Teaching Directed Numbers an Experiment

Teaching Directed Numbers an ExperimentAuthor(s): Tim RowlandSource: Mathematics in School, Vol. 11, No. 1 (Jan., 1982), pp. 24-27Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30213680 .

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Teaching directed numbers An experiment

by Tim Rowland, Homerton College, Cambridge

1. Introduction The purpose of this article is to report on a small-scale experiment in the teaching of addition and subtraction of directed numbers (integers, or whole numbers, positive and negative, to be precise) via the manipulation of coloured cubes. The method is described in Sections 2 and 3. My interest in the project arose after some very stimulating (for me) sessions on integers in Spring 1981 with a group of primary teachers attending the Mathematical Association Diploma course based at Homerton College. The reason for undertaking the experiment with children is the general lack of success of middle and lower ability groups, particularly with subtraction, using other methods. CSMS has recently reported on this'2, and their findings in this area will come as no great surprise to practising secondary schoolteachers. Broadly speaking, two methods are currently employed, based on different but related contexts. The first method is based on credits and debits in relation to, say, a bank balance or pocket money. Graham3 gives one of many possible variants. The second method is in the context of displacements on a number line, and is favoured by current school texts, e.g. SMP4. I believe that this method could be more satisfactory if it were based on proper composition of translations, following more general work with vectors. This might avoid the confusion between location and displacement on the number line, which is made explicit in SMP4 and discussed by Watson5. The lift model, the thermometer and tide levels are variants on this second theme.

2. The Model The model used here is based on the simple idea that

+2=3- 1=4-2=5-3= ... -3=1-4=2-5=3-6= ...

These differences are between natural (counting) numbers which are already familiar (although the second set of differences is not yet meaningful). The idea is captured in concrete form by using red and black cubes (balls, or counters of two contrasting colours do just as well). So by analogy with the above, -3 is embodied in a variety of ways by piles of cubes, e.g. one black and four red, two black and five red, and so on.

There is nothing strange or novel in this variety of equivalent representations - don't we expect children learning about fractions to appreciate that 2 = 4 = = - -? They can't add fractions until they do. Likewise, the notion of equivalent forms of an integer is firmly established in the lessons before we begin addition.

3. The Lessons In this small-scale experiment I decided to try the method with four fairly bright 11 year olds - Claire, George, Samantha and Simon, in their last year at a maintained junior school in north Cambridge. This choice was largely determined by the desire to work with children who had not already encountered, in any formal sense, either of the two other methods mentioned. The experiment is regarded as an initial feasibility trial. Clearly a comparative experiment with more typical groups - say parallel classes of 13 year olds - would have to be carried out before any general conclusions could be drawn. I would be interested to hear of others who have tried this approach in the classroom. It was intended to have five half-hour teaching sessions with the children, four in their school and one in the CCTV studio at Homerton. For practical reasons this was in fact reduced to four sessions over a two-week period in March 1981. The video film (of the last session) summarises the ideas of the first three lessons and then shows the two girls working on subtraction. This is their first attempt at it apart from a brief introduction in school.

Some excerpts, anecdotes and pitfalls from these lessons now follow:

Lesson 1 The red and black cubes (multilink) are tipped out of a plastic bag on to the table. The children haven't seen the apparatus before and proceed, without invitation, to make a dog, a tower, a cuboid. I give them about 10 min for this (and some time at the beginning of subsequent lessons) before explaining that we shall be working with various piles of cubes. I introduce the idea that a red cube and a black cube annihilate, or neutralise, each other, like soldiers from opposing armies (Papy's idea 4). So two piles are the same, or "equivalent", if we can reduce one to the other by removing (or adding) neutral pairs. Thus:

is equivalent to and to

(I believe that this pre-arithmetic stage introducing equivalence must not be rushed and give at least half an hour to it over the first two lessons.)

24 Mathematics in School, January 1982



The children consider questions such as:

- Make a bigger pile equivalent to

(we settled on the notation (1) for shorthand). - Make a smaller pile equivalent to (1); what is the smallest

possible pile? - Make some piles equivalent to (1); what is the smallest? - Make a pile equivalent to (7) with twice as many black cubes

as red (this added interest when the work was becoming routine).

At a more abstract level (we only had about 100 cubes) we discussed:

Is (12) equivalent to (12)? to (355)? (The second introduces a different way of looking at equivalence which the children used intuitively when the number of red and black cubes in a pile were close.)

- What are 0 and A if (600 )=(62' I found that on one occasion Simon lapsed into a multiplicative notion of equivalence, e.g. (2))=Q(10), presumably prompted by equivalent fractions. I regard the analogy between equivalent fractions and equivalent piles as an advantage (see Section 6).

Lesson 2 After revision of equivalence, we made a start on addition. We shall "add" two pairs together by combining them into a single pile.

