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.

Feedback please?

A Letter to My Algebra 2 Students on Complex Numbers

I'm glad to see somebody else taking this approach. At our school the fall of 9th grade focuses on algebra skills, but the spring semester focuses on the connections between algebra & geometry. One of the big concepts for the semester is understanding the graph of an equation. A few years ago I taught this course when we had it tracked, and I had our our "strong" students. I included a unit looking at transforming points both geometrically and algebraically. They quickly figured out how addition could accomplish translation, multiplication by a positive number could dilate (or if we just multiplied one coordinate, stretch it vertically or horizontally), and multiplication of a coordinate by -1 could reflect over an axis. I asked what seems to be missing and they said rotation. I agreed and noted that once we could rotate we could also handle reflections and stretching in other directions. So I asked what rotations could we do with our tools at hand, and the figured out we could rotate 180 degrees by multiplying both coordinates by -1.

ReplyDeleteAt this point I did more explaining that I would have liked. (I wish I had them do more discovery as your approach does), but I said that since multiplication by a positive doesn't rotate, multiplication by -1 rotates 180 degrees, and multiplication by other negative numbers could be broken down as two steps (the absolute value gave dilation, and the sign gave the rotation) we should look for other "directions" to multiply by. If we rotate 180 degrees twice we get back to where we start which agrees with squaring -1 to get 1, but what would happen if we rotate 90 degrees CCW twice. So we should get a number (not on the number line, but on the number plane) that when squared gives us -1 (actually two such numbers since rotating 90 degrees CW twice is the same).

From there we worked back and forth with the algebra and geometry and were able to figure out how to rotate by other amounts using complex numbers. It was great to see the algebra rules working nicely with the geometric rules. My goal was not so much for the students to learn about complex numbers, but to give them practice with algebra and to explore the interplay between algebra and geometry and do it in a way that was new and interesting.

Since then we've eliminated tracking. Now all 9th grade students take the same class. I've been pushing to include the unit on complex numbers and transformations, but my department is concerned that it will be too difficult for some students. They have agreed to consider it, though, for spring of 2014 if I can present a detailed unified approach to the semester that includes complex numbers. I've looked for a textbook that takes this approach, and have not yet found one. I'm quite interested in your ideas and experiences with it.

They're doing this without having done trig? Wow! I love your materials.

ReplyDeleteI wrote a poem about imaginary numbers. It doesn't include this aspect, but I thought you might be interested. (I've revised it recently, to correct the historical references. I'll post the new version soon.)

Yeah, and we're doing Trig in a few months, so the investment seems worthwhile.

DeleteVery interesting – very creative – hadn’t seen this introduction before. I really like this idea a lot.

ReplyDeleteDid you define (informally) the transformation (a,b) in terms of a rotation & dilation?

I find using the ordered pair to represent the transformation that is applied to an ordered pair a bit confusing. Could you call a transformation T0,1 (subscripts) or something like that instead of (0,1)?

Did you present the transformations as well as the connection to i on the same day? That seems like a lot. I think I would spend at least a couple of days on the transformations first.

I might use decimal estimates and avoid special triangle properties when introducing the transformation idea to keep it as simple as possible. I might use some exaggerated examples to get the ideas across - describe approximately what each of the following do: T1,500 T500,1 T-500,1 T500,501 T.001,.1, compositions of some of these, etc. I might provide a protractors or superimpose one over a grid and ask for approximate descriptions for transformations like T1,4 or T1,4 composed with T-2,3 etc. I might ask about the inverse transformation.

Here is a nice application. Draw a right triangle with legs of length 3 & 1. Label the smaller angle alpha. Draw a right triangle with legs of length 4 & 1 and label the smaller angle beta. What are possible side lengths for a right triangle with one angle measuring alpha + beta? (3 + i)(4 + i) answers the question.

Did you create the list of "introductions to complex numbers" at that other site? If you are still interested in creating a problem set showing why complex roots must come in pairs, I think you could use the framework in your letter quite nicely.

ReplyDeleteTo understand why they must come in pairs, you must understand what makes real number coefficients special. Special relative to what? Well, special compared to more general complex coefficients.

You could have them solve some quadratic equations with complex number coefficients. If you like, you could instead view them as quadratic equations with "transformation coefficients" to be solve for unknown transformations. For instance, you'd want to have them solve "x^2 = (a,b)" for the unknown transformation x. Fortunately, it is much more intuitive that you rotate half as far and dilate by the square root of the dilation (a,b) (exponential functions at work!) than to set it up with complex numbers and solve a simultaneous pair of nonlinear equations. You might even introduce them to the name "de Moivre".

Once they learn how to take the square roots of transformations (or complex numbers), you can throw a quadratic equation with complex coefficients at them. Explore the quadratic formula to see if it even makes sense, and once they realize it does if they can take the square root of a complex number, solve one or two. Then ask what happens when you look at the same equation but where you've conjugated the coefficients. Draw pictures of some pairs of roots for one equation and the conjugated equation to see the reflection of the pair of roots occurring. Then ask how the pairs of roots should be related for an equation and its conjugate when it has real coefficients. If you can get this far, I think students might be able to figure out for themselves that if there is a complex root of a real quadratic equation, the conjugate must also be a root.

And this will open up nice philosophical discussions: is there any reason the quadratic formula should work with complex coefficients? Is it shocking that you don't need to invent a "new type of number" to solve certain quadratic equations with complex numbers? Can every problem about translations be solved using the convenient algebraic notation and formalism of complex numbers and then "translated" back to the land of translations? If this were true, would there be an ontological difference between "translations" and "complex numbers"? When rotating in 3D, is there a corresponding "number" that behaves like rotations? etc. etc.

ReplyDeleteI like this a lot. Thanks!

DeleteI prefer starting with some historical/cultural background - inspired via http://www.youtube.com/watch?v=BDIv7r-X2kk&list=PL8C46A6764A8A21CD&index=6

ReplyDeleteThe historical background is fine, but not up my alley.

DeleteI start with letting kids explore the geometry by themselves.... http://untilnextstop.blogspot.de/2012/11/visualizing-complex-operations.html

ReplyDeleteI like this. I did this last year. And there was a cool "aha!" moment during that lesson, but it still felt unsatisfying to me.

DeleteI want my kids to find it to be both stunning and natural that multiplication by i represents a rotation of 90 degrees. Letting the kids just jump into the geometry satisfies my "stunning" requirement, but where does the conversation go from there? Why *should* i^2 = -1?

This intro (which I just finished with one of my Alg2 classes) is my attempt to plant the seeds so that the fact that i^2 = -1 also comes across as natural.

I'm pretty happy, though it was a bit rocky. The rockiness mostly had to do with the difficulty of some of the prerequisite ideas. Kids need to be up on rotations, dilations, the distance formula, multiplying binomials, and special right triangles. (Thankfully, though, those are all things that I need to teach anyway, so it's a well-connected topic.)

More blogging on this to come. There are lots of ways to open up complex numbers, but I think focusing on the weird connection between transformations and algebra is the way to go.