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Number Squares

This challenge is a bit different to my usual ones. I used it many years ago when I wanted some quite young children to do lots of adding in a more fun way.

You start with four numbers at the corners of a square. Then you add up the numbers at the two ends of each side and put the total in the middle of that side. So in my example 3 + 5 = 8, 5 + 4 = 9, 3 + 1 = 4, 1 + 4 = 5.
These four new answers give us the corners of a new square. The corners are 8, 9, 5 and 4.
These four new numbers are added up and the answers put in the centre of the edges of this new square. And so on and so on.

The diagram gets more and more complicated, growing as shown below:-









There is not much more to say, apart from have a go yourself. Use any starting numbers at the corners. Can you estimate what the size of the last four numbers will be?

What would happen if you used different shapes, for example pentagons or hexagons?

What would happen if you used subtraction, always taking the smaller from the bigger?

What would happen if you multiplied? Divided? What ...??

Why do this problem?

This is an engaging activity that involves both computational skills and organisational skills. I have found that quite young children (6-8 year olds) really enjoy it and do a great deal of work in performing many calculations without really realising it. Quite a few of the children use pentagons and hexagons to vary it a bit.

Possible approach

Doing a "demonstration" square with the class, making suggestions along the way, works well.

Having some pre-printed sheets available is helpful for some pupils, although just seeing the final shape on inner squares can be very confusing for some.

Key questions

Tell me about anything you have noticed.
What numbers did you start with?
Do you have any ideas about the number you might end up with in the middle?

Possible extension

You can reverse the process and see if children can come up with what the starting numbers could be to produce a certain specified result in the middle. Asking them how they would work this out leads to some interesting insights into children's methods of thinking through addition and subtraction. Older children (10-11 year olds) could try the reverse when it is addition taking place. The other thing is to try to predict the result in the middle when you have certain prescribed starting numbers.

Possible support

With some pupils it is useful to have an adult (more mature thinker) who can keep two fingers at the two places holding the numbers that the pupil is considering at that time.