Copyright © University of Cambridge. All rights reserved.

## One Wasn't Square

Mrs Morgan, the class's teacher, pinned numbers onto the backs of three children: Mona, Bob and Jamie.

"Now", she said, "Those three numbers add to a special kind of number. What is it?"

Michael put his hand up.

"It's a square number", he answered.

"Correct", smiled Mrs Morgan.

"Oh!" exclaimed Mona, "The two numbers I can see also add to a square!"

"And me!" called out Bob, "The two numbers I can see add to a square too!"

"Oh dear", said Jamie disappointedly, "the two numbers I can see don't add to a square! It's either $5$ too little or $6$ too big!"

What numbers did the three children have on their backs?

### Why do this problem?

This problem invites learners to explore square numbers and the relationship between them. It can be used to talk about a trial and improvement approach and also acts as an informal introduction to algebra.

Possible approach

You could introduce this problem by having three children come to the front of the class and pinning numbers on their backs, just so that the class becomes familiar with the context. For example, you could use the numbers $2$, $3$ and $6$. Ask each child to say something about the two numbers they can see, and invite the rest of the group to contribute statements about any of the
numbers.

Introduce the problem itself - you could make a note of the key pieces of information on the board for children to refer to. Without saying much more, encourage learners to work in pairs to try and solve the problem. Indicate that you will be interested in how they reach their solution so they need to be prepared to explain this in the plenary.

Bring the class together to talk about ways of approaching this task. You can highlight particularly efficient methods, for example those which use trial and improvement in a systematic way, or those that list all the square numbers and look for differences.

### Key questions

What is the gap between the two squares Jamie is talking about?

How can this help you work out the total of the two numbers Jamie can see?

What have you tried so far?

What numbers could each of the children be looking at? Why don't you try some out?

### Possible extension

Children could explore the gaps between square numbers up to $100$. What patterns can they discover?

Possible support

It might help for children to make a list of the square numbers, using a multiplication square if necessary. Finding some pairs of numbers that add to squares will get learners started on the problem.