*i*.

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.)

They enjoy the story, except sometimes they make comments about how John and Betty are freakishly precocious, and sometimes they wonder what is up with Betty's hair. We don't read the whole thing. Just up to where they have to invent

*i*.

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

*i*

^{2}= -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 2

*i*and

*-i*are located, and they can pretty much figure out where we should put complex numbers like 3+2

*i*and -1 - 3

*i*.

This lesson goes on to consider what we might mean by things like 5

*i*+ 6

*i*, 2(4

*i*), and 25

*i*/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.