Table Of ContentMATHEMATICS FOR
:-JUNIOR HIGH SCHOOL
1
VOLUME
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 Louisiana State University
J. A. Brown University of Delaware
Lenore John University of Chicago
B. W. Jones University of Colorado
P. S. Jones Urliversity of Michigan
J. R. Mayor American Association for the
Advancement of Science
P. C.~ osenbloom University oE Minnesota
Veryl Schult Supervisor of Mathematics,
Washington, D.C.
New Haven and London, Yale University Press
Copyright @ 1960,1 961 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.
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 w l t h 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 i n various parts
1
of the country in the ~chooly ear 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 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. 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~eedltionofthiavolumewaspreparedatawritingaeaaionheldatthe
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
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, Illinoia
B.H. Colvln, Weing Scientific Research Laboratorlee
3.A. Cooleg, Univeraity of TennesBee
Richard Dean, California Institute of Technology
H.M. 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
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 Milla, Hill Junior High School, Denver, Colorado
P .C. Rosenblcom, University of Minnesota
Elizabeth Roudebuah, Seattle Public Schoola, Seattle, 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 Public Schoole, Meetpert, Connecticut
O.C. Webbsr, University of Delaware
A.B. Willcox, Amherst college
CONTENTS
. . . . . . . . . . . . . . . . . . . . . . .
rnFACE
. . . . . . . . . . . . . . . . . . .
Nom To TEACHERS
. . . . . . . . . . . . .
1. WHATIS.M ATKEMATICS?
1- 1 . Mathematics as a Meth.od. o.f .R.ea.ao.nln.g .
1- 2 . Deductive Reasoning . . .
1- 3 From Arithmetic to Math.em.at.ic.s . . . .
1- 4. . Kind8 of Mathematic.s . . . . . . . . .
j
1- 5 . Mathematics Today . . . . . .
1- 6 . Mathematics aa a Vocation . . .
I
i$. Mathenatica In Other Vocation.s . . . .
1I-- . Mathematice for Recreation
1- 9 Highlighte of Flmt Yea.r .Ju.ni.or. H.ig.h .
School Mathematics
. . . . . . . . . . . . + . . . . . . . . .
2 NUMEfUTI.O N . . . . . . . . . . . . 13
2- 1 . ~istor~ofr N = ~ S . . . . . . . . . . . . 16
2- 2 . TheDecimalSystem . 18
2- 3 . Expanded Numerale and EZp.on.en.tl.al. N.ot.at.io.n. 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 to. B.as.e .S.ev.en. . . . 32
2- 7 . Numerals In Other Bases . . . . . 34
2- 8 . The Binar.y .an.d .Du.od.ec.im.al. S.ys.te.m.s . . . . . 36
2- g Summarg . . . . . . . . . . 45
SmpleQuestions forchapter2 47
. . . . . . . . . . . . . . . . . . .
UHom N.U MBeRS . . . . . . . . . . . . . 53
3- 1 . Counting Numbers . 53
3- 2 Commutative Properties for Whole Numbera . 54
3- 3.. Associative Properties for .Wh.ol.e .Nu.mb.er.s . . 56
3- 4 . The Distributive Property . . . . . . . 58
3- 5 . Set8 and the Closure .P.rop.e.rty. . . . . . . . 62
3- 6 . fnverae Operations . . . . . . 64
3- 7 Betweme88 and t.he. N.um.b.er .Li.ne. . . . . . . 66
.
3 - 8 TheNurnberOne . . . . . . . . . . . . . . 67
.
3- 9 . The Numbe.r .Ze.ro. . . . . . . . . . . . . . . 69
3-10 Summ~w . . . . 70
Answera to 'HOW Are You .Qu.es.ti.on.s . . . . . 71
Sample Questions for Chapter 3 72
.
+Included In etudent text only
vii
Chapter
. . . . . . . . . . . . . . . . .
4 NON-ME.T RICGEOmTRY . . . . . . . . .
4- 1 . Points. L.in.es.. .an.d .Sp.ac.e . . . . . . . . . .
4.2 . Planes . . . . . . . . . . . . . go
4- 3 Namea and Symbols . . . . . . . . . . . 82
.
4- 4 ~ntersectiono r sets . . . . . 84
.
