We've been working on perimeter and area lately and I usually find it to be fairly dull. This year has been a pleasant switch. I gave each student a set of tangrams (punched out from a file folder so that they hold up pretty well). Using the medium sized triangle and the two little triangles they had to create five regular polygons. This step alone made for a pretty good discussion.
Once they had created various polygons (triangle, square, rectangle, parallelogram, and trapezoid) they were to find the area and perimeter of each of them. I was surprised at how difficult it was for them to realize that all five shapes had the same area. They worked really hard to calculate the area for each one based on their measurements. It was fascinating.
After giving them some time to struggle with this we came back together as a group. We shared the different shapes because some students hadn't been able to figure out all five. Then we moved into looking at all of this algebraically. We compared the sides of each triangle and figured out that there are only 3 different lengths, the base of the larger triangle, the sides of the larger triangle or base of the smaller ones, and the sides of the smaller triangles. Using that knowledge we wrote one shape's perimeter as an algebraic expression. It was a challenge for them to write the rest that way, but they worked through it. This also helped them see that all five areas are the same. Then we finally had a discussion of the relationship between perimeter and are (the main goal, initially). Most students firmly believed that no shape could have a larger area than perimeter. This activity had shown them that the perimeter and area aren't always the same (shapes with the same perimeter can have different areas and vice versa). As a result, I immediately opened up a grid on our smartboard and had students try some different shapes and calculate the area and perimeter. They quickly found shapes that disproved their theory. Off they went, back to their seats with grid paper to see what else they could discover. They loved exploring this way (both with the tangrams and with the grid paper) and, I think, have a much better understanding of area and perimeter than my students have had in the past.