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TEACHING. - -
National Council of Teachers of Mathematics Principles and Standards for School Mathematics
Principles for School Mathematics
Standards for School Mathematics
NUMBER AND OPERATIONS
DATA ANALYSIS AND PROBABILITY
REASONING AND PROOF
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Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics
PREKINDERGARTEN Number and Operations:
KINDERGARTEN Number and Operations:
GRADE 1 Number and Operations Algebra: -
Number and Operations:
GRADE 2 Number and Operations:
Number and Operations Algebra:
GRADE 3 Number and Operations Algebra: -
Number and Operations: -
GRADE 4 Number and Operations Algebra:
Number and Operations:
GRADE 5 Number and Operations Algebra: -
Number and Operations:
Geometry Measurement Algebra:
GRADE 6 Number and Operations: -
Number and Operations: -
GRADE 7 Number and Operations Algebra Geometry: -
Measurement Geometry Algebra: -
Number and Operations Algebra: -
GRADE 8 Algebra: -
Data Analysis Number and Operations Algebra: -
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MathematicsFor Elementary Teachers TENTH EDITION A C O N T E M P O R A R Y A P P R O A C H
Gary L. Musser Blake E. Peterson William F. Burger Oregon State University Brigham Young University
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Irene, my wonderful wife of 52 years who is the best mother our son could have; Greg, our son, for his inquiring mind; Maranda, our granddaughter, for her willingness to listen; my parents who have passed away, but always with me; and Mary Burger, my initial coauthor's daughter. G.L.M.
Shauna, my beautiful eternal companion and best friend, for her continual support of all my endeavors; my four children: Quinn for his creative enthusiasm for life, Joelle for her quiet yet strong confidence, Taren for her unintimidated ap- proach to life, and Riley for his good choices and his dry wit. B.E.P.
VICE PRESIDENT & EXECUTIVE PUBLISHER Laurie Rosatone PROJECT EDITOR Jennifer Brady SENIOR CONTENT MANAGER Karoline Luciano SENIOR PRODUCTION EDITOR Kerry Weinstein MARKETING MANAGER Kimberly Kanakes SENIOR PRODUCT DESIGNER Tom Kulesa OPERATIONS MANAGER Melissa Edwards ASSISTANT CONTENT EDITOR Jacqueline Sinacori SENIOR PHOTO EDITOR Lisa Gee MEDIA SPECIALIST Laura Abrams COVER & TEXT DESIGN Madelyn Lesure
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Library of Congress Cataloging-in-Publication Data
Musser, Gary L. Mathematics for elementary teachers : a contemporary approach / Gary L. Musser, Oregon State University,
William F. Burger, Blake E. Peterson, Brigham Young University. -- 10th edition. pages cm
Includes index. ISBN 978-1-118-45744-3 (hardback)
1. Mathematics. 2. Mathematics–Study and teaching (Elementary) I. Title. QA39.3.M87 2014 510.2’4372–dc23 2013019907
Printed in the United States of America
10 9 8 7 6 5 4 3 2 1
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Gary L. Musser is Professor Emeritus from Oregon State University. He earned both his B.S. in Mathematics Education in 1961 and his M.S. in Mathematics in 1963 at the University of Michigan and his Ph.D. in Mathematics (Radical Theory) in 1970 at the University of Miami in Florida. He taught at the junior and senior high, junior college college, and university levels for more than 30 years. He spent his final 24 years teaching prospective teachers in the Department of Mathematics at Oregon State University. While at OSU, Dr. Musser developed the mathematics component of the elementary teacher program. Soon after Profesor William F. Burger joined the OSU Department of Mathematics in a similar capacity, the two of them began to write the first edtion of this book. Professor Burger passed away during the preparation of the second edition, and Professor Blake E. Peterson was hired at OSU as his replacement. Professor Peter- son joined Professor Musser as a coauthor beginning with the fifth edition.
Professor Musser has published 40 papers in many journals, including the Pacific Journal of Mathematics, Canadian Journal of Mathematics, The Mathematics Association of America Monthly, the NCTM’s The Mathematics Teacher, the NCTM’s The Arithmetic Teacher, School Science and Mathematics, The Oregon Mathematics Teacher, and The Computing Teacher. In addition, he is a coauthor of two other college mathematics books: College Geometry—A Problem-Solving Approach with Applications (2008) and A Mathematical View of Our World (2007). He also coauthored the K-8 series Mathematics in Action. He has given more than 65 invited lectures/ workshops at a variety of conferences, including NCTM and MAA conferences, and was awarded 15 federal, state, and local grants to improve the teaching of mathematics.
While Professor Musser was at OSU, he was awarded the university’s prestigious College of Science Carter Award for Teaching. He is currently living in sunny Las Vegas, were he continues to write, ponder the mysteries of the stock market, enjoy living with his wife and his faithful yellow lab, Zoey.
Blake E. Peterson is currently a Professor in the Department of Mathematics Educa- tion at Brigham Young University. He was born and raised in Logan, Utah, where he graduated from Logan High School. Before completing his BA in secondary mathe- matics education at Utah State University, he spent two years in Japan as a missionary for The Church of Jesus Christ of Latter Day Saints. After graduation, he took his new wife, Shauna, to southern California, where he taught and coached at Chino High School for two years. In 1988, he began graduate school at Washington State Univer- sity, where he later completed a M.S. and Ph.D. in pure mathematics.
After completing his Ph.D., Dr. Peterson was hired as a mathematics educator in the Department of Mathematics at Oregon State University in Corvallis, Oregon, where he taught for three years. It was at OSU where he met Gary Musser. He has since moved his wife and four children to Provo, Utah, to assume his position at Brigham Young University where he is currently a full professor.
Dr. Peterson has published papers in Rocky Mountain Mathematics Journal, The American Mathematical Monthly, The Mathematical Gazette, Mathematics Magazine, The New England Mathematics Journal, School Science and Mathematics, The Journal of Mathematics Teacher Education, and The Journal for Research in Mathematics as well as chapters in several books. He has also published in NCTM’s Mathematics Teacher, and Mathematics Teaching in the Middle School. His research interests are teacher education in Japan and productive use of student mathematical thinking during instruction, which is the basis of an NSF grant that he and 3 of his colleagues were recently awarded. In addition to teaching, research, and writing, Dr. Peterson has done consulting for the College Board, founded the Utah Association of Mathematics Teacher Educators, and has been the chair of the editorial panel for the Mathematics Teacher.
Aside from his academic interests, Dr. Peterson enjoys spending time with his family, fulfilling his church responsi- bilities, playing basketball, mountain biking, water skiing, and working in the yard.
ABOUT THE AUTHORS
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Are you puzzled by the numbers on the cover? They are 25 different randomly selected counting numbers from 1 to 100. In that set of numbers, two different arithmetic pro- gressions are highlighted. (An arithmetic progression is a sequence of numbers with a common difference between consecutive pairs.) For example, the sequence highlighted in green, namely 7, 15, 23, 31, is an arithmetic progression because the difference between 7 and 15 is 8, between 15 and 23 is 8, and between 23 and 31 is 8. Thus, the sequence 7, 15, 23, 31 forms an arithmetic progression of length 4 (there are 4 numbers in the sequence) with a common difference of 8. Similarly, the numbers highlighted in red, namely 45, 69, 93, form another arithmetic progression. This progression is of length 3 which has a common difference of 24.
You may be wondering why these arithmetic progressions are on the cover. It is to acknowledge the work of the mathematician Endre Szemerédi. On May 22, 2012, he was awarded the $1,000,000 Abel prize from the Norwegian Academy of Science and Letters for his analysis of such progressions. This award recognizes mathematicians for their contributions to mathematics that have a far reaching impact. One of Pro- fessor Szemerédi’s significant proofs is found in a paper he wrote in 1975. This paper proved a famous conjecture that had been posed by Paul Erdös and Paul Turán in 1936. Szemerédi’s 1975 paper and the Erdös/Turán conjecture are about finding arith- metic progressions in random sets of counting numbers (or integers). Namely, if one randomly selects half of the counting numbers from 1 and 100, what lengths of arith- metic progressions can one expect to find? What if one picks one-tenth of the numbers from 1 to 100 or if one picks half of the numbers between 1 and 1000, what lengths of arithmetic progressions is one assured to find in each of those situations? While the result of Szemerédi’s paper was interesting, his greater contribution was that the tech- nique used in the proof has been subsequently used by many other mathematicians.
Now let’s go back to the cover. Two progressions that were discussed above, one of length 4 and one of length 3, are shown in color. Are there others of length 3? Of length 4? Are there longer ones? It turns out that there are a total of 28 different arithmetic progressions of length three, 3 arithmetic progressions of length four and 1 progression of length five. See how many different progressions you can find on the cover. Perhaps you and your classmates can find all of them.
ABOUT THE COVER
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1 Introduction to Problem Solving 2
2 Sets, Whole Numbers, and Numeration 42
3 Whole Numbers: Operations and Properties 84
4 Whole Number Computation—Mental, Electronic, and Written 128
5 Number Theory 174
6 Fractions 206
7 Decimals, Ratio, Proportion, and Percent 250
8 Integers 302
9 Rational Numbers, Real Numbers, and Algebra 338
10 Statistics 412
11 Probability 484
12 Geometric Shapes 546
13 Measurement 644
14 Geometry Using Triangle Congruence and Similarity 716
15 Geometry Using Coordinates 780
16 Geometry Using Transformations 820
Epilogue: An Eclectic Approach to Geometry 877
Topic 1 Elementary Logic 881
Topic 2 Clock Arithmetic: A Mathematical System 891
Answers to Exercise/Problem Sets A and B, Chapter Reviews, Chapter Tests, and Topics Section A1
Contents of Book Companion Web Site
Resources for Technology Problems
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1 Introduction to Problem Solving 2 1.1 The Problem-Solving Process and Strategies 5 1.2 Three Additional Strategies 21
2 Sets, Whole Numbers, and Numeration 42 2.1 Sets as a Basis for Whole Numbers 45 2.2 Whole Numbers and Numeration 57 2.3 The Hindu–Arabic System 67
3 Whole Numbers: Operations and Properties 84 3.1 Addition and Subtraction 87 3.2 Multiplication and Division 101 3.3 Ordering and Exponents 116
4 Whole Number Computation—Mental, Electronic, and Written 128 4.1 Mental Math, Estimation, and Calculators 131 4.2 Written Algorithms for Whole-Number Operations 145 4.3 Algorithms in Other Bases 162
5 Number Theory 174 5.1 Primes, Composites, and Tests for Divisibility 177 5.2 Counting Factors, Greatest Common Factor, and Least
Common Multiple 190
6 Fractions 206 6.1 The Set of Fractions 209 6.2 Fractions: Addition and Subtraction 223 6.3 Fractions: Multiplication and Division 233
7 Decimals, Ratio, Proportion, and Percent 250 7.1 Decimals 253 7.2 Operations with Decimals 262 7.3 Ratio and Proportion 274 7.4 Percent 283
8 Integers 302 8.1 Addition and Subtraction 305 8.2 Multiplication, Division, and Order 318
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9 Rational Numbers, Real Numbers, and Algebra 338 9.1 The Rational Numbers 341 9.2 The Real Numbers 358 9.3 Relations and Functions 375 9.4 Functions and Their Graphs 391
10 Statistics 412 10.1 Statistical Problem Solving 415 10.2 Analyze and Interpret Data 440 10.3 Misleading Graphs and Statistics 460
11 Probability 484 11.1 Probability and Simple Experiments 487 11.2 Probability and Complex Experiments 502 11.3 Additional Counting Techniques 518 11.4 Simulation, Expected Value, Odds, and Conditional
12 Geometric Shapes 546 12.1 Recognizing Geometric Shapes—Level 0 549 12.2 Analyzing Geometric Shapes—Level 1 564 12.3 Relationships Between Geometric Shapes—Level 2 579 12.4 An Introduction to a Formal Approach to Geometry 589 12.5 Regular Polygons, Tessellations, and Circles 605 12.6 Describing Three-Dimensional Shapes 620
13 Measurement 644 13.1 Measurement with Nonstandard and Standard Units 647 13.2 Length and Area 665 13.3 Surface Area 686 13.4 Volume 696
14 Geometry Using Triangle Congruence and Similarity 716 14.1 Congruence of Triangles 719 14.2 Similarity of Triangles 729 14.3 Basic Euclidean Constructions 742 14.4 Additional Euclidean Constructions 755 14.5 Geometric Problem Solving Using Triangle Congruence
and Similarity 765
15 Geometry Using Coordinates 780 15.1 Distance and Slope in the Coordinate Plane 783 15.2 Equations and Coordinates 795 15.3 Geometric Problem Solving Using Coordinates 807
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16 Geometry Using Transformations 820 16.1 Transformations 823 16.2 Congruence and Similarity Using Transformations 846 16.3 Geometric Problem Solving Using Transformations 863
Epilogue: An Eclectic Approach to Geometry 877
Topic 1. Elementary Logic 881
Topic 2. Clock Arithmetic: A Mathematical System 891
Answers to Exercise/Problem Sets A and B, Chapter Reviews, Chapter Tests, and Topics Section A1
Contents of Book Companion Web Site Resources for Technology Problems
eManipulatives Spreadsheet Activities Geometer’s Sketchpad Activities
Technology Tutorials Spreadsheets Geometer’s Sketchpad Programming in Logo Graphing Calculators
Webmodules Algebraic Reasoning Children’s Literature Introduction to Graph Theory
Additional Resources Guide to Problem Solving Problems for Writing/Discussion Research Articles Web Links
Videos Book Overview Author Walk-Through Videos Children’s Videos
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W elcome to the study of the foundations of ele-mentary school mathematics. We hope you will find your studies enlightening, useful, and fun. We salute you for choosing teaching as a profession and hope that your experiences with this book will help prepare you to be the best possible teacher of mathematics that you can be. We have presented this elementary mathematics material from a variety of perspectives so that you will be better equipped to address that broad range of learning styles that you will encounter in your future students. This book also encourages prospective teachers to gain the ability to do the mathematics of elementary school and to understand the underlying concepts so they will be able to assist their students, in turn, to gain a deep understand- ing of mathematics.
