BMIndex.indd 29 7/31/2013 7:29:35 AM National Council of Teachers of Mathematics Principles and Standards for School Mathematics Principles for School Mathematics r EQUITY. &YDFMMFODF JO NBUIFNBUJDT FEVDBUJPO SFRVJSFT FRVJUZIJHIFYQFDUBUJPOTBOETUSPOHTVQQPSUGPSBMMTUVEFOUT r CURRICULUM. " DVSSJDVMVN JT NPSF UIBO B DPMMFDUJPO PG BDUJWJUJFTJUNVTUCFDPIFSFOU GPDVTFEPOJNQPSUBOUNBUIFNBUJDT BOEXFMMBSUJDVMBUFEBDSPTTUIFHSBEFT r TEACHING. &GGFDUJWF NBUIFNBUJDT UFBDIJOH SFRVJSFT VOEFSTUBOEJOHXIBUTUVEFOUTLOPXBOEOFFEUPMFBSOBOEUIFODIBMMFOHJOHBOETVQQPSUJOHUIFNUPMFBSOJUXFMM r LEARNING. 4UVEFOUT NVTU MFBSO NBUIFNBUJDT XJUI VOEFSTUBOEJOH BDUJWFMZCVJMEJOHOFXLOPXMFEHFGSPNFYQFSJFODFBOE QSJPSLOPXMFEHF r ASSESSMENT. "TTFTTNFOU TIPVME TVQQPSU UIF MFBSOJOH PG JNQPSUBOUNBUIFNBUJDTBOEGVSOJTIVTFGVMJOGPSNBUJPOUPCPUI UFBDIFSTBOETUVEFOUT r TECHNOLOGY. 5FDIOPMPHZJTFTTFOUJBMJOUFBDIJOHBOEMFBSOJOH NBUIFNBUJDT JU JOGMVFODFT UIF NBUIFNBUJDT UIBU JT UBVHIU BOEFOIBODFTTUVEFOUTMFBSOJOH Standards for School Mathematics NUMBER AND OPERATIONS DATA ANALYSIS AND PROBABILITY *OTUSVDUJPOBMQSPHSBNTGSPNQSFLJOEFSHBSUFOUISPVHIHSBEF TIPVMEFOBCMFBMMTUVEFOUTUP *OTUSVDUJPOBMQSPHSBNTGSPNQSFLJOEFSHBSUFOUISPVHIHSBEF TIPVMEFOBCMFBMMTUVEFOUTUP r VOEFSTUBOEOVNCFST XBZTPGSFQSFTFOUJOHOVNCFST SFMBUJPOTIJQTBNPOHOVNCFST BOEOVNCFSTZTUFNT r GPSNVMBUFRVFTUJPOTUIBUDBOCFBEESFTTFEXJUIEBUBBOEDPMMFDU PSHBOJ[F BOEEJTQMBZSFMFWBOUEBUBUPBOTXFSUIFN r TFMFDUBOEVTFBQQSPQSJBUFTUBUJTUJDBMNFUIPETUPBOBMZ[FEBUB r EFWFMPQ BOE FWBMVBUF JOGFSFODFT BOE QSFEJDUJPOT UIBU BSF CBTFEPOEBUB r VOEFSTUBOEBOEBQQMZCBTJDDPODFQUTPGQSPCBCJMJUZ r VOEFSTUBOE NFBOJOHT PG PQFSBUJPOT BOE IPX UIFZ SFMBUF UP POFBOPUIFS r DPNQVUFáVFOUMZBOENBLFSFBTPOBCMFFTUJNBUFT ALGEBRA *OTUSVDUJPOBMQSPHSBNTGSPNQSFLJOEFSHBSUFOUISPVHIHSBEF TIPVMEFOBCMFBMMTUVEFOUTUP r VOEFSTUBOEQBUUFSOT SFMBUJPOT BOEGVODUJPOT r SFQSFTFOUBOEBOBMZ[FNBUIFNBUJDBMTJUVBUJPOTBOETUSVDUVSFT VTJOHBMHFCSBJDTZNCPMT r VTFNBUIFNBUJDBMNPEFMTUPSFQSFTFOUBOEVOEFSTUBOERVBOUJUBUJWFSFMBUJPOTIJQT r BOBMZ[FDIBOHFJOWBSJPVTDPOUFYUT PROBLEM SOLVING *OTUSVDUJPOBMQSPHSBNTGSPNQSFLJOEFSHBSUFOUISPVHIHSBEF TIPVMEFOBCMFBMMTUVEFOUTUP r CVJMEOFXNBUIFNBUJDBMLOPXMFEHFUISPVHIQSPCMFNTPMWJOH r TPMWFQSPCMFNTUIBUBSJTFJONBUIFNBUJDTBOEJOPUIFSDPOUFYUT r BQQMZ BOE BEBQU B WBSJFUZ PG BQQSPQSJBUF TUSBUFHJFT UP TPMWF QSPCMFNT r NPOJUPSBOESFáFDUPOUIFQSPDFTTPGNBUIFNBUJDBMQSPCMFN TPMWJOH GEOMETRY REASONING AND PROOF *OTUSVDUJPOBMQSPHSBNTGSPNQSFLJOEFSHBSUFOUISPVHIHSBEF TIPVMEFOBCMFBMMTUVEFOUTUP *OTUSVDUJPOBMQSPHSBNTGSPNQSFLJOEFSHBSUFOUISPVHIHSBEF TIPVMEFOBCMFBMMTUVEFOUTUP r BOBMZ[F DIBSBDUFSJTUJDT BOE QSPQFSUJFT PG UXP BOE UISFF EJNFOTJPOBM HFPNFUSJD TIBQFT BOE EFWFMPQ NBUIFNBUJDBM BSHVNFOUTBCPVUHFPNFUSJDSFMBUJPOTIJQT r SFDPHOJ[F 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r BOBMZ[FBOEFWBMVBUFUIFNBUIFNBUJDBMUIJOLJOHBOETUSBUFHJFT PGPUIFST r VTF UIF MBOHVBHF PG NBUIFNBUJDT UP FYQSFTT NBUIFNBUJDBM JEFBTQSFDJTFMZ 7/31/2013 10:58:25 AM CONNECTIONS REPRESENTATION *OTUSVDUJPOBMQSPHSBNTGSPNQSFLJOEFSHBSUFOUISPVHIHSBEF TIPVMEFOBCMFBMMTUVEFOUTUP *OTUSVDUJPOBMQSPHSBNTGSPNQSFLJOEFSHBSUFOUISPVHIHSBEF TIPVMEFOBCMFBMMTUVEFOUTUP r SFDPHOJ[FBOEVTFDPOOFDUJPOTBNPOHNBUIFNBUJDBMJEFBT r DSFBUFBOEVTFSFQSFTFOUBUJPOTUPPSHBOJ[F SFDPSE BOEDPNNVOJDBUFNBUIFNBUJDBMJEFBT r VOEFSTUBOE IPX NBUIFNBUJDBM JEFBT JOUFSDPOOFDU BOE CVJME POPOFBOPUIFSUPQSPEVDFBDPIFSFOUXIPMF r SFDPHOJ[FBOEBQQMZNBUIFNBUJDTJODPOUFYUTPVUTJEFPGNBUIFNBUJDT r TFMFDU BQQMZ BOE USBOTMBUF BNPOH NBUIFNBUJDBM SFQSFTFOUBUJPOTUPTPMWFQSPCMFNT r VTF SFQSFTFOUBUJPOT UP NPEFM BOE JOUFSQSFU QIZTJDBM TPDJBM BOENBUIFNBUJDBMQIFOPNFOB Curriculum Focal Points for Prekindergarten through Grade Mathematics PREKINDERGARTEN Number and Operations:%FWFMPQJOHBOVOEFSTUBOEJOHPGXIPMF OVNCFST JODMVEJOHDPODFQUTPGDPSSFTQPOEFODF DPVOUJOH DBSEJOBMJUZ BOEDPNQBSJTPO Geometry: *EFOUJGZJOH TIBQFT BOE EFTDSJCJOH TQBUJBM SFMBUJPOTIJQT Measurement:*EFOUJGZJOHNFBTVSBCMFBUUSJCVUFTBOEDPNQBSJOH PCKFDUTCZVTJOHUIFTFBUUSJCVUFT Number and Operations: %FWFMPQJOH BO VOEFSTUBOEJOH PG EFDJNBMT JODMVEJOH UIF DPOOFDUJPOT CFUXFFO GSBDUJPOT BOE EFDJNBMT Measurement:%FWFMPQJOHBOVOEFSTUBOEJOHPGBSFBBOEEFUFSNJOJOHUIFBSFBTPGUXPEJNFOTJPOBMTIBQFT KINDERGARTEN Number and Operations:3FQSFTFOUJOH DPNQBSJOHBOEPSEFSJOH XIPMFOVNCFST BOEKPJOJOHBOETFQBSBUJOHTFUT Geometry:%FTDSJCJOHTIBQFTBOETQBDF Measurement:0SEFSJOHPCKFDUTCZNFBTVSBCMFBUUSJCVUFT GRADE Number and Operations BOE Algebra: %FWFMPQJOH BO VOEFSTUBOEJOHPGBOEáVFODZXJUIEJWJTJPOPGXIPMFOVNCFST Number and Operations:%FWFMPQJOHBOVOEFSTUBOEJOHPGBOE áVFODZXJUIBEEJUJPOBOETVCUSBDUJPOPGGSBDUJPOTBOEEFDJNBMT Geometry BOE Measurement BOE Algebra: %FTDSJCJOH UISFF EJNFOTJPOBM TIBQFT BOE BOBMZ[JOH UIFJS QSPQFSUJFT JODMVEJOH WPMVNFBOETVSGBDFBSFB GRADE Number and OperationsBOEAlgebra:%FWFMPQJOHVOEFSTUBOEJOHTPGBEEJUJPOBOETVCUSBDUJPOBOETUSBUFHJFTGPSCBTJDBEEJUJPOGBDUTBOESFMBUFETVCUSBDUJPOGBDUT Number and Operations:%FWFMPQJOHBOVOEFSTUBOEJOHPGXIPMF OVNCFSSFMBUJPOTIJQT JODMVEJOHHSPVQJOHJOUFOTBOEPOFT Geometry:$PNQPTJOHBOEEFDPNQPTJOHHFPNFUSJDTIBQFT GRADE Number and Operations:%FWFMPQJOHBOVOEFSTUBOEJOHPGBOEáVFODZXJUINVMUJQMJDBUJPOBOEEJWJTJPOPGGSBDUJPOTBOEEFDJNBMT Number and Operations:$POOFDUJOHSBUJPBOESBUFUPNVMUJQMJDBUJPOBOEEJWJTJPO Algebra:8SJUJOH JOUFSQSFUJOH BOEVTJOHNBUIFNBUJDBMFYQSFTTJPOTBOEFRVBUJPOT GRADE Number and Operations: %FWFMPQJOH BO VOEFSTUBOEJOH PG UIF CBTFUFOOVNFSBUJPOTZTUFNBOEQMBDFWBMVFDPODFQUT Number and Operations BOE Algebra: %FWFMPQJOH RVJDL SFDBMM PGBEEJUJPOGBDUTBOESFMBUFETVCUSBDUJPOGBDUTBOEáVFODZXJUI NVMUJEJHJUBEEJUJPOBOETVCUSBDUJPO Measurement:%FWFMPQJOHBOVOEFSTUBOEJOHPGMJOFBSNFBTVSFNFOUBOEGBDJMJUZJONFBTVSJOHMFOHUIT GRADE Number and Operations BOE Algebra BOE Geometry: %FWFMPQJOHBOVOEFSTUBOEJOHPGBOEBQQMZJOHQSPQPSUJPOBMJUZ JODMVEJOH TJNJMBSJUZ MeasurementBOEGeometryBOEAlgebra:%FWFMPQJOHBOVOEFSTUBOEJOHPGBOEVTJOHGPSNVMBTUPEFUFSNJOFTVSGBDFBSFBTBOE WPMVNFTPGUISFFEJNFOTJPOBMTIBQFT Number and Operations BOE Algebra: %FWFMPQJOH BO VOEFSTUBOEJOH PG PQFSBUJPOT PO BMM SBUJPOBM OVNCFST BOE TPMWJOH MJOFBSFRVBUJPOT GRADE Number and OperationsBOEAlgebra:%FWFMPQJOHVOEFSTUBOEJOHTPGNVMUJQMJDBUJPOBOEEJWJTJPOBOETUSBUFHJFTGPSCBTJDNVMUJQMJDBUJPOGBDUTBOESFMBUFEEJWJTJPOGBDUT Number and Operations:%FWFMPQJOHBOVOEFSTUBOEJOHPGGSBDUJPOTBOEGSBDUJPOFRVJWBMFODF Geometry:%FTDSJCJOHBOEBOBMZ[JOHQSPQFSUJFTPGUXPEJNFOTJPOBMTIBQFT GRADE Number and Operations BOE Algebra: %FWFMPQJOH RVJDL SFDBMM PG NVMUJQMJDBUJPO GBDUT BOE SFMBUFE EJWJTJPO GBDUT BOE áVFODZ XJUIXIPMFOVNCFSNVMUJQMJDBUJPO FMEndpaper.indd 16 GRADE Algebra:"OBMZ[JOHBOESFQSFTFOUJOHMJOFBSGVODUJPOTBOETPMWJOHMJOFBSFRVBUJPOTBOETZTUFNTPGMJOFBSFRVBUJPOT Geometry BOE Measurement: "OBMZ[JOH UXP BOE UISFF EJNFOTJPOBMTQBDFBOEàHVSFTCZVTJOHEJTUBODFBOEBOHMF Data AnalysisBOE Number and OperationsBOEAlgebra:"OBMZ[JOHBOETVNNBSJ[JOHEBUBTFUT 7/31/2013 10:58:25 AM WileyPLUS is a research-based online environment for effective teaching and learning WileyPLUS builds students’ confidence because it takes the guesswork out of studying by providing students with a clear roadmap: • • • what to how to it if they did it right It offers interactive resources along with a complete digital textbook that help students learn more With WileyPLUS, students take more initiative so you’ll have greater impact on their achievement in the classroom and beyond Now available for For more information, visit www.wileyplus.com ALL THE HELP, RESOURCES, AND PERSONAL SUPPORT YOU AND YOUR STUDENTS NEED! www.wileyplus.com/resources Student Partner Program 2-Minute Tutorials and all of the resources you and your students need to get started Student support from an experienced student user Collaborate with your colleagues, find a mentor, attend virtual and live events, and view resources www.WhereFacultyConnect.com Quick Start © Courtney Keating/iStockphoto Pre-loaded, ready-to-use assignments and presentations created by subject matter experts Technical Support 24/7 FAQs, online chat, and phone support www.wileyplus.com/support Your WileyPLUS Account Manager, providing personal training and support athematics M For Elementary Teachers TENTH EDITION A CONTEMPORARY APPROACH Gary L Musser t Blake E Peterson t William F Burger Oregon State University FMWileyPlus.indd Brigham Young University 7/31/2013 2:14:09 PM To: 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 approach to life, and Riley for his good choices and his dry wit B.E.P VICE PRESIDENT & EXECUTIVE PUBLISHER PROJECT EDITOR SENIOR CONTENT MANAGER SENIOR PRODUCTION EDITOR MARKETING MANAGER SENIOR PRODUCT DESIGNER OPERATIONS MANAGER ASSISTANT CONTENT EDITOR SENIOR PHOTO EDITOR MEDIA SPECIALIST COVER & TEXT DESIGN Laurie Rosatone Jennifer Brady Karoline Luciano Kerry Weinstein Kimberly Kanakes Tom Kulesa Melissa Edwards Jacqueline Sinacori Lisa Gee Laura Abrams Madelyn Lesure This book was set by Laserwords and printed and bound by Courier Kendallville The cover was printed by Courier Kendallville Copyright © 2014, 2011, 2008, 2005, John Wiley & Sons, Inc 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, scanning or otherwise, except as permitted under Sections 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc 222 Rosewood Drive, Danvers, MA 01923, website www.copyright.com Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030-5774, (201)748-6011, fax (201)748-6008, website http://www.wiley.com/go/permissions Evaluation copies are provided to qualified academics and professionals for review purposes only, for use in their courses during the next academic year These copies are licensed and may not be sold or transferred to a third party Upon completion of the review period, please return the evaluation copy to Wiley Return instructions and a free of charge return shipping label are available at www.wiley.com/go/returnlabel Outside of the United States, please contact your local representative 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) Mathematics Mathematics–Study and teaching (Elementary) I Title QA39.3.M87 2014 510.2’4372–dc23 2013019907 Printed in the United States of America 10 FMWileyPlus.indd 7/31/2013 2:14:09 PM ABOUT THE AUTHORS 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 Peterson 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 Education 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 mathematics 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 University, 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 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 responsibilities, playing basketball, mountain biking, water skiing, and working in the yard v FMWileyPlus.indd 7/31/2013 2:14:11 PM ABOUT THE COVER Are you puzzled by the numbers on the cover? They are 25 different randomly selected counting numbers from to 100 In that set of numbers, two different arithmetic progressions 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 and 15 is 8, between 15 and 23 is 8, and between 23 and 31 is Thus, the sequence 7, 15, 23, 31 forms an arithmetic progression of length (there are numbers in the sequence) with a common difference of Similarly, the numbers highlighted in red, namely 45, 69, 93, form another arithmetic progression This progression is of length 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 Professor 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 arithmetic progressions in random sets of counting numbers (or integers) Namely, if one randomly selects half of the counting numbers from and 100, what lengths of arithmetic progressions can one expect to find? What if one picks one-tenth of the numbers from to 100 or if one picks half of the numbers between 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 