This document discusses calculating the areas of squares that are tilted or oriented at an angle on a grid, rather than being perfectly horizontal or vertical. It presents some example tilted squares with different tilt values and asks the reader to look for patterns in their areas in order to make conjectures about a formula for the areas of tilted squares generally that could be proven.

1. Tilted Squares' printed from http://nrich.maths.org/
It's easy to work out the areas of squares drawn on a grid if they are oriented in the
usual way:
I wonder if you can find a quick and easy method to work out the areas of tilted
squares.
Here are some squares with a tilt of 1:
See the hint for suggested ways to calculate their areas.
Notice anything special about their areas?
Can you predict the areas of other squares with a tilt of 1?
What about squares with a tilt of 2? Or 3? Or 4? Or...?
Notice anything interesting?
Can you make any conjectures about the areas of tilted squares?
Can you prove your conjectures?