I've been thinking a lot about complex numbers over the past few weeks. I wasn't happy with any of the available introductions to complex numbers. I wanted to put transformations of the plane at the center of it all.
I've been experimenting in the classroom. I started by defining (0, 1) as a transformation as the unique rotation that takes (1, 0) to (0, 1). In other words, a 90 degree rotation. By similar reasoning, (-1, 0) is an 180 degree rotation. And then I asked kids to figure out what (0, 1) applied to (4, 1) would be. (They studied rotations in Geometry, and didn't need much review.)
Then we generalized further: Let (a, b) be the unique transformation that maps (1, 0) to (a, b). It's pretty intuitive that you can accomplish this in the plane with just a rotation and a dilation.
That's where things got interesting for us. This approach is still very much a work in progress, and my Thursday lesson flubbed. It's annoying, because I've got a bunch of kids going across the Atlantic on an exchange program and I didn't get to wrap things up for them. So I wrote this letter to tie together the loose ends.
I want to write more about this later, but at the moment this stands as my clearest statement of how I want to approach introducing complex numbers to kids.
A Letter to My Algebra 2 Students on Complex Numbers