This is an account of some sessions I did with children aged six and seven, exploring some aspects of different shapes using paths traced out by a beanbag being slid from person to person.
In the Hall, ask three people to sit on the floor. Give them a beanbag and tell them to slide it from one to another, and keep sliding it, so it goes from Sasha to Luke to Rufina, to Sasha to Luke to Rufina, to Sasha ….
Pretend the beanbag is covered in paint and marks out a trail. What shape is the path it traces out?
The beanbag marks out a triangle. If three more volunteers take their place do they produce a triangle as well? Does it look the same as the first one? Can they design a triangle to order? Can they make a big triangle? A small one? One where all the sides are equal? (Do we know a name for such a triangle?)
Looking at triangles in this sort of way has several advantages. It gives you a new insight; instead of being pictures on paper, or rigid shapes of card or plastic, we’re working with mental images, where we see a triangle as a path dependent upon the positions of the three points which are its corners. Change one – or more – of the corners and you change the triangle. We’re starting to see geometry not just as a static process but a dynamic one.
There are more questions waiting to be asked, some of them distinctly tricky. Can we make a triangle with a right-angled corner? Can we make one with two right-angled corners? (If not, why not?) Can we make a triangle to enclose as much floor space as possible? Can we make a triangle whose sides are big but which encloses a very small area? And we ought to ask whether it’s possible for three people to position themselves so that they don’t make a triangle.
Of course, neither you nor the children will want to stop with triangles. Four people will make quadrilaterals. What instructions do they need to make squares, oblongs, rhombuses, ….? If we fix the positions of two people where could two more sit so the four of them make a square?
In my next session we took the beanbag work into the Stars arena.
A very small modification moves the investigation into new and fascinating territory. Start with five people; ask them to sit down and slide the beanbag from one their next-door neighbour.
The path is a pentagon, and if the children spread themselves out equally – not necessarily an easy matter – the pentagon is regular.
Now ask what would happen if instead of sliding to the next person each child misses out their neighbour and slides to the next-but-one person instead. Can they visualise what shape will be traced out?
Ben’s diagram shows a five-pointed star. It’s quite a nice way of drawing a five-pointed star, in fact.
What will happen if we have seven people in the group and miss out one? A seven-pointed star is perhaps the natural prediction, and that’s what Craig gets. And if we miss out two people each time, Alice finds we get a different star, and a much more elegant one.
Any predictions for a six-person group, missing out one each time? Hands up, all those voting for a six-point star! The first time I did this, with Y2 pupils, was the only occasion in my life when I’ve actually seen someone’s jaw literally drop. A boy sat there with his hands open, waiting for the bag to reach him, and it went round and round and round, passing him each time. “I’m never going to get it”, he wailed! (There are some quite deep ideas in this statement.) Eden’s diagram (1) shows why.
What happens if you miss out two people rather than one? This time the path is even more limited; Eden’s diagram (2) shows it’s simply a straight line as the bag travels backwards and forwards between just two people.
There are lots of questions waiting to be asked. Why are things completely different with six people? What happens with other numbers?
The children were full of suggestions and ideas they wanted to explore. Nearly all used abstract diagrams from the start, but a couple preferred to draw people, while two others wanted to set their examples in a concrete setting (rabbits in a field and fish in an “Ekweriam”). I could see who chose to use a ruler, and who put explanations and descriptions on their diagrams.
Alexander was particularly keen to see what happened if he used a composite rule (miss out 1, then miss 2, then miss 1, miss 2, miss 1, …) with seven people. I certainly didn’t know what would happen, and our joint diagram delighted us when we found the final pattern included both the stars.
Someone explored all the patterns you can make with a group of nine people. Jo showed tremendous skill at spotting the group sizes and rules which generated triangles, squares, and even hexagons rather than producing stars. How much insight and understanding did this 7-yearold display to be able to create this diagram? “I’ve made a Hexagon!” she recorded with justifiable pride.
The Beanbag is one of my favourite themes. It’s instantly accessible and it’s a lovely example of the fact that mathematics does not consist of blocks of learning which are isolated from each other. Rather, they are linked so closely it can be impossible to disentangle them. Surprisingly, this investigation is at least as much about numbers as it is about geometry. It can be used with almost any group – I’ve used it with Classroom Assistants, B.Ed. students, teachers in disadvantaged schools in South Africa, and of course the 60 Y3 children whose recordings I’ve used throughout this article.
I’ve no intention of giving away too much. I had to explore everything for myself and enjoyed it immensely. Most of my first hazy guesses didn’t work out but it was well worth it in the end. It was wonderfully satisfying to find that the investigation brought together so many topics in mathematics.
