Learning and Teaching Project
Area and Perimeter
The task began with a homework problem. It is taken from Reconceptualizing Mathematics: Courseware for Elementary and Middle School Teachers, Center for Research in Mathematics and Science Education, 1998.

Is there a relationship between the area and the perimeter of a polygonal shape made with congruent square regions?  (For fixed area, find the minimum and maximum perimeter. For fixed perimeter, find the minimum and maximum area.) Squares must be joined complete-side to complete-side. The outside “boundary” should be a polygon. In particular, this would not permit a shape with a “hole” in the middle.