MATHEMATICS FOR. JUNIOR HIGH SCHOOL. VOLU-ME 1. PART 11 ... This
book may not ..... The second leason in 9-2 emphaeizes the use of percent for.
.MATHEMATICS FOR JUNIOR HIGH SCHOOL VOLU-ME 1 PART 11
School Mathematics Study Group
Mathematics for Junior High School, Volume
Unit 4
I
~ a t h e r n a t i c sfor Junior High School, Volume I Teacher's Commentary, Part I1
Prepared 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.Rosenbloom Very1 Schult
Louisiana State University University of Delaware University of Chicago University of Colorado University of Michigan American Association for the Advancement of Science University of Minnesota Supervisor of Mathematics, Washington, D.C.
Ncw Haven and London, Yale University Press
Copyright @ rg60, 1g61 by Yale University. Printed in the United States of America.
All rights reserved. This book may not be reproduced, in whole or in part, in any form, without written permission horn
the publishers. Finand support for the School Mathemaucs Study Group has been provided by the Nauonal Science Foundation.
Tho pnlirPinarg edltlon of th18 voltlme was prepared at a writing seanfon held at the Univeralty of Mlchlgan d u r i n g the surmner of 1959, baurd, in p a r t , on material prepared at the Plret SMSU urltlng reasion, hela a t Yale Univtreltg i n the Buamrer o f 1958. !Lbir revision was preparrd at Stanford Univer8ity i n the s u m e r of 1960, taking i n t o account the elamsmotn experience 111th the prelimintdition durlng the aaademic year 1959-60,
The follcwing i e a list o f all those who havs participated In the preparation of t h i s
volume. R.D. hnderson, bouiriaha State University
B.H. lunold, Vmgon State College S.A. Broun, Univereltr o f Delawam
Kenneth E. Brown, U . S . Office o i Bduoation MlldreO 8. Cole, K.D. Waldo Junlor High School, Aurora. I l l i n o i a
B.H. Colvin, Boclng S e l e n t l f l e Research Laboratorlsa Cooleg, University of Tenneeaee
J.A.
Richard Dean, California l m t i t u t r of lkchno3.ogg hhman, University of Wrfralo
H.M.
L. Roland Oeniae, Brentwood Junior High School, Brentwmd, New York
E. Qlenadlne [ilbb, Iowa State Teacher8 ~olligs Riahard m d , University of *land
Alice Hach. h o l m Public Schools, Reclne, Wisconsin
.
S .B Jackeon, U n i v e r s i t y o f Harylmd Lenore John, Urdverarity High 9cha01, Univeruitg of Chicago
B.W. Jones, Unlverslty of Colorado P.S. Joner, Univereltg of MlohLgan Houston Karma, Louisiana
State Unlverslty
Mlldred Keiffer, C i n o i n n a t l Alblio Sohoola, Cincinnati, Ohio
Hi&
LOvdjieff,
Anthong junior Him School, ninneapolim, Wnnerota
J.R. byor, American Association for the Advenoemtnt of Science
Sheldon &yere,
Bduoatlonal mmting Sarvloe
M e 1 M i l l s , IUll
Junior HI&
Sahool, Denver, Colorado
P .I!. Rosenbloom, Unlveroity of Minneuota Elizabeth Roudebueh, Seattle Publia School@, Seattle, Washington
Pew1 Schult, Washln@on Public Sohools, Washington, D.C.
Qsorge Sahaerer, Alexir I . w o n t High Sohaol, Wllnington, Dclawam Allen Shitlda, Univernlty of niehlgan Rothmll Stephens, Ktwx College John Ungner, Sohool Kathematios Study Omup, New Haven, Connecticut Ray Waluh, Yeetport public Soh-la, O,C.
Webbrr, Univer8ity of Ilslauafa
A.B.
Willoar, Amherst College
WertpoA, Conmetiout
Chapter
9.
10.
. .. .. .. .. .. .. .. .. .. 22g . . . . .. .. .. .. .. .. .. .. .. .. .. .. 262 265 Arithmetic Operatfons w i t h Decimals . . 267 mcimal Expansion . . . . . . . . . . . 269 Rounding . . . . . . . . . . . . . . . . 271 Percent and Decimals . . . . . . . . . . 8 1 Applications of Percent . . . . . . . . 275 Sample ~uestionsf o r Chapter 9 . . . . . . . . 281
RATIOS. PERCmS. AND DECIMALS 9- 1 Ratios and Proportion 9- 2 percent 9- 3 Ikclffla.1 Notation
.. . 9- 4 . 9- 5. 9- 6. . 9- 1. 9-
PARALLELS. PARAILELOORAMS. TRIANGLES. AND RIOIPT PRISMS 10- 1 Vertical and Adjacent Angles 10- 2 Three Lines 3n a Plane 10- 3 Parallel L i n e s and Correeponding Angles 10- 4 Converses (Turning a Statement round) 10- 5 Triangles 10- 6 Angles of a Triangle
. . . . . . . . . . . . .. .. .. .. .. .. 269 297 . . . . . . . . . 294 300 . 303 . . . . . .. .. .. .. .+. .. .. .. .. . 306 310 Parallelograms . . . . . . . . . . . . . lo-!! . Areas of Parallelograms and Triangles . 31 10"3 10- 9. Right Prisms . . . . . . . . + . . . . .322 Sample Questlona for Chapter 10 . . . . . . . . 324 11 . C I R C U S . . . . . . . . . . . . . . . . . . . . . 11- 1 . Circles and the Comrrasa . . . . . . . . 3 33 Interlore and ~ n t e r ~ e c t i o n s. . . . . . 341 Diameters and Tangents . . . . . . . . . 342 Arcs . . . . . . . . . . . . . . . . . . 34 Central Angles . . . . . . . . . . . . . 350 Length of a Circle . . . . . . . . . . . 353 Area of a Circle . . . . . . . . . . . . 3 b Cylindrical SolAds-Volume . . . . . . . 366 Cylindrical Solids.. Surface Area . . . . 367 Answers t o Review of Chapters 10 and 11 368 Questions for Chapter 11 . . . . . . . . 371 12. MATHSMATICAL SYSTElrIS . . . . . . . . . . . . . . . 381 12- 1 . New King of A d d i t i o n . . . . . * . . . 384 12- 2 . New Kind of Multiplication . . . . . . 387 12- 3 . What is an Operation? . . . . . . . . . 389 12- 4 . Closure . . . . . . . . . . . . . . . . 399 12- 5 . I d e n t i t y Element; Inverse of an Element 403 2 - 6 What is a Mathematical system? . . . . . 409 12- 7. Mathematical Systems Without Numbers . . 411 12- 8 . The Counting Numbers and the Whole Numbers . . . . . . . . . . . . . 12- 9. Modular Arithmetic . . . . . . . . . . . 41 413 Sample Questiona for Chapter 12 . . . . . . . . 422 .. .. .. .
A A
13.
1
. .. .. .. .. .. .. .. .. .. .. .. .. .. 429 430 431 . . . . . . . . . . . 13- 3 . Bar Qrapha . . . . . . . . . . . . . . . 436 13- 4 . C i r c l e Graphs . . . . . . . . . . . . . 438 13- 5. S u m a r l z i n g Data . . . . . . . . . . . . 440 13- 6. sampling . . . . . . . . . . . . . . . . 443 Sample Questions for Chapter 13 . . . . . . . . 445 MATHEMATICS AT WORK IN SCIENCE . . . . . . . . . . 451 14- 1 . The S c i e n t i f i c Seesaw . . . . . . . . . 451 STATISTICS AND GRAPHS 13- 1 Gathering Data 13- 2 Broken Cine Graphs
. .
14- 2
.
A Laboratory Experiment
. . . . . . . .. 452 454
Caution: Inductive Reasoning at Work! Graphical Interpretation 14- 5 Other Kinds of Levers 14- 6 The Role of Mathematics In S c i e n t i f i c Experiment Sample Questions for Chapter 1 4 14- 3. 4 - 4
...
. .. .. .. .. .. .. .. .. 455 458 . . . . . . .. .. .. .. .. .. .. .. 458 460
NOTE TO TEACHEXS
Based on t h e teaching experience of over 100 junior high school teachers in a l l parts of the country and t h e estimates of authors of the r e v i s i o n s (including junior high school teachers), it I s recommended that teaching time for Part 2 be as iollows: Chapter
Approximate number of days
18 12 13 11
10
a Total
69
Throughout the t e x t , problems, t o p i c s , and sections which were designed for the better students are I n d i c a t e d by an asterisk (*) Items starred in this manner should be uaed or omitted as a means of adjusting the approximate time schedule.
.
Chapter g
RATIOS, PERCENTS, A N D DEC IMALS The important concept of this chapter is ratio. When a physical or mathematical law can be written as a proportion, then this l a w can be used to deduce new Information from o l d . Thus in our f i r s t example, shadow length is p r o p o r t i o n a l to h e i g h t . Suppose we know the ratio of t h e measure of t h e shadow l e n g t h to the measure of a corresponding height. If we are given a n o t h e r shadow length w e can f i n d the corresponding height, or if we are given another height we can find the corresponding shadow length. The I d e a s behind the words r a t i o , percent, and decimal enable us to express old and familiar properties of rational numbers in a convenient way. A ratio may be expresaed by a r a t i o n a l number, A rational number may be expresaed as a percent or in decimal notation. Naturally, we have to spend a great deal of time h e l p i n g stueent s i n developing sufficient s k i l l with these new notations f o r the students to become really competent In us in& the notat ions Certainly there are important and interesting results to be established. No one w i l l call the theorem t h a t only r a t i o n a l numbers have repeating decimal expansions a t r i v i a l result. The correspondence of a point on the number l i n e w i t h a decimal expansion is an exciting and a far-reaching r e s u l t . On the o t h e r hand, decimals are slmply notatlolls and add only a l i t t l e to our understandfng of rational numbers as a mathematical system. Proportion, however, is a new concept, essentially having I t s origin in physical examples. The examples of this chapter will provide the students with situations where it is a natural t h i n g to introduce t h e notations of ratio, percent, and decimals as handy and e f f i c i e n t short c u t s of expression. It is our hope t h a t if a purpose is aeen for their introduction, the teaching of the technique of their use will prove easier.
.
Many teachers who have used the SMSG t e x t s fn grades 7 and 8 b e l i e v e that the attention given to decimale and percents and their applications is sufficient. Others are of the opinion that more tlme for the study of these topica is needed if their students are to achieve the degree of maatery of them which has become traditional. Special effort has been made In the revision of this chapter to bring about greater efficiency in the use of r a t i o , percents, and decimals. Additional expertence with r a t i o , percents, and decimals has been provided In exercises in the remainder of this t e x t and the t e x t prepared for use in grade 8. The p r i n c i p a l difference in treatment in the SMSG and the traditional t e x t s is t h a t in SMSO materials these t o p i c s are more closely related to the properties of number systems from a mathematical p o i n t of view, and lees time is given to a discussion of the social situations in which the various applications arise. Based on the teaching experience In the 12 centers f o r junior high school mathematics, the writing committee recommends t h a t 17 or 18 teaching days be used in teaching this chapter.
Ratios and Proportion Thia s e c t i o n should not take more than two class days. Class discussions may well Include some simple c a l c u l a t i o n e t h a t can be done In a few minutes. Most teachers will divide the rather long s e t of exercises into t w o assignments. The idea of r a t i o was Introduced in Chapter 6 . The purpose of t h i s section is to provide additional work with ratios and to give a meaningful introduction to proportion. Many applications are Included. It is recommended that the teacher emphasize t h a t when a mathematical or physical law can be written as a proportion, then this law can be used to obtain new information and that the uee of proportion involves only quite easy mathematical ideas -1.
.
The authors do not recommend that a p a r t i c u l a r method be used in solving problems involving two equal ratios
(proportions). Some answers can be obtained by mental computation. This ahould not be discouraged. In finding in the equation (proportion)
some studenta may wish to m u l t i p l y
f;3
and
one in some form so that the two fractions
n
5
by the number
n
and
3
have t h e
same denominator. In the l a e t step, remind the studenta of the d e f i n i t i o n of a rational number in Chapter 6. Other students may prefer to use the property that if , b f 0 and d 0, then ad = bc, and only then. The teacher will of course recognize this property as the familiar "product of the means equals the product of the extremes .' We have carefully avoided the fm i l i a r expression since it is a special device for a s p e c i a l situation and in t h i a form c a n n o t be so e a s i l y related to the mathematical structures which the student ehould constantly be making a part of his way of t h i n k i n g . Attention of the teacher I s called to the significance of only then in the statement of this property. The property as s t a t e d above and in Exercise *18 llluatratea what i a known as a necessary and sufficient condition, or an if and only if condition in mathematics. There are two separate properties t o be proved. The f i r s t I s proved in the t e x t and the proof of the aecond is left as a problem. See Exercise "18. In some clasaes the important distinction between necessary and sufficient, or if and only If, may be meaningful only to the better students. The authors strongly recommend t h a t the n o t a t i o n a:b::c:d not be uaed.
-
[pages 347-3501
Problems *14, *l5, *16, +17 are sometimes c a l l e d problems involving multiple ratios, It l a recommended t h a t these problems not be considered a separate type of problem situation. The teacher, however, may wish to give special attention in c l a s s to a problem 1 i k e these
.
-
Answers to Exercises 1.
9-1
Depends on h e i g h t of individuals. 2 x height.
Length of shadow i s
Ratio of number of glrls to total
5.
6.
5.
(b)
Ratio of number of boys to t o t a l is
(c)
Ratio of number of glrls to number of boys is
The following ratios are equal:
(a),
(c),
(a)
x=15
(b)
x=42
(c)
x=15
(dl,
(4.
2
J.
10 inches long
112.5 ft. tall
In the ratio 3:l
we have:
3 cups butter 2
cups augar
4$ 3
cup8 flour teaspoons vanilla
6 eggs It will now make 90 oookiea . To make 45 cookies you would rewrite the recipe In 1 the ratio 12:1 1%
cups b u t t e r
4
1
cup augar
I$ teaspoon vanllla
3
eggs
cups f l o u r
(a)
$1.65
(b)
45
(d)
316.8 feet or 317 to theneareat foot.
cents
( 4 5 c e n t a per dozen)
44 feet per eecond
914.
This is a starred section, to be used with some students if time permits. To make 24 pounda of a mixture of peanuts, cashews, and pecans i n a ratio of 5 to 2 to 1 the grocer would use 15 pounds of peanuts, 6 pounds of cashews, and 3, pounds of pecwa The answers to the questions in the discussion are: 1.
8 pounds of nuts.
2.
5 to 8 i a the ratio of peanuts to t o t a l .
3.
15
4.
Cashews
pounda of peanuts. 2 to 8
is r a t i o t o total
6 pounds of cashews. 5.
Pecans
1 to 8
is r a t i o of pecans to total
3 pounds of pecans. *15. 35 pounds of peanuts, of pecans.
6
50 pounds peanuts, pecans.
*17. 24
9-2.
14
lbs. of cashews,
30 pounds cashews,
7 lba.
20 pounds
inches.
Percent The concepts
df percent are introduced
b r i e f l y in Section 2. In Sections 7 and 8 of t h i s c h a p t e r , percent w i l l be treated more
f u l l y . In b o t h of the sections the meaning of percent is baaed a on t h e idea t h a t "a1'$ means = a x r(~r. All three "cases" of percent a r e introduced informally in Section 2 with numbers that are e a s i l y handled. However, you will n o t i c e that the three tt casesn of percent are not r e f e r r e d to in this textbook. Notice that the solutions of a l l problems are set up in t h e fonn a The method of solving f o r x should be the method t h a t 6= the p u p i l understands. Here we have used Property 1 from Section 1 and t h e property t h a t if ax = b then x = z- P u p l l s should be encouraged to use any method t h e y unde-rstand.