technique 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 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, arithmetic progressions of length four and 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 vi FMWileyPlus.indd 7/31/2013 2:14:12 PM Index I11 theorems, 310–11 zero as, 305 See also rational numbers interest (bank), 290–91, 396 interior angles See vertex/interior angles interior of the angle, 591 intermediate algorithms addition, 146, 163 long division, 153 multiplication, 150, 165 interquartile range (IQR), 444–47 interrelatedness of metric system, 657 intersecting arcs, 743 intersecting planes, 621 intersection or union of sets, 48, 87–88, 191–92, 494–95 intervals, data, 419–20 intuitive reasoning and algebra, 17 invalid argument, 885 inverse, 883 inverse property of addition integers, 310, 321 negative integers vs., 314 real numbers, 361 of multiplication fractions, 236 rational numbers, 348 real numbers, 361 invert and multiply, 237, 239–40, 349–50 Ionian numeration system, 65–66 IQR (interquartile range), 444–47 irrational numbers definition, 359 infinite number of, 361 number system diagrams, 359, 360 pi, 361 Pythagorean theorem and, 338 set of, 359 and square roots, 361 is an element of a set, 45 is divisible by, 178 is necessary and sufficient for, 884 is necessary for, 884 is not an element of a set, 45 isometries, 823–30, 846–52 about, 823, 848, 850–51 and congruence, 846–52 distance and, 846–48 glide reflection (See glide reflection) and parallel lines, 848 properties of, 848 reflection (See reflection) rotation (See rotation) rotations, 825–27 and similitudes, 833–35 size transformation, 833–35, 840, 852–53 symmetry, 830, 832 transformation congruence, 846–52 translation (See translation) translations/sliding transformations, 823–25 isosceles trapezoids, 582, 597–98 isosceles triangle(s) about, 569 child’s recognition of, 554–55 BMIndex.indd 11 congruent base angles of, 863 congruent medians of, 807–8 equilateral triangles as, 581 isosceles right triangles, 580, 674 mirror symmetry, 565 obtuse, 596 Pons Asinorum Theorem, 716 tessellation with acute isosceles triangle, 608 is sufficient for, 884 J Jefferson, Thomas, 431, 676 Johnson, C., 764 joules, 660, 664 K Kemeny, John, 169 key of a pictograph, 427 Killie’s Way solution, 353 kilograms, 657–58, 659 kiloliters, 656–58 kilometers, 653 kite(s) diagonal of, 724 diagonals of, 744–46 model and description, 582 one-to-one correspondence, 823 pairs of angles, 597 perimeter, 666–67 and perpendicular bisector of a line segment, 744–45 properties, 598 reflection symmetry, 823 rhombi as, 583 squares as, 583, 584 SSS congruence property, 724 and trapezoids, 583 Koch curve (or snowflake), 34, 733 Kovalevskaya, Sonya, 298 L lateral faces, 624 See also quadrilaterals lateral surface area, 686, 687, 690–91 latitude, 794 lattice method algorithm addition, 146–47, 163 least common multiple, 199 multiplication, 150, 165 lattice multiplication with decimals, 264–65 lattice, square See square lattice lattice, triangular, 556 law of detachment (modus ponens), 884–85 LCD See least common denominator LCM See least common multiple least common denominator (LCD), 224 least common multiple (LCM) about, 194–96 build-up method, 195–96 extending the concept, 196–97 greatest common factor and, 196–97 lattice method, 199 and least common denominator, 224 prime factorization method, 195, 196 set intersection method, 195, 196 Venn diagrams, finding with, 196, 199, 201 LeBlanc, Antoine (Sophie Germain), 31, 38 left-to-right methods, 132 legs of a right triangle, 674 Lehmer, D H., 188 Leibniz, Gottfried Wilhelm, 128, 644 length, 665–68 distance (See distance) of hypotenuse, 360, 371, 373 of a line segment, 568, 569, 591, 665–66, 760 measurement of, 650, 652–53 perimeter, 666–68 of PQ, 783 less likely, more likely, 489 less than addition inequalities, 368–70 integers, 327 rational numbers, 352 real numbers, 361 whole numbers, 116–17 definition, 116 and fractions, 216–17 general set formulation of, 59 inequalities, 364 multiplication (See less than multiplication) ordering, 59, 326–28 less than multiplication inequalities, 368–70 integers, 327 rational numbers, 352 real numbers, 361 whole numbers, 117 less than properties addition, 116 ordering, 327 real numbers, 361 solving inequalities, 368–70 transitive inequalities, 368–70 integers, 327–28 rational numbers, 352 real numbers, 361 whole numbers, 116 less than or equal to, 59, 116, 368–70 Let’s Make a Deal eManipulative, 517 Lilavat (Bhaskara), 242 linear equations, 803 linear functions, graphs of, 393–94 line graphs about, 423–24 and cropping, 465–66 misleading presentation, 464–66 pictorial embellishments, 428, 469–71 with three-dimensional effects, 467 when to use, 430 line (axis) of symmetry, 565 line plots, 417–18, aka dot plots line(s) about, 590–91 angle bisector, 724 axis of symmetry of a cube, 632 concurrent, 590 equations of, 795–800, 807–11 circumcenter of a triangle, 810 orthocenter of a triangle, 810 graphs (See line graphs) mapping with isometries, 848 number (See number lines) parallel (See parallel lines) perpendicular (See perpendicular bisector; perpendicular lines) in a plane, 590 points and, 590–91 point-slope equation of, 798–99 properties of, 591 reflection in, 826 skew, 622 slope (See slope) slope-intercept equation, 796–98 slope of, 787–88 in three-dimensional space, 622 transversal, 592–94, 603, 617, 722, 736 line segment(s) AA similarity property, 760 about, 595 adjacent, 575 bisector, 724 cevian, 297 collinearity test, 783–84 congruent, 719 connecting points on a graph, 393 construct a segment AB with Geometer’s Sketchpad, 601 coordinate distance formula, 783 copying, 743 defining, 591 diagonal, 573 diameter of a circle, 611 directed, 824–25 endpoints, 568, 591 equivalent, 824–25 Euclidean constructions, 743 height of obtuse triangle, 671 length of, 568, 569, 591, 665–66, 760 midpoint, 574 midpoint formula, 784, 786 midsegment, 683 model and description, 568 opposite (not adjacent), 575 parallel line segments test, 571 perpendicular bisector, 724 perpendicular line segments test, 573 properties, 591 same length, 568, 569 slope of, 787–88 square root of a whole number, 675 and translation, 846 of triangles, 617 liquid measures of capacity, 651–52 7/31/2013 7:29:34 AM I12 Index Literary Digest poll, 472 liters, 656 loan interest, 290–91 logic arguments, 884–87 arithmetic, 137–38 connectives, 882–84 logically equivalent statements, 883–84 statements, 878 long division base five algorithm, 165–66 base ten algorithm, 151–52 calculator activities, 137, 266 conversion from base ten a given base, 74, 75 decimals, 265 intermediate algorithm, 153 missing-factors approach, 151–54, 265 remainders, 151 repeating decimals, 267–68 scaffold method, 152–54, 166 standard algorithm, 154, 166 thinking strategy, 151–54 longitude, 794 longs, in base ten pieces, 68 Look for a Formula strategy, 413, 479 Look for a Pattern strategy additional problems, 23–24, 37 and coin toss probabilities, 510–12 combining with other strategies, 27, 28–29, 28–30, 413 counting factors, 190–91 decimals, 268 distance involving translations, 865 downward paths in grid, 22–23 exponents, 120, 324 find ones digits in 399 problem, 23–24, 37 integer addition, 311 integer multiplication, 319–20 long division, 268 negative exponents, 324–25 sum of counting numbers, 21–22 Loomis, Elisha, 760 lower quartile, 444–47 lowest terms fractions, 212–14, 224, 225, 235–36 rational numbers, 342–43 M machines, functions as, 382 magic hexagons, 98 magic squares additive, 20, 98, 112, 189, 260, 318 multiplicative, 113 magnifications (size transformations), 833–35, 840, 852–53 Make a List strategy, 24–26, 27, 28–30, 38 Mandelbrot, Benoit, 733, 755 mantissa, 266 mapping See transformation geometry BMIndex.indd 12 maps, geographical, 393 Marathe’s triangle, 100 marbles probabilities, 493–94, 502, 504–5, 506–8 mass, measurement of, 657–58 matching sets, 45 mathematical art, 820 “Mathematical Games” (Gardner), 334 Mathematical Sciences Research Institute (MSRI), 817 Mayan numeration system, 42, 63–64 Maya people, 42 mean (arithmetic average) about, 442–43 and bell-shaped curves, 451–54 definition, 442 extremes and means, 275 mean proportional, 763 mode, median, and, 443–44, 455 Oregon rain data, 455 and standard deviation, 448, 449–50 and variance, 447–48 and z-scores, 450 measurement, 647–60, 665–76, 686–92, 696–705 about, 648, 652 angles about, 592 dihedral, 621–22 directed, 826 isometry preservation of angle measure, 848, 852 regular n-gons, 605, 606–7, 610 vertex of the angle, 591–92 area, 650–51, 654–56 (See also area) capacity vs volume, 648, 651–52 central tendency (See central tendency measures) cylinders, 688–89 definition, 647 dimensional analysis for conversions, 659–60 dispersion (See dispersion measures) distance (See distance) English (See English system of measurement) estimation, 647–48 Fill ‘n Pour eManipulative, 200 holistic, 647–48 indirect, 732–33 informal (See nonstandard units of measurement) length, 650, 652–53 (See also length) mass, 657–58 metric system, 652–59 models (See measurement models) nonstandard units, 647–48, 649 number lines (See number lines) with square units, 668 standard units, 648, 650–60 surface area (See surface area) temperature, 652, 658–59 vertex/interior angles, 606–7, 610 vertex of the angle, 591–92 volume (See volume) weight, 649, 652 See also English system of measurement; metric system measurement division, 107–8 measurement models addition, 88 fractions, 223–24 integer, 306 integers, 305, 306, 308 missing-addend, 94–96 take-away, 93–94, 95 measures of dispersion See dispersion measures median(s) about, 441, 442 as arithmetic average of two middle scores, 442 and bell-shaped curves, 451–54 box and whisker plots, 444–47 for box and whisker plots, 447 centroid, 757 definition, 442 mean, mode, and, 443–44, 455 Oregon rain data, 455 of a triangle about, 751 collinear, 809 concurrent, 808–9 congruent, 808 ratio of, 808–9 members of a set, 45 memory function of calculators, 139 Memphis, Tennessee, 855 mental math addition, 131–34, 229 compatible numbers, 131–32 compensation, 132 decimals, 256–59 developing child’s ability, 135, 137 division, 131–34, 241–42 fraction equivalents, 285–86 fractions, 229, 241–42 halving and doubling, 143 left-to-right method, 132 multiplication, 131–34, 241–42 order of operations, 131–32 percents as fraction equivalents, 285–86 powers of 10, 132, 257 properties, 131 scaling up/down, 278 subtraction, 131–34, 229 whole numbers, 131–34 See also computational estimation; reasoning; thinking strategies Mere, Chevalier de, 484 meridians, 794 Mersenne number 267–1, 185, 198 meter prototype, 653 meters, 653, 654–56 metric system, 652–59 about, 652, 659 conversions to or from, 656, 657, 658, 659–60 conversions within, 653 converter diagram, 653, 656, 657 decimals, 653 and dimensional analysis, 659–60 English system compared to, 652–53, 654, 657, 658 measurements area, 654–56 length, 652–53 mass, 657–58 temperature, 658–59 volume, 656–57 prefix table, 654 unit tables, 653, 655, 657 world acceptance of, 676 See also powers of 10 metric tons, 658 midpoint Escher pattern from, 844 of line segments, 574, 591 of opposite edges of a cube, 632 of quadrilaterals, 598 of sides of the triangle, 768–70 midpoint formula, 784, 786 midquad theorem, 768–70 midsegment, 683 mild outliers, 456 mile, 650 milliliters, 656–58 millimeters, 653 million, 70 minuend, 94 Mira, 566, 761 misleading graphs and statistics, 460–72 about, 412, 460–61 bar graphs, 461–63, 469–71 circle graphs, 468–69 cropping, 465–66 and data analysis, 423 embedded graphs, 469, 471 line graphs, 464–66, 469–71 manipulation of x-axis, 423 pictographs, 467, 475 pictorial embellishments, 469–71 sampling bias, 471–72 scaling, 461–62 scaling and axis manipulation, 461–63, 464, 465, 466 three-dimensional effects, 466–67 missing-addend subtraction fractions, 228 integers, 313–14 whole numbers, 94–96, 163–64 missing factors See quotients mixed numbers, 216, 227, 235, 341–42, 353 mode and bell-shaped curves, 451–54 definition, 441 median, mean, and, 443–44, 455 Oregon rain data, 455 models black and red chips, 305–6, 319–20 blocks (See blocks) bundles of sticks, 68 chip abacus, 145–46, 148 fraction strips, 212 fuel gauge, 287 7/31/2013 7:29:34 AM Index I13 geometric shapes, 568, 569, 571 for integers, 305–6 measurement (See measurement models) part-to-whole model of fractions, 209 place value (See place value) region, 212, 223–24, 341 set (See set models) Use a Model strategy, 547, 636–37, 674–75, 686 modus ponens (law of detachment), 885 modus tollens (denying the consequent), 886–87 Moebius, A F., 598 Moore, R L., 637 Morawetz, Cathleen Synge, 637 more likely, less likely, 489 more than for ordering whole numbers, 59 MSRI (Mathematical Sciences Research Institute), 817 Multibase Blocks eManipulative, 76, 77, 78, 166, 167, 168 multibase pieces, 73 multi-digit addition, 92, 131 multiple-bar graphs, 420–21 multiple-circle graphs, 424–25, 430–31 multiple-line graphs, 423 multiple(s), 178, 195 See also least common multiple multiplication, 101–7 by 0, 1, 2, 5, or 9, 106–7 algorithms (See multiplication algorithms) base five, 164–65 as binary operation, 88 Cartesian product approach, 50, 497–98 clock arithmetic, 892 coefficients, 366, 368 compensation, 132 computational estimation, 134–35 cross-multiplication, 215, 217, 276–77, 352 decimals, 257, 263, 264–65 duplication algorithm, 157–58 estimation for fractions, 241–42 exponents, 117–18, 119 facts, 106–7, 164 fingers, using, 157 fraction of a fraction, 234 fractions, 233–37, 242, 244, 264 German low-stress algorithm, 158 greater than inequalities, 369 halving and doubling method, 143 identity (one), 104 integers, 318–23 lattice method algorithm, 150, 165 lattice, with decimals, 264–65 less than (See less than multiplication) magic squares, 113 mental math, 131–34, 241–42 mixed numbers, 235 of multi-digit numbers, 149–50 BMIndex.indd 13 negative integers, 318–19, 320, 327, 328 by a negative non-integer number, 327, 352, 361, 368–70 nonstandard algorithm, 157–58, 159 order of operations, 120–21 by powers of 10, 132, 257 probability and, 505–9 products (See products) properties (See multiplicative properties) rational numbers, 348–49, 352 real numbers, 361 rectangular array approach, 101–2, 235 repeated addition (See repeated addition approach to multiplication) Russian peasant algorithm, 157 of sets, 50 single-digit by multiple-digit numbers, 107 by special factors, 132 standard algorithms, 149–50, 165 thinking strategies, 106–7 tree diagram approach, 102 whole numbers, 101–7, 190–91, 263, 264–65 multiplication algorithms base five, 164–65 decimals, 263, 264–65 duplication, 157–58 fractions, 264 whole numbers, 149–50, 164–65 “multiplication makes bigger” misconception, 264 multiplicative compensation, 132 multiplicative identity (one), 104 multiplicative magic squares, 113 multiplicative numeration systems, 61–64, 69 multiplicative properties associative property clock arithmetic, 892 decimals, 257 fractions, 236 integers, 321–22 rational numbers, 348 real numbers, 361 whole numbers, 104, 107 cancellation property, 113, 323 closure property, 102, 236, 321, 348, 361 commutative property clock arithmetic, 892 decimals, 257 fractions, 234 integers, 321–22 in mental math, 131 rational numbers, 348 real numbers, 361 whole numbers, 102, 104, 106, 131 cross-multiplication property of ratios, 276–77 distributive property, over addition fractions, 237 integers, 321 rational numbers, 349 real numbers, 361 whole numbers, 105, 107, 131 distributive property over subtraction, 106, 237, 349 identity property, 104, 236, 321, 348, 361 inverse property, 236, 348, 361 probability property, 505–9 of zero, 106 zero divisor property, 323 musical scale, 282–83, 374 mutually exclusive events, 496 N Napier, John, 128 Napier’s “decimal point” notation, 259 NASA, 865 National Council of Teachers of Mathematics (NCTM), See also Common Core State Standard key concepts; NCTM Curriculum Focal Points; NCTM Standards natural numbers, 44 See also counting numbers NCTM Curriculum Focal Points pre-kindergarten, 44, 548 kindergarten, 4, 44, 86, 414, 548, 822 1st grade, 4, 44, 86, 130, 414, 548, 718 2nd grade, 44, 86, 130, 252, 646 3rd grade, 4, 86, 176, 208, 548, 718, 782, 822 4th grade, 4, 86, 130, 176, 252, 414, 646, 822 5th grade, 4, 130, 208, 252, 304, 340, 414, 548, 646 6th grade, 4, 44, 208, 252, 340, 414, 782 7th grade, 4, 252, 304, 340, 414, 486, 718, 822 8th grade, 340, 414, 548, 782 NCTM Standards pre-k–12 problem solving, pre-k–2 algebra, 86, 548 analysis and probability, 486 data analysis and probability, 414 geometry, 548, 718, 782, 822 measurement, 646 number and operations, 44, 86, 130, 176, 208 3rd–5th grades algebra, 86, 340, 548 analysis and probability, 486 data analysis and probability, 414 geometry, 548, 718, 782, 822 measurement, 646 number and operations, 44, 86, 130, 176, 208, 252 6th–8th grades algebra, 340 analysis and probability, 486 data analysis and probability, 414 geometry, 548, 718, 782, 822 measurement, 646 number and operations, 176, 208, 252, 304, 340 necessary conditionals and biconditionals, 884 Needle Problem, Buffon’s, 535 negation of a statement, 881 negative integers about, 305 addition, 309, 310 additive inverse vs., 314 division, 323–24 exponents, 324–26 multiplication, 318–19, 320, 327 number lines, 306 opposite vs., 314 negative numbers (non-integer) history of, 302 multiplication by, 327, 352, 361, 368–70 rational, 345 as real numbers, 359 subtracting mixed numbers with, 353 nets (patterns) for threedimensional shapes, 628–29, 687, 700 Newton, Sir Isaac, 17, 644 n factorial, 519 n-gons, regular angle measures in a regular, 606–7, 610 angles of, 605 central angle of, 605–6, 607 circles as, 611–12 convex nature of regular, 605 regular, 605 tessellations using, 610 See also polygons Noether, Emmy, 80 nondecimal numeration systems, 72–75 nonillion, 76 nonintersecting nonparallel lines, 622 nonnegative (principal) square root, 360–61 nonrepeating decimals, 359 See also irrational numbers nonstandard algorithms, 155–59 nonstandard units of measurement about, 647–48, 649 board foot, 708 horsepower, 664 joules, 664 parsecs, 665 Smoots, 660 nonterminating decimals, 268 nonzero numbers normal distribution, 452 notation absolute value, 315 angles, 574, 591 combinations, 521 7/31/2013 7:29:34 AM I14 Index notation (continued) congruence (is congruent to), 720 conversions with calculators, 325 coordinate system graphs, 391–93 decimals, 253, 259 directed line segments, 824–25 distance, 665, 666 divides, 178 English system of measurement, 650 equals sign, 364, 719 events, 490 expanded (See expanded form/ notation) expected value, 531 exponential functions, 396 exponents, 74, 119, 120 factorial, 186, 519 fractions, 210 function, 380–84 geometric shapes, 574–75 graphs on a coordinate system, 391–93 greater than or equal to, 116 greatest common factor, 191 in Hindu-Arabic numeration system, 70, 74 images of points under transformations, 846 integers, 305 least common factor, 194 less than or equal to, 116 line segments, 591 lines in three-dimensional space, 622 mean, 443 metric system units and prefixes, 653, 654, 655, 658 negative number, 302 negative signs and subtraction, 314 n factorial, 519 ordering whole numbers, 59 outliers, 445 parallel lines, 590 permutations, 519, 520 perpendicular lines, 592 pi, 361 planes in 3-d space, 622 points, 755 probability of an event, 490 radius, 383 sample space, 490 scientific, 139–40, 266, 324–26 subscripts for base, 72 subtraction, 314 sum of a + b, 87, 89 transformation geometry, 846 translations, 846 triangles, 574, 594 vertex of the angle, 574 volume, 697 See also inside back cover not equally likely outcomes, 493 nth percentile, 447 nth root, definition, 362 null set, 45 number-line model for fraction addition, 223–24 BMIndex.indd 14 Use Properties of Numbers number line(s) strategy, 175 base five, 163, 165 whole (See whole numbers) and box and whisker plot, zero (See zero) 444–45 See also counting numbers; and Cartesian Coordinate numeration systems System, 391–92 number sense, developing, 137 decimals, 253–54, 255–56 number sequence, 22 See also distance, 665–66 sequences fraction addition, 223–24 fractions, 210–11, 212, 216, 218 number systems See numeration systems integers, 306, 319, 326 number theory, 177–85, 190–98 line fractal self-similarity, 736 composite numbers, 177 metric converter diagram, 653, counting factors, 190–91 656, 657 Euclidean algorithm, 193–94, rational numbers, 341, 345 202 real numbers, 360, 590 greatest common factor, repeated-subtraction approach 191–94, 196–97, 199, 201, 202 to division, 110 least common multiple, transitive property of less than, 194–97, 199, 201, 224 116, 327–28–328 build-up method, 195–96 whole numbers prime factorization addition, 88 method, 195, 196 less than and addition set intersection method, property, 117 195, 196 ordering with, 59 tests for divisibility, 180–83 transitive property of “less See also prime numbers than,” 116 numerals number of a set, 57 about, 57 number(s) Braille, 67 about, 57–58 fractions as, 209–12 abundant, 199 irrational numbers, 338, amicable, 199 359–61, 371 base of a system, 68 word names for, 70, 72 betrothed, 199 See also numbers cardinal, 57 numeration systems, 60–64 compatible, 131–32 about, 64 counting (See counting additive, 61 numbers) Babylonian, 62–63 decimal (See decimals) base, 68–69 deficient, 199 binary, 75, 87–88, 891, 892 diagrams of relationships Braille, 67 between, 342, 351, 359, 360 Chinese, 66 even, 23, 380 comparison of, 64 fractions (See fractions) converting between bases, 73–75 frequency of, 418–20, 446, Egyptian, 60–61 450, 492 expanded notation (See identification, 57 expanded form/notation) inductive reasoning, 23 Hindu-Arabic (See Hinduinteger (See integers) Arabic numeration system) mixed numbers, 216, 227, 235, Ionian, 65–66 341–42, 353 Mayan, 42, 63–64 natural, 44 multiplicative, 62 negative (See negative integers; nondecimal, 72–75 negative numbers) pictographic, 60–61 numerals vs., 57 placeholder, 62 numeration systems (See positional, 42, 62 (See also numeration systems) place value) odd, 23, 24–25, 174 Roman, 61–62, 84 ordering (See ordering) subtractive, 61–62 ordinal, 57 tally, 60 pentagonal, 35 with zero, 63–64 perfect, 174, 199 See also numbers positive, 305 numerators, 206, 217, 218–19, 231, prime (See prime numbers) 239, 341 rational (See rational numbers) numerousness, 211 real (See real numbers) rectangular, 34–35, 380 O relatively prime, 198 objects Smith, 202 no particular order of square, 23 (combinations), 520–22 systems (See numeration ordered arrangement of systems) (permutation), 518–22 theory (See number theory) oblique circular cones, 626 oblique cylinders, 626 oblique prisms, 625, 698 oblique pyramids, 625 obtuse angles, 592, 595 obtuse equilateral triangles, 596 obtuse isosceles triangles, 596 obtuse scalene triangles, 596, 608 obtuse triangles, 594, 595 octagons, 575, 672 