Mathematics for Junior High School Volume 1 Part I

[Pages:260]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 ofthe 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, D.C.

New Haven and London, Yale University Press

Copyright @ 1960,1961by 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.

Financial support for che School Mathematics Study Group has been provided by the National Science Foundaaon.

Key ideas of Junior high school mathematics emphasized I n ' t h l s t e x t are: structure of arithmetic from an algebraic view: p o i n t ; the real number system as a progressing development;

;metric and non-metric relations in geometry. Throughout the 'materials theas ideas are associated w l t h t h e i r applications, .Important at this l e v e l are experience with and appreciation of abstract concepts, the role of d e f i n i t i o n , development of precise vocabulary and thought, experimentation, and proof. Substantial progress can be made on these concepts in the junior high school.

Fourteen experimental u n i t a f a r use In the seventh and

/ 1

eighth grades were written in by approximately 100 teachers

the summer of in 12 centers

1958 and t r i e d out i n various parts

1 of the country i n t h e ~ c h o o lyear 1958-59. 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 f o r grade 7 and a book of experimental units

f o r grade 8 . In the school year 1959-60, these seventh and

e i g h t h grade books were used by about 175 teachers in many

parts of the country,and then f u r t h e r revised in the summer of

1960.

Mathematice is fascinating to many persons because of its opportunities f o r creation and discovery as well as f o r i t s u t i l i t y . It is continuously and rapidly growing under the prodding of both Intellectual curiosity and practical applica-

t i o n s . 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 n o t the problems have routine

OP immediately determinable solutions. Recognition of these important factors has played a considerable p a r t in selection of content and method in thls t e x t ,

1

We firmly believe mathematics can and should be studied

i with success and enjoyment. ~t is o u r hope that this t e x t may

1 I

g r e a t l y assist all desirable goal.

teachers

who

use

it

to

achieve

thls

highly

1

~eprelim1na~eedltionofthiavolumewaspreparedatawritingaeaaionheldatthe

University of Michigan during the summer of 1959, baaed, in part, on m t e r l a l prepared at the flrat SWG writing session, h e l d at Yale University Zn the summer of 19%. This rev i a i o n was prepared at Stanford University In the summer of 1960, t a k i n g into account the cla5smam experience with the preliminary edition during the academic year 1959-60. The following i s a l i s t of all thoae who have participated i n the preparation of this

volume.

. R .D Anderaon, Louisiana State University

B.H. Arnold, Oregon State College J.A. Brown, University of Delaware Kenneth E, Brown, U.S. Offlce of Education Mildred B. Cole, K.D. Waldo Junior Wgh School, Aurora, I l l i n o i a B.H. Colvln, Weing Scientific Research Laboratorlee 3 . A . Cooleg, Univeraity of TennesBee Richard Dean, California I n s t i t u t e o f Technology H.M. k h m a n , University of Buffalo L . Roland Genise, Brentwood Junior High School, Brentwood, New York E. Glenadine Gibb, Iowa S t a t e Teachers College Richard Good, Universlty o f Maryland Alice Hach, Racine Public Schools, Racine, Wlaeonain S.B. Jackson, University of blaryland Lenore John, University High School, Unlvereity of Chicago

. B .U Jones, University of Colorado

P.S. 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 J.R. Mayor, AmerLcan Association for the Advanoement of Science Sheldon Meyers, Educational Testing Service Muriel M i l l a , H i l l Junior High School, Denver, Colorado

P .C. Rosenblcom, University of Minnesota Elizabeth Roudebuah, S e a t t l e Public Schoola, S e a t t l e , Washington

Very1 Schult, Washington Public Schools, Washington, D.C. 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 P u b l i c S c h o o l e , Meetpert, Connecticut O . C . Webbsr, University of Delaware A.B. Willcox, Amherst college

CONTENTS

rnFACE . . . . . . . . . . . . . . . . . . . . . . . Nom To TEACHERS . . . . . . . . . . . . . . . . . . .

j

I

1.

