Trapezoidal NumbersAuthor(s): Jim SmithSource: Mathematics in School, Vol. 26, No. 5 (Nov., 1997), pp. 46-47Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30215335 .

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Trapezoidal

Numbers

by Jim Smith

In presenting trapezoidal numbers to children I have tended to adopt a style of presentation, which I characterise as a 'negotiated understanding'. (Smith, 1991 and 1996). How- ever, as the intention here is to focus more on the mathemat- ics, I will begin with a slightly more formal definition.

The above pattern illustrates that 14 bricks can be arranged into a roughly trapezoidal pattern and that 14 will be counted as a trapezoidal number. Other trapezoidal num- bers can generated by following a set of rules;

1. Start with as many bricks as you like on the bottom row 2. Build upwards one complete row at a time, using one

less brick than you did on the previous row. 3. You must have at least two rows. 4. Keep going until you feel like stopping or you have only

two bricks on the top row.

Other authors have adopted different forms of 'definition' (see Feinburg - McBrian C, 1996 for example) which are based upon summing consecutive integers including 1. This allows the triangular numbers to be counted as a special case of trapezium numbers, which I argue to be incorrect. Whilst I am happy to accept that a square is a special rectangle, it does seem unlikely that a triangle can be a special trapezium with only three sides, none of which are parallel! However, some triangle numbers can make trapezoidal shapes, if appro- priately arranged; e.g. 15 and some triangle numbers cannot, e.g. 6.

Given this starting point, I ask pupils to investigate which numbers in the range say 1 to 100 are trapezoidal numbers and which are not trapezoidal numbers. Those which are not trapezoidal are named "Awkward Numbers" after their inventor (me).

46

The main activity is to focus on trying to find general results. If time is short, encourage pupils to look for results for trapezoidal numbers with two rows first, then three rows etc. (in a similar way to that suggested by Carol Feinburg - McBrian in her Activity 4).

Pupils can be encouraged to look for proofs and explana- tions of their general results. For example with three rows, beginning with 'n' in the first, the total would be;

n + (n + 1) + (n + 2) = 3n + 3 = 3(n + 1) Which must be a multiple of three greater than or equal to

nine, since n>1. There are some simple visual proofs accessible here, such

as: Which can easily be re-arranged by moving one brick to; and sheared to create a multiple of three; Previous pupils have discovered that 'if you have an odd

number of rows, the trapezoidal number is a multiple of the number of rows'. In other words, five rows generate multiples of five, seven rows multiples of seven, nine rows multiples of nine and so on.

Further Mathematics This is a classic example of a topic that provides differentia- tion by outcome. There is a very wide range of levels of mathematics that can be done here, from beginner looking at trapezoidal numbers, A Level student looking at proofs through to some advanced mathematics looking at the 'awkward' non-trapezoidal numbers.

If we consider representing the general trapezoidal num- ber as an arithmetic progression with a first term of a, common difference of 1 and having n rows, we can use the standard formula to sum an AP;

Sum = (n/2)(2a + (n - 1)d)

to represent the general trapezoidal number as

T = n(2a + n - 1)/2

We can prove that integer powers of 2 cannot be trapezoi- dal numbers by using proof by contradiction for if

2p = n(2a + n - 1)/2

Mathematics in School, November 1997

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then

2(p+ 1) = n(2a + n - 1)

but this is only possible if both n and 2a + n - 1 are powers of 2. Write n = 2q (with q>0 since n> 1) and 2a + n - 1 becomes 2a + 2q - 1 which should be a power of 2, but can readily be seen to be an odd number. Hence there is a contradiction and therefore integer powers of 2 (greater than

20) cannot be represented as trapezoidal numbers. Coming back for a moment to

T = n(2a + n - 1)/2

we can see that ifn is odd, 2a + n - 1 is even and thus divisible by two giving an integer, hence when n is odd T is a multiple of n.

However, this is not the end of the story as there are other awkward numbers which cannot be represented as trapezia with the definition adopted here. Some examples are 496 and 136. A correspondent from a school in Oxford once pointed out to me that the generalisation of non-trapezoidal numbers appears to be related to 'excessiveness'; non-trapezoidal numbers appear to be 0, 1 or 2 'excessive'. The notion of excessiveness is best illustrated by example;

14 has factors 1,2,7 and 14

The sum of the factors of 14 (ignoring 14) are 1 + 2 + 7 = 10 This result is 4 less than 14, so 14 is said to be 4 - excessive.

Let us try 136;

136 has factors 1, 2, 4, 8, 17, 34, 67, 136

Factor sum = 1 + 2 + 4 + 8 + 17 + 34 + 68 = 134

So 136 is 2 excessive and is (by coincidence?) not a trapezium number.

I have yet to prove or disprove the excessiveness conjec- ture, but it is interesting to note in passing that powers of 2 appear in the list of factors of non-trapezium numbers.

