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Sawyer

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http://www.m arco-learningsystems.com/pages/sawyer/Vision_in_Elementary_Mathematics.pdf

http://www.marco-learningsystems.com/pages/sawyer/sawyer.htm

Warwick Sawyer

Vision in ElementaryMathematics

Ch 3An unorthodox

point of entry

http://books.google.co m/books?id=0FY3KlWclvYC&lpg=PP1&dq=vision%20in%20elementary%20mathematics&pg=PA40#v=twopage&q&f=false

http://books.google.com/books?id=0FY3KlWclvYC&lpg=PP1&dq=vision%20in%20elementary%20mathematics&pg=PA40#v=twopage&q&f=falsehttp://www.marco-learningsystems.com/pages/sawyer/sawyer.htmhttp://www.marco-learningsystems.com/pages/sawyer/sawyer.htmhttp://www.marco-learningsystems.com/pages/sawyer/sawyer.htmhttp://www.marco-learningsystems.com/pages/sawyer/sawyer.htmhttp://www.marco-learningsystems.com/pages/sawyer/sawyer.htmhttp://www.marco-learningsystems.com/pages/sawyer/sawyer.htmhttp://www.marco-learningsystems.com/pages/sawyer/sawyer.htmhttp://www.marco-learningsystems.com/pages/sawyer/sawyer.htmhttp://www.marco-learningsystems.com/pages/sawyer/sawyer.htmhttp://www.marco-learningsystems.com/pages/sawyer/Vision_in_Elementary_Mathematics.pdfhttp://www.marco-learningsystems.com/pages/sawyer/Vision_in_Elementary_Mathematics.pdfhttp://www.marco-learningsystems.com/pages/sawyer/Vision_in_Elementary_Mathematics.pdfhttp://www.marco-learningsystems.com/pages/sawyer/Vision_in_Elementary_Mathematics.pdfhttp://www.marco-learningsystems.com/pages/sawyer/Vision_in_Elementary_Mathematics.pdfhttp://www.marco-learningsystems.com/pages/sawyer/Vision_in_Elementary_Mathematics.pdf

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It is a defect of most algebra books that they begin bydeveloping a lot of machinery, and it is a long time

before the learner sees what he can do with all thismachinery.

Notes

It is quite possible to use simultaneous equations as anintroduction to algebra . Within a single lesson, pupils who

previously did not know what x meant, can come, not merelyto see what simultaneous equations are, but to have somecompetence in solving them. No rules need to be learnt; thework proceeds on a basis of common sense.

Notes

A man has 2 sons. The sons are twins; they are thesame height. If we add the mans height to the

height of 1 son, we get 10 feet. The total height ofthe man and the 2 sons is 14 feet. What are theheights of the man and his sons?

Starting puzzle

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develop use of diagrams and symbols

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It is not enough for a child to hear an idea explained,and to understand it. Children need to spend sometime with the idea, to play around with it, to use itthemselves.

Textbooks for this reason contain masses of exercisesfor the pupils to work. Sometimes these are dull.Sometimes they go too quickly to difcult problems. Oneway to enliven the teaching of simultaneous equations,and to keep the work at the childrens own level, is to

get the pupils to invent the problems .

Notes

Think of a height for a man, and a height for hissons. Do not tell me these, but tell me, rst,what you get when you add the mans height tothe sons, and second, what you get when youadd the mans height to twice the sons height.

Notes

Some writers of textbooks get into great difculties, because they try to start with a denition.They will say perhaps. In algebra, letters such as x, m, s stand for unknown numbers. Thensomeone else will criticize this denition. In our example above, the child who made up the problemknew what m and s stood for; the other children did not know until they have solved the problem.Was m a known or an unknown number? Known to whom?Almost any short sentence you can say, attempting to dene what we mean by x, will be wrong.Not only that; it will be an ineffective approach to teaching. We do not give a baby a scienticdenition of dog - but babies know what dogs are. Much the best way to introduce children toalgebra is to show them algebra in action.

Notes

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We know that 1 man and 1 son take up 10 feet. By doubling, we see that 2 men and 2 sonsll 20 feet. In fact, we can make a long list of statements as in Figure 14.

An important device

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It should be noted that this is a logical process. Pupils ask Am I

allowed to do this? as if we were playing a game with certainrules. A pupil is allowed to write anything that is true, and notallowed to write anything untrue! These are the only rules of mathematics.

Notes

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Equationsinvolvingsubtractions

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Exercises

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Exercises

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Exercises

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Exercises

l d

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U i l d

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One of the things you nd in teaching is that you can give a veryclear explanation, and still the pupils - or some of them - will notunderstand. A poor teacher simply repeats the originalexplanation, like a robot. A good teacher tries to nd out exactlywhere the pupil failed to understand. This is the essence of teaching - try to guess what is happening in the pupils mind, andto lead the pupils thinking to the next stage. If we imagine thata pupil understands something, which in fact he does not, we arelike a man trying to lay bricks on a foundation of air.

Notes

A child can look at m+s=10 and simply see meaningless marks onpaper. We have to check carefully that the child can write theequation corresponding to any picture, and draw the picturecorresponding to any equation. This must be done often enoughfor pictures and equations to become closely associated in thechilds mind. Any time there is difculty in reasoning about anequation, the picture of it should be drawn.

Between pictures and equations

U titl d

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Vision in Elementary Mathematics

pbture2 picture 3Figure 20

be 7 and the son 3. In a puzzle we even have to consider that thema n might be 2 and the son 8 feet high. It is only when we cometo tho second statement in the puzzle, m*2s:14, that we findmost of ou r guesses fail to fit.

Children, of course, often guess their way to the solution. Dono t Bay, 'It is worthless unless yo u do it by the proper method.'Mathcmatics is meant to encourage thinking, an d guessing is onewe y of thinking. In fact, guessing helps to convey to a child then anlnf of the puzzle.If a child guesses m:7 and s:3, we askhlm to chcck whether this fits both equations. He finds, yes, itdOel 6t n * s : 10 , bu t no , it does no t fit rn * 2s : 14 . A few guessesh lp to drivo home the fact that a solution must fit a// the state-@tltl of tho puzzle.

