I was recently working with a Year 56 class who were being introduced to pentagonal numbers, a mathematical series produced by pentagons, a series which, as I discovered, can be explained by more than one definition.

The class had already spent some time on square numbers (any number multiplied by itself: a x a) and triangular numbers (the sums of these sequences: 1+2, 1+2+3, 1+2+3+4 etc).

As Karen Williams, their teacher at Woodside Primary, Oswestry, Shropshire, introduced the idea, I realised I had forgotten the procedure that produces the pentagonal number series. I thought I remembered that it starts with a regular pentagon, but I couldn’t remember much more than that.

I was quite surprised then when the pentagon Karen Williams demonstrated was not a regular one but the one shown. (Figure 1)

The series of pentagonal numbers were then introduced as a sequence of the addition of corresponding square numbers and triangular numbers, explained by geometrical diagrams. (Figure 2)

Pictures were drawn on the board and there was some discussion about these drawings. The pupils were encouraged to say what they observed and to make predictions.

The first four pentagonal numbers were then revealed to be: 2, 7, 15, 26. (figure 3)

The challenge was to draw out these patterns in their books and extend the sequence of numbers beyond their drawings.

It was interesting how well this task was tackled by children of different abilities. I realised that for some the business of doing all the drawings was almost a redundant exercise; they were able to see and describe how the pattern grew. For others, this drawing seemed a valuable way to help them grasp the features of the series.

There were several others, however, who lacked a clear sense of what it is that gives each of the square numbers their name. I wondered if they had ever been asked to discover for themselves which numbers were square, by arranging counters or cubes into square arrays. What was needed here, I felt, was some using and applying of square and triangular numbers in a more exploratory way. After working on their own for a while, I proposed that we all consider the following problem:

‘I am going to give each group 32 cubes. Your task is to find a way of arranging all these cubes into square and triangle patterns. You may have squares of any size and triangles of any size. Duplicates are allowed, but there must be no cubes left over.’

In this task, 1 is taken to be neither a square nor a triangular number.

Some pupils seized on their cubes and began arranging them into squares and triangles. Some simply watched. But there were several who ignored the cubes entirely and concerned themselves with calculations. This illustrates how it is possible to have a whole class working on a single activity, even though the children are working at different levels of attainment.

Within five minutes, most groups had produced at least one solution. There then followed a most impressive period in which the children, absorbed in the task, kept on discovering further solutions. We recorded each solution on the board and drew attention to the way the solutions used different numbers of each shape.

But the main benefit of this sharing came when someone described their solution without giving the actual numbers involved. The pupil said: “My way has three squares and one triangle. There are two small squares, and a medium-sized one and a medium-sized triangle.”

I translated these words into pictures and challenged the rest of the class to supply the missing numbers that would indicate the size of each shape: The example shown produced the drawing in Figure 4, which was then interpreted as Figure 5.

After about 20 minutes, the class had produced nine different solutions. A new one was discovered invariably through a process of reasoning about the numbers from an existing one. For example, 32 was seen as two lots of 16 which was immediately identified as a square number. But 16 can be partitioned as 10 and 6, both of which are triangular numbers.

Interestingly, no one thought to extend this idea to produce eight squares of 2 by 2. Some of their solutions are shown in figure 6.

After the lesson I decided to look for more information about pentagonal numbers. The internet produced some surprises. Typing “pentagonal numbers” into a search engine (www.google.com) produced a range of options.

Several of the sites (including www.madras.fife.sch.ukmaths) reveal a different way of generating pentagonal numbers spatially. The most common one is shown in figure 7. This produces the number sequence: 5, I2, 22, 35...

Another website (http:members.fortunecity.comjonhaysdigital.htm), however, revealed the story according to Pythagoras. This version has the sequence 7, I5, 26, 40, which is what the Woodside pupils were shown.

It is not a matter of right or wrong. The interest lies with the patterns themselves and the ways in which we perceive them; the ways we can describe that structure and use these descriptions to make a prediction.

However, looking at these two number patterns it occurred to me that the differences between each corresponding number in the two series can be accounted for by a simple adjustment to the spatial arrangement of combined square and triangular numbers.

If the base line of each triangle is removed, not only is the picture somehow more acceptable to the eye, but the two contradictory number sequences become the same. (Figure 8)

4 + 1 = 5

9 + 3 = 12

16 + 6 = 22

Number patterns generated from spatial patterns present excellent opportunities for investigation since they permit different levels of engagement.

I have used key stage 2 but the same ideas can easily be used at KS3. These are examples that could be placed under the heading “Properties of numbers and number sequences” at Year 6 and under “Algebra: sequences and functions” at Year 7.

It would be best aimed at pupils working at levels 4-6 since it is an algebraic sequence that arises from work with practical materials or drawings. Yet the benefits of having the whole class, at whatever level, having a single focus are immense.

Ian Sugarman teaches at Manchester Metropolitan University and is numeracy co-ordinator with NW Shropshire Education Action Zone