Yesterday I had my younger son (in 7th grade) read chapter 1 in Steven Strogatz’s new book Infinite Powers and we had a fun time talking about what he learned:
Using Steven Strogatz’s Infinite Powers with a 7th grader
Today I wanted to show him a hand waving overview of two of the more well-known ideas from calculus – finding tangent lines and finding areas under a curve.
I started with the tangent line problem:
He was struggling to remember some of the basic ideas about lines, so I broke the talk about tangent lines into two pieces to let him take his time remembering how to describe lines. Here’s the second part of the discussion:
With the tangent line discussion finished, we moved on to finding the area under a curve. To keep things simple, I stuck with the same function:
Mike's Math Page 
hi
perhaps you could discuss w him the tangent to a cycloid or the under a cycloid. there are plenty of ways of doing these, generally with nice geometric ideas, as the guys in the 16 hundreeds did it.