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If you had 36 cubes, what different cuboids could you make?

The Big Cheese

Investigate the area of 'slices' cut off this cube of cheese. What would happen if you had different-sized block of cheese to start with?

Making Boxes

Cut differently-sized square corners from a square piece of paper to make boxes without lids. Do they all have the same volume?

More Christmas Boxes

Why do this problem?

This well known challenge is one which gives pupils opportunities to further their understanding of volume and the factors which affect size. It covers that part of understanding where common sense may not be sufficient.

Possible approach

It would be good, after introducing the challenge, to allow those who wanted to cut out squared paper to help with the exploration.

Key questions

How did you find the number of cubes?
Is there any way in which you could change this challenge?

Possible extension

When exploring the volume - as distinct from the number of cubes - what about cutting a square $1.5$ by $1.5$ from each corner?
Suppose the $10$ by $10$ sheet had two smaller sheets cut from it? What would the total volume of the two boxes be?

Possible support

Some pupils will need assistance to cut off the squares accurately.