octahedrons, 624, 625 octillions, 76 odd counting numbers, 23, 24–25 and median, 442 nth Root and, 362 number of factors, 324 number of prime factors, 358 perfect number conjecture, 174 sum of odd numbers, 25–26 odds, 531–33 O’Neal, Shaquille, 314 one-column front-end estimation method, 135 ones digit and divisibility, 180–81, 183 as leaves on stem and leaf plots, 418–19 and patterns, 23, 24 and transitive property, 377 one stage tree diagrams, 512 one-to-one correspondence algebraic reasoning, 47 for comparing two whole numbers, 58 congruent polygons, 851 definition, 45 and reflection symmetry, 823 and simulation, 528 and symmetry, 823–24 between two sets, 44–45 operations in base five, 162–66 binary, 75, 87–88, 891, 892 compatibility with respect to, 131–32 order of, 120–21 real numbers, 361 on sets, 44, 47–51 whole number, four basic, 110 See also addition; division; multiplication; subtraction opposite angles, 596 opposite angles in quadrilaterals, 598 opposite line segments (not adjacent), 575 opposite of the opposite for rational numbers, 346 opposites of integers, 305–6, 308 of rational numbers, 346 opposite sides in quadrilaterals, 598 or connective, 882 ordered arrangement of objects (permutation), 518–20, 523–24 ordered data sets, 443, 518–20, 523–24 ordered pairs and Cartesian product of sets, 50 7/31/2013 7:29:34 AM Index I15 on coordinate plane, 785 fractions as, 216 functions as, 382–83 graph coordinates, 392 ratios, 274–75 relations and, 375 and scatterplots, 430 ordering counting chant, 58, 59 decimals, 255–56 and exponents, 117–20 fractions, 216–19 greater than (See greater than) inequalities, 368–70 integers, 326–28 less than (See less than) one-to-one correspondence, 45 PEMDAS mnemonic, 120 rational numbers, 351–53 real numbers, 361 and whole number operations, 116–17 whole numbers, 58–59 See also number lines order of operations, 120–21, 138 ordinal numbers, 57 Oregon Ducks football uniforms, 512 Oregon rain data, 455 organizing data See data organization and display; graphs origin of real-number lines, 391 orthocenter, 757 ounce, 652 outcome(s) about, 487 card decks, 488, 491, 523 certain, 491 certain or impossible, 491 coin tossing, 43, 487–88, 490, 510–12 dice throwing, 488, 490, 492–93, 500, 504 drawing gumballs, 496–97, 509 drawing marbles, 493–94, 502, 504–5, 506–8 equally-likely (See equallylikely outcomes) expected, 524 experiments with two outcomes, 510–12 favorable vs unfavorable, 532–33 and fundamental counting property, 503–4, 510–12, 518 impossible, 491 mutually exclusive, 496 not equally likely, 493 odds, 531–33 pairwise mutually exclusive simpler, 506 Pick Six wager, 524 possible, 488 spinning spinners, 488, 490, 496 tree diagrams of, 502–3 See also events; probability outliers about, 429–30 and box and whisker plots, 445, 447 BMIndex.indd 15 choosing not to use data from, 429 and interquartile range, 444 mild vs extreme, 456 and Scatterplot eManipulative, 437 and z-scores, 455 P pairs of angles, 597 pairwise mutually exclusive simpler events, 506 palindromes, 98 pan balance, 26–27, 85, 365–66, 367 parabola, 806 parallel lines about, 590, 595 angles associated with, 593 constructing, 749 Euclidean construction, 749 intersected by transversal lines, 592–94, 603, 617, 736 and isometries, 848 slopes of, 788 in three-dimensional space, 622 parallel line segments about, 568, 569 model and description, 568 notation, 590 sides of trapezoids, 583 parallel line segments, directed, 824–25 parallel line segments test, 571 parallelogram(s) about, 722 area of, 671, 672 congruent, 722 consecutive angles, 603 diagonal congruence, 766–67 Escher-type patterns, 832–33 identifying, 552–53 lateral faces of polyhedra, 624 mirror symmetry, 565 model and description, 571 pairs of angles, 597 perimeter, 666–67 properties, 598 properties of, 575, 581 as quadrilaterals, 729 rotation symmetry, 567 sides of, 766, 767 tessellations of a plane with, 830, 832 tessellation with, 608 translations, distance, and, 847 and trapezoids, 583, 678 parallel planes, 621, 626 parallel sides in quadrilaterals, 598 parallels of latitude, 794 parentheses on calculators, 138 parsecs, 665 Parthenon, Athens, Greece, 250 partition of a set, 377 part-to-part ratios, 275 part-to-whole model, 209 part-to-whole ratios, 275 Pascal, Blaise, 128, 484 Pascal’s triangle about, 22, 33 and coin toss probabilities, 510–12, 513 and combinations, 522, 523 comparing products of alternate numbers, 36 and different downward paths, 22 Fibonacci sequence and, 250 predicting sums of diagonals, 33 patterns chess, 557 Color Patterns eManipulative, 34 counting dots using, 14–17 Escher-type, 832–33 and Fibonacci sequence, 33, 35 generalizing patterns, 22 integer subtraction, 311 Pattern Blocks eManipulative, 223 tearing and piling up paper, 384 for three-dimensional shapes, 628–29 tiling, 552–53, 564 See also Look for a Pattern strategy; tessellations P(E), 490–97 PEMDAS (mnemonic for ordering), 120 pentagonal numbers, 35 pentagonal prisms, 625, 626 pentagonal pyramids, 624 pentagons, 605 pentominoes, 303, 588 percent grade, 292–93, 792 percentiles about, 447 for box and whisker plots, 447 percents, 283–91 and calculators, 285 and circle graphs, 424–26, 428, 469 compound interest, 290–91, 396 converting, 283–85 decimals and, 284 Draw a Diagram strategy, 283 fractions and, 284 free throw example, 292 percent grade, 292–93, 792 as ratios, 287–88 rounding, 290 solving problems, 286–91 equation approach, 288–89 grid approach, 287 proportion approach, 287–88 store discounts, 290 Percent to Fraction Table, 285 perfect number, 174, 199 perfect square, 122 perimeter, 666–68, 688 See also inside back cover period of a decimal, 268 permutations counting technique, 518–20 n factorial, 519 of r objects chosen from n objects, 520 use of, 523–24 perpendicular bisector angles, 745–46 and circumcenter of a triangle, 755–56, 811 Euclidean construction, 744–45 of line segments, 724 and reflection transformations, 827, 828, 849–50 rhombus properties and, 744–47 through a point not on a line, 748–49 perpendicular lines about, 592, 595 constructing, 748–49 diagonals, 583, 598 on the line, 748 slopes of, 789–90 in three-dimensional space, 622 through a point not on the line, 748–49 perpendicular line segments, 568, 569, 573 perpendicular sides in quadrilaterals, 598 perspective See misleading graphs and statistics Peter, Rozsa, 407 pi and area of a circle, 383, 673, 702 Buffon’s Needle Problem, 535 and great circle of a sphere, 692, 794 mnemonic, 370 notation, 361 perimeter of a circle, 668 volume of a sphere, 702–5 Pick Six wager, 524 Pick’s theorem, 679–80 pictogram numeration systems, 60–61 pictographs, 426–28, 430, 467, 475 pictorial embellishments to graphs, 426, 428, 469–71 pie charts See circle graphs pie charts or circle graphs, 424–26, 428, 430–31 pints, 652 Pioneer 10 (spacecraft), 664–65 Pixar Animation Studio, 585 pizza-cutting problem, 10–11, 28–30 placeholder for Babylonian numeration system, 62 place value addition algorithm, 146 decimals and, 255 and expanded notation, 69, 73–74 in Hindu-Arabic numeration system, 68–69 Hindu-Arabic system, 68–69 multi-digit numbers, 92 multiplication algorithm, 149 nondecimal systems, 72–75 for ordering fractions, 256 positional numeration systems, 42, 62–64 (See also HinduArabic numeration system) subtraction algorithm, 148 plane of symmetry, 631–32 plane(s) distance in coordinate plane, 783–86 intersecting, 621 7/31/2013 7:29:34 AM I16 Index plane(s) (continued) isometry types in, 850–51 parallel, 626 points in, 589 simple closed curve in the, 605 slope (See slope) tessellations and, 830, 832 in three-dimensional space, 620–21 See also transformation geometry Platonic solids/Euler’s formula, 624 Playing with Infinity (Peter), 407 point(s) about, 589–91 arrays of, 791, 793 of circles, 611 collinear, 590 corresponding, 626 distance, 748, 755–56 from point P to point Q, PQ, 665, 666, 742 between two, 590 image of, 823 image of P, 823 images of points under transformations, 846 lines and, 590–91 midpoint formula, 784, 786 noncollinear, 812 notation, 755 perpendicular line through, 748 on a plane, 780 properties of lines and, 591 and translations, 846 point-slope equation of a line, 798–99 poker chips model for integers, 305–6, 319–20 polar coordinates, 805 Polk, James K., 484 Pólya, George, 2, 121 Pólya’s four-step problem-solving process about, 2, 5–6, 30 and combining strategies to solve problems, 29–30 with Draw a Picture strategy, 10–12 with Guess and Test strategy, 7, 8–10 with Look for a Pattern strategy, 21–24 with Make a List strategy, 24–26 with Solve a Simpler Problem strategy, 26–27 with Use a Variable strategy, 12–14 See also problem-solving strategies polygonal regions, 608, 621, 623–25, 627 See also polyhedron/polyhedra polygon(s), 605–12 about, 605–6 angle measures in a regular, 605, 606–7, 610 angles of, 605 and angle sum in a triangle theorem, 594, 606–7 BMIndex.indd 16 arbitrary, 607 area of, 672, 679–80 center, 605 circles as, 611–12 congruence of, 851 congruent, 851–52 convex nature of regular, 605 equilateral and equiangular, 605 Euclidean construction, 757–59 Fermat prime, 759 Gauss’s theorem for constructible n-gons, 759, 764–65 graph of, 392–93 for measurement of an area, 650 one-to-one correspondence, 851 perimeter, 666–67 polyhedra made from, 625 regular, 605–6 tessellations with, 609–11 regular n-gons, 605–12, 757–59, 764–65 similar, 853–54 size transformations, 834 tessellations using, 610 See also quadrilaterals polyhedron/polyhedra, 623–25, 627 cubes, 623, 624–25 dodecahedron, 624 Euler’s formula, 624 hexahedron, 624 icosahedron, 624 octahedron, 624 Platonic solids/Euler’s formula, 624 prisms, 625 pyramids, 623, 624 regular, 624 rhombicuboctahedron, 626 semiregular, 625, 626 surface area, 686–88 tetrahedron, 624, 625, 633 three-dimensional aspect, 623–25 vertices, 623 Pons Asinorum Theorem, 716 pool shot paths, 863, 864–65 population, in statistics, 416 population predictions, 402 portability of measurement systems, 652, 659 positional numeration systems, 42, 61–64 See also place value position functions, 394–95 positive numbers integers, 305 rational, 345 as real numbers, 359 possible events, 488 See also outcomes pounds (weight), 652 Powell, J H., 472 powers of, 23 See also exponents powers of two, 75 powers of three, 23–24, 380 powers of 10 decimals, 254 dividing by, 257 Egyptian numeration system, 60–61 Hindu-Arabic numeration system, 68–69 multiplying by, 132, 257 Precious Mirror of the Four Elements (Chu), 302 prefixes for metric system, 654 prime factorization about, 179, 183–84 of composite number, 177–78 counting factors and, 190–91 divisibility tests for, 183 fundamental theorem of arithmetic and, 178, 190 greatest common factor, 192–94 least common multiple, 195, 196 prime factors counting factors vs., 190–91 even or odd number of, 358 fractions, 215 rational numbers, 342–43 test for, 183 prime number competition updates, 198 prime numbers, 177–84 2, nature of composites and, 177 and counting factors, 190–91 factorization (See prime factorization; prime factors) Fermat prime, 759 finding, 183–84 infinite number of, 197 largest known, 198 Mersenne number, 185, 198 prime meridian, 794 relatively prime, 198 Sieve of Eratosthenes, 177, 185, 186, 188 square root, 184 with their respective exponents, 190–91 triples, 188 twins, 186 primitive Pythagorean triple, 372 principal square root, 360–61 Principles See NCTM Principles for School Mathematics prisms about, 625, 627 Cavalieri’s principle, 703–5 surface area, 686–87 