WIXLIB

Chapter 1WHAT I3 MATHEMATICS?G e n e r a l RemarksThis chapter i s i n t e n d e d to give t h e pupil an appreciation f o rt h e Importance of rnathematlcs.I t s objectives are:I. To develop an understanding of what mathematics is asopposed to simple computation.11. To d e v e l o p an appreciation of the role of mathematicsin our culture.111. To motivate p u p i l s by p o i n t i n g o u t t h e need for mathematiclans and f o r mathematically trained people.S i n c e t h i s c h a p t e r is much different from o r d i n a r y textbookmaterial it will need a different treatment. The purpose o f thebuildchapter i s n o t t o teach many f a c t s or s k i l l s , but r a t h e r to -an enthusiasm f o r the study of mathematics. Good a t t c t u d e s willbe b u i l t i f you use i m a g i n a t i o n and enthusiasm i n g e t t i n g t h e s eo b j e c t i v e s across t o t h e p u p l l s . Since t h e material I s n o t to betaught f o r m a s t e r y , w e strongly recommend t h a t no t e s t be g i v e ncovering t h e contents of t h i s c h a p t e r ,Experience shows t h a t t h i s chapter c a n be covered within sixt o eight l e s s o n s .C e r t a k n l y n o more t h a n e i g h t days should b ed e v o t e d to it.Were s e v e n t h g r a d e r s are i n a new school situation and haveso many i n t e r r u p t i o n s during the first few days, some t e a c h e r s maywfsh t o precede this c h a p t e r with review exercises which a r e morefamiliar t o the pupils.Note t h a t Exercises 1-6, (Page 13) and E x e r c i s e s 1-7, fa age 14)are suggestions for background study t o be c a r r i e d on throughout-the year. These should be begun d u r i n g t h e first week, withp e r i o d i c r e p o r t s on progress by p u p i l s . Where guldance p e r s o n n e lare a v a i l a b l e , t h e i r services should be solicited t o h e l p t h eclass o u t l i n e a p l a n o f a c t i o n f o r t h e year.

It might be worthwhile t o have the pupils read this chapteragain a t the end of the year. The problems m i g h t also be solvedagain. They should be much eaaier t o aolve after the courae hasbeen completed.Encourage the more able student8 to solve the brainbuatersbut be ready to h e l p them if they have difficulties. Most pupilswill want to puzzle over the brainbusters f o r a few days. Forthie reason, only Individual help I s suggested u n t i l the timeseems appropriate f o r general class diacusslon.-as- a Method of Reasoning,1-1. Mathematics Page &. It might provide additional challenge to emphasize to thepupila that Exercises 1-1 and Exercises 1-2 are not easy. Moreover, no slmple formula for solution c a n be given. Some of thepupils (and many parental) will certainly f i n d the problems d i f f i c u l t and time-consuming at t h i s stage. You may not wish toassign all the problems in these two sectlone.-Answers to Exe cises1-1--Page-- 2:1. THO sons cross; one returns. Father crosses; other sonreturns. Two sons cross.2, No. They need a boat carrying 225 pounds. Solution asin 1 above,3. Man takea goose and returns alone. He takes fox andreturns with goose. He takea corn across river andreturns alone to p i c k up goose.4, Y e s , Thfs depends on the fact t h a t 8x 5y 2 hassolutions in integers, such as x -1, y 2 and x 4,y -6. The first means that if you fill the 5-gallonjug twice and empty it once Into the 8-gallon jug, youwill have 2 gallons left. The second solution meansthat if you fill the 8-gallon jug four times and use itto fill the 5-gallon jug 6 times, you will have 2 gallonsleft, Point o u t that the first s o l u t i o n is b e s t .[pages 1-21

