Prior to this lesson we had discussed the nature of a circle and reviewed its parts.
1. We rediscussed the formula for finding the area of a decagon. A=p x a/2
2. We worked out the problem for the geometric cabinet’s decagon. We talked carefully about the perimeter. (It was great that AR was observing the lesson; the older guys were better at explaining it carefully with a younger listener.)
3. I handed the older guys the circle and said how will we measure its area? The same way they said. AV was the first to realize there is not apothem in the language governing the circle. JV made the transition to radius. Then everyone thought it was good. BR and DW could tell by my face that this was going to get interesting. BR began looking around – on high alert. I handed LR the circle and asked him to find the perimeter. JV picked up the ruler. Difficulty ensued.
4. AR sidled up next to me and reminded me about the Sir Cumference book. Wouldn’t a string work? I said that it would but I had another way too. I took a ruler and drew a line on the chalkboard. I took the circle and made a mark with the chalk on the circle. I drew a beginning line on the chalk line and began to roll the circle until it came around to the chalk line again; I marked this. How big is the perimeter of the circle? BR snatched the ruler and measured. This was perplexing because it was larger than they thought it should be. So I did it again. Same distance. Then we discussed the word for perimeter is only for polygons. Circles have their own language. It is a circumference.
5. I then said, “hmmmmm, I wonder something.” I took the circle and laid it down at the staring point. I marked the diameter of the circle. I repeated myself three times. But there was a little left over. We tried a different sized circle beginning with measuring the circumference. Same little bit left over.
6. I took a compass up and measured the little bit and then wondered home many time the compass measurement would fit into one of the diameters of the circle. 7 times. Ahhhhh. so the left over it 1/7th of the diameter of the circle. So the whole circumference is 3 and 1/7 diameters. Ohhhhh. About this point AV begins to put the pieces together. He gets very still. I make it an improper fraction 22/7. He blurts out. That’s pi.
7. I divide it out and see that 1/7 is .142….. The kids really were having a hard time believing that pi was not magically derived, but rather was an observation by the Greeks. They named the universal thing peripharein; the first letter is Pi. This is why we call it pi.
8. Of course now we HAD to measure things and find their circumference. So off we went.
That was all for that day.
The next day.
9. We began again. All we had found in our hunt for the area of a circle had been how to find the circumference. We need to find the rest.
10. We wrote down the decagon’s formula again to guide us. Then we drew a circle and began to replace the terms for the decagon with circle language. A = c x r/2. JV mentioned that this was not the formula he had seen for the area of the circle. I agreed. But we all agreed that it was the right way of saying it.
11. HHHHmmmmmm. So we decided that we could replace the “c” with the parts of the formula. A= pi x d x r/2. Hmmmm.
12. Could we combine things now because diameter and radius were related? Hmmm – yes. D = 2r. Ok our new formula says: A = pi x 2r x r/2. After some math review here we realized that we couldn’t make it 3r. I was a little disappointed that AV didn’t realize the 2’s could cancel each other out.
13. I made that transition. Now A = pi x r x r. Ohhhh. BR, whose been doing quite a bit of cubing work, saw the square immediately. Now it says pi x r squared.
14. Off they went to find their papers from the previous day and now find the area.
PS: There is another layer to the proof. I did not do this proof. If anyone has photos of the layout, I would appreciate seeing them.