volume, 696–99 probability, 487–97, 502–12, 518–24, 528–35 additive probability, 506 additive property, 506 clearing counters game, 492 coin tossing (See coin tossing probabilities) complex experiments (See probability tree diagrams) conditional probability, 533–35 counting techniques, 518–24 combinations, 520–23 fundamental counting property, 503–4, 510–12, 518–19, 520–21 and Pascal’s triangle, 522, 523 permutations, 518–20, 523–24 use of, 523–24 definitions, 490 dice throwing, 488, 490, 492–93, 500, 504 and Draw a Diagram strategy, 43 drawing with/without replacement, 504–5 events (See events) expected value, 530–31 experimental, 492–93, 528–29 experiments with two outcomes, 510–12 frequency of a number, 418–20, 446, 450, 492 and fundamental counting property, 503–4, 510–12, 518–19, 520–21 intersection of sets, 494–95 Let’s Make a Deal eManipulative, 517 more likely, less likely, 489 and multiple actions, 493–95 and multiple objects, 491–93, 494–95 multiplicative probability, 505–9 odds, 531–33 outcomes (See outcomes) and Pascal’s triangle, 510–12, 513 and problem of the points, 484 properties additive probability, 506 fundamental counting property, 503–4, 510–12, 518–19, 520–21 multiplicative probability, 505–9 as ratios, 496–97 relative frequency of a number, 418–20, 446, 450, 492 simple experiments about, 487–89 computing probabilities in, 490–97 and counting techniques, 503–4 simulation, 528–30 theoretical, 492 unequally-likely outcomes, 493, 533, 535 union of sets, 494–95 Venn diagrams and, 533–35 probability of an event, P(E), 490–97 probability tree diagrams, 504–12 about, 509 and additive property of probability, 506–9 drawing with/without replacement, 504–5 expected value, 530 experiments with two outcomes, 510–12 marbles drawn with replacement, 506–8 and multiplicative property of probability, 505–9 7/31/2013 7:29:35 AM Index I17 simple experiments, 502–3 spinners with different color schemes, 508–9 problem of the points, 484 problem solvers, suggestions from, 6, 31 problem solving, 5–17, 21–28 algebraic, 14–17, 28–29 analyze data step, 415, 417 collect data step, 415, 416, 417, 422 formulate questions step, 415–16, 417, 422, 441 geometric with algebra, 780 with coordinates, 807–11 with transformations, 865 with triangle congruence and similarity, 765–70 importance of, interpreting data step, 417 Number Puzzles eManipulative, 19, 20, 98, 100 organize and display data step, 415, 416–17, 422–23 (See also data organization and display) percents, 287–89 Pólya (See Pólya’s four-step process) set method, 59 statistical, 415–17 strategies (See problem-solving strategies) suggestions for, 31 Tower of Hanoi eManipulative, 36 with Venn diagrams, 51, 79 See also algorithms; calculator activities; properties; solving equations problem-solving process See Pólya’s four-step problemsolving process problem-solving strategies about, 5–6, 30–31, 38–39 combining strategies, 2–3, 27, 28–30 Do a Simulation, 485, 529, 541 Draw a Diagram (See Draw a Diagram strategy) Draw a Picture (See Draw a Picture strategy) Guess and Test (See Guess and Test strategy) Identify Subgoals, 717, 774 Look for a Formula, 413, 479 Look for a Pattern (See Look for a Pattern strategy) Make a List, 24–26, 27, 28–30, 38 Solve an Equation, 339, 406, 810, 811 Solve an Equivalent Problem, 207, 247 Solve a Simpler Problem, 26–30, 38 Use a Model, 547, 636–37, 674–75, 686 Use a Variable, 12–14, 37, 313–14, 675 BMIndex.indd 17 Use a Variable with algebra, 14–17, 28–29 Use Cases, 303, 333 Use Coordinates, 781, 816–17 Use Dimensional Analysis, 645, 660, 711 Use Direct Reasoning, 85, 124 Use Indirect Reasoning, 129, 168, 169, 197, 358 Use Properties of Numbers, 175, 202 Use Symmetry, 821, 870 Work Backward, 16–17, 251, 298, 557 See also solving equations problems vs exercises, products about, 101 counting factors and, 190–91 cross-multiplication of fractions, 215 from decimal to fraction conversions, 257 exponents of the, 118 and “multiplication makes bigger” misconception, 264 order of operations, 120–21 in probability, 505 proofs, deduction about, 574 proper factors, 174, 199 proper subsets, 46, 47 properties addition property of equality, 367 additive probability, 506 associative (See associative property) cancellation (See cancellation property) closure (See closure property) commutative (See commutative property) distributive (See distributive property) division property of equality, 367 exponents, 363 fractions additive, 214, 225–27 density, 219 multiplicative, 235–37 fundamental counting property, 503–4, 510–12, 518, 520–21 geometric (See geometric properties) greater than, 361 identity (See identity property) integers additive, 309, 310, 321, 323 additive properties, 309–10 less than, 326–28 multiplicative, 321–22 ordering, 326–28 inverse additive, 310, 314, 321, 361 multiplicative, 236, 348, 361 less than (See less than properties) multiplicative (See multiplicative properties) ordering integers, 327 ordering rational numbers, 352 probability, 496 as problem solving strategy, 175 rational exponents, 363 rational numbers additive, 344, 345–46 density, 352 multiplicative, 348, 349 ordering, 352 transitive property for less than, 352 real numbers, 361 reflexive, 376 symmetric, 376–77 transitive (See transitive property) Use Properties of Numbers strategy, 175 whole numbers additive, 89, 90, 100, 131 multiplicative, 102, 104–7, 113, 131 transitive property of greater than and less than, 116 zero, 106 zero, 90 zero divisors, 323 See also geometric theorems; theorems proportion about, 276–79 and calculators, 289 divine proportion, 250 percent problem-solving, 287–88 scaling up/scaling down, 278 See also percents; ratios protractors, 591 Purser, Michael, 328 Pyramid of Cheops, 710 pyramids about, 624, 627 Cavalieri’s principle, 703–5 drawing, 634 surface area, 689–90 volume, 700–701 Pythagoras, 279 Pythagorean Proposition, The (Loomis), 760 Pythagorean theorem about, 674–75 application to nontriangular shapes, 734 and area of squares, 734, 740–41 and Babylonian method for diagonal of a square, 770 Bhaskara’s proof, 679 converse of, 767–68 eManipulative, 681 Euclid’s proof, 716, 760, 772, 869 Garfield’s proof, 684 and irrational numbers, 338 length of PQ, 783 length of the hypotenuse, 360, 371, 373 origin of, 770 and right triangles, 338, 360, 615, 674, 740–41 and surface area of a pyramid, 689 transformational proof, 867 and triangle inequality, 675–76 and Use a Variable strategy, 676 Pythagorean triples, 232–33, 357, 372 Q quadrants, 392 quadratic function graphs, 394–95 quadrilateral(s) about, 583 analysis of, 570–75, 579 angles, 570 arbitrary, 608–9 child’s recognition of, 554–55 congruence, 726 diagonal line segments, 573 diagonals of, 863 half-turn rotations, 863 kite (See kites) labeling shapes and parts of shapes, 574–75 midquad, 768–70 models and descriptions, 571, 582 opposite angles, 596–97 parallel line segments test, 571 parallelogram (See parallelograms) perimeter, 666–67 perpendicular line segments test, 573 properties of, 581 properties of angles, 596–98 rectangles (See rectangles) relationships among, 581–85 rhombus (See rhombus/ rhombi) SASAS congruence property, 726 SAS congruence property, 766 sides, 570 size transformations, 834 squares (See squares, geometric) tessellations with, 608–9 translation and, 824–25 trapezoids (See trapezoids) vertex, 570 quadrillions, 70, 76 quartile, upper/lower, 444–47 quarts, 652 quotients (missing factors) about, 108, 110 and calculators, 139 decimals, long division, and, 265 from decimal to fraction conversions, 257 denominators and numerators approach to division, 239 and “division makes smaller” misconception, 264 long division and, 151–54, 265 missing factor approach to division, 108–9, 151–54, 265, 892–93 ratio and, 274 simplified form, 350 7/31/2013 7:29:35 AM I18 Index R r objects chosen from n objects, 520 race track wagering, 524 radicals, 362 radicands, 362 radius about, 611, 626 and Archimedean method, 644 arc of radius, 743, 744–45 and area of a circle, 383, 672–73, 702 and circumference of a circle, 668 notation, 383 of a sphere, 626, 691–92 and surface area measures, 688, 690–91, 692 radius of a circle, 611 Ramanujan, Srinivasa, 19, 203 random digits, 529 Random Guess and Test strategy, 8, random-number table, 528–29 range about, 134 and dispersion/interquartile range, 444–47 and probability of an event, 491 range estimation, 134–35, 229, 242, 258 range of the function, 381 as subset of codomain, 381–82, 383–84 rates and proportions, 276, 278–79 ratios of, 274 rational exponents, 362–63 rational number(s), 341–53 adding the opposite, 346 addition, 343–46, 351–52 additive properties, 344, 345–46 comparing two, 342 cross-multiplication of inequality, 352 density property, 352 distributive property of multiplication over addition, 349 division, 349–51 equality of, 343 as exponents, 362–63 fractions as, 341–42 infinite number of fractions, 344–45 integers (nonzero) as, 341–42 multiplication, 348–49, 352 multiplicative properties, 348, 349 nonzero integers as, 341–42 number lines, 341 number system diagrams, 351, 359 products with fraction calculator, 348 properties additive, 344, 345–46 density, 352 multiplicative, 348, 349 ordering, 352 transitive property for less than, 352 BMIndex.indd 18 real numbers vs., 362 set models, 341 set of, 341 simplifying, 342–43 subtraction, 346–47 sums and differences with fraction calculator, 225, 348 ratio(s) about, 274–75 avoirdupois measures of weight, 652 cross-multiplication property, 276–77 definition, 274 density of a substance, 662 of distances, 852–53 and Draw a Diagram strategy, 43 and English system of measurement, 650–52 equality of, 275 equivalent ratios method for solving proportions, 277 extremes and means, 275 golden ratio, 250 and measurement systems, 650–52, 653, 654, 655 medians of a triangle, 808–9 metric convertibility of, 653 and metric system, 653, 654, 655 odds and, 531–33 part-to-part comparison, 275 part-to-whole comparison, 275 percents as, 287–88 probabilities as, 496–97 and rates, 274 scaling up/down, 278 whole-to-part comparison, 275 See also proportion ray and properties of lines, 591 See also rectangular array reading numbers, 70, 71, 72 Reagan, Ronald, 676 real number(s), 358–70 exponents, 365 geometric representation, 338 nth root, 362 number line, 360, 590 number system diagrams, 359, 360 properties, 361 rational numbers vs., 362 roots of, with calculators, 362, 364 set of, 359 See also decimals reasoning addition facts, 89, 90–92, 164 children and geometry, 546, 549–50, 556 direct reasoning, 10, 85, 124, 574, 884–87 facts for base five, 163–64 inductive, 23 logic, 137–38, 878, 882–87 logical arguments, 884–87 long division, 151–54 multiplication facts, 106–7 percents, 287 place-value, 255 similitude properties, 853 Use Direct Reasoning strategy, 85, 124 Use Indirect Reasoning strategy, 129, 168, 169, 197, 358 See also algebraic reasoning; indirect reasoning; logic; van Hiele Theory reciprocals, 237, 348 recognition—van Hiele level 0, 549, 552–56 rectangle(s) area of, 668–70 child’s recognition of, 549, 550, 554–55 diagonals of, 720 division eManipulative, 112 golden, 250 identifying, 552–53 mirror symmetry, 565 model and description, 571 pairs of angles, 597 as parallelograms, 581 perimeter, 666–67 properties, 598 properties of, 575 as road signs, 575 rotation symmetry, 567 squared, 316 squares relationship to, 581 rectangular array description of divides, 180–81 divides, 178 fraction multiplication, 235 multiplication, 101–2 rectangular numbers, 