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# More Numbers in the Ring

## More Numbers in the Ring

##### This problem is based on an idea taken from "Apex Maths Pupils' Book 2" by Ann Montague-Smith and Paul Harrison, published in 2003 by Cambridge University Press.

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Age 5 to 7

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*More Numbers in the Ring printable sheet*

Before doing this problem, it would be a good idea to look at Ring a Ring of Numbers.

Change the ring so that there are only three squares.

Can you place three different numbers in them so that their differences are odd?

Can you make the differences even?

What do you notice about the sum of each pair in each case?

Try with different numbers of squares around the ring.

What happens with 5 squares? 6 squares?

What do you notice?

This problem builds on Ring a Ring of Numbers. It encourages children to start from different examples and then begin to draw some more general conclusions based on their understanding of odd and even numbers.

It would be good to show the image in the problem to the class and ask what they notice, and whether they have any questions. Give them time to consider on their own, then to talk to a partner. Invite learners to offer their noticings and questions but try not to say anything more than "thank you" as they share their thoughts with everyone. Rather than answering any questions yourself,
encourage other members of the group to respond.

Use the ideas that have been offered to build up to introducing the task as stated and give pairs of children some time to try to make all odd differences or all even differences. You could invite pairs to record arrangements that work on the board as they find them and invite everyone to check that they are indeed solutions.

Once you have several ways on the board, invite learners to comment on what they notice. What do all the arrangements have in common? Is it possible to make an arrangement with three numbers with all odd differences? You can work through the rest of the problem in a similar way, drawing the whole class together as appropriate.

It is important to encourage the children to explain why the arrangements of odd/even numbers produce these results.

What happens when you put one more number in the ring?

What happens when you put two more numbers in the ring?

What happens when there is an odd number of numbers in the ring?

What happens when there is an even number of numbers in the ring?

The problem Number Differences makes a good follow-up challenge.

Some children will benefit from spending more time on the Ring a Ring of Numbers problem. Having digit cards to move around on a large piece of paper will also help and pupils might benefit from having sheets of blank rings so that they can try different combinations of numbers: