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# Twenty Divided Into Six

## Twenty Divided Into Six

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Age 7 to 11

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- Problem
- Getting Started
- Student Solutions
- Teachers' Resources

*Twenty Divided Into Six printable sheet*

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

She arranged the cards into six piles.

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

What was the total and how could this be done? Are you curious enough to find out?

This problem is one that can be accessed easily - everyone can make a start - and at the same time it is a great context in which to encourage children to persevere. Finding a full solution requires 'sticking power'! It offers opportunities for learners to practise addition and subtraction, along with some multiplication and division, and requires a systematic 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. 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.

What number must all the piles add to?

What is the total of each pile?

How do you know you have all the solutions?

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.

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.

Place four pebbles on the sand in the form of a square. Keep adding as few pebbles as necessary to double the area. How many extra pebbles are added each time?

Investigate the different shaped bracelets you could make from 18 different spherical beads. How do they compare if you use 24 beads?

How many different shaped boxes can you design for 36 sweets in one layer? Can you arrange the sweets so that no sweets of the same colour are next to each other in any direction?