step solutions for odd-numbered text exercises and for all items in the Chapter and Cumulative Tests, giving students a way to check their answers and ensure that they took the correct steps to arrive at an answer. It also provides practice tests with answers.
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Possibly the biggest challenge teachers face in the classroom is getting their pupils to think for themselves. When children learn to think independently, they are able to take control of their own learning. What's more, they become good at dealing with the many problems that life will inevitably throw their way – not only good at solving these problems, but at choosing the kind of thinking strategies that will help solve them.
This welcome boon for students of algebraic topology cuts a much-needed central path between other texts whose treatment of the classification theorem for compact surfaces is either too formalized and complex for those without detailed background knowledge, or too informal to afford students a comprehensive insight into the subject.
It's amazing how many images the world's photographers produce! Professional or not, images surround us in our everyday lives. What makes successful photographers stand out? What drives us to revisit the same images over and over?
Uncertainty is an inherent feature of both properties of physical systems and the inputs to these systems that needs to be quantified for cost effective and reliable designs. The states of these systems satisfy equations with random entries, referred to as stochastic equations, so that they are random functions of time and/or space.
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an alphabetical dictionary and handbook that gives parents of elementary, middle school, and high school students what they need to know to help their children understand the math they're learning. The book can also be used by students themselves and is suitable for anybody who is reviewing math to take standardized tests or other exams. Foreign students, whose English-language mathematics vocabulary needs to be strengthened, will also benefit from this book.
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Digital video from the Futures Channel. Synopsis: "To design buildings that don't fall down, you need to know how your materials will respond to forces such as gravity, wind, and earthquakes."
Running time 3:02 minutes.
Rating:
Related Resources
Usage Tip
Use of Resource:
How I use Structural Engineering - Video from the Futures Channel
This video comes with three lesson plans for grades 6-12 with details: Weight on a String
Subject: Mathematics
Topics: Algebra--Equations and Expressions; Powers, Roots and Scientific Notation
Grades: 9 - 12
Knowledge and Skills:
- Can evaluate expressions by substituting values for variables
- Can simplify expressions using correct order of operations
- Understands the function of grouping symbols in an expression
- Understands the concept "root"
Cantilevers
Subject: Mathematics
Topics: Algebra--Patterns, Functions and Relations; Linear Equations and Functions; Polynomial Equations and Functions
Grades: 6 - 12
Concepts:
- Function
Knowledge and Skills:
- Can explain the reasoning used to solve a problem
- Can plot a point in a two -dimensional coordinate system, given the coordinates, or determine the coordinates of a given point
- Can determine the equation of a linear function that closely matches a set of points (optional)
- Can determine the equation of a cubic function that closely matches a set of points (optional)
SkyHighScrapers
Subject: Mathematics
Topics: Algebra?Quadratic Equations and Functions
Grades: 8 - 12
Concepts:
- Quadratic
Knowledge and Skills:
- Can determine the equation of a quadratic function that closely matches a set of points
- Given the equation of a specific quadratic relation, can rapidly sketch its graph
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new, revised edition of the bestselling Speed Mathematics features new chapters on memorising numbers and general information, calculating statistics and compound interest, square roots, logarithms and easy trig calculations. Written so anyone can understand, this book teaches simple strategies that will enable readers to make lightning-quick calculations. People who excel at mathematics use better strategies than the rest of us; they are not necessarily more intelligent. With Speed Mathematics you'll discover methods to make maths easy and fun. This book is perfect for students, parents, teachers and anyone who enjoys working with figures and even those who are terrified of numbers!
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Facilitative Role of Graphing Calculators in Learning Col Algebr
Description: The purpose of this paper was to explore (1) the facilitative effect of the use of a graphing calculator (gc) on mathematical problem solving in the context of an adult (18]) college algebra class; (2) the connection between mathematicsMore...
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The purpose of this paper was to explore (1) the facilitative effect of the use of a graphing calculator (gc) on mathematical problem solving in the context of an adult (18]) college algebra class; (2) the connection between mathematics self-efficacy (mse) and college students' academic performance by using the gc as a tool in the college algebra class, (3) the effects of 6-week interventions designed to increase mse; and (4) the relationship between mse and individuals' learning styles. The results show that gc practice does benefit problem solving, particularly when the problem requires understanding of graphic representation. Higher accuracy and mse results showed that the use of gc successfully and positively promotes individuals' mse, and, specifically, algebra classes. On individuals' learning styles, visual learners do obtain a significant benefit from the use of gc's and higher mse when compared to balanced learners
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This comprehensive textbook is designed to take undergraduate and beginning graduate students of mathematics, science, and engineering from the rudimentary beginnings to the exciting frontiers of dynamical systems and their applications. It is a masterful exposition of the foundations of ordinary differential and difference equations from the contemporary viewpoint of dynamical systems and bifurcations. In both conception and execution, the authors implemented a fresh approach to mathematical narration. Fundamental ideas are explained in simple settings, the ramifications of theorems are explored for specific equations, and above all, the subject is related in the guise of a mathematical epic. With its insightful and engaging style, as well as its numerous computer-drawn illustrations of notable equations of theoretical and practical importance, this unique book will simply captivate the attention of students and instructors alike. 345 illustrations.80387971414
Book Description:Springer-Verlag New York Inc., New York, NY, 1996. Book Condition: New. Language: english. Focuses on the subject of dynamical systems. The fundamental ideas of dynamics and bifurcations are explained. Print On Demand. Bookseller Inventory # 5595454 InIn
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12th Grade Math Help
This is a representative list of topics covered in our Grade 12 Math program - however all programs will be customized for the individual student.
