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Accessible to all students with a sound background in high school mathematics, A Concise Introduction to Pure Mathematics, Third Edition presents some of the most fundamental and beautiful ideas in pure mathematics. It covers not only standard material but also many interesting topics not usually encountered at this level, such as the theory of solving cubic equations, the use of Euler's formula to study the five Platonic solids, the use of prime numbers to encode and decode secret information, and the theory of how to compare the sizes of two infinite sets.
New to the Third Edition side by sideThe Metric Handbook deals with all the principal building types from airports, factories and warehouses, offices shops and hospitals, to schools, religious buildings and libraries. For each type the book gives the basic design requirements and all the principal dimensional data, as well as succinct guidance on how to use the information and what regulations the designer may need to be aware of.
Third-Party JavaScript guides web developers through the complete development of a full-featured third-party JavaScript application. You'll learn dozens of techniques for developing widgets that collect data for analytics, provide helpful overlays and dialogs, or implement features like chat or commenting. The concepts and examples throughout this book represent the best practices for this emerging field, based on thousands of real-world dev hours and results from millions of users.
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GlobalShiksha introduces LearnNext which contains the complete syllabus for CBSE class VIII Mathematics and Science for current academic year. This package contains the lessons in audio-visual format, solved examples, practice exercises, experiments, tests and much more. The CD's are developed in such a way that the students can and understand the concepts effectively, clear the doubts with ease and score higher in exams. These multimedia lessons are marked with conceptual clarity, resourceful knowledge and practical effectiveness.
Features
Ø Students can experience a real-time classroom explanation
Ø The CD offers solution to every exercise with an option to revise and review your understanding of the chapter
Ø Audiovisual format is explained in a real-time classroom format as they can literally take part in the step-by-step solution process of all the exercises.
Chapters Covered
Maths
Ø Visualising Solid Shapes
Ø Practical Geometry
Ø Squares and Square Roots
Ø Algebraic Expressions and Identities
Ø Variations
Ø Understanding Quadrilaterals
Ø Exponents and Powers
Ø Comparing Quantities
Ø Introduction to Graphs
Ø Playing with Numbers
Ø Linear Equations in One Variable
Ø Mensuration
Ø Factorisation
Ø Cubes and Cube Roots
Ø Rational Numbers
Ø Data Handling
Science
Ø Materials: Metals and Non-Metals
Ø Sound
Ø Combustion and Flame
Ø Crop Production and Management
Ø Coal and Petroleum
Ø Microorganisms: Friend and Foe
Ø Synthetic Fibres and Plastics
Ø Force and Pressure
Ø Conservation of Plants and Animals
Ø Some Natural Phenomena
Ø Reproduction in Animals
Ø Light
Ø Chemical Effect of Electric Current
Ø Stars and the Solar System
Ø Cell-Structure and Functions
Ø Reaching the Age of Adolescence
Syllabus
LearnNext software is designed as per the guidelines of NCERT and the other State Boards of India. With the help of Subject Matter Experts, Instruction Designers & Online Tutors and path-breaking technology, we have developed an all-inclusive knowledge repository that is updated with the latest syllabus .
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Product Details
Ideal for all learning levels, the educator-endorsed and award-winning lessons will improve comprehension in multiple subjects, including algebra, geometry, physical science, world history, American government, writing, foreign languages and more with 400+ activities and 3,000+ exercises. Includes free access to Tutor.com and Encyclopedia Britannica® CD-ROM, and is suitable for grades 6-8.
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WJEC Linear and Unitised GCSE Mathematics
From first award in 2014 (first teaching on two-year courses September 2012) all centres in England will be required to follow linear GCSE specifications, whichever awarding organisation they use. Linear specifications are those where all examinations are taken at the end of the course.
This means that centres in England using WJEC GCSE Mathematics specifications MUST follow this linear specification.
GCSE Mathematics
The GCSE Mathematics Linear and Unitised specifications for teaching from September 2010 are now available to download under Specifications.
New WJEC Level 2 Certificate in Additional Mathematics
This qualification, for first teaching from September 2010, will provide a course of study for the most able candidates for GCSE Mathematics to be stretched and challenged. It will also provide an appropriate course of study for candidates who acquire GCSE Mathematics early.
Specification and specimen assessment materials are now available for download:
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Fundamentals of Mathematics is a work text that covers the traditional topics studied in a modern prealgebra course, as well as the topics of estimation, elementary analytic geometry, and introductory algebra. It is intended for students who
have had a previous course in prealgebra,
wish to meet the prerequisite of a higher level course such as elementary algebra, and
need to review fundamental mathematical concepts and techniques.
This text will help the student develop the insight and intuition necessary to master arithmetic techniques and manipulative skills. It was written with the following main objectives:
to provide the student with an understandable and usable source of information,
to provide the student with the maximum opportunity to see that arithmetic concepts and techniques are logically based,
to instill in the student the understanding and intuitive skills necessary to know how and when to use particular arithmetic concepts in subsequent material, courses, and nonclassroom situations, and
to give the student the ability to correctly interpret arithmetically obtained results.
We have tried to meet these objectives by presenting material dynamically, much the way an instructor might present the material visually in a classroom. (See the development of the concept of addition and subtraction of fractions in (Reference), for example.) Intuition and understanding are some of the keys to creative thinking; we believe that the material presented in this text will help the student realize that mathematics is a creative subject.
This text can be used in standard lecture or self-paced classes. To help meet our objectives and to make the study of prealgebra a pleasant and rewarding experience, Fundamentals of Mathematics is organized as follows.
Pedagogical Features
The work text format gives the student space to practice mathematical skills with ready reference to sample problems. The chapters are divided into sections, and each section is a complete treatment of a particular topic, which includes the following features:
Objectives
Each chapter begins with a set of objectives identifying the material to be covered. Each section begins with an overview that repeats the objectives for that particular section. Sections are divided into subsections that correspond to the section objectives, which makes for easier reading.
Sample Sets
Fundamentals of Mathematics contains examples that are set off in boxes for easy reference. The examples are referred to as Sample Sets for two reasons:
They serve as a representation to be imitated, which we believe will foster understanding of mathematical concepts and provide experience with mathematical techniques.
Sample Sets also serve as a preliminary representation of problem-solving techniques that may be used to solve more general and more complicated problems.
The examples have been carefully chosen to illustrate and develop concepts and techniques in the most instructive, easily remembered way. Concepts and techniques preceding the examples are introduced at a level below that normally used in similar texts and are thoroughly explained, assuming little previous knowledge.
Practice Sets
A parallel Practice Set follows each Sample Set, which reinforces the concepts just learned. There is adequate space for the student to work each problem directly on the page.
Answers to Practice Sets
The Answers to Practice Sets are given at the end of each section and can be easily located by referring to the page number, which appears after the last Practice Set in each section.
Section Exercises
The exercises at the end of each section are graded in terms of difficulty, although they are not grouped into categories. There is an ample number of problems, and after working through the exercises, the student will be capable of solving a variety of challenging problems.
The problems are paired so that the odd-numbered problems are equivalent in kind and difficulty to the even-numbered problems. Answers to the odd-numbered problems are provided at the back of the book.
Exercises for Review
This section consists of five problems that form a cumulative review of the material covered in the preceding sections of the text and is not limited to material in that chapter. The exercises are keyed by section for easy reference. Since these exercises are intended for review only, no work space is provided.
Summary of Key Concepts
A summary of the important ideas and formulas used throughout the chapter is included at the end of each chapter. More than just a list of terms, the summary is a valuable tool that reinforces concepts in preparation for the Proficiency Exam at the end of the chapter, as well as future exams. The summary keys each item to the section of the text where it is discussed.
Exercise Supplement
In addition to numerous section exercises, each chapter includes approximately 100 supplemental problems, which are referenced by section. Answers to the odd-numbered problems are included in the back of the book.
Proficiency Exam
Each chapter ends with a Proficiency Exam that can serve as a chapter review or evaluation. The Proficiency Exam is keyed to sections, which enables the student to refer back to the text for assistance. Answers to all the problems are included in the Answer Section at the end of the book.
Content
The writing style used in Fundamentals of Mathematics is informal and friendly, offering a straightforward approach to prealgebra mathematics. We have made a deliberate effort not to write another text that minimizes the use of words because we believe that students can best study arithmetic concepts and understand arithmetic techniques by using words and symbols rather than symbols alone. It has been our experience that students at the prealgebra level are not nearly experienced enough with mathematics to understand symbolic explanations alone; they need literal explanations to guide them through the symbols.
We have taken great care to present concepts and techniques so they are understandable and easily remembered. After concepts have been developed, students are warned about common pitfalls. We have tried to make the text an information source accessible to prealgebra students.
Addition and Subtraction of Whole Numbers
This chapter includes the study of whole numbers, including a discussion of the Hindu-Arabic numeration and the base ten number systems. Rounding whole numbers is also presented, as are the commutative and associative properties of addition.
Multiplication and Division of Whole Numbers
The operations of multiplication and division of whole numbers are explained in this chapter. Multiplication is described as repeated addition. Viewing multiplication in this way may provide students with a visualization of the meaning of algebraic terms such as 8x8x when they start learning algebra. The chapter also includes the commutative and associative properties of multiplication.
Exponents, Roots, and Factorizations of Whole Numbers
The concept and meaning of the word root is introduced in this chapter. A method of reading root notation and a method of determining some common roots, both mentally and by calculator, is then presented. We also present grouping symbols and the order of operations, prime factorization of whole numbers, and the greatest common factor and least common multiple of a collection of whole numbers.
Introduction to Fractions and Multiplication and Division of Fractions
We recognize that fractions constitute one of the foundations of problem solving. We have, therefore, given a detailed treatment of the operations of multiplication and division of fractions and the logic behind these operations. We believe that the logical treatment and many practice exercises will help students retain the information presented in this chapter and enable them to use it as a foundation for the study of rational expressions in an algebra course.
Addition and Subtraction of Fractions, Comparing Fractions, and Complex Fractions
A detailed treatment of the operations of addition and subtraction of fractions and the logic behind these operations is given in this chapter. Again, we believe that the logical treatment and many practice exercises will help students retain the information, thus enabling them to use it in the study of rational expressions in an algebra course. We have tried to make explanations dynamic. A method for comparing fractions is introduced, which gives the student another way of understanding the relationship between the words denominator and denomination. This method serves to show the student that it is sometimes possible to compare two different types of quantities. We also study a method of simplifying complex fractions and of combining operations with fractions.
Decimals
The student is introduced to decimals in terms of the base ten number system, fractions, and digits occurring to the right of the units position. A method of converting a fraction to a decimal is discussed. The logic behind the standard methods of operating on decimals is presented and many examples of how to apply the methods are given. The word of as related to the operation of multiplication is discussed. Nonterminating divisions are examined, as are combinations of operations with decimals and fractions.
Ratios and Rates
We begin by defining and distinguishing the terms ratio and rate. The meaning of proportion and some applications of proportion problems are described. Proportion problems are solved using the "Five-Step Method." We hope that by using this method the student will discover the value of introducing a variable as a first step in problem solving and the power of organization. The chapter concludes with discussions of percent, fractions of one percent, and some applications of percent.
Techniques of Estimation
One of the most powerful problem-solving tools is a knowledge of estimation techniques. We feel that estimation is so important that we devote an entire chapter to its study. We examine three estimation techniques: estimation by rounding, estimation by clustering, and estimation by rounding fractions. We also include a section on the distributive property, an important algebraic property.
Measurement and Geometry
This chapter presents some of the techniques of measurement in both the United States system and the metric system. Conversion from one unit to another (in a system) is examined in terms of unit fractions. A discussion of the simplification of denominate numbers is also included. This discussion helps the student understand more clearly the association between pure numbers and dimensions. The chapter concludes with a study of perimeter and circumference of geometric figures and area and volume of geometric figures and objects.
Signed Numbers
A look at algebraic concepts and techniques is begun in this chapter. Basic to the study of algebra is a working knowledge of signed numbers. Definitions of variables, constants, and real numbers are introduced. We then distinguish between positive and negative numbers, learn how to read signed numbers, and examine the origin and use of the double-negative property of real numbers. The concept of absolute value is presented both geometrically (using the number line) and algebraically. The algebraic definition is followed by an interpretation of its meaning and several detailed examples of its use. Addition, subtraction, multiplication, and division of signed numbers are presented first using the number line, then with absolute value.
Algebraic Expressions and Equations
The student is introduced to some elementary algebraic concepts and techniques in this final chapter. Algebraic expressions and the process of combining like terms are discussed in (Reference) and (Reference). The method of combining like terms in an algebraic expression is explained by using the interpretation of multiplication as a description of repeated addition (as in (Reference)).
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Practical
Linear Algebra -- A Geometry Toolboxforms the basis of a
first year course in Linear Algebra for non-math majors such as
engineers and computer scientists. In addition, this book provides
a solid foundation for work in computer graphics and computer aided
design. The authors emphasize a geometric and intuitive approach
that relies heavily on examples and illustrations rather than the
rigorous theorem-proof format used in standard texts.
Special
features include:
250
figures which are also available in electronic form,
150
numerical examples,
200
problems---many solutions are in the text and additional problems
are on a password-protected instructor's website,
supplementary
materials for instructors,
a
"WYSK" (What You Should Know) section closes each chapter,
providing a concise chapter summary which highlights the most
important points, giving students focus for their approach to
learning.
The figures
are not included as window dressing, in fact they play an
important role in bringing the reader to a robust understanding
of the mathematics. However they are not only instructional, they
are also fun!
For example,
on the right is a crazy Pacman path -- created with linear transformations
and a bit of coloration thanks to Postscript
Below, left
is an instructional tool used to demonstrate a rotation by 45 degree,
and below right is simply a fun example of what we can do with 3D
transformations.
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Arithmetic for carpenters and builders
This book was written for the purpose of presenting the subject of arithmetic, as used in the daily work of the carpenter and builder, in a simple form. It is intended for the practical man as well as for the beginner and the student.
The material is such that the work can be followed successfully by those who have had an eighth-grade education. Upon the completion of this study the student should be prepared to take up the problem of estimating the cost of buildings.
The author has not attempted to treat the subject exhaustively. Though the first chapters may seem elementary to some, they will furnish a much-needed review to others. Geometry is touched upon merely to serve as a foundation for the work in mensuration. Practical applications of geometric truths are emphasized, while rigid proofs and developments are omitted. Two chapters on the steel square, that most useful tool of the carpenter, are included.
Many of the problems do not admit of exact answers for the reason that the judgment of the student plays an important part in the solution. Different results will be obtained, depending upon the lengths and widths of the boards chosen and other similar details. Careful planning for the economical use of material is required, for this is an essential training for the work of the practical carpenter.
Students will do well to take advantage of this and every similar opportunity to develop their judgment. It is believed that the reader who follows this text carefully as a guide may learn much that may not strictly be classed as arithmetic. If the manner in which the subject is treated and the many applications made inspire the practical worker and the beginner to seek a better understanding of the fundamentals of carpentry, the author's most sanguine hopes will be justified.
As a text in vocational mathematics it is believed that this little volume should find a place in vocational and industrial schools, trade schools, manual training schools and night schools. It is also well adapted for use in correspondence instruction. The material has been used by the author in the Vocational Courses in Engineering at the Iowa State College and also in the extension classes of the Department of Engineering Extension in various cities in the state of Iowa.
No single book is the entire and original product of one man's mind; it consists rather of accumulated knowledge interpreted and adapted for a particular purpose. The author gratefully acknowledges assistance rendered knowingly and unknowingly by many others. Material borrowed from other sources is acknowledged in the text.
R. BURDETTE DALE.
TABLE OF CONTENTS
CHAPTER I
UNITS OF LENGTH, ADDITION, SUBTRACTION, MULTIPLICATION AND DIVISION OF UNITS OF LENGTH
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PEMDAS Widget is a simple but powerful calculator for scientists, engineers, students, and other professions to use when working on problems with equations.
PEMDAS stands for Parenthesis, Exponents, Multiplication, Division, Addition, and Subtraction, an acronym sometimes taught to children to help them remember the order of operations.
If anything in this document isn't clear, or you have questions/feedback in general, please don't hesitate to send an e-mail through the contact page!
Requirements
To use PEMDAS Widget, you must have:
A Mac computer with an Intel or PowerPC processor.
Mac OS X v10.5.8 Leopard or later *
1 MB of available disk space
* Although PEMDAS may work on Mac OS X 10.4 Tiger, PEMDAS is no longer tested on that Mac OS X version, so unfortunately is not recommended to use it.
The User Interface
The primary features of PEMDAS widget are the equation entry field and result field. PEMDAS also features a keypad as the default view, as shown below.
The keypad can be toggled to show the variable and equation history by clicking the Equation History toggle button. This view features an equation history and a variable list:
In both modes, the bottom bar features quick access to the Number Formatting Type selection button, and the Degrees/Radians selection button.
The bottom bar also has an info button, which is used to get to the preferences on the back of the widget. And when in the Equation History View, it has a resize handle, which can be used to resize the widget into a Mini Mode.
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Lesson Plan
Factoring Polynomials
Grade Levels
Commencement
,
9th Grade
Description
In this lesson, students will review multiplying two binomials together (FOIL) in the "Do-Now". Students will then learn how to factor a quadratic equation in the form of x2 + bx + c, when a is equal to 1.
Support Materials
SMART Board
This instructional content was intended for use with a SMART Board. The .xbk file below can only be opened with SMART Notebook software. To download this free software from the SMART Technologies website, please click here.
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Hi everyone! I need some urgent help! I have had many problems with algebra lately. I mostly have difficulties with step-by-step answers to my algebra problems. I can't solve it at all, no matter how much I try. I would be very relieved if someone would give me some help on this matter.
I really don't know why God made math, but you will be happy to know that a group of people also came up with Algebrator! Yes, Algebrator is a program that can help you crack math problems which you never thought you would be able to. Not only does it provide a solution the problem, but it also explains the steps involved in getting to that solution. All the Best!
Yes I agree, Algebrator is a really useful product. I bought it a few months back and I can say that it is the main reason I am passing my math class. I have recommended it to my friends and they too find it very useful. I strongly recommend it to help you with your math homework.
A extraordinary piece of math software is Algebrator. Even I faced similar difficulties while solving adding exponents, simplifying fractions and unlike denominators. Just by typing in the problem from homeworkand clicking on Solve – and step by step solution to my algebra homework would be ready. I have used it through several algebra classes - Algebra 2, Algebra 2 and Basic Math. I highly recommend the program.
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Saxon Algebra 2 Homeschool Curriculum
The
Saxon Algebra 2, Third Edition not only treats topics that are traditionally covered in second-year algebra, but also covers a considerable amount of geometry. Time
is spent developing geometric concepts and writing proof outlines. Students completing Algebra 2 will have studied the equivalent of one semester of
informal geometry. Applications to subjects such as physics and chemistry, as well as real-world problems, are also covered.
NOTE: Saxon Algebra 2 Third Edition is the last edition of Saxon Algebra 2
to include Geometry.
Algebra 2 Solutions Manual
Algebra 2 3rd Edition Solutions Manual*
Publisher: Saxon Homeschool
This is the solutions manual for Algebra 2 3rd Edition. It has the solutions [the work to solve the problems is shown] for the problems in the book. Also
included are answers for the practice set problems.
ISBN-13: 9781565771437
List $46.55
Sale Price $34.91
Algebra 2 Answer Key and Tests
Algebra 2 3rd Edition Answer Key and Tests
Publisher: Saxon Homeschool
This has the tests, answers to the tests, and answers to the practice and problem sets.
ISBN-13: 9781600321177
List $22.00
Sale Price $16.50
Saxon Teacher Algebra 2 Lesson and Test CDs
The Saxon Teacher Algebra 2 CDs contain over 110 hours of Algebra 2 content. Instruction for every part of every lesson, as well as complete solutions for every example problem, practice problem, problem set, and test problem in Algebra 2 is included.
There are 5 CDs in the set which cover 129 lessons and the test solutions.
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When to take linear algebra?
When to take linear algebra?
Quote by mathwonk
Knowing concepts of LA is not a requirement for taking many courses in later material, but it is a requirement for understanding them.
Amen!
(But understanding the concepts is not quite the same as being able to "talk the talk". It's years since I needed to write any math using the term "kernel", but I use the idea of what a kernel IS constantly.)
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Edwards Calculus program offers a solution to address the needs of any calculu
Calculus is a required, 3-semester course for all hard science majors such as mathematics, engineering, physics, statistics, computer science, and chemistry. One or more semesters of calculus are required for a number of other majors. The course can take many forms, but the following are the most common: Single Variable Calculus: This is usually a two-semester course that does not cover multivariable material. Calculus of a Single Variable 8e covers all the material usually taught in this 2-semester course. Multivariable Calculus - Calculus III. This may be taught as a separate course in which
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The practice book supplements the students class text book and provides thousands of graded practice questions. Not all are required to be completed. However, some students have trouble with certain concepts. The practice questions are designed to re-inforce existing knowledge and then by gradually increasing the level of difficulty, take students easily to the next level.
An activity kit is provided to each student. The kit consists of a number of shapes and activities that are done in the session, particularly to build a deep subject understanding that goes beyond being able to apply formulae.
Each student also has access to a number of online resources, provided through a dashboard. The online materials are not essential to the learning process, but can help enhance the learning experience.
Additional Assessments. A student can take additional assessments of chapters at their convenience. These assessments will also show up in their reports
Content used in class. All the materials used by the teacher in the class are available as a reference
Practice Exercises. Practice exercises are also available online. These are the same as those available in the Practice Book but can be accessed in multiple ways
Reports. There are a large variety of reports available online providing a detailed assessment of progress.
Additional Referral Materials. For each chapter, we provide links to a set of materials that we believe are useful in learning. These are typically video links or interesting stories or puzzles that engage children in the learning process
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Abstract reliability models can be developed and used to study the probabilitythat a particular system will not fail under various operating conditions forany specific periods of time. such reliability models are developed and used torepresent an idealized example of reality in order to explain the essentialrelationships involved. in this tutorial, mathematical models are stressed, asopposed to physical models, both of which are often used by the engineer. themodels developed here employ the language of mathematics, and like other models,may be descriptive and given an explanation of the system they represent.(author)
Out Of Print
Prices are plus sales tax when purchased within New York State and shipping charges, if applicable
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More About
This Textbook
Overview
This self-contained introduction to algebraic topology is suitable for a number of topology courses. It consists of about one quarter 'general topology' (without its usual pathologies) and three quarters 'algebraic topology' (centred around the fundamental group, a readily grasped topic which gives a good idea of what algebraic topology is). The book has emerged from courses given at the University of Newcastle-upon-Tyne to senior undergraduates and beginning postgraduates. It has been written at a level which will enable the reader to use it for self-study as well as a course book. The approach is leisurely and a geometric flavour is evident throughout. The many illustrations and over 350 exercises will prove invaluable as a teaching aid. This account will be welcomed by advanced students of pure mathematics at colleges and universities
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Title
Authors
Comments
This lesson focuses on a topic that deals with the relationship between two quantities, known as independent and dependent variables.
Document Type
Lesson Plan
Abstract
This lesson occurs towards the beginning of a unit on functions, a mathematical topic that deals with the relationship between two quantities, known as independent and dependent variables.The goal for this lesson will be to draw upon students' intuitive awareness of cause and effect, and to apply this knowledge to authentic, real-world situations. This will give students the theoretical and conceptual foundations they will need to grasp more complex and formulaic ideas in the unit. It will also promote meaningful learning as it will show students how mathematics is relevant to their everyday lives. Through this lessons tudents will be able to identify independent and dependent variables in the real world and express the relationship between specific independent and dependent variables
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Teach Me Calculus
I think that I'd like to re-learn calculus just for fun, since it wasn't a lot of fun the first time around and it all fell out of my head as soon as the coursework was done. Our Calc professors were fairly shitty at teaching the concepts, instead rattling through proofs while talking to the board in a Ben Stein monotone.
Anyway, do any of you have recommendations for good books on Calc or would you rather just teach it to me in this thread?
You are insane. If you want to have some fun I'd suggest drinking until you forget all about your little Calculus idea.
I'd make some joke about how much alcohol that would take, with regards to limits and his weight and some other cooked up information. But I don't remember anything from calculus(I didn't even too poorly in the class)
I took (and, yes, passed) Calculus I, II, and III at university no more than three years ago, but today I couldn't integrate an expression to save my life. My professors were all monotone sleep-inducers as well.
Strangely enough, this is actually something I'd like to re-learn too.
I dropped calc in high school because I was in danger of failing and the study hall that was during that time period was about to show the SW trilogy (only study hall I ever took). In college, I got an A+ in calculus, and all my profs wanted me to go into math (I split the difference with my business profs' pleading and majored in Economics).
