Mathematics for Junior ~igh School, Volume 2. Teacher's Commentary, Part II.
Prepared under the supervision of the. Panel on Seventh and Eighth Grades.
MATHEMATICS FOR JUNIOR HIGH SCHOOL.. .
I
I
1 -
MATHEMATICS
m p Y GROUP
School Mathematics Study Group
Mathematics for Junior H~ghSchool, Volume
Unit 8
2
Mathematics for Junior ~ i g School, h Volume 2 Teacher's Commentary, Part II
Prepared under the supervision of the
Panel on Seventh and Eighth Grades of the School Mathematics Study Group:
K. D. Andcrson J. A. Brown Lenore John
B. W. Joncs P. S . Jones J. R. Mayor P. C. Rosenbloom Very] Schult
Louisiana State University University of Delaware University of Chlcago University oCColorado University of Michigan American Association for the Advance menr oi7 Science Universiry of Minnesota Supervisor of Mathematics, Washngton, D.C.
New Haven and London, Yale University Press
rytio, 1961 by Yalc Universit>'. Cnp?rlght Printcii iri the U11irc.d 5tstcs of A~ncricn.
All rights rcscrved. This book may not be rcproduccd, in whole or 111 part, i11 any fbrm, without writtcn prrlnisrion f i r m the publishcrc. Fillancia1 support for the SchooI Marhrmatics Study Group has been provldcd b?, rhr N a t l i l ~ ~ a l Scicncc Foundation.
The preliminary e d i t i o n of t h l a volume was prepared at a writing s e s s i o n h e l d a t t h e U n i v e r s i t y of Hichigan durlng the summer of 1953. Revlslons were prepared a t Stanford Unlversitg i n t h e summer of 1960, taklng i n t o account t h e classroom experience w i t h t h e preliminary e d i t i o n during t h e academic year 1959-60. This e d l t i o n was prepared at Yale Unlversity i n the summer of 1961, again t a k i n g i n t o account t h e classroom experience w i t h the S t a n l ~ r dedition d u r i n g t h e academic year 1960-61. The following la a l i s t of a l l t h o s e who have participated i n t h e preparat:on of this volume.
R.D.
Anderson, Louisiana S t a t e University
B.H. Arnold, Oregon State College Brown, University of Delaware
J.A.
Kenneth E. Brown, U . S . Office of Education Mildred B. Cole, K.D. Waldo Junior Hlgh School, Aurora, Illinois B.H.
Colvln, Boeing S c i e n t i f i c Reaearch Laboratories
J.A.
Cooleg, U n i v e r s i t y of Tennessee
Richard Dean, California Inatitute of Technology
H.M. Gehman, University or Buffalo L. Roland Qenise, Brentwood J u n i o r Hlgh School, Brentwood, New York E. Glenadine Gibb, Iowa State Teachers College
Richard Good, University of Maryland Alice Hach, AM
Arbor P u b l i c S c h o o i s ,
Ann Arbor, Michigan
S.B. Jackson, University of Maryland Lenore John, U n i v e r s i t y High School, U n l v e r s i t y of Chicago B.W. P .S
Jones, University of Colorado
. Jones,
University of Michigan
Houston Karnes, Louisiana S t a t e University Mildred Kelffer, C i n c i n n a t i Public Schools, C i n c i n n a t i , Ohio
Nick Lovdjieff, Anthony Junior High School, Minneapolis, Minnesota
J.R. Mayor, American Association for t h e Advancement of Science Sheldon Meyers, Educational T e s t i n g S e r v l c e Muriel M l l l a , Hill J u n i o r High School, Denver, Colorado P.C. Rosenbloorn, U n i v e r s i t y or Minnesota E l i z a b e t h Roudebush ! d i t c h e l l ,
Seattle Public Schools, Seattle, Washington
Very1 Schult, Washington Public Schools, Washington, D.C. George Schaefer, Alexis I. DuPont Klgh School, Wilmlngton, Delaware Allen Shields, Unlversity of Michigan
Rothwell Stephena, Knox College
John Wagner, School Mathematics Study Group, New Haven, Connecticut Ray Walch, Weatport Public Schoola, Weatport, Connecticut G .C
A.B.
. Webber,
University of Delaware
Willcox, Amherat College
CONTENTS i !
NOTE TO TEACHERS
......................
ix
.............
219
CIiapLer
7.
PERI::UTATIO?JS AND SELECTI3NS General Remarks 7 - L The Pascal Triangle
.................. ............. . 7- 2 . Permutations . . . . . . . . . . . . . . . . ......... . 3 . Selections o r Combinations 7- Lk . Review of Permutatione and Selections . . . . Sample Q u e s t i o n s . . . . . . . . . . . . . . . . . . 6. PROBABILITY . . . . . . . . . . . . . . . . . . . . . ................ 1
8.1 .
Chance Events 3- 2 b p i r i c a l Probability 8- 3 P ~ ~ u b a b i l l toyr A or B 8- 4 Probablllty of A and B 8- 3 . Summary Sanple Q ~ e s t i o n s
..
.. .. .. .. .. .. .. .. .. .. .. .. ........... ...................
.................. 9. S I I . I I i A R TRIAI\JGLES A l K VARIATIOIJ . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . 9- 1 . Indirect PIeasurenent and Ratios . . . . . . . ............ 9- 2 T r l g o n o ~ . e t r i cRatfos 9- 2 . Reading a Table . . . . . . . . . . . . . . . 9- 4. Slope of a Line . . . . . . . . . . . . . . . 3- 2. Sltnilar Triangles . . . . . . . . . . . . . 9- 5 . Scale nrawlngs and I.la?s . . . . . . . . . . . 9 - 7. Kinds o r v a r i a t i o n . . . . . . . . . . . . . - 8. E i r e c t Yariation . . . . . . . . . . . . . . - 9 . Inverse 'Jariation . . . . . . . . . . . . . . 9-10. Other Types of Variation (optional) . . . . . 3-11. :urul~arj and R e ' r l e w . . . . . . . . . . . . . Saaple fmestions . . . . . . . . . . . . . . . . . . 1 7 . T.iO;.- ..iET..IC CE03;ETEY . . . . . . . . . IC- 1 . I n t r o d u c t i o n . . . . . . . . LC- 2 . Tetrahedrons ........ 10- 3 . S i n ~ p l e x e s . . . . . . . . . . 10- 4 . i l o d e l s of Gubes . . . . . . . 1 - 5. Polj.bed-ons . . . . . . . . . 10- ~5. One -Dimensional P o l y h e d r o n s . 1 0 - 7 . "TO-Dlxensional Polyhedrons . iO - 8. ?l-ree -iXriensional P o l y h e d m n s 10- 3 . Countiny VertLces, Ecges, and Eulerfs ?ornula . . . . . . . Sanp l? ?we s tions . . . . . . . . . . r
219 221 227
233 227 258
241 2k1 2i7
250 253 260
252
259 269
276 379 284 2% 291
296 300 303 307 312
315
Chapter
.............. . . . . . . . . .. .. .. .. .. .. .. .. .. .. .. . . .............. 11- 3 . RighC Prisms . . . . . . . . . . . . . . . . 11- 4 . Q b l i q u e Prisms . . . . . . . . . . . . . . . 11- 5 . PJI-arkids . . . . . . . . . . . . . . . . . . 11- 5 . Volumes of Pyramids . . . . . . . . . . . . . 11- 7 . Cones . . . . . . . . . . . . . . . . . . . . Sample Questions . . . . . . . . . . . . . . . . . . 2 . TiiE S P!IERE . . . . . . . . . . . . . . . . . . . . . . 12- 1 . I n t r o d u c t i o n . . . . . . . . . . . . . . . . 2 - 2 . Great and Srnall C i r c l e s . . . . . . . . . . . . 12- 3 . Properties of Great C i r c l e s . . . . . . . . . 12- 4 . L o c a t i n g P o i n t s on the S u r f a c e of the E a r t h . 12- . Volime and Surface Area of' a Spherical. S o l i d . 1 - 6 . Finding Lengths of Snall C i r c l e s . . . . . . Sanple Q u e ~ t i o n s . . . . . . . . . . . . . . . . . . 13 . I A T KOBODY ?i!OWS ABOUT ;sIATHEI:IATICS . . . . . . . . . 1 -1 Introduction . . . . . . . . . . . . . . . . 13- 2 . A Conjecture A b o u t Pllisxs . . . . . . . . . . 13- 3 . Distribution of P r i n e s . . . . . . . . . . . i3- 4. Pr~blemso n s p h e r e s . . . . . . . . . . . . . 13- 5 . A Problenz on C o l o r i n g \.laps . . . . . . . . . 13- 6 . Tb.e Travelling Salesnan . . . . . . . . . . . 1 3 - 7 . A p p l i e d Physical Problems . . . . . . . . . .
11
.
VOLUdES AXD SlJjiPACE AmAS General R e m e r l i s 11- 1 Areas of' Plane Figures 11- 2 Planes and Lines
13- 8
.
~onclus.ion
.................
NOTE TO TEACHERS
Based or. t h e t e a c h i n g expcrlcnce u f o v e r J 100 juni3r high s c h o o l t e a c h e r s in all parts of the c o u n t r y and t h e estimates of a u t h o - s
of the r e v i s i o n s ( i n c l u d i n g j u n i o r h i g h s c l ~ i o l
teachers), it is recomxend~dt h a t t e a c h i n g tlrne fcr l a ? t 2 be as follows: Chapter
Approximate nu:nber orl d a y s
Teac:*.ers a r e urged t o try n o t tc e x c e e d these a p p r ~ x l n a t c time all~smcntsso that pupils will n o t miss t h e c h a p t e r s at Lhe end of t h e c o u r s e . Some classes w i l l bc a S l e tu I ' l r i l s h z e r t a l n chapCers Tn l e s s t h a n t h e estimated tine, T l i r o i ~ ~ h o ut th e t e x : , p r o b l e r l s , t g p l c s , and s e c t i o n s !uY,lck~ were d e s i g n e d for t h e better s t u d e n t s a r e indicated 'cy an asterislc (*) I t e l l a st;arred In t h i s manner s h o u l d be u s e d or o i n i t t e d aa a neans of adjusting the a?proximste t i m e s c h e d u l e . Several s u p p l e m e n t a r y u n i t s arc available for s e l f - s t u d j by e x c e p t l o r i a l s t u d e n t s or for class u s e i n advanced sections.
.
Chapter 7
PEF2*WTkTIOF:S AND SELECTIONS
General Rem r k s
The s t u d y of pel-mutatlons and selec'Ylons ( o r comblnations) is a " f r e s h start" f o r many s t u d e n t s because n e a r l y zll t h a t Zs requ'red as a background is a wurking knowledge of t h e f o u r operations with numbers and an understanding of f o m u i a s . It m i g h t be well to c a l l this f a c t t o the a t t e n t i o r - cf certain p ~ p l l swho aay be a b l e b u t n o t always w'llin~. Committee 3rl-angernents a r e used in t h e beginning became t h e faniliarity of p u p i l s w i t k t h i s t y p e o f s i t u z t i o n ensures t h a t the o n l y new Ideas introduced a r e t h o s e of a p u r e l y m a t h e m t i c e l nature, Othcr v a r i e d and i n t e r e s t i n g applications o f t h e mathemati c a l p r i n c i p l e s Learned I n t h l s c h a p t c r a r c I ' o u d in t n e e x e r c i s e s . The authors p r e f e r tc:use the word " s e l e c t i o n " r a t h e r t h a n "combination" (although both words a r e f o u n d in t h e t e x t ) . It 13 bclicved t k a t "selec:ion" would be a nore n a t u r a l and meaningful t e r n f o r those working w i t h t h e s e ideas f o r t h e f i r s t t i m e . Pascal's t r i a n g l e , as introduced in S e c t i o n 1, n o t o n l y is an e x c i t i n g application of s e l e c t i o n s , b u t also serves o t h e r p u r p o s e s . It Is an exzellent device for h e l p i n g students r e c a l l quickly the number of s e l e c t i o n s t h a t can be made from n t h i n g s r at a time. Havllng t h i s Ilnformatior. about ~ e l e c t l o n sread1l.y available speeds up c l a s s d i s c u s s i o n and allows f a r development of concepts without i n t e r m p t i n g t h e t r a l n of thou@it with c o n s t a n t , and sometimes i r r e l e v a n t , calculations, Also the p a t t e r n s involved in t h e triangle s ~ g g e s lc e r t a i n i n d u c t i v e conclus ions about the runbe? r e l a t i o n s among t h e v a r i c u s e n t r i e s . Contrary t o what is prcbably the usual procedure, combinations ( s e l e c t l o n c ) a r e used as an ?ntroduct;lcn to permutations. By ordering t h e meabers oi? a subcommittee and a p p l y i n g the p r l n c l p l e s
learned in the study of s e l e c t i o n s , students a r e prepared to accept and u n d c r c t n n d t h e f o l l o w l a g p r o p e r t y : If n is a counting number, then t h e number o f permutations of any s e t of n numbers is n!. The st-ddy of pemnutationn In t h i s chapter. does not Include those
grou?s in which only a p a r t o f t h e items are repeated, such as the permutations of the l e t t e r s of the word, " ~ ~ s s l s s l p p ~ . " Some s e c t i o n s contain a l i s t o f Class Exercises which are planned for developmental c l a s s d i s c u s s i o n s and E r e not assignment exercises. The intention is n o t to l i m i t discussion t o the s p e c i f i c questions l i s t e d . It is e x p e c t e d t h a t t h e teacher will use this list of q u e s t i o n s merely as a guide. However, each q u e s t i o n is c a r e f u l l y ? l a c e d f o r develcpmental purposes f.n the o r d e r g l v e n and I s p o i n t e d t o w a r d a partis~laridea i n t h e develcpmental sequence. I f this material is r e l a t i v e l y unfamiliar t o the teacher, t is recomended t h a t t h e sequence b e adhered to as given. The exercises a t the end of each part of a sectlon are thought t~ be suf Pic lent for one ass lgnment It is hoped t h a t the study o f t h i s c h a p t e r will widen p u p i l s ' appreciation f o r mathematics as a s c l c n c e . The InvestZgation o f Pascal's t r i a n g l e begins wlth a r-early complete body of facts, i n c l u d e s the discovery and observation of quantitative r e l a t i o n s , inductively s u g g e s t s c e r t a i n p r i m i p l e s , v e r i f i e s these deductively, and d e r i v e s generalizations. It is thought t h a t t h e choice of Pascal's t r i a n g l e m i g h t be a good one because t h e body o f f a c t s is small, the quantitative relatiorls are of s very simple nature, t h e discovery of the p a t t e r n s involved is not difficult, and the number of experiments t h a t may be performed L s ample. It is a l s o hoped t h a t the study of t h l s chapter will prcve to be a source of intelleztual s a t i s f a c t i o n and furl f o r t h e pupiL and will h e l p t o s t r e n g t h e n his r e a l i z a t i o n of t h e power of mathematics. He may be impressed by the v a s t amount of hard work t h a t would b e required to determine the third t e r m in the two-thousandth l b e of Pascal's triangle i f h e were actually to write down all t h c
.
numbers i n a l l t h e rows. He should be e q u a l l y impressed by t h e ease w i t h which t h e same information may be obtained by using
F o r b e t t e r students, referehce may be made t o S e c t i o n 1 and t o P r o b l e m 6, Exercise 3b of t h e supplementary unit on ' ' ~ i n i t e Differences. !I A time allowance o f 13 class periods is s u g g e s t e d f o r t h i s chapter.