We started with what was quite obviously an enquiry into geometrical shapes, but gradually it dawned on me that it was just as much an arithmetical topic as a spatial one. You – or your pupils – may follow the same path as I did, or you may take a different route, but by the time you’ve found out everything you want to you’ll have used factors, multiples, and prime numbers. Whoever would have thought that the key to drawing stars lie in prime numbers?
And a real enquiry it was. I had to hypothesise, explore, evaluate, improve, amend, conjecture – all the thinking skills you want your pupils to explore. And yet it’s pretty well untouched in any school programme of study. Yes, any school. What age-group can you see yourself using these ideas with? After all, it worked with me at my level.
So would you use it with secondary children?
Or could you use the theme with primary pupils? In my final part I’ll give some examples what happened when I used some of these ideas with 6 and 7 year-olds.
So far I’ve mentioned drawing stars with 5, 6, 7, and 8 points. Sometimes you can compose a stars by overlapping shapes; other times you draw a single continuous zigzagging line.
Most of this was pretty new to me, and I figured that if I was going to get any insight into things I’d have to collect as much data as possible, and do so in a systematic manner.
So for the 9-point situation I sketched out 9 points equally spaced round a circle. Joining each to the next one gave me a regular nonagon, which isn’t a star at all. But when I joined each point to the next-but-one point, i.e. missing out one point each time, I got a genuine star. (A)
What if I miss two points each time? I get an equilateral triangle, and if I make all the three possible equilateral triangles I’ve got a new star – an overlap. (B)
And if I miss three points each time there’s a second zig-zag star. (C)
There aren’t going to be any more, because I can see that missing out four points each time (or five, or six, …) simply gives me shapes I’ve already discovered.
So my table of result now looks like this:
It had been a long time coming, but I was now indeed beginning to get some insights into what was happening. I still had plenty of questions –
are all stars either overlaps or zig-zags, or is there another type?
are there any numbers which will give no stars at all?
can I predict which types of stars, and how many, we’ll get for any number?
does every number after 6 give at least one zig-zag star?
can I predict how many zig-zags there’ll be for each number?
are there any numbers which will give more overlaps than zig-zags?
If you want some practical help, you can find circles with every number of dots up to 24 at http://nrich.maths.org/8506
The previous post was built on the observation that to draw a five-pointed star you use a single zig-zag line, and to draw a six-pointed star you must overlap two equilateral triangles. But there’s much more to find out.
The first time I explored these ideas I found some delightful surprises lying in wait, and the first one comes with seven-pointed stars. Actually there’s not one, but two seven-pointed stars you can draw, both by the single zig-zag line method. One version is – to my mind – elegant, and the other is fat and rather bloated
And what about eight? You may be beginning to expect that the superposition method will give an eight-pointed star, and if you use two squares that’s what you get.
But, wait a moment, there’s a second eight-pointed star – and this one is drawn by the single-line zig-zag method!
So we’ve already got a rich collection of results. With 5 points you get a zig-zag star; with 6 you get a star formed by overlap. 7 points gives you not one but two zig-zag stars, and 8 points produces both one zig-zag and one overlapping star. I’ll be very disappointed if you’re not already wondering about 9 pointed stars – zig-zags (and how many?), or overlap – or perhaps both?
It looks as if there are some rather seductive patterns to explore, doesn’t it?
More next time.
Every now and again you come across an idea that’s immensely rich in all the good things you want a topic to offer – accessibility, scope for exploration, coverage of a wide range of mathematical ideas, available to a wide range of ages and abilities, …. – but which is totally ignored in syllabuses and schemes of work.
Just what was wanted, in fact, when I was asked to do a series of sessions to be televised to a selection of schools in England and Pakistan. There was no chance of finding what each different group had covered, so I needed to use topics which were visual, accessible, exploratory – and which I could be pretty sure were not covered in their schemes of work.
I was working with secondary schools, but in different circumstances I’ve used most of the ideas equally well with primary children.
(There will be a short pause while you think of the topics you might have chosen in similar circumstances.)
Very high up on my list was the theme of Stars, and here’s where I got my starting point from.
Service was rather slow in Pizza Express and we discovered my wife and I were both doodling stars. Jill was drawing six-pointed stars, and all mine had five points. Jill drew hers by drawing an equilateral triangle and then another on top of it, turned through 180°.
Unlike hers, my pencil never left the paper. I drew one continuous line which turned at four points and eventually got back where it started.
Our sketches looked something like this:
A couple of questions immediately arose. Could our methods be interchanged? Could we produce a five-pointed star by overlapping two shapes, and could we draw a six-pointed star from one continuous line?
Now that gave us a lot of fun, and you may like to think about it for a while and try a few sketches.
Probably before long you’ve convinced yourself that the two methods are not interchangeable. But that only generates further questions, and the very first is to wonder what happens if you try stars with a different number of points.
So how are you going to draw a seven-pointed star? Overlap? Or zig-zag? And eight? And are any hypotheses developing?
I’ll post a second part soon.