And here I am, about 8 years after my last calc class, and I can't remember a damn thing about how to do calculus.
I've seen a few books called something like Calculus the easy way, but if you just want a normal textbook to teach yourself from, then Calculus, Early Transcendentals, by James Stewart, is decent.
On another note, if you're on a budget then you might be able to get a calc book for free at a university. It's not unusual for professors who are cleaning out their offices to put out stacks of free books.
I slept through my calculus classes but managed to get solid Bs in all the class.... All I did was flip open the textbook, looked at the example and the clarity came to me and I integrated equation.
To this day, I still can tell you that the area bound under the curve of function 1/x to the x and y axis will be 2. Except that I've forgotten how to properly integrate a 1/x as power rule don't apply here.
The best Calc teacher I ever had really was a teacher who had decided to go back to school for a better degree. She was our TA. We'd get to her class and she'd say "Okay, what did he go over?" .... "Okay, here's how you DO it....". That was one classy lady. She even spoke English!
Call me crazy but I love calculus. I didnt start appreciating calculus until I took PCHEM quantum mechanics. I was able to use calculus for real world problems, making me better appreciate the math. The way they teach calculus in math classes is quite boring.
I took through multivariable/diff eq, and I remember most of it (although to be honest since my later years of the calculus track were in physics focused classes, I can't tell you whether a differential equation is analytically solvable and without a table of generally solvable equations I can only guess blindly at a solution.)
Though there is no real point in being able to do most of these later skill sets in the absence of a book because you generally would need one anyway...
The best calculus professor I ever had was Michael Barnsley, who was very involved in fractal stuff in his research time. Mind you, he didn't make me understand the stuff much better than anyone else, but he was British and entertaining to hear speak.
I've had two assholes who would insult the class, mumbles, and goes off on a tangent all times. Hence me sleeping in the class.
My first calculus teacher was actually decent; would give time to help eveyone help the materials, but you had to be a geek; he just can't speak it in normal English.
Last calculus teacher I had, was (!) young, and very intelligent. However, he spoke nothing but mathematics and tough shit if you can't keep up.
Is there a law somewhere that calculus teacher must suck at teaching?
actually when i took calc at Bowling Green, my teacher was amazing. Best math teacher i ever had. he was from the Philipians but you could understand him. he was a graduate student that was going for his masters but he made everything so easy to understand. I was a complete slacker and i got an A.
I had a great calculus teacher in high school, and I was in a small class, so the learning experience was incredibly personal and rewarding. I got a 3 on the AP Calc exam, which I didn't think was too bad.
College was a completely different experience. Huge lecture halls, shitty professors; it was a nightmare. I actually did worse in calculus in college, and all the calc I learned in high school was "overwritten" by the one-size-fits-all way of learning large college classes are known for.
You are insane. If you want to have some fun I'd suggest drinking until you forget all about your little Calculus ideaWrong. Calculus is never useful, because there are always dorks like you out there doing it for me.
Wrong. Calculus is never useful, because there are always dorks like you out there doing it for me.
Seconded. I was going to post a reply saying that no, calculus is not a useful skill to have, but Wrao beat me to it. Everyone's either a calc person, or they're not. There's no middle ground. No one occasionally has to use calc to solve any kind of real-world problem. You either do calculus for your job or you forget about it. Hell, even algebra is pretty much like that.
I had the same teacher in HS for both geometry and calc. When I had him for geometry for freshman year, I don't think I quite understood exactly what type of effect he was trying to get with his persona. When I had him again for calc, his humor was actually funny, and he turned out to be one of the best teachers I had in HS. I'd been cleared to start straight into Calc 3 in college last year, but due to scheduling I was never able to take it. That'll fall to first semester this year. I just hope I remember enough to get me through. I'm sure I'll be fine, but I have that whole rusty feeling going.
Edit: Yontsey, who did you have, and do you know if he is still teaching? Or do you have any other profs you could recommend?
One concept that describes a basic use for calculus is that you can deal with things that do not vary linearly.
For example, you can calculate the area of a triangle using well known formulas because we assume the sides are straight lines. Using calculus, you divide the area into small strips, calculate the area of each strip, and add them all together.
I've actually done the Taylor Series during HS. If I remember correctly, that was the test that brought me back to the A range. Remembering what I had to do later for extra credit in order to keep the A will give me nightmares for years, however. Try putting your TI through binary calculus.
I don't know what your major was but it probably was not Engineering or Physics. In those majors you HAVE to know calc (an diffy-q) just to get by. So you are not likely to forget after having to learn it time and time again in most later classes. In CS, you probably don't need as much Calc though most accredited programs require you to take Calc to some degree... (Linear alegebra is very useful for some fields of CS though).
I would recommend NOT trying to re-learn calculus again (it can be boring) but instead go get a good physics book with lots of real-world calculus examples. These are what is lacking in many Calc classes and the reason IMHO that Math classes can be boring.... Try the Haliday and Resnick (sp?) series of Calc-based physics if you can find the old books on ebay those would be cheap and perfect.. Keep an old Calc book for reference..
Calculus is really useful is you're relatively handy and trying to figure out some stuff quick. I guess off of the top of my head I cannot think of an example, but I found myself applying calc 1 (though maybe unnecessarily, but it was cool at the time!!) to a ton of real-world situations while enrolled in it. Especially physics, circuit modeling, and optimizing measurements
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Essential Mathematics 9 (ICSE Board264 Our Price:224 You Save: 40
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Essential Mathematics 9 (ICSE Board) Book Description
About the Book :
Essential Mathematics is a series of two books for Classes 9 and 10. This series is based on the latest syllabus prescribed by the Council for the Indian School Certificate Examinations, New Delhi. Salient features of the books:Each chapter has a large number of solved problems to illustrate the concepts and methods. Stress has been laid on concept building. The text is lucid and to the point. In the exercises, problems are graded from simple to complex A list of important definitions, formulae and results are provided at the end of each chapter in the form of Points to Remember. Test your knowledge at the end of each chapter tests the childs learning.
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The book Essential Mathematics 9 (ICSE Board) by Paj Lewis
(author) is published or distributed by Ratna Sagar P. Ltd. [8183323677, 9788183323673].
This particular edition was published on or around 2007-1-1 date.
Essential Mathematics 9 (ICSE Board
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FIRST COURSE IN PROBABILITY leader is written as an elementary introduction to the mathematical theory of probability for readers in mathematics, engineering, and the sciences who possess the prerequisite knowledge of elementary calculus.
A major thrust of the Fifth Edition has been to make the book more accessible to today's readers. The exercise sets have been revised to include more simple, "mechanical" problems and new section of Self-test Problems, with fully worked out solutions, conclude each chapter. In addition many new applications have been added to demonstrate the importance of probability in real situations. A software diskette, packaged with each copy of the book, provides an easy to use tool to derive probabilities for binomial, Poisson, and normal random variables. It also illustrates and explores the central limit theorem, works with the strong law of large numbers, and more.
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The book covers the basic aspects of linear single loop feedback control theory. Explanations of the mathematical concepts used in classical control such as root loci, frequency response and stability methods are explained by making use of MATLAB plots but omitting the detailed mathematics found in many textbooks. There is a chapter on PID control and two chapters provide brief coverage of state...
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Mu Alpha Theta Fosters Interest in Mathematics
November 16, 1998
SIAM vice president for education Terry Herdman, who gave a talk on problems and curriculums in applied mathematics at the recent Mu Alpha Theta annual convention in Chicago, believes that SIAM members should be aware of the goals and accomplishments of this educational organization.
Mu Alpha Theta, a national honor society for mathematics students attending high schools and junior colleges, was founded at the University of Oklahoma in Norman, in 1957, by Richard V. Andree (who chaired the committee of the National Council of Teachers of Mathematics that originally conceived the idea) and his wife. The goals of the society are (1) to engender keen interest in mathematics, (2) to develop sound scholarship in the subject, and (3) to promote enjoyment of mathematics among high school and junior college students.
Mu Alpha Theta currently has approximately 50,000 student members in 1350 teacher-sponsored local chapters or groups; holds national, regional, and local meetings at which both students and mathematicians present mathematical material; and supports a variety of projects and contests. It also publishes The Mathematical Log, a quarterly journal that features understandable, audience-appropriate articles on mathematics as well as news about meetings, chapters, and members.
Herdman believes that SIAM members could be important role models for students as sponsors and supporters of local high school chapters. In addition, members of SIAM student chapters could benefit from interaction with local chapters of Mu Alpha Theta: SIAM student members might profit from learning how to mentor and guide younger students, or they might hone their presentation skills of mathematical topics in front of a younger student audience. Members interested in learning more about Mu Alpha Theta can contact: Mu Alpha Theta, 601 Elm Avenue, Room 423, Norman, OK 73019-0315; (405) 325-4489; matheta@ou.edu;
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Tuesday, 27 March 2012
Product Details: Paperback: 312 pages Publisher: Springer; Softcover reprint of hardcover 1st ed. 2007 edition (October 18, 2010) Language: English ISBN-10: 1441922326 ISBN-13: 978-1441922328 Product Dimensions: 9 x 6 x 0.7 inches Shipping Weight: 1 pounds (View shipping rates and policies) This is an undergraduate textbook on the basic aspects of personal savings and investing with a balanced mix of mathematical rigor and economic intuition lynda hardware tutorial with torrent download. It uses routine financial calculations as the motivation and basis for tools of elementary real analysis rather than taking the latter as given lynda hardware tutorial with torrent download. Proofs using induction, recurrence relations and proofs by contradiction are covered lynda hardware tutorial with torrent download. Inequalities such as the Arithmetic-Geometric Mean Inequality and the Cauchy-Schwarz Inequality are used lynda hardware tutorial with torrent download. 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. Tags: An Introduction to the Mathematics of Money Saving and Investing (9781441922328) David Lovelock, Marilou Mendel, Arthur L. Wright , tutorials, pdf, ebook, torrent, downloads, rapidshare, filesonic, hotfile, megaupload, fileserve
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Courses Offered
A self-paced course designed to review basic concepts of arithmetic and elementary algebra. Topics include fractions, decimals, percents, operations with real numbers, linear equations and inequalities, graphs and functions, solving linear systems, exponents, polynomials and problem solving. Grading is on a satisfactory/unsatisfactory basis. One class and one laboratory hour.
Prerequisite: MATH 106. Open to early childhood education and elementary/special education majors with sophomore standing, and to others with permission of the instructor. Not intended for first-year students. (Second semester/3 credits)
An introduction to mathematical concepts, their understanding and communication. Topics include visualization skills; basic shapes, their properties and relationships between them; communicating geometric ideas; the process of measurement; geometric concepts of length, area and volume; designing data investigations and making judgments under conditions of uncertainty. Emphasis is on developing a deep understanding of the fundamental ideas of elementary school mathematics. An integrated mathematics laboratory approach will be used, including the use of appropriate technology.
These courses promote students' understanding and appreciation of mathematics and develop quantitative and problem solving skills. Each course uses the computer to aid in exploration and computation. Various topics are offered each semester.
MATH 112 Applied Statistics (CORE—Foundation)
Prerequisite: MATH 099 or Level II placement on the Basic Math Skills Inventory or permission of the instructor. Not open to students who have received credit for ECMG 212, PSY 211 or SOC 261. (Summer and both semesters/3 credits)
Statistics with emphasis on applications. Topics covered include statistical measures, normal distribution, sampling theory, statistical inference, hypothesis testing and quality control, correlation, regression and analysis of variance. Students will use statistical software packages on the computer to explore topics in more depth.
Functions and graphs: polynomial,
exponential, logarithmic and trigonometric functions; analytic geometry.
Emphasis is on problem-solving, mathematical modeling and the use of
technology. Designed primarily as preparation for calculus.
MATH 200L Calculus Workshop
Prerequisite: A course in calculus at an institution other than Hood College, either by transfer or credit by exam. May be taken concurrently with the student's first mathematics class at Hood. This course is not open to students who have completed Math 201 at Hood. (Both semesters/1 credit)
An introduction to the topics and tools of calculus: differential equations and initial value problems, logarithmic graphs and mathematical modeling, slope fields, population models, Euler's Method. Mathematical and technical word processing software. Group projects and lab reports. This course is intended for students who plan to take courses beyond Calculus I at Hood, but who did not take MATH 201 here. Grading is on a satisfactory/unsatisfactory basis.
Prerequisite: MATH 201 or permission of the instructor. Students who did not complete MATH 201 at Hood must enroll in MATH 200L Calculus Workshop concurrently. (Both semesters/4 credits, six hours of integrated class work and computer laboratory)
Antiderivatives and the Fundamental Theorem of Calculus; distance, velocity and acceleration; the definite integral; uses of integrals and representations of functions; distribution and density functions; Taylor polynomials and infinite series. Emphasis is on problem solving, collaborative work, computer exploration, writing.
An introduction to basic concepts and techniques of discrete mathematics. Topics include logic, sets, positional numeration systems, mathematical induction, elementary combinatorics, algorithms, matrices, recursion and the basic concepts of graphs and trees. The relationship to the computer will be stressed throughout.
The study and application of the ideas and techniques of calculus to the solution of real-world problems. Emphasis is on qualitative, numerical and analytic methods of solution. Extensive use of the computer.
Developing and using mathematical models to analyze and solve real-world problems. Topics will include discrete and continuous, empirical and stochastic models. Students will use computer software for analysis and simulation and will complete individual and group projects.
An introduction to mathematical rigor and proof encountered in advanced mathematics. Topics include logic, sets, elementary number theory, relations, functions, limits, cardinality, the complex number system.
MATH 335 Teaching Assistantship in Mathematics
Prerequisite: Permission of the department. May be repeated for a maximum of 4 credits. (Either semester/1 or 2 credits)
An opportunity for students to serve as teaching and tutorial assistants for lower-division mathematics courses. Under the supervision of department faculty or The Josephine Steiner Center for Academic Achievement and Retention staff, assistants will aid students seeking to improve their mathematical skills. Grading is on a satisfactory/unsatisfactory basis.
An investigation of Euclidean and non-Euclidean geometries. Use of computer technology and independent work will be an integral part of the course.
MATH 339 Linear Algebra
Prerequisites: MATH 207 and MATH 202. (First semester/3 credits)
A modern introduction to linear algebra and its applications. Emphasis on geometric interpretation, extensive use of the computer. Linear systems, matrices, linear transformations, eigenvalues and dynamical systems.
A calculus-based course in the theory and application of modern probability and statistics. Topics will be chosen from the following: events and probabilities, random variables and distributions, expectation: means and variances, conditional probability and independence, generating functions and the Central Limit Theorem, hypothesis testing, point estimation, confidence intervals, linear models, ANOVA.
The study of selected topics in mathematics or computing, accomplished through reading, problem assignments and projects.
MATH 398 Mathematics Tutorial
Prerequisite: Permission of instructor. (Either semester/1-3 credits)
An opportunity to work with a faculty member and a small group of students in a semester- long program of directed study.
MATH 399 Internship in Mathematics
Prerequisites: 21 credits of mathematics at the 200 level or above and permission of the department. (Either semester/3 to 15 credits)
Supervised work in applied mathematics-related projects in a governmental, private-industrial or educational setting. In order to enroll in this course, a student must meet College internship requirements. Grading is on a satisfactory/unsatisfactory basis.
The study of the basic structures of modern abstract algebra: groups, rings and fields. Topics include cosets, direct products, homomorphisms, quotient structures and factorization. Applications may include symmetry groups, coding theory and connections with graph theory.
MATH 446/546 Operations Research
Prerequisite: MATH 320 or MGMT 312, or permission of the instructor. (Offered as needed/3 credits)
An introduction to real analysis and its development: infinite series, differentiability, continuity, the Riemann and Cauchy integrals, uniform convergence. Computer exploration and visualization are an essential part of the course.
The theory and applications of numerical computing: interpolation and curve-fitting, solutions of algebraic and functional equations, numerical integration, numerical solutions of differential equations.
MATH 470 Seminar: The History of Mathematics
Prerequisites: Senior standing and either MATH 440 or MATH 453, or permission of the department. (Second semester/3 credits)
A seminar in the history of mathematics. Students will use primary and secondary resources, both print and nonprint, to explore the history of mathematics from pre-history to the present.
MATH 111A The Mathematics of Daily Life (CORE—Foundation)
Prerequisite: Math 099 or Level II placement on the Basic Math Skills Inventory or permission of the department.(Either semester/3 credits)
This course introduces students to a wide range of applications of mathematics to modern life. Students will learn some surprisingly simple mathematical ideas that are fundamental in the working of the modern world. Among the topics of the course are: the mathematical tools that businesses use to schedule and plan efficiently; the number codes such as UPC, ZIP codes, and ISBN codes that help organize our lives; and the surprising paradoxes and complexities of elections.
MATH 111B The Mathematics of Democracy (CORE—Foundation)
Prerequisite (Either semester/3 credits)
Students in this course will study two basic questions about democracy –"How do we vote?" and "How do we allocate power?" – from a mathematical perspective. The mathematics reveals surprising paradoxes and complications in the answers to these questions. The course explores why we vote the way we do, what problems arise in voting, and what alternatives are being tried. It will also consider how we can divide the riches of society fairly – and even what the word "fairness" could possibly mean.
MATH 111G The Mathematics of Games and Sports (CORE—Foundation)
Prerequisite: MATH 099 or Level II placement on the Basic Math Skills Inventory or permission of the department (Offered once a year or by demand/3 credits)
This course examines the serious mathematics of fun. How often should one expect to see a perfect game in Major League Baseball? Why should you always split 8s in blackjack? How can a tournament among seven teams best be scheduled? Will women ever be faster than men in the highest levels of track performance? Is it better to bet on a color or a number in roulette? Students will explore all of these questions and more using mathematical tools such as probability, linear models, and graph theory. This class also uses computational tools to solve problems and analyze data.
MATH 112W Workshop Statistics (CORE—Foundation)
Prerequisite: MATH 099 or Level II placement on the Basic Math Skills Inventory. Not open to students who have received credit for ECMG 212, PSY 211 or SOC 261. (Second semester/3 credits)
An active-learning approach to introductory statistics. Emphasis is on collaboration, discovery, exploration, use of technology. Topics covered are the same as those in MATH 112: statistical measures, distributions, sampling, inference, confidence intervals, correlation, regression, analysis of variance. Students will use a statistical software package.
MATH 253 Multivariable Calculus
Prerequisite: MATH 202 or permission of instructor. Students who did not complete MATH 201 or 202 at Hood must enroll in MATH 200L Calculus Workshop concurrently. (First semester/4 credits)
Basic statistical methods as they apply to data and research in the human sciences and other fields. Topics include frequency distributions and their representations, measures of central tendency and dispersion, elementary probability, statistical sampling theory, testing hypotheses, non-parametric methods, linear regression, correlation and analysis of variance. Each student may be required to do a statistics project under the guidance of a cooperating faculty member in a specific discipline such as biology, economics, education, political science, psychology or sociology.
This course will examine high school geometry from a more sophisticated point of view, as well as exploring more advanced Euclidean and non-Euclidean geometrics. Topics covered may include analytic geometry, spherical geometry, hyperbolic geometry, fractal geometry and transfermational geometry. Labs in Geometer's Sketchpad will be an integral part of the course.
An examination of basic and advanced algebra concepts for teachers of mathematics. The course includes an introduction to the number theory and modern algebra topics that underlie the arithmetic and algebra taught in school. The focus is on collaborative learning, communication, and the appropriate use of technology, as well as on a deep understanding of algebraic theory.
MATH 505 Discrete Mathematics
(Either semester/3 credits)
Introduction to the basic mathematical structures and methods used to solve problems that are inherently finite in nature. Topics include logic, Boolean algebra, sets, relations, functions, matrices, induction and elementary recursion, and introductory treatments of combinatorics and graph theory.
MATH 507/407 Introduction to Graph Theory
Prerequisites: Enrollment in the High School Track of the M.S.in Mathematics Education program or an undergraduate degree in mathematics or permission of the instructor. (Second semester—odd years/3 credits)
A rigorous study of the theory of graphs, including simple and directed graphs, circuits, graph algorithms, connectedness, planarity and coloring problems.
MATH 509/409 Elementary Number Theory
Prerequisites: Enrollment in the High School Track of the M.S.in Mathematics Education program or an undergraduate degree in mathematics or permission of the instructor. (First semester—odd years/3 credits)
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Calculus: Early Transcendentals - 9th edition
Summary: The ninth edition continues to provide engineers with an accessible resource for learning calculus. The book includes carefully worked examples and special problem types that help improve comprehension. New applied exercises demonstrate the usefulness of the mathematics. Additional summary tables with step-by-step details are also incorporated into the chapters to make the concepts easier to understand. The Quick Check and Focus on Concepts exercises have been updated as well. Engine...show moreers become engaged in the material because of the easy-to-read style and real-world examples. ...show less
Chapter 5 Integration 5.1 An Overview of the Area Problem 5.2 The Indefinite Integral 5.3 Integration by Substitution 5.4 The Definition of Area as a Limit; Sigma Notation 5.5 The Definite Integral 5.6 The Fundamental Theorem of Calculus 5.7 Rectilinear Motion Revisited Using Integration 5.8 Average Value of a Function and its Applications 5.9 Evaluating Definite Integrals by Substitution 5.10 Logarithmic and Other Functions Defined by Integrals
Chapter 6 Applications of the Definite Integral in Geometry, Science, and Engineering 6.1 Area Between Two Curves 6.2 Volumes by Slicing; Disks and Washers 6.3 Volumes by Cylindrical Shells 6.4 Length of a Plane Curve 6.5 Area of a Surface of Revolution 6.6 Work 6.7 Moments, Centers of Gravity, and Centroids 6.8 Fluid Pressure and Force 6.9 Hyperbolic Functions and Hanging Cables
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Microsoft Math is a library of lessons, tutorials and tools to help students learn maths. It has a large array of real world maths problems in science as well as purely mathematical ones and covers basic…
For millions of business professionals and students who rely on the HP 12c series Financial Calculator, the 12c Platinum Software Version is now available for your PC. Built with the same layout, algorithms…
Advanced yet easy-to-use math calculator that immediately and precisely computes the result as you type a math expression. It allows multiples math expressions at same time. It also allows fractions and…
Advanced yet easy-to-use math calculator that immediately and precisely computes the result as you type a math expression. It allows multiples math expressions at same time. It also allows fractions and…
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• So, you think math is boring and there is nothing much anyone can do to make it "un-boring"?
• You think the average person can't really ever understand math, and "Who needs it?" if you have a calculator in hand!
Think Again!!
Introducing: Life of Fred Math
Never again hear the question which many math students have: "When are we ever gonna use this stuff?" or "Math is boring!"
No other textbooks are like these. Each text is written in the style of a novel with a humorous story line. Each section tells part of the life of Fred Gauss and how, in the course of his life, he encounters the need for the math and then learns the methods. Tons of solved examples. Each hardcover textbook contains ALL of the material – more than most instructors cover in traditional classroom settings. Includes tons of proofs.
Written by Dr. Stanley Schmidt with the intent to make math come alive with lots of humour, clear explanations, and silly illustrations that stick in the mind. The student will learn to think mathematically.
Completion of this series prepares student for third year college math.
IMPORTANT NOTE:
These books are designed to make your child THINK! and to learn on their own. It does not give step-by-step directions and answers to every question. They learn to apply to current questions the concepts previously taught. Upon completion they will understand how math works, why it works, and how to apply it. They will know the formulas and how to apply them in real-life situations, not just situations created for a textbook. This program relies heavily on reading comprehension and thinking, not rote spoon-fed learning that is quickly forgotten. Parents can learn these novel methods along with the student, but should not try to integrate the traditional rote/memorization methods.
Want to purchase all of the upper level Life of Fred Math Books at once? Purchase this kit and you will have all the upper level books currently available. Take your student from Pre-Algebra straight through College Level math.
This set can be started with any child who can add and subtract well and knows how to do long multiplication and division. This set would be considered a middle school math program, for students in 4th through 9th grades. When they are finished these books, they are ready to begin the High School Set. Many high school aged students would benefit by going quickly through these books to lay foundations that they might have missed.
This set can be started with any student who has completed Life of Fred Fractions and Life of Fred Decimals and Percents or has completed a Pre-Algebra program in another curriculum. This set could also be considered a College Prep set as it prepares the student for college level maths. After completing this set, the student is ready to go into the College Set.
