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Summing Squares

Age 14 to 16
Challenge Level

Why do this problem?

This problem offers an opportunity to visualise in three dimensions and gives practice in working with sequences that do not grow linearly. It provides a possible introduction to the formula for the sum of the first $n$ square numbers. By considering how the cuboids grow from the $n^{th}$ to the $(n+1)^{th}$ the foundations are laid for learning about proof by induction.

Possible approach

Set the scene for the problem - we are building up cuboids from a sequence of square prisms, adding on six square prisms each time.

Show an image of the first 3 by 2 by 1 cuboid and the second 5 by 3 by 2 cuboid. Give the students time to consider the first problem - is it possible to make the second cuboid by adding the six blue square prisms to it, without splitting any of them? Different people visualise this in different ways so if isometric paper and cubes are available, some students may wish to use them to share their ideas.

Allow plenty of time for students to share in pairs and then with the rest of the class their justifications for why at least one of the square prisms needs to be split. Encourage those who can do it by only splitting one to share their methods.

Next, set the problem of continuing the sequence by adding on the six pink square prisms. There is an image in the hint showing the new cuboid formed. Again, the question is whether any square prisms need to be split, and if so, how many. Explain that they are trying to find a way to describe how to assemble the next cuboid when six new pieces are added on, and give the students time to investigate this. Encourage them to record their thinking in a way that captures the way they "see" it.

Pause to reflect on what has been discovered so far:
$$6 \times 1^2 = 3 \times2 \times1$$
$$6\times(1^2+2^2) = 5\times3\times2$$
$$6\times(1^2+2^2+3^2) = 7\times4\times3$$

Ask them to predict how to continue this sequence, making reference to the cuboids involved.

Once students have a consistent way of making the next cuboid by adding on six square prisms, work can be done to express the dimensions and the volume of the $n^{th}$ cuboid. Small groups could produce a diagram and explanation to show how they would add pieces on to make the $(n+1)^{th}$ cuboid.

Finally, once a formula for the volume of the $n^{th}$ cuboid has been expressed, students can consider the relationship between what they have found and the sum of the first $n$ square numbers.

Key questions

Can you make the second cuboid by adding the six blue square prisms? Will you need to split any?
Can you visualise a way of making the third cuboid from the second, by adding the six pink prisms?
Can you visualise a way of making any cuboid from the previous one in this way?
Are you sure you will always be able to make the next cuboid?

Possible extension

Students could read this article about proof by induction.
Sums of Powers shows another way of considering the sum of the first $n$ square numbers.

Possible support

Lots of models of cuboids made from small cubes and lots of diagrams annotated with dimensions can help students to build up a picture of what is going on in this problem.

Try the problems Picturing Triangle Numbers and Picture Story for other tasks which think about sums of series in a very visual way.