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National Council of Teachers of MathematicsPrinciples and Standards for School Mathematics

Principles for School Mathematics

EQUITY. Excellence in mathematics education requires eq-uity high expectations and strong support for all students.

CURRICULUM. A curriculum is more than a collection of ac-tivities: it must be coherent, focused on important mathematics,and well articulated across the grades.

TEACHING. Effective mathematics teaching requires under-standing what students know and need to learn and then chal-lenging and supporting them to learn it well.

LEARNING. Students must learn mathematics with understand-ing, actively building new knowledge from experience and priorknowledge.

ASSESSMENT. Assessment should support the learning of im-portant mathematics and furnish useful information to bothteachers and students.

TECHNOLOGY. Technology is essential in teaching and learn-ing mathematics; it influences the mathematics that is taughtand enhances students learning.

Standards for School MathematicsNUMBER AND OPERATIONS

Instructional programs from prekindergarten through grade 12should enable all students to

understand numbers, ways of representing numbers, relation-ships among numbers, and number systems;

understand meanings of operations and how they relate to oneanother;

compute fluently and make reasonable estimates.

ALGEBRA

Instructional programs from prekindergarten through grade 12should enable all students to

understand patterns, relations, and functions;

represent and analyze mathematical situations and structuresusing algebraic symbols;

use mathematical models to represent and understand quanti-tative relationships;

analyze change in various contexts.

GEOMETRY

Instructional programs from prekindergarten through grade 12should enable all students to

analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical argu-ments about geometric relationships;

specify locations and describe spatial relationships using co-ordinate geometry and other representational systems;

apply transformations and use symmetry to analyze mathe-matical situations;

use visualization, spatial reasoning, and geometric modelingto solve problems.

MEASUREMENT

Instructional programs from prekindergarten through grade 12should enable all students to

understand measurable attributes of objects and the units, sys-tems, and processes of measurement;

apply appropriate techniques, tools, and formulas to deter-mine measurements.

DATA ANALYSIS AND PROBABILITY

Instructional programs from prekindergarten through grade 12should enable all students to

formulate questions that can be addressed with data and col-lect, organize, and display relevant data to answer them;

select and use appropriate statistical methods to analyze data;

develop and evaluate inferences and predictions that are basedon data;

understand and apply basic concepts of probability.

PROBLEM SOLVING

Instructional programs from prekindergarten through grade 12should enable all students to

build new mathematical knowledge through problem solving;

solve problems that arise in mathematics and in other contexts;

apply and adapt a variety of appropriate strategies to solveproblems;

monitor and reflect on the process of mathematical problemsolving.

REASONING AND PROOF

Instructional programs from prekindergarten through grade 12should enable all students to

recognize reasoning and proof as fundamental aspects ofmathematics;

make and investigate mathematical conjectures;

develop and evaluate mathematical arguments and proofs;

select and use various types of reasoning and methods of proof.

COMMUNICATION

Instructional programs from prekindergarten through grade 12should enable all students to

organize and consolidate their mathematical thinking throughcommunication;

communicate their mathematical thinking coherently andclearly to peers, teachers, and others;

analyze and evaluate the mathematical thinking and strategiesof others;

use the language of mathematics to express mathematicalideas precisely.

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CONNECTIONS

Instructional programs from prekindergarten through grade 12should enable all students to

recognize and use connections among mathematical ideas;

understand how mathematical ideas interconnect and build onone another to produce a coherent whole;

recognize and apply mathematics in contexts outside of mathematics.

REPRESENTATION

Instructional programs from prekindergarten through grade 12should enable all students to

create and use representations to organize, record, and com-municate mathematical ideas;

select, apply, and translate among mathematical representa-tions to solve problems;

use representations to model and interpret physical, social, andmathematical phenomena.

PREKINDERGARTENNumber and Operations: Developing an understanding ofwhole numbers, including concepts of correspondence, counting,cardinality, and comparison.Geometry: Identifying shapes and describing spatial relationships.Measurement: Identifying measurable attributes and comparingobjects by using these attributes.

KINDERGARTENNumber and Operations: Representing, comparing and orderingwhole numbers, and joining and separating sets.Geometry: Describing shapes and space.Measurement: Ordering objects by measurable attributes.

GRADE 1Number and Operations and Algebra: Developing under-standings of addition and subtraction and strategies for basicaddition facts and related subtraction facts.Number and Operations: Developing an understanding ofwhole number relationships, including grouping in tens and ones.Geometry: Composing and decomposing geometric shapes.

GRADE 2Number and Operations: Developing an understanding of thebase-ten numeration system and place-value concepts.Number and Operations and Algebra: Developing quick recall of addition facts and related subtraction facts and fluencywith multidigit addition and subtraction.Measurement: Developing an understanding of linear mea-surement and facility in measuring lengths.

GRADE 3Number and Operations and Algebra: Developing under-standings of multiplication and division and strategies for basicmultiplication facts and related division facts.Number and Operations: Developing an understanding offractions and fraction equivalence.Geometry: Describing and analyzing properties of two-dimensional shapes.