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9780534495015
ISBN:
053449501x
Pub Date: 2005 Publisher: Brooks/Cole
Summary: An increasing number of computer scientists from diverse areas are using discrete mathematical structures to explain concepts and problems. Based on their teaching experiences, the authors offer an accessible text that emphasizes the fundamentals of discrete mathematics and its advanced topics. This text shows how to express precise ideas in clear mathematical language. Students discover the importance of discrete ma...thematics in describing computer science structures and problem solving. They also learn how mastering discrete mathematics will help them develop important reasoning skills that will continue to be useful throughout their careers
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This GCSE Mathematics Studies Revision Guide provides students with essential and stimulating material to improve understanding of all the key topics on the specification and achieve exam success. Each topic includes a set of questions so that you can test your knowledge and understanding as your revision progresses, and answers to all questions are provided at the end of the book.
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Mathematics of Musical Signals 2: The Wave Equation
Department - LART Offered - Spring Course number - LMSC-P315
In this course, students explore the ways that symbolizing musical signals contributes to the design and development of sound. Students study the mathematics behind acoustic and electrical signals. This course continues the exploration of the mathematics behind musical signals that began in LMSC-P310. Students use mathematics to analyze musical signals. They evaluate complex waveforms using mathematics. And they apply mathematics to signals to understand transformation. Students explore resonance and the wave equation. In addition, students learn further how to describe and manipulate mathematically musical signals and their representations.
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This is the ultimate mathematics course. We will cover everything from Algebra I and work our way up to Calculus concepts with some great discussions along the way. Mathematics is a language and if you can become fluent in it (which if you complete this course, I promise you will be!), then that is simply one more step in your great knowledge bank. (Every student has the option to pick a certain semester or set of semesters to learn from, as I do know that this is a very long course. Thus, if there is simply a few things that you would like to work on, pick a semester and let me know. If you just want to get the whole feel of the course, I will be glad to start from the beginning.)
Please know that I am a bit more of a lecture type of teacher, and thus, you may hear me speak quite a lot. However, within mathematics, the main thing we try to accomplish is your knowledge of the basic priniciples, and thus, mathematics consists of repetition, repetition, repetition! Here is the basic outline of a daily lesson:
Get acquainted and review material from last lesson
Teach new material
Practice new material
Q/A Session
Assigned Work
Now, you will most likely learn a few new principles in one lesson, so steps 2 and 3 may be repeated a few times. Homework will be given daily, and you can usually just download the homework from clicking the links below. Tests will be at least once a week, and unit tests will be about every 3 - 4 weeks, unless changed by me. Attendance is crucial in this type of course! If you are not here for two to three days at a time, then we will not be able to effectively absorb the knowledge that is being presented to you, and thus, it will take longer to learn this. At the end of every semester, you will have a final exam which will cover all of what you have learned.
Below lists all semesters and lessons that will be involved with your course. Next to each lesson is a set number of problems. This is basically how much homework you will have for each lesson, and you will have until the end of the week to get all problems completed. Soon, each set of problems will have a hyperlink within them that goes to a page where the assignment is, and you can simply do the problems from there. These are listed so that potential students can examine how much work will be involved in each semester, and how you can estimate the approximate time that you will need to devote to this course.
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International Society for Bayesian Analysis (ISBA)
Promotes the development of Bayesian statistical theory and its application to problems in science, industry and government. News, history and minutes, archive of abstracts, information about the Reverend Thomas Bayes, and open positions in the field.Japanese Association of Mathematical Sciences (JAMS)
A scientific research organization whose main activity is to publish scientific journals in English, French or German, in particular Scientiae Mathematicae Japonicae. Journal and submission information, newsletter, and meetings. Also available in JapaneseMarc Chamberland
A mathematician at Grinnell College interested in differential equations and dynamical systems. Resources for the 3x + 1 problem and the Jacobian Conjecture include papers to download in PostScript format and information and proceedings for related conferences.
...more>>
Mathematics of Planet Earth
Mathematics of Planet Earth (MPE) "provides a platform to showcase the essential relevance of mathematics to planetary problems, coalesces activities currently dispersed among institutions, and creates a context for mathematical and interdisciplinary
...more>>
Minnesota Council of Teachers of Mathematics (MCTM)
An affiliate of the National Council of Teachers of Mathematics (NCTM). The site provides information about conferences, events, and programs; membership; listings of recommended math sites; MCTM highlights (Presidential Awardees, Shape of Space video,
...more>>
MOTIVATE - Univ. of Cambridge, UK
MOTIVATE is a project incorporating a series of videoconferences, run by the Millennium Mathematics Project at Cambridge. The objectives of MOTIVATE are: to enrich the mathematical experience of school students, to broaden their mathematical horizons,
...more>>
MSPnet: The Math and Science Partnership Network - TERC
The Math and Science Partnership (MSP) Program is a major research and development effort to understand and improve the performance of K-12 students in mathematics and science. MSPnet is their electronic community. Learn about individual partnership projects;
...more>>
National Council of Supervisors of Mathematics (NCSM)
A resource site for those interested in leadership in mathematics education. The site lists meetings and conferences, membership information and how to subscribe to the NCSM mailing list, publications, operations, information about the summer Leadership
...more>>
SyllabusWeb - Syllabus Press, Inc.
From the publishers of Syllabus Magazine, a technology magazine for high schools, colleges, and universities. Highlights of recent issues of the magazine and full text archives of all Press publications. The June 1995 issue covers telecommunications and
...more>>
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APPENDIX C
Description of Some
Foundations Courses
At Bloomsburg State University in
Pennsylvania a foundations course presupposes only arithmetic
and emphasizes the use of elementary mathematics in
decision making and problem solving. Game theory is used to
motivate the study of problem solving strategies. Hand
calculators are standard tools. Materials written for the course
have appeared as Mathematics in Daily Life by J. Growney
published by McGraw Hill. Exercises in the book take on a variety
of forms from those that require a brief application of a specific
concept or method to those which require discussion or multistep
application and evaluation of several ideas or procedures.
At Northern Illinois University a foundations course
presupposes two years of high school mathematics including one
year of algebra. The course aims to develop in the student a
competency in problem solving and analysis which is helpful in
personal decision- making; in evaluating concerns in the
community, state, and nation; in setting and achieving goals; and
in continued learning. A hand calculator is used throughout
the course. The mathematical content is one-third probability
and statistics and two-thirds logical statements and arguments,
geometry in problem solving, estimation and approximation
(inequalities, functions, average rates of change), and general
problem solving (including personal business applications).
Problem sets consist of routine and nonroutine exercises.
Materials written for the course have been published by
Kendall/Hunt as Mathematical Thinking in a Quantitative
World by L.R. Sons and P.J. Nicholls.
At the University of Tennessee the foundations course
"Algebraic Reasoning: Motivated by Actual Problems in Personal
Finance" presupposes two years of high school algebra and one
year of high school geometry. The course assumes use of a hand
calculator and places its emphasis "on the importance and
applicability of mathematics in real life." Problems used to
motivate algebraic concepts are relevant to most college
students' experience. Topics covered include borrowing money to
complete college, saving a lump sum for college education,
consolidation of debts, periodic payments, amortization
schedules and more! The course uses material developed locally
by J. Harvey Carruth.
At the \ulUniversity of Chicago a trio of faculty have
produced a series of ten-week (quarter) courses composed of
short mini-courses each devoted to a single theme. Funded by the
Sloan Foundation, the course development has involved the
production of units such as statistical analysis of literary style,
quantitative arguments and scientific method, the organization
of the brain, dynamical systems, and risk assessment and
epidemiology. Computer software was developed by the faculty
for use in the course as were other resource materials. J. Cowan,
S. Kurtz (Computer Science), and R. Thisted (Statistics) have
worked on the courses.
The Sloan Foundation also funded the development of a
quantitative methods course at SUNY Stony Brook taught by
D. Ferguson- the Department of Technology and Society. The
course builds mathematical models and uses these models to get
approximate answers to questions which arise out of human need.
Some examples of models are stock market simulation, drug
testing, life insurance, and quality control in production.
Computers are used in the course, and mathematical tools include
probability and linear programming. Quantitative methods are
portrayed as a "way of knowing" the world and mathematical
techniques are developed in the context of real problems. The
course uses three textbooks and notes, examples, and
laboratory activities developed locally. The textbooks are
Probability Examples by J. Truxal and N. Copp's {Vaccines: An Introduction to Risk both of which are in the New Liberal
Arts Monograph Series, and How to Model It: Problem Solving
for the Computer Age by A. Starfield, K. Smith, and A. Bleloch
which is published by McGraw Hill.
Trinity College (Connecticut) developed a course
"Essential Applications of Mathematics" aimed at enabling
students to "be conversant and comfortable with the reasoning
underlying such issues as environmental protection, the nuclear
arms race, the spread of AIDS, and the budget deficit." The
course focuses on five areas: numerical relations; proportions
and percents; data analysis, probability and statistics;
mathematical reasoning; and applications of algebra, geometry,
and functions. Microcomputers are used. Laboratories are a part
of the course. Materials have been developed locally. T.Craine
and L. Deephouse have been involved in the course development
which was linked to the adoption of a college mathematical
proficiency requirement.
At Dartmouth College L. Snell and R. Prosser have
developed with the participation of faculty at Grinnell,
Middlebury, and Spelman College a course called CHANCE. The
course develops concepts of probability and statistics only to the
extent needed to understand the applications. An important part
of the text material for the course is the journal Chance
started by Springer-Verlag in 1988. Computer simulations and
software packages are used. Units for study include maintaining
quality of manufactured goods in the face of variation, and
scoring streaks and records in sports. The New York Times
is also a resource for the course, and writing is a means used in
teaching the course.
Another course developed with the help of the Sloan
Foundation is "Case Studies in Quantitative Reasoning: An
Interdisciplinary Course" at Mount Holyoke College. H.
Pollatsek and R. Schwartz have divided the course into the three
units:
I. Narrative and Numbers: Salem
VillageWitchcraft;
II. Measurement and
Prediction: SAT Scores and GPA;
III. Rates of
Change: Modeling Population and Resources.
A three-hour weekly computer laboratory is an integral part
of the course's structure. The emphasis is on reasoning and ways
to construct and evaluate arguments. Besides study of
probability and statistics, the course involves simple algebra,
graph reading, linear and exponential models, rates of change,and
simulation. Students are required to write three substantial
papers in the course and do six laboratory reports besides
homework exercises. Resource materials consist of a readings
list rather than a specific textbook.
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Search Course Communities:
Course Communities
JOde: A Java Applet for Studying Ordinary Differential Equations
Course Topic(s):
Ordinary Differential Equations | Graphic Methods
A Java applet that allows one to interactively analyze (graphically) second order differential equations. It was designed to be a convenient aid in teaching differential equations, but is also useful for students. The webpage contains both a tutorial and complete instruction. It is a 2D representation that includes the graph.
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This course places quantitative skills and reasoning in the context of experiences that students will be likely to encounter. It emphasizes processing information in context from a variety of representations, understanding of both the information and the processing, and understanding which conclusions can be reasonably determined.
NOTE: This course is an alternative in Area A of the Core Curriculum and is not intended to supply sufficient algebraic background for students who intend to take Precalculus or the Calculus sequences for mathematics and science majors.
Expected Educational Results
As a result of completing this course students will be able to:
1. Solve real-world application problems using ratio, proportion, and percent.
2. Use geometric formulas and principles to solve applied problems.
3. Use logic to recognize valid and invalid arguments.
4. Apply fundamental counting principals and fundamental laws of probability to determine the probability of an event.
5. Compute and interpret measures of central tendency and variation.
6. Read and interpret data presented in various forms, including graphs.
7. Solve application problems involving consumer finance.
8. Students will create a scatter plot of data and determine if it is best modeled by a linear, quadratic, or exponential model.
9. Students will create models for data that is exactly linear and use the model to answer input and output questions in the context of applications.
10. Students will use the calculator to create models for data that is nearly linear, and use the model to answer input and output questions in the context of applications.
11. Students will use quadratic and exponential models to answer input and output questions.
General Education Outcomes
I. This course addresses the general education outcome relating to communication by providing additional support as follows:
a. Students improve their communication skills by taking part in general class discussions and in small group activities.
b. Students improve their reading skills by reading and discussing the textbook.
c. Unit tests, examinations, or other assignments provide opportunities for students to practice and improve mathematical writing skills. Mathematics has a specialized vocabulary that students are expected to use correctly.
II. This course addresses the general education outcome of demonstrating effective individual and group problem-solving and critical-thinking skills as follows:
a. Students will apply mathematical principles to solve a variety of application problems involving ratios, proportions, percentages, and probabilities.
b. Students will apply geometric principles to solve applied problems.
c. Students will solve problems involving consumer finance.
d. Students will use linear, quadratic, and exponential models to solve real-world problems.
III. This course addresses the general education outcome of using mathematical concepts to interpret, understand, and communicate quantitative data as follows:
a. Students will read, interpret data, and communicate results presented in various forms, including graphs.
b. Students will compute, interpret measures, and communicate results of analysis of central tendency and dispersion.
c. Students will create scatter plots, determine on an appropriate mathematical model for the data, and communicate results.
d. Students will use linear, quadratic, and exponential models to solve real-world problems and to communicate results.
IV.This course addresses the general education outcome of organizing information through the use of computer software packages as follows:
a. Students will use a graphing calculator or Excel to create scatter plots and decide on an appropriate mathematical model.
b. Students will use a graphing calculator or Excel to perform linear regression.
c. Students will use a graphing calculator or Excel to compute measures of central tendency and dispersion for a set of data.
ENTRY-LEVEL COMPETENCIES
The student is expected to have completed successfully the equivalent of Algebra II. Essential to success in MATH 1001 is mastery of the following:
1. Solving linear equations and inequalities.
2. Solving quadratic equations.
3. Understanding and using integral exponents.
4. Applying basic geometric concepts including the Pythagorean Theorem, the distance formula, areas and perimeters of rectangles, triangles, and circles.
5. Graphing linear relationships.
6. Computing slope.
7. Writing linear equations given appropriate information.
8. Creating a table of values and using it to graph a function.
9. Carrying out basic arithmetic with polynomials including factoring.
Assessment of Outcome Objectives
I. COURSE GRADE
The course grade will be determined by the individual instructor using a variety of evaluation methods. A portion of the course grade will be determined through the use of frequent assessment using such means as tests, quizzes, projects, or homework as developed by the instructor. Some of these methods will require the student to demonstrate ability in problem solving and critical thinking as evidenced by explaining and interpreting solutions. A portion of the evaluation process will require the student to demonstrate skill in writing both correct prose and correct mathematics. A comprehensive final examination is required. The final examination must count at least one-fifth and no more than one-third of the course grade. The final examination should include items which require the student to demonstrate problem solving and critical thinking.
II. DEPARTMENTAL ASSESSMENT
This course will be assessed on a regular assessment schedule determined by the discipline. An appropriate assessment instrument will be determined by the Math 1001 committee.
III. USE OF ASSESSMENT FINDINGS
The Math 1001 committee, or a special committee appointed by the Academic Group, will analyze the results of the assessment and determine implications for curriculum changes. The committee will prepare a report for the Academic Group summarizing its findings.
EFFECTIVE DATE: Fall 2011
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Fundamental Theorem of Calculus
In this lesson, Professor John Zhu gives an introduction to the fundamental theorem of calculus. He goes over the properties for the fundamental theorem of calculus as well as the definition of integral. He reviews four rules/ properties for calculus and performs a few example problems.
This content requires Javascript to be available and enabled in your browser.
Fundamental Theorem of Calculus
Simply evaluating integral at 2
bounds
Area under a curve
Accumulated value of
anti-derivative function
Fundamental Theorem of Calculus
Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.
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collection is a full course of material in the form of a textbook. The textbook is FHSST Mathematics textbook contains a total of 45 chapters to be used in grades 10, 11, and 12. At the end of this description is a complete Table of Contents.
The textbook is broken into 5 sections: basics (Chapter 1), Grade 10 (Chapters 2-16), Grade 11 (Chapters 17-34), Grade 12 (Chapters 35-45), and Exercises. In this collection, you will find folders for each of these sections and chapters are found within.
Description:Algebra 1 textbook answers and problem sets designed to illustrate all chapters covered in Algebra 1. All answers are illustrated with "motion lines" and explanations. Contributed by offering instant math help for struggling algebra students.
Description:This is a very fun Geometry and STEM wiki; I teach everything with arts and sciences and humanities. It is projects based learning and discovery learning; it is a work-in-progress and encourages collaborations across all subjects and worldwide…
Last Updated:Feb-01-2012
Subject(s):
Arts
Career & Technical Education Assignment/Homework
Asset: Article/Essay
...
This is a very fun Geometry and STEM wiki; I teach everything with arts and sciences and humanities. It is projects based learning and discovery learning; it is a work-in-progress and encourages collaborations across all subjects and worldwide…
Description:
START OF THIS PROJECT GUTENBERG EBOOK A FIRST BOOK IN ALGEBRA *
Last Updated:Jun-22-2012
Subject(s):
Mathematics
Mathematics > AlgebraBook: Text Book
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Description
The first half of a modern high school algebra sequence with a focus in seven major topics: transition from arithmetic to algebra, solving equations & inequalities, probability and statistics, proportional reasoning, linear equations and functions, systems of linear equations and inequalities, and operations on polynomials. Students enrolled in this course must take the WA State High School End of Course Algebra Assessment if they have not attempted it once already. Prerequisite: Must be working toward a high school diploma.
Intended Learning Outcomes
Select and justify functions and equations to model and solve problems
Solve problems that can be represented by linear functions, equations, and inequalities
Solve problems that can be represented by a system of two linear equations or inequalities.
Solve problems that can be represented by quadratic functions and equations
Solve problems that can be represented by exponential functions and equations
Know the relationship between real numbers and the number line, and compare and order real numbers with and without the number line
Recognize the multiple uses of variables, determine all possible values of variables that satisfy prescribed conditions, and evaluate algebraic expressions that involve variables
Interpret and use integer exponents and square and cube roots, and apply the laws and properties of exponents to simplify and evaluate exponential expressions
Determine whether approximations or exact values of real numbers are appropriate, depending on the context, and justify the selection
Use algebraic properties to factor and combine like terms in polynomials
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Appendix D. Exact Solutions of Polynomial Equations
Introduction
For thousands of years the solution of polynomial equations was one of the most important and productive problems in algebra, and indeed in all of mathematics. Apart from the question of solution methods, the study of quadratic, cubic, and quartic equations and their geometric equivalents led to great advances in the concept of number over a period of more than two millennia, including irrational, negative, and complex numbers, and the limitations of numbers constructible with ruler and compass. Later on, the problem of quintic and higher order equations led to what has been called higher algebra or abstract algebra, that is, to the study of structures such as groups, rings, fields, vector spaces, and algebras of many other kinds, further greatly expanding the concept of number, and many other concepts. All of these structures and many others were then gathered together into category theory, which studies all of the mappings within and between all such objects, providing insight not only into the algebraic structures themselves, but also into algebraic geometry, the study of geometric objects by means of algebraic structures defined on them.
Furthermore, the solution of quadratic equations turned out to be fundamental in elementary physics in the period from Galileo to Newton. (Going beyond the elementary level requires calculus, as explained, for example, in Ken Iverson's book Elementary Analysis, which can be used as a continuation from this study of algebra.) For example, in elastic collisions, where both momentum and energy are conserved, the conservation laws are respectively linear and quadratic in form, so that the solution which preserves both reduces to the two solutions of a quadratic equation. If m and v0 are two-element vectors of masses and initial velocities, we must find a vector v1 such that:
(+/m*v0)=(+/m*v1)
(+/m**:v0)=(+/m**:v1)
One solution, v0, represents the state before the collision, and the other solution represents the state after the collision.
The solution of polynomial equations is fundamental to computer graphics, computer-aided design, and robotics.
The original problem of solving polynomials was to find solutions that could be expressed exactly in terms of the five basic algebraic operations, + - * % %:, starting from integers. The body of this textbook has been concerned only with numeric solutions, but this Appendix reviews the classical results and methods for solving quadratic, cubic, and quartic equations.
General Considerations
We know that every polynomial is a product of linear terms, of the form */x-r possibly multiplied by a constant that has no effect on the roots. We can divide the coefficient vector c of a polynomial by its last element _1{c, giving a new coefficient vector for a polynomial with 1 for the coefficient of the highest power of x present. We call this a monic polynomial.
The monadic case of the polynomial verb p. allows us to convert between coefficient vectors and a representation of a polynomial with boxed multiplier and list of roots. For example:
Each linear polynomial of the form x-r has a root r, so the product polynomial has a root for each of the r values. That is, at each of those values, at least one term has the value 0, so the product also has the value 0. These are the only roots, because at any other value of x, the factors are all non-zero and therefore the product is non-zero.
Furthermore, we know that the constant term in a polynomial */x-r is the product of all of the roots, by construction. The second-highest order term is the sum of all of the roots, again by construction. In the case of a quadratic polynomial, we saw in Chapter 14 that the coefficients of the polynomial are (r0*r1),(-r0+r1),1, possibly multiplied by a constant. Thus the polynomial with roots 2 3 has coefficients 6 _5 1, as above. A cubic polynomial has a term composed of the sum of products of pairs of roots, that is, (r1*r2)+(r0*r2)+(r0*r1). A quartic polynomial has a term composed of sums of products of triples of roots, and so on.
These terms represent functions of the roots with the special property that interchanging roots does not change the value of the function, unlike, say r0-r1, which is negated by interchanging r0 and r1. Functions with this property are called symmetric. Their study was fundamental in the proof that quintic and higher-order polynomials do not have solutions using only the functions + - * % %:, known as solutions in radicals.
Quadratic Equations
Although many mathematicians in Babylonia, Egypt, Greece, India, and China developed partial methods for solving quadratic equations, the general method to be described here is due to Muhammad ibn Musa al-Khwarizmi of Persia in the 9th century. It was introduced into Europe in the 12th century, first appearing in a Hebrew work by Abraham bar Ḥiyya ha-Nasi, then in Latin translations of al-Khwarizmi and ha-Nasi. However, the full solution including negative and complex roots was not recognized until the 19th century.
Given a three-element coefficient vector c from which we can make the polynomial p=.c&p., where the last element of c is not 0, how can we go about solving the equation 0=p x for its two roots, r0 and r1? We can begin by dividing by the last element of c to give an equivalent monic polynomial.
We can graph p, and find approximate roots by inspection.
We can use the iterative rootfinder described in Chapter 12.
We can look for integer roots by factoring the constant term.
We can graph the sum and product terms r0+r1 (a straight line) and r0*r1 (a hyperbola) on r0 and r1 axes, and look for roots where the two curves meet, as in the following graph.
Graph
None of these methods is entirely satisfactory. What we would like is an expression for the roots in terms of the coefficients of the polynomial. There is a particular case where this is very easy, the case where the two roots are equal. We have
(x-r)*(x-r)
(x^2)+(-2*r*x)+(r^2)
Setting 0=.*:(x-r) and solving gives us
0=*:x-r
(%:0)=x-r
r+0=x
r=x
as desired. Graphically, we can see that this is the case where the curve just touches the x axis at the point (r,0), as in the following graph:
Graph
Usually the graph of a quadratic polynomial in real numbers intersects the x axis in two distinct points, or not at all. But in those cases, adding a constant to the polynomial would move the graph up (for a positive constant) or down (for a negative constant), so there must be an intermediate value that would move the graph to just touch the x axis. Let us call that number n. Then adding n,0 0 to c, or setting p1=.n+p, has this result in the graph.
Graph
We now know that p1 is the square of x-r for some still unknown r, but we also know that
(1{c)=2*r
((1{c)%2)=r
Using this r, we can proceed as follows:
0=p1 x
n=n+p x
(%:n)=x-r
(r+%:n)=x
Now in fact n has two square roots, one of which is returned by the %: function. We call this the principal square root, or the principal value of the square root function. For positive y, %:y is the positive square root, and there is also a negative root. For negative y, *:y is 0j1*%:-y. Its negation is also a square root of y. So to get both square roots of y, it suffices to write 1 _1*%:y. (For complex numbers in general other than real numbers, %:z is the square root with positive imaginary part. The other square root is the negation of this principal square root.) Inserting this into the solution above above gives:
(r+1 _1*%:n)=x
It now remains to find n. We know that the square of x-r is (x^2)+(_2*r*x)+r^2. So if we have (x^2)+(_2*r*x)+k we want to add s=.(r^2)-k, or equivalently, in terms of values we know (*:b%2)-k, which results in the square of the monomial. We then know how to take the square roots of (*:b%2)-k, and it is easy to arrange the results in a convenient form for calculation. This method of solution is called completing the square.
Given a polynomial in the form (a*x^2)+(b*x)+c, we therefore proceed as follows:
This result is correct within the limit of precision of computer arithmetic. Examining the formula for the solutions shows that these roots are of the form:
-b%a
1
(*:b)-4*a*c)
5
(1+1 _1*%:5)%2
1.61803 _0.618034
The first of these is known as the Golden Ratio. It plays an important role in art, mathematics, biology and other areas.