I-5.1iIt3Most pupils will need paper and pencil f o r this one, Asthey attempt the solution in front of the class it is agood idea to minimize the help from others in t h e class.If an error 18 made, another student should be selected t opresent h i a solution. It should be pointed out that acroaafng neceasltatss a landing.5 CI cross river; returns.Ml M2 M3.C1 C2 C3C croaa river; C1 returns.2 3M1 Me cross river; M1 C2 return.CtM1 M3 cross river; C 3 returns.C1 C2 cross river; C1 returns.C1 C3 cross river.67,Balance the two groups of 3 marbles each. If they balancethen it is only necessary to balance the remaining twomarbles t o f i n d the heavy one. If the t w o groups of 3marbles do n o t balance, take the heavier group. Of the 3marbles in the heavier group balance any 2 marbles. Ifthey balance, the remaining marble is the heaviest one.If the 2 marbles do not balance, the heaviest w i l l be 1of the 2 on the balmce.In aolving the problem is it practical to t r y out all t h epossible waye the dominoes may be placed on the board?This would be difficult because there are more than65,536 ways t o cover the whole board, The solution maybe found in another way:H o w many squares are there altogether on the board?(64)How many squares muet be covered? (62)What is special about the t w o square8 next to eachother? (They sre of different colors. )What I s speclal about the two opposite corners?co hey are the same color. )

If you place any number of domlnoes on the board can yousay anythlng about the kinds of squares which will beHow doea t h i s compare with the kinds of squareswhich you are supposed to cover? Do you have to makeeven one experiment in order t o get the answer to theoriglnal problem? Can two squares of t h e same color becovered with one domino? The answer t o t h l a questionahould help some of the students reason why the solutionI s impoaa lblecovered?.1-2. Deductive Reasonlnq.Pam 3 . After t h e concept of deductive reasoning has been I n t r o duced, it i a s t i l l necessary t o give the students an understandingof I t s importance. Examplea of s c i e n t i f i c advances in which deductive reaaoning was used might be Elnstelnts discovery of the2atomic energg formula E MC , apace travel, satellites, d i g i t a lcomputer development, nuclear energg, e t c .-- - Page -4:Anawers to Exercises 1-2I.After each person receivea one pencll, the 5th pencilmust go to one of the 4 people.2.(a) Yes.(b) Someone g e t s a t least 3 pencils.3.367.4.Tell the class that the f i r s t two to enroll might betwins.7.Only 1.8.6 committees,9.12 committees.h he answerm a t take leap year i n t o account .)1.[pages 3-45

to Mathematlcs.1 - From Arithmetic Page 5 . Before considering Gauss's discovery of summing an a r i t h m e t l c series, you may wish to offer the f o l l o w i n g as an example ofa s c i e n t i f i c experiment i n which mathematics i s used.Make a pendulum by fastening apsilingweight to a string 40 inches long.A t t a c h the s t r i n g t o an object thateye hookw i l l not move, The pendulum ehouldswing freely. Set the penduluminto motion. Count t h e number oftimes it swings back and forth in30 seconds. Shorten the string by5 inches and repeat t h i s experiment.fShorten again by 5 inches and do it0i,again. Make a table of your ob---Length of s t r i n gIn kncheeNo. of gwingsi n 30 secondsDoes there aeem to be a relationshfp between the length ofthe pendulum and the time? C a n you predict t h e number of s w i n g si n 30 seconds If the string is 20 inches long? How does thenumber of swings depend upon t h e length of t h e atring?You have actually repeated an experiment done by Galileoabout 400 years ago. Galileo w a s a famous Italian scientist whol i v e d i n the years 1564-1642. He g o t the idea for the experimentby watching a hanging light fixture swing back and f o r t h , Hetimed t h e swing by means of his pulse beat. He was one of thef i r s t a c l e n t i s t s to show how important it was to Investigateproblems by the experimental method.

