Here’s a neat visual proof that builds on the formula for circumference of a circle to intuitively derive the formula for the area of a circle.

It visualizes a solid disc (the filled-in circle) as a bunch of concentric circles, each of which can then be cut and straightened out, forming a triangle. The area of a triangle is well-known to be half of the area of a rectangle, or $\frac{1}{2}bh$, so substituting in $b = 2 \pi r$ and $h = r$, we have:

This visualization hints at a different lens for viewing the world, through which you see a whole, static object being built up over many slices of time from many smaller parts. I.e., the lens of calculus.

Isaac and I went through this with pencil-and-paper, as well as physically cutting up some rubber bands. He seemed to understand the idea well enough.

I think we can expose kids to interesting ideas early on, as long as we take an intuitive approach, without getting too lost in a forest of abstract symbols. This approach introduces the larger theme that math is about exploring relationships, building up and discussing arguments about why things are, as opposed to being just about computing numbers so you can “get the right answer”, so someone will give you a good grade.