We can now use the expression derived above to create an explicit function definition for solving quadratic equations given the coefficient vector c, as follows:
qe=.3 : 0
('c';'b';'a')=.y
((-b)+(1 _1*%:(*:b)-4*a*c))%2*a
)
For example:
c=._12 1 1
qe c
3 _4
c p. 3 _4
0 0
For an equation with real coefficients, the character of the solution depends on the sign of the expression (*:b)-4*a*c, the argument of the square root function. If it is positive, there are two real roots. If it is 0, there are two equal real roots. If it is negative, there are two complex roots. This expression is therefore called the discriminant, because it discriminates among these cases. For example, if we first define disc as the discriminant function:
Cubic Equations
The cube of a positive real number is positive, and the cube of a negative real number is negative. It follows that any cubic polynomial takes on both positive and values for real arguments of sufficient magnitude, where the cubic term is much greater in magnitude than the other terms. The polynomial thus must be zero for some real argument, having at least one real root. It may have three, as shown in the following graph.
Graph
The simplest cubic equation is 1=x^3. It has the three roots
1 _0.5j0.866025 _0.5j_0.866025
If we set w1=._0.5j0.866025 and w2=._0.5j_0.866025, we get the following relationships:
w1=w2^2
w2=w1^2
1=w1*w2
_1=w1+w2
Thus w1 and w2 are the roots of the quadratic equation 0=_1 0 1 p.x . Using the quadratic formula above we get the expression 0.5*_1+1 _1*%:_3, which yields these complex cube roots of 1.
There is no method for completing a cube, that is, for rearranging a cubic into the form (x-r)^3. What we have to do instead is to find a quadratic relationship among its coefficients (and thus among its roots) that we can solve and substitute back into the cubic. This method was worked out by Scipione del Ferro and Tartaglia, and published by Gerolamo Cardano in 1545. The full solution for negative and complex roots was not recognized until much later. However, the appearance of complex numbers in expressions for real roots was essential to the development of complex arithmetic, raising questions in the 16th century that were not fully resolved until the 19th.
To solve a cubic equation 0=cv p. x, first we want to simplify the form of the coefficient vector cv=.d,c,b,a. As before, we divide through by the last element of cv to produce a monic polynomial. Then we make the substitution t-(2{cv)%3 for x in this monic polynomial, which has the result that the new coefficient vector has the form d,c,0 1. This form is called a depressed cubic. Note that:
(t-(2{cv)%3)=x
t=x+(2{cv)%3
For example, starting from the monic coefficient vector cv, let us define:
A significant amount of routine algebraic manipulation has been omitted above. It can be treated as an exercise, or the student can verify the equivalence numerically by assigning appropriate values to the variables and executing each expression. For example:
The other two solutions of the cubic equation require finding the other two cube roots of u^3 and v^3, which turn out to be w1 and w2 times the primary cube roots. We must be careful to match the resulting four values in pairs, observing the condition 0=p+3*u*v. The result is
x1=.(w1*u)+(w2*v)
x2=.(w2*u)+(w1*v)
These pairings work because 1=w1*w2, so (u*v)=(w1*u)*(w2*v) and (u*v)=(w2*u)*(w1*v).
However, if the discriminant is negative, using the cube root function 3&^: does not in general match the appropriate cube roots. We must use the relationship
p+3*u*v=0
3*u*v=(-p)
v=(-p)%(3*u)
There is one other case to consider, if u is 0, in which case p is 0. Then we cannot use the relationship above, but must use
Again, these are excellent approximations of the roots, with errors less than 1e_15.
Quartic Equations
Gerolamo Cardano's student Lodovico Ferrari discovered a method for solving quartic equations by means of an auxiliary cubic equation in 1540, before Cardano published the solution of the cubic. Ferrari's method uses the concept of completing the square twice. The condition for completing the second square results in the auxiliary cubic. Several other methods have been discovered since using techniques such as factorization of the polynomial, Galois theory, and algebraic geometry.
The first two steps are as for the cubic equation, to reduce the quartic to monic form (divide by the coefficient of x^4) and then to depressed form (substitute t=.x-b%4, where b is the coefficient of x^3). The new coefficient vector will have the form cvd=.ed,dd,cd,0 1. In terms of the original coefficient vector cv=.e,d,c,b,a, these have the values
If dd=0, the polynomial can be rewritten as (ed,cd,1)p. x^2, that is, as an easily-solved quadratic in x^2. If ed=0, one of the roots is 0, and we can write an equation for the other roots as (dd,cd,0 1) p. x, that is, as a depressed cubic solvable by methods discussed above.
At this point we are given that 0=(ed,dd,cd,0 1) p.x for certain unknown values of x, or that
Any of the three solutions will serve our purpose. Each solution of the cubic allows us to complete the second square, resulting in a pair of quadratic equations. Each such pair matches the roots of the quartic in pairs, in one of three possible arrangements:
Equations of Higher Degree
The methods described above fail to work for quintic (fifth-degree) equations and for all higher degrees. When one tries Ferrari's method on quintic polynomials, one does indeed get an auxiliary equation, but it is of the sixth degree, and so is of no help. The Abel-Ruffini theorem established that equations of higher degree cannot be solved in radicals, and Galois theory greatly clarified the relationship between the coefficients and the roots of any polynomial. However, these topics are far beyond the scope of this textbook.
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Free-Standing Mathematics Qualifications (FSMQ)
The Free-Standing Mathematics Qualifications are a group of separate qualifications, that are neither a GCSE nor an A-level. They are meant to bridge the gap between GCSE Mathematics and AS level Mathematics, and are typically taken by students who take their GCSE Mathematics a year early, and study the syllabus for the FSMQ in year 11.
AQA
At foundation level, there are three qualifications: 'Managing Money,' 'Working in 2 and 3 Dimensions' and 'Making Sense of Data'.
At intermediate level, there are again three qualifications: 'Calculation Finances,' 'Handling and Interpreting Data' and 'Using Algebra, Functions and Graphs'.
At Advanced level, there are four different qualifications: 'Using and Applying Statistics,' 'Modeling with Calculus,' 'Using and Applying Decision Mathematics' and 'Working with Algebraic and Graphical Techniques'.
Qualifications
There are three levels, 1 to 3, which are foundation, intermediate and advanced. The intermediate qualification is roughly the equivalent to GCSE Mathematics, however the advanced qualification is a mixture of AS topics, and is about the same as one AS-module.
Some modules in both AQA and OCR FSMQ may be used towards the AS Use of Mathematics as well.
UCAS
UCAS points are awarded on achievement in the FSMQ, with 20 points for an A, 17 points for a B, 13 for a C, 10 for a D and 7 for an E. Unlike GCSEs, it is impossible to gain the A* grade, with the highest grade being an A.
Formulae
Unlike other examinations, for some FSMQs, formula sheets or booklets are not provided, and candidates are expected to recall all formulae in the syllabus.
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The following resources may be of use in learning to understand mathematics and how it relates to a particular discipline.
HELM
HELM (short for 'Helping Engineers Learn Mathematics') is a project funded by HEFCE through
FDTL4 (2002-2005) and involved creating resources to aid the mathematical understanding of undergraduates in Engineering.
It involved five Universities including Manchester.
METAL
MathCentre
Free resources to support the transition from school mathematics to
university mathematics in a range of disciplines
Includes detailed notes, quick sheets,video, pod casts etc. covering a range
of mathematical topics.
MathAid
Some Interactive web based pdf documents, Interactive web based pdf documents from the University of Plymouth. The
files include notes, exercises and quizzes and so allow students to test
their understanding of the material with immediate feedback.
Packages include Basic Algebra, Trigonometry, Calculus etc. and also
Applications of Mathematics in Science and Engineering.
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Academic Links
Mathematics
We strive to educate students in all levels of mathematics. Our goal is to teach students to be fluent in numbers and also to understand the nuances of mathematics and its role in our culture. We believe that mathematics is a universal language and mastering its many aspects is necessary to be a productive global citizen.
The Math department at University Prep combines a traditional curriculum and innovative teaching styles. In the Middle School we fuse the two by using an inquiry, project-based approach in the 6th and 7th grade, which is supplemented by traditional skill building. This allows students to learn concepts through a variety of methods, ensuring that the lessons are accessible to all students. It also means that students can also do the math that they understand so well.
In the Upper School, we follow a very traditional algebraic curriculum focusing on the fundamentals. In geometry, however, our program is a technology-based geometry program, which allows our students to learn geometry and model mathematical ideas through dynamic sketches. This experience gives the students the ability to think at a deeper level about mathematics than the static approach of more traditional geometry. This perspective is one that stays with them throughout our math sequence, as algebra and geometry come together in Pre-Calculus and Calculus.
We use a variety of assessments in all of our classes. In Middle School, especially in 6th and 7th grade, assessment tends to be project based. While students still take the traditional tests and quizzes, there are many more opportunities for them to delve deeper into the material using a more hands-on approach. Some recent projects have included, writing a math children's book to solidify prior knowledge, learning percents through sports and creating a Sports Center video, using Geometer Sketchpad to create tessellation designs, designing and drawing their dream house to learn proportions, and collecting and analyzing their daily garbage to understand statistics.
In the Upper School, most of our major assessments are traditional tests and quizzes. However, we do many other types of assessment to allow students to show what they know in a variety of ways. These include posters, presentations, experiments with write-ups, and oral tests. In the Upper School, classes end the semester with a cumulative final exam.
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Calculus is the study of change, in the same way that geometry is the study of shape and algebra is the study of equations. A course in calculus is a gateway to other, more advanced courses in mathematics devoted to the study of functions and limits, broadly called mathematical analysis. Calculus has widespread applications in science, economics, and engineering and can solve many problems for which algebra alone is insufficient.
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Summary: This chapter gives a detailed analysis of how teaching with variation is helpful for students' learning of algebraic equations by using typical teaching episodes in grade seven in China. Also, it provides a demonstration showing how variation is used as an effective way of teaching through the discussion after the analysis.
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A fascinating tour through parts of geometry students are unlikely to see in the rest of their studies while, at the same time, anchoring their excursions to the well known parallel postulate of Euclid. The author shows how alternatives to Euclids fifth postulate lead to interesting and different patterns and symmetries, and, in the process of examining geometric objects, the author incorporates the algebra of complex and hypercomplex numbers, some graph theory, and some topology. Interesting problems are scattered throughout the text. Nevertheless, the book merely assumes a course in Euclidean geometry at high school level. While many concepts introduced are advanced, the mathematical techniques are not. Singers lively exposition and off-beat approach will greatly appeal both to students and mathematicians, and the contents of the book can be covered in a one-semester course, perhaps as a sequel to a Euclidean geometry course.
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0136020 Mathematics With Graph Theory
Adopting a user-friendly conversational -- and at times humorous -- style, these authors make the principles and practices of discrete mathematics as much fun as possible while presenting comprehensive, rigorous coverage.
Examples and exercises integrated throughout each chapter serve to pique student interest and bring clarity to even the most complex concepts. Above all, the book is designed to engage today's students in the interesting, applicable facets of modern mathematics
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Themes
Nuzedd
Statistics Quick Reference was designed by a qualified statistics Instructor. Each of the concepts was explained in detail, followed by an example for better understanding.
Please consider purchasing the donate version of the app to support the developers.
The topics include:
1)Basic Terms and...
This app was developed by a qualified Statistics tutor. A must have app for all Statistics students. Almost all the topics that you will need to get through your statistics course are explained in detail and simple manner.
Benefits of PRO version:
In addition to the topic covered in the free...
This is an Ad-free version of Geometry Formulas.
Please consider purchasing the app to support the developers.
Get all the Geometry formulas and concepts on your phone. This app is particularly designed to help students to check out the geometry formulas and concepts, just in few taps. The app...
Get all the Geometry formulas and concepts on your phone. This app is particularly designed to help students to check out the geometry formulas and concepts, just in few taps. The app is particularly designed to consume the least memory and processing capability.
The concepts include:
(A) Basic...
Word Problems Made Simple
a....
Word Problems Made Simple Pro...
This is an Ad-free version of Algebra Cheat Sheet.
Please consider purchasing the app to support the developers.
Algebra Quick Reference Pro provides you with the quick reference of formulas in Algebra.
Topics include:
1. Basic Properties and Facts (includes properties of radicals, exponents,...
Trigonometry Quick reference guide has all the important formulas and concepts covered to help the student review them on regular basis.
Topics include:
a) Definitions - Right Triangle Definition and Unit Triangle definition
b) Facts and properties of trigonometric functions i.e., Domain, Range...
Forget taking down calculus formulas on a paper!
Calculus Quick Reference lists down all the important formulas and evaluation techniques used in calculus which makes it easier for you to memorize and apply them in solving problems.
The topics include:
1) Limits (Basic Properties, Basic Limit...
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1.4 Variables and Equations
Numbers are abstract creatures that we use to represent how many "things" we have. Variables, in turn, can abstractly represent numbers. When there's a relationship between different variables in the form of a pattern, we can develop an equation to represent this relationship.
We started talking about counting numbers, mathematical creatures that you've worked with since you were just a little kid, and then wound talking about irrational numbers, which are numbers that you probably won't run into in the day-to-day physical world. Together, the counting numbers, integers, rational numbers, and irrational numbers, make up all of the real numbers. These are the types of numbers that we'll be using for this course.
Now that we have a sense of our abstract numbers, can we move up a level of abstraction? This is the type of question that the mathematician always asks. By representing objects from the real world as numbers, such as having 2 tables or having 3 chairs, we can use the language of mathematics to answer questions about quantity. If we combine 2 tables with 3 tables, then instead of thinking about the tables, we can first just think about adding the 2 and the 3 to get 5 and then translate back to the specific object, namely 5 tables. Perhaps we'll receive a similar benefit if we can think of numbers as specific objects and develop an abstract representation of the numbers themselves.
To see one real world benefit, let's say you're selling Apps for the iphone and you make 30 cents per App. If you sell a total of 1000 Apps, you'd make $
dollars. If you're a business person, and have to figure out how much you make on a monthly basis, you'd probably want to have some sort of spreadsheet or computer program that will compute the profit for you.
You can let x, known as a variable, represent the number that you sell without specifying a particular number –that's why "x" is more abstract than a number – and then in your spreadsheet program, use .3x to represent your profit; you then tell the program the number that you sell and you let it do the computation for you.
Using variables, we can then create equations that can then be used to model a real world situation. We can think of an equation as a way to describe a process using the language of mathematics. As an example, if you wanted your profit to be $300 and wanted to know how many units you'd need to sell in order to achieve this profit, then you could set up the equation:
300=.3x
In other words, "Find the unknown quantity x, so that .3x is equal to our desired profit of $300". We've translated a real world problem into the mathematical language: now using the rules of algebra, one branch of mathematics, we can solve for the unknown variable. In this case, we can divide both sides by .3 to get that x = 1000.
In this book we'll learn many different techniques that we can use to solveequations. But, before solving an equation, it's super important to understand how to translate the problem from English to mathematics. While there are no hard and fast rules as to how to do this, some key tips are:
Know what it is that you're actually looking for
Start with a "rough" equation – one that contains both English and math
Draw a picture if possible, labeling unknown quantities
Explore!
You're trying to construct a rectangular pen for your dear pig Wilbur. You'd like the length of the pen to be 4 more than the width so that Wilbur has ample room to run around. If x represents the width of the pen and you have 100 feet of wire available to construct the pen, then which equation represents the relationship between the unknown width and the other information given?
From our picture, we see that the two widths are each x while each length is x+4. Adding up the total number of x's (four of them) plus the non-x values (8) gives us the total perimeter, which is 100. This tells us that the equation:
4x+8=100
can be used to describe the relationship. Once we've translated the problem into the language of mathematics, we'll be able to use tools from algebra to solve for the unknown.
Truth be told, the above example probably doesn't have much of a real world application. In fact, to be completely honest, most problems that you'll see in any introductory algebra course don't have much "practical" value. However, these basic concepts are the building blocks of more advanced problem solving techniques that are used in the real world to solve some very, very complicated problems.
As an example, consider an airline looking to maximize their profit. They need to figure out how many planes they should have, the number of routes, what to charge customers, among a host of other variables. Do you think that they're just going to "guess" what number to assign each of the unknown variables in order to maximize their profit? Of course not! They use some very advanced techniques from algebra to figure out all of their unknowns (potentially hundreds), in order to maximize profit.
After working through some problems in this section, which will give you a chance to practice translating some English statements into mathematical ones, you'll begin to notice that one helpful tool in developing equations is being able to visualize what's going on. In the next section, we're going to study the number line, which will connect the concepts of numbers, variables, and representing them both visually.
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Uses full-strength commercial software. Nydick and Liberatore use Lingo (for math programming,) Expert Choice (for decision analysis,) and Extend (for simulation.) These packages provide a more "real world" experience by using software that is designed specifically to solve problems in practice. This in turn gives them a better understanding of how management science techniques can be used. The three software packages are packaged with the text on a CD.
Narrowed topics. This text includes only three major concepts: math programming, decision analysis, and simulation.
Flexible approach for math programming formulations: In these chapters, professors can choose between algebraic formulations and the Lingo modeling language.
Strong author support to help professors teach with the revolutionary approach. Videos available of authors teaching entire course.
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Search Course Communities:
Course Communities
Lesson 10: Extraction of Roots
Course Topic(s):
Developmental Math | Quadratics
This lesson introduces quadratic equations and graphs. Equations of the form (ax^2 + c = 0) are solved via extraction of roots. Later application problems involving volume and surface area and compound interest (problems of the form (a(x - p)^2 = q ) are presented.
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Algebra is related to mathematics, but for historical reasons, the word "algebra" has three meanings as a bare word, depending on the context. The word also constitutes various terms in mathematics, showing...
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Writing an algebraic expression in mathematics involves any combination of variables or letters and numbers. Understand the concept of algebra and its flexible expressions with insight from a math teacher in this video on mathematics.
Expert: Jimmy... Christopher Rokosz[more]
This video clip focuses on rational and irrational Numbers, reciprocals, and the sign of products, quotients, sums and differences. This video clip provides clear and definitive examples for each topic. Throughout the clip, questions are provided to ...help guide the learner through the covered material. (14:45)[more]
This video clip covers Algebra 1 topics dealing with representing functions with tables and graphs. Input and output tables, dependent and independent variables, domain and range, and graphing functions are also discussed in this video. This video cl...ip provides clear and definitive examples for each topic. Throughout the clip, questions are provided to help guide the learner through the covered material. (11
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Authors
With the emphasis the Common Core State Standards (CCSS) places on modeling, Modeling With Mathematics: A Fourth Year Course (Fourth Year 2e) addresses these modeling requirements while exploring how mathematics is ever-present in solving real-world problems. Intended for students fulfilling a fourth year math credit, the Fourth Year 2e program helps solidify students' understanding by providing a different kind of learning experience. With Fourth Year 2e students model real-world applications with a functions approach netting a deeper grasp of the important concepts necessary for success on the forthcoming Common Core assessments.
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Assessment Rules
Curriculum Design: Outline Syllabus
Number theory: division with remainder; highest common factors and the Euclidean algorithm; lowest common multiples; prime numbers; the Fundamental Theorem of Arithmetic and the existence of infinitely many prime numbers; applications of prime factorization.
To introduce students to mathematical proofs; to state and prove fundamental results in number theory; to generalize the notion of congruence to that of an equivalence relation and explain its usefulness; to generalize the notion of a highest common factor from pairs of integers to pairs of real polynomials.
Educational Aims: General: Knowledge, Understanding and Skills
University mathematics has a rather different feel from that encountered at school; the emphasis is placed far more on proving general theorems than on performing calculations. This is because a result which can be applied to many different cases is clearly more powerful than a calculation that deals only with a single specific case.
For this reason we begin by taking a look at the language and structure of mathematical proofs in general, emphasizing how logic can be used to express mathematical arguments in a concise and rigorous manner.
We then apply these ideas to the study of number theory, establishing several fundamental results such as Bezout's Theorem on highest common factors and the Fundamental Theorem of Arithmetic on prime factorizations.
Next, we introduce the concept of congruence of integers. This, on the one hand, gives us a simplified form of integer arithmetic that enables us to answer with ease certain questions which would otherwise seem impossibly difficult; and on the other it leads naturally to the abstract idea of an equivalence relation whichhas applications in many areas of mathematics.
Finally, we show how the idea of a highest common factor can be generalized from the integers to the polynomials.
Set theory: understand and be able to use basic set-theoretic notation.
Logic: understand the use of truth tables, and be able to set them up; be able to express mathematical statements symbolically, using quantifiers and connectives, and to negate them; master the three main methods of proof (direct, contraposition and contradiction); be able to present simple mathematical proofs.
Number theory: be able to perform division with remainder, and prove the underlying theorem; know what is meant by the highest common factor of a pair of integers, and understand how to compute it using the Euclidean algorithm and prime factorization; be able to state and prove Bezout's theorem, and know how to apply it to derive related results; know what the lowest common multiple of a pair of integers is, its relation to the highest common factor, and how to compute it; know what is meant by a prime number, and how to find prime numbers using the Sieve of Eratosthenes; be able to state and prove the Fundamental Theorem of Arithmetic, and to prove that there are infinitely many prime numbers; be familiar with various applications of prime factorization.
Congruences: know what it means that two integers are congruent modulo a given number; be able to solve linear congruences, and understand the proofs of the underlying theorems; be able to state, prove, and apply the Chinese Remainder Theorem.
Relations: know what is meant by a relation, be able to decide whether or not a relation is reflexive, symmetric, or transitive; know what is meant by an equivalence relation; know what is meant by an equivalence class and a congruence class; be able to define the sum and the product of two congruence classes, and show that it is well-defined; be familiar with various applications of the arithmetic of congruence classes; be able to construct the integers from the natural numbers, the rational numbers from the integers, and the complex numbers from the real numbers.
Polynomials: know the division algorithm; know what is meant by the highest common factor of a pair of polynomials, and how to compute it using the Euclidean algorithm.
Learning Outcomes: General: Knowledge, Understanding and Skills
The student will be familiar with some fundamental concepts in logic and elementary number theory, will be able to understand and present simple mathematical proofs, and will know how to apply the theory to calculate highest common factors and solve equations involving congruences.
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Graphing Calculactor Manual For The Ti-83, Plus Ti-84 And The Ti-89 -accompany The Triola Statistics Series - 07 edition
Summary: TI-83/84 Plus and TI-89 Manual is organized to follow the sequence of topics in the text, and it is an easy-to-follow, step-by-step guide on how to use the TI-83/84 Plus and TI-89 graphing calculators. It provides worked-out examples to help students fully understand and use the graphing calculator21.44 +$3.99 s/h
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McGraw-Hill's GED Mathematics
(McGraw-Hill's GED297.431329
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About the Book
Create your own path to GED success with help from McGraw-Hill's GED test series
The newly revised McGraw-Hill's GED test series helps you develop the skills you need to pass all five areas of the GED test.
Presented in a clear, appealing format, these books offer many opportunities for test practice and explain the essential concepts of each subject so you can succeed on every portion of the GED exam. The series covers: Language Arts, Reading * Language Arts, Writing * Mathematics * Science * Social Studies
"McGraw-Hill's GED Mathematics" guides you through the GED preparation process step-by-step. A Pretest helps you find out your strengths and weaknesses so you can create a study plan to fit your needs. The following chapters introduce you to math concepts on which hundreds of GED questions are based. Then check your understanding of these ideas with the Posttest, presented in the GED format. You can then see how ready you are for the big exam by taking the full-length Practice Test.
"McGraw-Hill's GED Mathematics" includes: Clear instructions to show you how to use number grids and coordinate plane grids Instruction and frequent practice with the "Casio fx-260" calculator Problem-solving strategies to help you understand word problems Easy-to-follow lessons to develop essential math skills in whole numbers, decimals, fractions, percents, ratios, data analysis, geometry, and algebra
With "McGraw-Hill's GED Mathematics," you will sharpen your study skills for test success!
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James Stewart, author of the best-selling calculus textbook series, and his coauthors Lothar Redlin and Saleem Watson, wrote "Trigonometry, 2/e, ...Show synopsisJames Stewart, author of the best-selling calculus textbook series, and his coauthors Lothar Redlin and Saleem Watson, wrote "Trigonometry, 2/e, International Edition" to address a problem they frequently saw in their classrooms: Students who attempted to memorize facts and mimic examples-and who were not prepared to "think mathematically." With this text, Stewart, Redlin and Watson help students learn to think mathematically and develop true, lasting problem-solving skills. Patient, clear, and accurate, "Trigonometry, 2/e, International Edition" consistently illustrates how useful and applicable trigonometry is to real life
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Prepare for Pre-Stats/ GMAT/ SAT Stats 101For the things of this world cannot be made known without knowledge of mathematics
 
Course introduces you to basic numerical and statistics skills to get you up-to-date on the basic prerequisites on statistics. Brushing up on basic numerical skills is necessary to succeed in taking statistics for Business, Psychology, Sociology and so on. This course will enable you to to sharpen numerical skills in case you haven't taken a formal mathematics course or a data management course. This course has been especially designed for adult learners.