34–35, 380 repeated-addition approach to multiplication, 101–2 rectangular numbers, 34–35, 380 rectangular regions, 621 “reducing” fractions, 214, 215 Rees, Mina, 479 reflection(s) about, 828 compass constructions, 838 distance preservation, 847 eManipulative, 845 glide (See glide reflections) isometries, 827–30 notation, 828, 846 perpendicular bisector and reflection transformations, 827, 828, 849–50 reflection image, 823, 829, 838 reflection symmetry, 565–66, 631–32, 823 reflex angles, 592, 595 reflexive property of relations, 376 region model, 212, 223–24, 341 regression lines, 430 regrouping/exchanging in addition, 91, 92, 145, 146 in subtraction, 147–48 regular polyhedra, 624 regular tessellations, 609–11 Reid, Constance Bowman, 203 relations, 375–78 arrow diagrams, 375 equivalence, 377 and functions, 375 number system diagrams, 342, 351, 360 partition, 377 reflexive property, 376 symmetric, 376–77 transitive property, 377 See also functions relationships—van Hiele level 2, 550, 579–84, 589 relative amounts, 210, 430 relative comparisons of English and metric systems, 654 relative complement or difference of sets, 49 relative frequency of a number, 418–20, 446, 450, 492 See also probability relatively prime numbers, 198 relative vs absolute in circle graphs, 468 remainders, 110 repeated addition approach to multiplication about, 101 and calculators, 112 clock arithmetic, 892 with fractions, 233 negative integers, 318–19 properties, 104 whole numbers, 101 repeated subtraction, division by, 110 repeating decimals, 267–69 calculator, 267–68 fraction representation, 268–69 and fractions, 284 long division algorithm, 267–68 nonterminating, 268 rational numbers, 358 real numbers, 358 repetend, 267, 359 repetend, 267, 359 replacement, probability and, 504–5 Reuleaux triangle, 870 “reversals stage “ of children, 64 reverse Polish notation, 137 Rhind Papyrus, 206 rhombicuboctahedrons, 625, 626 rhombus analyzing diagonals with slopes, 789–90 coordinate geometry, 808 rhombus/rhombi child’s recognition of, 550 congruence, 721–22 diagonals of, 574, 744–46, 748–49 identifying, 553 as kite, 583 mirror symmetry, 565 model and description, 571 pairs of angles, 597 as parallelograms, 581, 766–67 perimeter, 666–67 and perpendicular bisector of a line segment, 744–45 properties, 598 properties of, 575 as road signs, 575 rotation symmetry, 567 squares relationship to, 581 7/31/2013 7:29:35 AM Index I19 Rick, Killie, 353 Riese, Adam, 259 right angles, 568, 569, 595 right circular cones, 626, 690–91, 692, 703 right circular cylinders, 625, 688–89, 703 right distributivity of division over addition, 131 right pentagonal prisms, 625 right pentagonal pyramids, 624 right prism surface area and volume, 703 right rectangular prisms, 686–87, 697 right regular pyramids, 624, 703 right square prisms, 625 right triangle prisms, 625, 698 right triangle(s) about, 595 geometric representation of real numbers, 360 hypotenuse, 360, 674 isosceles right triangles, 580, 674 model and description, 569 and Pythagorean theorem, 338, 360, 615, 674, 740–41 square for finding area of, 674–75 vertex/interior angles, 594, 784 See also Pythagorean theorem rigid motion, 823 See also isometries rise over the run, 787 Rivest, Shamir, and Adlemann (RSA) algorithm, 328 road signs, 575 Robinson, Julia Bowman, 124 rods, 650 Roman numeration system, 61–62, 84 rotation distance and, 847 distance preservation, 847 eManipulative, 840 Escher-type drawings, 832–33 finding images with compass and straightedge, 836 half-turn, 863 image, 863 isometries, 825–27 notation, 826, 846 rotation of tessellations, 608–9 rotation symmetry, 566–67, 612, 632, 830 rounding to compatible numbers, 136–37 computational estimation, 135–37, 154, 258, 290 decimals, 258 division, 154 down, 136 a five up, 136 to nearest even, 136 percents, 290 truncate, 136 up, 136 whole numbers, 135–37 RSA algorithm, 328 Rudin, Mary Ellen, 637 Russian peasant algorithm, 157 BMIndex.indd 19 S same length line segments, 568, 569 sample, in statistics, 416, 471 sample space about, 487–88, 506 additive property of probability, 506 Cartesian product of sets, 497–98 conditional probability, 533–35 counting techniques in place of, 518–24 dealing with large, 495 equally-likely outcomes, 495–96 examples of, 493, 495 not equally-likely probability, 496 properties of probability, 496 simple experiments, 487–88, 490–97 See also tree diagrams sampling bias, 471–72 SAS (side-angle-side) congruence property and ASA congruence property, 773 for kites, 724 for quadrilaterals, 766 and rotation transformation, 847 for triangles, 721, 724 similarity property, 731, 769–70, 774 SASAS congruence property for quadrilaterals, 726 scaffold division, 158 scaffold method, long division, 152–54, 166 scale factor, 834 scale factor, size transformation, 853 scalene triangles, 569, 570, 596, 608 scaling and axis manipulation, 461–63, 464, 465, 466 scaling up/scaling down (ratio), 278 Scatterplot eManipulative, 433, 434, 437, 438 scatterplots, 428–30 School Mathematics Study Group, 871 scientific notation and calculators, 139–40 characteristic, 266 exponents, 324–26 mantissa, 266 standard notation vs., 325 scratch addition, 155 segment construction, 760 See also line segments self-similarity, 400, 733–34, 736 semiregular polyhedra, 626 semiregular tessellations, 610 sequences about, 22 arithmetic, 379 common difference, 379 common ratio of, 379 counting numbers, 21–23 Fibonacci (See Fibonacci sequence) geometric, 379 identifying (See Look for a Pattern strategy) term of, 22, 378–79 set intersection method greatest common factor, 191–92 least common multiple, 195, 196 set method of problem solving, 59 set models integers, 306 rational numbers, 341 sample space (See sample space) take-away, 93–94, 95 whole numbers, 87–88, 93–94, 101, 108 set notation elements in a finite set, 58 intersection of sets, 494–95 set-builder, 45, 46, 49, 50, 87–88 set of fractions, 210 solution sets, 364 union of sets, 494–95 set(s), 45–51 about, 50 and arrow diagrams, 375–76 as basis for whole numbers, 45–51 binary operations using, 87–88 Cartesian product, 50, 497–98 combinations and, 520–22 complement of, 49 continuous set of numbers, 421 counting numbers, 44 DeMorgan’s laws, 53, 54 difference or relative complement of, 49 disjoint, 47 and Draw a Diagram strategy, 43 elements/members of, 45 empty, 45 equal, 45 equivalent, 45 finite, 46, 58 fractions concept of, 209–16 ordering, 216–19 functions (See functions) infinite, 46–47 inherent rules regarding, 45 integers, 305 intersection or union of, 48, 87–88, 191–92, 494–95 irrational numbers, 359 matching, 45 mean of a data set, 443 notation (See set notation) null, 45 number of, 57 operations on, 44, 47–51 ordered pair, 50 partition of, 377 probability and, 494–95 proper subset, 46, 47 rational numbers, 341 real numbers, 359 relations (See relations) sample space (See sample space) set difference, 88 solution, 364 subsets (See subset of a set) theory union or intersection of, 48, 87–88, 191–92, 494–95 universal, 46 Venn diagrams of, 46, 47–51 sextillions, 76 shape identification, 549, 553–55 See also geometric shapes sharing division, 107–8 Shells and Starfish (Escher), 820 Shiing-Shen Chern, 817 Shi-Ku Chu, 302 side-angle-side See SAS side-side-side See SSS sides of angles, 568, 569 sides of a parallelogram, 766, 767 sides of parallelograms, 766, 767 sides of quadrilaterals, 570 sides of triangles about, 569–70 congruent, 722 copying an angle, 743 included, 722 midpoint of, 768–70 obtuse scalene triangle, 596 perpendicular bisector, 755–56 Sierpinski triangle (or gasket), 36, 734, 741 Sieve of Eratosthenes, 177, 185, 186, 188 Sieve of Eratosthenes eManipulative, 185, 186, 188 similarity AA similarity property, 731, 760 about, 729 fractals and self-similarity, 733–34 geometric problem-solving with, 768–70 SAS similarity property, 731, 769–70, 774 self-similarity, 400 similar shapes, 853–54 size transformations, 852–53 SSS similarity property, 731, 774, 852 and surface area formulas, 703 similitudes properties, 853 of shapes, 852–53 size transformations, 833–35, 840, 852–53 simple closed curve, 605 simple experiments about, 487–89 computing probabilities in, 490–97 and counting techniques, 503–4 and tree diagrams, 502–3 simplified method for integer subtraction, 312 simplifying fractions, 212–14, 224, 225, 235–36 quotients (missing factors), 350 7/31/2013 7:29:35 AM I20 Index simplifying (continued) rational numbers, 342–43 subtraction, 312 Simpson’s paradox, 292 simulation about, 528–30 Do a Simulation strategy, 485, 529, 541 eManipulative, 536, 538, 539, 541 simultaneous equations, 799–802 size transformations, 833–35, 840, 852–53 skew lines, 622 slant height, 689–91, 703 sliding transformation, 823–25 slope for analyzing diagonals, 789–90 collinearity, 788 in coordinate plane, 787–90, 791 percent grade, 792 point-slope equation of a line, 798–99 slope computations, 787 slope intercept equation of a line, 796–98 slope ratio, 275 theorems, 788, 789–90, 796–98, 799 Smith number, 202 Smoot, Oliver, 660 solution of an equation, 16 solution sets, 364 solutions of simultaneous equations, 799–802, 804 Solve an Equation strategy, 339, 406, 810, 811 Solve an Equivalent Problem strategy, 207, 247 solving equations about, 16 algebraically, 14–17, 364–68 balancing method, 364–68 Cover Up method, 16–17 Guess and Test method, 16–17 inequalities, 368–70 percents, 288–89 simultaneous equation solutions, 799–802, 804 Solve an Equation strategy, 339, 406, 810, 811 Solve an Equivalent Problem strategy, 207, 247 Solve a Simpler Problem strategy, 26–30, 38 transposing method, 368 Work Backward strategy, 16–17, 251, 298, 557 See also problem solving; problem-solving strategies sorting shapes into categories, 553–55 special factors, multiplying by, 132 sphere(s) about, 627 Archimedean method for deriving volume, 644 Cavalieri’s principle, 703–5 center, 626 circumference, 668, 680 diameter, 626 BMIndex.indd 20 great circle of a sphere, 630, 692, 794 surface area of, 691–92 volume, 702–5 spinning spinners probabilities, 488, 490, 496, 508–9 spreadsheet activities on the Web site Base Converter, 76, 77, 78 Circle Graph Budget, 438 Coin Toss, 498 Consecutive Integer Sum, 18 Cubic, 403 Euclidean, 202 Function Machines and Tables, 390 Roll the Dice, 498 Scaffold Division, 158 Standard Deviation, 457 Spruce Goose (flying boat), 280 square(s), geometric 8-by-8, 96 acre, 650 additive magic, 20, 98, 112, 189, 260, 318 analysis of, 574–75 analytic vs holistic analysis, 550, 556 area measures, metric system, 654–56 area of, 734, 740–41 child’s recognition of, 549, 550, 554–55 diagonal of, 770 for finding area of a right triangle, 674–75 foot, 650 identifying, 552–53 inch, 650 as kites, 583, 584 mile, 650 mirror symmetry, 565 model and description, 571 multiplicative magic, 113 oblique prism, 625 pairs of angles, 597 as parallelograms, 581 perimeter, 666–67 properties, 598 properties of, 575, 596–98 as rectangles, 581, 584 as rhombus, 581 right square prism, 625 as road signs, 575 rotation symmetry, 567 squared square, 316 yard, 650 squared differences and variance, 447–48 square dot paper, 555–56, 568 squared rectangles, 316 squared squares, 316 square foot, multiples of, 650 square lattice diagonal of a rectangle, 720 eManipulative, 601 geoboard (See geoboards) as “ideal” collection of points, 589 and Pythagorean theorem, 675 square dot paper, 555–56 square of a number of 2, 358 counting numbers, 23 perfect, 122 square pyramid surface area, 689–90 square root and calculators, 