Mathematical reasoning for elementary teachers 7th global edition by TLOng

Mathematical reasoning for elementary teachers 7th global edition by TLOng Mathematical reasoning for elementary teachers 7th global edition by TLOng Mathematical reasoning for elementary teachers 7th global edition by TLOng Mathematical reasoning for elementary teachers 7th global edition by TLOng Mathematical reasoning for elementary teachers 7th global edition by TLOng v

Mathematical Reasoning for Elementary Teachers For these Global Editions, the editorial team at Pearson has collaborated with educators across the world to address a wide range of subjects and requirements, equipping students with the best possible learning tools This Global Edition preserves the cutting-edge approach and pedagogy of the original, but also features alterations, customization, and adaptation from the North American version Global edition Global edition Global edition seventh edition Long • DeTemple • Millman Mathematical Reasoning for Elementary Teachers SEVENTH edition Calvin T Long • Duane W DeTemple • Richard S Millman This is a special edition of an established title widely used by colleges and universities throughout the world Pearson published this exclusive edition for the benefit of students outside the United States and Canada If you purchased this book within the United States or Canada you should be aware that it has been imported without the approval of the Publisher or Author Pearson Global Edition LONG_1292062363_mech.indd 6/25/14 2:34 PM A01_LONG2365_07_GE_FM.indd Page 04/06/14 9:30 PM f-w-155-user /205/AW00196/9781292062365_DETEMPLE/DETEMPLE_MATHEMATICAL_REASONING_FOR_ELEMENT Mathematical Reasoning FOR ELEMENTARY TEACHERS This book presents the mathematical knowledge needed for teaching, with an emphasis on why future teachers are learning the content, as well as when and how they will use it in the classroom The Seventh Edition teaches the content in context to prepare today’s students for tomorrow’s classroom Continuing from the Common Core, the following eight Standards for Mathematical Practice are designed to teach students to SMP Make sense of problems and persevere in solving them The Common Core State Stan2 Reason abstractly and quantitatively dards for Mathematics include Construct viable arguments and critique the reasoning of others Standards for Mathematical Model with mathematics Use appropriate tools strategically Practice (SMP), which have been integrat6 Attend to precision ed throughout this text It’s important for Look for and make use of structure Look for and express regularity in repeated reasoning future teachers to know what will be expectThe Standards for Mathematical Practice elaborate and reinforce the importance of Pólya’s four ed of them when they are in the classroom, principles of problems solving In particular, special attention is given to the fourth principle, to and these SMP references ensure that future Page 42 teachers be both familiar and comfortable p quantities and their operation including differing units, such as cm, cm , cm , Fahrenheit versus with these mathematical practices Instances Celsius temperature, and so on Computations with different units can cause a real change in a problem M03_LONG0999_07_SE_C03.indd Page 126 28/08/13 8:55 AM f-402 /204/AW00123/9780321900999_DETEMPLE/DETEMPLE_MATHEMATICAL_REASONING_FOR_ELEMENTAR where an SMP applies are called out with an Unfortunately, you will see an example of a disaster in the paragraph immediately after SMP “Mathematically proficient students make sense of quantities and their relationships in problem icon and highlighted text situations Quantitative reasoning entails habits of creating a coherent representation of the prob• • SMP lem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them .” Page 527 COOPERATIVE INVESTIGATION Numbers from Rectangles Material Needed Directions One rectangle of each of these shapes for each student: Step Use the rectangles to determine whether or not there are representations of each of the numbers 0, 1, 2, , 39 as a sum of the numbers 1, 2, 4, 8, or 16, with each of the latter group of numbers used at most once Step For each representation determined in step 1, record the numbers (rectangles) used by placing a or a in the appropriate columns of the record sheet The rows for 0, 1, 18, 19, and 20 have been done for you (a) Do all the numbers from through 39 have such a representation? 16 (b) What additional numbers could be represented if you had a 32 rectangle? (c) Describe any interesting patterns you see on your record sheet One record sheet like this for each student: 16 0 0 20 0 0 21 18 0 38 19 0 1 39 16 1 0 Page 152 COOPERATIVE INVESTIGATIONS begin each chapter, offering content-related games and puzzles that motivate the chapter These can be easily adapted for use in the elementary classroom A01_LONG2365_07_GE_FM.indd Page 7/9/14 7:58 PM s-w-149 /205/AW00196/9781292062365_DETEMPLE/DETEMPLE_MATHEMATICAL_REASONING_FOR_ELEMENT INTO THE CLASSROOM Ann Hlabangana-Clay Discusses the Addition of Fractions I use red shoelace licorice to introduce adding fractions It is flexible and tangible for small fingers to demonstrate whole to part To start the lesson, I give each student one whole red shoelace licorice I ask them to spread it out from end to end and use it to measure a starting line and an 1 ending line To find + , I give each student a length shoelace and have them measure it against the whole Each student also gets a length shoelace to measure against the whole and to compare 1 to the After comparing the and the , I have the students connect the two shoelaces together 2 and share their findings with their partner Page 310 Into the Classroom 15 (Writing) (a) How would you use colored counters to help students understand that - (- 4) = 4? (b) How would you use colored counters to help students understand that - (- n) = n for every integer n (positive, negative, or 0)? Our completely updated INTO THE CLASSROOM feature provides insights from real elementary and middle school teachers related to various topics throughout the text, as students in this course are 16 (Writing) The definition of absolute value is often confusing to students On the one hand, they understand that the absolute thinking ahead to when they will have classrooms of their own value of a number is always positive On the other hand, the definition states that ͉ n ͉ = - n sometimes How would you Along with the feature, we have added new Into the Classroom explain this seeming contradiction? problems into the problem sets These problems pose questions that Page 256 will help future teachers consider how they might clarify subtle and M09_LONG2365_07_GE_C09.indd Page 469 26/05/14 7:42 PM f-w-155-user /205/AW00196/9781292062365_DETEMPLE/DETEMPLE_ often misunderstood points for their future students A IM P • • Integrating Mathematics and Pedagogy (IMAP) videos, available in MyMathLab, enable future teachers to see elementary and middle school students working out problems and explaining their thought processes IMAP videos often help future students understand why they need to understand the elementary and middle school mathematics at a deeper conceptual level VI DEO Page 299 Responding to Students 29 Larisa, a second grader, is asked whether the following segments drawn on a page are parallel: Responding to Students exercises give insight into the mathematical