Squares and Next-door Squares


Why do this problem?

This problem, although not difficult to start, can lead learners into generalising about the difference between successive squares. Learners will need plenty of $1$ cm squared paper, and you will need some larger squares if you want to demonstrate to a group. They will also need scissors and possibly coloured pencils and glue to make a permanent record.

Possible approach

You could start by showing some square numbers cut from squared paper and asking why they are called "square numbers". You could either cut squares from squared paper with larger squares or download and enlarge this sheet. Can the group think of any other square numbers?
 
Alternatively, if the group already knows something about square numbers you could go straight to placing one square on top of the next one and seeing the difference. Remember to put two sides and the corner between together. Then the difference can be cut off, laid out in a line, and compared with the differences between other 'next-door' squares.
 
After this learners could work in pairs on the actual problem from a printed sheet or a computer, so that they are able to talk through their ideas with a partner. They will need $1$ cm squared paper or two copies of this sheet. This coloured version might be helpful for some learners.
 
At the end of the session all could come together again and discuss their findings. Can they say what kind of numbers all the differences are? What do the differences look like if they are arranged in order?

Key questions

What is the name of these numbers cut out of squared paper?
Can you tell me why that is their name?
What other square numbers can you think of?
Did you put two sides and the corner between together?
How long is the bit you have cut off if you arrange it in a line?
What kind of numbers are these?
What do the differences look like if they are arranged in order?
Have you discovered anything else?

Possible extension

Learners could try to generalise from the results of this problem and possibly reach the formula $(a + 1)^2$ = $a^2$ + 2a +1.

Possible support

Suggest just beginning the problem, arranging the squares in order, which is quite easy, or try this Stage 1 problem - Odd and even.