Time for some detective work!

A tray is to be made from a special sheet of paper which is 30cm by 21cm.

Squares will be cut from each of the corners of the paper and the edges folded up to form the tray.

What should be the size of the cut out squares if the tray is to have the largest possible volume?

Use algebra, let the side of the cut out square be x.

Use some scrap A4 paper to try out your ideas.

Use a spreadsheet to record your working.

Discuss your thoughts with someone else.

Draw a graph of your results.

Work systematically.

Did you find the maximum volume of the tray?

You can claim a Transum virtual trophy if you got the correct answers:

Length of sides of cut-out squares: | cm to three significant figures. |

Maximum volume of tray: | cm^{3}
to three significant figures. |

Note that you do not have to get the answer right to prove you have done a good investigation. Your teacher is the best person to give you feedback on your work.

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Extend your investigation to include sheets of paper of different sizes.

What if the tray needed a lid?

Think about the surface area, the minimum volume or other shape trays.

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A mathematical investigation is quite different to other mathematical activities. The best investigations are open ended and allow students to choose the way they work and how they record their findings. It is one of the few occasions when 'going off on a tangent' is not only acceptable but actively encouraged (within reason).

Students may ask for 'the answers' but this supposes that the activity is
closed. Investigations can always be extended by varying the initial
instructions or asking the question 'what if...?'. Sometimes students point out
that the instructions are ambiguous and can be interpreted in different ways.
This is fine and the students are encouraged to explain how they interpreted the
instructions in their report.

Some students may benefit from a writing frame when producing the reports
of their investigations. Teachers may suggest sections or headings such as
Introduction, Interpretation, Research, Working and Conclusion or something
similar.