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Thursday, September 30, 2010

A Whole New Kind of Number

I think I've finally got my introduction to imaginary numbers and the complex plane to a point where, let's say, the students can make room in their brain for the idea. They still don't like it, but they leave with some sense of what the heck we mean when we say 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 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.

Monday, September 20, 2010

Radical Comes from Radix Which Means Root in Latin*

Factoid from the title care of Justin Lanier (which, Justin factoid, is pronounced the opposite of "La-Far.") It is unclear if radix has anything to do with radishes. Radishes are root vegetables, right? I'm thinking yeah.

So anyway I thought I'd throw this out there and see if anyone's got anything better. My irrational and complex numbers unit is pretty anemic. The way we have it calendared, we also kind of have to race through it. This intro lesson is my effort to give them something to grab onto. I'm open to suggestions.

The first three are new. The last two are what I used last year (I tried it as a puzzle - see if you can figure out the patterns and determine the missing values - some of them really liked it, and some of them sat and freaked out for 20 minutes because they couldn't find the cube root function in their calculator.) I think I'm going to give them all of it this year, and instruction will be a combo of encouraging them to think/play/struggle and good old d.i.

*I said Greek first. Oops.