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Geoboards
You may like to use the
Geoboard interactivity to investigate this problem. To get a square board, just click the small circle of dots to the left of the letters.
On a $4$ by $4$ geoboard (say)  how many different sized squares can you make using rubber bands?
How could you make a square with NO pins along a side (an edge) and just the 4 pins at the corners (vertices)?
The basic unit of measurement is one square unit (as shaded in the diagram).
How can you make a square whose area is 2 square units?
Can you make a square with an area of 3 square units? 

This problem is written to stimulate exploration on a geoboard and the hope is that you and your pupils will find further questions to investigate.
The problem is a nice way of raising pupils' awareness of "tilted" squares on a grid and of challenging the frequentlyheard cry of "but that's a diamond"! It would be interesting to set children off on tackling this problem, perhaps in pairs, and to listen to the discussion which follows. If, after a short time, noone has found a tilted square, you could cut out a square from paper, and, after
ascertaining from the class what shape it is, you could stick it on the board in different orientations, asking each time whether the shape has changed.
Having found all the different squares, investigating their areas is a natural followup which is made accessible by the grid itself.