In mathematicswe often u s e the inductive method t o discoversomething. Then we use deductive reasoning to prove that it istrue.John Freedrich Karl Gauss was born in Brunawick, Germany, in1777. He died i n 1855 at the age of 78. The p u p i l s may be interested in n o t i n g t h a t his lifetime almost spanned the years from theAmerican Revolution to the Civil War.Many mathematicians consider Gauss as one of the three greate a t mathematicians of a l l times, the other two being Newton andArchimedes.In this age of space exploration'lt is interesting to n o t ethat Gauss developed powerful methods of calculating o r b i t s ofcomets and planets. His interests extended a l s o to such fieldsas magnetism, gravitation, and mapping. In 1833 Gauss inventedthe e l e c t r i c telegraph, which he and h i s fellow worker, WilhelmWeber, used as a matter of course in sending messages.In 1807 Gauss was appointed Director of the GdttingenObservatory and Lecturer of Mathematics at G8ttingen University.In l a t e r years the greatest honor t h a t a G e r m a n mathematiciancould have was to be appointed t o the professorahip which Gausshad once held.Thla sectlon deals with Gauss's discovery of the known methodof s m i n g an arithmetic series. It dramatizes how some pupils(and mathematicians) apply insight to f i n d i n g a a o l u t i o n to aproblem. Your better students should be told t h a t there aremethods other than G a u a a l s for f i n d i n g the sum of a seriee ofnumbers. Some students might be encouraged t o discover methodsof their o m for addfng number series quickly.The "middle number" method is one t h a t may be used. Thisscheme can be used f o r an even or an odd number of integers. Thefollowing examples may be used to explain this method a thestudents who have tried to discover other methods.

In t h i s s e r i e st h e m i d d l e number ( 4 ) is the average of t.hei n d i v i d u a l numbers of the s e r i e s . T h e sum is the product of t h emiddle umber (4) and the n u n b e r of i n t e g e r s in the series o r4 x 7 25.Some pupils may p r e f e r t o t h i n k of t h e series as(1 7) (2 6) (3 5) 4 (4 4 ) c ( k 4) ( 4 4) 4 7 4 28.orExampleE.1 t 3 4 5 6 7 -1-8 ?In this case t h e "middle number" is halfway between h and 5,11Then t h e p r o d u c t (%) x 8 36 is seen to g i v e t h e correct.LSum.It may seem more p l a u s i b l e here to write t h e sum as( 1 8) ( 2 7) ( 3 6) (4 5) C l e a r l y , Gauss's method is t o b e p r e f e r r e d in t h i s c a s e .Answers to Exercises 1-3--page6:-1. A n o t h e r method is this: 2 4 2 3,1 5 3 3.T h a t is, the sum is t h e same as:3 3 3 3 3 5 3 1 5 .This c a n b e c a l l c d t h e "averaging method."2.E i t h e r method works.Gauss method:8 x 5 '.I?-Averaging method:3.11.16 x 8 6k.5 x 4 20.Here t h e r e is an even number of quantitiess o t h a t t h e "averaging method" must bemodified t o g i v e 8 e i & t f s o r 8 x 8 64.(a)1!.(b)( )916.(d)T h e sum of the f i r s t 'In" consecutive odd numbers q u a l sthe square o f "nu.(e)64.

8.9,Y e s , provided that the numbefs are In arithmetic progreaaion; that is, there 18 the same difference between eachp a i r of adjacent numbers,Yes. If we start with 1 there are 200 Integers in theseries giving usv.If we s t a r t with 0 therea r e 201 integers in the series g i v i n g usw.The products of like factors are equal. The method a l s omay be used in a a e r l e e if we select a number other than1 or 0 as a starting point. Some of the better atudentamay investigate whether the method works in other numberseries,10. (a) If you add 1 t o the quantity, the sum up to anynumber is equal t o the next number. Hence, thesum plus 1 is equal to 2 256 s 512. Therefore,the sum is 511.(b) This is more In the s p i r i t of Gauss:Sum I 2 4 2562 x sum 2 4 256 512 Subtracting: Sum 511.11. Sum 2 (6 18 486)3 x sum 6 18 486 1458Subtracting: 2 x ( t h e sum) 1458 - 2 1456Sum, 728.,.Kind8--of Mathematics.Page 7. DiBcussion of t h i s section should emphasize the dynamiccharacter of mathematics. It is n o t a "dead" subject as manyparents believe,1-4.

