Some years ago I suddenly had to do some maths with some boys who were a bit turned off about it. If it had been today in England they would have said, "It's not cool!"

There were two small PE hoops nearby and some small bean bags. I put down the hoops as you see:

I collected eight of the bean bags. "Do they really have beans in?" I asked. They did not know and neither did I. Never mind.

I suggested that we put them in the hoops. Four ended up being in the blue hoop, six in the red hoop so that two were in the overlap.

We went on to talk about how many were in the blue and how many were in the red and how the ones in the middle seemed to be counted twice. Try this for yourself.

We tried putting the bean bags in the hoops in a different way and each time we counted how many were in each of the two hoops.

Well it was time to use the yellow hoop that had been around:

I suggested we made sure that there were four in the blue, five in the red and six in the yellow. So we all tried and then ...?

Well have a go at this one.

Now the investigation is to take this much further. Try to find as many ways as you can for having those numbers $4$, $5$ and $6$ using just eight objects. I guess you'll need to record your results somehow so that you do not do the same ones twice!

Have you found yourself using some kind of 'system' or 'method' for going from one arrangement to the next? Try to explain it if you have.

When you're pretty sure you cannot find any more, check yours with a friend and see if there are any new ones!

As always we then have to ask "I wonder what would happen if ...?"

This month it's very easy to invent new ideas, for example, "I wonder what would happen if I used a different number of objects?" You could go about this in order and try six objects and then seven, you've done eight so move on to nine ...

Any other ideas?