7-1. The Pascal Triangle In this s e c t i o n , as in those t o f o l l o w , t h e ideas a r e developed through carefuily s e l e c t e d c l a s s d i s c u s s i o n e x e r c i s e s . The questions in these exercises should be answered by s t u d e n t s as t h e y mantpulate o b j e c t s cr w r i t e symbols f o r the elements of a s e t . Tabulating and counting a r e important key n o t i o n s t o be used t h r o u g h o u t t h e section. Many students will p r e f e r to u s e semiconcrete r e p r e s e n t a t i o n s p r o v i d e d by writing symbols on paper b u t some use might a l s o be made of c o n c r e t e o b j e c t s t o be grouped and counted. Perhaps t h e t i t l e f o r t h i s s e c t i o n w i l l not be very u s e f u l u n t l l the s t u d y of the section has been completed. If questions a r e aslced about the title, no harm will be done in displaying a t r i a n g l e of t h r e e o r f o u r rows d u r i n g the f i r s t dZscussion session. Some of the i n t e r e s t i n g properties of Pascal's t r i a n g l e might even be p o i n t e d o u t at t h a t t i m e One of t h e important ideas to be b r o u g h t out i n this study is t h a t if 3 o b j e c t s have been s e l e c t e d from 5 then a selection o f 2 o b j e c t s has a l s o been made. The symmetry of Pascal's t r l a n g l e is based on t h i s p r o p e r t y . The student s h o u l d clearly understand t h a t t h e number of ways o f selecting 17 o b j e c t s from a s e t of 26 o b j e c t s is exactly the same as t h e number of ways of selecting 9 o b j e c t s from a s e t of 26 o b j e c t s
.
.
C pages
281- 288 ]
S t u d e n t s should be encouraged to observe that each entry in Pascal's t r i a n g l e is t h e sum o f t h e two nwnbers n e a r e s t to it on t h e preceding line. The better students, by t h e end o f t h e c h a p t e r a t l e a s t , may be able to s t a t e why this p r o p e r t y is true. One way of showing t h i s property follows. Consider 5 lines of ?ascalls triangle:
5 t h i n g s taken t h r e e at a time t h e sum of t h e number o f combinations of 4 t h i n g s taken Why is t h e number of combinations of
at a time and the number of combinations o f 4 things taken 3 at a time? Think of 5 l e t t e r s A , B, C, D, %. I? we s e l e c t 3 of the 4 l e t t e r s B, C, D, E, then of course, we have p a r t of t h e s e t s of 3 of t h e 5 l e t t e r s A , 3 , C, D, E. T h e number w e 2
have is
(:)
.
The ones w e have n o t c o u n t e d a r e t h e groups of
3
is one l e t t e r of the group. Any group of 3 i n c l u d i n g A c o n s i s t s o f A and 2 o f t h e 4 l e t t e r s B, C, D, E . The 4 nw-ber of groups of 2 frorn four letters is ( 2 ) . Hence,
of which
A
Or in more general terms, let us try t o show
[pages 281-2883
-+, ,n
'
-
3
1)
+
(n
1)
( n - l)(n
-
2)(n
3)
1 . 2 - 3
+
(n
-
2)
.g
But
We have proved only a special case. O t h e r s p e c i a l cases could be pruved s l r n i l a r l y . Tn o r d e r to prove tine general property,
we would have to use mathematical inductLon. The b e t t e r s t u d e n t s may a l s o b e Interested in other properties of Pascal's t r i a n g i e . The sum of the numbers in any row i s always
a power o f 2 ; n o t conslderlng t h e I t s , the greatest common fact07 i n any prime-numbered row I d the number of the row*; the third term in each row can be predlcted by u s b g the f o r m u l a ,
and the fmrth term in each row can be predicted by
uslng the formula, PascaL1s triangle
,
Reference w i l l be made t o
In the s e c t i o n s to follow and even though no
s t u d e n t observes these p r o p e r t i e s now, several may do so as the work progresses , P a r t i c u l a r l y in t h l s f l r s t s c c t l o n the treatment is intended t o be q ~ l t eInformal. It would be better t h a t no formulas be introduced i n t 3 the general c l a s s discussions. However, the s t a t e ment of gecerallzations whlch can be illustrated In s p e c i a l caves should be encouraged.
[pages 281-2863
Answers -t o Class Exercises 7-la I.
EA,D,EJ
2.
Yes
7*
B
8.
4
9.
10.
3
11.
b3
Or
12.
{B,c,D)
13.
A
and
14.
10
15.
6
16.
4
17.
Ec,A~;
and
D
a r e on t h e same number of committees.
tC,D3;
IC,B3;
or
5
1
7 D
W,EI
Answers t o Exercises 7-la 1.
2.
LA,B,C,E],
(a)
IA,B,c,DI,
(b)
1
(c)
I A ~ ,( ~ 3 ,
(d)
Same.
(4
1
(a)
1
(b)
4
( 4 IKI,
1
IC3,
, CMI,
IA,B,D,E), ID],
IN]
[pages 281 -283
3
CA,C,D,EJ,
CB,C,D,E)
(d)
The three-menber committee is ccmgosed o f n l l t h e club members who are n u t Ln the corresponding one-memjer co~rlmittec.
3.
(e)
L
(f)
IKyLl,
I
, [K,M,N],
K
{K,L,M]
CK,Il,MI,
K
(L,M),
[L,N~,
CM,N~
(a) T h e r e is one committee with t h r e e nembers, namely, [ P , Q , R ] ; t h e r e are t h r e e committees wl th two members, nernely, I Q , R ) , I P , R ) , IP,Q]; t h e r e a r e three cflmmittees w i t h one member, namely, (PI, , (R]
.
{u,v],
[v] .
4.
The conunittees a1.e
5.
If t h e c l u b r.ember is Z, t h e o n l y c o m l t t e e is
{u],
{z] .
Answers -t o Class 3 x e r c i e e s 7-lb 1.
One c o m l t t e ? with four members.
2.
One committee with no members at a l l .
11.
One committee w i t h no merrbem TKO committees w i t 5 one member. One committee w i t h t w o members
.
.
5.
1, 1. The second 1 0 t h r e e members each.
6.
6
7.
Two vacanc l e s
8.
AISO
9.
4
.
t e l l s how many ccmrnittees w i t h
Four.
6.
[pages 283-285
I
10,
Three vacancies.
11.
3,
Answer8 2
3.
4,
4.
4.
to Exercises
7-lb
members
(a) The number of possible committees with formed from a club of 4 members.
2
( b ) The number of p a s s i b l e committees with. formed from a club of 3 menbers.
1 member
(a) The number of p o s s i b l e committees w i t h formed from a club o f 6 menbers .
4
members
( b ) The number of possible committees w i t h formed from a club of 5 members.
3
members
(a)
The f i r s t
(b)
The first
1 5 in
the
6th
row.
5 in t h e
5th
row.
5. (a,b) The left-hand column answers Part column Part (b).
(a), the right-hand
I! !
(c)
By adding 10 e n t r y and the
and
5
"5"
is the r i g h t - h a n d one in the f i f t h
(the
"10"
is the right-hand
POW).
(d)
There a r e
2G
IA,B,cI,
list:
o f them; t h e following is an incomplete
IA,B,DI,
IA,B,Fl,
IA,B,EI,
~A,C,D),
etc.
( e ) Yes;
20 = 10 + 10.
The f i r s t exercises should enable the s t u d e n t s tc recognfze t h a t in c e r t a i n s i t u a t i o n s o r d e r or arrangement of elements in a
-
While A , B, C is one s e l e c t i o n of t h r e e elements, t h e s e three elements may b e ordered or arranged in six d i f f e r e n t ways. Again the s t u d e n t s s h o u l d be urged to write out all o f these arrangements o r permutations. It is not t o o tedious t o w r i t e o u t a l l of the arrangements o f 4 elementa but, beyond f o u r , time s p e n t in making s u c h llsts would be better used in o t h e r nays. Particularly the experience o f l i s t i n g the 24 permutations of 4 l e t t e r s s h o u l d convince t h e student of t h e value in being systematic i n constructing t h e list. However, no p a r t i c u l a r systematic method, as s u c h , is recommended f o r use. Tn t h i s s e c t i o n the development o f formulas is begun. These formulas In themselves a r e n o t so important as a t o o l f o r problem solving as i s the process of developing the formulas and t h e experlence in using symbols, such as, n, (n 1 , and (n - r) , t o express generalizations. s e t is important.
-
[pages 287-303
1
It may be pointed out that one flnds the symbol well as t h e symbol
' I P ~ , ~t t
" n ~ r " as
in mathematical literature to represent
the number of permutations of n elements o f a s e t taken r a t a time. In permutation problems, whether or n o t an element o f a s e t can b e repeated in arrangements must be lmown. The f o r m u l a s , of course, apply t o permutations w i t h o u t repetitions. One o f t h e i n t e r e s t i n g applications is in counting the automobile l i c e n s e p l a t e s which can be made by c e r t a i n rules of s e l e c t i o n and arrangements f o r 10 d l g i t s and/gr 26 l e t t e r s . F o r example, in 1960 I n t h e s t a t e o f Illinois more than 3 million cars were licensed. The license p l a t e s d i s p l a y e d numerals ~f s e v e n d i g i t s . What new method will p r o v i d e f o r l i c e n s e plates f o r 4,000,000 cars and yet involve fewer d i g i t s and l e t t e r s on a g f v e n license p l a t e ? A t t h e close of this section teachers m i g h t encourage individual s t u d e n t s t o submit plans that I l l i n o i s m i g h t use and ask the class to discuss the advantages and disadvantages o f several of the plans. Teachers and students will observe that t h e Multiplication Property has been used before it is introduced. T h i s order was chosen so t h a t the p r o p e r t y would b e better accepted by s t u d e n t s as reasonable and important.
Answers -to Class Exercises 7-2a
First way
B
Secretary E
Second way
E
B
Treasurer 2.
[pages 287-303]
-
Answers t o Exercise3 7-2a
1.
2.
(a)
(a)
Chairman
A,
Chairman Chairman Chairman Chairman Chairman
A,
F,
Treasurer H;
H,
Treasurer
Secretary H, Secretary Secretary Secretary
A, A,
F,
Runners
Runners
Runners
1 2 3 4 P R S T
1 2 3 4 R P S T R F T S
1 2 3 4 S P R T
1 2 3 4 T P R S
S P T R
T P S R
P T S R
R T P S
S S S S
T R f S
P T R S
R S P T R S T P R T S P
P S R T P S T R
R R T T
P T P H
T P R P
T R S P T S R P
T S P R
Presents
Presents
1 2 3
1 2 3
1 2 3
A B C
B A C
C A B
A C R
B C A
C B A
Presents
In 6 d i f f e r e n t w a y s .
7.
F;
Treasurer A; Treasurer H; Treasurer F; Treasurer A .
Runners
P A T S
1;.
F, F, H, H,
Secretary Secretary
23.
There is only one c o r r e c t way.
[pages 289-2911
No.
Answers 1.
7.
to Exercises
7-2b
(a) 3 (b)
2
(a)
n
(b) n
(c)
-
3
2
1
(c)
Yes. There will always be one less chafr f o r the second person t o choose because one c h a i r is filled with the first person,
(d)
n
(e)
n
(f)
n
. (n (n (n
1) I)
1)
. .
(n (n
2) 2)
.
(n
-
3)
Answers to Exercises 7-2c
1.
2.
(a)
6.5.4.3-2.1
(b)
7-6.5-4*3*2.1
(c)
10.9-8.7.6.5*4*3.2-1
(d)
15-14.13.12*11*10*948*7.6.5-443.2.1
(a)
7
(b)
6
-
(61)
(c)
10
(5:)
(d)
12
Ipages 294-297 1
I9!)
.
Ol!)
-
Answers to Exercises 7-2d 1.
7 . 4 = 2 8
[pages 297-3003
- Exercises
'(-2e
Answers t o
P49,7 2.
4.
6.
7.
(a)
12.11610
(b)
(12*11*10)~(~*8.7~6-5*4*3.2.1)
(c)
(12m11-10)
(3)
20-19*18.17
(b)
(20.19-18-17) (16*15.14- 1 3 . 1 2 - 1 1 ~ 1 0 ~ 9 ~ 8 - 7 ~ 6 ~ 5 * 4 ~
(c)
20!
{n
-
[a)
Use
(b)
Hence,
[a)
26-25.24423.22
>
'n,4
-
-
7.. (
n
-
[n
-
r
( ~ o t et h a t t h e rlumber
1
=
r.)
P)!
n!
pn,r -
( b ) 26! =
60.50.24
1600.
+
r + 1) + 1
26.25.24-23.22 - 13*:2*1,1
>
1 Z ! = P12,12
=
P20,20
0
'n,r = ( n - 1) n of f a c t o r s here is
(c
lo.
# ( 9 - 8 - 7 - 6 * 5 *03m2.1) 4
Smallest
n
is
3
8.
P8 9
[pages 301-3021
=
91%l3 days.
li
=
1,680
7-3. Selec.t;icns o r Combinations - The s t u d e n t s should now be raady to develop formulas f o r use in determining t h e number o f p o s s i b l e selections of n o b j e c t s taken r a t a time. Their worlc w i t h Pascal's triangle in 7-1
and with permutations i n 7-2 is of c o u r s e v development in t h i s sect! on.
~ Lmportant y f o r the
We u s e t h e n a t a t i o n ,
(
, which is
m o s t widely used in c u r r e n t mathematical literature,
However, it is well a l s o Ccr t h e teecher to c a l l a t t e n t i o n t o o t h e r symbols used f o r t h e same i d e a s , s u c h a s , 'I %,r I1 and 'lnC r."
It may be d e s i r a b l e to g i v e several mare examples, ic special Cases,
o f the l'ormula
n(n
-
l)(n
-
2)..,.*(to 1.!
r factors)
before attempting to develop it in t h e g e n e r a l c a s e . In some classes t h e t e a c h e r may choose to designate the r - t h f a c t o r of the numerator a s {n - r + 1) T h l s s h o u l d be done only if students can understand why the p a r t i c u l a r choice is made, I n p r a c t l c e more people count r f a c t o r s in the nunerator t h a n t h i n k what the last f a c t o r is by u s l r l g ( 3 r + 1) S t u d e n t s who h z ~ ecompleted this c h a p t e r w L t h l l t t l e d i f f i c ~ l t y and considerable interest should be encouraged to consider additlona1 generalizatFons aSout Pascal's t r i a n g l e , some cf whLch are s u g g e s t e d in the next to the last paragraph of t h i s Commentary, SectIan 7-1. Another problem in which some s t u d e n t s may b e i n t e r e s t e d is associ2ted w i t h the sequence of trlengular numbers.
.
-
[pages 303-3081
.
The dots tn t h e following designs a r e arranged in equilateral triangular fashion. (a)
How many d o t s a r e t h e r e in each design shown? (1, 3 , 6 , 10, 15, 21)
i n the answer to ( a ) r e l a t e d to the (Third entry in each row.)
( b ) How are the numbers
Pascal T r i a n g l e ? (c)
If more designs of the same type and of successively l a r g e r size were drawn, how many dots would be in t h e eightieth diagram?
(3240)
( 500,500)
Id)
How many d o t s Fn t h e thousandth diagram?
(e)
I n what s p o r t IS t h e f o u r t h diagram used? a
4
0
4
. .
. .. 0
o ow ling)
.
a
v
w
a m .
. . * a
.
. a m . *
I
*
I
e .
.
. t .
.
I
.
a . . . . .
.
1, 3, 6, 1 , 5 , 2 are c a l l e d triangular numbers. See t h e Supplementary Unit on F i n i t e D i f f e r e n c e s The ability of t h e mathematicfan t o p r e d i c t the f u t u r e w i t h a maximum degree o f c e r t a i n t y is a n o t h e r way in which t h e power of mathematics may be demonstrated. The formula n n + l
The numbers
.
may be used by t h e s t u d e n t t o make a p r e d i c t i o n as t o how many dots will have to be mde t o construct t h e one-thousandth diagram slmilzr to t h o s e associated w i t h the sequence o f t r i a n g u l a r numbers. This a c t i v i t y , while seemingly of little s i g n i f i c a n c e , is parallel to the work of the mathematician who p r e d i c t s that an airplane, s t i l l on t h e drawing boards, will, f i v e years hence, carry a payload of five tons a t a speed of 1650 miles per hour a t an altitude of 50,000 f e e t znd, at t h e end of t h e flve-year p e r i o d be shown c o r r e c t in every d e t a i l . Mathematics, indeed, m i g h t be r e f e r r e d to as a "crystal b a l l " !