&.
[pages 353-3561
.
8
In t h e example of the c l a s s of 11 the fraction
44
girls and
14 boys,
indicates percent e a s i l y because the
i denominator is 100. The sum of 11 and 25 l4 is one. The sum of the ratlos 44 and is one, and the sum of 44$ and
,
&
is 10M or one. Exercises 9-2a emphasize writing percents first as fractions with 100 as the denominator, and then with the $ aymbol. stands fop one or the Another point of emphasis I s t h a t 1whole of a quantity.
5w
-
Answeps to Exercises 9-2a 1 1. (a)
m
(b)
2.
One
(f) (9)
& &
7.
(d)
The ratios t o t a l
(a)
Mortgage
2@;
clothing 1%;
recreation ; (b)
me
end the percents total
1,
1w.1
food 3M; operating ; health and aavinge and insurance l@. taxes
car
;
;
8 answers total low.
The second leason in 9-2 emphaeizes the use of percent for purgoaea of comperlson, and for giving information In more usable form. Pupils should learn to calculate mentally 1$ of a quantity, and alao 1 Frequent oral practice should include euch calculations .
-
Answers t o J3xerciaes 9-2b 1.
(a)
6*
2.
(8)
5@
or
3.
(4 6%
Or
&
(b)
4M
(b)
decreased
(b)
higher than both
hint
22 4 x = m
- 3 .
Decimal Notation Presumably a l l s t u d e n t s w i l l have had an introduction to decimals in sixth grade, An attempt is made here to extend t h e Place Value Chart which is commonly used in e a r l i e r grades to Include notation uaed in the expanded form. Thus we have the place nam'e in words, declmal n o t a t i o n , and as a power of ten. The purpose of t h i s section is to review base and place value, reading and writing r a t i o n a l numbers in decimal form, and to extend the notation for the expanded form. The teacher could illustrate changing a r a t i o n a l number into o t h e r b a s e s . Example 1. Write 1 as a duodecimal. 1 There are 12 t w e l f t h s in one unit, so in of a unit, 12 there a r e T , or 3 twelfths, and none l e f t over.
I
II I
I I
-
Example 2. Write as a duodecimal. There are 12 t w e l f t h s in one unit and 60 t w e l f t h s in 60 5 units, so In of a unit, there are g twelfths or ;7 twelfths. The twelfths d i g i t I s "7" and there is 1 of a twelfth l e f t over. There are 12 one-hundred-forty-fourths 1 of a t w e l f t h , t h e r e are 12 In one t w e l f t h of a unit, so in 2 or 6 one-hundred-forty-fourths and none left over. The one-hundred-forty-fwths d i g i t is "6"
r,
.
-
Answers . to Exercises 1.
9-7
(a)
65.87
(e)
0.0026
(b)
436.19
(f)
0.30005
(c)
50.24
(€5)
(d)
300.04
0,483
[pages 362-3641
&) .10
a number equals the number,
1
(P) 4.
2%
(T) 5 .
Any ratlonal number may be w r i t t e n in decimal notation.
may be repreaented as
5.
(F) 6. If one-third of a c l a s s is g i r l s , the boya make up of it. (F) 7.
6@
The sum of three tenths and three hundredths is three thousandths.
(T) 8.
~f
5. =
60
~9
then x
= 12.
(T) 9 .
.O1 dlvided by .01 equals 1. (one over ten to the f i f t h power) may be written (P) 10. 1 10
(F) -11. If Bob weighs half as much
as h i s father, the r a t i o
showing the comparison of Bob's w e i g h t is 2 : l .
(T) 12.
8
wefght t o his father's
is another name for the number
3%.
( T ) 13. An increase i n the price of an i t e m from $20 to $28 is an increase of (F) 14.
4M.
If a c l a s s has a t o t a l of 32 p u p i l s , 20 of them boys, the number of boys i e 6M of the number of p u p i l s In the c l a s s .
(T) 15. Five percent of $ 7.54% of $100. ( P ) 16. 62.5% and
1
I s the same amount of money aa
are names f o r two different numbers.
IT. Matchinq Choose items from Column B which make the st:atements in Column A c o r r e c t . Place the letters of your choices from Column B in the spaces to the l e f t in Column A .
283 Column A b -
1.
Column B
The d i g i t g occupies the place I n the decimal numeral 3284 ,569
.
d
2.
and b are twonumbera and b f 0, the of a t o b is the quotient
If a
9.
- 3. a
ratlo
lo@
repeating
25 - 4
The decimal numeral 473 .45 rounded to the nearest t e n t h
25 4
is
473 . 4
The decimal numeral f o r the 1 r a t i o n a l number 8 is a
a block of six d i g i t s
-
.
propor'tlon
h k -
6. 2.54 cm. equals
e -
8. Another name for the number one
-
9.
millimeters.
The declmal numeral f o r the 2 repeats rational number 7
-
fs
m 10. -
percent
the d i g i t
decimal
7.
thousandths
473 05
9384.562 9 occupies the
-
5.
thousands
In the decimal numeral place.
1 4.
(a)
Isastatementofequality of two r a t i o s . A
...
The number 0.45E is times the number 0.09w
... .
5
I . Mu1 tlple Choice 1.
2.
SIX
$ 3 5 F is
(a)
$210.00
(d)
$2100.
(b)
$ 21 .OO
(e)
None of these
(c)
$
2.10
1.
If of t h e number answer is More than
30
(b)
More t h a n
3
(c)
More than
45
(d)
More than
500
(e)
None of these. of
but lesa than
90,
5.
b u t less than
but less than
450. 1000, 2.
c -
$320 -00 is computed, the answer is
(a)
$16
(a)
$1.60
(b)
$160
(e)
None of these
(c)
$3 .Po
3.
d
In a class of 42 p u p i l s there are 25 boys. The number of boys is what percent (nearest whole p e r c e n t ) of t h e number of p u p i l s ?
(5)
5%
(e)
None of these
(4 5W 5.
b
5400 is computed, the c o r r e c t
but less than
(a)
3. If
4.
p e r c e n t of
4.
a
In the c l a s s of 42 pupils there are 17 g i r l s . The number of g i r l s is what percent (rounded t o the n e a r e s t terlth of a percent ) of the number of p u p i l s ? (a)
40.q
(d)
40.7%
(e)
None of these
6. hMch of the following repeat
a single d i g i t when
written a s decimals?
7.
cornrnlseion. If he s e l l s worth of merchandise, he should receive An agent is paid
(a)
8750
(dl
Sb.75
(b)
$75
(e)
None of the answers is
correct
8.
.
10
7-
b -
4 of the 8 gamea already played. If it wlns the next two gamea, what percent of the games will it then have won? A ball club won
(a)
8W
(dl
5@
(b)
7M
(4
4M
(4 6@ g.
$1500
8.
c
~f Tomwas auccesaful In
13 o u t of 20 tries in p r a c t i c i n g free throws, which of the f o l l o w i n g represents h i s accuracy?
(a)
87%
(dl
2M
(b)
65%
(4
1%
(c
33%
The product of
046 x .3
Is
(a)
0.0138
(d)
13.8000
(b)
0.1380
(e)
138.0000
(c)
1.3800
10.
a
11.
12.
The quotient problem
1 . 4 is the answer to the division
(a)
3.6
(dl
.36
(b)
3.6
(4
36
(4
036
What is another way of w r i t i n g (a)
6.S
(c,
(b)
6%
(dl
t h a t is another way of writing
1
.*
d
13.
1
.065?
-6% ls?
14. John s e l l s magazines and may keep 2% he c o l l e c t s . If his sales are $3.50
(e)
11.
of' the money he may keep:
None of the answe~sis correct . 14. b
-
IV. Problems 1.
What commission wlll a real e s t a t e agent receive for s e l l i n g a house f o p $15,400 if his rate of commission is
5$?
m
2.
The sale p r i c e on a dresa i a $22.80 and the marked price showing on the price t a g I s $30.00. What was the rate of discount? 2!!E
3.
An increase in rent of
of the preaent rent will add $4.50 to the monthly rent that Mr. Johnson will pay. hhat Is the monthly rent that M r . Johnson now pays?
4.
3@
of the family Income f o r food. If the monthly income of the family is $423, what amount of money is allowed for food f o r the month? A family budget allows
$126.go 5.
Dorothy was 5 f e e t t a l l (to the nearest inch) ,,[hen school opened In September. In June her height was 5 feet 3 inches. What is the percent of increase in her height? 2%-
6. (a)
Write
4(1)
+ 3 (&) + 5 ( 3 )
in positional
4.305
notation. (b)
Write
0.12
in expanded fonn.
7. Find the decimal to the nearest 8.
(a)
Round
(b)
Round
(&)
+
$(2)
0.4
tenth f o r
14.657 t o the nearest t e n t h .
14.7
5.93
5.9349 t o t h e n e a r e s t hundredth.
21 . 9. Find the decimsl to the nearest hundredth f o r 1 3
1.62 -
10. A t a c e r t a i n time of the day a man 6 feet tall casts a shadow 8 feet long. A t the same time a nearby tree c a s t s a shadow of 40 feet. Row t a l l is the tree?
30 feet 11.
What is the greatest p o s s i b l e error in a measurement that is made to the nearest centimeter? one-ha1 f centimeter
12. How much l a the difference of the sum of 0.75 a n d t h e sumof 0.5 and 0.125? 1
1.05 and
The decimal numeral f o r the rational number
0.09m
...
1.175
111
is
14. Calculate the measure (in cm.) of the periroeter of a square whose edges measure 3 O cm. 120
-
15.
Calculate the area (in sq. cm.) of the I n t e r i o r of a square whose edges measure 30 cm. 900 sq. cm.
16.
C a l c u l a t e the volume (in cu. cm.) of the interior of a
cube whose edges measure
30 cm.
27,000
cu. em.
17. If Bobts shadow is 80 inchea long when Johnfs shadow is 120 inchea long, how t a l l is John if Bob is 30 inches tall?
-
45 in.
18. A cookie recipe using 1$ cups of flour w i l l y i e l d 48 cooklea. If Janet wante to bake 24 cookiea, how much flour should she use?
1
CHAPTER 10
PARALLELS, PARALLELOGRAMS, TRIANGLES, AND RIGHT PRISMS tl
Informal geometry" as presented i n this chapter, is concerned w i t h the discovery of geometric principles through experimentation and, where feasible, t h e verification of empirical conclusions by deductive reasoning. Students perform as scientists in collecting data and then perform a s mathematlciana in the analysis and interpretation of the data they have obtained. Data needed t o formulate a statement of a geometric property are obtained by measurement, with protractor or ruler, or by superimposing one figure on another. Pupils a r e Introduced to the use of deductive reasoning as a method f o r a s c e r t a i n i n g what is true about a geometric figure, arguing from p r e v i o u s l y stated p r i n c i p l e s and definitions. We reserve for a later tfme the systematic organization of geometry as a deductive system, starting with postulates and undefined terms, and developing theorems and definitions on t h i s basis. The s p e c i f l c purposes f o r which this chapter were planned are these: 1,
To develop awareness of the ideas of points, l i n e e , and planes, and their Intersections.
2.
To give the pupils experience in verification of experlmental r e s u l t s by informal deductive argument on the basis of p r e v i o u s l y stated p r i n c i p l e s .
3.
To introduce c e r t a i n geometric concepts and r e l a t i o n s as l i s t e d below.
The major topics are: 1.
Some angle relationships in the figure formed by parallel lines and t h e i r transversal.
2
The angle and side relationships in a triangle.
3.
The angle and side relationship6 in a parallelogram.
4.
Areas of parallelograms and triangles.
5.
Volumes of ri@t prisms.
Some Wneral Obaervationa and Sup;gestions As in other chapters, precise terminolow is emphasized throughout t h e t e x t material. It is necessary to make this emphasis because many of the words t h a t are casually used by eeventh graders are not as clearly understood by t h e i r uaers as we would like t o have them. A t thia l e v e l , the consequences of casual language are not always serious but may become ao as students proceed in t h e i r mathematical s t u d i e s . A l l of the terminology developed i n p ~ e v l o u achapters should be used whenever such usage clarifies and simplifies geometric statements but care should be exercised that in our attempts to be as exact as possible, we do not make a complicated thing out of what may be, eaaentially, a very simple idea. To avoid thie situation, it is suggested that meanings be given first in words of comon usage and then in the more precise terminologg. The translation from common uaage to precise uaage then becomee an exercise in analytical thinking. Ideally, once a word that is coannonly used is pre-empted for a special meaning in a new vocabularg, the new meaning must be adhered to from t h a t point on. In actual fact, however, it is often d i f f i c u l t to convince a aeventh grader that he should do thia. In t h i s caae we should accept, for the time being, his way of speaking, evaluate and discuss what ideas he is attempting to present, and then encourage him t o rephrase his statements in more precise words. Since students o f t e n learn b e s t by imitation and habft formation, it is suggested t h a t t h e teacher become thoroughly familiar with the new terminology and use it at every posaible occasion. Through the simultaneous use of both t h e common and the preciae ways of speaking it is hoped the student will become more and more proficient in the latter and grow to appreciate its value u n t i l he eventually uaes it as a matter of course.
Some particular terminology uaed in the t e x t may need further clarification. "~ntersectin the empty s e t ' ' and "are p a r a l l e l " as applied to lines have the same meaning and may be used i n t e r changeably where the linets are in the same plane. "Intersect in the empty a e t " or "are skewtt have the same meaning when applied to lines that are not in the same plane. The f r o n t edge of the ceiling and the side edge of the f l o o r are skew and have no p o i n t of Intersection. Which of these phrases t o use in a particular c o n t e x t depends on the conclusions one wishes to draw. If the question is, "What are all the possible intereectiona of two lines in space?'' one of the. possible intersections is "the empty s e t . " This i s a phrase that might seem preferable to "parallel lines have no intersection," On the other hand, if the question i s "HOW are t h e opposite sides o f a parallelogmm related?" then "they are parallel" might seem preferable to "they Intersect in the empty set." Use your own judgment in matters of t h i s kind. It may be noted that the authors a t times say "the lines do not intersect" even though thia is not s t r i c t l y so in the precise s e t terminology. We say "the lines may Intersect in the empty set." Since the phrase "do not intersect" i s 80 commonly used, it aeems desirable to use it here. It is a l s o thought t h a t there is some advantage in presenting aome I d e a s in more general terms, which s e t language permits. The chapter Includes a few deductive developments of a more or less informal nature. One of the problem8 arising in such a development is that pupils usually f a l l to appreciate the need for justifying statements with reasons prevloualy adjudged acceptable to the group as a whole. One proposal that might impress them wlth the fact t h a t o n l y previoualy stated and accepted properties, definitions, and reasons should be used is t o suggest that football and basketball games would be much more confusing if i n each game the mlea were changed without consulting anybody and that new rules be made up as the game goes along! It might be an interesting game but hardly a fair one!
