Cable DirectedNumbers

8/10/2019 Cable DirectedNumbers

1/4

The Ground from Which Directed Numbers Grow

Author(s): John CableSource: Mathematics in School, Vol. 1, No. 1 (Nov., 1971), pp. 10-12Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30210683.

Accessed: 26/07/2013 00:47

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at.http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of

content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact [email protected].

.

The Mathematical Associationis collaborating with JSTOR to digitize, preserve and extend access to

Mathematics in School.

http://www.jstor.org

This content downloaded from 128.226.37.5 on Fri, 26 Jul 2013 00:47:59 AMAll use subject to JSTOR Terms and Conditions
http://www.jstor.org/action/showPublisher?publisherCode=mathashttp://www.jstor.org/stable/30210683?origin=JSTOR-pdfhttp://www.jstor.org/page/info/about/policies/terms.jsphttp://www.jstor.org/page/info/about/policies/terms.jsphttp://www.jstor.org/page/info/about/policies/terms.jsphttp://www.jstor.org/page/info/about/policies/terms.jsphttp://www.jstor.org/page/info/about/policies/terms.jsphttp://www.jstor.org/stable/30210683?origin=JSTOR-pdfhttp://www.jstor.org/action/showPublisher?publisherCode=mathas


2/4

T h e ground

f r om

whi h

directednum ers

r ow

by John Cable

1.

Most teachers introduce directed numbers by means

of concrete situations,

such as temperature

scales and

displacements

on a line, and I for one

believe this

approach

to be sound. Indeed, if the usual treatment

is

open to criticism,

it is because the teacher

is in too

much hurry to

reach the rules for addition

and other

operations and

hence leaves the concrete

situations too

early. Be that as it may, the purpose of this article is

to

examine

in some detail some of the concrete situations

and ideas that

underlie directed numbers.

We may bear

the

following points in mind:

(i)

these ideas and situations are not merely

launching pads for the flight into symbolic

manipulation,

but provide some of the contexts to

which

directed numbers and their arithmetic are later

to be applied;

(ii) weaker pupils,

i.e. the lower part of the total

ability

range, may well confine their use of directed

numbers

to very elementary applications;

(iii) the complexities

to which we

shall draw

attention may

perhaps account for some of the

difficulties pupils find;

(iv) it may be

that the richness of structure which we

hope to uncover may be worth the conscious

attention

of at least the more able pupils.

2. First we may

note the existence of

SCALES with

negative as well as positive parts.

B T T E R Y

-30

-20

-

10

-O

---10 1 I I I

--20

-

--c

1 0

--30

(What happens if you

change the leads over?)

Accelerometer

from spacecraft

Even

the weakest mathematician is surely capable of

reading

such a scale, and no doubt every syllabus will

include this skill either explicitly or implicitly.

3. Of slightly

greater difficulty is the reading

of two

such scales simultaneously as with co-ordinates.

x x

The

first stage of work with co-ordinates is simply

plotting of points

and reading of co-ordinates

(e.g.

describing shapes

over the telephone).

The

generalisation from the first quadrant to other

quadrants is plausible,

and incidently facilitates the

treatment of symmetrical figures.

4. It will be noted

that the symbols -2,

+3, etc., in the

above

situations are merely labels for points. There

is

no

obvious call to add them or perform other

operations because there are no

corresponding

operations that are naturally performed

on points. The

matter

may be emphasised by a change of notation.

If

for the moment we confine attention to

integer points,

then

we may as well use letters of the alphabet

as

numbers:

...-D

-C -B -A

0

+A +B +C

There is no urge to write

B + -C = -A

(Incidentally, this notation may be elaborated. When

you reach the

end of the alphabet, you

may continue

AA, AB, AC ...

AZ, BA ... Can this go

on for ever? If

you

go into two dimensions, can you use letters

as

co-ordinates?)

5.

A further elementary use of directed numbers is for

CHANGES and ERRORS.

Example

1.

Draw a line on a piece of paper and ask

your

friends to estimate its length in mm. Then

measure

it properly. Record the various estimates and

the errors:

Name

Estimate (mm) Error (mm)

Alan 40 +8

Brian 25 -7

etc.

Example 2.

Population

now Pop. 10 years ago Change

Anglia

53 48

+5

Bretony 42 59 -17

etc.

