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7. "Changing Shapes With Matrices" ISBN 9780962167430; 80 pp.; 8 1/2x11"; Paper on SALE
$4.50; also on
Don's 2-disk CD set with all his materials @$70.95
In this new, Don shows how young people can
do matrix transformations
Valorie, age 11, made up a matrix that caused a change (or transformation)
in the shape of a dog, similar to the one D'Arcy Thompson talked about
with fish in his 1917 classic 'On Growth and Form', also shown in
the book 'The Art of Graphics for the IBM PC' written in 1986!
Exciting stuff!
Preface:
Why transformations and why matrices?
A Map to Transformations
Chapter 1:
Plotting points - graphing linear equations
Chapter 2:
Grocery store arithmetic to multiply matrices
Chapter 3:
Steps to do a transformation and a
point-by-point restatement of Valerie's work
Chapter 4:
Questions and other student work
Chapter 5:
Some special matrices
Appendix 1:
Selected answers
Appendix 2:
Transformations without matrices
Appendix 3:
Graph paper to copy
Appendix 4:
Computer programs to do the transformations
Appendix 5:
Bibliography
Appendix 6:
The 81-2x2 matrices using only 1's, 0's or
-1's, and their rules
Bruce Artwick, in his "Flight Simulator" computer program (now
sold by Microsoft), changes the heading , pitch and bank of an airplane, by
rotating it around three different axes, as well as translations and scaling, in
moving the airplane. He uses matrices to do these
transformations and for us to simulate the plane's flight. Don shows how young
people can make these 2-D changes. |
MATH 1201
APPLIED MATHEMATICS FOR THE HEALTH SCIENCES I (2 cr.)
This introductory math course is designed specifically for students in Associate Degree healthcare programs. You will practice mathematical techniques and develop problem solving skills that you will use in the advanced math and science courses in your program. You will gain mathematical fluency in such areas as polynomials, algebraic inequalities, rational functions, exponential equations and graphs, and logarithmic models. |
Pre-Algebra Proficiency Program
Overview
The Catchup Math Pre-Algebra Proficiency Program reviews the course content covered in Pre-Algebra
textbooks and is divided into six sequential sections. Pre-tests (quizzes) are given for each section,
and students are assigned lessons based on incorrect answers. Re-teaching includes text lessons (in both
English and Spanish), videos, activities, and practice problems with tutorial solutions. A student takes
repeat quizzes and receives lesson-prescriptions until passing each section. Teachers can review student
work written on our online whiteboard. Material from earlier courses and sections is often included and
even repeated.
Section 1
Absolute Value, Adding and Subtracting Negatives, Graphing on the Coordinate Plane, Inequalities, Integers,
Mean, Median and Mode, Multiplying a Fraction by a Fraction, Multiplying and Dividing with Negatives,
Number Line, Order of Operations, Simplest Form of a Fraction, Word Problems. |
Overview
Intended for students of mathematics as well as of engineering, physical science, economics, business studies, and computer science, this handbook contains vital information and formulas for algebra, geometry, calculus, numerical methods, and statistics. Comprehensive tables of standard derivatives and integrals, together with the tables of Laplace, Fourier, and Z transforms are included. A spiral binding that allows the handbook to lay flat for easy reference enhances the user-friendly design.
Author Biography
Alan Davies is a professor of mathematics at the University of Hertfordshire and has been teaching mathematics to engineers, scientists, and mathematicians for more than 30 years. He is the author of An Introduction to Computational Geometry for Curves and Surfaces. Diane Crann is a mathematics graduate with more than 10 years of experience in organizing mathematics-related activities for people of all ages, including the Royal Institution Mathematics Masterclass Series.
Independent Publishers Group is a serious publishing enterprise with a reputation for integrity, expertise, and personalized service. |
Find a Laveen CalculusI have taught probability at Daniel Webster College and Hesser College. I have taught differential equations at all levels all the way through graduate school at several different universities. A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders. |
Fresno, TX AlgebraThomas and received an A in the course. Linear Algebra is the study of matrices and their properties. The applications for linear algebra are far reaching whether you want to continue studying advanced algebra or computer science. |
2/5 - Unit 7 test on Thursday. Today, you were given a vocabulary review sheet, an exponent properties sheet and a Unit 7 review packet. You got the sheets done in class today. Look over the review packet. Complete what you know and circle those problems you would like covered in class tomorrow.
2/4 - Section 8-6: HW #2. Test on Thursday.
1/31 - Optional Homework to help you better prepare for next week's test: Optional Homework
1/30 - No homework
-----------------------------Homework for 2nd Marking Period Below----------------------------
-
1/29 - Hope you did well on your exams. See you tomorrow.
1/23 - I will be running a review on Friday @ 12:30 in room 252. Come with questions.
1/22 - Tonight, you should look at each packet and identify those concepts that are giving you the most trouble. We will field questions tomorrow.
1/21 - Keep working on your packets. We will be going over questions tomorrow and Wednesday.
1/18 - You now have Unit 4 Word Problems review as well as a Vocabulary sheet. Answers will be posted on Ms. Garruto's site on Monday.
1/17 - Section 8-6: Multiplying a Polynomial by a Monomial Worksheet.
You have also been given the Unit 3. That makes three. Again, look over each one and identify what sections of which you may have questions.
1/16 - Worksheet: Scientific Notation - HW #2
1/15 - You now have Review Packet #2. Look it over, complete what you can do and circle those problem of which you may have a question.
10/9 - Hope you did well on today's Unit 2 test. If you missed it, make sure to make arrangements for a make up. Only homework is yellow sheet (Perfect 10) which is due on 10/16 (next Tuesday). Remember: you can hand that in as many times as you want. I will mark what's correct that period.
9/14 - Unit test will be Thursday, September 20. You should have completed much of your review packet. If you did not take advantage of last night and today in class, continue to prepare for the test. Good holiday to those who observe.
9/13 - Today, you received your Unit 1 Test Study Guide. Refer to the answer file on Ms. Garruto's site to see how you are doing. Do not just copy the answers down. You are not getting credit for the study guide and you will not be helping yourself prepare for the test next Thursday.
9/6 - From the packet today, complete the sections marked homework. Use sections 2-1 + 2-7 of your textbook for reference. If you need some more help, go to Username: studentv On right hand side (box "QL #") next to word "GO", type in 1285 and click on "GO". |
Parametric Differentiation
In this lesson, our instructor John Zhu gives an introduction to parametric differentiation. He explains parametric differentiation and in relation to position, speed and acceleration. He works through several example problems.
This content requires Javascript to be available and enabled in your browser.
Parametric Differentiation
Differentiate the 2 parametric
parts separately
Divide
Simplify
Parametric Differentiation
Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture. |
Mathematics - a set of over 100 digital, interactive presentations (whiteboard enabled), covering the vast majority of compulsory topics for the age of 17 – 19 years. Thousands of high quality interactive pages including illustrations, animations, films, 3D models, tests and activities place difficult mathematical problems into easy to relate to real-life context. Striking multimedia presentations and carefully designed activities support preparation for exams and classroom tests |
Lulu Marketplace
Elements of Algebra
This book is an edited reprint of Part I of J. Hewlett's 1822 English translation. You can preview this book. This edition is in hardcover.
The paperback edition
is available from Tarquin books.
Leonhard Euler (1707--1783) is one of the most influential and prolific mathematicians of all time. His Elements of Algebra is one of the first books to set out algebra in the modern form we would recognize today. However, it is sufficiently different from most modern approaches to the subject to be interesting for contemporary readers. Indeed, the choices made for setting out the curriculum, and the details of the techniques Euler employs, may surprise even expert readers. It is also the only mathematical work of Euler which is genuinely accessible to all. Euler's style is unhurried, and yet rarely seems long winded.
Ratings & Reviews
An excellent book. Build quality was very good for the price, and it was delivered quickly. The author's explanations were different from those used in the classes that I remember attending, many times making things much clearer for me. |
Specification
Aims
To develop a basic understanding dynamical systems theory, particularly those
aspects important in applications.
To describe and illustrate how the basic behaviours found in dynamical systems
may be recognized and analyzed.
Brief Description of the unit
Dynamical systems theory is the mathematical theory of time-varying systems; it
is used in the modelling of a wide range of physical, biological, engineering,
economic and other phenomena. This module presents a broad introduction to the
area, with emphasis on those aspects important in the modelling and simulation of
systems. General dynamical systems are described, along with the most basic sorts
of behaviour that they can show. The dynamical systems most commonly encountered
in applications are formed from sets of differential equations, and these are
described, including some practical aspects of their simulation. The most regular
kinds of behaviour---equilibrium and periodic---are the most easy to analyze
theoretically; linearization about such trajectories are discussed (for periodic
behaviour this is done using the Poincaré map.)
Much more complex behaviours, including chaos, may be found; these are described by
means of their attractors. The linearization approach can be extended to these, and
leads to the concept of Lyapunov exponents.
In applications it is often important to know how the observed behaviour changes
with changes in the system parameters; such changes can often be sudden, but
frequently conform to one of a relatively small number of scenarios: the study of
these forms the subject of bifurcation theory. The simplest bifurcations are
discussed.
Learning Outcomes
On successful completion of this course unit students will
understand the general concept of a dynamical system, and the significance
of dynamical systems for modelling real world phenomena;
be able to analyze simple dynamical systems to find and classify regular
behaviour;
appreciate some of the more complex behaviours (including chaotic),
and understand some of the features of the attractors characterizing such
behaviour;
be familiar with some of the simpler bifurcation scenarios, and how they
can be analyzed. |
Fully aligned with the core curriculum standards of the NJ Department of Education, this fourth edition of our popular test prep provides the up-to-date instruction and practice that eighth grade students need to improve their math skills and pass this important high-stakes exam.
The book includes a full-length diagnostic test and a full-length practice test based on official exam questions. Each test comes complete with detailed explanations of answers, allowing you to focus on areas in need of further study.
Our TestWare CD features both of the book's tests in a timed format with automatic scoring, detailed answer explanations, and diagnostic feedback. Instant reports help you zero in on the topics and types of questions that give you trouble now, so you'll succeed when it counts! REA's test-taking tips and strategies offer an added boost of confidence and ease anxiety before the exam.
Whether used in a classroom, at home for self-study, or as a textbook supplement, teachers, parents, and students will consider this book a "must-have" prep for the NJ ASK 8 Mathematics exam.
About the Author
Mel Friedman is Lead Mathematics Editor at REA. After teaching mathematics at the high school level for twelve years, he went on to teach at a number of colleges and universities, the most recent being Kutztown University in Kutztown, Pa. and West Chester University in West Chester, Pa. In addition to his work as a math consultant, he served as a test-item writer for Educational Testing Service and ACT, Inc. He received a Bachelor of Arts degree in math from Rutgers University, and an M.S. in math from Fairleigh Dickinson University.
Stephen Hearne is a professor at Skyline College in San Bruno, California, where he teaches Quantitative Reasoning. He earned his Ph.D. degree from the University of Mississippi in 1998, specializing in Quantitative Psychology. For the past twenty years, Dr. Hearne has tutored students of all ages in math, algebra, statistics, and test preparation. He prides himself in being able to make the complex simple.
Penny Luczak received her B.A. in Mathematics from Rutgers University and her M.A. in Mathematics from Villanova University. She is currently a full-time faculty member at Camden County College in New Jersey. She previously worked as an adjunct instructor at Camden County College as well as Burlington County College and Rutgers University |
Learn Geometry NOW! - Geometry for the Person Who Has Never Understood Math!
Synopsis
Geometry is that section of math that deals with lines and shapes instead of just numbers. If you have trouble understanding geometry, you're not alone! Plenty of people have had trouble with geometry, even those skilled at other branches of mathematics.
So if you can't look at a triangle without flinching, don't worry-—help is on the way. Throughout the book, you will go over the major geometrical concepts, from points and lines to three-dimensional solids. Geometry isn't impossible; you can and will learn it. Just believe in yourself, and you might find that geometry isn't as difficult as you thought.
NOW Books is an imprint of Minute Help Press. Each enriching book in the NOW series teaches readers how to improve their lives in about an hour--whether it's breaking a habit, publishing a book, or finding a job, the NOW series has something for you. Search for "NOW Books" to find other titles in the series, and follow @MinuteHelp to find out about upcoming books.
Found In
eBook Information
ISBN: 9781610429597 |
Features:
Math Calculator is an expression calculator
You can input an expression including variable x, for example, log(x), then input a value of 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.,...
Advanced yet easy-to-use math calculator that immediately and precisely computes the result as you type a math expression. It allows multiples math expressions at same time. It also allows fractions and defining your own variables and functions....
The Machinist's Calculator has been developed to quickly solve common machine shop trigonometry and math problems at a price every machinist can afford! As a machinist or CNC programmer, you often have to use trigonometry to calculate hole...
madly calculator aero is a new software utility designed as a math calculator. It assures basic mathematical functions but also an entire set of trigonometric and complex functions such as : sin, cos, tan, logarithm, root and more. This is an AS3 simple math calculator that is XML driven and resizable. XML Settings: - component width and height - maximum characters to display Enjoy this new release from Oxylus Flash. Simple and easy to use flash calculator component thatPHP Calc is the most basic calculator written in php. This is a 5 minute script that is powerful because you can make it calculate anything. It just takes some imagination and some knowledge of math. Feel free to change the script to make it |
Synopsis
Geometric measure theory provides the framework to understand the structure of a crystal, a soap bubble cluster, or a universe. Measure Theory: A Beginner's Guide is essential to any student who wants to learn geometric measure theory, and will appeal to researchers and mathematicians working in the field. Morgan emphasizes geometry over proofs and technicalities providing a fast and efficient insight into many aspects of the subject.
New to the 4th edition: * Abundant illustrations, examples, exercises, and solutions. * The latest results on soap bubble clusters, including a new chapter on "Double Bubbles in Spheres, Gauss Space, and Tori." * A new chapter on "Manifolds with Density and Perelman's Proof of the Poincaré Conjecture." * Contributions by undergraduates.
Found In
eBook Information
ISBN: 9780080922409 |
William Lowell Putnam Mathematical Competition
This annual Putnam exam competition, open to undergraduates, is held on
the first Saturday of December. More than 4000 students from over 500
colleges and universities in the U.S. and Canada take part in this, the
best known and most prestigious mathematics competition in America. The
test consists of 12 mathematics problems (each worth 10 points) in which
the emphasis is less on knowing a vast amount of mathematics and more
on seeing through to the heart of a problem. In 2012, there were 4,277
contestants in the U.S. and Canada. Fewer than 5% received scores of 32
or higher; and a median score was 0 points out of a possible 120.
All students with interests in the mathematical sciences are strongly
encouraged to participate. The problem-solving skills developed through
practicing for and participating in the competition should prove useful
both in course work and in later life.
Credit for preparing for and
participating in the competition is available through a Mathematics 191 seminar in Advanced Problem Solving which is offered every fall semester. The Fall 2013 seminar (Math 191 section 1) will be taught by Professor Alexander Givental on Tuesdays and Thursdays from 12:30-2. |
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
The Sullivan/Struve/Mazzarella AlgebraSeries was written to motivate students to "do the math" outside of the classroom through a design and organization that models what you do inside the classroom. The left-to-right annotations in the examples provide a teacher's voice through every step of the problem-solving process. The Sullivan exercise sets, which begin with Quick Checks to reinforce each example, present problem types of every possible derivation with a gradual increase in difficulty level. The new "Do the Math" Workbook acts as a companion to the text and to MyMathLab® by providing short warm-up exercises, guided practice examples, and additional "Do the Math" practice exercises for every section of the text.
10. Graphs of Quadratic Equations in Two Variables and an Introduction to Functions
10.1 Quadratic Equations in Two Variables
10.2 Relations
Putting the Concepts Together (Sections 10.1—10.2)
10.3 An Introduction to Functions
Chapter 10 Activity: Discovering Shifting
Chapter 10 Review
Chapter 10 Test
Appendix A: Table of Square Roots
Appendix B: Geometry Review
Answers to Selected Exercises
Graphing Answers Section
Applications Index
Subject Index
Photo Credits
New to this edition
New and Updated Features
Quick Check Exercises, which follow every example, are now numbered as the first problems in each section's exercise set to make them easier to assign as homework. Quick Checks provide the platform for student to get "into the text." By starting their homework with these Exercises, students will be directed to the instructional material in that section, increasing their confidence and ability to work any math problem–particularly when they are away from the classroom. Answers to the Quick Check exercises have been included in the back of the text.
The exercise sets have been updated and re-designed in a 2-column format for better organization and more visual appeal.
The Annotated Instructor's Edition contains annotated answers placed next to their respective exercises.
Content Changes
The former Chapter 8, Introduction to Graphing and Equations of Lines, and Chapter 9, Systems of Linear Equations and Inequalities, have been moved to Chapters 3 and 4, allowing instructors to cover "everything linear" early in the course.
The former Section 8.7, Variation, is now Section 7.9, so that content on variation appears in the same chapter as rational expressions.
New to the Supplements Package
The Do the Math Workbook offers a collection of 5-Minute Warm-Up exercises, Guided Practice exercises, and Do the Math exercises for each section in the text. These worksheets can be used as in-class assignments, as an in-lab study assignment or for homework.
The Videos on DVD offer a lecture for every section of the text. All videos include optional subtitles in English and Spanish.
MyMathLab enhancements include:
Substantially increased coverage of exercises (including Chapter Review exercises) to give students more opportunity for practice.
Authors in Action Videos, made with Camtasia, take students into the classrooms of authors Michael Sullivan, Katherine Struve, and Janet Mazzarella.
Video lectures and chapter test solutions on video, now with optional subtitles in English and Spanish.
Translating Word Problems Animations to help students practice the translation step of solving word problems.
An Interactive English/Spanish glossary that offers definitions of important mathematical terms in both English and Spanish.
Features & benefits
Sullivan Examples and Showcase Examples provide students with superior guidance and instruction when they need it most–when they are away from the instructor and the classroom.
Sullivan examples feature an active two-column format in which annotations are provided to the left of the algebra, mirroring the way that we read. The annotations explain what the authors are about to do in each step instead of what was just done.
Showcase Examples provide how-to instruction in an easy-to-understand, 3-column format. The left column describes a step, the middle column provides a brief annotation, as needed, to explain the step, and the right column presents the algebra.
Placed at the conclusion of most examples, the Quick Check exercises provide students with immediate reinforcement and instant feedback to determine their understanding of the concepts presented in the examples.
Students learn algebra by doing algebra. Throughout the textbook, the exercise sets are grouped into eight categories–some of which appear only as needed:
"Preparing For..."problems are located at the opening of a section. These test students' grasp of the prerequisite material for each new section.
Quick Check Exercises immediately follow the examples, allowing students to practice and apply what they have just learned. These exercises are also assignable as homework, so students can easily refer back to the relevant example for extra help.
Building Skills exercises are drill problems that develop the students' understanding of the procedures and skills in working with the methods presented in the section. These exercises are tied to the section objectives and are often linked back to an example.
Mixed Practice exercises offer comprehensive skill assessment by asking students to relate multiple concepts or objectives. These exercise sets may also include problems from previous sections so that students must first recognize the problem-type before using the appropriate technique to solve it.
Applying the Concepts exercises ask students to apply the mathematical concepts to real-world situations.
Extending the Concepts exercises go beyond the basics, using a variety of problems to sharpen students' critical-thinking skills.
Explaining the Concepts problems ask students to explain the concepts in their own words.
Graphing calculator exercises are optional and may appear at the end of a section's exercise set.
Study Skills features are a regular theme throughout the book, anticipating students' needs and providing the voice of an instructor.
Section 1.1: Success in Mathematics introduces the basics of study skills, such as what to do during the first week of the semester; what to do before, during and after class; how to use the text effectively; and prepare for an exam.
In Words features help students understand definitions and theorems by putting them in plain English, just like an instructor would do in class.
Work Smart features identify common errors to avoid and encourage students to work more efficiently.
Work Smart: Study Skills boxes appear throughout the book to remind students to stay organized and to manage their time in the most effective way possible.
Test preparation features help students make the most of their study time.
Chapter Tests reflect the levels and types of exercises that students are likely to see on an exam, providing a valuable study tool that decreases anxiety and stress.
The Chapter Test Prep Video CD provides step-by-step solutions to every problem from the Chapter Tests, and is included with every new copy of the book. These videos provide guidance and support when students need the most help: the night before an exam.
The Big Picture: Putting It Together chapter openers summarize the concepts and techniques previously presented and then relate this material to the concepts about to be presented.
Author biography
Mike Sullivan, III is a professor of mathematics at Joliet Junior College. He holds graduate degrees from DePaul University in both mathematics and economics. Mike is an author or co-author on more than 20 books, including a statistics book and a developmental mathematics series. Mike is the father of three children and an avid golfer who tries to spend as much of his limited free time as possible on the golf course. |
Math - Step by Step Regents Answers
Home Instruction Schools
Step by step Math Regents Answers.
Have it your way! Finally, regents answers you can understand!
Math Regents Exams with Step by Step Answers
If you are interested in an explanation of how to arrive at the correct
answers to past Algebra and Geometry Regents exams, this
is the place. We have worked some of them out for you and converted the
files into PDF format.If you come across any errors, or if you have
any questions, please feel free to email us [email protected]
Another first! Here are step-by-step answers for Books 1,2, and 3 of the March 2007
New York State Grade 8 Mathematics Test and Books 1 and 2 of the 7th grade test and Books 1,2 and 3 of the 6th grade test. The actual tests can be found at the
NYS Education Department website.
Click below on the appropriate booklet for the step-by-step answers. |
Description of this Textbook
Covers the NCEA Level 1 Achievement Standards 1.30 to 1.36 and replaces the popular Year 11 Graphics Study Guide. It features brief, clearly explained theory, examples, illustrations and numerous activities for student practice. Use throughout the year to support classroom work, to help with internal assessments and to revise for end-of-year exams. Extra material online. |
Standards in this domain:
Apply geometric concepts in modeling situations
G-MG.1. Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).★
G-MG.2. Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).★
G-MG.3. Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).★
(IA) Use diagrams consisting of vertices and edges (vertex-edge graphs) to model and solve problems related to networks.
IA.8. Understand, analyze, evaluate, and apply vertex-edge graphs to model and solve problems related to paths, circuits, networks, and relationships among a finite number of elements, in real-world and abstract settings.★ |
More About
This Textbook
OverviewRelated Subjects
Meet 6, 2002
Superb
This is an excellent introduction to the mathematics of computer graphics for readers with a basic knowledge of linear algebra, written in mathematical English with standard math notation. Each chapter has a few exercises. Suitable as an assigned textbook for an introductory computer graphics math course.
1 out of 1 people found this review helpful.
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Anonymous
Posted January 9, 2012
Swell
Ive not read the hole thing yet but it seems really good so far:)
Was this review helpful? YesNoThank you for your feedback.Report this reviewThank you, this review has been flagged. |
L3 Geography has chapters covering:natural processes,cultural processes,skills and ideas as well as full descriptions of the internal achievement standards. It includes a pull out 2013 style exam paper reflecting the new curriculum, a Revision Tracker to optimise study and an exa ...
L3 Classics Roman has chapters covering:Aeneid,juvenals satires,art and architecture,augustus,roman religion, as well as full descriptions of the internal achievement standards. It includes a pull out 2013 style exam paper reflecting the new curriculum, a Revision Tracker to opti ...
L3 Classics Greek has chapters covering:Aristophanes,Greek vase painting,Alexander the great,Greek science,Socrates, as well as full descriptions of the internal achievement standards. It includes a pull out 2013 style exam paper reflecting the new cirriculum, a Revision Tracker ...
L3 History New Zealand has chapters covering:differentiation,integration,real and complex numbers,graphs and equations relating to conics as well as full descriptions of the internal achievement standards. It includes a pull out 2013 style exam paper reflecting the new curriculum ...
L1 History has chapters covering:historical sources,causes and consequences of historical events,events of significance to New Zealand as well as full descriptions of the internal achievement standards. It includes pull out 2011/2012 exam papers, a Revision Tracker to optimise st ...
Dragon Maths 6 is a write-on student workbook that contains a full mathematics programme for most Year 8 students. It gives comprehensive coverage of work at mathematics curriculum Level 4. It fully covers the Advanced Multiplicative / Early Proportional Stage (stage 7) of the Nu ...
Sustainability is what will save planet earth and its inhabitants. Students therefore need to understand and talk with confidence about issues such as carbon offsetting and virtual reality footprints. They need to share a vision for a sustainable future and ready themselves for ... |
Listed here are the Learning Outcomes for Calculus I. Periodic
assessment of how well these outcomes are being achieved contributes
to the Institute's process for reviewing and continuously improving
its academic programs and course offerings. Each semester, data is
collected on a subset of these outcomes in the form of 1) direct
assessment through scores achieved on particular questions on exams,
and 2) indirect assessment through student responses to questions
included in the end-of-term surveys. Your feedback through the online
surveys is an important part of this process and we hope you will make
every effort to complete the course surveys when they become available
near the end of the semester.
Learning Outcomes for Calculus I
Upon completing this course, it is expected that a student will be
able to do the following:
1. Mathematical Foundations:
Limits of Indeterminate Forms: Explain the concept of a limit
and evaluate elementary examples of indeterminate forms.
Continuity: Demonstrate a working knowledge of continuity for
functions of one variable.
Derivative--First Principles: State and apply the fundamental
definition of the derivative, understand its relationship to the
tangent line, and recognize when a function is not differentiable. |
So you clicked on this thread. Clearly you have some motivation to become better at Math. Well, this is the thread for you!
[SIZE="3"]The Keys to Math Wizardry[/SIZE]
1.Start Small. No matter what your subject is, complex partial differential equations to high school algebra, you will not get anywhere by starting at the back of the book or with the hardest problems you can find. Harder problems are built on concepts and techniques that the problem creator assumes you already know. Thus, it is your job to first learn the concepts and techniques. Read the book, and do simple problems first. They will reinforce what you have learned already and help you build the confidence to tackle ever harder problems.
2.Read the book. The best way to learn the concepts, and expose yourself to the theorems, is by reading the book. I know, most of us HATE to read the book, and thus go right to the problems, but if you want to solve problems you need to understand the concepts behind them. Read slowly, and if you don't understand a word or a sentence, go find out what it means. It could be the reason you don't understand why or how the next theorem introduced works or has any relevance at all.
3.Do the homework. Do the homework again. Do the homework again and again. The only way you will learn problem solving skills is to solve problems. You will see that even if you did the problem yesterday, you might not be able to solve it that easy today. Of course, the goal is to get to the point where you can solve the problem without much difficulty.
4.Go to class. The instructor might be able to explain things much better than the book can. If you have any questions you were not able to resolve yourself, class is the time to ask. Do not be afraid that your question is stupid, because there are probably many more students in the class stuck on the same thing who do not possess the courage to actually speak up about it.
5.Reference. By nature, some textbooks and some authors just suck. They are unclear and seem to pull solutions out of some magic mathematical hat. The good news is, you do not have to stick to that text if you feel that way. Ask your instructors, or other people who have been or are in your boat, and see if they can recommend you any books that have a knack for explanation, or a comprehensive solutions guide to a variety of pertinent problems. (They might even let you borrow!) For instance, one of the best calculus books around is "The Calculus Lifesaver" by Adrian Banner, and I have yet to find a course which uses this book as its main text.
6.Study Together. Go find a friend who is taking your course or one like yours and study together. Multiple researchers have shown that learning is reinforced when you have to explain it to someone else, plainly because you really have to know what you are talking about to get it across. Also, you will benefit if you are on the student end of such peer instruction. Think about it as someone giving you pieces to your puzzle while you are giving pieces to them for their own. Sometimes all it takes is a certain someone to say just the right words which makes it all click in your head. You will undoubtedly make good friends in the process.
7.Get Help. Usually there will be someone or somewhere at your school which provides tutoring services or assistance programs. It is not shameful to ask for help. There is no reason to struggle because of something as ridiculous as your ego. Go see your teaching assistants or your instructor. If they have office hours, show up every time you are stuck. It really helps to get to know those giving you your grades...that is...as long as you are nice.
i find that in the more complicated forms of calculus and algebra, there are a limited number of possible scenarios that every problem is a member of. ive got a good teacher that would do examples of all the possible question types
^^^ Throughout the 12 (and counting) years of schooling i have done, i have never once had a teacher which i considered "good". I always had to figure everything out for myself which didn't really benefit me in the long run. I only hope to "god" that once i start university ill get some decent professors who can properly prepare me for exams.
Mathematics is always challenging for everybody. It is a great humbler. This being the case, I don't understand how being a "wizard" is even possible.
In any case...
Treat and learn mathematics as a tool. Do science, use math. Put simply - adding for the sake of adding is silly, but adding up your dollars makes adding worthwhile.
And projecteuler.net is pretty cool.
the planes wheels are simply the point of contact against the tarmac however they are frefloating and will spin to any velocity until they fail... assuming the wheels can spin to an infinate velocity the plane will pull itself forward regardless of the backwards motion of a dynamic tarmac because the prop is attached to the fuselage of the plane
there IS friction on the wheels bearings or whatever plane wheels use to negate friction to spin at a high velocity however...for assuming the tarmac is retreating at a rate of 30 meters a second and the prop is providint a rearwards thrust propelling the plane at 30 meters a second the wheels would be spinning at a rate that would propell the plane at 30 meters per second were the tarmac static. the wheels were they the point of propulsion as in a car...would be need to spin twice as fast to achieve the same velocity. however as a planes propulsion is a turbofan/prop/jet that is not its point of contact whilst a land vehicles propulsion is the same as its point of contact, should lead to a conclusion that the plane wil propell itself forward inducing the wheels to simply spin at twice the speed were the tarmac static
See? That was easy. One step, you will think so linearly that math becomes easy.
Experience to validate: I did this and I've gotten A+ in every math course and calculus at university.
You either are not being taught good mathematics courses, or have been taught a fucking amazing decades-long computer science course covering chaos, multivariable nonlinear analysis, spectral theory, number theory, advanced group theory, spherical harmonics, Green's functions, complex nonlinear fourier analysis, order theory, proof theory, model theory and hypercomplex analysis to name but a few.
the only "way" (i guess you could say) that .999... is equal to the value of 1.0 is in the display of calculators that just simply round to the nearer value.
the math that was done above does not prove anything at all, especially that any repeating value is equal to anything but its self. Mostly because that math was *wrong*
here:
Quote:
let x=.999999...
okay, we'll let x equal that
10x=9.99999....
okay now we have an equation, lets solve for x...
Subtract x from both sides
9x=9
wrong, subtracting x from both sides is done as follows:
10x=9.99999....
(10x) - (x) = (9.99999....) - (x)
9x = 9.999999 - x
this proves nothing, or shows nothing.
Therefore, x=1
also, incorrect...
if you wanted to solve for x, you just would have had to divide by "10" (or whatever coefficient is in front of the variable there), so you would get:
10x = 9.99999....
(10x)/10 = (9.999..)/10
now... here is where I would like you all to listen...
everybody... PLEASE TAKE OUT your Ti-83+ or Ti-89 calculators...
enter each of the following in and see what you get as an answer:
9/10
9.9/10
9.99/10
9.999/10
9.9999/10
9.99999/10
9.99999999999999999999/10
as you can see, the only reason that last entry is "equal" to one is because the value and amount of the final value that it increases by per iteration is so so so (let me fully express the word "so") infitesimally small, that the calculator reached its bounds of sigfigs and rounds it up... thats it. done.
you can spend eternity doing those calculations above by hand and you'd never reach the value of one (1), you would literally get an infinite amount of *.9s*
I wish it wasn't so time consuming and frustrating to practice math. The key with all types of math is to practice like a motherfucker. Things become second nature when you do them for the 9002nd time...
Therefore, 9x=9, as the right hand side of the equation we simplified above.
The reason this isn't intuitive is because the .9 represented literally repeats infinitely (not just a lot) and we're inherently not equipped to deal with infinity intuitively.
wrong, since
10x = 9.9999
subtract .9... from both sides leaves you with not 9x = 9 but:
9.000001x = 9re wrong here.
I'll put it this way. .999...-.999...=0youre too much of a lazy, self-absorbed idiot to realize that I am wrong so you shell it off as you "not wanting to argue" and refer me to wikipedia
even though I've clearly refuted each of your points in a logical manner to continue this debate, not argument. grow up kid.
"10-.999...= 9.00...01"
That is not the problem in question--it's also not the right answer unless there are a finite amount of nine. I've given you a proof for this, and you have tried to point out flaws in the individual step, where there are none. Math is not always intuitive: it's the proof that counts. You have not refuted my points, you've failed to understand them.
9.999...-.999...=9 is the problem you have said is inaccurate, as a part of my proof.
The other side is 10x-x, which is always 9x no matter the value for x.
Do you not accept those?
If you do, I can prove validly that .999 repeating for infinity=1. If you do not, speak to a math teacher, because I'm done.
The formal proof (or the most popular one anyways), is representing 0.999... as a geometric series, and showing it converges exactly to 1. |
is a challenging and rigorous course which develops analytical and problem-solving skills through learning and applying mathematical techniques and methods. The course will be of interest to those who enjoy the satisfaction of solving problems. The Pure (or core) Mathematics course includes work involving aspects of algebra, co-ordinate geometry, trigonometry, calculus, numerical methods and vectors, The applied units take those skills and apply them in the areas of Statistics and Mechanics.
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This is a well-regarded qualification that has the potential to open up many doors in the future. It will support anyone who is studying any subject which involves some work with figures or data especially Science subjects, Economics and Psychology.
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Many A level students continue their studies at degree level in this or a related subject. It is necessary for the further study of Mathematics at University and highly desirable for many other subjects, particularly the Sciences. Many Universities also require the study of Further Mathematics for Mathematics degrees.