 
Brush up on your basic Math/ Statistics skills now with an expert
 
Why should you enroll in this course:
You will be able to improve your numerical skills
Course will prepare you with the prerequisites for taking statistics courses (non-calculus) in college and university
Course brushes some important concepts that will be a repeat on most business stats and psych stats 101 undergrad courses in college
What's in the box:
Mixture of PPT and video lectures with relevant references to the worksheets
Email help is also available throughout the course
 
Course outline:
 
Factorials
Combinations
Permutations
Fractions
Percents
Number Properties
Operations
Probability
Statistics
 
About the Instructor
Chirayu K Trivedi Canada
Mr. Sheru is a coordinator for a company called My York Tutor. He holds the degree of BA-Mathematics for Commerce Graduate(Honors) and has 5+ years of tutoring experience at the high school and university level. He has trained more than 1000 students in past 5 years. He has tutored both high school Mathematics, including data management and calculus, and university courses, including business statistics and psychology statistics.
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determine if integrating a unit on functions would benefit students. Previous studies have shown that integrating science and mathematics increases students' understanding of certain topics in science. TypicallyEach year thousands of students are tracked into mathematics classes. In these particular classes, students may struggle or find their mathematics skills less academically able than their classmates and give up on the tasks that are introduced to
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About the book
Mathematics for the International Student: Mathematics HL has been written to reflect the
syllabus for the two-year IB Diploma Mathematics HL course. It is not our intention to define the
course. Teachers are encouraged to use other resources. We have developed the book independently
of the International Baccalaureate Organization (IBO) in consultation with many experienced
teachers of IB Mathematics. The text is not endorsed by the IBO.
This second edition builds on the strengths of the first edition. Many excellent suggestions were
received from teachers around the world and these are reflected in the changes. In some cases
sections have been consolidated to allow for greater efficiency. Changes have also been made in
response to the introduction of a calculator-free examination paper. A large number of questions,
including some to challenge even the best students, have been added. In particular, the final chapter
contains over 200 miscellaneous questions, some of which require the use of a graphics calculator.
These questions have been included to provide more difficult challenges for students and to give
them experience at working with problems that may or may not require the use of a graphics
calculator.
The combination of textbook and interactive Student CD will foster the mathematical development
of students in a stimulating way. Frequent use of the interactive features on the CD is certain to
nurture a much deeper understanding and appreciation of mathematical concepts.
The book contains many problems from the basic to the advanced, to cater for a wide range of
student abilities and interests. While some of the exercises are simply designed to build skills,
every effort has been made to contextualise problems, so that students can see everyday uses and
practical applications of the mathematics they are studying, and appreciate the universality of
mathematics.
Emphasis is placed on the gradual development of concepts with appropriate worked examples, but
we have also provided extension material for those who wish to go beyond the scope of the
syllabus. Some proofs have been included for completeness and interest although they will not be
examined.
For students who may not have a good understanding of the necessary background knowledge for
this course, we have provided printable pages of information, examples, exercises and answers on
the Student CD. To access these pages, simply click on the 'Background knowledge' icons when
running the CD.
It is not our intention that each chapter be worked through in full. Time constraints will not allow
for this. Teachers must select exercises carefully, according to the abilities and prior knowledge of
their students, to make the most efficient use of time and give as thorough coverage of work as
possible.
Investigations throughout the book will add to the discovery aspect of the course and enhance
student understanding and learning. Many Investigations could be developed into portfolio
assignments. Teachers should follow the guidelines for portfolio assignments to ensure they set
acceptable portfolio pieces for their students that meet the requirement criteria for the portfolios.
Review sets appear at the end of each chapter and a suggested order for teaching the two-year
course is given at the end of this Foreword.
The extensive use of graphics calculators and computer packages throughout the book enables
students to realise the importance, application and appropriate use of technology. No single aspect
of technology has been favoured. It is as important that students work with a pen and paper as it is
that they use their calculator or graphics calculator, or use a spreadsheet or graphing package on
computer.
The interactive features of the CD allow immediate access to our own specially designed geometry
packages, graphing packages and more. Teachers are provided with a quick and easy way to
demonstrate concepts, and students can discover for themselves and re-visit when necessary.
Instructions appropriate to each graphic calculator problem are on the CD and can be printed for students.
These instructions are written for Texas Instruments and Casio calculators.
In this changing world of mathematics education, we believe that the contextual approach shown in this
book, with the associated use of technology, will enhance the students' understanding, knowledge and
appreciation of mathematics, and its universal application.
Using the interactive student CD
The interactive CD is ideal for independent study. Frequent use will nurture
a deeper understanding of Mathematics. Students can revisit concepts taught in
class and undertake their own revision and practice. The CD also has the text of
the book, allowing students to leave the textbook at school and keep the CD at
home.
The icon denotes an Interactive Link on the CD. Simply 'click' the
icon to access a range of interactive features:
spreadsheets
video clips
graphing and geometry software
graphics calculator instructions
computer demonstrations and simulations
background knowledge (as printable pages)
For a complete list of all the active links on the Mathematics HL CORE second edition CD,
click here.
For those who want to make sure they have the prerequisite levels of understanding
for this course, printable pages of background informations, examples, exercises and
answers and provided on the CD. Click the 'Background knowledge' icon on
pages 12 and 248.
Graphics calculators: Instructions for using graphics calculators are also given on
the CD and can be printed. Instructions are given for Texas Instruments and Casio calculators.
Click on the relevant icon (TI or C) to access the instructions for the other type of
calculator.
Note on accuracy
Students are reminded that in assessment tasks, including examination papers, unless
otherwise stated in the question, all numerical answers must be given exactly or to
three significant figures.
HL and SL combined classes
HL Options
This is a companion to the Mathematics HL (Core) textbook. It offers coverage
of each of the following options:
Topic 8 – Statistics and probability
Topic 9 – Sets, relations and groups
Topic 10 – Series and differential equations
Topic 11 – Discrete mathematics
In addition, coverage of the Geometry option for students undertaking the IB Diploma
course Further Mathematics is presented on the CD that accompanies the
HL Options book.
Supplementary books
A separated book of WORKED SOLUTIONS give the fully worked solutions for every question
(discussions, investigations and projects excepted) in each chapter of the
Mathematics HL (Core) textbook. The HL (CORE) EXAMINATION PREPARATION & PRACTICE
GUIDE offers additional questions and practice exams to help students prepare for the
Mathematics HL examination. For more information email
info@haesemathematics.com.au.
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Introduction to the PascGalois JE Applets
So what is PascGalois JE? The PascGalois project was started in the late 1990's
as a new and innovative way to visualize concepts in an introductory abstract algebra course,
primarily group theory. Although this is still its main function the project has branched out into
areas of number theory, discreet mathematics, dynamical systems and combinatorics. In 2004, the software was
totally rewritten in Java so that it could run on any operating system, not just Windows. Furthermore,
we revamped the user interface to make the program easier to use and built a rule-based calculation
engine so that the program will support more types of group structures. Since the first release of
PascGalois JE we have revised the user interface several times and have added many new features,
such as three-dimensional viewing of two-dimensional automata, advanced element counting,
generalized update rule input, and probability density graphing of two-dimensional automata.
As with any program, when you add more and more options the user interface gets more complicated
and the ease of use decreases. We have tried to make this program as easy to use a possible by
using a tabbing system.
Although we still use the program for our first course in abstract algebra you may feel that the
PascGalois JE program has become more of an undergraduate research tool instead of a teaching tool.
This is why we have also constructed a series of easier to use applets. The applets restrict
the options that the user has and as the series progresses the applets introduce new options at each
stage. By the end of the series the user has used most of the facilities on the PascGalois JE
program.
So which should you use? The sequence of written labs we have constructed use the
PascGalois JE program and the sequence of web-based labs embed the applets in the lab itself.
So if you want to use the full set of
features from the start you should use this sequence of labs with the PascGalois JE
program. On the other hand, if you are finding the PascGalois JE program too cumbersome to use
you should use our web-based sequence of labs. There are a few things you should note about the applets.
First, each of the applets have corresponding applications that can be downloaded from our
web site. Second, applets can not access the user's hard drive nor can they access the computer
clipboard. This means that if you use the applets you will not be able copy images to a word processor
nor will you be able to save the current settings of the program. So if you are writing up a lab
report and using the PascGalois JE program you will be able to transfer images and information
over to your word processor. On the other hand, if you are using the web-based labs with the applets
the only way to transfer the image to your word processor is to open the the applet in full screen mode
and do a screen capture (Alt+PrintScreen in Windows) and then paste this into your word processor.
So what does it do? It graphs one and two-dimensional cellular automata over finite group
structures. Here is an easy example of a one-dimensional cellular automaton.
Consider Pascal's triangle and its construction using the "adding the two entries above"
rule. That is, put 1's down the diagonals and for each entry inside the triangle add the
two entries above it. You will get the following,
Another way to think about this is to consider the first row (the single 1) as a starting point
(or seed) with 0's going out infinitely in both directions. That is,
... 0 0 0 1 0 0 0 0 0 ...
In the language of cellular automata we call this time-step 0. The next row, or time-step 1,
is taken from the first row by adding each two consecutive entries together, obtaining
... 0 0 0 1 1 0 0 0 0 ...
we do it again for time-step 2,
... 0 0 0 1 2 1 0 0 0 ...
time-step 3,
... 0 0 0 1 3 3 1 0 0 ...
time-step 4,
... 0 0 0 1 4 6 4 1 0 ...
and so on.
Now we will go a little further, we will take each of the entries mod a particular
number n. For example, let n = 3, generate Pascal's triangle and then mod each entry by 3.
You should note that this is exactly the same as if were were to generate Pascal's triangle
using addition mod 3. That is, start with your seed of
... 0 0 0 1 0 0 0 0 0 ...
Now do the addition rule but after each addition take the result mod 3. For time-steps 1 and 2 it makes
no difference we still get
... 0 0 0 1 1 0 0 0 0 ...
and
... 0 0 0 1 2 1 0 0 0 ...
Now for time-step 3 we have,
... 0 0 0 1 0 0 1 0 0 ...
and time-step 4,
... 0 0 0 1 1 0 1 1 0 ...
and so on. The point is that we are still constructing Pascal's triangle we are just using
the group operation of Z3 instead of addition (which is the group operation of Z).
Yes, we are going yet another step further. Numbers are nice but in this form it is a bit
difficult to see patterns. So what we will do now is color each entry of the triangle a color that
corresponds to the group element. If we color 0 red, 1 black and 2 green and graph about 100 rows
or time-steps we get the following image.
Clearly we have some structure here. What you will see in this sequence of labs is that these triangles
can hold far more information about the structure of the group.
The PascGalois Project and the PascGalois JE program derive their names from Pascal and Galois since the
first visualizations were using Pascal's triangle with group theoretic operations (the Galois portion of the
name). The PascGalois JE program is capable of producing automata using many other group structures as well as
other seeds and update rules, it is not restricted to Pascal's triangle.
For this lab we will stick to Zn and Pascal's triangle.
Before going into the software let's get a feel for what it is doing.
Your first assignment is to create some of these triangles by hand.
Exercises:
By hand generate the first 10 rows of Pascal's triangle.
By hand generate the first 10 rows of Pascal's triangle, mod 2.
By hand generate the first 10 rows of Pascal's triangle, mod 3.
By hand generate the first 10 rows of Pascal's triangle, mod 4.
By hand generate the first 10 rows of Pascal's triangle, mod 5.
By hand generate the first 10 rows of Pascal's triangle, mod 6.
For each of the triangles mod 2, 3, 4, 5 and 6, above use different colors
to color in the triangle. We have constructed an
empty triangle to help make this easier.
We would suggest using some type of painting program to fill in the triangles
and then copy anf paste the result into your lab report.
Note that you should only color the triangles that point upward.
An Introduction to the First Applets
To get you started we are going to look at just the first sequence of applets.
The first applets are restricted to graphing one-dimemsional automata using
the Pascal's triangle update rule over only one class of groups. So they
will create images like the one above but for a specific group. For example,
the applet to the right will graph the one-dimemsional automata over
Zn. There are other applets that will use Dn,
Sn, .... The applets in this series have facilities for color
scheme alteration, zooming and element counting. When we embed an applet into
the labs we will usually have links to a full screen version and the help
system for the applet. The full screen versions will open the applet up in another
window that is resizeable so you can enlarge the viewing area. Furthermore, you
may wish to use the full screen mode to copy and paste images from the applets into
your lab reports. You can also click the full screen link several times to open
several of these applets at once.
Some basics about triangle generation
We are going to start out with Pascalís triangle modulo 2, 3, 4, 5 and 6.
Since this applet graphs Zn we need to input the
value of 2 for n.
At the top of the applet you will see an "n =" box simply input a 2 into this
box. (2 is the default entry so you will probably not need to do anything here)
We also need to give the program seed values. This series of applets uses a
default of two seeds, mainly because most of the groups encountered in an introductory
abstract algebra course have either one or two generators. We can force the
program to use only one seed by leaving one of the seed entries blank. So
to start with a seed of 1 (like our examples above) we would put a 1 in one of the
seed boxes and leave the other box blank. Change the seed entries so that there is a
1 in one box and nothing in the other.
The number of rows tells the program how many rows (or time-steps) of the
automaton should be generated after the seed row. So if this is set
to 100 the image will actually contain 101 rows. We will leave this entry
at 100.
The creation and graphing of one of these triangles can be a little time
consuming if there are a large number of rows to create, so when we
wrote the software we did not have the automaton regraph every thime a change
was made. So to create or update an image you must click the Refresh/Apply
tool button in the upper right corner of the applet. Try this. Notice that the image
appears in the box on the left and the color correspondence in the
box on the right. Also note that the division bar between these two boxes
is movable.
Change the number of rows to 9 and regraph the image. Compare this
image with your mod 2 Pascalís triangle.
Move your mouse over the image and notice what appears in the status bar
at the bottom. The program tells you what location the element is in as
well as what the element is. Furthermore, if you hover over a location for a
second or two a tool tip will appear with the same information.
Change n to 3 and regraph the image. Compare this
image with your mod 3 Pascalís triangle. Do the same with n set to 4, 5
and 6.
Now set the number of rows to 200 and regraph the images with n set to
2, 3, 4, 5 and 6. Do you notice any patterns? If so, how would you describe them?
Some basics about colorings
Often our choice of coloring affects the type and/or amount of structure we observe in Pascalís and other
related triangles. We will see that altering the colors often reveals hidden structure in the images.
When you generate a triangle the program will use the default color settings and color each element a
unique color (up to 60 elements and then it rotates the color scheme). The program allows you to change
the color of any element as well as group sets of elements together with the same color. We will look at a few
examples below. There is also a feature where you can drag and drop colors from one window to another.
Unfortunately, with the applets you will not be able to save your color schemes
since applets do not have access to the user's hard drive. The PascGalois JE
application does have the ability to save and load the color schemes that you create.
Generate Pascalís triangle modulo six, keep the number of rows at the default 100.
Just to see how to change the color of an individual color do the following.
Double-click the element 3 either on the image or the element list.
At this point the color chooser dialog box will appear.
Select a purple color and click OK.
You will notice that the color has changed in the color correspondence
box. Now click the Refresh/Apply toolbar button. Notice what happened to the triangle
image.
To reset the colors to the default scheme select the Colors > Reset to Default
Color Scheme in the menu.
Now we will highlight colors. Select both 3 and 5 from the color correspondence.
To select multiple items simply hold down the Control key and click all the
items of interest. Now select Colors > Highlight Elements from the menu.
You will notice that the elements that were selected are now
colored red and the other elements are black. Letís change these colors
before refreshing the image.
Double-click either the 3 or the 5 and select the color yellow.
Now double-click any of the black colors
and select a gray color. Notice that all of the colors in the respective groups
changed when you made
a single change. This is because the elements are now linked together.
The 3 and 5 are considered a
set and the 0, 1, 2 and 4 are a set. Refresh the image.
To ungroup the colors select Colors > Reset to Default Color Scheme from
the menu and refresh the image.
We will use another type of color grouping, subset grouping. In the
color correspondence window select
the numbers 0 and 3. Select Colors > Group Elements from the menu.
Note that 0 and 3 are now the
same color. Now select 1 and 4 and then select the Colors toolbar
button followed by Group Elements.
Finally select 2 and 5 followed by the Colors toolbar button and
then Group Elements. Refresh the graph.
Reset the colors to the default scheme and refresh the image.
You can also use the PascGalois triangle itself to select colors.
Put the mouse over a section of red
(the element 0) and double click. The color selector will appear for the element 0.
Select a different
color and click OK. Note that the new color is in the color correspondence box.
Refresh the image to see the new triangle.
Another way to change an elementís color is to right-click on the element
either in the color correspondence
box or on the image and a small popup menu will appear with
two options, Set Element Color
and Set Element Color to Transparent. If you select the transparent
option the color box will simply
be a rectangle with an X through it. Refresh the image to see the change.
You can also change the
background color by selecting Colors > Set Background Color.
A few other things to note about color changes. There are several other
grouping options under the
Colors menu, we will use these in later labs. Also, there are options
to undo and redo color scheme
changes. The program will keep up to 20 changes for each color scheme.
There are also facilities to
add, remove and rename color schemes. If you do add color schemes
you can select the different color
schemes using the drop-down selector over the color
scheme window.
Some basics about zooming
If you donít have Pascalís triangle modulo six on the screen please
regenerate it, keep the number of rows at the default 100.
Select Zoom > Zoom In from the menu. Notice that the mouse pointer has
changed when you are
in the triangle window. Click somewhere inside the triangle.
Click several more times to see what happens.
Select Zoom > Zoom Out from the menu. Notice that the mouse pointer has
changed again. Click
somewhere inside the triangle. Click several more times to see what happens.
Select Zoom > Reset Zoom to Full View from the menu. Notice that the mouse
pointer has not changed but the triangle has zoomed out to its fullest.
Select Zoom > Turn Off Zoom from the menu. Notice that the mouse pointer
has changed back to its default.
The default zoom is 2X. This can be changed by selecting Zoom > Zoom Factor
from the menu followed by the desired zoom factor.
Select Zoom > Zoom Box from the menu.
Click and drag over a portion of the triangle. Note that the area will be
shaded. Release the mouse
button and the program will zoom in on the selected portion.
The program may need to alter the
bounds of the selection but you will get at least what you selected.
If you are in the process of zooming with the Zoom Box feature and
notice that your area is not what
you want you can cancel the zoom by pressing the right mouse button before
releasing the left button. Give this a try.
Reset the zoom to full view and turn the zooming off.
There are also facilities to undo and redo zooms. The program will keep
a maximum of 50 zooms in memory.
Some basics about element counting
These applets also have a feature for counting the number of elements
within a given selected region. This option was put in mainly for fractal
dimension explorations in a dynamical systems course. We will not be using
this feature in our sequence of abstract algebra labs so we will not
go into this feature. If you are interested in fractals and fractal dimension
please read the section on counts in the help system.
Other applets in this first series
The applet above was for generating triangles over Zn.
We also have applets in this first seires to generate applets over
Un (the integers under multiplication mod n),
Zn X Zm (the Cartesian product of
Zn and Zm both under addition)
Dn (symmetry group for a regular n-gon),
Q (the Quaternions),
Sn (group of permutations on n letters),
Qn (generalized Quaternion groups) and
Cn (the dicyclic group).
These applets have the same features as the Zn applet,
the only difference is in the notation for the seed values. For a
detailed description of the element notation for these applets please see
the help system for the applet.
An Introduction to the Other Applets
As we pointed out in the introduction there are a sequence of applets
that keep adding more and more features. As the labs progress we will
be using these applets. We have a general applet page on this site that lists all of
the applets but we will give a quick description of the features of them here.
The Single Group Viewers. These are the ones in this first series. They are restricted
to a single class of groups and the Pascal's triangle update rule. They have options
for zooming, color manipulation and element counting.
Viewers with Group and Seed Options. There are two applets here one the uses
two seed values and one that uses a seed table. They both add a group
selection facility so that you can select the class of group you wish to work with.
Hence you do not need two seprate applets to work with two different groups.
The group selection also adds two types of groups not found in the first series.
One is a user defined structure where the user can input their own
group structure using an operation table. Actually, the operation table does not
even need to define a group. The other is an advanced group input which
allows the user to input any Cartesian product or quotient group that can be derived
from the built-in group structures or the user input structures. The applet
that uses two seed values is just like the first series and the seed table applet
allows the user to input more than one or two seed values. The update rule for these
applets is still the Pascal's triangle update rule and they have all of the other
options that the first series has.
Viewers with Group, Seed and Update Rule Options. There are two applets
in this group as well. The first has all of the options of the seed table applet
above but adds the ability to graph finite automata as well. The second adds on to
this by allowing the user to change the update rule.
Viewers with Full Options. There are three applets in this group. These have
almost all of the options that the PascGalois JE application has to offer. Some things
that PascGalois JE will do but these applets will not are three-dimensional
viewing of two-dimensional automata and advanced counting of sub-triangles and
sub-pyramids. The three applets in this group are the 1-D Automaton Viewer, the
2-D Automaton Viewer and the 1-D and 2-D Automaton Viewer. The 1-D Automaton Viewer
graphs only one-dimensional automata (like all of the previous applets) but it adds
an advanced counting, period and death calculation facility as well as a group
calculator. The 2-D Automaton Viewer is for the creation of two-dimensional
automata over finite groups. It has many of the same features as the
1-D Automaton Viewer but will graph levels (or time-steps) of two-dimensional
automata.
Superimposer. This applet is a specilaized applet that is used in
one of the Abstract Algebra labs.
Group Calculator. This applet is simply a group operation calculator
with the added features of subgroup generation and coset generation.
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Mathematics for the Liberal Arts Student
Fred Richman, Carol L. Walker,
Robert J. Wisner, James W. Brewer
This text is for a one or two semester terminal course in
mathematics. Such a course allows for leisurely exploration in place of
drill---it is a course in mathematics appreciation. For this audience one
must constantly keep in mind the Hippocratic admonition, "First, do no
harm." The authors believe that the spirit of mathematics can be
communicated by means of simple ideas and problems without scaring or boring
the students.
The history of the subject is integrated into the text. We maintain an
historical consciousness throughout, but no attempt is made to offer a
comprehensive history of mathematics, nor to include complete biographies of
mathematicians. We introduce major mathematicians when their stories relate
to the material at hand. Notes at the end of each chapter provide another
opportunity to put a human face on mathematics.
Also included are short encounters, one page treatments of various topics.
Conversion between Celsius and Fahrenheit. Ladders for approximating the
square roots of 2 and 3, etc. Collapsing compasses. The four color problem.
We try to make the chapters as independent as possible, so that instructors
can choose whatever pleases them. If the teacher doesn't like the material,
the students certainly will not. We also strive to avoid superfluous
generality, preferring illustrations to formulas. It is sometimes difficult
for a mathematician to realize that an idea, depending on a parameter, may
be understood in general by examining it for 5, yet be totally opaque when
stated in terms of n. As John Stuart Mill said, "Not only may we reason
from particulars to particulars without passing through generals, but we
perpetually do so reason."
The typical college freshman will have all the prerequisites for the book.
Nevertheless, an appendix is included for review of basic arithmetic
concepts and notations for those students who are a bit rusty on these
points.
The text meets the guidelines for two semesters of liberal arts mathematics
established for the State of Florida's universities and community colleges.
It has been used for both semesters at Florida Atlantic University for the
past two years, and at New Mexico State University for the first semester
for the past year.
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Welcome to the Benet Academy Mathematics Department. You will find the list of courses offered by our department in the curriculum guide.
The goals of the Mathematics Department are organized around the unique Benet Academy philosophy of being college preparatory, Catholic, and Benedictine. With this philosophy in mind, the department goals are:
1.) to prepare students for admission to, and for success in, institutions of higher learning;
2.) to prepare students for everyday life by teaching the skills necessary for problem solving and decision making in a mathematical setting;
3.) to provide students with an appreciation for, and ability to use, electronic media in a technological world;
4.) to help students realize the value of intellectual honesty and other characteristics of good citizenship.