MrthEmetlclans regularly guess solutions of their problems.Eyftggaetlc methods of solution are needed when problems aretoo canPllcctcd for the answer to be guessed. One can weanchllCtea EWRy flom pure guesswork by posing problems in which

the auaberr ere too large fo r the solution to be guessed. Th e childthea lgcr why rome system for solving is necessary.EuFFerl then a ehlld ha s learnt to associate equations and

pletuf ar }Ed t ure the pictures properly - that is to say, he doesne t erlug ften eceldental details of the picture. We make surethct be een eerly v r to written equations (illustrated if necessaryby pletulcr) the :lmplc arguments that were used to solve the

60

An Unorthodox Point of Entryfirst problem in this chapter. We should no t be surprised if hefinds difficultyin the section headed 'An Important Device'.Some children pick up the ideas of this section very quickly.Others require much time to think about them. There are twoideas involved, an d both are liable to cause difficulty.First, achild may no t se e that a statement such as rn*s:10 necessarilycarries with it the truth of other statements such as 2nr*2s:20an d 7m*7s:70. Second, even after having understood this firstpoint, a child may not know which of allthese possible statementsis the one that will help solve a particular problem. An d indeed,it is no t surpriding that an understanding of this second pointshould often come slowly, fo r it is an example of an advanced typeof thinking - what on e might call strategic planning. Th e childha s to say to himself, 'Here is this equation. I could multiply itby 2, or by 3, or by 4, or by many other numbers. Which of al lthese willgive me a result that combines nicely with the otherinformation I have about this puzzle?'To answer such a questionrequires a fairly wide mental horizon. A child cannot be expectedto make a wise choice about which operation to perform whilehe is still puzzled about the details of the operations themselves.No r can this wise choice be expected until the child has solvedenough problems successfully for him to see the process of solutionas a whole. It is, therefore, quite in order fo r a teacher to se tproblems (at any rate fo r the average pupil) with a hint, 'multiplyequation (i) by 4'.

Th e power of strategic planning is the most important elementin the solving of problems, and it is the one that is least taught.Rote teaching bypasses it completely. Th e pupil is told what to do,so he never learns to weigh on e method of attack against another.A pupil ca n only learn good judgement by seeing the effects ofbad judgement. It is instructive to work on e problem by five orsix different methods, some of which ma y lead to blind alleys andhave to be abandoned, some of which may give the correct solu-tion but only after laborious wanderings, an d some of whichgive the solution with a minimum of effort.

There should be plenty of discussion of what method to use,an d different methods should be tried out. This is emphasized,because a teacher may feel that it is a waste of time to explore

6t

mpicture I

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note the constraint andfreedom arising from

the equation m+s=10students should notargue from accidentaldetails of the picture

Children, of course, often guess their way to the solution. Do not say, It is worthless unless youdo it by the proper method. Mathematics is meant to encourage thinking, and guessing is one wayof thinking. In fact, guessing helps to convey to a child the meaning of the puzzle . If a childguesses m=7 and s=3, we ask him to check whether this ts both questions. He nds, yes, it doest m+s=10, but no, it does not t m+2s=14. A few guesses help to drive home the fact that a

solution must t all the statements of the puzzle.

Notes

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Vtslon in Elementary Mathematics

approaches that lead to no solution. It is by no means wastedtlme. We only appreciate and understand a good method if wehovo saen the consequences of using bad methods.

So much for strategic planning; we return to the first and smallerdifficulty, that of the child who sees no connexion betweenrz*s:10 and2m*2s:20.

We have already made some effort to treat this pictorially. But,of course, in teaching it is not sufficient to show a picture once.It has to be pulled out many times, and in the end its messagesinks into the learner's mind. A picture is, in fact, a most valuableway of reminding someone of a sustained chain of thought. Yourhow the picture and ask, 'Do you remember the discussion wehad about this?'

Our earlier picture pointed out that if a man and his son fittedinto a space of l0 feet, by simply repeating the drawing you could8 e a way of filling 20 feet.

Figure 2l

FhgE trl rhowr that, if l0 feet holds a man and a son, 2O feetwlU hold t nen end 2 sons. But perhaps it does not show this asoleedy Er lt mlght. If we were asked to illustrate '2 men and 2aoaafr th aatural thing to draw would be as in Fig;ne 22.

An Unorthodox point of Entryrt is not immediately obvious that this collection just fills 20 feet.Collection C contains 2 men and 2 sons, as does collection B,but the arrangement is different. C suggests 2m*Zs where Bsuggests m*s*m*s. So, while it is immediately evident to theeye that B takes twice as much room as A, it is not immediatelyevident that C does. It can be appreciated by the mind, if we recoC-

nize that the change in the order of the men and the sons, as liego from B to C, does not affect the total space occupied.A demonstration with movable objects, such as bricks or rods,

might help children to appreciate the connexion between A, B,and C.

At any rate, the pictorial approach should help to bring a childsome distance towards feeling (and remembering) that the state-ment, 'If m+s:10, then 2m*2s:20' is a reasonable one.

We supplernent the pictures by experiments in arithmetic.Different pupils are asked to think of 2 numbers that add up to10. the first number any pupil has thought of we call m for sliort,the second s. So for each pupil, rn +s:10. We then ask round theclass, 'What did you get for 2rn*s? What for 2m*2s?

What for2m*3s2' The answers are tabulated like this:

Jack2m* s 182m*% 202m*3s 22

Mary Anne13 t920 2027 2t

Bilt15

2025

,0

rt now stands out very clearly that 20 occurs all the way acrossthe middle row. The class may try to find 2 numbers adding to r0that make 2nr*2s different from 20. They will fail to nno anexception (except any due to faulty arithmetic).

- Further, in looking at the tabre above, the children will noticehow each column goes up by equal steps. Jack has lg, 20, Zzrising by 2, Mary has 13, ZO, Z7 rising by 7, and so on. Andtheother pupils will often spontaneously say, .Jack was thinking of8 and 2,' and in the same way thcy wil discover what num6ersthe others chose. rf they do this by guesswork, they may noticethat the second number, .r, is the same as the step by whictr thenumbers rise; for example, Jack had s:2 and his numbers fg,20,22fise by steps of 2. And in this way they will be led back to

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A child may not see that a statement such as m+s=10 necessarily carries with it the truth of other statements suchas 2m+2s=20 and 7m+7s=70.

Notes

Collection C contains 2 men and 2 sons, as doescollection B, but the arrangement is different. Whileit is immediately evident to the eye that B takes twiceas much room as A, it is not immediately evident thatC does.

