Why do this problem?
is one that concentrates on the length of the edges of cuboids rather than on the faces or vertices. It can help children to understand the nets of 3D shapes. It is a good precursor to lessons on finding the volume of cuboids.
You could introduce this investigation with the 2D rectangles as is done in the text of the problem. Alternatively, you could start by showing the group two cuboids that you have made up beforehand. Have the nets available so the children can see how they are folded into the cuboid. This sheet
of two nets might be useful. (You will need two copies
to show both the nets and the made-up cuboids.)
When you have discussed the problem and what needs to be done children could work in pairs so that they are able to talk through their ideas with a partner. Supply $1$ cm$^2$ paper, scissors and sticky tape for making the models.
When introducing the third length, point out that the work done beforehand has not been wasted - the cuboids already made are just part of a now enlarged family. Some children may notice this for themselves.
Making a list or table is an important and useful way of checking that all possibilities have been included. This could be done at the end of the lesson. Show the group how to do this methodically so that none are included twice, such as $3 \times4 \times2$ and $2 \times4 \times3$. One way to do this is to list in order of the size of the numbers so that the above example will always be
recorded as $2 \times3 \times4$.
Later, carefully constructed model cuboids along with their nets could be made for display.
The cuboids can also be sketched on isometric dotty paper.
Which would be a good cuboid to start on?
Where do the two rectangles that are the same size go on the net of the cuboid?
Learners could investigate what how many cuboids will be possible when another length is added. Can they predict what would happen for five lengths?
$1$ length - $1$ cuboid
$2$ lengths - $4$ cuboids
$3$ lengths - $?$ cuboids
$4$ lengths - $?$ cuboids
Some children might find it easier to construct cuboids out of drinking straws and plasticine, or special construction kits.