I start by asking them to place some real numbers on a number line.
Then I ask them to think about the lengths of sides of different squares. We try several fractions and terminating decimals to try to find one that we can square and get a 2, and we are unable to find one.
So that's why people needed to invent irrational numbers: to solve this problem. We just define radical 2 to be the number that gets you a 2 when you multiply it by itself.
Then we read through this story. I have them read the slides popcorn style (reader of this slide chooses reader of the next slide.)
The story gets them up to: i is the number people had to invent because there aren't any real numbers we can square and get -1. And if i2 = -1, it stands to reason that we can define i as the square root of -1.
This is the best, most grabby part of the lesson: I put the number line back up, and say
So if i is a number...where do we put it?
Stop and wait and let the room be silent for a little while. They're considering things, and deciding against them. They sometimes suggest putting it at both 1 and -1, but of course they don't really know. So I say:
i isn't on the line. But it is on the board.
Then I carefully measure with the thumb and finger of one hand the distance between 0 and 1, turn my hand, and put i the same distance above 0. Then they can tell me where 2i and -i are located, and they can pretty much figure out where we should put complex numbers like 3+2i and -1 - 3i.
This lesson goes on to consider what we might mean by things like 5i + 6i, 2(4i), and 25i/5. Having the graph to refer to really helps. It sets us up nicely for powers of i tomorrow, too.
With all three groups today, there was a moment of "ick." "I don't like this." "This is weird." I tried hard to acknowledge and legitimize that feeling. I told them that feeling of discomfort is normal when you're making room in your mind for a brand new idea. I likened it to that saying "Pain is weakness leaving the body."
Except I said that weird feeling is ignorance leaving the brain. They seemed to like that.