We have also sought to present this material in a man- ner consistent with the recommendations in (1) The Mathematical Education of Teachers prepared by the Conference Board of the Mathematical Sciences, (2) the National Council of Teachers of Mathematics’ Standards Documents, and (3) The Common Core State Standards for Mathematics. In addition, we have received valuable advice from many of our colleagues around the United States through questionnaires, reviews, focus groups, and personal communications. We have taken great care to respect this advice and to ensure that the content of the book has mathematical integrity and is accessible and helpful to the variety of students who will use it. As al- ways, we look forward to hearing from you about your experiences with our text.
GARY L. MUSSER, email@example.com BLAKE E. PETERSON, firstname.lastname@example.org
Unique Content Features Number Systems The order in which we present the number systems in this book is unique and most relevant to elementary school teachers. The topics are covered to parallel their evolution historically and their development in the elementary/middle school curriculum. Fractions and integers are treated separately as an extension of the whole numbers. Then rational numbers can be treated at a brisk pace as extensions of both fractions (by adjoining their opposites) and integers (by adjoining their appro- priate quotients) since students have a mastery of the concepts of reciprocals from fractions (and quotients) and opposites from integers from preceding chapters. Longtime users of this book have commented to us that this whole numbers-fractions-integers-rationals-reals
approach is clearly superior to the seemingly more effi- cient sequence of whole numbers-integers-rationals-reals that is more appropriate to use when teaching high school mathematics.
Approach to Geometry Geometry is organized from the point of view of the five-level van Hiele model of a child’s development in geometry. After studying shapes and measurement, geometry is approached more formally through Euclidean congruence and similarity, coordinates, and transformations. The Epilogue provides an eclectic approach by solving geometry problems using a variety of techniques.
Additional Topics Topic 1, “Elementary Logic,” may be used anywhere in a course.
Topic 2, “Clock Arithmetic: A Mathematical System,” uses the concepts of opposite and reciprocal and hence may be most instructive after Chapter 6, “Fractions,” and Chapter 8, “Integers,” have been completed. This section also contains an introduction to modular arithmetic.
Underlying Themes Problem Solving An extensive collection of problem- solving strategies is developed throughout the book; these strategies can be applied to a generous supply of problems in the exercise/problem sets. The depth of problem-solving coverage can be varied by the number of strategies selected throughout the book and by the problems assigned.
Deductive Reasoning The use of deduction is pro- moted throughout the book The approach is gradual, with later chapters having more multistep problems. In particular, the last sections of Chapters 14, 15, and 16 and the Epilogue offer a rich source of interesting theo- rems and problems in geometry.
Technology Various forms of technology are an inte- gral part of society and can enrich the mathematical understanding of students when used appropriately. Thus, calculators and their capabilities (long division with remainders, fraction calculations, and more) are introduced throughout the book within the body of the text.
In addition, the book companion Web site has eMa- nipulatives, spreadsheets, and sketches from Geometer’s
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Sketchpad®. The eManipulatives are electronic versions of the manipulatives commonly used in the elementary classroom, such as the geoboard, base ten blocks, black and red chips, and pattern blocks. The spreadsheets contain dynamic representations of functions, statistics, and probability simulations. The sketches in Geometer’s Sketchpad® are dynamic representations of geomet- ric relationships that allow exploration. Exercises and problems that involve eManipulatives, spreadsheets, and Geometer’s Sketchpad® sketches have been integrated into the problem sets throughout the text.