These elementary

problems on Changes

and Errors

do

not involve addition or other operations on directed

numbers. The

pupil may certainly

have to do

subtraction of

the essentially positive

numbers that

occur

in other columns of the tables, but he is not

asked to combine two of the directed numbers.

6. Nevertheless,

changes can be combined,

and the

natural

way of combining them leads to addition

of

directed numbers.

Easiest to picture are changes of

position,

or DISPLA CEMENTS.

Simple Displacement Game

//+\

I

I

I I I I I

ISTA'

I

I I

I

I I

FINISH

FINISH

10

This content downloaded from 128.226.37.5 on Fri, 26 Jul 201300:47:59 AMAll use subject to JSTOR Terms and Conditions
http://www.jstor.org/page/info/about/policies/terms.jsphttp://www.jstor.org/page/info/about/policies/terms.jsphttp://www.jstor.org/page/info/about/policies/terms.jsp


3/4

Place your counter

at Start. Spin the

spinner. If it

reads +2, move two places to the right; if -2, two

places left.

This game does

not itself require the

notation of

addition; it merely

introduces the idea

of performing

one

displacement after another.

7. Addition arises

if you start to think about the game

rather

than actually playing it. Imagine you were to

spin -2 followed

by +3; where would you

finish? What

single displacement would be equivalent? Write:

-2 + +3 = +1.

If the pupil

shows no interest

in writing

-2

+ +3 = +1, then I suppose the only honest course

is

to drop the matter

and console yourself

with the

thought that

he is not yet ready for this piece of

abstraction.

On

the other hand, if he takes it without resistance,

you can press

the matter for all it is worth,

which is

quite

a lot, because the set of displacements displays

the full additive

structure of directed numbers, that is

it forms a commutative

group. This means

you can ask

questions like:

Is +2 + -3 the same as -3+ +2?

If several displacements are performed one after the

other,

does it make any difference in what order they

are done?

Find

x

so that +2 + x = -5.

Incidentally, there is no need to introduce the

operation

of subtraction in connection with equations

like

+2 + x = -5.

The device of adding the inverse (e.g. -2 is the

inverse of +2)

will

provide for all needs.

8.

The displacement game may be played in two

dimensions on

a grid. Moreover, having

introduced

things like

+

-1

as labels for displacements, one may proceed

to draw

iourneys like.

+2

+1

-1

\ 2

\ 1

and to ask: What

single displacement is equivalent

(the

short-cut)?

It is to be noted

that the addition sign

here does not

signify

addition of directed

numbers.(It

is addition of

2-dimensional

displacements, or, if you

prefer, of

column matrices.)

I am not sure whether this

ambiguity

does harm or not.

9.

One thing that does seem unnecessarily confusing

is

to use directed numbers simultaneously as labels for

points and as labels for

displacements.

-1

+2

-1

-3 -2 -1

0

+1 +2 +3

Let us again make

a change of notation.

Continue to

let

+2, -1, etc. refer to points, but label displacements

as

(R2),

(L1),

etc.

R stands

for right ; L for

left .

On the one hand, we still have the additive

structure

among displacements:

(R2) +

(L3)

=

(Ll)

etc.

It

is also true that, if you start at point -3, and make

a displacement

(R7), you finish at point

+4. But there

is

little

temptation to write

-3 + (R7) = +4.

If points and

displacements share a common

system

of

labelling, there may be a

temptation

to add

a

displacement-label to a point-label:

-3 + +7 = +4.

This sort of thing

may perhaps be acceptable

later,

when

addition

is secure, but it undoubtedly

is

something of a hybrid, and

does

not make

for clarity.

10. In two dimensions there is less temptation

to add a

displacement matrix

to a pair of co-ordinates

because

they are written

differently. However,

the principle

remains

that, if you wish to concentrate on addition of

displacements,

it is distracting to have

co-ordinates

around

as well, and better to use a bare grid, as we did

above.

11. Of course, just because a child can write

-1

+ +5 = +4

as symbolising

a result about displacements, it does not

follow

that he has grasped addition of directed

numbers in all its fullness.

