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
Perhaps start by exploring the problem
Picturing Triangle Numbers, which develops the formula
for the nth triangular number using a pictorial approach.
Ask students to imagine that they are building cubes of
different sizes from smaller cubes.
How many small cubes will be required to make a 1 by 1 by 1
cube?
How many small cubes will be required to make a 1 by 1 by 1
cube and a 2 by 2 by
2 cube?
How many small cubes will be required to make a 1 by 1 by 1
cube and a 2 by 2 by
2 cube and a 3 by 3
by 3 cube?
...
Discuss anything that they have noticed, and ask for
predictions for constructing a set of all the cubes up to 10
by 10 by 10.
Are they surprised by the patterns they have noticed?
One way of representing this result is using the image
provided in the problem. Hand out copies of
this
worksheet, and ask students to spend some time thinking
about how the image relates to the formula. Encourage them to
use multilink cubes or draw diagrams to show how the sum of
cubes is represented in the image.
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 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.
Possible support
Try the problems
Picturing Triangle Numbers and
Picturing Square Numbers.