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## Twenty Divided Into Six

Katie had a pack of twenty cards numbered from $1$ to $20$.

She arranged the cards into six unequal piles.

The numbers on the cards in each pile added to the same total.

What was the total and how could this be done?

### Why do this problem?

This problem is one that can be accessed easily - everyone can make a start. It offers opportunities for learners to practise addition and subtraction, along with some multiplication and division, and requires a systematic approach.

Possible approach

You could start by asking the group to work on the problem in pairs with digit cards numbered from $1$ to $20$ without saying very much else at this stage. (You could print and cut out cards from

this sheet if you do not have enough to go round.) Learners might find it useful to make jottings on mini-whiteboards or paper as they explore the
problem.

After some time, draw the class together to find out how they are getting on. Invite some pairs to share their approach so far with the whole group. Some children might be using trial and improvement, some may have worked out what the total of each pile needs to be and then used trial and improvement. You may need to talk about how they work out the total of each pile if this does not come
up naturally. Can they think of a quick way of doing it without a calculator?

They could then continue to work in pairs on the problem. After the initial calculations the problem is a fairly simple one of adding and building the piles but there are many ways of doing it. It would be interesting and instructive to listen to the way that the various pairs are working on the problem, and you may like to gather solutions as a whole class on the board.

In a plenary at the end of the lesson, you could talk about how they have found the different solutions and you may want to ask whether they think they have got them all. This might be a good opportunity to share ways of working systematically so that they could convince you they would be able to find every solution.

Key questions

What number must all the piles add to?

What is the total of each pile?

How do you know you have all the solutions?

### Possible extension

Learners could find as many completely different solutions to this problem as possible and some children will be able to suggest a way to find them all.

### Possible support

If you want to focus on finding all possibilities, some learners might benefit from using a calculator so they are not held up by the mental arithmetic.