The mathematics and computer science curricula are designed to accommodate and benefit all Benet Academy students. Students whose interests and strengths lie in areas other than mathematics, as well as those who wish to pursue a mathematics related college program or career will find a mathematics course of study that meets their needs and abilities. The department utilizes proficiency exams as well as ongoing teacher evaluations to ensure appropriate student placement in the curricula. Classes in the mathematics department range from first year algebra to college level mathematics and computer science courses where college credit may be earned.
In addition, for students with an extracurricular interest in mathematics, the math team participates in academic competitions throughout the school year. We are proud that the math team's "season" has consistently culminated in successful participation in the ICTM state competition.
If you have any questions about the mathematics or computer science curricula, please contact me.
Curriculum Guide
The new Benet Academy curriculum guide contains general academic information and policies, course descriptions, course planning guide, graduation requirements, test information, and more. The guide contains a search feature so you may easily find the information you are looking for by typing keywords into the "Find" box located in your Adobe Reader. You will need the free Adobe Reader to view the guide. Copy and paste this link in your browser to download the Reader:
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Eigenvalues, Eigenvectors, and Differential EquationsWilliam Cherry April 2009The concepts of eigenvalue and eigenvector occur throughout advanced mathematics. They are often introduced in an introductory linear algebra class, and when introduced there
Math 2700 (Cherry) Homework for the Week of January 2327Reading. Read sections 1.11.4 of your textbook. It is extremely important that you develop condence in rowoperation calculations. This will be important over the entire course. Also, the terminolog
Math 2700 (Cherry) Homework for the Week of January 30 February 3Quiz Monday, January 30. We will begin class on Monday, January 30 with a quiz covering sections 1.1 and 1.2.Reading. Read sections 1.51.7 of your textbook.Web HomeworkYou have three web
Math 2700 (Cherry) Homework for the Week of February 610Quiz Monday, February 6. We will begin class on Monday, February 6 with a quiz covering sections 1.3, 1.4 and 1.5Reading. 1.71.9 of your textbook.Web HomeworkYou have two web assignments this wee
Math 2700 (Cherry) Homework for the Week of February 1317 2024 27 March 2Reading. Finish reading Chapter 2. We will skip section 2.7, at least for now.Quiz. We will have a quiz on Friday, March 2 on matrix inversion (nding the inverse of a matrix).Web HomeworkY
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Guidelines for Working Together with Academic Integrity
It is not good that students should be alone
Though certain assignments are meant specifically to be done in a group,
you are encouraged to work with others on out-of-class
assignments unless otherwise indicated. In fact, I strongly encourage
you to find 2-3 other students and agree to meet at regularly-scheduled
times each week to study together. Some reasons for doing this
(some of these may apply to you more than others, depending upon
your innate mathematical ability):
If you view math as a set of skills to learn, then you are
embarking on a very difficult venture indeed when you set down
to learn in a detached way the myriad of skills that go with
a specific course. As it happens, these skills are generally
based upon a relatively small collection of ideas/concepts that,
if mastered, make the skills much easier to apprehend. You
will find it easier to master these fundamental ideas if you
work at learning the vocabulary — using the mathematical terms
specific to the course with classmates. You should talk about
the concepts, identifying carefully what they are and what they
are not. (For instance, a function is not the same thing
as a graph.) No section is likely to emphasize more than two
concepts, and often these are repeats (concepts that have
appeared earlier).
If you spend time in demonstrations to others in your group
of your solutions to various problems,
you will find frequently that another has solved
a problem substantially differently than you.
Real learning begins when you grapple
together over why different methods lead to the same or different
answers.
You may find that another group member can solve a problem you've
tried but has left you stymied.
Even if you tend to get all of the problems, there is value in
explaining things to others. Every teacher you know will tell
you that they thought they knew a subject well before teaching
it, but gained new insights/understanding as they taught it.
Studying together provides opportunity for Christian interaction.
We can encourage one another, bear one another's
burdens/frustrations, and share our gifts (and griefs) with others.
One day you will seek employment. Companies are looking for people
who, while individually competent, have good teamwork experiences
and skills. Now is the time to develop those skills.
In the final analysis, each student is responsible for his or her
own education
Having said all of this, remember that you are individually accountable
for your learning. (Exams are not group efforts.) The end result must
be that you are able to discuss (usually in writing) the
mathematics necessary to the course. Mathematics is not a spectator
sport! Work on a problem by yourself before seeking help, identifying
specifically the place you get stuck. When you get help, ask for the
least amount of information necessary to get you going again.
Don't work with someone who can't resist telling you the entire solution
everytime you ask a simle question.
Work that is turned in for a grade (except work explicitly designated as
group work) should be written up independently. If you do this
in proximity of study partners resist the temptation to
"copy" or to lean too heavily on someone else
(neither of which is not allowed), mistakenly thinking that you would
have been able to do the work on your own.
You may borrow someone's idea for solving a problem,
but the words must be yours. All assignments (except for projects
specifically assigned in groups) are to be written up separately on
your own.
Give as much attention to presenting your solutions in
a coherent manner (using mathematical symbols as part of your
sentence structure) as you give to actually solving the problem,
since
Writing up a solution is an important step in learning the material
well.
Written work is the main source of your grade, and practice
makes perfect. Concientious work on daily assignments will
lead to quality work on tests.
Choose Wisely
If you choose to work with a group, try to pick a group that is
Fun.
Pick some people with whom you enjoy studying. There is no reason
that studying can't be fun. (But don't choose someone with whom
you have too much fun INSTEAD of studying.)
Balanced.
Do not work in a group that is too unbalanced.
Certainly there are times when one person in a group has the key insight
that solves a problem, and it is also true that different people have
different strengths and weaknesses.
Bbut if the same person is always having all the
insights, or if you are never contributing
to the group effort, then you should probably seek out a different group.
Balance also extends to preparation. Do not get answers or hints from
someone who has completed the work before you have even begun. Ideally,
each member of the group should come with roughly the same amount of
background work done.
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Next: Lesson 1
Previous: Chapter Review Exercises
Chapter 4: Inverse Functions and Trigonometric Equations
Chapter Outline
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Chapter Summary
Description
In this chapter, students will relate their knowledge of inverse functions to trigonometric functions, and will apply the domain, range and quadrants of the six inverse trigonometric functions to evaluate expressions.
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Fundamentals of Algebraic Modeling : An Introduction to Mathematical Modeling with Algebra andimmons, Johnson, and McCook provide an applied text for intermediate algebra students not continuing in math and science. Students are encouraged to develop and test mathematical models in a variety of real-world applications such as personal finance and home decorating. The use of technology is introduced, although not required, through end-of-chapter lab exercises.
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Created for the independent, homeschooling student, Teaching Textbooks has helped thousands of high schoolers gain a firm foundation in upper-level math without constant parental or teacher involvement.
Extraordinarily clear illustrations, examples, and graphs have a non-threatening, hand-drawn look, and engaging real life questions make learning pre-algebra practical and applicable. Textbook examples are clear while the audiovisual support includes lecture, practice and solution CDs for every chapter, homework, and test problem. The review-method structure helps students build problem solving skills as they practice core concepts and rote techniques.
Teaching Textbooks' new Pre-Algebra Version 2.0 edition now includes automated grading! Students watch the lesson on the computer, work a problem in the consumable workbook, and type their answer into the computer; the computer will then grade the problem. If students choose to view the solution, they can see a step-by-step audiovisual solution.
Teaching Textbooks Pre Algebra 2.0 includes the following new features:
Automated grading
A digital gradebook that can manage multiple student accounts and be easily edited by a parent.
Over a dozen more lessons and hundreds of new problems and solutions
Interactive lectures
Hints and second chance options for many problems
Animated buddies to cheer the student on
Reference numbers for each problem so students and parents can see where a problem was first introduced
What exactly comes in this kit?
What is included in this kit? And what is different about this new version vs. the old one? Is it compatible to the older version?
asked 1 year, 10 months ago
by
Anonymous
Dallas, GA
on Teaching Textbooks Pre-Algebra Kit, Version 2.0
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0out of0found this question helpful.
1 answer
Answers
answer 1
This kit will include the Student Textbook, Tests and Answer Keys and the CD-ROM set. The 2.0 version contains additional teaching and lesson materials with interactive CD-ROMs. It is not compatible with the older edition.
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Note to students
Do your older friends ever tell you they wish they had worked harder when they were in the classes that you are in now? Do you have friends in college who say they are struggling because they weren't challenged enough by the adults around them? It is hard sometimes to stay focused in school, when there are other things that seem more interesting and relevant than what you are learning in class today.
It may surprise you to know that the national organization Achieve ( reports that, when students are surveyed two years after high-school graduation, 75% express the wish that they had worked harder and had been challenged more by the adults around them. Amazingly, more than 60% say they wish they had taken more challenging mathematics courses in high school.
You are at a time in your life when you can take advantage of opportunities that those surveyed students did not. One way to do that is by challenging yourself to take lots of math. Taking rigorous mathematics courses can be a key to success in just about any career you might want to pursue when you leave school. The work isn't done only in your junior or senior year of high school, either. As you explore the resources on this site, notice how important ideas of middle school mathematics lay the foundation for your future success in Algebra I, Geometry, Algebra II, Precalculus, Calculus, and Statistics.
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Welcome to the Mathematics 9 and 10 instructional video series from Frances
Kelsey Secondary School. To use these videos, just look at the list of topics
below and click on the title that you would like to see.
If you are completely new to solving equations, or would like a review, then
click on the videos in the order that they are listed below. Each video assumes
that you know the strategies explained in the videos listed above it.
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Differential Equations, Student Solutions Manual: An Introduction to Modern Methods and Applications
Differential Equations: An Introduction to Modern Methods and Applications is a textbook designed for a first course in differential equations commonly taken by undergraduates majoring in engineering or science. It emphasizes a systems approach to the subject and integrates the use of modern computing technology in the context of contemporary applications from engineering and science. Section exercises throughout the text are designed to give students hands-on experience in modeling, analysis, and computer experimentation. Optional projects at the end of each chapter provide additional opportunitites for students to explore the role played by differential equations in scientific and engineering problems of a more serious nature.
Ott and Longnecker's AN INTRODUCTION TO STATISTICAL METHODS AND DATA ANALYSIS, Sixth Edition, provides a broad overview of statistical methods for advanced undergraduate and graduate students from a ...
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Synopses & Reviews
Publisher Comments:
If you are looking for a quick nuts-and-bolts overview, turn to Schaum's Easy Outlines
Schaum's Easy Outline of College Mathematics is a pared-down, simplified, and tightly focused review of the topic. With an emphasis on clarity and brevity, it features a streamlined and updated format and the absolute essence of the subject, presented in a concise and readily understandable form. Graphic elements such as sidebars, reader-alert icons, and boxed highlights stress selected points from the text, illuminate keys to learning, and give you quick pointers to the essentials. Expert tips for mastering college mathematics Last-minute essentials to pass the course Easily understood review of college mathematics Supports all the major textbooks for college mathematics courses Appropriate for the following courses: Introduction to Mathematical Modeling, Pre-Calculus, Discrete Mathematics, Trigonometry, College Algebra, Calculus
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Polynomials, functions, and trig, oh my! From the author of two bestselling Complete Idiot's Guides ' comes a book aimed at high school and college students who need course help or a brush-up. It follows a standard precalculus curriculum, includes sample problems, and will help students make sense of their textbooks. Difficult topics, such as quadratic equations, logarithms, graphing trig functions, and matrix operations are presented ...
Your step-by-step solution to mastering precalculus Understanding precalculus often opens the door to learning more advanced and practical math subjects, and can also help satisfy college requisites. Precalculus Demystified , Second Edition, is your key to mastering this sometimes tricky subject. This self-teaching guide presents general precalculus concepts first, so you'll ease into the basics. You'll gradually master functions, graphs of ...
Get the confidence and math skills you need to get started with calculus Are you preparing for calculus? This hands-on workbook helps you master basic pre-calculus concepts and practice the types of problems you'll encounter in the course. You'll get hundreds of valuable exercises, problem-solving shortcuts, plenty of workspace, and step-by-step solutions to every problem. You'll also memorize the most frequently used equations, see how to ...
Over the last thirty years, many influential church leaders and church planters in America have adopted various models for reaching unchurched people. An 'attractional' model will seek to attract people to a local church. Younger leaders may advocate a more 'missional' approach, in which believers live and work among unchurched people and intentionally seek to serve like Christ. While each of these approaches have merit, something is still ...
The fun and easy way to learn pre-calculus Getting ready for calculus but still feel a bit confused? Have no fear. Pre-Calculus For Dummies is an un-intimidating, hands-on guide that walks you through all the essential topics, from absolute value and quadratic equations to logarithms and exponential functions to trig identities and matrix operations. With this guide's help you'll quickly and painlessly get a handle on all of the concepts — ...
Besides being an important area of math for everyday use, algebra is a passport to studying subjects like calculus, trigonometry, number theory, and geometry, just to name a few. To understand algebra is to possess the power to grow your skills and knowledge so you can ace your courses and possibly pursue further study in math. Algebra II For Dummies is the fun and easy way to get a handle on this subject and solve even the trickiest algebra ...
This edition includes the most recent Algebra 2/Trigonometry Regents tests through June 2012. These ever popular guides contain study tips, test-taking strategies, score analysis charts, and other valuable features. They are an ideal source of practice and test preparation. The detailed answer explanations make each exam a practical learning experience. In addition to practice exams that reflect the standard Regents format, this book reviewsA no-nonsense, practical guide to help you improve your algebra II skills with solid instruction and plenty of practice, practice, practice Practice Makes Perfect: Algebra II presents thorough coverage of skills, such as handling decimals and fractions, functions, and linear and quadratic equations, as well as an introducing you to probability and trigonometry. Inside you will find the help you need for boosting your skills, preparing for ...
When the numbers just don't add up… Following in the footsteps of the successful The Humongous Books of Calculus Problems , bestselling author Michael Kelley has taken a typical algebra workbook, and made notes in the margins, adding missing steps and simplifying concepts and solutions. Students will learn how to interpret and solve problems as they are typically presented in algebra courses—and become prepared to solve those problems ...
This edition includes the most recent Integrated Algebra The book reviews all pertinent math topics, including sets, algebraic language, linear ...
From radical problems to rational functions -- solve equations with ease Do you have a grasp of Algebra II terms and concepts, but can't seem to work your way through problems? No fear -- this hands-on guide focuses on helping you solve the many types of Algebra II problems in an easy, step-by-step manner. With just enough refresher explanations before each set of problems, you'll sharpen your skills and improve your performance. You'll see ...
Tough Test Questions? Missed Lectures? Not Enough Time? Fortunately, there's Schaum's ...
Stewart ...
People are hungry to make a difference in their community, yet most don't know where to start. In fact, 'serving the least' is often one of the most neglected biblical mandates in the church. Barefoot Church shows readers how today's church can be a catalyst for individual, collective, and social renewal in any context. Whether pastors or laypeople, readers will discover practical ideas that end up being as much about the Gospel and personal ...
*Immoderate Greatness* explains how a civilization's very magnitude conspires against it to cause downfall. Civilizations are hard-wired for self-destruction. They travel an arc from initial success to terminal decay and ultimate collapse due to intrinsic, inescapable biophysical limits combined with an inexorable trend toward moral decay and practical failure. Because our own civilization is global, its collapse will also be global, as well ... ...
Predictive Analytics: Microsoft® Excel Excel predictive analytics for serious data crunchers! The Microsoft Excel MVP Conrad Carlberg shows ...
Simply put, the revised, 11th edition of Statistics for Business and Economics is powerful. The authors bring more than twenty-five years of unmatched experience to this text, along with sound statistical methodology, a proven problem-scenario approach, and meaningful applications that clearly demonstrate how statistical information informs decisions in the business world. Thoroughly updated, the text's more than 350 real business examples, ...
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Math and Dosage Calculations for Medical Careers teaches the skills and techniques medical assistants,
nurses, pharmacy technicians, and paramedics need to calculate the amount of medication they should administer to patients. Students learn to calculate dosages based on ratio proportions, fraction proportions, the formula method, and dimensional analysis.
1 Fractions and Decimals
2 Percents, Ratios, and Proportions
3 Systems of Weights and Measures
4 Equipment for Dosage Measurement
5 Drug Orders
6 Drug Labels and Package Inserts
7 Methods of Dosage Calculations
8 Oral Dosages
9 Parenteral Dosages
10 Intravenous Dosages
11 Calculations for Special Populations
12 Specialized Calculations
Appendices
A. Comprehensive Evaluation
B. Answer Key
Glossary
Credits
Index
Reference Cards
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Purposes
The first main goal of this course is to
connect the
mathematics you have learned (and some you haven't) with the history
you have learned (and some you haven't). The second main goal
is
to connect the mathematics you have learned together.
Grading
Your grade in this course will be based
on four
main components. One-fifth each will be determined by daily
reading reactions and by a midterm essay exam. Three-tenths
each
will be determined by a research paper in history and mathematics and
by a final essay exam. More components may be integrated into
this evaluation. Any alternate grading proposal must be
discussed
before and submitted by January 29 (the second Friday of the course).
Reading
This book is much more a history book
than a mathematics
book. It reads like "what mathematics was going on during all
the
history I learned about in humanities?" The book begins at
the
dawn of human mathematics and runs through the second world
war.
You have reading assignments for each class day, roughly ranging from 7 to 15 pages. Our entire
reading will come from our book, and we will complete the entire book
by the end of the course. Each day (with three exceptions -
the
first day, midterm day, and the final day) you are required to email
reading reactions before class. These reading reactions must
include
reactions to at least five topics in the reading. They must
be
written in intelligible English. Each one will be evaluated
out
of 5 points, with points deducted for fewer than five points being
addressed. They must be time-stamped by 9:00a that
day.
They must be in the body of the email - no attachments. I
take no
responsibility for email that gets spam filtered. To help
avoid
this - be sure to include a subject beginning with "390". I also will do no
detective
work to determine who gets credit for them - sign your
submissions. Finally, you will only receive credit for your
reading reactions for that day if you are in class that day.
By
getting your reactions an hour before class, I hope to incorporate some
of them into the class session that day. With 45 of you, I
cannot
promise to do this for everyone.
Exams
There will be a midterm and a final
exam. Both
will be essay exams and involve analysis of the mathematical and
historical content of our investigation. Both will be written
in
class using laptop computers provided by you. I will accept you not
using a
computer for the exam, but I expect your work will be far lesser
quality by doing so - less written, no editing capability, illegible
&c, and I will not compensate in grading. All exams
will be
graded to the same standard. The final will
naturally be more lengthy. Both will include a variety of
questions and allow for some choice of which questions to
answer.
More details will be provided as we approach the exams.
Research Paper
You will write a 1200-2000 word research paper
on a topic in the history of mathemaitcs. Papers will be graded
in three main
aspects:
writing, historical content, and mathematical content.
Stories
about mathematicians will not suffice as mathematical content, and a
date and name will not suffice for historical
content. I
will assign a signed letter evaluation in each of these three aspects
and then average them. The final paper will be a
substantial research paper on a topic not covered in
class.
Selecting the topic by the deadline will be worth 5%, the annotated
bibliography will be worth 20%, the draft will be worth 30%, and the
final paper will be worth 45%. For those
seeking a
teaching certification, the topic must be from the NCTM/NYS standards
at
their level of anticipated certification. For those who are
not,
the topic must be from a post-secondary class they have already (or are
currently) taken. The topic should be a topic
of no
more than a week at either level (one point on NYS standards would be
typical). Due dates are indicated in the schedule below.
For
those wishing to satisfy oral research requirement, may present
research paper during a GREAT day (The most important are 4, 14, 16, 22, 28, 10, 12, 24, 27) math. history session.
Other components
More aspects for evaluation may include,
but are not
limited to problem sets related to the material, connections to
curriculum (at pre-K - 16 levels), paper on mathematical notation.
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. Of
course, you are always welcome to approach me outside of class to
discuss these issues as well.
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
miss class
because of observance of religious holidays the opportunity to make up
missed work. You are responsible for notifying me no later
than February 1 of plans to observe the holiday.
Schedule
January 20 Course Introduction
January 22 1.1 / 1.2
January 25 1.3
January 27 -2.1.2
January 29 2.1.3-
February 1 -2.2.2
February 3 2.2.3-
February 5 -3.1.2 Research Project
Topic Due
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GraspMath Learning Systems Algebra 1 (16 ) DVD Series
Please Note: Pricing and availability are subject to change without notice.
This Algebra 1 DVD tutor series was made to give students clear, concise explanations of topics in a manner that the student may enjoy while acquiring the skills necessary to be successful in current and future mathematical courses. Throughout each tape segment, pertinent definitions, theorems, and steps are shown and clearly explained. To reinforce these mathematical ideas, they are continually referenced and reviewed as examples are worked. Careful attention has been given to the NCTM Standards.
An ample number and variety of examples are solved on each tape segment by the lecturer. Each example solved is thoroughly explained step by step for the student. The examples on a single tape are graded in difficulty. Students are first exposed to examples that increase their confidence and problem-solving skills; then they are eased gradually through more difficult problems.
Algebra I consists of 89 video segments. Each segment is approximately 15 minutes in length. Topics iinclude algebraic expressions, exponents, real numbers, solving equations and inequalities and applications, polynomials, factoring, rational expressions, graphing linear equations and inequalities in two variables, relations and functions, solving systems of linear equations, radicals, and solving quadratic equations.
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The Practice of Statistics long stands as the only high school statistics textbook that directly reflects the College Board course description for AP* Statistics. Combining the data analysis approach with the power of technology, innovative pedagogy, and a number of new features, the fourth edition will provide students with the most effective text for learning statistics and succeeding on the AP* Exam.
This program provides keystrokes and screenshots of all TI Technology. In addition, activities, designed to illustrate important statistical concepts using the TI-Nspire, are available on their companion website. Click for additional information.
Statistics Through Applications, Second Edition (STA 2e) is the only text written by high school teachers for high school students taking an on-level (non-AP*) statistics class. Designed to be read, this book takes a data analysis approach that emphasizes conceptual understanding over computation, while recognizing that some computation is necessary. The focus is on the statistical thinking behind data gathering and interpretation.
The high school statistics course is often the first applied math course students take. STA engages students in learning how statisticians contribute to our understanding of the world and helps students to become more discerning consumers of the statistics they encounter in ads, economic reports, political campaigns, and elsewhere.
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If you rely heavily on a basic calculator when working with numbers, it's important to be aware of what calculators actually do.
A basic calculator ("dumb calculator") will only do what you tell it to do, whether right or wrong. Any calculator (including a "programmable" calculator) is nothing more than a machine which completes its functions automatically - like a small robot.
Think of it in terms of this simple analogy:
If you use a light switch to turn on the lights in a building, the lights come on when the switch is turned to "on". If you turn the switch to "off", the lights go off. It always works that way.
Now suppose the building is full of explosive gas because of a gas leak. When the light switch is turned on, the building will explode.
The simple switch mechanism will not protect you from a possible explosion. The switch goes on when you turn it on, no matter what.
You could call the light switch a "dumb light switch" because it just does what you tell it to do - nothing more or less.
That is whybasiccalculators are called "dumb calculators".
Unless a calculator automatically takes into account "PEMDAS" Order, it will give you an incorrect answer if you make logical errors in your reasoning . . . using the calculator to perform a series of calculations in the wrong order.
No calculator will protect you from yourself. Every calculator will do exactly what you tell it to do - no matter what.
Students ultimately fail when they use a basic calculator ("dumb calculator") to avoid learning the principles of arithmetic.
Often students will get in the habit of using calculators for all their computations, beginning at a very young age . . . frequently with the encouragement of parents, or the local school system.
They grow to believe (trust, have faith) that calculators will always give them the correct answer.
Multiple-step arithmetic problems.
Multi-step computations associated with algebra and advanced math courses often involve long chains of numbers, brackets, exponents, and fractions.
When completing multi-step computations, students who rely heavily on basic ("dumb") calculators begin to arrive at incorrect answers on a regular basis. This even occurs when a student's problem solving logic and mathematical approach are 100% correct.
These students routinely state that their answers are correct because they used a calculator.
Read over the following examples. Keep in mind that a calculator is nothing more than a machine. It is up to the machine operator to use it correctly. It is up to the user to understand what a calculator actually does depends on how each number is entered.
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Algebra
Around 825 A.D. in Baghdad, Mohammed ibn-Musa al-Khwarizimi wrote a book called "Kitab al-jabr wa al-muqabalah", which means "The science of restoration and reduction" and stood as the major algebraic work of the period. The word "algebra" comes from this title ("al-jabr"), since this was the first textbook used in Europe for this subject. The word "algorithm" is a distortion of al-Khwarizmi's name.
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What this wiki offers
High quality content written by math graduate students. The content is reviewed and can be updated on the fly with your comments and feedback.