questions and procedures that children will come up with on their own, and offer ways to respond to them B A C D After a brief pause, she says, “Yes, they are because they don’t meet.” How would you respond to Larisa? Page 469 A01_LONG2365_07_GE_FM.indd Page 6/30/14 3:57 PM s-w-149 ~/Desktop/30:6:2014 TH EDITION GLOBAL EDITION Mathematical Reasoning FOR ELEMENTARY TEACHERS C a l v i n T D u a n e W Richard S Long DeTemple Millman Boston Amsterdam Delhi Columbus Cape Town Mexico City Indianapolis Dubai Sao Paulo New York London Sydney Madrid San Francisco Milan Hong Kong Munich Seoul Upper Saddle River Paris Singapore Montréal Taipei Toronto Tokyo A01_LONG2365_07_GE_FM.indd Page 7/9/14 7:58 PM s-w-149 /205/AW00196/9781292062365_DETEMPLE/DETEMPLE_MATHEMATICAL_REASONING_FOR_ELEMENT Editorial Director, Mathematics: Christine Hoag Senior Manufacturing Controller, Global Edition: Trudy Kimber Editor in Chief: Anne Kelly Content Development Manager: Bob Carroll Senior Acquisitions Editor: Marnie Greenhut Senior Content Developer: Mary Durnwald Development Editor: Lenore Parens Associate Marketing Manager: Alicia Frankel Content Editor: Christine O’Brien Marketing Assistant: Brooke Smith Editorial Assistant: Christopher Tominich Senior Author Support/Technology Specialist: Joe Vetere Senior Managing Editor: Karen Wernholm Image Manager: Rachel Youdelman Senior Project Editor: Patty Bergin Procurement Specialist: Carol Melville Digital Assets Manager: Marianne Groth Associate Director of Design, USHE EMSS/HSC/EDU: Andrea Nix Associate Media Producer: Nicholas Sweeny Program Design Lead: Barbara T Atkinson Head of Learning Asset Acquisition, Global Edition: Laura Dent Production Management, Composition, and Illustrations: Aptara®, Inc Assistant Acquisitions Editor, Global Edition: Karthik Subramanian Interior Design: Nancy Goulet/Studio Wink Assistant Project Editor, Global Edition: Mrithyunjayan Nilayamgode Cover Design: Shree Mohanambal Inbakumar/Lumina Datamatics Ltd Media Producer, Global Edition: M Vikram Kumar Cover Image: © hxdbzxy/Shutterstock For permission to use copyrighted material, grateful acknowledgment is made to the copyright holders on page 849, which is hereby made part of this copyright page Many of the designations used by manufacturers and sellers to distinguish their products are claimed as trademarks Where those designations appear in this book, and Pearson was aware of a trademark claim, the designations have been printed in initial caps or all caps Pearson Education Limited Edinburgh Gate Harlow Essex CM20 2JE England and Associated Companies throughout the world Visit us on the World Wide Web at: www.pearsonglobaleditions.com © Pearson Education Limited 2015 The rights of Calvin T Long, Duane W DeTemple, and Richard S Millman to be identified as the authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988 Authorized adaptation from the United States edition, entitled Mathematical Reasoning for Elementary Teachers, 7th edition, ISBN 978-0-321-90099-9, by Calvin T Long, Duane W DeTemple, and Richard S Millman, published by Pearson Education © 2015 All rights reserved No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without either the prior written permission of the publisher or a license permitting restricted copying in the United Kingdom issued by the Copyright Licensing Agency Ltd, Saffron House, 6–10 Kirby Street, London EC1N 8TS All trademarks used herein are the property of their respective owners The use of any trademark in this text does not vest in the author or publisher any trademark ownership rights in such trademarks, nor does the use of such trademarks imply any affiliation with or endorsement of this book by such owners The author and publisher of this book have used their best efforts in preparing this book These efforts include the development, research, and testing of the theories and programs to determine their effectiveness The author and publisher make no warranty of any kind, expressed or implied, with regard to these programs or the documentation contained in this book The author and publisher shall not be liable in any event for incidental or consequential damages in connection with, or arising out of, the furnishing, performance, or use of these programs ISBN 10: 1-292-06236-3 ISBN 13: 978-1-292-06236-5 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library 10 Typeset in Times Ten by Aptara® Printed and Bound in Great Britain by CPI Group (UK) Ltd Croydon, CR0 4YY A01_LONG2365_07_GE_FM.indd Page 04/06/14 9:30 PM f-w-155-user Dedication /205/AW00196/9781292062365_DETEMPLE/DETEMPLE_MATHEMATICAL_REASONING_FOR_ELEMENT To the memory of my good wife and constant helpmate, Jean C.T.L To my wife, Janet, and my daughters, Jill and Rachel D.W.D To Sandy, for her loving support R.S.M A01_LONG2365_07_GE_FM.indd Page 04/06/14 9:30 PM f-w-155-user /205/AW00196/9781292062365_DETEMPLE/DETEMPLE_MATHEMATICAL_REASONING_FOR_ELEMENT A01_LONG2365_07_GE_FM.indd Page 04/06/14 9:30 PM f-w-155-user /205/AW00196/9781292062365_DETEMPLE/DETEMPLE_MATHEMATICAL_REASONING_FOR_ELEMENT Contents Preface 13 Thinking Critically 27 1.1 An Introduction to Problem Solving 29 1.2 Pólya’s Problem-Solving Principles and the Standards for Mathematical Practice of the Common Core State Standards for Mathematics 34 • Guess and Check • Make an Orderly List • Draw a Diagram • Pólya’s Problem-Solving Principles and the Standards for Mathematical Practice of the Common Core Standards for Mathematics 1.3 More Problem-Solving Strategies 45 • Look for a Pattern • Make a Table • Use a Variable • Consider Special Cases • Pascal’s Triangle 1.4 Algebra as a Problem-Solving Strategy 57 • Use a Variable • Use Two Variables 1.5 Additional Problem-Solving Strategies 68 • Working Backward • Eliminate Possibilities • The Pigeonhole Principle 1.6 Reasoning Mathematically 76 • Inductive Reasoning • Representational Reasoning • Mathematical Statements • Deductive Reasoning Chapter Summary 87 Chapter Review Exercises 90 Sets and Whole Numbers 93 2.1 Sets and Operations on Sets 95 • Venn Diagrams • Relationships and Operations on Sets • Using Sets for Problem Solving 2.2 Sets, Counting, and the Whole Numbers 104 • One-to-One Correspondence and Equivalent Sets • The Whole Numbers • Representing the Whole Numbers Pictorially and with Manipulatives • Ordering the Whole Numbers • Problem Solving with Whole Numbers and Venn Diagrams 2.3 Addition and Subtraction of Whole Numbers 115 • The Set Model of Whole-Number Addition • The Measurement (Number-Line) Model of Addition • Properties of Whole-Number Addition • Subtraction of Whole Numbers • Take-Away Model • Missing-Addend Model • Comparison Model • Number-Line Model 2.4 Multiplication and Division of Whole Numbers 127 • Multiplication of Whole Numbers • Multiplication as Repeated Addition • The Array Model for Multiplication • The Rectangular Area Model for Multiplication • The Skip-Count Model for Multiplication • The Multiplication Tree Model • The Cartesian Product Model of Multiplication • Properties of Whole-Number Multiplication • Division of Whole Numbers • The Repeated-Subtraction Model of Division • The Partition Model of Division • The Missing-Factor Model of Division • Division by Zero Is Undefined • Division with Remainders • Why Does Quotient with Remainder (the Division Algorithm) Work? • Exponents and the Power Operation Chapter Summary 145 Chapter Review Exercises 149 