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Is There a Theorem?

Draw a square. A second square of the same size slides around the first always maintaining contact and keeping the same orientation. How far does the dot travel?

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Pick's Theorem

Stage: 3 Challenge Level: Challenge Level:1

* How many different figures can be described as $(4, 0)$?

* What do you notice about $(4,0)$ figures?

* Choose another particular value for $(p,i)$ and explore different shapes.

* Have you tried drawing shapes with the same area?

* What do you notice about those figures whose areas are the same?

* What ways are there of increasing the area by $1$ unit?

* Draw more figures; tabulate the information about their perimeter points ($p$), interior points ($i$) and their areas ($A$).