The ability to use the discussion pages of any page in this wiki to dialogue with us or with other students about mathematics!
Your feedback is essential to the success of this wiki project. If you found these pages helpful for your preparation for the final exam, we'd love to hear about it in the Discussion page! Of course, we also want to hear what you would like to see improved.
Resources by Topic
Please note that this section is a work in progress. Many questions have not been classified yet, and tags may be wrong or misleading. However, we still want to offer this additional resource of browsing by topic, because working through several relevant questions in a row is a great way of mastering a topic with which you're not yet comfortable.
Caution is needed when working through problems on the same topic from a different course. While problems are often applicable cross-courses, sometimes questions from a particular course require a technique that was not covered in another course, even though both courses covered the same general topic.
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Unit specification
Aims
This module aims to engage students with a circle of algorithmic techniques and concrete problems arising in elementary number theory and graph theory.
Brief description
Modern Discrete Mathematics is a broad subject bearing on everything from logic to logistics. Roughly speaking, it is a part of mathematics that touches on those subjects that Calculus and Algebra can't: problems where there is no sensible notion continuity
or smoothness and little algebraic structure. The subject, which is typically concerned with finite—or at the most countable—sets of objects, abounds with interesting, concrete problems and entertaining examples.
Intended learning outcomes
Students should develop the ability to think and argue algorithmically, mainly by studying examples in elementary number theory, graph theory and combinatorics.
On completion of this unit successful students will be able to:
understand proofs by induction thoroughly and be fluent in their construction;
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Algebra
Book Description
Part of a two-volume set which presents students with a complete introduction to algebra, this textbook describes the basic notions, including rings, groups, modules and fields, and the main theories pertaining to these notions.
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Loci: Developers
Open Standards, Web-Based Mathlets: Making Interactive Tutorials Using the HTML5 canvas Element
Abstract
Interactive math tutorials, often called mathlets, are designed to provide a more visceral learning experience than traditional textbook methods and to enhance intuitive understanding of complex ideas by allowing users to alter parameters that influence visual scenes. We describe methods for creating such tutorials using the HTML5 canvas element. First, we discuss some motivations for writing such mathlets, then walk-through the process of creating a mathlet with canvas. Then, we compare canvas to alternatives, explaining our decision to use it, and provide links to other demonstrations and resources.
This article requires JavaScript to be enabled in your browser. The article discusses browser support for the HTML5 canvas element.
This article uses jsMath, which requires JavaScript, to process the mathematics expressions. If your browser supports JavaScript, be sure it is enabled. Once the jsMath scripts are running, clicking the "jsMath" button in the lower right corner of the browser window brings up a panel with configuration options and links to documentation and download pages, including instructions for installing missing mathematics fonts.
Copyright 2013. All rights reserved. The Mathematical Association of America.
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logarithms, fractions, fraction/decimal conversions, degrees to decimal conversion, reciprocals, factorials, grads, hyperbolics, polar/rectangular conversions. Fraction feature allows operations with fractions and mixed numbers. Two-variable statistics allow you to enter, delete, insert, and edit individual statistical data elements. Calculator runs on solar and battery power.
TI-34 MultiView is ideal for middle school math, pre-Algebra, Algebra I and II, trigonometry, general science, geometry, and biology. MultiView display shows fractions as they are written on paper. View multiple calculations on a four-line display and easily scroll through entries. Enter multiple calculations to compare results and explore patterns on the same screen. Simplify and convert fractions to decimals and back again. Integer division key expresses results as quotient and remainders. Toggle Key lets you quickly view fractions, decimals and terms including Pi in alternate forms. Functions include previous entry, power, roots, reciprocals, variable statistics and seven memories. Scientific calculator also features user-friendly menus, automatic shutoff, hard plastic color-coded keys, nonskid rubber feet, impact-resistant cover with a quick-reference card, and dual power with solar and battery operation.
Handheld graphing calculator lets you visualize in dynamic graphing with the high-resolution full-color, backlit display. Ideal for Algebra, Trigonometry, Geometry, Statistics, Business/Finance, Biology, Physics, Chemistry, and Engineering. Touchpad navigation works more like a laptop computer. You can even transfer class assignments from this handheld to your PC or Mac computer. Calculator also lets you color-code equations, objects, points and lines and make faster, stronger connections between equations, graphs and geometric representations on screen. 3D Graphing lets you graph and rotate manually and automatically. The thin, lightweight TI-Nspire CX runs on the included rechargeable battery and includes a wall adapter, software, a USB Unit-to-unit cable, and USB Unit-to-Computer cable.
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Lecture 1: Simple Equations
Embed
Lecture Details :
Introduction to basic algebraic equations of the form Ax=B
Course Description :
This is the original Algebra course on the Khan Academy and is where Sal continues to add videos that are not done for some other organization. It starts from very basic algebra and works its way through algebra II.
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Even though contemporary biology and mathematics are inextricably linked, high school biology and mathematics courses have traditionally been taught in isolation. But this is beginning to change. This volume presents papers related to the integration of biology and mathematics in high school classes.
The first part of the book provides the rationale for integrating mathematics and biology in high school courses as well as opportunities for doing so. The second part explores the development and integration of curricular materials and includes responses from teachers.
Papers in the third part of the book explore the interconnections between biology and mathematics in light of new technologies in biology. The last paper in the book discusses what works and what doesn't and presents positive responses from students to the integration of mathematics and biology in their classes.
Co-published with the Center for Discrete Mathematics and Theoretical Computer Science beginning with Volume 8. Volumes 1-7 were co-published with the Association for Computer Machinery (ACM).
Readership
High school teachers, education specialists, graduate students, and research mathematicians interested in mathematics and biology education.
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Math Strategies
Course Description
Math Strategies is a Response to Invention aimed at assisting students who have been identified through ISAT and MAP testing data as having deficits in math. Math Strategies uses a concrete- representational- abstract (CRA) approach to learning using discovery activities in the Math Elevations curriculum. Activities will increase students' fact fluency and individual math deficits.
Enduring Understandings
Mathematics can help us make more informed decisions, work efficiently, solve problems, and appreciate its relevance in the world. Geometric methods can help us to make connections and draw conclusions from the world in which we live. Functions and number operations play fundamental roles in helping us to make sense of various situations. Using prior knowledge, appropriate technology, and logical thinking, we can analyze data and effectively communicate the reasonableness of solutions. Multiple mathematical approaches and strategies can be used to reach a desired outcome. Algebraic models, patterns, and graphical representations are tools that can help us make meaningful connections to
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What is Algebra?
Date: 08/24/97 at 21:56:12
From: Kathryn Bowser
Subject: Algebra
Ok, I know this may seem stupid but I really haven't started to do
actual algebra in school. I was wondering if u might write back and
explain from ground on if it not to much trouble. thanx.
Kathryn Bowser
Date: 08/25/97 at 14:50:50
From: Doctor Ken
Subject: Re: Algebra
Hi Kathryn -
Wow, that's a pretty tall order. It would be pretty difficult for me
to try to explain high school algebra from the beginning! But there's
nothing wrong with you asking the question.
Let me start by trying to say what algebra is. This is kind of hard,
because different people mean somewhat different things when they say
algebra (a professional mathematician might mean something different
from your teacher, for instance, but they're sort of talking about
different aspects of the same kind of thing).
But here's my simplistic view: algebra is the class where you learn
how to work with "unknown quantities."
For instance, let's look at a simple math problem: Kathryn has 22
dollars, which is 4 more dollars than Momma Bowser. How much money
does Momma Bowser have?
In algebra, you learn to write "equations" to solve problems like
these. Here's an equation for that problem: 22 = 4 + M . M represents
the amount of money Momma Bowser has. In algebra, you learn how to
solve problems like that, to get the answer: M = 18.
If you want more algebra stuff, you might try looking at our Algebra
sections on the Web:
and
Good luck!
-Doctor Ken, The Math Forum
Check out our web site!
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Overview
Table of contents
Introduction; Part I Limits 1. The Concept of Limit2. Special Limits 3. Continuity; Part II Differentiation 4. Definition of the Derivative and the Derivatives of Some Simple Functions; 5. Rules of Differentiation 6. Additional Derivatives Part III Integration,7. The Indefinite Integral and Basic Integration Formulas and Rules; 8. Basic Integration Techniques 9. The Definite Integral; Part IV Applications of the Derivative and the Definite Integrals, 10. Application of the Derivative 11. Applications of the Definite IntegralAppendix: Differentiation Formulas; Appendix: Integration Formulas; Answer Key; Worked Solutions
Author comments
William D. Clark, Ph.D., has been a professor of mathematics at Stephen F. Austin State University for more than 30 years.
Sandra Luna McCune, Ph.D., is Regents Professor currently teaching as a mathematics specialist in the Department of Elementary Education at Stephen F. Austin State University (SFASU).
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Activities After the Peer Group Meeting
Submit written responses to the following to
your instructor for evaluation.
Consideration of the Solutions
For each of the problems addressed in the Peer
Group Meeting, think about the following points.
Does the solution make sense to you? What does it mean? Are
the units correct? The numerical value of the solution is of
limited value in this regard. It is much more important to consider
the solution stated in terms of the symbols used to represent
values of the data. (This is a perfect opportunity to correct
errors.)
How could the problems addressed be generalized or otherwise
changed in an interesting way? How would the solutions change with
changes in the data, if they would change at all? For example, all
of the problems addressed are no more than two-dimensional. Would
the solutions change in an interesting way if the problems were
presented in three dimensions rather than two dimensions?
Provide new solutions for changes you suggest in the problems
that were addressed in this Peer Group Meeting.
Do the problems addressed here have any importance in a
larger context? For example, do they provide some basis for
addressing more "real world" problems in chemistry, engineering,
medicine, or biology?
Wider Considerations
What did you learn from participation in this exercise? What
was new to you? What requirements found in performing the exercise
were surprises to you?
Did the mathematics involved in solving the problems pose a
problem for you? Did this exercise convince you that proficiency in
mathematics is a prerequisite for doing professional work in the
field of your choice? If not, why not?
How do the problems addressed in this module relate to
medicine, biology, engineering, and chemistry? Can the problems
addressed here be generalized to problems in fields other than
physics? Do the generalizations involve more mathematics than were
necessary here, or less?
A careful reading of this module implies that notation, the
way one chooses to write stuff, is important in understanding,
especially for mathematical stuff. What do you think of this
implication
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Tmsca Middle School Calculator Test 4
You are here because you browse for Tmsca Middle School Calculator Test 4. We Try to providing the best Content For pdf, ebooks, Books, Journal or Papers in Chemistry, Physics, mathematics, Programming, Health and more category that you can browse for Free . Below is the result for Tmsca Middle School Calculator Test 4 query . Click On the title to download or to read online pdf & Book Manuals
Study Guide for the Middle School Mathematics Test
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VISTA VERDE MIDDLE SCHOOL
2012-2013%20Course%20Descriptions.pdf. VISTA VERDE MIDDLE SCHOOL. MATH: 6th grade math is a comprehensive course aligned with the California will take the IUSD Middle School Placement Test in order to determine math placement in higher level thinking skills that are specific to social science, for example: thinking projects, comprehension questions, and open-ended projects.ΗΤΤΡ://ΙUSD.ΟRG/VV/DΟCUΜΕΝΤS/2013/2012-2013%20CΟURSΕ%20DΕSCRΙΡΤΙΟΝS.ΡDF
Middle School Math Expectations the Catholic Diocese of Richmond
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Christopher.%20C.%20MTEL%20Prep%207.pdf. MIDDLE SCHOOL MATHEMATICS - Bridgewater State University. Middle School Math Practice Test with detailed solutions. Algebra The Middle School Mathematics test includes multiple-choice items and two open. Modeling and solving problems using quadratice relations, functions, and systems. 10.ΗΤΤΡ://
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Linear Algebra With Application - 8th edition
Summary: Updated and revised to increase clarity and further improve student learning, the Eighth Edition of Gareth Williams' classic text is designed for the introductory course in linear algebra. It provides a flexible blend of theory and engaging applications for students within engineering, science, mathematics, business management, and physics. It is organized into three parts that contain core and optional sections. There is then ample time for the instructor to select the material that...show more gives the course the desired flavor.Part 1 introduces the basics, presenting systems of linear equations, vectors and subspaces of R(n) (make sure it is superscript n), matrices, linear transformations, determinants, and eigenvectors. Part 2 builds on the material presented in Part1 and goes on to introduce the concepts of general vector spaces, discussing properties of bases, developing the rank/nullity theorem, and introducing spaces of matrices and functions.Part 3 completes the course with important ideas and methods of numerical linear algebra, such as ill-conditioning, pivoting, and LU decomposition.Throughout the text the author takes care to fully and clearly develop the mathematical concepts and provide modern applications to reinforce those concepts. The applications range from theoretical applications within differential equations and least square analysis, to practical applications in fields such as archeology, demography, electrical engineering and more. New exercises can be found throughout that tie back to the modern examples in the text.Key Features of the Eighth Edition:-- Updated and revised throughout with new section material and exercises included in every chapter. -- Each section begins with a motivating introduction, which ties material to the previously learned topics. -- Carefully explained examples illustrate key concepts throughout the text. -- Includes such new topics such as QR Factorization and Singular Value Decomposition.-- Includes new app ...show less
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Discrete Mathematics Mathematical Reasoning and Proof with Puzzles, Patterns, and Games
9780471476023
ISBN:
0471476021
Pub Date: 2005 Publisher: John Wiley & Sons Inc
Summary: Did you know that games and puzzles have given birth to many of today's deepest mathematical subjects? Now, with Douglas Ensley and Winston Crawley's Introduction to Discrete Mathematics, you can explore mathematical writing, abstract structures, counting, discrete probability, and graph theory, through games, puzzles, patterns, magic tricks, and real-world problems. You will discover how new mathematical topics can ...be applied to everyday situations, learn how to work with proofs, and develop your problem-solving skills along the way. Online applications help improve your mathematical reasoning. Highly intriguing, interactive Flash-based applications illustrate key mathematical concepts and help you develop your ability to reason mathematically, solve problems, and work with proofs. Explore More icons in the text direct you to online activities at Improve your grade with the Student Solutions Manual. A supplementary Student Solutions Manual contains more detailed solutions to selected exercises in the text
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8 Teaching and Learning Functions
Mindy Kalchman and Kenneth R. Koedinger
As functional relationships become more complex, as in the growth of a population or the accumulation of interest over time, solutions are not so easily calculated because the base changes each period. In these situationsHow Students Learn: History, Mathematics, and Science in the Classroom
8
Teaching and Learning Functions
Mindy Kalchman and Kenneth R. KoedingerAs functional relationships become more complex, as in the growth of a population or the accumulation of interest over time, solutions are not so easily calculated because the base changes each period. In these situations,
OCR for page 352
How Students Learn: History, Mathematics, and Science in the Classroom
algebraic tools allow highly complex problems to be solved and displayed in a way that provides a powerful image of change over time.
Many students would be more than a little surprised at this description. Few students view algebra as a powerful toolkit that allows them to solve complex problems much more easily. Rather, they regard the algebra itself as the problem, and the toolkit as hopelessly complex. This result is not surprising given that algebra is often taught in ways that violate all three principles of learning set forth in How People Learn and highlighted in this volume.
The first principle suggests the importance of building new knowledge on the foundation of students' existing knowledge and understanding. Because students have many encounters with functional relationships in their everyday lives, they bring a great deal of relevant knowledge to the classroom. That knowledge can help students reason carefully through algebra problems. Box 8-1 suggests that a problem described in its everyday manifestation can be solved by many more students than the same problem presented only as a mathematical equation. Yet if the existing mathematics understandings students bring to the classroom are not linked to formal algebra learning, they will not be available to support new learning.
The second principle of How People Learn argues that students need a strong conceptual understanding of function as well as procedural fluency. The new and very central concept introduced with functions is that of a dependent relationship: the value of one thing depends on, is determined by, or is a function of another. The kinds of problems we are dealing with no longer are focused on determining a specific value (the cost of 5 gallons of gas). They are now focused on the rule or expression that tells us how one thing (cost) is related to another (amount of gas). A "function" is formally defined in mathematics as "a set of ordered pairs of numbers (x, y) such that to each value of the first variable (x) there corresponds a unique value of the second variable (y)."2 Such a definition, while true, does not signal to students that they are beginning to learn about a new class of problems in which the value of one thing is determined by the value of another, and the rule that tells them how they are related.
Within mathematics education, function has come to have a broader interpretation that refers not only to the formal definition, but also to the multiple ways in which functions can be written and described.3 Common ways of describing functions include tables, graphs, algebraic symbols, words, and problem situations. Each of these representations describes how the value of one variable is determined by the value of another. For instance, in a verbal problem situation such as "you get two dollars for every kilometer you walk in a walkathon," the dollars earned depend on, are determined by, or are a function of the distance walked. Conceptually, students need to understand that these are different ways of describing the same relationship.
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How Students Learn: History, Mathematics, and Science in the Classroom
Good instruction is not just about developing students' facility with performing various procedures, such as finding the value of y given x or creating a graph given an equation. Instruction should also help students develop a conceptual understanding of function, the ability to represent a function in a variety of ways, and fluency in moving among multiple representations of functions. The slope of the line as represented in an equation, for example, should have a "meaning" in the verbal description of the relationship between two variables, as well as a visual representation on a graph.
The third principle of How People Learn suggests the importance of students' engaging in metacognitive processes, monitoring their understanding as they go. Because mathematical relationships are generalized in algebra, students must operate at a higher level of abstraction than is typical of the mathematics they have generally encountered previously. At all levels of mathematics, students need to be engaged in monitoring their problem solving and reflecting on their solutions and strategies. But the metacognitive engagement is particularly important as mathematics becomes more abstract, because students will have few clues even when a solution has gone terribly awry if they are not actively engaged in sense making.
When students' conceptual understanding and metacognitive monitoring are weak, their efforts to solve even fairly simple algebra problems can, and often do, fail. Consider the problem in Figure 8-1a. How might students approach and respond to this problem? What graph-reading and table-building skills are required? Are such skills sufficient for a correct solution? If students lack a conceptual understanding of linear function, what errors might they make? Figure 8-1b shows an example student solution.
What skills does this student exhibit? What does this student understand and not understand about functions? This student has shown that he knows how to construct a table of values and knows how to record in that table coordinate points he has determined to be on the graph. He also clearly recalls that an algorithm for finding the slope of the function is dividing the change in y(Δy) by the change in x(Δx). There are, however, significant problems with this solution that reveal this student's weak conceptual understanding of functions.
Problem: Make a table of values that would produce the function seen on page 356.
First, and most superficially, the student (likely carelessly) mislabeled the coordinate for the y-intercept (0, 3) rather than (0, –3). This led him to make an error in calculating Δy by subtracting 0 from 3 rather than from –3. In so doing, he arrived at a value for the slope of the function that was negative—an impossible solution given that the graph is of an increasing linear function. This slip, by itself, is of less concern than the fact that the
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How Students Learn: History, Mathematics, and Science in the Classroom
BOX 8-1
Linking Formal Mathematical Understanding to Informal Reasoning
Which of these problems is most difficult for a beginning algebra student?
Story Problem
When Ted got home from his waiter job, he multiplied his hourly wage by the 6 hours he worked that day. Then he added the $66 he made in tips and found he had earned $81.90. How much does Ted make per hour?
Word Problem
Starting with some number, if I multiply it by 6 and then add 66, I get 81.9. What number did I start with?
Equation
Solve for x:
x * 6 + 66 = 81.90
Most teachers and researchers predict that students will have more difficulty correctly solving the story or word problem than the equation.4 They might explain this expectation by saying that a student needs to read the verbal problems (story and word) and then translate them into the equation. In fact, research investigating urban high school students' performance on such problems found that on average, they scored 66 percent on the story problem, 62 percent on the word problem, and only 43 percent on the equation.5 In other words, students were more likely to solve the verbal problems correctly than the equation. Investigating students' written work helps explain why.
Students often solved the verbal problems without using the equation. For instance, some students used a generate-and-test strategy: They estimated a value for the hourly rate (e.g., $4/hour), computed the corresponding pay (e.g., $90), compared it against the given value ($81.90),
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and repeated as needed. Other students used a more efficient unwind or working backwards strategy. They started with the final value of 81.9 and subtracted 66 to undo the last step of adding 66. Then they took the resulting 15.9 and divided by 6 to undo the first step of multiplying by 6. These strategies made the verbal problems easier than expected. But why were the equations difficult for students? Although experts in algebra may believe no reading is involved in equation solving, students do in fact need to learn how to read equations. The majority of student errors on equations can be attributed to difficulties in correctly comprehending the meaning of the equation.6 In the above equation, for example, many students added 6 and 66, but no student did so on the verbal problems.
Besides providing some insight into how students think about algebraic problem solving, these studies illustrate how experts in an area such as algebra may have an "expert blind spot" for learning challenges beginners may experience. An expert blind spot occurs when someone skilled in an area overestimates the ease of learning its formalisms or jargon and underestimates learners' informal understanding of its key ideas. As a result, too little attention is paid to linking formal mathematical understanding to informal reasoning. Looking closely at students' work, the strategies they employ, and the errors they make, and even comparing their performance on similar kinds of problems, are some of the ways we can get past such blind spots and our natural tendency to think students think as we do.
Such studies of student thinking contributed to the creation of a technology-enhanced algebra course, originally Pump Algebra Tutor and now Cognitive Tutor Algebra.7 That course includes an intelligent tutor that provides students with individualized assistance as they use multiple representations (words, tables, graphs, and equations) to analyze real-world problem situations. Numerous classroom studies have shown that this course significantly improves student achievement relative to alternative algebra courses (see The course, which was based on basic research on learning science, is now in use in over 1,500 schools.
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FIGURE 8-1
student did not recognize the inconsistency between the positive slope of the line and the negative slope in the equation. Even good mathematicians could make such a mistake, but they would likely monitor their work as they went along or reflect on the plausibility of the answer and detect the inconsistency. This student could have caught and corrected his error had he
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acquired both fluency in interpreting the slope of a function from its equation (i.e., to see that it represents a decreasing function) and a reflective strategy for comparing features of different representations.
A second, more fundamental error in the student's solution was that the table of values does not represent a linear function. That is, there is not a constant change in y for every unit change in x. The first three coordinates in the student's table were linear, but he then recorded (2.5, 0) as the fourth coordinate pair rather than (3, 0), which would have made the function linear. He appears to have estimated and recorded coordinate points by visually reading them off the graph without regard for whether the final table embodied linearity. Furthermore, the student did not realize that the equation he produced, , translates not only into a decreasing line, but also into a table of numbers that decreases by for every positive unit change in x.
At a surface level, this student's solution reflects some weaknesses in procedural knowledge, namely, getting the sign wrong on the y-intercept and imprecisely reading x-y coordinates off the graph. More important, however, these surface errors reflect a deeper weakness in the student's conceptual understanding of function. The student either did not have or did not apply knowledge for interpreting key features (e.g., increasing or decreasing) of different function representations (e.g., graph, equation, table) and for using strategies for checking the consistency of these interpretations (e.g., all should be increasing). In general, the student's work on this problem reflects an incomplete conceptual framework for linear functions, one that does not provide a solid foundation for fluid and flexible movement among a function's representations.
This student's work is representative of the difficulties many secondary-level students have with such a problem after completing a traditional textbook unit on functions. In a study of learning and teaching functions, about 25 percent of students taking ninth- and eleventh-grade advanced mathematics courses made errors of this type—that is, providing a table of values that does not reflect a constant slope—following instruction on functions.8 This performance contrasts with that of ninth- and eleventh-grade mathematics students who solved this same problem after receiving instruction based on the curriculum described in this chapter. This group of students had an 88 percent success rate on the problem. Because these students had developed a deeper understanding of the concept of function, they knew that the y-values in a table must change by the same amount for every unit change in x for the function to be linear. The example in Figure 8-1c shows such thinking.
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FIGURE 8-1
Problem: Make a table of values that would produce the function seen above.
This student identified a possible y-intercept based on a reasonable scale for the y-axis. She then labeled the x- and y-axes, from which she determined coordinate pairs from the graph and recorded them in a table of values. She determined and recorded values that show a constant increase in y for every positive unit change in x. She also derived an equation for the function that not only corresponds to both the graph and the table, but also represents a linear relationship between x and y.
How might one teach to achieve this kind of understanding? The goal of this chapter is to illustrate approaches to teaching functions that foster deep understanding and mathematical fluency. We emphasize the importance of designing thoughtful instructional approaches and curricula
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that reflect the principles of How People Learn (as outlined in Chapter 1), as well as recent research on what it means to learn and understand functions in particular. We first describe our approach to addressing each of the three principles. We then provide three sample lessons that emphasize those principles in sequence. We hope that these examples provide interesting activi ties to try with students. More important, these activities incorporate important discoveries about student learning that teachers can use to design other instructional activities to achieve the same goals.
