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MATH32012 - Commutative Algebra
Requisites
Brief Description
The central theme of this course is factorisation (theory and practice) in commutative rings; rings of polynomials are our main examples but there are others, such as rings of algebraic integers.
Polynomials are familiar objects which play a part in virtually every branch of mathematics. Historically, the study of solutions of polynomial equations (algebraic geometry and number theory) and the study of symmetries of polynomials (invariant theory) were a major source of inspiration for the vast expansion of algebra in the 19th and 20th centuries.
In this course the algebra of polynomials in n variables over a field of coefficients is the basic object of study. The course covers fairly recent advances which have important applications to computer algebra and computational algebraic geometry (Grbner bases - an extension of the Euclidean division algorithm to polynomials in 2 or more variables), together with a selection of more classical material.
Aims
The course unit will deepen and extend students' knowledge and understanding of commutative algebra. By the end of the course unit the student will have learned more about familiar mathematical objects such as polynomials and algebraic numbers, will have acquired various computational and algebraic skills and will have seen how the introduction of structural ideas leads to the solution of mathematical problems.
Learning Outcomes
On successful completion of this course unit students will be able to demonstrate
facility in dealing with polynomials (in one and more variables);
understanding of some basic ideal structure of polynomial rings;
appreciation of the subtleties of factorisation into prime and irreducible elements;
ability to compute generating sets and Gröbner bases for ideals in polynomial rings;
ability to relate polynomials to other algebraic structures such as algebraic varieties;
ability to solve problems relating to the factorisation of polynomials, irreducible polynomials and polynomial equations in several variables.
Future topics requiring this course unit
None, though the material connects usefully with algebraic geometry and Galois theory.
Teaching & Learning Process (Hours Allocated To)
Assessment and Feedback
Coursework: weighting 20%
End of semester examination: two hours weighting 80%
Further Reading
[1,2] are are useful general references on algebra though neither covers more advanced content of the course such as Grbner bases. For Grbner bases see [3]. A treatment of factorisation as well as Grbner bases, with many exercises, is given in [4]. [5] is a new textbook which combines Grbner bases with more traditional material and contains some practical examples. |
MTH 115
Math for Teachers Grades P-6
Course info & reviews
Service course . Topics include problem solving, numeration, computation, number theory, and rational numbers. Designed to provide content background for teaching mathematics in elementary grades. Successful completion of this course may require an examination in basic mathematics. Open only to early childhood or middle childhood major...
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MTH 1150
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My instructor was great, the book helpful though I did like the e version better it was nice to be able to book mark pages cut text as well as many of the other options for the e book. |
Free & inexpensive math CURRICULUM materials and resources
I will try to collect here a list of free or cheap math curriculum resources. They could be either a complete or partial curriculum, or curriculum-related, such as scope and sequence or diagnostic tests related.
Free downloadable materials
Mathematics Enhancement Programme (MEP) - School curriculum materials Entire workbooks to download (PDFs) for FREE for grades 1-10 for non-commercial purposes. These are good quality, too, and not just machine-made worksheet collections. Some homeschoolers use MEP for a complete curriculum. Includes some interactive materials. Provided by Centre for Innovation in Mathematics Teaching.
XYZ Online Textbooks
Complete textbooks for Introductory Mathematics (prealgebra), Introductory Algebra, and Intermediate Algebra that are available for free online reading, evaluation, and even printing. For each section of the book, the website also shows links to corresponding math videos from MathTV.com.
Shmoop.com Prealgebra
Free learning guides (tutorials) for all prealgebra topics with interactive practice problems, step-by-step examples, graphs, and real-world applications. This can be used for an online pre-algebra textbook.
AlgebraFree.com
Includes a free algebra 1 textbook called Llevada's Algebra 1, downloadable as PDF files by chapter, containing over 6,000 exercises, plus over 70 videos with lectures and solved exercises.
Curriculum Improvement Project - PASS books
Free, yet complete, math books in downloadable form (PDF files) for algebra, pre-algebra, liberal arts mathematics, and other math courses from the Parallel Alternative Strategies for Students (PASS) Building General Math Skills. These books are written to help students with various learning needs and are presented in an easy-to-understand format. Teacher guides are also available.
Pre-algebra Study Guide worksheets
Over 100 very nice, compact prealgebra worksheets from the Glencoe Parent and Student Study Guide. The sheets have a short, instructional part that reviews the concept, several problems, and answers upside down in the end of each sheet.
She Loves Math
Online math lessons geared for girls, from counting through calculus. By math tutor and "mathaholic" Lisa Johnson.
Understanding algebra A complete online algebra text for Algebra I by James W.
Brennan. Also a printable version.
Calculus Made Easy
A free download of an old textbook, acclaimed to be a lively introduction to calculus, with clarity and simplicity.
Free videos, animated math courses, or online math practice sites
Khan Academy
Possibly the web's biggest and free site for math videos. What started out as Sal making a few algebra videos for his cousins has grown to over 2,100 videos and 100 self-paced exercises and assessments covering everything from arithmetic to physics, finance, and history.
BrightStorm Math
Over 2,000 free videos covering all high school math topics from algebra to calculus. Registration required (free).
MathVids.com
Free math videos grouped by math topic. Basic math, middle school, high school, and college levels. Also includes links to further resources.
Virtual Nerd
Video tutorials for prealgebra, algebra 1, algebra 2, and intro physics. This will also include practice problems and quizzes sometime during 2010-2011 school year. Includes both a free and paid (premium) versions.
Sophia
A large online library of short video lessons and quizzes. The videos are recorded by several different tutors, who vary in their methods and teaching styles, so you have the chance to learn the same lesson from several perspectives.
Algebra 2 Video Lessons
This site includes online video lessons and other support materials to accompany Holt Algebra 2 textbook. Click on the Homework help, choose a chapter, and then view the lessons.
Mathematics Illuminated
Mathematics Illuminated is a 13-part, integrated-media resource created for adult learners and high school teachers. The series covers the broad scope of human knowledge through the study of mathematics and its relevance in the world today. It reaches beyond formulas and computations to explore the math of patterns, symmetry, relationships, multiple dimensions, and more, all the while uncovering the secrets and hidden delights of the ever-evolving world of mathematics. This course us videos, online texts, web interactive activities, and group activities.
The Futures Channel
A website featuring short movies that show how algebra or science is used in the real world, plus supplemental activities & lesson plans. On the website you can view the movie clips and activities for free, or purchase a DVD that has them.
Miscellaneous math materials
Mathematics Benchmarks, Grades K-12
These are from Charles A. Dana Center at The University of Texas at Austin. The benchmarks describe the content and skills necessary for students on any given grade (K-6), or by strands (K-6 and 7-12). You can use these to have an idea of what topics to cover on any grade. |
Thinking Mathematicallyitzer continues to raise the bar with his engaging applications developed to motivate students from diverse majors and backgrounds. Thinking Mathematically, Fifth Edition, draws from the author's unique background in art, psychology, and math to present math in the context of real-world applications.Students in this course are not math majors, and they may never take a subsequent math course, so they are often nervous about taking the class. Blitzer understands those students' needs and provides helpful tools in every chapter to help them mas... MOREter the material. Voice balloons appear right when students need them, showing what an instructor would say when leading a student through the problem. Study tips, chapter review grids, Chapter Tests, and abundant exercises provide ample review and practice.The Fifth Edition's MyMathLabreg; course boasts more than 2,000 assignable exercises, plus a new question type for applications-driven questions that correlate to every section of the textbook. Chapter Test Prep Videos show students how to work out solutions to the Chapter Tests; the videos are available in MyMathLab and on YouTubetrade;.
11.5 Probability with the Fundamental Counting principle, Permutations, and Combinations
11.6 Events Involving Not and Or; Odds
11.7 Events Involving And; Conditional Probability
11.8 Expected Value
12. Statistics
12.1 Sampling, Frequency Distributions, and Graphs
12.2 Measures of Central Tendency
12.3 Measures of Dispersion
12.4 The Normal Distribution
12.5 Problem Solving with the Normal Distribution
12.6 Scatter Plots, Correlation, and Regression Lines
13. Mathematical Systems
13.1 Mathematical Systems
13.2 Rotational Symmetry, Groups, and Clock Arithmetic
14. Voting and Apportionment
14.1 Voting Methods
14.2 Flaws of Voting Methods
14.3 Apportionment Methods
14.4 Flaws of Apportionment Methods
15. Graph Theory
15.5 Graphs, Paths, and Circuits
15.2 Euler Paths and Euler Circuits
15.3 Hamilton Paths and Hamilton Circuits
15.4 Trees's love for teaching mathematics was nourished for nearly 30 years at Miami Dade College, where he received numerous teaching awards, including Innovator of the Year from the League for Innovations in the Community College and an endowed chair based on excellence in the classroom. In addition to Thinking Mathematically, Bob has written textbooks covering introductory algebra, college algebra, algebra and trigonometry, and precalculus, all published by Prentice Hall. When not secluded in his Northern California writer's cabin, Bob can be found hiking the beaches and trails of Point Reyes National Seashore, and tending to the chores required by his beloved entourage of horses, chickens, and irritable roosters. |
Applied Mathematics - 4th edition
Summary: Applied Mathematics is a comprehensive text designed to benefit students in various fields of study. Text content emphasizes the application of mathematics to a variety of vocational and technical areas. The text uses realistic applications to develop problem-solving skills and provide an understanding of the importance of math in the real world. --This text refers to an alternate Hardcover edition.
Acceptable Fourth Edition, Text. Pre-loved books for the budget-conscious consumer. With more than 50 years' experience, we aim to please with immediate shipping and fast, friendly service. All orde...show morers ship on |
Study Guide for the Semester 1 Exam- PreCalculus
Start Preparing Early:
Go back and review your notes from Modules 1 – 5.
Go back and review your old exams. Rework problems that you missed and make sure you
understand the mistake you made.
Each chapter in your book as "Study Aides". Use these as extra practice.
Review the Graded HW from each lesson.
Review the Practice Exercises that were assigned in each lesson.
Think about EVERYTHING you know about the following topics:
1. Odd Functions, Even Functions
2. Inverses
3. Composition Functions
4. Shifting Functions
5. What are the steps for creating a STAT Plot based on a table of data?
6. What are the steps for finding the linear regression equation using the calculator?
7. Vertical Asymptotes, Horizontal Asymptotes
8. Zero of a function, Root of a function
9. Imaginary Roots
10. Maximum or Minimum of a Function
11. Possible Rational Roots
12. Complex Conjugate
13. Change of Base Formula for Logarithms
14. Properties used to expand or condense logarithmic expressions
15. Solving a logarithmic equation
16. Solving an exponential equation
17. Domain/Range of Logarithmic Functions
18. Domain/Range of Exponential Functions
19. The acronym ASTC
20. The acronym SOH-CAH-TOA
21. Csc(x), cot(x), sec(x)
22. How do you obtain a good graph for y=csc(x) using the graphing calculator?
23. When graphing trig functions, what is amplitude? Period? Horizontal Shift? Vertical Shift?
24. How do you find the vertical asymptotes of a trig function?
25. Given a trig equation, how do you solve it algebraically? how can you solve it graphically?
26. What does it mean to "verify the identity"
27. Explain some of the strategies you can use to simplify trig expressions.
28. Sum and Difference Formulas
29. Reviewing the HW problems in 5.05, what are some of the ways we used the sum/difference
formulas in the graded assignment?
30. Pythagorean Identities
31. Inverse Trigonometric Function
32. Review the HW to find out how to find the EXACT value of an inverse trig function.
33. Review the HW to find out how to use the graphing calculator to find the value of an inverse trig
function |
Elementary and Intermediate Algebra - 4th edition
Summary: Ideal for lecture-format courses taught at the post-secondary level, ELEMENTARY AND INTERMEDIATE ALGEBRA, Fourth Edition, makes algebra accessible and engaging. Author Charles ''Pat'' McKeague's passion for teaching mathematics is apparent on every page, and with many years of experience teaching mathematics, he knows how to write in a way that you will understand and appreciate. His attention to detail and exceptionally clear writing style help you to move through each new concept w...show moreith ease, and real-world applications in every chapter highlight the relevance of what you are0840064195293.96 |
Synopses & Reviews
Publisher Comments:
Commutative Algebra is best understood with knowledge of the geometric ideas that have played a great role in its formation, in short, with a view towards algebraic geometry. The author presents a comprehensive view of commutative algebra, from basics, such as localization and primary decomposition, through dimension theory, differentials, homological methods, free resolutions and duality, emphasizing the origins of the ideas and their connections with other parts of mathematics. Many exercises illustrate and sharpen the theory and extended exercises give the reader an active part in complementing the material presented in the text. One novel feature is a chapter devoted to a quick but thorough treatment of Grobner basis theory and the constructive methods in commutative algebra and algebraic geometry that flow from it. Applications of the theory and even suggestions for computer algebra projects are included. This book will appeal to readers from beginners to advanced students of commutative algebra or algebraic geometry. To help beginners, the essential ideals from algebraic geometry are treated from scratch. Appendices on homological algebra, multilinear algebra and several other useful topics help to make the book relatively self- contained. Novel results and presentations are scattered throughout the text.
Synopsis:
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by Springer Verlag, |
Practical Math Success - 3rd edition
Summary: This book is geared toward anyone wishing to overcome math anxiety. Updated and re-evaluated by math experts to ensure the most current lessons and practice exercises, this resource includes: essential math basics and tips for test-takers. |
first-year calculus roughly in the order in which it was first discovered. The first two chapters show how the ancient calculations of practical problems led to infinite series, differential and integral calculus and to differential equations. The establishment of mathematical rigour for these subjects in the 19th century for one and several variables is treated in chapters III and IV. Many quotations are included to give the flavor of the history. The text is complemented by a large number of examples, calculations and mathematical pictures and will provide stimulating and enjoyable reading for students, teachers, as well as researchers. |
Short description
Grade 9 & 10 math eBooks comprise three principle sections. These are, notably: (Read more) math eBooks are produced such as that as well as a Publications Guide, and three principle publications corresponding to the principle sections (Number and Algebra, Geometry and Measures and Statistics) there are individual modules produced within each principle section which are published as eBooks.
Elementary and Circle Geometry is a module within the Geometry and Measures principle section of our Grade 9 & 10 publications. (Less) |
Prerequisite recommended: Mathematics 20S and/or a good knowledge of basic arithmetic skills – addition, subtraction, multiplication and division – as well as fractions, decimals and percent.
Description: This course will review, in a problem-solving context, the basic concepts of whole numbers, fractions, decimals and per cent in the first six weeks. Students will then study algebra, geometry and trigonometry up to an intermediate level, concluding with graphing basic functions, spreadsheets and trigonometry. It is highly recommended that anyone planning to take Applied 40S take this course. |
approac... read more
Elementary Mathematics from an Advanced Standpoint: Geometry by Felix Klein This comprehensive treatment features analytic formulas, enabling precise formulation of geometric facts, and it covers geometric manifolds and transformations, concluding with a systematic discussion of fundamentals. 1939 edition. Includes 141 figures.
A Modern View of Geometry by Leonard M. Blumenthal Elegant exposition of the postulation geometry of planes, including coordination of affine and projective planes. Historical background, set theory, propositional calculus, affine planes with Desargues and Pappus properties, much more. Includes 56 figuresAnalytical Conics by Barry Spain This concise text introduces analytical geometry, covering basic ideas and methods. An invaluable preparation for more advanced treatments, it features solutions to many of its problems. 1957 edition.
Product Description:
approach. An introductory chapter leads to discussions of projective geometry's axiomatic foundations: establishing coordinates in a plane; relations between the basic theorems; higher-dimensional space; and conics. Additional topics include coordinate systems and linear transformations; an abstract consideration of coordinate systems; an analytical treatment of conic sections; coordinates on a conic; pairs of conics; quadric surfaces; and the Jordan canonical form. Numerous figures illuminate |
hard to understand at first but after awhile you get used to it He doesnt understand questions very well and take him awhile to understand what youre asking he makes a lot of mistakes in his examples He writes really fast and you basically have to scribble it down as fast as possible which makes it hard to listen and comprehend whats going on
He is a very nice person, but he isnt very good with explaining the material. Hard to understand. His exams are short and not so hard but confusing. If you miss class u will miss a lot because he covers a lot in each lecture. He gives many examples in class and exams are very similar to those examples. dont miss the DGD's because they help a lot.
If you struggle with math, don't take it with this prof. I know a lot of people complain about Li's accent, but it's really not the problem... He just can't teach. All he does is write problems on the board and expects you to know how it's done. If you can't learn this way please for the love of god, do not take this prof.
He's a nice guy but horrible prof. He is the hardest marker, one mistake will make you lose majority of your marks for that question. He asks complicated questions at times and the class avg was 40% for both midterms(so bad he said the final can be our final mark). Avoid him, switch out.
I attended every class and by the end of the semester started experiencing rage toward him. Never ever ever ever again will I permit myself to take a course with him. He does not TEACH. Throws examples at you with minimal transitional 'explanations' and sends you home. Thank god for an amazing TA! Never, ever, ever, ever again!
Horrible prof. He does not know how to teach. I would have really appreciated to have a math TEACHER not a prof who simply copies his own examples on the board.. I recommend that you read the text before attending class--his notes are only a source of examples. If I were you I'd try switch out of this prof's course while I still had the chance!
Good prof, takes his midterm questions from the examples done in class. People like to complain about and blame his accent for their problems in the course but if you just pay attention and get over the fact that english is not his first language he's not hard to understand. show up to class and pay attention and you'll do well.
Going to class is beneficial since he uses his own examples and notes. He often explains the concepts more clearly than the textbook does. His exams are fairly easy if you take the time to learn the material. His accent makes him hard to follow sometimes but you can usually keep up with him.
I don't know why he decided to teach the course in some random order; makes the whole thing messy and impossible to follow properly. Will give you the worst examples possible, skipping steps and simply copying his notes, while making mistakes. Goes so fast copying his notes that you won't catch a thing, and also unreadable. Course is easy tho.
He's a little hard to understand sometimes because of his accent, but he is a good prof overall. He will always answer questions if people don't understand something in the lectures and basically tells you what to study for the exams. Do the textbook questions he assigns and pay attention in class and you'll do fine.
Well, you will have hard time in the beginning. His accent needs more careful lol. His notes are good. Do the textbook exercises! I was almost failing the course but I did very good in FINAL and he gave me A!! So work hard! Very useful when it comes to his office hours!
I have failed Calc II before. Why? Because I just had a really bad attitude towards math plus stupid choices of not studying.I knew I HAD to pass it this time. I admit it took a while to get confidence. Weixuan Li presents math by answering: How, Why do we do this calculation, & What YOU need to know how to do! It was clear, and AWESOME.
He gives by far the worst notes. Terrible explanations, usually does examples by skipping important steps. Asking questions is a no no, would just erase the board and rewrite it without answering the question. Class average for midterms was so low (~35%) that your final is your mark. Sometimes funny. |
Elementary and Intermediate's students are visual learners, and Angel/Runde offers a visual presentation to help them succeed in math. Visual examples and diagrams are used to explain concepts and procedures. New Understanding Algebra boxes and an innovative color coding system for variables and notation keep students focused. Short, clear sentences reinforce the presentation of each topic and help students overcome language barriers to learn math. TheAngel author teammeets the needs of todayrs"s learners by pairing concise explanations... MORE with the new Understanding Algebra feature and an updated approach to examples. Discussions throughout the text have been thoroughly revised for brevity and accessibility. Whenever possible, a visual example or diagram is used to explain concepts and procedures. Understanding Algebra call-outs highlight key points throughout the text, allowing readers to identify important points at a glance. The updated examples use color to highlight the variables and important notation to clearly illustrate the solution process. For all readers interested in algebra.
2.4 Solving Linear Equations with a Variable on Only One Side of the Equation
Mid-Chapter Test: Sections 2.1—2.4
2.5 Solving Linear Equations with the Variable on Both Sides of the Equation
2.6 Formulas
2.7 Ratios and Proportions
Chapter 2 Summary
Chapter 2 Review Exercises
Chapter 2 Practice Test
Cumulative Review Test
3. Applications of Algebra
3.1 Changing Application Problems into Equations
3.2 Solving Application Problems
Mid-Chapter Test: Sections 3.1—3.2
3.3 Geometric Problems
3.4 Motion, Money, and Mixture Problems
Chapter 3 Summary
Chapter 3 Review Exercises
Chapter 3 Practice Test
Cumulative Review Test
4. Graphing Linear Equations
4.1 The Cartesian Coordinate System and Linear Equations in Two Variables
4.2 Graphing Linear Equations
4.3 Slope of a Line
Mid-Chapter Test: Sections 4.1—4.3
4.4 Slope-Intercept and Point-Slope Forms of a Linear Equation
Chapter 4 Summary
Chapter 4 Review Exercises
Chapter 4 Practice Test
Cumulative Review Test
5. Exponents and Polynomials
5.1 Exponents
5.2 Negative Exponents
5.3 Scientific Notation
Mid-Chapter Test: Sections 5.1—5.3
5.4 Addition and Subtraction of Polynomials
5.5 Multiplication of Polynomials
5.6 Division of Polynomials
Chapter 5 Summary
Chapter 5 Review Exercises
Chapter 5 Practice Test
Cumulative Review Test
6. Factoring
6.1 Factoring a Monomial from a Polynomial
6.2 Factoring by Grouping
6.3 Factoring Trinomials of the Form ax2 + bx + c, a = 1
6.4 Factoring Trinomials of the Form ax2 + bx + c, a ≠ 1
Mid-Chapter Test: Sections 6.1—6.4
6.5 Special Factoring Formulas and a General Review of Factoring
6.6 Solving Quadratic Equations Using Factoring
6.7 Applications of Quadratic Equations
Chapter 6 Summary
Chapter 6 Review Exercises
Chapter 6 Practice Test
Cumulative Review Test
7. Rational Expressions and Equations
7.1 Simplifying Rational Expressions
7.2 Multiplication and Division of Rational Expressions
7.3 Addition and Subtraction of Rational Expressions with a Common Denominator and Finding the Least Common Denominator
7.4 Addition and Subtraction of Rational Expressions
Mid-Chapter Test: Sections 7.1—7.4
7.5 Complex Fractions
7.6 Solving Rational Equations
7.7 Rational Equations: Applications and Problem Solving
7.8 Variation
Chapter 7 Summary
Chapter 7 Review Exercises
Chapter 7 Practice Test
Cumulative Review Test
8. Functions and Their Graphs
8.1 More on Graphs
8.2 Functions
8.3 Linear Functions
Mid-Chapter Test: Sections 8.1—8.3
8.4 Slope, Modeling, and Linear Relationships
8.5 The Algebra of Functions
Chapter 8 Summary
Chapter 8 Review Exercises
Chapter 8 Practice Test
Cumulative Review Test
9. Systems of Linear Equations
9.1 Solving Systems of Equations Graphically
9.2 Solving Systems of Equations by Substitution
9.3 Solving Systems of Equations by the Addition Method
9.4 Solving Systems of Linear Equations in Three Variables
Mid-Chapter Test: Sections 9.1—9.4
9.5 Systems of Linear Equations: Applications and Problem Solving
9.6 Solving Systems of Equations Using Matrices
9.7 Solving Systems of Equations Using Determinants and Cramer's Rule
Chapter 9 Summary
Chapter 9 Review Exercises
Chapter 9 Practice Test
Cumulative Review Test
10. Inequalities in One and Two Variables
10.1 Solving Linear Inequalities in One Variable
10.2 Solving Equations and Inequalities Containing Absolute Values
Mid-Chapter Test: Sections 10.1—10.2
10.3 Graphing Linear Inequalities in Two Variables and Systems of Linear Inequalities
Chapter 10 Summary
Chapter 10 Review Exercises
Chapter 10 Practice Test
Cumulative Review Test
11. Roots, Radicals, and Complex Numbers
11.1 Roots and Radicals
11.2 Rational Exponents
11.3 Simplifying Radicals
11.4 Adding, Subtracting, and Multiplying Radicals
Mid-Chapter Test: Sections 11.1—11.4
11.5 Dividing Radicals
11.6 Solving Radical Equations
11.7 Complex Numbers
Chapter 11 Summary
Chapter 11 Review Exercises
Chapter 11 Practice Test
Cumulative Review Test
12. Quadratic Functions
12.1 Solving Quadratic Equations by Completing the Square
12.2 Solving Quadratic Equations by the Quadratic Formula
12.3 Quadratic Equations: Applications and Problem Solving
Mid-Chapter Test: Sections 12.1—12.3
12.4 Factoring Expressions and Solving Equations That Are Quadratic in Form
12.5 Graphing Quadratic Functions
12.6 Quadratic and Other Inequalities in One Variable
Chapter 12 Summary
Chapter 12 Review Exercises
Chapter 12 Practice Test
Cumulative Review Test
13. Exponential and Logarithmic Functions
13.1 Composite and Inverse Functions
13.2 Exponential Functions
13.3 Logarithmic Functions
13.4 Properties of Logarithms
Mid-Chapter Test: Sections 13.1—13.4
13.5 Common Logarithms
13.6 Exponential and Logarithmic Equations
13.7 Natural Exponential and Natural Logarithmic Functions 818
Chapter 13 Summary 829
Chapter 13 Review Exercises 832
Chapter 13 Practice Test 835
Cumulative Review Test 836
14. Conic Sections
14.1 The Parabola and the Circle
14.2 The Ellipse
Mid-Chapter Test: Sections 14.1—14.2
14.3 The Hyperbola
14.4 Nonlinear Systems of Equations and Their Applications
Chapter 14 Summary
Chapter 14 Review Exercises
Chapter 14 Practice Test
Cumulative Review Test
15. Sequences, Series, and the Binomial Theorem
15.1 Sequences and Series
15.2 Arithmetic Sequences and Series
15.3 Geometric Sequences and Series
Mid-Chapter Test: Sections 15.1—15.3
15.4 The Binomial Theorem
Chapter 15 Summary
Chapter 15 Review Exercises
Chapter 15 Practice Test
Cumulative Review Test
Appendices
A. Review of Decimals and Percent
B. Finding the Greatest Common Factor and Least Common Denominator
C. Geometry
D. Review of Exponents, Polynomials, and Factoring
Answers
Applications Index
Subject Index
Allen R. Angel received his AAS in Electrical Technology from New York City Community College. He then received his BS in Physics and his MS in Mathematics from SUNY at New Paltz, and he took additional graduate work at Rutgers University. He is Professor Emeritus at Monroe Community College in Rochester, New York where he served for many years as the chair of the Mathematics Department. He also served as the Assistant Director of the National Science Foundation Summer Institutes at Rutgers University from 1967–73. He served as the President of the New York State Mathematics Association of Two Year Colleges (NYSMATYC) and the Northeast Vice President of the American Mathematics Association of Two Year Colleges (AMATYC). He is the recipient of many awards including a number of NISOD Excellence in Teaching Awards, NYSMATYC's Outstanding Contributions to Mathematics Education Award, and AMATYC's President Award. Allen enjoy tennis, worldwide travel, and visiting with his children and granddaughter.
Dennis Runde received his BS and MS in mathematics from the University of Wisconsin—Platteville and Milwaukee, respectively. He has a PhD in Mathematics Education from the University of South Florida. He has been teaching for twenty years at State College of Florida, Manatee, and Sarasota Counties and for ten years at Saint Stephen's Episcopal School. Besides coaching little league baseball, his other interests include history, politics, fishing, canoeing, and cooking. He and his wife Kristin stay busy keeping up with their three sons–Alex, Nick, and Max. |
McGraw-Hill's Conquering the ACT Math
You can read this ebook online in eb20 format without having to download anything.
more
WE WANT TO HELP YOU SUCCEED ON THE ACT* MATH SECTION
If math is the hardest part of the ACT for you, we're here to help. McGraw-Hill's Conquering ACT Math has been specially designed and created by experienced ACT coaches. They'll give you test-smart strategies for answering every kind of ACT math question. You'll also get intensive practice with every question type to help you build your test-taking confidence. With McGraw-Hill's Conquering ACT Math, you'll have everything you need to get test-ready-and achieve your best ACT math score. |
More editions of Information guides: A survey of subject guides to sources of information produced by library and information services in the United Kingdom (British Library research & development report):
The International Mathematics Tournament of Towns is a problem solving competition in which teams from different cities are handicapped according to the population of the city. Ranking only behind the International Mathematical Olympiad, the competition had its origins in Eastern Europe (as did the Olympiad) but is now open to cities throughout the world.
This book, of 169 pages, contains a unique record of the problems and solutions of the Tournament from 1993 to 1997. They have been translated from the Russian, with solutions composed in English by an international panel of mathematicians including Andy Liu of Canada and others. [via]
More editions of International Mathematics Tournament of Towns, Book 4: 1993-1997 (Enrichment Series, Volume 15):
This book introduces senior students aspiring to Olympiad competition to particular mathematical problem solving techniques. The book contains formal treatments of methods with which students may be familiar, or may introduce the student to new, sometimes powerful, techniques. [via]
More editions of Methods of Problem Solving, Book 1 (Enrichment Series, Volume 9): |
Algebraic Thinking
Moderator
Good afternoon and welcome to today's chat with NCTM President Cathy Seeley. Today's topic is algebraic thinking.
Here's our first question:
Question from:
Chico, California
One thing that might help develop some algebraic thinking is to present more problems that students have to be able to solve with any variation of the original given constraints. For example, if the question were "The pool manager Jim always keeps 100 gallons of water in his pool. At the end of each day he checks the pool to see how much water is in it. On Monday, only 82 gallons of water was left in the pool. How many gallons of water does Jim need to add?" The kids can figure this out with subtraction fairly easily. However, if there were more parts to the question like, "On Tuesday there was only 52 gallons left. How many gallons does Jim need to add now?" And then a final part to the question, "Write an equation that will tell Jim how many gallons to add at the end of any given day." This type of thinking will help them to develop the concept of variables and equations, and it is done in a fairly simple way.