[pages 303-308
I
Answers
i*
2 Zxercises
7-3a
(a)
The number of' comLLnzdt;ions or 6 o b j e c t s taken 2 a t a time; t h e number ui' permutations of 8 o b j e c t s t a k e n zt s t i m e ; fifty-twc f a c t o r i a l ; the number of combinations of 52 t h i n g s t a e n 13 zt a t i m e ; t h e number of pemutaticns of 9 things taker1 7 at 2 t F m e ; e i g h t - f o u r t k - s .
(a) 15 and
15
(d)
3
a d
3
(b)
35
and
35
(el
28
and
28
(c)
56
and
56
(
s F-
=
.
The d i s c u s s i o n in S e c t l o n 1 concerning t h e number
committees determined by inclusion a n d e x c l ~ s l o ns h o u l d help to explain t h e r e s u l t . of
-
The p r o d l ~ c t n ( n 1) is t h e number of ways t h a t n t h i n g s can be arranged 2 a t a t i m e , Thlts includes all pairs [AB, X C , AD, 8 A , CA, DA, . . . ) and dlsticguishes betweer AB and Bk, AC and C A , e t c . Hence, each of t h e p o s s F b l e selections is counted twlce. Thm, t h e number of distinct
.,.,
s e l e c t i o n s i~
-Q+Q.
---
Answer8 to Exercises 7-3b
4.
6.
13.
(a)
(i2)
)
%ice as many,
=
10 weeks and
63.
li = 66
12
Note t h a t
2
132 =
Pig, :,
days more.
26
-
1 = 63,
4.6-3) (-----. 7
and t h a t
[pages 307-3081
D
I
7-4.
-
Review of Permutations&a
Selections
Answers t o Exercises 7-4
-
b
or
6.5.4-3.2-1
1.
Pt,6
2.
2
3.
*'36
4.
3 0 7 . 5 or 105 c a r s .
5*
P5,5 o r
or 6
3
or
6*
or
words.
720
outfits.
96495 o r 4560 hand-shakes 7-
5!
or
orders.
120
14m13'12m11'10
1.2.3.4.5
Or
or
6!,
or
8
7
7'
'8,3
8.
26
9.
9m10.10-10*10 o r
26 o r
lo.
Permutations.
11.
Selections:
676
6 or
2002
or 336
committees.
simals.
c a l l letters, 90,000
or
license plates,
8 - 7 * 6 * 5 - 4 , or
6720
pa.tterns.
I (5)
or
!
. 6 5.4 .
or
56
each.
2
(a)
10
Ib)
[pages 309-3103
10
s e t s of
5
books
Sample Questions
(Ariswe~sa r e a t t h e end.)
I.
PL11 i n w i t h the pi-ope- number.
2.
In how many ways can s i x s t u d e n t s be s e a t e d in s i x chairs?
3.
What is t h e second sumber in the
14th row z f t h e Pascal
Trtarlgle as it was shown ir, t h i s c h a p t e r ? JI
.
s t a t i o n s . Low rnany different kinds of t i c k e t s a r e needed tu p r o v i d e t i c k e t s between any kdo of the s t a t i o n s , If t h e same k3.nd of ticlcet is good ir. e i t h e r d i r e c t i o n ? On a c e r t a l n rallway there are
15
5.
In a b a s e t a l l league o f 10 teams, each team plays 17 games w i t h each of 5hc o t h e r s . How aany g m e s are played eacn season?
6.
How many committees w i t h group cf nine men?
'{.
How many n'mbers are there in the 13th T r i a n g l e as showrl Ln t h i s chapter?
8.
12 rrienibers in a club, how many members will be on the zommittee t h a t can be mad^ up in the Largest numter of ways? Yow many wzys c a n t h i s be done?
If t h e r e a r c
4 members can
be formed from a
row 3f the Pascal
9.
I n how many ways can t h e f'rst t h ~ e eplaces In a race be decided if t h e r e are 8 dogs In the race? ( N O ties take place .)
10.
Four students a r e chooslng among 8 colleges. I n how many ways can the s t u d e n t s choose a c o l l e g e t o attecd?
Answers t o Sample Questions 1.
(a)
L
le)
3
Chapter 8
PROBABILITY
NFne or ten teaching days a r e recommended f o r t h i s c h a p t e r .
8-1.
Chance Events
Those r e s p o n s i b l e f o r t h e p r e p a r a t i o n of th:s textbook b e l i e v e t h a t t h e elementary notions o f permutations, s e l e c t i o n s , and probability are an important part or the g e n e r a l education of all
j u n i o r high school students. In each of t h e s e c t l o n s and especially in Section 8-2 on empirical probability an effort has been made t o g i v e t h e student some n o t i o n of the g r e a t and increasing i m p o r t ance of probability in o t h e r d i s c i p l i n e s , b o t h in science and tn the s o c i a l sciences. The t e a c h e r may choose t o add t o this d i s cussion In c l a s s . Consideration o f the importance of probability serves as a motivating device if not extended t o o far. The s t u d e n t s , however, probably will have more i n t e r e s t in a c t u a l l y working problems than in a lengthy t r e a t m e n t of t h e significance of probability. It is preferable t h a t l i t t l e emphasis be p l a c e d on use or formulas. The problem s i t u a t i o n s should be s i m p l e enough f o r the s t u d e n t to analyze each situation and determine t h e probability by counting possible outcomes d i r e c t l y and working with t h e "samples " of a t o t a l population where d a t a on a sample only Is available. More will be accomplished by the student if he understands the b a s i c idea t h a n if he is merely able to c a t e g o r i z e situations and a p p l y
an a p p r o p r i a t e formula. t o construct tables'' t o a s s i s t them in counting. It is important that t h e students understand that each outcome l i s t e d In such a table is considered equally likely to occur. Suggestions f o r e a s y or
In
some situations students may be encouraged
n
'
systematic ways t o t a b u l a t e p o s s i b l e outcomes F n t o s s i n g colns
are g l v e c in t h e t e x t , A particular method is not important b u t the development of a systematic way of tabulating is important. If students are encouraged t o make t h e i r own models, such as cubes, c i r c l e s with s p i n n i n g p o i n t s , and so or!, f o r expermenting, they should understand t h a t such models a r e n o t n e c e s s a r i l y "honest. " Whenever a model Is r e f e r r e d to in an example or problem,! it I s always assumed t o be h o n e a t . The n o t a t i o n P ( A ) l a introduced. Whenever posslble t h i s notation should be used so that t h e students galn familiarity i w i t h it. It I s important that the teacher utilize correct terminology t o instill in the students the value of doing s o . Teachera should not, nowever, i n s i s t on exacting n o t a t i o n s or terminology if by doing so the student is d i s t r a c t e d from an important idea. The n o t i o n that the "law of averages" a f f e c t s outcomes m y be difficult to d i s p e l . Burton W. Jones * s book, Elementary Concepts of Mathenatica, pages 209-210, is an excellent teacher reference on t h i s matter. It was felt that the use of d i c e should be left to t h e teacher's discretion. A cube w i t h f a c e s , numbered 1, 2, 3, 4 , 5, and 6 or l e t t e r e d A , B, C, D, E, and F serves t h e purpose equally well. Different c o l o r s applied t o each face may a l s o be used. I t m y be t h a t "tossing coins" wFll be met with some objections. Dlsks w i t h d i f f e r e n t colored sides nay be used instead of c o i n a . It may be stressed that games of chance are u e d only because they serve as good models for discussing elementam probability. H i s t o r i c a l l y , interest in gambling led t o the first i n t e r e s t in probability. The p r o p e r l y t h a t t h e r e a r e 2n posaible outcomes if a c o i n is t o s s e d n t u e s may be developed inductively and I s interesting as well as u s e f u l . Meaning 1s added t o the p r o p e r t y if other simllar situations a r e considered, such as the p o s s i b l e outcomes if a cube is t o s s e d n times. I
i
[pages 311-3141
I n answering most probabllity questions r e l a t e d t o this property, t h e enswer is t h e same whether the coins ( o r t h e cubes) are tossed n times or in groups o f n. Some class discussfon o f t h i s would be a p p r o p r i a t e . One group of p u p i l s m l g h t be asked t o solve some of the problems, thinking of a coin being tossed n times, while o t h e r p u p i l s think of tossing n c o i n s a t one time, Results could then be compared. It is very important t h a t students understand the limitations of p r o b a b i l i t y , particularly in the elementary and somewhat oversimplified situattons in which we are using it. The paragraphs in the t e x t r e f e r r i n g to the limitations should be carefully discussed In c l a s s . Two p o i n t s cannot b e overemphasized. If 15 c o i n s a r e tossed in s u c c e s s i o n and by rare chance t h e f i r s t 1)f show heads, still t h e probability t h a t t h e fifteenth coin will show heads is
1 3. N o t e that this q u e s t i o n is quite d i f f e r e n t from t h e probability of obtaining; 15 heeds if
15
c o i n s a r e tossed, which 1s
(;)I5
The second p o i n t of caution is that if 100,000,000 coins are tossed, or even more, we can never be s u r e that one-half of them will be heads. Finally, because t h e s e elementary t o p i c s used to introduce , p r o b a b i l i t y a r e s o c l o s e l y associated with their applications, students are apt t o t h i n k of probabllity as befng the applications ( t o s s i n g coins, selecting m a r b l e s , drawing names or numbers) rather than as a part of mathematics used to descrlbe these games, and ! social o r business s i t u a t i o n s . P r o b a b i l i t y is a very important branch of modern mathematics. Experts i n probabllity a r e c a l l e d ; p ~ o b a b i l i s t s (pronounced p r o b t a b i l - l s t s ) . The rewards f o r p r o b a b i l i s t s today in research and in t e a c h i n g are very g r e a t Indeed. P r o b a b i l i t y I s an excellent f l e l d of w o ~ kf o r a young scientist of p r o e i s e to chrrose. ,
/
I
[pages 311-3141
Answers
3.
6.
(a)
-
Exercises 8-la
I Do. The p r o b a b i l i t y of g e t t i n g a head t o show is T. We should expect 4, 5, or 6 heads to show in 9 tosses.
(b)
$
(c)
Bo
(a)
NO
(b)
If
.
(b)
P
occur. 9.
10.
There a r e
Yes-
2'7
P (even number) = 4 (a)
(z)
5
=
2 5.
g4
~ ( w h l t e )=
red
from
OF
= 36
. 0
anlnanals)
2 5
= 3v or
1 P
Yes
No
( ~ o t h s rmay be teacher. )
12.
(d)
Red card and a black card = 20.
(e)
The sum of ( b ) , ( c ) ,
(a)
Two coins a t the same time.
and (d) is the same as ( a ) .
Ist c o i n
2nd coin
H T
H
H T
H T
T
Three o u t o f four possibilities have at least one coin with a head. (b) Yes. (c)
8-4.
One in each hand.
When they are tossed separately b o t h events can happen. P( A ) + P (B) = P (c) only when the t w o events are mutually exc lus i v e .
- -A - and -
P r o b a b i l i t y of
3
Again It is recommended that the treatment of t h i s new
situation be informal and that the s t u d e n t be encouraged t o "think through" a problem rather than t o use a formula. It should be clear to him t h a t t h e probability that event A and event B b o t h occur is less than P ( A ) and l e s s t h a n P ( R ) For example, i f t h e p r o b a b i l i t y that Joe will g e t an A I n mathematics is 0.9 and the probability that he will get an A i n French is 0.7 the probability that he will get an A in b o t h s u b j e c t s is less t h a n t h a t he will g e t at least one A . A p o i n t of d i s t i n c t i o n between P ( A or B) and P ( A and B) is t h a t in the former, where A and B are mutually exclusive, we r e f e r t o a s i n g l e t o s s of a c o i n , a sFngle s p i n of a wheel, o r a single selection, while in the l a t t e r at l e a s t 2 t o s s e s of a c o i n , 2 s p i n s of a p o i n t e r , or 2 s e l e c t i o n s are involved,
--
--
[pages 333-3361
.
a few exmples we ask the s t u d e n t to conclude t h a t if A and B are independent,
With
events
P( A
and
3) = P(R)
P(B)
.
course, no proof of t h i s p r o p e r t y in t h e t e x t . It is important t h a t t h e student understand t h a t P ( A and B) is t h e product o f P(A) and P ( B ) only if A and B are independent. Events A and 3 are independent o n l y if t h e occurrence o f A o r t n e non-occurrence of A have no e f f e c t whatsoever on R. An example of two e v e n t s which ere n o t independent mlght be t h e s e l e c t i o n of t w o marbles from a box c o n t a i n I n g n marbles, in which t h e f i r s t marble s e l e c t e d Is not replaced There I s ,
,3f
b e f o r e the ~ e c o n ds e l e c t i o n or d r a w i n g .
Again the material is presented in a way intended t o encourage t h e student to t h i n k through a problem situation and t o u s e his best judgment r a t h e r than to apply s formula-
-
~ n s w e r sto Exercises 8-4 1.
(a) Yes.
2.
(a)
If A,B,C F(A
(b)
One e v e n t has no e f f e c t on t h e o t h e r .
2 3
and
and D a r e independent events, then B and C and D) = P ( A ) x P ( B ) x P(C) x P ( D )
(assuming A
was played f i r s t . )
[pages 333-337 1
.
4.
both
(a)
(b)
P(both on green)
white
(c)
5. 6.
on red) =
white and
.3
2 6 =
F1
and blue) =
white)
1 6
=
1
1
=
1 51 = 6
4 . 4
1
16
=9 q = B i
~4 * $ = zWith . combinations q = s = z where
4
(2)
o r selec t l o n s :
is number of p a i r s of t w o whites
and
(9 *)
7.
(a)
Yes
8.
(a)
&
9.
(a) Not Independent
( d)
Independent
(b)
Independent
(e)
Independent
(c)
Not independent
13.
is the number of p o s s i b l e p a i r s .
Event
A:
The f i r s t man wlll s o l v e t h e problem.
Event
B:
The second man will s o l v e t h e problem.
(a)
Neither man can s o l v e t h e problem.
not
A
and n o t
1
B) = 7
.7
[pages 337-339
-- 3i;. 7
1
(b) Either man o r b o t h men may s o l v e the problem. P(A
not P(A
B)
and n o t A
and
and
6)
B)
=
2 7
=
1 7
-
7 - 14 3 = 56 3 p
Total T h F s c o u l d have been f o u n d by
1
7 - 29
-35-36'
11.
The probability is equal to one (the d e s i r e d committee may be chosen in one way) d i v i d e d by t h e total number o f ways a committee o f t h r e e can be chosen.
12.
There a r e 26 ways that t h e s i x c o i n s may have heads and t a i l s appear. There is only one way that will n o t heve at
l e a s t one head.
at
( 6 tails.)
l e a s t one head)
=
Therefore,
zb-1
6 4 - 1 - 6 3 64 -
nm
7
13.
(a) 1
14.
L i s t the possible l e n g t h s f o r a t r i a n g l e :
lo, 9 , 10, 9, 10, 3, 10, 9,
8 7
10,
7
10,8,
6 5
10,
l o , 9, 4
lo,
10, 9 , 3 10, 9, 2
a,
6 8, 5 4 8, 3
1 0 , 8,
10, 7 , 6 10, 7, 5 10, 7 , 4
10, 6 , 5
There are
15.
50 such combinations.
Method I.
(a)
This is a p r o b l e m where replacements have n o t been made. Us e c omb i n a t ions
.
T h e r e are
)
There a r e
( 10) ways of drawlng any kind of p a i r .
Thus,
ways o f drawing even numbers.
F(2
even numbers) =
- = 5 = 9'
P(A
and
.
Method 11. (a)
Using
B) = P ( A )
P(B) ,
(b)
The sum o f two even numbers is a n even number;
t h e r e f o r e the information in ( a ) is a s t a r t . The sum of two odd numbers is a l s o an even number; t h e r e f o r e we need p a i r s o f odd numbers. There a r e
(2)
= 10
p a i r s o f odd numbers.
The probability of
drawing t w o odd numbers is the same as drawing two
evens, that is
2
V'
S i n c e we c a n satisf'y t h e requirements of the problem e l t h e r by drawing two even o r two odds we add t h e probabilities.
