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.