A01_LONG2365_07_GE_FM.indd Page 04/06/14 9:30 PM f-w-155-user /205/AW00196/9781292062365_DETEMPLE/DETEMPLE_MATHEMATICAL_REASONING_FOR_ELEMENT CONTENTS Numeration and Computation 151 3.1 Numeration Systems Past and Present 153 • The Egyptian System • The Roman System • The Babylonian System • The Mayan System • The Indo-Arabic System • Physical Models for Positional Systems 3.2 Algorithms for Addition and Subtraction of Whole Numbers 163 • The Addition Algorithm • The Subtraction Algorithm 3.3 Algorithms for Multiplication and Division of Whole Numbers 173 • Multiplication Algorithms • Division Algorithms 3.4 Mental Arithmetic and Estimation 183 • The One-Digit Facts • Easy Combinations • Adjustment • Working from Left to Right • Estimation • Front-End Method • Rounding • Approximating by Rounding 3.5 Nondecimal Positional Systems 193 • Base-Five and Base-Six Place Value Including Conversion to the Decimal System and Operations • Addition, Subtraction, and Multiplication in Base Six Chapter Summary 203 Chapter Review Exercises 206 Number Theory 209 4.1 Divisibility of Natural Numbers 211 • Divides, Divisors, Factors, Multiples • Prime and Composite Numbers • The Divisors of a Natural Number • Two Questions about Primes • There Are Infinitely Many Primes • Determining Whether a Given Natural Number Is Prime 4.2 Tests for Divisibility 223 • Divisibility of Sums and Differences • Divisibility by 2, 5, and 10 • Divisibility by 4, 8, and Other Powers of • Divisibility by and • Combining Divisibility Tests • Summary of Useful Divisibility Tests • Applications of Divisibility • Illustrating Factors and Divisibility with a Manipulative 4.3 Greatest Common Divisors and Least Common Multiples 231 • GCD Method 1: Greatest Common Divisors by Intersection of Sets • GCD Method 2: Greatest Common Divisor from Prime Factorizations • GCD Method 3: Greatest Common Divisor from the Euclidean Algorithm • An Application of the Greatest Common Factor • The Least Common Multiple • LCM Method 1: Least Common Multiples by Intersection of Sets • LCM Method 2: Least Common Multiples from Prime Factorizations • LCM Method 3: Least Common Multiples by Using the Euclidean Algorithm • An Application of the LCM Chapter Summary 243 Chapter Review Exercises 245 Integers 247 5.1 Representations of Integers 249 • Absolute Value of an Integer • Criteria for the Representation of the Integers • Representing Integers with Colored Counters • The Addition-by-0 Property with Colored Counters • Taking Opposites with Colored Counters • Mail-Time Representations of Integers • Number-Line Representations of Integers 5.2 Addition and Subtraction of Integers 258 • Addition of Integers • Addition of Integers by Using Sets of Colored Counters • Addition of Integers by Using Mail-Time Stories • Addition of Integers by Using a Number Line • Subtraction of Integers • Subtraction of Integers with Colored Counters A01_LONG2365_07_GE_FM.indd Page 04/06/14 9:30 PM f-w-155-user /205/AW00196/9781292062365_DETEMPLE/DETEMPLE_MATHEMATICAL_REASONING_FOR_ELEMENT CONTENTS • The Equivalence of Subtraction with Addition of the Opposite • Subtraction of Integers by Using Mail-Time Stories • Subtraction of Integers by Using the Number Line • Ordering the Set of Integers 5.3 Multiplication and Division of Integers 273 • Multiplication of Integers • Multiplication of Integers by Using Loops of Colored Counters • Multiplication of Integers by Using Mail-Time Stories • Multiplication of Integers by Using a Number Line • Division of Integers • Multiplication and Division with Colored-Counter Arrays Chapter Summary 284 Chapter Review Exercises 286 Fractions and Rational Numbers 289 6.1 The Basic Concepts of Fractions and Rational Numbers 291 • Fraction Models • Equivalent Fractions • Fractions in Simplest Form • Common Denominators • Rational Numbers • Ordering Fractions and Rational Numbers 6.2 Addition and Subtraction of Fractions 307 • Addition of Fractions • Proper Fractions and Mixed Numbers • Subtraction of Fractions 6.3 Multiplication and Division of Fractions 317 • Multiplication of a Fraction by an Integer • Multiplication of an Integer by a Fraction • Multiplication of a Fraction by a Fraction • Division of Fractions • Algorithms for Calculating the Division of Fractions • Reciprocals as Multiplicative Inverses in the Rational Numbers 6.4 The Rational Number System 332 • Properties of Addition and Subtraction • Properties of Multiplication and Division • Properties of the Order Relation • The Density Property of Rational Numbers • Computations with Rational Numbers • Estimations • Mental Arithmetic Chapter Summary 344 Chapter Review Exercises 348 Decimals, Real Numbers, and Proportional Reasoning 349 7.1 Decimals and Real Numbers 351 • Representations of Decimals • Multiplying and Dividing Decimals by Powers of 10 • Terminating Decimals as Fractions • Repeating Decimals and Rational Numbers • The Set of Real Numbers • Irrationality of 12 • Real Numbers and the Number Line 7.2 Computations with Decimals 366 • Rounding Decimals • Adding and Subtracting Decimals • Ordering Decimals and the Real Numbers • Multiplying Decimals • Dividing Decimals 7.3 Proportional Reasoning 374 • Ratio • Proportion • Applications of Proportional Reasoning 7.4 Percent 387 • Percent • Solving the Three Basic Types of Percent Problems • Percentage Increase and Decrease • Compound Interest • The Mathematics of Growth Chapter Summary 399 Chapter Review Exercises 401 Z05_LONG2365_07_GE_IDX.indd Page 858 04/06/14 5:17 PM f-w-155-user /205/AW00196/9781292062365_DETEMPLE/DETEMPLE_MATHEMATICAL_REASONING_FOR_ELEMENT www.downloadslide.net 858 INDEX Open-Top Box Problem, 431 Operations and/or, 80 binary, 115 sets and, 95–96 Operator, 319 Opposite, 248, 249 subtracting by adding the, 266 Opposite of the opposite of an integer, 253 Ordered pair, 130, 419 Ordering decimals, 368 fractions and rational numbers, 299–301 integers, sets of, 269–270 real numbers, 368 whole numbers, 108–109 Orderly list strategy, 39–40 Ordinal numbers, 104–105 Orientation-preserving transformations, 592 Orientation reversing, 592 Origin, coordinate system, 420 Or operations, 80 Outcome, 748 Outliers, 718 P Pantozzi, R., 498 Paper-folding constructions, 667–668 Investigating Triangles via Paper Folding, 452–453 Parallelism, condition for, 436 Parallel lines constructing, 658–660 defined, 435 proof, 436 in space, 491 Parallelograms, area of, 535–536 Parallel planes, 491 Partial-difference algorithm, 168 Partial-products algorithm, 175 Partial-sum algorithm, 166 Partition model of division, 133, 134 Partitive division, 134 Pascal, B., 51 Pascal’s identity, 774 Pascal’s triangle, 50–53 Patterns See also Tilings activities for teaching, 610–611 border, 611–614 dot representations to discover number, 79–80 frieze, 611 Pascal’s triangle and, 52–53 periodic, 611 problem solving and looking for, 45–47 wallpaper, 611, 614 Pennies, decimal representation using, 353 Pentagonal arch, 477 Pentahedron, 493 Percentiles defined, 735 determining, of a population, 735 determining, of a population in an interval, 736 determining, of a sample, 735 z scores and, 733–736 Percents calculating, 390–391 compound interest, calculating, 392–393 cost of debt, calculating, 394 defined, 387–388 expressing decimals as, 388 expressing fractions as, 389–390 expressing, as decimals, 388 expressing, as fractions, 388–389 increases and decreases, computing, 392 population growth, calculating, 