For instance,

if you revise with him the

work he did

earlier

on

Co-ordinate

Patterns, in which he noticed

that

the points (4,0) (3,1) (2,2)

(1,3)

and (0,4) all lay

on a line while their co-ordinates all displayed the

pattern

x+y=4

and if you then

ask him whether x + y =

4 continues to

hold

for the point (-1, +5), he may well fail to see how

the co-ordinates -1 and +5 may be added since they

are not displacement

numbers. (We leave

the resolution

of this difficulty as an exercise for the teacher.)

12.

Multiplication of directed numbers is considerably

more

difficult than addition. There will be pupils who

drop out between

addition and

multiplication.

(Subtraction

is probably more difficult

still.)

Multiplication

does not arise very obviously

from

displacements because one does not multiply

displacements

together. Still less does one multiply

points

together, or displacements by points. Before a

child begins to multiply directed numbers,

he must, I

believe, develop

the notion of a directed number as a

COMPARISON

FACTOR. That is, he must acquire the

habit

of describing one thing as -2 (or +3, etc.) times

another.

13.

Actually this can arise nicely in connection with

Enlargements.

After

you have done positive enlargements, e.g. draw

flag B to be 2 times as high as flag A,

A

you can try negative enlargements:

11



4/4

flag C is -2 times as tall as flag A.

The factors +2 and -2 in this context are usually

known as the Scale Factors of the enlargements, but

scale factors are particular cases of Comparison

Factors.

14. It is probably

helpful

if the pupil has done some

conscious work on essentially positive comparison

factors. Cuisenaire Rods provide a good start:

Pink

L. Green

ed

Write: PINK = 2

x

RED

Red

RED =

1

x PINK

L

Gen.

=

1c

x RED

White

etc.

And play with these RELATION DIA GRAMS:

2 Red

White

32

4

laLight

Green

Pink

(Fill in the remaining comparison factors.)

15. Similar work can be done on comparing directed

objects.

-2

At IB( IC

B = -2 xA

etc.

16. There are, however, further difficulties with

multiplication, even after the idea of a directed

comparison factor is established. Consider again

displacements on a line:

+6

-3

-6

Is one to jump in straight away and observe that

displacement

+6 is -2 times displacement -3, and

hence write

+6 = -2 x -3

This may have its advantages, but it is another hybrid:

multiplication of a comparison factor and a

displacement-number.

Let us change the notation again. Let displacements

be symbolised by

(R2), (L2), etc., as before,

while comparison factors are

(S2), (Op2), etc.

S standing for same and Op for

opposite .

Then we have many relations:

(Op2) x

(L3)

= (R6)

(Op3)

x

(L2) = (R6)(S3)

(S3) x (R2) = (R6)

(S2) x (R3) = (R6)

(Please play around with the notation for a while.)

But in no case are we multiplying together two

elements of the same kind. (So, for example, the

question of commutativity does not arise.)

17. A purer kind of multiplication involves comparison

factors only.

Let A, B, C, etc., be displacements or other directed

objects, which, however, we shall not label by means

of directed numbers.

After successfully doing problems on the comparison

factors relating such objects when the objects are

visibly present on the page, one may graduate to more

abstract problems like this:

If B is +3 times A, and C is -2 times B, how do A and

C

compare?

The problems may be illustrated by a relation diagram:

+3

C

The answer is, of course, that C is -6 times A. The

number -6 has been derived from +3 and -2. This

relationship between the comparison factors is called

multiplication , and one writes

-6 = +3 x -2

or one could write

(S3) x (Op2) = (Op6)

or one may say that multiplication of comparison

factors is defined by the relation: if B = x.A, and

C = y.B, then C = yx.A.

Multiplication of comparison factors provides a pure

multiplicative structure: if you exclude the zero

comparison

factor, you have a group.

However, it is all fairly abstract. First we have the

elements (e.g. displacements) A, B, C, etc. Then we

have comparison relations +3, etc., between pairs of

elements.

Finally, multiplication is a process whereby two of

these comparison relations combine to form a third.

Some pupils may never manage all that. Yet, until a

pupil has, it is doubtful if he can claim to understand

multiplication of directed numbers.

In

the next issue:

Professor Zeeman writes on why mathematics is

the most original and most creative of all the

sciences.

Dr Flynn, Gordano Comprehensive School,

Portishead, puts his views on streaming.

A piece of C. S. E. coursework is assessed and there

is a letter on the Open University Foundation

Course and its possible effects on the teaching of

mathematics.

12


Documents

Cable DirectedNumbers