Thus,
P(thzt the sum of the two numbers is even) 2 4 = 9+ 9 = 9 *
Thls naturally could have been s o l v e d by finding t h e number of ways of o b t a i n i n g a sum which was odd. The
of
o b t a i n i n g a sum which is odd) =
3.
(Each odd number--there are 5--must be matched w i t h an even number--there are 5. Thus, the numerator is
5 x 5
= 25.)
L i s t all t h e ways t h a t the sum of t h e t w o numbera is d i v i s i b l e by 3. 1
and
2
4 and 8
1 and 5 1 and 8 2 and 4 3 and 6
3
7
6 7 8
and 4 and 2
the sum
5
is d i v i s i b l e by
2
5 5
and and and and
9
and and
9 8
and
10
3) =
10
7 10
3
=
1 T.
(d) (e)
the the
less than 2 0 )
su?l is
=
1.
pairs have a sum more t h a n
20)
= 0.
(d) cannot fail and ( e ) cannot happen.
Note:
BRA LNSUSTER . (a)
one
of t h e p a 5 r being a d i m e ) equals t h e number o f pairs w i t h a dime divided by the number of p a i r s with two heads. There a r e 3 p a i r s with a dime--DP, DN, and Q . There a r e ( 42 ) = 6 p o s s i b l e p a i r s w i t h t w o
(b)
one
.
heads.
of t h e p a i r a dtrne)
3
1
6 ' T'
one
of t h e three heads a dime) equals t h e number of ways a dime my be a member of the t h r e e c o i n s with heads showing d i v i d e d by the number of ways t h e r e can be t h r e e heads shoving. There a r e ( 3 * ) = 3 ways t h a t
a dime may be one o f the c o i n s . There -re
one 17.
=
!F
(3)
=
(DPN, DPQ, DNQ)
.
ways t h a t t h r e e heads may show.
of t h r e e heads a dlme)
3
= T.
BRAINBUSTER.
(a)
Probability equals the number of ways t h a t (DQ?) may occur with a l l heads showing divided by t h e number of ways that ( ? ? ? ) may occur w i t h a l l heads showing. p
(3
= T 3= ~ A (3)
(b)
Probability e q u a l s t h e number of ways that (c?) may occur with heads divFded by t h e number o f ways t h a t may occur with heads showing. (??) p
s
T(( 21)Z m4 = 25 .
[pages 3140-341)
(c)
-
~ ( 2heads and one pair is a d h e ) .
a dfme)
=
~ ( 2 heads)
one
of t h e
~ ( 2heads) =
one
of p a l r a d i m e ) =
4
- (=2 )m = 5 *
~ ( 2heads and one a dime) = (d)
A:
3
coins come up heads.
Event
B:
2
of these
and
3) = P ( A )
P(A
and
3) =
3
I
16 '5=i5*
Event
P(A
2
are a dime and a quarter.
P(B) .
x . m3 = m3 o
Answers to Exercises
[pages 340-3421
T o t a l possibilities = 605.4-302,
$ + 23 = 1.
The summust be
or 720.
1 since
(3b)
Probability =
1 m.
is just another
way of saying (not 3 a ) , and we know that the probability that an event will o c c u r plus the probability that an event will not o c c u r is 1. 2
7
if y o u include the p o s s i b i l i t y of standing between them.
$
if you do not include t h i s possibility.
(a1
9
[cl
17
(a)
0.4 x 0.6 = 0.24
(d)
Panthers lose the Bear1s game and win the Nationals game. 0.6 x 0.6 = 0 . 3 6 ,
[pages 343-3451
I
l . Z;l = 5
16.
(a)
17.
2.2-2.2.2.2
(,)
6
+ ($
+
6
= 2
r,)6
=
+
4
Snme may think of t h i o problem as
6 6 (4 + (,I
+
6 (,)
+
(,I6
=
1 + 6 + 1 5 + 20
6 -t 1. = 64. You may wish t o poir,t o u t t h a t t h i s is t h e s i x t h row of Pascal ' s triangle. -t
13
-t
Sample Questions
True-False --
-T
1.
I f t h e probability o ? winning t h e f o o t b a l l game next F r i d a y is lose.
T
2,
5
t h e team is aore likely to win than to
I f a box c o n t a i n s t w o r e d and two b l a c k marbles, t h e p r o b a b l l i t y of p l c k i E g a red marble 1s the same as the
probabllity o f picking a black marble on t h e f l r s t draw, $lrnes, we expect t o have heads
Y
3.
If a perm3 is t o s s e d 20 show exactly 10 times.
-F
4.
P r o b a b i l i t y is always a nunber l e s s t h a n
T
5.
-T
6.
If two pennies are t o s s e d 100 times, we can expect two heads to show on about 25 t o s s z s 3 The probability of "x" winning the race is 7' "xM has less than an even chance or winning.
.
15
3.
-T -T
7.
Decision making can be based on a mowledge of' chance.
8.
A s t a t e m e n t , such as, a weather f o r e c a s t statement,
-F
9.
An honest coln is one t h a t when t o s s e d and allowed to fall freely will always show heads.
-T -F
10.
P r o b a b i l i t y i s sometimes used by govermmcnta.
11.
If
-F
12.
I f two c o i n s a r e tossed, t h r e e outcomes are p o s s i b l e : two heads, two t a l l s , a head and a t a l l . Thus, the p r o b a b i l i t y of t o s s f n g two heads is o n e - t h i r d .
-T
13.
If f i v e coins a r e tossed, the probabllity t h a t
are t o s s e d
5 s a chance
times, we c a n expect t h a t t w o heads will s h o w on e x a c t l y 25 t o s s e s , t w o pennies
100
t w o are
heads is t h e same as the probability t h a t t h r e e are talls ,
-F
14.
Drawing a spade and drawing an ace from a deck of cards a r e mutually exclusive events.
T
15.
If events A and B are independent the probability o f the event: A and B , 13 the product of the probability 3f A and t h e p r o b a b i l i t y of B.
A
Multiple Choice
-D
1.
Suppose you t o s s a c o i n f i v e times and each
shows h e a d s .
time it The p r o b a b i l i t y that heads will show on
the next t o s s is
...
E,
None of the above 1s correct
.
A
2.
Mark h a s three cards. One card has t h e l e t t e r A p r i n t e d on it, a n o t h e r t h e l e t t e r C, and t h e t h i r d has a T . If he mixes (shuffles) t h e cards w i t h t h e l e t t e r s face down and t u r n s them over one a t a time, what is t h e probability that they will be t u r n e d o v e r in t h e o r d e r CAT? A.
zI
3
B E.
None of the above is
correct,
C
3.
A weathermants f o r e c a s t s are c o r r e c t 20 times in one month and not correct 10 times in t h e same month.
Based on t h i s information, the estimate cf t h e measure of chance t h a t h i s future f o r e c a s t s will be c o r r e c t
E.
None of the above is
correct
-3
4.
.
A drawer contains 10 b l u e stockings and 10 red s t o c k i n g s . What is t h e smallest number of stockings
you need to take to be sure you have a p a i r of t h e same c o l o r s t o c k i n g s ?
-A
E
5.
6.
There are two roads from M i l l t o w n t o Luck and t h r e e roads from Luck t o Turtle Lake. How many d i f f e r e n t ways can John t r a v e l from Milltown to Turtle Lake?
If s i x honest colns are
D.
2
E.
None o f these.
tossed
freely, how many possible outcomes axle t h e r e ?
E.
-D
and allowed t o f a l l
7.
None of the above is correct,
Each face of two r e g u l a r t e t r a h e d r o n s is p a i n t e d a particular c o l o r , red, b l u e , yellow, and green. If the t e t r a h e d r o n s are r o l l e d what i s t h e chance t h a t both solids will s t o p on t h e white faces? 1 J. A'
T
D.
3 7
1
B'
F
E.
None of t h e above is correct
.
Computation Questions Compute t h e measures of chance f o r t h e s e e v e n t s .
( )
-
8.
A box c o n t a i n s a red marble, green marble and yellow marble. If you draw t w o marbles w l t h o u t looking, what a r e t h e chances t h a t you draw t h e red and g r e e n marble in any o r d e r ?
9.
)
-
Three jackets are in a dark c l o s e t . One jacket belongs to you, one t o Bill, and one to Bob. If you r e a c h i n t o
the c l o s e t and take out t w o jackets, what is t h e chance that you g e t your own jacket?
( 8 ) lo. -
How m a n y equally likely events are p o s s i b l e if a r e g u l a r octahedron (an e i g h t - s lded s o l i d ) is r o l l e d ?
1 -
How many outcones are p o s s l b l e if two r e g u l a r octahedrons are rolled simultaneously ( a t the same t l m e ) ?
($1
Three c a r d s a r e numbered like the ones below.
-
12.
Without looking a t the numbers you are t o draw any two cards. What is t h e chance that t h e sum o f the t w o numbers on t h e cards you draw is odd? Assume that t h e probability of a f a m i l y havlng a boy I s
72' (a)
and having a girl is a l s o
1
y.
What is the probability of a fanily having t h r e e
boys in succession? (b)
What is t h e p r o b a b i l i t y
in
a famlly w i t h t h r e e
children of n o t having three g i r l s ? (c)
What is the p r o b a b i l i t y o f a famlly having t w o boys and a girl, in t h a t o r d e r ?
(d)
What is t h e probability of a family w i t h t h r e e c h i l d r e n having two boys and a g i r l , i n any o r d e r ?
(e)
What 1s t h e probability of a family with four c h i l d r e n having two boys and two g i r l s , in any order?
-
(f)
mat
(g)
'hat
(h)
Occasionally one reads in t h e r-ewspaper of a family that has twelve c h i l d r e n , a l l o f them boys. In families t h a t have t w e l v e c h i 1 d r e n , how often is t h i s l i k e l y to happcn?
is t h e p r o b a b i l i t y o f a family with four c h l l d r e n heving f o u r girls?
is t h e probability in a famlly of four childrer. o f not having half b o y s , t a l f g i r l s ?
BIBLIOGRAPHY 1.
INTRODUCTORY PROBABILITY AND STATISTICAL INFERENCE. An Experimental Course. New York: Commission on Mathematics, College Entrance Examination Board, 1959. See e s p e c i a l l y Chapter 4 , A n Intuitive Introduction to P r o b a b i l i t y , and Chapter 1, What a r e P r o b a b i l i t y and S t a t i s t i c s ?
2.
Jones, B.W. ELEMEPjTARY CONCEPTS OF MATHEMATICS, New York: The Macmillan C o . , 1947. See e s p e c i a l l y Chapter VI, P e r m u t a t i o n s , Combinations and P r o b a b i l i t y .
3.
M o s t e l l e r , F., Rourke, R . E . K . , Thomas, G.B. Jr., PROBABILITY AM> STATISTICS, Reading, Mass : Addison-Wes ley P u b l i s h i n g CO., I n c . , 1961.
.
Chapter 9 SIMILAR TRIANGLES AND VARIATION
Introduction This c h a p t e r involves ideas f r o m b o t h algebra and geometry. In t h e i n t e r p l a y , each contributes t o t h e other. The central geomctrfcal concept here is that o f s l m l l a r triangles, end the fundamental algebraic ideas c e n t e r around v a r i a t i o n and proportion. These ideas are i n t r o d u c e d f i r s t by dL~cu33ionof congruent right triangles and the trigonometric r a t i o s beginning w i t h the
development of the Idea t h a t each of thc ratios
5, ,
$
depends
only on the angle between the X-axis and the ray which connects the p o i n t (x,y) w i t h the origin. Then the rat:os are deflned by name in terms cf these l e t t e r s and f l m l l y in terms of the s i d e s of a Fight t r i a n g l e . A p p l i c a t i o n s t o measurement are given, and here some review of t h e c h a p t e r on r e l a t i v e e r r o r and m a t e r i a l in the Seventh Grade c o u r s e may be needed. A f t e r f i x i n g ideas f o r t h e r i g h t triangle, the concept of similarity of triangles in general is developed. Then f o l l o w two applications: o l o p c , which is merely a v a r i a n t of the t r i g o n o n e t r i c tangent, and scale drawlngs and maps which apply and extend the idea of similarity. Then, finally, the connection w i t h direct and Inverse v a r l a t i o n is t h e more algebraic phase of the material already covered. Not only are the various topics in t h i s chapter inti mat el.^ i n t e r r e l a t e d b u t there are many r e l a t i o n s h i p s t o and a p p l i c a t i o n s of previous material: graphs, approximate measurement, right triangles, ratio, equations The first part of this chapter has been drastically revised, chiefly in the d i r e c t i o n of concentrating f i r s t on a careful
.
development o f the right t r i a n g l e rattos b e f o r e moving into t h e more g e n e r a l n o t i o n of similar triangles. Recommended t e a c h i n g time: 15 days,
-1.
I n d i r e c t Measurement
and Ratios
The o b j e c t of t h i s s e c t i o n is to develop t h e idea that the
values of t h e t h r e e ratios
,
,
X
depend only on t h e angle
between the X-axis and t h e ray which connects t h e p o i n t (x,y) w i t h the origln and n o t on t h e p a r t i c u l a r p o i n t (x,y) on that ray which is chosen. To this end, various examples in terms of coordinates a r e used. For Class mercises 9-la these suggestions a r e g i v e n . Probably the f i r s t five problems should be a cooperative class endeavor. The s i x t h can be done by each member of the c l a s s a t his s e a t and t h e i r r e s u l t s t a b u l a t e d and compared. With some of t h e remaining problems some members of t h e class may need much h e l p , but p r o b a b l y the b e t t e r ones c a n do most o f the work unalded.
Answers -t o Class Exercise 9-la
[pages 347-349
I
1
I
,
I
I
(b)
Yes, they are equal.
( 4 3.
3
5'
Yes, they are equal.
6,
(a)
x = 6
7.
(a)
R i g h t triangle.
8.
(a)
hypotenuse
(b)
[pages 349-3501
yes.
10. ( a )
Right triangles
(b)
Yes. Because OB, --+ t h e same ray OB.
(c)
f
(d)
Yes. They are a l l equal.
(e)
Yes.
11. F o r
OD,
and
OF
a r e determined by
is equal for the three triangles ;
A
AOR
r
= x
similarly,
4
T*
OD Z 12.8
Answers t o Exercises 9-la
[pages 350-351 1
and
OF
B
19 -2.
(c) Answers wlll vary. The r 4 a t i u s shou16 be the n a m e , b u t these may vary it points whose coordinates a r e n o t Integers a r e u s e d . Ermrs Fn measurement will a f f e c t pupils1 answers.
-X
AAOB
10. 4 12
15 6
6 12
9
A
9
(b)
Yes,
they are e q u a l .
(c)
Yes,
they a r e equal.
(d) Yes,
they a r e e q u a l .
(e)
X
5.2
A c o D10.4 - 7 AEOD
-
me r e s u l t s
should agree w i t h t h e t a b l e . .
5'2 (or *= 3
about
1.73)
[pages 351-353
1
( a ) right triangle
45'
(b)
since triangles.
(4
1
AAQB
(d) Should be equal t o
6
(e)
NC.
1r1
ACmST,
Thus,
y
=
1
and
and
are I s o s c e l e s , right;
ACQD
1.
x = l.
x = -1 fi
Answers t o Ciass Exercises 9 - l b
I.
-
- -
-
(a)
AC,
(b)
They are congruent by property A . S . A . - -
(c)
DR,
CE,
BR,
FS.
CR,
AAOB
L -
DS.
-ba
2b
AC OD
E
AEOF
Tz
3b
b c
2b
ZF 3b
3F
-ac 2a
E
3a
E
[pages 353-354
1
3.
(a) Yes. Yes.
Yes.
(c) Yea.
Yes.
(b)
4.
Multiplication property of 1.
(a) 4b
Answers to Exercises 1.
2.
Yes.
9-lb
(a)
&I (e)
(b)
OB % 13.9
(a)
g % -9 8 or 1
(b)
08 S 13.9 2
19 ' 8 and
$2
s. .
answers for l ( c ) and l ( d ) 3.
60'.
9.8
x
(4 f
A l i t t l e less than
~ 8 1 6 4feet.