Cathy Seeley:
This is exactly the kind of thinking that helps students develop the ability to use algebraic thinking to make generalizations. And this is just what we want students to do with algebra—to think beyond specific examples to more general cases. I prefer this idea to just learning recipes to solve problem 'types' (like coin problems, age problems, mixtures, etc.).
Question from
Missoula, Montana
Is algebra more than what most of us learned with x's and y's and many homework problems? You seem to imply that.
Cathy Seeley:
Algebra can be a powerful set of tools for representing situations, analyzing mathematical relationships, making generalizations and solving problems. It can extend well beyond the limited types of problems once filling traditional algebra texts to serving as a set of approaches in a student's mathematical toolkit. Today, algebra can be used to deal with data and make predictions. It can be used to model sophisticated situations from science, social studies or economics, to name a few. If we do our job well to develop algebraic thinking across the grades, American students will never be heard to say that they had no use for algebra. Rather, they will incorporate algebraic techniques into their broader mathematical thinking to deal with everyday life as well as advanced applications in mathematics and science.
Question from
Fayetteville North Carolina
One of the red flags I see in developing algebraic thinking is how we develop students' conceptual understanding of equality, especially in the early years. We must allow students to discuss and explore all aspects of what equality is and is not in order to break away from the idea that math is all about "the answer." Number relationships and balanced equations may not boil down to a nice little number for an answer, and it's important in the developing years for students to understand this.
Cathy Seeley:
I couldn't agree with you more. Equivalence/equality is undoubtedly one of the most important, connecting ideas in school mathematics. As you note, too many students see the equal sign as what precedes the answer. Developing this concept of equivalence calls for lots of experiences with materials as students are developing their conceptual understanding of numbers and operations. More important, it calls for teachers to help students connect their experiences with the mathematical idea(s) they are developing, in this case, equivalence or equality.
Question from
Lanham, Maryland
A major mistake is when we don't tell our first grade students the truth about the numbers. We should tell them the whole story about the numbers. We should tell them about positive and negative numbers both and also tell them that the real numbers are a part of all the numbers, and there are more numbers to learn (complex numbers). I have seen many students who think 2 minus 4 can't be done by asking, "If we have only two things, how can we take four out of them?" This line of thinking is because of not introducing students to negative numbers and algebraic thinking early. I hope we also show the power of algebra to our young students by letting them use it to solve problems. Thanks for asking.
Cathy Seeley:
Thanks for sharing this perspective. I recently visited a second-grade classroom where the (excellent) teacher was leading students in solving problems using subtraction. I heard her tell students to remember that you can't take a bigger number away from a smaller one. Unfortunately, this message, if sent often enough, can plant a future misunderstanding related to negative numbers and to the meaning of subtraction. Better to keep coming back to what the numbers represent and what the problem calls for in terms of a solution based on the particular situation. I'm not sure I'd advocate negative numbers with first-graders, but it's an interesting concept.
Question from
Omaha, Nebraska
I am a community college math teacher. Approximately 90 percent of the students who take our placement test (COMPASS, through ACT) assess into a developmental class in at least one of the areas: math, reading, English.
Yesterday I had a student wanting into my Intermediate Algebra class who assured me he had passed Algebra II in high school this spring. He could not combine like terms (he thought 5x - 5 = x), he could not graph a linear equation, he could not factor x2 - 5x -6, and so on. I don't really mind, since it does mean job security for me, but what is the disconnect between grades and knowledge?
Cathy Seeley:
This is a troubling phenomenon. My opinion is that the disconnect between grades and knowledge is more about how we teach than about how we grade. It is quite likely that this student, like others, passed a test at one time that called for these skills. I am guessing that this learning was temporary and superficial if it was based on lecture and memorization. Unless we connect algebraic skills to meaningful situations, build conceptual understanding and facilitate the use of algebraic thinking to solve problems, we risk perpetuating the study of algebra as a vast set of meaningless abstract rules and procedures that, once learned, may be quickly forgotten, never to return. The key, I believe, lies both in what content we choose to teach, and in how we engage students in their learning; watch for my next two President's Messages on this one-two punch. (By the way, in my recent teaching experience in a French system in West Africa, I noted what I had read elsewhere—that other countries do not use factoring as a general method for solving quadratic equations.) Thanks for bringing up this important and all too common problem.
Question from
Murfreesboro, Tennessee
I feel that students need to practice critical thinking skills in the elementary grades. This can be done throughout the curriculum, not just in the mathematics courses.
Cathy Seeley:
I couldn't have said it better myself. Not just starting earlier, but incorporating critical thinking across the curriculum. I believe that the different ways we teach problem solving and critical thinking in various content areas can help students access learning in different ways, based on their strengths.
Question from
Riverside, California
What is algebraic thinking?
We think of algebra as an art of symbol manipulation (e.g., let "x" be an unknown.) But I sense that teachers of algebra are not offering sufficient training in axiomatic thinking.
To prove the point, not a single algebra teacher in my experience could tell an inquisitive student, why (-a)x(-b) is PLUS (a)x(b).
Sure, there are hokey, plausibility arguments, like, "Why not?" or "Symmetry demands it." In fact, the axioms of distributivity, associativity, etc. come into play. Also, the real explanation may be too subtle for the classroom, which the teacher should acknowledge.
MORAL: Although the result may sound elementary, its justification may not be. The teacher should have the discernment to know how much to teach and how much to omit.
Cathy Seeley:
Your point brings up an important issue and one of NCTM's primary goals: providing high-quality professional development. This year's professional development focus of the year on algebraic thinking supports the kind of teacher understanding that you describe. Some universities and regional educational facilities offer high-quality professional development that gets to the heart of algebraic thinking. Unfortunately, such experiences are not accessible to all teachers. A rich set of resources to help teachers expand their deep understanding of algebraic concepts can be found throughout this year on NCTM's Web site. Look for the magnifying-glass icon on the home page and throughout the site to lead teachers to articles, books, and online resources. Teachers can use these individually, or, ideally, with colleagues as starting points for professional growth.
Question from
University Park, Pennsylvania
I think we need to clarify what we mean by "algebraic thinking." I believe too much emphasis is placed on patterns (which is an important idea). As I look at the Japanese curricular materials, what seems to be missing in the typical US approach is the emphasis on writing equations. We rarely discuss the importance of expressing our mathematical thinking using mathematical expressions and equations, nor trying to interpret mathematical thinking expressed as an equation or expression. Thus, for too many children expressions like 2x(5+3) simply means to calculate and find the answer.
This is just one simple example to illustrate the need for mathematics teachers to critically re-evaluate what we mean by "algebraic thinking." Representing our thinking through mathematical symbolisms must be an important part of algebraic thinking.
Cathy Seeley:
Your comments reinforce the importance of being able to use multiple types of representations for a situation, as called for in Principles and Standards for School Mathematics. The symbolic representation is an important part of a student's algebraic development. I hope you would agree that this representation needs to be well grounded in understanding what lies behind the symbols. By incorporating algebraic opportunities throughout the curriculum and across grades, we can help students develop all types of representational skills.
Question from
Archbald, Pennsylvania
When it comes to middle school math in the United States (specifically grade 8), what percent of the students do you think should be taking an Algebra 1 course as an elective instead of taking the traditional 8th grade math course, which includes some algebra concepts?
Cathy Seeley:
I don't think there is a set percent of students we can identify. Of more importance is the question of what is the nature of the middle school curriculum? A middle school program that addresses central features of Principles and Standards in School Mathematics, including such key ideas as proportionality and the transition to algebra, can clearly be beneficial to many students, much more so than what we might consider 'traditional' eighth-grade mathematics. We are no longer in a time, as we were when I started teaching middle school mathematics 35 years ago, when the middle school curriculum is simply a repeat of K-6 arithmetic. Now that we have so much to offer, there is far less reason to accelerate students into algebra. At the same time, if a student is to be able to take calculus in high school, or another course beyond the level of pre-calculus, they must either start the high school sequence in grade 8 or find a way to double up during high school. In summary, the major reason to accelerate: opening up options for advanced mathematics study to all students, especially those not typically represented in such courses. Cautions against accelerating: make sure the student is motivated to continue mathematics study every year in high school; make sure there are good course offerings for students in 11th and 12th grade; ensure that students have the benefit of the rich mathematics (especially proportionality) that should be part of the middle school program.
Question from
Honolulu, Hawaii
I think the ideas around equality or equivalence in the early grades are very important, as the question from North Carolina implied. In fact, young children can handle much more sophistication in ideas than we might have expected. This causes us to think about early mathematics in different ways.
Cathy Seeley:
Absolutely. As we rethink both priorities and opportunities at the elementary grades, we can likely not only build a foundation for algebraic thinking, but also strengthen students' development of concepts related to number, operations, and other mathematical strands.
Question from
Seattle
I attended the 6–8 and 9–12 Navigations E-workshops, which were very good in showing algebraic thinking across the grades and the development across the grades.
Cathy Seeley:
Great! This model for professional development is one we think holds a lot of promise, and the Council will continue to explore how we can use it to meet the needs of more teachers. In whatever form our professional development takes, this notion of developing this important thread of algebraic thinking across the grades is critical, starting with a student's earliest school experience.
Question from
Turners Falls, Massachusetts
Are we perpetuating an artificial segregation of math content (algebra, geometry, etc.) by emphasizing "algebraic thinking" as a concept? Why not place the emphasis on mathematical thinking?
Cathy Seeley:
Mathematical thinking is very appropriate for our broader attention. However, the term may be too broad. While all teachers might agree that they should help students learn to think, many elementary teachers, in particular, don't see where algebra fits in with their teaching. By focusing on what skills and knowledge build toward algebraic understanding, teachers can see more easily how these might fit with their teaching. For example, a teacher might agree that working with patterns, developing the notion of equivalence, and exploring relationships are important, even though they might not have thought of these as topics related to algebra. By making explicit what we mean by algebraic thinking, we can increase the likelihood that this type of experience can be part of what students do in school. But, as you observe, the bigger picture demands that we not teach algebraic thinking in isolation from other strands, but rather capitalize on our attention to algebraic thinking to find ways to connect it with other important parts of the curriculum toward a student's broader mathematical thinking.
Question from
Rochester, Michigan
I have two favorite problems that can help accomplish your goal. These are mathematical computations that are very down-to-earth and real-life, and they are things that everyone should have been empowered to solve by the time they leave elementary school. They are arithmetic but very much lead to algebraic thinking.
1. You and your best friend went on a weeklong vacation and agreed to split the costs down the middle. Over the week, each of you paid for various joint expenses (like gasoline, meal checks, motel bills) and kept track of what you had paid. At the end of the vacation, it turns out that you had paid A dollars and your friend had paid B dollars. Assume that A is less than B. How much money must you give to your friend at this point to even things up? This can be solved with algebra (in more than one way), by example, verbally, or with a picture, among other approaches. Multiple correct answers are possible.
2. (a) Exactly how many days old are you today? (b) Use your answer to (a) to figure out what day of the week you were born on. (The solution gets into the algebraic ideas of modular arithmetic.)
Problems like these make math be lively, relevant, and interesting, and they start leading students' thought patterns in the right directions.
Cathy Seeley:
Thanks for sharing these ideas. Adapted for appropriate grade levels, problems like these can be engaging tasks as part of a comprehensive approach to incorporating algebraic thinking in the elementary grades.
Question from
Hopewell, Virginia
As a first-year high school algebra teacher I am struggling to find ways to connect with students. Nearly 40 percent of my students are repeaters or students with learning disabilities, and keeping them from giving up on themselves is my main challenge. I have been moving around the room, asking students to stand up and act out parts of equations and mathematic properties, or even simply come up to write an answer on the board and then defend it. Still I hear the kids saying things like "I'm just dumb," or "I'll never learn this," or "Even if I knew how, it wouldn't do me any good." Are there any strategies out there to help engage students with limited motivation?
Cathy Seeley:
I think you have identified a critical factor—engaging students. I think that such engagement can precede motivation. By choosing tasks that allow students to tackle challenging and interesting problems, we can set up a classroom where students work in small groups to come up with solutions, discussing, justifying, and even arguing about approaches and consequences. Some of the National Science Foundation curriculum projects, both for middle school and high school, provide such tasks. This type of activity fits beautifully with the functions-based approach advocated in NCTM's Principles and Standards for School Mathematics.
Question from
Ontario, Canada
At what point do we introduce algebra to students who still have not mastered basic numeracy skills? Students who have troubles with operations using fractions are not likely going to understand or be successful with basic algebra. I have students in my grade 10 class that still use calculators to add and subtract fractions. They cannot even grasp the concept of solving for an unknown.
Cathy Seeley:
It would certainly be ideal to have all students be proficient in arithmetic before progressing to algebra. However, a few years ago I had an experience teaching a ninth-grade algebra class that caused me to re-examine my beliefs about necessary prerequisites for learning algebra. One particular student, whom I call Crystal, could not do fraction operations and asked if she could use a fraction calculator in the algebra class. I quickly discovered that, in spite of her arithmetic deficiency, Crystal was an outstanding algebraic thinker, as long as she had her fraction calculator to help her get answers to fraction problems. To make a long story a little shorter, eventually Crystal was motivated by her success in algebra to go back and learn fractions. She continued through precalculus in high school and went on to graduate from college and graduate school. We need to be careful not to let our own beliefs about how mathematics must be organized get in the way of allowing all students the opportunity to show us what they can do. Even though computational proficiency helps in higher-level mathematics, there is no evidence that students who are weak in some areas of computation cannot succeed in algebra or higher-level mathematics.
Question from
Athens, Georgia
What place does memorization of facts have in developing algebraic thinking? While we want to develop higher-order thinking and understanding of concepts, are there not certain facts students must simply "know?" And how do calculators in the classroom fit in to this?
Cathy Seeley:
This continues the discussion of the previous question. I absolutely agree that there are facts that students should know. I also recognize that there may be students who are otherwise ready to proceed to algebra or higher-level mathematics, even though they may not know all these facts (or skills). Technology offers us a way for both the teacher and student to see what students can do beyond what they have learned. And, as in the case of students like Crystal, sometimes success in more challenging mathematics can motivate learning of the things we wish they had known before. Engagement and opportunity can absolutely lead to motivation and success.
Question from
Raleigh, North Carolina
There are many well-respected theories out there on multiple intelligences. If a given student tends to think "geometrically" or visually, are we justified in forcing them to repeat algebra classes until they pass? If we are justified (I hope we are), then how do we get them to make this leap?
Cathy Seeley:
You've identified an important issue. In fact, students do think in different ways, and this might be one argument in support of a more integrated high school curriculum where algebra and geometry are not approached in isolation. But regardless of how you organize the curriculum, students should have opportunities to develop their thinking along different lines. When we teach algebraic problem-solving skills, this can increase students' repertoire of approaches beyond what may be their first line of attack. At the same time, a student who thinks geometrically or visually may benefit from a more visual approach to algebra. This is one of the most exciting advances in the teaching of secondary mathematics—that we can represent situations in multiple ways and can approach the solving of problems in different ways. Using the power of graphing technology and a functions-based approach to algebra can allow students to solve problems from either a symbolic or a graphical approach. There is no reason why any person who is reasonably successful in other content areas cannot be successful in mathematics if we offer multiple avenues toward mathematical understanding.
Question from
Newnan, Georgia
So which is better—a continuous mathematics course and integrated math course, or separate courses (Algebra I, Algebra II, etc)? How is the continuous different from integrated?
Cathy Seeley:
You folks in Georgia are certainly dealing with this issue. While many districts in the United States (and essentially all in Canada and elsewhere) have implemented an integrated approach to secondary mathematics, Georgia may be the first state to try to adopt a requirement for integrated high school mathematics in all schools (at least it's the only one that comes to mind). In any case, there are many apparent advantages to such an approach, not the least of which is the opportunity for students to connect otherwise isolated pieces of mathematical knowledge and skills. But the flip side is that this is a huge change for many schools, and the professional development and materials support needs are tremendous. If the state chooses to go this direction (or for schools outside of Georgia considering such a move), this professional development support and finding appropriate instructional materials are needs that must be addressed, as well as the need for ongoing support for teachers in terms of planning time, collaboration time, and so on.
The benefits may be significant and may be worth the effort, but it is important to recognize what is being asked of teachers and to acknowledge that making such a change on such a large scale may be quite challenging. For an individual teacher or school to adopt an integrated approach may be much more doable and may lead to more visible results more quickly. Regardless of whether you teach an integrated program or a course-by-course program, the essential things are what content you choose and how you actively engage students in their learning.
Moderator Thank you all for your stimulating participation this afternoon. We had far more questions submitted live during the hour (and in advance) than Cathy could answer today. She will review all questions submitted and add several answers for the final chat transcript, which will be posted on the NCTM Web site Monday or Tuesday.
Thank you again.
Cathy Seeley:
Thanks to all of you for your energetic participation. You've kept me typing as fast as I can! I've enjoyed this professional interchange, and I look forward to reflecting on as many other additional questions as I can. Be sure to check the NCTM Web site for resources on Algebraic Thinking, the Professional Development Focus of the Year. Look for the magnifying glass icon.
Thanks for your interest, and I'll see you at our next chat!
Moderator
The following questions are representative of those submitted in advance or during the hour of the online chat. Time restrictions prevented Cathy from answering all the questions submitted for this chat.
Question from 7. Jakarta, Indonesia
What is the best way to teach algebra in grades 1, 2, and 3 in elementary school?
Cathy Seeley:
This question is way too big for a short response. The best source of information NCTM has to offer on our Professional Development Focus of the Year can be found on the NCTM Web site, accessible from many paths. Look for the magnifying glass icon on the home page and throughout the Web site. You will be delighted at the wealth of resources on incorporating algebraic thinking across the grades. One resource you will find there is the algebra standard in Principles and Standards for School Mathematics, as well as the excellent series of Navigations publications, which includes algebra books for each grade band (Go to: ). We are also seeing an increasing number of online professional development programs related to algebraic thinking, although with so many available, you should choose any such program carefully.
Question from
Columbia, Maryland
Introducing mental math, mathematical properties, number patterns, representation of objects or numbers and generalization of the concept at an early age/grade.
Cathy Seeley:
These are important components of algebraic thinking. Mental math is tremendously important and helps students predict answers in thinking ahead about problems they are tackling. Patterns help students develop generalizations, which lie at the heart of thinking algebraically. And we now know that being able to represent situations in many ways is not only an important ability, but can help students deepen their understanding of the situation and develop mathematical ideas at the same time.
Question from
San Jose, California
It seems to me that schools are pushing the expectations for traditional Algebra I classes earlier and earlier. I have noticed children are having a harder time grasping concepts, and I feel that this is due partly to their cognitive immaturity. At what age is it cognitively appropriate to introduce these concepts?
Cathy Seeley:
This is an important and timely question. The issue of accelerating students into Algebra I earlier and earlier brings several problems, not the least of which is cognitive maturity. For example, what happens to the critical content of the middle grades, in particular proportional reasoning and increasingly sophisticated ways of dealing with data and statistics? I worry that when we accelerate students into algebra too soon, they may miss this. Also, unless we have good options at the eleventh and twelfth grades, why are we accelerating the students? In addition, if the student is not highly motivated to continue high school study through every year in high school, considering calculus, or possibly another advanced course, what is the benefit of accelerating the student? Furthermore, unless prevented by the school or district, some or even many of these students may stop their mathematics study before twelfth grade, which is a disaster for any student going to college. Finally, I think there may be questions of at what age students are developmentally ready to deal with the level of abstraction called for in a formal algebra course. While this may vary from student to student, we need to seriously question the value of pushing algebra ever further down into the middle school and elementary school. Rather, the NCTM focus on algebraic thinking gives us many ways to incorporate age-appropriate ideas that incorporate algebraic thinking from preschool on.
Question from
Dallas, Texas
I have taught math from grade 6 through AP Calculus. In my opinion, algebraic thinking should be taught at the lower grades with the use of manipulatives, such as use of the hands-on algebra that uses scales and colored pawns. Many of the students in high school still lack a sense of what equality means and hence are quick to violate it. Also, it would help a great deal if properties were introduced in the lower grades with their correct names. I have been accused of personally inventing the distributive property by a Pre-AP Geometry student who had never heard of it by the 9th grade, and of being a fool for not knowing what the "popcorn rule" is, as in her words, even the 4th grade teachers know what that is.
Further, a student who can spell Mississippi and knows its capital should also be able to state "subtract 5 from both sides" instead of talking about arbitrarily "moving it to the other side." Vertical consistency in notation beginning at the early grades would help a great deal also. Everyone who has taught Algebra knows the problems that come up with the use of "slashy fractions," that is, when 1/2 evolves into the ambiguous 1/2x or worse 1/2x + 3.
Another thing that would help is in the area of reduction of fractions and factoring. If the lower grades learned to factor the GCF from the numerator and denominator then cancel, their students would not be as confused about canceling binomial terms in a rational expression involving polynomials.
I think NCTM can help by promoting vertical team meetings that involve all levels and grades, not just the MS/HS Pre-AP/AP teachers.
Cathy Seeley:
Your comments address a range of issues. I'll start with the last: I think meeting across grade levels is one of the most important things that can happen within a school system. Such meetings are most constructive when all involved both share with and learn from each other. Often, I have found that secondary teachers benefit from seeing some of the powerful mathematical ideas addressed before they see students. And all teachers benefit from seeing what content their students may later deal with in school. I think the most important content directions we can give elementary teachers is to teach for a balance of understanding, skills, and applications and to do whatever is necessary to actively engage students in their own learning. I also think that we can do more toward maintaining mathematical precision in terms of definitions and language, but only if terms are attached to a sound understanding of what they represent. And by the way, I have never heard of the "popcorn rule" either.
Question from
Oakland, California
I wonder if you agree with trying to level the playing field by publicizing the resources available at zero or nominal cost over the Internet.
I am speaking of Web sites like AOL@School, Hotmath.com, Quickmath.com, etc. There are many others. These can relieve students of math anxiety when it comes to absorbing new concepts in algebra.
If teachers incorporate them into homework assignments, then more students might be able to keep up in class and "get it" more easily. Especially those without math help at home or tutors.
Thank you!
Cathy Seeley:
I think the best way to level the playing field is to provide all students with the opportunity to be actively engaged in learning mathematics that will serve them in the future. In looking at a couple of such Web sites, it appears that they provide help on doing homework exercises from common textbooks or answering questions. This kind of help may be quite useful for some purposes, especially if there is limited assistance at home. However, I also hope our mathematics teaching extends beyond this kind of assistance toward the rich classroom experiences that help students learn mathematics deeply in a way that will stay with them through their future mathematics study.
Question from Auckland, New Zealand
New Zealand has had a strand of Algebra in the mathematics curriculum since 1992, for students from age 5 up. This strand emphasizes algebra as relationships. For the first few years the equal sign and > and < are part of that focus. There is a continuing strand on patterning.
More recently a National Numeracy Project has been introduced for ages 5–14. This includes dealing with part-whole relationships in numbers for ease in mental calculation. For example 19 + 7 is done more easily as 20 + 6. We hold that the thinking behind this, which differs for different numbers and different operations, constitutes algebraic thinking.
My colleague Murray Britt and I have been doing research on the extent to which students can generalize from such examples to the correct use of algebraic notation to express this relationship as a variable.
Anyone interested in learning more about this can see Chapter 5 in my report available on
professional/2003Y7_9NPReport.pdf.
For a description of the algebraic thinking involved, write to me and I will send you a copy of a paper in press with Educational Studies in Mathematics.
k.irwin@auckland.ac.nz
Cathy Seeley:
Thanks for sharing this resource. I think the American mathematics curriculum can benefit a lot when we examine what is done outside the United States. I particularly like your early emphasis on the important ideas of equivalence and relationships. These are indeed central ideas to the development of algebraic thinking.
Question from
Archbald, Pennsylvania
Do you feel that calculators should be used on a daily basis in middle school math classes, specifically grades 7 and 8?
Cathy Seeley:
I think calculators should be available for students to use at these grades, understanding that the teacher helps students decide when and how to use them. It is critical that students learn to make these decisions. Teachers can help by identifying when calculators should not be used ("Put your calculators away; we're going to do some mental math.") and when they can be helpful (as in solving complex problems). We also need to make sure that we capitalize on the availability of calculators by giving students challenging problems that go beyond the limitations of what we can do with students when they do not have such access. Simply giving students long lists of computational exercises and then giving them calculators to do them defeats the purpose of having this tool available.
Question from
Oak Park, Illinois
This is not terribly profound, but it is a part of the puzzle of developing algebraic thinking preK–12: Students need lots of experiences in varied real-world contexts of creating data tables (especially 2-variable T-charts or T-tables) and thoroughly examining the patterns in the data. Then they need to graph the data and again thoroughly analyze those patterns. Then they should relate the patterns in the graph to the patterns they saw in the table.
I have done many such activities with students of varying backgrounds in grades 3 through 8. With those students who are ready, we examine the patterns again and use the two variables to create expressions and then equations.
Cathy Seeley:
This is a wonderful summary of the power of learning how to represent situations in multiple ways. It provides students with the opportunity to see algebra as the study of patterns and relationships, which sets the stage beautifully for their increasingly sophisticated development of algebraic thinking and, eventually, symbolic procedures.
Question from
Chicago
There is a challenge with students coming into high school and understanding very few concepts of algebra. There is a program in CPS to have students who scored lower than 50 percent on the Iowa Basic Skills test to take a double Algebra course as 9th graders. Part of this is Algebra Problem Solving (IMP or Mathscape). These programs do not seem to emphasize any real algebra equation work that the future tests call for. Are problem solving curricula really beneficial if the goal is to pass tests that requires students to do equations quickly?
Cathy Seeley:
Problem-solving curricula are absolutely essential to prepare our students well for success in algebra and the courses that follow it. However, the flip side is not true. Teaching only equation-solving skills without adequate attention to understanding and problem solving is short-sighted and likely to backfire on students as they move deeper into the secondary mathematics program. Of course, balance is the key, and an ideal algebra program will include not only problem solving, but developing conceptual understanding and skills development as well.
Question from
Ft. Lauderdale, Florida
The first order of business is to ensure that the K–5 teachers have enough content knowledge to prepare the youngsters. Content knowledge is lacking in most K–5 teachers. Mathematics has evolved over the last decade, however, some teachers are still teaching for the industrial era. They believe that computation is the basis of elementary education. Some elementary teachers are not comfortable teaching mathematics. NCTM could offer online courses to help elementary teachers acquire the needed content knowledge with which to help the K–5 students develop a strong solid base on which middle and high school teachers can build.
Cathy Seeley:
Teaching algebraic thinking beginning at the elementary level presents increased demands on elementary teachers' understanding of mathematics at a deep level. Professional development is a priority for NCTM. Online courses offer great potential for delivering professional development to teachers who might not otherwise have access to it.
Question from
Fairburn, Georgia
Many teachers don't know what "algebraic thinking" looks like in the elementary classroom. If they did, they would feel more comfortable when they are told to include it in their curriculum.
Cathy Seeley:
This is another reason why professional development plays such an important role. Teachers of mathematics at all levels, not just elementary, need to make a lifelong commitment to their professional growth. For elementary teachers, experiencing the kind of algebra that is conceptual and relevant can be a liberating experience, not to mention the benefits for guiding their students' learning.
Question from
Marion, New York
We can help elementary students look for patterns through the combined creation of tables, graphs, and equations. Even young children can extend patterns to create tables, learn to graph the results, and describe the pattern of the table and the graph in words. Students can think and question critically in an algebraic context when they are given the opportunity to extend a problem through patterns and see it through a visual model. For example: My third graders create tables and graph the story of the Gingerbread Man. If the Gingerbread Man begins to run as soon as the oven is opened, and runs 2 feet per second, the wife begins to run 2 seconds after the oven is opened and she runs 2 feet per second, the wolf begins to run 5 seconds after the oven is opened and he runs 4 ft per second does the Gingerbread Man ever get eaten? They pull amazing stuff off the tables and graphs.
Cathy Seeley:
This is a nice way to develop informal ideas of algebra. Thanks for sharing your example!
Question from
Tempe, Arizona
People who share these views are usually "burned at the stake," but several years of my 39 years of teaching math I taught with the Saxon series. Students liked it and did better on college entrance exams! It was an incremental development and geometry was integrated throughout. Students could go at their own pace. Independent study was much easier. It de-emphasizes the sacred role of the teacher. Students wanted their own copy of the text to use in college. (Most students would like to burn their own copy of the math text they had to use!)
And yet it was banned from use in some areas because it didn't have colored pictures and it supposedly did follow the standards. If students do better in college because of it and like it much better, then maybe the standards need to be revised or replaced.
Cathy Seeley:
Thanks for your honest and direct comments. You are correct that the Saxon text has not been adopted in many states and systems. Teachers tend to have strong feelings either for or against the program, and, frankly, results are rather mixed. I think that if students are to take responsibility for their own learning and become independent algebraic/mathematical thinkers, they need to come to rely more on themselves than either the textbook or the teacher. I continue to believe in the importance of the teacher, not for telling students things, but for structuring a classroom where learning is likely to happen. A textbook is only a tool, and it cannot address the wide range of problems students will encounter. It may help students deal with certain types of problems, and that's great. My preferred model of teaching would have students engaged in a variety of activities, including a good dose of small-group work on solving engaging problems that draw students into figuring out how they will use mathematics to find solutions. The role of the teacher is critical not only in structuring and facilitating their work, but in asking good questions that push students' thinking farther than they thought they could go. One of the strengths of NCTM as an organization is the diversity of views of its members. When we openly discuss our different points of view, we can all become better educators.
Question from San Francisco
I am a Math Learning Specialist. I help learning-disabled students really understand math. Why do so few math teachers understand the ways in which various learning disabilities hamper students' abilities to master math?
