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This practical challenge invites you to investigate the different squares you can make on a square geoboard or pegboard.

Tiles on a Patio

How many ways can you find of tiling the square patio, using square tiles of different sizes?


Place four pebbles on the sand in the form of a square. Keep adding as few pebbles as necessary to double the area. How many extra pebbles are added each time?

Wrapping Presents

Age 7 to 11
Challenge Level

Wrapping Presents


Choose a box and work out the smallest rectangle of paper needed to wrap it so that it's completely covered.

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Why do this problem?


This activity is excellent for introducing children to both surface area and to nets. It opens up the whole idea of nets that are not necessarily the ones that you would find in text books.


Possible approach


To give more purpose to this investigation, you might like to show the children some special colourful wrapping paper to be used for preparing some boxes to be used in a display or some other appropriate purpose. Explain that because there is so little of this paper, they will first need to find out exactly how much is needed to wrap each box.

This activity is suitable for working in pairs (which of course will require less special paper at the end). Supply the children with lots of newspaper and a selection of boxes to choose from. Spend a little time in comparing the boxes. Ask them to predict which box they think will need the most wrapping paper, the least paper, two that might need the same amount etc. Encourage them to experiment with wrapping the boxes in various ways and to cut off parts of the sheet of newspaper, keeping it rectangular. Have the pairs demonstrate their solution to others and compare the shapes and areas of their papers. Discuss the predictions they made earlier. The piece of newspaper can then be used as a template for cutting the same rectangle from the special wrapping paper. Perhaps the children could first try to fit the templates together over the special paper in such a way that little is wasted.

Key questions

Tell me about the paper shape you have.
Explain to me how each face of the box is covered.


Possible extension


An extension to the investigation would be to tie ribbons (or join bands of paper) around the parcels, then take them off and compare the lengths. First, ask the children to find a parcel they thought would use the same length of ribbon, one that would use less, the one that would take the longest piece etc.

For more extension work

Go to the Auditorium steps here.

Possible support

Some pupils may need to simply cut out rectangles to match each face of the box and then experiment in ways of fixing them together so that they work.