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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?
Geoboards with one square