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ED065335.Pdf DOCUMENT RESUME ED 065 335 SE 014 398 TITLE Mathematics for Elementary School Teachers: The Rational Numbers. INSTITUTION National Council of Teachers of Mathematics, Inc., Washington, D.C. PUB DATE 72 NOTE 453p. AVAILABLE FROMNCTM, 1201 Sixteenth Street, N.W., Washington, D.C. 20036 ($6.00) EDRS PRICE MF-$0.65 HC Not Available from EDRS. DESCRIPTORS Arithmetic; *Elementary School Mathematics; Elementary School Teachers; *Fractions; Mathematics Education; *Rational Numbers; *Teacher Education; Textbooks IDENTIFIERS Film Supplements ABSTRACT This book is an extension of the 1966 film/text series (ED 018 276) from the National Council of Teachers of Mathematics and was written to accompany 12 new teacher-education films. It is strong enough, however, to also serve alone as a text for elementary school teachers for the study of rational numbers. The 12 chapters corresponsing to the films were written separately by committee members with various methods of presentation. Aspects of rational numbers covered include a rationale for their introduction; the four operations with positive, decimal, and negative rational numbers; measurement; and graphing. WO II 0 O O I II 0 0 I _ o -1`,---.4.77777:- sr .,- Ne.17-,r t,i',:).;..pce, ''''.?;'71rf; 1" s "77.7--77" ' .:. .",-is., .,` '',1 ,...., t;:. L, P. 1,:i.,:;),'''',,';:% :,2, iEl .1. '''.........a.......lii Mathematics for Elementary School Teachers THE RATIONAL NUMBERS NATIONAL COUNCIL OF TEACHERS OF MATHEMATICS 1201 Sixteenth St., NW, Washington, D.C. 20036 2 "PERMISSION TO REPROD UCE THIS COPY RIGHTED MATERIAL IIY MICROFICHE ONLY HAS BEEN GRANTED BY TO ERIC AND ORGANIZATIONS OPERATING UNDER AGREEMENTS WITH THE US. OFFICE OF EDUCATION. FURTHER REPRODUCTION OUTSIDE THE ERIC SYSTEM REQUIRES PER- MISSION OF THE COPYRIGHT OWNER." Copyright ® 1972 by THE NATIONAL COUNCIL OF TEACHERS OF MATHEMATICS, INC. Ail Rights Reserved Library of Congress Catalog Card Number: 724769 PRINTED IN THE UNITED STATES OF AMERICA 3 HISTORY AND ACKNOWLEDGMENTS In 1966 the NCTM released for general distribution by Universal Education and Visual Arts a series of films, produced by Davidson Films under a National Science Foundation Grant, entitled "Mathematics for Elementary School Teachers." The films and an accompanying text were supervised and produced by a committee headed by Harry D. Ruderman. The topics covered by the films and text were: 1. Beginning Number Concepts 2. Development of Our Decimal Numeration System 3. Addition and Its Properties 4. Multiplication and Its Properties 5. Subtraction 6. Division 7. Addition and Subtraction Algorithms 8. Multiplication Algorithms and the Distributive Property 9. Division Algorithms 10. The Whole-Number SystemKey Ideas The success of the films and text was immediate. Soon the NCTM was besieged by teachers, teachers of teachers, and school systems to carry the series forward to include, at least, the rational numbers. President Donovan A. Johnson named a committee consisting of Julius H. Hlavaty, chairman; Robert B. Davis, Abraham M. Glicksman, Leon A. Henkin, Donovan R. Lichtenberg, Joseph Moray, Harry D. Ruderman, David W. Wells, and Lauren G. Woodby to prepare a propsal for a new series of films. In 1967 the NCTM, on the suggestion of this committee, signed a precedent-setting contract with General Learning Corporation, Davidson Films, and Harry D. Ruderman as director for the production of twelve teacher-training films (for the elementary level) and thirty short, single- concept films for students. These two series of films, under the title "Elementary Mathematics for Teachers and Students," were produced and released for distribution in 1970. This book was written to accompany the twelve teacher-training films. iii History and Acknowledgments While it is complete and can stand on its own as a text for elementary school teachers for the study of rational numbers, it is intended to be a strong aid in the use of the films. The committee named above, with Harry D. Ruderman as executive director, was given the responsibility for developing scripts for the films, following through with production of the films, and preparing the text materials. Lauren G. Woodby was the film teacher, and Joseph Moray was the classroom teacher. Many of the film scripts were prepared by members of the committee. However, the entire committee had an oppor- tunity to react to all the scripts and all the films. Each chapter in this text was written by a member of the committee. No editorial attempt has been made to impose on all the chapters any rigidly uniform method of presentation. To expedite completion of the text, the Board of Directors designated Jack E. Forbes, Harry D. Ruderman, and Julius H. Hlavaty (chairman) as an editorial committee. Thanks are expressed to