(164.35)
They are n o t the same as the
6
9-2.
(a)
About
5 feet.
(b)
About
13
(c)
1 3 ~= 1p2
5*
169
25.
=
(5.1)
feet.
+ 144 +
Trigonometric Ratios
In t h i s s e c t i o n we give t h e u s u a l names t o the r a t i o s dealt with in the previous s e c t i o n . A table f o r 30°, 45', 60' is c o n s t r u c t e d and used. A s i d e from t h e construction of t h i s table, the p u r p o s e of t h e Class Exercise 9 - 2 is t o f i x in mind t h e d e f i n i t i o n s of t h e trigonometric r a t i o s "forwards and backwards1' and to g i v e s t u d e n t s some p r a c t i c e I n p i c k i n g out the p r o p e r ratio f o r a given problem. T h i s is Intended only as a very b r i e f int~oductiont o the methods and the simplest problems of trigonometry. The purpose of t h i s s e c t i o n is to f l x t h e Ideas of t h e previous s e c t i o n and to add t h e i n t e r e s t which computation of t h i s kind induces.
A n s w e r s to Class Exercise
1.
(a)
10
(b)
About
9-2 (c)
17.3
(d)
L tan L tan
60'
2
1-73
45O 2 1.00
t a n A 3 0 ~ 2 .57 2.
bout
5
L
60'
s
45'
2 .TO
(a)
10
(e)
(b)
A ~ O U 8.7 ~
(f )
cos
(c)
sin
(g)
(d)
sin sin
cos cos
L 60° L 45O
3 -87
L
Z .50
30'
5
.70
[pages 358-3603
L 30°
.50
% -87
3.
(a)
They appear to be equal.
(b)
They appear to be equal.
4.
m(L
sinLBOA
3 0 ~ )
,
tanLBOA
30
.50
-57
60
45
.70
1 .OO
45
60
.87
1,73
30
cos
L BOA
m(L
( ~ e a dup)
5.
(a)
Tangent
(c)
5
=
g =
tan
Read up f o r cosine only
L BOA
6(tan
L BOA)
y = 6(*57) y 2 3.42
6.
7.
8.
(a)
Sine
(c)
Tangent
(b)
Cosine
(d)
Cosine
(a)
TR
(b)
ST
(a)
EF
(4
(b)
ED
(d)
C pages
BOA)
ED
EF
360-3 62 1
10.
(a)
To find c os l n e
3 ,
w e t h e tangent; t o f i n d
OB,
usethe
(b)
To find sine .
0
u s e the cosine;
AB,
use the
(c)
To f i n d
03,
use the s i n e ;
BA,
use t h e tangent; t o find
OA,
use the s i n e ; to f i n d
.
t o find
t o flnd
GA,
use t h e
tangent.
11.
(a) To f i n d
OB,
use the
sine. (b) To f i n d cosine. (c)
u s e t h e cosine; to f i n d
To find 03, tangent.
-
kR,
use t i e
OA, u s e t h e
-
Answers t~ Exercises 9-2 1.
tan
60O
=
AP
v' Hence,
50tan60°=BP.
= BP
~ h u s , B P % 50(1.73) 2 E6.5. 2.
Let
x
denote the heieht it touches on the building.
s i n 45'
=
z.Hence,
x = 25 s i n 4 5 O X 2 5 ( . 7 0 ) 2 17.5.
3.
Since cos 45' = s i n 45', p r e v i o u s problem.
4.
tan30° 2 (a)
ED
Then
the answer is the sane as I n t h e
Hence,
ED = 150
t a n 30'
r
lso(0.577)
86.6.
4
'In 450 = l e n g t h of t h e hypotenuse-
o f the hypotenuse is equal t o
[pages 362-364
1
Hence, t h e l e n g t h
b
The l e n g t h of the hypotenwe ie equal t o
sin 60' s i n 30'
w - .866 2
1.731
.TOO
tan 60' tan 30°
1.732 0.574
3.02.
We have seen from previous constructions that when we double the angle we do n o t double the value of the ratio.
m(L ABD)
= 30
m ( l DBC) = 30
BD 1 BD
1 :
1
9-3.
because
rn(L APB)
% 17.3
Rezding -a Table
It is important t h a t p u p f l s learn to read a mathematical t k b l e . They have already had some experience in domg t h i s ( s e e t ~ b L eof squares and s q u a r e r o o t s of numbers in Chapter 4). The process of readlng a t a b l e does n o t cause them much d i f f i c u l t y but the s p e c i a l construction of the table of trlgonomeeric ratios makes
[pages 364-3671
careful i n s t r u c t i o n here necessary. An Important by-product is the emphasis on the co-function relationship and the d e f i n i t i o n of the r a t i o called t h e cotangent. The approximate nature of t h e numbers, correct to ten-thowandths to be found in t h e table should be emphasized. It m i g h t be pointed out that this table was n o t computed from measurements (although w i t h modern Instruments this would be possible) but that more advanced mathematics gives o t h e r ways of constructing such a table. Notice that angle measurements '0 and 90' are avoided. The teacher is here referred to the Teachersr Commentary on Chapter 4, Section 5 . The reason f o r avoiding these here i s the c o n f l i c t between the definition in terms of r i g h t triangles, which here breaks down, and t h e definition ln terms of coordinates of points on rays which not only applies in defLnlng trigonometric functions of 0' and 90' but obtuse angles and more general angles as w e l l . Some students, however, may ask what are t h e values of t h e functions f o r '0 and 90' and the teacher may wish to be prepared 4 to answer. For an angle of oO, t h e ray which we have c a l l e d OB lies along the X-axis and if we choose the p o i n t 3 to be (1,0) we have y = 0 , x = 1, r = 1 and, hence, s i n 0' = 0 = 0, cos '0 = 1 = 1, tan o0 = 0 = 0. For the angle of 90' t h e ray w i l l lie along the Y-axfs and the point B may be taken to be ( 01 ) Here x = 0 , y = 1, r = I, and it is easy to find that: s M
go0
=
0, cos go0 = 1.
But the t h i r d function, tan go0, presents special d i f f i c u l t i e s since the d e f i n i t i o n would give 1 which Is n o t a number. Thus, there is no number which 1s equal to tan 90'. In o t h e r words, the tangent of a right angle does not e x i s t .
[pages 364-3671
to Exercises
Answers 1.
2.
1
3.
9-3
(a)
0.1736
(f) 0.8391
(b)
0.1763
(g)
1.1918
(c)
0.6561
(h)
1,7321
(d)
0.8910
(1) 2.7475
(e)
0.9903
(3)
0.9994
( a ) yes (b)
Yes
(4
Yes
(a)
The t a n g e n t of an angle is greater than zero b u t not always l e s s than one.
(b)
The tangent of an angle increases w i t h the size o f the angle between lo and 89'.
(c)
The dffferences between consecutive t a b l e readings varies throughout t h e t a b l e .
(d)
The differences between the tangents of two consecutive
-
angles are smaller f o r smaller consecutive angles than f o r larger consecutive angles. 4.
6.
(a)
52.99
(c)
0.2582101
or
0.2562
(b)
89.9586
(d)
0.2835165
or
0.2836
3
= cot
L
CAB = 1.25.
In the t a b l e , c o t 39' 2 1.2349 and Hence,
7.
tan
L
c o t 1.25 2 39'.
KLM = 3.0.
Hence,
L KLFl
Z 78'.
[pages 369-3701
cot
38'
2 1.2799.
8.
t a n L ~ m = i . 5 . Hence,
fKLM556°.
This 1s more than half of the previous angle.
9.
200 = tan 59' =
10.
or
X
(2)
(3)
(4) ( b ) No.
59'.
or x = 200(0.6009)
*O0
1.6643
(a) ( 1
X m =c o t
cot
L
PO&
= tan
L PQO
mus, Z
120.
=
g.
t a n L P O Q =PQ m.
1
tan
I+
l
L POQ
L PO0
WOP.
ap
By (1) and (31, c o t
sin
l 'W OP '
m PQ =
=
PQ and
I
PO& =
tan cos
L POQ
=
L POQ' CP
m-
-
These a r e n o t recZprocals of each o t h e r . 11.
Let
x
3 p.m.
stand f o r t h e distance in miles to the l i g h t h o u s e at Then
tan 52'
=
5.
x = 30 tan 52'
2 30(1.28) Z 38.4.
be the d i s t a n c e i n m i l e s between the s h i p and t h e lighthouse a t 5 p.m. Then 30 = cos 59'. Hence, 30 2 48.7 30 % r = Let
r
cos 52O
,616
(Cther trigonometric ratios could be used. Some pupils may find the measurement o f the other angle, 3a0, and use trigonometric r a t l o s o f t h i s angle The answers, however,
.
should be about the same .) 12.
(a) 60'. Because a l l are congruent and the sum of their measures is 180.
[pages 370-3723
S i d e s of an equilateral t r i w l g l e
L
COB 2
are congruent. Angles 3f an e q u i l a t e r a l triangle
L 3CO
are congruent
- -
The mid-point of a segment separates the segment i n t o t w o congruent segments.
OA E AC
ACAB
ACAB
OAB
CAB
(c)
mL
S .A .S
.
180
[d) mLI)EA = 30'
tan 60'
*14.
(a) (DF)
=
=
2
%
J-T Z 7
(DE)
.
They a r e corresponding angles of congruent triangles Lt is c o n g r u e n t to an angle, and the sum of t h e i r measures is 180.
ORB = 90'
( c ) sin 6n0=-
.
+
.--
2
-
(90
2
+
0.06Go
1.7321
(EF)
= 12 + l2
[pages 372-373
60) = 30.
1
s i n 45'
(b)
=
fi 7
1.4142
2
In right triangle AD = tan 32o
ABD
In right triangle
q 0.7071
AD = 100 s i n 60'
ADC
DC = c o t 3 2 ' m
1~0(0.8660)= 86,6
DC
=
(86,6)(~.6003)
87 t o the n e a r e s t Integer.
DC
=
138 t o t h e nearest i n t e g e r .
AD
=
=
9-4.
The S l o p e of a Line
This section is included because of Its c l o s e connection w i t h the tangent of an angle, with e a r l i e r work on graphing, and w i t h the l a t e r m a t e r i a l on variation. There is n o attempt to find the slope of any lines except t h o s e through t h e o r i g i n , b u t in view of t h e more general definition o f slope, I t is i m p o r t a n t to connect the slope with the tangent of t h e angle r a t h e r than the r a t i o In g e n e r a l , the s l o p e of a l i n e is defined t o be the tangent o f the angle whfch the line makes with the X-axis. For instance, the llne y = x + 3 makes an angle o f 45' with the X-axis ; hence, i t s slope 1s tan 45' = 1. The values o f the ratlos f v a r y along t h l s line as may be seen from the following table:
5.
r pages
373-375
1
The s l o p e of the l i n e y = mx -t b is rn, the coefficient of x . Because these more general considerations are b e s t deal', with l a t e r , we c c n s i d e r vnly slopes of lines through the o ~ i g i nbut students should not be t a u g h t th-lnB here which t h e y will have to unlearn later. Later the connectFon between slope and d i r e c t v a r f a t i o n w i l l be e x p l o i t e d . The mention of "grade" is only incidental.
-
-Answers to Class Exercl.ses 9-4 3 1. (a) t a n L A o B = ~ -A-
2.
3.
(a) y
(a)
=$
-
oo
y=?x
(b) y = -3j+ (c)
tan
(d)
tan
OD
[pages 374-376 1
L
COR
=
I .q
ROR
=
4
-
Answers to Exercises 9-4 1.
(a)
y =
q1x
1 (b) Y = 3 x
he
l i n e d e s c r i b e d herme
is the X - a x i s .)
2.
(a)
$
( b ) About
3.
(a)
&
(b)
4
(a)
About
0.05
( c ) Abotlt
(b)
bout
0.05
(dl
(e)
0.0349
(b)
3.0349 (according t o t a b l e . a l i t t l e more then s l r l P O . )
[c)
About
5.
3.5
27'
3cU
About
5
3'
or
1
Actually t h e
tan
2"
is
feet
(d) A b o ~ t 3.570
6.
9- 5 .
(a)
About
54'
(b) About
IOU3
feet
SJmllar Triangles
Our conce-n in this chapter is to g i v e a r a t h e r precise d e f i n i t i o n of slmllar triangles, w l t h o u t the r e s t r i c t i o n t o rigkt t~iangles. Tt is easy for t e a c h e r and s t u d e n t (and t e x t author!) t o use t h e t e r n "cor-.esponding anglest' and ''ratlou o f correspondi n g p a i r s of sides" r a t h e r l o o s e l y . Cccaslonal use and rcfcrence
t o t h e precise definition given in t h i s sect'on I s quite d e s i r a b l e . Students may f Lnd the topic of' s i m i l a r triangles quite difficult
Just because the words and terms we o f t e n use a r e used too carel e s s ly
.
Again, of course, t h e r e fs no i n t e n t to provtde p?oofs, or even informal arguments, for t h e p m p e r t i e s stated in t h i s s e c t i o n . The experience provided in S e c t i o n 9-1 should make t h e conclusions a p p e a r reasonable. T h a t is t h e b e s t we can hope f o r in t h i s kind of treatmen?. The l o g i c a l p o i n t s i n v o l v e d in p r o v i n g these p r o p e r t i e s are among t h e most difficult and s u S t l e of a l l elementary geometry and are l e f t f o r the formal geometrg c o u r s e , u s u a l l y
s t u d i e d in Grade 10. Class Exercises 9 - 5 involve a d d i t i o n a l computations l i k e that in the text to assist the pupil in understanding the definition of s i m i l a r t r i a n g l e s . The corresponding angles of the figures are congruent as before, but the r a t i o of the corresponding sides varies f o r each p a l r . If the teacher f e e l s t h e need of more examples along t h i s l i n e they can be s u p p l i e d by extending the flgure in the f i r s t p a r t of this s e c t i o n . In f a c t , w i t h the figure given, the triangles COD and EDF may a l s o be compared.
--
Answers to Class Exercises 9-5 1.
(a)
m(f A T )
=
m ( L B 1 ) =75
3C,
(b) A t C ' = 8 (c)
2 = 5
5 m,
3 1 B l C f = H(5) = TT.
AlCl
cannot be determined
s i n c e there is i n s u f f i c i e n t i n f o n a t i o n given. (d)
AIB'
cannot
( e ) m(L A T )
be detemnined--insufficient information.
= 30,
Information.
m(L C f )
cannot be determfncd--insufficient
(a)
A l t e r n a t e d e f i n l t l o n 1.
(b)
The t h i r d p a i r of corresponding sides are not necessarily
congruent. (c)
Alternate d e f i n i t i o n 2.
(d)
I n s u f f i c i e n t information t o make t h e claim t h a t AABC
-
AA'BvC'
.
( e ) Alternate definition 2. (a)
No; f o r example, take a square and a rectangle which is
not
square; the corresponding angles are equal, but the corresponding sides are n o t proportional.
(b)
a
No; consider two parallelogmms whose sides are pro-
portional; t h e i r corresponding angles need not be congruent. (a)
When
A I B t = 6,
AtCl = 12
and
B ' C q = 14
(c)
When
Bt C T = 5,
A ' B 1 = l5
and
30 ATCt = 7
Y e s , The triangles will be similar because the angles o f one t r i a n g l e are congruent to the corresponding angles of t h e o t h e r triangle. Yes. If two angles of a t r i a n g l e are sponding angles of a second triangle, f i r s t triangle will correspond t o and t h i r d angle of the second t r i a n g l e . 1 and L 2 are b o r n , the measure'
L
180
-
( m ( ~1) + m(L
21)
2
[pages 382-383 ]
congruent t o c o r r e the third angle o f t h e be congruent t o the ( If t h e measures of of L 3 must be
Answers t o Exercises 1.
AB (a) E
= AD E .Corresponding
sides o f
are
(b)
proportional. A 3 = AEAC Same rcacon.
(c)
- =
AD
ne
(dl
AB . z Same AB . They #m
reason. are n o t
corresponding s I d e s of similar b e and are t h e r e f o r e n o t proportlonal
I
.