Notes

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Vtslon in Elementary Mathematics

approaches that lead to no solution. It is by no means wastedtlme. We only appreciate and understand a good method if wehovo saen the consequences of using bad methods.

So much for strategic planning; we return to the first and smallerdifficulty, that of the child who sees no connexion betweenrz*s:10 and2m*2s:20.

We have already made some effort to treat this pictorially. But,of course, in teaching it is not sufficient to show a picture once.It has to be pulled out many times, and in the end its messagesinks into the learner's mind. A picture is, in fact, a most valuableway of reminding someone of a sustained chain of thought. Yourhow the picture and ask, 'Do you remember the discussion wehad about this?'

Our earlier picture pointed out that if a man and his son fittedinto a space of l0 feet, by simply repeating the drawing you could8 e a way of filling 20 feet.

Figure 2l

FhgE trl rhowr that, if l0 feet holds a man and a son, 2O feetwlU hold t nen end 2 sons. But perhaps it does not show this asoleedy Er lt mlght. If we were asked to illustrate '2 men and 2aoaafr th aatural thing to draw would be as in Fig;ne 22.

An Unorthodox point of Entryrt is not immediately obvious that this collection just fills 20 feet.Collection C contains 2 men and 2 sons, as does collection B,but the arrangement is different. C suggests 2m*Zs where Bsuggests m*s*m*s. So, while it is immediately evident to theeye that B takes twice as much room as A, it is not immediatelyevident that C does. It can be appreciated by the mind, if we recoC-

nize that the change in the order of the men and the sons, as liego from B to C, does not affect the total space occupied.A demonstration with movable objects, such as bricks or rods,

might help children to appreciate the connexion between A, B,and C.

At any rate, the pictorial approach should help to bring a childsome distance towards feeling (and remembering) that the state-ment, 'If m+s:10, then 2m*2s:20' is a reasonable one.

We supplernent the pictures by experiments in arithmetic.Different pupils are asked to think of 2 numbers that add up to10. the first number any pupil has thought of we call m for sliort,the second s. So for each pupil, rn +s:10. We then ask round theclass, 'What did you get for 2rn*s? What for 2m*2s?

What for2m*3s2' The answers are tabulated like this:

Jack2m* s 182m*% 202m*3s 22

Mary Anne13 t920 2027 2t

Bilt15

2025

,0

rt now stands out very clearly that 20 occurs all the way acrossthe middle row. The class may try to find 2 numbers adding to r0that make 2nr*2s different from 20. They will fail to nno anexception (except any due to faulty arithmetic).

- Further, in looking at the tabre above, the children will noticehow each column goes up by equal steps. Jack has lg, 20, Zzrising by 2, Mary has 13, ZO, Z7 rising by 7, and so on. Andtheother pupils will often spontaneously say, .Jack was thinking of8 and 2,' and in the same way thcy wil discover what num6ersthe others chose. rf they do this by guesswork, they may noticethat the second number, .r, is the same as the step by whictr thenumbers rise; for example, Jack had s:2 and his numbers fg,20,22fise by steps of 2. And in this way they will be led back to

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We supplement the pictures by experiments in arithmetic. Different pupils are asked to think of 2numbers that add up to 10. The rst number any pupil has thought of we call m for short, the second s.So for each pupil, m+s=10. We then ask round the class, What did you get for 2m+s? What for 2m+2s?

What for 2m+3s? The answers are tabulated ...

Notes

It appears from the table that, if we know m+s=10 we are able to say what 2m+2s is,but we cannot predict the values of 2m+s and 2m+3s. One way to test understanding ofthis work is to give a pupil a list of expressions, such as 3m+s, 3m+2s, 3m+3s, 3m+4s,

4m+s, 4m+2s, etc., and ask him to pick out those whose values are xed when we knowm+s=10, and to say what these values are.

It is just as important to realize that the value of 3m+2s cannot be deduced from that ofm+s, as it is to know that 4m+4s can.

Notes

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y

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Graphs rest on one of the simplest and most useful ideas inmathematics. A graph will often help us to solve a problem, andchildren should l earn to regard graphs as a useful aid, and to

sketch a graph whenever occasion arises. It is a great pity whengraphs are taug ht ponderously and unattractively, as a subject inthemselves, so t hat pupils come to dislike and avoid graphical work.

Sawyer (2003) p.270

Web View

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CHAPTER ELEVENGraphs

Gnapns rest on one of the simplest and most useful ideas inmathematics. A graph will often help us to solve a problem, andchildren shou ld learn to regard graphs as a useful aid, and tosketch a grap h whenever occasion arises. It is a great pity whengraphs are ta ught ponderously and unattractively, as a subjectin themselves, so that pupits come to dislike and avoid graphicalwork.

The introd uction to graphical work can be quite informal. Aclass of youn g children are confronted with the following prob-lem. There ar e two aeroplanes, a faster one that flies at l0 milesa minute and a slower one that flies at 5 miles a minute. Theslower plane h as a start of 20 miles, and the faster one is chasingit. When will the faster plane overtake the slower one?

It is quite natural to start drawing pictures (see Figure 160).

0 2 3 minutes

Graphs

(as shown by the dotted lines) and they will predict that over-taking occurs after 4 minutes, corresponding to the point on thegraph where the dotted lines cross. It is not even necessary to usegraph paper. A rough sketch on paper or a blackboard may wellbe sufficient to suggest the dotted lines, if the drawing is reason-ably accurate. Squared paper may be used, but ifso one shouldnot make a meal of it. Minute attention to the mechanics ofplotting points is definitely out of order. The aim is to picturothe problem quickly, and to predict when the chase will end.Once we have guessed that 4 minutes is the required time, it iseasy to check that this is correct. For in 4 minutes the fasterplane will cover 40 miles; the slower one will cover 20 miles, andas it had a start of 20 miles, will also be at the 40 mile mark.The graph thus appears as an aid to guessing, which is essentiallywhat it is.

The children may make up similar problems for themselves,and draw the graphs. Without being told by anyone, they willsoon come to realize the useful fact that the graph of an objectmoving at a fixed speed is always a straight line.

Young children delight in codes and secret messages. This isanother direction from which graphs can be approached. Thecode is a very simple one, as in Figure 161.