An occasional reminder about "making up rules as you go along" is usually sufficient to make the p o i n t desired. Frequently students a r e asked t o m a k e a general statement about a property or the reaulta obtained through experiment. In the t e x t such statements are p a r t i a l l y written so t h a t the grammatical form of the statement I s suggested without h i n t i n g t o o strongly a t the mathematical i d e a s Involved. Before considering these, students should be encouraged to formulate their own statements of p r i n c i p l e s and propertlea but such statements should be very c l o s e l y examined t o ensure that the meaning is p r e c i s e and clear. When a statement seems s a t i s f a c t o r y to all, then show pupils the formulation in the answer section f o r comparison. They then may use these statements as models in f u t u r e work in this chapter. About a week p r i o r t o the i n t r o d u c t i o n of this chapter, pictures and a r t i c l e s p e r t a i n i n g to p l a n e geometry could be placed on the classroom b u l l e t i n board to arouse curiosity and supplement the h f s t o r i c a l facts b r i e f l y mentioned in the first s e c t i o n . Here are t w o puzzles t h a t w i l l add interest to the display: 1.
Drawlng a t r i a n g l e is easy but can you draw a t r i a n g l e so t h a t each of t h e dots lettered A, 8 , and C are midpoints of its sldes?
2.
A very t h r i f t y cabinet-
maker wished to construct a t a b l e t o p two feet square out of a piece of plywood shaped as in the figure. . He was able to do this with only two sawlngs, If you are as clever as the cabinetmaker, you can do t h e erne.
Pictures and geometric diagrams suitable f o r posting are not always easy to find but some good ones can be obtained from such magazines aq Scientific American , Fortune , Popular Scfence , Mechanlcq , J i f e , and, of course, aome technical magazines. A pattern of "oak tag" paper might be posted along with a completed model f o r the constmotion of a regular tetrahedron as an a c t i v i t y in equilateral triangles.
The following hints are given about applications for the purpose of motivation. These may be supplemented by material found in the introductory chapters of plane geometry t e x t s t h a t are available.
There is never an over-supply of good thlnicera. The world needs people who can begin w i t h a body of f a c t s , relate them, and think through to l o g i c a l conclusions. The study of geometry helps to t r a i n such p e o p l e . In t h e a i r c r a f t industry there is a g r e a t demand f o r workers trained In geometry because there is a comiderable amount of geometric howledge involved In the conatmotion of an airplane. The main problem 18 t o learn how air will flow about an airplane of given shape moving in a given direation at a given apeed. From t h i s the lifttng force and the air resistance may be calculated. The parallelogram of forces may be wed f o r an illustration. In order t o f i n d the single force equivalent to two forces acting simultaneously at a point we can draw a diagram like t h i s in which the given forces are represented in magnitude and direction by t h e segments AB and E. We complete the parallelogram, and the diagonal AD g l v e s the magnitude and direction of t h e r e s u l t a n t f o r c e . Geometry is a l s o used to A B determine t h e forces i n an e l e c t r o mametfc f i e l d , and why rubber i s elastic, and how an oll company shotald schedule its production. In the theory of relatfvity and in t h e deslgn of agricultural experiments completely different concepts of space are used. Today, the physfcist, the chemist, the b i o l o g i s t , the engineer, the economist, the psychologist, and t h e rnZlitary strategist use geometry in ways f a r removed from some of those w h i c h were not even discussed or dreamed of only f i f t e e n
-
&?7
years ago.
-
10-1. Vertical and Adjacent Angles Concepts to be developed:
--
1.
When the intersection of t w o lines i a not the empty set, four angles and f o u r half-planes are determined.
2.
'huo a n a e s which have a common ray and whose i n t e r i o r s have
no points in common are c a l l e d adjacent angles.
3.
Non-adjacent angles determined by the i n t e r s e c t i o n of two lines [in a single point] are called vertical a n d e s .
4.
When t w o l i n e s intersect In a p o i n t , they determine four different h a l f - p l a n e s . Two of these half-planes i n t e r s e c t i n the i n t e r i o r of one angle in a p a i r of vertical anglea. The i n t e r i o r of t h e other angle i n t h e p a i r of vertical angles I s the intersection of the remaining t w o half-planes.
5.
The angles in a pair of v e r t i c a l angles are ~ ~ W r U e n tt,h a t is, are equal In measure.
Suggest ions
Review very briefly the possible intersectiona of two l i n e s in space using t w o meter sticks, pointers, or pieces of c o a t hanger wire to represent a pair of lines intersecting i n space. If t h e intersection i s not the empty s e t , both wires should be grasped In one hand a t their p o i n t of intersection so t h a t the other hand i s free t o indicate parts of t h e figure. U s e this same device to suggest adjacent angles and vertical angles. By t h i s procedure, s t u d e n t s will be encouraged to think of these i d e a s in terms of "general" space and not j u s t t h a t p o r t i o n of space represented by t h e chalkboard o r paper. A l s o , i t l a often more convenient and more time-saving to carry a geometric f i g u r e to the ::tudents in t h i s way than t o have t h e students carry themselves to the same figure drawn on t h e chalkboard. Pupils may say " ~ n g l e A i s a v e r t i c a l angle". This i s not good terminology because we always refer t o vertical angles i n pairs. A similar remark holds f o r adjacent angles. " ~ d j a c e n t angles" should be used only in a phrase which includes some mention of a pair of angles. In Problem 5 ( e ) it is not necessary to formalize the subtraction axiom. The notions involved i n t h e deduction required should already be understood and accepted by the group from previous experience w l t h number sentences.
In p a r t ( e ) of Problem 6, the reason for m(L x ) = m(L z) should be developed as a statement of the equallty of two numbers and that this p a r t i c u l a r equality is true because m ( L x ) and m(L Z ) are "two names for the same number.'' Anawers to Questions in Section 10-1
.
Yes The half -plane containing point D Interaects the half-plane containing point C to form the Interior of L CAD. Yea. The half-plane containing B intersects the half-plane c o n t a i n i n g D to form the interior of angle BAD.
Answers to Exercises 10-1 1.
(b)
Lm, L m L Ja,L m L J K M r LLm L= L JKL and L MKN
(c)
Two pairs of vertlcal angles are formed.
(8)
(b) (c)
2-
3
4.
(a)
(a)
40
=
m(L JKM)
m(L W )
40 =
m(l
LKN)
=
m(L
JKL)
(b)
140 =
(c)
The angles of a pair of v e ~ t i c a langles seem to have equal measures
(d)
They do appear equal.
140
.
Property 1: When two lines intersect, the two angles in each pair of vertical angles which are formed have equal measure, or are congruent.
180
(c)
120,
120
180
(d)
110,
110
sun of m(L JKM) and rn(F JKL) 180 = sum of m(L JKM) and m(L NKM) Then m(L X ) = m(L NKM) since they b o t h must be names f o r the same number. 180
=
Subtract m(L y) from 180.
rn(L X) m ( L z)
=
180
=
180
m(L
= m
X)
from
180.
Subtract
m(L z >
- rn(L y ) ;
- m(L (L 2)
y)
Y e s , two l i n e s which are skew. The edge of the floor at the f r o n t of the room and the edge of the ceiling at one side.
---
-
10-2. Three Lines in g Plane Concepts to be developed: 1.
A l i n e which intersects two or more lines in d i s t i n c t points
is c a l l e d a transversal.
2.
In any figure consisting of l i n e s whose I n t e r s e c t i o n s a r e not empty, there are a t l e a s t two pairs of v e r t i c a l angles and at l e a s t four pairs of adjacent angles.
3.
Two angles t h e sum of whose measures is
180 degrees are
called supplementary angles. 4.
Supplementary angles are not necessarily adjacent.
5.
Two d l f f e r e n t angles whose interiors l i e on the same s i d e of a transversal such t h a t a complete r a y of one is contained in a r a y of the other, are c a l l e d c o ~ r e s p o n d i n qangles. [pages 402-403, 4041
It will prove helpful to make a demonatration model constructed of three f l a t sticks free to turn on pivoted connections made with rubber bands, small round-head b o l t a , or of clinched tacks or nails.
On the chalkboard represent parallel lfnea In other than hor3zontal positions at first. Otherwise, studenta get the habit of t h i n k i n g of parallel lines as being horlzontal and have difficulty seeing them in complicated figures where t h e parallel l i n e s are not h o r i z o n t a l . Colored chalk lends interest to a chalkboard drawlng and I s e f f e c t i v e in calling special a t t e n t i o n to certain portions of a figure. Pupila should draw figures according to the directions i n the exercises even though the figures appear in the t e x t . This activity helps to f i x t h e features of a drawing i n their minds. Students! drawings frequently are t o o small. In the exercises of thla s e c t i o n lines should be represented by segments about two to three inches long. Free-hand sketches are acceptable if done carefully. Problem 6 of the exercises I s of great importance and by no means ahould be omitted. The basic definition included in this problem requires t h a t it be done by a l l pupils.
-
Answers t o Questions in S e c t i o n 10-2. Yes;
yes;
no.
1
1.
2.
3.
I 4.
5.
(a)
Pour pairs
(c)
Yes
(d)
No. Since tl and t2 are parallel they do not Intersect. To be a transversal of t2 and tj, line tl must I n t e r s e c t both l l n e a .
(a)
Trlangle
(b)
Six pairs
(c)
Twelve pairs
(a)
Line t is the transveraal and intersects A, I and J 2 . If the lines are extended any l i n e may be considered as a transversal of the other two.
(b)
Blght angles; twelve if
(c)
Four pairs; six p a i r s if
(d)
The angles in any pair of v e r t i c a l angles have equal measures.
(e)
Yea, because
(f)
No, adjacent angles may or may not be congruent.
(a)
E i g h t pairs; 12 If you consider the intersection of and A*.
(b)
Sum of measures is
(a)
Yes
(b)
No
(c)
Yes
R Q
either
.
S
Lh
and
and J 2
1
Lf
and
intersect
.
e2 intersect.
are vertical angles.
180.
s t r a i g h t nor reflex angle has been
def lned )
(dl
m u g)
(e)
Yes
=
100
m(L e )
(f) Yes [pages 405-407]
=
100
m(L r)
=
80
6.
*7.
L g
(a)
L d
(b)
Yes
(4
No
(d)
Four pairs
and
(4
NO
(f) The measure of each angle
go.
is
Yes
( a ) Each shows two line8 with their transversal. h here are others,) (b)
The figures differ in the number of pairs of lines whose intersection consists of one p o i n t . h here are others. )
(c)
A l l three l i n e s may be p a r a l l e l .
(d)
The intersections of three d i f f e r e n t lines in the same plane may consist of 0 31 2 or 3 points. -9
*8.
J
With three planes in space the cases a r e : ( a ) A 1 1 pass t h r o u g h a p o i n t .
h he
corner where t w o
walls and t h e c e i l i n g meet. ) (b)
Two planes parallel b u t both intersected by the t h i r d ( f l o o r , c e i l i n g and one wall).
(c)
Each p a i r of planes intersect but their l i n e s of intersection a r e parallel. (1n one type of house, t h e planes of the roof and the plane of the attic floor. )
(d)
Three parallel planes (a stack of shelves).
10-3. Parallel Lines and correspond in^ Angles Concepts to be developed:
--
1. When in the same plane a t r a n s v e r s a l intersects t w o l i n e s and t h e corresponding angles have unequal measures, then the t w o l i n e s will i n t e r s e c t .
[pages 407-4091
301 2.
When in the aame plane a transversal i n t e r s e c t s two lines and the corresponding angles have equal measures, then the two lines do not I n t e r s e c t .
The emphasis should be on the chlldrents discovering the
relationships by t h e i r own observations. Be careful not to "kill" their interest by telling them t h e answers. Ask l e a d i n g questions and t r y to draw t h e answers out of the atudents. I f class time i s short, t h e experiments In this s e c t i o n may be speeded up if each student is assigned the measuring f o r two cases and then reports his findings for t a b u l a t i o n on the chalkboard. It m i g h t be well to carry out other experiments in t h i s same manner. If an overhead projector is available, by a l l means use it for this section. O n one p i e c e of plastic f i l m , draw with chinamarking pencil l i n e r2 and transversal t. On another piece draw l i n e rl. By superimposing the second piece on the f i r s t and projecting the Image on the chalkboard, l i n e rl can be rotated to d i f f e r e n t positions through Point A and observations as to angle measures and intersections can be tabulated on the chalkboard. Since the figure is projected on the chalkboard, marks may be erased without obliterating the figure.
i
Answers to Class Exercises and Discussion 10-3 1.
Y e s , they i n t e r s e c t on t h e l e f t side of
2.
Yes,
3.
If the Third column, t h i r d e n t r y is "the empty set," corresponding angles are unequal, rl and r2 Intersect in a point-to the right or left of the transversal t. If the corresponding angles are equal, rl and r2 are parallel.
rl and
r2
t.
intersect on the right s i d e of
[pages 409-8101
t.
Intersection of
rl
and
r2
ri&t side
empty s e t left side
5.
measures,
6.
parallel
Intersect
Answers to Exercises 10-3. 1.
r
ntersectlon of
I 1
and
2
(a)
below
(b)
parallel
(c)
above
(d)
below
(e)
parallel
(f)
above
(g)
be1 ow
(h)
parallel
(i)
above
'
[pagee 410-4111
!
303 2.
Aa
3.
90.
in classroom (opposite edges of ceiling, e t c .). Minimum distance should occur when the transversal
is perpendicular to the l i n e s . (This idea is mark f u l l y developed in Section 9-7.)
10-4. Converses urnin in^ a Statement Around) Concepts to be developed:
--
then" sentence may be formed by 1. A converse of an " i f interchanging the " i f t ' part and the "then1' part.
2.
of a statement may be t m e or false regardless of whether the original statement is true or f a l s e . A converse
Caution students that the meanings of words used in the sample statements of the t e x t are understood to be t h o s e in t h e widest use. "Mary and Sue" are not names f o r ships! Care must be taken not to refer to a converse as the converse because for any one statement there may be several d i f f e r e n t converaea, To form a converse i t is only necessary to exchange one fact of the "If1' clause with one fact of the "then" clause and where these clauses contain more t h a n one fact, there will be several converses. It is felt that the " t u r n i n g around" device is as far as one needs to go f o r seventh graders. It might prove helpful t o clarify the meaning of "true" or "false" before tackling the exercises. Without going i n t o d e t a i l s of logic, it should be sufficient to propose that a "falsen statement is one that Is not always true and a statement is not considered true if at l e a s t one counter-example can be found. For instance, l e t us examine the statement: "The set o f whole numbers is closed under subtraction.'l As a counter-example one might say, '"There I s no whole number which added to five gives three and, therefore, ( 3 5 ) is not a name for a whole number." This one counter-example is a l l t h a t is needed to deny the q u o t a t i o n . Emphasize t h a t only one counter-example is needed to prove a statement f a l s e .
-
Whlle one counter-example can be used to show that a statement is Palae, I t is a great d e a l more d i f f i c u l t to show that a statement is t r u e . In thia s e c t i o n we do n o t expect that the students or teacher will p o v e t h a t statements are t r u e . In Section 6 a geometric proof is given for the property of the sum of the angles of a t r i a n g l e , and it is suggested In t h e exercises that students try t o discover proofs fop several other p r o p e r t i e s . In t h i s section only inductive arguments can be given to i n d i c a t e that a statement is true. Students should be reminded t h a t even though thousands of cases were investigated and a l l wepe in agreement w i t h the statement or property, still the property would not necessarily be t r u e . Also, f a i l u r e t o f i n d a counter-example w i l l not constitute a proof, since it is always conceivable that someone else might find such an example. In "if -- then" statements, the "if" clause is considered to be the postulate or postulates. In mathematics it is recognized t h a t even a formal proof i a based on postulates and that the property to be proved is true only if the postulates used in t h e proof are true. Mathematicians do n o t recognize any absolute t r u t h in mathematics. A mathematical agreement, baaed on p o s t u l a t e s which may not necessarily even be in agreement with experience, is considered a proof, provided deduction is used c o r r e c t l y . Again, it I s recognized t h a t the v a l i d i t y of t h e proof does not determine the t r u t h of the property. The t r u t h of a property 1s completely dependent on the t r u t h of the p o ~ t u l a t e s .