Cathy Seeley:
Unfortunately, many teacher education programs are limited in the amount of time they can spend learning about special needs students of all kinds. One of the roles you can play is to connect classroom teachers with resources, including yourself, on how they can better meet the needs of their students. It's great to have someone in a resource role who understands both mathematics (at a very deep level) and also special needs students.
Question from
Centreville, Virginia
I love this topic! I am constantly trying to find ways to incorporate algebraic thinking in my classroom both within formal mathematics discussions and in the thinking that is essential when students are solving practical applications of algebraic concepts to real life. I think that weaving the strands throughout mathematics is very important in grade school and middle school. I do, however, support the structure of separate courses that exist in many of the schools throughout the United States. This approach allows students to concentrate their attention on one strand so that the topic can be studied in depth. I taught mathematics for 15 years in California, and the schools in which I taught had this form of "traditional mathematics." However, I tutored students who were attending schools that had adopted "integrated mathematics." Their understanding of mathematics was shallow and they felt scattered. One student in particular, who was actually naturally very talented in mathematics, was very frustrated. Her conclusion was that she was just dumb and unable to understand math. She gave up somewhere in her sophomore year and couldn't wait to unload that course. I kept trying to assure her that she was actually very talented in mathematics, but her test and quiz scores were lower than she wanted and only furthered her self-assessment in this field. I was very sad for her. Her brother, on the other hand, went to a different school that had a traditional approach. He was also talented in mathematics but was lazier than his sister. He still excelled in the subject and continues to be confident that he is good in mathematics. What was sad for me was that both students were extremely capable, but only one believed it. I attribute that, in part, to the way they experienced mathematics in the classroom. The girl's understanding was a mile wide but an inch deep. The boy's was narrower, but he had a deep understanding of algebra and algebraic thinking that allowed him to solve complex problems and feel successful in mathematics. The breadth of his knowledge would undoubtedly grow as he took more courses.
These are not the only students with whom I have had this experience. I have actually tutored many students in both integrated mathematics courses and in traditional courses and have seen the same results across the board. It is, to me, striking.
I actually had many students come to our high school having been in an integrated program with a similar experience. They had very little exposure to in-depth algebraic thinking and a smattering of knowledge about different strands of mathematics. They generally demonstrated an inadequate knowledge of Algebra when tested for placement in Geometry and would be enrolled in Algebra I for the school year.
Perhaps with an excellent textbook and an excellent instructor, integrated mathematics would be able to allow ALL students to be successful. I don't know. Certainly there will always be those brilliant students who could learn mathematics on their own without regard to the methodology implemented. What I have seen generally, though, is that otherwise talented students feel frustrated and inadequate. It seems like a well-intentioned program is failing them in this country.
Cathy Seeley:
Thanks for your participation in this chat. You raise an important issue. I honestly think that we don't have adequate information to determine whether an integrated program works better than a traditional one at the secondary level. The rest of the world uses such an approach. Perhaps one of the issues is getting clarity about what outcomes we want to see. If we are evaluating students on traditional equation-solving skills, then it may well be that students coming from a non-traditional program might not perform the skills at the same level at the same time as more traditionally prepared students. However, if we evaluate the ability to use algebra, for example, to solve diverse problems, I think we would all agree that many of our students over the years have not understood how to apply the skills they learned to the problems they encountered. I found this over and over again in teaching and tutoring students at this level. And this was the case when algebra was taught only to the more successful students. If we are now to help more/all students learn algebra and higher-level mathematics, I think we must be open to different ways of teaching and even different priorities in terms of the content we address and the organization of the curriculum.
Question from
West Liberty, Ohio
I've long been a believer in an integrated approach to mathematics, combining algebra, geometry, statistics, and data analysis. We have not taught a formal geometry class at my high school for at least the past 25 years. The state of Ohio seems to be moving in this direction also with the state standards, indicators, and the Ohio Graduation Test.
Cathy Seeley:
There continue to be success stories where school systems, and now states, may be moving in this direction which is widely used outside of the United States.
Question from
Oakland, Maine
I think there is a misunderstanding of what "algebra" is with teachers, students, and parents. Many teachers are reluctant to use the term algebra in the early development of mathematical thinking, and children begin to think of algebra as a monster that is too scary to conquer. Parents are always willing to jump on the "My child needs to be taking algebra" bandwagon, without looking at the development of the child's mathematical thinking and realizing the background is rich with algebra.
I feel teachers need to use terminology and make connections to algebra at an earlier stage of a student's mathematical development. If this happens kids will begin to work with concepts more fluidly, and parents will realize that algebra is not a separate subject but part of understanding of what we call "math."
Cathy Seeley:
It would be great if students didn't think of their mathematics experience as a set of isolated topics. That's just the goal of incorporating algebraic thinking as a strand within a balanced mathematics program PK-12.
Question from
Towson, Maryland
What can we do about the biggest textbook publishing houses that continue to produce and peddle those books? How do we "push" school districts to consider alternative programs such as IMP or CORE or Connected Mathematics?
Cathy Seeley:
I think we may be focusing on the wrong battle. If we pay attention to what mathematics we want to teach and help teachers grow through professional development that lets them learn how to engage students in learning, then I trust teachers to demand tools that support that learning. Publishing is all about supply and demand.
Question from
Culpeper, Virginia
The current math standards of many of our states create barriers to offering integrated secondary math courses. For example, in Virginia, we were beginning to offer math this way, but in most cases have backed away, since our state standards are fairly traditional and require tests in Algebra I, Geometry, and Algebra II, each separately.
Cathy Seeley:
Our accountability systems do influence what we teach. But I would argue that if we teach a rich, balanced problem-solving-focused program that includes skills, concepts and applications, our students can do fine, even on low-level skills-based tests. (See our chat from last month on accountability)
Question from Jefferson, Ohio
Our guidance counselor says that the colleges and universities are not on the same page. They want to know whether a student has had Algebra II, Trig, Calculus, etc. How can you change everyone's thinking?
Cathy Seeley:
The state of New York, for one, has offered integrated mathematics for many years with no negative impact on students entering college. I have found guidance counselors generally quite open to meeting the needs of students, but often uninformed about how to do that in mathematics. As mathematics educators, we have a responsibility to work together with counselors, providing accurate information from the universities and colleges most often selected by your students. It may be appropriate to engage in conversation with mathematics faculty from these post-secondary institutions to get a clear message of what will work. If there are, in fact, barriers at that level, solutions will start by connecting and communicating.
Question from
Princeton, New Jersey
How can assessment be modified to encourage the integration of topics and early attention to algebraic thinking? Do assessment strands such as Numbers, Measurement, Geometry, Data, and Algebra help or hinder?
Cathy Seeley:
I think the strands do not in and of themselves hinder learning. I will return to the notion that if we teach a good, rich, balanced mathematics program, our students can do fine on pretty much any test they encounter.
Question from:
36. Taichung, Taiwan
I'm currently teaching in an American school in Taiwan and we still use the Algebra I, Geometry, and Algebra II courses. My own children, however, went to school in Arizona where those courses were integrated. My son seemed to pick up on the various concepts great and had a great score on his SAT while my daughter, on the other hand, struggles to understand math and had a very hard time with the SAT test. She reviewed for the test quite a bit and was unable to have any concrete knowledge or comprehension of the algebra concepts that were still Algebra I concepts but the more advanced ones.
My first year here, I taught the middle school math where the math is integrated, streaming up from elementary as well as the 8th graders that were placed in Algebra I. Having done both at the same time, I did note that it was very beneficial for my 8th graders to learn the algebra concepts—although they had a hard time understanding them—and then see them all the way throughout the book for the rest of the year. I continually heard comments such as, "Oh, I get it now. I never understood that before." It was easier to teach Algebra as a coherent class and much less time consuming in preparation. I also found that the more advanced 8th graders who were in the Algebra I class did fine on the formula memorizing and plugging in numbers, but they had a difficult time with problem solving and the pace we needed to keep in that class. And our block program didn't allow us to delve into the problem solving very much. I've moved into the 5th grade classroom this year, and they were just introduced to the concept o =
"n." You should have heard my room; you'd have thought Martians had landed. Once they calmed down, they could do the math, they just had to learn to rearrange the equation so they'd be ready later on.
Previously, I taught kindergarten. They were quite open to problem solving because they didn't have any preconceived notions of how things had to work. I had my kids really doing multiplication and division before the year was over as well as carrying and borrowing, but they weren't afraid of it because it was encased in games.
I've noticed that we tend to want the kids to know more per grade level as the years progress so we can do more with them in the later years, but if they don't spend enough time with some very concrete things at a young age, they have no real concept of mathematics for the more abstract things. Unfortunately, we are in a time crunch that makes it very difficult to spend the time with the concrete manipulative—the way I would like to at least.
Cathy Seeley:
Students learn in different ways, and there are many variables that affect their learning. As teachers, we must continue to offer a variety of learning experiences and give students opportunities to deal with the mathematics themselves, not just watching the teacher present the "rule du jour." I agree with you that our curriculum is still too crowded. We need to take the time to talk with teachers above and below our grade level to identify where we should spend more time and where we can spend less. We certainly need to free ourselves from the belief that we have to teach every page, or even every chapter in a textbook. Developing lasting understanding, as you observe, takes time. And I would argue that this time is not only worth it, but necessary.
Question from
Vilas, North Carolina
What kinds of staff development is offered to help teachers gain a deeper understanding of the content in algebra? Where can teachers find extensions/projects to differentiate instruction in the algebra strands in all grades?
Cathy Seeley:
I would refer you to your local and regional educational centers, as well as the universities in your state. North Carolina has done some nice work in assessment and mathematics over the years. In terms of NCTM, check out the Algebraic Thinking Focus on the Web site. Look for the magnifying glass icon, and you will discover a wealth of resources. The Navigations and Illuminations examples are among many others.
Question from
Naperville, Illinois
There is increasing pressure for all students at the eighth grade to take algebra. I put the emphasis on all, because there appears to be little or no concern on ability or prep for the course. It is just implied that students do better on tests when they have had algebra. The concern tends to be with math test scores and not with the content or subject matter that should be presented to the students at any level. What are your thoughts on this aspect of the mathematics curriculum?
Cathy Seeley:
I think we must be careful if we move all students into a formal algebra course. Whether we do this or offer a rich middle school curriculum that develops a strong base of algebraic thinking, we should not shortcut the development that leads to symbolic understanding. For example, one of the most important concepts that leads to success in algebra, in my opinion, is a solid understanding of proportionality well beyond the simple study of ratios, proportions, and percent. This critical connecting concept needs to be well developed, and often, when the middle school curriculum is compressed, students do not develop this understanding. I have shared in other responses my concerns about what happens to these students in high school. The main thing is to prepare all students well for success in four years of academic mathematics so that they can succeed in college.
Question from
Ridgefield, New Jersey
What are your thoughts on how much algebra should be taught at the middle school level and also being from an urban area where many students are low level? How do we catch them up?
Cathy Seeley:
There is increasing evidence that many students thought to be low-level, especially when they come from a background of poverty, are in fact victims of lack of access to educational opportunities. The best thing we can do is to hook them into engaging tasks where they get to show us what they can do. I have found in situations like this that we can often discover new stars among students that nobody thought could succeed.
Question from
Mobile, Alabama
I have children that are very weak in fundamental skills such as facts and procedures. As we all are aware, high-stakes state tests are trying to push me to objectives that I feel won't be fully understood if that groundwork is not laid down properly. When administration comes in and sees a lot of broad framework and understanding being done, they get nervous. They do not see the DIRECT OBJECTIVES being covered. An incentive bonus at the end of the year is at stake for test grades for all the teachers here. After the long-winded introduction—Is there a point at which you can feel safe that the students will be motivated to go back and work on facts and operations while you're pushing ahead to stuff—like simplifying fractions for instance—or do you feel that something that low level has to be tackled before moving on? I read the Crystal story, but this is more elementary.
Cathy Seeley:
This is a real and challenging problem. First, I continue to believe we must teach a mathematics program that has integrity. We cannot be pulled down to teaching in ways that we think do not serve students well for their future. If we teach a sound mathematics program, grounded in understanding, driven by challenging and engaging problems and inclusive of computational development, our students will do fine on the tests, even low-level tests. And do not underestimate the potential of a calculator to allow students access to problems that would otherwise be beyond their skill level. If motivated, students can learn these higher-level skills. A comment Crystal made could be made by an elementary student with this kind of success as well—when the opportunity arose for her to learn fraction operations AFTER succeeding in algebra, she observed: "The time has come; I'm wasting too much time using my calculator to do fractions." I'm not suggesting giving up and handing out calculators for all computation. But we must allow students the opportunity to solve problems. The biggest difference in test performance among groups of high-performing and low-performing students is not found in low-level facts and procedures. Rather, the differences show up in the more complex problem-solving situations that many students have never experienced.
Question from
North Chicago area, Illinois
I think one of the major issues we are faced with as a 'Nation at Risk' is how to present mathematics problems in a way that encourages students (not discourages them) to think and interpret what is being asked for themselves—and then validate their non-linear thinking. Too often, teachers approach a problem with a particular agenda and do not encourage creative solutions to problems. Here's an example of what I mean:
In a 3rd grade lesson about place value, the opening question for individual reflection looks like this "With the follow numbers, what is the largest number that can be made and what is the smallest? (given 8, 5, 3, 0, 1, 9) What teachers sometimes do is put so many constraints on problems that students are encouraged not to think creatively? (The teacher WANTS 985,310 and 103,589 so she/he adds statements to the original – carefully crafted – question, like, since there are 6 digits you need a 6-digit number in your answer … whereas students might come up with all sorts of other creative notions (exponents or decimals or mathematical symbols).
Or
Talking about patterns 2,000 1,200 800 600 500 ___ ___ and having a student describe how we divide the difference by 2 each time (which is clever but since the teacher has a particular agenda, the answer is not accepted and dismissed in search of the 'correct' answer that was being sought (1/2 of the difference is being subtracted). Both are correct and furthermore making the connection between should be celebrated and emphasized!!!
I think this does a disservice to students' development of mathematical thinking. I don't think there is ever a moment where a teacher should not take every opportunity to point out connections between concepts. I am keenly aware that not all teachers of math are acutely aware of all these connections but shouldn't that be a priority of ours to help make meaningful connections and/or set up meaningful connections in the years to come? Do you have any thoughts on the relative importance of this notion?
Cathy Seeley:
Increasingly, I believe that the role of the teacher is to ask good questions, not tell students answers (or even hints). Far better than minute coaching are questions like: How do you know? Why do you think so? Would that still be true if you had twice as many? And so on. I think you are correct that students need opportunities to express their developing algebraic/mathematical thinking, even if that goes in a different direction than the teacher expected. The key is for the teacher to help the student connect what the student has done with the appropriate mathematics being used so that the student has the possibility of using what he or she has learned in another situation.
Question from
Savannah, Georgia
I teach honors Algebra II and a couple different levels of precalculus. Though I have the students use their calculator often in class to enhance their understanding of many different concepts, I still mix in non-calculator quizzes and portions of tests that require good computational skills and a command of operations on fractions. Is this overkill or a necessity for students who should be preparing for calculus courses and need a strong algebraic foundation?
Cathy Seeley:
I think it is helpful to reinforce skills previously taught, and I definitely think some non-calculator work is appropriate, especially for mental math. However, I also think we need to put in perspective the level of computational proficiency that is actually necessary for success in higher-level mathematics. It may not be a useful way to spend time in algebra II to have students do pencil-and-paper long division, for example, when they are unlikely to need this algorithm in the future, and when there are more important concepts they need to practice. But the teacher can choose how to balance the program so that students' needs are best addressed.
Question from
Hopkinsville, Kentucky
Has the NCTM studied the types of courses and curriculum used in England, Spain, or even South American countries. I often had foreign students, and their problem solving skills left my students in the dust.
Cathy Seeley:
There is mixed evidence about problem-solving expertise in other countries. One of the complaints of some international mathematics educators is that their students are better at skills than at creatively solving problems. However, there are definitely some interesting programs in problem solving, especially from England and the Netherlands. (I'll have to check out Spain and South America a bit more…) In the Unite States, our most traditional skills-based secondary programs probably did leave many students ill equipped to handle more complex problems. Consequently, many of the newest mathematics curriculum programs have incorporated the more innovative aspects of some of these programs in their focus on problem solving.
Question from
Eldersburg, Maryland
I am struggling with helping teachers understand the importance of teaching alternative algorithms as a way of facilitating an understanding of algebra. In order to be able to manipulate an algebraic equation, very young children can think about and discover many ways to solve a regrouping with addition problem or subtraction, multiplication, and division. They learn to think about pure computation in a multitude of ways. How can we help elementary teachers see this as a valuable part of their instruction?
Cathy Seeley:
I think that seeing other approaches to solving problems is valuable, as is experiencing other ways to perform computational skills. Other countries often use different computational algorithms than we do. Among other benefits, students can see that mathematics is not magic, but rather, efficient ways to deal with numbers and problem situations. Professional development can help teachers themselves experience alternative algorithms, which can provide them with insights into student thinking.
Question from
Saint George, South Carolina
How do you get students out of a find-the-answer mode and into understanding that problems can have multiple solutions?
Cathy Seeley:
We have conditioned students through our teaching techniques to look for quick answers. The best way to combat this kind of student thinking is by teaching differently. When we engage students in interactive solving of problems that may not easily be solved and/or that may have more than one possible correct solution, over time they can learn to trust their own thinking, rather than trying to guess which rule to use.
Question from
Cleveland
How should algebraic thinking best be developed for middle school students? Many people believe that a traditional algebra course, focused on rules and manipulations, should simply be taught earlier. Shouldn't middle school algebra look different than that?
Cathy Seeley:
YES!!! Middle school mathematics should be a time of rich exploration of new ways of thinking. It should also be a time of powerful connections between elementary work with numbers and operations leading to symbolic awareness and the ability to make generalizations. The most important connecting idea at this level, in my opinion, is the development of proportional reasoning. If this is done well, students can see how quantities can grow proportionally, leading naturally into the study of linear relationships. Whether we offer algebra as a course in middle school, or whether we develop algebraic thinking that sets students up for success in an algebra course later, middle school mathematics needs to be a rich, balanced program, with an emphasis on representing and solving problems.
Question from
Macomb, Illinois
Can you say some more about "early algebra," especially the need to think in abstractions and understand the varied meaning of the equal sign? For example, in the 70s the program Developing Mathematical Processes had young children working with equations such as W + D = B + C. Perhaps we should try to work with these ideas even before the push to use numbers and counting to verify answers.
Cathy Seeley:
There are many ways for students to explore patterns at an early age, especially patterns dealing with equivalence, such as the one you suggest. Students can explore equivalence with balance scales (I believe these were an important tool in the DMP program; they used a nifty two-piece blue plastic balance that was nearly indestructible), representing what they find with pictures and, possibly, basic symbols. The key idea is that this kind of mathematical exploration can plant important seeds that can develop into algebraic understanding if nurtured well.
Question from
Reston, Virginia
Robert Gagne offered a hierarchy of learning that progressed from verbal skills through problem solving - with intermediate steps such as concept learning, analysis, synthesis and principle learning in between. Similarly, wouldn't we do well to pursue a generally acceptable definition of algebraic thinking? The more precisely we can define our objectives the better our chances of developing strategies for achieving them.
Cathy Seeley:
It is often helpful to rely on a theory of learning that lays out developmental or instructional stages. In the case of algebraic thinking, there have been several articles and other publications that address this development. One excellent source is Principles and Standards for School Mathematics. Other NCTM resources can be found through the NCTM Web site, especially through the articles and links included as part of the Algebraic Thinking Focus (look for the magnifying-glass icon).
Question from Fairfield, Connecticut
At what grade level can you start to involve students in critical thinking?
Cathy Seeley:
If you ask my friend, Denise, an outstanding kindergarten teacher, she would tell you that you can involve students of any age in appropriate critical thinking. Asking students to justify their thinking and make predictions are just a couple of ways we can do that.
Question from
Lexington, Kentucky
You wrote, "The rest of the world, including our colleagues in Canada, teach mathematics, not as separate courses, but as a continuous program from elementary through secondary school. In the United States, some schools offer an alternative, such as an integrated program that incorporates algebra as a strand blended with geometry and other advanced topics." I would be interested in knowing about comparisons of the "successes" of such approaches in developing algebraic reasoning and conceptual understanding.
Cathy Seeley:
In terms of international comparisons, the TIMSS studies are the best source of information. For the integrated secondary programs in the United States supported by the National Science Foundation, you can visit the Web site for the dissemination of information about these five programs at: The biggest challenge in answering your question is that we don't have widely accepted measures for algebraic reasoning and conceptual understanding to use in comparing various approaches.
Question from
Acton, Indiana
As we strive to achieve the goals of "No Child Left Behind" and raising standards/expectations of our students in the middle school environment, it seems that these two ideas do not always mesh very well. Many schools have either increased the number of students enrolled in Algebra I courses in the middle grades and/or have begun teaching Algebra I and geometry courses. Unfortunately, some of these courses have led to algorithmic teaching of basic algebra skills instead of an understanding of a variety of mathematical concepts such as those outlined in the Principles and Standards for School Mathematics.
Would many students in the middle grades be better served if an emphasis were placed on developing understanding a wide range of concepts and standards instead of offering a traditional algebra course?
There area several factors that influence the necessity to offer an algebra course and/or a geometry course in the middle grades, including demands from the local school board and parents as well as restrictions of coursework due to limitations of textbooks. Should there be another course offering developed for the middle grades, which would include the development of student understanding in the areas of algebra, geometry, data analysis, number sense, and measurement?
Cathy Seeley:
This seems like an important balance for all students to experience during the middle school years. Even if we teach an algebra course, I would hope that these elements would precede the course, and also would be incorporated appropriately in the algebra course itself.
Acton, Indiana
What is the best practice to identify students that would be best served by a traditional algebra course?
Cathy Seeley:
Increasingly, I am coming to question whether any students are well served by a traditional algebra course, if by that you mean a course focused on learning abstract rules and procedures. Even students who have been successful in such courses in the past could well have benefited from experiences with graphical and other representations, as well as focusing on using algebra to solve complex problems, rather than primarily solving certain types of story problems.
Question from
Besancon, France
What are the main weaknesses that have been identified in student achievement in the algebra field, and at which levels? What does NCTM suggest to try to overcome these problems?
Cathy Seeley:
For many years, students who study algebra have had difficulty applying what they learn to solve problems. Teachers have searched for ways to 'teach story problems.' As we try to teach algebra to more students, this problem is exacerbated, especially in traditional algebra classrooms focused on abstract rules and procedures. Principles and Standards for School Mathematics calls for a much richer vision of algebra, building on the use of functions to represent and analyze relationships. In such a program, many or all students can learn a range of problem-solving approaches that serve them well regardless of the type of problem.
Question from
Sigourney, Iowa
I use an integrated curriculum and use a lot of explorations. The students do very well with it, but I am having difficulties getting that knowledge to transfer to textbook problems. Any ideas about how to ease this transfer?
Cathy Seeley:
If your students are successful in a good integrated program, they are likely learning important knowledge and skills. In terms of transferring to textbook problems, I'm not sure which types of problems you mean. If you mean routine equations to solve, this is an important skill that may call for more practice than they have had, connected to what they know about how equations help in solving problems. If you mean typical types of word problems (coins, age, etc.), you will need to decide how much time to spend on such problems based on your curriculum and whatever test is used in your accountability system. Otherwise, it may be reasonable to question to what extent traditional textbook problems are useful. But I continue to believe that a strong program that balances skills, conceptual understanding, and problem solving is the best preparation we can give students for whatever types of tests they face.
Question from
Accra, Ghana
What will be Algebra III, Algebra IV, Algebra V, etc.? I think algebraic concepts pervade the mathematics curriculum from K to university. So segmentation might not be useful. Rather, we must find ways to teach algebraic concepts appropriately to different groups of students and help them to make sense of those concepts as they relate to the students' daily lives. For example, commutative law remains commutative at all levels K to university. How can we teach it to K students so that they can recognize it in terms of a + b = b + a at the middle school level? Can they also apply the concept in real-life situations? I think these are some of the fundamental issues to be concerned with.
Greetings from Ghana.
Cathy Seeley:
Hello to Ghana. I agree with you that segmenting the curriculum in this way does not serve students well. There are many ways to connect mathematics, including algebra, to students' lives both in and outside of school. Many elementary teachers give students experiences dealing with commutativity as a concept, even if they may not give it a name. Developing many properties and characteristics of equivalence is definitely worth the investment at the elementary level.
Question from
Springfield, Missouri
When parents ask, how do we respond to the question: What is so important about algebra anyway?
Cathy Seeley:
Algebra has long been a gatekeeper, and the evidence is solid that it is critical as a first step toward college. Since many students don't know themselves whether they are likely to attend college, we need to start them along this path. But this in itself is not enough to drive us to call for algebra for all students. Algebra, as described by today's vision in Principles and Standards for School Mathematics, is a powerful set of tools that helps students represent relationships, make generalizations, analyze situations, make predictions, and, most importantly, solve many kinds of problems in mathematics, other disciplines, and life outside of school. Today we know how to connect algebra to engaging and useful situations. Invite parents to experience this kind of problem experience in your classroom (ideally) or through materials you send home.
Question from
Kingfield, Maine
As a high school mathematics teacher, I see many students that come in to my classroom without basic number sense. Would you think that before we can work on algebraic thinking, we need to work on number sense first, or do you think they can (or even should), be emphasized simultaneously?
Along those lines, it would seem that part of the answer to that question depends on the resources of the school and district itself. If developing algebraic thinking at the elementary and middle grades requires the purchase of manipulatives and other items not normally in a math class budget, then it would seem to be prohibitive for smaller and more rural districts.
Thoughts?
Cathy Seeley:
Absolutely, number sense is critical as a primary goal of elementary mathematics. I believe that incorporating algebraic thinking can support this number sense. For example, as we develop the idea of equivalence, students can learn multiple ways to represent a number. As students explore patterns with numbers, they are laying the foundation for more sophisticated algebraic patterns later. These goals can and should be connected. I might argue a bit with your statement that manipulatives and other tools might not normally be in a math class budget. These are as essential to the teaching of mathematics as maps and globes to the teaching of social studies or science equipment to the teaching of science. There are ways to economize, but basic counting, place-value and geometric manipulatives, to name a few, should be a priority even in small schools.
Question from
Indianapolis, Indiana
Do you have some data indicating how parents can support their children in fostering algebraic thinking?
Cathy Seeley:
There are many ways for parents to help their students developing algebraic skills. Asking students to notice patterns, formulate generalizations and make predictions is a great way to do this throughout elementary and middle school. This can start with simple situations like the sequence of lights in a traffic light and can extend to noticing patterns of house numbers. Asking students to give many names for a number and talking about equivalence can also help. Having students justify their thinking any time they are working with a mathematical situation is helpful. Questions like "How do you know?" "Why do you think so?" "What else can you say about the situation?" can also be useful.
Question from
Naperville, Illinois
Mathematics programs: Where does one find an unbiased evaluation of mathematic programs? That is, one that looks at the aspects of the curriculum and gives an honest refection of how the programs meet the goals and objectives of NCTM.
Cathy Seeley:
This is the question of the hour (and day and year). Gathering and packaging this type of information is a priority of many efforts right now, both within and outside of NCTM. Recent efforts by the U.S. Department of Education and the Mathematical Sciences Education Board have told us that there simply is not enough information to make such judgments at this point. Individual programs are often accompanied by related data, but it is important to look at studies across districts and schools. Instead, we can look at recommendations such as those presented in Principles and Standards for School Mathematics that are based on more focused studies of effective practices |
Hi All, I am in need of aid on simplifying fractions, graphing, ratios and relations. Since I am a beginner to College Algebra, I really want to learn the bedrocks of Pre Algebra fully. Can anyone recommend the best place from where I can begin reading the basics? I have a midterm next week.
Believe me, it's sometimes quite hard to learn a topic alone because of its difficulty just like difference between quadratic and cubic relationship. It's sometimes better to request someone to teach you the details rather than understanding the topic on your own. In that way, you can understand it very well because the topic can be explained systematically. Fortunately, I encountered this new program that could help in understanding problems in algebra. It's a not costly quick hassle-free way of learning math concepts. Try using Algebrator and I guarantee you that you'll have no trouble solving algebra problems anymore. It shows all the pertinent solutions for a problem. You'll have a good time learning algebra because it's user-friendly. Give it a try.
Even I made use of Algebrator to understand the basic principles of Remedial Algebra a month back. It is worth investing in the purchase of Algebrator since it offers powerfultutoring in Algebra 2 and is available at an affordable rate.
Algebrator is the program that I have used through several math classes - Intermediate algebra, Basic Math and Intermediate algebra. It is a really a great piece of algebra software. I remember of going through problems with multiplying fractions, subtracting fractions and graphing lines. I would simply type in a problem homework, click on Solve – and step by step solution to my math homework. I highly recommend the program. |
Advanced Differential Equations
This course introduces students to the modern theory and methods of ordinary and partial differential equations. It builds on the classical foundations of differential equations studied in second year. Ordinary and partial differential equations form an essential part of the mathematical background required for engineering and the physical sciences. A large number of real-life problems can be modelled using differential equations, making the subject one of the most widely applicable areas of mathematics. The course concentrates on some fundamental analytical and numerical methods for applied differential equations arising from the mathematical modelling of physical, chemical and biological systems.
