Bean bags for Bernard's bag

How could you put eight beanbags in the hoops so that there are four in the blue hoop, five in the red and six in the yellow? Can you find all the ways of doing this?

Age

7 to 11

Challenge level

Exploring and noticing Working systematically Conjecturing and generalising Visualising and representing Reasoning, convincing and proving

Being curious Being resourceful Being resilient Being collaborative

Problem

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?

Printable NRICH Roadshow resource.

Student Solutions

We did not have any solutions sent in for this activity, maybe it was done very practically and not much was recorded. The recording is not always the most important aspect of a piece of mathematics.

However, we would still love to hear from you if you have worked on this problem. Teachers, you may like to send in a summary of your pupils' work. Please email us: primary.nrich@maths.org

But a little time on and we got this reponse - thank you.

We are Dominic and Sam (year 5) from St Nicolas CE Junior School, Newbury.

We noticed that you had no solutions last month to this problem, so here is our solution.

We started with 2 hoops, red and blue. We had 6 bean bags in the red hoop and 4 in the blue. We worked out that we had to add the number of bags in each together and subtract the number of bags to get 2, which is the number of bags that go in the middle (where the hoops overlap). Then if there are 6 in the red hoop, 2 are in the middle so 4 are not in the middle.

On the other way, if there are 4 in the blue, 2 are in the middle and 2 are not. For 3 hoops, we had 0 in the middle first and then 1 in the middle, next 2 and lastly 3. This way we didn't have two solutions the same. We worked out 14 solutions and we think we found them all.

From James Dixon Primary School in Bromley in March 2013

Hi its James Dixon Primary School here. Well we found 5 different solutions to Dominic and Sam. At first we thought we had the same results because our maths group also found 14 solutions, but we were mistaken. We only had 9 the same as them and five different ones. That means together with St Nicholas School we have found 19 solutions!

Here are our other 5 solutions.

Written by Farrah and Kieran on behalf of the Year 5 Aspen class Wednesday maths group.

Extra five solutions written out by Inthujan and Sowkhaetul

Thank you James Dixon School for these extra solutions, well done.

Teachers' Resources

Bean Bags for Bernard's Bag

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.

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 ...?"

Any other ideas?

Why do this problem?

This activity is especially good to do with a small number of pupils. This will give them confidence to talk with each other and discuss their ideas. It is amazing how many different ways of approaching the problem they will find.

One teacher commented that "ALL children loved this! It was good for developing understanding of Venn diagrams and recording in a systematic way".

Possible approach

Introducing this activity as described in the problem with hoops and bean bags would be a good idea, if possible. This might mean going outside, taking over the school hall, or pushing all the tables to the edge of the classroom to create floor space. You could start with just two hoops and eight bean bags, and ask one child to put the bean bags in the hoops however they like. Ask questions about this distribution of bean bags, such as the number in each hoop and the number in the overlap. It's important at this stage to listen to the pupils so that you can assess their understanding of the overlapping area.

Introduce a third hoop and set up the challenge. Invite children to work in pairs or groups to find at least one solution. Give them the choice as to what they use in terms of equipment: some may want hoops, some may be happy to draw representations. However they work at the problem, encourage them to record their solution/s somehow so they don't forget.

Once some solutions have been found, bring the group together again to share (and check!) the answers they've got so far. At this point, you can challenge them to find ALL the solutions. Give them a chance to talk in their pairs about how they might do this before sharing some suggestions. Draw attention to the methods which use some sort of system, so that there is less chance that solutions will be missed out.

This investigation would make an engaging and attractive display, and you could encourage learners to describe the system they devised so that anyone looking at the display would be able to follow it.

(This activity would also be worth doing as staff CPD.)

Key questions

How many bean bags are in this space?

Are there the correct number in the red hoop etc.?

Possible extension

Simply ask the pupils to suggest extensions by asking "I wonder what would happen if we ...?"

Possible further work which leads to material for the exceptionally mathematically able

Go to Plants teachers' notes

Possible support

You could provide this sheet for those having difficulty in recording. It contains the three hoops printed six times.

Or search by topic

Number and algebra

Geometry and measure

Probability and statistics

Working mathematically

Advanced mathematics

For younger learners