184 definition divide and average method, 371 irrational, 371 primes and, 184 principal, 360–61 square units, 668 See also squares, geometric squeeze method for decimals, 371 SSS (side-side-side) congruence property angles, 743–44 converse of Pythagorean theorem, 767–68 kites, 724 parallelograms, 766 and SAS congruence property, 772 triangles, 722–24, 765, 767 similarity property, 731, 774, 852 stages of a tree diagram, 512 stages of learning, van Hiele, 546 standard algorithms addition, 145, 163 division, 154, 166 long division, 165–66 multiplication, 149–50, 165 subtraction, 147, 164 standard deviation, 448–50 and bell-shaped curve, 452–53 calculator activities, 448–49 and mean, 448, 449–50 unbiased, 460 and variance, 447–48, 449 and z-scores, 450 standard meter prototype, 653 standard units of measurement See English system of measurement star, five-pointed, with Geometer’s Sketchpad, 616 statements, logical, 881–84 state standards See Common Core State Standard key concepts statistics, 415–31, 440–55, 460–72 about, 412 central tendency, 441–44 graphs (See graphs; misleading graphs and statistics) interquartile range, 444–47 lower quartile, 444–47 mean (See mean) median (See median) mode, 441, 443–44, 451–54, 455 quartile statistics, 447 sampling bias, 471–72 standard deviation (See standard deviation) statistics question, 415–16, 417, 422, 441 trends, 412 upper quartile, 444–47 variance, 447–48, 449 See also entries beginning with “data” stem and leaf plots, 418–19, 446 step functions, graphs of, 398 Stevin, Simon, 259 Stiller, Lewis, 557 stock market, 400 straight angles, 592, 595 straightedge constructions Gauss’s Theorem, 759, 764–65 impossible, 750 quadrilateral, 836 reflection image, 838 regular polygons, 757–59 straightedge properties, 742, 743 strategy, See also Pólya’s four-step problem-solving process; problem-solving strategies subscripts, 72 See also entries beginning with “base” subset of a set combinations, 520–22 definition, 46 equilateral triangles as, 581 events as, 487 of populations, 416 proper, 46, 47 range as subset of codomain, 381–82, 383–84 substitution method of solving simultaneous equations, 804 subtract-from-the-base algorithm, 148–49 subtraction about, 93–96 by adding the complement, 156 adding the opposite approach, 312–13 algorithms (See subtraction algorithms) base five algorithm, 163–64 blocks/pieces, 148–49, 156, 160 cashier’s algorithm, 156 chip abacus, 148 clock arithmetic, 891–92 comparison approach, 95–96 compensation, 132 decimal algorithm, 262, 264 decimals, 262, 264 difference (See difference) difference of sets, 49 equal-additions method, 132, 229 estimation for fractions, 229 facts, 94, 163–64 fractions, 227–28, 347 integers, 311–14 left-to-right method, 132 mental math, 131–34, 229 missing-addend approach, 94–96, 163–64, 228, 313–14 mixed numbers, 353 nonstandard algorithm, 156 notation, 314 order of operations, 120–21 pattern, 311 place value, 148 rational numbers, 346–47 real numbers, 361 regrouping, 147–48 repeated subtraction approach to division, 110 simplified method, 312 7/31/2013 7:29:35 AM Index I21 standard algorithm, 147, 164 take-away approach, 93–94, 95, 228, 311–12, 891–92 unlike denominators, 228, 347 whole numbers, 93–96, 147–49, 163–64, 262, 264 subtraction algorithms base five, 163–64 decimals, 262, 264 equal-additions, 156 standard, 147, 164 subtract-from-the-base, 148–49, 156, 160 whole numbers, 147–49 subtraction compensation, 131 subtractive principle of Roman numeration system, 61–62 subtrahend, 94 successive differences, 30 sufficient conditionals and biconditionals, 884 sum angle sum in a triangle, 594, 606–7 consecutive whole numbers, 12–13 of counting numbers, 13–14 of even numbers, 26 first n counting numbers, 21–23 with fraction calculator, 225, 348 infinite geometric series vs., 233 notation, 87, 89 of odd numbers, 25–26 order of operations, 120–21 of a plus b, 87, 89–90 of probabilities of ten consecutive Fibonacci numbers, 189 thinking strategies for, 90–92 summands (addends), 87, 133 See also missing-addend subtraction supplementary angles, 592, 595 surface area, 686–92 cones, 690–91, 692 cylinders, 688–89 formulas (See inside back cover) lateral surface area, 686, 687, 690–91 prisms, 686–87 pyramids, 689–90 spheres, 691–92 survey of college freshmen and Draw a Diagram strategy, 43, 79 symbols, list of See inside back cover symmetrical distribution, 451 symmetry axis of rotational symmetry, 632 of circles, 611–12 equivalent and inequivalent patterns in the plane, 836 in Escher’s art, 820 glide reflection, 832 isometries, 830, 832 isometries and, 823 line (axis) of symmetry, 565 BMIndex.indd 21 and Mira, 566, 761 plane of symmetry, 631–32 property, 376–77 reflection, 565–66, 631–32, 823 rotation, 566–67, 612, 632, 830 tessellations and glide reflection symmetry, 832 translation, 830, 832 Use Symmetry strategy, 821, 870 symmetry (geometric), 565–67 symmetry patterns, 836 symmetry transformations, 830, 832 Systematic Guess and Test strategy, 8, system of equations (simultaneous equations), 799–802 T table(s) addition, 91 angle measures in a regular n-gon, 610 for base five operations, 164 bias survey sources, 472 box volumes, 397–98 decimals and fractions, 257 English System units, 650, 651, 652 exponential functions, 396 exponential growth, 380 functions as, 382 function values of t, 395 geometric shape properties, 575, 598 geometric shapes, 568, 569, 571, 595, 627, 703 images of points under transformations metric system prefixes, 654 numeration system summary, 64 operations on sets, 50 outcome frequency, 492 percents and fractions, 285 polyhedron, 624 random-number, 528–29 sampling bias sources, 472 sequences, arithmetic and geometric, 379 stem and leaf data, 418, 419, 446 summary of graft uses, 430 truth tables, 881–87 unit cubes, 380 visual comparison of data with box and whisker plots, 447 whole-number properties, 105 tablespoon, 652 take-away subtraction clock arithmetic, 891–92 fractions, 228 integers, 311–12 whole numbers, 93–94, 95 tally numeration system, 60 tangent line to a circle, 755 Taylor, Richard, 174 Tchebyshev, 188 teachers and teaching child’s “reversals stage ,” 64 of geometry, 817 Killie’s [teacher’s] Way, 353 simulating addition thinking strategies, 93 “Ten Commandments for Teachers” (Pólya), See also van Hiele Theory teaspoon, 652 temperature measures, 652, 658–59 “Ten Commandments for Teachers” (Pólya), terminal side of a directed angle, 826 terminating decimals, 254 term of a sequence, 22, 378–79 tessellation(s), 608–11 about, 608 dual of a, 614 eManipulative, 615, 618 Escher-type patterns, 820 and glide reflection symmetry, 832 with hexagons, 609–10, 612 plane with parallelograms, 830, 832 regular, 609–11 with regular polygons, 609–11 rotation procedure, 608–9 rotation transformation of a triangle, 832–33 semiregular, 610 tracing and rotation, 608–9 tests for divisibility, 180–83 tetrahedron, 624, 625, 633 tetromino, 11–12, 586 Theon of Alexandria, 775 theorem(s) additive cancellation integers, 310 rational numbers, 346 cross-multiplication fraction inequality, 217 rational-number inequality, 352 decimals fractions and, 255 multiplying/dividing by powers of 10, 257 deductive reasoning, 550 definition, 118 dividing decimals by powers of 10, 257 dividing rational numbers, 350 divisibility tests, 180, 181, 182, 183 division of fractions, 239 of exponents, 118–19 factors counting factors, 190 greatest common factor, 192, 193 least common factor, 197 prime factor test, 184 fractions addition, 218–19 cross-multiplication of fraction inequality, 217 dividing, 239 equality, 214 with repeating decimals, 268, 269 subtracting, 228 with terminating decimals, 255 Fundamental Theorem of Arithmetic, 178, 190 geometric (See geometric theorems) greatest common factor, 192, 193 integers addition, 310–11 multiplication, 321–22 least common factor, 197 multiplication, 217, 321–22, 352 multiplying decimals by powers of 10, 257 odds of an event, 532 opposite of the opposite for rational numbers, 346 permutations, 519, 520 prime factor test, 184 primes, infinite number of, 197 Pythagorean (See Pythagorean theorem) rational numbers, 342, 344, 346 real numbers, 358 subtraction of fractions, 228 See also properties theoretical probability, 492 thermal imaging, 865 thinking strategies addition facts, 90–92 facts for base five, 163–64 long division, 151–54 multiplication facts, 106–7 See also mental math; reasoning thousand, 70 three-dimensional effects (graphs), 466–67 three-dimensional shapes, 620–27 curved, 625–27 dihedral angles, 621–22 lateral surface area, 686, 687, 690–91 nets (patterns) for, 628–29, 687 planes in, 620–21 polyhedra, 623–25 skew lines, 622 volume, 648, 651–52, 656–57 See also volume three-dimensional space, Cartesian coordinates in, 792, 794–95 TI-34 MultiView calculator, 137–40 tiling patterns, 552–53, 564 Todd, Olga Taussky, 542 ton, 652 Tower of Hanoi eManipulative, 36 tracing for tessellation creation, 608–9 transformation, 823 transformation geometry, 823–35, 846–54, 863–65 about, 854 applied problems, 864–65 clockwise/counterclockwise orientation, 826, 830 composition eManipulative, 858, 862 congruence isometries, 846–52 polygons, 851 shapes, 851–52 7/31/2013 7:29:35 AM I22 Index transformation geometry (continued) Escher-type patterns, 820 functions as, 383–84 geometric problem-solving using, 865 glide axis, 828 half-turn, 863 image of P, 823 isometries (See isometries) midsegment proof, 878 midsegment theory, 878 notation, 846 rotation, 825–27 similarity, 854 similitudes, 833–35, 840, 852–53 symmetry (See symmetry) transformations congruence, 831, 846–52 isometries, 823–30 congruence and, 846–52 making Escher-type patterns, 832–33 notation, 846 perpendicular bisector and reflection transformations, 827, 828, 849–50 properties, 852 similitudes, 833–35, 840, 852–53 size transformations, 833–35, 840, 852–53 solving problems with, 865 symmetry (See symmetry) transitive property, 377 of greater than, whole numbers, 116 less than inequalities, 368–70 integers, 327–28 rational numbers, 352 real numbers, 361 whole numbers, 116 translation(s) distance and, 824–25 distance preservation, 846–47 notation, 846 notations, 846 quadrilaterals, 824–25 sliding transformations, 823–25 symmetry, 830, 832 transposing method for solving equations, 368 transversal lines, 592–94, 603, 617, 722, 736 trapezoid(s) analysis of, 582–83 area of, 671–72 isosceles trapezoids, 582 and kites, 583 midsegment, 683 model and description, 582 pairs of angles, 597–98 and parallelograms, 583, 678 perimeter, 667 perpendicular diagonals, 583 properties, 598 as road signs, 575 tessellation with, 608 tree diagrams about, 502 for multiplication, 102 BMIndex.indd 22 one stage, 512 and probability, 502–3 triangle relationships, 581 two stage, 512 See also probability tree diagrams trends and bar graphs, 430 and line graphs, 430 and multiple-circle graphs, 424–25, 428, 430–31 multiple-line graphs for, 423 and pictographs, 430, 467, 475 statistics and, 412 triangle congruence, 719–24 AAS property, 724 about, 719–20 alternate interior angles, 594 ASA property, 721–22, 724, 772, 773 correspondence, 719–20 distance and, 847–48 equality vs., 720 Euclid on, 724 geometric problem solving, 765–70 and isometries, 846–52 paired vertices/correspondence, 719–20 reflections and, 847 reflections preserving distance, 847–48 SAS property, 721, 724 sides of the triangle, 722 and similar triangles, 730, 765–68 SSS property, 722–24, 765, 767 triangle inequality, 675 triangle(s) 5-con pairs, 724 