2.
same conditions would hold, namely, corresponding s l d e s o f s i r r i l a r triangles a r e p r o p o r t i o n a l .
110, the
are , the rat-lo o f p a i r s of corresponding s i d e s a r e equzl.
If t w o
Id)
AD n #A%z
A=
They a r e n o t p a i r s o f correspondA
i n g s i d e s of As, t h e r e f o r e they do n o t form equal ratios.
C
6.
1 AE3 L ABC and L ADE '1 ACD . Since
they
are corresponding angles of p a r a l l e l l i n e s w i t h AB and as t r a n s v e r s a l s . BAC S L EAD s i n c e it is the same angle. S i n c e the t h r e e angles o f AAED are congruent t o the correspondingangles of AABC, the two t r i a n g l e s axle similar. L AED corresponds t o f ABC.
7.
AABC A A I B ' C 1 (S.A.S.). Using t h e arguement f r o 3 Problem 6 , A A f E t F t Is similar t o AAtBfCr. 8 Therefore AABC is slmilar to O A ' B t C '
C
B
A
A'
.
8.
This can be s h o w n i n t h e
saxe way as
c' in P r o b l e m 7 , except
E i a now l o c a t e d s o t h a t is l o c a t e d so t h a t A 1 C 1 = ~ ( A ' F) '
t h a t the point
and
Fl
and
Thus,
A'B'
-
A'C'
AIBl
= 3 ( A t ~ )l
.
AD),
= ~ ( A C ) .
( ~ o t e : t h e multiplication p r o p e r t y of cquality is used several times in g3lng from the second t o the t h i r d step
.
Similarly,
(b)
9-6.
AB
=
A'B'
with
~(AB),
replacing
AtB1
Thus,
B'Ct = ~ ( B c )
.
The same method may be used t o show that A'C1 = ~ ( A c ) and BIGt = ~ ( B c ) .
S c a l e Drawings and Naps
This s e c t i o n is t o be t h o u g h t o f as an application of similar triangles, not an exhaustive t r e a t i s e on t h e c o n s t r u c t i o n of s c a l e drawings and maps. This is the reason f o r t h e b r i e f t r e a t m e n t . The fundamental ideas h e r e should already be familiar to t h e s t u d e n t but t h e l r sharpening is made p o s s i b l e by the t h e o r y t h a t came b e f o r e , A t e a c h e r may p r e f e r to glve a little more woxlk on t h e r e a d i n g of maps but it s h o u l d n o t be carried much further lest it d i s r u p t t h e continuity of t h e chapter and g i v e t o o much promlnence t o problems f a r d i s t a n t f r o m s i m i l a r ffgures. P e r h a p s s p e c i a l comment should be made about the n o t a t i o n 1 Inch = 1 mile, This is r e a l l y a correspondence--in no sense is it equality. But t h i s is the usual way of expressing t h i s corres~ondenceand students should become f a m i l i a r w i t h it. There should be no confusion s i n c e obviously one inch Is n o t the same as one m i l e .
Answers 1.
Class E x e r c i s e s 9-6
(a)
About
17 cm.
(b)
About
340
feet
[pages 384-387 I
(c)
172 = 152 3402 =
2.
3.
4.
(a)
About
(b)
10.0
(a)
25
(b)
18.8 inches inches (d)
5
50 mn.
(e)
2:
(c)
%
inches
(f)
2.5
(a)
130
feet
(d)
80 feet
(b)
20
feet
(e)
(c)
80
feet
(f) 200
Answers 1.
+ 82 or 289 = 225 + 64 3o02 + 1602 or 115,600 = 90,000 +
850
inches
inches
inches em.
5 feet
-
feet
Exercises 9-6
4
2.
miles
5
Inches
inches
( ~ e a s u r e m e n t sare n o t as indicated on drawing.)
[pages 387-388
I
25,600.
4.
( ~ o to t scale
given in t h e text.)
5 inches
A
The measurement o f 50 miles.
I\C is
0
62
Inches which represents
I
I 6a
ma 18.9
7.
(a)
1 *00 OoO
bout
6
4
miles. feet
(b)
He passes the p i t c h e r t s box about
(c)
About
105
bout
42.8
(d)
scale given In the text. ) ( N O ~to
feet
feet
[pages 388-3903
3
f e e t away.
8.
(a)
corresponding angles o f t h e two trisngles are congruent me
.
(k )
The approximate measurements on the s ~ a 3 edrawing a r e as i n d r c a t e d in t h e figure. Since 5 I n c h e s correspnnds t o 103 r e e t , t h e distances
A
d e s i r e d a r e approximately: B = 50 feet, AS = 95 f e e t . Hzre some review o f Cllapter 5 I s in order. Since one cannot measure inches t o any greater precis Lon than 0.1, t h e r e l a t i v e error in t h e answer c o u l d be as much approxfmately as 1 foot. Hence, t h e r e 1s no potnt i n c a l c u l a t i n g it t o more declmal p l a c e s .
5"
S
B
-
(a) AB 5 inches A C = 3-q3 inches BC = 3 inches
c
A
( ~ o drawt to
scale as
required by problem. )
10.
(a)
AC
--
BC
=
AB
10
Lnches
:7
inches
6
(b)
m(L
(c)
C i t y B is northwest of Clty A . h he compass d i r e c t i o n is about 323', b u t angles g r e a t e r than 180' have not been dLscussed. )
inches
(b) and ( c ) a r e sm.e as f o r Problem 9 .
BAC) X 37
11.
Here the measurements would be approximately those of the figure If on t h e nap 5 inches corresponds t o 100 f e e t . (a)
BC
(3)
SA %
%
135 feet.
45 f e e t .
AR 2 55
*L2.
feet.
Without a scale drawing, one may use the f i g u r e given and n o t i c e t h a t BC is t h e hypotenuse of a right triangle w i t h sides
100
and
40 -k 50
90, respectively. Hence, the l e n g t h of t h e hypotenuse =
If we let x denote the d i s tance SA, use slmilar trlangles t o g e t
Similarly,
for
be notlced t h a t
o r it rray AR = LOO - x ,
AR,
9-7.
Kinds
of
Variation
The concept of v a r l a t l o n can be v e r y appropriately introduced by use of tables o f data and graphs as w e l l as in connection w i t h similar triangles. An attempt has been made in t h i s section to use examples from science. It is hoped that one o f the c o n t r f b u tions of these two books f o r use in Grades 7 and 8 can be t h e encouragement of t h e p r a . c t i c e o f using applications o f the mathematics t o s c i e n c e as well as t o t h e more traditional business and s o c i a l topFcs Here s e v e r a l types of variation a r e used so t h a t some d i f f e r ences smong them can be observed, In t h e s e c t i o n s t o follow s p e c i f i c attention w i l l be given to direct and Inverse v a r l a t l o n and in an o p t i o n a l s e c t i o n t o still other t o p i c s . Attention is c a l l e d t o t h e graph of y = ax t o show the relation of the second p a r t o f the c h a p t e r beginnrng w i t h t h i s s e c t i o n to t h e f i r s t p a r t o f t h e c h a p t e r on similar r i g h t triangles. Only very s p e c i a l kinds of variation a r e considered in t h i s and t h e following s e c t i o n s . The s taternent ''y varies as x v a r i e s " cvuld i n c l u d e any table of values of x and y in which not all the v a l u e s of y a r e e q u a l . The s t a t e m e n t "y increases as x increases'' could be made a b o u t any of the following
.
r e l a t lonships :
y = x + 3 , y = x
with
x positive,
g
= f iwith x
positive.
But a c t u a l l y t h e t e r m "y v a r i e s directly as xl' o r even " y varies as xl1 is d e f i n d t o mean a v e r y s p e c i a l kind o f relationship; namely, when x i e m u l t i p l i e d by any number, the corresponding v a l u e of y Fs multiplied by t h e same number. The same idea is expressed by writing that "x and y a r e p r o p o r t l o n n l . :1 By t h i s t i m e in thrs c h a p t e r , s t u d e n t s should be familiar w i t h t h l s id.ea from t h e d i s c i ~ s s i o nof eimilar triangles, Though t h e terminology i s not used, we a r e r e a l l y here consider-
sotne s p e c i a l arid simple k i n d s of functions. Hence, n o t only are t h e s e sect:ons i m p o r t a n t in themselves and f o r t h e f r connection w i t h trig
the previocs p a y t of t h i s c h a p t e r but as an Tntroduction t o t h e idea of f w c t i o n which becomes s o important in l a t e r ratnematics. Dzvotees o f Wirmie t h e Pooh may remember the l i n e :
h he more it s c o w s , t i d d l e t y Fom, The mare it go?;, tiddlety €om, On snowing
.
"
T h i s right; be i n t e r p r e t e d t o mean a much mare coritplrx r e l a ' i T o n a h i p than a n y we have considered h e r e ; t h e r a t e of snowing 4 s p r o g o r t i o n a 1 t o t h e aflourt it has already snowed. This is wh2t i s czlFcd " l n c r e a s in& exponentially" and example of which is cclntinuous
i n t e r e s t , a type o f i n t e r e s t which is now beginning to appeer in banking p r a c t i c e
.
Answe?s ----to C l a s s E x e r c l s c s 9-7
w Amount in pounds c Cost In dollars
'
(a)
c o a t increases
(b)
0.6
or
0
1
2
5
3 ' 4
6
$ 0 $0.60$1,20 $1.80;
8
9 1 0
$
4 0 3.00$3.60$4.20$4,80 5+0!'60C
c = 0.6 w
(c)
0.60
[ pages 392-393
7
3
2.
(a)
If P = 4 0 , v s 94.
(b)
The volume decreases.
(c)
If
P
= 10,
v 2 37'0.
(d)
Answers
The pressure decreases.
Exercises
9-7
1.
(a) S
increases
(b) w
increases
[pages 393-3951
(a)
d increases, t tends to increase. Due to d i f f i c u l t y in measuring the t h e 4 the graph does n o t give a true picture of t h e actual relationship. Yes.
As
0.6
(b) The prediction can s a f e l y be s t a t e d as more than seconds, s
h
1
2
3
4
100
100
100
100
100
T
25
11.1
5
xg 2
6.2
4
6
7
8
100 5
100 4
9
2.8
2.0
1.6
9
100
10
100
100
x
x 1.2
5 1
9-8.
Dlrect Variation
v a r i a t i o n is t h r o u g h reference t o t h e graph of a stralght l i n e . As x increases, y i n c r e a s e s . This is t r u e when n e g a t i v e values are considered ran x as well as positive v a l u e s , The increase in y f o r an increase o f 1 i n x is constant. Thls constant can be c a l l e d t h e c o n s t a n t o f proportlonallty. The c o n s t a n t of proportionality is a l s o the ratio o f the zhange In y, between t w o p o i n t s on t h e gmph o f a s t r a i g h t l i n e o r between two of t h e elements in a s e t of d a t a , to t h e change i n x. If a value of y i s multiplied by k t h e v a l u e of t h e corresponding x of an ordered p a i r ( x , y ) is a l s o multiplied by k. The relation of d i r e c t v a r i a t i o n can be expressed by the number sentence y = ax ~ G T 2 # 0 . Except f o r the nmber p a i r ( 0 , C ) t h e r a t i o of number pairs, , which s e t i s f l e s t h e e q u a t i c n , As a . Since f o r number p a l m in which t h e relation is dircet variation, = a, t h e set sf numbers y and t h e s e t of nmbers x are said to b e p r o p o r t i o n a l . A l s o , a is t h e s l o p e o f t h e iine y = ax. P r o p e r t l e z like these enumerated in t h e above paragraph can be d i s c u s s e d carefully and in some detail f o r direct v a r i a t i o n . T3en In considering Inverse and o t h e r kinds of variation it may be h e l p f u l to t e s t these p r c p e r t i e s to see if they h o l d in t h e new s i t u z t i o n . One of the ways t o talk about d i r e c t
-
-
Answers t o Questions -. In S e c t i o n 9-8
Multiply, the answer is $9.00. c = (u. 6 0 ) (w) Multiplication. Slope is 0 ~ 6 0 . Now (0.60)(10) = 6.00
.
which is the i n c r e a s e
In c o s t .
2 i s t h e change in c o s t for each u n i t change in w ( 5 0 )( 1 ) = ( 5 0 )( 2 ) = ( 5 0 ) (3.5) (50)(t) =
50: 100:
miles 5~ 100 miles
50
weight
.
one hour.
two hours.
175: 175 miles in t h r e e and one-half hours. Sot: 5 0 t m i l e s in t h o u r s .
=
C pages
396- 397 I
Answers -t o Class Exercises 3-8 1.
(a)
3 k = g
2.
(a)
t = 3 %
(b)
2 = 0,32n
(4
Yes
(dl
Yes
(e)
32
or
32n 100
d =
t =32n)
(for
Answers to Exercises 9-8 1.
.
(a)
t
(b)
d = 0.339n
(a)
n(2)
33.91
=
or
2
or
33, n .=A-
feet
(b) d = 2n
3.
5
(c)
no,
(a)
i
will Increase,
d
For 2,
k3
For 3,
k4 = 12
Yes.
n
(b)
12f
=
#
not decrease. i
increases
= 2
0; that is,
n
cannot be zero.
7.
x -y3 == k(k) (-12)
:=k
8.
(a) Yes, when
decreases, (b)
9.
b
increases, a a decreases.
increases, when
a = lob
d = k t 240 = ( k ) ( 6 ) 40 = k
d = 40t
[pages 399-4001
b
(a)
Yes,
[b)
y
Is halved.
(c)
x
is multiplied by
(d)
Yes.
12.
(a)
x
13.
(a)
sample:
11.
5
i
y
I s doubled.
10.
is h a l v e d .
f
M
= 5280
m,
end
f
whers m I s the number of m i l e s is t h e number of f e e t .
L
( m ) ( ~ )where ,
=
is t r i p l e d .
y
(b)
rr,eters and
M
l s the number o f c c n t i -
c
is t h e number of meters,
9-9. Inverse Vzriation In the study of any concept in mathe.mties it is desirable to observe aituatlons l r i which the properties associated with the
concept do n o t hold as well as to s t u d y situations in which the p r o g e r t i e s h o l d , For t h i s reason, if no o t h e r , it is highly desirable tc i n t r o d u c e i n v e r s e as w e l l as direct; v a r l a t l o n . In a d d i t l o n t h e r e a r e many tmportant r e l a t i o n s in s c i e n c e and In social s ltuat ions which are illustra'cions of i n v e r s e variaSion. A number of these a r e Included in t h e exerc t s e s . It has been s a i d thzt t h e arithmetic of Grades 1 through k is concerned w i t h t h e relation x + y = a, while t h e arithmetic o f Grades 5 through 8 I s concerned with the r e l a t i o n xy = a. Thls statement is based on th? Important p l a c e in upper grade mathematics of such t o p i c s as m u l . t i p 2 i c a t i o n a?d d i v i s lun, area, p e r c e n t p r o b l e m , and r a t e problems. I n t e r e s t i n g interpretations of h v c r e e variation may b e made in ar-y of these situatFons. T h i s reiati~nof i n v e r s e v a r i a t i c n can be expressed by the number sentence
xy
=
a,
a
#
0,
or by
y
=
5. In e i t h e r X
case
no value can be o b t a i n e d f o r y lf x is 0 , o r f o r x if y is 3 . n h i s introducc~ a new s i t u a t i o n wl Lh graphs. The graph
c f xy = 3 does n o t cross ',he Y-axis or t h e X - a x i s , Hence, t h e grspn is in two parts and is said to be d~scontinuous. It pay be w e l l t o pint o u t th2.t y can be determined f 3 r x, any p o s l t l ~ ~ e
number, no matter hor.; sma:l, even f o r x = 10-lOOO. ~ i n i l a r l y ,a value can be determined f o r y Y f xs any nega-t;ive umber, nc rnatter now close t h e number is t o zero. In hzgher zithematics t h e stxdents w i l l lezmn that t h e I f n e s x = 0 and y = O are aslynptotes of :he hyperbola xy = a and tSat the hyperbola s: s a i d t o zppraach t h e ares asymptotically. Here again, o n l y a v e r y speclal kind o f relationship is c o n s i d e r e d . Ws c m l d say " y increases as x decreasest' about any hY t h e f o l l o w i n g r e l 2 t i o n s h i p s :
asswnlng in the last t w o that x is a p o s i t i v e real number. B u t h e r e we mazn by "y v a r l e s InverzeLy as x" t h a t t h e p r o p o r k i o n of 5ncrcase uT y is equal t o t h e p r o p o r t i ~ nof d e c ~ e a s cof x ; that I s , If y Is r n ~ r l t t i p l e dby a n-mber, the corresporidLng x i e Qlvlded by t h a t same number. It xould probsbly be confusjne to 2 s l a s s to note that "y varies inversely az x" means t h e sane as "y varies d i r e c t
ly a s
i" b u t -
t h e teacher shculd
Anslwex8s to Jixercises
c o r v i n c e himelf of t h i s f a c t .