GRADE 4Number and Operations and Algebra: Developing quick recall of multiplication facts and related division facts and fluency with whole number multiplication.

Curriculum Focal Pointsfor Prekindergarten through Grade 8 Mathematics

Number and Operations: Developing an understanding of decimals, including the connections between fractions and decimals.Measurement: Developing an understanding of area and determining the areas of two-dimensional shapes.

GRADE 5Number and Operations and Algebra: Developing an under-standing of and fluency with division of whole numbers.Number and Operations: Developing an understanding of and fluency with addition and subtraction of fractions and decimals.Geometry and Measurement and Algebra: Describing three-dimensional shapes and analyzing their properties, includingvolume and surface area.

GRADE 6Number and Operations: Developing an understanding of andfluency with multiplication and division of fractions and decimals.Number and Operations: Connecting ratio and rate to multi-plication and division.Algebra: Writing, interpreting, and using mathematical expres-sions and equations.

GRADE 7Number and Operations and Algebra and Geometry: Devel-oping an understanding of and applying proportionality, includ-ing similarity.Measurement and Geometry and Algebra: Developing an understanding of and using formulas to determine surface areasand volumes of three-dimensional shapes.Number and Operations and Algebra: Developing an under-standing of operations on all rational numbers and solving linear equations.

GRADE 8Algebra: Analyzing and representing linear functions and solving linear equations and systems of linear equations.Geometry and Measurement: Analyzing two- and three-dimensional space and figures by using distance and angle.Data Analysis and Number and Operations and Algebra:Analyzing and summarizing data sets.

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athematicsF o r E l e m e n t a r y Te a c h e r s

A CONTEMPORARY APPROACH

Gary L. MusserOregon State University

William F. Burger

Blake E. PetersonBrigham Young Univeristy

MathematicsME I G H T H E D I T I O N

John Wiley & Sons, Inc.

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To:

Irene, my supportive wife of over 45 years; Greg, my son, for his continuing progressin life; Maranda, my granddaughter, for her enthusiasm and appreciation of her love;my parents, who have both passed away, but are always on my mind and in my heart;and Mary Burger, Bill Burgers wonderful daughter.

G.L.M.

Shauna, my eternal companion and best friend, for making me smile along thiswonderful journey called life; Quinn, Joelle, Taren, and Riley, my four children, forchoosing the right; Mark, Kent, and Miles, my brothers, for their examples and support.

B.E.P.PUBLISHER Laurie RosatoneACQUISITIONS EDITOR Jessica JacobsASSISTANT EDITOR Michael ShroffSENIOR PRODUCTION EDITOR Valerie A. VargasMARKETING MANAGER Jaclyn ElkinsCREATIVE DIRECTOR Harry NolanSENIOR DESIGNER Kevin MurphyPRODUCTION MANAGEMENT SERVICES mb editorial servicesSENIOR PHOTO EDITOR Lisa GeeEDITORIAL ASSISTANT Jeffrey BensonMEDIA EDITOR Stefanie LiebmanCOVER & TEXT DESIGN Michael JungCOVER IMAGE BY Miao Jin, Junho Kim, and Xianfeng David GuBICENTENNIAL LOGO DESIGN Richard J. Pacifico

This book was set in 10/12 Times New Roman by GGS Book Services and printed and bound byRRDJefferson City. The cover was printed by RRDJefferson City.

This book is printed on acid-free paper.

Copyright 2008 John Wiley & Sons, Inc. All rights reserved. No part of this publication may bereproduced, 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 1976United States Copyright Act, without either the prior written permission of the Publisher, or authorizationthrough payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc. 222 RosewoodDrive, Danvers, MA 01923, website www.copyright.com. Requests to the Publisher for permission shouldbe addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ07030-5774, (201) 748-6011, fax (201) 748-6008, website http://www.wiley.com/go/permissions.

To order books or for customer service please, call 1-800-CALL WILEY (225-5945).

ISBN-13 978-0470-10583-2

Printed in the United States of America

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About the Authors

v

GARY L. MUSSER is Professor Emeritus from Oregon State University. He earned bothhis B.S. in Mathematics Education in 1961 and his M.S. in Mathematics in 1963 at theUniversity of Michigan and his Ph.D. in Mathematics (Radical Theory) in 1970 at theUniversity of Miami in Florida. He taught at the junior and senior high, junior college,college, and university levels for more than 30 years. He served his last 24 years teachingprospective teachers in the Department of Mathematics at Oregon State University.While at OSU, Dr. Musser developed the mathematics component of the elementaryteacher program. Soon after Professor William F. Burger joined the OSU Departmentof Mathematics in a similar capacity, the two of them began to write the first edition 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. ProfessorPeterson 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, CanadianJournal of Mathematics, The Mathematics Association of America Monthly, the NCTMs The Mathematics Teacher, theNCTMs The Arithmetic Teacher, School Science and Mathematics, The Oregon Mathematics Teacher, and The C...