Available in 2014
On successful completion of this course, students will: 1. Have a broad overview of ordinary and partial differential equations as well as an appreciation of the application of analysis and linear algebra in studying differential equations. 2. Have the skills to build mathematical models of relevant real-world problems based on differential equations. 3. Be able to solve these differential equations using appropriate computer software if necessary, and to interpret the solutions. 4. Understand the concepts of accuracy, consistency, stability and convergence of numerical schemes for solving differential equations. |
uses simplified language about mathematics to promote active and independent learning; strengthening critical thinking and writing skills. A ? six-step? approach to problem-solving, numerous tips, and clear, concise explanations throughout the book enable users to "understand" the concepts underlying mathematical processes. Beginning with the foundations of the mathematical process, some of the topics covered are: whole numbers and decimals; integers; fractions; percents; measurement; area and perimeter; interpreting and analyzing data; symbolic representation, linear and nonlinear equations; powers and logarithms; formulas and applications; higher-degree equations; absolute values and inequalities; slope and distance; basic concepts in geometry; and an introduction to trigonometry. This book can serve as a valuable reference handbook for engineering technicians, nurses, dieticians, job trainers, home-schooling professionals, and others who require a basic knowledge of non-calculus mathematics. |
Stewart's clear, direct writing style in SINGLE VARIABLE CALCULUS guides you through key ideas, theorems, and problem-solving steps. Every concept is supported by thoughtfully worked examples |
Digital geometry is about deriving geometric information from digital pictures. The field emerged from its mathematical roots some forty-years ago through work in computer-based imaging, and it is used today in many fields, such as digital image processing and analysis (with applications in medical imaging, pattern recognition, and robotics) and of course computer graphics. Digital Geometry is the first book to detail the concepts, algorithms, and practices of the discipline. This comphrehensive text and reference provides an introduction to the mathematical foundations of digital geometry, some of which date back to ancient times, and also discusses the key processes involved, such as geometric algorithms as well as operations on pictures an introduction to differential methods in physics. Part I contains a comprehensive presentation of the geometry of manifolds and Lie groups, including infinite dimensional settings. The differential geometric notions introduced in Part I are used in Part II to develop selected topics in field theory, from the basic principles up to the present state of the art. This second part is a systematic development of a covariant Hamiltonian formulation of field theory starting from the principle of stationary action.
In the series of volumes which together will constitute the Handbook of Differential Geometry we try to give a rather complete survey of the field of differential geometry. The different chapters will both deal with the basic material of differential geometry and with research results (old and recent).All chapters are written by experts in the area and contain a large bibliography. In this second volume a wide range of areas in the very broad field of differential geometry is discussed, as there are Riemannian geometry, Lorentzian geometry, Finsler geometry, symplectic geometry, contact geometry, complex geometry, Lagrange geometry and the geometry of foliations. Although this does not cover the whole of differential geometry, the reader will be provided with an overview of some its...
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Buy APPLICATIONS OF CONTACT GEOMETRY AND TOPOLOGY IN PHYSICS by KHOLODENKO ARKADY L and Read this Book on Kobo's Free Apps. Discover Kobo's Vast Collection of Ebooks Today - Over 3 Million Titles, Including 2 Million Free Ones!
Although contact geometry and topology is briefly discussed in V I Arnol'd's book Mathematical Methods of Classical Mechanics (Springer- Verlag, 1989, 2nd edition), it still remains a domain of research in pure mathematics, e.g. see the recent monograph by H Geiges An Introduction to Contact Topology (Cambridge U Press, 2008). Some attempts to use contact geometry in physics were made in the monograph Contact Geometry and Nonlinear Differential Equations (Cambridge U Press, 2007). Unfortunately, even the excellent style of this monograph is not sufficient to attract the attention of the physics community to this type of problems. This book is the first serious attempt to change the existing status quo. In it we demonstrate that, in fact, all branches of theoretical physics can be...
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The geometry of power exponents includes the Newton polyhedron, normal cones of its faces, power and logarithmic transformations. On the basis of the geometry universal algorithms for simplifications of systems of nonlinear equations (algebraic, ordinary differential and partial differential) were developed.The algorithms form a new calculus which allows to make local and asymptotical analysis of solutions to those systems.The efficiency of the calculus is demonstrated with regard to several complicated problems from Robotics, Celestial Mechanics, Hydrodynamics and Thermodynamics. The calculus also gives classical results obtained earlier intuitively and is an alternative to Algebraic Geometry, Differential Algebra, Lie group Analysis and Nonstandard Analysis.
This introduction to modern geometry differs from other books in the field due to its emphasis on applications and its discussion of special relativity as a major example of a non-Euclidean geometry. Additionally, it covers the two important areas of non-Euclidean geometry, spherical geometry and projective geometry, as well as emphasising transformations, and conics and planetary orbits. Much emphasis is placed on applications throughout the book, which motivate the topics, and many additional applications are given in the exercises. It makes an excellent introduction for those who need to know how geometry is used in addition to its formal theory.
Buy The Advanced Geometry of Plane Curves and Their Applications by C. Zwikker and Read this Book on Kobo's Free Apps. Discover Kobo's Vast Collection of Ebooks Today - Over 3 Million Titles, Including 2 Million Free Ones!
In the series of volumes which together will constitute the Handbook of Differential Geometry a rather complete survey of the field of differential geometry is given. The different chapters will both deal with the basic material of differential geometry and with research results (old and recent). All chapters are written by experts in the area and contain a large bibliography.
$37.50
Amazon MarketplaceReal Algebraic Geometry: Proceedings of the Conference held in Rennes, France, June 24-28, 1991 (Lecture Notes in Mathematics) (English and French Edition)
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Ten years after the first Rennes international meeting on real algebraic geometry, the second one looked at the developments in the subject during the intervening decade - see the 6 survey papers listed below. Further contributions from the participants on recent research covered real algebra and geometry, topology of real algebraic varieties and 16thHilbert problem, classical algebraic geometry, techniques in real algebraic geometry, algorithms in real algebraic geometry, semialgebraic geometry, real analytic geometry. CONTENTS: Survey papers: M. Knebusch: Semialgebraic topology in the last ten years.- R. Parimala: Algebraic and topological invariants of real algebraic varieties.- Polotovskii, G.M.: On the classification of decomposing plane...
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The differential geometric formulation of analytical mechanics not only offers a new insight into Mechanics, but also provides a more rigorous formulation of its physical content from a mathematical viewpoint.Topics covered in this volume include differential forms, the differential geometry of tangent and cotangent bundles, almost tangent geometry, symplectic and pre-symplectic Lagrangian and Hamiltonian formalisms, tensors and connections on manifolds, and geometrical aspects of variational and constraint theories.The book may be considered as a self-contained text and only presupposes that readers are acquainted with linear and multilinear algebra as well as advanced calculus. a collection of research surveys on the Distance Geometry Problem (DGP) and its applications. It will be divided into three parts: Theory, Methods and Applications. Each part will contain at least one survey and several research papers. The first part, Theory, will deal with theoretical aspects of the DGP, including a new class of problems and the study of its complexities as well as the relation between DGP and other related topics, such as: distance matrix theory, Euclidean distance matrix completion problem, multispherical structure of distance matrices, distance geometry and geometric algebra, algebraic distance geometry theory, visualization of K-dimensional structures in the plane, graph rigidity, and theory of discretizable DGP: symmetry and complexity. The second...
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Differential geometry has a long, wonderful history. It has found relevance in areas ranging from machinery design to the classification of four-manifolds to the creation of theories of nature's fundamental forces to the study of DNA. This book studies the differential geometry of surfaces with the goal of helping students make the transition from the compartmentalized courses in a standard university curriculum to a type of mathematics that is a unified whole. It mixes together geometry, calculus, linear algebra, differential equations, complex variables, the calculus of variations, and notions from the sciences. Differential geometry is not just for mathematics majors. It is also for students in engineering and the sciences. The mix of ideas offer students the opportunity to visualize |
Pearson Mathematics 7 Essentials Edition Student Book
From time to time, not all images from hardcopy texts will be found in eBooks, due to copyright restrictions. We apologise for any inconvenience.
Description
Pearson Mathematics 7 Essentials Edition Student Book is a streamlined edition of the Pearson Mathematics 7 Student Book, providing the same strong pedagogy and up-to-date research to comprehensively cover the Australian Curriculum requirements. Avoiding additional content and concentrating on essential theory, you and your students can focus on the national curriculum's outcomes. This student book is an uncluttered, clean and concise making it manageable for students, plus it is more economical, lighter and thinner than other Pearson Mathematics books. This text is compatible with Pearson Mathematics' bridging workbooks, homework programs and teacher companions.
Lighter and thinner than other Australian Curriculum textbooks, making Pearson Mathematics Essentials Editions more manageable than other books
Target audience
Suitable for Year 7 students.
Series overview
Focus on the fundamentals with Pearson Mathematics Essentials Edition. These streamlined editions of the Pearson Mathematics student books use the same strong pedagogy and up-to-date research to comprehensively cover the requirements of the Australian Curriculum. We've simplified the text to give you the 'nuts and bolts', covering all the theory and all the questions at a more affordable price. |
Problem Solving Approach to Mathematics - With CD - 10th edition
Summary: The new edition of this best-selling text includes a new focus on active and collaborative learning, while maintaining its emphasis on developing skills and concepts. With a wealth of pedagogical tools, as well as relevant discussions of standard curricula and assessments, this book will be a valuable textbook and reference for future teachers. With this revision, two new chapters are included to address the needs of future middle school teachers, in accordance to the NCTM Focal Poin...show morets document. ...show less
With CD!VeryGood
South Beach Bookstore Miami Beach, FL
0321570545 |
Intermediate Algebra
9780321358356
ISBN:
032135835X
Edition: 2 Pub Date: 2006 Publisher: Addison-Wesley
Summary: This student-focused text addresses individual learning styles through the use of a complete study system that starts with a learning styles inventory and presents targeted learning strategies designed to guide students toward success in this and future college-level courses. Students who approach math with trepidation will find that Intermediate Algebra, Second Edition, builds competence and confidence. The study sy...stem
Carson, Tom is the author of Intermediate Algebra, published 2006 under ISBN 9780321358356 and 032135835X. Thirty three Intermediate Algebra textbooks are available for sale on ValoreBooks.com, twenty eight used from the cheapest price of $0.01, or buy new starting at $15.15College Station, TXShipping:StandardComments:032135835x. This book is brand new, unread, and is in excellent condition. There is some scuffing... [more]032135835x. This book is brand new, unread, and is in excellent condition. There is some scuffing on the back of the book. Orders received before 3: 30pm CST ship daily Monday [more]
.[ |
The aim of this book is to examine the geometry of our world and, by blending theory with a variety of every-day examples, to stimulate the imagination of the readers and develop their geometric intuition. It tries to recapture the excitement that surrounded geometry during the Renaissance as the development of perspective drawing gathered pace, or more recently as engineers sought to show that all the world was a machine. The same excitement is here still, as enquiring minds today puzzle over a random-dot stereogram or the interpretation of an image painstakingly transmitted from Jupiter.The book will give a solid foundation for a variety of undergraduate courses, to provide a basis for aMore... geometric component of graduate teacher training, and to provide background for those who work in computer graphics and scene analysis. It begins with a self-contained development of the geometry of extended Euclidean space. This framework is then used to systematically clarify and develop the art of perspective drawing and its converse discipline of scene analysis and to analyze the behavior of bar-and-joint mechanisms and hinged-panel mechanisms. Spherical polyhedra are introduced and scene analysis is applied to drawings of these and associated objects. The book concludes by showing how a natural relaxation of the axioms developed in the early chapters leads to the concept of a matroid and briefly examines some of the attractive properties of these natural structures.Less...
Preface
Combinatorial Figures
Drawing figures
Modeling figures
The circuits of a figure
Combinatorial Geometries
The definition of a combinatorial geometry
Lines and planes of a combinatorial geometry
Two families of combinatorial geometries
Sketches of planar geometries and figures
Models of some combinatorial cubes
Subgeometries of a combinatorial geometry
Isomorphic combinatorial geometries
Planar Geometries and Projective Planes
The intersection of coplanar lines
Projective planes
Subgeometries of a projective plane
An extended Euclidean plane
Pencil and roller constructions
Coordinatizing an extended Euclidean plane
Sylvester's Theorem and the Fano plane
Non-Planar Geometries and Projective Spaces
Projective spaces
Subgeometries of a projective space
Desargues' Theorem and projective spaces
Extended Euclidean space
Coordinatizing extended Euclidean space
Perpendicular lines and planes
Pythagoras' Theorem, length and angle
The existence of a tetrahedron
Perspective Drawings
The definition and basic properties of a perspective drawing
Perspective drawings in a combinatorial geometry
Practical perspective drawing methods
Vanishing points
Perspective drawings of a box
Perspective rendition from an ideal viewpoint
Binocular Vision and Single Image Stereograms
Binocular vision
Binocular vision and random-dot stereograms
Single-image stereograms
Scene Analysis
Perspective drawings of intersections of planes
Scenes and other planar figures
The existence of boxes
Completing a partial scene
Two problems
Distortion and Anamorphic Art
Viewing perspective drawings
Anamorphic art
Distortion in 3-point perspective drawings
Distortion in 2-point perspective drawings
Distortion in 1-point perspective drawings
Distortion in affine projections
Planar Bar-And-Joint Mechanisms
Bar-and-joint models
Four-bar linkages
Rocking and rotation in 4-bar linkages
Coupler curves
Parallelogram and kite linkages
Plagiographs
Cognate linkages
Approximate and exact linear motion
Non-Planar Hinged-Panel-Mechanisms
Hinged-panel models
Four-panel cycles
Panel cycles of at least five panels
Polyhedral models
Graphs, Models and Geometries
The definition of a graph
Connected graphs and acyclic graphs
The rigidity of a one-story building
Hinge graphs and developments
Graphs that are also geometries
Spherical Polyhedra
The definition of a polyhedron
Planar graphs and three-connected graphs
The definition of a spherical polyhedron
Vertices, edges, and faces of a spherical polyhedron
The existence of spherical polyhedra
D�rer's melancholy octahedron
Scene Analysis and Spherical Polyhedra
Perspective drawings of a spherical polyhedron
Some applications of the theorem
Perspective drawings of a wider class of unions of polygonal regions
. Matroids
The definition of a matroid
Geometries, graphs, and matroids
Bases of a matroid
The rank function of a matroid
Isomorphism and representable matroids
Projective and affine geometry
Dual matroids
Restrictions and contractions of a matroid
Minors of a matroid
Paving matroids |
Functional analysis is the branch of mathematics dealing with spaces of functions. It is a valuable tool in theoretical mathematics as well as engineering. It is at the very core of numerical simulation. In this class, I will explain the concepts of convergence and talk about topology. You will understand the difference between strong convergence and weak convergence. You will also see how these two concepts can be used. You will learn about different types of spaces including metric spaces, Banach Spaces, Hilbert Spaces and what property can be expected. You will see beautiful lemmas and theorems such as Riesz and Lax-Milgram and I will also describe Lp spaces, Sobolev spaces and provide a few details about PDEs, or Partial Differential Equations. Workload: 4-6 hours/week Taught In: English Subtitles Available In: English Mooc available at: Workload: 2-3 hours/week Taught In: English Subtitles Available In: English MMOC available at:
In |
I hope that someone out there can provide me with some help. I am working my way through the small paperback book "Taxicab Geometry - An Adventure in Non-Euclidean Geometry" by Eugene F. Krause with a topics class of high school pre-math anal jrs and srs. And I would like to pick the brains of those who have used it before. What I'd like is a summative type activity to evaluate and access. This could be in the form of a paper, project, test, or whatever has worked for you. We are trying to focus on writing across the entire curriculum and they have done some "mathematically meaningful writing" so it could be a written. But it doesn't have to be...
At Newport, we in the math department have defined mathematically meaningful writing as (in correct paragraph form) 1) state and explain the problem, 2) explain your process (including any false starts), 3) state any conclusions and/or answers and justify them (Why are they the right answers? Are there other less correct answers?), 4) reflect on what your learned and what you did, and finally 5) prepare it like a 5-star restaurant would prepare a meal-with class. |
Mathematics
It is the mission of the A. E. Wright Middle School mathematics department through the coordinated efforts of its faculty to provide a program such that all students will be inspired to achieve and excel in the highest level of math possible. The goal of the mathematics department is to maintain our high state scores by continuing to teach the state standards at the highest level. With the implementation of technology in our instruction, we are using authentic real-world problems in our curriculum.
We believe in placing your child in a math class that will allow him/her to find success and enjoy mathematics. To ensure this, we revaluate the children annually. All incoming 6th graders take a test for placement.
Course Descriptions
6th Grade College Prep/High
By the end of sixth grade, students have mastered the four arithmetic operations with whole numbers, fractions, and decimals; they accurately compute and solve problems. They apply their knowledge to statistics and probability in calculating the range, mean, median, and mode of data sets. Students conceptually understand and work with ratios and proportions; they compute percentages (e.g. tax, tips, interest). Students know about π and the formulas for the circumference and area of a circle. They use letters for numbers in formulas involving geometric shapes and in representing an unknown part of a ratio. They solve 1-step linear equations.
7th Grade College Prep/High
By the end of grade seven, students are adept at manipulating numbers and equations and understand the general principles at work. Students understand and use factoring of numerators and denominators and properties of exponents. They know the Pythagorean theorem and solve problems in which they compute the surface area and volume of basic three-dimensional objects and understand how area and volume change with a change in scale. Students make conversions between units of measurement. They know and use different representations of fractional numbers (fractions, decimals, and percents) and are proficient at changing from one to another. They increase their facility with ratio and proportion, compute percents of increase and decrease, and compute simple and compound interest. They graph linear functions and understand the idea of slope and its relation to ratio.
8th Grade Pre-Algebra
By the end of eighth grade, students understand, use, and connect a variety of techniques for solving linear equations, inequalities, and systems of equations in applied contexts. They understand the meaning of variables, expressions, equations, and inequalities, and their applications in problem solving. Students graph and interpret the graphs of a variety of functions to visually represent connections to real-world problems. Students solve problems involving scale drawings and similar figures that connect geometry and algebra.
Algebra
Symbolic reasoning and calculations with symbols are central in algebra. Through the study of algebra, a student develops an understanding of the symbolic language of mathematics and the sciences. In addition, algebraic skills and concepts are developed and used in a wide variety of problem-solving situations. Students with less than a B average, will be recommended to retake Algebra in high school.
Accelerated Course Expectations
6 Accelerated:
In this comprehensive course, students work with a full range of topics that are needed for the successful study of mathematics. Emphasis is placed on the strengthening of skills in the use of fractions, ratios, decimals and percents. Variables are introduced early in the curriculum to familiarize the student with algebraic concepts. Operations involving inverse operations are used throughout the curriculum. Problem solving strategies, critical thinking and connections to technology are woven throughout the course. Placement is based on the
test given to all incoming 6th graders.
7 Accelerated:
In this comprehensive course, students work, write and simplify numerical expressions as well as introductory algebraic expressions. They focus on developing problem-solving strategies, especially on writing equations that require the use of signed numbers and several inverse operations for solutions. Emphasis is placed on the use of variables in equations and expressions. Statistics, non-routine problem solving, cooperative learning, functions, geometric formulas, ratios and proportions are explored. Prerequisite: an A average in 6 acc. and teacher recommendation.
8 Accelerated/Algebra Honors:
This honors course is designed to sharpen skills that use variables with real numbers. Students are introduced to polynomials, algebraic functions and the quadratic formula. There is an emphasis on learning strategies for solving word problems. This is a high school level course. Prerequisite: an A average in 7acc. and teacher recommendation.
Math is Fun: This site has clear descriptions and demos of concepts. There also are many good math games.
Aleks.com: Aleks is an online program that can be used to supplement mathematics and language arts learning. Most students will increase their understanding dramatically with regular usage.
Math Fun and Games
Cool Math: A large collection of math games to keep your brain thinking. |
Variables and Expressions (for Holt Algebra 1, Lesson 1-1) A variable is a letter that represents a value that can change. A constant is a value that ...
Name: _____ Cause u0026 Effect Write the cause and effect for each sentence. 1. Tim forgot his math book, so he was unable to complete his homework.
2 Individual Results with a Group Math Assessment Assessment is the key to instruction and intervention. It helps educators understand studentsu0027 performance, determine if they ...
Honoring the Text and Academic Authors Associationu0027s 2008 Textbook Excellence Award and McGuffey Longevity Award winners as well as staff and volunteers who have shown ...
3rd Grade Top Pick Curriculum Language Arts Option 1 MOVING BEYOND THE PAGE - Age 7-9 Full Year Curriculum Package Prerequisites: The student is able to read and comprehend ... |
Curriculum & Instruction: Functions&Algebra for Teachers
EDCI 200 Z05 (CRN: 60951)
3 Credit Hours
Jump Navigation
About EDCI 200 Z05
This course builds on the prior course Mathematics as a Second Language. Participants will obtain deep understanding of the concept of a function, utilize functions in problem solving, appreciate the pervasiveness of the function idea in the K-8 mathematics curriculum as well as everyday life, and engage in a variety of problem-solving activities that relate directly to the K-8 mathematics classroom. Topics include functions, graphs, inverse functions, linear functions, the algebra and geometry of straight lines, solving linear equations and inequalities, and an introduction to nonlinear functions. |
Supplies for Grade 7 – IB-2 for Sept 2012. Maths. Textbook - $50.00 EC rental for the year or $250.00 EC to purchase. If the textbook is rented, it must be returned on the last day of school in good condition or you will be billed the full amount
This workshop is designed for teachers who are new to the IBMaths Studies course and aims to help delegates familiarise themselves with the make up and philosophy of the course. ... For a good textbook treatment of . conditional probability. and .
The IB learner profile is used as a guide in developing and implementing the curriculum. ... Holt Geometry textbook available in hard copy, disk or online. Methodology. This course will utilize both inquiry and deductive thinking.
IB Math SL is intended to be comprehensive rather than rigorous, however, students can expect to be challenged. ... Students may use any notes they have (but not the textbook) to answer this homework quiz. A homework quiz will be graded out of ten points. 1
International Baccalaureate. Primary Years Programme. ... A choice of Chinese Maths or English Maths. ... PYP students access a wide range of learning resources and materials and do not follow one specific textbook. ( Collaborative learning.
In MYP mathematics, the four main objectives support the IB learner profile, ... from their own and other subject groups teachers working in isolation multiple sources and resources for learning a textbook-driven curriculum students investigating, ...
[See also holdings for newer editions, if any]. Textbook recommended for a few of our courses/modules. May be out of print now. MMM114 Materials Science I. (a+, Oct) check on MMM Book List. MMM211 Materials Science ... ENM101 Engineering MathsIB. (a, Oct) check on ENM Book List. and later modules.
In particular we are an International Baccalaureate world school. ... and the students start to use their GCSE textbook in Year 9. ... Mathematics is a very popular and successful subject in our Sixth Form with two thirds of our students taking A level Maths, Further Maths or IB options.
Maths Assessment Guidelines. ... These can include questions from the textbook or homework book, test revision, ... In line with both the new International IGCSE Maths course and IBMaths courses, revision tests will consist of two main sections.
Students will follow advanced IB&M modules whose content extends beyond the basic textbook knowledge. The MSc in IB&M modules reflect state ... The MSc in International Business and ... if your subjects include an equivalent of the IB/EB Maths Higher or Maths Standard/Methods and you ...
Your IB MYP programme is designed to help you develop into an excellent young adult in many different ways. ... Maths in MYP aims to help you develop strong reasoning and problem-solving skills in real-life contexts. ... A Mathematics textbook?
... maths; two sciences; ancient or modern history or geography; and a modern or ancient language. ... however concern was expressed about buying a textbook linked to a particular specification. ... 'A' levels and IB.
The information contained in these notes is correct to my knowledge and by no means copied from any textbook, therefore, ... The (.ib model for the voltage controlled current source comes in useful for the 'common ... This time I'll go through the maths a lot faster, and let you work it out ...
Should I Do More to Upgrade My Maths? What Study Skills Will I Need? ... e.g. IB) you will be taking ... Maths for Economics, Oxford University Press also covers the required topics, and will be a textbook for module EC121. You should aim to be familiar with the following topics before you ...
Noted that there isn't an obvious good textbook for the course, ... using past questions from IB higher level Physics papers ... Noted that you can't do Engineering or Physics at uni without Higher level maths, which is not easy.
[ This part provides two extra sets of questions for each exercise in the textbook, namely Elementary Set and Advanced Set. You may choose to complete any . ONE. ... Mark the angle between the line segment IB and the plane ABCDEF. [ Copy of the figure is provided in the Appendix. ] 5.
There exist strong beliefs among them that if you do not follow the textbook, ... at the same time as some have great ambitions and want to enter the International Baccalaureate (IB) programme. ... This is more maths. Before, it was almost as kindergarten.
Maths. Music. Physical Education. Science. ... A textbook will be used to resource the section on the French Revolution. Ancient Civilizations: ... An end of term assessment will follow the IB pattern.
in International Business endows its graduates with solid academic knowledge in Business Administration, ... Basic maths: introductory course. ... The students may choose among the following - There is no single adopted textbook.
This was written by a Victorian teacher who has worked with the IB structure: "In Victoria, three VCE Year 12 subjects have always seemed to work well. The current 4 Year 11 subjects (Maths Methods, two implementations of ... and textbook and other resource developers will 'do their ...
Bored by Elizabeth I but could avoid questions on her in the exam. European History – big syllabus (as for IB today but not modern A levels ... So I never got on with maths because I used to say, 'Hang on a minute, I need ... In fact in today's textbook, you would see much more ...
The International Baccalaureate Program is a rigorous academic program for students in their final two years of high school. ... Biology HL Group 5 Maths and computer science. Math Studies SL. ... Beside the textbook, students will use resources such as magazine articles, songs, games, ...
The International Baccalaureate Program is a rigorous academic program for students in their final two years of High School. ... Biology SL Group 5 Maths and computer science. Math Studies SL. ... Beside the textbook, students will use resources such as magazine articles, songs, games, ...
... school, grades, maths. Easy! Choose the correct sources. Your essay must include a mixture of sources from: websites, books ... Keep in mind that the examiners are IB teachers from your chosen subject ... • Provided a foot note/ citation for any method found in a textbook or reference ...
Elements was widely considered the most successful and influential Mathematics textbook of all time, ... A complex number a + ib represents a point in a plane. ... Russian maths genius Perelman urged to take $1m prize bbc.co.uk, Wednesday, 24 March 20102.
... from the list of approved Diploma Programme subjects—normally one of the student's six chosen subjects for the IB diploma. ... school, grades, maths. Easy! Choose the correct sources. Your ... if an essay involves something to do with waves then nothing is gained by giving textbook ...
Grade 1 maths, increase of 3% in ... multigrade or otherwise), textbook availability, teacher training for textbooks and knowledge of subject, whether teachers and principals live in the ... ...
Parents listen and interact as their children read to them, demonstrate experiments, use the computer, play maths games together ... International Baccalaureate Organisation (IBO) National Center for School Curriculum and Textbook Development (NCCT ...
can you unambiguously write down what LCP states (see 148 of the AS textbook)? remember this is a . predictive. ... those who do Maths Statistics will be able to explain that: e.g. dichloromethane gives M(g)+( peaks at 85, 87 and 89 in a 9:6:1 ratio.
Textbook in preparation where nuclear physics and particle physics in integrated. ... THE PROFILE OF OUR SCHOOL IS MATHS-SCIENCES OVER ALL THE FOURTH GRADES. Several times, with several persons ... More of Particle Physics I teach within IB Diploma Programme.
... as the Education Governance and Management Specialist in MoNE to prepare a policy on the use of English in science and maths. ... Victoria Certificate Education (VCE) and International Baccalaureate (IB), while ... The national curriculum strongly recommends that textbook writers select and ...
... asked the class which of us enjoyed Maths and preferred translating from English to Latin rather than from Latin to English (I was ... I had been amazed by a sentence in my history textbook mentioning Julius Caesar's De Bello Gallico as something `you may one day read in the ... (ib.). I had ...
... the IB Middle Years and Diploma programmes and the Pre-U examination. ... or superior to the exemplar material in the course textbook. ... Marks of one candidate wrongly aggregated. Further maths grade raised from B to A on 22nd October, by which time student had lost his University place.
IB. 3. Visits. i. Oversees Trips. ii. Hong Kong Trips. 4. Initiatives. i ... the History curriculum prepares the students to perform basic tasks such as how to use the index and glossary of a textbook. ... Maths Lesson; Racial Instruction; History lesson; Look at school life in Nazi Germany ... |
This eBook introduces the subject of logarithms and exponentials, from the basic definition of logarithm, through the laws of logarithms, undertaking an assessment and an appreciation of exponential graphs, looking at the linear form of exponentials interspersed with a series of questions and worked examples |
I know this isn't exactly the sort of question usually asked here, but I was hoping someone could help anyway...
I'm a "TA" at my university, but a very good teacher, so I am actually teaching Calculus 3 this semester. We are using Calculus: Early Transcendentals (6e) by Stewart.
I am "supposed" to cover chapters 12-16. But I think I'm going too slow; I'm only in section 13.3 (Arc Length and Curvature) right now.
I have a few questions to ask on how the class should be taught...
How much explanation would you give on definitions/theorems, in the way of proof, for example? Obviously I'm not proving every little thing (like properties of dot product for example lol...), but I really feel it is important for them to see the reasoning, the logic behind these things. I really don't like just stating something and moving on.
Are there any sections that can/should be skipped? I'm thinking 13.4 (Velocity and Acceleration Vectors) can mostly be avoided... maybe a quick mention of the relationship , but definitely skip stuff like Kepler's laws of motion.
Just looking ahead to chapter 14, Functions of 2 Variables and Partial Differentiation... How much detail is really necessary here? I mean limits, yes... the fact that you now have infinitely many paths on which to approach a point as opposed to just from the left or right. But in general proving that a particular limit DOES exist seems to be quite difficult, without resorting to the characterization (which I don't really want to spend time on...). But other than that, there are a ton of PAGES in chapter 14, but the material is not difficult. There's just a lot of it, and I don't want to continue falling further and further behind...
So... I don't know. What would you guys do?
February 24th 2011, 11:09 AM
wilbursmith
Well
Iīll give you my perspective.
