### Polydron

This activity investigates how you might make squares and pentominoes from Polydron.

### Tiles on a Patio

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

### My New Patio

What is the smallest number of tiles needed to tile this patio? Can you investigate patios of different sizes?

# Geoboards

## 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 frequently-heard 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, no-one 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 follow-up which is made accessible by the grid itself.