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Section 1
Objectives
Test 1
Section 2
Objectives
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Section 4
Objectives
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Objectives
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Objectives
Final
Exam
80
0-100
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ometry
ISBN: 11
Extending 0-536-08809-8
652Chapter Perspective
The beauty of mathematics is accentuated through the lens of geometry. Geometric designs in buildings, such as the Alhambra in Granada, Spain shown in this photo, reveal symmetry and give insight into this important mathematical idea. The intriguing world of patterns, designs, and tessellations brings to life some fascinating relationships among mathematics, nature, and art. Also, whether you are thinking of an astronomer describing the changes in the positions of stars in the sky, or a physicist describing the movement of particles in an accelerator, you notice that motion plays an integral part in natural events. The idea of transformations helps us give useful mathematical descriptions of motions in our world. In this chapter, we focus on gures, relationships, and patterns in space, including geometric descriptions of motion, tessellations, and special polygons to extend the development of your spatial sense.
Big Ideas
Proportionality: If two quantities vary proportionally, that relationship can be represented as a linear function. Patterns: Relationships can be described and generalizations made for mathematical situations that have numbers or objects that repeat in predictable ways. Orientation and Location: Objects in space can be oriented in an innite number of ways, and an objects location in space can be described quantitatively. Shapes and Solids: Two- and three-dimensional objects with or without curved surfaces can be described, classied, and analyzed by their attributes. Transformations: Objects in space can be transformed in an innite number of ways, and those transformations can be described and analyzed mathematically.
Connection to the NCTM Principles and Standards
The NCTM Principles and Standards for School Mathematics (2000) indicate that the mathematics curriculum in geometry for grades PreK8 should prepare students to apply transformations and use symmetry to analyze mathematical situations; use visualization, spatial reasoning, and geometric modeling to solve problems (p. 41).
Connection to the PreK8 Classroom
In grades PreK2, students use their own physical experiences with shapes, such as tting pieces into a puzzle, to learn about slides, turns, ips, and symmetry. In grades 35, students are ready to mentally manipulate shapes to make predictions, learn the mathematical language to describe their predictions, then verify their predictions physically. In grades 68, students further develop spatial sense by analyzing motions of geometric gures and by putting polygons together to form patterns such as tessellations.
ISBN: 0-536-08809-8
653654
C H A P T E R 11
EXTENDING GEOMETRY
Section
11.1
Translations, Rotations, and Reections Connecting Transformations and Symmetry Transformations That Change Size Transformations That Change Both Size and Shape
In this section, we investigate the geometric transformations that do and do not affect a gures shape or size. We use mathematical ideas and symbols to describe, analyze, and compare the different types of transformation. You will have opportunities to use special computer software to create and move the geometric gures.
Transformations
Essential Understandings for Section 11.1
The orientation of an object does not change the other attributes of the object. Congruent gures remain congruent through translations, rotations, and reections. Shapes can be transformed to similar shapes (but larger or smaller) with proportional corresponding sides and congruent corresponding angles. Some shapes can be divided in half where one half folds exactly on top of the other (line symmetry). Some shapes can be rotated around a point in less than one complete turn and land exactly on top of themselves (rotational symmetry).
In Mini-Investigation 11.1, you are asked to use some motions that are related to geometric transformations.
Draw a picture to illustrate each motion that you used. Technology Extension: Use geometry exploration software (GES) to create a triangle or quadrilateral. Explore the GES options for sliding, turning, and ipping the gure. (See Appendix B.) Describe how GES may be helpful in studying different ways to move gures.
M I N I - I N V E S T I G A T I O N 11 . 1
Solving a Problem
In what ways (slide, turn, ip, or combinations) can you move pentomino (a) to test whether it matches pentominos (b), (c), (d), and (e)? (Use a tracing if one will help.)
(a)
(b)
(c)
(d)
(e)
Translations, Rotations and Reections
ISBN: 0-536-08809-8
In Mini-Investigation 11.1, you may have observed that an object may be slid, turned, or ipped without changing its shape or size. With these three simple655
P
P B
A F I G U R E 11 .1
A gure and its translation image.
motions, or combinations of these motions, we can move a gure anywhere in space to match another gure that is congruent to it. We use these motions to describe transformations in the following subsections. Slides or Translations. You encounter the motion of slides often in everyday life: a notebook sliding across a desk, a window frame sliding up and down, or a drawer sliding in and out. In Figure 11.1, the blue pentomino shape is the image of the yellow pentomino shape after all the points in the plane have been slid in the direction and distance indicated by the slide arrow (or directed segment) from point A to point B. To visualize the slide, trace the yellow pentomino outline and the slide arrow. Then, slide point A on the tracing paper along the arrow to point B. (If you trace the entire arrow, you will not accidentally turn the tracing paper.) The pentomino shape you traced on your paper should now coincide with the blue pentomino shape, which is the image of the slide. The physical motion of sliding, mathematically called a translation, is dened as follows.
Denition of a Translation
If each point P in the plane corresponds to a unique point in the plane, P , such that directed segment PP is congruent and parallel to the directed segment AB, then the correspondence is called the translation associated with the directed segment AB and is written TAB .
The denition indicates that when a gure is translated along a directed segment AB, each point, P, of the gure slides to point P , in the same distance and direction that point A slides to get to point B. To describe further the effect of a translation, we write TAB(P) = P to say that P is the image of P under the translation associated with the directed segment AB. When considering the translation TAB, point B becomes the image of point A. If we want to indicate a slide from B to A, we write TBA.
ISBN: 0-536-08809-8
Turns or Rotations. Turns in the real world include food rotating on a turntable in a microwave oven, doorknobs turning, and a Ferris wheel turning around its large axis as the seats turn on their individual axes. The blue pentomino shape
Mathematics for Elementary School Teachers, Fourth Edition, by Phares O'Daffer, Randall Charles, Thomas Cooney, John Dossey, and Jane Schielack. Published by Addison Wesley. Copyright 2008 by Pearson Education, Inc.
656
C H A P T E R 11
EXTENDING GEOMETRY
P
C
P
F I G U R E 11 . 2
A gure and its rotation image.
P
shown in Figure 11.2 is the image of the yellow pentomino shape after all the points in the plane have been turned around point C in the direction and angle measure indicated by the turn angle, PCP . T visualize the turn, trace the yellow pentomino outline and PC of the turn angle. o Place the point of your pencil on point C and hold point C still while you turn the tracing paper in the direction and angle measure of the turn angle until the PC drawn on your tracing paper coincides with P C on the page. The pentomino shape you traced on your paper should now coincide with the blue pentomino shape, the image of the turn. In mathematics a turn is called a rotation, and is dened as follows.
Denition of a Rotation
C P
Consider a point C and an angle with measure A from - 180 to 180. If each point P in the plane corresponds to a unique point in the plane, P , such that m PCP A and PC = P C, then the correspondence is called the rotation, with center C and angle A, and is written RC, A .
The denition indicates that when a gure is rotated around a center point C, each point, P, of the gure turns around C through an angle, a, to point P , which is the same distance from C as P. When considering the rotation R C, a, we write R C, a(P) = P to say that P is the image of P under the rotation associated with the center C and angle a. In some cases, as in Figure 11.3, the center of the rotation, C, is in the interior of a gure, and the rotation image of a point P of the gure about that center is another point P of that same gure. When this happens, the gure is rotated onto itself. We will consider this special case in more detail on pp. 663664. Flips or Reections. Flipped images in the real world appear when you look in mirrors, turn transparencies over on the overhead projector, or press an inked stamp onto a piece of paper. The blue pentomino shape in Figure 11.4 is the image of the yellow pentomino shape after all the points in the plane have been ipped over line /. To visualize the ip, trace the yellow pentomino outline and line /, marking point L on line /. Lift your tracing paper, ip it over, and replace it on the page by laying the traced line / over line / on the page with point L on the tracing paper
F I G U R E 11 . 3
RC,90(P)
P
P L
P
F I G U R E 11 . 4
ISBN: 0-536-08809-8
A gure and its reection image657
matching point L on the page. The pentomino shape that you traced should now coincide with the blue pentomino shape, the image of the ip. The physical motion of ipping (mathematically called a reection) can be dened as follows.
Denition of a Reection
If each point on line / corresponds to itself, and each other point P in the plane corresponds to a unique point P in the plane, such that / is the perpendicular bisector of PP , then the correspondence is called the reection in line /, and is written M/ .
P P
The denition indicates that when a gure is reected about a line /, each point, P, of the gure is ipped to point P , so that line / is the perpendicular bisector of PP . With this notation, we write M /(P) = P to say that P is the image of P under the reection in /. In some cases, as in Figure 11.5, the reection image in line / of a point P of a gure is another point P of that same gure. When this happens, the gure is reected onto itself. We will consider this special case in more detail on p. 661. In Example 11.1, you are asked to identify the motion that produced the given image.
F I G U R E 11 . 5
M/(P)
P
Example
11.1
Identifying Transformations
Identify which transformation (translation, rotation, or reection), if possible, that would change each yellow gure shown to the corresponding blue image. Justify your answers and use correct notation to show the effect of the transformation on point A.
D B C B C C
A B
A
D
A
D B
A
D C (a) B B A A
ISBN: 0-536-08809-8
(b) C C D D B C (d) A A D D B C
(658
C H A P T E R 11
EXTENDING GEOMETRY
SOLUTION
In part (a), the transformation is a reection because the orientation of the gure has changed. In the original rectangle, reading from A clockwise around the gure gives ABCD. In the blue image, reading clockwise from A gives A D C B . We write M line of reflection (A) = A . In part (b) the transformation is a rotation. Its orientation hasnt changed, but the segments that were horizontal now arent. We write R C, angle of rotation(A) = A . In part (c) the transformation is a translation. If A is connected to A , B to B , C to C and D to D , the segments formed are parallel and congruent. All the points moved the same distance in the same direction. We write TBB(A) = A . In part (d) the blue image of the yellow gure cant be produced with a single transformation. The orientation of the gure has changed, so it must have been reected. But the image doesnt appear where it would if a mirror had been placed between the gures. Thus it has been translated and reected.
YOUR TURN
Practice: Identify the type of transformation that would transform each yellow gure to the corresponding blue image. Use correct notation to show the effect of the transformation on point A.
A
A
A (a) A (b)
A
A
A A (c) (d)
Reect: Which of the transformations in the practice problem changed the orientation of the points in the gure? When given a gure and a transformation such as a reection, rotation, or a translation, there are various techniques that can be used to nd the image of the gure under the specied transformation. Example 11.2 illustrates some ways this can be done659
Example
11.2
Using Transformations
Find the image of quadrilateral ABCD under the following transformations: a. TMN b. R O,45
M A 45 D C
c. M /
N B
O
SOLUTION
a.
M A A
N B B
D
D A
C B
C
b.
O
45
D A B D
C
C
c.
A
B
D D
C C
A
B
ISBN: 0-536-08809-8
Henrys Solution: I drew ABCD on paper and then traced it on another sheet of paper. For (a) I made two dots on my tracing paper and slid one dot from M to N, using the other dot to ensure that the direction didnt change. For (b) I made a dot on the tracing paper to place on point O and another dot on the side of angle O. Then I rotated the tracing paper so that the dot on the side of the angle turned through an angle of 45. Then I marked points on the original paper for A , B , C , and D .660
C H A P T E R 11
EXTENDING GEOMETRY
For (c) I marked two dots on line / and then traced ABCD and the dots. Then I ipped the tracing over line /, keeping the traced dots on the original. Then I marked points on the original paper for A , B , C , and D . Lenas Solution: I used GES and the instructions for creating translations, rotations, and reections in Appendix B.4, p. 871, to complete (a), (b), and (c).
YOUR TURN
Practice: Copy ^ ABC and use a method of your choice to nd the image of ^ ABC under the following transformations. a. TRS b. R O,90 c. M /
R A O 90 B C S
Reect: What is true about the gure and its image under each of the preceding transformations? Combinations of Motions. As indicated in the solution to Example 11.1, no single translation, rotation, or reection transformed gure ABCD to gure A B C D in part (d). It required a translation and then a reection over the line of translation. This combination motion is called a glidereection. In MiniInvestigation 11.2 on the next page, you will see the results of following a reection with a reection, a rotation with a reection, and a translation with a rotation. All two-motion combinations of rotations, reections, or translations other than the translation followed by a reection can be replaced by a single rotation, reection, or translation. You will also have an opportunity to explore combinations of motions in Exercises 39, 40, 46, and 49 at the end of this section. Transformations and Congruence. Experiences with motions in the real world suggest that translations, rotations, reections, and glidereections dont change the size or shape of a gure because the points in the plane maintain their relative distances from one another. In other words, distance is preserved. Mathematicians call a transformation that preserves distance and other characteristics a congruence transformation, or an isometry. The properties of polygons and transformations suggest that translations, rotations, reections, and glidereections are isometries. Thus, in an isometry, not only distance but also betweenness, angle measure, and size and shape are preserved. That is, when one of these properties is determined in a gure, it is the same in the transformed image of the gure. Thus a gure can be repositioned anywhere in space without changing its size or shape through the use of no more than the four types of transformation that we have discussed. As a result, we can state the transformation denition of congruence on the next page661
Talk about which of those combinations of motions you could have completed in just one motion. Explain how.
M I N I - I N V E S T I G A T I O N 11 . 2
Technology Option
Use GES or other tools to describe the result of the following combination of motions on the gure shown: a. Reect ABC over line m and then reect the resulting image over line n. b. Rotate ABC 90 counterclockwise around point P and then reect the resulting image over line n. c. Translate ABC on line PQ from C to P and then rotate the resulting image 180 clockwise around point Q.
A
B P C
Q
m
n
Denition of Congruence, Using Transformations
Two gures are congruent if and only if there exists a translation, rotation, reection, or glidereection that sets up a correspondence of one gure as the image of the other.
An interesting illustration of how transformations can be used to solve problems is the crossword puzzle problem in Exercise 56 at the end of this section.
(a) Original figure
Connecting Transformations and Symmetry
Symmetry in the Plane. The idea of symmetry plays a central role in art, interior design, landscaping, and architecture, as well as in mathematics. When a plane gure can be traced and folded so that one half coincides with the other half, as shown in Figure 11.6, we have modeled the congruence transformation we have referred to as a reection, in which each point in one half of the gure corresponds in a special way to a unique point in the other half of the gure and each point on the line of reection corresponds to itself. In this case, we say that the gure has reectional symmetry. The fold line is called the line of symmetry.
(b) FoldDo the halves match?
ISBN: 0-536-08809-8
F I G U R E 11 . 6
Folding paper to illustrate reectional Student Work: Transformations
Connection to the PreK8 Classroom
Elementary school students are asked to test their understanding of important mathematical concepts, such as transformations, by applying logical reasoning to questions about the concept. Below are three student responses to the following question.
Elizabeth took the parallelogram below and performed some slides, ips, and turns with it. When she nished, she claimed she had a rectangle. Is it possible that Elizabeths claim was correct? Why or why not?
Analyze the work for students A, B, and C.
Give your interpretation of
Student A
what each student was thinking.
Which of the student
explanations shows the deepest understanding of the concepts of slides, ips, and turns? Why?
What other reactions do
you have to the student work?
Student B
Student C
ISBN: 0-536-08809-8
Source: 2006 Heinemann Publishing. Question ID = 12, #1, 2, 3.
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F I G U R E 11 . 7
Testing for reectional symmetry with a plastic reector.
Source: GeoReector Mirror courtesy of Learning Resources, Inc.
A commercially produced piece of plexiglass, shown in Figure 11.7, can also be used to identify reectional symmetry. If a plastic reector can be placed on a gure so that the reection of half the gure ts exactly on the other half, the line of the reector is a line of reectional symmetry. Another type of symmetry is rotational symmetry. To test a gure for rotational symmetry, we can use a trace and turn test, as shown in Figure 11.8. The red gure is traced in blue on a piece of tracing paper or clear plastic. Then, keeping the blue gure directly on top of the red gure and holding the center point xed, turn the blue tracing until it again coincides with the red gure. Since the blue tracing ts exactly on the red gure after rotating it 90, we say the red gure has 90 rotational symmetry. The blue tracing also ts exactly on the red gure after rotating it 180, 270, and 360. Since a tracing of any gure will t back on the gure after a rotation of 360, we concern ourselves only with angles of rotation a, where a 6 360. Also, if a gure has a rotational symmetry, a tracing of it will also coincide with the gure if rotated na, where n is any nonzero integer. Because of this, we usually use the smallest possible angle a through which the tracing can be rotated to t back on the gure to describe the rotational symmetry of the gure. We say that the gure has
A
A
ISBN: 0-536-08809-8
F I G U R E 11 . 8
Using tracing paper to test for rotationalA
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C H A P T E R 11
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a rotational symmetry. The point that is held xed is called the center of rotational symmetry. When a gure has 180 rotational symmetry, we say that the gure has point symmetry about its center of rotation. In a gure with point symmetry, the center of the half turn (180 rotation) is the midpoint of a segment connecting any point of the gure and its image after the turn. To connect transformations and symmetry, we observe that the trace and turn test for rotational symmetry models the congruence transformation we have referred to as a rotation, in which each point of the gure corresponds in a special way to another point of the gure, except for the xed point, which corresponds to itself. We apply these ideas about symmetry in Example 11.3.
Example
11.3
Describing Symmetry Properties
Describe the symmetry properties of the gure on the left.
SOLUTION
The star shown has reectional symmetry. There are ve lines of symmetry, as shown below. b. A tracing can be made to coincide with the star after it is rotated 72 about its center, so the star has 72 rotational symmetry.
YOUR TURN 2
a.
Practice: Describe the symmetry properties of the following gures:
1
3
5
4
(a) Rhombus
(b) Equilateral triangle
(c) Pinwheel
Reect: differ.
Explain how the symmetry properties of a rhombus and a parallelogram
In Chapter 10, we learned that triangles and quadrilaterals could be classied according to side and angle properties. Classes of triangles or quadrilaterals thus formed were given special names, such as equilateral triangles and parallelograms. Triangles and quadrilaterals also may be classied according to symmetry properties, but it isnt common practice to give special names to the classes of gures formed. In Exercise 58 at the end of this section, you will be asked to classify triangles and quadrilaterals according to their symmetry properties.
Connection to the PreK8 Classroom
Just as symmetry is basic to an understanding of the universe, so are ideas of symmetry necessary to an understanding of mathematics. Students in the primary grades fold and cut paper to make hearts, pumpkins, and other gures having lines of symmetry. Ideas of symmetry are gradually expanded as students progress through elementary school665
B A P
C
Transformations That Change Size
Size Transformations. The idea of same shape but not necessarily same size plays an important role in everyday life. For example, people enlarge photographs as shown with the buttery pictures below. They also enlarge blueprints and draw large circuit diagrams on a wall and shrink them to the size of a microchip. What guarantees that these larger and smaller images retain the essential characteristics of the originals? One way is to use a special transformation that enlarges or shrinks the size of a gure along specied lines by multiplying distances by a given factor. The physical action of enlarging or shrinking, mathematically called a size transformation, is dened as follows (with reference to Figure 11.9).
B A B A P C P
C
O F I G U R E 11 . 9
Using a size-changing transformation to enlarge and shrink a gure.
Denition of a Size Transformation
If point O corresponds to itself, and each other point P in the plane corre r (OP) for r>0, then the sponds to a unique point on OP such that OP correspondence is called the size transformation associated with center O and scale factor r and can be written SO,r .
The denition indicates that when a gure is shrunk or enlarged from a center O, by a factor r, the image of each point P of the gure is determined by multiplying OP by r to produce OP on OP . P is the image of P. With this notation, we can write SO,r(P) = P to say that P is the image of P under the size transformation with center O and scale factor r. The enlarging and shrinking lines are determined by a point called the center of the size transformation. The multiplier that enlarges or shrinks the lengths is called the scale factor. For example, the blue gure, A B C P , in Figure 11.9 is the image of parallelogram ABCP when it is enlarged by using point O as the center and 2 as the scale factor. Note that OA is 2 times OA, OB is 2 times OB, OC is 2 times OC, and OP is 2 times OP. The yellow gure, A B C P , is the shrunken image of ABCP when 1 is 2 the scale factor instead of 2. In this case, OA is one-half OA, OB is one-half OB, OC is one-half OC, and OP is one-half OP. Size transformations can also be carried out with a computer drawing program, as illustrated in Figure 11.10.
ISBN: 0-536-08809-8
Buddy Mays/CORBIS
Steve Kaufman/CORBIS666
C H A P T E R 11
EXTENDING GEOMETRY
Original image
Transformed image
F I G U R E 11 .1 0
Effecting a size transformation by using a computer drawing program.
Source: 2006 Corel Corporation Ltd. Box shot(s) reprinted with permission from Corel Corporation.
Size Transformations and Similarity. To relate size transformations to similarity, we rst consider some of the size properties of transformations. Measuring the sides BC B of the parallelograms in Figure 11.9 reveals that AB = AC . This result suggests that B the ratios of lengths of sides of a gure and the ratios of the lengths of corresponding sides in its image are equal. Because of this property and the fact that size transformations also preserve betweenness and angle measure, size transformations also preserve shape. Sometimes a gure can be made to correspond to a gure with the same shape by a single size transformationbut not always. In Figure 11.11 on p. 671, the original triangleon the left in part (a)was rst reected in line r, then this reected image was subjected to a size transformation to produce the blue image of the triangle shown in part (b). Similar gures were described, with focus on proportional variation, on p. 384 and on p. 621. The idea that a combination of transformations is needed to transform a gure into a gure of different size and shape is used in the following denition of similarity.
Denition of Similarity, Using Transformations
Two gures are similar if and only if there exists a combination of an isometry and a size transformation that generates one gure as the image of the other.
ISBN: 0-536-08809-8
In Example 11.4, a size transformation along with the problem-solving strategies of draw a picture and write an equation are used to solve a practical problem667
r (a) F I G U R E 11 .11
(b)
Transforming a triangle into another triangle that is the same shape.
Example
11.4
Problem Solving: The Poster Problem
The service organization at Lonnies school is planning a fundraiser with stars as the theme. Lonnie can only nd a pattern for a star that is 8 centimeters high, which is too small for the posters. How can Lonnie enlarge the star pattern so that it is 20 centimeters high?
SOLUTION
Trace the center O of a size transformation and the 8-centimeter star pattern in the lower left corner of the paper. The height needs to be changed from 8 to 20 centimeters, so the scale factor is 20 , 8, or 2.5. Draw the rays from O through
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O668
C H A P T E R 11
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the tips of the small star pattern and measure 2.5 times the distance from O to a small star tip along these rays and mark the tips of the larger star. Connect every other point to form the 20-centimeter star pattern.
YOUR TURN
Practice: Use a size transformation to nd an image of the pentagon that is threefourths as wide as the following pentagon.
Reect: Could you use a size transformation to nd an image that is congruent to the original shape? Explain your answer and use mathematical notation to symbolize the transformation.
Transformations That Change Both Size and Shape
Topological Transformations. Transformations that represent shrinking, stretching, or bending a curve or surface without tearing it or joining points are called topological transformations. These transformations can change both the size and shape of a gure, as shown in Figure 11.12. Note that the images have no holes or breaks that were not also in the originals. The fact that connectedness doesnt change in a topological transformation leads to other important characteristics that remain constant, such as the characteristic of having no ends. For example, squares, triangles, and all other polygons can be transformed by a topological transformation into a circle. If one gure can be transformed to another by a topological transformation, the two gures are topologically equivalent. For example, a rubber band is topologically equivalent to a compact disk, and a bagel is topologically equivalent to a coffee mug with a handle. Also, a schematic diagram of a circuit is topologically equivalent to a diagram to scale of the actual circuitry; how the components and wires are connected and their relative locations are what is preserved. However, a picture frame and a grocery sack are not topologically equivalent. If the picture frame were made of modeling clay, it could never be formed into the shape of a grocery sack by just stretching or squeezing the clay. Some parts would have to be joined or the clay broken apart, neither of which is allowed in a topological transformation.
(a) Topological transformation of a line. F I G U R E 11 .1 2
(b) Topological transformation of a circle.
ISBN: 0-536-08809-8
Topological transformation of a line and a circle669
In Mini-Investigation 11.3 you are asked to analyze topological transformations further.
Make a chart showing your classication. Talk with classmates and compare your charts.
M I N I - I N V E S T I G A T I O N 11 . 3
Using Mathematical Reasoning
How would you classify the digits 09 into topologically equivalent gures?
In Exercise 54 at the end of this section you are asked to use a topological transformation to solve an interesting problem, the castle court problem.
Connection to the PreK8 Classroom
Many young children can distinguish topological properties such as connectedness, separation, inside, and outside before they can distinguish characteristics of rigid geometry such as length, angle measure, or relative position. Elementary school students extend their spatial visualization abilities by drawing on balloons and watching what changes and what doesnt change as the balloon is inated and deated. The topological concepts of a gure being connected and separating a plane are used later in classifying shapes and creating formal denitions for polygons and other geometric gures.
Problems and Exercises
for Section 11.1
8.
figure image
A. Reinforcing Concepts and Practicing Skills
1. What information is necessary for identifying a specic slide? 2. How is the information in Exercise 1 related to the mathematical notation for a translation? 3. What information is necessary for identifying a specic turn? 4. How is the information in Exercise 3 related to the mathematical notation for a rotation? 5. What information is necessary for identifying a specic ip? 6. How is the information in Exercise 5 related to the mathematical notation for a reection? In each diagram in Exercises 710, identify the motion or combination of motions that would produce the image. Justify your choices. 7.
image
9.
image
figure
10.
image
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figure figure670
C H A P T E R 11
EXTENDING GEOMETRY
For Exercises 1122, use tracing paper, graph paper, a geoboard, or geometry exploration software to nd the image of the quadrilateral obtained from each transformation. 11. TMN 12. TDC 13. TRS 14. TAD 15. R O,a 16. R C,180 17. R Q,-90 18. R O,180 19. M l 20. M r 21. M s 22. M t
t E A Q D P r C s S M O B
24. Describe the rotational symmetry, if any, of each of the gures in (a) through (f ). Use the trace and turn test if helpful. a. b.
c.
N
d.
R
23. Describe the reectional and rotational symmetry properties of each object. Use a plastic reector, paper folding, or GES when needed. a. Propeller
e.
b. Flag f.
c. Advertising logo
ISBN: 0-536-08809-8
25. Which uppercase letters of the alphabet have reectional symmetry671
26. Which uppercase letters of the alphabet have rotational symmetry? 27. Which uppercase letters of the alphabet have point symmetry? 28. Use a compass and a straightedge to construct the image of point A under a rotation with center C and indicated by the angle a shown in the gure. (See Appendix D for construction techniques.)
A
C
B
O
A
a. Find the image of AB under the size transformation SC,3 . Then, nd the values of the ratios in (b)(d). CA CB AB b. c. d. CA CB AB e. Find the image of AB under the size transformation SC,0.5 . Then, nd the values of the ratios (f ) through (h). AB CA CB f. g. h. CA CB AB 35. Explain how to use the transformational denition of similarity to test whether each pair of gures is similar.
B
29. The pattern that follows is called a frieze pattern. Imagine that it continues indenitely in each direction, and tell as much about the symmetry of the pattern as you can.
A D E
C
30. If you place a plastic reector on line /, you form a word from the half-word shown. Explain why this is true, and create another half-word.
B
F (a) C G H A (b) F E D
31. Decide whether the pair of objects in each part of the gure are topologically equivalent. Support your answer.
(a)
B. Deepening Understanding
36. Describe the symmetry properties of the following geometric shapes or gures:
(b)
ISBN: 0-536-08809-8
32. Identify two physical objects that are topologically equivalent to each of the following: a. Ball b. Donut c. Sack with two handles 33. Classify the letters of the alphabet into groups of topologically equivalent gures. 34. Copy the following gure to nd the images and ratios specied and verify your answers:
(a) (b)
(d) (672
C H A P T E R 11
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37. Make a chart showing how the isometries are alike and how they are different. Consider the following characteristics: a. Reverse orientation b. Points that map into themselves c. Mathematical notation d. Role of segments, angles, and lines 38. Figures that correspond under a translation, rotation, or reection are said to be translation congruent, rotation congruent, or reection congruent. Position two congruent gures on a piece of paper to satisfy each combination of congruences listed. Use tracing paper, mirrors, or GES to justify your choices. a. All three congruences b. Translation and rotation congruences only c. Translation and reection congruences only d. Rotation and reection congruences only e. Translation congruence only f. Rotation congruence only g. Reection congruence only h. None of the three congruences 39. Find the image of the quadrilateral when it is subjected to the glidereection TRS followed by M4 . Does it RS matter whether you translate rst and then reect, or reect rst and then translate? Justify your answer.
R F G H
41. What different types of symmetry, if any, do you see in these automobile manufacturers logos?
(a)
(b)
(c)
(d)
S E
42. Create three logos for an imaginary corporation as follows: a. One is to have rotational symmetry but no reectional symmetry. b. One is to have reectional symmetry but not rotational symmetry. c. One is to have both rotational and reectional symmetry. 43. Draw, if possible, a quadrilateral that has a. a line of symmetry but no rotational symmetry. b. rotational symmetry but no line of symmetry. c. both reectional and rotational symmetry.
40. Use tracing paper, geoboards, or GES to nd the images obtained from the following combinations of motions, where, for example, TAB*TCD indicates a translation from A to B followed by a translation from C to D.
C. Reasoning and Problem Solving
44. Copy the following gures on dot paper and nd the original shape if the shaded shape is its image under the given transformation:
A A B
P E r A B s D (a) TAB C
G F E
B C D M N (c) MMN
O
(b) RF,90
(a) RO,30 * RO,60
(b) Mr * Ms
C
A
B
(c) TAB * TBC
45. Give a type of triangle, if possible, that has a. no lines of symmetry. b. exactly one line of symmetry. c. exactly two lines of symmetry. d. exactly three lines of symmetry. e. more than three lines of symmetry. 46. Gina showed Maria the following GES sketch. Gina claimed that she created gure XYZ by using a673
combination of two motions applied to the original gure ABC. The dashed lines show the result of the rst motion. Describe the combination of motions that Gina could have used. Explain how XYZ could result from ABC in just one motion.
P
X A B Z C P Q Y
Copy the sketch and use this information to recreate gure W on it. Are P and W congruent? Explain. 51. The Recreation Room Construction Problem. A building contractor had instructions from the architect to make a rectangular recreation room larger by adding 4 feet to both the length and the width. Is the new room rectangle similar to that of the original room rectangle? Explain and support your conclusion. 52. The Mirror Problem. What is the smallest size of mirror, hung appropriately, that would allow Julie to see her total height? Explain with a diagram. 53. The Flower Garden Problem. A gardener made a model showing the layout of a planned ower garden that was quadrilateral-shaped with dimensions w, x, y, and z. If she builds the actual garden with dimensions 25w, 25x, 25y, and 25z, can she be sure the model and the actual garden are the same shape? Explain. 54. The Castle Court Problem. When the gatekeeper at a castle was asked how to get to the inner chamber, he said, After you enter, always stick to the right-hand wall. Use the topological transformation of the map of the inner court in the gure for this exercise to decide whether the gatekeeper is correct. In following this path, does the visitor go through all the halls of the inner court? If the gatekeeper had said stick to the lefthand wall, would the visitor have reached the inner chamber?
m n
47. Give a type of quadrilateral, if possible, that has a. no lines of symmetry. b. exactly one line of symmetry. c. exactly two lines of symmetry. d. exactly three lines of symmetry. e. exactly four lines of symmetry. 48. How can GES be used to show whether a transformation preserves distance? Create various gures to illustrate your techniques. 49. Matisto told his classmates that he obtained gure K from a gure A by using the following two GES transformations: TXY (A) = D and R (0,0),90(D) = K.
K
Y X
a. Copy the grid and use this information to locate gure A on it. Note that X is located at (0, - 1) and that Y is located at (2, 0). b. Is A congruent to K?
ISBN: 0-536-08809-8
50. Lana told her classmates that she got figure P from a gure W by using the size transformation S(0,0),2(W ) = P674
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55. Draw a topological transformation of a map of the main oor of a house with which you are familiar. Would following the gatekeepers instructions in Exercise 54 take you through each room of the house and return you to the front door? 56. The Crossword Puzzle Problem. Crossword puzzles are designed in such a way that, if C is the center of the square, the transformation R C,180, generates an image of the colored squares in the blank puzzle that is exactly the same as those in the original blank puzzle. Copy the following crossword puzzle template and color in at least 16 squares to design a crossword puzzle having that characteristic.
60. If you have access to LOGO computer software, try the following translation: Input numbers for the distance D and the direction angle A and check to see if the following program will translate the gure HALF-ARROW to a new position. Then create a new gure, and try the translation again.
TO SLIDE :A :D HALF-ARROW PU RT :A FD :D PD HALF-ARROW END TO HALF-ARROW FD 25RT 30 BK 10 FD 10 LT 30 BK 25 END
61. If you have access to LOGO computer software, try the following rotation: Input numbers for the distance D from the center and the angle A telling how far to turn, and check to see if the following program will rotate the figure HALF-ARROW to a new position. Then create a new gure and try the rotation again.
TO TURN :D :A HALF-ARROW PU BK :D FD PD FD 2 BK 2 PU RT :A FD :D PD HALF-ARROW END TO HALF-ARROW FD 25RT 30 BK 10 FD 10 LT 30 BK 25
57. The Box Pattern Problem. A manufacturer found that the machines in his factory could make boxes most efciently if the six-square box pattern had four squares in a row and had rotational symmetry. He asked a staff engineer to analyze all 35 possible box patterns and present a report on the patterns that had the needed characteristics. He asked the engineer to give reasons to assure him that all such patterns had been included. Prepare the engineers report for the manufacturer. 58. How would you classify different types of triangles and quadrilaterals according to their symmetry properties? Make a chart to show your classications. 59. How many other pentominos, like the one shown below, have one line of symmetry? Draw a picture of each pentomino.
62. If you have access to LOGO computer software, try the following ip: Input numbers for the distance D and the direction angle A, and check to see if the following program will ip the gure HALF-ARROW to a new position. Then create a new gure and try the ip again.