Itīs imossible to cover all the material in Stewart (great book by the way) so youīll have to skip alot, thatīs a fact. It is also one of the problems young teachers face during their first years of their career, not enough time but alot to cover.
1. There are so many university-whatever homepages on the internet with lessonplans and stuff. Check them out to see whatīs really important. Hereīs just one example. You can always ask previous teachers at your university what, how and when.
2. The review sections in Stewart are quite good. They usually pic up the most important stuff. Try to cover as much as possible from there.
3. Make a semester plan for yourself, donīt be too time-optimistic.
4. Whatever you cover, do it properly. Donīt blame yourself if thereīs not enough time to cover everything. Thereīs a reason why you already teach at at a university, youīre good and thatīll keep your students motivated. The more they understand after your lectures the more they will study on their own.
Donīt know if this made sense but just get that plan and then concentrate on what your good at. |
Hi guys
You are both welcome. I also suggest searching the web for sites on math using StumbleUpon.
Hi Bob
You said that like it is something bad. What did you mean byThat's not how I got it, but okay. I got it now, so... I found another one, but I have it bookmarked on my laptopYes. It is a website that allows you to find pages related to your interestThanks a Lot! I just downloaded Algebra of LogicYou're welcomebarbie19022002 wrote:
HI Stefy, little late to join the discuss...but thanks for..I installed it in my google chrome..
Installed what?
stefy wrote:
Yes. It is a website that allows you to find pages related to your interest.
Could you please nameI spent time searching. However, there are two google features worth knowing: 1) Say you want to search only for pdf files, write the keywords and add filetype:pdf. For example google: c++ quick reference filetype:pdf 2) You can specify what domain you want to search in, for example: square root site:mathsisfun.com Or say you want to search only in .edu sites, then write: my_keyword site:.edu |
Placement and Testing Program
General Education Courses
Purpose:
The Department of Mathematics' general education courses aim to prepare students to communicate logically, become proficient at quantitative reasoning, and to think critically and analytically. The courses intend to enable students to deal comfortably with the basic notions of number and chance and to acquire mathematical tools to help reach their career objectives.
University Level Courses:
Mat 101, Mathematics for Elementary Teachers, develops students' understanding of the mathematics needed to teach elementary school. Topics include sets, functions, logic, the real number system, number theory, and problem solving. The course is mainly for early childhood, elementary education, and special education majors.
Mat 103, Introduction to Mathematics, is a liberal arts introduction to the nature and beauty of mathematics. Topics include logic, graph theory, probability, and decision theory. An additional topic is chosen from finance, geometry, and statistics.
Mat 105, College Algebra and Trigonometry, is a unified course in algebra and trigonometry. Topics include the study of polynomial, exponential, and logarithmic functions, systems of linear equations, and trigonometry.
Mat 107, College Algebra, is a thorough treatment of college algebra. Topics include polynomial, exponential, and logarithmic functions, systems of linear equations, and an introduction to linear programming.
Developmental Courses:
Many students find that the reasoning required in the above courses is quite different from that of their high school courses and also find that their arithmetical and/or algebraic skills need strengthening. We offer, therefore, two courses, Mat 001 and Mat 000, described below, to help students prepare for university level mathematics courses.
Mat 001, Fundamental Skills in Arithmetic, is designed to strengthen basic arithmetic skills and to introduce the elements of algebra. Students, in general, are placed in Mat 001 if their math SAT is less than 440. A student (other than an Early Childhood, Elementary, and Special Education major) must complete this course and the subsequent course Mat 000 with a grade of C- before enrolling in a 100 level mathematics course. An Early Childhood, Elementary, or Special Education major with an SAT math score less than 480 must complete this course with a grade of at least C- before enrolling in Mat101.
Mat 000, Fundamentals of Algebra, aims at strengthening basic algebraic skills. A student (other than an Early Childhood, Elementary, and Special Education major) with a SAT math score greater than or equal to 440 and less than 480 must successfully complete this course with a grade of at least C- before enrolling in a 100 level mathematics course (see note below).
The Mathematics Placement Exam:
Although students are placed into their mathematics courses by SAT score, they are given the opportunity to 'test-out" of their initial placement.
There are three different placement tests: Arithmetic, Basic Algebra, and College Algebra and Trigonometry. All three tests are approximately 50 minutes in length and have a multiple-choice format.
The Arithmetic Placement Test is for students who wish to "test out" of Mat 001. It measures basic arithmetic and reasoning skills pertaining to whole numbers, fractions, decimals, percents, ratio and proportion, and some geometry. A passing score on this test allows students to enroll in Mat 000 with the exception of Early Childhood, Elementary, and Special Education majors who may enroll in Mat 101.
The Basic Algebra Test is for students who wish to "test out" of Mat 000. It measures fundamental algebraic skills pertaining to linear, quadratic, and rational expressions, factoring, exponents, and inequalities. A passing score on this test allows students to enroll in Math 101, Mat 103, Mat105, Mat107, orMat 121.
The College Algebra and Trigonometry Test, is for students who wish to "test out" of Mat 110. It measures more advanced algebra skills and knowledge of basic trigonometry. Topics include polynomial, exponential, rational, and logarithmic functions, graphing, and trigonometric functions and identities. A passing score on this test allows students to enroll in Math 161, Calculus I.
Students may take a challenge exam to test out of their SAT placement. This exam is given during freshmen summer orientation or on the second and third nights of classes each semester. Additional times on which the test is given are posted outside the Department of Mathematics office, Room 101, 25 University Avenue. Test results are normally made available on the same day as the exam. Scores are posted at the same site and are forwarded to advisors and others responsible for scheduling.
Advanced Placement Policy:
Course credit for success on AP exams in mathematics is awarded as follows:
AP Test : Score on AP Test
3
4
5
Calculus AB
MAT 108
MAT 161
MAT 161
Calculus BC
MAT 161
MAT 162
MAT 162
Statistics
MAT 121
MAT 121
MAT 121
If placed in a calculus class because of an SAT score, the student must still pass a departmental examination administered during the first day of classes before being allowed to continue.
Dr. Peter Zimmer, Director of the Department of Mathematics' Developmental Education Program can answer questions or provide additional information regarding the testing program. He may be contacted at (610-436-2696) or by e-mail (pzimmer@wcupa.edu).
Summary of the Placement and Testing Program
Department of Mathematics
First-Year Students:
Students with SAT Math scores of 440 or below are placed in MAT 001; those with scores greater than 440 and less than or equal to 480 are placed in MAT 000. A score above 480 entitles a student to enroll in a 100 level mathematics course. For those students who require MAT 101, i.e., early childhood, elementary, or special education majors, the remedial mathematics course is MAT 001 for students with an SAT math score of 480 or below.
Students are not placed in remedial math in the same semester as remedial English. Those that require both remedial English and remedial math will first be scheduled for remedial English and placed in remedial math the following semester.
Students have the opportunity to "test out" of remedial mathematics courses by taking the required placement exam.
Students who intend to take Mat161, Calculus I and have an SAT math score of 590 or less are placed in Mat110, Pre Calculus. Students who wish to "test out" of this placement must pass the College Algebra and Trigonometry placement test.
Transfer Students:
Students who receive transfer credit for a 100 level mathematics course are not required to take a remedial mathematics course. Those who did not receive such credit are required to pass the appropriate mathematics placement test before enrolling in any mathematics course.
NOTE: A recent University study has shown those students with SAT Math scores of 530 or below tend to do better in university level mathematics courses if they first satisfactorily complete the appropriate remedial math course. We therefore encourage students whose SAT math score is above 480 but below 530 to take the recommended mathematics placement exam. After assessing their own individual performance on the exam, students may choose to take the recommended remedial math course to help assure success in university level mathematics courses. |
Hi fellow students , I heard that there are certain software that can help with us studying ,like a tutor substitute. Is this really true? Is there a software that can help me with algebra ? I have never tried one thus far , but they are probably not hard to use I assume. If anyone has such a program, I would really appreciate some more information about it. I'm in Intermediate algebra now, so I've been studying things like easiest way to find common multiple and it's not easy at all.
I have a solution for you and it might just prove to be a better one than buying a new textbook. Try Algebrator, it covers a pretty comprehensive list of mathematical topics and is highly recommended. With it you can solve various types of problems and it'll also address all your enquiries as to how it came up with a particular answer. I tried it when I was having difficulty solving questions based on easiest way to find common multiple and I really enjoyed using it.
I have tried out numerous software. I would without any doubt say that Algebrator has assisted me to come to grips with my problems on function composition, unlike denominators and ratios. All I did was to merely key in the problem. The answer showed up almost immediately showing all the steps to the solution . It was quite easy to follow. I have relied on this for my math classes to figure out Remedial Algebra and Algebra 1. I would highly recommend you to try out Algebrator.
Y'all have got to be pulling my leg ! How could this not be general information or promoted in periodicals? How can I acquire additional info for trying Algebrator? Pardon someone for being a tad bit skeptical , but do you believe if someone can get a test version to exercise this software ? |
book in the Math Made a Bit Easier series by independent math tutor Larry Zafran. It contains 50 abridged lesson plans covering basic algebra and geometry, for a target audience of tutors, parents, and homeschoolers. Each lesson plan includes all of the components of a typical classroom lesson such as aim, motivation, warm-up exercises, demonstrative examples, questions for thought and discussion, and connections to earlier and later material.
This book is intended to be used in strict conjunction with the fourth book of the series (Basic Algebra and Geometry Made a Bit Easier: Concepts Explained in Plain English). The book assumes that the instructor actually knows the material him/herself, but could benefit from having a general guideline to follow. The author makes a point of identifying the concepts which most students tend to find easy or difficult, including suggestions on how to help with the latter.
The book includes an introduction describing how the book can be put to best use, as well as a section on how to effectively work with students who are struggling with the material. The author explains that for the vast majority of students, the root of the problem can be traced back to never having fully mastered basic math concepts and skills. The book's lessons make frequent reference to reviewing earlier books in the series as needed so that the student masters all of the prerequisite material.
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He is a dedicated student of the piano, and the leader of a large and active group of board game players which focuses on abstract strategy games from Europe.
He presently lives in Cary, NC where he works as an independent math tutor, writer, and webmaster. |
From angles to functions to identities - solve trig equations with ease
Got. With just enough refresher explanations before each set of problems, you'll sharpen your skills and improve your performance. You'll see how to work with angles, circles, triangles, graphs, functions, the laws of sines and cosines, and more!
100s of Problems! * Step-by-step answer sets clearly identify where you went wrong (or right) with a problem * Get the inside scoop on graphing trig functions * Know where to begin and how to solve the most common equations * Use trig in practical applications with confidence less |
Basic College Mathematics - 4th edition
Summary: Elayn Martin-Gay firmly believes that every student can succeed, and her developmental math textbooks and video resources are motivated by this belief. Basic College Mathematics, Fourth Edition was written to help readers effectively make the transition from arithmetic to algebra. The new edition offers new resources like the Student Organizer and now includes Student Resources in the back of the book to help students on their quest for success Annotated Instructor's Edition. Book has the same contents as the student edition, but includes answers. 4th Edition Ships same or next day. Expedited shipping takes 2-3 business days; standard sh...show moreipping takes 4-14 business days. ...show less
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wen/Nelson's ELEMENTARY TECHNICAL MATHEMATICS, Ninth Edition is a well-respected, extremely user-friendly text. It emphasizes essential math skills and consistently relates math to practical applications so students can see how learning math will help them on the job. The applications are drawn from a wide array of technical fields, making the text useful to a broad range of students. Annotated examples and visual images are used to engage students and assist with problem solving. Comprehensive and well-organized, this text engages students, p... MOREroviding them with a solid foundation in mathematical principles that will help them to succeed in the current course and beyond. |
hands to explore and build proficiency and eventually to replicate... I've previously taken regular calculus classes with engineers and won... This is not the same at all. We were solving real problems every day...Students work through problems using an online interactive textbook D...When teams become stuck on a problem Chiel or a teaching assistant m...
hands, to explore and build proficiency, and, eventually, to replicate and build on recent math models used in the biological sciences. The course is cross-listed as both a biology and biomedical engineering class.
"I've previously taken regular calculus classes with engineers and wondered what would the classes ever be useful for," said Kate Coyle, a biology major who completed the Dynamics class and graduated this semester. "Labs I've had in biology and physics show you the protocol and the expected result.
"This is not the same, at all. We were solving real problems every day."
Students work through problems using an online interactive textbook, Dynamics of Biological Systems: A Modeling Manual Chiel wrote and the computer programming language Mathematica, which scientists worldwide rely on to build mathematical models of complex systems. Chiel's book is available free to students as well as teachers who may want to use it as is or as a model for their own classes.
When teams become stuck on a problem, ,Chiel or a teaching assistant makes suggestions, gives clues and tries to coax out the answer. After success, teachers quiz individuals about how they found the solution and what they'd learned.
The class of 30 is spread out among hexagonal tables. Teams power up their laptops and go to work. Each day the teachers rotate to a different group of students, and after each class they compare notes on who has mastered the skills and who needs extra help, Gill said.
When the second half of the semester begins, teams choose a mathematical model that was recently published in a scientific journal, begin reconstructing and analyzing it and then writing in detail what they learn. The students then extend the model to answer new questions that they ask themselves, and write up results as if they were writing for a scientific journal.
Coyle and her teammates Valencia Williams and Joshua DeRivera focused on a pa |
A software to calculate expression, roots, extremum, derivateve, integral, etc.Features: Math Calculator is an expression calculator. You can input an expression including variable x, for example, log(x), then input a valueof x; You can also input an expression such as log(20) directly.Math Calculator is an equation solver Math Calculator can be used to solve equations with one variable, for example, sin(x)=0. Math Calculator is a function analyzer Math Calculator has the abilities of finding maximum and minimum.Math Calculator is a derivative calculator and calculus calculator. You can use this program to calculate derivative and 2 level derivate of a given function.Math Calculator is an integral calculator. Math Calculator has the ability of calculating definite integral. |
Investigating Prealgebra - 02 edition
ISBN13:978-0534453091 ISBN10: 0534453090 This edition has also been released as: ISBN13: 978-0030226243 ISBN10: 0030226244
Summary: Investigating Prealgebra is a flexible worktext that supports a variety of teaching methods and prepares students for their first course in algebra. The book engages students in active learning, allowing them to discover knowledge and describe their understandings. Through a blend of lecture-type Discussions and constructivist Investigations, this book provides a comprehensive foundation in prealgebra concepts.
Introduction to the Chapter The Order of Operations Formulas Paired Data and Line Graphs Properties of Equality Equations With More Than One Operation Read, Reflect, and Respond: Learning Styles Glossary and Procedures Review Test
3. INTEGERS
Introduction to the Chapter Introduction to Integers Addition of Integers Subtraction of Integers Multiplication and Division of Integers Introduction to Polynomials Read, Reflect and Respond: Negative Numbers Glossary and Procedures Review Test
4. FRACTIONS
Introduction to the Chapter Multiples and Factors Introduction to Fractions Multiplication of Fractions Division of Fractions Addition and Subtraction of Fractions Mixed Numbers Arithmetic with Mixed Numbers Solving Equations Read, Reflect, and Respond: The Value of Mathematics in Daily Life Glossary and Procedures Review Test
Introduction to the Chapter Introduction to Ratios and Rates Proportions Percents Applications of Percents Circle Graphs Read, Reflect, and Respond: The Value of Mathematics for Citizenship Glossary and Procedures Review Testorgasorus Books, Inc. MO Wentzville, MO
PAPERBACK Fair 0534453090 Student Edition. Missing up to 10 pages. Moderate wear, wrinkling, Curling or creasing on cover and spine. May have used stickers or residue. Fair binding may have a few l...show moreoose or242 |
A First Course in Computational Algebraic guide to computing in algebraic geometry with many explicit computational examples introducing the computer algebra system Singular. This quick guide is designed for young students with some background in algebra who wish to perform their first experiments in computational geometry. It provides a compact presentation of the basic theory, with particular emphasis on explicit computational examples using the computer algebra system Singular. A First Course in Computational Algebr... MOREaic Geometry is designed for young students with some background in algebra who wish to perform their first experiments in computational geometry. Originating from a course taught at the African Institute for Mathematical Sciences, the book gives a compact presentation of the basic theory, with particular emphasis on explicit computational examples using the freely available computer algebra system, Singular. Readers will quickly gain the confidence to begin performing their own experiments. |
Algebra and Trigon students develop insight into mathematical ideas. The authors' attention to detail... MORE and clarity, as in James Stewart's market-leading Calculus text, is what makes this text the market leader. Choose the algebra textbook that's written so you can understand it. ALGEBRA AND TRIGONOMETRY reads simply and clearly so you can grasp the math you need to ace the test. And with Video Skillbuilder CD-ROM, you'll follow video presentations that show you step-by-step how it all works. Plus, this edition comes with iLrn, the online tool that lets you sign on, save time, and get the grade you want. With iLrn, you'll get customized explanations of the material you need to know through explanations you can understand, as well as tons of practice and step-by-step problem-solving help. Make ALGEBRA AND TRIGONOMETRY your choice today. |
Math Principles for Food Service Occupations, 5th Edition
Math Principals for Food Service Occupations teaches readers that the understanding and application of mathematics is critical for all food service jobs, from entry level to executive chef or food service manager. All the mathematical problems and concepts presented are explained in a simplified, logical, step by step manner. It is a book that guides food service students and professionals in the use of mathematical skills to successfully perform their duties as a culinary professional or as a manager of a food service business. Now out in the 5th edition, this book is unique because it follows a logical step-by-step process to illustrate and demonstrate the importance of understanding and using math concepts to effectively make money in this demanding business. Part 1 trains the reader to use the calculator, while Part 2 reviews basic math fundamentals. Subsequent parts address math essentials in food preparation and math essentials in food service record keeping while the last part of the book concentrates on managerial math. New to this 5th edition, "Chef Sez", quotes from chefs, managers and presidents of companies, are used to show readers how applicable math skills are to food service professionals. "TIPS" (To Insure Perfect Solutions) are included to provide hints on how to make problem solving simple. Learning objectives and key words have also been expanded and added at the beginning of each chapter to identify key information, and case studies have been added to help readers understand why knowledge of math can solve problems in the food service industry. The content meets the required knowledge and competencies for business and math skills as required by the American Culinary Federation124.95
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is for engineering students and a reference for practising engineers, especially those who wish to explore Python. This new edition features 18 additional exercises and the addition of rational function interpolation. Brent's method of root finding was replaced by Ridder's method, and the Fletcher-Reeves method of optimization was dropped in favor of the downhill simplex method. Each numerical method is explained in detail, and its shortcomings are pointed out. The examples that follow individual topics fall into two categories: hand computations that illustrate the inner workings of the method and small programs that show how the computer code is utilized in solving a problem. This second edition also includes more robust computer code with each method, which is available on the book Web site. This code is made simple and easy to understand by avoiding complex bookkeeping schemes, while maintaining the essential features of the method. less |
Offering a uniquely modern, balanced approach, Tussy/Gustafson/Koenig's DEVELOPMENTAL MATHEMATICS FOR COLLEGE STUDENTS, Third Edition, integrates the best of traditional drill and practice with the best elements of the reform movement. To many developmental math students, algebra is like a foreign language. They have difficulty translating the words, their meanings, and how they apply to problem solving. Emphasizing the "language of algebra," the text's fully integrated learning process is designed to expand students' reasoning abilities and teach them how to read, write, and think mathematically. It blends instructional approaches that include vocabulary, practice, and well-defined pedagogy with an emphasis on reasoning, modeling, communication, and technology skills. The text's resource package--anchored by Enhanced WebAssign, an online homework management tool--saves instructors time while also providing additional help and skill-building practice for students outside of class.
N. Tuchina A way to Success: English for University Students ( Uploaded - Ryushare ) |
More About
This Textbook
Overview
This is the second volume in a series of innovative proceedings entirely devoted to the connections between mathematics and computer science. Here mathematics and computer science are directly confronted and joined to tackle intricate problems in computer science with deep and innovative mathematical approaches.
The book serves as an outstanding tool and a main information source for a large public in applied mathematics, discrete mathematics and computer science, including researchers, teachers, graduate students and engineers. It provides an overview of the current questions in computer science and the related modern and powerful mathematical methods. The range of applications is very wide and reaches beyond computer |
The International Mathematical Olympiad (IMO) is an annual international mathematics competition held for pre-collegiate students. This book is an amalgamation of the booklets originally produced to guide students intending to contend for placement on their country's IMO team.
Foreword
Combinatorics
A Quick Reminder
Partial Fraction
Geometric Progressions
Extending the Binomial Theorem
Recurrence Relations
Generating Functions
Of Rabbits and Postmen
Solutions
Geometry
The Circumcircle
Incircles
Exercises
The 6-Point Circle?
The Euler Line and the Nine Point Circle
Some More Examples
Hints
Solutions
Glossary
Solving Problems
Introduction
A Problem to Solve
Mathematics: What is it?
Back to Six Circles
More on Research Methods
Georg P�lya
Asking Questions
Solutions
Number Theory 2
A Problem
Euler's �-function
Back to Section 4.2
Wilson
Some More Problems
Solutions
Means and Inequalities
Introduction
Rules to Order the Reals By
Means Arithmetic and Geometric
More Means
More Inequalities
A Collection of Problems
Solutions
Combinatorics 3
Introduction
Inclusion-Exclusion
Derangements (Revisited)
Linear Diophantine Equations Again
Non-taking Rooks
The Board of Forbidden Positions
Stirling Numbers
Some Other Numbers
Solutions
Creating Problems
Introduction
Counting
Packing
Intersecting
Chessboards
Squigonometry
The Equations of Squares
Solutions
IMO Problems 2
Introduction
Aus 3
Hel 2
Tur 4
Rom 4
Uss 1
Revue
Hints Aus 3
Hints �Hel 2
Hints �Tur 4
Hints � Rom 4
Hints � Uss 1
Some More Olympiad Problems
Solutions |
Job hunters--or anyone who wants to practice taking self-assessment tests--look no further: This step-by-step guide gives you the key skills for taking numerical reasoning tests. Now routinely used in recruitment, these tests can be daunting. Smith … see full wiki
An excellent resource if all you need is a quick refresher, of limited value if you need deeper tutoring
This book is basically a review of all the mathematics the typical child learns from kindergarten through eighth grade. Each chapter begins with a list of the terms used in the chapter and this is followed by an explanation of the concepts and then examples of how the concepts are applied. Each chapter concludes with a set of practice exercises followed by the solutions. The topics covered are:
*) The basic algorithms of addition, subtraction, multiplication and division applied to signed integers, decimal numbers and fractions. *) Problems involving rates of change. *) Percentages and many ways in which they are applied. *) Ratios and proportions *) Interpreting data that appears in tables, reaching conclusions from the data.
If you are faced with the prospect of taking an exam over the ability to perform numerical reasoning and you are uncertain about your ability to apply these concepts, then this is a book that will be invaluable if all you need is a quick refresher. If your problems are more deep-seated with a lack of fundamental understanding, then something that offers greater depth of explanation will be needed. |
The most helpful favourable review
The most helpful critical review
15 of 15 people found the following review helpful
5.0 out of 5 starsGreat revision tool...
thin and does the job very well. I've begun using it to teach from now. Just ordered 30 for my class. Comes along with a teachers book that is very similar but goes further in that it has worked solutions for all of the exercises.
Can be used along with Course Companion which is the actual answer to a revision book from the same series, and that comes with a CD for an electronic version of the book. Seeing it's done by the examing board it cant be beat, unless you compare it with D.Rayner's books and they ensure A*'s.
This book proved to be very helpful while preparing for my GCSE Maths Higher Tier in 2009. It highlights the most common mistakes pupils make, so you can keep these in mind and avoid them on the actual exam, which saves you points! As a more of a home-based learner (I attended evening classes just once a week), I can tell this book explains everything very clearly and having the answers at the end, you can check if you understood everything properly. I found it very useful when self-studying, plain and understandable. I was reluctant at first and unsure if it wasn't just a waste of money as there are so many books of this kind out there. This one though was well worth the money. Highly recommended.
*UPDATE*Thanks to this book I got an A grade. It's worth every penny!!
Great to use along side your textbook, with fantastic explanation to help you access the A/A* material. I'd definitely recommend getting this book as early on in the course to help you progress and secure the high grade!
This book offers a range of questions that come up in Maths GCSE Higher paper. The questions are annotated with examiners comments that identify the way in which previous students have failed to provide a correct answer and gives clues to what the examiner is actually looking for. The use of this book will help to obtain those extra few marks that make the difference between an A and an A* grade.
This book follows the systems of examples then tests with answers and examiner comments and tips. It is as good as revision books ghet, they are just something to support the horrible task of revision and this one does it gently. Not much you can say really about revision books but they do help so are worth getting. This one is fine and I am working through it with a friends daughter who needs some help. I actualy like maths but how it is taught in school puts so many people off this subject, don't blame the kids for their grades ask if the teachers are up to the job! Having been involved in Ofstead insoections maths is very poorly taught in most schools and these books help fill the gaps for those who want to raise their game to a C or above. |
Mathematics in Action : Prealgebra Problem Solving book of the Mathematics in Action series, Prealgebra Problem Solving, Fourth Edition, illustrates how mathematics arises naturally from everyday situations through updated and revised real-life activities And The accompanying practice exercises. Along with the activities And The exercises within the text, MathXL® and MyMathLab® have been enhanced to create a better overall learning experience For The reader. Technology integrated throughout the text helps readers interpret real-life data algebraically, numerically, symbolically, and g... MOREraphically. The active style of this book develops readers' mathematical literacy and builds a solid foundation for future study in mathematics and other disciplines.
1. Write formulas for and calculate volumes of rectangular prisms (boxes) and right circular cylinders (cans).
2. Recognize geometric properties of three-dimensional figures.
What Have I Learned?
How Can I Practice?
Chapter 7 Summary
Chapter 7 Gateway Review
Chapter 8. Problem Solving with Mathematical Models
Activity 8.1 A Model of Fitness
Objectives:
1. Describe a mathematical model as a set of verbal statements.
2. Translate verbal rules into symbolic equations.
3. Solve problems that involve equations of the form y = ax + b.
4. Solve equations of the form y = ax + b for the input x.
5. Evaluate the expression ax + b in the equations of the form y = ax + b to obtain an output y.
Activity 8.2 Comparing Energy Costs
Objectives:
1. Write symbolic equations from information organized in a table.
2. Produce tables and graphs to compare outputs from two different mathematical models.
3. Solve equations of the form ax + b = cx + d.
Activity 8.3 Mathematical Modeling
Objectives:
1. Develop an equation to model and solve a problem.
2. Solve problems using formulas as models.
3. Recognize patterns and trends between two variables using a table as a model.
4. Recognize patterns and trends between two variables using a graph as a model.
What Have I Learned?
How Can I Practice?
Chapter 8 Summary
Chapter 8 Gateway Review
Appendix Learning Math Opens Doors: Twelve Keys to Success
The Consortium for Foundation Mathematics is a team of fourteen co-authors, primarily from the State University of New York and the City University of New York systems. Using the AMATYC Crossroads standards, the team developed an activity-based approach to mathematics in an effort to reach the large population of college students who, for whatever reason, have not yet succeeded in learning mathematics. |
resource provides a bridge to calculus with the introduction of sequences and series. Lessons in algebra, functions, trigonometry, analytic geometry, and graphical analysis begin with rules followed by exercises. Answer key included.
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This resource provides a bridge to calculus with the introduction of sequences and series. Lessons in algebra, functions, trigonometry, analytic geometry, and graphical analysis begin with rules followed by exercises. Answer key |
From Counting to Calculating, part of the Mathematics for the Majority series produced by The Schools Council Project in Secondary School Mathematics, is aimed at expanding teachers' knowledge about the use of number in calculations in order to pass on this understanding to higher secondary students within their lessons. This…
Number Appreciation, part of the Mathematics for the Majority series produced by The Schools Council Project in Secondary School Mathematics, is aimed at expanding teachers' knowledge about the number system in order to pass on this understanding to higher secondary students within their lessons. This book compliments From…
Mathematics From Outdoors, part of the Mathematics for the Majority series produced by The Schools Council Project in Secondary School Mathematics, is aimed at expanding teachers' knowledge and is designed to help teachers to develop classroom activities that will interest students and enable them to link the mathematics they…
Machines, Mechanisms and Mathematics, part of the Mathematics for the Majority series produced by The Schools Council Project in Secondary School Mathematics, is aimed at expanding teachers' knowledge, and is written in the belief that much of mathematics has been developed to solve practical problems.
Many practical applications…
Luck and Judgement, part of the Mathematics for the Majority series produced by The Schools Council Project in Secondary School Mathematics, is aimed at expanding teachers' knowledge, concentrating upon a practical approach to the teaching of probability and statistics.
The book has two parts, the first concentrating upon…
Crossing Subject Boundaries, part of the Mathematics for the Majority series produced by The Schools Council Project in Secondary School Mathematics, is a book aimed at teachers, encouraging collaboration across different subject areas.
Each chapter focuses upon a different area of the curriculum and describes aspects of mathematics…
At the time in the…
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• Introductory material, and…
Keys to Chemistry Book 2 continued from Book 1 and took the study of chemistry in secondary school up to O level standard. The book is a reflection of chemistry teaching in the 1970s, following the Nuffield developments in the previous decade.
Contents
1. Particles in motion
2. Atoms, relative atomic masses and moles
3. Atoms,…
This book first published in 1977 is about graphs, their drawing, their interpretation, their development and their use. It discusses the teaching of graphs from their early introduction and as far as the beginnings of integral and differential calculus. The book also places the teaching of graphs in an historical context as it reviews…
This book from the Schools Council discusses the aims of the Mathematics for the Majority project and includes contributions from teachers who were actively concerned with teaching mathematics at this standard. It is written in two parts:
Part 1 discusses the philosophy and approach relevant to the learning of mathematics by the…
This Schools Council publication addresses assignment cards and how they can play an important part in mathematics teaching. The book discusses methods of preparing them and gives numerous examples.
