After this year of teaching the material, I've come to the conclusion that I need to do 2 D figures such as triangles and rectangles first and include both circumference and area of circles. This might make the transition into volume and surface area for 3 dimensional figures easier.
I hate to admit I made a mistake when I did not take into account the best order to teach this material. Ohh well, now to look at how to help reteach the material. I think I'm going to spend time introducing circles, the formula for area and circumference.
|Picture 1 - square|
First you cut the piece of paper into a square since a circle is x^2 + y^2 = r^2 and the r is the radius or the distance from the center of the square to the edge. So you cut a piece of paper that is exactly the length from the center of the square to the edge after folding the square so there are four lines creased into it. (Picture 1)
|Picture 2 - With first set of cuts|
|Picture - 3 With second set of cuts|
This is based on Archimedes theorem that if you have a polynomial with enough sides, you end up with a circle. Out of all the years, I've done this, only one or two students figured it out. Usually after 10 minutes of letting them struggle, I sit over at the side and slowly work through the process so students can wonder over and check things out.
I love challenging them. Give it a try.