My younger son and I started talking about circles this morning. Even before the first example problem in our Prealgebra book there’s a sketch of the proof that the area of a circle is Seeing that proof, I closed the book and decided to walk through it with my son for today’s project.
To start I asked him what he already knew about circles – turns out that he’s heard a lot about circles. He even thinks to find the area of a circle by chopping it up into pieces:
In the next part we got a little sidetracked by some algebra. That’s fine. Instead of putting all of the focus on the geometry proof, I decided to take a little detour to try to make the algebra a bit more clear. I suspect that my son’s algebra mistakes here are pretty common mistakes. Hopefully we’ll get through some of these difficulties with a bit more practice.
Now back to the geometry. We chop up the circle into 16 pieces now and try to rearrange the shapes into a (near) rectangle. Our drawings probably aren’t exactly right, but at least you get the idea that the shape is starting to look more and more like a rectangle. With the extra algebra practice from the last video, the formula for the area of a circle now falls into place.
Maybe the next project is comparing the famous “proof” that and seeing if he can see where things are going wrong. For today, though, a nice little proof and hopefully some valuable algebra practice. Fun morning.