We do this with various pairs of piles and later record what is happening, e.g.:

is recorded +

George sees fairly quickly how to dispense with the apparatus by adding number-pairs. I am happy about this but encourage them to work with the cubes so long as they are useful. What if we do the same addition with equivalent piles?

(6)+( )=0(4).

The answers are equivalent, since (6) can be reduced to (5). This is verified with other examples.

G. Aretakis

( 4)=!() 5 ) 12) ( 2)/

1)=) +(3) d Lesson 3 (I had considered introducing the symbols '2, -3, etc. as labels for certain pairs of cubes, but this would be to deny any previous encounters with the notation.)

We begin by looking at a Celsius thermometer and at mark- ings above and below the zero. Our particular thermometer does not distinguish between them, but the children are familiar with -3 for "three below zero" and I suggest we write '3 for "three above zero" (we say, negative three, positive three. In these lessons we did not in fact return to the thermometer model).

Positive numbers are represented by black cubes, negative

numbers by red. Thus '2 is 00, or (2), and-3 is OOO, or (0). But there are, of course, many equivalent representations, e.g.

2-= A pile like represents 0.

For the purposes of addition, as we saw in the last lesson, any one of these representations will do. Thus, *2 +-3 can be cal- culated (using the smallest piles) by combining (g) and (0) to give (2), which represents -1 since it is equivalent to (0)

+2 +-3 =-1

We practise this physically, using the apparatus. Later we record how the cubes were used, like this (Samantha):

+1 +-3

0() + ( ) = -2

-14++10

(0) + (100 (10 =-4

4' ++6

4+) ( 10) = 10

-3 +-2

0+ =-5

The lesson concludes with a brief preliminary discussion of subtraction:

- What happens if I take away ifrom

from ?

- How about (1) - (1)? (They match black with black and red with red and get (1).)

- Try (2)-(4). ("I can't take four black from two black.") - Make a pile bigger than (2) but equivalent to it.

How about (6) - (4)?

Lesson 4 (recorded on videofilm) The lesson begins with Simon and George demonstrating equivalence and addition. Their recall is good.

Then Samantha and Claire tackle subtraction:

+6-72=+4 -6 - -2 =-4

-7--4 =-3

These examples present no difficulty since they are taking away a pile of cubes of one colour from a larger pile of the same colour. They match (their word) the cubes in the second pile with a subset of the first, which they remove.

-7 --4 =-3

They then try +3--2 ?

Not surprisingly, they falter at this, since cubes of different colours cannot be matched as in the previous examples. In fact

Mathematics in School, January 1982 25



Claire arrives at the answer '5 after about 7 min, after guidance from me along the following lines:

- How did we do (4)_(21) at the end of the last lesson?

Samantha responds by adding the piles, but is corrected by Claire, who then matches cubes of the same colour and gives (2) as the answer, and Samantha agrees. (I regret my choice of example! Aware of the time, I stubbornly return to the original question *3- -2, conscious that I ought instead to continue along the lines of the end of Lesson 3.)

- What about then, if we can only match cubes oo o

of the same colour?

Claire suggests adding two neutral pairs to the -2 pile, and after moderate encouragement from me adds another, so the problem becomes:

She matches the black cubes, but is still left with five red cubes to "take away" from an empty pile!

I suggest (one could question the extent of my direction of their investigation) that instead of adding neutral pairs to the two red cubes they try the same idea with the three black

cubes. Claire does so, giving answer *5.

and the

(Incidentally, Claire's flexibility and insight is a surprise to me, even after three half-hour sessions with the group.) - Now try -1 -'2.

Claire uses her procedure with the cubes:

and writes -3.

I ask Samantha to try +3--1. She has 0 0 _ , and arbi-

O

trarily adds just one red cube to the +3 pile,

giving

Claire: "That's wrong. She should have put in a black one too." Samantha agrees, and gives

I then ask Claire to try -2--5 and Samantha *2 -*4. They do these confidently and accurately, and the lesson ends.

*3--2 =+5 -1 -'2=-3 -2 --5=*3 +4 -+2 = +2-+4 =-2

4. Evaluation The four children enjoyed the lessons, an important factor in an area of mathematics not renowned for its appeal to children. Of course, one cannot discount the novelty of the experience for them, the smallness of the teaching group and my personal interest in their achievement. Nevertheless, I attribute the enjoyment in part to the attractiveness of the apparatus, and to the fact that apparatus was used throughout.

Addition was straightforward for the children (who, though young, were admittedly able) and after some experiment they arrived at a procedure for subtraction which was both meaningful and successful. There seems to be no contradiction here to my belief in the basic feasibility of the method as a teaching approach.