9-9 -
(c)
The t i m e 5s halved.
(d)
r
decreases, such that the p r o d ~ i c tof still 10C.
r
and
t
Is
(b)
(n)( a ) = 240 where b o t h
n
md
s
a r e countlng numbers
only.
(b)
4.
If D As W of W
is doubled, W is halved. increases, D decreases, such t h a t t h e product and D is s t i l l 36.
(r)(p) =200
If the number If the number
6
p
r
is doubled, the number is doubled, the number
r p
is halved. is h a l v e d .
For d i r e c t variation the r a t l o of the variables is constant.
For inverse v a r i a t i o n , the product of the variables is a constant.
[pages 403 - 404 1
9.
is n o t a c o n s t a n t .
The product of a and b
No.
is a c o n s t a n t .
The table suggests t h a t
r e l a t e d by t h e equation
a
ar.d
The r a t i o b
a-e
1 a = Tb.
10* (a)
No.
(0)
No.
(b}
No. If eIther f a c t o r Ls z e m , the product would be zer3. S i n c e t h e p r o d u c t c a n n o t b e z e r o , n e i t h e r factor can be zero.
c
Yes.
(d)
Inverse.
They are related by
y = 2x
[ pages 40h- 435 1
2
.
9-10.
-
Other Types of Varlatlon ( ~ p t i o n a l )
In b e t t e r c l a s s e s part1 c u l a r l y , the teachers may w i s h to g l v e a t l e a s t brief a ' c t e n t l o n t3 o t h e r types of v a r i a t l . o n . For thls reason v a r l a t l g n of one number in an c r d e ~ e dp a i r as the square o f the other nunber and, analognusly, v a r i a t i o K of one of c palr, " h v e r s e l y as the square" of the o t h e r are Introduced. A l s o reference is made t o the graph of y = sin x for x = 10, 20, , 80. T h i s last t o p i c m i g h t be considered In o r d e r to r e f e r a g a i n t o some of t h e makeria; e a r l l e r In t h e chapter. Unfort~natel q , students do n o t have a t t h l s time the necessary bnckgraund f o r the s t u d g o f t h e Int e r e s t L n g and exceedingly inportant p r o p e r t i e s ol" t;he graph of y = s i n x,
...
[pages 405-407
1
This s e c t i o n m i g h t be assigned f o r s e l f s t u d y by t h e s t u d e n t s who have made t h e most r a p i d progress w l t h the e a r l l e r parts o f the chapter, o r t o those s t u d e n t s who show the g r e a t e s t interest in f u t u r e s c i e n t i f i c study.
-
-
Answers t c Questions in S e c t i o n 9-10
If the number If t
t is doubled, the number d is tripled, d is multiplied by 9.
..
the body falls 1600 ft. In 10 seconds.
Gallleo:
Born 1564
-
Died 1642.
Answers t o Fixerclses 9-lOa 1,
(a)
4
2.
(a)
S
(b)
S
sq. in.,
= e
6
faces,
24
sq. in.
2
varies as the square of
e.
(c1
[pages 407-4083
I s multlplled by
4.
,
(e)
(1) If
e = 3,
s = 9
The mean of the r a t i o s
d
is
26.4
t d = 26.4t 2
Yes, the theoretical c u r v e f i t s .
[pages 408-409
1
Yes,
d -
seems t o remaln constant according t o h e r data.
t2
x marks the o b s e ~ l r a t l o n s
.
marks p o i n t s d e s c r i b e d by
d = 26.4t
(3)
E = 4 V
(b)
E =
2
.
(c)
2
4(6)2
=
144
16
=
4v?
4
=
v2
v = 2
5.
..
(a) ?100 B T ; =x~ , x = - .5 14 (b)
5
x ( 7 0 ) = 25,
.'.
2%
3
lbs, is r e q u i r e d .
is r e q u i r e d .
(d)
+(65)*
C
=
C
= 105%
j!
$10.57.
o r cast
=
140*,
C o s t = $14.07.
1 2
(4 c
= j+
c = % (75)2
=
Enough c a n be
bought to sow t h e p l o t .
6.
( a ) d = k t2
(c)
1 4 4 = 9k
16
= k
d = 16t2 (d)
d = 16(5)' d = 400
..
Answers $& Exercises
the distance is
9-lob
(1) P
varies d i r e c t l y as
(2) S
varies as the square
of
(3) V of
e.
e.
varies as t h e cube e.
for
P
=
for
s
=
for
4e
6e V = e3
2
[pages 409-411 1
400
ft.
4.
=+,1
(e)
If
P-10 1 S = 3TT 2 38 v = 15,625 % 16
9-11. Summary The committee t h a t prepared t h i s c h a p t e r has no recommendation on the use of t h e swnmary. It is included f o r t h e convenience o f teacher and s t u d e n t s . The review exercises could be used as test questions, additional assignments, or general class summary discussion.
-
Answers t o Exercises 9-11
1.
y = m 16 = k(2) k = 8 and
.'.
y = 2x
y = &
y = 2(5) = 10
3.
2
y = h 16 = k ( 2 1 2 k 4 and y = 4x2 2 y = 4(5) = 100
-
rn
[pages 411-4151
(a)
d
=
5
= k(10)
kp
k = 7
solving for k 1 d = -p 2
and
replacing d pull rnus t be
14 = lp P p = 28,
.' . (b)
d =
1 pint
.
?I( l 4 )
=
1 16
gal.
=
7
7,
and
and
28
d,
k with ibs ,
d i s t a n c e will be
OP
lbs. p e r sq, in. f o r 1 g a l . ('7) (16) l b s , p e r sq. in. f o r
7
14
and
$.
in.
PV = K 1 ( 7 ) ( ~ =) K = 7 , If V = - lo then P (1 ~ =) 7 .
P
=
(16)(7) = 112, hence,
112
l b s . per sq. in. I s t h e pressure
exerted by a c e r t a i n amount of hydrogen I n a 1 g a l . jug if it is l a t e r enclosed in a halfp i n t jar.
In x
+
y = k,
p r o d u c t of
x
not vary I n v e r s e l y as
y
does
and
y
1s n o t
xy
=
Inverse variation,
x because t h e
constant.
k
Direct v a r i a t i o n y = kx a is t h e constant of proportionality. 2
V = m h If
..
r is multiplied by v = r ( 5 r )2 h V
=
5 and h
Is unchanged
25(mzh)
hence, the volume is mult~plied by 2% is the constant
5.
11.
Using t h e r e l a t i o n
h =
$ ~5 we
The graph is a half-llne,
The graph is t h e right h a l f o f t h e parabola
[page 4161
have :
Sample Questions
n o t intended t o provide a s i n g l e The f o l l o w i n g questions are t e s t f o r Chapter 9 .
They a r e intended only t o h e l p t h e teacher in
preparing h i s own t e s t s .
True-False --
-T
1,
Triangle
-AB -
AtBt -
-F
2.
3.
and
if
~(LA~B~C.).
~ ( ~ A B =c )
L
2 POQ = 7
sin40°=
1 c o s 400'
In the figure below
-T
AfBfCt
In t h e figure
sin
-F
BC
i s similar to t r i a n g l e
ABC
6.
PQ = 6.
The p o i n t
( 4 , 5)
iles on the l i n e whose equation 2s
The p o i n t
( 4 , 5)
l i e s on the line j o i n i n g
(1% 1 5 )
( 0 , 0)
and
F -
7.
The tangent o f a n a n g l e is always a n u m b e r b e t w e e n
I.
and
-T
8.
0
The height of a t r e e may be c a l c u l a t e d if t h e length of its shadow and t h e angle of e l e v a t i o n of the s u n are
mown.
-F
9.
Any two right t r i a n g l e s a r e sim3.lar.
-T -F -T
10.
If
F = 14a, F
11.
If
F = IC
12.
If x x Is
varies i n v e r s e l y w i t h 9, then y = 2 when
y,
-F
13.
I
v a r i e s d i r e c t l y as
then
Z'
y
with
F
v a r i e s d i r e c t l y as
varies inversely as
x
a.
6.
and x = 18.
x
y
is
4
when
varies inversely
y.
T -
14.
If the measures of two angles o f one triangle are equal, respectively, t o t h e measures of two angles of another triangle, then the t h i r d angles a l s o have equal measure.
-F
15.
If t w o s i d e s o f one t r i a n g l e have lengths equal, respect i v e l y , t o t h e lengths of two s i d e s of another t r i a n g l e , then t h e third s i d e s must also have equal l e n g t h s .
-F
16.
The s i n e of an angle varies directly w i t h the measure o f the angle.
Multiple Choice Write the l e t t e r of the best answer.
-C
1.
If one a c u t e angle of a r f g h t triangle has t h e same measure as an acute angle of another right triangle, t h e n the triangles a r e A.
congruent
D.
all of the above
B.
isosceles
E,
c a n n o t tell from t h i s
C.
similar
information
-D
-C
2.
3.
The lengths of the s i d e s of a triangle a r e 6 , 8, and 9 i n c h e s . Whlch of t h e following a r e t h e l e n g t h s of t h e s i d e s of a t r i a n g l e s i m i l a r to it? A.
3,
4,
6
inches
D.
18,
B.
2,
3,
4
inches
E.
ncne of the above
C.
9, 12, 16 i n c h e s
The e q u a t i o n of t h e l i n e through Is
B.
-C
-A
-D
4,
5.
6.
4
y = g ~
E.
24,
27
(0, 0)
inches
and
(4, 5)
none of the above
The slope o f t h e line whose equatfon is
y
=
x
is
D.
it does n o t have any
E.
none of t h e above
The s l o p e a f t h e X-axis is A.
0
D
it does n o t have any
3.
1
E.
none of t h e above
C.
y = o
The slope of t h e Y-axis is
D
it does n o t have a n y
E.
none of t h e above
C -
7.
The s i n e o f an angie of a r i g h t triangle
A.
depends on the l e n g t h of one s i d e only
B.
depends o n the l e n g t h of the hypotenuse only
C.
depends on the measure of the angle only
D.
all of the above I
E.
-D
8.
none o f t h e above
I n t h e drawing we c a n find t h e number
-B
9.
x
if we know
A.
AC
and
BC
B.
AC
and
m(LACB>
G.
AC
and
~(LABC)
D.
all of t h e above are c o r r e c t
E.
none of the above is correct
A
If you spend t w o dollars buying beans, t h e
c o s t per
pound of t h e beans and t h e number of pounds purchased
provide an example of
d i r e c t variation
B -
10.
B
.
inverse v a r l a t i o n
C
.
v a r i a t i o n as the square
D.
i n v e r s e square variation
E.
none of the above
If t =
p
varies inversely w i t h
2
then what Ls
p
when
E.
t and p t = lo?
2n o t
=
6 when
enough information
Completion ?or Problems 11-16 u s e t h i s figure:
11.
The coordinates o f p o i n t
C
8 12. -
The coordinates of p o i n t
D are
are
>
(5, (
4
, 4).
17.
The measure of each or the t h r e e angles of an i s o s c e l e s , rlght trimgles a r e , and
18.
A right triangle has a
15'
angle,
-
The measure of
t h e o t h e r acute angle is
19.
The s l o p e of the l i n e j o i n i n g the p o i n t s
and
(Q,
5)
2s
(2, 1)
3 20 152
20.
In t h e figure
1
tan A = 3.8 BC = ,
and
AC
has l e n g t h
For Problems 21-25 use t h i s figure:
9
21.
9 -
22.
LA sin L A
5
23.
tan
LB
4 fl
25.
cot
L A=
T
m
tan
= =
=
40
feet.
F o r Problems 26 and 27 use t h i s figure:
16
26.
If gn
=
12,
Q t R 1 = 15,
and
P r Q T = 20,
=
18, Q I R 1 = 27,
and
PR
then
PQ = 30
27.
If &R PI R ' =
57.73
28.
In the f i g u r e
=
--
4
1.7321 7 -
Given: s h 30°
=
tan 30'
1.7321 2
c o s 30'
S
feet.
115.46
29.
Inverseiy 3 0 .
,
I
I
-1 I
I I
I 1 Stream
-
I
I
--------
S
The d i s t a n c ~between R and S is The d i s t a n c e between S a ~ dT is
In P r o b l e m 18, RT
If
-
T r e p r e s e n t tress on opposite s i d e s ~f
md
stream.
then
20,
l k
.500i3
.5773.
=
y =
a T,
then
y
=
varies
with
x.
the
100 feet.
Chapter 10
NON -KETRIC GEOPETRY Many teachers have found t h a t when Chapter 10 Is taught b e f o r e C h a p t e r 11 t h a t Chapter 11 is much easier for t h e pupils, in fact that Chapter 10 is s o h e l p f u l that s e v e r a l days teaching tlme on Chapter 11 can be saved. After a very c a r e f u l t r y o u t o f this chapter over e two-year p e r i o d , the a u t h o r s o f t h e t e x t strongly recommend that Chapter 10 be i n c l u d e d as a r e g u l a r p a r t of the course. The material of t h f s c h a p t e r will be .,zw to a l m o f t all pupils. There a r e a number of reasons for includip-g it in e i g h t h grade mathematics. Among t h e s e are: h e l p s develop s p a t i a l intuition and understanding. err,phaslzes in a n o t h e r c o n t e x t t h e r o l e of mathematics r e d u c i n g things t o t h e i r s l m p l e s t elements. a f f o r d s o t h e r ways of l o o k i n g a t o b j e c t s i n the world ebout us and r a i s e s fundamental questicns about them.
3.
It It in It
4.
It i l l u s t r a t e s types o f mathematical (geometric) reason-
5.
i n g and approaches t o problems. It gives an interesting i n s i g h t into the meaning o f dlmens ion.
1. 2.
The general purposes of this chapter a r e slrnilar to those f o r C h a p t e r 4 o f Volwr,c I. It is suggested t h a t teachers read the b e g i n n i n g of t h e Commentary f o r Teachers of t h a t c h a p t e r . In this c h a p t e r t h e r e is more emphasis on t h e use ~f m o d e l s in understanding s p a t i a l geonetry. There is correspondingly less err,phasis on " s e t s " as such. This chapter is more geometry than !I sets." However, the terminology of s e t s , i n t e r s e c t i o n s , and unions is used tinroughout this c h a p t e r . The s t u d e n t s s h o u l d already have some farniliarty w i t h these ideas. A qulck run-over of t h e ideas of Chapter 4 of V ~ l u m eI would be u s e f u l as background f o r
.
However, if s e t s , i n t e r s e c t i o n s , unions, and some geometrical terms can be explained as you go along, t h i s , m t e r i a l s h o u l d be teachable w i t h o u t much s p e c i f i c background on the p a r t of s t u d e n t s . The use of terms like "simplex" and t h e d L s t i n c t i o n s between 2- and 3-dimensional polyhedrons should c o n t r i b u t e tc precision of
most s t u d e n t s
t h o u g h t and language.
Time. -
The m a t e r i a l of t h l s u n i t is recommended f o r s t u d y f o r p e r i o d of about two weeks. If class i n t e r e s t is h l g h , some e x t r a
time could p r o f i t a b l y be used. Reading. The t e x t is w r i t t e n w i t h t h e i n t e n t i o n that t h e pupils read (and study) it, There may be occasional passages w h i c h some p u p i l s will n o t follow e a s i l y . Pupils should be encouraged t o read beyond these passages when they occur, and then go back and s t u d y them. Teachers may flnd it p r o f i t a b l e t o r e a d some of t h e material w i t h t h e p u p i l s .