394 tips, calculating, 393 Perimeter of circles (circumference), 540–541 defined, 539 Periodic patterns, 611 Periodic (repeating) decimals, 357–360 Permutations defined, 770, 771 formulas for the number of r-permutations, 772 notations, 772 solving problems with, 774–779 Perpendicular bisector, constructing, 660–662 Perpendicular lines condition for, 438 constructing, 658–660 defined, 438, 461 proof, 438 Pi (p), 540, 541 Estimating p with Geometric Probability, 787 Pick, G., 552 Pictographs, 695, 702–703 Pie charts or graphs, 695, 701–702 Pigeonhole Principle, 71–72 Place value Babylonian, 155–156 base-ten, 159 defined, 155 Indo-Arabic, 159 manipulatives to study, 159–161 Mayan, 157–158 NCTM standards, 164 Place-value cards adding with, 166 defined, 166 multiplying with, 175–176 subtracting with, 169, 170 Place-value diagrams adding with, 166 defined, 166 subtracting with, 169, 170 Planes figures, 456, 457–466 half, 473 intersecting, 491 parallel, 491 rigid motion of, 587 in space, 491 transformation of, 587 Platonic solids, 495 Points collinear, 457, 458–459 defined, 457 distance between, 459 of intersection, 458 noncollinear, 457 plotting, 420–421 Point-slope form, 426 Point symmetry, 610 Pólya, G., 34, 35 Pólya problem-solving principles, 34 See also Problem-solving strategies carry out the plan, 36 devise a plan, 35 diagram strategy and, 40–42 guess and check strategy and, 36–39 look back, 36 orderly list strategy and, 39–40 understand the problem, 35 Polygonal curves classification of, 475 defined, 474 sides/edges, 474 vertices, 474 Polygons angle of, 476 area of, 535–538 constructing regular, 665–666 defined, 474 equiangular, 482 equilateral, 482 exterior angle of, 476 five-pointed star, measuring the angles of, 478 Getting Rhombunctious! Folding Paper Polygons, 640 inscribed, 665 interior angle of, 476 irregular, with tilings, 623–624 lattice, 538 names of, 475 regular, 482–485 relative frequency, 732 sum of angle measures in a convex polygon, 476–477 sum of interior angle measures of general, 478 tiling the plane with congruent polygonal tiles, 625 total turns, 479 Polyhedra defined, 493 edges, 493 Euler’s formula, 496–498 examples of, 499 faces, 493 prism, 494–495 pyramid, 494 regular, 495–496 types of, 493 vertices, 493 Population(s) defined, 728 means and standard deviations, calculating, 729–731 Using Samples to Approximate Characteristics of Populations, 738–739 Positional notation, 164 Positional systems See Numeration systems Positional values in base five and six, 194 in base ten, 159 in decimal system, 354 Positive integers, 248, 249 Possibilities, eliminating, 69–71 Z05_LONG2365_07_GE_IDX.indd Page 859 04/06/14 5:17 PM f-w-155-user /205/AW00196/9781292062365_DETEMPLE/DETEMPLE_MATHEMATICAL_REASONING_FOR_ELEMENT www.downloadslide.net INDEX Possibility tree, 760 Power operation for whole numbers, 138 Preimage, 587 Premises, 82 Primality, test for, 218–219 Prime divisors of n, 218 Prime factorization defined, 215 greatest common divisor from, 232–233 least common multiples from, 236–237 number of, 217 of 600, determining, 215 Prime numbers building blocks of natural numbers, 213 defined, 213 determining, 218–219 via rectangular arrays, 210 Prime-power factorization of n, 215 Principal, 393 Principles and Standards for School Mathematics See National Council of Teachers of Mathematics (NCTM), Principles and Standards for School Mathematics Prisms, 494–495 surface area of right, 569–570 volume of oblique, 563 volume of right, 562–563 Probability basic definitions, 747–755 conditional, 764–766 counting principles and, 759–767 defined, 746 event E, 748, 749 events, complementary, 753–755 events, independent, 766 events, mutually exclusive, 751–753 events, non-mutually exclusive, 753 expected values, 784–786 experimental, 746, 749–750 factorials and, 760–761 function, 748–749 geometric, 786–787 NCTM standards, 747 odds, 783–784 outcome, 748 permutations and combinations, 770–779 sample space, 748 simulations, 787–789 theoretical, 746, 750–751 trees, 764 Problem solving introduction to, 29–32 NCTM’s standards on, 29, 36 sets and, 99 whole numbers and, 109–110 Problem-solving strategies algebra, 57–64 deductive reasoning, 82–83 diagrams, using, 40–42 eliminating possibilities, 69–71 guess and check, 30–32, 36–39 inductive reasoning, 76–78 orderly list, 39–40 Pascal’s triangle, 50–53 patterns, looking for, 45–47 Pigeonhole Principle, 71–72 Pólya principles, 34–36 representational reasoning, 78–82 special cases, 50–54 tables, using, 47–49 variables, 49, 57–64 working backward, 68–69 Product Cartesian product model, multiplication, 129–131 defined, 128 divisibility by, 226 Proof angle measure in regular polygon, 483 conjecture and, 78 by contradiction, 82 equidistance property of angle bisector, 664 equidistance property of perpendicular bisector, 661 of irrational numbers, 361–362 parallel lines, 436 perpendicular lines, 438 Pythagorean theorem, 553–556 sum of angle measures in a convex polygon, 476–477 sum of angle measures in a triangle, 465 Thales’ theorem, 650 Proper divisor, 211 Proper fractions, 310–312 PropertiesSee also specific properties addition-by-0, 252–253 addition of area, 532–533 additive identity property of zero, 117, 118, 260, 333 additive inverses, 250, 333 angle-angle (AA), 675 angle-angle-angle (AAA), 675 angle-angle-side (AAS), 651–652 angle-side-angle (ASA), 650–651 congruence, 532–533 corresponding angles, 463–464 cross-product, 295 existence of negatives, 260 of integer addition, 260 of integer multiplication, 275 of integer subtraction, 266 multiplicative identity, 131, 132, 275, 335 multiplicative inverse, 335 order relation, 335–336 of rational number addition, 332–333 of rational number multiplication, 334–336 sets and, 99–101 side-angle-side (SAS), 647–650, 677–678 side-side-side (SSS), 643–645, 676–677 Venn diagrams for verifying, 100–101 of whole-number addition, 117–119 of whole-number multiplication, 131–132 Proportional reasoning applications of, 379–383 constant of proportionality, 380 defined, 374 diagrams, use of, 381 proportions, 377–379 ratios, 374–377 Proportions conditions for, 378 defined, 377–378 determining, 378 859 proving property of, 378–379 y is to x, 380 Protractors for measuring angles, 461 Ptolemy, C., 311 Pyramid algorithm, 177, 178 Pyramids, 494 right regular, 571–572 slant height, 571 surface area of, 571–572 volume of, 563–565 Pythagoras, 412, 453, 553 Pythagorean theorem applications of, 554–555 converse of, 555–556 defined, 553 proving, 553–554 Q Quadrants, 420 Quadratic functions, 429–430 Quadrilaterals classification of, 480–481 exploring, 481–482 midpoint figure of, 678–679 Quartiles defined, 695, 717 determining, 717–718 lower, 717 upper, 717 Quotients, 135, 177 R Radius, circle, 441 Ramanujan, S., 226–227 Random numbers, 729 Random samples, 728–729 Random sequence of digits, 729 Range data set, 716 of functions, 411, 412 Rate problem, solving, 62 Rational numbers addition of, 332–333 additive inverse, 333 comparing, 300–301 computations and examples, 337–339 defined, 298 density property, 336–337 multiplication of, 334–335 multiplicative inverse, 335 negatives, 333 opposite, 333 ordering, 299–301 order relation, 335–336 reciprocals and multiplicative inverses, 326–328 repeating (periodic) decimals as, 358–360 represented by terminating decimals, 356 set of, 299 subtraction of, 334 Ratios defined, 374 determining, 375–377 examples of, 375 in simplest