Time to take Pythagorean Theorem and the idea of square roots to a new level.
I love the patterns discovered so clearly while working on Theodorus' Wheel in class.
After they make the wheel, I ask them to imagine them as part of an art project. They make hats, baby carriages, snail shells, Princess Lea's hairdo etc.
Here is what Theodorus' Wheel is:
The Wheel Of Theodorus, or Square Root Spiral, is a rather simple geometric construction that allows you to construct line segments with length of the square root of any integer. Observe the figure below.
The construction is fairly simple. Starting with a 1-1-root 2 right triangle, use the hypotenuse as one leg, and add a perpendicular segment of length 1 to create another right triangle. Pythathagoras says that the resulting triangle has a hypotenuse of root 3. And so it goes. And it creates this very nice spiral effect. The main point is that complex numbers provide a very convenient and very elegant way to express the geometric relationships in the figure.