TO FLIP :A :D HALF-ARROW PU LT :A BK :D RT 90 PD FD 60 BK 120 FD 60 LT 90 PU BK :D RT 180 - :A PD REVERSEHALF-ARROW END TO HALF-ARROW FD 25RT 30 BK 10 FD0 1 LT 30 BK 25 END
TO REVERSEHALFARROW FD 25 LT 30 BK 10 FD 10 BK 30 END
D. Communicating and Connecting Ideas
63. Historical Pathways. In 1872, Felix Klein published Erlanger Programm, a paper that described a new view of geometry as the study of those properties, such as the commutative property of transformation combination, that are preserved by certain types of transformations. In your group, answer the following questions and devise a way to support your answer675
a. b. c. d.
Does TAB*TPQ = TPQ*TAB? Does Mr*Ms = Ms*Mr? Does RA,180*RB,180 = RB,180*RA,180? Does RC,a*RC,b = RC,b*RC,a?
64. Making Connections. Write conjectures about the connections between the following: a. Reections and line symmetry b. Rotations and turn symmetry
Section
11.2
Geometric Patterns and Tessellations Polygons That Tessellate Combinations of Regular Polygons That Tessellate Other Ways to Generate Tessellations Tessellations with Irregular or Curved Sides
In this section, we explore tiling patterns that consist of geometric gures that t without gaps or overlaps to cover a plane. We use characteristics and properties of geometric gures to decide which polygon shapes can be used to tile a oor by themselves and which combinations of polygon shapes can be used to tile a oor. We also explore tiling with irregular gures.
Geometric Patterns
Essential Understandings for Section 11.2
Some shapes or combinations of shapes can be put together without overlapping to completely cover the plane. There are a nite number of ways one type of regular polygon or a pair of types of regular polygons can be put together to completely cover the plane.
Mini-Investigation 11.4 gives you an opportunity to look at the general idea of tiling a oor and which polygons can be used to do so.
Draw tilings by using tracing paper to convince someone that your solution is correct. Verify your solution in another way and describe tilings you have seen in real-world situations.
M I N I - I N V E S T I G A T I O N 11 . 4
Solving a Problem
Which of the tiles shown can a manufacturer advertise correctly as tiles that can be used by themselves to tile a rectangular oor? (Note: Partial tiles may be used at the sides to exactly ll the rectangle.)
60 90 108 120
ISBN: 0-536-08809-8
Equilateral triangle
Square
Regular pentagon
Regular hexagon676
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EXTENDING GEOMETRY
(a) F I G U R E 11 .1 3
(b)
Interesting geometric patterns.
Source: (a) Weyl, Herman, Symmetry. Copyright 1952 Princeton University Press. Reprinted by permission of Princeton University Press; (b) Adapted from Branko, Grnbaum, and G. C. Shephard, Tilings and Patterns An Introduction, p. 5. Copyright 1987 by W. H. Freeman and Company. Used with permission.
Geometric Patterns and Tessellations
From oor tiling to Escher art, the world is full of geometric patterns. The Greek wall pattern shown in Figure 11.13(a) is an example of a design repeating some basic element in a systematic manner, commonly known as a pattern. Another example is the interesting Mexican strip pattern shown in Figure 11.13(b). Note the translations in both patterns: the reectional symmetry in the top part of the pattern in Figure 11.13(a), and the rotational symmetry in Figure 11.13(b). We now dene a special type of pattern.
Denition of a Tessellation
A tessellation is a special type of pattern that consists of geometric gures that t without gaps or overlaps to cover the plane.
Figure 11.14 shows an example of a tessellation. Because this pattern involves the use of regular octagons and squares, we say that a combination of these gures will tessellate the plane or, more simply, will tessellate. Sometimes the word tiling is used to mean tessellation, and the word tile is used to mean tessellate. In Mini-Investigation 11.4, you may have found that an equilateral triangle will tessellate the plane. As indicated by the tessellations in Figure 11.15, the gures in a tessellation can be curved and do not need to be polygons.
F I G U R E 11 .14
Example of a tessellation.
Source: Adapted from Branko, Grnbaum, and G. C. Shepard, Tilings and PatternsAn Introduction, p. 5. Copyright 1987 by W. H. Freeman and Company. Used with permission.
F I G U R E 11 .1 5
Curved tessellations.
Source: Adapted from Branko, Grnbaum, and G. C. Shepard, Tilings and PatternsAn Introduction, p. 5. Copyright 1987 by W. H. Freeman and Company. Used with permission677
Connection to the PreK8 Classroom
Analyzing tessellations enhances students understanding of the various properties of different polygons, including angle measures, congruences, and symmetries. Creating tessellations provides students with an interesting context in which to apply their knowledge of translations, rotations, and reections.
Polygons That Tessellate
The process of nding which polygons will tessellate the plane is an example of what makes mathematics challenging and interesting. It also enables us to broaden our ideas about polygons and transformations by applying them in analyzing tessellations. Lets take a brief look at part of this process. In Mini-Investigation 11.4, you may have drawn the correct conclusion that an equilateral triangle, a square, and a regular hexagon can each be used to form a tessellation. Such a tessellation, made up of congruent regular polygons of one type, all meeting edge to edge and vertex to vertex, is called a regular tessellation. Now lets consider tessellations involving nonregular polygons. Because the sum of the measures of the angles of a triangle is 180 and the sum of the measures of the angles of a quadrilateral is 360, we can show that these types of gures can be made to t around their vertices to tessellate the plane. We conclude that any triangle or quadrilateral will tessellate the plane. A natural extension of that idea is to investigate whether pentagons and hexagons tessellate the plane. You may have found in Mini-Investigation 11.4 that a regular pentagon does not tessellate the plane but that a regular hexagon will. Some nonregular pentagons and hexagons, such as regions A and B in Figure 11.16, will also tessellate the plane. No convex polygon with more than six sides will tessellate.
B
F I G U R E 11 .1 6
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A
Nonregular pentagons and hexagons that tessellate.
Source: ODaffer/Clemens, Geometry: An Investigative Approach, Second Edition, Figure 4.15 from p. 95, 1992. Reprinted by permission of Pearson Education, Inc678
C H A P T E R 11
EXTENDING GEOMETRY
In Mini-Investigation 11.5 you are asked to verify a generalization about which regular polygons will not tessellate the plane.
Write a convincing argument in support of your conclusion.
M I N I - I N V E S T I G A T I O N 11 . 5
Using Mathematical Reasoning
How could you use the fact that the measure of an interior angle of a regular hexagon is 120 to convince someone that no regular polygon with more than six sides will tessellate the plane?
Combinations of Regular Polygons That Tessellate
Piet Hein has stated, We will have to evolve problem-solvers galore, for each problem they solve creates ten problems more [Hein (1969), p. 32]. In this spirit, the question about whether combinations of regular polygons can tessellate the plane naturally arises from the question about single-polygon regular tessellations that we answered earlier. Figure 11.17 depicts two combinations of regular polygons that do tessellate the plane. The pattern of polygons in Figure 11.17(a) is a special kind of tessellation, dened as follows.
Denition of a Semiregular Tessellation
A tessellation formed by two or more regular polygons with the arrangement of polygons at each vertex the same is called a semiregular tessellation.
The tessellation in Figure 11.17(b) has two different arrangements at vertices and is not a semiregular tessellation. Because a semiregular tessellation is a special kind of tessellation, another natural question arises: How many different semiregular tessellations are there? This question is explored more fully in ODaffer and Clemens (1992), in which they established that there are only 21 ways to arrange regular polygons around a point, with no gaps or overlaps. Of these 21 ways, only 8 can be extended to form a semiregular tessellation. Thus we conclude that only eight semiregular tessellations are possible. These tessellations are shown in Figure 11.18.
44 333 44 333 333 44 (a) F I G U R E 11 .17 34 A 3 3B 4
34 4A 6
(b)
ISBN: 0-536-08809-8
Two types of tessellations involving the combination of regular polygons679
(a)
(b)
(c)
(d)
(e) F I G U R E 11 .1 8
(f)
(g)
(h)
The eight possible semiregular tessellations.
The symbol 4, 8, 8 or 4, 82 can be used to describe tessellation (h) in Figure 11.18 because the arrangement around each vertex is square, octagon, octagon. The other tessellations can be symbolized in a similar manner. Thus we symbolize tessellations a), c), and f ) in Figure 11.18 as 34, 6; 32, 4, 3, 4; and 3, 122. The problem in Example 11.5 can be solved using ideas about semiregular tessellations. The problem-solving strategies make a model, draw a diagram, and use reasoning are helpful when solving the problem.
Example
11.5
Problem Solving: The Ancient Temple Floor
A oor restoration specialist was called upon to restore the oor tile in an ancient temple. He knew that the oor had been tiled only with equilateral triangles and regular hexagons and was a semiregular tessellation. The only remaining section of original tile is shown. Can the specialist use this information to restore the oor to its original appearance?680
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Working Toward a Solution
Understand the problem What does the situation involve? What has to be determined? What are the key data and conditions? What are some assumptions? Develop a plan What strategies might be useful? Are there any subproblems? Should the answer be estimated or calculated? What method of calculation should be used? Implement the plan How should the strategies be used? Restoring a tiled oor in an ancient temple What the original tiling looked like The only remaining section of tile is pictured. Only equilateral triangles and hexagons were used in the semiregular tiling. Assume that the tiling must be formed by extending the available section. Use a model; draw a diagram; use reasoning Deciding what arrangements will t around a point No estimation or calculation is needed. Not applicable Use a model with cutout triangles and hexagons, or a diagram to extend the arrangementfor example, the following:
Then, use reasoning to decide whether this arrangement is a semiregular tessellation. What is the answer? Look back Is the interpretation correct? Is the calculation correct? Is the answer reasonable? Is there another way to solve the problem? (Continue this process in Your Turn to look for other tessellations and solve the problem.) Yes. The diagram meets the conditions. Not applicable Look for all possibilities and decide. We could nd the number of degrees in the angles of the polygons and use these measures to look for different ways to arrange them around a point.
YOUR TURN
Practice: Complete the solution to the example problem. Reect: What if the example problem didnt specify that the oor tiling was a semiregular tessellation? Would that change the solution? Explain.
Other Ways to Generate Tessellations
Using a Kaleidoscope. A three-mirror kaleidoscope provides a model that uses reections to create tessellations of the plane. To make a three-mirror kaleidoscope, you can fasten three mirrors of the same size together to form the vertical faces of an equilateral triangular prism. When you place an equilateral triangle piece of construction paper with a pattern on it in the base of the kaleidoscope and peer over the edge, as in Figure 11.19, an interesting geometric pattern appears. If we investigate the semiregular tessellations further, we nd that only those with lines of symmetry that form a superimposed tessellation of equilateral triangles681
F I G U R E 11 .1 9
Viewing tessellations through a three-mirror kaleidoscope.
Source: Photo: Farnsworth Kaleidoscope Images.
can be produced with a three-mirror kaleidoscope. The other tessellations can be produced with other types of kaleidoscopes, based on their lines of symmetry. Using Computer Software. The following LOGO procedures will create a portion of the tessellation of hexagons shown in the next gure:
TO HEXAGON :SIDE REPEAT 6 [FD :SIDE RIGHT END 60]
TO TESSHEX PENUP BACK 80 LEFT 90 PENDOWN REPEAT 4 [HEXAGON 30 RIGHT 120 FD 30 LEFT 60 HEXAGON 30 FD 30 LEFT 60] END
Similar procedures can be created easily to produce a portion of a regular tessellation of triangles or squares. In fact, writing LOGO procedures to produce the semiregular tessellations is both possible and interesting. Exercise 46 at the end of this section asks you to do so.
Tessellations with Irregular or Curved Sides
The Moors, who occupied Spain from 711 until 1492, were forbidden by their religion to draw living objects. To compensate, they mastered creative design, as indicated by the drawings of two of the simpler designs, shown in Figure 11.20, with which they decorated the walls of the Alhambra in Granada. M. C. Escher, a Dutch artist whose drawings, including the one shown in Figure 11.21, have long been a special source of enjoyment for people interested in mathematics, made the following remarks about the Moors inuence on his art:
ISBN: 0-536-08809-8
This is the richest source of inspiration I have ever struck: nor has it yet dried up. [A] surface can be regularly divided into, or lled up with, similar-shaped gures (congruent) which are contiguous to one another, without leaving any open spaces. The Moors were past masters of this. [Escher (1960), p. 11]682
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F I G U R E 11 . 2 0
Drawings from the Alhambra.
Source: M. C. Eschers Drawings from the Alhambra. 2006 The M. C. Escher CompanyHolland. All rights reserved.
F I G U R E 11 . 2 1
A classic drawing by M. C. Escher.
Source: M. C. Eschers Day and Night. 2006 The M. C. Escher CompanyHolland. All rights reserved.
Interestingly, techniques are readily accessible for creating curved tessellation art. The basic approach is to begin with a shape that will tessellate and alter it, often using transformations, to produce the effect wanted. In the following subsections, we describe some ways to do so. Replacing Polygon Edges with Curved Lines. The basic gure for the tessellation in Figure 11.22 was created by starting at the midpoint of each edge of the basic quadrilateral, cutting out a piece, rotating it 180 about the midpoint, and taping it back on the original gurecalled a cut and turn method. A similar method uses only tessellating polygons with congruent adjacent sides to make a basic gure for a curved tessellation. As shown in Figure 11.23, it involves starting with a tessellating polygon, cutting out a piece that contains a complete side, rotating it about a corner, and taping it along the adjacent side. Another way to replace each edge of a square or rectangle with a curve is shown in Figure 11.24. In this procedure, the basic gure of the tessellation was created by cutting out a piece that includes a side of the basic rectangle and sliding the piece so that the side683
Tape on here.
Cut out this piece of the quadrilateral. Turn piece about this point.
F I G U R E 11 . 2 2
The cut and turn method.
Rotate around this point Cut out
Cut out Rotate around this point (b) Cut out, rotate, and tape
(a) Start with a square, draw cutout lines. F I G U R E 11 . 2 3
(c) Decorate the figure
The cut and turn method with special polygons.
Rectangle
Slide piece to left
Slide piece up
Tessellating shape
ISBN: 0-536-08809-8
F I G U R E 11 . 2 4
The cut and slide method684
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Design
Decorated design F I G U R E 11 . 2 6
The interior design method.
on the cutout piece coincides with its opposite sidecalled a cut and slide method. Notice how the transformation we have called a translation is involved in this procedure. Making a Design in the Polygon Interior. An appropriate design drawn in the interior of a polygon can produce interesting curved tessellations, as shown in Figure 11.26. When creating such a tessellation, it is helpful to begin with a rough sketch of the tessellation you want to produce and work backward to develop the interior design. Example 11.6 demonstrates the creation of a curved tessellation and then gives you an opportunity to use one of the methods just described.
Example
11.6
Making a Curved Tessellation
Use one of the methods described to make a curved tessellation from a tessellation of equilateral triangles.
SOLUTION
Figure
Decorated figure YOUR TURN
Practice: Use a method not used in the example problem to make a curved tessellation from a tessellation of equilateral triangles.
ISBN: 0-536-08809-8
Reect: What type of curved tessellation can you make with one method you could have used that you cant make with another Tessellations from Transformations
Connection to the PreK8 Classroom
Experiences with tessellations enable elementary school students to integrate art with mathematics, as illustrated in Figure 11.25. They provide an interesting environment in which students can apply what they have learned about polygons and their angles. Geometric design fosters creative thinking and helps students further develop their spatial visualization abilities.
What idea about
tessellations of polygons is this activity based upon?
Use the rst example to
explain why this procedure works to produce a gure that tessellates the plane.
What transformation is
used in carrying out this procedure?
F I G U R E 11 . 2 5
Excerpt from a sixth-grade mathematics textbook
ISBN: 0-536-08809-8
Source: Scott ForesmanAddison Wesley Mathematics, Grade 6, p. 516. 2004 Pearson Education. Reprinted with permission.
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Problems and Exercises
for Section 11.2
A. Reinforcing Concepts and Practicing Skills
1. Describe or draw an arrangement of polygons that has a characteristic that prohibits it from being a tessellation. 2. Describe or draw an arrangement of polygons that has a characteristic, different from the one selected in Exercise 1, that prohibits it from being a tessellation. 3. How many equilateral triangles will t about a point when tessellating the plane? How do you know? 4. How many rectangles will t about a point when tessellating the plane? How do you know? 5. How many quadrilaterals will t about a point when tessellating the plane? How do you know? 6. Use the idea that all parallelograms tessellate to show that the following triangle tessellates the plane:
16. Give an example of a regular polygon, other than a pentagon, that wont tessellate the plane. 17. Could you tile a oor by using only pentagons such as the ones shown? Use tracing paper and show enough of the tessellation to convince someone of your conclusion.
7. How many different types of regular tessellations are there? Describe them. 8. Give an example of a tessellation made of squares that is not a regular tessellation. 9. Give an example of a tessellation made of equilateral triangles that is not a regular tessellation. 10. Why does a regular pentagon not tessellate the plane? 11. Are the curved tessellations in Figure 11.15 regular tessellations? Explain. 12. Will a rhombus tessellate the plane? How do you know? 13. Will the nonconvex quadrilateral shown in the next gure tessellate the plane? Devise a way to convince someone that your answer is correct.
(a)
(b)
18. Why wont a regular octagon tessellate the plane by itself? Describe a combination of a regular octagon and another regular polygon that will tessellate the plane. 19. Use symbols to represent semiregular tessellations (b), (d), and (g) in Figure 11.18. 20. How many different semiregular tessellations are there? 21. Why is the tessellation shown below not a regular tessellation?
22. Why is the tessellation shown not a regular tessellation? 14. Why is it impossible for a regular polygon with more than six sides to tessellate the plane? 15. If this pattern were placed in a three-mirror kaleidoscope, what tessellation do you think would be produced? Describe the tessellation as accurately as possible, and draw a picture to explain your thinking687
23. Why is the tessellation shown not a semiregular tessellation?
tessellate the plane. Draw an example of each of these types of tessellating gures. 31. Make a curved tessellation from one of the following geometric gures: a. A tessellation of equilateral triangles b. A tessellation of squares c. A tessellation of general quadrilaterals 32. A vertex gure is formed by connecting the midpoints of the edges that form the vertex of a tessellation, as shown in the following gure:
24. Why is the tessellation shown not a semiregular tessellation? A vertex gures tessellation is created by drawing all the vertex gures of a tessellation. Draw the vertex gures tessellation for the tessellation of regular hexagons. Describe this tessellation. 33. Adapt the LOGO procedure on p. 681 to produce a portion of a tessellation of a. squares. b. equilateral triangles. 25. Draw the arrangement of regular polygons around a vertex described by 6, 6, 3, 3. Explain why it cant be extended to form a semiregular tessellation of the plane. 26. What type(s) of symmetry, if any, is present in the completed tessellation suggested in Figure 11.22? Describe the symmetry as accurately as possible. 27. What type(s) of symmetry, if any, is present in the completed tessellation suggested in Figure 11.24? Describe the symmetry as accurately as possible. Figures for Exercise 28
B. Deepening Understanding
28. Trace each tessellation shown in the gure for this exercise, recognizing that it actually covers the entire plane. In how many ways can you slide, turn, and ip each tessellation to match the original tessellation? Devise ways to describe these slides, turns, and ips. 29. The dual of a tessellation is a tessellation created by connecting the centers of neighboring polygons in the tessellation. a. Draw the dual of the tessellation of regular hexagons. b. Draw the dual of the tessellation of equilateral triangles. c. Describe any relationships that you see between the results of parts (a) and (b). 30. Every pentagon with a pair of parallel sides and every hexagon with three pairs of equal parallel sides will
(a)
ISBN: 0-536-08809-8
(b688
C H A P T E R 11
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34. Write a LOGO procedure that will produce a gure consisting of an equilateral triangle and a square with a common side. Such a conguration might be used as a part of a procedure to produce the semiregular tessellation 33, 42.
40.
C. Reasoning and Problem Solving
35. Of the 12 types of pentomino, nd at least 2 that will tessellate the plane. Show a portion of the tessellation. 36. Of the 35 hexominos, nd at least 2 that will tessellate the plane and show a portion of the tessellation. 37. A heptiamond is a gure formed by seven connected, nonoverlapping, and congruent equilateral triangles, each touching others only along a complete side. Of the 24 heptiamonds, all but 1 will tessellate the plane. Choose a heptiamond and show how to tessellate the plane with it. 38. Which tessellations should the following design make when placed in a three-mirror kaleidoscope? Explain.
41.
42.
A demi-regular tessellation is a tessellation of regular polygons that has exactly two or three different polygon arrangements about its vertices. There are 9 demiregular tessellations with exactly two vertex arrangements, and 5 that have exactly three vertex arrangementsa total of 14 demiregular tessellations in all. Four demiregular tessellations are shown in exercises 3942. List all their vertex arrangements. 39.
43. Is the following tessellation a semiregular or a demiregular tessellation? Explain why689
44. This tessellation of pentagons was made by rst drawing a tessellation of hexagons and then dividing each hexagon into ve congruent pentagons. Trace the tessellation of hexagons that was used, and use observation to give some of the characteristics of the hexagon.
48. The Kaleidoscope Problem. A geometer made a kaleidoscope out of three mirrors put together to form a 454590 right triangle. He asserted that he could put a pattern in the kaleidoscope that would produce a semiregular tessellation. His colleague disagreed. Who was correct, the geometer or his colleague? Justify your conclusion. 49. The famous tessellation shown was created by Johannes Kepler. Describe the types of gures that make up the tessellation.
45. The following are instructions showing how to create a tessellation of nonregular hexagons. Use this method to create a tessellation of nonregular hexagons that have opposite sides parallel and congruent. Describe your technique. (See gure below.) 46. How would you instruct the LOGO turtle to have it produce semiregular tessellation (b) in Figure 11.18? a. Write a paragraph describing the instructions that you would give the turtle. b. Use GES or a computer drawing program to create a tessellation. What functions of the technology did you nd most useful in completing your tessellation? 47. The Roman Bath House Tiling Problem. A oor restoration specialist was called upon to restore the tile oor in a Roman bath house. She knew that the oor had been tiled only with equilateral triangles and squares and that it was a semiregular tessellation. The only remaining pieces of the original tile are shown.
D. Communicating and Connecting Ideas
50. The following pattern shows what appears to be a tessellation of squares, regular pentagons, regular hexagons, regular heptagons, and regular octagons. How would you convince someone that it is a fake tessellation of regular polygons?
Can you help the specialist by drawing a diagram of what the original tiled oor looked like? Is there more than one possibility? Explain. Figure for Exercise 45
M M
ISBN: 0-536-08809-8
Start with any parallelogram
Create two new sides on each of a pair of opposite sides.
Form a tessellation by sliding, and rotating 180 about the midpoint, M, of a side690
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51. Discuss what is communicated to you by a tessellation such as the one shown in Exercise 49 or by an Escher design such as the one shown in Figure 11.21. How do you feel when you see it? What message, if any, does it convey to you about mathematics? 52. Historical Pathways. The following pattern is a marble design found in a thirteenth-century Roman church:
the vertical and horizontal lines are squares on the hypotenuse of the right triangle. How does this relate to the puzzle gure in Figure 10.34(b), p. 623?
a. Is it a tessellation? Why or why not? b. Describe at least two lines of symmetry of the design. 53. Making Connections. Illustrate the connection between tessellations and the Pythagorean Theorem by explaining how the dissection of the following tessellation by the vertical and horizontal lines proves the Pythagorean Theorem. Assume that the squares that make up the tessellation are the squares on the legs of a right triangle and that the squares formed by
Section
11.3
Special Polygons
Golden Triangles and Rectangles Star-Shaped Polygons
Star Polygons
In this section, we examine some special topics that involve polygons. We explore and generalize some properties of the pentagram, the golden rectangle, and star polygons.
Essential Understandings for Section 11.3
Polygons shaped like stars can be uniquely described by their sides and angles. Special triangles and rectangles can be described uniquely by the ratios of pairs of their sides691
Mini-Investigation 11.6 gives you an opportunity to make some interesting historical connections.
Talk about why the ancient Greeks, who chose the pentagram as a sacred symbol because of its special beauty, called ^BFG a golden triangle.
M I N I - I N V E S T I G A T I O N 11 . 6
Making a Connection
How many different pairs of segments can you nd in the following pentagram with a ratio of lengths equal to the golden ratio, F 1.618?
B x x F x G x
Estimation Application
Golden Shape The golden rectangle, in which the ratio of its length to its width is approximately 1.618, is purported to be the most aesthetically pleasing of all rectangular shapes. Choose two rectangular shapes in your environment that you think are the closest to golden rectangles, and estimate their golden ratios. Measure to see how good your estimates were.
A 1 B A
C
J I
H
E
D
Golden Triangles and Rectangles
In the pentagram shown in Mini-Investigation 11.6, an isosceles triangle that forms one of the points of the star, such as ^BFG, is called a golden triangle because the ratio of its longer side to its shorter side is the golden ratio, f = (1 + 15)> 2 L 1.618. We know that the measure of an interior angle of the pentagram, such as ABC, is 108, so we can conclude from the equations 2x + a = 108, 4x + a = 180, and b = 2x, that x = a = 36 and b = 72. Using this information, try to identify 9 other golden triangles in the pentagram. The golden triangle is interesting but hasnt received as much attention as the often-discussed and used golden rectangle. Mathematicians, artists, architects, and others have long considered the golden rectangle an especially pleasing formin nature, in buildings, and in works of art. To construct a golden rectangle, we start with a unit square, as in Figure 11.27(a), and nd the midpoint M of one side. Next, we consider MB. Then, as shown in Figure 11.27(b), we use M as the center and MB as the radius and strike an arc on the extension of DC at E. Finally, we construct EF DE and complete the golden rectangle AFED. To verify that the ratio of the length to the width of rectangle AFED is the golden ratio, and that AFED is a golden rectangle, we use the Pythagorean Theorem and show that d = 15 . It then follows that DE = 1 + 15 = (1 +2 15) . 2 2 2 This length of DE is approximately (1 + 22.236) , or 1.618, and AFED is a golden rectangle. Note that 1.618 is an approximation of the golden ratio, referred to on p. 563.
1
D
M (a) 1 B
C
A
F
1
d
1
D
1 2
M
1 2 (b)
C d
E
ISBN: 0-536-08809-8
F I G U R E 11 . 2 7
Construction of a golden rectangle692
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V1
Star Polygons
Using Circles to Produce Star Polygons. The pentagram shown in MiniInvestigation 11.6 leads to consideration of another interesting type of polygon, called a star polygon. Star polygons are frequently used in advertising logos, artistic designs, quilts, and other decorative situations. The star polygon shown in Figure 11.28 is constructed from ve equally spaced points on a circle. Beginning at V1 , we go in one direction and draw a segment to every second point, returning to V1 . The segments drawn to form the star polygon are V1V3 , V3V5 , V5V2 , V2V4 , and V4V1 . Because the star polygon shown in Figure 11.28 was constructed from ve points with segments connecting every second point, it is denoted {5}. This star polygon is a nonsimple polygon having ve ver2 tices with angles of equal measure and ve congruent sides and is called a regular star polygon.
V5
V2
V4
V3
F I G U R E 11 . 2 8
A regular star polygon.
Roman Soumar/CORBIS
Mini-Investigation 11.7 lets you explore the properties of star polygons and make some generalizations.
Talk about which are regular polygons and which are the same star polygonand why. Technology Extension: If you have access to LOGO, use it to construct the star polygons with eight points and formulate possible generalizations.
M I N I - I N V E S T I G A T I O N 11 . 7
Finding a Pattern
What possible relationships can you discover by constructing the following star polygons? a. E 8 F 4 b. E 8 F 6 c. E 6 F 2 d. E 6 F 7
ISBN: 0-536-08809-8
Mini-Investigation 11.7 and the analysis of the star polygons with eight points suggest the following generalizations693
Generalizations About Star Polygons
n = the number of equally spaced points on the circle. d = the dth point to which segments are drawn.
n E n F is the same as the star polygon E n - d F . d n The gure produced for E n F and E n - 1 F is not a star polygon, but is a regular n-gon. 1
The star polygon
The n-sided star polygon are relatively prime.
E n F exists if and only if d Z 1, d Z n - 1, and n and d d
If fraction a is equivalent to a lowest terms fraction n , where E n F is a star polygon, b d d then E a F , sometimes called an improper representation of a star polygon, b represents an n-sided star polygon.
To illustrate the generalizations for star polygons, note, from the rst bulleted statement, that E 5 F is the same star polygon as E 5 F , even though they are 2 3 produced differently. From the second bulleted statement, we see that E 5 F and E 5 F 1 4 are each regular pentagons. In the third and fourth bulleted statements, we see that E 7 F ts all the requirements and represents a seven-sided star polygon, and that 3 E 14 F is an improper (taken from the improper fraction terminology) representa6 tion of a seven-sided star polygon. In subsequent discussion, we will omit the improper designation, and simply agree that symbols such as E 7 F and E 14 F repre3 6 sent the same star polygon. In Mini-Investigation 11.7, you may have discovered some other interesting results when you used the procedure suggested by the notation E n F . The following generald izes one possible discovery and gives some examples. To interpret E a F write the fraction a in lowest terms, say n , and determine what b b d gure E n F produces. E a F produces the same gure, but in a different way. So E a F can d b b be a star polygon, a regular polygon, and in the special case of E 2 F , a line segment. 1 For example, using E 16 F , we see that 16 is equivalent to 8 , and since E 8 F is an 6 6 3 3 eight-sided star polygon, E 16 F is also a representation of an eight-sided star 6 polygon. As a second example, consider E 9 F and E 9 F . 9 is equivalent to 3 , and 9 is 6 3 6 2 3 equivalent to 3 . Since E 3 F and E 3 F are triangles, we might say that E 9 F and E 9 F also 1 1 2 6 3 represent triangles. Finally, since 4 is equivalent to 2 , and E 2 F produces a line 2 1 1 segment, E 4 F also represents a line segment. 2 The following Web site has a very useful little device that produces star polygons. You insert the values for n and d, and it automatically draws the star polygon: In Exercise 40 at the end of this section, you study star polygons further. In Example 11.7, we apply some of the ideas about star polygons.
Example
11.7
Identifying Star Polygons
Describe the star polygons having seven vertices and verify that all of them have been identied.
ISBN: 0-536-08809-8
SOLUTION
There are two star polygons with seven vertices, namely,
E 7 F and E 7 F , as follows: 2 3694
C H A P T E R 11
EXTENDING GEOMETRY
Note that E 7 F and E 7 F form regular polygons, not star polygons; E 7 F and E 7 F are 1 6 5 4 the same star polygons as those shown here. So there are only two star polygons with seven vertices.
YOUR TURN
7 2
7 3
Practice: Describe the star polygons with nine vertices and verify that you have identied all of them. Reect: Explain how the process for producing star polygon E 7 F is different 2 from the process for producing the star polygon E 7 F even though the nal pictures 5 are the same.
Dent angle
30
Star-Shaped Polygons
Point angle
(a) 360 n 120 60
(b) F I G U R E 11 . 2 9
Examples of star-shaped polygons.
If we stop after only considering star polygons that can be produced by sequentially connecting points on a circle with line segments, we miss a lot of very interesting star-shaped gures. For example, a quiltmaker might want to use star-shaped gures like those shown in Figure 11.29. To compare these six-pointed gures with six-pointed star polygons, we observe that E 6 F and E 6 F produce regular hexagons, E 6 F and E 6 F produce triangles, and E 6 F 1 5 2 4 3 produces a straight line. Thus the polygons shown in Figure 11.29 are not star polygons, nor can they be produced by connecting points on a circle with line segments and then erasing the interior parts of the segments. However, we can use the term star-shaped polygon to describe a nonconvex symmetric gure like the one shown in Figure 11.29(a) or (b) that isnt a star polygon. Star-shaped polygons have n star-tip points, 2n congruent sides, n congruent point angles with measure a, and n congruent dent angles b such that b = A 360 B + a, or a = b - A 360 B . Exercise n n 48 at the end of this section asks you to verify this relationship. A six-pointed, starshaped polygon with point angle 30 is denoted by 630 . Figure 11.30 compares a star polygon and star-shaped polygon. It illustrates the importance of precise denitions, which are required to differentiate two different but closely related ideas. Although the basic shape of the ve-pointed stars shown are the same here, that isnt always the case when an n-pointed star polygon and an n-pointed star-shaped polygon are compared. This information about the relationship between dent angles and point angles is useful in constructing star-shaped polygons that meet our specications. For example, if you want to produce a star-shaped polygon with a specic point angle for a special quilt, you can quickly calculate the measure of the dent angle for that gure. Conversely, if you want to make a star-shaped polygon with a specic dent angle for tessellation or other695
5 (a) Star polygon 2 Nonsimple, nonconvex 5 vertices 5 sides 5 point angles F I G U R E 11 . 3 0
(b) Star-shaped polygon 5 36 Simple, nonconvex 10 vertices 10 sides 5 point angles
Comparison of a star polygon and a star-shaped polygon.
design purposes, you can quickly calculate the measure of the point angle for that gure. Example 11.8 demonstrates how to construct a desired star-shaped polygon.
Example
11.8
Problem Solving: The Star Design
An artist wants to make a painting that includes a ve-pointed star-shaped polygon with a fairly thin point angle of 18. How could the artist accurately construct the star-shaped polygon 518 ?
SOLUTION
The measure of the point angle is 18, so the dent angle is construct the star-shaped polygon as follows:
A 360 B + 18 = 90. We 5
a. Plot ve equally spaced points on a circle. b. Connect two of the points, A and B, and construct 45 angles at those two points. Then produce the 90 angle at point D. c. Use a compass to nd the other dent angle points.
B 45 A D 45 45 D B
A
45
YOUR TURN
Practice: Construct a six-pointed star-shaped polygon with a point angle of 30.
ISBN: 0-536-08809-8
Reect: If you want a dent angle of 120, how will you perform the construction to nd the dent angle vertex D696
C H A P T E R 11
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The results in Example 11.9 allow us to investigate how star-shaped polygons can be used in tessellations. The problem-solving strategies draw a picture and use logical reasoning are helpful here.
Example
11.9
Problem Solving: The Quilt
A quiltmaker wanted to make a quilt from six-pointed star-shaped polygons and either squares or equilateral triangles. Is this combination possible? If so, what would the quilt pattern look like?