It is i m p o r t a n t a l s o to p o i n t o u t h e r e ( a n d t h r o u g h o u t thec o u r s e ) t h a t c e r t a i n important ingredients a r e commoli to a l l t h emany varieties o f mathematics. The method of l o g i c a l reasoning,t h e u s e and manipulation of abstract symbols, t h e i n s i s t e n c e onp r e c i s i o n of thought and clarity o f expression, the emphasls ongenera: results--these a r e some characteristics which need to bestressed whenever p o s s i b l e .Probability--page- - - 9:T h i s s e c t i o n g i v e s only a v e r y b r i e f introduction to p r o b a bility. Although students may become interested a t this p o i n t a n dattempt more complex problems, it would bo b c t t e r if they waited.A c h a p t e r on probability is i n c l u d e d in V o l u n11. Some s u p p l e mentary u n i t s also a r e available.-Answers to Exercises 1-&--page 9 :3.One o u t of f o u r o-r-r1One o u t of two o r 7.'I.One o u t of 52 or5.Four o u t of 5 22.Iq.- 41o r 52 3.-T h e probability may be t h o u g h t of as a ratio ofthe number of p o s s i b l e f a v o r a b l e s e l tci o n st o t a l number of a l l p w e c t l o n s .5 eThe p u p i l s s h o u l d be reminded t h a t t o say his c h a n c e s a r e6.7.8.1 o u t o f 13 does not mean that he will necessarily drawa n a c e in t h e f i r s t 13 draws.One o u t of six. A d i e h a s 6 s i d e s and o n l y o n e side hast w o dots.There are four possibilities In a l l , o n l y one of whlch1is favorable. Hence the probability is T.One out of?,6. The moreadvanced s t u d e n t s should r e a s o nt h a t t h e r e are 36 p o s s i b l e combinations. A t a b l e may b ec o n s t r u c t e d t o show t h e possibilities. A possible diagramis t h e following.[pages 8-91

Number ofPossibilities1. 3435361st d i e1 1 1 1 1 1 2 2 2 . . . 6662nd d i e1 2 3 4 5 6 1 2 3 . . . . 4 56only one possible way of making t w o ones243567 8 9Since there i61the probability is 56.9.The p o s s i b i l i t i e s are e a s i l y enumerated.Number ofPossibilities1st coln2nd coin3rd coin4H H H HH H T TH T H T1235 6 7 8T T T TH H T TH T H T1For 3 heads to come up, the probability l a 8.Forexactly 2 heads the probability is 3 out of 8 o r g3. Forat least two heads the chance is 4 out of 8 or 1NoteCt h a t thla equals the probability of exactly two heads1plus the probability of 3 heads (B).-.---6)Class Activities 1-6 and 1-?--paps 13-14:The exercises suggested in Section 1-6 and Section 1-7 shouldbe undertaken as year-long p r o j e c t s t o be reported on periodically.Much of t h i s information would be good bulletin-board materisl.A general aim is to make the pupils alert t o the current newsrelating to mathematics and mathematicians. A s the year progressesthey should gain an increasing appreciation of the i m p o r t a n t r o l emathematical thought is playing in our present civilization.The National Science o u n d a t i o n , Washington 25, D.C., publ i s h e s pamphlets which contain fnformatlon on the number ofmathematicians and where they are employed,[pages 13-14 I

