It's my first time teaching circle formulas. Teaching circumference was easy: First, we went around the room and measured a bunch of circular objects, recording both their diameters and circumferences. Then, we put the values on the board in a table and calculated the ratios between the measurements ("how many times bigger?") and observed that it was always just over 3. Then, we practiced estimating circumferences given diameters. Then the kids learned that the exact ratio is pi, and they practiced finding circumferences, both exact (in terms of pi) and calculator approximations, when a diagram shows them only the radius.
Then... I gave them this. It's probably not special, but it worked well for us!
1. What is the area of the shaded square?
2. What is the approximate area of the 1/4 slice of the circle? How did you get this?
3. What is the approximate area of the whole circle? How did you get this?
After individual attempts and a brief whole-class discussion, they then practiced estimating areas of various other circles using the same technique. (On their paper, again, so they can each think at their own paces.)
We have not yet discussed the circle area formula. I am insistent on this. Area formula has to come after estimation, so that they can remember why it makes sense that (3/4 * r^2)(4 slices) leaves you with a circular area of approximately 3*r^2, or pi*r^2 to be exact.
Yay 7th-graders for being good sports and being open to a world where we develop formulas together!