5. Subtraction of Integers Any approach to the integers must in part be judged on its success in relation to subtraction2. Children first meet subtraction (of natural numbers) by physically matching and taking away real objects - counters, sheep and so on. This concept of subtraction is extended using Dienes' apparatus to exploit the place-value notation. Again, units are physically taken from units, longs from longs, after decomposing (if necessary) the number subtracted from. Subtraction is seen as an operation complementary to addition, in the sense that 5-2= 0 and 2 + E = 5 have the same solution (truth set).

Now it is true in the additive group of integers (as opposed to natural numbers) that if 2 + l = 5 then -2 +(2 + 0)=-2 + 5. Using the associativity of addition, and the properties of in- verses and the identity 0, we then see that we can perform subtraction by adding the additive inverse. Indeed, in the formal sense this is the only viable definition of subtraction. But I regard it as rather subtle, and discontinuous initially, for children whose previous experience of subtraction is of taking something away from something else, or of building up one set or number to make another. Hence my approach to subtraction in the lessons, which differs from that of Nuffield' where the additive inverse approach is introduced from the outset. Clearly the "add the inverse" viewpoint can (and should) be arrived at with coloured cubes eventually, when experience leads to the insight, e.g. -5 -'2 is:

Clearly we have to add two red cubes to the five we had in order to provide two black cubes (later removed) with which we can match the two we wanted to take away.

This point was made even clearer by a group of primary teachers with whom I discussed the method. Their approach to subtraction was a sort of integers counterpart of "equal addition". Cubes were added to both numbers to make the subtrahend zero, e.g.

This makes the point (adding the inverse) neatly and may in fact be easier to understand instrumentally. One could, how- ever, transfer much of the debate about decomposition (equivalence) versus equal addition wholesale to this situation.

6. Conclusion A list of books and articles which suggest an essentially similar teaching approach to the integers is given in references Watson5 to Gibbs "2. Each varies from the present approach in some respect, e.g. adherence to the concrete model, emphasis on equivalence, treatment of subtraction.

As a formal development of the integers as equivalence classes of ordered pairs of integers, it can be found in many under- graduate texts on number systems, e.g. Griffiths and Hilton". The article by Stewart and Tall3 is also very relevant. I consider that the embodiment using piles of coloured cubes makes this "proper" treatment of the integers accessible to children in the lower secondary years.

The analogy with the construction of the rationals as ordered

pairs of integers m is interesting. It has always been conven-

tional to introduce the positive rationals (fractions) to children before the integers. Arguably, this (the rationals) is a more complex construction for the following reasons:

(a) Equivalence of fractions is defined in terms of multiplica- tion as opposed to addition.

(b) Addition of fractions can only be performed after suitable equivalent representatives have been found, e.g. +1 12 The latter part of Lesson 2 illustrates that we can add integers using any representative piles of the integers

26 Mathematics in School, January 1982



we are adding, in particular the smallest (or "canonical" ') piles as we did in Lesson 3.

(c) Subtraction of fractions also demands a prior search for suitable representatives of both fractions. With integers we only need an alternative representative - a "big enough pile" - of the integer we are subtracting from, e.g.

-2-*3=(~0)(3 =(3)- (o)= (+)=(-5.

References 1. Kiichemann, D. (1980) "Children's Understanding of Integers", Mathe-

matics in School, 9, 2. 2. Hart, K. (ed.) (1980) Children's Understanding of Mathematics 11-16, John

Murray. 3. Graham, J. D. (1980) Letter to the Editor, Mathematics in School, 9, 5.

4. School Mathematics Project, Book C, Chapter 2, (CUP). 5. Watson, F. R. (1975) "Conversation Piece" Mathematical Education for

Teaching, 2, 2. 6. Sawyer, W. W. (1964) Vision in Elementary Mathematics, Pelican. 7. Wain, G. T. and Woodrow, D. (ed.) (1980) Mathematics Teacher Education

Project Tutor's Guide, Blackie. 8. Nuffield Foundation (1969) Computation and Structure 4, Chambers and

Murray. 9. Auckland, K. et al. (1977) Primary Mathematics, Globe Education.

10. BBC TV Colliding Numbers. From Maths Today, available from Gateway Education Films.

11. Bartolini, P. (1976) "Addition and Subtraction of Directed Numbers" Mathematics Teaching, 74.

12. Gibbs, R. A. (1977) "Holes and Plugs" Mathematics Teaching, 81. 13. Stewart, I. and Tall, D. (1979) "Calculations and Canonical Elements,

Part 1", Mathematics in School, 8, 4. 14. Papy (1964) Mathimatique Moderne (Chap. 20), Didier. 15. Griffiths, H. B. and Hilton, P. J. (1970) Classical Mathematics, Chap. 23,

Van Nostrand.

