Time for some detective work! Here is an investigation for you.
Maxvoltray
A tray is to be made from a special sheet of paper which is 20cm 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 some scrap A4 paper to try out your ideas.
Discuss your thoughts with someone else.
Work systematically.
Use a spreadsheet to record your working.
Use algebra, let the side of the cut out square be x.
Draw a graph of your results.
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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.
Younger 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.