ADDRESSING THE THREE PRINCIPLES
Principle #1: Building on Prior Knowledge
Principle 1 emphasizes the importance of students and teachers continually making links between students' experiences outside the mathematics classroom and their school learning experiences. The understandings students bring to the classroom can be viewed in two ways: as their everyday, informal, experiential, out-of-school knowledge, and as their school-based or "instructional" knowledge. In the instructional approach illustrated here, students are introduced to function and its multiple representations by having their prior experiences and knowledge engaged in the context of a walkathon. This particular context was chosen because (1) students are familiar with money and distance as variable quantities, (2) they understand the contingency relationship between the variables, and (3) they are interested in and motivated by the rate at which money is earned.
The use of a powerful instructional context, which we call a "bridging context," is crucial here. We use this term because the context serves to bridge students' numeric (equations) and spatial (graphic) understandings and to link their everyday experiences to lessons in the mathematics classroom. Following is an example of a classroom interaction that occurred during students' first lesson on functions, showing how use of the walkathon context as an introduction to functions in multiple forms—real-world situation (walkathon), table, graph, verbal ("$1.00 for each kilometer"), situation-specific symbols ($ = 1 * km), and generic symbolic (y = x * 1)—accomplishes these bridging goals. Figures 8-2a through 8-2c show changes in the whiteboard as the lesson proceeded.
Teacher
What we're looking at is, we're looking at what we do to numbers, to one set of numbers, to get other numbers…. So how many of you have done something like a walkathon? A readathon? A swimathon? A bikeathon?
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[Students raise their hands or nod.] So most of you… So I would say "Hi Tom [talking to a student in class], I'm going to raise money for such and such a charity and I'm going to walk ten kilometers."
Tom
OK.
Teacher
Say you're gonna sponsor me one dollar for every kilometer that I walk. So that's sort of the first way that we can think about a function. It's a rule. One dollar for every kilometer walked. So you have one dollar for each kilometer [writing "$1.00 for each kilometer" on the board while saying it]. So then what I do is I need to calculate how much money I'm gonna earn. And I have to start somewhere. So at zero kilometers how much money do I have Tom? How much are you gonna pay me if I collapse at the starting line? [Fills in the number 0 in the left-hand column of a table labeled "km"; the right-hand column is labeled "$".]
Tom
None.
Teacher
So Tom, I managed to walk one kilometer [putting a "1" in the "km" column of the table of values below the "0"]….
Tom
One dollar.
Teacher
One dollar [moving to the graph]. So I'm going to go over one kilometer and up one dollar [see Figure 8-2a].
FIGURE 8-2a Graphing a point from the table: "Over by one kilometer and up by one dollar." The teacher uses everyday English ("up by") and maintains connection with the situation by incorporating the units "kilometer" and "dollar."
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FIGURE 8-2b The teacher and students construct the table and graph point by point, and a line is then drawn.
[Students continue to provide the dollar amounts for each of the successive kilometer values. Simple as it is, students are encouraged to describe the computation—"I multiply two kilometers by one to get two dollars." The teacher fills in the table and graphs each coordinate pair. [The board is now as shown in Figure 8-2b.]
Teacher
Now, what I want you to try and do, first I want you to look at this [pointing to the table that goes from x = 0 to x = 10 for y = x] and tell me what's happening here.
Melissa
You, like, earn one dollar every time you go up. Like it gets bigger by one every time.
Teacher
So every time you walk one kilometer you get one more dollar, right? [Makes "> 1" marks between successive "$" values in the table—see Figure 8-2c.] And if you look on the graph, every time I walk one kilometer I get one more dollar. [Makes "step" marks on the graph.] So now I want to come up with an equation, I want to come up with some way of using this symbol [pointing to the "km" header in the left-hand column of the table] and this symbol [pointing to the "$" header in the right-hand column of the table] to say the same thing, that for every kilometer I walk, let's put it this way, the money I earn is gonna be equal to one times the number of kilometers I walk. Someone want to try that?
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FIGURE 8-3 Sample computer screen. In this configuration, students can change the value of a, n, or b to effect immediate and automatic changes in the graph and the table. For example, if students change the value of b, just the y-intercept of the curve will change. If students change a or n to a positive value other than 1, the degree of steepness of the curve will change. If students change the value of a to a negative value, the curve will come down. All graphic patterns will be reflected in the table of values.
Students must employ effective metacognitive strategies to negotiate and complete these computer activities. Opportunities for exploring, persevering, and knowing when and how to obtain help are abundant. Metacognitive activity is illustrated in the following situation, which has occurred among students from middle school through high school who have worked through these activities.
When students are asked to change the parameters of y = x2 to make it curve down and go through a colored point that is in the lower right quadrant, their first intuition is often to make the exponent rather than the coefficient negative. When they make that change, they are surprised to find that the graph changes shape entirely and that a negative exponent will not
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satisfy their needs. By trying a number of other possible alterations (persevering), some students discover that they need to change the coefficient of x2 rather than the exponent to a negative number to make the function curve down. It is then a matter of further exploration and discovery to find the correct value that will make the graph pass through the point in question. Some students, however, require support to discover this solution. Some try to subtract a value from x2 but are then reminded by the result they see on the computer screen that subtracting an amount from x2 causes a downward vertical shift of the graph. Drawing students' attention to earlier exercises in which they multiplied the x in y = x by a negative number to make the numeric pattern and the graph go down encourages them to apply that same notion to y = x2. To follow up, we suggest emphasizing for students the numeric pattern in the tables of values for decreasing curves to show how the number pattern decreases with a negative coefficient but not with a negative exponent.
Following is a typical exchange between the circulating teacher and a pair of students struggling with flipping the function y = x2 (i.e., reflecting it in the x-axis). This exchange illustrates the use of metacognitive prompting to help students supervise their own learning by suggesting the coordination of conclusions drawn from one representation (e.g., slope in linear functions) with those drawn from another (e.g., slope in power functions).
Teacher
How did you make a straight line come down or change direction?
John
We used minus.
Teacher
How did you use "minus"?
Pete
Oh yeah, we times it by minus something.
Teacher
So … how about here [pointing at the x2]?
John
We could times it by minus 2 [typing in x2 • -2]. There! It worked.
Without metacognitive awareness and skills, students are at risk of missing important inconsistencies in their work and will not be in a position to self-correct or to move on to more advanced problem solving. The example shown earlier in Figure 8-1a involves a student not reflecting on the inconsistency between a negative slope in his equation and a positive slope in his graph. Another sort of difficulty may arise when students attempt to apply "rules" or algorithms they have been taught for simplifying a solution to a situation that in fact does not warrant such simplification or efficiency.
For example, many high school mathematics students are taught that "you only really need two points to graph a straight line" or "if you know it's a straight line, you only need two points." The key phrase here is "if you know it's a straight line." In our research, we have found students applying
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FIGURE 8-4
that two-point rule for graphing straight lines to the graphing of curved-line functions. In the example shown in Figure 8-4, an eleventh-grade advanced mathematics student who had been learning functions primarily from a textbook unit decided to calculate and plot only two points of the function y = x2 +1 and then to join them incorrectly with a straight line. This student had just finished a unit that included transformations of quadratic functions and thus presumably knew that y = x2 makes a parabola rather than a straight line. What this student did not know to perform, or at least exercise, was a metacognitive analysis of the problem that would have ruled out the application of the two-points rule for graphing this particular function.
Summary of Principle #3 in the Context of Operating on y = x2. The general metacognitive opportunities for the computer activities in our curriculum are extensive. Students must develop and engage their skills involving prediction, error detection, and correction, as well as strategies for scientific inquiry such as hypothesis generating and testing. For instance, because there are innumerable combinations of y-intercept, coefficient, and exponent that will move y = x2 through each of the colored points, students must recognize and acknowledge alternative solution paths. Some students may fixate on the steepness of the curve and get as close to the colored points as possible by adjusting just the steepness of the curve (by changing either the exponent or the coefficient of x2) and then changing the y-intercept. Others may begin by selecting a manageable y-intercept and then adjust the steepness of the curve by changing the exponent or the coefficient. Others may use both strategies equally. Furthermore, students must constantly be pre-
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dicting the shapes and behaviors of the functions with which they are working and adjusting and readjusting their expectations with respect to the mathematical properties and characteristics of linear and nonlinear functions.
SUMMARY
Sometimes mathematics instruction can lead to what we refer to as "ungrounded competence." A student with ungrounded competence will display elements of sophisticated procedural or quantitative skills in some contexts, but in other contexts will make errors indicating a lack of conceptual or qualitative understanding underpinning these skills. The student solution shown earlier in Figure 8-1a illustrates such ungrounded competence. On the one hand, the student displays elements of sophisticated skills, including the slope formula and negative and fractional coefficients. On the other hand, the student displays a lack of coordinated conceptual understanding of linear functions and how they appear in graphical, tabular, and symbolic representations. In particular, he does not appear to be able to extract qualitative features such as linearity and the sign of the slope and to check that all three representations share these qualitative features.
The curricular approach described in this chapter is based on cognitive principles and a detailed developmental model of student learning. It was designed to produce grounded competence whereby students can reason with and about multiple representations of mathematical functions flexibly and fluently. Experimental studies have shown that this curriculum is effective in improving student learning beyond that achieved by the same teachers using a more traditional curriculum. We hope that teachers will find the principles, developmental model, and instructional examples provided here useful in guiding their curriculum and teaching choices.
We have presented three example lessons that were designed within one possible unifying context. Other lessons and contexts are possible and desirable, but these three examples illustrate some key points. For instance, students may learn more effectively when given a gradual introduction to ideas. Our curriculum employs three strategies for creating such a gradual introduction to ideas:
Starting with a familiar context: Contexts that are familiar to students, such as the walkathon, allow them to draw on prior knowledge to think through a mathematical process or idea using a concrete example.
Starting with simple content: To get at the essence of the idea while avoiding other, distracting difficulties, our curriculum starts with mathematical content that is as simple as possible—the function "you get one dollar for every kilometer you walk" (y = x).
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Focusing on having students express concepts in their own language before learning and using conventional terminology: To the extent that a curriculum initially illustrates an idea in an unfamiliar context or with more-complex content, students may be less likely to be able to construct or invent their own language for the idea. Students may better understand and explain new ideas when they progress from thinking about those ideas using their own invented or natural language to thinking about them using formal conventional terms.
A risk of simplicity and familiarity is that students may not acquire the full generality of relevant ideas and concepts. Our curriculum helps students acquire correct generalizations by constructing multiple representations for the same idea for the same problem at the same time. Students make comparisons and contrasts across representations. For example, they may compare the functions y = .5x, y = 2x, and y = 10x in different representations and consider how the change in slope looks in the graph and how the table and symbolic formula change from function to function. We also emphasize the use of multiple representations because it facilitates the necessary bridging between the spatial and numerical aspects of functions. Each representation has both spatial and numerical components, and students need experience with identifying and constructing how they are linked.
As illustrated earlier in Figure 8-1a, a curriculum that does not take this multiple-representation approach can lead students to acquire shallow ideas about functions, slope, and linearity. The student whose response is shown in that figure had a superficial understanding of how tables and graphs are linked: he could read off points from the graph, but he lacked a deep understanding of the relationship between tables and graphs and the underlying idea of linearity. He did not see or "encode" the fact that because the graph is linear, equal changes in x must yield equal changes in y, and the values in the table must represent this critical characteristic of linearity.
The curriculum presented in this chapter attempts to focus limited instructional time on core conceptual understanding by using multiple representations and generalizing from variations on just a few familiar contexts. The goal is to develop robust, generalizable knowledge, and there may be multiple pathways to this end. Because instructional time is limited, we decided to experiment with a primary emphasis on a single simple, real-world context for introducing function concepts instead of using multiple contexts or a single complex context. This is not to say that students would not benefit from a greater variety of contexts and some experience with rich, complex, real-world contexts. Other contexts that are relevant to students' current real-world experience could help them build further on prior knowledge. Moreover, contexts that are relevant to students' future real-world experiences, such as fixed and variable costs of production, could help them
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in their later work life. Since our lessons can be accomplished in anywhere from 3 to 6 weeks (650 minutes), there is sufficient time for other activities to supplement and extend students' experience.
In addition to providing a gradual introduction to complex ideas, a key point illustrated by our lessons is that curriculum should be mathematically sound and targeted toward high standards. Although the lessons described here start gradually, they quickly progress to the point at which students work with and learn about sophisticated mathematical functions at or beyond what is typical for their grade level. For instance, students progress from functions such as y = x to y = 10 – .4x in their study of linear functions across lessons 1 to 3, and from y = x2 to y = (x – 2)2 + 4 in their study of nonlinear functions across lessons 4 to 8.
We do not mean to suggest that this is the only curriculum that promotes a deep conceptual understanding of functions or that illustrates the principles of How People Learn. Indeed, it has important similarities, as well as differences, with other successful innovations in algebra instruction, such as the Jasper Woodbury series and Cognitive Tutor Algebra (previously called PUMP), both described in How People Learn. All of these programs build on students' prior knowledge by using problem situations and making connections among multiple representations of function. However, whereas the Jasper Woodbury series emphasizes rich, complex, real-world contexts, the approach described in this chapter keeps the context simple to help students perceive and understand the richness and complexity of the underlying mathematical functions. And whereas Cognitive Tutor Algebra uses a wide variety of real-world contexts and provides intelligent computer tutor support, the approach described here uses spreadsheet technology and focuses on a single context within which a wide variety of content is illustrated.
All of these curricula, however, stand in contrast to more traditional textbook-based curricula, which have focused on developing the numeric/ symbolic and spatial aspects of functions in isolation and without particular attention to the out-of-school knowledge that students bring to the classroom. Furthermore, these traditional approaches do not endeavor to connect the two sorts of understandings, which we have tried to show is an essential part of building a conceptual framework that underpins students' learning of functions and ultimately their learning in related areas.
ACKNOWLEDGMENTS
Thanks to Ryan Baker, Brad Stephens, and Eric Knuth for helpful comments. Thanks to the McDonnell Foundation for funding.
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Are the Honours math courses at Waterloo similar to the high school calculus/vectors and advanced functions type of stuff?
Calc/Vectors would prepare you for math courses in university, but they're generally a lot harder and concepts are at a much deeper level. There's a lot less of those easy solve the equation type questions, and a lot more proofs (how formulas are derived, and the relationships between different things and why they make sense)
Mathematics/Chartered Accountancy University of Waterloo Class of 2016
Of course it's going to be challenging. Your Calculus portion in grade 12 will obviously be useful for Calc I that you take in 1st year. Youll also be taking Linear Algebra in 2nd term, which relates to the Vectors portion a little. You also have to take an Algebra course in 1st term that involves a lot of proofing, number theory, stuff you don't really do in HS. Not that sure about upper year math courses.
UW Software Engineering 2015 Check out my blog, where I talk about school, coop terms, and other random stuff :)
Is this the sort of thing which is covered in Waterloo's Math 147 and 148 courses?
If it is, these university "advanced calculus" notes do not look anything like the high school calculus/vectors or advanced functions courses. In fact these "advanced calculus" notes look like a completely different foreign language to me!The University of Toronto is the best school in the country, especially for math. The Fields Institute, a renowned centre for mathematics research, is nearby, and they do a ton of research at the university. Waterloo is also good, but it seems that UofT's not getting the commendation it deservesFrom attempting to read these "advanced calculus" notes (linked in an earlier post) and trying to do some of the easier proofs (without looking at the answers), so far I find I don't quite "get it".
My impression is that it seems like they are trying to minimize (or avoiding) actual calculations in the proof heavy stuff.
I'll look at these notes further, and see whether I have any great interest in abstract mathematical proofs. So far it seems kinda "meh" to me.
Lately I've also been reading some course notes from a quantum mechanics course at University of Toronto. These introductory notes on quantum theory are somewhat more interesting to me, that I've spent a bit of time trying to figure out more of the calculus math behind the calculations (like differential equations).
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Financial functions, including time value of money, cash flow and amortization
This Texas Instruments TI-84 Plus graphing calculator, 10-Unit classroom pack comes with ten calculators, a poster, transparency, and a guidebook to help teachers instruct students on how to use graphing calculators to solve complex mathematical equations.
Memory and Display The display on the calculator can show up to eight lines at once so that charts, triangles, and graphs can all be seen easily. It can also display up to sixteen characters to make long calculations easy. In addition, there is a built-in flash memory to help you store certain equations or results so that you can quickly recall them for later use. The calculator is also fully compatible with the TI-83 Plus, so learning it is a breeze if you have already used the TI-84.
Preloaded Software In addition to being able to help chart graphs and do other mathematical equations, this calculator has software preloaded as well. There are thirteen full handheld software applications already installed, including Cabri Jr, which makes the TI-84 an even better value.
Graphing Styles With some of the more complex calculations, there may be a need to chart more than one graph. The TI-84 has seven different graphing styles so that you can easily differentiate between multiple graphs. This keeps you organized as you work and helps to prevent confusion.
Financial Functions The TI-84 isn't just for graphing or doing complex math problems for students. It can also be used by economics teachers and students because it has several financial functions built into the software, as well. This makes it a very versatile device that can be used across multiple academic disciplines
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1998 | Series: Revision Guides7,"ASIN":"1841460303","isPreorder":0},{"priceBreaksMAP":null,"buyingPrice":4.7,"ASIN":"1847622577","isPreorder":0},{"priceBreaksMAP":null,"buyingPrice":4.85,"ASIN":"1841462306","isPreorder":0}],"shippingId":"1841460303::srKG8bl3EG5Neh9fjGlvs5GaDchWnqwTkoRSP1p2Wj2ncw8PxkoSXvQh63ihlkPLH2FyqTG2BbgyeIr%2BPeT2Xh8ElJwzC8l3,1847622577::gcs4cgaXomwQayhHxMTXbCaNZs%2F2zGBUFoa783JoXUu03HctF%2FtgcLzcwVoBpqMsX6wiMKiy2f10%2FGclK5aESp%2BCUsmUY1Ru,1841462306::CaQPLp5oVrpxdsq32Ry7iEEYYARBFrclAzInvfe491qBZsSFZCkULh2%2BCCeDoGRyzmSQ%2BOHR41T6Yf4W96WwjRK7cS362L CGP manage to produce guides which contain all the hard facts kids need for their exams, but which present them in a friendly, digestible format accessible to both children and adults. This comprehensive guide includes sections on numbers, algebra, shapes and statistics, presented with the usual CGP humour and cartoons. It is useful both as a reference book (when are shapes congruent and similar? What are the 8 simple rules of geometry?) and as a work-your-way through-it revision guide.
I purchased this book roughly 6-7 weeks before the actual mathematics exam, thinking that it was too late to revise the majority of the year 9 curriculm. But I found it simple, fun and easy to understand because of its unique teaching methods. For instance it had useful pictures, strange jokes and questions at the end of each chapter to test your knowledge. All of which proved to be extremely helpful, due to the amount of time I had to revise.
My son was having problems in maths due to lessons missed after illness. I had very little knowledge of modern secondary level maths so this book was perfect to help us both to tackle some tricky new topics.
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The Philosophy of Mathematics: An Introductory Essay by Stephan Körner A distinguished philosopher surveys the mathematical views and influence of Plato, Aristotle, Leibniz, and Kant. He also examines the relationship between mathematical theories, empirical data, and philosophical presuppositions. 1968 editionProblem Solving Through Recreational Mathematics by Bonnie Averbach, Orin Chein Fascinating approach to mathematical teaching stresses use of recreational problems, puzzles, and games to teach critical thinking. Logic, number and graph theory, games of strategy, much more. Includes answers to selected problems. 1980History of Mathematics, Vol. II by David E. Smith Volume II of a two-volume history — from Egyptian papyri and medieval maps to modern graphs and diagrams. Evolution of arithmetic, geometry, trigonometry, calculating devices, algebra, calculus, more. Problems, recreations, and applications.
History of Mathematics, Vol. I by David E. Smith Volume 1 of a two-volume history — from Egyptian papyri and medieval maps to modern graphs and diagrams. Non-technical chronological survey with thousands of biographical notes, critical evaluations, contemporary opinions on over 1,100 mathematiciansPopular account ranges from counting to mathematical logic and covers the many mathematical concepts that relate to infinity: graphic representation of functions; pairings and other combinations; prime numbers; logarithms and circular functions; formulas, analytical geometry; infinite lines, complex numbers, expansion in the power series; metamathematics; more. 216
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Summary of part 1
Broadly speaking, the first five weeks cover Chapters 1-14 of the course textbook.
I will also include a proof of Cantor's result that the set of algebraic
numbers is denumerable (see page 181) - this will be the climax of my part of the
course! The following sections and subsections will be omitted (although some of
this material is likely to feature in weeks 6-12):
Week 6 Test
Here are the
solutions and marking scheme for the 2012
test, and here is my feedback on how it went.
You can now reclaim your papers from your feedback supervisors.
Problem Sheets
Problems for work in Feedback Classes are listed below. Answers
(including partial attempts!) should be submitted for
marking. Handing-in procedures and weekly deadlines should be
negotiated directly with supervisors, who may (or may not) wish to
focus on starred questions. Throughout this particular semester, marks
are awarded for effort, rather than accuracy.
Students are responsible for ensuring that they are properly prepared
by downloading these pages well in advance of the relevant week. The
time and place of each supervision class (with name of supervisor) will
be posted on the 1st year notice board at the start of the semester.
Examinations
Here is a another sample, with solutions
(and marking scheme) to help with revision of the first 6 weeks' work.
Lecture Notes
Here are my handwritten lecture notes for each of the first five weeks. They
approximate what I write on the board, but will not be helpful to students
who do not attend. Everything they contain is also in the course text, where
thorough explanations are given.
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What do you want from your history of mathematics course? Not enough new teachers of the subject ask this question, and fewer still know that most authors will tell you what they are presenting. Howard Eves gives you his opinion in the introduction to his classic text: "In the belief that a college course in the history of mathematics should be primarily a mathematics [his italics] course, an effort has been made to inject a considerable amount of genuine mathematics into this book. It is hoped that a student using this book will learn much mathematics, as well as history."
This text does indeed introduce the reader to the history of mathematics, but it is not overly historical in the sense that newer texts [those of Katz and Suzuki, for example] are. For the most part the notation is modern, as are the explanations. For the novice teacher, this works very well; with more experience one might like to see the mathematics as it was actually done at the time. All the usual topics are here. In fact, one could argue that they are the usual topics because Eves made them so! The book itself is a wonderful read. The emphasis, however, is on biography and anecdotes, not as much on mathematics. Oh, but what anecdotes! After all, this is the man who wrote the wonderful In Mathematical Circles series of books. It is the stories and trivia that will stick with the reader. For future mathematics teachers, this is definitely not a bad thing, as these stories can be used to motivate and entertain students in the mathematics classroom. The book also contains illustrations, maps, pictures of mathematicians, and pages of famous works. These not only enhance the readability of the text but can also be used by future mathematics teachers in their classrooms.
Mathematics is not done in isolation and Eves' text recognizes that. Throughout the text we find relationships between mathematicians and their times and the new "Cultural Connections" contribute to this emphasis. "Cultural Connections" are new to this edition and are written by the author's son, historian Jamie Eves. In these brief essays the younger Eves gives an overview of the history of each time period, including social and political topics. This places this text in the middle ground between books that pay little attention to background history and the relationship of mathematics to the broader culture and books [like those of Calinger and Cooke] that excel in this area.
What most people love about Eves' book [and this reviewer is no exception] are the "problem studies" and "essay topics" at the end of each chapter. While the problem studies are not always directly connected with the history contained in the chapter, they do fit the time period. These problems are non-trivial, but they allow the student to investigate for him/herself some interesting mathematics. They can also be done in the classroom. This is where Eves does "inject a considerable amount of genuine mathematics into this book." The problem studies and essay topics are a great help to the beginning history of mathematics instructor who wishes to assign projects to students but has yet to develop a collection of her own. These are also useful for other mathematics courses where the instructor wishes to interject some history into the course. One can find an appropriate problem study for most any course in the standard undergraduate curriculum. Instructors should be warned, however, that the book does include a fairly complete set of answers to the problem studies.
If you are teaching a history of mathematics course for the first time, you should give serious consideration to this text. You can feel comfortable in the knowledge that hundreds of your colleagues have taught successful courses using the book. If you have taught history of mathematics before, likely you have seen this book, but if not, get it and read it. You will use the anecdotes forever. If you do not teach history of mathematics, but want to enliven your class with ready-made historical projects, look no further.
Gary S. Stoudt, Professor of Mathematics, Indiana University of Pennsylvania
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