Instructions To Master -12th Grade Math
Step 1
Monitoring to your progress and ask questions from the start of your 12th Grade Math semester. If you have problems with this homework, write down the specific numbers. Ask for help as soon as you have trouble and do not wait until the day before to a test. Teachers are a set more likely to help you if you have the material ready at hand and try to prepare ahead of time.
Take the time to read the objects in the textbook even if you think that you understand it. Many students rely severely on the teachers lecture, but math textbooks now often contain color-coded examples, suitable charts and tricks for remembering formulas. The writers gear to the text books toward various learning styles, so you might pick something up that you missed from the teacher.
Step 4
Figure out how concepts are mutually similar and different from each other. For example compare the concepts we learnt on Algebra 1, 2 when compared with 12th Grade Curriculum. Then, determine the differences between the two skills. This will help you remember formulas in addition to graphs so that you retain the information in a "big-picture" way as you enter 12th grade.
Step 5
When a test draws near, use this chapter reviews and chapter tests of the book to practice quizzing yourself. Assigned to homework problems alone are almost never enough practice to do well on a test. In addition, tackle problems that are harder than ones your teacher assigned or that you expect to be on the test.
12th Grade Math
12th grade math, cover all topics covered in earlier grades, but in advanced level. It is very important to be well prepared and comfortable with concepts in this grade to get ready for Higher studies. You need to have a good knowledge of topics like: Algebra, Calculus, Geometry, Probability and Statistics.
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Functions and Their Graphs. Using Technology: Graphing a Function. The Algebra of Functions. Portfolio. Functions and Mathematical Models. Using Technology: Finding the Points of Intersection of Two Graphs and Modeling. Limits. Using Technology: Finding the Limit of a Function. One-Sided Limits and Continuity. Using Technology: Finding the Points of Discontinuity of a Function. The Derivative. Using Technology: Graphing a Function and Its Tangent Line. Summary of Principal Formulas and Terms. Review Exercises.
3. DIFFERENTIATION.
Basic Rules of Differentiation. Using Technology: Finding the Rate of Change of a Function. The Product and Quotient Rules. Using Technology: The Product and Quotient Rules. The Chain Rule. Using Technology: Finding the Derivative of a Composite Function. Marginal Functions in Economics. Higher-Order Derivatives. Using Technology: Finding the Second Derivative of a Function at a Given Point. Implicit Differentiation and Related Rates. Differentials. Portfolio. Using Technology: Finding the Differential of a Function. Summary of Principal Formulas and Terms. Review Exercises.
4. APPLICATIONS OF THE DERIVATIVE.
Applications of the First Derivative. Using Technology: Using the First Derivative to Analyze a Function. Applications of the Second Derivative. Using Technology: Finding the Inflection Points of a Function. Curve Sketching. Using Technology: Analyzing the Properties of a Function. Optimization I. Using Technology: Finding the Absolute Extrema of a Function. Optimization II. Summary of Principal Terms. Review Exercises.
Antiderivatives and the Rules of Integration. Integration by Substitution. Area and the Definite Integral. The Fundamental Theorem of Calculus. Using Technology: Evaluating Definite Integrals. Evaluating Definite Integrals. Using Technology: Evaluating Definite Integrals for Piecewise-Defined Functions. Area between Two Curves. Using Technology: Finding the Area between Two Curves. Applications of the Definite Integral to Business and Economics. Using Technology: Consumers' Surplus and Producers' Surplus. Summary of Principal Formulas and Terms. Review Exercises.
Functions of Several Variables. Partial Derivatives. Using Technology: Finding Partial Derivatives at a Given Point. Maxima and Minima of Functions of Several Variables. The Method of Least Squares. Using Technology: Finding an Equation of a Least-Squares Line. Constrained Maxima and Minima and the Method of Lagrange Multipliers. Double Integrals. Summary of Principal Terms. Review Exercises. Answers to Odd-Numbered Exercises.
Index.