Topics as diverse as the Binary system, punched card matrices, vectors and topology feature as do geometrical and statistical assignments.… |
Franklin Park, IL Geometry just adding and multiplication but computing eignevalues, too, or the Gauss elimination. In other classes, as Crystallography or Statics, it was mainly 3-D vectors. In a computer science class, we used the MatLab to operate with matrices. |
Definition:
The study of Mathematics provides a foundation for the learning of science, technology, and for the interpretation of quantitative information in other subjects. It teaches students how to reason logically and develop skills useful in everyday life.
Current Standards Adopted:
The Common Core State Standards (CCSS) for Mathematics were adopted October 2010.
Accountability:
Student accountability for the Common Core State Standards grades 3 through high school begins 2014 - 15. |
About:
Basic Properties of Real Numbers: Symbols and Notations
Metadata
Name:
Basic Properties of Real Numbers: Symbols and Notations
ID:
m18872Topics covered in this module: understand the difference between variables and constants, be familiar with the symbols of operation, equality, and inequality, be familiar with grouping symbols, be able to correctly use the order of operations. |
Mathematics 2
Covers the mathematics necessary to perform calculations in, and create models for, the real world of Science and Engineering. Specifically, it will demonstrate how to do mathematics in a three-dimensional world. The course describes the fundamental ideas of calculus of functions of one and two variables, differential equations and linear algebra. It continues from MATH1110 to complete a first year of Mathematics suitable for Science and Engineering students, and others for whom Mathematics is a tool. |
Teach21 Project Based Learning This Pool Is Too Cool Algebra II
View
YouEntry Event:
Invfactor higher order polynomials by applying various methods including factoring by grouping and the sum and difference of two cubes; analyze and describe the relationship between the factored form and the graphical representation.
Knowledge:
Know the various methods of factoring higher order polynomials.
Reasoning:
Decide which method of factoring higher order polynomial is best to use under different situations.
Analyze and describe the relationship between the factored form and the graphical representation.
Skills:
Use the factored form of higher order polynomials to find the graphical representation and vice versa.
solve quadratic equations over the set of complex numbers: apply the techniques of factoring, completing the square, and the quadratic formula; use the discriminate to determine the number and nature of the roots; identify the maxima and minima; use words, graphs, tables, and equations to generate and analyze solutions to practical problems.
Knowledge:
Know the various methods of solving quadratic equations.
Know that the discriminate can be used to find the number and nature of the roots.
Reasoning:
Decide which method of solving quadratic equations is best to use under different situations.
Analyze graphs and tables.
Skills:
Apply the techniques of factoring, completing the square, and the quadratic formula to solve quadratic equations.
Identify the maxima or minima of a quadratic function.
Use words, graphs, tables, and equations to generate solutions to practical problems.
identify a real life situation that exhibits characteristics of change that can be modeled by a quadratic equations; pose a questions; make a hypothesis as to the answer; develop, justify, and implement a method to collect, organize and analyze related data; extend the nature of collected, discrete data to that of a continuous function that describes the known data set; generalize the results to make a conclusion; compare the hypothesis and the conclusion; present the project numerically, analytically, graphically and verbally using the predictive and analytic tools of algebra (with and without technology).
Knowledge:
Know that a real life situation that exhibits characteristics of change can sometimes be modeled by a quadratic equation.
Reasoning:
Pose a question.
Make a hypothesis as to the answer.
Develop, justify, and implement a method to collect, organize and analyze related data.
Skills:
Extend the nature of collected, discrete data to that of a continuous function that describes the known data set.
Generalize the results to make a conclusion.
Compare the hypothesis and the conclusion.
Product:
Present the product numerically, analytically, graphically and verbally using the predictive and analytic tools of algebra (with and without technology).The teacher will engage students in online demonstrations of numerical, algebraic, and graphical representations of quadratic functions.
Students search and navigate the Internet to find relevant information related to quadratic functions. Students use graphing utilities to analyze relationships between the various representations. Students communicate their problem solving methods using presentation software.
Daily Writing Journal that includes accomplishments and a reflection of lessons learned
21C.O.9-12.2.TT4 - Student uses technology tools and multiple media sources to analyze a real-world problem, design and implement a process to assess the information, and chart and evaluate progress toward the solution.
The teacher provides opportunities for students to generate and analyze multiple representations of real-world data with and without the use of technology.
Students use manipulatives to model various forms of a quadratic function. Students generate multiple representations of data in a variety of ways, sometimes using technology tools such as CBR's, and solve problems related to the data.21C.O.9-12.3.LS5 - Student exhibits positive leadership through interpersonal and problem-solving skills that contribute to achieving the goal. He/she helps others stay focused, distributes tasks and responsibilities effectively, and monitors group progress toward the goal without undermining the efforts of others.
Use words, graphs, tables, and equations to generate solutions to practical problems
Extend the nature of collected, discrete data to that of a continuous function that describes the known data set
Generalize the results to make a conclusion
Compare the hypothesis and the conclusion
Driving Question:
How can mathematics be used to determine the design requirements of an efficiently heated pool?
Assessment Plan:
Construction of a Scale Model Product: It has been several days since your engineering division at Hi-Tech Pools, Inc. received the matting project. The head engineer reports to your group that the President of Hi-Tech Pools, Inc. is skeptical of your project. He does not believe that it is possible to construct a mat of uniform width around a rectangular pool, so that the area of the mat is the same as the area of the pool's surface. The Chief Engineer is requesting that your engineering division construct a two-dimensional scale drawing, using rational number measurements that represent an example of a pool with a mat meeting these specifications. The scale drawing must be constructed on cardboard, poster board, or cardstock paper. After completing the scale drawing, cut the mat into pieces in such a way that the pieces can be laid on top of the pool's surface to prove that the areas are the same. Prepare a short presentation for the Chief Engineer that would simulate his presentation to the President of Hi-Tech Pools, Inc. Include a numerical justification of your solution and the method that you used to determine the dimensions of the mat and pool.
Demonstrating and Applying the Derived Formula: In only a few days, your engineering team will make their presentation to the design branch of Hi-Tech Pools, detailing the derivation of a formula, accompanied by a graphical representation to find the width of the mat, if you are given the length and width of any rectangular pool. The Chief Engineer is requesting from each of you, a persuasive essay that demonstrates the use of your formula, shows a drawing for your Scale Model Product, includes the widths of mats for a minimum of two different size pools, shows a drawing of your team's alternate heating system and justifies why your team's alternate heating system would be more appealing to the consumer.
Project ScenarioMajor Group Products
Construction of a Scale Model Product: Presentation of a two-dimensional scale drawing model using rational number measurements that includes methods used to determine the dimensions of the mat and pool. (Could be used as a practice presentation. Requires the use of visual manipulatives that assists students in the Culminating Assessment.)
Culminating Assessment (Project Scenario): Multimedia presentation, research summary that includes formula derivation, a drawing used to determine the formula, and an alternative heating solution. All reasoning is supported by sound mathematical evidence.
Major Individual Projects
Demonstrating and Applying the Derived Formula: Persuasive essay that demonstrates the use of the formula, shows a drawing for your Scale Model Product, includes the widths of mats for a minimum of two different size pools, shows a drawing of their team's alternate heating system and justifies why their team's alternate heating system would be more appealing to the consumer.
19. Some real life situations that exhibit characteristics of change can be modeled by a quadratic equation
X
20. How to simplify radicals
X
Resources:
No Data Entered
Manage the Process:
Before the project begins:
Divide students into teams of 3 or 4 students for major group products and projects.
Prepare a Resource/Learning Center for differentiating and tiering. Include the following possible tips or hints (mathematical knowledge students will need to know to complete this project) in the project Resource/Learning Center:
As a homework assignment at the end of each day, each student will use a word processor to keep a daily writing journal that includes accomplishments and a reflection of lessons learned. All entries will be in complete sentences.
Launch the Project.
Driving Question: How can mathematics be used to determine the design requirements of an efficiently heated pool?
Entry Event: InvDistribute the Project Scenario to each student graphical representationDistribute Hi-Tech Pools Team Roles descriptions to each student. For groups of 4 students, two of the students can share the responsibilities of Design Engineer or Research Engineer. As an assignment, each team submits a Team Contract. Examples and ideas for writing contracts can be found at and searching "employment agreement contracts."
Students will work in pairs on Identifying Linear, Quadratic and Cubic Functions Activity. This activity helps students identify data sets as linear, quadratic and cubic by using first, second and third differences. Before given this activity, students will need to know how to enter an equation, graph an equation, use the list function, use regression, enter the regression equation and graph the regression equation on a graphing calculator. Directions are included in the activity.
Think-Pair-Share discussion questions after the activity should include the following:
What do common differences represent?
What would happen if you continue to find common differences after you have constant differences?
What are the regression equations for the following tables? Graph each regression.
Use one of the following activities to provide opportunities for students to generate data that models the height of an object falling due to the force of gravity. Students will work in teams to acquire data using motion detectors. They will analyze the data to find a function.
Student Directions: On your own paper, write the title, "Factoring Quadratics Investigation using Algebra Tiles," and your team's name. Write the activity name. Follow the directions for each activity. Copy and answer all questions. Use a straight edge to copy the algebra tiles solutions to your paper. For each investigation, write the expanded polynomial and its equivalent factored form.
Student Directions: On your own paper, write the title, "Factoring Quadratics Practice using Algebra Tiles," and your team's name.Write the quadratic. Write the equivalent factored form. Use a straight edge to copy the algebra tiles solutions to your paper. Do the first 10 investigations.
Additional Entry Point for Differentiation: Use and build the scale drawing out of Algebra Tiles. Start in the center frame of the screen and build the scale pool with one xy-tile. Build the scale mat around the pool using x-tiles, y-tiles and 1-tiles. Check the scale pool and mat to see if it meets the design requirements. Move the mat pieces on top of the pool. Move the sliders until the scale pool and mat meets the design requirements. Determine the possible measurements of the pool.
Explore the Vertex Form.
Essential Question: Why is one form of a polynomial expression more useful than another?
Students will work in teams to investigate and explore the vertex form.
Student Directions: On your own paper, write the title, "Explore the Vertex Form," and your team's name. Write the activity name. Follow the directions for each activity. Copy and answer all questions in activities A, B & C.
Think-Pair-Share discussion questions should include the following:
Why is it helpful to change a quadratic function to parabolic form?
What is the difference in the graphs of parabolas when there are two zeroes, one zero and no zeroes?
What is the difference in the general forms of quadratics when there are two zeroes, one zero and no zeroes?
Explore Completing the Square.
Essential Question: Why are the coordinates of the vertex of a quadratic function important?
Students will work in teams to investigate and explore completing the square.
Students will take notes from as an introduction to the method of completing the square and converting quadratic functions into vertex form by completing the square. Algebra tiles can be used to model this process. Use -> Mathematics -> Intermediate Algebra -> Quadratic Equations -> Web Lesson 497 to explore applications of completing the square.
Student Directions: On your own paper, write the title, "Complete the Square," and your team's name. Follow the directions. Submit your record of notes and the two tables for assessment.
Practice completing the square and converting from standard form to vertex form. Check by graphing both equations.
Use real-world examples from textbooks or Internet resources to identify the vertex of a quadratic function as relative extrema.
Think-Pair-Share discussion questions should include the following:
Why would you need to find the coordinates of the vertex of a quadratic function?
Is there another name for a quadratic function?
Explore the Discriminant.
Essential Question: Why is it important to evaluate the discriminant?
Students will work in teams to investigate and explore the discriminant.
Student Directions: On your own paper, write the title, "Explore the Discriminant," and your team's name. Use the method of completing the square to derive the quadratic formula. Discuss the information gained from the different values of the discriminant, b2 – 4ac.
Explore the graph of a quadratic function when the value of the discriminant is less than zero.
Think-Pair-Share discussion questions should include the following:
Why would you need to find the value of the discriminant?
What are the situations for the different values of the discriminant?
What do each of these situations tell you about the nature of the roots?
Differentiation: Classroom format includes a mix of whole group, collaborative group, paired and individual activities. Quadratic functions are modeled in a wide variety of ways using physical and virtual manipulatives, graphing technology and Internet web sites. All explorations offer a variety of entry points. A Resource/Learning Center is provided that includes materials to meet the needs of all learners. Step-by-step instructions should be provided for the special needs student. |
Quick Overview
Student Book
Better examples, better questions
Clear, student-friendly design
Frequently Asked Questions in every chapter help students and parents
Lots of opportunities to check understanding before, during, and after the lesson
Two glossaries: instructional words and mathematical words
Available in hardcover text or eBook
Better Examples. Better Questions.
Clean design with fewer words to help students focus on learning math
Multiple worked examples explain the thinking and reasoning using student language to model good communication
Questions gradually increase in difficulty to build students' confidence
Better metacognitive questions allow students to self-assess and develop flexible thinking while practising basic skills
Student Success Adapted Program provides ready-to-use support for struggling students for every lesson in the student book
3-in-1 Teacher's Resource (print, CD, online) provides suggestions in every lesson for assessment and supporting diverse student needs, including suggestions for providing extra challenge and extra support.
Provides 100% curriculum coverage
Developed by leading Canadian mathematics educator, Dr. Marian Small
` |
Framework Alignment
* Insert Course Name
* * * * * *
6.11.01 Recognize, represent, order,
compare real numbers, and locate
real numbers on a number line (e.g.,
π, √ 2, √ 5, 2/3, -1.6).
6.11.02 Represent numbers in
equivalent forms (e.g.,
fraction/decimal/percent,
exponential/logarithmic,
radical/rational exponents, absolute
value, scientific notation).
6.11.03 Use matrices to organize data
6.11.04 Apply rules of order to real
–number expressions
6.11.05 Simplify or test expressions
by applying field properties
(commutative, associative,
distributive), order properties
(transitive, reflexive, symmetric),
and properties of equality for the
set of real numbers.
6.11.06 Apply number theory to the
solution of problems ( prime,
composite prime factorization , GCF,
LCM, divisibility rules
6.11.07 Determine the effects of
operations on the magnitudes of
quantities ( multiplication, division,
powers, roots)
6.11.08 Determine the appropriate
solution , including rounding, from a
context( rounding up, down, to the
nearest integer)
6.11.09 Solve problems involving
estimates or data (e.g., use averages
to estimate the cost of a job that
includes labor and materials).
6.11.10 Perform numerical
computations with real numbers
6.11.11 Perform numerical
computations with non-real complex
numbers
6.11.12 Solve problems using simple
matrix operations (addition,
subtraction, multiplication, scalar
multiplication).
6.11.13 Set up , evaluate, or solve
single –and multi- step number
sentences and word problems with
rational numbers using the four basic
operations
6.11.14 Determine the most cost
effective option using single- and
multi- step calculations and then
comparing results
6.11.15 Judge the reasonableness of
solutions and find mistakes in
calculations, logic, and formula
application
6.11.16 Simplify numerical problems
involving absolute value.
6.11.17 Set up, evaluate, or solve
number sentences or word problems
involving ratios and proportions with
rational numbers ( scale drawing, unit
rate, scale factor, rate of change
6.11.18 Set up , evaluate, or solve
common problems involving percent (
sales tax, tip, interest, discount,
markup, commission, compound
interest.
6.11.19 Set up, evaluate, or solve
problems stated in terms of direct
and inverse variation or simple
quantities
7.11.01 Change from one unit to
another within the same system of
measurement, including calculations
with mixed units (e.g., 3½ hours plus
4 hours and 20 minutes; 2½ feet
minus 16 inches).
7.11.02 Change from one unit in one
system of measurement to a unit in
another system of measurement,
given a conversion factor.
7.11.03 Determine and calculate to an
indicated precision the length, width,
height, perimeter/circumference,
area, volume, surface area, angle
measures, or sums of angle measures
of common geometric figures or
combinations of common geometric
figures.
7.11.04 Describe the general trends
of how the change in one measure
affects other measures in the same
figure (e.g. length, area, volume).
7.11.05 Determine the linear
measure, perimeter, area, surface
area, and volume of similar figures.
7.11.06 Determine the ratio of
perimeters, areas, and volumes of
figures
7.11.07 Use measures expressed as
rates (e.g., speed, density), measures
expressed as products (e.g., person-
days), and dimensional analysis (e.g.,
converting ft/sec to yards/min) to
solve problems.
8.11.01 Simplify or identify equivalent
algebraic expressions (e.g.,
exponential, rational, logarithmic,
factored, polynomial).
8.11.02 Represent mathematical
relationships using symbolic algebra
8.11.03 Identify essential
quantitative relationships in a
situation, and determine the class or
classes of functions (e.g., linear,
quadratic, exponential) that model
the relationships
8.11.04 Determine a specific term, a
finite sum, or a rule that generates
terms of a pattern
8.11.05 Model and describe slope as a
constant rate of change
8.11.06 Evaluate variable expressions
and functions
8.11.07 Identify an equation of a line
or an equation of a line of best fit
from given information (e.g., from a
set of ordered pairs, graphs, tables).
8.11.08 Recognize and describe the
general shape and properties of
functions from graphs, tables, or
equations (e.g., linear, absolute value,
quadratic, exponential, logarithmic).
8.11.09 Identify slope from an
equation, table of values, or graph
8.11.10 Interpret the role of the
coefficients and constants on the
graphs of linear and quadratic
functions, given a set of equations.
8.11.11 Analyze functions by
investigating domain, range, rates of
change, intercepts, and zeros.
8.11.12 Create and connect
representations that are tabular,
graphic, numeric, and symbolic from a
set of data.
8.11.13 Represent quantitative
relationships graphically, and
interpret the meaning of the graph
or a specific part of the graph as it
relates to the situation represented
by the graph
8.11.14 Model problems using
mathematical functions and relations
(e.g., linear, non-linear).
8.11.15 Interpret the graph of a
system of equations and inequalities,
including cases where there are no
solutions
8.11.16 Solve linear equations and
inequalities, including selecting and
evaluating formulas.
8.11.17 Solve systems of equations
and inequalities.
8.11.18 Solve quadratic equations
over the complex number system,
including selecting and evaluating
formulas.
8.11.19 Solve problems that include
nonlinear functions, including
selecting and evaluating formulas
(i.e., absolute value, trigonometric,
logarithmic, exponential).
8.11.20 Identify, interpret, and write
equations for circles and other conic
sections.
8.11.21 Recognize and apply
mathematical and algebraic axioms,
theorems of algebra, and deductive
reasoning.
8.11.22 Identify equivalent forms of
equations, inequalities, and systems
of equations.
9.11.01 Apply the Pythagorean
theorem.
9.11.02 Identify and represent
transformations (rotations,
reflections, translations, dilations) of
an object in the plane, and describe
the effects of transformations on
points in words or coordinates.
9.11.03 Determine how changing the
scale factor affects the size and
position of a figure in the plane.
9.11.04 Classify plane figures
according to their properties.
9.11.05 Identify, apply, or solve
problems that require knowledge of
geometric properties of plane figures
(e.g., triangles, quadrilaterals,
parallel lines cut by a transversal,
angles, diagonals, triangle inequality).
9.11.06 Identify a three-dimensional
object from different perspectives.
9.11.07 Identify the relationship
between two-dimensional patterns
(e.g., nets) and related three-
dimensional objects (e.g., cylinders,
prisms, cones).
9.11.08 Identify two- and three-
dimensional figures that would match
a set of given conditions.
9.11.09 Solve problems that involve
calculating distance, midpoint, and
slope using coordinate geometry.
9.11.10 Identify, apply, and solve
problems that require knowledge of
geometric relationships of circles
(e.g. arcs, chords, tangents, secants,
central angles, inscribed angles).
9.11.11 Graph, locate, and identify
points on a coordinate system.
9.11.12 Solve problems involving
similar figures.
9.11.13 Solve problems using triangle
congruence.
9.11.14 Describe how two or more
objects are related in space (e.g.,
skew lines, the possible ways three
planes might intersect).
9.11.15 Identify relationships
between circles and other objects in
the plane (e.g., inscribed circles,
concentric circles, internal/external
tangency).
9.11.16 Recognize and apply the
conditions that assure congruence
and similarity.
9.11.17 Recognize and apply
mathematical and geometric axioms,
fundamental theorems of geometry,
and deductive reasoning.
9.11.18 Identify a counter example to
disprove a conjecture.
9.11.19 Determine distances and angle
measures using indirect measurement
(e.g., properties of right triangles,
Law of Sines, Law of Cosines).
9.11.20 Solve problems using 45˚-45˚-
90˚ and 30˚-60˚-90˚ triangles.
9.11.21 Identify graphs of a given
trigonometric function (sin x, cos x)
using its characteristics (e.g., period,
amplitude).
9.11.22 Define, identify, and evaluate
trigonometric ratios.
9.11.23 Use trigonometric identities
(sin^2x + cos^2x = 1)
10.11.01 Read, interpret, predict,
interpolate, extrapolate, and use
information from a variety of graphs,
charts, and tables.
10.11.02 Translate from one
representation of data to another
(e.g., a bar graph to a circle graph).
10.11.03 Solve problems involving
Venn diagrams.
10.11.04 Find an unknown value in a
dataset given information about
descriptive statistics.
10.11.05 Calculate, interpret, and use
measures of central tendency and
dispersion.
10.11.06 Compare two or more data
sets on measures of central tendency
and dispersion
10.11.07 Compute the probability of
an event composed of single or
repeated trials with or without
10.11.08 Compute probabilities for
compound events.
10.11.09 Determine geometric
probability based on area.
10.11.10 Apply counting techniques
(e.g., permutations, combinations,
Fundamental Counting Principle |
A scholarly article on math symbols. Scroll down the screen to find "The Origin of Some Mathematical Symbols" in table form. Includes the meaning and name of the person who introduced it and the year it was introduced. From the Encyclopedia of Math.
An entire math course online, complete with illustrations, text, and downloadable course materials. The website is visually engaging, modern, and friendly in its approach to classical math concepts. Sections include: The Primes, Combinatorics Counts, How Big is Infinity?, Topology's Twists and Turns, Other Dimensions, The Beauty of Symmetry, Making Sense of Randomness, Geometries Beyond Euclid, Game Theory, Harmonious Math, Connecting with Networks, In Sync, The Concepts of Chaos. Adobe Reader is required to download and view PDF documents used in this course.
Find lessons on introductory through advanced algebra topics. Also provides practical tips, hints, examples of common mistakes, and an extensive selection of word problems. Browse the listing or use the "search" box at the top of the page.
"The goal of this nonprofit site is to help high school students meet the New York State Regents requirements in English, Mathematics, Science, and Social Studies." Include Algebra, Geometry and Algebra 2/Trig. Provided by Oswego City School District.
"Webmath is a math-help web site that generates answers to specific math questions and problems, as entered by a user, at any particular moment. The math answers are generated and displayed real-time, at the moment a web user types in their math problem and clicks 'solve.' In addition to the answers, Webmath also shows the student how to arrive at the answer." From Discovery Education.
"Mysteriously beautiful fractals are shaking up the world of mathematics and deepening our understanding of nature." Video and articles. Includes information on Benoit Mandelbrot as well as interactive "Design a Fractal" activities and a Mandelbrot Set explorer. From PBS NOVA. |
Wolfram Alpha Launches Problem Generator To Help Students Learn Math
If you're studying math or science, you are probably pretty familiar with Wolfram Alpha as a tool for figuring out complicated equations. That makes it a pretty good tool for cheating, but not necessarily for learning. Today, the Wolfram Alpha team is launching a new service for learners, the Wolfram Problem Generator, that turns the "computational knowledge engine" on its head.
The Problem Generator – which is available to all Wolfram Alpha Pro subscribers now – creates random practice questions for students, and Wolfram Alpha then helps them find the answers step-by-step.
Right now, the Generator covers six subjects: arithmetic, number theory, algebra, calculus, linear algebra and statistics. The difficulty of the questions can be tuned down for students in elementary school and tuned up for those in college calculus classes. As the company notes in today's announcements, the material for students in elementary and secondary schools closely follows the Common Core Standards initiative.
Using the tools is pretty straightforward. Students (or their teachers) choose which topic they want to study and the difficulty level (beginner, intermediate or advanced) and the system will generate a problem.
The team notes that the tool uses Wolfram Alpha's natural language processing capabilities to try to understand the students' answers to "ensure that all students can learn and express themselves in their own unique way." This may actually be the highlight of this service. Too often, after all, similar tools force a very rigid way of answering complex math questions on their students and when they make a mistake, it's not clear if the answer is wrong or if the student just got the syntax wrong.
If a student can't find the answer after three tries, Wolfram Alpha can show a step-by-step solution. The Problem Generator also allows teachers to easily create printable quizzes with multiple-choice tests.
CrunchBase
Wolfram Alpha
FoundedMay 2009
Total FundingNot available
OverviewWolfram Research is building a computational knowledge engine called Wolfram|Alpha for the web to be launched in May 2009. The product will contain data in various fields including physical sciences, technology, geography, weather, cooking, business, music, etc. in order to provide answers to questions that users input. Its language interface will accommodate variations in how users frame their questions, … |
The Calculus Tutor DVD Series will help students understand the fundamental elements of calculus- -how to take algebra and extends it to include rates of change between quantities. Concepts are introduced in an easy to understand way and step-by-step example problems help students understand each part of the process.
This lesson introduces the concept of the improper integral. Grades 9-12. 14 minutes on DVD. |
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The idea is to be able to quickly access the pertinent capabilities of Wolfram|Alpha relevant for specific subject areas. Currently, these subject areas are Algebra, Calculus, and Music Theory. But the company says it plans to add apps for other subjects - "for every major course, from elementary school to graduate school," - including those fields outside math and science.
The Wolfram Course Assistant apps guide users through coursework in order to help them solve problems - not just provide answers. As with any query you enter via Wolfram|Alpha, a lot of additional information is generated in order to help you understand the answer - and the context of the answer.
Each app is organized according to some of the major curriculum units of a particular subject area. And within each of these units, are sub-sections that cover some of the types of problems learners are likely to face. But rather than a study guide that only provides a stock set of questions and answers, these new apps actually solve problems, computing the solution to whatever is asked, by using the Wolfram|Alpha technology.
The apps are built using the Wolfram|Alpha computational engine, its API and the Mathematica-based development pipeline.
The Algebra and Music Theory apps cost $1.99. The Calculus app is $2.99 |
Not Your Typical Algebra Workbook Algebra Puzzles uses games, puzzles, and other problem-solving activities to give students fresh, new ways of exploring learned concepts. While reviewing essential concepts and vocabulary for pre-algebra and algebra, the book helps students visualize and think more deeply about these abstract ideas. The perfect antidote to algebra anxiety.
Related Information
See all of our Math and Puzzles for more teacher supplies like the Algebra Puzzles CTP2569 Math.
Visit our teaching supplies publisher and manufacturer departments to see more products we carry from Creative Teaching Press .
This product is also listed in our teacher supplies catalog as product codes CTC2569,CTCCTP2569. |
PRACTICAL MATH APPLICATIONS, 3E offers users math skills needed for business and personal applications. The text begins with a comprehensive review of the basic math functions (addition, subtraction, multiplication, and division) and progresses to fractions and decimals. Once the students have mastered the basics, they are introduced to practical applications that develop critical thinking skills. These applications include bank records, purchasing and pricing merchandise, payroll, taxes, insurance, consumer credit, and interest (simple and compound). This easy-to-follow, step-by-step approach allows students to work at their own pace. Numerous self-help tips, practice activities, and self-assessments are provided so that each student feels competent in their newly acquired skill before moving on to the next118.95
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Mathematics in Social Choice and Decision Making
1. Analyze and interpret a preference schedule and the possible results using different
voting methods.
2. Analyze a given weighted voting system. Calculate the Banzhaf power index and
the Shapley-Shubik index for a given weighted voting system.
3. Describe the goal of a fair-division problem and calculate a discrete fair division
for a small number of players and objects when: 1. each player has an equal share; 2. the players all have different shares.
4. Calculate the apportionment of seats in a representative body when the individual
population sizes and number of seats are given, using the methods of Hamilton, Jefferson,
Webster, and Hill-Huntington.
5. Analyze a two-dimensional game matrix, to determine if saddle points exist and
produce the players. best strategies.
Text
None.
Grading Procedure
We discuss the important problem of social choice. How does a group of individuals,
each with his or her own set of values, select one outcome from a list of possibilities?
This problem arises frequently in a democratic society and even in authoritarian institutions
where decisions are made by more than one person. While "majority rule" is a good
system for deciding an election involving just two candidates, there is no perfect
way of deciding an election when three or more candidates are running. Group decision
making is often a strategic encounter, and citizens need to be aware of the difficulties
that can arise when some participants have an incentive to manipulate the outcome.
We consider decision-making bodies in which the individual voters or parties do not
have equal power. In particular, we will look at weighted voting systems such as the
electoral college, stockholders in a corporation, or political parties in a national
assembly, in which the voters cast different numbers of votes. While the notion of
power is central in political science, it is typically difficulty to quantify. We
will find that a voter's power in such a system may not be proportional to the number
of votes that he or she is entitled to cast. We describe two well-known indices for
measuring power in weighted voting systems that will enable us to assess the fairness
of weighted voting systems.
The idea of fairness in decision making becomes most explicit. Here we describe some
fair-decision schemes in which a group of individuals with different values can be
assured of each receiving what he or she views as a fair share when dividing up objects
like cakes or the goods in an estate. An important theme of this chapter is finding
procedures that produce "envy-free" allocations, in which each person gets a largest
portion (as he or she values the cake or other goods) and hence does not envy anybody
else.
We discuss the apportionment problem, which is to round a set of fractions to whole
numbers while preserving their sum; of course, the sum of the original fractions must
be a whole number to start. Apportionment problems occur when resources must be allocated
in integer quantities--for instance, when college administrators allocate faculty
positions to each department.
