Mathematics for Junior High School Volume 1 Part I

1 Mathematics FOR :- Junior high School Volume 1 - - PART I School Mathematics Study Group Mathematics for Junior high School , Volume Unit 3 Mathematics for Junior high School , Volume : Teacher's Commentary, Part I Preparrd under the supervision of the Panel on Seventh and Eighth Grades of the School Mathematics Study Group: R. D. Anderson J. A. Brown Lenore John B. W. Jones P. S. Jones J. R. Mayor P. C. ~osenbloom Veryl Schult Louisiana State University University of Delaware University of Chicago University of Colorado Urliversity of Michigan American Association for the Advancement of Science University oE Minnesota Supervisor of Mathematics , Washington, New Haven and London, Yale University Press Copyright @ 1960, 1961 by Yale University. Printed in the United States of America. All rights reserved. his book may not be reproduced, in whole or in part, in any form, without written permission from the publishers.

2 Financial support for che School Mathematics Study Group has been provided by the National Science Foundaaon. Key ideas of Junior high School Mathematics emphasized In 'thls text are: structure of arithmetic from an algebraic view- :point; the real number system as a progressing development; ;metric and non-metric relations in geometry. Throughout the 'materials theas ideas are associated wlth their applications, .Important at this level are experience with and appreciation of abstract concepts, the role of definition, development of precise vocabulary and thought, experimentation, and proof. Substantial progress can be made on these concepts in the Junior high School . / Fourteen experimental unita far use In the seventh and eighth grades were written in the summer of 1958 and tried out 1 by approximately 100 teachers in 12 centers in various parts 1 of the country in the ~chool year 1958-59.

3 On the basis of I teacher evaluations theee unita were revised during the summer of 1959 and, wlth a number of new units, were made a part of sample textbooks for grade 7 and a book of experimental units for grade 8. In the School year 1959-60, these seventh and eighth grade books were used by about 175 teachers in many parts of the country,and then further revised in the summer of 1960. Mathematice is fascinating to many persons because of its opportunities for creation and discovery as well as for its utility. It is continuously and rapidly growing under the prodding of both Intellectual curiosity and practical applica- tions. Even Junior high School students may formulate mathematical questions and conjectures which they can test and perhaps settle; they can develop systematic attacks on mathematical problems whether or not the problems have routine OP immediately determinable solutions.

4 Recognition of these important factors has played a considerable part in selection of content and method in thls text, 1 We firmly believe Mathematics can and should be studied i with success and enjoyment. ~t is our hope that this text may 1 greatly assist all teachers who use it to achieve thls highly I desirable goal. 1 ~eprelim1na~eedltionofthiavolumewasprepa redatawritingaeaaionheldatthe University of Michigan during the summer of 1959, baaed, in part, on mterlal prepared at the flrat SWG writing session, held at Yale University Zn the summer of 19%. This re- viaion was prepared at Stanford University In the summer of 1960, taking into account the cla5smam experience with the preliminary edition during the academic year 1959-60. The following is a list of all thoae who have participated in the preparation of this Volume . R .D . Anderaon, Louisiana State University Arnold, Oregon State College Brown, University of Delaware Kenneth E, Brown, Offlce of Education Mildred B.

5 Cole, Waldo Junior Wgh School , Aurora, Illinoia Colvln, Weing Scientific Research Laboratorlee Cooleg, Univeraity of TennesBee Richard Dean, California Institute of Technology khman, University of Buffalo L. Roland Genise, Brentwood Junior high School , Brentwood, New York E. Glenadine Gibb, Iowa State Teachers College Richard Good, Universlty of Maryland Alice Hach, Racine Public Schools, Racine, Wlaeonain Jackson, University of blaryland Lenore John, University high School , Unlvereity of Chicago B .U . Jones, University of Colorado Jones, University of Michigan Houston Kames, Louisiana State University Mildred Keif'fer, Cincinnati Public Schools, Cincinnati, Ohio Nick Lovdjiefr, Anthony Junior high School , Mnneapolla, Minnesota Mayor, AmerLcan Association for the Advanoement of Science Sheldon Meyers, Educational Testing Service Muriel Milla, Hill Junior high School , Denver, Colorado P.

6 C. Rosenblcom, University of Minnesota Elizabeth Roudebuah, Seattle Public Schoola, Seattle, Washington Very1 Schult, Washington Public Schools, Washington, QeoPge Schaefer, Alexis I. DuPont high School , Xilinington, Delaware Allen Shielda, University of Mlchigan Rothwell Stephens, Knox College John Wagner, Sohool Mathematics Study Group, New Haven, Connecticut Ray Walch, Weatport Public Schoole, Meetpert, Connecticut Webbsr, University of Delaware Willcox, Amherst college CONTENTS .. rnFACE .. Nom To TEACHERS .. 1. WHATISMATKEMATICS? 1- 1 . Mathematics as a Method of Reaaonlng 1- 2 . Deductive Reasoning .. 1- 3 . From Arithmetic to Mathematics .. 1- 4. Kind8 of Mathematics .. j 1- 5 Mathematics Today I 1- 6 . Mathematics aa a Vocation .. Mathenatica In Other Vocations .. 1- I- i$ . Mathematice for Recreation .. 1- 9 . Highlighte of Flmt Year Junior high School Mathematics .