-
Materials. In teaching s u b j e c t matter like this t o j u i o r high school s t u d e n t s , modeis a r e o f considerabie u s e . Familiarity and facility w i t h models s h o u l d increase teachlng effectiveness as w e l l as improve b a s i c understandings. The s t u d e n t s should be encouraged t o use cardboard or oak t a g f o r t h e i r models of tetrahedrons and c u b e s . They will be asked to draw on the surfaces o f t h e s e l a t e r . It is suggested t h a t each student contribute one model of a r e g ~ l a rt e t r a h e d r o n and one of a cube for l a t e r class use. These will be ceeded in the study of polyhedrons. Each s t u d e n t s h o u l d k e e p a model of a cube and a model of a tetrahedron at home f o r u s e in homework. There a r e several suggestions f o r making models from b l o c k s of wood. The boys in t h e class should be encouraged t o make these.
[pages
417-4181
10-1. Introduction
10-2.
Tetrahedrons
It is probably d e s i r a b l e f o r t h e teacher t o follow t h e i n structions i n t h e t e x t t o make in class e l t h e r a model o f a r e g u l a r t e t r a h e d r o n o r a model o f a non-regular t e t r a h e d r o n or both. Have each s t u d e n t leave one model o f a tetrahedron w i t h you. Have them keep one model o f a t e t r a h e d r o n t o use at home. Some boys m i g h t like t o make models of t e t r a h e d r o n s by sawing wood blocks,
Answers to Questions The edges of
(PQRS)
The faces o f
(PW) are
(fa-
(PQ),
are
(PQR),
(PR),
(PSI,
(QR),
(PIIS),
(pa),
and
(Qs),
and
(W).
Answers to Exercises 10-2
1.
( ~ o n s t r utci o n )
3.
The measure or t h e new angle
must n o t be more than t h e sum of the measures of angles BAC and BAD n o r less than t h e i r differences Otherwise, it won1t f i t . his could be tried in c l a s s . ) DAC
.
10-3.
Simplexes
The plural o f simplex is a l s o correctly written as "simplices" W e choose "simplexes" (which is similar to v e r t e x and v e r t i c e s )
.
because the term simplex itself is a new word which t h e students should learn t o use w i t h ease. [pages 417-4231
In C h a p t e r 4 we regarded points, l i n e s , and planes as o u r "building blocks." Here (with some p r i o r n o t i o n of lines and p l a n e s ) we will use p o i n t s , segments, t r i a n g u l a r r e g i o n s , and s o l i d tetrahedrons as o u r " b u i l d i n g b l o c k s . I t The d i s c u s s i o n about " t a k i n g p o i n t s between" and "dimens ion1' p r o b a b l y will need to be read more than once by p u p l l s . A f t e r one g e t s t h e general ideas, the m a t e r i a l is quite readable. A class discussion of these concepts p r i o r t o the reading of t h e t e x t by t h e pupils may be a desirable practice. The discuss i c n
or "betweenness" i l l u s t r a t e s an tmportant f a c t
a b o u t reading some mathematics. One sometimes has an easier t i m e r e a d i n g f o r general ideas and t h e n rereading to fill in d e t a i l s .
Answers to -Class Exercise 10-3 I.
(a and b)
The f i g u r e s a r e g i v e n wlth this p r o b l e m in t h e
student text.
I n ( a ) the figure is a t r i a n g l e . In ( b ) t h e f i g u r e r e p r e s e n t s the union of a trlangle and :ts i n t e r i o r .
2.
(c)
IJo; 2 (in drawing t h e s i d e s of t h e triangle and in s h a d i n g t h e i n t e r i o r of t h e triangle.)
(a)
The union o f t h e edges of t h e tetrahedron
(b)
The mSon o f t h e f a c e s (including t h e edges and v e r t i c e s ) of the t e t r a h e d r o n (ABGD)
(c)
S o l i d tetrahedron.
.
(d) No; 3.
3
(in ( a ) , (b), and ( c )
.)
(a) The p o i n t i t s e l f (b)
0
[pages 421-5251
(ABCD).
Answers t o Exercises 10-3 1.
(a)
6 and 7 .
3
The b e t t e r students might be encouraged t o draw figures
f o r each of t h e s e problems, as:
8,
10-4.
Place two tetrahedrons t o g e t h e r s o t h a t t h e i n t e r s e c t i o n is exactly one face of each. (see t h e third of the f o u r figures in a row at t h e beginning of S e c t i o n 10-5.)
Models -of Cubes
[pages 425-4271
Answers to Exercises 10-4
I,
Constructions
2.
There are a number o f p o s s i b l e answers
. ..
(PABD) and (PBCD) c o u l d be two of t h e
if one f a c e is l a b e l e d as on t h e 12
right.
.
Y e s , you canmake t h e comparison. The t o p v e r t i c e s of the s i x pyramids would correspond t o t h e s i x points in t h e middle of t h e faces of t h e cube. 24;
10-5. Polyhedrons
In c l a s s it would be a gcod idea to h o l d or fasten models of tetrahedrons t o g e t h e r t o i n d i c a t e v a r i o u s 3-dimensional polyhedrons. A l s o , b l o c k s o f wood w:th some c o r n e r s and edges sawed o f f (and maybe w i t h a wedge-shaped p i e c e removed) a r e good models o f 3dimensional polyhedrons. Models o f tetrahedrons s h o u l d b e h e l d t o g e t h e r t o i n d i c a t e how the i n t e r s e c t i o n o f two could be exactly an edge of one b u t n o t a n edge of t h e o t h e r . h hey would n o t i n t e r s e c t IIn i c e l y . " ) Other examples can a l s o be shown here.
[pages 427-4311
Answers t o Exercises 10-5
1.
(a)
( o n e possible answe-r)
(BD),
11.
D r a w t h e segments
5.
(a) D r a w segments (BD), (BF) o r ( K D ) .
(CE)
,
(BH),
and
(HE).
(DF),
(FYI),
and e i t h e r
6.
P W ) , (XYZ), (PRZ)
(a)
(b)
(FPW),
(XYQ), ( Y Z Q ) , ( z w ) , ( X Z P ) ,
IFXYZ), (FXPQ),
(FXZP) and
10-6.
(XPQ),
.
and
One-Dimensional
(FPRZ)
.
(FXYQ),
( ~ z Q ) ,IFZW),
Polyhedrons
The students may enjoy drawing a l l s o r t s o f simple closed polygons. Some should be in a plane. But the s t u d e n t s a l s o should
be encouraged t o draw some s i m y l e c l o s e d polygons on surfaces o f cubes, tetrahedrons, wooden b l o c k s , and so on. It may be d e s i r a b l e to have a little practice in r e p r e s e n t i n g on a plane a few 1-dimensional polyhedrons which are not contained
i n a plane. Answers t o Questions
The polygonal paths from P to S are PS, PRS, P R Q S , PQS, P W . Five in all. Other simple c l o s e d polygons c o n t a i n i n g BE and GA are:
BEDFGAB BEDCHFGAB
BEGAHCB BEGAHFDCB
BEDFGAHCR
-
Answers t o Exercises 10-6 1.
2
3
paths from A t o 3. s i m p l e closed polygons.
[pages 433-436
1
2.
(a)
7
(b)
PGP,
(c)
P($RSP
PEP,
PWP,
WQ, PWP,
PRQSP,
PwQP.
(we a r e naming t h e v e r t i c e s in o r d e r r e t u r n i n g to a s t a r t i n g p o i n t . )
3.
PABCDEPQ,
5.
There are many ways of doing it.
PCBAFEDQ,
PEDCBAFQ
(there are o t h e r s )
10-7. Two-Dimensional Polyhedrons T h i s section gives a g o ~ do p p o r t u n i t y t o draw many 2-dimension-
a1 simplexes and polyhedrons and t o i l l u s t r a t e again s e t s of 2simplexes which i n t e r s e c t n i c e l y . Some drawings arid i l l u s t r a t i o n s should be in the plane and some should be on s u r f a c e s o r 3-dimensional polyhedrons.
Answers t o Questions
S t u d e n t s are asked if they see a r e l a t i o n s h i p among t h e f a c e s , edges, and v e r t i c e s o f certain polyhedrons. It would be d i f f i c u l t t o di-scover Euler' s formula from j u s t three c a s e s . The r e l a t i o n s h i p w i l l be developed in detail in S e c t i o n 10-9.
Answers --t o T a b l e and Exercises 10-7 1.
Tetrahedron Cube Cube (with s lmplexes )
F 4 6
E 6 12
12
18
[pages 436-439
3
and many other p o s s i b i l i t i e s .
and many o t h e r possibil2ties.
and many o t h e r possibilities.
No, there won't be f o u r . The union of t h e f i r s t three must itself be a 2-simplex. Yes, a tetrahedron
.
6.
6 12
s e t s of 4-simplexes each.
original
edges,
and
6
s e t s of
4 new
edges e a c h .
This can b e done in many ways We have f i r s t subd i v i d e d it along AB into two flgures like the second in t h e t e x t o f t h i s s e c t i o n .
.
*8.
(a)
True
(b)
False.
T h i s Is true o n l y of a 1-simplex whose endpoints
a r e a v e r t e x of t h e 2-simplex and a p o i n t of t h e o p p o s i t e 1-simplex which is not a v e r t e x o f t h e 2-simplex. (c)
False. Thls is t r u e o n l y o f a 2-simplex containing a 1-simplex (edge) o f the 3-simplex and a p o i n t of t h e o p p o s i t e edge which is n o t a v e r t e x of t h e 2-simplex.
10-8.
Three -Dimens lonal Polyhedrons
'
In t h i s s e c t i o n and the next s e c t i o n t h e r e a r e many f i n e opportunities t o u s e t h e models whlch t h e pupils have prepared. It is s u g g e s t e d that the class be d i v i d e d into three o r f o u r groups and t h a t each group t a k e s e v e r a l models o f regular 3 inch tetrahedra and fasten them t o g e t h e r ( w i t h ' cellulose t a p e o r p a s t e ) t o produce some rather p e c u l i a r looking 3-dimensional p o l y h e d r o n s . They can also f a s t e n models of cubes t o g e t h e r t o produce odd l o o k i n g polyhedrons.
Again t h e r e is a good opportunity here to encourage some of t h e boys i n the class to make models of 3-dimensional polyhedrons out o f b l o c k s of wood. S t a r t with a block o f wood and saw comers
o r edges o f f o f it and possibly notches ost of it. The surface o b t a i n e d will be a s i m p l e s u r f a c e as long as no holes are punched
t h r o u g h t h e solid. Be sure t h a t the surface is made up only of flat p o r t i o n s . A surface like t h i s could be c o v e r e d with paper and then colored o r drawn on. It could then be re-covered w i t h p a p e r f o r f u r t h e r w e . It is i n t e r e s t i n g that no matter how t h e block is c u t up (without holes) e v e r y simple c l o s e d polygon on I t w i l l separate the s u r f a c e i n t o j u s t two pieces. Also t h e models can be used f o r examples of EulerTs formula In t h e n e x t s e c t i o n , The f a c e s , edges, and vertices can be counted,rather e a s i l y on objects like t h i s . Make a model of a "square doughnut" o u t of 8 cubes as suggested in t h e t e x t . The surface wlll not be a simple s u r f a c e .
-
Answers t o Exercises 10-8 1 and 2.
Constructions.
3.
Three:
4.
Two:
[pages
441-4451
5.
( a ) Five of t h e many p o s s i b i l i t i e s are
as
follows:
Intersections: A
and
B
(3) (4)
face face Pace face
(5)
vertex
( 1) (2)
A
and
C
B
No.
(c)
Yes.
and
C
face face
edge
$
edge
B
vertex vertex
a
$ represents the empty (b)
B
set.
A
and
B
$ edge
$ B
B
and
C
T h e intersection o f cube # 7
with cube # 6
is a f a c e and t h e i n t e r s ec t r o n of cube # 7 w i t h cube # 1 is an edge. Thus, the f i g u r e is not a "simple surface. I I
10-9.
Counting Vertices, Edges, and Faces - - E u l e r l s Formula
Use models of t e t r a h e d r o n s and cubes in explaining the material of t h i s s e c t i o n t o t h e c l a s s . The counting of f a c e s , edges, and v e r t i c e s on any simple surrace s h o u l d b e i n t e r e s t i n g and informative f o r t h e p u p i l s . Use models made of wood f o r some of the examples. U s e t h e model of the "square doughnutf' of the l a s t s e c t i o n t o h e l p pupils c o u n t faces on this type of surface. The t e a c h e r or student should subdivide the surface of a model of a cube in an irregular way and count faces, v e r t i c e s , and edges as i n d i c a t e d in t h e text. I n traditional s o l i d geometry t e x t s a polyhedron 1 s u s u a l l y d e f i n e d as a s o l i d formed by p o r t i o n s o f plane surfaces, c a l l e d faces of t h e polyhedron. It i s assumed that the faces a r e enclosed by polygons. A theorem, c r e d i t e d t o t h e German mathematician Euler, is u s u a l l y included in the t e x t . The theorem s t a t e s that lf t h e number of edges o f a polyhedron is denoted by E, the nwnber of vertices by V , and the number of faces by F, then E + 2 = V + F. I n this s e c t i o n we s e e t h a t the formula does n o t hold in general f o r surfaces which a r e not s i m p l e . The traditional treatment usually does not make a p o i n t of thls limitation of t h e
[pages 445-4481
f o r m . ~ l a . Actually the "scuare doughnut" d e s c r i b e d In the text s a t i s f i e s t h e traditional definition of ?olyhedrons t o which
reference was made. In Chapter 13 It i s noted that E u l e r l s f o r n u l a rvaz known by Descartes who l i v e d some one hundred y e a x before Euler.
Regular Pol3;hedrons Some of the b e s t s t u d z n t s will enjoy a study of why t h e r e
are only f i v e regular polynedrons . Recall t h a t a regular polyhedron is d e f i n e d to be a polyhedron a l l o f whose f a c e s w e congment ~ e g u l a rp o l y g o n s . One of the most' s u r p r i s i n g f a c t s of s o l i d geornet~jis that t h e r e a r e only f i v e r e g u l a r polyhedra. S i n c e t h e usual proof of t h i s f a c t u s e s p o l y h e d r a l a n g l e s , t h e impression is often given that it is a metric p r o p e ~ t y . But it is not s u c h a p r o p e r t y . T h i s can be shown by g i v i r ~ ga. p r o o f depe:lding only on E u l e r 7s i'orrnula: 1. V + F - B = 2 and the fu l b w i n g t h r e e o t h e r as sumpt-ifins : 2 Each vertex is t h e end-point of t h e sane number of edges.
3.
Eazh face is bounded by t h e same number of edges.
4.
Each f a c c has at Least tllree edges and each v e r t e x i s
the end-point o f at l e a s t three edges. Yere is t h e p r o o f . Assumfng P r o p e r t y 2, we ma.y l e t r stand f o r t h e number of edges for each v e r t e x . Then t h e p r o d u c t rV c o u n t s e a c h edge twice, s i n c e each edge has two end-points. Hence,
or V = -2r3' Assuming Property 3, we may l e t s stand for t h e number c f edges w h i c h each f a c e has. Then the product SF counts e a c h edge twice, s i n c e each edge is on the boundary of two faces. Hence,
(4
rV=2E,
Ib 1
sP=2E,
[pages
or
F = -2E S '
445-4481
Next, assuming Eulerts f o r m u l a , we replace V f o m ~ l aby t h e expressions In [ a ) and (b) and have 2E + 2E - E = 2, r 2 E 2( +~ - 1) = 2 . S
and
in thts
F
t h a t is,
S
that is,
1 If r'
and
I were b o t h less t h a n or equal to 7, 1
( d ) would be denied.
Either
But
1
>
1 means
4
Either
(el
B u t Property 4,
than o r eqcal t o (f)
3.
A t l e a s t one of
Property
Ilence,
1
> r
1. is greater than B
or
I;
r.
Hence,