2*p

oa.8*twtFlgute lol

Suppose I wish to convey to tomeono that I am thinking of thopoint P in Figure 161. If I belln at O, to reach P I have to travol3 steps across tho papc, rnd 2 up. We always begin at Or lbcsouth-west corner of thc peper. Tho two numberc, 3 and 2, com.pletely fix the position of P, \lVo agree that the number a$oss L

271

=r1

IFigure 160

Tho f,pt llno shows the situation at the beginning of the chase.The othcr lin os ahow the positions of the planes after l, 2,3,4minutos. In Flguro 160 the positions of the planes have beens h own only for tho ffrst three times, corresponding to 0, l, and2minutes. Alro ady ohildren will notice that the marks lie in lino

270

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"L,-+" I

http://books.google.com/books?id=0FY3KlWclvYC&lpg=PP1&dq=vision%20in%20elementary%20mathematics&pg=PA270#v=twopage&q&f=falsehttp://books.google.com/books?id=0FY3KlWclvYC&lpg=PP1&dq=vision%20in%20elementary%20mathematics&pg=PA270#v=twopage&q&f=falsehttp://books.google.com/books?id=0FY3KlWclvYC&printsec=frontcover&dq=vision+in+elementary+mathematics&hl=en&ei=gXHsTMblCsiHcb6-9fIO&sa=X&oi=book_result&ct=result&resnum=1&ved=0CC0Q6AEwAA#v=onepage&q&f=falsehttp://books.google.com/books?id=0FY3KlWclvYC&printsec=frontcover&dq=vision+in+elementary+mathematics&hl=en&ei=gXHsTMblCsiHcb6-9fIO&sa=X&oi=book_result&ct=result&resnum=1&ved=0CC0Q6AEwAA#v=onepage&q&f=falsehttp://books.google.com/books?id=0FY3KlWclvYC&printsec=frontcover&dq=vision+in+elementary+mathematics&hl=en&ei=gXHsTMblCsiHcb6-9fIO&sa=X&oi=book_result&ct=result&resnum=1&ved=0CC0Q6AEwAA#v=onepage&q&f=falsehttp://books.google.com/books?id=0FY3KlWclvYC&printsec=frontcover&dq=vision+in+elementary+mathematics&hl=en&ei=gXHsTMblCsiHcb6-9fIO&sa=X&oi=book_result&ct=result&resnum=1&ved=0CC0Q6AEwAA#v=onepage&q&f=falsehttp://books.google.com/books?id=0FY3KlWclvYC&printsec=frontcover&dq=vision+in+elementary+mathematics&hl=en&ei=gXHsTMblCsiHcb6-9fIO&sa=X&oi=book_result&ct=result&resnum=1&ved=0CC0Q6AEwAA#v=onepage&q&f=false

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CHAPTER ELEVEN

Graphs

Gnapns rest on one of the simplest and most useful ideas inmathematics. A graph will often help us to solve a problem, andchildren should learn to regard graphs as a useful aid, and tosketch a graph whenever occasion arises. It is a great pity whengraphs are taught ponderously and unattractively, as a subjectin themselves, so that pupits come to dislike and avoid graphicalwork.

The introduction to graphical work can be quite informal. Aclass of young children are confronted with the following prob-lem. There are two aeroplanes, a faster one that flies at l0 milesa minute and a slower one that flies at 5 miles a minute. Theslower plane has a start of 20 miles, and the faster one is chasingit. When will the faster plane overtake the slower one?

It is quite natural to start drawing pictures (see Figure 160).

0 2 3 minutes

Graphs

(as shown by the dotted lines) and they will predict that over-taking occurs after 4 minutes, corresponding to the point on thegraph where the dotted lines cross. It is not even necessary to usegraph paper. A rough sketch on paper or a blackboard may wellbe sufficient to suggest the dotted lines, if the drawing is reason-ably accurate. Squared paper may be used, but ifso one shouldnot make a meal of it. Minute attention to the mechanics ofplotting points is definitely out of order. The aim is to picturothe problem quickly, and to predict when the chase will end.Once we have guessed that 4 minutes is the required time, it iseasy to check that this is correct. For in 4 minutes the fasterplane will cover 40 miles; the slower one will cover 20 miles, andas it had a start of 20 miles, will also be at the 40 mile mark.The graph thus appears as an aid to guessing, which is essentiallywhat it is.

The children may make up similar problems for themselves,and draw the graphs. Without being told by anyone, they willsoon come to realize the useful fact that the graph of an objectmoving at a fixed speed is always a straight line.

Young children delight in codes and secret messages. This isanother direction from which graphs can be approached. Thecode is a very simple one, as in Figure 161.

2*p

oa.8*twtFlgute lol

Suppose I wish to convey to tomeono that I am thinking of thopoint P in Figure 161. If I belln at O, to reach P I have to travol3 steps across tho papc, rnd 2 up. We always begin at Or lbcsouth-west corner of thc peper. Tho two numberc, 3 and 2, com.pletely fix the position of P, \lVo agree that the number a$oss L

271

=r1

IFigure 160

Tho f,pt llno shows the situation at the beginning of the chase.The othcr linos ahow the positions of the planes after l, 2,3,4minutos. In Flguro 160 the positions of the planes have beenshown only for tho ffrst three times, corresponding to 0, l, and2minutes. Alroady ohildren will notice that the marks lie in lino

270

'::L--L-'.'ttt

"L,-+" I

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Vision in Elementary Mathematicsalways given first, and the number lrp second. This has to beremembered, and of course a little practice is needed to fix it inthe memory. Our code name for the point p is thus (3,2).

It may need to be emphasized that all code names for pointsare based on our starting at the corner, O. Sometimes, when twoor more points are named in code, children tend to measure on

from thepoint

last named, or they may be in doubt whether thisis what we intend them to do. However, that is not our intention.For instance in Figure 162, aftet marking (3,2) atp, children may

oFigure 162

want to mark (2,1) at R, since R is 2 across and I up from P. Butwe always work from O. The point (2,1) is not R but p. The codename of R is (5,3).

Children can have a lot of fun with these codes. A child can draw

Figure 163

a picture, turn lt into code, then other children can try to turn thecode back into the picture.

For examplo, Flgure 163 is intended to represent a cat's face.272

Graplw

The code instructions might read, 'Join (2,1) to (4,1) to (5,2) to(5,5) to (4,4) to (2,4) to (1,5) to (1,2) to (2,1).' If a teacher hasfollowed these instructions and drawn the cat's head on theblackboard a discussion might follow as to where the cat's eyesought to be. Children do not need to come to the board. They cancall out their suggestions in code, for example, 'The eyes shouldbe at (2,3) and (4,3)'.