-
Answers t o Questions in S e c t i o n 10-4 (b)
Not always
1,
Yes;
No
2.
Yea;
Yes
3
Statement "a" Statement " b"
is true. is false
.
[pages 412-4131
Answers to Exercises 10-4 1.
One of many examples is:
No. 2.
3
(a)
false
(d)
true (if no amputees)
(b 1
true
(e)
true
(c
true
(f) t r u e
(a)
If Blackie is a cocker-spaniel, then Blackie is a dog,
(true)
If it is a holiday in t h e U . S . , then it is J u l y 4th. ( f a l s e ) I f Robert is t h e tallest boy i n h i s class, then he is t h e t a l l e s t boy i n h i s school. ( f a l s e )
If an animal has four l e g s , t h e n t h e animal is a horse, ( f a l s e )
If an -animal has thick f u r , then the animal is a bear. (false) If Mark is Susan's brother, then Susan i s Mark's sister. ( t r u e ) true
(c)
false
(e) fqlse
true
(d)
true
(f) true
If two lines are parallel and are intersected by a transversal, a p a i r of corresponding angles a r e
congruent
.
The angles are congruent.
similar results
.
True
[pages 414-4151
Classmates should have
6.
If in a plane, a transversal intersects t w o lines and the two lines intersect in a point, then the angles in a pair of correspondfng angles formed ape unequal in measure. (true)
7.
Yes. If a figure is a simple closed curve, then I t is a c i r c l e . (false) If a figure is a c i r c l e , then I t I s a simple closed curve . ( true )
10-5. Triangles Concepts to be developed:
accordance with a common property.
1.
A set is determined in
2.
There are three set8 of triangles determined according to the measures of their s i d e s .
( a ) The s e t of isosceles triangles has as members triangles which have two sides t h a t are equal in length. (b)
The s e t of scalene triangles is the set of triangles in which no t w o s i d e s have the same measure.
(c)
The set of equilateral triangles I s the set of triangles which have three sides equal in length.
3
An angle and a side of a triangle are said to be opposite each o t h e r if t h e i r intersection contains duet the endpoints of the segment referred to as side.
4.
If two sides of a triangle are equal in length, the angles opposite these sides have equal measures.
5.
I f t w o anglea of a triangle have equal meaaures, then the sides opposite these angles are equal in length.
,Suggestions '
Remind students that a s e t should be "well-defined." There must be no doubt or question a8 to what l a or l a not a member of a set. This does not mean, however, t h a t a definition of s e t should be attempted. The s e t or beautiful paintings in the world 1s not a "well-defined" set because what l a beautiful i s
c o n t r o v e r s i a l . The set of triangles is not a "well-defined" set if I t is not clear whether spherical triangles are to be included. In o u r t e x t , s i n c e all trlangles are understood t o be plane f i g u r e s the s e t of t r i a n g l e s is c l e a r l y d e f i n e d . In other words a s e t is well-defined if and only if there is no doubt as to whether or not a t h i n g belongs i n the set. This idea is important In defining the s e t of isosceles triangles and emphasizes t h e need for precise and complete terminology. Students should t r y t o answer q u e s t i o n s in the t e x t as they reach them and not read ahead f o r the answers. Answers are Included in the text so t h a t , if desired, a l l or a p o r t i o n of the text may be assigned f o r reading outside of c l a s s . Have a number of soda straws measured and creased before c l a s s begins. This exercise may seem easy to most pupils but even the brighter ones will jump t o incorrect conclusions about a straw d i v i d e d into three pieces of 2", 3'' , and 5" . Sodastraw figures g i v e t h e c l a s s a n opportunity t o handle triangles i n regions of space other t h a n the chalkboard or the drawing paper. Make c e r t a i n that the soda-straw figures are understood t o be merely representations of t r i a n g l e s and are not actual triangles.
Answers to Questions In Section 10-5
Figures (b) and (f) are triangles. (a)
Segment joining two p o i n t s is missing.
( c ) and ( d ) (e)
Four points instead of three.
Two p o i n t s joined by a curve which is not a s t r a i g h t l i n e segnent.
[pages 417-419 3
Common property: Common property:
l e a s t two s i d e s are equal in length, A 1 1 three sides have equal measures No; no; no At
Soda-straw figures. (a) scalene (b)
e q u i l a t e r a l and isosceles
(c)
isosceles
(d)
isosceles
(e)
equilateral and isosceles
Concerning first drawing (Triangle ABC): Yes, because A and C are the only polnts shared by angle B and aide No, because points A and C are not the only p o i n t s shared by angle C and E . Angle C and a i d e K, because p o f n t s A and B are the only p o i n t s shared by angle C and s i d e E. Concerning second drawing (~riangle ABC ) : Isosceles, because at l e a s t t w o sides have equal measures. They are equal . The measures of at least two angles in each triangle are
z,
.
equal Property 3: If two sides of a triangle are equal in length, then the angles opposite these sides have equal measures. Concerning third drawing (Tpiangle ABE ) : They are equal. The sides oppoalte the equal anglee are equal. Isosceles triangles, If two a n g l e s of a triangle are equal in measure, then the sides opposite these angles are equal in length. Yes. They are converses.
[pages 420-421
Answers to Claas Discussion Problema 10-5 1.
Yea.
Advantagee:
(1) Useful when only angle measures are known.
(2) Ruler may not be available. Disadvantages : (1) Protractors not as generally available as rulers. (2)
Angle measure l a often more difficult to estimate
than linear measure. 2.
1.
LA)
=
~(LB)
1.
Given
=
m(E)
2.
Converse of Prope~ty3
3.
Given
4.
Converse of Property 3
5.
A l l names for the same
2.
m(E)
3.
~ ( L c =) ~ ( L B ) m(G) = m(E) = m(E) m(s)
4.
5.
m(E)
=
number
.
Answers to Exercises 10-5
4.
Not necessarily, because an equilateral t r i a n g l e must have a l l three sides equal and an isoaceles t r i a n g l e does not satisfy thls requirement.
5.
Yes, becauae an equilateral triangle does satisfy the requirements f o r being isosceles by having two sides of the aame length. "only two aides equal In length" is not part of the requirement for isosceles t r i a n g l e .
-
[pages 422-4231
310
6.
P
opposite
Q opposite R
7.
opposite
PR
PQ
( a ) No. The outside two s e c t i o n s , even if h e l d end t o end in a straight line, would just reach from one end of the 3-inch section to the other. Thus no t r i a n g l e would be formed. (b)
No. If the end piecea were f o l d e d toward each other, they would not meet.
(c)
The sum of the measures of any two sides of a triangle must be greater than the measure of the t h i r d side.
10-6. Angles of a Trian~le Concepts to be developed:
--
1.
The sum of the measures of the angles of a triangle in a plane is 180.
2
A property may be suggested by inductive reasoning but
it is
proved by deductive reasoning.
3
The proof of P r o p e r t y 4 applies to any triangle in a plane.
Suggestions Since the c l a s s exercises require the f o l d i n g and c u t t i n g of triangular shapes, much class time may be saved by supplying the c l a s s with a q u a n t i t y of such shapes already c u t out Instead of spending time in t h e d i s t r i b u t i o n and collecting of scissors. By using a paper cutter and cutting several piecea of paper at a time, a sufficient supply can be e a s i l y obtained. Triangles should be fairly l a r g e and as d i f f e r e n t as possible. [ pages 4 23-4 24 ]
311 Whether the mounting and pasting suggested in the t e x t is 'done in c l a m or not is optional. I n f a c t , t h f s p a r t i c u l a r a c t i v i t y coul-d b e a good "lesson clincher" if made a part of the homewopk assignment. However, the tearing and f o l d i n g f o r the purposes of experiment should be done In class as suggested. The angle sum for a triangle is f i r s t developed experimentally and then proved deductively. To some pupils it may seem a i l l y to prove deductively what is perfectly obvious i n t u f t i v e l y One way to meet this c r i t i c i s m I s to shake these s t u d e n t s ' faith in the lnfalllbllity of t h e i r i n t u i t i o n . Here a r e a few o p t i c a l i l l u s i o n s , quickly and e a s i l y drawn on the chalkboard, t h a t m i g h t do the trick:
.
and have the same length. and A;, do n o t bulge a p a r t . The t o p hat i a as wide as I t is t a l l . In this paragraph a r e mentioned other ways helpful in emphasizing the need f o p v e r i f i c a t i o n of "obvioustt or intuitive conclusions. Test students' ability to distinguish between two sharp points by touch alone ( t r y this on f i n g e r tips first and then on the back of the neck), or ask what t h e b i l l would be f o r , shoeing a horae at one penny f o r the f i r s t nail, two f o r the second, four f o r the t h i r d , eight f o r the f o u r t h , e t c . f o r f o u r shoes w i t h eight nails t o the shoe. Tell about t h e n a t i v e of China who f o r seventy years, uponawakening in the morning, noted t h a t the f i r s t person he saw was another Chinese. On t h e eve of his seventy-firat birthday, after having made the same obaervatian
approximately 25,915 times, he went t o sleep certain in the knowledge that the f i r s t person he would see in the morning would be another Chinese. It was a Russian! Spend some time with the class discussing why the deductive proof applies to a l l planar triangles and why the conclusions drawn from empirical data In Exercises 1 and 2 apply only t o the particular triangles t e s t e d . T t is recognized that Problem 4 of the classroom exercises w i l l be d i f f i c u l t , particularly the attempt to decide on a legitimate reason f o r each s t e p . Thie l e the only example of a formal deductive proof given in the chapter, howaver, an8 it ie hoped that the teacher nil1 go through it w l t h the claes, though not expecting real masterg. It might be helpful t o r e o a l l f o r the clarrs t h e idea of "making up the rules after the game ha8 rrtarted" a8 suggested In the seoond paragraph, Page 4 of t h i ~connnentaq.
Answers to Question8 in Section 10-6
1.
(a)
180, yes
(b)
The sum of the measures of is
2.
(e)
L
2,
and
L BAC
180.
the sum appears to be
(f) Yes 3.
1,
180
(8)
Yes
(a)
Yes, because of Property
(b )
Copresponding angles 2a.
(c)
Vertical angles.
(d)
Were drawn so a a to have equal meaaures
(e)
The measures in the sum on the left are equal t o the measuree in the sum on the r i g h t .
.
Yea.
(f) By definition of "sum."
2a.
Yes
.
Converse of Property
Property 1.
.
313 ( g ) Angles on one side of a line. (h) Two names f o r the same number a s indicated In steps ( e ) and ( g)
(I) The t r a n s l a t i o n of the symbols in step ( h ) i n t o words.
Answers to Exercises 10-6.
1.
60
2.
(a)
60
(c)
5
(b) 30
(dl
20
65 and 65,
3.
Yes, e i t h e r
4.
~ ( L A B c=) ~ ( L D E F ) Property
5.
(a)
80
Property
or
50 and
80.
4.
4
(d)
60
Property
3
(e)
4
Property
4
and
4
-I
6.
(b)
91
Property
4
(c)
40
Property
3
(a) Yes, Property
1
Yes, Property
2
(b) (c)
and converse Property 3 .
or converse of Property
2a
I.
m(Ly)
=
m(Ln)
1.
Property
2.
m(L y)
=
m(L
2.
Property 2 or converse of Property 2a
3.
Two names f o r t h e
U)
same
[pages 428-4301
1
number,
m(L y ) .
Assignletters to other angles a f the f l g u r e t h a n these to which letters have already been assigned.
mILg)
1.
As drawn
is parallel
2.
Property
3.
P r o p e r t y of Problem 6
4.
Angles on one side of a l i n e .
1.
m(Lgr) =
2.
line
r
3
miLb)
4.
m u g ' )
m(L
C )
=m(La)
+m(La) + = 180
2a
5.
rn(Lgl) + m ( L a ) + m ( L c ) = m ( L g ) + m ( L b ) + m ( L c ) Each measure In the left hand sum I s a measure in the right hand sum.
6.
m(L
g)
+ m(Lb) + m(L
c ) = 180
Two names for the same number.
1
1 1
10-7.
Parallelo~ams Concepts to be developed: 1.
The shortest segment from a point A to a l i n e r lles on the line through point A perpendicular to l i n e r. The length of this s e p e n t I s c a l l e d the distance from pofnt A
to l i n e 2.
r.
If and A2 are parallel lines, then the distance from is the same as t h e distance from any a point on A 1 to other point on kl to A 2 .
x2
3.
In a plane a line perpendicular to one of two parallel l i n e s I s perpendicular t o the other also.
4.
The distance between two p a r a l l e l l i n e s is the length of any segment of a line perpendicular to both glven lines and
having an endpoint on each.
5.
A polygon is a simple closed curve which I s the union of l i n e
segments.
6.
A quadrilateral
7.
Opposite sides of a quadrilateral are those which have no point in common. (They do not intersect.)
8.
A parallelogram is
is a polygon w i t h four aides. a polygon with five sides.
A pentagon I s
a quadrilateral whose opposite s i d e s are
parallel.
1 !
9. The opposite sides of 10.
a parallelogram are congruent.
The i n t e r f o r of a parallelogram may be separated by a
diagonal into two triangular regions which are equal i n area and have t h e same shape.
1
i
!
Sumestions. It is intended that as much as possible of the development be done in c l a s s . Note t h a t the class material a c t u a l l y includes Problems 1 and 2 of the Exercises 10-7. Because in developing Property 5 the drawing of parallelograms by the c l a s s would take up a large amount of time, it would
be advisable to have prepared an ample supply of parallelogram
cut-outs c u t on t h e paper c u t t e r . It would be possible to use these same cut-outs f o r Problem 5 of the exercises.
m(L AC
140, m(L TCA) = 90, m(L TBA) = 30, is the shortest. No segments shorter than AC, The s h o r t e s t segment from a p o i n t A to a p o i n t of a line r the segnent from A which is perpendicular t o r Y e s . All a r e right angles because of Properties 2, 2a, 3 and 4. TDA)
=
is
.
Yes, The intersection of opposite sides of a quadrilateral is the empty set. o p p o s i t e DC, opposite BC f o r both quadrilateral and parallelogram. The opposite sides are c o n g r u e n t . Y e s , Yes. Y e s , t h e area of the r e g i o n bounded by one of the triangles is e q u a l to one-half the area of the reglon bounded by the parallelogram because the t r i a n g u l a r region is one of two equal parts of the r e g i o n bounded by the parallelogram.
AD
Answers to Exercises 10-7 1.
A s in classroom
2.
As in
classroom
[pages 431-4363
5.
The triangular pieces are equal in area. A diagonal sepapates the region of a parallelogram into two triangular regions which are equal in area and have the same shape. (congruent )
6.
Yes, in a plane, lines perpendicular to one of two parallel l i n e s are perpendicular to the o t h e r .
.
(a)
m(G)
=
(b)
m(E)
=
(c)
One p a i r of opposite e l d e s may be equal in length. The remaining pair must be unequal in length.
(a)
If, in a plane, a line is perpendicular to one of several parallel l i n e s , then it i a perpendicular to the others.
+8.
