Picture Story
Why do this problem?
This problem encourages visualisation of a three dimensional idea in a two dimensional context. It offers a visual demonstration of the sum of cubes which can lead to a proof. By extending the picture, ideas of proof by induction could be introduced.
Possible approach
This printable worksheet may be useful: Picture Story.
Perhaps start by exploring the problem Picturing Triangle Numbers, which develops the formula for the nth triangular number using a pictorial approach.
Are they surprised by the patterns they have noticed?
Can they draw similar images or extend the existing image to represent the sum of the first 7, 8, 9, 10 cubes? Does this support their predictions above?
Can they now deduce the general formula for the sum of the first n cube numbers? In small groups, they could develop pictures to support a proof of their generalisation, which could be presented to the rest of the class.
Key questions
Where are the cube numbers in the picture?
How does the picture show $(1+2+3+4+5+6)^2$?
Could you draw similar pictures for other sums of cubes? Can you always draw such a picture?
Possible support
Try the problems Picturing Triangle Numbers and Picturing Square Numbers.
Possible extension
The problem Summing Squares offers a similar visual proof idea in three dimensions for the sum of the first n square numbers. Students could read the article Proof by Induction to find out more about this important method of proof.
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