This type of work does not lead to any particular mathemat-ical result, but it makes children familiar with the system ofspecifying points by numbers. The learned name for thesenumbers is 'the coordinates of the point', and coordinates arewidely used in mathematics. The name need not be mentioned tochildren; 'coordinate' is a long word for a simple idea. But allkinds of questions can be discussed. If you were at (3,4) andmoved across (to the east), what would be the code names of thepoints you passed through? If you started at (5,1) and moved up(north), what points would you pass through? If you began at O

and moved north east, what points would you pass through?What do you notice about the numbers for them?

The code idea can also be used for actual messages. The dotson the graph paper can be made to form letters.

Figure 164 shows th lcttt f,, The codc instructions might read,'Join (l,l) to (1,5) to (3,1) to (3,4) to (2,3) to (1,3). Also join(2,3) to (4,1).'

In the diagrams go fri We havo not used points on the edgeof the squared papef. SonCtlmee we may want to mention such

273

n

\ /

\ /o

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Vision in Elementary Mathematicspoints. In Figure 165 if we start at O, to reach I we have to go 3across

Graphslent form, for example, that you can get the second number bysubtracting the first number from 6.)

In order to state this result briefly, we bring in abbreviations.Let x be short for'the number we go across to get to a point',and y for 'the number we go up to get to the point,. That is, thecode name of the point is (x,l) ; x is short for . the first number inthe bracket, y is short for the second number in the bracket'. Butwe have noticed that, for the points A, B, C...G on the line,these two numbers add up to 6. That is xfy:6, for each ofthese points.

This raises two questions. (i) If we take any other point on theline, shall we findforittoothat x*t:6? (ii) Canwefindanypointnot on the line that has x*l:6? Needless to say, we do notanswer these questions. We ask the children to investigate. Theymight suggest that we try the point fI, which is (3*,2*), or Kwhich is (5t,*), or other points on the line. In each case, we findthat xfy:f. With H, for example, x:3+ and, y:2!, and thesenumbers do add up to 6. So far as

we canjudge

by experiment,every point on the line has xll:6. What, then, about thesecond question; can x+/:6 ever happen when the point(x,y) is not on the line? Again the children should suggest pointr,and try what happens. In Figure 167,4 points, p, e., R,^S have

oFl:uro 167

been chosen off the llne, P b (1,j) and for p the 2 numbers addup to 9. For Q, which lr (t,?) wc get the sum 7. For R, which is(1,4) we get the sum 5. FOt 0, whlch is (2,1) we get the sum 3. Ifchildren take many pohil they wil probably notice that for

275

B

OAFigure 165

and 0 up. So I is written (3,0). To get to B from O, we go 0across and 2 up, so B is written (0,2). lf we want to mention Oitself, we write (0,0), for to get from O to O, wc go 0 across ando up.

THINGS TO NOTICEIn Figure 166 we have drawn a line and marked several pointson it, dotted about in a rather random order.

Figure 166

Consldcr tho code names of a few of these points. The point I is(6,0); .B lc (2,4); C is (5,1); D is (0,6); .E is (1,5); ^F is (3,3); G is(4,2). Do wo notice anything about these numbers? Childrenusually notice fairly quickly that the two numbers inside thebracket add up to 6. (They may state this result in some equiva-

274

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FA \ G

t'r

x

Pn \

a

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Vision in Elementary Mathematicspoints above the line, the sum x*y is always more than 6, whilefor points below the line x*y is always less than 6; only on theline is x*/ equal to 6. They may also notice that the nearer thepoint is to the line, the nearer x*/ is to 6.

So the statement x+y:6 is very closely connected with theline in Figure 166. It is true when the point (x,y) lies on that line,and only then is it true. So mathematicians use x+,r:6 as a wayof specifying that line. You can tell a child that if he went up to amathematician in the street and said, 'I am thinking of the linex+y:6', the mathematician would know exactly what the childmeant. He would know the chilcl was thinking of the lineshown in Figure 166.

Different children may express what they notice in differentways. Instead of saying'The two numbers add up to 6,' they maysay, 'If you take one number away from 6 you get the other.'This of course expresses the same observation in a slightly differ-ent way. fn our picture language of Chapter 5, it does not matter

whether you say that the boy standing on the stool is as tall as theman (b*s:m for short), or that the boy's height subtractedfrom the man's gives the height of the stool (nt-b:s for short),or that the stool's height taken from the man's gives the boy's(m-s:b for short). All three statements correspond to the samepicture; they are different ways of saying the same thing. If achild says to the mathematician, 'f am thinking of the linel:6-x' or 'f am thinking of the line x:6-y', the message iscxactly the same as that conveyed earlier.

We began with the line drawn on graph paper, and from it wearrlvcd at the equation x-1-y:6. Sometimes it is necessary to goin ths opposite direction. A mathematician may say to us, 'I amthinking of xly:4'. What has he in mind? Suppose we take anypoint of the graph paper (see Figure 168), say ,4, which is (4,3).As 4-F3*=7, the statement x*y:4 is not true for A.

We put n blob of yellow paint on I (shown in Figure 168 by ahollow circle). Yellow is for points that fail the test, black forthose thEt paes. Next we try B, which is (3,1). Now 3 and I doadd up to 4, ao the statement x*l:4 is true for .8. We put ablob of black paint on B. For C also, which is (2,2), we findx+y:4 to bo true, so C is painted black. We try D, which is (2,3).

276

Graphs

As 2 and 3 do not add up to 4, D gets a yellow mark. But E,being (1,3), has two numbers which do add up to 4, so E ispainted black. If we continue in this way, we eventually obtaina drawing in black on a yellow background. The black pointsgive us the graph of x*y:4. These are the points for which thestatemel'rt is true that the number across and the number up give4 when added.

oFigure 168

We need not restrict ourselves to whole numbers. A point suchas 4 with code name (3r,+), is marked black.