WORKING TOWARD A SOLUTION
The quiltmaker needs to decide which type of six-pointed star-shaped polygon to use. Drawing a picture and reasoning indicate that the dent angles of the starshaped polygon must be either 90 (to t a square), as shown in part (a) of the next gure or 120 (to t two equilateral triangles), as shown in part (b). Then the quiltmaker can calculate the measure of the point angles, if needed.
120
(a)
(b)
The following quilt pattern meets the conditions of the problem for the starshaped polygon shown in part (a) of the preceding gure. The measure of the point angle is 30.
YOUR TURN
Practice: Draw the quilt pattern for the polygon shown in part (b) of the gure.
ISBN: 0-536-08809-8
Reect: Why cant you use a dent angle of 60, which would hold a single equilateral triangle, to produce a third quilt pattern solution to the problem697
Connection to the PreK8 Classroom
Making geometric designs has benets for elementary school students. It provides an opportunity to learn and extend ideas of geometry. It also provides a path to success for students who may not be as successful in other areas of mathematics. It promotes active involvement and is an excellent activity for integrating mathematics and art.
Problems and Exercises
for Section 11.3
10. How are the processes for constructing star polygons E 5 F and E 5 F alike? How are they different? What can 2 3 you conclude about the resulting star polygons? 11. What is the number of degrees in the point angle of star polygon a. E 5 F ? b. E 8 F ? c. E 9 F ? 2 3 2 For Exercises 1216, tell if the gure is a star polygon, a polygon, or other, and how many sides it has. 12. E 9 F 13. E 8 F 3 3
A. Reinforcing Concepts and Practicing Skills
1. Which of the following is shaped most like a golden rectangle? Explain. a. a 5 * 7 photo b. a 3 * 5 card c. a 20 * 35 picture frame 2. Find an object or part of the place where you live that is close to being a golden rectangle. Measure it to the nearest centimeter and check. What is the ratio of the length to the width? 3. Perform the construction of a golden rectangle described in Figure 11.27. Then, measure to the nearest millimeter and calculate the ratio AF> FE. By how much does the ratio you calculated differ from the golden ratio, which is approximately equal to 1.618? 4. What are the dening characteristics of a golden triangle? 5. Name two pairs of segments in the following pentagram for which the ratio of their measures is the golden ratio.
Q
E 15 F 9 16. E 10 F 5
14.
15.
E 12 F 5
17. Give an example to illustrate that the star polygons E n F 1 n and E n - 1 F , where n is the number of equally spaced points on the circle, are regular polygons. For Exercises 1820, nd the number of degrees in the point angle of each gure. 18. E 5 F 19. E 8 F 20. E 6 F 2 2 2 21. How are star polygons and star-shaped polygons alike? How are they different? 22. How does the number of sides of the star polygon E 9 F 4 differ from the number of sides of the star-shaped polygon 920 ? For Exercises 2326, calculate the measures of the point angles of the star-shaped polygons shown. 23.
P H
D
E
R F
G T S
ISBN: 0-536-08809-8
8. Why is the polygon represented by E 8 F not a star 6 polygon? 9. Describe the polygons produced by interpretation of the following symbols: a. E 8 F b. E 8 F 4 6 c.
6. Describe the measures of the angles of a golden triangle. 7. How many sides does the star polygon E 5 F have? 2
110
E6F 2
d.
E6F 7698 24.
C H A P T E R 11
EXTENDING GEOMETRY
29.
140
70
25. 30.
75 40
26. 31. A wallpaper designer wants to use a star-shaped polygon with eight points and a point angle of 15. What should be the dent angle of this polygon?
40
B. Deepening Understanding
32. A graphics designer wants to use a star-shaped polygon with 10 points and a point angle of 12. What should be the dent angle of this polygon? 33. A quiltmaker thought that a six-pointed star-shaped polygon with a point angle of 30 would go together with squares to make a quilt. Was she correct? Explain. 34. How many star polygons are there with 12 sides? Give symbols for these polygons and verify your answer. 35. How many star polygons are there with 10 sides? Give symbols for these polygons and draw a picture of each. 36. Use GES and inductive reasoning to make a generalization about the sum of the measures of all the point angles in any nonsymmetric ve-pointed star. 37. Use GES and try several different examples to check the generalization about the relationship between the point angle and the dent angle in a star-shaped polygon. 38. For the pentagram shown in the gure in MiniInvestigation 11.6, it has been said that the ratio of the length of the side of the large outer pentagon to the length of the side of the smaller inner pentagon is f2. Do you agree? Support your decision. 39. Construct a star-shaped polygon with eight points and a point angle measure of 45.
ISBN: 0-536-08809-8
For Exercises 2730, calculate the measures of the dent angles of the star-shaped polygons shown. 27.
20
28.
10
C. Reasoning and Problem Solving
40. The number of numbers less than n and relatively prime to n is found by using the formula699
nc1 - a
1 1 b d * c1 - a b d , p1 p2
Figure for Exercise 44
where p1 , p2 , are prime factors of n. Show how to use this formula to help nd the number of star polygons there are with 36 sides. 41. A student who liked algebra claimed that the following was a true statement: If the fraction n is equivalent to d n d n n-d n - d the gure produced for E d F is a regular n gon. Give two examples to illustrate the truth of n-d the statement or a counterexample to show that it is false. 42. Describe three different six-pointed star-shaped polygons. If the area of the smallest is 1, estimate the areas of the others. How might you check your estimates? 43. Use the gure shown and the ideas about the sum of the exterior angles of a polygon to prove that what you discovered in Exercise 36 is true.
(a)
q
g p f e j t d a
b h c i
r
s (b)
ISBN: 0-536-08809-8
44. Analyze the tessellations shown on the right. i. What regular polygons are used? ii. What are the measures of the point and dent angles of the star-shaped polygons that are used? iii. Are all vertices surrounded alike? If not, what different vertex arrangements are there? 45. The Wallpaper Design Problem. A wallpaper designer wants to use the tessellation shown in the gure as the base for some decorative wallpaper. To make the tessellation, he needs to make accurate templates for the 9-gon and the star-shaped polygon. What directions (angle measures, side lengths, and the like) would you give him for making the templates? 46. If the ratio of a to b in the square shown in the gure for this exercise (see p. 700) is the golden ratio, prove that the shaded rectangle is a golden rectangle. (Hint: Use properties of similar triangles.) 47. The ancient Greeks found that the proportion l (l + w) holds only for the length and width of a w= l golden rectangle. If the width in the proportion is 1, show that the length is (1 +2 15) .
Figure for Exercise 45700
C H A P T E R 11
EXTENDING GEOMETRY
Figure for Exercise 46
a b
b a
51. Historical Pathways. The problem of constructing tessellations that utilized star-shaped polygons challenged early Islamic artists. The following design is adapted from Islamic art found in a Russian mosque. Describe the different geometric gures used in this tessellation. If you assume that the ve-pointed star is a pentagram star, draw some conclusions about the angle measures of the other gures.
a b
b
a
48. How would you use the following partial picture of a star-shaped polygon inscribed in a regular n-sided polygon to verify that b = (360) + a, where b is the measure n of the dent angle and a is the measure of the point angle?
360 n x
52. Making Connections. Show a connection between geometry and art by making and coloring a tessellation that uses the following polygons:
x
x
x {418} Regular pentagon
53. Making Connections. Connect algebra and geometry by nding an algebraic expression for the ratio of the length to width of golden rectangle AHCD. Do not use approximate values for square roots.
D. Communicating and Connecting Ideas
49. Discuss how you would produce an accurate drawing of a six-pointed star-shaped polygon with a point angle of 60. Describe your procedure in detail. 50. Work in a small group and devise a procedure to test whether people really do think that the golden rectangle is the most aesthetically pleasing rectangle. Write a paragraph explaining your procedure and the results.
A
B
H
1
1
D
1 2
M
1 2
F
C701
Section
11.4
The Regular Polyhedra Prisms and Pyramids Cylinders, Cones, and Spheres Symmetry in Three Dimensions Visualizing Three-Dimensional Figures
In this section, we classify and dene various types of three-dimensional gures. We also explore properties of these gures, such as symmetry, and relationships involving them. Finally, we describe and analyze different ways to view three-dimensional gures.
Three-Dimensional Figures
Essential Understandings for Section 11.4
All polyhedra can be described completely by their faces, edges, and vertices. There is more than one way to classify most solids. Polyhedra can have reectional or rotational symmetry. Solid gures can be viewed from different perspectives.
The Regular Polyhedra
Vertex Face
Edge F I G U R E 11 . 31
Parts of a polyhedron.
A polyhedron is a collection of polygons (triangles, rectangles, pentagons, hexagons, and the like) joined to enclose a region of space. A polyhedron has faces, edges, and vertices, as shown in Figure 11.31. The prex poly- means many or several and the sufx -hedron indicates surfaces or faces. The ve polyhedra shown in Figure 11.32 can be modeled from cardboard, Styrofoam, or connected drinking straws and are called regular polyhedra. The ancient Greeks used prexes that indicated the number of faces to name these polyhedra. Tetrameans four, octa- means eight, dodeca- means twelve, and icosa- means twenty. Because hexa- means six, a cube may be called a hexahedron. The regular polyhedra shown in Figure 11.32 are also called the Platonic solids because Plato associated earth, air, re, water, and creative energy with these ve solids and used them in his description of the universe. These regular polyhedra might be thought of as basic threedimensional building blocks of geometry because they are the only polyhedra that have edges of equal length and the arrangement of polygons at all vertices the same. A nonregular polyhedron with two different vertex arrangements is shown in Figure 11.33.
A
B Regular tetrahedron
ISBN: 0-536-08809-8
Regular hexahedron or cube
Regular octahedron
Regular dodecahedron
Regular icosahedron Three triangles surround vertex A. Four triangles surround vertex B. F I G U R E 11 . 3 3
F I G U R E 11 . 3 2
Regular polyhedra.
A nonregular poly702
C H A P T E R 11
EXTENDING GEOMETRY
F I G U R E 11 . 3 4
Geometric forms in the radiolarian skeleton.
Source: Darcy W. Thompson, On Growth and Form. In J. T. Bonner (ed.), abridged edition, Cambridge University Press, England, 1971, p. 168. Reprinted with permission of Cambridge University Press.
Evolving scientic theories of the universe long ago replaced Platos description, but nature hasnt ignored the Platonic solids. Skeletons of tiny sea creatures shown in Figure 11.34 and called radiolarians, made of silica and measuring only a fraction of a millimeter in diameter, have the form of the octahedron, icosahedron, and dodecahedron. Also, several mineral crystals and some viruses take the form of these and other polyhedra. Mini-Investigation 11.8 gives you an opportunity to explore further the regular polyhedra.
Write an equation that expresses the relationships you found.
M I N I - I N V E S T I G A T I O N 11 . 8
Finding a Pattern
When you complete the following table and look for a pattern, what relationships among vertices, faces, and edges do you nd?
Polyhedron V F V F E
Tetrahedron Cube Octahedron Dodecahedron Icosahedron
? ? 6 20 12
? ? 8 12 20
? ? 14 32 32
? ? 12 30 30
ISBN: 0-536-08809-8
The formula you discovered in Mini-Investigation 11.8 is called Eulers formula, named after the Swiss mathematician, Leonhard Euler (17071783). Example 11.10 involves the use of this formula703
Example
11.10
Using Eulers Formula
a. Is the polyhedron shown in the next gure a regular polyhedron? Why or why not? b. Does Eulers formula hold for it? Explain.
A
SOLUTION
B
It isnt a regular polyhedron because, although the faces might all be congruent equilateral triangles, the arrangements of polygons at all vertices arent the same: Five sides meet at vertex A, and four sides meet at vertex B. b. In the polyhedron, V = 7, F = 10, and E = 15; 7 + 10 = 15 + 2, so Eulers formula, V + F = E + 2, holds.
YOUR TURN
a.
Practice: Is the second polyhedron shown to the left a regular polyhedron? Why or why not? Does Eulers formula hold for it? Explain. Reect: Do you think that Eulers formula holds for all polyhedra? Explain.
Many different polyhedra can be created simply by slicing off parts of one of the ve regular polyhedra. Figure 11.35 shows the results of such slicing. The idea of
(a) Truncated cube
(b) Truncated octahedron
(c) Cube octahedron
F I G U R E 11 . 3 5
ISBN: 0-536-08809-8
(a) Slicing off the corners of a cube can change the original square faces to regular octagons; (b) slicing off the corners of a regular octahedron can change the original triangular faces to regular hexagons; (c) slicing off the corners of a cube or an octahedron at the midpoint of each edge changes each of these polyhedra to a cube octa704
C H A P T E R 11
EXTENDING GEOMETRY
slicing or taking a cross section of a solid has many practical applications. For example, X-ray tomography is the use of computer graphic techniques to create a three-dimensional object solely from data about planar slices of the object. Contour maps, temperature analyses of materials, and biological analyses are other important uses of slicing. Example 11.11 involves analyzing the polyhedron that results when the faces of a regular polyhedron are sliced in a certain way.
Example
11.11
Analyzing a Sliced Polyhedron
Describe the number and type of faces of the polyhedron formed when every corner of a tetrahedron is sliced as shown to the left. Show that Eulers formula holds.
SOLUTION
The polyhedron has 4 hexagonal and 4 triangular faces. It has 12 vertices, 8 faces, and 18 edges. As 12 + 8 = 18 + 2, Eulers formula, V + F = E + 2, holds.
YOUR TURN
Practice: Complete the example problem by slicing off the corners of a cube. Reect: Do these examples prove that Eulers formula holds for any polyhedron formed by cutting off the corners of another polyhedron?
Prisms and Pyramids
A prism is a polyhedron with a pair of congruent faces, called bases, that lie in parallel planes. The vertices of the bases are joined to form the parallelogram-shaped lateral faces of the prism. Adjacent lateral faces share a common edge called a lateral edge. An altitude of a prism is a segment that is perpendicular to both bases with endpoints in the planes of the bases.
base lateral edge lateral face base Prism
altitude
As shown in Figure 11.36, a prism is named by the shape of its bases. If the lateral edges of a prism are perpendicular to its bases, the prism is a right prism. If the lateral edges are not perpendicular to the bases, the prism is an oblique prism. A pyramid is a polyhedron formed by connecting the vertices of a polygon to a point not in the plane of the polygon. The polygon is called the base of the pyramid, and the point is called the vertex of the pyramid. Since the vertex is also the highest point of the pyramid, relative to its base, it is also called the apex of the pyramid. The remaining faces are triangles and are called the lateral faces of the pyramid. The segment from the vertex perpendicular to the base is called the altitude of the pyramid705
(a) Right square, or rectangular, prism F I G U R E 11 . 3 6
(b) Right triangular prism
(c) Right pentagonal prism
(d) Oblique square prism
Prisms.
vertex
(a) Triangular pyramid F I G U R E 11 . 3 7
(b) Square, or rectangular, pyramid
(c) Pentagonal pyramid
(d) Oblique square pyramid
lateral face slant height altitude base Right Regular Pyramid
Pyramids.
As shown in Figure 11.37, a pyramid is named by the shape of its base. When the base of a pyramid is a regular polygon, the lateral faces are isosceles triangles, and the altitude is perpendicular to the base at its center, the pyramid is called a right regular pyramid. In such a pyramid, the height of an isosceles triangular lateral face is called the slant height. In contrast to the pyramid shown in the cartoon on p. 706, the pyramids built by the ancient Egyptians were spectacular. Mini-Investigation 11.9 suggests that you broaden your perspective on the applicability on Eulers formula.
M I N I - I N V E S T I G A T I O N 11 . 9
Draw a square pyramid and show that Eulers formula holds for it.
Using Mathematical Reasoning
Do you think that Eulers formula holds for prisms and pyramids? Copy and complete a chart like the following to help you investigate this question and support your conclusion.
Name of Polyhedron Number of Number of Number of Base Edges Vertices V Faces F Number of V F E 2? Edges E (Yes or No)
ISBN: 0-536-08809-8
Triangular prism Square prism n-gon prism Triangular pyramid Square pyramid n-gon pyramid
3 ? n 3 ? n
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?706
C H A P T E R 11
EXTENDING GEOMETRY
The New Yorker Collection 1963 Robert Weber from cartoonbank.com. All Rights Reserved.
Cylinders, Cones, and Spheres
If you imagine prisms, pyramids, and polyhedra with thousands of faces, these solid gures come very close to the solids with curved surfaces shown in Figure 11.38. In the cylinder and cone, the shaded circles are called bases. The radius, r, of the cylinder or cone is the radius of its base. The height, h, of a cylinder is the perpendicular distance from one base to the other. The line through the centers of the bases of a cylinder is called the axis of the cylinder. The height, h, of the cone is the perpendicular distance from the vertex, or apex, V, to the base. The line through the center of the base and the vertex is called the axis of the cone. Point O is the center of the sphere, and r is the radius of the sphere. From ice cream cones to cans to architecture
r
V
h
h
O
r
r (a) Circular cylinder F I G U R E 11 . 3 8
r (b) Circular cone (c) Sphere
ISBN: 0-536-08809-8
Cylinders, cones, and spheres707
to the earth and moon, the everyday importance of these building blocks of geometry with curved surfaces is apparent. Along with prisms, pyramids, and other polyhedra, cylinders, cones, and spheres are used to model real objects. Example 11.12 asks you to compare the characteristics of different three-dimensional gures.
Example
11.12
Analyzing Cylinders, Cones, and Spheres
a. How are a cylinder and a prism alike? b. How are a cylinder and a prism different?
SOLUTION
They both have pairs of congruent, parallel bases. Each has a dimension called height. They are both three-dimensional gures. b. The bases of a prism are polygons; the bases of a cylinder are not polygons. A prism has lateral faces that are polygons; a cylinder has a lateral surface that is not made up of polygons.
YOUR TURN
a.
Practice: How are a cone and a pyramid alike? How are they different? Reect: Which polyhedra are most like a sphere? Support your answer.
Symmetry in Three Dimensions
In Section 10.1, we discussed the symmetry of two-dimensional gures. Symmetry is also an important characteristic of some three-dimensional gures. The apple shown in Figure 11.39, in its ideal form, has reectional symmetry. The vertical slice that separates the apple into two symmetric parts goes through a plane of symmetry of the apple. You might think of this plane as a mirror. If half the apple were held against the mirror, it together with its reection would appear to be a whole apple.
ISBN: 0-536-08809-8
F I G U R E 11 . 3 9
Reectional and rotational symmetry in an ideal apple.
Mathematics for Elementary School Teachers, Fourth Edition, by Phares O'Daffer, Randall Charles, Thomas Cooney, John Dossey, and Jane Schielack. Published by Addison Wesley. Copyright 2008 by Pearson Education, Inc.
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C H A P T E R 11
EXTENDING GEOMETRY
(a) Plane of reflectional symmetry, parallel to the faces F I G U R E 11 . 4 0
(b) Plane of reflectional symmetry, containing two edges
(c) Axis of rotational symmetry, through the center of opposite faces
(d) Axis of rotational symmetry, through midpoints of pairs of opposite edges
(e) Axis of rotational symmetry, through opposite vertices
Symmetry of a cube.
Slicing an ideal apple horizontally provides an example of rotational symmetry. The part of the apple showing a star can be rotated 72 about an axis through its center to appear to be in the same position. Thus we say that the piece of apple has 72 rotational symmetry. The apple piece can be rotated through 72 ve times before returning to its original position. Thus we say that the apple piece has rotational symmetry of order 5. The axis through the center represents the axis of rotational symmetry. A cube has several planes of reectional symmetry. One plane of symmetry is parallel to a pair of faces of the cube and intersects edges at only one point, as shown in Figure 11.40(a). Because a cube has three different pairs of faces, there are three of this type of plane of symmetry. Another plane of symmetry contains a pair of edges of the cube, as shown in Figure 11.40(b). As there are six different pairs of edges, there are six of this type of plane of symmetry. A cube also has several axes of rotational symmetry. One type of axis goes through the center of opposite faces, as shown in Figure 11.40(c); there are three of this type of axis. Another type of axis goes through the midpoints of a pair of opposite edges of the cube, as shown in Figure 11.40(d); there are six of this type of axis. A third type of axis goes through opposite vertices of the cube, as shown in Figure 11.40(e); there are four pairs of opposite vertices, so there are four of this type of axis. Visualizing reectional and rotational symmetry properties of the cube is a starting point for analyzing symmetry properties of other solid gures in order to solve practical problems. Example 11.13 shows how to solve a practical problem by applying threedimensional symmetry ideas. The problem-solving strategies draw a diagram or make a model (with drinking straws) or both are helpful when solving this problem.
Example
11.13
Problem Solving: Planning a Sculpture
An artist wants to build a sculpture that includes a tetrahedral object with rods welded to its exterior. She decided to weld the rods so that they would look like extensions of all the axes of symmetry of the tetrahedron. What instructions would help her decide where to weld the rods?
WORKING TOWARD A SOLUTION
ISBN: 0-536-08809-8
One type of axis of rotational symmetry passes through a vertex and the center of a face opposite this vertex, as shown on the left, indicating where some rods should be welded as shown on the facing page709
YOUR TURN
Practice: a. Give additional instructions to tell the artist where rods should be welded. b. Suppose that the artist wanted to weld sheets of metal to show extensions of all planes of symmetry of the tetrahedron. Use the gure on the left to help give her instructions about where to weld them. Reect: A plane of symmetry of a tetrahedron contains an edge and a midpoint of an opposite edge. How many different such planes are there? Explain.
Visualizing Three-Dimensional Figures
One component of spatial sense is representing a three-dimensional gure with a twodimensional picture. There are different types of two-dimensional representations. Visualizing Polyhedra from Their Patterns. A pattern or planar net for a polyhedron is an arrangement of polygons that can be folded to form the polyhedron. For example, each of the patterns shown in Figure 11.41 can be folded to make a cube. They are 3 of 35 possible hexominos, which are made from six squares that are always connected by at least one common side.
F I G U R E 11 . 4 1
Hexomino patterns for a cube.
In Exercise 48 at the end of this section, you are asked to look at pictures of hexominos and visualize how they can be folded to make a cube. Example 11.14 shows how to visualize other polyhedra from patterns.
Example
11.14
Visualizing Polyhedra from Patterns
What polyhedron could be made with the pattern shown on the left?
SOLUTION
If we visualize folding the pattern, we see that a tetrahedron could be formed.
YOUR TURN
Practice: What polyhedra could be made with these patterns?
ISBN: 0-536-08809-8
(a)
(b)
Reect: Do you think that other patterns can be folded to make a tetrahedron? Explain, using an example if possible710
C H A P T E R 11
EXTENDING GEOMETRY
Top End
Side Object
F I G U R E 11 . 4 2
End, top, and side views of an object.
Visualizing Three-Dimensional Figures from Different Views. Architects, engineers, and others often need to represent a three-dimensional real-world object on a two-dimensional piece of paper. To give a total picture of the object as they visualize it, they turn it to show an end view, a side view, and a top view and draw a picture of each view. Figure 11.42 illustrates this approach. Computer programs called computer-aided design (CAD) are now used to create such drawings. Some of these CAD programs can receive a three-dimensional image and create a two-dimensional drawing from it. Also, similar technology can receive a twodimensional drawing and create a three-dimensional image from it. These types of CAD programs are valuable aids to those involved in areas such as architecture, building, and manufacturing. In Example 11.15, we look at an object from three different viewpoints.
Example
11.15
Drawing End, Side, and Top Views
Draw the end view, side view, and top views of the iron casting shown.
SOLUTION
End view
Side view
Top view
YOUR TURN
Practice: Draw the end, top, and side view of the object on the left.
ISBN: 0-536-08809-8
Reect: Can you draw a gure accurately after seeing only the side view and the end view? Explain Different Views of Three-Dimensional Figures
Connection to the PreK8 Classroom
To develop space perception skills, students need to learn to look at actual twodimensional objects, or drawings of objects on a page, and visualize what the object would look like from different points of view. Study the page from a middle school mathematics textbook shown in Figure 11.43 and answer the questions.
Do you need to see all
three views (front, side, and top) in order to determine the original gure? Why or why not?
Can two gures have the
same front, side, and top views? Explain.
When would these views of
a three-dimensional object be useful?
F I G U R E 11 . 4 3
Excerpt from a middle school mathematics textbook.
ISBN: 0-536-08809-8
Source: Page 603 from Scott ForesmanAddison Wesley Middle School Math, Course 1 by R. I. Charles, J. A. Dossey, S. J. Leinwand, C. J. Seeley, and C. B. Vonder Embse. 1998 by Pearson Education, Inc. publishing as Prentice Hall. Used by permission.
711712
C H A P T E R 11
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Visualizing Three-Dimensional Figures and Their Shadows. In the transformations discussed in Section 11.1, the dimensions of an object did not change. Transforming a three-dimensional gure into a two-dimensional shadow involves the use of shadow geometry. In such transformations, straightness is preserved, but length may change, as Trixie discovers with her sunbeam.
HI & LOIS, King Features Syndicate. Example 11.16 asks you to visualize the different shadows that could be made with a given three-dimensional object.
Example
11.16
Connecting Shadows to Three-Dimensional Objects
What shadow gures can you make using a cylinder with a circular base?
SOLUTION
Some possible answers are as follows: a. A circle, with a light source on the line through the center of the circle. b. A rectangle, with a light source on a line perpendicular to and bisecting the segment joining the centers of the bases. c. A nonrectangular parallelogram, if the light source creates a beam that is bisecting but not perpendicular to the segment joining the centers of the bases.
YOUR TURN
Practice: What are some shadow gures you can make with a circular cone? Reect: What shapes of shadows can you not make with a cylinder? Why?
Visualizing Three-Dimensional Figures from Perspective Drawings. From daily observation, you have probably noticed that an object nearer to you appears larger than an object of the same size that is farther away. Not until late in the thirteenth century, however, did artists, many of whom were mathematicians, begin to explore this idea of perspective and develop ways to attain a two-dimensional representation of the actual appearance of three-dimensional objects. One technique for depicting three dimensions on a at surface is to base the orientation of the picture on a vanishing point. In a drawing in which the front surface of the object or scene is in a plane parallel to the plane containing the picture, lines that move away from and should appear parallel to the viewer are drawn to meet at713
(a) Draw the front (b) Choose a vanishing of the box parallel point and use a ruler to the plane of to connect the the paper. vertices of the box to the vanishing point. F I G U R E 11 . 4 4
(c) Draw the back of (d) Erase the lines the box with edges that are no longer parallel to the necessary. corresponding edges of the front of the box.
A rectangular box drawn with one-point perspective.
the vanishing point. Such representations are called perspective drawings and are illustrated in Figure 11.44. Example 11.17 gives you an opportunity to practice using a vanishing point to make a perspective drawing.
Example
11.17
Using a Vanishing Point to Make a Perspective Drawing
Use the technique shown in Figure 11.44 to make a perspective drawing of a hexagonal prism.
SOLUTION
(a) Draw the hexagonal base parallel to the plane of the paper.
(b) Choose a vanishing point and connect it to each vertex of the hexagon.
(c) Draw the other base of the prism with appropriate parallel edges.
(d) Erase unnecessary lines.
YOUR TURN
Practice: Draw the prism from a different perspective by putting the vanishing point in a different place on the paper. Reect:
F I G U R E 11 . 4 5
ISBN: 0-536-08809-8
How would you use this technique to draw a hexagonal pyramid?
A cube represented by a perspective drawing on isometric dot paper.
Another technique for creating perspective drawings is the use of isometric dot paper, as shown in Figure 11.45. With isometric dot paper, the front surface of the object is not parallel to the plane containing the picture714
C H A P T E R 11
EXTENDING GEOMETRY
Problems and Exercises
for Section 11.4
10. A truncated cube 11. A truncated rectangular prism 12. Show that Eulers formula holds for the truncated octahedron and cube octahedron in Figure 11.35(b) and (c). 13. Describe the different planes of reectional symmetry for each of the gures shown in Exercise 7. Assume that the prism in (a) is a right prism with equilateral triangular bases and that the pyramids in (b) and (c) are right regular pyramids. 14. Describe the different axes of rotational symmetry for each of the gures shown in Exercise 7. Assume that the triangular faces of gure (a) are equilateral triangles and the nontriangular face of gure (c) is a square. 15. One type of axis of rotational symmetry and one type of plane of reectional symmetry are shown in the following square prism. How many different planes of symmetry and different axes of rotational symmetry does the square prism have?
A. Reinforcing Concepts and Practicing Skills
1. a. What is the difference between a regular tetrahedron and a nonregular tetrahedron? b. Describe or sketch a nonregular tetrahedron. 2. Complete Exercise 1 for an octahedron. 3. Name some real-world objects that can be modeled by the following geometric gures: a. a cube b. a rectangular prism other than a cube c. a triangular prism d. a pyramid e. a cylinder f. a cone g. a sphere 4. Name or describe some other hexahedrons besides a cube. 5. a. How many pairs of bases does a right rectangular prism have? b. Is there any other prism with more than one pair of faces that can be identied as bases? 6. Make a sketch of the following: a. a circular cylinder with radius of approximately 3 cm and height of approximately 20 cm b. a circular cylinder with radius of approximately 10 cm and height of approximately 3 cm c. a circular cone with radius of approximately 3 cm and height of approximately 20 cm d. a circular cone with radius of approximately 10 cm and height of approximately 3 cm 7. Show that Eulers formula holds for each of the following polyhedra:
(a)
(b)
(c)
Copy and complete the following table:
Polyhedron F V F V E
a. b. c.
? ? ?
? ? ?
? ? ?
? ? ?
Show that Eulers formula holds for the polyhedra described in Exercises 811. 8. A right pentagonal prism 9. A hexagonal pyramid
16. a. Describe the different types of planes of symmetry and axes of symmetry of a regular octahedron. b. How many planes and axes of symmetry does a regular octahedron have? c. Make a sketch and compare the symmetry properties of a regular octahedron and a cube. 17. Draw or describe a pyramid that has ve planes of symmetry. 18. Describe the characteristics that a pyramid must have for it to have an axis of rotational symmetry. 19. Describe the characteristics that a prism must have for it to have an axis of rotational symmetry. 20. A regular tetrahedron has two different types of axes of rotational symmetry and one type of plane of reectional symmetry, as shown in (a)(c) in the gure for this exercise on p. 715. Copy and complete the table (also on p. 715), giving the number of axes of rotational symmetry of each order and the number of different planes of symmetry for a regular tetrahedron. 21. Which of the patterns shown on p. 715 can be folded to make an open-top box? 22. A pentomino is a gure formed by ve connected, nonoverlapping congruent squares, each touching others only along a complete side. Of the 12 different15
pentominos, how many can be folded to make an open-top box? 23. a. Draw a pattern for a rectangular prism that is not a cube. b. Draw another pattern for the same rectangular prism. 24. a. Draw a pattern for a rectangular pyramid. b. Draw another pattern for the same rectangular pyramid. 25. Each solid gure shown in the gure below is sliced by a plane. Identify the cross sections formed.
26. Sue built a gure with ve cubes. The top and side views of the gure look as follows:
Top view
Side view
Draw or describe the gure she made. 27. Draw or describe all the gures you could make by joining ve cubes that would have the following top view:
Table for Exercise 20
Number of Axes of Rotational Symmetry
Polyhedron
Regular tetrahedron
Order 2
Order 3
Order 4
Order 5
Order 6
Total
Number of Planes of Symmetry
Figure for Exercise 20
(a) Axis of rotational symmetry through a vertex and the center of the opposite face
(b) Axis of rotational symmetry through a pair of opposite edges
(c) Plane of rotational symmetry intersecting an edge and the midpoint of the edge opposite that edge
Figure for Exercise 21
Figure for Exercise 25
(a)
(b)
(a)
ISBN: 0-536-08809-8
(b)
(c)
(d716
C H A P T E R 11
EXTENDING GEOMETRY
28. Draw the end, side, and top views of the following object:
29. Describe at least four different shapes of shadows that you can make with a cube. 30. Make a perspective drawing of an isosceles triangular pyramid. Draw it so that all edges appear, using dotted lines if necessary. 31. Use isometric dot paper to draw a perspective representation of a shape that could be made with three cubes.
b. Keeping the front face of the box in the same position, move the vanishing point closer to the upper left corner of your paper and make a perspective drawing. c. Repeat with the vanishing point near the upper right corner of the paper, then near the lower right corner, and nally near the lower left corner. d. Compare the drawings and make a conjecture about the effect that the position of the vanishing point has on the perspective. 39. How could you show a cube from two different perspectives on the isometric dot paper shown in the gure below?
B. Deepening Understanding
32. What is the resulting polyhedron if a. the center points of the faces of an octahedron are connected? b. the center points of the faces of a tetrahedron are connected? 33. How can you form a tetrahedron by connecting selected vertices of a cube? Explain how you know that the edges of this tetrahedron are congruent. 34. Draw and describe one of the six identical polyhedra that are formed when a point at the center of a cube is connected to each of the vertices of the cube. 35. Why are the regular polyhedra good models for dice? 36. Compare the patterns in Exercises 23(a) and 24(a). a. How are they alike? b. How are they different? c. How does each illustrate the denition of prism and of pyramid? 37. An antiprism is like a prism except that the lateral faces are triangles and the bases have been rotated. a. Show that Eulers formula holds for the following square antiprism:
C. Reasoning and Problem Solving
40. Answer the following questions to determine why Eulers formula continues to hold for the polyhedron formed by cutting corners off of an octahedron, as in Figure 11.35(b): a. For an octahedron, V = ____, F = ____, and E = _____. b. When you slice off one corner of the octahedron, you (gain or lose) ____ vertices, (gain or lose) ____ faces, and (gain or lose) ____ edges. c. Therefore, the total change in V is ____; the total change in F is ____; and the total change in V + F is ____. d. The total change in E is ____. e. What does the comparison of the total change in V + F to the total change in E tell you? 41. Use the reasoning from the previous exercise to determine why Eulers formula would hold for the polyhedra formed by slicing off the corners of any prism. 42. Show how to slice a cylindrical piece of cheese into eight congruent pieces with three slices. 43. What do you think the new exposed surfaces will look like when the following cube is separated into two parts by a slice that goes through the marked midpoints of its edges? Justify your answer.
b. How many faces, edges, and vertices would a triangular antiprism have? Does Eulers formula hold? 38. a. Place a vanishing point in about the middle of your paper to make a perspective drawing of a rectangular box17
44. The Toy Blocks Problem. A toy factory manager has eight hundred 8-inch cubic blocks that have been painted red. She wants to produce small blocks by cutting each 8-inch cube into sixty-four 2-inch blocks. To utilize her machines as efciently as possible, she wants to make the fewest number of cuts possible. She needs unpainted blocks to package in a block set and blocks painted on exactly two faces to package in another set. Prepare a report for the manager indicating a. the fewest cuts needed to produce 64 cubes from one block. b. the total number of unpainted blocks and blocks painted on two faces that can be produced. Justify your conclusions in the report. 45. The Polyhedra Sculpture Problem. A sculptor found the following table of information about the dihedral angles (angles formed by adjacent faces) of the regular polyhedra. He knows that polyhedra that will t all the way around an edge with no gaps can be used to make a sculpture he is planning.
