Yesterday we had about a 30 min drive and I had the boys open up to a random page in this book for a few short discussions in the car:
There were some fun topics that were accessible for kids, but then Newton’s method came up. Ha ha – not really drive time talk 🙂
It did seem like it could be a fun project, though, so I took a crack at it today. The goal was not computation, but mainly just the geometric ideas. Here’s how we got started:
Next I asked the boys if they could find situations in which Newton’s method wouldn’t work as nicely as it did in the first video. They were able to identify a few potential problems:
Now I had both kids draw their own picture to play out what would happen when you used Newton’s method to find roots. I think there’s a lot of ways to used the exercise here to help older kids understand ideas about tangent lines and function generally. I mostly let the kids play around here, though, and the results were actually pretty fun:
Finally, we went to Mathematica to see some situations in which Newton’s method produces some amazing pictures. Here we switch from real-valued functions to complex valued functions. Since I wasn’t going into the details of now Newton’s method works, rather than using some easier to understand code, I just borrowed some existing code from here:
The boys were amazed by the pictures. For example, (and this is one we looked at with the camera off) here’s a picture showing which root Newton’s method converges to depending on where you start for the function 
Definitely a fun project. Even if the computational details are a bit out of reach, it is fun to share ideas like this with kids every now and then.
Mike's Math Page 