form, 376 Z05_LONG2365_07_GE_IDX.indd Page 860 04/06/14 5:17 PM f-w-155-user /205/AW00196/9781292062365_DETEMPLE/DETEMPLE_MATHEMATICAL_REASONING_FOR_ELEMENT www.downloadslide.net 860 INDEX Ratio table, 381 Rays defined, 460 endpoint, 460 union of, 460 vertex, 460 Real numbers defined, 350, 360 NCTM standards, 351 number line and, 362–363 order of, 368 set of, 360–363 Reasoning See also Proportional reasoning deductive, 82–83 direct reasoning, rule of, 82 indirect reasoning, rule of, 82 inductive, 76–78 NCTM standards, 76, 78–79 representational, 78–82 Reasoning and Proof Standard, 76, 78 Reciprocals and multiplicative inverses, 326–328 Rectangles, area of, 535 Rectangular area model, multiplication, 129 Rectangular Arrays, Primes and Composites via, 210 Reflections consecutive, across parallel lines, 594–595 description and definitions, 590–592, 594 three successive, 596 two, in distinct lines, 595 two successive, 593 Reflection (line or bilateral) symmetry, 608–609 Reflex angles, 460 Region Interior to Two Simple Curves and External to Both of the Curves, 473–474 Regions counting, in the plane, 473 defined, 473 simple closed curves, 473 Regular polygons, 482–485 Regular polyhedra, 495–496 Regular tilings, 621–622 Relationships, variables to express, 406 Relative frequency, 731–732 Remainders, division, 136–137, 177 Repeated addition, multiplication as, 127–128 Repeated-subtraction model of division, 133–134 Repeating decimals, 357–360 Representational reasoning, 78–82 Representations decimals, 352–354 even and odd numbers, 212–213 integers, 249–255 integers as sums, 219 Rhombus, 658, 660 Right angles, 460, 461 Right circular cones, 500 surface area of, 572 Right cylinders, 500 surface area, 569, 570–571 volume of, 562–563 Right prisms, 494 surface area of, 569–570 volume of, 562–563 Right regular pyramids, 571–572 Rigid motions classification of general, 596–598 defined, 587 examples of, 436–437, 587–588 glide-reflections, 592–593, 594 orientation reversing, 592 reflections, 590–592, 594 rotations (turns), 589–590, 594 translations (slides), 588–589, 594 Rodriguez, T M., 477 Roman numeration system, 154–155 Rotations (turns), 589–590, 594 Rotation (turn) symmetry, 609–610 Rounding approximating by, 189–190 decimals, 366 defined, 188–189 5-up rule, 189 Rules direct reasoning, 82 for exponents, 139, 140 5-up rule, 189, 366 indirect reasoning, 82 invert and multiply, 317, 325, 326 Russian peasant algorithm, 175 S Samples biased, 728 defined, 728 random, 728–729 Using Samples to Approximate Characteristics of Populations, 738–739 Sample space, 748 Scaffold algorithm, 177, 178 Scale factor, 574, 674 Scratch method, 163 Semiregular tilings, 622–623 Set models discrete, 293 of fractions, 293 of whole-number addition, 115–117 Sets associativity, 99 builder notation, 95 Cartesian product of, 130 commutativity, 99 complement of set A, 96 complements, finding, 97 defined, 94, 95 denoting, 95 describing, 95–96 disjoint, 98 distributive property, 100 element of, 95 empty, 97, 100 equal, 97 equivalent or matching, 105–106 finite, 106 infinite, 106 integer, ordering, 269–270 intersection of, 97–98 listing in braces, 95 member of, 95 one-to-one correspondence, 105 other terms for, 95 problem solving using, 99 properties, 99–101 of rational numbers, 299 of real numbers, 360–363 relationships and operations, 97–98 subsets, 97 transitivity, 99 union of, 98 Venn diagrams, 96 word description, 95 Sharing, division by, 134 Short division, 179 Sides of angles, 460 of polygonal curves, 474 Side-angle-side (SAS) property congruent triangles, 647–650 similar triangles, 677–678 Side-side-side (SSS) property congruent triangles, 643–645 similar triangles, 676–677 Sieve of Eratosthenes, 218 Similarity indirect measurement with, 676 principle of measurement, 574–575 transformations, 600–601 Similar triangles angle-angle (AA) property, 675 angle-angle-angle (AAA) property, 675 defined, 674 problem solving with, 678–680 side-angle-side (SAS) property, 677–678 side-side-side (SSS) property, 676–677 Simple closed curves, 471–473 Simple closed surface, 492 Simple curves, 471 Simplest form, fractions in, 296 Simulations, 787–789 SI system See Metric measurements 68–95–99.7 rule for normal distribution, 733 Size (dilation) transformations, 599 Skew lines, 491 Skip counting, 128 Skip-count model, multiplication, 129 Slide arrow, 588 Slides (translations), 588–589, 594 Slope, 422–424 Slope-intercept form, 426, 427 Small squares, decimal representation using, 352 SMP See Standards for Mathematical Practice (SMP) Solids in space, 491–493 Solution set of equations, 410 Space curves in, 491–493 figures in, 490–500 half, 491 lines in, 491 planes in, 491 solids in, 491–493 surfaces in, 491–493 Special cases, as a problem-solving strategy, 50–54 Spheres, 492 surface area of, 573–574 volume of, 565 Square numbers, defined, 211 Z05_LONG2365_07_GE_IDX.indd Page 861 04/06/14 5:17 PM f-w-155-user /205/AW00196/9781292062365_DETEMPLE/DETEMPLE_MATHEMATICAL_REASONING_FOR_ELEMENT www.downloadslide.net INDEX Squares, Areas, and Side Lengths, 350 Stacked short division, 214 Standard deviation alternative calculation of, 722 computing, 721 defined, 695, 720 examples of, 721–723 population means and calculating, 729–731 Standardized form, distribution, 733–734 Standards for Mathematical Practice (SMP) argument construction, 554 border patterns, 613 construction software programs, 668 description of, 42–43 estimation and modeling, 187–188 fraction division, 323 geometric figures, 454 integer division, 278 integer representation, 250 mathematical models, 675–676 measurements, 527 precision, 212 proportional reasoning, 374 side-side-side (SSS) property, 644 slope, 424 tangent line, 442 Statements, mathematical “and” operations, 80 conditional, 80–81 defined, 80, 80–81 if then, 80–81 “or” operations, 80 Statistical inference defined, 695, 727 distributions, 731–733 population means and standard deviations, 729–731 populations and samples, 728–729 z scores and percentiles, 733–736 Statistics See also Data Blackwell, work of, 701 data, measuring the center and variation of, 711–723 data, organizing and representing, 695–704 descriptive, 728 NCTM standards, 694, 695 Stem-and-leaf plots, 695, 696–697 Sticks in bundles, 160 Straight angles, 460 Straightedge, 457 Strayton, M., 216 Strings and Loops, 746 Strips, decimal representation using, 352 Subsets, 97 Subtraction algorithm for whole numbers, 168–170 in base five, 197–198 in base six, 200 comparison model, 120, 121 of decimals, 366–368 defined, 120 of fractions, 312–313 of integers, 263–270 minuend, 120 missing-addend model, 120, 121 models, identifying, 122 number-line (measurement) model, 120, 121 of rational numbers, 334 subtrahend, 120 take-away model, 120, 121 of whole numbers, 120–123 with exchanging, 169–170 without exchanging, 169 Subtraction algorithm for whole numbers, 168–170 Subtrahend, 120 Sum(s) defined, 115, 116 divisibility of, 223 representing integers as, 219 Summands, 116 Supplementary angles, 462 Surface areas defined, 561 problem-solving strategy, 568 of pyramids, 571–572 of right circular cones, 572 of right cylinders, 569, 570–571 of right prisms, 569–570 of spheres, 573–574 volume versus, 561 Surfaces in space, 491–493 Symmetries center of, 609 circular, 609 defined, 608 NCTM standards, 608 point, 610 reflection (line or bilateral), 608–609 rotation (turn), 609–610 T Tables functions as, 413 ratio, 381 using, 47–49 