AA similarity property, 731, 760 acute, 594, 595 altitude, 751 analysis of, 569–70, 579 angles (See angles of triangles) area of, 670–71, 672, 678, 686 array, 525–26 base, 671 centroid, 757 centroid of a, 808–9 cevian line segments, 297 child’s recognition of, 549 circumcenter of a, 811 circumscribed circles, 755–56, 758 compass properties, 742 congruence (See triangle congruence) coordinate system, 807–11 equilateral (See equilateral triangles) fractals and self-similarity, 733–34 geometric problem solving, 768–70 harmonic, 233 height, 671 and Hero’s formula, 678 hypotenuse, 360 identifying, 552–55 inequality theorem, 675–76 inscribed circles, 755–56, 757 isosceles (See isosceles triangles) legs, 674 Marathe’s, 100 median, 751, 808–9 midsegment, 768–70 models and descriptions, 570 notation, 574, 594 obtuse, 594 obtuse scalene, 596 orthocenter, 757 Pascal’s (See Pascal’s triangle) perimeter, 667 and perpendicular bisector, 744–45, 755–56, 811 properties of, 807–11 relationships among types of, 580–81 Reuleaux, 870 right (See right triangles) right triangle prisms, 625, 698 as road signs, 575 rotation transformation, 832–33 scalene, 569, 570, 596, 608 sides (See sides of triangles) Sierpinski, 36, 734, 741 similarity (See triangle similarity) similitudes, 852–53 square lattice and, 675 tessellations with, 608 vertex, 297, 566, 569, 570, 671 triangle similarity, 729–34 about, 729–31, 768–70 fractals and self-similarity, 733–34 indirect measurement, 732–33 properties, 731–32 and segment length construction, 760 and similitudes, 852–54 triangular dot paper, 556 triangular geoboard, 556 triangular lattice, 556 triangular numbers, 32 triangular prism, 625 trillion, 70, 76 triples, prime, 188 triples, Pythagorean, 684 troy ounces, 652 true equations, 16 truncate (rounding), 136 truncated cube, 625 truncated polyhedra, 625 truncated tetrahedron, 625 truth tables, 881–87 twin prime conjecture, 174 twin primes, 186 two-column front-end estimation method, 135 two stage tree diagrams, 512 U Ulam’s conjecture, 174 Ulam, Stanislaw, 542 unbiased standard deviation, 460 unequally-likely outcomes, 493, 533, 535 union of two rays, 591 See also vertex of the angle union or intersection of sets, 48, 87–88, 191–92, 494–95 unitary fractions, 231 unit cubes, 380 unit distance, 666 unit fraction exponents, 362 unit fractions, 206 units, in base ten pieces, 68 units of measurement, 647, 650 universal set or universe, 46 University of Oregon football uniforms, 512 unknowns See variables unlike denominators addition, 224 division, 239 subtraction, 228, 347 upper quartile, 444–47 V valid argument, 884–85 value absolute, 315 discrete, 421 expected, 530–31 place (See place value) van Hiele-Geldorf, Dieke, 546 van Hiele, Pierre, 546, 549 van Hiele Theory, 546, 549 about, 546, 549 level 0—recognition, 549, 552–56 level 1—analysis about, 549–50, 567–68, 589 line segments, 568–69 quadrilaterals, 570–75 triangles, 569–70 level 2—relationships, 550, 579–84, 589 level 3—deduction, 550 level 4—axiomatics, 550 variable(s) about, 12 coefficients, 366, 368 concept of variables in algebra, 14, 16, 17 exponent as, 395 greatest integer function, 401 Use a Variable strategy, 14–17, 313–14 Use a Variable with algebra, 14–17, 28–29 variance and standard deviation, 447–48, 449 vector geometry, 877 Venn diagrams complement of a set, 49 conditional probability, 533–35 DeMorgan’s laws, 53, 54 difference of sets, 49 eManipulative, 52, 55 freshmen survey problem, 79 GCF and LCM, finding, 196, 199, 201 intersection of sets, 48 problem solving with, 51, 79 quadrilateral relationships, 584 of sets, 46, 47–51 triangle relationships, 580 union of sets, 48 vertex arrangement of polyhedra, 630 in tessellations, 610, 619–20 vertex figure, 614 7/31/2013 7:29:35 AM Index I23 vertex/interior angles about, 605–6 alternate interior angles, 593–94 angle bisector construction, 746 and angle sum in a triangle, 594, 606 interior angles on the same side of the transversal, 603 measurement of, 606–7, 610 of n-gons, 605–7, 610 polyhedrons, 623 of quadrilaterals, 575 and radius of a circumscribed circle, 811 right triangles, 594, 784 and tessellation, 608, 610 and transformations, 826 vertex of the angle about, 568, 574 adjacent angles, 591, 595 measurement of, 591–92 polyhedrons, 623–25, 627 tessellations, 608 vertex of the base, 623–25, 627 vertex of the quadrilateral, 570 vertex of the triangle, 569, 570, 608, 610 vertical angles, 592, 595 vertical axis See y-axis vertical line test, 399 Vieta, Francois, 259 volume, 696–705 capacity vs., 648, 651–52 Cavalieri’s principle, 703–5 cones, 701–2, 703 cylinders, 699–700 formulas (See inside back cover) holistic measures, 647–48 prisms, 696–99 pyramids, 700–701 spheres, 702–5 units of measurement, 656–57 and water displacement, 707 von Neumann, John, 121 W Wallis, 370 water displacement and volume, 707 Weierstrass, Karl, 298 weight measures, 649, 652 Wells, H G., 412 BMIndex.indd 23 ordinal, 57 whole number(s), 57–59, 131–40, pentagonal, 35 145–55, 162–66 primes with their respective about, 44 exponents, 190–91 addition properties about, 87–93, 105, 133 additive, 89, 90, 100, 131 algorithms, 145–47, 163, multiplicative, 102, 104–7, 262 113, 131 decimals, 262 transitive property of additive properties, 89, 90, greater than and less 100, 131 than, 116 algorithms, 145–54 zero, 106 addition, 145–47, 163, 262 reading, 70, 71, 72 division, 151–54, 165–66, rectangular, 35 265–66 rounding, 135–37 multiplication, 149–50, separating decimal portion of a 164–65, 263, 264–65 numeral from, 70 subtraction, 147–49, 262, set models 264 addition, 87–88 amicable, 199 division, 108 calculator computations, 137–40 multiplication, 101 cardinal, 57 subtraction, 93–94 closed set, 89 sets as basis for, 45–51 decimal point and, 253 sharing division, 107–8 decimals and, 253 square roots, 184 division, 107–10, 151–54, subtraction 165–66, 265–66 algorithms, 147–49 elements in a set, 57–58 decimals, 262, 264 estimation, 134–39 missing-addend approach, expanded notation/form, 69 94–96, 163–64 exponents, 74, 117–20 take-away approach, factors, 177 93–94, 95 a divides b, 178 transitive property of greater four basic operations, 110 than and less than, 116 greater than and greater than triangular, 32 or equal to, 59 unclosed set, 94 greatest common factor, 191–94 in Use a Variable strategy, identification, 57 12–13 intermediate algorithm for writing, 70, 71, 72 multiplication, 150, 165 zero, property of, 106 least common multiple, 194–97 whole-to-part ratios, 275 less than, 59, 116–17 Who Wants to Be a Millionaire?, 705 measurement division, 107–8 Wiles, Andrew, 174 mental math, 131–34 Work Backward strategy, 16–17, multiplication, 101–7, 190–91, 251, 298, 557 263, 264–65 World Cup Soccer multiplicative properties, 102, Championships, 612 104–7, 113, 131 World Series, 538 naming, 70, 72 writing numbers, 70, 71, 72 n factorial, 519 number lines, 59 number system diagram, 342, X 351, 360 x-axis numerals and, 57 of bar graphs and histograms, and numeration, 57–64 421, 422–23 ordering, 58–59, 116–17 polar coordinates, 805 reversing information to distort graph, 462–63 and scaling, 461–62, 464, 465, 466 x-coordinate about, 392, 785 equations of lines, 795–800 midpoint formula, 784, 786 simultaneous equations, 799–802 Y yard, 650 y-axis of bar graphs and histograms, 421 of coordinate system, 392 of histograms, 421 reversing information to distort graph, 462–63 y-coordinate about, 392, 785 equations of lines, 795–800 midpoint formula, 784, 786 orthocenter of a triangle, 810 simultaneous equations, 799–802 slope-intercept equation, 796–98 See also slope y-intercept point-slope equation of a line, 798–99 slope-intercept equation of a line, 796–98 Young, Grace Chisholm, 334 Young, William, 334 Z zero adding, 91 as additive identity, 90 in clock arithmetic, 891 division property of, 109 divisors property of integers, 323 as exponent, 120 as identity property, 90 as integer, 305 in Mayan numeration system, 63–64 multiplication property of, 106 repetend and repetend not zero, 359 zero factorial, 519 zero pair, 305 z-scores, 449–50, 453–55 7/31/2013 7:29:35 AM List of Symbols Symbol { } {x | } ∈ ∉ {} or ∅ = ≠ ∼ # ⊂ U ∪ ∩ A − ( a, b ) × n( .) < > ≤ ≥ 3five (35 ) am ≈ | | n! GCF LCM abcd a:b % a |a | a2n π n a1/n BMEndpaper.indd 15 Meaning set braces set builder notation is an element of is not an element of empty set equal to is not equal to is equivalent to (sets) is a subset of is not a subset of is a proper subset of universal set union of sets intersection of sets complement of a set Page 45 45 46 46 46 46 46 46 46 46 47 47 48 49 50 difference of sets ordered pair Cartesian product of sets number of elements in a set is less than is greater than less than or equal to greater than or equal to three base five exponent ( m ) is approximately divides does not divide 50 51 51 60 61 61 61 61 75 77, 122 140 186 186 n factorial greatest common factor least common multiple repeating decimal ratio percent negative number opposite of a number absolute value negative integer exponent square root pi (3.14159 ) nth root 192 198 202 281 288 299 321 322 332 342 379 379 380 nth root of a 381 Symbol a m /n f (a ) x P( E ) n Pr nC r P ( A|B ) AB ∠ABC ΔABC AB AB m l AB m ( ∠ABC ) l⊥m ↔ ≅ ∼ AB TAB ]ABC RO ,a Ml M AB SO ,k TAB ( P ) RO ,a ( P ) Ml (P ) M l (TAB ( P )) HO ∼ ∧ ∨ → ↔ ⊕, − , ⊗, ÷ a ≡ b mod m Meaning mth power of the nth root of a image of a under the function f mean probability of event E number of permutations number of combinations probability of A given B line segment AB angle ABC triangle ABC line AB the length of segment AB m is parallel to l ray AB measure of angle ABC l is perpendicular to m correspondence is congruent to is similar to directed line segment from A to B translation determined by AB directed angle ABC rotation around O with directed angle of measure a reflection in line l reflection in line containing AB size transformation with center O and scale factor k image of P under TAB image of P under RO ,a image of P under M l image of P under TAB followed by M l half-turn with center O negation (logic) conjunction (and) disjunction (or) implication (if-then) biconditional (if and only if) clock arithmetic operations a is congruent to b modulo m Page 381 399 483 513 547 549 562 591, 619 592, 620 592, 622 618 619 619 619 620 621 756 756 767 867 867 868 869 870 875 877 890 890 890 890 908 927 928 928 929 930 939–40 94 7/30/2013 3:03:51 PM Geometry Formulas for Perimeter, Circumference, Area, Volume, and Surface Area Rectangle P = 2a + 2b A = ab Cube V = s3 S = 6s3 Square P = 4s A = s2 Right Prism V = Ah S = 2A + Ph Triangle P=a+b+c A = 12 bh Parallelogram P = 2a + 2b A = bh Trapezoid P=a+b+c+d A = 12 ( a + b )h Right Regular Pyramid V = 13 Ah S = A + 12 Pl Right Circular Cylinder V = Ah = (πr2)h S = 2A + ch = 2(πr2) + (2πr2)h Regular n-gon P = ns A = 12 rP Right Circular Cone V = 13 Ah = 13 (πr2)h S = A + 12 Cl = πr2 + πr h2 − r Circle C = 2πr A = πr2 Sphere V = 34 r3 S = 4πr2 Right Rectangular Prism V = abc S = 2(ab + ac + bc) BMEndpaper.indd 16 7/30/2013 3:03:54 PM ...BMIndex.indd 29 7/31 /2013 7:29:35 AM National Council of Teachers of Mathematics Principles and Standards for School Mathematics Principles for School Mathematics r EQUITY. &YDFMMFODF... 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,... Wiley representative for a demonstration and further details FMPreface.indd 19 8/1 /2013 12:05:35 PM ACKNOWLEDGMENTS During the development of Mathematics for Elementary Teachers, Eighth, Ninth,