. . . . . . . . . . . . . WHATISMATKEMATICS? . 1- 1 Mathematics as a Method of Reaaonlng . . . . . . . . . . 1- 2 Deductive Reasoning . . . . 1- 3 From Arithmetic t o Mathematics . . . . . . . . 1- 4. Kind8 of Mathematics . . . . . . . . . . . 1- 5 Mathematics Today . . . . . . . 1- 6 Mathematics aa a Vocation . . . . Mathenatica In Other Vocations i$. . . . . . 1I- Mathematice f o r Recreation . 1- 9 Highlighte of Flmt Year Junior High . . . . . . . . School Mathematics

. . . . . . . . . . . . + . . . . . . . . . 2 NUMEfUTION

13

. . . . . . . . . . . . . 2- 1 ~ i s t o r ~ofr 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 k a e seven

26

. . . . 2- 6 Changing from Base Ten t o 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

. . . . . . . . . . . . . . 3- 1 Counting Numbers

. . 3- 2 Commutative Properties for Whole Numbera

. 3- 3 . Associative Properties for Whole Numbers

. . . . . . . . . . 3- 4 The Distributive Property

. . . . . . . . 3- 5 Set8 and the Closure Property

. . . . . . . . . . . . . 3- 6 fnverae Operations

. . . . . . . 3- 7 Betweme88 and the Number Line . . . . . . . . . . . . . . . 3 - 8 TheNurnberOne . . . . . . . . . . . . . . . 3- 9 The Number Zero

. . . . . . . . . . . . . . . . . . . 3-10 Summ~w

. . . . Answera t o 'HOW A r e You

Questions

. . . . . . . . . . Sample Questions for Chapter 3

53 53

54

56 58 62

64 66

67 69

70

71

72

. +Included In etudent text only

vii

. . . . . . . . . . . . . . . . . . . Chapter 4

. . . . . . . . . . . NON-METRICGEOmTRY . . . . . 4- 1 Points. Lines. and Space . . . . . . . . . . . . . . 4.2 Planes . . . . . . . . . 4- 3 Namea and Symbols . . . . . . . 4- 4 ~ n t e r s e c t i o nor sets

. 4- 5 ~nteraectionsof Lines and Planes

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82

84 86

4- 6 Segmenta

. . . . . . . . . . . . . . . . 4- 7. Separations . . . . . . . . . . . 4- 8 Angles and Triangles . . . . . . . . . 4- 9 One-to-one Correspondenoe . . . . . . . . . . . . 4-10 SimpleClosedCumres . . . . . . . . . . Sample Questions for Chapter 4

88 90

92

94 97 99

5 .

. . . . . . . FACTORING AND PRIMES

. . . . . . . . 5- 1 Primes . . . . . . . . . 5- 2 Factors . . . . . 5- 3 D i v i s i b i l i t y . 5- 4 Greatest Common Factor

. 5- 5 Remindera in Division

. . . . . . . . . 5- 6 Review . . Least Comon Multiple

.I. . . . . . . . . Summary

Sample Questions f o r Chapter 5

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105

1?08 114 117 121 125 130 134 139

6 .

. . . . . . . . . . . .. .. .. .. .. .. . . THERATIONALNUMBERSYSl%M

. . . . . . . . . . Overview

. 6- 1 Hiatory of Fractions

. . . 6- 2 ~ational umbers . 6- 3 Properties of Rational

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Numbera

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6- 4

. * . . . . . . . . . . 6- 5 . . . . . 6- 6 . . . . . . . . 6- 7

Re~lpmcala Ualng the Number Line Multlplicatlon of Rational Numbers

D i v i s i o n of Rational Numbere

6- 8. Addition and Subtraction of Ratloml Numbers

. . . . . . . . 6- 9 and 6-10. Ratio and Decimsls . . . . . . . .. .. .. .. .. .. .. .. .. .. 6 6.11 Orderlng

Sample mest i o n s i o C~hapter

7 .

. . . . . . . . . . . . . . . . . . MEASUREMENT

. . . . . . . . . . . Introduction . . . 7- 1 Counting and Meaauring . 7- 2 Subdivision and Measurement

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7- 3 Subdividing U n i b of Measurement

. . . . . . . . . . . . . . . 7- 4 Standard Unite . 7- 5 Precision of Measurement and the Oreateat

. . . . . . . . . . . . . Possible Error . . . . . . . . . . . . 7- 6 Measurement of Angles . . . . . . . . . . Sample Queatiom f o r Chapter 7

. 212 8.

ARRA. VOLUME. WEIGHTAND TIME

. . . . . . . 8- 1 Rectangle . . 8- 2 Rectangular P r i s m . . . . 8- 3 Other Meaeums

Sample Questtom f o r Chapter

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23 245 252

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