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A Square of Numbers

Can you put the numbers $1$ to $8$ into the circles so that the four calculations are correct?


square of sums

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Why do this problem?

This problem entices learners as it is straight-forward to understand what is required. However it is not as simple as it looks! It will test children's understanding of the properties of numbers and the operations of addition, subtraction, multiplication and division. It also presents a good opportunity to talk about working in a systematic way.

Possible approach

It would be a good idea to have the problem on the interactive whiteboard, or for you to draw it on the board, so that it can be referred to easily. Pose the challenge orally for the class and then give them some time to talk in pairs about how they might go about solving it. Share a few ideas among the whole group, listening out for those that indicate clear reasons for their suggestions. You might begin to list some possibilities for each circle based on what the class suggest. Learners could then have a go, either on mini-whiteboards, or using this sheet of the problem for working on. Explain that you will want to know how they went about solving it, not just the answer, so you could ask them to keep a record of what they try.

In the plenary, invite some children to describe what they did to solve the problem, emphasising that there isn't one right way to go about it, but perhaps there are some ways that are more efficient than others? (You could label the circles with letters, or colour them using different colours, to help discussion.) Many children might have started with a trial and improvement approach, which is very helpful, whereas others might have combined this with a system, for example trying the largest number in a particular circle first, then the next largest etc.

Key questions

Which numbers could go here? Why?
Where could the two largest even numbers go? Why?
Where could the $1$ go? Why?
How will you keep track of what you have tried?

Possible extension

Some children will enjoy finding all the different solutions and justifying that they haven't missed any out. You could also challenge them to make a similar problem which uses different numbers or puts the operations around a square in a different order.

Possible support

Having a copy of the problem on this sheet  will be helpful for many children and giving them numbered counters to move around makes it easy to correct mistakes.