Regular polyhedron
Cube Tetrahedron Octahedron Dodecahedron Icosahedron
a. Which deltahedra are also regular polyhedra? b. Draw a picture of a deltahedron with six faces that is not a regular polyhedron. 48. Can every hexomino be folded to make a cube? Explain. 49. Could the following two gures describe actual objects? Explain your conclusions.
(a)
(b)
50. The Dollar Bill Problem. A magician claimed that he could make a tetrahedron out of a $1 bill. He began to fold the bill along the solid lines as shown. Do you believe that he could really do it? Explain your conclusion.
Degree Measure of Dihedral Angle
90 7032 10928 11634 13811 Midpoint
a. Use the table to determine which regular polyhedra could be used in his sculpture. b. Could he use two tetrahedra and two octahedra in combination for his sculpture? Why or why not? 46. Construct a tetrahedron from a sealed envelope, as follows: a. Find point C so that ^ ABC is equilateral. b. Make a cut DE parallel to AB. Discard the other part of the envelope. c. Fold AC and BC in both directions. d. Pinch the envelope to join D and E. e. Tape the opening to make a tetahedron.
A D
51. Without counting all the faces, edges, and vertices, how would you convince someone that Eulers formula still holds for the gure that remains when one corner of a cube is cut off? Make a sketch.
D. Communicating and Connecting Ideas
52. In a small group, discuss differences, if any, between mathematical and everyday meanings of each of the following terms: a. Sphere b. Cube c. Pyramid d. Prism Why is it important to have precise meanings for terms in mathematics? 53. Historical Pathways. The Greek philosopher Plato (430347 B.C.), in his book Timaeus, associated the cube with earth, the tetrahedron with re, the octahedron with air, and the icosahedron with water. He associated the dodecahedron with what was used to create the universe. To develop this cosmology, Plato used the relationships between these polyhedra. Follow the instructions given in order to view two of those relationships. a. Draw a cube as follows: Draw square ADHE in front and a partially dashed square BCGF in back of it.
C
ISBN: 0-536-08809-8
B
E
47. A deltahedron is a polyhedron with equilateral triangle faces718
C H A P T E R 11
EXTENDING GEOMETRY
B A D
C
F E H
G
Join vertices A and B, D and C, E and F, and H and G.
b. Now use a colored pencil to join all midpoints of adjacent faces of the cube. What polyhedron is formed? c. Now draw another cube and use a colored pencil to join vertices B, D, E, and G with all possible segments. What polyhedron is formed? 54. Making Connections. Put a familiar object in a bag and have another person visualize it only by feeling, using names of polyhedra or other geometric ideas to describe it. Discuss the value of appropriate mathematical language in doing this activity.
Chapter Summary
Reections on the Big Ideas and Essential Understandings: Questions and Answers
S E C T I O N 11 .1
What are some different types of transformations? (pp. 654661) An object can be mapped onto any other gure congruent to it through the use of one or more of three types of transformations called isometries: translations, rotations, and reections. How can we use congruence transformations, or isometries, to describe symmetry? ( pp. 661664) When a gure can be traced and folded so that one half exactly coincides with the other half, it has reectional symmetry about the line of symmetry. When it can be turned less than 360 about a xed point called a center of rotational symmetry to t back on itself, it has rotational symmetry. When a gure ts back on itself after a halfturn, it has point symmetry. What kind of transformations can change a gures size? (pp. 665668) A size transformation enlarges or shrinks a gure along lines determined by a point called the center of the size transformation and according to a multiplier called the scale factor. A combination of one or more isometries and a size transformation can map a gure onto any other similar gure. What kind of transformations can change both a gures size and shape? (pp. 668669) Topological transformations preserve the connectedness of curves and surfaces but can change both size and shape. What is meant by geometric patterns and tessellations? (pp. 676677) A design repeating some basic element in a systematic manner is called a pattern. A tessellation is a special type of pattern that consists of
geometric gures that t without gaps or overlaps to cover the plane. Which polygons will tessellate the plane by themselves? (pp. 677678) Equilateral triangles, squares, and regular hexagons are the only regular polygons that tessellate by themselves. Every triangle and quadrilateral will tessellate. Several nonregular pentagons and hexagons will tessellate. No convex polygon with more than six sides will tessellate, but some nonconvex polygons with more than six sides will tessellate. Can combinations of regular polygons tessellate the plane? (pp. 678680) Many different tessellations can be formed with a combination of regular polygons. Of interest are semiregular tessellations, formed by two or more polygons with the arrangement of polygons at each vertex the same. There are eight different semiregular tessellations of the plane. What other interesting ways are there to generate tessellations? (pp. 680681) A three-mirror kaleidoscope or computer software such as GES or LOGO can be used to generate tessellations using translations, rotations, and reections. How can tessellations with irregular or curved sides be formed? (pp. 681685) Techniques include replacing the edges of a tessellating polygon with curved lines (using rotation in the cut and turn or translation in the cut and slide methods) or making a design in the interior of a tessellating polygon. What are some characteristics of golden triangles and rectangles? (p. 691) In both a golden triangle and a golden rectangle, the ratio of its longer side to its shorter side is the golden ratio, f = (1 +2 15) L 1.618.
S E C T I O N 11 . 2
S E C T I O N 11 . 3 SUMMARY
719
What are some generalizations that can be made about star polygons? (pp. 692694) If n = the number of equally spaced points on the circle and d = the dth point that segments are drawn to, (a) the star polygon n {n} is the same as the star polygon {n - d}; (b) the polyd n n gons {1} and {n - 1} are regular polygons; and (c) the n-sided star polygon {n} exists only if d Z 1, d Z n - 1, d and n and d are relatively prime. What are some characteristics of star-shaped polygons? (pp. 694697) Star-shaped polygons have n startip points, 2n congruent sides, n congruent point angles with measure a, and n congruent dent angles b such that b = (360) + a. They differ from star polygons in n that they are simple polygons, whereas star polygons are nonsimple.
S E C T I O N 11 . 4
What are regular polyhedra? (pp. 701704) A polyhedron is a collection of polygons joined to enclose a region of space, forming a gure with faces, edges, and vertices. A regular polyhedrons faces are congruent regular polygons, and the arrangement of these polygons is the same at all vertices of the polyhedron. The ve regular polyhedra include the regular tetrahedron, the cube, the regular octahedron, the regular dodecahedron, and the regular icosahedron. What are some other important polyhedra? (pp. 704705) Prisms and pyramids are important types of polyhedra. A prism has two congruent, parallel bases
joined by quadrilateral lateral faces. A pyramid has one base and triangular lateral faces joined at a common vertex called the apex. What important three-dimensional gures are there other than polyhedra? (pp. 706707) Cylinders, cones, and spheres are three-dimensional gures with curved surfaces, rather than polygonal faces as in polyhedra. Cylinders are like prisms, in that they have two congruent, parallel bases. Cones are like pyramids, in that they have one base and an apex. A sphere might be considered like a regular polyhedron with a very large number of faces. How can we apply the idea of symmetry to threedimensional gures? (pp. 707709) A three-dimensional gure that can be sliced into two congruent parts along a plane of symmetry is said to have reectional symmetry. A three-dimensional gure that can be rotated less than 360 about an axis of rotational symmetry until it matches itself is said to have rotational symmetry. How can three-dimensional gures be visualized? (pp. 709713) Three-dimensional gures can be represented in two dimensions in a variety of ways. A pattern, or planar net, can be used to show an arrangement of polygons that can be folded to form a polyhedron. A three-dimensional object can be turned to show only its end view, its side view, or its top view. Shadows can also produce two-dimensional representations of threedimensional objects. Finally, perspective drawings based on a vanishing point can be created to represent threedimensional gures.
Terms, Concepts, and Generalizations
S E C T I O N 11 .1
ISBN: 0-536-08809-8
Image (p. 655) Slide arrow (p. 655) Directed segment (p. 655) Translation (p. 655) Rotation (p. 656) Reection (p. 657) Glidereection (p. 660) Congruence transformation (p. 660) Isometry (p. 660) Congruence (p. 661) Reectional symmetry (p. 661) Line of symmetry (p. 661) n rotational symmetry (p. 663) Center of rotational symmetry (p. 664) Point symmetry (p. 664) Size transformation (p. 665) Center of size transformation (p. 665) Scale factor (p. 665)
Similarity (p. 666) Topological transformation (p. 668) Topologically equivalent (p. 668)
S E C T I O N 11 . 2
Pattern (p. 676) Tessellation (p. 676) Regular tessellation (p. 677) Semiregular tessellation (p. 678)
S E C T I O N 11 . 3
Golden triangle (p. 691) Pentagram (p. 691) Golden rectangle (p. 691) Star polygon (p. 692) Regular star polygon (p. 692) Star-shaped polygon (p. 694)
S E C T I O N 11 . 4
Polyhedron (p. 701) Regular polyhedra (p. 701)
Eulers formula (p. 702) Prism (p. 704) Right prism (p. 704) Oblique prism (p. 704) Pyramid (p. 704) Right regular pyramid (p. 705) Cylinder (p. 706) Cone (p. 706) Sphere (p. 706) Reectional symmetry (p. 707) Plane of symmetry (p. 707) Rotational symmetry (p. 708) Axis of rotational symmetry (p. 708) Pattern or planar net (p. 709) Shadow geometry (p. 712) Vanishing point (p. 712) Perspective drawings (p. 713)
Mathematics for Elementary School Teachers, Fourth Edition, by Phares O'Daffer, Randall Charles, Thomas Cooney, John Dossey, and Jane Schielack. Published by Addison Wesley. Copyright 2008 by Pearson Education, Inc.
720
C H A P T E R 11
EXTENDING GEOMETRY
Chapter Review
Concepts and Skills
1. Describe the ve regular polyhedra and explain how they are named. 2. Give the dimensions of a picture frame that is a golden rectangle. 3. Show that Eulers formula holds for a square pyramid. 4. Draw two patterns of squares that can be used to form cubes. 5. Draw the front, side, and top views of the following object: 15. Consider the following gure:
a. If it is drawn on a balloon that can be stretched, shrunk, and twisted, draw two different shapes that could be produced. b. If it is made of wire, draw two different shapes that its shadow could take.
Reasoning and Problem Solving
6. Draw a cube in perspective. 7. Draw a tetrahedron in perspective. 8. Draw a sketch of the star polygon {8}. Give another sym3 bold for this same star polygon. Is it a regular polygon? 9. An eight-point star-shaped polygon has a point angle of 36. What is the measure of its dent angle? 10. Describe the symmetry properties of the following gures: 16. If you marked the midpoints of the edges of a cube and sliced off all its corners through the midpoints of its edges, how many and what type of faces would the truncated gure have? 17. Do you believe the following generalization about the measure of a point angle of a star polygon is true? The measure of a point angle of the star polygon E n F is d (|n - 2d|180) . Support your belief in writing, giving n evidence. 18. How do the axes of rotational symmetry of an octahedron compare to the axes of rotational symmetry of a cube?
(a)
(b)
11. How many planes and axes of rotational symmetry does a right rectangular prism have? 12. Name three regular polygons that will tessellate the plane. 13. Draw a small portion of a semiregular tessellation. a. Tell why it is semiregular. b. What is true about a tessellation that utilizes a combination of regular polygons but isnt semiregular? 14. Use tracing paper to nd the image of the quadrilateral shown for the following transformations:
B E D A C
a. TAB 4 c. MAB e. TAB*RB,180
b. R A,90 d. SE,2
19. The Fancy Quilt Problem. Stacy wants to make a quilt that uses a 12-pointed star-shaped polygon that tessellates with equilateral triangles. To make the star, she needs to know the measure in degrees of the point angles and the dent angles of the star-shaped polygon. If possible, supply this information and sketch the quilt pattern. If not, explain what additional information you need in order to do so REVIEW
721
20. a. How are congruence transformations and size transformations alike? b. How are they different? 21. Draw two topologically equivalent gures that look as different as you can make them. Justify why they are topologically equivalent. 22. Measuring Trees Problem. Jared wants an estimate of the height of some of the trees near his lake house. Use ideas of size transformations to design a plan for him to use his height to estimate accurately a trees height without directly measuring the tree. 23. The Open-Top Box Pattern Problem. A manufacturer found that the machines in his factory could make open-top boxes most efciently if the vesquare box pattern had no more than two squares in a row and had reectional symmetry. He could also manufacture the pattern more easily if it would tessellate the plane. Assume that you are the engineer given
the task of analyzing all 12 possible open-top box patterns, nd a pattern or patterns that meet the conditions, and verify that you have found the correct pattern or patterns.
Alternative Assessment
24. Use paper cutouts or GES to make a tessellation based on the following: a. Slide images b. Turn images c. Flip images Write a paragraph for each one, describing your procedure. 25. Choose ve different types of familiar rectangular objects that you estimate to be in the shape of a golden rectangle and analyze them to see how close they are to being a golden rectangle. Devise a technique to rate the closeness of rectangles to golden rectangles. Art1Reading ArtPracticing Iconography Axia CollegeReading Art2Reading ArtPracticing IconographyThe primary role of artists throughout time has been to capture the world and document happenings by painting pictures and portraits. The art that
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Running head: THE PERFECT ERA1Perfect Era Paper Casey Aker University of PhoenixNovember 15, 2009The Perfect Era During the early part of the 18th century, colonial expansion spread colonists westward and south. Because of this expansion many colonist
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Thomas Jefferson and HistoryMaster subtitle style Casey Harris Click to edit11Thomas Jefferson Bio. Born in 1743 in Albemarle county, Virginia Studied at the college of William and Mary and read law He contributed his pen more than his voice At age 33
Chapter1 Introduction:WhatThisBook isAboutCOPYRIGHT 2008 Thomson South-Western, a part of The Thomson Corporation. Thomson, the Star logo, and South-Western are trademarks used herein under license.IntroductoryAnecdote:OilBidding Young geologist prepa
SUMMER SESSION 2006/7 Examination for the BA (HONOURS) ACCOUNTING AND FINANCE PATHWAYS BUSINESS SCHOOL OPTION FINAL YEAR 5B3030 AUDIT AND ASSURANCEJOINT UNIT LEADER/AUTHORS: NEIL RODDA AND GRAHAM HOLT30th April 2007 5.00 pm 8.00 pmInstructions Time all
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06043682Audit And AssuranceYear 3INDEPENDENCE AND THE PROVISION OF OTHER NON-ASSURANCE SERVICES TO ASSURANCE CLIENTS Introduction The intention of this essay is to critically discuss and explicate the fundamental aspects that surround the audit and ass
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IntroductionIn this essay, we are going to examine the auditors role in respect to the services provided by auditors. Auditors tend to work on the ethics and standards put forward by the standard bodies in order to make sure the work carried by auditors
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The following report will explain and critically discuss the issue of auditor independence with regard to the provision of other services. There are two kinds of auditing, that is internal and external (independent) auditing. Internal auditing helps provi
Azeem Mohammad 06128671Audit and Assurance The following essay will explain and critically discuss the issue of auditor independence with regard to the provision of other services. There are two kinds of auditing, that is internal and external (independe |
Content: Group theory takes up about half of Algebra I. We study in
particular groups, subgroups and homomorphisms from one group to
another, with many examples of each idea. Groups occur in nature as the
symmetry groups of geometric or other mathematical or physical
constructions; we treat the notion of an abstract group G acting on a
set S. A key structural element of algebra introduced in this module
is the notion of a normal subgroupH of a group G and the
associated quotient groupG/H. We will also study group
actions. These have many applications including Sylow's theorem, which
we shall see is in some sense a partial converse to Lagrange's theorem.
We next study quadratic forms. A quadratic form is a homogeneous
quadratic polynomial expression
in several variables.
Quadratic forms occur in geometry as the equation of a quadratic cone,
or as the leading term of the equation of a plane conic or a quadric
hypersurface. By a change of coordinates, we can always write q(x)
in
the diagonal form. For a quadratic form over R, the number of positive or
negative diagonal coefficients
ai is
an invariant of the quadratic form which is very important in
applications.
We discuss a square matrix
matrix M as an endomorphism
of a
vector space V. We study Jordan canonical form of 2x2
and
3x3
matrices.
The general
case will be
treated in MA245 Algebra II.
Aims: To provide a further introduction to abstract group theory,
building upon the material in year 1 from Foundations and taking in some
of the classical theorems on finite groups.
To develop upon first year linear algebra, paying particular attention
to canonical forms of linear maps, matrices and bilinear forms.
To make students familiar with some important techniques in linear
algebra and group theory which are used in other modules.
Objectives: By the end of the module students should be familiar with: the
isomorphism theorems for groups and applications; quotient groups;
Cayley's theorem; group actions and lots of applications, including the
class equation and Sylow's theorem; the theory and computations of the
the Jordan normal form of matrices and linear maps; bilinear forms,
quadratic forms, and choosing canonical bases for these. |
Book Description: Inside the Book:Preliminaries and Basic OperationsSigned Numbers, Frac-tions, and PercentsTerminology, Sets, and ExpressionsEquations, Ratios, and ProportionsEquations with Two Vari-ablesMonomials, Polynomials, and FactoringAlgebraic FractionsInequalities, Graphing, and Absolute ValueCoordinate GeometryFunctions and VariationsRoots and RadicalsQuadratic EquationsWord ProblemsReview QuestionsResource CenterGlossaryWhy CliffsNotes?Go with the name you know and trust...Get the information you need—fast!CliffsNotes Quick Review guides give you a clear, concise, easy-to-use review of the basics. Introducing each topic, defining key terms, and carefully walking you through sample problems, this guide helps you grasp and understand the important concepts needed to succeed.Master the Basics–FastComplete coverage of core conceptsEasy topic-by-topic organizationAccess hundreds of practice problems at CliffsNotes.com
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We will provide students with experiences that will enhance their ability to understand mathematics and the mathematical procedures necessary to make informed judgments on issues, to act as wise consumers, and to come to logical determinations as they pertain to personal and professional endeavors.
The purpose of the mathematics curriculum at Assabet Valley is to:
Provide a rigorous and relevant course of studies to meet the needs of all students Prepare all students for all graduation requirements of the Commonwealth of Massachusetts and of Assabet Valley Regional Technical High School Instruct students in how to reason, solve problems and to produce a level of competence required for their high school years and their adult lives Provide a rigorous and relevant course of studies to meet the needs of all students
Our curriculum provides students with the opportunities to develop a foundation from which they can pursue a profession and/or further their education at a higher level, such as two or four year colleges. |
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Understanding Mathematics: From Counting to Calculus is an easy-to-read, non-technical book—beginning with counting and building up to more advanced math topics like calculus. Hundreds of detailed examples, shaded step-by-step boxes, and pictures emphasizing important concepts are included.
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Understanding Mathematics is a comprehensive, easy-to-read, non-technical discussion covering a broad range of math topics ranging from counting to calculus. These math fundamentals, normally taught from first grade through college, are covered in detail with a special emphasis on real world examples and practical applications. Most importantly, this text provides an explanation of both the "why" and the "how" of math. The author conferred with teachers and students for feedback while writing the book, creating an end product that is a concise and easy-to-follow, step-by-step process of studying arithmetic, fractions, percentages, negative numbers, exponents, algebra, solving equations, functions, graphing, geometry, trigonometry, logarithms, complex numbers, calculus, and derivatives. Your student is sure to grasp the fundamentals of all these topics with the many bright illustrations and simple language contained within this 350-page book.
This book is perfect for homeschool families, any student who is struggling with understanding math concepts, and also adults who remain uncomfortable using math or explaining it to others. You will find it extremely useful for many years! Act now and order Understanding Mathematics |
Algebra II
Chapter 8: Sequences and Series - Qest on Friday, May 3. The students explore finding sums (series) of sequences, whether geometric or arithmetic. These can be found in many places in the world, such as building a pattern with bricks, etc. and needing to know how many total bricks are needed for the project.
Chapter 10: Introduction to Limits in Calculus: This includes introducing key vocabulary, working with function continuity, and learning differnt techniques to calculate the limit of a graph at a given value of x. -Quest on Thursday, May 16.
Pre-Algebra
Chapter 10-12: The students will be reviewing topics in Chapters 10-12. We will be practicing these on IXL, an online program that keeps track of students' progress and provides feedback. The IXL standards assigned are listed in the downloads and will be graded on May 15th. The students will be reviewing Surface Area and Volume of 3D figures, Probability & Graphs, Angles, and Transformations. |
Written by best-selling GCSE Maths authors, these are the books that students and teachers love now structured to support the new two-tier GCSE Maths specifications.These are the resources you can trust at a price you can afford! [via]
Mathematics for Edexcel GCSE - Main texts. The Main Texts provide thorough preparation for students on either a 1-year or 2-year course with clear, student-friendly language used throughout. Each chapter consists of: *Fully worked examples with explanatory notes and commentary *Carefully graded exercises *What you need to know - a summary of key facts and skills *A review exercise consisting of exam-style questions which reflect how Edexcel assess the work, plus lots of pass exam questions. Section reviews give further opportunities to consolidate skills acquired over a number of chapters. Exam practice sections contain compilations of exam and exam-style questions organised for non-calculator paper and calculator paper practice, in preparation for the exams. [via] |
More About
This Textbook
Overview
The many diverse articles presented in these three volumes, collected on the occasion of Alexander Grothendieck's sixtieth birthday and originally published in 1990, were offered as a tribute to one of the world's greatest living mathematicians. Grothendieck changed the very way we think about many branches of mathematics. Many of his ideas, revolutionary when introduced, now seem so natural as to have been inevitable. Indeed, it is difficult to fully grasp the influence his vast contributions to modern mathematics have subsequently had on new generations of mathematicians.
Many of the groundbreaking contributions in these volumes contain material that is now considered foundational to the subject. Topics addressed by these top-notch contributors match the breadth of Grothendieck's own interests, including: functional analysis, algebraic geometry, algebraic topology, number theory, representation theory, K-theory, category theory, and homological |
Highlights of Calculus
Highlights of Calculus is a series of short videos that introduces the basic ideas of calculus - how it works and why it is important. The intended audience is high school students, college students, or anyone who might need help understanding the subject.
In addition to the videos, there are summary slides and practice problems complete with an audio narration by Professor Strang. You can find these resources to the right of each video.
View the complete course at:
License: Creative Commons BY-NC-SA
More information at
More courses at
Highlights of Calculus is a series of videos that introduce the fundamental concepts of calculus to both high school and college students. Renowned mathematics professor, Gilbert Strang, will guide students through a number of calculus t...
Calculus is about change. One function tells how quickly another function is changing. Professor Strang shows how calculus applies to ordinary life situations, such as:
* driving a car
* climbing a mountain
* growing to ...
Calculus finds the relationship between the distance traveled and the speed — easy for constant speed, not so easy for changing speed. Professor Strang is finding the "rate of change" and the "slope of a curve" and the "derivative of a f...
At the top and bottom of a curve (Max and Min), the slope is zero. The "second derivative" shows whether the curve is bending down or up. Here is a real-world example of a minimum problem:
What route from home to work takes the shortest...
Professor Strang explains how the "magic number e" connects to ordinary things like the interest on a bank account. The graph of y = e^x has the special property that its slope equals its height (it goes up "exponentially fast"!). This i |
Major Features of the Text
A Balanced Approach: Form, Function, and Fluency
Form follows function. The form of a wing follows from the function of flight. Similarly, the form of an algebraic expression or equation reflects its function. To use algebra in later courses, students need not only manipulative skill, but fluency in the language of algebra, including an ability to recognize algebraic form and an understanding of the purpose of different forms.
Restoring Meaning to Expressions and Equations
After introducing each type of function — linear, power, quadratic, exponential, polynomial — the text encourages students to pause and examine the basic forms of expressions for that function, see how they are constructed, and consider the different properties of the function that the different forms reveal. Students also study the types of equations that arise from each function.
Maintaining Manipulative Skills: Review and Practice
Acquiring the skills to perform basic algebraic manipulations is as important as recognizing algebraic forms. Algebra: Form and Function provides sections reviewing the rules of algebra, and the reasons for them, throughout the book, numerous exercises to reinforce skills in each chapter, and a section of drill problems on solving equations at the end of the chapters on linear, power, and quadratic functions.
Students with Varying Backgrounds
Algebra: Form and Function is thought-provoking for well-prepared students while still accessible to students with weaker backgrounds, making it understandable to students of all ability levels. By emphasizing the basic ideas of algebra, the book provides a conceptual basis to help students master the material. After completing this course, students will be well-prepared for Precalculus, Calculus, and other subsequent courses in mathematics and other disciplines.
Changes Since The Preliminary Edition
NEW — four new chapters on summation notation, sequences and series, matrices, and probability and statistics.
The initial chapter reviewing basic skills is now three shorter chapters on rules and the reasons for them, placed throughout the book.
NEW — Focus on Practice sections at the end of the chapters on linear, power, and quadratic functions. These sections provide practice solving linear, power, and quadratic equations.
NEW material on radical expressions in Chapter 6, the chapter on the exponent rules.
NEW material on solving inequalities, and absolute value equations and inequalities, in Chapter 3, the chapter on rules for equations. |
Visual Linear Algebra
Visual Linear Algebra is a new kind of textbook—a blend of interactive computer tutorials and traditional text. The computer tutorials provide a lively learning environment in which students are introduced to concepts and methods and where they develop their intuition. The traditional sections provide the backbone whose core is the development of theory and where students' understanding is solidified. Although the design of Visual Linear Algebra is novel, the goals for the book are quite traditional. Foremost among these is to provide a rich set of materials that help students achieve a thorough understanding of the core topics of linear algebra and genuine competence in using them.
Tutorials and traditional text. Visual Linear Algebra covers the topics in a standard one-semester introductory linear algebra course in forty-seven sections arranged in eight chapters. In each chapter, some sections are written in a traditional textbook style and some are tutorials designed to be worked through using either Maple or Mathematica.
About the tutorials. Each tutorial is a self-contained treatment of a core topic or application of linear algebra that a student can work through with minimal assistance from an instructor. The thirty tutorials are provided on the accompanying CD both as Maple worksheets and as Mathematica notebooks. They also appear in print as sections of the textbook.
Geometry is used extensively to help students develop their intuition about the concepts of linear algebra.
Applications. Students benefit greatly from working through an application, if the application captures their interest and the materials give them substantial activities that yield worthwhile results. Ten carefully selected applications have been developed and an entire tutorial is devoted to each of them.
Active Learning. To encourage students to be active learners, the tutorials have been designed to engage and retain their interest. The exercises, demonstrations, explorations, visualizations, and animations are designed to stimulate students' interest, encourage them to think clearly about the mathematics they are working through, and help them check their comprehension.
I like the book a lot and in fact have been watching for it ever since I participated in a visual linear algebra workshop. The geometric material and the computer images are very worthwhile. This text raises the bar extremely high for visual treatments.
You can tell that it has profited from lots of pedagocial thinking and discussion by the way topics are introduced and sequenced--very "clean" and appealing.
Frederick Gass, Miami University
This is indeed a new kind of textbook, and it tackles with the difficult task of naturally incorporating computers into the standard linear algebra course in order to enhance student participation and understanding. This reviewer believes that this job has been done very well. |
Yes, it matters. You will have homework assigned out of the current edition, so unless you somehow get a hold of the 2011 edition's problem sets, you're not going to have the same problems, in the same order, which matters for homework. (sometimes the problem is the same but the numbering is scrambled; other times, the problem is tweaked or entirely altered or removed). |
idea. The only reason that this book gets 2 stars is because it has a decent answer section.
Reviewed by a reader
We've used this book in our freshman course in Mathematics (Linear Algebra) at University of Copenhagen, Denmark. It's very good as a introduction to Mathematical Proofs too.
Schaum's Outline of Modern Abstract Algebra (Schaum's)
Editorial review
Reviewed by Alex "supermanifold", (MTL)
y, for instance). It's just a matter of scoping them out carefully, and dishing out the money (for photocopies, even).
Reviewed by "torybug", (College Park, Maryland)
ve a few bucks.
Reviewed by "kem2070", (Seattle, WA USA)
y exercises (to test your understanding) have the answers. Some have an answer, some have a partial answer, some have a hint, and some have nothing. This is a little aggravating, but it does not take away from the book.
Reviewed by "pawntep", (BANGKOK, Thailand)
I am an undergrad student in Computer Science. The content in this book is terse and very cohesive. And its cohesiveness is what I like most. Each successive chapter is developed rigorously upon previous chapters. A lot of proofs of most
Algebra for College Students
Editorial review
Intended for a course that blends intermediate and college algebra topics written at an intermediate algebra level, the goal of this text is to provide a sound transition between elementary algebra and more advanced courses in mathematics
Understanding Algebra: Revised
Editorial review
This text features outstanding pedagogy, cumulative exercise sets, end-of-chapter key ideas for review, special boxed features called "Pointers for Better Understanding", annotated worked examples with "concept capsules", and chapter test
Schaum's Outline of Basic Mathematics for Electricity and Electronics (Schaum's)
Editorial review
Elementary Algebra: Structure and Use
Editorial review
This text is intended for a beginning or elementary algebra course offered at both two- and four-year schools. Elementary Algebra Structure and Use is an introductory text for students with either no background in algebra or for those stu
Math for Physics
Reviewed by a reader
Take care that you buy the right edition. I entered the isbn and came up with this book but it ended up being an old edition so I had to get a list of corrections from my professor.
Reviewed by a reader
Well I resented having to buy this book for my physics class during summer. It's not a required book during the fall or spring semesters. Basically this book is a condensation of calculus with physics applications. In addition to being ov
Introduction To Discrete Math
Editorial review
Intended for a one- or two-term discrete math introductory course, Introduction to Discrete Mathematics is designed to fulfill a general education requirement or a prerequisite for computer science courses. Written in an informal and conv
Standard Basic Math and Applied Plant Calculations
Editorial review
Prospective plant operators and engineers are provided with a review of math basics, with emphasis on free-hand calculations and mental-estimating method and are acquainted with typical on-the-job plant problems.
Reviewing Math
Editorial review
For the returning adult to community college or introductory student, this book encourages students to approach math differently. Unusual design includes 3-dimensional stereograms and math playing cards. Answers to problems (on same page)
Pre-Algebra: A Review
Editorial review
PRE-ALGEBRA-A REVIEW is a workbook designed for use in any PreAlgebra course. A student can study typical elementary statistics problems, review geometric perimeter and area applications, perform metric conversions and algebraic translati
Math Activities for Young Children: A Resource Guide for Parents and Teachers (College Custom Series)
Reviewed by a reader
Elementary Algebra Review
Editorial review
This workbook is designed for use in any elementary algebra course or by any student needing to retrace typical elementary algebra problems. Upon completion of these review problems, the student should feel comfortable taking any entrance
Intermediate Algebra, Form A
Editorial review
Revised to accommodate a stronger emphasis on graphing, this second edition introduces graphing and graphing techniques, functional notation, a transitional approach to graphing parabolas, and a formal development of functions. Also new t
Arithmetic and Algebra Again (Schaum's Paperbacks)
Reviewed by a reader
Reviewed by a reader
Mind Over Math: Put Yourself on the Road to Success by Freeing Yourself from Math Anxiety
ssor to teach me the subject. However I have discovered that is not true and that self learning college algebra is better for me. This is one of the most liberating experiences of my life and I am grateful to the authors of this book.
Reviewed by Shari H. Goforth, (Northern California)
This book was recommended to me by a college counselor who has high praise for the authors. If I may paraphrase: The book is eloquently written, the format is very well put together and usable. The authors discuss the aversion to math tha |
Applied Optimization
In this lecture our professor will walk you through Applied Optimization questions in Calculus. You will learn through a series of real world problems that begin with a quick sketch, Interval Analysis, Rewriting in One Variable, investigation of Maxima and Minima, and Critical Points before reaching the Optimal Result. Five fully worked out examples round out this lecture.
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Applied Optimization
Remember to get
your function down to a function of a single variable.
Remember to check
the endpoints as well as the critical points!
Include units on
your final result.
Applied Optimization
Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture. |
Computational Science
Objectives
Introduction to numerics and computational science. The role of applied mathematics in modeling scientific phenomena. Simulation and computational science.
Contents
The numeri cal simulation of real world phenomena is a challenging task, which requires a well balanced interplay between mathematics, informatics, and the application areas. This course will provide an introduction into numerics
and computational science (or scientific computing), with topics ranging from basic principles in numerical mathematics and numerical analysis to examples of modern high performance algorithms. Along (simplified examples) from application areas as, e.g., mechanics and biomechanics, an overview on computational science will be combined with an introduction into topics as mathematical modelling, the analysis of methods and the development of scientific software for numerical simulation. This will be done by combining theoretical with practical exercises.
Section 2. Orientation info.
Your are browsing the content Course description of the topic People directory. You arrived from. |
There's no such thing as too much practice. This reproducible program builds skills incrementally. By inviting students to "show what they know" in a variety of new formats, these stimulating lessons will enable struggling students to actually enjoy the learing process. As in all of the binder programs, the dual emphasis is on (1) mastery of the basics... more...
This textbook provides a general overview of realistic mathematical models in life sciences, considering both deterministic and stochastic models and covering dynamical systems, game theory, stochastic processes, and statistical methods. Each mathematical model is explained and illustrated individually with an appropriate biological example. more...
