[Math] Lecture on Fractals for Middle School Students

Question

I'm going to have a one-hour lecture for middle school students next Monday. It will be about fractals. The students know virtually nothing about this subject.
I'll show some fractal images and a few short films which I've found on youtube, discuss on the ways a fractal can be constructed, and introduce a software. But I'll need other things, too:

1. Some serious mathematical content.
2. Some questions to propose to the students for further study.
   Could you please help me with these?

Thanks in advance.

Answer 1

Your task is both a challenge and an opportunity: they will be unfamiliar with complex numbers, but perhaps you could motivate the utility of complex numbers. I might try to introduce them to the computation of a Julia set, at first entirely computationally, showing them how $z$ grows under repeated computation of znew = zold² + c, all in terms of coordinates and distance from the origin (without mentioning complex numbers). They need not know any programming language to understand a simple iterative loop. Once they see how some starting points $z$ scoot off to infinity, and others hang around the origin, they can appreciate it would be natural to color each point according to its scooting-to-$\infty$ speed. And then they could understand how to make a Julia set:
     
     _{(Image from cgtutor)}

With this understanding secured, you might be able to introduce complex numbers.

For motivating applications, you could easily connect to the use of fractals in computer graphics in movies (Lord of the Rings; The Hobbit, etc.):
  
  _{(Image from LifeInWireframe)}