Take away, use of term, 168 Take-away model, 120, 121 Tangent line, 441–442 Tangrams, 517–519 Temperature Celsius scale, 526 converting from Celsius to Fahrenheit, 527 Fahrenheit scale, 526 Terminal side, angle, 465–466 Terminating decimals as fractions, 355–357 Tessellations See Tilings Tests divisibility, 223–228 primality, 218–219 Tetrahedron, 493, 496 Thales’ theorem, 649–650 Theaetetas, 495 Theano, 412 Theorems addition principle of counting, 759 addition principle of probability, 759 alternate interior angles, 464 angle measure in regular polygon, 483 circumscribed circle of a triangle, 662 classification of general rigid motions, 597 closure property for subtraction of integers, 266 compound interest, calculating, 392–393 861 conditional probability, 765 conditions for proportions, 378 converse of the Pythagorean theorem, 556 defined, 78 density property of rational numbers, 336–337 distance change under dilation, 599 distance formula, 421–422 divisibility tests, 223–228 divisors of natural numbers, 216 equidistance property of angle bisector, 663 equidistance property of perpendicular bisector, 661 Euclidean algorithm, 233–234 formulas for the number of r-combinations, 773 formulas for the number of r-permutations, 772 fundamental, of arithmetic, 214–215 Gauss-Wantzel constructability theorem, 666 greatest common divisor from prime factorizations, 232–233 inscribed circle of a triangle, 664 irrationality of 12, 361–362 isosceles triangle theorem, 649 isosceles triangle theorem, converse, 651 Jordan curve, 472 multiplication principle of counting, 761 multiplication property of independent events, 766 multiplying and dividing decimals by powers of 36, 355 number of permutations of a set of objects, 761 opposite of the opposite of an integer, 253 Pascal’s identity, 774 point-slope form, 426 prime divisors of n, 218 primes, number of, 217 probability of complementary events, 754 probability of compound events, 765 probability of mutually exclusive events, 752 probability of non-mutually exclusive events, 753 Pythagorean, 553–556 rational numbers and periodic decimals, 360 rational numbers represented by terminating decimals, 356 real numbers and the number line, 363 regular tilings, 622 similarity principle of measurement, 574–575 slope-intercept form, 426 subtracting by adding the opposite, 266 subtraction is equivalent to addition of the negative, 334 sum of angle measures in a convex polygon, 476–477 sum of angle measures in a triangle, 465 sum of interior angle measures of general polygons, 478 Thales’, 649–650 three successive reflections, 596 tiling the plane with congruent polygonal tiles, 625 total turns, 479 triangle inequality, 646–647 two-point form of the equation of a line, 428 two reflections in distinct lines, 595 vertical angles, 462–463 Z05_LONG2365_07_GE_IDX.indd Page 862 04/06/14 5:17 PM f-w-155-user /205/AW00196/9781292062365_DETEMPLE/DETEMPLE_MATHEMATICAL_REASONING_FOR_ELEMENT www.downloadslide.net 862 INDEX Theoretical probability, 746, 750–751 Tiles defined, 621 to represent whole numbers, 107 Tilings (tessellations) congruent polygonal, 625 defined, 621 Escher-like designs, 625–628 examples of, 619–620 irregular polygons and, 623–624 regular, 621–622 semiregular, 622–623 Tips, calculating, 393 Total turns, 479 Traditional algorithm, 166 Transformational geometry, 587 Transformations equivalent, 588, 593 identity, 588, 593 orientation-preserving, 592 of the plane, 587 rigid motions, 587–598 similarity, 600–601 size (dilation), 599 Transitive property, 336 Transitivity, 99 Translations (slides), 588–589, 594 Translation vector, 588 Transversal, 458 Trapezoids, area of, 537 Tree model, multiplication, 129 Triangles altitude of, 439, 440–441 area of, 536–537 centroid of, discovering, 679–680 circumscribed circle of, 662 classification of, 479–480 congruent, 641–652 Exploring Toothpick Triangles, 647 Finding Special Points of a Triangle with Paper Folding, 662–663 inequality, 646–647 Investigating Triangles via Paper Folding, 452–453 isosceles, 435, 649, 651 measure of angles in, 464–465 medial, 678 median of, 679 Pascal’s, 50–53 side, 435 similar, 674–680 six parts of, 642 Triangular numbers, 59, 116–117 Trichotomy, law of, 270 Trichotomy property, 336 Truth tables, 80 Turn angle, 589 Turn arrow, 589 Turn center, 589 Turns (rotations), 589–590, 594 Turn (rotation) symmetry, 609–610 Two-point form of the equation of a line, 428 U Union of sets, 98 Unit analysis, 527–528 U.S Customary system of measurements See Customary measurements Unit rate, 375 Units See also Customary measurements; Metric measurements of area, 531 defined, 213, 291 measurement, 516 nonstandard, 531–532 Unit squares, decimal representation using, 352 Upper quartiles, 717 Using Samples to Approximate Characteristics of Populations, 738–739 V Values See also Positional values defined, 411 van Hiele levels, 454–455 Variability data, 695 measures of, 716–719 Variables defined, 58 describing generalized properties/ patterns using, 406 to determine general formulas, 59–60 domain, 408 expressing formulas using, 407 expressing relationships using, 406 Gauss’ insight, 58 generalized, 406 as a problem-solving strategy, 49, 57–64 role of, identifying, 407 symbols for, 406 as unknowns in equations, 407 Venn, J., 96 Venn diagrams, 96 problem solving using, 110 for verifying properties, 100–101 Vertex figure, 621 Vertical angles, 462–463 Vertical-line test, 411 Vertices common, pyramid, 494 polygonal curves, 474 polyhedral, 493 rays, 460 Visualizations, choosing good, 703 Voderberg, H., 625 Volume of cones, 563–564 customary, 520–521 metric, 524–525 of oblique cylinders, 563 of oblique prisms, 563 of pyramids, 563–565 of right cylinders, 562–563 of right prisms, 562–563 of spheres, 565 versus surface area, 561 W Wallpaper patterns, 611, 614 Wantzel, P., 666 Gauss-Wantzel constructability theorem, 666 Weight customary, 526 difference between mass and, 525 metric, 526 Wells-Heard, S., 556 Whole numbers addition algorithm for, 163–168 addition of, 115–119 defined, 94, 106 determining, 106–107 division algorithm for, 177–179 division of, 133–140 even, 212–213 manipulatives to represent, 107–108, 121–122 multiplication algorithm for, 174–177 multiplication of, 127–133 odd, 212–213 ordering, 108–109 problem solving with, 109–110 properties of addition, 117–119 properties of multiplication, 131–132 subtraction algorithm for, 168–170 subtraction of, 120–123 Word description, sets and, 95 Working backward strategy, 68–69 Working from left to right, 185–186 X x-axis, 420 x-coordinates, 420 Y y-axis, 420 y-coordinates, 420 y-intercept, 426 Z z curve, 733 Zero, 106 additive identity property of, 117, 118, 260, 333 angle, 460 division by, 135–136 as an exponent, 139 Zeroth row, 51 z scores calculating, 735 defined, 734 percentiles and, 733–736 Z05_LONG2365_07_GE_NCTM.indd Page 867 7/2/14 8:39 PM s-w-149 /205/AW00196/9781292062365_DETEMPLE/DETEMPLE_MATHEMATICAL_REASONING_FOR_ELEMENT www.downloadslide.net NCTM Principles and Standards for State Mathematics D ecisions made by teachers, school administrators, and other education professionals about the content and character of school mathematics have important consequences for both students and for society These decisions should be based on sound professional guidance Principles and Standards for School Mathematics is intended to provide such guidance The Principles describe particular features of high-quality