Automatic sequences are sequences which are produced by a finite automaton. Although they are not random they may look as being random. They are complicated, in the sense of not being not ultimately periodic, they may look rather complicated, in the sense that it may not be easy to name the rule by which the sequence is generated, however there exists... more...
Tips for simplifying tricky operations Get the skills you need to solve problems and equations and be ready for algebra class Whether you're a student preparing to take algebra or a parent who wants to brush up on basic math, this fun, friendly guide has the tools you need to get in gear. From positive, negative, and whole numbers to fractions, decimals,... more...
Algebra may seem intimidating?but it doesn't have to be. With Teach Yourself VISUALLY Algebra , you can learn algebra in a fraction of the time and without ever losing your cool. This visual guide takes advantage of color and illustrations to factor out confusion and helps you easily master the subject. You'll review the various properties of numbers,... more...
Describes the evolution of several socio-biological systems using mathematical kinetic theory. Specifically, this book deals with modeling and simulations of biological systems whose dynamics follow the rules of mechanics as well as rules governed by their own ability to organize movement and biological functions. more... |
This text for upper-level undergraduates and graduate students examines the events that led to a 19th-century intellectual revolution: the reinterpretation of the calculus undertaken by Augustin-Louis Cauchy and his peers. These intellectuals transformed the uses of calculus from problem-solving methods into a collection of well-defined theorems about limits, continuity, series, derivatives, and integrals. Beginning with a survey of the characteristic 19th-century view of analysis, the book proceeds to an examination of the 18th-century concept of calculus and focuses on the innovative methods of Cauchy and his contemporaries in refining existing methods into the basis of rigorous calculus. 1981 edition.
Table of Contents for The Origins of Cauchy's Rigorous Calculus
1. Cauchy and the Nineteenth-Century Revolution in Calculus
2. The Status of Foundations in Eighteenth-Century Calculus
3. The Algebraic Background of Cauchy's New Analysis
4. The Origins of the Basic Concepts of Cauchy's Analysis: Limit, Continuity, Convergence |
Prealgebra
Students who may be ready for Algebra 1 in Grade 8 take Pre-Algebra in Grade 7. The course covers the background required for Upper School level work in algebra and geometry, but it is acknowledged that not all students will master this background in one year. Topics include whole numbers, integers, rational numbers, decimals, fractions, proportions, percents, geometry, and graphs. The course emphasizes the symbolic language of mathematics and encourages the transition from concrete to abstract thinking through the use of algebra to solve problems involving equations and inequalities. Additional time is allotted for a math lab, which provides opportunities for students to reinforce math concepts through a hands-on approach. At the conclusion of the course, students either move on to Algebra 1 or extend their study of Pre-Algebra. |
Key Facts to Know in Algebra 1
The main focus (a.k.a. target) for the year in an Algebra 1 course is the STAR Test.
Mr. L wants all students to achieve at their highest level, and to do that requires commitment, dedication, time, and perseverance. Success on the STAR Test does not happen by magic, but rather by knowing key facts, techniques, and algorithms. Here are items Mr. L wants you to know for success in Algebra 1:
Statement of the Quadratic Formula
Proof of the Quadratic Formula
Six ways to write the direction of a straight line
Four ways to write the equation of a straight line; you need to know the equation as well as the name of the equation
4 Responses to "Key Facts to Know in Algebra 1"
I am a new high school math teacher. I teach Alg. 1, 1A, and 1B. to slow and difficult students.
I am looking for answers on how I can address my classes and get them motivated, lean Algebra 1, and be respectful at the same time.
Since these are low achiever, what can I do to make things more interesting and have them engage in the learning process without disrespect.
Hi Bryan,
If you visit my companion website, you will find several free activities that you can download. You can also download many handouts from conferences. This will give you a taste of the types of things that thousands of other teachers have used in their classrooms with success. Best wishes,
- Mr. L
Hi Leslie,
I think many students may need "a fresh start" with another teacher and another approach. Students can flounder when only given a symbolic approach to algebra, but may succeed when offered a more conceptual approach with less symbolic manipulation and cumbersome arithmetic. An approach that can be successful is to show "Algebra in the Real World", so that students can see how math relates to things around them.
Best wishes,
- Mr. L |
text offers comprehensive, in-depth, and precise coverage of college algebra topics, incorporated into a framework of tested teaching strategy and combined with carefully selected pedagogical features. The book preserves the integrity of the mathematics, yet does not discourage students with material that is confusing or too rigorous. The authors teach students to master difficult problems quickly and to develop skills they'll need in future courses and in everyday life. |
Student Solutions Manual for Cohen/Lee/Sklar's Precalculus, 7th
Book Description: Contains fully worked-out solutions to all of the odd-numbered exercises in the text, giving students a way to check their answers and ensure that they took the correct steps to arrive at an answer. |
Glen Oaks Statistics available upon request.Algebra I will help you learn to use and understand the following five fundamental concepts:
1. Simplify expressions
2. Solve equations/inequalities
3 |
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Self taught mechanical drawing and elementary machine design
SELF TAUGHT MECHANICAL DRAWING AND ELEMENTARY MACHINE DESIGN
A Treatise Comprising the First Principles of Geometric and Mechanical Drawings Workshop Mathematics, Mechanics, Strength of Materials, and the Design of Machine Details, including Cams, Sprockets, Gearing, Shafts, Pulleys, Belting, Couplings, Screws and Bolts, Clutches, Flywheels, etc. Prepared for the Use of Practical Mechanics and Young Draftsmen.
The demand for an elementary treatise on mechanical drawing, including the first principles of machine design, and presented in such a way as to meet, in particular, the needs of the student whose previous theoretical knowledge is limited, has caused the author to prepare the present volume. It has been the author's aim to adapt this treatise to the requirements of the practical mechanic and young draftsman, and to present the matter in as clear and concise a manner as possible, so as to make "self-study" easy. In order to meet the demands of this class of students, practically all the important elements of machine design have been dealt with, and, besides, algebraic formulas have been explained and the elements of trigonometry have been treated in a manner suited to the needs of the practical man.
In arranging the material, the author has first devoted himself to mechanical drawing, pure and simple, because a thorough understanding of the principles of representing objects greatly facilitates further study of mechanical subjects; then, attention has been given to the mathematics necessary for the solution of the problems in machine design presented later, and to a practical introduction to theoretical mechanics and strength of materials; and, finally, the various elements entering in machine design, such as cams, gears, sprocket wheels, cone pulleys, bolts, screws, couplings, clutches, shafting, flywheels, etc., have been treated. This arrangement makes it possible to present a continuous course of study which is easily comprehended and assimilated even by students of limited previous training.
Portions of the section on mechanical drawing was published by the author in The Patternmaker several years ago. These articles have, however, been carefully revised to harmonize with the present treatise, and in some sections amplified. In the preparation of the material, the author has also consulted the works of various authors on machine design, and credit has been given in the text wherever use has been made of material from such sources.
General Principles. In designing machinery it is frequently desirable to give to some part of the mechanism an irregular motion. This is often done by the use of cams, which are made of such form that when they receive motion, either rotary or reciprocating, they impart to a follower the desired irregular motion.
The follower is sometimes flat, and sometimes round. When the follower is round it is usually made in the form of a wheel or roller, so as to lessen the wear and the friction. The follower may work upon the edge of the cam, or if round, it may work in a groove formed either on the face or on the side of the cam.
The working surfaces of cams with round followers are laid out from a pitch line, so called, which passes through the center of the follower. The shape of this pitch line determines the work which the cam will do. The working surface of the cam is at a distance from the follower equal to one-half the diameter of the follower. This principle of a pitch line holds good whether the cam works only upon its edge like the one shown in Fig. 139, or whether it has an outer portion to insure the positive return of the follower. This outer portion is frequently made in the form of a rim of uniform thickness around the groove.
Design a Cam Having a Straight Follower Which Moves Toward or From the Axis of the Cam, as Shown in Fig. 136. Let it be required that the follower shall advance at a uniform rate from a to b as the cam makes a half revolution, this advance being preceded and followed by a period of rest of
a twelfth of a revolution of the cam.
Divide that half of the cam during the revolution of which the follower is to be raised from a to b, in this case the half at the right of the vertical center line, into a number of equal angles, and divide the distance from a to b into the same number of equal spaces. Mark off the points so obtained onto the successive radial lines as indicated by the dotted lines, and at the points where these dotted lines intersect the radial lines draw lines at right angles to the radial lines to represent the position of the follower when these radial lines become vertical as the cam revolves.
A period of rest in a cam is represented by a circular portion, having the axis of the cam as its center. In order, therefore, to obtain the required periods of rest, the distances of a and b from the center are marked off upon the radial lines c and d, these lines being made a twelfth of a revolution from the vertical center line, and lines representing the follower are drawn at these points as before. To get the return of the follower the space from c to d is divided into a number of equal angles, and the distance from e to f is divided off to represent the desired rate of return of the follower. In this case the rate of return is made uniform, so the distance ef is spaced off equally. The distance of these points from the axis is marked off upon the radial lines between c and d, and lines representing the follower are drawn.
A curved line, which may be made with the aid of the irregular curves, which is tangent to all of the lines representing the follower, gives the shape of the cam.
Fig. 137 shows a cam having the conditions as to the rise, rest and return of the follower the same as the one shown in Fig. 136, the follower, however, being pivoted at one end.
Draw the arc ab representing the path of a point in the follower at the vertical center line, and divide that part of the arc through which the follower rises into the same number of equal spaces as the half circle at the right of the vertical center line is divided into angles. Through these points draw lines, as shown, representing consecutive positions of the working face of the follower. The various distances of the follower from the axis of the cam are now marked off upon the corresponding radial lines as before. Lines to represent the follower are now drawn across each of these radial lines, at the same angle to them that the follower makes with the vertical center line when at that part of its stroke corresponding to the particular radial line across which the line representing the follower is being drawn. A curved line passing along tangent to all of these lines gives the shape of the cam as before.
Design a Cam with a Round Follower Rising Vertically. In Fig. 138 the follower has the same uniform rise, and the same periods of rest as before.
A cam with a round follower is less limited in its capabilities than one with a straight follower; in the one here shown the follower on its return drops below the position in which it is shown. That part of the cam during which the conditions are the same as in the others is divided off and the position of the center of the follower upon the radial lines is obtained in the same manner as before. That part of the cam representing the return of the follower is divided into such angles as desired, and the distance through which the follower is to drop as the cam revolves through each of these angles is marked off upon the proper radial line. A curved line which is now made to pass through all of the points so obtained gives the pitch line of the cam.
In drawing such a cam it is not always necessary to fully draw the working faces. The pitch line and the method of obtaining it being shown, a number of circles representing consecutive positions of the follower may be drawn. This will usually be sufficient. The side view of the cam, which in a case like this would naturally be made in section, will give opportunity to show any further detail that may be desired.
Design a Cam with a Round Follower Mounted on a Swinging Arm. Fig. 139 shows such a cam, all of the conditions as to rise, rest and return of the follower being the same as in the cam shown in Fig. 138. The cam is divided into the same angles as before, and the position of the follower is laid out on these radial lines as though it moved vertically. |
Applied Numerical Analysis
9780321133045
ISBN:
0321133048
Edition: 7 Pub Date: 2003 Publisher: Addison-Wesley
Summary: The seventh edition of this classic text has retained the features that make it popular, while updating its treatment and inclusion of Computer Algebra Systems and Programming Languages. Interesting and timely applications motivate and enhance students' understanding of methods and analysis of results. This text incorporates a balance of theory with techniques and applications, including optional theory-based section...s in each chapter. The exercise sets include additional challenging problems and projects which show practical applications of the material. Also, sections which discuss the use of computer algebra systems such as Maple®, Mathematica®, and MATLAB®, facilitate the integration of technology in the course. Furthermore, the text incorporates programming material in both FORTRAN and C. The breadth of topics, such as partial differential equations, systems of nonlinear equations, and matrix algebra, provide comprehensive and flexible, coverage of all aspects of numerical analysis |
Elsevier Science and Technology, July 2009, Pages: 744
Partial Differential Equations and Boundary Value Problems with Maple presents all of the material normally covered in a standard course on partial differential equations, while focusing on the natural union between this material and the powerful computational software, Maple. The Maple commands are so intuitive and easy to learn, students can learn what they need to know about the software in a matter of hours- an investment that provides substantial returns. Maple's animation capabilities allow students and practitioners to see real-time displays of the solutions of partial differential equations. Maple files can be found on the books website.
- Provides a quick overview of the software w/simple commands needed to get started - Includes review material on linear algebra and Ordinary Differential equations, and their contribution in solving partial differential equations - Incorporates an early introduction to Sturm-Liouville boundary problems and generalized eigenfunction expansions - Numerous example problems and end of each chapter exercises
Articolo, George A. Dr. George A. Articolo has 35 years of teaching experience in physics and applied mathematics at Rutgers University, and has been a consultant for several government research laboratories and aerospace corporations. He has a Ph.D. in mathematical physics with degrees from Temple University and Rensselaer Polytechnic Institute. |
The class meets six times a week: four times in lecture, once in conference,
and once in the computer laboratory. You are responsible for any and all
material discussed in lecture, conference, and lab.
Aside from the 6 hours that you spend in class each week, you should devote
at least another 8-10 hours to studying on your
own: reading the book, reading and organizing your notes, solving problems.
Conferences:
In the Friday conference sessions, you will meet with the Peer Learning
Assistant (PLA) for the class. You will be able to ask the PLA questions on the material
covered and homework. The PLA may lso give you in-class assignments and
review course material.
Homework: Written Homework: Problems will be assigned for each section of the
book covered and will be posted on the class web page. It is necessary to do,
at a minimum, the assigned problems so that you can learn and understand
the mathematics. You should do additional problems for further practice.
Working the exercises will help you learn, and give you some perspective
on your progress. You are welcome to
discuss homework problems with one another but you must write up your homework
solutions on your own. Be mindful of your academic integrity.
Your homework will be collected at the beginning of each Monday's class.
Late homework will not be accepted.
If you must miss Monday's class, you should have your work turned in before class time in order for it to be graded.
Do not wait until the weekend to start your homework.
Work on the problems daily.
Your work should be very legible and done neatly. If the work is not
presentable, and is illegible, you will not receive credit for it. Please staple
the sheets of your assignment together. Do not use paper torn out of spiral
bound notebooks. In the upper right hand corner of your assignment you
should write your name, the class section number, and the list
of book sections for the assignment.
Discipline yourself to write clear readable solutions,
they will be of great value as review.
You
need to show both your answer and the work leading to it. Merely having
the right answer gets no credit - we can all look them up in the back of the
book.
Online Homework: There will be homework using the online tool
WebWork. This is the same software that you used for the Math Placement Exam
that you took during the summer. There will occasionally be 7-15 questions
on WebWork that must be done. Go to
Do not use the WebWork system to email for help on problems; such an
email will be sent to all the professors and assistants for all the sections of
MA1022! Instead, see Prof. Weekes or your PLA as soon as possible.
Quizzes:
Each Monday, there will be a 15-20 minute in-class quiz emphasizing the
most recently covered topics. If you miss a quiz for any reason (illness, travel,
etc.), you will receive a score of zero. However, don't worry, the lowest quiz
score will be dropped. Make-up quizzes will, thus, never be given.
Labs:
Each week, students will meet in the computer lab (SH003) with the
Instructor's Assistant (IA) who is Jane Bouchard. We will use the computer
algebra system, Maple V, as a visual and computational aid to help you explore
the mathematical theory and ideas of the calculus. You will not be given
credit for a lab report if you did not attend the lab.
There are no make-up labs.
The first lab will be on Oct. 31st/Nov. 1st.
The final lab will be on Dec. 5th/Dec. 6th.
Final Exam/Basic Skills Test:
On Wednesday 12th December from 7-9 pm, you will have a 2 hour comprehensive final examination. Make arrangements now so that there are no
con
flicts with the final exam.
The Final Assessment will consist of two parts. The first part is the Final
Exam which is used in determining your course average score as detailed before.
The other part is the Basic Skills Exam. You cannot pass the course if you do not pass the
Basic Skills Exam.
Students with failing averages in the course are given grades of NR, whether
or not they passed the Basic Skills exam.
If you pass the Basic Skills component, then your course average will be used
by the professor to determine your grade for the course.
If you fail the Basic Skills Exam, yet have what the instructor determines
to be a course average high enough to pass the course, you will be given a
grade of I (incomplete). You will be given the opportunity to re-take the
Basic Skills exam at a later date; if you pass it, you will receive the grade
that is based on your course average.
Mathematics Tutoring Center:
The Mathematics Tutoring Center is available for any WPI student taking a
course in calculus, differential equations, statistics, and linear algebra.
Stratton Hall 002A
Monday-Thursday 10am-8pm and Friday 10am-4pm.
No appointment needed - drop in any time!
Academic Dishonesty
Please read WPI's
Academic Honesty Policy
and all its pages. Make note of the examples of
academic dishonesty; i.e. acts that interfere with the process of evaluation
by misrepresentation
of the relation between the work being evaluated (or the resulting evaluation)
and the student's actual state of knowledge.
Each student is responsible for familiarizing him/herself with
academic integrity issues and policies at WPI.
All suspected cases of dishonesty will be fully investigated.
Ask Prof. Weekes if you are in any way unsure whether your proposed actions/collaborations will be considered academically honest or not.
Students with Disabilities
Students with disabilities who believe that they may need accommodations in this class are encouraged to contact the Disability Services Office (DSO), as soon as possible to ensure that such accommodations are implemented in a timely fashion. The DSO is located in the Student Development and Counseling Center and the phone number is 508-831-4908,
e-mail is DSO@WPI.
If you are eligible for course adaptations or accommodations because of a disability (whether or not you choose to use these accommodations), or if you have medical information that I should know about please make an appointment with me immediately. |
Properties: Evaluate The learner will be able to
evaluate mathematical and algebraic expressions using the following properties: associative, commutative, identity, substitution, inverse and zero properties.
Data Collection: Organize The learner will be able to
collect, organize ,and display data with appropriate notation in tables, charts, and graphs (scatter plots, line graphs, bar graphs, and pie charts).
Figures: Two-/Three-Dimensional Objects The learner will be able to
use appropriate vocabulary to precisely explain, classify, and comprehend relationships among types of two- and three-dimensional objects by applying their defining properties.
Mathematical Reasoning: Explain The learner will be able to
apply many different methods to describe and communicate mathematical reasoning and concepts such as words, numbers, symbols, graphical forms, and/or models.
Area/Volume/Length: Differences The learner will be able to
identify the differences and relationships between length, area, and volume (capacity) measure in the metric and U.S. Customary measurement systems. |
STP Mathematics for Jamaica Grade 8
by
Sue Chandler - Ewart Smith
STP Mathematics for Jamaica is an, up-to-date, Mathematics course created by the STP Mathematics author team and Jamaican experts in Mathematics education and tailored to the needs of Lower Secondary students of Jamaica.
Top page |
Common Core Standards: 7.EE.4, 6.EE.6, 6.EE.7 Concepts: substitution, creating equations or formulas, solving equations, use of variables Student Activity: Students will discover an equation that will allow them to calculate the cost of a cell plan with variables involved using algebraic concepts. Students will have to do research to discover the costs. |
Teach21 Instructional Guide Algebra I
Students will create and use systems of linear equations to solve real world problems.
<
Content Standards and Objectives
Objective ID
Objectives
M.O.A1.2.9
create and solve systems of linear equations graphically and numerically using the elimination method and the substitution method, given a real-world situation.
21st Century Skills
Learning Skills & Technology Tools
Teaching Strategies Culminating Activity
Evidence of Success
Information and Communication Skills:
21C.O.9-12.1.LS3 - Student creates information using advanced skills of analysis, synthesis and evaluation and shares this information through a variety of oral, written and multimedia communications that target academic, professional and technical audiences and purposes.
The teacher will engage students in investigations into the representation of a system of linear equations using a variety of methods including the use of graphing calculators.
Students create and solve systems of linear equations through a variety of methods that include the use of graphing calculators. Students communicate the real-world meaning of the algebraic solution to systems of linear equations.
The teacher provides opportunities for students to generate and analyze multiple representations of real-world data with and without the use of technology. The teacher provides opportunities for students to work in small groups to analyze representations of systems of linear equations to determine the solution.
Students work in small groups to create representations of real-world data as systems of linear equations. Students work in small groups to analyze the meaning of an algebraic solution of a system of linear equations in the context of real-world situations.
Personal and Workplace Skills:
21C.O.9-12.3.LS2 - Student independently considers multiple perspectives and can represent a problem in more than one way, quickly and calmly changes focus and goals as the situation requires, and actively seeks innovations (e.g. technology) that will enhance his/her work.
The teacher presents real-world situations and establishes objectives and benchmarks to guide students through completion of both individual and group activities.
Students work collaboratively to investigate the relationship between graphs of lines and their linear equations.In real-world problem-solving situations, students identify the necessary information and create systems of linear equations.Students interpret the solution in the context of the problem-solving situation.
Performance Objectives (Know/Do)
Know:
The solution of a system of linear equations can be determined from its graphical representation.
The solution of a system of linear equations can be determined using the substitution method.
The solution of a system of linear equations can be determined using the elimination method.
A system may have zero solutions, one solution, or infinitely many solutions.
Do:
Model real-world data with systems of linear equations in two variables.
Determine the solution of a system of linear equations graphically, by means of substitution, and using elimination.
Interpret the solution to a system of two linear equations in a real-world situation.
Graph systems of linear equations using the graphing calculators.
Interpret the solution to a system of linear equations in the context of real-world situations.
Big Idea
Systems
Enduring Understandings
Sometimes the "correct" mathematical answer is not the best solution to real-world problems.
Essential Questions
How can systems of linear equations model real-world situations?
What are the limits of mathematical representation and modeling?
When is the "correct" answer not the best solution?
Learning Plan & Notes to Instructor
Notes:
The intent of this unit is to develop in students an understanding that systems of linear equations can be used to solve real world problems.
In order for all students to successfully complete the learning activities, academic prompts and Differentiated Instruction should be integrated throughout the unit.
· Interest can be developed through a variety of real-world problem situations.
· Readiness can be determined by teacher observation and/or pre-assessment.
· Learning styles and readiness levels can be addressed through visual presentations, such paper-and pencil graphs and calculator-generated graphs; kinesthetic presentations, such as those utilizing graphing calculators; auditory presentations, such as group reporting and teacher-directed instruction.
· Content can be differentiated by readiness as students are provided with open-ended academic prompts designed to provide insight into the current level of student understanding.
· Process can be differentiated through the use of whole class, small group, paired and individual activities.
· Product can be differentiated as students are prompted to demonstrate their understanding with written explanations and by creating visual displays.
· Scaffolding can be provided through the use of exit slips, which inform the teacher of the immediate learning needs of individual students.
Unit Summary:
In the context of a real-world situation, students create two linear equations that model the conditions. Students act out/model the situations that are simultaneously imposed in the problem. Students progress to using tables and graphs to model the situation. They find values that make both equations true and come to understand that the intersection of the two lines provides a solution to the system of equations and to the real-world problem.
In small groups, students investigate solutions to systems of linear equations. They interpret and describe solutions to systems that have no solution (inconsistent), one solution, or infinitely many solutions (redundant).
As students work in small groups to solve problems, watch for student ability to collect data in tables, write linear equations in slope-intercept form, and determine the solution.
Students then progress to algebraic methods of solving systems of linear equations in two variables: the substitution method and the elimination method. These methods allow students to determine the exact solution of a system without graphing or constructing a table.
The elimination method provides an opportunity to review the value/usefulness of "standard form." Students develop their skills in using and choosing among these methods as they use systems of linear equations to model real-world situations (see Air Traffic Controller,
Asking students to create Frayer models of these three methods of solving systems of linear equations (graphs, substitution, elimination), gives them an opportunity to compare and contrast the advantages and disadvantages of each.
You are the drum major of your high school band. The band director has created a half-time routine.
You and the tuba player keep colliding. Create a visual model of the situation for the band director and describe possible solutions.
Dollar Detective
You are a detective. The manager of a large movie theater suspects that his employees may be dishonest and has asked you to investigate. The computer records indicating the number of adults and the number of children who purchased tickets have been lost. The turnstile indicates that 7423 people bought tickets for movies on Friday night and 8559 tickets were sold on Saturday. Adult tickets cost $7 and student tickets cost $3. Store receipts are $41,463 on Friday and $50,501. Create a presentation of your results to the store manager.
You are the chairman of the annual sports banquet committee at your school. You have found two possible sites: one site has a high room rent and a low cost per meal; the second site has a high cost per meal charge and a low room rent. Make recommendations to the other members of the banquet committee. Create specific examples to illustrate your explanation. |
Discovering Math: Advanced Probability DVD This program introduces and develops concepts of probability, such as discrete and continuous variables, and dependent and independent events. It also discusses various methods of determining probabilities, as well as their applications.
9 - 12
DVD
$59.95
Fractions Mastery DVD An 11-lesson program that teaches about fractions and how to work with fractions to build computation and problem-solving skills. |
Next: Sequences
Previous: Ordinary Differential Equations
Chapter 8: Infinite Series
Chapter Outline
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Chapter Summary
Description
This chapter introduces the study of sequences and infinite series. The properties presented describe the behavior of a sequence or series, including whether a sequence approaches a number or an infinite series adds to a number. |
Algebra 1 Complete Home School Kit with Solutions Manual, 3rd Edition Algebra 1, Algebra 2, Advanced Mathematics, and Calculus each contain a series of lessons covering all areas of general math. Advanced Mathematics is a comprehensive precalculus course that includes advanced algebra, geometry, trigonometry, discrete mathematics, and mathematical analysis. Each lesson in the Saxon math program presents a small portion of math content (called an increment) that builds on prior knowledge and understanding. Home School Kits for the upper grades math programs includes a textbook, Home School Packet (with test forms, textbook answers, and test answers), and a Solutions Manual. |
Mathematics Course Descriptions
Students with an ACT math subscore of 18 or lower are required by the state to take Bridge Math. Students must take a math course every year.
Algebra I
Last Updated on Tuesday, 06 March 2012 11:47 |
Grade 9; one term; one credit
This course includes the use of the language of algebra, operations with real numbers, solutions of linear equations and inequalities, graphing linear equations and inequalities, solutions of systems of equations, problem solving computations with polynomials, factoring polynomials, performing operations with algebraic fractions, solving expressions containing radicals, and solving quadratic equations.
Geometry
Last Updated on Tuesday, 06 March 2012 11:47 |
Grade 9,10; one term; one credit
This course content includes the properties, relationships and geometric reasoning pertaining to 1-, 2-, and 3-dimensional figures. Topics developed include the properties of lines, planes, angles, polygons, polyhedrons, circles, cones, cylinders and spheres, the concepts of congruency and similarity as applied to polygons, perimeter, area and volume of geometric figures, methods of reasoning, types of proofs, and coordinate geometry.
Advanced Algebra and Trigometry
Last Updated on Tuesday, 06 March 2012 11:50 |
Grade 11,12; one term; one credit
This course extends Algebra II concepts as well as placing an emphasis on trigonometry. This class is intended to further prepare students for success in Precalculus. Students scoring high enough in Algebra II may skip this course and move directly to Precalculus.
Prerequisite: Algebra II
Precalculus
Last Updated on Tuesday, 06 March 2012 11:48 |
Grade 11, 12; one term; one credit.
Precalculus includes properties, graphs, inverses and application of trigonometric functions. The algebra transformation and application of linear, exponential, logarithmic, polynomial and other special functions are studies. Sequences and series are expanded to explore the concept of limits. Graphing utilities are used frequently.
Calculus
Last Updated on Tuesday, 06 March 2012 11:44 |
Grade 11,12; one term; one credit
Calculus is a more in-depth and rigorous approach to the study of functions and their graphs, limits of functions, the derivative and applications of differentiation (maximum and minimum, related rates, velocity, acceleration), the integral and applications of integration, differential equations and geometry of conic.
Prerequisites: Pre-Calculus with "C" or better and teacher recommendation.
Bridge Mathematics
Last Updated on Wednesday, 21 March 2012 13:18 |
Bridge Math was developed by the state of Tennessee to help and encourage seniors who have scored below an 18 on the ACT (math section). Skills are developed in an environment that promotes learning beyond skill and drill techniques. New skills are introduced in conjunction with appropriate mathematical concepts and are related to previous learning. Multiple problem solving strategies are taught. |
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Course: CS 103, Spring 2007 School: Vanderbilt Rating:
Document PreviewfaceThis book is meant for students of engineering or science who need both a useful programming language for solving problems in their disciplines and an introduction to ideas from the discipline of computer science. It is a provided free of cha
Department of Mechanical EngineeringInterpolationES 140 Section 5 Fall 2006Department of Mechanical EngineeringDefinition - InterpolationInterpolate-To estimate a missing value by taking an average of known values at neighboring points.
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ANSC 120 / SAAS 20 Visual Inspection and Health Management of Swine Lab Report 10 October 30, 2007 Section Number_3:00 5:00_ Name_Russ Caudill_ 1. What is the function of the farrowing crate for sows AND how does it fit into the health management of
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Lecture 1 Introduction to U.S. Women of Color FeminismsApril 5, 2007 Women's Studies 60 Dr. Mireille Miller-YoungGoals of Lecture Provide a brief history of the women's movement and the rise of feminism and women of color feminism. Introduce som
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Revolutions of 1848What forces drove the revolutions of 1848? How were they similar or different from the French Revolution of 1789? Keep in mind the context created by the Industrial Revolution. Emergence of the middle class who challenge the exist
I.ReviewA. Revolutions of 1848, the ideals of 1789model of French Revolution, nationalism, revolt of the growing working class. Promises of 1789 have been betrayed. B. International Rivalriesfollowing the Vienna Conference. Initially it is the mid
In what ways did World War II put Hitler's ideology into practice? How did the Nazi state define citizenship? What was the purpose of appeasement?I.OriginsA. World War I ReviewCauses are complicated: Franz Ferdinand, Wilhelm II's blank checkB
I.Primo Levi and the CampsA. Background1. 2. The Italian Jew: from Torino Scientist-PhD in physics3. 1941: Nazis take over; sent to Auschwitz in February 1944 after being transferred from a camp in ItalyB.Main Insights1. Dehumanization and
What prompted the mass uprisings in both east and west during 1968? Did the demonstrators achieve their goals? What long-term consequences did the events of 1968 have?I.Divided EuropeA. West Economic Miracle-vast growth1. Marshall Plan-money fu
HIST 4A Fall 2007 Week X TA: Jessica Elliott Timeline Review for Final ExamOPTIONAL Timeline Review for Final ExamThis is a guide for making an OPTIONAL timeline of significant historical events and people from the second half of the course. If
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This product is made available subject to the terms of GNU Lesser General Public License Version 2.1. A copy of the LGPL license can be found at http:/ -Third Party Code. Additional copyright notices and license terms a
THE WORLD: CHAPTER SIXTEENQUESTIONS: How were empires the agents of change in the sixteenth and seventeenth centuries? Empires resulted in migration as well as exchange of technology, plants and animals, ideas, religion, politics, and art. The excha
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Chapter Eighteen: Mental Revolutions: Religion And Science In The Sixteenth And Seventeenth CenturiesWhat was Armesto trying to convey with the story of the Christian missionary in the Andes? The behavior of missionaries twenty indigenous population
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Peter Zywiak 2/27/08 Engr 166 Sec. 04 Intro: Using the MacLaurin Series expansion it is possible to expand the cosine function into a formula. This formula can be expanded to an infinite series. By using this series the cosine function is expressed i
Ranking Task ActivityOne more capacitor network 1F 30VWhat is the charge stored on each?5F2F What is the total energy stored? 3F 4FRemember:Qon a capacitor=Cof that capacitor Vacross that capacitor6FWhich of these diagrams represent the |
Book Description: ELEMENTARY STATISTICS: A STEP BY STEP APPROACH is for introductory statistics courses with a basic algebra prerequisite. The book is non-theoretical, explaining concepts intuitively and teaching problem solving through worked examples and step-by-step instructions. In recent editions, Al Bluman has placed more emphasis on conceptual understanding and understanding results, along with increased focus on Excel, MINITAB, and the TI-83 Plus and TI-84 Plus graphing calculators; computing technologies commonly used in such courses. The 8th edition of Bluman provides a significant leap forward in terms of online course management with McGraw-Hill's new homework platform, Connect Statistics – Hosted by ALEKS. Statistic instructors served as digital contributors to choose the problems that will be available, authoring each algorithm and providing stepped out solutions that go into great detail and are focused on areas where students commonly make mistakes. From there, the ALEKS Corporation reviewed each algorithm to ensure accuracy. The result is an online homework platform that provides superior content and feedback, allowing students to effectively learn the material being taught.
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AIEEE Syllabus 2012
UNIT 1: Sets, Relations and Functions
Sets and their Representations, Union, intersection and complements of sets,
and their algebraic properties, Relations, equivalence relations, mappings, oneone,
into and onto mappings, composition of mappings.
UNIT 2: Complex Numbers
Complex numbers in the form a+ib and their representation in a plane. Argand
diagram. Algebra of complex numbers, Modulus and Argument (or amplitude) of
a complex number, square root of a complex number. Cube roots of unity,
triangle inequality.
UNIT 3: Matrices and Determinants
Determinants and matrices of order two and three, properties of determinants,
Evaluation of determinants. Area of triangles using determinants, Addition and
multiplication of matrices, adjoint and inverse of matrix. Test of consistency and
solution of simultaneous linear equations using determinants and matrices
UNIT 4: Quadratic Equations
Quadratic equations in real and complex number system and their solutions.
Relation between roots and co-efficients, nature of roots, formation of quadratic
equations with given roots; Symmetric functions of roots, equations reducible to
quadratic equations - application to practical problems
UNIT 5: Permutations and Combinations
Fundamental principle of counting; Permutation as an arrangement and
combination as selection, Meaning of P(n,r) and C(n,r). Simple applications.
UNIT 6: Mathematical Induction and Its Applications
AIEEE Notes Mathematics Chapter 05 Chapter wise notes for AIEEE and other entrance examinations such as AIPMT and IITJEE. These Notes are available for FREE downloads under this section.
