# Starting with transformations for complex numbers

I’m searching for a way to teach complex numbers that starts with transformations and then brings in Algebra from there. What I’m envisioning is that we’ll become fluent in the arithmetic of transformations with rotations, translations and dilations and that the kicker will be that arithmetic with i gives a shockingly succinct way to say all that.
In other words, in most approaches that I’ve seen, the kicker with complex numbers is that we learn to be more imaginative and open to weirdness. I like that, but I think I would like it more if the focus was on the shocking efficacy of algebra for representing rotations and dilations.
I think that this is going to be a mess with the Regents class that I teach, simply because I’m not investing in all the tools that I would need to actually pull this off. Today I had everything going great. Kids were developing the regular 1, i, -1, -i cycle with 90 degree rotations represented by (0, 1)^n. It was all great, and right then I felt this pressure to introduce i, and then I did, and in a lot of ways it undercut what I had spent the last half hour building.
What I’d really like to do is to murk around in right triangle trigonometry for a while before introducing i. I’d like to define a class of transformations T_(a,b) that uses dilations and rotations to map the point (1, 0) to (a, b). I’d like to develop a general rule for the composition of these transformations, and then, only then, I’d like to say: “Wait. What if we made the following formal moves: i will represent (0, 1), (1, 0) will just be 1, and, in general, (a, b) will be a + bi?”
Then, I think, we could have a revalatory moment for these kids, where everything just falls together in an absurdly clean way…as long as you’re willing to swallow that i^2 = -1.
At that point, I don’t think that I’d have to ask kids to swallow the idea that i^2 = -1. I think that they’d demand it.
Anyway, that’s what I’d like to try to do. I’m not sure that I’ll be able to pull it off, but I’d like to give it a shot.

# Kate Nowak’s epic Credit Cards/Exponential Functions Unit

This packet really builds up to $e$ and where that number comes from. It’s designed for 3 days.

Here tis.

# Dan Meyer’s Three-Act Problems and Exponential Functions

There’s two good ones that I know of. Maybe there are more:

I think that these will show up when I’m doing Geometric Sequences, before we prove a bunch of stuff about exponents and powers.

# Negative bases and exponential functions

Graph $y=(-1)^{x}$.

I’d like to do this problem after we know about rational functions. We’ll start with integer inputs and then move to rational inputs, where we quickly run into the sort of domain problem we have with radicals and negatives.

Maybe even better: let’s play with this function again once we know about complex numbers and try to graph this function again. There’s another host of problems waiting for us there, even conceived as a real –> complex function.

# Irrational numbers raised to irrational powers

Use the laws of exponents to evaluate $\sqrt{2}^{\sqrt{2}^{\sqrt{2}}}$.

[Source: Wikipedia. See below.]

# Bouncing balls and line of best fit

“A basketball is dropped from different heights, and the height of the first bounce is measured each time. What is the relationship between the height of the drop and the height of the bounce?”

Click me! NOW.

# Dice and linear regression

OK, now here’s a regression problem that I can get behind. Via CME:

“Roll 1 die. Write down what you roll. Roll 2 dice. Write down the sum of all the numbers that you roll. Roll 3 dice. You know what to do. Roll 4. Write down the sum.”

“Predict: how much will you roll with 5 dice? 10 dice? 123 dice?”

A variant:

“Roll 1 die 3 times. How many 6’s do you roll? Roll 2, three times. How many 6’s do you roll? Continue onwards and upwards.”

Do I have to mention the connection to probability?