We discuss the mathematical field called game theory, which describes situations involving
two or more decision makers having different goals. Game theory provides a collection
of models to assist in the analysis of conflict and cooperation. It prescribes optimal
strategies for games of total conflict in which one player's gain is equal to the
other player's loss. It also provides insights into more cooperative situations in
which players are trying to coordinate their choices, as well as encounters of partial
conflict that involve aspects of both competition and cooperation. |
This review is from: Introduction to Topology: Third Edition (Paperback)
This book is ideal for self-study. If you have not had the luxury of taking a topology course during your undergraduate studies, but you need to know some topology and you have to study it by yourself, this is the book you need. It is very readable and it explains carefully every concept. However, it is just an introductory text and it contains only basic material. You don't have to invest a lot of time to study the material in this book: let's say 40-60 hours of study are enough to grasp everything. I reccomend it especially to those graduate students of applied mathematics, finance, statistics or economics, who need to use some basic result from topology in their work.
This review is from: Introduction to Topology: Third Edition (Paperback)
I was not a mathematics major, and only in recent years have I ventured into abstract mathematics. I was motivated to learn about topology as an aid to understanding a particular 3-D earth modeling application.
I read Introduction to Topology in three stages: as a review of set theory and metric spaces (chapters 1 and 2), then as an introduction to topology (chapter 3), and lastly as a detailed look at two important topological properties, connectedness (chapter 4) and compactness (chapter 5). I had previously read (and reviewed) another book titled Metric Spaces by Victor Bryant, but Mendelson´¿s book was my first serious look at topology.
My reading of Mendelson´¿s 200-page text required about 100 hours, substantially longer than the 40 to 60 hours estimated by an earlier reviewer. No solutions are provided for the section problems, which are generally of the form ´¿Prove that ´¿.´¿.
The first chapter provides a concise overview of set theory and functions that is essential for Mendelson´¿s subsequent set-theoretic analysis of metric spaces and topology.
The second chapter is a solid introduction to metric spaces with good discussions on continuity, open balls and neighborhoods, limits from a metric space perspective, open sets and closed sets, subspaces, and equivalence of metric spaces. Chapter 2 concludes with a brief introduction to Hilbert space in a section titled ´¿an infinite dimensional Euclidian space´¿.
The third chapter introduces topological spaces as a generalization of metric spaces, and many theorems are largely restatements of the metric space theorems derived in chapter 2. I was thankful for this approach.
Mendelson begins chapter 3 by demonstrating that 1) open sets and neighborhoods are preserved in passing from a metric space to its associated topological space and 2) the existence of a one to one correspondence between the collection of all topological spaces and the collection of all neighborhood spaces.
He then reminds us that in a metric space we can say that there are points of a subset A arbitrarily close to a point x if the metric d(x, A) = 0. In characterizing this notion of ´¿arbitrary closeness´¿ in a topological space, Mendelson introduces the closure of A, the interior of A, and the boundary of A. Other topics included topological functions, continuity, homeomorphism (the equivalence relation), subspaces, and relative topology. The final sections in chapter 3 on products of topological spaces, identification topologies, and categories and functors were more difficult.
In chapter 4 the initial sections (connectedness on the real line, the intermediate value theorem, and fixed point theorems) were largely familiar. But thereafter I became bogged down with the discussions of path-connected topological spaces, especially with the longer proofs involving the concepts of homotopic paths, the fundamental group, and simple connectedness.
Chapter 5, titled Compactness, was even more abstract and difficult, with topics like coverings, finite coverings, subcoverings, compactness, compactness on the real line, products of compact spaces, compact metric spaces, the Lebesgue number, the Bolzano-Weierstrass property, and countability. I will definitely need to look at another text or two before I can handle more advanced topics.
I suspect that a reader familiar with analysis would have substantially less difficulty with the last two chapters.
In summary, Introduction to Topology quite useful for self-study. Mendelson´¿s short text was intended for a one-semester undergraduate course, and it is thereby ideal for readers that either require a basic introduction to topology, or need a quick review of material previously studied. The last two chapters on connectedness and compactness are substantially more difficult, but are still accessible to the persistent reader.
This review is from: Introduction to Topology: Third Edition (Paperback)
since, for some reason, my school didn't offer any topology course, I decided to study topology on my own. It was very fortunate that I found this book in the library. That was right after I took my first analysis course. But I could understand most of the book at that time. After reviewing basic set theory, the author discussed metric spaces, and then he motivates the definition of topological spaces. This is great, I think, becuase many of introductory topology books often give the definition of topological spaces with any motivation. However it is very important to motivate each concept in mathematics especially in introductory level. And this book does this. And as I did, this book is even good for indivisual study. However, you can get almost no geometricl flavor of topology from this book. For example, there is only one section in one chapter in which the author discusses the fumdamental group. Thus, after all this is the best introduction to "point-set topology". So if you don't know almost anything about topolosy, I strongly recommend this book. And one more thing. If you are still wondering if you should buy this one, just look at the price!
This review is from: Introduction to Topology: Third Edition (Paperback)
I know that some people don't like Dover, but I think Dover is great, and Mendelson's Introduction of Topology is an example of why.
Although the book is very short (around 150 pages), it covers the basics of topology very thoroughly and should prepare the reader for the considerably more abstruse Spanier's Algebraic Topology or other texts of such ilk.
If you are a recreational topologist, or are simply tryinging to figure out which way is up in your first topology course, this is for you. |
AMS Chelsea PublishingThis text consists purely of exercises and solutions. This seems like a difficult way to learn combinatorics (or any mathematics for that matter) but it is surprisingly effective. Lovasz starts off with simple problems that anyone can solve and quickly moves to more advanced problems. Lovasz's text covers more material than any other introductory combinatorics text I've seen.
For a more traditional (expository) introduction, consider Brualdi's Introductory Combinatorics.
9 of 10 people found the following review helpful
5.0 out of 5 starsBuy from AMS; new print available6 Nov 2008
By S. Oum - Published on Amazon.com
This book is certainly excellent; moreover, you can now buy cheaper because it is in print again by AMS. ISBN: 0821842625 |
Course Description (From UGa Bulletin): To appropriately select and use technology in mathematics instruction with an emphasis on the organization and design of materials for secondary mathematics courses.
Prerequisites for EMAT 4700/6700:
Textbook:
None. Readings as posted . . .
Time:
Place:
Course outline
The following software will be used:
1. Geometer's Sketchpad 4.07 or 5.0
2. Graphing Calculator 3.5 or 4.0
3. Excel
4. Microsoft Word
5. Firefox
6. Dreamweaver
7. FTP tools if needed
Projects/Course Requirements.
Objectives
• To use application software and technological tools to solve mathematical problems, engage in mathematical investigations, create mathematical demonstrations, and construct new ideas of mathematics for yourself.
• To analyze the affordances of software applications and its connections to the mathematics and how to take this into account when planning activities and lessons.
• To design mathematical activities and lessons that capitalize on the affordances of technology.
• To communicate mathematical ideas that arise from computer investigations using word processing and web technologies.
• To communicate mathematical ideas via the computer applications.
• To become familiar with recent issues in the literature regarding the use of technology in mathematics education.
The University of Georgia seeks to promote and ensure academic
honesty and personal integrity among students and other members
of the University Community. A policy on academic honesty has
been developed to serve these goals. All members of the academic
community are responsible for knowing the policy and procedures
on academic honesty. |
Conventional calculus is too hard and too complex. Students are forced to learn too many theorems and proofs. In Free Calculus, the author suggests a direct approach to the two fundamental concepts of calculus — differentiation and integration — using two inequalities. Regular calculus is condensed into a single concise chapter. This... more...
The book "Single variable Differential and Integral Calculus" is an interesting text book for students of mathematics and physics programs, and a reference book for graduate students in any engineering field. This book is unique in the field of mathematical analysis in content and in style. It aims to define, compare and discuss topics in... more...
The approximation of functions by linear positive operators is an important research topic in general mathematics and it also provides powerful tools to application areas such as computer-aided geometric design, numerical analysis, and solutions of differential equations. q-Calculus is a generalization of many subjects, such as hypergeometric... more...
Aimed toward researchers, postgraduate students, and scientists in linear operator theory and mathematical inequalities, this self-contained monograph focuses on numerical radius inequalities for bounded linear operators on complex Hilbert spaces for the case of one and two operators. Students at the graduate level will learn some essentials that may... more...
From differentiation to integration - solve problems with ease Got a grasp on the terms and concepts you need to know, but get lost halfway through a problem or, worse yet, not know where to begin? Have no fear! This hands-on guide focuses on helping you solve the many types of calculus problems you encounter in a focused, step-by-step manner. With... more...
Projects for Calculus is designed to add depth and meaning to any calculus course. The fifty-two projects presented in this text offer the opportunity to expand the use and understanding of mathematics. The wide range of topics will appeal to both instructors and students. Shorter, less demanding projects can be managed by the independent learner,... more...
Boost Your grades with this illustrated quick-study guide. You will use it from high school all the way to graduate school and beyond. Includes both Calculus I and II. Clear and concise explanations. Difficult concepts are explained in simple terms. Illustrated with graphs and diagrams. Search for the words or phrases. Access the guide anytime, anywhere... more... |
for Dummies
A plain-English guide to the basics of trig Trigonometry deals with the relationship between the sides and angles of triangles... mostly right ...Show synopsisA plain-English guide to the basics of trig Trigonometry deals with the relationship between the sides and angles of triangles... mostly right triangles. In practical use, trigonometry is a friend to astronomers who use triangulation to measure the distance between stars. Trig also has applications in fields as broad as financial analysis, music theory, biology, medical imaging, cryptology, game development, and seismology. From sines and cosines to logarithms, conic sections, and polynomials, this friendly guide takes the torture out of trigonometry, explaining basic concepts in plain English and offering lots of easy-to-grasp example problems. It also explains the "why" of trigonometry, using real-world examples that illustrate the value of trigonometry in a variety of careers.Tracks to a typical Trigonometry course at the high school or college levelPacked with example trig problemsFrom the author of "Trigonometry Workbook For Dummies" "Trigonometry For Dummies" is for any student who needs an introduction to, or better understanding of, high-school to college-level trigonometryVery good. No dust jacket as issued. Trade paperback (US). Glued...Very good. No dust jacket as issued. Trade paperback (US). Glued binding. 368 p. Contains: Illustrations. For Dummies (Lifestyles Paperback). Audience: General/trade. BOX# 071010A: This book is in Very Good condition, and the CD (is not) includes: This book has (0) pages contain HIGHLIGHT and (0) pages with UNDER-LINES, NOTES, and ANSWERS () pages are Creases. Hard Cover tips or Paperback Covers and Stem or binding is (damaged); The Jacket Cover is (N/A). This is an Ex-Library s: (N/A). SPECIAL NOTES (). If this book is not as the above condition, return at 4045 NW 185 ST. Miami Gardens, FL 33055. For full refund no need for email or call.
Description:New, Publisher overstock, may have small remainder mark....New, Publisher overstock, may have small remainder mark. Excellent condition, never read, purchased from publisher as excess inventory.
Description:New. A plain-English guide to the basics of trigFrom sines and...New. A plain-English guide to the basics of trigFrom sines and cosines to logarithms, conic sections, and polynomials, this friendly guide takes the torture out of trigonometry, explaining basic concepts in plain English, offering lots of easy-to-gras |
Find a Glen EllynUse of a graphing calculator can be a blessing and a curse, and it's essential that students recognize how calculators can both help and hinder them in their quest to understand mathematics, the true universal language.For most of my adult life, I've played guitar. While I can strum or finger-pi... |
College Mathematics for Business, Economics, Life Sciences, and Social Sciences - With Mymathlab - 11th edition
Summary: Designed to be accessible, this book develops a thorough, functional understanding of mathematical concepts in preparation for its application in other areas. Concentrates on developing concepts and ideas followed immediately by developing computational skills and problem solving. Features a collection of important topics from mathematics of finance, algebra, linear programming, probability, and descriptive statistics, with an emphasis on cross-discipline principles ...show moreand practices. For the professional who wants to acquire essential mathematical tools for application in business, economics, and the life and social sciences. ...show less
Only lightly used. Book has minimal wear to cover and binding. A few pages may have small creases and minimal underlining. Book selection as BIG as Texas.
$34.50 +$3.99 s/h
Good
Books Revisited Chatham, NJ
Possible retired library copy, some have markings or writing.
$141 |
They look okay. They are mostly word problems which is better than 50 "find the derivative of" or "solve the following equation" type problems. But, it doesn't seem to be terribly focussed on proofs. It is more along the lines of "graph this", "determine that", "give a formula for...." Actually, Gelfand's Algebra is similar in that it is probably more non-proofs than proofs, but there are some good ones in there and the reason it is like that is simply because it has to bridge the gap between someone that cannot prove any theorems to eliciting that out of the student. In the later chapters, Gelfand is almost entirely dwelling (and I am talking about the exercises, here, not just the text) on the large project of actually proving inequalities between the arithmetic, geometric and harmonic means. There is a problem in the middle of the book to show that a polynomial of degree n is determined by n+1 of its values. He has as problems both to prove that the square root of two and to prove that the square root of three are irrational. (He provides a solution to the first and leaves the second unsolved for the student to do.) And, that's just "Algebra II" which is probably appropriately compared to the earlier parts of Exeter's problem sets.
Of course, the next question is whether or not you can get a 14 year old, say, to do all this. Maybe that is a lot more dubitible for Gelfand's book than it is for Exeter's program. I don't know given how selective Exeter is. But, then again, who knows what kind of students go through Gelfand's Correspondence Program and if they even do those problems at all? What I do know is that my oldest is 10 and he was jamming on the arithmetical problems in the beginning of Gelfand's book back when he was nine. He has been doing the Singapore series which we put him back into because we didn't want to take the chance that he was too young to sustain progress through Gelfand. |
FREEonline how to Video Tutorials, examples, and solutions for math problems on everything, literally all aspects of math, AKA quantifiable logic, a philosophy constrained to universal consistency and quantitative analysis, not to mention of course algebra, trigonometry, calculus, etc. Select the subject area you are interested in and get help or how to tutoring right now, 24 hours per day, 365 days per year. Enjoy!
Below are explanations of mathematical tricks you can use to perform some operations very quickly or more easily than brute force iterative process, i.e. out smarting the problem. |
Welcome to the Qualifying Quiz!
The Qualifying Quiz is the centerpiece of the Mathcamp application: it's a set of 7-10 problems, designed to be fun, challenging, and thought-provoking for you, and to give us a good read on how you think about math. We assume no background outside of the standard high-school curriculum, but the problems are anything but standard: you'll have to think hard to solve them. If the problems stick with you, nag you in the shower, and make you want to learn more, then that's a good indication that Mathcamp's classes will be fun for you.
The 2014 Qualifying Quiz is not yet available; it will be posted by February 1st. It will be similar in format and difficulty to previous Mathcamp Qualifying Quizzes, so we invite you to check out the 2013 Quiz (below).
Instructions
We call it a Quiz, but it's really a challenge: a chance for you to show us how you approach new problems and new concepts in mathematics. What matters to us are not just your final results, but your reasoning. Correct answers on their own will count for very little: you need to justify all your assertions and prove to us that your solution is correct. (For some tips on writing proofs, see proof tips.) Sometimes it may take a while to find the right way of approaching a problem. Be patient: there is no time limit on this quiz.
We don't expect every applicant to solve every problem: in the past, we have sometimes admitted people who could do only half of them, occasionally even fewer. However, don't just do four or five problems and declare yourself done! The more problems you attempt, the better your chances. The problems are roughly in increasing order of difficulty, but the later problems often have some easier parts. We strongly recommend that you try all the problems and send us the results of your efforts: partial solutions, conjectures, methods -- everything counts. None of the problems require a computer; you are welcome to use one if you'd like, but first read a word of warning about computers.
If you need clarification on a problem, please contact us. You may not consult or get help from anyone else. You may use books or the Web to look up definitions, formulas, or standard techniques, but any information obtained in this way must be clearly referenced in your solution. Please do not try to look for the problems themselves: we want to see how well you can do math, not how well you can use Google. Any deviation from these rules will be considered plagiarism and may disqualify you from attending Mathcamp. Read more about our policy on getting help.
You may handwrite or type your Quiz. For those interested in typing mathematics beautifully, we also offer you the LaTeX source file for the Qualifying Quiz. Please don't let it distract you from solving the problems, though! For more information on LaTeX, take a look at our LaTeX tutorial for The QQ. No matter how you write up your solutions, we require that you upload them with your application as PDF files; visit our PDF tutorial if you need help.
Finally: don't write your name on your Quiz. (Really!) Instead, write your applicant ID number on your Quiz. (You will receive an applicant ID number when you start your online application, on the welcome page.) Why not your name? When we grade Quizzes, we want to be thinking just about the math. So we're doing an experiment this year in which we evaluate all applicants just by ID number and then uncover the names later. We appreciate your help with the experiment.
Good luck and have fun!
The 2013 Quiz Problems
A teacher asks 100 students to help her with a math contest that she's organizing: each student is supposed to come up with one or two problems to include in the contest. All 100 students do as requested, which takes the teacher by surprise: she actually needed only n problems, with n < 100, so now she has to decide which problems to use. She announces that she will call on the students one by one, at random, and collect all the problems from each student she calls on. Once she has n problems, she will stop. (If the last student wrote two problems and the teacher only needs one of them, she'll pick one at random.)
Angela thought of one problem, Bill thought of two. Bill tells Angela, "Because I wrote more problems than you, I have a better chance of getting a problem on the contest." Angela says, "But the teacher is picking people completely at random! My chances of getting picked are as good as yours." Who is right? Does it depend on n? Does it depend on how many problems the other students came up with? (Be sure to explain any special cases.)
Catherine only thought of one problem, but she overhears Bill talking to Angela and gets worried. She quickly composes another problem, just in case. Does this change her chances of having a problem on the contest? Does it change Bill's or Angela's chances, and if so, in what direction? (Note: you do not need to compute the exact probabilities.)
The subscript ! sometimes indicates that a string of numbers is to be interpreted in factorial base: the i-th number from the right ranges from 0 to i and tells you what multiple of i! to add. For example,
20301! = 2·5! + 0·4! + 3·3! + 0·2! + 1·1! = 240 + 18 + 1 = 259.
Warm-up: show that a number is even if and only if its factorial base representation ends in 0. State and prove a similar condition for divisibility by 3.
The number 27 has the property that its binary representation 110112 ends with its factorial base representation 1011!. Show that there are infinitely many numbers with this property.
Charles and Lilly are playing a game. They start with an empty pot, to which a piece of candy is automatically added at the beginning of every turn. The player whose turn it is then has a choice: he/she can either take all the candy in the pot or pass. (For instance, if Lilly goes first and takes the pot on her first turn, she gets one piece of candy. If she passes and Charles takes the pot on the next turn, he gets two pieces, etc.) If a player decides to take the pot, he/she must pass on his/her next k turns. The winner is the first player to collect n pieces of candy.
Charles unwisely offers Lilly the choice of going first or second. Which should she choose in order to be sure of winning, and how should she play? (Her choice may depend on n and/or k. You need to describe Lilly's strategy completely and prove that Charles cannot win against it, no matter what he does. We suggest you start with the case k = 1.)
Show that any m x n rectangle can be chopped up into binary blocks no two of which are the same. For example, a 5x5 rectangle can be cut into binary blocks of dimensions 4x4, 4x1, 1x4, and 1x1. (Note that a 4x1 block is considered different from a 1x4 block: rotation is not allowed.)
If m = 1 or m = 2, show that you get the same set of binary blocks from your m x n rectangle no matter how you arrange the cuts. (In other words, the set of blocks is uniquely determined by m and n in this case.)
On the other hand, if m = n = 5, there is more than one set of binary blocks that works. Can you demonstrate this?
Ideally, we'd like to find a set of rectangular blocks with integer sides (not necessarily powers of two) such that (i) every m x n rectangle can be cut into blocks from this set that are all different, and (ii) the blocks obtained in this way are uniquely determined by m and n. As we just saw, the set of binary blocks satisfies condition (i) but not (ii). Can you find a different set of blocks that satisfies both conditions?
(Hard!) Our results in (b) and (c) raise the question: for what values of m and n is the set of binary blocks unique? Try to find a necessary and sufficient condition for uniqueness. (You'll need to prove that if m and n satisfy the condition, uniqueness is guaranteed, whereas if they don't satisfy it, there are at least two different sets of blocks that work. If the general version seems too hard, try for some partial results: are there classes of numbers m and n for which you can prove that there is only one set of binary blocks? Are there classes of numbers for which you can prove that there is more than one?)
A farmer living on the xy-plane wants to fence off a rectangular field with sides parallel to the axes and area at least 1. A malicious king tries to stop the farmer by preemptively placing stakes at infinitely many points in the plane (thereby staking his own claim to those points). It's a constitutional monarchy, so the king can't just claim every point for himself: the law dictates that each point where he places a stake must have a circle of non-zero radius around it that contains no other stakes. (These unclaimed circles around each stake can be of different sizes and can be as small as the king likes.) Can the king place his stakes in such a way that the farmer can't find a rectangle of area 1 with no stakes anywhere in its interior or boundary?
Isosceles triangle ABC has AB = AC and ∠BAC = α. The
triangle is originally situated in the plane with A at (0,0) and
midpoint M of side BC at (1,0). You are allowed to move the
triangle by reflecting it across any of its edges. Your goal is to
come up with a sequence of moves that switches A and M – that is, brings A to (1,0) and M to (0,0).
Show that such a switch is impossible if α=60° or 90°.
If α = 45°, show that such a switch is still impossible, but that the triangle can come close to its destination: we can simultaneously get A within distance 0.1 of (1,0) and M within distance 0.1 of (0,0). Can we get even closer? How close?
(Open question!) What else can you say about the possible values of f (0)? Are there values of f (0) for which more than one such function exists? If so, how many different functions are there? Partial results are welcome. |
...
More About
This Book
the 2007 QTS Standards.
Table of Contents
Mathematics background
Interest in mathematics
Perceived competence and confidence in mathematics
Mathematics test
Answers to test questions
Targets for further development
Revision and further |
8th Grade Math Curriculum
Posted: October 10, 2001
Last Updated: May 3, 2006
Rationale
The eighth grade mathematics curriculum is designed to expose and facilitate learning of the basic mathematical
concepts, knowledge, and skills necessary for the successful participation and application in the secondary level
math classes.
Course Description
The eighth grade student's mathematics course is designed to review and acquire a solid foundation, which includes
the knowledge of numerical operations, mathematical systems, geometry, number theory, probability and statistics,
measurements, and algebraic expressions.
Grade Classification: 8
Duration: 1 Year
Performance Required Alignment to Instructional Strategies Assessment Level of
Show-Me Standards Performance
Required
General Objective # NO,1,A Performance Content
The student will read, write,
and compare numbers 3.3
MA 5
1. Specific Objectives:
The student will: MA, NO, 1, In cooperative learning groups students Students will complete a
A. compare and order A, 8 will discuss the values of rationals and constructed response
rationals and percents percents and plot them on a number question taken from the
including finding their line. Assessment Annotations for
approximate location on a the Curriculum
number line. Real Numbers by Richard Powers Frameworks.p.195
Benchmark B problem #2
Ordering Rational Numbers by Richard
Page 1 of 15
Last Updated: 4/17/2010
Powers
General Objective #NO, 1, B Performance Content
The student will represent 3.4
and use rational numbers. MA 1
2. Specific Objectives:
The student will: MA, NO, 1, Discuss methods of fractional Students will create a
B, 8 representation and computation performance event in the
1. use fractions to solve including shortcuts. Practice with form of a take home
problems. selected response quizzes. test(including an answer
key) that compares the cost
MAP Released Items by Richard of items from several
Powers stores. The performance
event will include topics
such as fraction off,
percentage off, sales tax,
etc.
2. use decimals to solve MA, NO, 1, Discuss methods of decimal Students will create a
problems. B, 8 representation and computation performance event in the
including shortcuts. Practice with form of a take home
selected response quizzes. test(including an answer
key) that compares the cost
of items from several
stores. The performance
event will include topics
such as fraction off,
percentage off, sales tax,
etc.
3. use percents to solve MA, NO, 1, Analyze the use of proportions and Students will create a
problems. B, 8 equations calculating percents including performance event in the
but not limited to Commission, form of a take home
Discount/Sale Price, and Percent of a test(including an answer
number. Practice with selected response key) that compares the cost
quizzes. of items from several
stores. The performance
event will include topics
such as fraction off,
percentage off, sales tax,
etc.
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General Objective # NO, 1, Performance Content
C
The student will compose 3.6 MA 1
and decompose numbers.
3. Specific Objectives:
The student will: MA, NO, 1, Interpret among the equivalent Students will take a free
1. make conversions among C, 8 representations of fractions, decimals and constructed response
fractions, decimals,and and percents, and make conversions. exam.
percents.
2. make conversions among MA, NO, 1, Re-introduce students to expanded Students will take a free
expanded notation, scientific C, 8 notation and exponential notation. Also, and constructed response
notation and standard form. engage the students in the actual exam.
multiplication of large and small
numbers and promote the discovery of
patterns to aid in the computation.
General Objective # NO, 1, Performance Content
D
The student will classify and 1.10 MA 5
describe numeric
relationships.
4. Specific Objectives:
The student will: MA, NO, 1, Analyze the rules of divisibility and Students will complete a
1. use factors and multiples D, 8 apply them in the evaluation of the selected response quiz.
to describe relationships LCM and GCF.
between and among numbers
including but not limited to MAP Released Items by Richard
LCM, GCF, and divisibility Powers
rules.
2. use factors and multiples MA, NO, 1, Using multiple methods, provide Students will complete a
to justify characteristics of D, 8 extensive examples of how to create the selected response quiz.
numbers including but not factorization using only prime numbers
limited to prime and after facilitating a conversation on the
composite numbers, and nature of prime and composite numbers.
prime factorization.
MAP Released Items by Richard
Powers
General Objective # NO,2,B Performance Content
The student will describe
effects of operations 3.4, 4.1 MA 1
5. Specific Objectives:
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The student will: MA, NO, 2, Using a variety of techniques such as Given problems with errors
1. describe the effects of B, 8 rules, money, temperature, etc., students the student will be expected
multiplication and division will calculate and analyze products and to find the error and
on integers. quotients. describe the effect the
wrong process had on the
solution.
General Objective # NO, 2, Performance Content
C
The student will apply 1.6, 1.10 MA 5
properties of operations.
6. Specific Objectives:
The student will: MA, NO, 2, Review the order of operations for Students will complete a
1. apply properties of C, 8 rational numbers, and design problems selected response quiz. The
operations to rational to solve including problems with quiz will contain problems
numbers, including order of inverse operations. including order of
operations and inverse operations and inverse
operations. Order of Operations by Richard Powers operations.
General Objective # NO, 2, Performance Content
D
The student will apply 1.6, 3.4
operations on real numbers. MA 5
7. Specific Objectives:
The student will: MA, NO, 2, Given a list of perfect squares and Via a selected response
1. apply the relationship D, 8 exam students will evaluate
square roots, the teacher will facilitate a
between squares and square discussion and provide examples of how squares and square roots of
roots to solve a problem. to approximate the values of nonperfect perfect squares and
squares. approximate the values of
square roots of non perfect
Squares, Square Root, Cubes , and Cube squares, and solve problems
Roots by Richard Powers involving them.
2. apply the relationship MA, NO, 2, Given a list of perfect cubes and cube Via a selected response
between cubes and cube D, 8 roots, the teacher will facilitate a exam students will evaluate
roots to solve a problem. discussion and provide examples of how cube and cube roots of
to approximate the values of nonperfect perfect squares and
squares. approximate the values of
cube roots of non-perfect
Squares, Square Root, Cubes , and Cube cubes, and solve problems
Roots by Richard Powers involving them.
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General Objective # NO, 3, Performance Content
C
The student will compute 1.10, 3.3 MA 1
problems.
8. Specific Objectives:
The student will: MA, NO, 3, Using manipulatives the students will Students will complete a
1. apply all operations on C, 8 discover the rules of operations on all selected response test over
rational numbers. rational numbers and make applications the applications of all
with this knowledge. operations on rational
numbers.
Adding Integers by Richard Powers
General Objective # NO, 3, Performance Content
D
The student will estimate and 3.3, 4.1
justify solutions. MA 1
9. Specific Objectives:
The student will: MA, NO, 3, Based on previously answered questions Given problems in a
D, 8 the students will be presented with selected response format
1. estimate the results of all estimation strategies that help them find students will determine the
operations on rational the reasonableness of an answer. solution through estimation.
numbers and percents.
2. justify the results of all MA, NO, 3, Based on estimates the students will Students will pick a
operations on rational D, 8 defend the estimates and provide three problem and its solution
numbers and percents. reasons. and defend the
reasonableness of the
solution.
General Objective #NO, 3, E Performance Content
The student will use
proportional reasoning. 3.3
MA 1
10. Specific Objectives:
The student will: MA, NO, 3, Facilitate a discussion on rates and Students will be able to
1. solve problems involving E,8 provide example of how to convert rates determine a rate based on a
rates into unit rates including unit pricing. table or graph, and use this
rate to calculate an amount.
2. solve problems involving MA, NO, 3, Assist students in measuring on a scale Using a variety of
proportions, such as scaling E,8 drawing and making conversions, as assessments (selected and
and finding equivalent ratios. well as using proportions to solve for a closed ended constructed
missing value. response), the students will
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solve problems using
proportions including but
not limited to scale
drawing.