6, m(K) 12,
m(E)
=
3
=
4
--
Let 1. 2.
3.
5. 6.
t
L2
be any l i n e p a r a l l e l to
II.
is perpendicular to
1.
Aa drawn
and intersects /I
2.
Perpendiculars form
m(L a)
=
90
r i g h t angles
drawn Property 2a Two names for the aame number m ( L b ) = 90 6. If two l i n e s interTherefore, t is perpens e c t t o form a right dicular to d l angle, they are
4
As
parallel J2
perpendicular,
[pages 436-437 1
*9.
(a)
A J O F contains A J I D , A J H E , D I H E , D I C) F, and E H G F. B M E K contalns B L I K and L 13 E I. I G C E is t h e remaining parallelogram in thla lbsting.
(b)
E,
- EH,
(c)
LI, ME.
(d)
,
FG.
-
EC.
-
10-8. Areas of Pa~allelogramaand Triangles Concepts to be developed:
--
1.
Angles of a parallelogram at opposite vertices are congruent.
2.
Consecutive angles of a parallelogram are supplementary
angles.
3.
If one angle of a parallelogram is a right angle, the others must be also.
4.
A rectangle is a parallelogram w i t h one right angle.
5.
The area of a parallelggram may be determined, given the l e n g t h of a base and the length of the altitude t o that base.
6.
Any side
7.
The area of a triangle rnay be considered as half of the area of a parallelogram whose base is the base of the triangle and whose altitude is the altitude of the triangle t o that base.
8.
Any s i d e of a triangle rnay
of a parallelogram may be considered its base.
be considered as a base, and for
each base there l a a corresponding a l t i t u d e . Sugge st ions
Chalkboard drawings of triangles and parallelogcame should be so made t h a t rarely do the altitudes discussed l i e on a v e r t i c a l line. Every e f f o r t should be made to have pupils r e a l i z e that the altitudes of figures do not always extend in a vertical direction. It will also be evident t h a t baaea do not
319
neceesarily l i e on horizontal lines. Closely related to this Is the fact that any one of three altitudes and t h e i r corresponding bases may be used t o find the area of a scalene t r f a n g l e and t h a t there are, except for a rhombua or a square, two ways to calculate the area of a parallelogram. Consult the answer s e c t i o n f o r comments on t h e development of exercises in the t e x t . Particularly n o t e t h a t a parallelogram with one rlght angle is a rectangle. If time permits, thla m i g h t be a good jumping off place f o r a discussion of what is "necessary and sufficient" In a d e f i n i t i o n and perhaps why in some statements the "if and only if" phraseology is used.
Answers to Questions in Section 10-8
The oppoelte sidea have the same measure.
(E, DC)
;
(AD,
L ABC, L E D ,
BC)
CDA.
There are four.
There are four transversals. m(L A ) + m(L B ) = 180 The angles of a parallelogram a t two consecutive vertices are The angles of a parallelogram at t w o opposite vertices are
If L A f a a r i g h t angle, all of the angles of the parallelogram are right angles. Y e s , it is a rectangle. If L A and L B are not r i g h t angles and L B i s a n a c u t e angle, then L A must be obtuse since m(L A ) + m(L B) = 180. Ql lies on extended because, as our figuroe shows, L A, which is an angle of h AQD, is congruent to the angle a t B + determined by BC and extension of AB. D Q QlC is a parallelogram. L D and L DQQt are consecutive angles and, therefore, supplementary. Since rn(L DQQt) = 90, then m(L D) = 90. m(E) = rn(&&l) because each represents the sum of measures
-
I 1
respectively equal
.
[pages 438-4431
The number of aquare units of area of a pa~allelogram I s the product of the number of linear units in the base and the qumber of linear units I n the altitude to the base. The number of square units in the area of a triangle is onehalf the product of the number of linear units I n the base and the number of l i n e a r units In the a l t i t u d e to this base.
-
-
Answers to,Exercises 10-8 1.
2.
(a) 2
sq. ft.
(d)
72
(b)
6 aq.yd.
(e)
105
(c)
80 sq. cm.
(a) 10 sq. in.
(d)
136 sq. in.
sq. cm.
(e)
( b ) 32
4.
7.
(c)
2'8 a q . yd
(a)
15,000 s q . ft.
(c)
11,250
(a)
AB
65
sq. ft. sq. m.
sq. ft.
sq. ft.
5"
DS
=
ABCD
2I' 10 sq. in.
23' Area of ABCD
4" 10 sq. In.
FJ
Area of
*
(b)
AD
(c )
Yes, they do agree
RB
#
.
[pages 443-445]
8,
It may be a trapezoid or a f i g u r e having no opposite sides p a r a l l e l .
(a)
No.
(b)
No.
By definition, a rectangle is a special kind of parallelogram whose angles all have a measure
of (c)
90.
has two pairs of parallel sides. T h i s is all t h a t is needed t o define a paralleloA square
Yes.
gram.
9.
11.
(d )
Y e s . Because parallelograms, rectangles, and squares are a l l four-sided (quadrilateral) figures.
(a)
AB
(b)
CB a
(c)
AC
(d)
They do agree, but are not the same.
(a)
b
Ib)
h
(4 (4
b x
(e)
A = (b
A
CD
1;"
q1
AF
2
221
BF
r~
=
'J
I'
A
=
$
sq.in.
A
a
G$
sq. in.
A
sq. in.
+x
+
= bh -t
x)h, xh
and by the dist~ibutiveproperty,
xh
(f) A = ?
[pages 445-4483
&RST = A r e a of QUSV less the area of and QW. The area of RUS and QTV = xh T + xh Area of QRST (bh + xh) xh, A bh
(h) Area of
--
10-9,
-
-
RUS
xh.
R i a t Prisms
--
Concep$s to be developed: 1.
D e f i n i t i o n of a r i g h t prism as a figure obtained from two polygons of same size and shape l y i n g in parallel planes.
2.
Meanings of terms edge, face, vertex, base, height, as a p p l i e d to a r l g h t prism.
3.
Method of o b t a i n i n g volume of a r i g h t prism from the height and the area of the base.
Su@;glestions S e c t i o n 10-9 has been written from a q u i t e i n t u i t i v e point of vfew. In particular, no explicit discussion has been given of t h e concepts of l i n e s perpendlcular t o planes or of perpendicular planes, although both are s t r i c t l y involved in the idea of a right prism. One p o i n t should be brought out clearly by the teacher in classroom d l a c u s s l o n . In the case of a right prism, if stood on one base, the upper base i a "directly above" the lower base. In f a c t , if we imagine t h e prism in this position s l i c e d horizontally (that is, parallel to the planes of the bases) then each such c u t is d i r e c t l y above the lower base. Thus, the interior of the prism can be thought of as a series of layers or slabs p i l e d v e r t i c a l l y on each other. The fact that such a prism, modeled in wood, can be decomposed by simply c u t t i n g it into slabs of wood of the same size and shape and piling them on each other, is t a c i t l y used in the dlscusaion of the volume of the interior of a r i g h t prism. For there it is taken for granted that If we construct a s l a b made of cubes (or parts of oubea) j u s t covering the base, then the top of this slab also just fits inside the priam. It was f e l t that t h i s fact could better be emphasized by
the teacher In c l a s s discussion than to try to develop it In detail in the student + 6 manual
.
Answers t o Exercises 10-9 1.
2.
(a)
64 cubic feet
(b)
18 cubic cm.
(c)
30
(a)
c u b i c in.
30
cubic feet
(b)
640 cubic metera
(c)
126 cublc in.
3.
135 square feet f o r each column.
4.
(a) 8+
3
square feet.
(b)
575
square feet.
(c)
37$
cubic f e e t .
5.
V = B h
6.
21
7.
180 cubic inches.
8.
I.$ cubic
10.
square inches.
Shape of base unknown.
feet.
m e a f Prism
I
Faces
Vertices
Triangular
5
6
Pentagonal
7
10
Hexagonal
8
12
Octagonal
10
16
-
Sample Questions For Chapter 10
This is not a c h a p t e r test. Teachers should c a r e f u l l y select items from t h e following and prepare items of t h e i r own in making up a chanter t e s t .
F
1.
If one angle of an isosceles triangle has a measurement of 66*, one of the other two angles m u s t have a measurement of 66O.
T
2
A statement may be
T
3.
P a l r s of corresponding congruent angles are formed when a transversal intersects two parallel lines.
T
4.
A statement and I t s converse may b o t h be true.
F
5.
The intersection set of three l i n e s in a plane must be three points.
T
6.
If a t r i a n g l e has two s i d e s which are congruent, then it has two angles which are congruent.
F
7.
All isosceles t r i a n g l e s have the same shape regardless of size.
T
8.
The sum of the degree measures of the three angles of a triangle is equal to 180.
F
9.
An equilateral triangle is a l s o a scalene t r i a n g l e .
F
10.
The converse of a false statement is always false.
F
11.
If a t r i a n g l e has only t w o sides congruent, it can have three angles congruent.
T
12.
An equilateral triangle is also an
T
1
In the f i g u r e at the r i g h t , A, B, and C are symbols f o r the vertices of the triangle.
T
14.
true while i t s converse is false.
fs08CeleS
A
A l l parallelog~ama are quadrflaterals.
triangle.
In the four-sided figure at the r i g h t , if m ( E ) = m(@) and if and are parallel, then the flgure i a
AD
a parallelogram
.
A triangle may never include two angles whose measures
are
90.
If four sides of a parallelogram are congruent, then the figure is always a square
.
In the figure at the r i g h t , if and are parallel, then WXYZ is a parallelogram. W/
w' / / 7
It is possible to draw a triangle whoae sides measure 4 inches, 2 inches, and 1 i n c h . The flgure shown at the r i g h t conslats of two p a i r s of parallel iinea. Use the f i g u r e in marking the two f o l l o a n g ~ t a t e ments true or f a l s e . b
"'t
22.
If
rn(Lw() =
m(L8) then a l l t h e angles s h o w n i n
the f i g u r e are congruent. 23.
The triangles shown below all have t h e same area.
24.
If one s i d e of an i s o s c e l e s t r f a n g l e has a measure of 8, one of the o t h e r two sides must have a measure of
8.
Corresponding angles have i n t e r i o r s on the same side of the transversal.
If one of a pair of v e r t i c a l angles measures 50, other one of the same pair would measure 130.
In the drawing at the r i g h t , if the measure of angle A is 80, then the measure of angle B is 80. A
28.
C
When a line I n t e r s e c t s two other l i n e s in d i s t l n c t points, it is c a l l e d a t r a n s v e r s a l of those l i n e s .
the
327 Multiple Choice
a
1.
If In the same plane a transversal intersects two lines and the corresponding angles are congruent, then the two lines are..
.
(a) parallel lines.
c
b
2.
3.
(b )
skew l i n e s .
(c )
perpendicular
(d)
intersecting lines.
(e)
none of the above answers is c o r r e c t .
lines.
If the measure of one angle of a scalene t r i a n g l e I s 50, which of the following statements is always true? (a)
One of the other angles has a measure of
90.
(b)
One of the o t h e r angles has a measure of
50.
(c)
The sum of the measures of the o t h e r t w o angles I s 130.
(d)
Two of t h e a i d e s are equal.
(e)
One of the other angles has a measure
In the figure shown at the right, how many transversals intersect lines rn and n?
(a) 1 (b)
2
(4
3
(dl
4
(4
5
501
(c)
2
(b
1
None
(e
None
(f)
None
(g 1
1
None (b) The only inverses are those l i a t e d below. 74 is the inverse of (h)
501
1
2
(9)
1
(C
(c)
.>.
is the inveree of
74.
501.
Is t h e i n v e r s e of 2. is the inverse of 1.
None.
No; if t h e r e are two i d e n t i t i e s (P and Q) f o r a given operation, then consider the result when P is combined w i t h Q. Since Q is an i d e n t i t y , the r e s u l t must be P. But since P is a l s o an i d e n t i t y , the result must be Q. Thus, P and Q must be the same element since each equals the r e s u l t of combining P and Q.
Answers to Exercises 12-5b 1.
(a)
(b)
lx
=
61, 6), 61, 61, 6),
x = I
1 2x r 1 3x = 1 4 = 1 5x =- 1
(mod (mod (mod (mod (mod
I, 5
Each is its own i n v e r s e .
not possible not possible n o t possible x = 5
rnultiplicatlve inverse of a
b
a
multiplicative inverse of a
1
2
3
14.2
2
2
3
2 + 2 = 1
3
2
3
2
2
3
2
2 + 3
3 3
2
3 + 3
4
3
2
4 + 3
1
4
4
1 + 4 = 4
2
4
4
2 + 4 =
3
2 - 4 m j
3
4
4
3 + 4 -
2
3 . 4 ~ 2
4
4
4
4 + 4 = 1
b + a
=
= =
3
1 * 3 = 3 2.3'1
4
3
4
2 * 2 = 4
1
3.2'1
3
4 . 2 ~ 3 1
4
.
4 4~
4 - 4 E 1
- -
additive inverse of
4.
(mod
5)
a
b
-a
(a) no (b) no
(4
no
(d) yes, except divfaion by zero.
5.
6.
3,
4,
51
(a)
[O,
1,
(b)
El,
53,
(53
(c)
E2,
43,
(a)
{A,
B],
ll, 511 E53 [C, ~ 1 , [A,
(b) yes,
2,
D
(c)
Ec,
(dl
Ec* DI
Dl
[pages
557-5591
additive
inverse of
a
If you wish, you might b r i n g up t h e general problem of d e f i n i n g an operation which is inverse to a given operation * deffned on a s e t , If there is an identity element e f o r *, if every element of the s e t has an Inverse element in t h e s e t , and if * is associative, then (the Inverse of t
could be written a b. o p e r a t i o n for *. Hence +
a
*
I
Then
*
b)
*
1
a
will be the inverse
b = ( t h e inverse of
b)
*
a.
For example: suppose a and b are r a t i o n a l numbera, b / 0, I and * is the multiplication operation, then is dlvislon I ( t h e inverse operation) and is the inverse of b. Hence :
--
12-6. What is Mathematical System? Here the mathematical system Is given an informal d e f i n i t i o n and is followed by diacusalon in terms of previous examplea and some new ones. Here the teacher should n o t try to be too formal. Teaching Suggestions In Section 12-3, it was pointed o u t that a table can l i s t a s e t and describe an operation defined on t h a t s e t . Thus, a table really describes a mathematical system, and not merely a n operation. I l l u s t r a t e by discussing t a b l e s ( a ) - ( e ) of Section 12-3, and by showing that each table does describe a mathematical system (a set and one or more operations defined on t h a t set i n each case, it will be one operation). In Example 1, Part ( c ) (egg-tfmer arithmetic), remind the pupils of the symmetry test for c o m u t a t i v i t y discovered in Problem 2 of Exercises 12-3. The table for egg-timer arlthemetlc I s symmetric, s o the o p e r a t i o n is commutative. Have the class d e c i d e on a word for the operatfon in Table ( c ) of this section. ("Twiddle" is sometimes used.)
--
Answers to Exercises 12-6 1.
Each one of Tables ( a ) , (b), ( c ) describes a mathematical system. For Table (a), the s e t is (A, B); the operation is 0 . POP Table (b), the set I s [P, Q, R, S); the operation Is * For Table ( c ) , the s e t I s A, , , \ 1; the
.
-
operation I s 2.
(4 Q
(a) A
0 (4 0 (b)
Id)
3.