the persons named above, to whom the NCTM and the profession owe a debt of gratitude for the films and the present text. With grateful acknowledgment of their special contributions, the following should be signalized: Harry D. Ruderman, who has now directed with great success and special devotion two major projects for the NCTM Francis Keppell, president of General Learning Corporation, whose insightful interest kept the project afloat and helped it to a conclusion Jack W. Davidson and his staff, who contributed not only high technical skill but personal and creative input toward the high quality of the films Joseph Moray, who was classroom teacher for both film projects mentioned above and, in addition, wrote all the scripts for the first film series, in which he was also the studio teacher Lauren G. Woodby, who was the studio teacher for the second film series James D. Gates, executive secretary of the NCTM, who was a large factor in the negotiations of the original contract and the production of this text Charles R. Hucka, associate executive secretary of the NCTM, and his editorial staff, who made large contributions toward the actual production of this book JULIUS H. HLAVATY iv CONTENTS HISTORY AND ACKNOWLEDGMENTS iii 1 BEYOND THE WHOLE NUMBERS 1 David W. Wells and Stuart A. Choate The Need for Numbers beyond the Whole Numbers / Physical Models for Rational Numbers / Strategies for Introducing Rational Numbers / Summary 2 FRACTIONS AND RATIONAL NUMBERS 25 Leon A. Henkin Problems Giving Rise to Rational Numbers / Fractions as Numerals;Equivalent Numerals / Why We Use Equivalent,Fractions in Computation / Computing Equivalent Fractions / Summary ADDITION OF RATIONAL NUMBERS 55 Joseph Moray Regions and Addition / Finding a Common Denom- inator / Finding Common Denominators to Compute Sums / Least CommonMultiple / PrimeFactoriza- tion / Properties of Addition of Rational Numbers / Closure Property for Addition / Addition Is Commu- tative / Addition Is Associative / Addition Property of 0 / Other Properties of Addition / Mixed Numerals and Sums / Summary 4 SUBTRACTION OF RATIONAL NUMBERS 86 Joseph Moray Extension of Subtraction to Rational Numbers / Sub- traction Algorithms / Equal Additions / Properties of Subtraction / The Role of 0 in Subtraction / "Shifting of Terms" in Subtraction / Summary Contents 5 MULTIPLICATION OF RATIONAL NUMBERS 112 Harry D. Ruderman Preparing the Student to Figure Out for Himself: As- sociating a Rectangular Region with an Indicated Product / Computing Products of Rational Numbers without a Traditional Algorithm / Arriving at the Tra- ditional Algorithm for Multiplying with Fractions / An Important Consequence of the Algorithm Properties of the MultiplicationofRational Numbers / Some Shortcuts Based on Properties of Multiplication / "Of" and "Times" / Alternate Approaches to Multiplication with Fractions Using the Number Line / Summary of Key Ideas 6 DIVISION OF RATIONAL NUMBERS 147 Harry D. Ruderman A Strategy for Enabling a Student to Figure Out for Himself How to Compute Quotients / Interpreting a Fraction as a Quotient / Interpreting the Remainder When Dividing Whole Numbers / Properties of 0 and 1 in Division / Properties of Division of Rational Num- bers / Alternate Approaches to Dividing withFrac- tions / Summary of Key Ideas 7DECIMALS: ADDITION AND SUBTRACTION 179 David W. Wells and Philip L. Cox Introducing the Extension of Decimal Numeration / Naming Rational Numbers by Decimals / Can All Rational Numbers Be Named as Decimals? / Comput- ing Sums of Rational Numbers with Decimals / Com- puting Differences with Decimals / "Ragged Decimals" Pro and Con / Summary 8DECIMALS: MULTIPLICATION AND DIVISION 213 Donovan R. Lichtenberg Multiplying with Decimals / Division with Decimals / ExtensionoftheDecimal ConceptInfinite Dec- imals / Summary 9MEASUREMENT 239 Lauren G. Woodby Discrete and Continuous Quantities / The Measure- mentProcess / Order / ArbitraryUnits / Needfor Rational Numbers / AreaMeasurementArbitrary Units /Weight MeasurementArbitrary Units / Ratio / Standard Units / The Geoboard / Indirect Measurement of Length / Two Indirect-Measurement Instruments / Units of Area / Direct Measurement of Area vi rot Contents 10NEGATIVE RATIONALS 267 Robert B. Davis The Game "Guess My Rule" / A First Experience with Negative Numbers / Mathematics and Legisla- tion / A Concrete Model for Negative Integers / The Addition of Negative Rationals / Negative Rationals / The Question of "Order" / An Informal Look at the Structure of This New Number System / A Formal Development of the Negative Integers / Summary 11 OPERATIONS EXTENDED TONEGATIVE RATIONALS 297 Abraham M. Glicksman Subtraction Involving Negative Rational Numbers / Multiplication Involving Negative Rational Numbers / Division Involving
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