Odd Squares

This problem is a wonderful example of a context in which a proof is accessible to children via an image with algebra not being required. It would be a good choice to try with your pupils once they are familiar with square numbers. As well as encouraging visualisation, it gives learners opportunities to conjecture, justify and generalise.

You could introduce this problem orally. Ask each learner to think of a number and to go through the operations mentally. Invite everyone to jot down the result and share what they have with a neighbour. It might be a good idea to encourage pairs to check each other's arithmetic too! Ask pairs to talk about anything they notice about their two numbers. You could share some of these observations with the whole group. Go through the process again, asking each child to choose a different starting number and again, to note down the end result. Do all four numbers that each pair now has share any properties?

At this stage, you could collect results on the board for all to see. What do the class notice about all the numbers? Give them time to discuss why they think the result is always even in pairs or small groups. This is a chance for them to offer some suggestions, however 'polished' the explanation might be.

At this point, reveal the diagrams (or draw them on the board as you go through the steps again). Without saying anything else, give the group time to discuss further. Ask each pair or small group to develop an oral explanation which they can share with everyone. As a whole group then, you can create an explanation together which uses the pictures to prove, whatever the starting number, the result is always even. Learners may want to include further images. (It's important that learners understand that this will be the case whatever the starting number. The image given happens to be for a starting number of $5$. We can't draw images for every possible starting number so how do we know the result will always be even? This is the key to generalisation and proof.)

It would be great to try and capture this for a display. You could jot down the steps of the explanation on the board as the children build it up and then the final version could be put up on the wall with the problem itself and the images. It would be good to display any other proofs which the class has come up with alongside the visual proof as well.

What do you notice about the result each time?

Is this always going to be the case? How do you know?

Can you describe what is happening in the images?

What can you say about the pattern of dots on each side of the red line in the third image?

If there was a fourth picture, what could it look like?

What is the starting number in the picture?

Can you draw a similar series of pictures for different starting numbers?

Picturing Triangle Numbers is another problem which focuses on visual proof. Although it leads into algebra, many children will be able to offer written or oral proofs.

Some learners might find it useful to use counters or cubes to represent the numbers and therefore to build up a picture of what is going on in this way. This will also allow them physically to take away the diagonal line which is crossed out in the third image if that helps.