,
B
(1)
A B
(f)
S
(31
(8)
p
(k) A
(h)
\
(I)
R
The operation o is not commutative, since Table ( a ) is not symmetric. The owerations * and are both commutative, slnce both Tables (b) and ( c ) a r e symmetric.
-
4.
There is no i d e n t i t y element for t h e operation o . There fs no element e , such t h a t both of the equations A o e = A and B o e = B are correct. The element R is the identity element f o r t h e operation * The row of Table (b) with "R" in the l e f t column is the same as the top raw, and the column with llR" at the t o p is t h e aame aa the l e f t column. The element A is the identity element f o r the operation . The f i r s t row and column of Table ( c ) are the same as t h e top row and left column respectively.
.
-
5.
(a)
S
(b)
S
(4
Q (f) Q
(4 R
(g)
R
(h)
(dl
\ \
(1)
\
(J)
\
-
Each of the operations * and seems to be assoclatfve since, in each of the cases we have t r i e d , the corresponding expressions are equal. To prove the operations are associative, we would have to examine a l l caaea and show t h a t the corresponding expressions are equal. To prove an operation is not a s s o c l a t l v e , a person would have to f i n d one example where the corresponding expreaslons are not equal. BRAINBUSTER. ( a ) The element 2 cannot be combined with 2 by t h e operation * ( t h a t l a , 2 * 2 is not
defined)
.
(b)
2 1 ie not uniquely defined. Many results are possible when 2 and 1 are comblned.
(c)
The set given by this table I s 11, 2, 3 , 41. But it I s not possible t o combine every p a i r of elements (e .g 3 and 3 ) We do not have an operation defined on the s e t .
.
---
.
Mathematical Systems Without Numbers
12-7.
Skills and Understandinp;~ 1.
To recognize a mathematical ~ y a t e mwhen it i s described in worda.
2.
For systems without numbers: To recognize the elements of the set; t o recognize the operation; t o recognize an i d e n t i t y element; to recognize the inverse of an element.
Teaching S u ~ e s t i o n s Each p u p i l should have his own rectangle to manipulate, such as, a 3" x 5" card. Do n o t use square cards. Be sure t h a t each pupil labels h i s rectangle correctly so t h a t comparisons between different pupils are poaaible. Check especially that each corner of the card I s l a b e l e d with the same letter on both sides. Streas
that t h e c a ~ dis used only to represent a geometric figure -a closed rectangular r e g i o n . Some of the geometric concepts of Chapters 4 and 9 should be reviewed h e r e . It cannot be repeated t o o o f t e n that t h e changes of p o s i t i o n of a rectangle a r e the elements of the s e t in the mathematical system discussed in this s e c t i o n . One of these changes is something that is "done"; t h a t is, it is a physical activity, but It is an element of t h e s e t -- it is not the operation of the system. The operation of the system Is much more e l u s i v e . Any operation d e f i n e d on the set must be a way or combining any two of these physical actlvlties (changes) to get a definite thing. The particular operation we have chosen combines two of these changes by doing the ffrst one and then the o t h e r . The r e s u l t ( d e f i n i t e thing) obtained is one of the changes, b u t t h e o p e r a t i o n is the way of combining them, t h a t is: First do ..., and then do
...
For ease in grading pupils1 written work it is essential that a l l students use the same n o t a t i o n in Exercises 5 and *6. One possible notation is described in the answers. Discussion of Exercises 12-7 3.
In proving associativity, "all cases" must be considered. There is one case f o r each triple of (not necessarily d i f f e r e n t ) elements of t h e set on which the o p e r a t i o n i s defined. For the operation ANTH, there are 4 elements In the set, so there w i l l be 4 4 4 = 64 t r i p l e s ; t h a t is, 64 cases m u s t be considered to prove the a s s o c i a t i v e property. and * 6 , For e a s e in grading wrltten work it is essential t h a t all s t u d e n t s use the same notation in these exercises. One p o s s i b l e notation i s described in the answers.
[ pages 562- 567 ]
Answers to Exercises 12-7 1.
2.
AKPHI
I
v
H
R
I
I
V
H
R
V
V
1
R
H
H
H
R
I
V
(a) V (b) V
(4 v (d)' V
(4 3.
1
(a) Yea (b)
Yes
(c)
Yes, the operation is assoclatlve. A ppoof would require that 64 cases be checked. Each p u p i l should check two or three; do not attempt to check a l l cases. See dlscusslon, Exercises 12-6.
I is the I d e n t i t y .
(d)
Yes.
(e)
Y e s . Each element is its own Inverse.
I (b)
Yes
(d)
Yee.
(e)
Yes.
I (c)
yes
A l l cases can be checked (there are 8 cases in.a l l ) . See discussion of Problem 3 , mercitses 12-6.
I is the identity element.
(f) Yes. Each element l a i t s own inverse.
5.
1
I
R
S
T
U
V
R
R
S
I
U
V
T
S
S
I
R
V
T
U
ANTH
The operation i a not comutative (R ANTH T { T ANTH R) I is the i d e n t i t y element. Each of I, T, U, V is its own inverae element; R and 3 are invereee of each other.
+6.
Notation:
I:
Leave the square In place.
R1:
Rotate clockwise
1
of the way around.
R2:
Rotate clockwf se
1
of the way around.
Rj:
Rotate clockwise
$
of the way around.
H:
F l i p the square over, ualng a horizontal a x l a .
V:
F l i p the square over, uslng a vertical a x i s .
Dl : F l i p the square over, uslng an axis from upper left to lower right.
D2:
F l i p the square over, ualng an axis from lower
left to upper right. Note: It was suggested that a aquare card not be ueed. This ppoblem is included t o ehow w b such a suggestion was made.
ANTH
is the Identity element. The operation is commutative ( R ~ANTH H # H ANTH I l l ) . I
not
-
12-8. The Counting Numbers and the Whole Numbers This section has problems which lead the pupils to conclude that the counting numbers and the whole numbers each form a mathematical system. It I s pointed out t h a t the dlstrlbutive property with which the pupil is familiar cornea from the abstract discussion of t h i s property. The pupils should not be expected to duplicate the abstract definition. One of the objectives of the section Is to show a w a g t o p u l l together the concept o f systems. Some of the stte of numbers considered in ordinary arithmetic are: the rational number^, the whole numbera, the counting numbers, the even numbers, e t c . Discussion of Exercises 12-8
4..
One p o s s i b l e model of the mathematical system in this exercise is as follows: L e t A = [l, 21, B = {l,
2, 33, C - El, 2, 43, D = (1, 2, 3 41. Then, from the tables in the problem, the operation is intersection and the operation o is union. Each of these operations distributes over the o t h e r . [pages 570-571 1
Answers to Exercises 12-8 1.
(a)
Since the sum of two counting numbers is alwaya another counting number and the product of two counting numbers is always a count-lng number, the s e t is c l o s e d under addition and multiplication.
(b)
Both the commutative property and the aaaoc1atlve property hold f o r addition and multiplication. Examples:
Commutative:
2
+3
=
3
+ 2;
4 ~ 6 = 6 x 4 Associative:
3
+ (4 +
7) = (3
+ 4 ) + 7;
3 x (6 x 8) = (3 x 6) x 8.
2.
(c )
There l a no identity element for addition. The i d e n t i t y element f o r multiplicatlon I s 1; f o r every counting number n, n 1 = n = 1 n.
(d)
The counting numbers are not closed under subtraction or dfviaion.
(a)
The s e t of whole numbers l a closed under addition and multiplication.
(b)
Both operations are commutative and aaaociative.
(c)
There is an i d e n t i t y element for addition. It is zero; f o r any whole number n, n + 0 = n = 0 + n. The number 3 is the identity element f o r
multiplication. The anewers are the same as f o r 1 (a), ( b ) , ( c ) except t h a t there is an i d e n t i t y element f o r addition In the whole number system and n o t in the counting number system.
3.
(a) Three examples are: 2(3 5 ( 7 -t 10) = ( 5 . 7 ) + ( 5 (1 . 1) -k (1 1). (b)
4.
.
3 ) + (2 10); l(1 + 1) = 4 ) = (2
-
.
Exercises 12-7.
Yes, here are 3 d i s t r i b u t e s over
illustrations t h a t
*
0:
A *
( B O C ) = A = ( A * B ) O
a *
(BOB)
(A*c)
= B u ( B ~ B 0) ( B * B ) C * ( B O D ) = ~ = ( C * B ) O(c*D)
(b)
Yes, here are
3
distributes over
5.
illustrations that
o
*.
A o ( B * c ) = A = ( A O B ) *
( A O C )
s o (B*B)= B =
(BOB)"
(BOB)
C O ( B * D ) = D = ( C O B ) *
(COD)
(a) Closed; commutative; associative;
i d e n t i t y ; only the number
1
1
is the
has an inverse.
(b)
Closed; commutative; associative; no identity; no inverses.
(c)
Closed; commutatlve; associative; 0 is t h e i d e n t f t y ; only the number 0 has an inverse.
(d)
Closed; commutative; a s s o c i a t i v e ; no i d e n t i t y ;
no inverses. (e)
4);
Addition doea not distribute o v e r multiplication; for example, 2 + ( 3 4 ) = 14 f 30 = (2 + 3 ) (2 + 4 ) .
See discussion,
(a)
+
Closed; commutative; associative; 0 is the Identity; o n l y the number 0 has an inverse.
(f) Not closed; dornmutatfve; not associative; no identity; no inverses.
6.
(a)
Both sets are closed under the operations.
operations are commutative and associative. systems involve the same set. (b)
Both Both
has an identity and 5 ( b ) does not. A l s o , the sets are different in these two systems.
The system
5(a)
*7.
Many reeults are p o s s i b l e , of course.
+8.
(a) Yea. We are asked to consider the two expressions a * (b o c ) and ( a + b ) o ( a * c), and find whethep or not they are always equal. For example, using a = 8, b = 12, c = 15, , 8 * (12 o 15) 8 60 = 4. (8 * 12) o (8 * 15) = 4 o 1 = 4.
-
(b)
Yes. We are asked to coneider tho t w o expreaeions a o (b * c ) and (a o b) (a o c), and find whether or not they are always equal. For example, using a 8, b = 12, c = 15, 8 o (12 * 15) = 8 o 3 24 ( 8 o 12) * (8 o 15) = 24 * 120 = 24.
-
-
12-9. Modular Arithmetic In thia section, the number l i n e is used to provide a picture of hovr equivalence classes of whole numbers can be developed. A t thia time it may be wise to re-read the first paragraphs of Section 12-1. We use the term ''multiple" to mean "multiple by a
whole number. It Problems which may be used f o r motivation t o explain the meaning of modular systems include the ordinary 12-hour clock, the days of the week, and the months of the year. For example, "Today I s Tuesday; what day will it be six days from now?'' Answer: Monday; this is mod 7. "1t I s 4:30 o~clock. What time will it be 10 hours from now?" Answer: 2:TO; t h i s l a mod 12
.
Modular arithmetic may be thought of as a mathematical system with two o p e r a t i o n s . Section 12-1 discussed modular addition and Section 12-2 dlacuased modular multiplication. The two operations together allow us t o use the distributive property; thus, the whole numbera form a system under modular additlon and multiplication. In modular arithmetic only a f i n i t e number of symbols 1s needed because i n f i n i t e l y many whole numbers are represented by each symbol , Other interesting highlights are: A product of non-zero factors may be zero in some systems. There may be many replacements f o r x In a number sentence to make it true.
Answers to Exercises 12-9
(~ncouragethe pupils to look for patterns and to use what they have previously learned about systems to make the t a b l e s . )
[pages 573-5751
Mod 5:
Yes;
mod 8:
Yes
Mod 5:
Yes;
mod 8:
Yes
Mod 5 :
Yes; mod 8:
Yes
Mod 5: 2 and
1
3 no inverse.
and 4 are their own inverses; are inverses of each other; 0 has
Mod 8: Only 1, 3, is its own Inverse.
5,
7 are inverses; each
Yes; mod 8: No. 2 x 4 G 0 (mod 8), 4 x 2 = 0 (mod 81, 4 x 4 r 0 (mod 81, 4 x 6 0 (mod 8), 6 x 4 = O (mod 8). Mod 5 :
2
(d)
0
0
(4
4
5
(f)
1
~ ( g ) 1.
Any power
of 6 ends in 6 .
( h ) Not d e f i n e d in this
system.
[pages 575-577 1
9.
(a)
4;
What number added to
3 gives 7?
The set is closed under eubtraction mod
11.
(a)
3,
(b)
3,
(c)
0
K
2 .
8, 13 and others 7, 11 and others
(add
5)
(add
4)
and a l l multiples of 5 is a countlng number.
5.
of the form 5K,
(d)
Any even number
(e)
3,
or any odd number greater than 3 .
(f)
1,
3
(d)
Anyevennumber
5 and
3 , 5, numbera )
(f) 1,
7,
so on (all odd numbers).
9, 11, 13 and so on ( a l l odd
[page
5771
Sample Questions
- False
Part I. True
T
1.
Operations can be described by t a b l e s .
T
2.
A symbol can be made to mean anything providing we define it.
F
3.
The identity f o r multiplication i n ordinary arithmetic is zero.
F
4.
The 3dentity f o r addition in ordfnary arithmetic is one.
T
5.
The additive inverse of
T
6.
In ordinary arithmetic, with the s e t composed of a l l the rational numbers except zero, the inverse of division is multigllcation.
P
7.
Allmathematical systems are sets ofnumbers.
F
8.
1h m o d 5
T
9.
The s e t mod 4.
arithmetic, 10, 1,
2,
31
2
0/;5
in the mod 4
=
system l a
2 (mod5).
I s closed u n d e ~subtraction
Part If. Computation Answers :
Find the sums:
+ 2)
1.
(9
2.
( 5 + 4 + 3 ) (mod 6 )
(mod 12)
Find the differences:
3
(5
4.
(3
-
2) (mod 6 )
5 ) (mod 7)
Find the products:
5.
1(3 + 7 ) x 6 1 (mod 9)
6.
3* (mod 8)
11 0
2.
Answers :
Find t h e quotients: 2
7.
J(mod5)
8m
7O
4 0
(mod 11)
P a r t 111. Multiple Choice The table below descrlbea a mathematical system. be used f n answering questions 1, 2, and 3 below.
1.
Which one of t h e following statements is t r u e ? are starred)
.
A.
2.
It i a to
The s e t {A, B, C , D] respect to the operstlon
is
not
(~nawers
closed with
o.
.
*B.
The operation
o
is commutative
C.
The operatfon element
o
does not have an identity
.
The operation
o
is
E.
None of the above.
.
not
assoclative.
The i d e n t i t y f o r the operation A.
D
B.
B
"C.
C
D.
Both
E.
None of the above.
A
and
B
o
la:
3.
In themathematical system: A.
Only
B has an inverse.
B.
Only
D has an Inverse.
C.
Only
A
D.
None of the elementa has an inverse.
*E.
4.
6
C
have inversea.
A l l the elements have inverses.
For what modulus A.
Mod9
B.
M O ~ 6
c,
~ o d8
*D.
Mod 7
E,
5.
and
m
is
2
-
5
=
4 (mod m )
true?
None of the above.
For the system consisting of the s e t of odd numbers and the operation of multiplication:
.
A.
The system is not c l b s e d
B,
The system is not commutative.
C.
The system has no Identlty element
*D.
None of the above is correct.
E.
A l l of the above are correct.
For t h e system consisting of the set of even numbers and the operation of addition: A.
*B. C.
The system is not closed. The system has an identity element. The system has an inverae f o r addition for each
element.
