Why do this problem?
This problem offers opportunities for visualising, and for consolidating the formula for working out the volume of a cuboid, while at the same time challenging students to conjecture, test out ideas and compare different strategies for arriving at an optimum solution.
This printable worksheet may be useful: Cuboid Challenge.
"Visualise a 20 by 20 square sheet of paper. Now imagine cutting out a square of side 4cm from each corner, and folding up the flaps. Turn to your neighbour and check you agree what the dimensions of the resulting box will be. Try to do this without writing anything down or using your hands to gesture."
Bring the class together and confirm the dimensions, and the resulting volume.
Allow some time for the students to work in pairs to explore the effect of different sized cuts on the volume of the resulting box. Can they find the largest possible volume?
Collect the results and list them in an ordered table. Ask about any noticeable patterns or trends. What is the maximum volume that anyone has found? Can we improve on this?
There are different possibilities for this:
some students may use trial and improvement, possibly using calculators;
some students may draw up a spreadsheet and use trial and improvement;
some students may draw up a graph and look for maximum values.
Encourage some students to try each method so that you can bring the class together to compare and discuss their results.
Ask each pair of students to choose a different sized starting square and find the cut which produces the maximum volume.
Collect these on the board and ask for any comments, and what patterns they notice.
How can you be sure you have found the maximum volume?
Can you convince someone else?
Instead of starting with square sheets of paper, students may investigate rectangular ones. In order to make pattern spotting easier, you may wish to organise this this in some way, for example giving different groups of students sets of rectangles (such as rectangles where the length is twice the breadth, three times the breadth, four times the breadth etc.)
Offer students 20 x 20 square grids
and encourage them to make different sized boxes, working systematically and recording their results as they work.