For my Algebra 2 final exam I wanted to give students a chance to experience putting ideas together to learn something new. I also didn't want them to freak out. So I gave them a short intro to the Mandelbrot set (that was basically ripped off from this article).
A bunch of students thanked me for it, which was nice. They thought it was cool, and not boring. I sent them the link to the Jonathan Coulton song after the final:
Good news: Lots of them were able to tell if a complex number is in the Mandelbrot set and used previous knowledge (function notation, composition of functions and multiplying complex numbers) to learn something new. It also made for a more interesting final.
I wonder how this would work as a full fleshed-out lesson next year. I'm not sure that the Mandelbrot Set is really helpful for leading students to create the math that they ought to be learning. Meaning, I'm not sure how much would be gained from showing them the graph of the set and seeing what questions interest them. Anyway, here's the doc: