Charlie has been exploring squares with vertices drawn on the points of a square dotty grid.

Unfortunately, he rubbed out some of his work and only left behind one side of each square.

**Can you recreate the squares he drew?
Is there more than one possibility?**

Could any line joining two points be the side of a square whose vertices lie on grid points?

How can you be sure?

Alison has been drawing squares and their diagonals. Here are some of the diagonals she drew:

**Can you recreate the squares she drew from her diagonals?
Is there more than one possibility?**

Can you find a method to draw a square when you are just given the diagonal?

Could any line joining two points be the diagonal of a square whose vertices lie on grid points?

Can you find a way to help Alison decide whether a given line could be the diagonal of such a square?

Charlie and Alison played around with rhombuses next.

Charlie said "Whenever I join two points to make a line, I can use my line as a side of several different rhombuses".

Do you agree with him?

When you are given a line, is there a quick way to work out how many rhombuses can be drawn using that line as one of the sides?

Alison said "When I draw a rhombus, it shares its diagonal with infinitely many other rhombuses."

Do you agree with her?

Not all lines can be the diagonal of a rhombus. Is there a quick way to decide which lines could be the diagonal of a rhombus?