Complex Numbers

I wanted to do something cool with complex numbers- basic operations are rather boring, especially since it was review from last year. So, I thought about fractals and decided to have students practice magnitude, adding and multiplying complex numbers in the context of the Mandelbrot set. It's a simple iteration that makes a pretty picture, seemed perfect to me!

I had a bit of a false start because I assumed students had seen iterations before or at least used subscripts to name terms in a sequence. False assumption! After some back tracking and translating, they understood the methodology. I did a few examples with them first, asking them to predict what types of results could occur.  After exploring some key points I let them in on the magnitude greater than 2 trick.  We also listened to the Mandelbrot song (beware- make sure to use a clean version) which both provides additional information and gives the formula in the form of a catchy tune!

Between the two classes they managed to check 81 points, which seems like a lot, but doesn't actually make the most spectacular picture. If I do this again I will assign a set number of points to each student (10?) and give them time to work in class but also expect some work at home. Then we would get a more filled in picture.

Or, I could have each student choose their own iterating function, playing around on a fractal app until they got a picture they liked. They could then check 10 points by hand to show what the different colors mean. If I used a color printer these could be nice posters.

Even if it wasn't beautiful math art this year, it gave some motivation for operations with complex numbers and they got in a good amount of practice.

Following fractals we solved quadratics using Sam's awesome scale of efficiency. I moved quickly to complex roots of quadratics and then into polynomials of higher degree when students showed they had mastery of quadratics. Students enjoyed making up challenging problems for each other to solve. Asking them to make up problems was a great way to figure out/reinforce that a polynomial with real coefficients will always have imaginary roots in conjugate pairs.

The final challenge was coming up with an assessment. I decided not to have students solve a quintic with complex roots on the test but instead had several questions which focused on one step of the process at a time. I'm still undecided if that was the right choice. Students generally did well on the exam and it was easier for me to pinpoint their errors with shorter problems. It also gave students the opportunity to show me that they knew how to do a later step even if they messed up an earlier one because they were separate problems. As I'm typing this I am realizing that the best thing to do would have been a scratch off hint bank! Then students could scratch off a hint if need be and continue the problem to show mastery of later steps. (The benefit of scratch off hints is I would know who used the hints.) So many good ideas for next year, I really hope I get to teach the course again and remember to look back at this post!