You will notice that once again we have a straight line for ourgraph, and it has the same direction as the earlier line, fromnorth west to south east. At this point, all sorts of questionsshould spring to children's minds. If we took some other straightline, running from north west to south east, would we get anequation 'x*.y:something' for it? What about lines in otherdirections? Does every equation have a straight line for itsgraph? The more children can be encouraged to propose andinvestigate such quosllons tha botter it will bs for their mathe-matics, and the morg intorecting tho work will be. Finding out isexciting; being told lr dull.We may exporlmant wlth a llne ln n different direction, forexample the lina ihown ln Flgure I59.We may tabulate ar followtt

PointABCDENumberacou(*) 0 t 2 3 4Numberup(/) I 2 3 4 5

What can we notice abgut thero numbers? You may find that tho277

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Vision in Elementary Mathematicsearlier work with x*y:6 and xly:4 has put the children'sminds in a groove. They become obsessed with adding x and y.They will say 'For A, x*y is 1 ; for B it is 3 ; for C it is 5. Oh yes,we are getting the odd numbers.'Now the trouble, so to speak, isthat this is perfectly true. We do get the odd numbers. But thisis something which often happens in trying to solve a puzzle;

we notice something, which is not quite what we want, and thisblinds us to some much simpler observation, which in fact would

E

o Figure 169solve our pazzle for us. In this particular case, we may draw adividend from the guessing game described in Chapter 4. Regardx as 'the number called out', and y as 'the number answered'.You call l, f answer 2. You call 4, I answer 5; what am I doing?I am adding I to the number you say. The rule that fits the tableebovo is in fact y:x+l. Further evidence can be obtained byconrldering fractions; for example (+,1+), (3*,4+), (l$,2,-!) are3 moro points on this line.

A lraph can be drawn to illustrate any of the guessing games

doscrlbed on page 79. There may be a slight practical difficultywlth rcmo of these; in games 4 and 5 for example rather largenumbor! occur, and it may be rather inconvenient to graph these.It is not lmpossible; such graphs can be made, if a large sheetof papor b ured, and the children have enough patience andcourago. Tho teccher should use his judgement. It is bad to dowork whloh the pupils find so tedious that the intellectual excite-ment of tho lnvertlgation becomes destroyed. On the other hand,children should trot be given the impression that something ted-

278

Graphsious is impossible. We could most certainly draw the graph ofgame number 5 if it became necessary in some industrial orscientific problem, where we were determined to obtain a solu-tion. In a class of children, there may be some enthusiastic orobstinate pupils, who find some degree of difficulty a challenge.They perhaps might draw the more tedious graphs, and show the

results to the rest of the class. This procedure corresponds to theneeds of modern society. We cannot all be experts on everything.An efficient society has many specialists, but encourages thesespecialists to work harmoniously together, and to keep the wholecommunity infonned of what they are learning.

All the graphs we have mentioned in this chapter so far havebeen straight lines, and most of the graphs that arise from guess-ing games are likely to be straight lines. There is a danger thatchildren will jump to the (false) conclusion, that all graphs arestraight lines. They may then fall into the habit of getting twopoints on the graph, then putting a ruler against these and draw-ing a straight line. It is therefore essential, fairly early in theteaching of graphs, to raise the question, .Does every equationlead to a straight line?' and to give at least one example of anequation that does not. A simple example would be the equationxI:12. Here the test is: do the two numbers used in writing thepoint multiply together to give 12? Thus (2,6) would pass thetest, since 2x6:12, but (3,5) would fail and get a yellow blob,since 3 x 5 is not 12. It is easily seen that ths following points alipass the test; (1,12), (2,6), (3,4), (4,3), (6,2), (12,1). It is quiteclear from Finro 170 that thoco polnh do not llc on a straightline. Thero aro ofcourso othcr polntr on tho graph besides thoseghoryn in Figuro 170, for oxemplo (E,1*) and(1f,8), nottomention(48,*)

and(0.1,120).

If we plot many poltttt, thy wlll bogtn to appear like thecontinuous curvo lhown ln Flaure l?1.

CIRCLBT IEEAIDBD AS GRAPHSA simple curve can havr r felrly complicated equation. A circle,for example, is a very wcll.known surve, but we have to write afairly long equation to tpe{:lfy a circle on graph paper. Such

279

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Vision in Elementary Mathematicsequations can be found by a quite straightforward procedure,which we do not want to discuss at the moment, as it woulddistract pupils from the basic ideas of graphs.*

Figurc 171

For oxample, the equation x2+y2+25: l0x* lOy represents acirclc. It h not too easy to guess numbers for x, y that 'pass the

rTho cquttlon of the circle with centre (a,D) and radius r is in fact:xz + y2 + az + b2 :2ax *2by * i

If we replaco dr Ar end r by particular numbers wo can obtain as manyexamples of thc oquetlons of circles as rve wish.

280

Graphstest'. Pupils, however, can check for themselves the statementthat all the points listed here do qualify, and should be markedin black: (5,0); (0,5); (5,10); (10,5); (2,1); (1,2); (g,t); (1,8);(9,2); (2,9); (8,9); (9,8). For example, totest that (9,g) passes thitest, wc replace x by 9 and y by g in the equation.Then x2f-yz+25:l0xf lOybeconres 8lJ-64+25:90+80

Figure 172

which is a true statement, since it says 170:170. The otherpoints may be tested in the same way. These points are shown inFigure 172 and. it will be seen that they do suggest the shape of acircle. There are of course many other points that pass the test;but most of them involve rather awkward numbors. For instance(6.4), (9.8) qualifies, not to spoak of a polnt whlch is approx-imately (8.535), (t,465).

You may notico c corteln dlffcrence betwoon the circle justdrawn and tho

curvo for ,fl-12 whlch wc had carlier. If we-arelooking for ri polnt that pcttt the tilt JU*12, and we think itwould be a good ldea to chooao, !Ey, 2 for x, then we have nochoice for y; we muat tako 6 for y, Fut lt is not so for the circle.As we have seen (2,1) e.nd (1,9) both pass the 1*1 yzayz{25: lOx* 10y. Even after deEldlng to choose 2 for x, we have morethan one possibility for /, f,o the rltuation here is rather differentfrom that in the gamu cE hapter 4, in which you catt oui-anumber and I answor, Fo! here lf you choose Zfor x,I -Jr"v

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Vision in Elementary Mathematicseither 1 or 9 fory. On the other hand, xl:l2is exactly like thosegames. If you call out a number for x,I divide 12 by that numberto find y. In shorthand, y:l2lx

It is naturally easiest to sketch graphs when we have such adefinite rule. We usually make a table like this:

and read off the points that qualify, (1,12), (2,6) and so on. Thismethod leads us directly to the points that pass the test and haveto be marked in black. It does not mention explicitly the pointsthat fail, but it is to be understood that all other points are to bemarked yellow. For there is a definite rule that gives y when x hasbeen fixed. This is the correct value, the only value that makes thepoint (x,y) pass the test.