Conclusions

There are probably many ways to define trapezoidal num- bers, each leading to different generalisations. The particular version offered here provides mathematical activity for a wide range of ability and opportunities for tackling proofs in a variety of ways. Pupils in a number of schools in the UK and in the USA have been studying trapezoidal numbers and found this to be an accessible and motivating activity. Stu- dents on teacher training courses have also found the activity to be an appropriate mathematical challenge and there are still unresolved problems for readers to attempt. I-]

References Feinburg-McBrian C, 1996, The Case of Trapezoidal Numbers, Mathemat-

ics Teacher 89 (January 1996): 16-27 Smith D N (Jim), 1991, Trapezium Numbers, Mathematics in School,

Mathematical Association (UK), (January 1991). Smith D N (Jim), 1996, Getting Started: a collection of mathematical

activities for secondary classroom, ISBN 0906588340, Mathematical Association, Leicester, UK.

Author D N (Jim) Smith, Mathematics Education Centre, Sheffield Hallam University, Sheffield S10 2NA, UK e-mail: d.n.smith@shu.ac.uk

LETTERS LETTERS LETTERS LETTERS

Dear Editors, A neater solution to the second trisection problem featured in the January issue (Oliver, 1997) is to draw the diagram below and use the fact that angles

a

Y

,

in the same segment of a circle are equal.

Tony Barnard Kings College, London

Dear Editors, This year's May edition (Vol. 26, No. 3) was most enjoyable, I look forward to the next History issue. Readers of Patricia Rothman's article Meditations on Women in the History of Mathematics may be interested in two other books not mentioned in the references.

Grinstein, L. and Campbell, P. (Eds.) Women of Mathematics. Greenwood Press. ISBN 0-313- 24849-4. This is a biographical dictionary of women Mathematicians and contains excellent bibliog- raphies.

Kramer, E. The Nature and Growth of Modem Mathematics. Princeton. ISBN 0-691-02372-7. Tak- ing a themed rather than a chronological approach this is a very detailed account of the History of Mathematics particularly as it relates to issues and

Mathematics in School, November 1997

concepts currently being researched. Edna Kramer includes many biographies of women Mathemati- cians.

I hope this is helpful,

Barrie Westgate Lincoln

REVIEW Common Threads: Women, Mathematics and Work Mary Harris Trentham Books 1997 1 85856 015 2 226 pages, 250mm x 168mm, softback a13.95

This long awaited book will be greeted with affection by mathematics teachers everywhere who recall Mary Harris' exhibition Common Threads. For these and for those who are too new to mathematics teaching to have seen the exhibition or who were unlucky enough to have missed it, the book will be an enduring reminder of the significance of Harris' work to mathematics education. As with the exhibi- tion Harris is concerned to explore the gender divide within mathematics. Her choice of textile activities as a counterpoint was inspired; simultaneously it both comforts, and shocks. At first glance it appears so innocent. As mathematicians we know instinctively that needle-workers must be skilled in a range of mathematics but it took Harris' exhibition to show us the complexities of this mathematics and shock us into appreciating the sophistication of it.

The book presents the story of the exhibition. Of course it should not be hoped that the book would

be a pictorial record of the exhibition but the book is perhaps a bit sparsely illustrated and sadly, not in colour (readers might like to refer to MiS September 1988 for colour illustrations). However the illustra- tions are used to good effect and a favourite of frieze patterns demonstrated with baby socks is included.

The first four chapters set the educational scene relating the history that brought UK education to view mathematics as a male discipline with needlework as women's work. By comparison with the rest of the book they are written as if to fulfil others' expectations of what a scholarly book on mathematics should include. I have to say that once the author abandons "the conventions of academic writing" on page 92 and starts to report her own (hi)story the book comes alive and recreates the feelings that were generated by the exhibition itself. I found myself wondering if it was some male editor, or reviewer, who persuaded Harris to include the early chapters as if her own narrative had to be justified in a masculine academia. From this point in chapter five the book is simply lifted to a higher plain. Harris tells us of the motivation for the exhibition and how various aspects of it came to be. She skilfully weaves in the reactions of teachers and pupils to both the exhibition and to the learning materials that she introduced. For me the strength of her story is the richness of what is reported here. We are treated to the most wonderful tapestry of mathe- matics that grew out of her inspiration; Harris inter- laces her narrative with a rich strand of reporting by teachers who used her activities.

Harris goes on to review the overseas tour of the 'new' Common Threads Exhibition, exploring issues of ethno-mathematics. Often Harris was able to ac- company the exhibition to run workshops to interpret Common Threads and its theoretical background and to discuss the pedagogical issues that it raised. Perhaps the following quote which refers to the exhibition's time in Uganda illustrates the power of

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