After finding the formula for the circumference of a circle, my 6th graders were ready to work on finding the area of a circle.
I asked them to draw a circle on notebook paper, any size, but not too small. Then I gave each a centimeter cube to trace one face onto their paper to remind them how much the area of one square centimeter covered. (You can see it drawn on top right pic below.)
Question 1: Give me a guess — only by looking — what is the area of your circle, in square centimeters? Please write that number on your paper, label it “guess.”
Question 2: Now use whatever tools you need, give me a better answer. I know some of you already know the formula for the area of a circle, but you may not use it unless you can tell me where it came from. (No one even tried.)
I can’t tell you how happy I was to see all the different ways the the kids had tried to approximate area. Their perseverance humbled me. A few students drew triangles, knew to use “base times height divided by two,” but erroneously used radius as height.
I had the kids fold their circles like this.
They cut out the pieces — turning every other piece 180 degrees — and glued them together. Cristian said, “First we had a circle, then triangles, then a rectangle. That’s crazy!”
Never once did I answer any of the questions. I just asked them. I began with, “What is the area of this rectangle or parallelogram?” Step by step, as a class, the kids walked this equation all the way to Area = pi x radius^2. :-)
This morning, two days after the activity, I did a My Favorite No to see if they remembered: 31 out of 33 students got the correct circumference formula; 24 of 33 got the correct area formula. Bonus if they showed me how the rectangle model helped explain the area of a circle.