# Ring of Numbers

Choose four of the numbers from 1 to 9 to put in the squares so that the differences between joined squares are odd.

Choose four of the numbers from this list: 1, 2, 3, 4, 5, 6, 7, 8, 9 to put in the four 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?

Image

Here .doc .pdf are some sheets for recording your solutions.

##### 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.

Try putting one of the numbers in any square to start with. What numbers could go on each side of it?

When you add the numbers in two joined squares, what kind of number do you get?

You might like to print off this sheet of blank rings to help you try out some different numbers.

Oli from Oakmeeds School began the first part of this question where we had to make odd differences between pairs of numbers.

You need odd, even, odd, even as odd + even make odd. Each pair has an odd and an even.Rukmini from Hopscotch Nursery also said:

When the differences are all odd, the sums are all odd.

Rukmini then went on to say:

To make the differences even, you need the numbers 2, 4, 6, 8. Then the sums are also even.

Absolutely right - well done to both Oli and Rukmini. What about the order of the numbers 2, 4, 6 and 8 in the ring? Does it matter? I'll leave you all to ponder on that.

**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 of addition and subtraction.

### Possible approach

It would be good to have the image of the 'ring' on the interactive whiteboard, or projected onto a screen. Begin by writing 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 use this sheet of blank circles .doc .pdf to try out their ideas. As they find arrangements that work, you could invite them to record them on the board on some pre-drawn rings.

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 .doc .pdf 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.