UNIT 7: Binomial Theorem and its Applications
Binomial Theorem for a positive integral index; general term and middle term;
Binomial Theorem for any index. Properties of Binomial Co-efficients. Simple
applications for approximations.
UNIT 10: Integral Calculus
Integral as an anti-derivative. Fundamental integrals involving algebraic,
trigonometric, exponential and logarithmic functions. Integration by substitution,
by parts and by partial fractions. Integration using trigonometric identities.
Integral as limit of a sum. Properties of definite integrals. Evaluation of definite
integrals; Determining areas of the regions bounded by simple curves.
UNIT 11: Differential Equations
Ordinary differential equations, their order and degree. Formation of differential
equations. Solution of differential equations by the method of separation of
variables. Solution of homogeneous and linear differential equations, and those
of the type d2y/dx2 = f(x)
UNIT 12: Two dimensional Geometry
Recall of Cartesian system of rectangular co-ordinates in a plane, distance
formula, area of a triangle, condition for the collinearity of three points and
section formula, centroid and in-centre of a triangle, locus and its equation,
translation of axes, slope of a line, parallel and perpendicular lines, intercepts of
a line on the coordinate axes.
The straight line and pair of straight lines
Various forms of equations of a line, intersection of lines, angles between two
lines, conditions for concurrence of three lines, distance of a point from a line
Equations of internal and external bisectors of angles between two lines,
coordinates of centroid, orthocentre and circumcentre of a triangle, equation of
family of lines passing through the point of intersection of two lines,
homogeneous equation of second degree in x and y, angle between pair of lines
through the origin, combined equation of the bisectors of the angles between a
pair of lines, condition for the general second degree equation to represent a pair
of lines, point of intersection and angle between two lines.
Circles and Family of Circles
Standard form of equation of a circle, general form of the equation of a circle, its
radius and centre, equation of a circle in the parametric form, equation of a circle
when the end points of a diameter are given, points of intersection of a line and a
circle with the centre at the origin and condition for a line to be tangent to the
circle, length of the tangent, equation of the tangent, equation of a family of
circles through the intersection of two circles, condition for two intersecting
circles to be orthogonal. Conic Sections
Sections of cones, equations of conic sections (parabola, ellipse and hyperbola) in
standard forms, condition for y = mx + c to be a tangent and point(s) of
tangency.
UNIT 13: Three Dimensional Geometry
Coordinates of a point in space, distance between two points; Section formula,
direction ratios and direction cosines, angle between two intersecting lines. Skew
lines, the shortest distance between them and its equation. Equations of a line
and a plane in different forms; intersection of a line and a plane, coplanar lines,
equation of a sphere, its centre and radius. Diameter form of the equation of a
sphere.
UNIT 14: Vector Algebra
Vectors and Scalars, addition of vectors, components of a vector in two
dimensions and three dimensional space, scalar and vector products, scalar and
vector triple product. Application of vectors to plane geometry.
UNIT 15: Measures of Central Tendency and Dispersion
Calculation of Mean, median and mode of grouped and ungrouped data.
Calculation of standard deviation, variance and mean deviation for grouped and
ungrouped data.
UNIT 16: Probability
Probability of an event, addition and multiplication theorems of probability and
their applications; Conditional probability; Bayes' Theorem, Probability
distribution of a random variate; Binomial and Poisson distributions and their
properties.
UNIT 17: Trigonometry
Trigonometrical identities and equations. Inverse trigonometric functions and
their properties. Properties of triangles, including centroid, incentre, circumcentre
and orthocentre, solution of triangles. Heights and Distances
UNIT 18: Statics
Introduction, basic concepts and basic laws of mechanics, force, resultant of
forces acting at a point, parallelogram law of forces, resolved parts of a force,
Equilibrium of a particle under three concurrent forces, triangle law of forces and
its converse, Lami's theorem and its converse, Two parallel forces, like and unlike
parallel forces, couple and its moment.
UNIT 19: Dynamics
Speed and velocity, average speed, instantaneous speed, acceleration and
retardation, resultant of two velocities. Motion of a particle along a line, moving
with constant acceleration. Motion under gravity. Laws of motion, Projectile
motion.
PHYSICS
UNIT 1: UNITS AND MEASUREMENT
AIEEE Notes Mathematics Chapter 05 Chapter wise notes for AIEEE and other entrance examinations such as AIPMT and IITJEE. These Notes are available for FREE downloads under this section.
UNIT 2: DESCRIPTION OF MOTION IN ONE DIMENSION
Motion in a straight line, uniform and non-uniform motion, their graphical
representation. Uniformly accelerated motion, and its applications
UNIT 4: LAWS OF MOTION
UNIT 5 WORK, ENERGY AND POWER
Concept of work, energy and power. Energy - kinetic and potential. Conservation
of energy and its applications, Elastic collisions in one and two dimensions.
Different forms of energy.
UNIT 6: ROTATIONAL MOTION AND MOMENT OF INERTIA
Centre of mass of a two-particle system. Centre of mass of a rigid body, general
motion of a rigid body, nature of rotational motion, torque, angular momentum,
its conservation and applications. Moment of Inertia, parallel and perpendicular
axes theorem, expression of moment of inertia for ring, disc and sphere.
UNIT 7: GRAVITATION
Acceleration due to gravity, one and two-dimensional motion under gravity.
Universal law of gravitation, variation in the acceleration due to gravity of the
earth. Planetary motion, Kepler's laws, artificial satellite - geostationary
satellite, gravitational potential energy near the surface of earth, gravitational
potential and escape velocity
UNIT 11: HEAT AND THERMODYNAMICS
Thermal expansion of solids, liquids and gases and their specific heats,
Relationship between Cp and Cv for gases, first law of thermodynamics,
thermodynamic processes. Second law of thermodynamics, Carnot cycle,
efficiency of heat engines.
UNIT 12: TRANSFERENCE OF HEAT
Modes of transference of heat. Thermal conductivity. Black body radiations,
Kirchoff's Law, Wien's law, Stefan's law of radiation and Newton's law of cooling.
UNIT 13: ELECTROSTATICS
Electric charge - its unit and conservation, Coulomb's law, dielectric constant,
electric field, lines of force, field due to dipole and its behaviour in a uniform
electric field, electric flux, Gauss's theorem and its applications. Electric potential,
potential due to a point charge. Conductors and insulators, distribution of charge
on conductors. Capacitance, parallel plate capacitor, combination of capacitors,
energy of capacitor.
UNIT 14: CURRENT ELECTRICITY
Electric current and its unit, sources of energy, cells- primary and secondary,
grouping of cells resistance of different materials, temperature dependence,
specific resistivity, Ohm's law, Kirchoff's law, series and parallel circuits.
Wheatstone Bridge with their applications and potentiometer with their
applications.
UNIT 15: THERMAL AND CHEMICAL EFFECTS OF CURRENTS
UNIT 16: MAGNETIC EFFECTS OF CURRENTS
Oersted's experiment, Bio-Savart's law, magnetic field due to straight wire,
circular loop and solenoid, force on a moving charge in a uniform magnetic field
(Lorentz force), forces and torques on currents in a magnetic field, force between
two current carrying wires, moving coil galvanometer and conversion to ammeter
and voltmeter.
UNIT 19: RAY OPTICS
Reflection and refraction of light at plane and curved surfaces, total internal
reflection, optical fibre; deviation and dispersion of light by a prism; Lens
formula, magnification and resolving power; microscope and telescope.
UNIT 5: Chemical Energetics and Thermodynamics
Energy changes during a chemical reaction, Internal energy and Enthalpy,
Internal energy and Enthalpy changes, Origin of Enthalpy change in a reaction,
Hess's Law of constant heat summation, numericals based on these concepts.
Enthalpies of reactions(Enthalpy of neutralization, Enthalpy of combustion,
Enthalpy of fusion and vaporization).
Sources of energy (conservation of energy sources and identification of
alternative sources, pollution associated with consumption of fuels. The sun as
the primary source).
First law of thermodynamics; Relation between Internal energy and Enthalpy,
application of first law of thermodynamics.
Second law of thermodynamics : Entropy, Gibbs energy, Spontaneity of a
chemical reaction, Gibbs energy change and chemical equilibrium, Gibbs energy
available for useful work.
UNIT 8: Rates of Chemical Reactions and Chemical Kinetics
Rate of reaction, Instantaneous rate of reaction and order of reaction. Factors
affecting rates of reactions - factors affecting rate of collisions encountered
between the reactant molecules, effect of temperature on the reaction rate,
concept of activation energy, catalyst. Effect of light on rates of reactions.
Elementary reactions as steps to more complex reactions. How fast are chemical
reactions?
Rate law expression. Order of a reaction (with suitable examples). Units of rates
and specific rate constants. Order of reaction and effect of concentration (study
will be confined to first order only). Temperature dependence of rate constant -
Fast reactions (only elementary idea). Mechanism of reaction (only elementary
idea). Photochemical reactions.
UNIT 17: Transition Metals including Lanthanides
Electronic configuration : General characteristic properties, oxidation states of
transition metals. First row transition metals and general properties of their
compounds-oxides, halides and sulphides.
General properties of second and third row transition elements (Groupwise
discussion).
Preparation and reactions, properties and uses of Potassium dichromate and
Potassium permanganate.
Inner Transition Elements: General discussion with special reference to oxidation
states and lanthanide contraction.
UNIT 23: Organic Compounds Containing Halogens
(Haloalkanes and Haloarenes)
Methods of preparation, physical properties and reactions.
Preparation, properties and uses of Chloroform and Iodoform,
UNIT 24: Organic compounds containing Oxygen
General methods of preparation, correlation of physical properties with their
structures, chemical properties and uses of Alchols, polyhydric alcohols, Ethers,
aldehydes, ketones, carboxylic acids and their derivatives, Phenol, Benzaldehyde
and Benzoic acid - their important methods of preparation and reactions. Acidity
of carboxylic acids and phenol effect of substituents on the acidity of carboxylic
acids.
UNIT 25: Organic Compounds Containing Nitrogen
(Cyanides, isocyanides, nitrocompounds and amines)
Nomenclature and classification of amines, cyanides, isocyanides,
nitrocompounds and their methods of preparation; correlation of their physical
properties with structure, chemical reactions and uses - Basicity of amines
UNIT 26: Synthetic and Natural Polymers
Classification of Polymers, natural and synthetic polymers (with stress on their
general methods of preparation) and important uses of the following :
Teflon, PVC, Polystyrene, Nylon-66, terylene, Bakelite
UNIT 28: Chemistry in Action
UNIT 29: Environmental Chemistry
Environmental pollutants; soil, water and air pollution; major atmospheric
pollutants; acid rain, Ozone and its reactions causing ozone layer depletion,
effects of the depletion of ozone layer, industrial air pollution.
List of Colleges
List of NATA Participating Institution in the Country as on March 06, 2009.....
Our Curriculum
The Syllabus designed by us is entirely for Architecture Aptitude only, of various Architecture.......
Our Team
We at THE FUTURIST are a team of architects/ designers and technocrats. We mould
and prepare students.... |
In Pursuit of the Unknown: 17 Equations That Changed the World
by Ian Stewart Publisher Comments
Most people are familiar with historys great equations: Newtons Law of Gravity, for instance, or Einsteins theory of relativity. But the way these mathematical breakthroughs have contributed to human progress is seldom appreciated. In In Pursuit of the...Cartoon Guide to Statistics
by Larry Gonick Publisher Comments... (read more)
Trigonometry (Homework Helpers)
by Denise Szecsei Publisher Comments
Homework Helpers: Algebra emphasizes the role that arithmetic plays in the development of Algebra and covers all of the topics in a typical Algebra 1 class, including: * Solving linear equalities and inequalities * Solving systems of linear equations... (read more)
The King of Infinite Space: Euclid and His Elements
by David Berlinski Publisher Comments
Geometry defines the world around us, helping us make sense of everything from architecture to military science to fashion. And for over two thousand years, geometry has been equated with Euclids Elements, arguably the most influential book in |
This workbook is written as an aid to learn the skills necessary for
success in college level Algebra classes. You may use a single lesson to help
review a particular topic, or you may want to work your way through the
entire workbook. Topics presented here are at the level of Math 091
(Elementary Algebra), as taught in the UW Colleges, but the workbook does not
include all topics that would be covered in the course. For the best
preparation for Intermediate Algebra (Math 105), you should take and complete
the Math 091 course (or one of its equivalents).
Whether you are reviewing previously learned material or studying Algebra
for the first time, this material should help you build a mathematical
foundation for any further studies. The concepts presented here are not
intended to cover all possible course topics, but rather only to strengthen
foundations. You should strive for a mastery of all topics in this workbook,
for later successes will depend on these basic skills. By 'mastery', I expect
that you can not only do most of the exercises correctly, but also that you
can recognize mistakes in your own work.
The single most important factor for success will be your own personal
effort. Math is not a mystical subject; rather it is based on common sense.
With practice and guidance, you should soon be able to judge your own work
for correctness. Indeed, one of the best signs of mathematical mastery is the
ability to find and correct your own errors.
A final few words of advice will get you off to a good start in learning
Algebra: Take responsibility for your own learning by working carefully
through the entire workbook. Ask questions whenever you do not understand a
concept. Attend classes regularly. Remember, at all times, that learning is
an active process that requires your participation. |
The following are small programs that may (or may) not be useful.
They are listed here as it is likely that their primary reason for existing is for use in teaching.
Some are Java applets (or standalone Java programs) developed using processing.
These will work in a browser (that supports Java) or can be downloaded to a Linux, MacOSX, or Windows machine.
Others are Lua scripts developed on the iPad application Codea.
These are designed to be run within Codea on an iPad, though it may be possible to adapt them to run using Love2D (see this discussion for details).
Getting code onto the iPad can be somewhat tricky, see this discussion for methods.
The Java applets were originally released under the GPL.
The Codea code is released under the CC0 to the extent allowed (some pieces are code that I've adapted from others; see Codea for specific details).Manifolds occur throughout mathematics -- and hence other sciences -- as spaces in which something interesting happens or spaces in which something interesting is to be found.
Often the set of solutions to some problem forms a manifold, or the set of possible configurations of some system forms a manifold.
By studying manifolds and by developing a set of tools with which to study particular manifolds, we can gain considerable insight into those problems where manifolds occur.
The key property of a manifold is that it locally looks like ordinary Euclidean space.
Therefore, things that work on small patches of Euclidean spaces can often be made to work on manifolds.
The most important of these things is calculus.
Indeed, one can regard manifolds as the right places to do calculus and many of the questions involving manifolds have their origin in considering a particular theorem of calculus.
This course is designed to be an introduction to the theory of manifolds, looking at particular examples and developing simple techniques for studying themFourier analysis involves rewriting functions of position as functions of frequency.
In many situations, the behaviour of a system is much simpler when it is viewed as dependent on frequency rather than position.
It is therefore a Good Thing to have the output of the system written in terms of frequency.
However, it is often much easier to find the values of that system in terms of position.
The techniques of Fourier theory allow us to make the best of both: we can read off the values in terms of position and then use Fourier analysis to rewrite these in terms of frequency.
In this course we shall be studying the mathematical tools and ideas behind Fourier theory.The lectures will be delivered as PDF presentations.
I shall endeavour to make the notes available the day before the lecture so that students can "follow along".
There may be additional notes written during the lectures.
These will be posted shortly after the lecture finishes.
I shall often prepare a little more than I shall actually give, any extra will be taken up in the next lecture or deferred to the wiki.
The notes will be available in several different layouts.
It is important to know which is which.
Beamer.
This is what will actually appear on the screen during the lectures.
You must never print this version.
As each "transition" results in a new page, this can easily exceed 100 pages.
Trans.
This is a "one frame per page" version of the above.
If you intend to "follow along" with the lecture on your own computer then this is probably the best.
However, I still strongly urge you not to print it.
Handout.
This is a more condensed version in terms of space.
By putting 4 slides on a page the total number of pages is significantly reduced.
If you want to print something then print this version.
Annotated.
It will probably take a little experimenting to find the best way to present the annotated version.
As a first go, the PDF contains just the annotated pages.
For most pages, this should be sufficient to locate it within the main presentation.
For some it may be useful to have the previous page included as well.
Let me know if this, or something else, would be useful. |
The purpose of this book is to actively involve the reader in the heuristic processes of conjecturing, discovering, formulating, classifying, defining, refuting, proving, etc. within the context of Euclidean geometry. The book deals with many interesting and beautiful geometric results which have only been discovered during the past 300 years such as the Euler line, the theorems of Ceva, Napoleon, Morley, Miquel, Varignon, etc. Some original results are also presented which have not been published elsewhere before.
Many problems lend themselves excellently for exploration on computer with dynamic geometry programs such as Cabri and/or Sketchpad, although it is not essential to have these programs. The reader should be well acquainted with high school Euclidean and transformation geometry, as well as trigonometry. The book is addressed primarily to university or college lecturers involved in the under-graduate or in-service training of high school mathematics teachers, but may also interest teachers who are looking for enrichment material and gifted high school mathematics pupils.
219 pp., 3rd Revised version 2009, Paperbound, ISBN: 978-0-557-10295-2. The book is now available as a downloadable PDF or as printed paperback and payments can be made via Credit Card or PayPal for International Customers. Follow the link below for more information or to preview and buy the book: |
Steps to Achieving the Highest Levels of Mathematics
Overview
Because of the sequential nature of learning mathematics, course placement is based on student effort, teacher recommendation, standardized test scores, and parent/student input. To remain on grade-level, students need to participate in three math credits during their freshman and sophomore years, especially to complete Algebra I, Parts I and II and Euclidean Geometry. Students will need a graphing calculator; all are encouraged to use the Texas Instruments series. The TI-81-86 graphing series is recommended for Algebra I pt. I, and beyond. All students are strongly encouraged to study mathematics beyond tenth grade at the most challenging levels.
Mathematics: Meeting New Challenges Grade 7
Mathematics in grade seven continues to develop the topics learned in elementary school, with an emphasis on problem solving using algebraic skills. Students review basic understanding of fractions and decimals; a sound grasp of multiplication facts is a must. Students participate in the American Mathematics Competition Examination, as well as taking the NECAP and MAP assessments. Based on NECAP scores, districts assessments, and teacher recommendation, students can enroll in Algebra I: Part 1 and earn high school credit.
Pre-algebra Grade 8
Qualifying students may participate in this course in grade 7. Topics covered include numbers and operations, powers and exponents, order of operations, scientific notation, prime factorization, rational and irrational numbers, ratios, and percent of change. In algebra, students identify inverse operations, distributive property, graphing equations, and inequalities, scale models, identify congruent and similar figures, transformations, Pythagorean Theorem, ratios, circles, and classify and sketch solids. In data analysis and probability, students find outcomes and odds, and draft appropriate data displays. Problem-solving is integrated throughout the course. District assessments include NECAP and MAP. The eighth grade participates in the American Mathematics Competition-8.
Algebra I: Part I 1 credit Grades 8-12 Math credit
Prerequisite(s): Pre-algebra and teacher recommendation
This course develops ability in the real number system. Combining like terms and balancing equations leads to expertise in solving and graphing linear functions. Students use graphing calculators extensively as an aid in learning about slope as a rate of change and in solving linear systems. Problem solving skills are integrated into all topics and students complete the first seven chapters of the text, Algebra I (McDougall, Littell-Larson Ed.)
Algebra I: Part II 1 credit Grades 9-12 Math credit.
Prerequisite(s): Algebra I: Part I and teacher recommendation
This course extends a student's ability to work with algebraic expressions and functions. The rules for exponents are learned, leading students to more sophisticated equations (quadratic, exponential, radical) and graphs (parabolas, exponentials, hyperbolas). Graphing calculators are used extensively to understand and learn graphing translations. Factoring and division techniques are developed in order to solve higher degree equations. Permutations and combinations are taught as part of more advanced probability activities.
Euclidean Geometry 1 credit Grades 9-12 Math credit
Prerequisite(s): Algebra 1: Part I
Students demonstrate high level reasoning by writing proofs and solving problems dealing with points, lines, angles, triangles, quadrilaterals, circles, and other shapes. Students study area, perimeter, and volume and the connections between the three dimensions. Students will complete a variety of two- and three-dimensional hands-on projects. The texts used are Geometry (Houghton, Mifflin) and Flatland (Abbott).
Algebra II 1 credit Grades 10-12 Math credit
Prerequisite(s): Algebra I: Part II
This course extends the study of previous algebra courses and assumes a strong working knowledge of those topics. It encompasses the study of functions including logarithmic functions, irrational and complex numbers, polynomial equations, analytic geometry, conic sections, and series and sequences. The text is Algebra II (McDougall, Littell).
Pre-calculus 1 credit Grades 11-12 Math credit
Prerequisite(s): Algebra II and Geometry
This course completes the preparation for college-level calculus. Students develop skills in advanced function analysis and the use of these functions for modeling applications. Concepts in trigonometry are extended to include circular and inverse functions. Analytic trigonometry is studied to apply in vector, parametric, and polar applications. Conic sections are reviewed and extended as are topics in discrete mathematics. Students use graphing calculators on a daily basis.
AP Calculus 1½ credits Grades 11-12 Math credit
Prerequisite(s): Pre-calculus and teacher recommendation
This full-year course prepares students for The College Board Advanced Placement Examination, level AB, which is equivalent to one semester of college calculus. This course includes a brief review of the algebraic and transcendental functions and a study of topics in the differential and integral calculus. Students taking this class participate in the AP Calculus exam. The primary text is Calculus: Graphical, Numerical, Algebraic, 3rd ed. (Finney, Demana, Waits, Kennedy).
Math Workshop 1 credit Grades 9-10 Elective credit
Prerequisite(s): Teacher recommendation
This class is designed to prepare a student for algebra at the high school level. The course is a review of necessary and fundamental arithmetic skills. Students learn to cope with the frustrations of mathematics and pursue a variety of strategies to unlock the principles and procedures of mathematics. Organizational and study skills are emphasized in this elective course.
Introductory Algebra 1 credit Grades 9-10 Math credit
Prerequisite(s): Teacher recommendation
In this course, students investigate algebra using different methods and strategies to solve problems. Algebraic concepts are applied to real-world situations. Students learn the language of algebra to solve simple equations, work with algebraic expressions, and communicate mathematical thinking to others. Students use graphing and other data analysis to organize and understand mathematics. The text used is Algebra: Concepts and Applications (Glencoe McGraw-Hill).
Discovering Geometry 1 credit Grades 10-12 Math credit
Prerequisite(s): Algebra I: Part I or Introductory Algebra
This course is an introduction to the practical aspects of geometry. Topics include the properties of angles, triangles, quadrilaterals, circles and other two-dimensional shapes, area, perimeter, volume, ratio and proportion, geometric construction, and right angle trigonometry. Students complete a variety of two- and three-dimensional hands-on projects. The text used is Discovering Geometry: An Intuitive Approach (Key Curriculum).
Consumer Math 1 credit Grades 11-12 Math credit
Prerequisite(s): Teacher recommendation
Topics in this course relate to problems that consumers face in everyday life. Topics include: housing, income and expenses, taxes, consumer credit, banking and loans, insurance, and investments. Solid arithmetic skills are important. A calculator is strongly recommended. The text is Consumer Mathematics (Houghton, Mifflin). |
Construction Math Review Courses
How well do you know construction math? Being able to do quick, accurate calculations doesn't just make it easier to successfully complete your California State Contractors License exam - it's also something you need to have a profitable, successful career. With a focus on practical ideas and real-world examples, our construction math review course can teach you everything you need to become a successful professional. It's never been easier or more affordable to get the math skills you need to succeed as a contractor or construction professional.
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Learn basic construction math and the best ways to solve real-world problems
Fractions, decimals, and other construction math concepts in easy-to-understand lessons
Conversions between different measurement systems
Key formulas for areas, volumes, etc.
Advanced topics like application of volume problems, on-center problems, and how to calculate rise, slope, and pitch |
We will follow these topics pretty closely, though in a different order. This class should prepare students for the techniques involved in algebraic geometry and algebraic number theory. In particular, by understanding rings and their spectra, students will gain a geometric intuition for algebra, and an algebraic intuition for geometry.
Text, Exercises, and Lecture
We will be approaching these topics in the following way: we will be going through much of the book "Introduction to Commutative Algebra", by Atiyah and MacDonald, hereafter referred to as "A-M". This text covers almost all of the above topics, though we might digress occasionally to discuss some noncommutative topics. The real meat of the book is in the exercises. We will attempt to go through Chapters 1,2,3,5,6,7,8,9 of this book. Students will be required to work on a large number of exercises (essentially all of the exercises) from each of these chapters. Students are expected to work in small groups in order to facilitate this work.
Half of classtime will be devoted to lecture. Lecture will provide background, examples, and motivation, to accompany the technical material. The other half of classtime will be devoted to a discussion of the exercises. Students will be expected to present their solutions, and attempts at solutions. Every enrolled student will be required to speak on his or her solutions, at various times.
The exercises in A-M vary greatly in difficulty: students are not expected to solve every problem, but an earnest attempt is expected. Many problems rely on the results of other problems. It is nearly impossible to go through the problems in A-M, while skipping some along the way.
Grades
Grades will be determined by the quality of their problem solutions, as evidenced by written work and presentation. A take-home final will be given as well, covering the material throughout the course.
Administration
All details of the course will be given on the SlugMath wiki. The URL for this course is [1]. Students should create an account on the wiki by going to the wiki, and creating an account. Students should not edit the wiki, outside of discussion pages, without permission. Students can discuss problems and their solutions, using the discussion pages for each week. Discussion and collaboration is highly encouraged.
For additional information and advice, Marty can be contacted easily by e-mail at weissman AT ucsc DOT edu. Office hours are readily available by appointment.
Week 1
Days
MW, Sept. 29, Oct. 1.
Admin
Introductions. Syllabus. About the book and exercises. Grading. The Wiki.
Week 8
Week 9
Days
MW, Nov. 24, (26).
Problems
Students lecture on A-M, Ch. 6.
Week 10
Days
MW, Dec. 1,3.
Problems
Students lecture on A-M, Ch. 7
Math
Here are some Image:Math203Week10.pdf. Notes by the inventor can be found here. However, these notes are difficult (for me) to follow since I am not sure what ordering the author chooses for monomials. |
Townsend, MA Algebra wrote many papers and tutorials presented at simulation conferences that presented these concepts and applications. I also understand very well the use of determinants to solve simultaneous linear equations, and the development of inverse or transpose matrices. I am qualifiedHe is presently taking Calculus as a junior in high school.Algebra I encompasses the following concepts:
Variables that may represent numbers whose values are not yet known. Evaluating expressions based on the basic properties of arithmetic operations - addition, subtraction, multiplication, division, and exponentiation. Equations displaying properties of equality and inequality |
KS2 Revise and Shine is a simple eight-week revision programme with key information and practice test questions. The unique revision structure and highly visual content help to make revision effective...
Ensure top marks and complete coverage with Collins' brand new IGCSE Maths course for the Cambridge International Examinations syllabus 0580. Provide rigour with thousands of tried and tested questions using...
Studies show that children concentrate and perform better on basic maths exams when the questions are printed on a pleasant coloured background. This book combines that visual learning technique with...
Provides practice exercises based on the Level 3 Common Entrance Maths examination at 13+. Featuring a range of rigorous and challenging exam-style questions, this title includes exercises that are arranged... |
id: 05878659
dt: j
an: 2011c.00198
au: Roberts, Sally K.; Tayeh, Carla
ti: All these rays! what's the point?
so: Math. Teach. M. Sch. 16, No. 7, 408-413 (2011).
py: 2011
pu: National Council of Teachers of Mathematics (NCTM), Reston, VA
la: EN
cc: B50
ut: geometric concepts; geometry; elementary algebra; preservice teacher
education; mathematics skills; equations; mathematical concepts; middle
schools; secondary school mathematics
ci:
li:
ab: Summary: Every semester, the authors encounter students who are attracted
to the visual and spatial aspects of geometry. They have other students
who consider geometry to be challenging for the very same reasons.
Students are confounded not only by the fact that geometry relies on
visual interpretations but also because it has a language of its own
and because the rules can change depending on underlying assumptions.
Some students in their teacher-candidate classes testify that they are
good at algebra but never did well in geometry. The "good at algebra"
comment usually translates to being able to complete the symbol
manipulation and to "plug and chug" numbers into an equation to return
a correct answer. Deriving an equation or using algebra to generalize a
solution to a problem is a very different story. Students repeatedly
say, "Just give me the formula, and I can solve the problem," with
apparent disregard for the fact that discovering the algebraic equation
to solve the problem is the true demonstration of algebraic
understanding and proficiency. To help their teacher candidates prepare
to meet future classroom expectations, the authors begin their geometry
courses by introducing a series of carefully sequenced investigations.
Teacher candidates move beyond "Just give me a formula" to a deeper
understanding by exploring the algebra found in geometric models.
Integrating geometry and algebra helps students view mathematics as a
connected whole, bringing meaning to algebra through geometry.
(Contains 3 figures.) (ERIC)
rv: |
Book Description: This general review covers equations, functions, and graphs; limits, derivatives; integrals and antiderivatives; word problems; applications of integrals to geometry; and much more. Additional features make this volume especially helpful to students working on their own. They include worked-out examples, a summary of the main points of each chapter, exercises, and where needed, background material on algebra, geometry, and reading comprehension |
97815595306Each Mathercise book is a set of 50 blackline-master activities designed by Discovering Geometry author Michael Serra as warm-up activities for the start of class. Each activity takes 10 minutes and includes one reasoning exercise, one solving exercise, one sketching or graphing exercise, and space for a review exercise of your own design. |
Topic: this is a mathematics education question (but applies to other sciences too).
Assumtions
Assumptions
Specializing early
Topic: this is a mathematics education question (but applies to other sciences too).
Assumtions |
Course objectives: The purpose of this course is to introduce
students to the techniques and concepts of modern numerical analysis. The
course may be taken alone to provide an introduction to the ideas of numerical
analysis in the context of the simplest problems of analysis and algebra,
or,the student may continue with CSE/MATH 456 for a more complete
introduction and some more advanced applications. Numerical analysis is
the study of algorithms for computing numerical answers to mathematical
problems. We shall investigate algorithms for a variety of basic problems,
studying their reliability, efficiency, and computer implementation. In
comparison to 455 and 456, the course CSE/MATH 451 is more of an overview
of numerical algorithms, with less emphasis on analysis of the algorithms.
(Credit will not be given for both 451 and 455.)
Classroom and lab: On Mondays and Wednesdays the class will meet
in 115 McAllister, the McAllister
Technology Classroom. This room is equipped an X-windows terminal connected
to a high quality projection system, so that computer demonstrations can
be included in the class time. On Fridays we will meet in the High
Performance Computing Classroom in 215 Osmond. This room contains X-windows
terminals for all the students, and at least part of most Friday classes
will be devoted to guided computer explorations.
Prerequisites: Single variable and multivariable calculus; matrix
algebra, and a working knowledge of computer programming.The main computer
languages used will be FORTRAN and Matlab. Other languages may be acceptable
as well.
Grading: There will be two midterm exams, each worth 20% of the
grade, a final exam worth 25%, and homework and lab assignments worth 35%.
Course topics:
Floating point computation, error propagation
Rootfinding for nonlinear equations
Numerical solution of linear systems
Interpolation and approximation
Numerical integration
These topics correspond roughly to chapters 1, 2, 3, 4, 6, and 7 of the
text. |
Mathematics (SparkNotes.com) - WebCT.com & iTurf Inc.
Over 100 guides for mathematics ranging from pre-algebra topics to advanced work in calculus, written by students and recent graduates of Harvard University. The site also includes message boards for beginner, high school, and advanced math, calculus,
...more>>
Mathematics - Student Helpmate - Chris Divyak
Search or browse this archive of questions about algebra, calculus, geometry, statistics, trigonometry, and other college math; then pay for access to answers. To submit your own problem to Student Helpmate, type your question or upload it as a file;
...more>>
mathepower.com - Markus Hendler
A series of calculators for solving problems, for classes 1-10. Primarily in German, but with an English version. Themes include: fractions, geometry, equations, arithmetical operations, trigonometry.
...more>>
Mathepower - Markus Hendler
Online calculators for most calculations covered in the first ten years of school, from basic operations through the first stages of algebra. Available in English, German, and French.Math Fundamentals Problem of the Week - Math Forum
Math problems for students working with concepts of number, operation, and measurement, as well as introductory geometry, data, and probability. The goal is to challenge students with non-routine problems and encourage them to put their solutions into Homework Help
Email contact for homework help in pre-algebra, algebra I and II, college algebra, geometry, trigonometry, pre-calculus, and calculus. Site also contains a math history timeline; math dictionary; some basic differentiation and integration rules; trigonometricMath Index - George Mason University
A glossary of math terms with illustrated examples of problems and tips for understanding concepts, from addition of positive and negative numbers to the definition of the derivative. Part of the DAU (Defense Acquisition University) Math Refresher course.
...more>>
Math Kangaroo - Maria Omelanczuk
Math Kangaroo's international "not-selective competition" recognizes all students who participate in its 75 minute-long multiple choice test: "Each participant is seen as a winner and receives recognition and gifts on the test day in March." Open to any
...more>>
mathlab.com - Mehrdad Simkani
A Java applet of unmarked straightedge and collapsible compass tools to virtually draw the lines and circles fundamental to Euclidean geometry. The help page explains how to use the Euclid applet, and contains descriptions for constructing found in propositions
...more>>
Math Lessons - Cynthia Lanius
A variety of algebraic and geometric projects and puzzles that help students find mathematical patterns and formulas. The Million $ Mission explores exponential growth, a Fractal Geometry Unit teaches the secrets of fractals, the Mathematics of Cartography
...more>>
The Math Less Traveled - Brent Yorgey
A blog "dedicated to exploring beautiful mathematics." Posts, which date back to March, 2006, have included "Proof by animation," "The Nuclear Pennies Game," "Recounting the Rationals," "Predicting Pi," "Square roots with pencil and paper," "Irrationality
...more>>
MathMagic on the Web - Alan A. Hodson; The Math Forum
MathMagic was a K-12 telecommunications project developed in El Paso, Texas. The intent of the project was to provide motivation for students to use computer technology while increasing problem-solving strategies and communications skills. MathMagic posted
...more>>
Math Message Board - David Manura
A searchable Web-based discussion area for questions, answers, and talk on any math topic that allows for input of mathematical notation, diagrams, and HTML into message postings. Subject areas include General Questions; Elementary Math; Algebra/Geometry/Trig;
...more>>
Math Mistakes - Michael Pershan, editor
Compilation, analysis, and discussion of the mathematical errors that students make. Pershan, who teaches high school math in New York City, has posted a new error on this blog every day since June, 2012.
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MathNotations - David Marain
Math investigations, challenges, problems of the day, and standardized test practice that emphasize the development of conceptual understanding in mathematics. Marain's blog, which dates back December of 2006, also features dialogue on issues in mathematics |
Computational Introduction To Number Theory And Algebra - 05 edition
Summary: Number theory and algebra play an increasingly significant role in computing and communications, as evidenced by the striking applications of these subjects to such fields as cryptography and coding theory. This introductory book emphasises algorithms and applications, such as cryptography and error correcting codes, and is accessible to a broad audience. The mathematical prerequisites are minimal: nothing beyond material in a typical undergraduate course in calculus...show more is presumed, other than some experience in doing proofs - everything else is developed from scratch. Thus the book can serve several purposes. It can be used as a reference and for self-study by readers who want to learn the mathematical foundations of modern cryptography. It is also ideal as a textbook for introductory courses in number theory and algebra, especially those geared towards computer science students |
(Original post by Dreamweaver)
I was in exactly the same position as you a couple of weeks ago. Chapter 2 mixed is painful. Thankfully, the actual exam questions don't seem to be too bad.
Yeah the last 2 chapters are hard to self teach. Livemaths seems really good for this. Matrices aren't as bad as Vectors (IMHO) so it might be worth starting with those although vectors do pop up in one or two of the questions. How are you finding the integration?
(Original post by JohnyTheLad)Yeah it's like WTF at the beginning. Get the edexcel FP3 book. It explains most bits well. for a unit vector, you basically write the vector out and divide it by its modulus.
Scalar dor product -> you get a number
Vector cross product -> you get a vector
Both have useful applications, i.e. finding the area of a triangle etc..
No, not just death. I want it to be locked up in permanent spiked chastity, forced to worship the feet of the many women who were forced to go through D1 and be caned and whipped eternally. And as for D2, we can have it castrated and forced to become a sissy maid. |
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Math110 College Algebra
4 credits
Section 1MWF1:10 - 2:00MC 420
Th1:00 - 1:50MC 420(note this time!!)
Dr. Mark Saegrove
Office MC 525Ph. 796-3657Home Phone 1-608-735-4789
CATALOG COURSE DESCRIPTION:
Review of basic algebra, second degreeproblems and to be able to communication solutions and explore options.
3.Life Value Skills:
A. Develops an appreciation for the intellectual honesty of deductive reasoning; a mathematician's work must stand up to the scrutiny of logic, and it is unethical to try to pass off invalid work.
B. Understands the need to do one's own work, to honestly challenge yourself to
master the material.
4. Cultural Skills:
A.Learns to read, write and manipulate mathematical notation
B.Experiences mathematics as a culture of its own, with its own language and modes of thinking.
5.Aesthetic Skills:
A. Develops an appreciation for the austere intellectual beauty of deductive reasoning.
B. Develops an appreciation for mathematical elegance.
General Course Objectives: This course is designed to cause the student to learn traditional college algebra concepts and problem solving skills.It should serve to prepare students for Math 180, Math 230, Math 265, or Math 270.
Prerequisite: Acceptable placement score or C grade in Math 001 or equivalent (typically high school algebra). See me right away if you have a question about your math background as it relates to this requirement.
A valid verifiable excuse must be presented in order to make up missed exams or quizzes."
Extra Help: If you find that you need extra help, see me right away. Tutoring can be made available from the Learning Center if necessary320, 796-3085) within ten days to discuss your accommodation needs.
Note: accommodation for special test-taking needs will be made only after these needs are confirmed in writing by Mr. Wojciechowski. |
Description
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Identifier:
mathphyspapers02stokrich
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Evidence reported by scanner-julie-l for item mathphyspapers02stokrich on Mar 20, 2006; no visible copyright symbol and date found; stated date is 1883; the country of the source library is the United States; not published by the US government. |
getting started
getting started
I am in 12th grade and currently in AP calc. I am kinda bored with it and was wonering if there is any good online material that can let me learn calc faster and go beyond it into hiogher maths? thanks
The book I have isn't the best at explaining things, but I guess I can manage because I have never had trouble with math. I am not trying to take some magic pill, just my class is full of tards for the most part who get tripped up with simple things like limits. Also reccomendations for other maths would be appreciated.
Try checking out a book on linear algebra (matrices, vector spaces, and so forth). It's a freshman or sophomore-level college course that has enough ties to Calc that you'll be able to understand it with some patience.
The alternative would be to study Calculus III, which doesn't really depend on Calculus II for anything. It's actually pretty easy; I took it in high school. |
Mathematics and Its History
(Undergraduate Texts in MathematicsAcceptable 3rd ed. 2010New:
New BRAND NEW BOOK! Shipped within 24-48 hours. Normal delivery time is 5-12 days.
AwesomeBooksUK
OXON, GBR
$108.42
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About the Book
From a review of the second edition:
"This book covers many interesting topics not usually covered in a present day undergraduate course, as well as certain basic topics such as the development of the calculus and the solution of polynomial equations. The fact that the topics are introduced in their historical contexts will enable students to better appreciate and understand the mathematical ideas involved...If one constructs a list of topics central to a history course, then they would closely resemble those chosen here."
(David Parrott, Australian Mathematical Society)
This book offers a collection of historical essays detailing a large variety of mathematical disciplines and issues; it 's accessible to a broad audience. This third edition includes new chapters on simple groups and new sections on alternating groups and the Poincare conjecture. Many more exercises have been added as well as commentary that helps place the exercises in context. |
"The mathematical sciences particularly exhibit order, symmetry, and limitation; and these are the greatest forms of the beautiful." -Aristotle
The mission of the Mathematics Department of Miami Dade College, Kendall Campus, is to support the college in its efforts to provide a high quality education by keeping the diversity of our learners and their needs at the center of our decision-making with regard to curriculum, instruction, support, and assessment. Specifically, the Kendall Campus Mathematics Department supports a vision that is based upon the vital importance of the first years of collegiate mathematics education-an education that prepares the learner for a future role in society by engaging him or her in instruction that includes fluency, understanding, reasoning, and communication skills needed to comprehend mathematics and its applications. Scientifically and technologically literate, the learner becomes a resourceful problem solver ready to make educated personal and professional decisions, and amply prepared to contribute to the improvement of our global society.
"Transire Summ Pectus Mundoque Potiri"
(Transcend Human Limitations and Master the Universe)
Our faculty and staff are here to assist students in every possible way. Please feel free to browse through our department site and find out about the services we provide for students |
Get the grade you want in algebra with BEGINNING & INTERMEDIATE ALGEBRA by Gustafson, Karr, and Massey! Written with you in mind, the authors guide you through formerly difficult concepts with their clearly written language and helpful learning tools. Prepare for exams with numerous resources located online and throughout the text such as Guided Practice exercises, online homework problems, and Chapter Summaries in an easy-to-read grid format. Use this text, and you'll learn solid mathematical skills that will help you both in future mathematical courses and in real life |
Product Details:
CSET Foundation-Level Mathematics 110, 111 Includes 11 competencies/skills found on CSET Mathematics Subtests I and II and 125 sample-test questions. This guide, aligned specifically to standards prescribed by the California Department of Education, covers the sub-areas of Algebra; Geometry; Number Theory; and Probability and Statistics.
Description:
This guide, aligned specifically to standards prescribed by the Colorado
Department of Education, Includes 21 competencies/skills found on the PLACE Science test and 125 sample test questions on scientific inquiry and connections, physical science, life science, and Earth and ...
Description:
Are you ready to teach? Don''t let a certification exam
delay your career. Practice for the real exam with this 40 question and 3 constructed response practice test that covers the core content found on the CSET Foundational Level ... |
A student may NOT take this course
for credit if he/she already has credit for a college algebra or calculus
course.
General:
Math 167 is a preparatory
course for calculus. Students interested in taking calculus but feel that
they are not ready for it should take this class. For a few majors
Math 167 is the prefered mathematics course. To check which majors require
calculus or precalculus, go to "List of Majors by
Math Course requirement".
Prerequisite:
None.
Description:
The study of elementary functions, their
analysis and application. Included are investigations of polynomial, rational,
exponential, and trigonometric functions. |
MTH:160B COLLEGE ALGEBRA WITH CALCULATOR TI-83: NON-TECHNICAL MAJORS
MATHEMATICS DEPARTMENT'S STUDENT OBJECTIVES AND ASSIGNMENTS
A Student Should Be Able To:
Introduction to Graphs and the Graphing Calculator { REVIEW }* · Plot points by hand and using a grapher
· Graph equations by hand and using a grapher
· Find the point(s) of intersection of two graphs
FINISH THE ASSIGNMENT OF ALL ODD EXERCISES
Chapter R Basic Concepts of Algebra { REVIEW }
R.1 The Real-Number System · Identify various kinds of real numbers
· Use interval notation to write a set of numbers
· Identify the properties of real numbers
· Find the absolute value of a real number
FINISH THE ASSIGNMENT OF: (ask your professor)
R.2 Integer Exponents, Scientific Notation, and Order of Operations · Simplify expressions with integer exponents
· Solve problems using scientific notation
· Use the rules for order of operations
FINISH THE ASSIGNMENT OF ALL ODD EXERCISES
1.1 Functions, Graphs, and Graphers · Determine whether a correspondence or relation is a function
· Find function values, or outputs, using a formula
· Find the domain and the range of a function
· Determine whether a graph is that of a functions
· Solve applied problems using functions
FINISH THE ASSIGNMENT OF ALL ODD EXERCISES
1.3 Modeling: Data Analysis, Curve Fitting, and Linear Regression · Analyze a set of data to determine whether it can be modeled
by a linear function
· Fit a regression line to a set of data; then use the linear model
to make predictions
FINISH THE ASSIGNMENT OF ALL ODD EXERCISES
1.4 More on Functions · Graph functions, looking for intervals on which the function
is increasing, decreasing, or constant, and determine
relative maxima and minima
· Given an application, find a function formula that models the
application; find the domain of the function and function values, and then graph the function
· Graph functions defined piecewise
· Find the sum, the difference, the product, and the quotient
of two functions, and
determine their domains
FINISH THE ASSIGNMENT OF ALL ODD EXERCISES
1.5 Symmetry and Transformations · Determine whether a graph is symmetric with respect to
the x-axis, the y-axis, and the origin
· Determine whether a function is even, odd, or neither
even nor odd
· Given the graph of a function, graph its transformation
under translations, reflections, stretchings, and shrinkings
FINISH THE ASSIGNMENT OF ALL ODD EXERCISES
1.7 Distance, Midpoints, and Circles · Find the distance between two points in the plane and find
the midpoint of a segment
· Find an equation of a circle with a given center and radius, and
given an equation of a circle, find the center and the radius
· Graph equations of circles
FINISH THE ASSIGNMENT OF ALL ODD EXERCISES
2.3 Zeros of Quadratic Functions and Models · Find zeros of quadratic functions and solve quadratic
equations by completing the square and by using
the quadratic formula
· Solve equations that are reducible to quadratic
· Solve applied problems
FINISH THE ASSIGNMENT OF ALL ODD EXERCISES
2.4 Analyzing Graphs of Quadratic Functions · Find the vertex, the line of symmetry, and the maximum
or minimum value of a quadratic function using the
method of completing the square
· Graph quadratic functions
· Solve applied problems involving maximum and minimum
function values
FINISH THE ASSIGNMENT OF ALL ODD EXERCISES
3.1 Polynomial Functions and Modeling · Use a grapher to graph a polynomial function and find its
real-number zeros relative maximum and minimum
values, and domain and range
· Solve applied problems using polynomial models
· Fit linear, quadratic, power, cubic, and quartic polynomial
functions to date and make predictions
FINISH THE ASSIGNMENT OF ALL ODD EXERCISES
3.2 Polynomial Division; The Remainder and Factor Theorems · Perform long division with polynomials and determine whether one polynomial is a factor of another
· Use synthetic division to divide a polynomial by (x - c)
· Use the remainder theorem to find a function value f(c)
· Use the factor theorem to determine whether (x - c) is a factor of f(x)
FINISH THE ASSIGNMENT OF ALL ODD EXERCISES
3.3 Theorems about Zeros of Polynomial Functions · Factor polynomial functions and find the zeros of their multiplicities
· Find a polynomial with specified zeros
· Find a polynomial function with integer coefficients, find the rational
zeros and the other zeros, if possible with this level of algebra
FINISH THE ASSIGNMENT OF ALL ODD EXERCISES
4.1 Composite and Inverse Functions · Find the composition of two functions and the domain of the
composition; decompose a function as a composition
of two functions
· Determine whether a function is one-to-one, and if it is, find a
formula for its inverse
· Simplify expressions of the type (f • f -1)(x) and (f -1 • f)(x)
FINISH THE ASSIGNMENT OF ALL ODD EXERCISES
4.3 Logarithmic Functions and Graphs · Graph logarithmic functions
· Convert between exponential and logarithmic equations
· Find common and natural logarithms using a grapher
FINISH THE ASSIGNMENT OF ALL ODD EXERCISES
4.4 Properties of Logarithmic Functions · Convert from logarithms of products, powers, and quotients to
expressions in terms of individual logarithms, and conversely
· Simplify expressions of the type loga(ax) and alogax FINISH THE ASSIGNMENT OF ALL ODD EXERCISES
5.1 Systems of Equations in Two Variables · Solve a system of two linear equations in two variables by graphing
· Solve a system of two linear equations in two variables by using
the substitution and the elimination methods
· Use systems of two linear equations to solve applied problems
FINISH THE ASSIGNMENT OF ALL ODD EXERCISES
5.2 Systems of Equations in Three Variables · Solve systems of linear equations in three variables
· Use systems of three equations to solve applied problems
· Model a situation using a quadratic function
FINISH THE ASSIGNMENT OF ALL ODD EXERCISES
5.3 Matrices and Systems of Equations · Solve systems of equations using matrices
FINISH THE ASSIGNMENT OF ALL ODD EXERCISES
5.4 Matrix Operations · Add, subtract, and multiply matrices when possible
· Write a matrix equation equivalent to a systems of equations
FINISH THE ASSIGNMENT OF ALL ODD EXERCISES
5.5 Inverses of Matrices · Find the inverse of a square matrix, if it exists
· Use inverses of matrices to solve systems of equations
FINISH THE ASSIGNMENT OF ALL ODD EXERCISES
6.1 The Parabola · Given an equation of a parabola, complete the square, if necessary,
and then find the vertex, the focus, and the directrix and
graph the parabola
FINISH THE ASSIGNMENT OF ALL ODD EXERCISES
6.2 The Circle (and the Ellipse [ OPTIONAL ]) · Given an equation of a circle, complete the square, if necessary,
and then find the center and the radius and graph the circle
· Given an equation of an ellipse, complete the square, if necessary,
and then find the center, the vertices, and the foci and graph
the ellipse
FINISH THE ASSIGNMENT OF: (ask your professor)
6.3 The Hyperbola [ OPTIONAL ] · Given an equation of a hyperbola, complete the square, if necessary,
and then find the center, the vertices, and the foci and graph
the hyperbola
FINISH THE ASSIGNMENT OF: (ask your professor)
6.4 Nonlinear Systems of Equations · Solve a nonlinear system of equations
· Use nonlinear systems of equations to solve applied problems
FINISH THE ASSIGNMENT OF ALL ODD EXERCISES
Chapter 7 Sequences, Series, and Combinatorics
7.1 Sequences and Series · Find terms of sequences given the nth term
· Look for a pattern in a sequence and try to determine a general term
· Convert between sigma notation and other notation for a series
· Construct the terms of a recursively defined sequence
FINISH THE ASSIGNMENT OF ALL ODD EXERCISES
7.2 Arithmetic Sequences and Series · For any arithmetic sequences, find the nth term when n is given
and n when the nth term is given, and given two terms,
find the common difference and construct the sequence
· Find the sum of the first n terms of an arithmetic sequence
FINISH THE ASSIGNMENT OF ALL ODD EXERCISES
7.3 Geometric Sequences and Series · Identify the common ratio of a geometric sequence, and find
a given term and the sum of the first n terms
· Find the sum of an infinite geometric series, if it exists
FINISH THE ASSIGNMENT OF ALL ODD EXERCISES
PROJECT #7
7.4 Mathematical Induction [ OPTIONAL ] · List the statements of an infinite sequence that is defined by a formula
· Do proofs by mathematical induction
FINISH THE ASSIGNMENT OF: (ask your professor)
7.7 The Binomial Theorem · Expand a power of a binomial using Pascal's triangle or factorial notation
· Find a specific term of a binomial expansion
· Find the total number of subsets of a set of n objects
FINISH THE ASSIGNMENT OF ALL ODD EXERCISES
A Descartes' Rule of Signs · Use Descartes' rule of signs to find information about the number
of real zeros of a polynomial function with real coefficients
FINISH THE ASSIGNMENT OF: (ASK YOUR PROFESSOR)
* { REVIEW } topics covered in prerequisite courses and skills needed for this course. Students are expected to brush up on these topics to the point of knowing the concepts at a problem solving conversational level every day of the course.
** [ OPTIONAL ] means your professor will determine the need for covering this OPTIONAL topic based on the types of projects selected in your professor's syllabus. |
Alevel Core 2
This chapter explores negative areas. It covers understanding that areas below the x axis are negative, calculating areas under a curve, some or all of which may be under the x axis. Before attempting this chapter you must have prior [...]
This chapter explores sine and cosine graphs. It covers sketching sine and cosine graphs, finding angles which have same sine or cosine values, and solving simple equations such as sin x = 0.6 You must have prior knowledge of basic [...]
This chapter explores trigonometric equations. This is the first part of this part. The chapter covers solving harder trigonometric equations of the type for example 2x = 0.4 or cos (x – π/3) = √2. Before attempting this lesson you [...]
[...]
In this chapter we shall be exploring the cosine rule. In trigonometry we can use the cosine rule to find the missing sides and angles in a triangle. You must have some basic knowledge of trigonometry and bearings to appreciate [...]
This chapter covers the sine rule used in trigonometry. We shall be exploring how to use the sine rule to find missing sides and angles. You must have some basic knowledge of trigonometry. If we know the sides/angles in a [...]
For this lesion you must have some basic knowledge of integration. In this maths entry we shall be evaluating the area under the curve between certain limits. There are some applications where you will be required to find the area [...]
Expanding very large powered numbers can be quite tedious, since the expansion requires a lot of expanding over and over again. For example trying to expand (x+1)5 and (x+1)7, this would take a very long time and cause lot's of [...] |
Product Details
Uncommon Mathematical Excursions by Dan Kalman
This book presents an assortment of topics that extend the standard algebra-geometry-calculus curriculum of advanced secondary school and introductory college mathematics. It is intended as enrichment reading for anyone familiar with the standard curriculum, including teachers, scientists, engineers, analysts, and advanced students of mathematics. The book is divided into three parts each with a specific theme. In the first part, all of the topics are related to polynomials: properties and applications of Horner form, reverse and palindromic polynomials, identities linking roots and coefficients, among others. Topics in the second part are all connected in some way with maxima and minima. They include a new idea about an old approach to Lagrange multipliers, optimization as a method of proof, and some unusual max/min problems. In the final part calculus is the focus. Here the reader will find a limit-free development of differentiation, visually appealing treatment of envelopes and asymptotes, a rumination on the subject's surprising power and simplicity, and other topics. The book is particularly recommended for professional development and continuing education of secondary and college mathematics teachers. For more information, visit |
Lecture 26: Defining a plane in R3 with a point and normal vector
Embed
Lecture Details :
Determining the equation for a plane in R3 using a point on the plane and a normal vector
Course Description :
Matrices, vectors, vector spaces, transformations. Covers all topics in a first year college linear algebra course. This is an advanced course normally taken by science or engineering majors after taking at least two semesters of calculus (although calculus really isn't a prereq) so don't confuse this with regular high school algebra. |
Find a Picacho, AZ ScienceTo understand what it is all about we must understand what Calculus involves: derivatives and integrals which are functions derived from functions. So far math has been about numbers. Now the student must learn to see it from the perspective of functions: polynomial, rational, radical, exponential and trigonometric functions. |
Proofs in Algebra: Properties of Equality
In this lesson our instructor talks about proofs in algebra and properties of equality. She talks about addition property of equality, subtraction property of equality, multiplication property of equality, division property of equality, and addition property of equality using angles. She the talks more about the reflexive property of equality, symmetric property of equality, transitive property of equality, substitution property of equality, and the distributive property equality. She discusses the two column proof and does three proof examples. Four complete extra example videos round up this lesson.
This content requires Javascript to be available and enabled in your browser.
One way to organize deductive reasoning is by using a two-column proof
Proofs in Algebra: Properties of Equality
Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture. |
Understandable Statistics (Hardcover)
9780618949922
ISBN:
0618949925
Edition: 9 Publisher: Houghton Mifflin Company
Summary: This algebra based text is a thorough yet approachable statistics guide for students. The new edition addresses the growing importance of developing students' critical thinking and statistical literacy skills with the introduction of new features and exercises |
Mathematics in Education
FindGraph is a graphing, curve-fitting, and digitizing tool for engineers, scientists and business. Discover the model that best describes your data. FindGraph is a comprehensive graphing, curve fitting, and digitizing tool. FindGraph offers 12...
Easy CurveFit - A very easy tool to fit curve includes linear and nonlinear. Easy CurveFit - A very easy tool to fit curve includes linear and nonlinear. Easy CurveFit is a very easy tool to fit curve. Only three steps you should do, and...
Crocodile Mathematics is user-friendly mathematical modeling software for secondary school geometry and numeracy. Crocodile Mathematics is user-friendly mathematical modeling software for secondary school geometry and numeracy. Using this...
A powerful mathematical expressions editor. A powerful mathematical expressions editor. Formulator is aware of presentation and semantics face of mathematics, and so it allows not only to visually edit, but also to calculate simple Content MathML...
Description: This script provides a GUI-based approach to stopping and starting the Folding@home service for Intel Mac users who already have fah6 installed and configured correctly. Description: This script provides a GUI-based approach to...
An antenna design and electronics/electrical tool package. An antenna design and electronics/electrical tool package. It takes you step by step through the design of the following types of antennas:- Dipole- Fat Dipole- Yagi- J-Pole- Super J-Pole-...
A&G Grapher is a powerful graphing application that allows you to draw any 2D and 3D mathematical equation. A&G Grapher is a powerful graphing application that allows you to draw any 2D and 3D mathematical equation. It can be used by...
A problem analyzer is now available. A problem analyzer is now available. It generates an simple report with of statisics and information about the optimization problem and relevant warnings about the problem formulation are included. A solution...
Timez Attack is an educational game that kids can use to practice the multiplication tables. Timez Attack is an educational game that kids can use to practice the multiplication tables. You will help the character you have chosen (a male or...
This simple demo allows the user to experiment with a wide variety of 2-dimensional triangle groups. This simple demo allows the user to experiment with a wide variety of 2-dimensional triangle groups. In the process the users gains a direct...
Calclipse™ is a math tool written in Java™. Calclipse™ is a math tool written in Java™. Calclipse comprises a scripting framework containing a software library for evaluating mathematical expressions. The Calclipse...
Although a large number of tools for the three-dimensional visualization of molecular structures exists, it often proved convenient to tailor a visualization tool specifically for a certain purpose. Although a large number of tools for the...
Realzone Table is a simple and truly useful utility that will help you study, learn, and double-check your mathematical tables. Realzone Table is a simple and truly useful utility that will help you study, learn, and double-check your mathematical...
A native OSX application for viewing analog and digital simulation waveforms from VCD files as well as transaction level modeling (TLM) traces. A native OSX application for viewing analog and digital simulation waveforms from VCD files as well as...
A minimalistic tool for molecular phylogenetics able to recover phylogenies from nucleotide sequences. A minimalistic tool for molecular phylogenetics able to recover phylogenies from nucleotide sequences. PhyloCoco uses the likelihood or the...
An antenna analysis program. An antenna analysis program. Any type of antenna may be analyzed. The physical design of the antenna is entered (such as the lengths of wires and elements). For a given frequency, the feedpoint impedance is calculated,...
POLYMATH Educational is a computational system that has been created for educational or professional purpose. POLYMATH Educational is a computational system that has been created for educational or professional purpose. Polymath Educational...
Bell Curve is a very useful and small windows application wich serves to help senior students to understand the curve mechanism and introduce them to the Normal Curve. Bell Curve is a very useful and small windows application wich serves to help...
This is a product which can be found most useful by scientists, engineers, professors, and students. This is a product which can be found most useful by scientists, engineers, professors, and students. This calculator follows classical approach...
Partial Derivatives is a simple program that finds approximate numerical values for the 1st and 2nd order partial derivatives of a function at a given point. Partial Derivatives is a simple program that finds approximate numerical values for the... |
Math trainer soft/freeware
Hey, does anyone knwo of a program that can train you in different basic fields of mathematics, and preferrably keep track of performance statistics, so pin-pointing individual weaknesses would be easier?
I would find this kind of a program very useful and more motivating than simply going through problems in books. |
PARKSIDE HIGH SCHOOL MATHEMATICS DEPARTMENT
PARKSIDE HIGH SCHOOL MATHEMATICS DEPARTMENT
Course Title: MAT1L
Course Description:
This course emphasizes further development of mathematical knowledge and skills to prepare students
for success in their everyday lives, in the workplace, and in the Grade 11 workplace course.
This course is organized in three strands related to money sense, measurement, and proportional
reasoning. In all strands, the focus is on developing and consolidating key foundational mathematical
concepts and skills by solving authentic, everyday problems.
Students have opportunities to further develop their mathematical literacy and problem solving skills,
and to continue developing their skills in reading, writing, and oral language through relevant and
practical math activities.
Course Level: Level 1 MAT2L
Course Description:
This course emphasizes the extension of mathematical knowledge and skills to prepare students for
success in their everyday lives, in the workplace, and in the Grade 11 Mathematics Workplace
Preparation course.
This course is organized in three strands related to money sense, measurement, and proportional
reasoning. In all strands, the focus is on strengthening and extending key foundational mathematical
concepts and skills by solving authentic, everyday problems.
Students have opportunities to extend their mathematical literacy and problem solving skills, and to
continue developing their skills in reading, writing, and oral language through relevant and practical math
activities.
Course Level: Level 23E
Course Description:
This course enables students to broaden their understanding of mathematics as it applies in the
workplace and daily life. Students will solve problems associated with earning money, paying taxes, and
making purchases; apply calculations of simple and compound interest in saving, investing, and
borrowing; and calculate the costs of transportation and travel in a variety of situations. Students will
consolidate their mathematical skills as they solve problems and communicate their thinking.
Course Level: Level 34E
Course Description:
This course enables students to broaden their understanding of mathematics as it is applied in important
areas of daily living. Students will use statistics in investigating questions of interest and apply principles
of probability in familiar situations. The students will also investigate accommodation costs and create
household budgets; solve problems involving estimation and measurement.
Course Level: Level 41P
Course Description:
This course enables students to develop an understanding of mathematical concepts related to
introductory algebra, proportional reasoning, and measurement and geometry through investigation, the
effective use of technology, and hands-on activities. Students will investigate real-life examples to
develop various representations of linear relations, and will determine the connections between the
representations. They will also explore certain relationships that emerge from the measurement of three-
dimensional figures and two-dimensional shapes. Students will consolidate their mathematical skills as
they solve problems and communicate their thinking.
Course Level: Level 1 70% EQAO: 10% Culminating Activity & Final Exam:2P
Course Description:
This course enables students to consolidate their understanding of linear relations and extend their
problem-solving and algebraic skills through investigation, the effective use of technology, and hands-on
activities. Students will develop and graph equations in analytic geometry; solve and apply linear systems,
using real-life examples; and explore and interpret graphs of quadratic relations. Students will investigate
similar triangles, the trigonometry of right triangles, and the measurement of three-dimensional figures.
Students will consolidate their mathematical skills as they solve problems and communicate their
thinking.
Course Level: Level 2 MBF3C
Course Description:
This course enables students to broaden their understanding of mathematics as a problem-solving in the
real world. Students will extend their understanding of quadratic relations; investigate situations
involving exponential growth; solve problems involving compound interest; solve financial problems
connected with vehicle ownership; develop their ability to reason by collecting, analysing, and evaluating
data involving one variable; connect probability and statistics; and solve problems in geometry and
trigonometry. Students will consolidate their mathematical skills as they solve problems and
communicate their thinking.
Course Level: Level 3 MAP4C
Course Description:
This course enables students to broaden their understanding of real-world applications of mathematics.
Students will analyse data using statistical methods; solve problems involving application of geometry
and trigonometry; solve financial problems connected with annuities, budgets, and renting or owning
accommodations; simplify expressions; and solve equations. Students will reason mathematically and
communicate their thinking as they solve multi-step problems. This course prepares students for college
programs in areas such as business, health sciences, human services, and for certain skilled trades.
Course Level: Level 41D
Course Description:
This course enables students to develop an understanding of mathematical concepts related to algebra,
analytic geometry, and measurement and geometry through investigation, effective use of technology
and abstract reasoning. Students will investigate relationships, which they will then generalize as
equations of lines, and will determine the connections between different representations of a linear
relation. They will also explore relationships that emerge from the measurement of three-dimensional
figures and two-dimensional shapes. Students will reason mathematically and communicate their
thinking as they solve multi-step problems.
Course Level: Level 1 EQAO: worth 10% Final Exam: worth2D
Course Description:
This course enables students to broaden their understanding of relationships and extend their problem-
solving algebraic skills through investigation, effective use of technology and abstract reasoning. Students
will explore quadratic relations and their applications; solve and apply linear systems; verify properties of
geometric figures using analytic geometry; and investigate the trigonometry of right and acute triangles.
Students will reason mathematically and communicate their thinking as they solve multi-step problems.
Course Level: Level 2F3M
Course Description:
This course introduces basic features of the function by extending students' experiences with quadratic
relations. It focuses on quadratic, trigonometric, and exponential functions and their use in modelling
real-world situations. Students will represent functions numerically, graphically, and algebraically; simply
expressions; solve equations; and solve problems relating to applications. Students will reason
mathematically and communicate their thinking as they solve multi-step problems.
Course Level: Level 3, University/CollegeCR3U
Course Description:
This course introduces the mathematical concept of the function by extending students' experiences with
linear and quadratic relations. Students will investigate properties of discrete and continuous functions,
including trigonometric and exponential functions; represent the functions numerically, algebraically, and
graphically; solve problems involving applications of functions; investigate inverse functions; and develop
facility in determining equivalent algebraic expressions. Students will reason mathematically and
communicate their thinking as they solve multi-step problems.
Course Level: Level 3 MDM4U
Course Description:
This course broadens students' understanding of mathematics as it relates to managing data. Students
will apply methods for organizing and analysing large amounts of information; solve problems involving
probability and statistics; and carry out a culminating investigation that integrates statistical concepts and
skills. Students will also refine their use of mathematical processes necessary for success in senior
mathematics. Students planning to enter university programs in business, social sciences, and the
humanities will find this course of particular interest.
Course Level: Level 4, University Preparation
Course Evaluation:
(a) Regular tests or assignments based on work done to date Project: 15% Final Exam: worth 15 MHF4U
Course Description:
This course extends students' experience with functions. Students will investigate the properties of
polynomial, rational, logarithmic, and trigonometric functions; develop techniques for combining
functions; broaden their understanding of rates of change; and develop facility in applying these concepts
and skills. Students will also refine their use of their use of the mathematical processes necessary for
success in senior mathematics. This course is intended both for students taking the Calculus and Vectors
course as a prerequisite for university program and for those wishing to consolidate their understanding
of mathematics before proceeding to any one of a variety of university programsV4U
Course Description:
This course builds on students' previous experience with functions and their developing understanding of
rates of change. Students will solve problems involving geometric and algebraic representations of
vectors and representations of lines and planes in three-dimensional space; broaden their understanding
of rates of change to include the derivatives of polynomial, sinusoidal, exponential, rational, and radical
functions; and apply these concepts and skills to the modelling of real-world relationships. Students will
also refine their use of the mathematical processes necessary for success in senior mathematics. This
course is intended for students who choose to pursue careers in fields such as science, engineering,
economics, and some areas of business, including those students who will be required to take a university
level calculus, linear algebra, or physics course |
...ractice Problems Involving Computation 1 Assume you have a black hole of initial mass M0 and some specified initial spin and that it accretes matter at the innermost stable circular orbit Write a computer program to calculate the dimensionless spin parameter j a M J M 2 of the hole as a function...
...omputability and Modeling Computation What are some really impressive things that computers can do Land the space shuttle and other aircraft from the ground Automatically track the location of a space or land vehicle Beat a grandmaster at chess Are there any things that computers can t do Yes...
...omputation Calculators and Common Sense A Position of the National Council of Teachers of Mathematics Question Is there a place for both computation and calculators in the math classroom NCTM Position School mathematics programs should provide students with a range of knowledge skills and tools...
1 0 1 Sample Computational Problems Factoring a polynomial To factor a polynomial place the insertion point inside or to the right of the polynomial select Factor from the Compute menu Example 1 5x5 5x4 1 0 2 10x3 10x2 5x 5 5 x 2 3 1 x 1 Finding the roots of a polynomial To nd the roots of a polynomial place the insertion point inside or to the rig...
...age 1 of 4 Models of computation indicates problems that have been selected for discussion in section time permitting Problem 1 In lecture we saw an enumeration of FSMs having the property that every FSM that can be built is equivalent to some FSM in that enumeration A We didn t deal with FSMs... |
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