General Objective # AR,1,B Performance Content
The student will create and
analyze patterns. 1.6, 3.6 MA 4
11. Specific Objectives:
The student will: MA, AR, 1, Teacher will model the use of patterns Students will design and
B, 8 and how to determine the next terms in justify pictorial and
1. generalize patterns a numerical pattern as well as graphical representations of
represented graphically using discovering the rule to determine the nth patterns including writing
words or symbolic rules, term. algebraic expressions or
including recursive notation. number sentences.
Patterns and Sequences by Richard
Powers
General Objective # AR, 1, Performance Content
C
The student will classify 1.6
objects and representations. MA 4
12. Specific Objectives:
The student will: MA, AR, 1, Given multiple representations of Students will compare and
1. compare and contrast C, 8 patterns students will analyze the contrast patterns based on
various forms of similarities and differences. data given in multiple
representations of patterns. formats such as tables,
graphs etc.
General Objective # AR, 1,D Performance Content
The student will identify and
compare functions. 1.6, 3.6 MA 4
13. Specific Objectives:
The student will: MA, AR, 1, Facilitate a discussion on the definition Using tables, graphs and
1. compare properties of D, 8 and characteristics of linear functions. equations, the student will
linear functions between or verify whether a linear
among tables, graphs, and function is represented. The
equations. student will also compare
intercepts, slopes, etc.
General Objective # AR, 2, Performance Content
A
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The student will represent 1.6, 3.1 MA 4
mathematical situations.
14. Specific Objectives:
The student will: MA, AR, 2, Analyze patterns represented in various The student will complete
1. use symbolic algebra to A, 8 forms, and discuss the development of a a constructed response quiz
represent and solve problems symbolic rule to find any term in the in which he/she will
that involve linear pattern. generate patterns and their
relationships including symbolic rule.
recursive relationships. Understanding and Writing Equations
by Richard Powers
MAP Released Items by Richard
Powers
General Objective # AR, 2,B Performance Content
The student will describe and 3.6 MA 4
use mathematical
manipulation.
15. Specific Objectives:
The student will: MA, AR, 2, Based on previously learned skills Students will complete an
1. generate equivalent forms B, 8 students will create and simplify linear exam in which they
for linear expressions. expressions by applying properties of simplify expressions and
operations and combining like terms. solve equations including
but not limited to changing
between slope intercept and
standard form
General Objective #AR, 3, A Performance Content
The student will use
mathematical models. 1.6, 3.6 MA 4
16. Specific Objectives:
The student will: MA, AR, 3, Teacher will model different ways to Students will choose
1. model and solve problems, A,8 represent and solve problems. another appropriate
using multiple representation of a problem
representations such as as well as solve, and
graphs, tables, equations or analyze the process,
inequalities. solution and graph of an
equation and inequality.
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General Objective #AR, 4, A Performance Content
The student will analyze
change. 1.6, 4.1 MA 2,
MA 4
17. Specific Objectives:
The student will: MA, AR, 4, Teacher will facilitate a discussion that Students will analyze the
1. analyze the nature of A,8 analyzes the nature of change in linear result of changes in slope
changes (including slope and relationships. and intercepts including
intercepts) in quantities in negative slope, and rate of
linear relationships. change.
General Objective #GSR, 1, Performance Content
A
1.6, 3.6 MA 2
The student will describe and
use geometric relationships.
18. Specific Objectives:
The student will: MA, GSR, 1, Using concept circles students will Students will take a series
1. describe, classify and A, 8 analyze triangles and their relationships. of selected and constructed
generate relationships Also students will solve problems by response questions.
between and among triangles writing and solving equations including
using their defining Pythagorean Theorem.
properties including Classifying Polygons by Richard
Pythagorean theorem. Powers
2. describe, classify and MA, GSR, 1, Using concept circles students will Students will take a series
generate relationships A, 8 analyze quadrilaterals and their of selected and constructed
between and among relationships. Also students will solve response questions.
quadrilaterals using their problems by writing and solving
defining properties. equations
Classifying Polygons by Richard
Powers
3. describe, classify and MA, GSR, 1, Using concept circles students will .Students will take a series
generate relationships A, 8 analyze polygons and their of selected and constructed
between and among relationships. Also students will solve response questions.
polygons using their defining problems by writing and solving
properties. equations
Classifying Polygons by Richard
Powers
4. describe, classify and MA, GSR, 1, Using concept circles students will Students will take a series
generate relationships A, 8 analyze 3-dimensional objects and their of selected and constructed
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between and among types of relationships. response questions
3-dimensional objects using
their defining properties Cross sections by Richard Powers
including a)Pythagorean
Theorem b) cross section of
a 3-dimensional object
results in what 2-
dimensional shape.
General Objective #GSR, 1, Performance Content
B
1.6, 3.6 MA 2
The student will apply
geometric relationships.
19. Specific Objectives:
The student will: MA, GSR, 1, Students will create, analyze and solve Students will take a series
1. apply relationships B, 8 problems using similar polygons of selected and constructed
between corresponding sides including solving for a missing side or response questions
and corresponding areas of angle.
similar polygons to solve
problems.
General Objective #GSR, 2, Performance Content
A
3.6 MA 2
The student will use
coordinate systems.
20. Specific Objectives:
The student will: MA, GSR, 2, Students will create similar triangles Students will participate in
1. use coordinate geometry A, 8 and put in vertices to make right a performance event in
to analyze properties of right triangles when given the hypotenuse. which they will graph the
triangles. Also students will calculate area. missing vertex to form a
right triangle, and solve for
The Coordinate System by Richard the missing side and/or
Powers perimeter and area.
2. use coordinate geometry MA, GSR, 2, Students will graph points and
to analyze properties of A, 8 determine which quadrilateral results.
quadrilaterals. Also students will discuss attributes and
calculate areas.
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Area and Perimeter of Rectangles by
Richard Powers
General Objective #GSR, 3, Performance Content
A
3.6 MA 2
The student will use
transformations on objects.
21. Specific Objectives:
The student will: MA, GSR, 3, Following teacher led discussion and Students will take a series
1. reposition shapes under A, 8 examples; students will complete of selected and constructed
formal transformations such guided and independent practice. response questions
as reflection, rotation, and Students will also analyze
translation. transformations and identify which has
occurred in an example.
Transformations by Richard Powers
General Objective #GSR, 3, Performance Content
B
3.6 MA 2
The student will use
transformations on functions.
22. Specific Objectives:
The student will: MA, GSR, 3, After completing dilations students will Students will take a series
1. describe the relationship B, 8 hypothesize about and evaluate the of selected and constructed
between the scale factor and areas based on scale factor including the response questions
the areas of the image using use of a table of values for linear, area,
a dilation and volume.
(stretching/shrinking).
Transformations by Richard Powers
General Objective #GSR, 3, Performance Content
C
1.6 MA 2
The student will use
symmetry.
23. Specific Objectives:
The student will: MA, GSR, 3, Students will determine if a regular Students will take a series
1. identify the number of C, 8 polygon has rotational symmetry and of selected and constructed
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rotational symmetries of describe the symmetry. response questions
regular polygons.
Rotational Symmetry by Richard
Powers
General Objective #GSR, 4, Performance Content
A
3.3 MA 2
The student will recognize
and draw 3-dimensional
representations.
24. Specific Objectives:
The student will: MA, GSR, 4, Students will compare an isometric Students will create an
1. create isometric drawings A, 8 drawing with a mat plan. isometric drawing from a
from a given mat plan. given mat plan.
Isometric Drawings and Mat Plans by
Richard Powers
General Objective #GSR, 4, Performance Content
B
3.1 MA 2
The student will draw and
use visual modes.
25. Specific Objectives:
The student will: MA, GSR, 4, Teacher will model the use of perimeter Students will manipulate
1. draw or use visual models B, 8 and area of complex polygons with shapes to create a new
to represent and solve manipulatives and discuss the shape or change perimeter
problems. similarities and differences. or area. Also students will
view a visual model and
pick the corresponding
result. (tent or cube)
General Objective #M, 1, B Performance Content
The student will identify 1.6 MA 2
equivalent measures.
26. Specific Objectives:
The student will: MA, M, 1, B, Students will convert within cubic units Students will take a quiz of
1. identify the equivalent 8 including but not limited to yd to ft and selected and constructed
volume of measures within a m to cm. response questions.
system of measurement (m ³
to cm ³)
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General Objective #M, 2, B Performance Content
The student will use angle 1.4, 3.2 MA 2
measurement.
27. Specific Objectives:
The student will: MA, M, 2, B, Based on discussion of reflex angles Students will perform
1. use tools to determine the 8 and prior knowledge of protractors, measurements of reflex
measure of reflex angles to students will measure angles that are angles.
the nearest degree. greater than 180 degrees
Angle Measure by Richard Powers
General Objective #M, 2, C Performance Content
The student will apply 3.4, 4.1 MA 2
geometric measurements.
28. Specific Objectives:
The student will: MA, M, 2, C, Teacher will demonstrate the use of Student will pick a three-
1. describe how to solve 8 formulas and how to evaluate them to dimensional object and
problems involving surface find surface area, volume, or any part of explain how to find surface
area and/or volume of a the formula including finding errors of area and volume.
rectangular or triangular computation.
prism, or cylinder.
Surface Area by Richard Powers
General Objective #M, 2, D Performance Content
The student will analyze 1.7, 3.8 MA 2
precision.
29. Specific Objectives:
The student will: MA, M, 2, D, Following a discussion on the precision Students will have to
1. analyze precision and 8 (rounding to the smallest place), the choose the most precise
accuracy in measurement students will determine the number of measurement and determine
situations and determine significant digits. the number of significant
number of significant digits. digits.
Using Significant Digits by Richard
Powers
General Objective #M, 2, E Performance Content
The student will use 1.6, 1.10 MA 2
relationships within a
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measurement system.
30. Specific Objectives:
The student will: MA, M, 2, E, Students will make conversion Students will take a quiz of
1. convert square or cubic 8 including but not limited to ft to yd selected and constructed
units to equivalent square or squared. response questions.
cubic units within the same
system of measurement.
General Objective #DP, 1, A Performance Content
The student will formulate 1.2 MA 3
questions.
31. Specific Objectives:
The student will: MA, DP, 1, Students will create and analyze a Teacher observation of
1. formulate questions, A, 8 survey and determine if it is biased or survey the student created
design studies, and collect not. and analyzed.(Social
data about a characteristic. Studies)
Displaying Data by Richard Powers
General Objective #DP, 1, C Performance Content
The student will represent 1.8, 3.6 MA 3
and interpret data.
32. Specific Objectives:
The student will: MA, DP, 1, C, Students will use provided data to Students will take an exam
1. select, create, and use 8 choose and create a representation of selected and constructed
appropriate graphical including but not limited to all types of response questions.
representation of data graphs, stem and leaf, histogram, Venn
(including scatter plots). diagram with which the students will
create 5 questions that can be answered
by using the representation.
Displaying Data by Richard Powers
General Objective #DP, 2, A Performance Content
The student will describe and 3.4 MA 3
analyze data.
33. Specific Objectives:
The student will: MA, DP, 2, Using appropriate data representations Students will take an exam
1. find, use and interpret A, 8 such as box and whisker, students will of selected and constructed
measures of center, outliers, find, use and interpret values including response questions.
and spread, including range mean without outliers, median, mode,
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and inter quartile range. spread, outliers, and most typical.
Displaying Data by Richard Powers
MAP Released Items by Richard
Powers
General Objective #DP, 2, B Performance Content
The student will compare 3.6 MA 3
data representations.
34. Specific Objectives:
The student will: MA, DP, 2, B, Determine the best choice for Given a set of data the
1. compare different 8 representing data and which students will choose and
representations of the same representation is misleading. construct an approprite
data and evaluate how well representation along with a
each representation shows Displaying Data by Richard Powers representation that is
important aspects of the data. misleading.
General Objective #DP, 3, A Performance Content
The student will develop and 3.5 MA 3
evaluate inferences.
35. Specific Objectives:
The student will: MA, DP, 3, Determine whether a correlation is Students will graph two sets
1. make conjectures about A, 8 positive, negative, or none. Also of data and determine the
possible relationships students will approximate a fitted line. correlation. If appropriate
between 2 characteristics of the students will put in a
a sample on the basis of Displaying Data by Richard Powers fitted line and make
scatter plots of the data and predictions.
approximate lines of fit.
General Objective #DP, 4, A Performance Content
The student will apply basic 3.5 MA 3
concepts of probability.
36. Specific Objectives:
The student will: MA, DP, 4, Students will describe how theoretical Students will take a quiz of
1. make conjectures (based A, 8 and experimental probability are selected and constructed
on theoretical probability) different as well as how they can work response questions.
about the results of together to help us find the solution to
experiments. problems. These problems range from
simple probability to dependent events
Page 14 of 15
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to finding the probability of an
experiment based on theoretical
probability.
Experimental and Theoretical
Probability by Richard Powers
Page 15 of 15
Last Updated: 4/17 |
Susanna Epp's DISCRETE MATHEMATICS: AN INTRODUCTION TO MATHEMATICAL REASONING provides a clear introduction to discrete mathematics and mathematical reasoning in a compact form that focuses on core topics. Renowned for her lucid, accessible prose, Epp explains complex, abstract concepts with clarity and precision, helping students develop the ability to think abstractly as they study each topic. In doing so, the book provides students with a strong foundation both for computer science and for other upper-level mathematics courses. |
First Steps for Math Olympians: Using the American Mathematics Competitions
By J. Douglas Faires
This book will promote interest in mathematics by providing students with the tools to attack problems that occur on mathematical problem-solving exams, and specifically to level the playing field for those who do not have access to the enrichment programs that are common at the top academic high schools. The book can be used either for self-study or to give people who want to help students prepare for mathematics exams easy access to topic-oriented material and samples of problems based on that material.
About the Author
J. Douglas Faires received his BS in mathematics from Youngstown University in 1963. He earned his PhD in mathematics from the University of South Carolina in 1970. Faires was a Professor at Youngstown State University since 1980. He has been actively involved in the MAA for many years. For example, he was Governor of the Ohio Section from 1997-2000. He was a member of the MAA's Strategic Planning Committee for the AMC. Faires was a past president of Pi Mu Epsilon and he was a member of the Council for many years. He was the National Director of the AMC-10 Competition of the American Mathematics Competitions. Faires was the recipient of many awards and honors. He was named the Outstanding College-University Teacher of Mathematics by the Ohio Section of the MAA in 1996; he also received five Distinguished Professorship awards from Youngstown State University and an honorary Doctor of Science degree in May 2006. Faires has authored and coauthored numerous books including Numerical Analysis (now in its eighth edition!), Numerical Methods (third edition), and Precalculus (fourth edition). |
Wavelet theory evolved as mathematicians from areas such as harmonic analysis, functional analysis, and approximation theory brought their specialties together to develop the foundational results and construct algorithms for use in applications. Through development of an undergraduate course on wavelets, the organizer came to realize that the very manner in which wavelet theory came into being is an effective way to present the material to undergraduates. Constructing discrete wavelet transforms in an ad hoc manner (1) shows students that real-world problems are typically solved by using different areas of mathematics, (2) solidifies ideas from sophomore calculus and linear algebra, (3) establishes the computer as an effective learning tool, (4) provides strong motivation for taking upper level classes such as real analysis, (5) allows students to learn about a current topic and its uses in real-world applications. |
This physics-exploration applet allows the user to experiment with different roller coaster track designs, then test. Friction and mass are modeled. Includes hot links to explanations of various phys... More: lessons, discussions, ratings, reviews,...
An applet essentially mimicking a graphing calculator, this is used in a number of activities from the same author. Graph functions, experiment with parameters, distinguish between functions by graphi... More: lessons, discussions, ratings, reviews,...
Move a point around the unit circle to generate the graph of sine, cosine, or tangent. The appropriate measurement is displayed in the circle and translated onto the function graph. (After going to t... More: lessons, discussions, ratings, reviews,...
Introduction
You know how to find the length of the third side of a right triangle when the lengths of the two other sides were given, but how does a person determine the length of a side if t... More: lessons, discussions, ratings, reviews,...
Solving sine equations. This shows a sine curve that can be manipulated and a horizontal red bar that moves up and down. Students are given equations to solve, like sin x = -.5. They graph sin x and tOn this online calculator calculate mathematical expressions and complex numbers. You can do matrix algebra and solve linear systems of equations and graph all 2D graph types. You can also calculate z |
England unveils new "real life" maths curriculum
Following reports that many students enter the workplace struggling with maths, England has announced plans to beef up instruction -- including algebra, probability, statistics and advanced calculation -- to focus on the use of maths in "real situations". The country also is launching new exams tied to maths to encourage students to study the subject up to age 18 and better enable them to compete for jobs. |
Basic Math For Management Professionals ...
(Paperback)
The lack of mathematical knowledge is a major obstacle for many marketing and management professionals. Without a solid foundation in accounting, finance, mathematics or economics, these people often become confused and frustrated, leading to situations where they are unable to play a strategic role in the higher echelons of a company.This is a simple and fun to read book which provides an introduction to the underlying mathematical concepts in marketing and management, in easy to understand terms. Written in a conversational style, this delightful book provides the tools in order to fully understand mathematical concepts, using realistic approaches, real life examples and illustrations to gently introduce these concepts to the reader. The book also includes relevant non-mathematical issues such as price sensitivity, product distribution and sales estimates |
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More About
This Textbook
Overview
The objective of the book is for the reader to learn where approximation methods come from, why they work, why they sometimes don't work, and when to use which of many techniques that are available, and to do all this in a style that emphasizes readability and usefulness to the numerical methods novice. Each chapter and each section begins with the basic, elementary material and gradually builds up to more advanced topics.
Related Subjects
Meet the Author
JAMES F. EPPERSON, PhD, is Associate Editor of Mathematical Reviews for the American Mathematical Society. He was previously associate professor in the Department of Mathematics at The University of Alabama in Huntsville and assistant professor at the University of Georgia in Athens. He earned his doctorate at Carnegie Mellon University in Pittsburgh and his undergraduate degree from the College of Engineering at the University of Michigan, Ann Arbor |
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This course examines how numerical methods are used by engineers to translate the language of mathematics and physics into information that may be used to make engineering decisions. Often, this translation is implemented so that calculations may be done by machines (computers). Upon successful completion of this course, the student will be able to: Quantify absolute and relative errors; Distinguish between round-off and truncation errors; Interconvert binary and base-10 number representations; Define and use floating-point representations; Quantify how errors propagate through arithmetic operations; Derive difference equations for first and second order derivatives; Evaluate first and second order derivatives from numerical evaluations of continuous functions or table lookup of discrete data; Describe situations in which numerical solutions to nonlinear equations are needed; Implement the bisection method for solving equations; List advantages and disadvantages of the bisection method; Implement both Newton-Raphson and secant methods; Describe the difference between Newton-Raphson and secant methods; Demonstrate the relative performance of bisection, Newton-Raphson, and secant methods; Define and identify special types of matrices; Perform basic matrix operations; Define and perform Gaussian elimination to solve a linear system; Identify pitfalls of Gaussian elimination; Define and perform Gauss-Seidel method for solving a linear system; Use LU decomposition to find the inverse of a matrix; Define and perform singular value decomposition; explain the significance of singular value decomposition; Define interpolation; Define and use direct interpolation to approximate data and find derivatives; Define and use NewtonĺÎĺĺÎĺs divided difference method of interpolation; Define and use Lagrange and spline interpolation; Define regression; Perform linear least-squares regression and nonlinear regression; Derive and apply the trapezoidal rule and Simpson's rule of integration; Distinguish Simpson's method from the trapezoidal rule; Estimate errors in trapezoidal and Simpson integration; Derive and apply Romberg and Gaussian quadrature for integration; Define and distinguish between ordinary and partial differential equations; Implement Euler's methods for solving ordinary differential equations; Investigate how step size affects accuracy in Euler's method; Implement and use the Runge-Kutta 2nd order method for solving ordinary differential equations; Apply the shooting method to solve boundary-value problems; Define Fourier series and the Fourier transform; Find Fourier coefficients for a given data set or function and domain; Describe the finite element method for one-dimensional problems. (Mechanical Engineering 205 |
Identify the essential content and process readiness indicators for student success in Algebra I
Identify virtual manipulatives and interactive applets that target the essential skills and knowledge aligned with each of the content readiness indicators
Analyze virtual manipulatives and interactive applets according to given criteria including: alignment with mathematics learning goals, instructional strengths and limitations, ease of use, and availability of support materials
Use virtual manipulatives and interactive applets in activities that target the essential skills and knowledge required to meet the essential algebra readiness indicators
Develop activities that use virtual manipulatives and interactive applets to target the essential skills and knowledge required to meet the essential algebra readiness indicators
*This course is open to all certified WV educators who are teaching mathematics as well as school administrators. **This course is recognized as a technology course and will meet the technology coursework requirments for recertification. ***This course is approved for one course of the three course General Math bundle for Special Education teachers who are working toward becoming "Highly Qualified."
Course Syllabus
Getting Ready for Algebra Using Virtual Manipulatives
Course Description
There is substantial evidence to suggest that a solid foundation in algebra provides a gateway to the higher levels of mathematics necessary for success in higher education, technological or scientific occupations, and business applications. Given this reality, as well as the increased focus on accountability and high academic standards, many schools and districts have instituted policies that require all students to complete algebra as a requirement for high school graduation.
In response to the accountability measures outlined in the No Child Left Behind Act of 2001, the Southern Regional Education Board (SREB) worked with a panel of teachers and experts from the Educational Testing Service (ETS) to develop 17 Algebra I readiness indicators, including the 5 "process" indicators and the 12 "content and skills" indicators. This course is structured around the 12 content and skills readiness indicators and will introduce a collection of virtual manipulatives that will help curriculum planners and classroom teachers meet the demand to prepare students for Algebra I.
Prerequisites
This is an introductory workshop for teachers, technology specialists, curriculum specialists, professional development specialists, and other school personnel who integrate technology into mathematics instruction. Participants are expected to have a set of baseline skills in both mathematics and technology. The prerequisite skills and knowledge are as follows:
Technological
Participants are expected to have basic technology skills and regular access to computers. Specifically, participants should be proficient with browsing the Internet, using email, and saving and accessing computer files.
Mathematics Content/Standards
This online workshop addresses the mathematics skills and knowledge that are necessary for students to be successful in algebra as described in the SREB report, Getting Students Ready for Algebra I, and the National Council of Teachers of Mathematics' (NCTM's) Principles and Standards for School Mathematics (PSSM 2000).
Participants should have a working knowledge of the expectations outlined in the NCTM Algebra Standard, which states:
"Instructional programs from pre-kindergarten through grade 12 should enable all students to:
understand patterns, relations, and functions,
represent and analyze mathematical situations and structures using algebraic symbols,
use mathematical models to represent and understand quantitative relationships, and
analyze change in various contexts" (PSSM p. 37).
Additionally, participants should have specific understanding of the algebra goals and expectations for students in grades 6-8 as outlined in NCTM's Principles and Standards for School Mathematics (PSSM 2000) on pages 222-231.
Goals
identify virtual manipulatives and interactive applets that target the essential skills and knowledge aligned with each of the content readiness indicators,
analyze virtual manipulatives and interactive applets according to given criteria including: alignment with mathematics learning goals, instructional strengths and limitations, ease of use, and availability of support materials,
use virtual manipulatives and interactive applets in activities that target the essential skills and knowledge required to meet the essential algebra readiness indicator,s
develop activities that use virtual manipulatives and interactive applets to target the essential skills and knowledge required to meet the essential algebra readiness indicators.
Assessment and Course Requirements
Each session includes readings, an activity, and a discussion assignment, which participants are required to complete.
Course Products
As a final product, participants will create a lesson plan that incorporates a virtual manipulative or online tool into the curriculum.
Discussion Participation
Students will be evaluated on the frequency and quality of their discussion board participation. Students are required to post a minimum of three substantial postings each session, including one that begins a new thread and one that responds to an existing thread. Postings that begin new threads will be reviewed based on their relevance, demonstrated understanding of course concepts, examples cited, and overall quality. Postings that respond to other students will be evaluated on relevance, degree to which they extend the discussion, and tone.
Session Two: Number and Operations Indicators
The activities in this and all other sessions will help you make connections between readiness indicators, instructional strategies, and virtual manipulatives. You will first engage in the activities as learners and then discuss the activities from both the learning and teaching perspectives.
Download the "Visualizing Fractions" activity and complete the assignment.
Session Five: Algebra and Functions Indicators
For this activity and the remaining activities in this session, you may want to refer to the attached list called "Notation for Functions," which contains the proper notation for identifying various functions within the online tools.
Session Six: Summary and Final Project
Read "Mathematically Appropriate Uses of Technology." This reading discusses some of the issues that mathematics educators face in deciding which technology tools can improve student achievement and learning.
Participants will also complete their final project which is a plan, where they are going to select a virtual manipulative and describe a plan for using it to address one of the SREB algebra readiness indicators with middle school students. |
for students without in-depth mathematical training, this text includes a comprehensive presentation and analysis of algorithms of time ...Show synopsisText for students without in-depth mathematical training, this text includes a comprehensive presentation and analysis of algorithms of time-dependent phenomena plus beam, plate, and shell theories. Solution guide available upon request |
Maths for Science
Sally Jordan, Shelagh Ross, and Pat Murphy
Building from the foundations of math--numbers, fractions, and units and scales of measurement--the book leads students through a range of widely used skills and concepts (e.g., equations, logarithms, differentiation, and probability and statistics), providing a complete course of essential math for science. |
The book supplied with this course is written on the level of understanding that will be expected of one having completed the prerequisite courses. It will be included in the cost of the course. This book will be necessary to learn the subject and to follow the lessons with a better understanding of the course and the expectations if the course instructors.
The introduction to the course and what will be taught as well and any definitions of words are included. The problems set forth in this book will be clearly stated and explanations are available for reference.
Each lesson will be introduced, explained and the appropriate set of problems will be offered for the student to process and complete in a timely manner. At the end of each lesson a quiz will be given to determine the comprehension of that particular lesson. If further assistance is needed that will be provided.
The lessons will begin with basics such as sets, equations, vectors, partial orderings, functions and integers. They will give examples in the way of problems, how they are processed, the definition and the solution. There may be a particular theory that is applied to that equation which will be explained.
The basic equations and step by step instructions will lay out the problems, from n3 to more complicated problem structures. Some of the subjects may include Homomorphism's, Cosets and Quotient Modules and Characterization of Free Modules.
The workbook will have all the necessary instructions and solutions for the student to follow along. The workbook can be downloaded and printed or a hard copy may be sent.
There are many supplemental resource materials available on line for those who want to use them. Some can be downloaded for free and others can be viewed on line. |
MATH 413
Fundamentals of Algebra II
Course info & reviews
This course will cover topics concerning the properties of the real number system, the solutions of equations and inequalities, the algebra of rational expressions, exponents and radicals, an introduction to quadratic equations, functions and graphs, and the solution of systems of linear equations. Prerequisite(s): MATH 0313 or an appr... |
books.google.com - This market-leading introduction to probability features exceptionally clear explanations of the mathematics of probability theory and explores its many diverse applications through numerous interesting and motivational examples. The outstanding problem sets are a hallmark feature of this book. Provides... first course in probability
A first course in probability
This market-leading introduction to probability features exceptionally clear explanations of the mathematics of probability theory and explores its many diverse applications through numerous interesting and motivational examples. The outstanding problem sets are a hallmark feature of this book. Provides clear, complete explanations to fully explain mathematical concepts. Features subsections on the probabilistic method and the maximum-minimums identity. Includes many new examples relating to DNA matching, utility, finance, and applications of the probabilistic method. Features an intuitive treatment of probability—intuitive explanations follow many examples. The Probability Models Disk included with each copy of the book, contains six probability models that are referenced in the book and allow readers to quickly and easily perform calculations and simulations.
Page 520
User ratings
THe book is prescribed at IIM Ahmedabad in the course "Quantitative MEthods II" So its a basic tool for learning probability
Review: A First Course In Probability
User Review - Michael - Goodreads
As the saying goes, probability is inherently challenging, not trivial like calculus. If your feelings about calculus are a bit different, you might find this book not so much a "first course" as a confusing deathmarch. This ISBN is the 2nd Edition.Read full review
References to this book
References from web pages
JSTOR: A First Course in Probability Overall, my students and I like A First Course in Probability. Whereas the students are sometimes intimidated by the difficulty of the problems (some of ... links.jstor.org/ sici?sici=0162-1459(199809)93%3A443%3C1242%3AAFCIP%3E2.0.CO%3B2-N
About the author (2002) |
Wiki
Saxon math, developed by John Saxon, is a teaching method for incremental learning of mathematics. It involves teaching a new mathematical concept every day and constantly reviewing old concepts. Early editions were deprecated for providing very few opportunities to practice the new material before plunging into a review of all previous material. Newer editions typically split the day's work evenly between practicing the new material and reviewing old material. Its primary strength is in a steady review of all previous material, which is especially important to students who struggle with retaining the math they previously learned.
In all books before Algebra 1/2 (the equivalent of a Pre-Algebra book), the book is designed for the student to complete assorted mental math problems, learn a new mathematical concept, practice problems relating to that lesson, and solve a varied number of problems which include what the students learned today and in select previous lessons -- all for one day's class. This daily cycle is interrupted for tests and additional topics. In the Algebra 1/2 book and all higher books in the series, the mental math is dropped, and tests are given more frequently.
The Saxon math program has a specific set of products to support homeschoolers, including solution keys and ready-made tests, which makes it popular among some homeschool families. It has also been adopted as an alternative to reform mathematics programs in public and private schools. Saxon teaches familiar algorithms and uses familiar terminology, unlike many reform texts, which also contributes to its popularity.
My husband and I have been homeschooling now for 12 years. It has not been an easy road, but the benefits have certainly been worth it: well adjusted kids who are excelling academically. While some homeschoolers choose to "unschool" and simply follow their students' lead, we followed a more traditional route, using various resources and a variety of textbooks. Early on we changed curriculum companies year to year in our search for just the right approach or writing … more |
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