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Ring a Ring of Numbers

Choose four of the numbers from this list: $1, 2, 3, 4, 5, 6, 7, 8, 9$ to put in the squares below so that the difference between joined squares is odd.
Only one number is allowed in each square. You must use four different numbers.
What can you say about the sum of each pair of joined squares?

What must you do to make the difference even?
What do you notice about the sum of the pairs now?
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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. To find out more about this book, and order a copy go to the CUP website.

Why do this problem?

This problem provides a context in which children can recognise odd and even numbers, and begin to think about their properties. It also offers practice in addition and subtraction.

Possible approach

It would be good to have the interactivity on the interactive whiteboard, or projected onto a screen. Begin by placing any four numbers in the ring and asking questions about them, for example:
  • Which pair of numbers has a total of ...?
  • Which pair of numbers has a difference of ...?
  • Which pair of numbers has the highest/lowest total?
  • Which pair of numbers has the greatest/least difference?
These questions will help children become familiar with the vocabulary of the problem and so you can then lead into the main activity. Having asked the question, give pairs of children chance to find at least one way of making odd differences. They could be working at computers and/or using this sheet of blank circles and/or using digit cards. You could then test some of these using the interactivity, and record the arrangements that work on board. Once you have several ways on the board, invite learners to comment on what they notice. What do all the arrangements have in common? 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. You could make drawings like these using paired joined squares to help them understand.

Key questions

What do you notice about the numbers in the ring when the difference between joined pairs is odd?
What do you notice about the numbers in the ring when the difference between joined pairs is even?
Can you explain why?

Possible extension

More Numbers in the Ring allows children to investigate different numbers of numbers in the ring.

Possible support

Some learners might benefit from having counters or other objects to help with their addition and subtraction.