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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
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.