D.
A l l of the above are c o r r e c t .
.
None of the above is c o r r e c t .
%
7.
A mathematical syatem c o n s i s t s of several things.
Which of the following is always necessary in a mathematic a 1 system? A.
Numbers
B.
An i d e n t i t y element
C
.
+D. E.
The commutative property
One or more operations None of the above
Use the mathematical system as described below in answering Questions 8, 9, and 10. The s e t of elements in o u r system is t h e set of changes of a rectangle. The elements are:
I means leave
H means f l i p on the h o r i z o n t a l axis
alone.
V
means f l i p
on the v e r t i c a l
ax1s
The following I s an i l l u s t r a t i o n of o u r operation V * H means do change V and t h e n do change Thus V * H = R.
8.
H * H Is: A.
H
"B.
I
E.
None of the above.
R means turn halfway around its center
*
;
H.
9.
I * R *A.
10.
is:
R
B.
V
C.
I
D.
H
E.
R " H
( H w v ) * V A.
I
B.
V * V
*C.
H * I
is:
D.
V
E.
None of the above.
Bibliography
1.
2.
3.
Allendoerfer, Carl B., and Oakley, Cletus 0. PRINCIPLES OF MATHEMATICS. New York: McGraw-Hill Book Company, 1955. For S e c t i o n s 1, 2 and 9 use pages 66-68. For Section 7 use 71-73. Andree, Rlchard V. SELECTIONS FROM MODERN ABSTRACT ALGEBRA. New York: Henry Holt and Company, 1958. For Sections 1, 2 and 9 use Chapters 1 and 2. For S e c t i o n 7 use pages 78-86. Jones, Burton W. "Miniature Number Systems," TKE MATHEMATICS TEACHER. Washington, D C : National Council of Teachers of Mathematics , April, 1958. pp 226-231 O r e , Oystein. NUMBER THEORY AND ITS HISTORY. New York: McGraw-Hill Book Company, 1948, pp 209-340. Uspensky, d V., and Heaslet, M. A . ELEMENTARY NUMBER THEORY. New York: McGraw-HI11 Book Company, 1938,
..
4.
5.
.
p p . 126-325.
.
.
.
Chapter 13
STATISTICS AND GRAPHS
Ii
Introduction II Traditional seventh grade material gives some time to graphs ! but treats them as an Isolated topic. In thls t e x t , graphs are : introduced a a an integral part of s t a t i a t i c s . Statistics, as a : t o p f c for seventh graders, is new. Only the most elementary phases of s t a t i s t i c s are given but the student I s expected to become aware of thls branch of mathematics. In addition t o the three common types of s t a t i s t i c a l graphs presented In the t e x t , the concept of mean, median and mode as meaaures of central tendency, and of average deviation and range as measures of spread are introduced. m e emphasis I s on understanding t h e meanings of these measures rather than developing skill in finding the measures. Students of this age are beginning to f i n d a need for understanding elementary statistics. Work in social studies includes current events and some of t h l s material contains graphs and s t a t i s t i c s t h a t students should understand. School drives often make use of student-made graphs. The collection and organization of data is touched upon and opportunity for organization ie Included in the exercises. The teacher may find i t of value t o have students c o l l e c t , organize ' and graph data of their own. Every school provides i t s own ! material. The number of studenta enrolled, the s i z e and number of class sections, and the number of students that participate in various a c t i v i t i e s are some areas i n which s t a t i s t i c a l data ape usually available. The interpretation and ability t o read graph8 are probably of more importance than the a b i l i t y to make graphs. The teacher should stress t h i s phase of the work with graphs. raaking graphs is one a c t i v i t y t h a t helps the student learn to read graphs. 1
The t e x t provides some work in finding the mean, median, mode, range and average deviation as an aid to understanding t h e meaning of these terms. The work on sampling is very general. P u p i l s may be interested in reporting examples of sampling round in newspapers and news magazines. Additional material t h a t can be used f a found fn the annual reports of large corporations and in the material released by the National Industrial Conference Board. The Conference Board w i l l send material at your request. (see the ~ibliography,) You and some of your s t u d e n t s might be Interested in the informative and pleasant book, How to Lie with S t a t i a t l c s , This chapter should require about 10 days t o teach,
----
13-1. Gathering Data This section introduces the word "data" and develops i t s meaning. The organization of data and reading a s t a t l a t l c % l table are utilized both as ideas and as a means in understanding data.
Answers t o Exercises 13-1 1.
The general trends in the data show t h a t the population has been i n c r e a a l n g since 1790. The per cent of increase has been lesa since 1860 than before that t h e .
2.
From 1800 to 1810. H i g h immigration rate and beginning of Westward Movement
3
From 1930 to 1940. The depression of the 19301s.
4.
Population increase was due in part to large number of inmigrants from Ireland. Famine was due to f a i l u r e of p o t a t o crop in I r e l a n d in 1845, 1846, and 1847.
.
[page 579-582 I
6.
These may be arranged from largest to smallest or amalleat to l a r g e s t .
7.
This information can be found in almanacs or local publications.
8.
(a)
1935 1936 193i 193
1939 1940 1941
1942 1943
28,781
36,329
70,756 51,776
1945 1946 1947 1948
108,721 147,292 170,570
34,956 50,244 67,893 W99
1944
23,725 28,551
38,119
1949 1950 1951 1952
1953 1954 1955
188,317 249,187 205,717 265,520
170,434 208,177
237,790
(b)
23,725 28,551 28,781 34,956 36,329 38,119
50,244
9.
51,776 67,895
&;:2;
108,721 147,292 170,434
170,570
188,317 20 ,717 20 ,177 237,7iO 249,l 7
2
265,520
Number of automobiles sold by the manufacturer:
13-2. Broken Llne Graphs Broken l i n e papha are u t i l i z e d whenever the change t h a t occurs in some item I s to be emphasized. The teacher should be sure atudent~know how to read these graphs. One of the best ways of l e a r n i n g to read graphs is to make them.
Some observations on making graphs are noted b e l o w . It would be helpful if the atudent listed theae point8 before making graphs. The points are:
1,
Plan before making any marks on the paper. This includes planning room f o r t i t l e , scales and any names needed.
2,
Make grapha
3.
Print.
k.
Use rulers to draw l i n e s that should be straight and to enclose the graph with line eegments f o r a finished appearance
aB
large as space pemnite.
.
5.
Use a suitable scale. This is found by dividing the largest number to be paphed by the number of unlts available on the scale.
6.
L i n e graphs should be started at the left edge.
Values between points should be interpreted cautiously. Changes are not instantaneous but s i n c e they are irregular, no c e r t a i n t y can be attached to a reading between pofnts. These readings will be fair approximations much of the time.
Answers t o Claas Discussion Questions 13-2 1
.
More between 1900 and 1910 than between 1800 and 1810. This I s seen more easily from the graph than from t h e table of data.
2.
No.
3
The piece of the broken l i n e from 1810 to 1820 would be
horizontal. 4.
1945: 140 million; 1895: 68 to 70 million. Increase 70 t o 72 million.
5.
About
170 million.
Answers Exercises 13-2
1.
(a) 20
(d)
30 or 50
(b)
25
(e)
lo0,ooo
(c)
10,000
2.
Enrollment in Franklin Junior High School, 1952-1956
YEARS
3.
(a) Graph f s on the next page.
Unsucce aaful Candidate
(b)
Year
Elected
Hoover (R) F. D. Roosevelt F. D. Roosevelt F. D. Rooaevelt F. D. Roosevelt rnJJnan(Dl
Eisenhower (R) Eisenhower (R) Kennedy (D)
(D) (D) (D) (D)
(D) Ikmocrat ( R) Republican
smith (D) Hoover (R) Landon (R) Willkle ( R ) Dewey (R) Dewey (R) Stevenson (D) Stevenson (D) Nixon (R)
PRESIDENTIAL
ELECTION
-,---REPUBLICAN
DEMOCRATIC
YEARS
PARTY PARTY
This graph is dependent on l o c a l s t a t i s t i c e .
13-3
Bar Graphs -
Bar graphs are used to compare data about similar Items. Since data that indicate change can be considered as a comparison of almilar items, moat l l n e graphs could also be shown as bar graphs. Bar graphs c a m o t always be displayed as llne graphs. It is s u i t able t o ahow the growth of one school In either type of graph but the graph of the enrollment of different schools in a d i s t r l c t is sultsble only for a bar graph. Line graphs emphasize change; bar graphs emphasize comparison. P r l n c i p l e a of graph construction t h a t were given for T h e graphs are applicable to bar graphs, One additional p o i n t needs to be made. Bars should be the same width; spaces should be the same width but I t is not necessary for the spaces to be t h e same width as the bars. Color adds a great deal t o graphs.
Answers
to Exercises
13-3
1.
3,260,000 x $3.25 = $10,595,000
3.
Between 1958 and 1959.
Table shows increase is
368,000. 4.
Accidental Deaths, 1956
Fires
............... ............... ............... ...............
I-
....... b
Roilrood ::::::: Accidents """'
NUMBER OF PEOPLE
KILLED
H i a e s t P o i n t in Selected S t a t e s
5. Alabama
Alaska
Arizona Arkansas
Callforn
Colorado 0
I
1
4
8
16
I2
Feet in Thousands
6.
Games Won
-
National League
Pittsburgh
St. Louia Milwaukee
Los Angelea Sari Prancisc
Cincinnati ~h iiadelphia
Chicago
Gmes
Won
[page 590 I
20
24
13-4
Circle Graphs
Circle graphs can be used only when the d a t a are concerned with the whole of something and the divisions of thfs item. Xt is possible t o consider such data as a comparison of parts and use a bar graph to show the relationships. The comparison of parts I s shown equally well by circle graphs. C i r c l e graphs emphasize the unity of the item graphed. The students may need some review both in the use of a prot r a c t o r and in using percents before they will be able to work with circle graphs. It may be advisable to have students select from the four graphs in the Exercises. P o s s i b l y , some of the class could do two while the rest of the class do the o t h e r two.
-
Exercises 13-4 1.
School-Related Accidents, 1949
IN
SCHOOL BUlLDlNOS
ON
PLAYGROUNDS
School -Related Accidents, 1956
PLAYGROUNDS
Money for Stage Curtains Percent
Number of Degree s
4.
Homes of Washington Jr. High Pupils In Relation t o School Number of Percent 50
Degrees
180
TWO MttES
MORE
MORE THAN BUT LESS
13-5 Sumarizirq Data The approach t o descriptive statistics used here is one of trying to describe a set of numbers by j u s t two numbers, one to give an idea of the magnitudes of the numbers and the other t o show how the items vary or scatter. The averages ( m e a n , mode, and median) give the idea of the magnitude o f the numbers in the s e t ; the range and the average deviation from the mean show how the numbers vary or scatter. The arithmetic mean and the median are used so frequently in newspapers, magazines, and other media that it is most important for everyone to understand thoroughly the two concepts. The short-cut of Problem 6 in mercises 13-5; is a standard short -cut which the pupils may f i n d useful. No attempt at a rigorous proof for I t should be made.
In statistics a deviation I s technically defined so as to involve positive and negative numbers but there is no need to introduce negative numbers here since in computing the average d e v l a t l o n one averages the absolute values of the deviations.
Answers to m e ~ c l s e s13-5a
85
1.
Mode:
2.
(a) Arithmetic mean:
3.
(b)
Median:
(a)
Mean:
(b)
3
(4
7
913 = 83
85 $6100
(d) No, because the mean is larger than such a large percentage of the salaries. (e)
If there is an even number 2n of items (as there are 10 in thia problem) the median after arranging the items according to size is taken t o be the average between the nth and (n + 1)th items.
(f) Median is better than mean, slnce the mean glvea the impression that the salariea are higher than they a r e . The mean is affected by the large salary of $12,500, but the median I s not.
4.
(a) Mean: (b)
Median:
56.5 49
-
Answers to mercises 13-5b 1.
In making tables of ages of pupils in each grade. In tabulating data on distances from school, e t c . between 15 and 17.
2.
Median:
3.
Temperature
60
-
90
-
50 55
Frequency (NO. in temperature range)
54
1
59
1
64
2
94
1 21
T o t a1
-2 .
21 = 1%.1
Consequently the middle number w i l l be t h e
11th one. There muet be ten temperatures below the median temperature and ten above. In order to find the temperat u r e which corresponds t o 11, we count f r o m the top of the frequency colunm, To obtain 11 we must use 5 of the frequency 7. Hence the median is nearer 74 than 70. This may be checked by counting f r o m the h t t o m of the table.
Answers 1.
to mercf eee
13-5c
In 1953 and 1954.
3.
Mean is 44.5. The deviations from I t are 0.5, 0.5, 4 1.5. The average deviation l a or 1.
4.
Average or mean
1.5,
is 83.4
Deviations from the mean are 1.6, 1.4, 4.6, 7.4, 6 . 6 , 0.6, 3.4, 1.4, 0.6, 0.4 Average of these deviations from the mean is 2.8.
.
5.
Mean is 84. Deviations are 10, 0 , 16, 10, 14, 14, 12, 0, 8 , 12. Average deviation from the mean is 9.6.
+6.
Suppose the guessed average I s 45. Then the deviations of t h e scores are 5, 2, 1 and 2, 3 , 4, 6. The sum of the deviations f o r scores leaa than 45 is 8. The sum of the
deviations for scores greater than
45
is
15.
Since the sum of the deviations for the acores m t e r than 45 is more %&m the &urn of the devlatlona for the scores smaller than 45 we add 1 to the guessed mean to g e t the correct mean, 45 + 1 = 46. =
=
1.
-
3
-
Sampling
Although the theory of random sampling is much too complicated to be presented to the pupils, sampling has become such an important part of engineering, buslnesa, p o l i t i c s , and other fields that it is i m p o r t a n t f o r everyone to have some acquaintance with the I d e a of aampl l n g Members of the c l a s s may f i n d it Interesting to bring in the results of some of the public opinion polls reported in newspapers and magazinee, especially around election time. You may wish to discuss some examplee of b i a s e d sampling. The numbera found in Problem 2 are close t o the actual results. It is impossible to come cloaer with the percents given to o n l y the nearest t e n t h of a percent, since 0.1 % makes a difference of more than 48,000 v o t e s . For your information, hvman received 24,105,812; Dewey, 21,970,065; Thurmond, 1,169,021; Wallace, 1,157,172 v o t e s .
.
-
-
Answera to Exercises 13-6 1.
The nature of this problem results in answers that vary.
2.
(a) Trmman Dewey
Thurmond
Wallace (b)
21,730,000 24,170,000
980,000 1,950,000
Tmnml
24,120,000
D ~ U ~ Y
21,980,000
Thurmond
1,170,000
Wallace
1,170,000
[pages 606-607 I
Sample T e s t Items U s e these data f o r Problema 1-5 ~ f r o e aDebt of
(4 Period
the United States for the Years 1861 to 1955 ( ~ i v-year e periods )
(b) Millions
of Dollars
(c
Bllliona of Ikllare
(a) Ikviations f r o m Mean
I.
( a ) What (c)?
In column In other words, whlch groas debt appears most
l a the mode of the aet of figures
often? (b) 2.
Does this number tell you very much about the total debt picture of the United States Treasury since 18611
(a) Find the median of the
s e t of figures in column ( c )
which will give you the median national debt from 1861 to 1955. Thia means t h a t you will need t o f i n d the middle figure between 1 and 274. In order t o do t h i s , list the numbers fpom column ( c ) In order, and flnd the middle one on the l i s t by counting up from the bottom or down from the top. (b) Does the median tell you much about the t o t a l debt over this period of years? Do you see that it is necessary to f l n d the mean in order to take into consideration the vePy large
numbers in the table?