7 2 . NUMEfUTION ..+.. 13 .. 2- 1 . ~istor~r of N=~S 16 2- 2 . TheDecimalSystem .. 18 2- 3 . Expanded Numerale and EZponentlal Notation . 20 .. 2- 4 . Numerals in Base Seven 22 2- 5 . omp put at ion in kae seven .. 26 .. 2- 6 . Changing from Base Ten to Base Seven 32 2- 7 . Numerals In Other Bases .. 34 .. 2- 8 . The Binary and Duodecimal Systems 36 2- g . Summarg .. 45 SmpleQuestions forchapter2 .. 47 UHom NUMBeRS .. 53 3- 1 . Counting Numbers .. 53 3- 2 . Commutative Properties for Whole Numbera . 54 3- 3. Associative Properties for Whole Numbers . 56 3- 4 . The Distributive Property .. 58 3- 5 . Set8 and the Closure Property .. 62 3- 6 . fnverae Operations .. 64 3- 7 . Betweme88 and the Number Line .. 66 3-8 . TheNurnberOne .. 67 .. 3- 9 The Number Zero 69 3-10 . Summ~w .. 70 .. Answera to 'HOW Are You Questions 71 Sample Questions for Chapter 3 .. 72 +Included In etudent text only.

8 Vii Chapter 4 . NON-METRICGEOmTRY .. 4- 1 . Points. Lines. and Space . Planes .. 4- 3 Namea and Symbols go 82 .. 4- 4 ~ntersection or sets 84 .. 4- 5 ~nteraections of Lines and Planes 86 .. 4- 6 Segmenta 88 .. 4- 7. Separations 90 .. 4- 8 Angles and Triangles 92 .. 4- 9 One-to-one Correspondenoe 94 4-10 . SimpleClosedCumres .. 97 .. Sample Questions for Chapter 4 99 .. 5 . FACTORING AND PRIMES 105 .. 5- 1 Primes 1 8 .. 5- 2 . Factors 10 .. 5- 3 Divisibility 114 .. 5- 4 . Greatest Common Factor 117 .. 5- 5 . Remindera in Division 121 .. 5- 6 . Review 125 .. Least Comon Multiple 130 .. Summary 134 .. Sample Questions for Chapter 5 139 .. 6 . THERATIONALNUMBERSYSl%M .. Overview .. 6- 1 . Hiatory of Fractions .. 6- 2 . ~ational umbers .. 6- 3 . Properties of Rational Numbera .. 6- 4 . Re~lpmcala 6- 5 . Ualng the Number Line *.. 6- 6 . Multlplicatlon of Rational Numbers 6- 7.

9 Division of Rational Numbere .. 6- 8. Addition and Subtraction of Ratloml Numbers .. 6- 9 and 6-10. Ratio and Decimsls . Orderlng .. Sample mes tions io~ Chapter 6 .. 7 . MEASUREMENT .. Introduction .. 7- 1 . Counting and Meaauring .. 7- 2 . Subdivision and Measurement .. 7- 3 . Subdividing Unib of Measurement .. 7- 4 . Standard Unite 7- 5 . Precision of Measurement and the Oreateat Possible Error .. 7- 6 . Measurement of Angles .. Sample Queatiom for Chapter 7 .. 8 . ARRA. Volume . WEIGHT AND TIME 217 8- 1 . Rectangle .. 8- 3 Other Meaeums 212 8- 2 Rectangular Prism 23 .. 245 .. Sample Questtom for Chapter 6 252 NOTE TO TFACHERS I Based on the teaching experience of nearly 200 Junior thigh School teachere In all parta of the country and the estimates ! of' the authors of the revision ( including Junior high School teachem), it is recommended that teaching time for Part 1, be as follows : Chapter Approximate number of days 7 15 14 15 12 17 13 13 Total 106 Teachers are urged to trg not to exceed theae approximate time allotments so that pupile will not miss the chapters at the end of the courae, Some classes will be able to finish certain chaptere Ln leas than the estimated time.

10 Throughout the text, problems, topics and section8 which were designed for the better students are indicated by an aaterisk (*) . Items starred in thla mer should be used or omitted aa a means of adjusting the approximate time schedule. Chapter 1 WHAT I3 Mathematics ? General Remarks This chapter is intended to give the pupil an appreciation for the Importance of rnathematlcs. Its objectives are: I. To develop an understanding of what Mathematics is as opposed to simple computation. 11. To develop an appreciation of the role of Mathematics in our culture. 111. To motivate pupils by pointing out the need for mathe maticlans and for mathematically trained people. Since this chapter is much different from ordinary textbook material it will need a different treatment. The purpose of the chapter is - not to teach many facts or skills, but rather to build -- an enthusiasm for the study of Mathematics .