Course Options We recognize that the structure of the mathematics for elementary teachers course will vary depending upon the college or university. Thus, we have organized this text so that it may be adapted to accommodate these differences.
Basic course: Chapters 1-7 Basic course with logic: Topic 1, Chapters 1–7 Basic course with informal geometry: Chapters 1–7,
12 Basic course with introduction to geometry and mea-
surement: Chapters 1–7, 12, 13
Summary of Changes to the Tenth Edition
Mathematical Tasks have been added to sections throughout the book to allow instructors more flex- ibility in how they choose to organize their classroom instruction. These tasks are designed to be investigated by the students in class. As the solutions to these tasks are discussed by students and the instructor, the big ideas of the section emerge and can be solidified through a classroom discussion.
Chapter 6 contains a new discussion of fractions on a number line to be consistent with the Common Core standards.
Chapter 10 has been revised to include a discus- sion of recommendations by the GAISE document and the NCTM Principles and Standards for School Mathematics. These revisions include a discussion of steps to statistical problem solving. Namely, (1) formulate questions, (2) collect data, (3) organize and display data, (4) analyze and interpret data. These steps are then applied in several of the examples through the chapter.
Chapter 12 has been substantially revised. Sections 12.1, 12.2, and 12.3 have been organized to parallel the first three van Hiele levels. In this way, students will be able to pass through the levels in a more meaningful fashion so that they will get a strong feeling about how
their students will view geometry at various van Hiele levels.
Chapter 13 contains several new examples to give stu- dents the opportunity to see how the various equations for area and volume are applied in different contexts.
Children’s Videos are videos of children solving math- ematical problems linked to QR codes placed in the margin of the book in locations where the content being discussed is related to the content of the prob- lems being solved by the children. These videos will bring the mathematical content being studied to life.
Author Walk-Throughs are videos linked to the QR code on the third page of each chapter. These brief videos are of an author, Blake Peterson, describing and showing points of major emphasis in each chapter so students’ study can be more focused.
Children’s Literature and Reflections from Research margin notes have been revised/refreshed.
Common Core margin notes have been added through- out the text to highlight the correlation between the content of this text and the Common Core standards.
Professional recommendation statements from the Common Core State Standards for Mathematics, the National Council of Teachers of Mathematics’ Principles and Standards for School Mathematics, and the Curriculum Focal Points, have been compiled on the third page of each chapter.
Pedagogy The general organization of the book was motivated by the following mathematics learning cube:
The three dimensions of the cube—cognitive levels, representational levels, and mathematical content—are integrated throughout the textual material as well as in the problem sets and chapter tests. Problem sets are organized into exercises (to support knowledge, skill, and understanding) and problems (to support problem solv- ing and applications).
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We have developed new pedagogical features to imple- ment and reinforce the goals discussed above and to address the many challenges in the course.
Summary of Pedagogical Changes to the Tenth Edition
Student Page Snapshots have been updated.
Reflection from Research margin notes have been edited and updated.
Mathematical Structure reveals the mathematical ideas of the book. Main Definitions, Theorems, and Properties in each section are highlighted in boxes for quick review.
Children’s Literature references have been edited and updated. Also, there is additional material offered on the Web site on this topic.
Check for Understanding have been updated to reflect the revision of the problem sets.
Mathematical Tasks have been integrated throughout.
Author Walk-Throughs videos have been made avail- able via QR codes on the third page of every chapter.
Children’s videos, produced by Blake Peterson and available via QR codes, have been integrated through- out.
Key Features Problem-Solving Strategies are integrated throughout the book. Six strategies are introduced in Chapter 1. The last strategy in the strategy box at the top of the second page of each chapter after Chapter l contains a new strategy.
Mathematical Tasks are located in various places throughout each section. These tasks can be presented to the whole class or small groups to investigate. As the stu-
dents discuss their solutions with each other and the instructor, the big mathematical ideas of the sec- tion emerge.
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Technology Problems appear in the Exercise/Problem sets throughout the book. These problems rely on and are enriched by the use of technology. The tech- nology used includes activities from the eManipulaties (virtual manipulatives),
spreadsheets, Geometer’s Sketchpad®, and the TI-34 II MultiView. Most of these technological resources can be accessed through the accompany- ing book companion Web site.
Student Page Snapshots have been updated. Each chapter has a page from an elementary school textbook relevant to the material being studied. Exercise/Problem Sets are separated into Part A
(all answers are provided in the back of the book and all solutions are provided in our supplement Hints and Solutions for Part A Problems) and Part B (answers are only provided in the Instructors Resource Manual). In additio