Many books teach graphs by this method, and readers come tothink of drawing graphs as a matter of making tables like the onejust used. These readers may then find themselves puzzled whenthey meet a more complicated equation, such as the equation fora circle. The explanation, by which points that pass the test aremarked black and those that fail are marked yellow, covers allpossibilities. Starting from this idea, we can see why a table is

Graphspoints passes, (2,6), since 6:1212 is true. All the others fail,since 3 is nol l2l2 and 4 is not 1212, and so on. So we can markour graph paper as in Figure 173. We get a vertical line, inwhich one point, (2,6) is black. All the others are yellow. If wethen turn our attention to the points (3,1), (3,2), (3,3), (3,4), (3,5),etc. and mark the results in, we shall get another vertical line,containing

one blackpoint

with all the rest yellow. We continuelike this, getting many vertical lines. Each line contains oneblack point, and these black points lie on the graph of t:l2lx.These black points are the ones that could be read off directlyfrom the table for y:l2lx. The table does in fact give the quick-est and most practical way of drawing this graph. The discussionof black and yellow points is helpful for understanding what ismeant by the graph of a more complicated equation.

Most beginners at graphs are asked to draw the graph l:x2,From the table:

we see that the graph contains the points (0,0), (l,l), (2,4),(3,9\.If we go on, we can see that the graph in Figure 174 also contains(4,16), (5,25), (6,36), and so on. The numbers for y get largevery

Ftnre l?{rapidly, so that a vcry tall, thln plece of graph paper is needed.Also the points just mOntlOned lle very far apart, and it is not

283

x123456y126432.42

x0123y0149

-Figure 173

sometlmo appropriate. With a definiterule likey:121* wemayconsider tho po lnt s (2,3), (2,4\, (2,5), (2,6), (2,7), (2,5 +), (2,6*), aadso on. For ell of these, x, the first number, is 2. Only one of these

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Vision in Elementary Mathematicseasy to draw the curve through them. In Figure 174, instead ofgoing on beyond (3,9) we have put in extra points correspondingto x:*, x:12, and, x:2i. These points are @, r;, (t|,2f), and(2r,6+), and they help us to see how the curve goes.

Several mild surprises occur in the sketching of such curves,and these help to relieve the tedium of routine practice in making

graphs. Suppose, for example, that after sketching !:x2 we gootr to y==.r2 *1-2x. The table for this is:

The table for y:vz *ut'

and we see a connexion between the two tables. If in the first tablewe disregard 1 and |, the first two entries for y, the remainingnumbers in the y row are exactly the same as those in the table

for y:yz. How does this show itself in thegraphs? (See

Figuret7s.)

-?-------

J

Figure 175

If the graph Bre drawn on fairly thin paper, the children will findthat thoy can hold the papers against the window, so that thegraph of ./:*l is exactly covered by part of the graph ofy:2s2J.l-2X,

284

Graphs

The equatiofi !:2s2!l-2x has been written in that order,rather than as y:x2-2x* l, to avoid negative numbers Cominginto the calculation. For example, if we work out y-x2-2x*lfor x:ln we get l-2+1, and some children may find difficultyin carrying out the instruction, 'Begin with 1. Take away 2 . . ,'.Negative numbers do not come in at any stage of this calculation,

so this work can be done before the idea of negative number hasbeen reached.The graph of y:1zLr4-4x may be studied in this connexion.

With this graph you can cover either of the graphs in Figure 175.If you draw the gfaph of y:lox-x2, taking values of x from

0 to 10, you get an arch like the curve in Figure 176. Ifyou draw

Figure 176

this graph and turn it upside down, you will ffnd that part of itwill exactly cover the curvo /-.x2. ln fact, all tho four graphs wehave considcrcd rccently atc th6 samc ctltu ln dfurent positions.Nor does it stop thero. Wc ctrn flnd many other equations thatgive this curvo. We ecn ruekc up e cultablo oquation in the fol-lowing way. First thlak sf Bny number, ray 2. Then think of someother number, say 6, end wrlt lt wlth the lctter x; this gives us 6x.Finally take x2, Wc now heve 3 lngrodients, 2, 6x, and xz. Jointhese together in any wey yu llkc by plus and minus signs. Youmight take xz+,6x*2 ot 6r#2:x2 or x2-6x-2, for example.You will find that ye;l"l-6"r p2 or y:6x*2-xz s1 y:y2-6x-2 has a graph thct can bo pluced to cover some part of thegraph y:xz. We may aot get s very interesting graph. For

285

t+ 22t 33t 4* t2+4 46I e

t+ 22+ 32+ 462 e

tcO+1v t + 0

x 0 * Iv 0 + 1

!'x" !-t(2+l-2tI

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Vision in Elementary Mathematicsinstance, if we take our first example above, y:x2*6x12, thisleads to the table:

Figure 177

and itsgraph

looks like Figure 177. You will find that it can beplaced to cover the part of the graph of y:az that goes throughthe points (3,9), (4,16), (5,25), and (6,36).

Sometimes you may get only a tiny piece of graph. Forexample!:3-2x-x2 gives us the curve in Figure 178 joining (0,3) to

Figure 178

(r,O), *o hardly 3ct a look at it before it is over the edge of thepaper. And thh ralses a question which we shall consider in the

286

Graphs

next chapter; could we not perhaps bring along some morepieces of paper and follow the curve after it has left our originalpiece?

The curve that goes with y:yz and y:1gr- x2 and the otherequations we have been considering is called a parabola. It occursoften in everyday life, for instance as the path of a ball thrown

into the air on a calm day, the curve of the reflector on a caf,headlamp, the shape of the jet of water from a hose. This lastexample encourages us to believe that it may be possible to followthe curve beyond the edge ofthe paper, for ajet ofwater ceasesto move in the shape of an arch when it hits an obstacle, but if theobstacle is removed, then the water continues in a smooth curve.Perhaps the edge of the paper represents an obstacle which wemay succeed in removing. Children may fnd it stimulating toconsider this idea for themselves. The graph y:lOx-x2 shownin Figure 176 is very suitable for discussion. Where do the chil-dren think this curve would go if we continued it to the right?Where would it go if continued to the left? The experimentindicated in Figure 175 may give them some ideas about whereX:x2 would go if it continued to the left.