mathematics education The Standards describe the mathematical content and processes that students should learn Together, the Principles and Standards constitute a vision to guide educators as they strive for the continual improvement of mathematics education in classrooms, schools, and educational systems You may read the online version of Principles and Standards for School Mathematics at http://standards.nctm.org Principles for School Mathematics • Equity Excellence in mathematics education requires equity—high expectations and strong support for all students • Curriculum A curriculum is more than a collection of activities: it must be coherent, focused on important mathematics, and well articulated across the grades • Teaching Effective mathematics teaching requires understanding what students know and need to learn and then challenging and supporting them to learn it well • Learning Students must learn mathematics with understanding, actively building new knowledge from experience and prior knowledge • Assessment Assessment should support the learning of important mathematics and furnish useful information to both teachers and students • Technology Technology is essential in teaching and learning mathematics; it influences the mathematics that is taught and enhances students’ learning The Content Standards Number and Operations Standard Instructional programs from prekindergarten through grade 12 should enable all students to: • Understand numbers, ways of representing numbers, relationships among numbers, and number systems • Understand meanings of operations and how they relate to one another • Compute fluently and make reasonable estimates Algebra Standard Instructional programs from prekindergarten through grade 12 should enable all students to: • • • • Understand patterns, relations, and functions Represent and analyze mathematical situations and structures using algebraic symbols Use mathematical models to represent and understand quantitative relationships Analyze change in various contexts Geometry Standard Instructional programs from prekindergarten through grade 12 should enable all students to: • Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships • Specify locations and describe spatial relationships using coordinate geometry and other representational systems • Apply transformations and use symmetry to analyze mathematical situations • Use visualization, spatial reasoning, and geometric modeling to solve problems Z05_LONG2365_07_GE_NCTM.indd Page 868 7/2/14 8:39 PM s-w-149 /205/AW00196/9781292062365_DETEMPLE/DETEMPLE_MATHEMATICAL_REASONING_FOR_ELEMENT www.downloadslide.net Measurement Standard Instructional programs from prekindergarten through grade 12 should enable all students to: • Understand measurable attributes of objects and the units, systems, and processes of measurement • Apply appropriate techniques, tools, and formulas to determine measurements Data Analysis and Probability Standard Instructional programs from prekindergarten through grade 12 should enable all students to: • Formulate questions that can be addressed with data and collect, organize, and display relevant data to answer them • Select and use appropriate statistical methods to analyze data • Develop and evaluate inferences and predictions that are based on data • Understand and apply basic concepts of probability The Process Standards Problem Solving Standard Instructional programs from prekindergarten through grade 12 should enable all students to: • • • • Build new mathematical knowledge through problem solving Solve problems that arise in mathematics and in other contexts Apply and adapt a variety of appropriate strategies to solve problems Monitor and reflect on the process of mathematical problem solving Reasoning and Proof Standard Instructional programs from prekindergarten through grade 12 should enable all students to: • • • • Recognize reasoning and proof as fundamental aspects of mathematics Make and investigate mathematical conjectures Develop and evaluate mathematical arguments and proofs Select and use various types of reasoning and methods of proof Communication Standard Instructional programs from prekindergarten through grade 12 should enable all students to: • • • • Organize and consolidate their mathematical thinking through communication Communicate their mathematical thinking coherently and clearly to peers, teachers, and others Analyze and evaluate the mathematical thinking and strategies of others Use the language of Mathematics to express mathematical ideas precisely Connections Standard Instructional programs from prekindergarten through grade 12 should enable all students to: • Recognize and use connections among mathematical ideas • Understand how mathematical ideas interconnect and build on one another to produce a coherent whole • Recognize and apply mathematics in contexts outside of mathematics Representation Standard Instructional programs from prekindergarten through grade 12 should enable all students to: • Create and use representations to organize, record, and communicate mathematical ideas • Select, apply, and translate among mathematical representations to solve problems • Use representations to model and interpret physical, social, and mathematical phenomena SOURCE: NCTM Principles and Standards for School Mathematics, http://standards.nctm.org Reprinted with permission Z06_LONG2365_07_GE_IBC.indd Page 870 7/2/14 8:39 PM s-w-149 /205/AW00196/9781292062365_DETEMPLE/DETEMPLE_MATHEMATICAL_REASONING_FOR_ELEMENT www.downloadslide.net Common Core State Standards for Mathematics The Common Core State Standards for Mathematics (CCSS-M) “ define what students should understand and be able to in their study of mathematics.” The complete statements of the CCSS-M can be obtained at http://www.corestandards.org/Math The standards are partitioned into two parts, beginning with the Standards for Mathematical Practice which are listed below These are followed by the Standards for Mathematical Content, which details the level of mathematical knowledge, skills, and understandings that should be reached according to grade level Standards for Mathematical Practice The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their students These practices rest on important “processes and proficiencies” with longstanding importance in mathematics education The first of these are the NCTM process standards of problem solving, reasoning and proof, communication, representation, and connections The second are the strands of mathematical proficiency specified in the National Research Council’s report Adding It Up: adaptive reasoning, strategic competence, conceptual understanding (comprehension of mathematical concepts, operations and relations), procedural fluency (skill in carrying out procedures flexibly, accurately, efficiently and appropriately), and productive disposition (habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy) Make sense of problems and persevere in solving them Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution They analyze givens, constraints, relationships, and goals They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution They monitor and evaluate their progress and change course if necessary Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, “Does this make sense?” They can understand the approaches of others to solving complex problems and identify correspondences