 How much fruit is in that slice of fruit? In other words, what is the area of a polygon? If the polygon is drawn on a grid, how can you find the area by counting the grid points? In this session we drew a wide variety of different polygons, and each time we computed the number of grid points on the boundary, the number of grid points in the interior, and the area of the polygon. Then we asked the questions:

• Draw as many polygons as you can with zero grid points in the interior. Do you notice a pattern?
• Now do the same with one grid point in the interior.
• Finally do the same with two grid points in the interior.
• Is there a pattern in the data? Can you describe all the patterns in a single formula?
• Why does your formula work? First start with a simple rectangle and argue that it always works.
• Argue the formula works for the next simplest shape: a right triangle
• How can you adapt this argument to work for more complicated shapes?
• Try the formula on as huge and complicated a polygon as you can!