Graphs naturally find a place in a book on vision in algebra.since graphs give us something to look at when we think aboutalgebra. We do not learn graphs in order to draw neat graphs inan examination. We learn them rather to hclp ourselves to thinkabout algebra. Whenever we tneet an oxprossion in algebra, weshould think'What would its graph look llkc?'and wo shouldsketch a qulck, rough llttlo graph, Jurt onough to show how thething bchavor. At f,rt, thoro $sphr wlll not toll us much, but aswe keep drawlng thom, wc $rll [nd w gat a fccling for them.There is a spoclal rortot! fo lneludng $aphc at this particularstage of this book, It lr ln order to rfk thc question posed a litfleearlier; can wo follow r graph beyond tho edge ofthe paper? Thegl:aph y:lOx-.rl rnehu the edle of the paper when x:10,since.r:10 makor l-0, io thc $aph seems to end at the point(10,0). What happonr Fwr 1o beyond x:10 and consider.r:11?When r:ll, y-110-l2l utd wo cannot deal with this withoutintroducing negatlvc ltlunbdl, If wc say that 110-121 is -11,we have to plot tho polnt (l 1, - I l). Are we justified in doing this

287

rcOl23y 2 I 1829

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Vision in Elementary Mathematicsand where should this point be? These questions will be discussedin Chapter 12.

INVESTIGATING GRAPHS

As has already been mentioned, children should be encouraged

to pose questions about graphs and to experiment freely. In thecourse of these experiments they may run into one or two diffi-culties. Curiously enough, difficulties tend to arise not with themost complicated but with the simplest situations. A thing canbe so simple that it becomes, so to speak, invisible. For example,if a line is drawn through the points (0,2), (1,3), (2,4) childrenreadily enough spot the law y:2i12. With the line through(0,1), (1,2), (2,3), they manage to spot y:x+I. But then theystick at the line shown in Figure 179 which goes through the

polnts (0,0), (1,1), (2,2\, and (3,3). They notice surely that in eachbracket the two numbers are the same, but they may comerlowly to the law y:x that expresses this fact. The reason may bethet 'adding 2' and 'adding 1' are familiar operations in arith-motlc, which they quickly recognize. 'Leaving alone' is not anoporgtlon that needs to be taught, so this idea may not come to achild's nlind so readily. In tenns of a guessing gafilo y:a corres-ponds to 'Wratever number you call out, f answer the samenumberr. ln terms of a test to bc passed,.y:x means that for eachpoint, tho two numbers in the bracket must be the same. Clearly(0,0) and (l,l) and (2,2) pass this test.

Anothor llne that may cause difficulty is shown in Figure 180.The points on thh line include (0,2), (1,2\, (2,2), (3,2), and (4,2).

288

Graphs

What is there to be said about these? We could of course con-tinue this line to get (5,2), (6,2), and so on. In fact we could geta point described by '(any number you like, 2),. So to get on thisline we can go any amount we like across, then 2 up. So if ("r,y) is

on this line, x may be anything at all, but y must be 2. What isthe test, then, for being on the line? There is no point in lookingat x, for x may be anything at all. We need only examine thisecond number, y.If y is 2, the point is on the line. If y is not2, the point is not on the line. So y:2 is our test; (17*,2) is onthe line, because the second number is 2; (3,4) is not on the linebecause 4 is not 2.

Flgurr ltt

Similar remarks epply tO the lino in Figure lgl. The pointsmarked on this lino oro (tr0), (l,l), (3,2), (3,3), (3,4), and (3,5).You will be on this llnc lf you ltart from 0, go 3 across, and anynumber up. The numbof eetot. must be 3. So the equation for

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Vision in ElementarY Mathematics

this line is x:3. To tell whether a point is on this line or not weonly need to look at the first number, x, and see whether it is3 or not. You may notice that this equation, x:3, will give greattrouble to anyone who has been brought up by the rule' 'Make atable showing the values of y for x:0, x:1, x:2, and so on.'For we cannot make any such table; y is not even mentioned in

the equation x : 3. This is another exampleshowi n g the advantage

of the 'black and yellow' approach over tlre 'make a table'procedure.

CHAPTER TWELVE

Negative Numbers

Mlrner,tA,rIcs grows gradually from one idea to another, andit is extremely difficult to say at what stage a new idea has beenintroduced. At the present moment it is hard to say whether thisbook has yet explained the idea of negatiue number. We havecertainly been on tho brink of it several times. For instance,in Chapter 10, we had pictures to illustrate the addition of *2cand- 3c. Yet we have nevergone the wholeway;wehaveneverreached the stage where the instruction 'Think of a number'might receive the reply, 'I am thinking of -3'.

On page 251 we met'the multiplication table of * and -' butthis did not involve the idea of negative numbers. Indeed it isremarkable that the multiplication table of plus and minus wasknown about two thousand years before negative numbersbecame generally accepted.* It was already known in antiquitythat n-3 times z-2 was the same as n2-Sn*6, but this wasalways on the understanding that n was a number not less than 3.For example, z might be 10, as in our discussion on pago 247 randthe result that l0-3 times 10-2 is 100-50*6 would havc beenquite familiar to a mathcmatician in anclont Babylon. It eays that7 times 8 is 56 and no novol ldoa lc lnvolvod, It dld not occur toanyono for ccnturier to wondor whcthcr tny tcnlo could be madeof taking 0 for r, to thst n-3 tlmil n-2 oquals nz-5n*6would lead ur to thc lder thrt -2 tlou -3 ohould equal *6.Indeed this llnc of thoulht l! hlthly unnttural. We pictured 10- 3times 10-2 ao l0 lowl, 3 ol whloh wfo oossed out, each rowcontaining l0 obJcatr, 2 of wbleh woro crossed out. If we are topicture 0-3 tim$ 0-t lU thf wry, we have to imagine thatthere are no rows to bqln wlth, and that we cross orit 3 rows;each row containe no ohfget, end thon 2 objects have to be re-moved from it. It ll nqt lulprlcing that this line of thought

. See Bell, Dcvtlopnnl Al'Mathematics, pp. 34, 62, and 96.29t