Still catching up - this was today's lesson...
This lesson's scaffolding is based off of some ideas in the book Where Mathematics Comes From, by George Lakoff. I found this book to be fascinating, as it speaks directly to the question of whether math is a human discovery or a human invention. The book is firmly on the side of invention, explaining how a set of basic "grounding metaphors" create a foundation on which we've built the rest of mathematics. If you're interested in these ideas, I highly recommend reading it. Whether or not you agree with the conclusions, it'll really get you thinking.
In the book, they discuss the metaphor that is used to understand where the imaginary number i comes from. Essentially, you can think of multiplying by -1 on the real number line as a 180 degree rotation. Once you accept this metaphor, then it opens the question: multiplying by what number gives you a 90 degree rotation? If you multiply by that number twice, that means you've rotated 90 degrees twice. Since this is equal to 180 degrees, multiplying by the mystery number twice gives the same result as multiplying by -1. Therefore, the mystery number must be the square root of -1. Whoa. I thought that was an interesting way of looking at it. It allows you to then see why the imaginary axis is perpendicular to the real axis. It also allows you to see why the powers of i follow that 4 number pattern which students need to know. You can go around in a circle, stopping every 90 degrees at an axis. If you start from 1 on the real axis, you'll pass through 1, i, -1, -i, then back to 1. Also, this had an added benefit: one student asked what would happen if you rotate 180 degrees twice. He realized that this would be the same as multiplying by -1 twice. When I pointed out that this rotation would get you back to where you started from, I think his jaw dropped a bit - he may have finally understood a reason that explains why a "negative times a negative is a positive"!
So I designed a handout for students to work through these ideas, we discussed them, and I finished with about 20 minutes of lecture to tie it all together and do some exaxmples. I love this lesson, because I feel like it allows for some deep learning, and the mathematics are not that tough, so almost all students can follow it pretty easily.
This lesson will be posted on ILoveMath.