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?

## Suggestions

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

Use some scrap A4 paper to try out your ideas.

Discuss your thoughts with someone else.

Draw a graph of your results.

Work systematically.

## Results

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: cm3 to three significant figures.

Check

Note that you do 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.

The solutions to this and other Transum puzzles, exercises and activities are available here when you are signed in to your Transum subscription account. If you do not yet have an account and you are a teacher, tutor or parent you can apply for one by completing the form on the Sign Up page.

## Extension

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.

Investigations Home

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.

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