3.
Find the mean of the set of numbers in column ( c ) t o the nearest whole number.
4.
Find the deviations f r o m the mean and then the average deviation to the neare a t whole number.
5.
-plain
the lapge deviations above the mean.
Azbllc School IQwollments in United S t a t e a
Kindergarten and Grades 1--8
(Enrollment l a given in millions for the yeare ahown on the horizontal base line. )
(a)
What kind of graph l a the graph In t h i s problem?
(b) What was the enrollment in 19107 (c)
19201
1930? 1958~
In which two years was the enrollment the erne?
(d) Estimate the enrollment in 1925. (e)
In which ten-year period d i d the enrollment decrease?
(f) If the graph continues in a straight line f o r the yeare 1955, 1958, and 1960, what would be the predicted enrollment in 19601
School Enrollment Year: 1950
ELEMENTARY
(a)
STUDEITS
What fractional p a r t of the total enrollment is in
elementary school? in eecon&ry 8chooX? in college? (b)
Account for such a large group of children enrolled
in elementary school. (c)
If you were asked to draw t h i s circle graph, how would you find the size of each cloeed curve?
(d)
If there were about 29,OOQ,OOO students enrolled In achool in 1950, how many were In the elementam grades? In college? I n secondary school?
--
Answers
w e s t e d Test Items
1.
(a)
2.
( a ) Median
3.
Mean = 49,
Mode = 1
-
to
1
(b)
No.
(b)
No.
nearest whole number of billlone.
4.
k v i a t i o n e from mean are 46, 47, 47, 47, 47, 48, 48, 48, 48, 48, 48, 25, 28, 33, 20, 6, 210, 208, and 225, respectively. Average deviation is 67.2.
5.
World War 11; Defense Program aince World War 11.
6. (a) Broken line graph. (b)
17 million, 19 mllllon, 21 million, 23 million.
(c)
1940, 1950.
(d)
About
20 million.
(e) 1930 - 1940. (f) 24 million.
(b)
Compulsory school attendance and increase I n b i r t h rate.
(0)
36'.
72O,
2 5 2 ' .
(d) 20,300,000; 2,~ 0 , 0 0 0 ; 5,800,000.
The National Industrial Conference Board, 460 Park Avenue, New York 22, N. Y. This organization will send business atatistica and graphs free. Huff, Zlarrell, HOW TO LIE WITH STATISTICS. New Yo*: W . W. Norton and Company, ma., 1954. 142 p.
Chapter 14
MATHEMATICS AT WOFN IN SCIENCE
-
14-1. The $ c i e n t i f i c Seesaw Much of the science that I s taught in the schoola t o d a y I s d e s c r i p t i v e science, and it omits or neglects the studies of a quantitative nature by which one learns t o carry out the main steps of a scientific experiment, namely: (a)
The scientist observes what happens and c o l l e c t s and
studies the quantitative data. ( b
To explain the f a c t s he has obaepved, he s t a t e s his hunch as a hypothesis which expresses the pattern he sees in the data.
(c)
He makes predictions, telling what will be seen if certain o t h e r observations in the experiment are made.
(d)
He goes back to the equipment and t e s t a hie p r e d i c t i o n . If it works in a reasonable number of cases, he c a n s t a t e his dlacovery as a theory or princlple. No amount of t e a t s will 'prove" the principle, but a rather large number of tests will show that the p r i n c i p l e is very l i k e l y to be true. The p r i n c i p l e is usually s t a t e d with mathematical terms and symbols.
The purpoaes of this chapter are: (a)
to illustrate by a simple experiment the t y p i c a l inductive method of science;
(b )
t o give pupils experience in collecting mathematical
data in an expeptment in science; (c)
to help pupils Bee the importance of the qvantitatlve aspect of the phyfical experiment; t
(d)
to f u r n i s h applications of some mathematical skills
The chapter is rather s h o r t and probably can be completed in or 5 days.
. 4
-
14-2. A Laboratory Experiment The equipment I s simple. If standard units of known weight are not available, new marbles f ~ o mthe same manufacturer or new pennies can be used. Marbles a r e colorful and about right aa . units of weight, b u t they roll when they a r e dropped. A meter stick or yardstick I s needed to serve as t h e balance rod or seesaw.
The law of the l e v e r is one that can be discovered r a t h e r e a s i l y if measurements are c a r e f u l l y made; and y e t it is fundament a l in science. It i s used i n much equipment that w e see around us such as crowbars, tongs, and s c i s s o r s . It is recommended t h a t each pupil take part in the experiment The teacher may f i n d it profitable t o divide the c l a s s into groups of four or five pupils so t h a t each group can carry out i t s own experiment. A special committee of pupils should be appointed in advance to secure the necessary equipment and to prepare it for use.
Square pieces about 8 Inches on the s i d e may be c u t from the thin p l a s t i c covers or bags used in packaging. The four corners may be tied together with s t u r d y thread which i s then looped around the meter s t i c k or y a r d s t i c k . The s t i c k may be suspended by the same kind of thread. To simplify reading distance measurements, t h e s t i c k should be balanced a t the center p o i n t before t h e bags are suspended. A thumbtack a t the proper p l a c e will help balance the s t i c k i f i t i s o u t of balance i n t h e beginning. The problem of f i n d i n g a support from which to suspend the meter s t i c k (balance b a r ) may take some ingenuity to solve. It may be p o s s i b l e t o borrow standards from the science department or to manufacture hooks by using c o a t hangers. As a l a a t resort some t e a c h e r s have used another s t i c k r e s t i n g across the backs of t w o c h a i r s as a support for the balance. Be sure to give complete and clear instructions In advance ao that each pupil understands what is to be done.
In the group of pupil8 working together on t h e experiment, two pupils may adjust the distances and the others check the measurements and keep the record. As the experiment proceeds, the pupils may rotate reaponsibilltiea so that each has an opportunity to do the complete experiment.
TABLE I w d
w D
(a)
(b)
(4
(dl
(4
(f)
(g)
(hl
(1)
(J)
10 12 10 12
10 12 20
10 12
10 12
10
12
10 12
6
24
15
8
5
12 12 10
10 12
8
10 12 24
10
5
10 12 15
As the pupils are doing the experiment and recording measurements in Table I, help them discover two relationships: if the value o f w is increased, the correaponding diatmoe, 8 , f'rom the fulcrum l a decreased; also, if a value of the w e i g h t I s doubled, the corresponding distance from the fulcrum i s only h a l f aa much; if the weight I s made three times as much, the distance is only one-third as much, and so on. The product w and d a8 well a a W and should be approximately 120.
of
O In each column in Table I
The teacher should be aware that t h e weight of an object means the gravitational force exerted on it by the earth. Although we have n o t belabored the d i s t i n c t i o n between mass and weight, I t seem8 worthwhile to speak precisely in this respect. In particular, the students should be encouraged not to confuse the weight of an object with the object Itself, as I s frequently the case in ordinary conversation. Tables 11, 111, and IV euggest ways of beginning the measurements in order t o g e t a lmge number of readings and also to help bring out certain relationships. Also, reading the meter stick should increaee the pupilst familiarity with this kind of measurement.
TABLE I1
TABLE 111
(a)
.
.
(b)
(4
(d)
(4
TABLE
IV
(f)
--
(g)
Caution : Induct;i v e Reasoning; a t Work! After recording a l l the measurements In their tables, pupils should be able t o see and formulate In t h e i r own words the law of the lever, namely t h a t the product of the measures of one weight and t h e corresponding distance from the fulcrum is equal to the product of the measures of the other weight and t h e corresponding distance from the fulcrum. The f o m l a wd = should be discovered by the pupils. 14-15
[pages 612-6131
Answers to Exercises 14-3 1
(a:
Equal masses are at equal distances from the fulcrum.
(b)
D is h a l f as much as it was.
(c)
D is twice a s much as i t was. twice as large as the o r i g i n a l
(d)
The product of the measures of the d i s t a n c e and of the weight on one side equals the product of the measures of the distance and of the weight on the other side. wd
=
The new D.
D is
Fm.
2.
Table V
3.
Suggest t h a t pupils check the cases with the amallest numbers.
The purpose of Table V is to provide opportunity f o r pupils to use arithmetic and apply the law to specific caaes. The l a s t four figures in the t a b l e show t h a t if a sufficiently long lever is used, an object of tremendous,weight can be l i f t e d .
14-4. Qraphical Interpretation Graphing is a s k i l l that needs to be used both in mathematics and science. This experiment gives a s e t of figures which can be graphed e a s i l y . Be sure that the pupils know how to prepare t h e axes and how to mark t h e scale on each a x i s . They may need some p r e p a r a t o r y practice in plotting points and probably will find it d i f f i c u l t to draw a smooth curve through the p o i n t s t h a t they plot. !The points for which one of the p a i r of numbers is to be found are: ( 4 , 3 0 ; (20, 6); (16, 7 ; ($, 18).
[pages 614-6153
The teacher may wish t o follow this graphing with several other graphs, for practice, which y i e l d similar curves, aa WD = 48 or = 90. The hyperbda is such an Interesting curve that the better students should be encouraged t o investigate it and i t s uses further. It may be appropriate to tell the p u p i l s t h a t the hyperbola I s the curve of fnverae variation. It results whenever the product of two quantities l a a constant as in 120 or D = 120 WD = 120, which can also be written W = -5.
T.
I
Answers t o Exercises 14-4 1.
( a ) As the value of w I s increased, the corresponding value of d decreases. (b)
A
I s increased, the correaponding value of the
d
is decreased. (It may be appropriate to point out that t h i s is an inverse relationship.)
w
20)
(b)
(8,
15)
(4
(209
6)
(6, 20)
(b)
(4,
30)
(4
(151
8)
2.
(a) ( 6 ,
3.
(a)
4.
(a) (10, 2 5 ) (b)
5.
5.
I s not.
(15, 8) 1s.
5) is not.
(c)
(5,
(d)
(20, 15) I s not.
These values are approxim~t1ons. For example, 17.1, a l i t t l e mare than 17 or approximately 17 are all correct in (a).
(4
(7,
17.1)
(dl
117,
(bj
(8,
15)
(4
(5.7,
(4
(13.3,
See g r a p h .
9)
i
(f) 3
7) 21) 5.2)
I
ORAPH OF WD:BB FROM THE OATA OF TABLE
P
8.
As the value of
w decreases, the corresponding value
increases. As the value of w of d decreases. of d
9.
Increases, the corresponding value
The curves have similar shapes.
Both ahow inverse
variation. In a l l t h i s work, help pupils to become aware of the importance of mathematics in science and of the need for mathematical data to carry out experiments. Other Kinds ---
of Levers Other types of levera are ~ I m p l ymentioned in passing. Pupils who are interested may study t h e m i n d i v i 8 u ~ l ybut it is not in keeping with the a p l r i t of the chapter to spend c l a s s t h e i n a thorough Investigation of a l l types of levers,
14-5.
---
14-6. The Role of Mathematics
in Scientific
EXperiment This discussion of the lever d o e ~not introduce new mathematics. The i n t e n t here i s t o suggest In a small way some of the many important roles which mathematics plays In our
c i v i l i z a t Ion. In quantitative studies of all kinds, that is, studies in which quantitfes are important and are carefully measured, the mathematics of arithmetic and measurement is invaluable, The pupils saw thia in measuring distances, comparing weights, and making observations of the data. In the search for a general pattern or for a new combination of facts, t h e medium of inductive reasoning 18 used. Frequently the reasoning process is described in mathematical language which is p a r t i c u l a r l y appropriate f o r desc~lbinggeneral c h a Y a c t e r i ~ t i ~ ~ and properties.
Ordinarily, the general rule itaelf 1s described b e s t by a mathematical formula. In interpreting the general pattern which has been described in mathematical language, graphing 1s often effective. We have ereen In our discussions of the lever t h a t the use of a graph give8 a convenient picture of a great deal of
information. A t many places in the discuasfon of the lever, use was made of deductive reasoning. Smetfmes, in fact, it is possible to develop a general law in mathematical terns entirely by using deductive reasoning and mathematical operations baaed on certain physical assumptions. Throughout the entire fibre of our s c i e n t i f i c a c t i v i t y and general civilization, mathematics plays a domlnant role in analysis, description, interpretation, and generalization. The historical examples of the text a l l I n v o l v e simple mathematics which the p u p i l s should be able to appreciate. You may wish to emphasize the increasingly complex nature of presentday s c i e n t i f i c discoveries, many of which use new kinds of advanced mathematics. Claee study of the current news of science might reveal a number of good Illustrations. The value of the chapter lies in the understanding which I t provides of the way mathematics works with science. There I s no particular value in requiring pupils to solve many numerical examplee using the principle of the lever unless they need practice in computation. It is d i f f i c u l t to test, in the usual way, t h e ideas the chapter I s designed t o present; but a few sample questions f o r the chapter are included.
Answers t o Exercises 14-6 1.
Some'seesaws are made so that they can be s h i f t e d on the fulcrum in order to alter the dlatances of the riders Prom the fulcrum and still pemlt them to s i t at the ends of the seesaw, In this way, riders of different weights can balance each other without moving from the ends of the seesaw.
[pages 619-6211
2.
g
3.
75 pounds
4.
1 7T ounces
6.
With a lever on a fulcrum a t t h e proper place, a 90 pound g i r l can lift 1,000 pounds, For example:
yd.
4b IN. I 1000 1
50 IN,
7.
About
8.
37.5
9.
1100 pounds
lo.
2
ft. f r o m t h e fulcrum
(1.8 ft.)
pounds
(a) and
(b)
Sample Test Questions (~nswersare at ~ n d )
1.
If two objecta at the opposite ends of a seesaw balance, and the weight of one 18 halved, what muat be done t o the o t h e r to preserve the balance?
2.
If two boys on the oppoaite s i d e s of a seesaw balance, and one moves twice as f a r as he was from the fulcrum, while the other does not move, what happens t o the seesaw?
Find the mi~singvalue In each case:
boo4
8 FT.
A
24 FT] ?
[pages 621-6221
I
115 LB.[
.p
140
9.
1
P
A
1-
L8.1
3 F Z 50 ~ TON
12 FT.
Ld
6 IN.
P
25 IN.
I
~ W OL D . ~
A
Tell how to balance a 60 pound boy a n d h i s 40 pound sister if she sita 6 ft. from the center support of the seesaw.
10,
Complete t h e table
11.
If WD = 30, what happens t o D as W What happens to W as D increa~es?
lncreaaea?
12.
If WD = 75 what is D when W when D is 10?
13.
Which of the points below are on the graph of
WD
=
151
What is
7)
(4 (8, 8)
(10, 6)
(b)
S t a t e one or two other formulas (similar t o
WD = 100) in which one value decreaees as the other increaaes.
Answers to Sample Questions 1.
Halve the weight of the other or double the comesponding distance from the fulcrum.
2.
It l a out of balance and the boy who d i d not move I s higher than the other,
3.
1000 lb.
4.
60
5.
1 2 tons
8.
135 in.
9.
Seat the boy
-s
10.
W
633
(a) (99 14.
I s
inches
W D
4 ft
. from the fulcrum.
12
36 decreases.
4 W
4:
3
864
decreases.
11.
D
3
(a)
14.
rt = 100 In = 100 r p = 100 (where any conatant may be used where 100 appears. )