4- 5 . ~nteraecti.on.so .f .Li.ne.s .an.d .Pl.an.es. . . . . . 86
4- 6 Segmenta . . . . . . . . . . . . . . . . 88
4- 7. Separations . . . . . . . . . . . 90
4- 8 Angles and Triangles . . . . . . . . . 92
4- 9 . One-to-one Corresponden.oe. . . . . . . . . . 94
4-10 SimpleClosedCumres . . . . . . . . . . 97
Sample Questions for Chapter 4 99
. . . . . . . . . . . . . . . . .
5 FACTORING AND PRIM.ES. . . . . . . . . . . . . . . . . 105
5- 1 . Primes . . . . . . . . . . . . . . . . . .
5- 2 . Factors . . . . . . . . . . . . . . . 11°08
5- 3 . Divisibility . . . . . . . . . . 114
5- 4 Greatest Common Factor . . . . . . . . . . 117
.
5- 5 Reminder.a .in. D.iv.is.io.n . . . . . . . . . . . 121
.
5- 6 . Review . . . . . . . . . . . 125
.I.. Least Co.mo.n .Mu.lt.ip.le. . . . . . . . . . . . 130
Summary . . . . . . . . . . 134
Sample Questions for Chapter 5 139
. . . . . . . . . . . . . .
6 THERATIONAL.NU.M.BE.RS.Y.Sl%.M. . . . . . . . . . . . . .
Overview . . . . . . . . . . .
.
6- 1 Hiatory of Fraction.s . . . . . . . . . . . .
.
6- 2 . ~ational umbers . . . . . .
6- 3 . Properties of. R.a.tio.na.l .Nu.mb.er.a . . . . . . .
6- 4 . Re~lpmcala * . . . . . . . . . .
6- 5 . Ualng the Number Line . . . .
6- 6 . Multlplicatlon of Rational Numb.er.s . . . . .
6- 7 Division of Rational Numbere
6- 8. Addition and Subtraction of R.a.tlo.m.l .Nu.mb.er.s
6- 9 .a nd 6-10. R.at.io .an.d .D.eci m.sls. . . . . . . . .
6.11 Orderlng . . . . . . . . . .
6
Sample mes tions i o C~ha pter
. . . . . . . . . . . . . . . . . . . . .
7 MEASUREMENT . . . . . . . . . . . . . . . . . . .
Intro.d uction . . . . . . . . . .
7- 1 . Counting and Meaauring . . . . . . . .
7- 2 . Subdivision and Measurement . . . . .
7- 3 . Subdividing Unib. o.f .Me.as.ur.em.en.t . . . . . .
7- 4 . Standard Unite
7- 5 Precision of Measur.em.en.t .an.d .th.e .O.re.ate.at. .
. Possible Error . . . . . . . . . . .
7- 6 Measurement of Angles . . . . . . . . . .
Sample Queatiom for Chapter 7
. . . . . . . . . . . .
8 ARRA. V.O LUME. WEIGH.TA. ND. T.IM.E . . . . . . . . . . . . 217
8- 1 . Rectangle . . . . . . . . . . . . . 212
8- 2 . Rectangular Prism. . . . . . . . . . . . . 23
8- 3 Other Meaeums . . . . . . . . . . 245
6
Sample Questtom for Chapter 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.
Throughout the text, problems, topics and section8 which
were designed for the better students are indicated by an
.
aaterisk (*) Items starred in thla m e r s hould 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 t o teach many facts or skills, but rather to -b-u ild
an enthusiasm for the study of mathematics. Good attctudes will
be built if you use imagination and enthusiasm i n getting these
objectives across to the puplls. Since the material I s -not to be
taught for mastery, we strongly recommend that -no test be given
covering the contents of this chapter,
Experience shows that this chapter can be covered within six
to eight lessons. Certaknly no more than eight days should be
devoted to it.
Were seventh graders are in a new school situation and have
so many interruptions during the first few days, some teachers may
wfsh t o precede this chapter with review exercises which are more
familiar to the pupils.
Note that Exercises 1-6, (Page 13) and Exercises 1-7, fa age 14)
are suggestions for background study to be carried on throughout
the year. These should be begun during the first week, with
periodic reports on progress by pupils. Where guldance personnel
are available, their services should be solicited t o help the
class outline a plan of action for the year.