Other Editions of Calculus for the Managerial, Life, and Social Sciences - With CD:
| 677.169 | 1 |
Precalculus: Graphs and Models - 5th edition
Summary: The Graphs and Models series by Bittinger, Beecher, Ellenbogen, and Penna is known for helping students ''see the math'' through its focus on visualization and technology. These texts continue to maintain the features that have helped students succeed for years: focus on functions, visual emphasis, side-by-side algebraic and graphical solutions, and real-data applications. With the Fifth Edition, visualization is taken to a new level with technology. The authors also in...show moretegrate smartphone apps, encouraging readers to visualize the math. In addition, ongoing review has been added with new Mid-Chapter Mixed Review exercise sets and new Study Guide summaries to help students new instructors edition. May contain answers and/or notes in margins. Includes CD-ROM! Ships same day or next business day. Free USPS Tracking Number. Excellent Customer Service. Ships from TN
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New
Mccool Books Denham Springs, LA
Please read before purchase>> annotated teacher edition with publisher notation review copy on cover New Book no writing or marks includes all Students content and all answers. text INCLUDES Graphing ...show moreCalculator Manual >>No access code or other supplements. ship immediately - Expedited shipping available ...show less
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Initially developed by NJCATE and a team of math, science, communications and technology faculty, this learning module employs the NJCATE Integrated Curriculum Model to integrate core and technical material. Accessing...
In this animated and interactive object, students read how to use the IMCONJUGATE() function to convert complex numbers to their conjugate in rectangular form. Target Audience: 2-4 Year College Students
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How to Solve Word Problems in Geometry
How to Solve Word Problems in GeometryDawn B. Sova | McGraw-Hill | 3999-33-16 | 363 pages | English | PDFThe easiest way to solve the hardest problems! Geometry's extensive use of figures and visual calculations make its word problems especially difficult to solve. This book picks up where most textbooks leave off, making techniques for solving problems easy to grasp and offering many illustrative examples to make learning easy. Each year more than two million students take high school or remedial geometry courses. Geometry word problems are abstract and especially hard to solve–this guide offers detailed, easy-to-follow solution procedures. Emphasizes the mechanics of problem-solving. Includes worked-out problems and a 61-question self-test with answers.Download No Mirrors below, please! Follow Rules! ****
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The Mathematics of Games and Gambling (2nd edition).
The Mathematics of Games and Gambling (2nd edition) Edward W.
Packel Published by The Mathematical Association of America The Mathematical Association of America (MAA) is a professional society that focuses on undergraduate mathematics education. Members include teachers at the college and high school level; graduate and undergraduate students; and mathematicians and scientists. (2006) 192
pp., hard cover, ISBNISBN abbr. International Standard Book Number
This book begins with the history of many gambling-related games
and activities and then brings out the elementary probability theoryprobability theory
Branch of mathematics that deals with analysis of random events. Probability is the numerical assessment of likelihood on a scale from 0 (impossibility) to 1 (absolute certainty). behind each of these games and activities. It can be divided into two
parts: Chapter 1 to Chapter 3 and Chapter 4 to Chapter 7. The first part
is suitable for readers who are interested in games for leisure and
gambling purposes but do not have a strong mathematics background. The
second part involves more mathematics and probability, such as counting
methods and probability distributions Many probability distributions are so important in theory or applications that they have been given specific names. Discrete distributions With finite support
The Bernoulli distribution, which takes value 1 with probability p
. High school students will find
this book a good preparation for doing an elementary statistics and
probability course at university level. This book is also a good
resource for first year university students, in particular in
mathematics and statistics, to understand the theories behind the games.
The author explains the notions and axioms of probability without
technical language. Fair dice and cards are used to demonstrate
probability calculations and the odds of one event against another one.
The mathematical expectation or the expected pay-off of a game is
extremely important to readers as the players or the gamblers can use
this value to judge whether the game is fair, or is biased in favour of
themselves or their opponents.
Counting methods are essential in probability calculations. The
author distinguishes between permutations and combinationspermutations and combinations: see probability. permutations and combinations
Number of ways a subset of objects can be selected from a given set of objects. In a permutation, order is important; in a combination, it is not. . He also
demonstrates the selection of outcomes with and without replacement
using poker, bridge, and Keno type games. Personally, I think the
binomial distributionbinomial distribution n. The frequency distribution of the probability of a specified number of successes in an arbitrary number of repeated independent Bernoulli trials. Also called Bernoulli distribution. and the normal approximation to binomial
probabilities are probably the most difficult mathematics for most
readers of this book. But I would say the gambler's ruin problem is
the most interesting topic to readers, as it presents the cases when the
player will be ruined in a repeated game with different winning
probabilities.
This second edition provides a number of websites and online
resources for games and is updated with popular games such as online
poker. It is a good reference for the mathematics of games and gambling.
COPYRIGHT 2007 The Australian Association of Mathematics Teachers, Inc.
No portion of this article can be reproduced without the express written permission from the copyright holder.
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