between different approaches Reason abstractly and quantitatively Mathematically proficient students make sense of quantities and their relationships in problem situations They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize—to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents—and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects SOURCE: From the Common Core, The Standards for Mathematical Practice © Copyright 2010 National Governors Association Center for Best Practices and Council of Chief State School Officers All rights reserved Z06_LONG2365_07_GE_IBC.indd Page 871 7/2/14 8:39 PM s-w-149 /205/AW00196/9781292062365_DETEMPLE/DETEMPLE_MATHEMATICAL_REASONING_FOR_ELEMENT www.downloadslide.net Construct viable arguments and critique the reasoning of others Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments They make conjectures and build a logical progression of statements to explore the truth of their conjectures They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples They justify their conclusions, communicate them to others, and respond to the arguments of others They reason inductively about data, making plausible arguments that take into account the context from which the data arose Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades Later, students learn to determine domains to which an argument applies Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments Model with mathematics Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace In early grades, this might be as simple as writing an addition equation to describe a situation In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas They can analyze those relationships mathematically to draw conclusions They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose Use appropriate tools strategically Mathematically proficient students consider the available tools when solving a mathematical problem These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator They detect possible errors by strategically using estimation and other mathematical knowledge When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems They are able to use technological tools to explore and deepen their understanding of concepts Attend to precision Mathematically proficient students try to communicate precisely to others They try to use clear definitions in discussion with others and in their own reasoning They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context In the elementary grades, students give carefully formulated explanations to each other By the time they reach high school they have learned to examine claims and make explicit use of definitions Z06_LONG2365_07_GE_IBC.indd Page 872 7/2/14 8:39 PM s-w-149 /205/AW00196/9781292062365_DETEMPLE/DETEMPLE_MATHEMATICAL_REASONING_FOR_ELEMENT www.downloadslide.net Look for and make use of structure Mathematically proficient students look closely to discern a pattern or structure Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have Later, students will see * equals the well remembered * + * 3, in preparation for learning about the distributive property In the expression x2 + 9x + 14, older students can see the 14 as * and the as + They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems They also can step back for an overview and shift perspective They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects For example, they can see - 3(x - y)2 as minus a positive number times a square and use that to realize that its value cannot be more than for any real numbers x and y Look for and express regularity in repeated reasoning Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation (y - 2)/(x - 1) = Noticing the regularity in the way terms cancel when expanding (x - 1)(x + 1), (x - 1) (x2 + x + 1), and (x - 1)(x3 + x2 + x + 1) might lead them to the general formula for the sum of a geometric series As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details They continually evaluate the reasonableness of their intermediate results Connecting the Standards for Mathematical Practice to the Standards for Mathematical Content The Standards for Mathematical Practice describe ways in which developing student practitioners of the discipline of mathematics increasingly ought to engage with the subject matter as they grow in mathematical maturity and expertise throughout the elementary, middle and high school years Designers of curricula, assessments, and professional development should all attend to the need to connect the mathematical practices to mathematical content in mathematics instruction The Standards for Mathematical Content are a balanced combination of procedure and understanding Expectations that begin with the word “understand” are often especially good opportunities to connect the practices to the content Students who lack understanding of a topic may rely on procedures too heavily Without a flexible base from which to work, they may be less likely to consider analogous problems, represent problems coherently, justify conclusions, apply the mathematics to practical situations, use technology mindfully to work with the mathematics, explain the mathematics accurately to other students, step back for an overview, or deviate from a known procedure to find a shortcut In short, a lack of understanding effectively prevents a student from engaging in the mathematical practices In this respect, those content standards which set an expectation of understanding are potential “points of intersection” between the Standards for Mathematical Content and the Standards for Mathematical Practice These points of intersection are intended to be weighted toward central and generative concepts in the school mathematics curriculum that most merit the time, resources, innovative energies, and focus necessary to qualitatively improve the curriculum, instruction, assessment, professional development, and student achievement in mathematics Z05_LONG2365_07_GE_IDX.indd Page 864 04/06/14 5:17 PM f-w-155-user /205/AW00196/9781292062365_DETEMPLE/DETEMPLE_MATHEMATICAL_REASONING_FOR_ELEMENT www.downloadslide.net Z05_LONG2365_07_GE_IDX.indd Page 865 04/06/14 5:17 PM f-w-155-user /205/AW00196/9781292062365_DETEMPLE/DETEMPLE_MATHEMATICAL_REASONING_FOR_ELEMENT www.downloadslide.net Z05_LONG2365_07_GE_IDX.indd Page 866 04/06/14 5:17 PM f-w-155-user /205/AW00196/9781292062365_DETEMPLE/DETEMPLE_MATHEMATICAL_REASONING_FOR_ELEMENT www.downloadslide.net Z05_LONG2365_07_GE_NCTM.indd Page 869 7/2/14 8:39 PM s-w-149 /205/AW00196/9781292062365_DETEMPLE/DETEMPLE_MATHEMATICAL_REASONING_FOR_ELEMENT www.downloadslide.net Z05_LONG2365_07_GE_IDX.indd Page 863 04/06/14 5:17 PM f-w-155-user /205/AW00196/9781292062365_DETEMPLE/DETEMPLE_MATHEMATICAL_REASONING_FOR_ELEMENT www.downloadslide.net ... /205/AW00196/9781292062365_DETEMPLE/DETEMPLE _MATHEMATICAL_ REASONING_ FOR_ ELEMENT Mathematical Reasoning FOR ELEMENTARY TEACHERS This book presents the mathematical knowledge needed for teaching, with an emphasis on why future teachers. .. United States edition, entitled Mathematical Reasoning for Elementary Teachers, 7th edition, ISBN 978-0-321-90099-9, by Calvin T Long, Duane W DeTemple, and Richard S Millman, published by Pearson... A01_LONG2365_07_GE_FM.indd Page 6/30/14 3:57 PM s-w-149 ~/Desktop/30:6:2014 TH EDITION GLOBAL EDITION Mathematical Reasoning FOR ELEMENTARY TEACHERS C a l v i n T D u a n e W Richard S Long DeTemple Millman