4

Solutions

to Puzzles, Pastimes, Problems on page 15

1. Symmetrical Patterns

Fig. 1.5 Fig. 1.6

117\

Fig. 1.7

(i) Figure 1.5. (ii) Figure 1.6. (iii) Lines of symmetry, 1, 3, 2, none, 2, 6, 2. (iv) Figure 1.7. (v) A dodecagon with 12 angles of 1500, and six sides of length double that of the other six sides. The second ring forms a larger regular dodecagon with sides double the length of those of the original dodecagon. Interesting patterns can be made by placing rings around the other shapes.

2. Pythagoras Again (i) If 4x-4 is the hypotenuse, then x2+(3x+4)2=(4x-4)2, i.e. 6x2= 56x, which has solutions x = 0, or x = 28/3, when 3x + 4 = 32, and 4x -4 = 100/3, and the sides of the right-angled triangle are proportional to (28, 96, 100), i.e. (7, 24, 25).

But when 3x +4 is the hypotenuse, x2+(4x - 4)2=(3x + 4)2, i.e. 8x2= 56, which gives the non-zero solution x= 7, 4x -4= 24, 3x +4=25, and so this triangle has sides, (7, 24, 25) and it is similar to the first triangle.

(ii) Any odd number is 2p + 1, whose square is 4p2+ 4p + 1 = (2p2+2p)+(2p2+2p+ 1). But 2p2+2p+1=(p+1)2+p2 and

2p2+2p=2p(p+ 1) and 2p+ 1=(p+ 1)2-p2, and so m=p+1 and n =p.

(iii) 2(m2+n2)=(m +n)2+(m - n)2. (iv) 5=12+22 1/2(12+32) 13=22+3 2=1/2(12+ 52)

17= 12+4 2= 1/2(32+52) 29= 22+52= 1/2(32+ 72) 37= 12+62= 1/2(52+7) 53 =(22+72)= /2(52+92)

(v) (m2+n2)2=(m2- n2)2+(2mn)2. (vi) If k= 1, 2(m2+n2)=(m+n)2+(m -n)2. When k=3,

23(m 2 + n 2)= 22X 2(m2 +n2)= 22(m +n)2+22(m -n)2 and similarly for odd values of k =2p+1, 22p+(m2+n2)= 22b(m +n)2+22p (m - n)2.

When n is even, 2p, 22(m2+ n2)=(2+m)2+(2Pn)2 (vii) Take a= 2mn, b= m2- n2, c=m2+ n2. Then 59 cannot be

a or c, as it is not even and is not the sum of two squares. So m2 -n2= 59, and m-n= 1, m+n=59, m=30, n=29. The only Pythagorean Triad is (59, 1 740, 1 741), where 592= 3 481= 1 740+1 741.

60 cannot be the hypotenuse as it is not the sum of two squares. If m2- n2=60, m2+n2 must be even, and a, b, c will have a common factor. So we try 2mn = 60, which has four sets of integral solutions (30, 1), (15, 2), (10, 3) and (6, 5), which produce the triads (60, 899, 901), (60, 221, 229), (60, 91, 109) and (60, 11, 61).

65 could be the hypotenuse, since 12+82=42+72=65. Hence the triads (16, 63, 65) and (56, 33, 65). But if m2-n2=65=1 x 65=5x 13, and so (m, n)=(33, 32) or (9, 4), yielding two more triads, (2 112, 65, 2 113) and (72, 62, 97).

3. Queer Quadratics (i) Not a true quadratic, as the terms in x2 cancel: x =0 is the only solution. (ii) x = 1/2 is the only solution. (iii) An identity which is satisfied by all values of x. (iv) x = 0 is the only solution. (v) No solution, as 4 cannot equal 3. (vi) No solution, since a2+b2 cannot be zero, unless a=b=O, which is precluded, as neither a nor b can be zero.

4. Filling the Minibuses If x, y, z are the numbers of 10, 12, 15 seaters needed, then 10x +12y +15z = 120, which shows that z must be even. Try z= 0 or 2 or 4. Then 10x+12y= 120, and possible solutions for (x, y) are (12, 0), (6, 5) and (0, 10) of which none is practic- able as x, y and z cannot exceed 5. Similary when z=2, (3, 5) only is possible, and when z= 4, only (0, 5) is possible. Hence (x, y, z) can be (3, 5, 2) or (0, 5, 4), and the latter uses the fewest vehicles.

When 10x + 12y+ 1.z= 121, there is only one solution, (4, 3, 3).

5. Magic Numbers 8547 x 13= 111 111= 5 291 x 21= 3 x 7 x 11 x 13 x 37. So 8 547=3x7 x 11x37, and 5 291=11 x 13x37.

Mathematics in School, January 1982 27



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Teaching Directed Numbers an Experiment