II!11A survey of college requirements in the v o c a t i o n s s t u d e n t smay choose should be especially valuable at this stage. We hopeit may eaee the difficult task of effective guidance in choosingt h e i r high school courses.One word of c a u t i o n is perhaps warranted here. The purposeof these sections on mathematics today 1s to call a t t e n t i o n to itscentral r o l e in the pupil's daily life. The necessity for a minimum howledge of mathematics is to be stressed. It is n o t t h e-----Ai n t e n t i o n t o recruit people for careers as rnathernatlcians.--1-8. Mathematics for Recreation.Page 15, Brlng out the fascination of mathematics as a leisurea c t l v i t y or hobby. ncourage students in f i n d i n g recreationalmathematics from the booka available at school m d from c u r r e n tmagazines or rotogravure a e c t i o n of newspapers.Choose auch problems now and then, throughout the year, at atime when the class needs a change of pace. These k i n d s o fproblems can be used profitably with the class period before alengthy vacation.Dlseuasion of Kijnigsberg Bridges Problem.By experimenting you can show that it is impossible t o passi j u s t once over every bridge if there are more than two p o i n t siIwhere an odd number of r o u t e s come together. Since there aretIPour p o i n t s where an odd number of route8 come together, t h i s.,, I make8 it impoesible to walk over each bridge once and only once.You may w i s h to consult the supplementary unit on the KGnigsbergBridges f o r further ideas.j---Answer to Exercise 1-8--page 16:1The first figure can be drawn if you s t a r t at e i t h e r ofthe vertices where an odd number of segments come together. The second figure has no such vertices so itcan be drawn by starting at any vertex. The t h i r d f i g u r ehas four vertices where an odd number of segments cometogether ao it cannot be drawn without l i f t i n g yourp e n c i l or retracing a line.[pages 15-16]

1-9, Highlights --of First-Year J u n i o r High School Mathematics.-Page 16,--This s e c t i o n is intended to give the student a generaloverview of the scope of the first-year course in junior highschool mathematics. It is intended to broaden h i s ideas of mathematics by discussing very b r i e f l y the t o p i c s that will be studied.This, together with the i n t r o d u c t i o n of logical r e a s o n i n g andprobability earlier in t h i s chapter, w i l l h e l p d i s p e l the generalidea t h a t mathematics is computation only.Reading this chapter again later in t h e year may s t r e n g t h e nthe student's understanding of what mathematics is,

Chapter 2IntroductionFor t h i s unit l i t t l e background is needed except f a m i l i a r i t yw i t h the n u n h e r symbols and t h e b a s i c operations w i t h numbers.The purpose of the unit is t o deepen the pupilis understmding ofthe decimal notation f o r whole numbers, especially with r e g a r d t oplzce value, and thus to h e l p him d e l v e a little d e e p e r into t h ereasons for the procedures, which he already knows, for c a r r y i n gout the a d d i t i o n and multiplication operations. One of t h e bestways t o accomplish this is t o c o n s i d e r systems of n m b e r n o t a t i o n su s i n g bases c t h e r than t e n . S i n c e , in using a new base, t h ep u p i l must necessarily l o o k a t the reasons for "carrying" and theother mechanical p r o c e d u r e s in a new light, h e s h o u l d g a i n d e e p e rinsight i n t o t h e decimal s y s t e m . A c e r t a i n amount of computationin o t h e r systems is necessary to "fix1' these ideas, but s u c hcomputation should n o t be regarded as an end in itself. Some ofthe p u p i l s , however, may e n j o y d e v e l o p i n g a c e r t a i n proficiencyi n u s i n g new bases I n computing.Perheps the most i m p o r t a n t reason for i n t r o d u c i n g ancients p b o l i s m s f o r numbers is to c o n t r a s t them with our decimal system,in which not only t h e symbol, but its p o s i t i o n , has significance.I t should b e shown, as c t h e r systems are presented, t h a t positlonhas some s i g n i f i c a n c e i n them a l s o , The Roman System h a s a s t a r tin this d i r e c t i o r : in that XL represents a different r?wriber f r o mLX, but t h e s t a r t was s v e r y primitive one. The Babylonians alsomade use o f p o s i t i o n , b u t l a c k e d a symbol f o r

Mathematics for Junior High School, volume : Teacher's Commentary, Part I Preparrd under the supervision of the Panel on Seventh and Eighth Grades of the School Mathematics Study Group: R. D. Anderson J. A. Brown Lenore John B. W. Jones P. S. Jones J. R. Mayor P. C. osenbloom Veryl Sch