# Local Linearity: The Big Reveal

Precal is currently doing a quick, basic intro to Calculus. We talked a little bit about limits (we can do them with graphs and with charts), we talked about continuity (three part definition, removable vs nonremovable), and today we talked about local linearity.

Disclaimer: I took this wholesale from an AP Summer Institute instructor.

I started by throwing this up on the projector:

I ask them to tell me everything they can about this function. I typically give them two or three minutes to write everything down. (If I’d been paying attention to the fundamental five, I’d probably have turned this into a small group discussion. “Make a list with your neighbor…” I didn’t, though, and since this was the very start of the lesson, I’m ok with it.)

Sometimes they need a little prompting with their ideas, but they come up with things you’d expect: it’s a line, it’s red, it has a slope of 1, it’s y-intercept is 0, it’s y=x, it’s increasing, it’s continuous, it’s odd. Someone asked me if each square was one. I shrugged – they’ve gotten used to that by this point in the year.

I ask them how much we can see. “Not much.”

I click on a point so that they can see the scale.

I ask again, how much can we see. “Barely anything! Oh, there are probably all of these curves, aren’t there?”

I zoom out, so they can see this:

They’re reactions were perfect. “Oh, wow!” “I knew it!” “You tricked us!” One girl was out of the classroom while we did this. When she walked back in, her friends told her, “You missed it! That was the graph! She tricked us!” I mention the phrase local linearity, and we’re off.

We start talking about slope. We don’t know how to find the slope of a function, so what can we do? One class wanted to find the average slope, so I had to nudge them towards finding the slope at a point. Can we do that? “Yes.” Cool, how? “Uhm….we could zoom in so that it looks like a line!” Yes, yes we can.

So we do. We find the slope at 0, at $\frac{\pi}{2}$, at $\pi$. I point out that it would be nice if we could just have a function whose output was the slope, so that we didn’t have to do all of these approximations over and over again. They agree – yes, that would be nice. (I do love it when they play along…which usually happens when I’ve set up the need before giving them something…but it also requires them to take the bait.)

We switch to $y = x^2$. I give each of them an x-value or two, and they have to find the slope by zooming in, finding two points near their x-value, and then finding the slope between them. Maybe it would be better to stick with $\sin x$, but you have to do a lot of points to see the shape, and I have really small classes.

After they all have their slopes, we put them in a table on Desmos. In one class, they did a great job finding their slopes, and it was a line. In the other, it didn’t work out so well, so we had to redo some points. Eventually, they got it, though. That would have been a spot where I could have gotten some really good estimation conversations going (why do you think that point is wrong? positive slope, decreasing function, etc.), but I didn’t. Maybe next year. They were able to figure out which points were wrong, though, and fix them without me, so I’ll take it.

If we have a bunch of points in a line, we can find the line of best fit. Desmos is really great for this – we typed in $y = mx + b$, created m and b sliders, and went to town. They got $y = 2x$ pretty quickly.

At this point, I mention the word derivative. In the advanced class, I throw up the limit definition of the derivative and talk about it. We don’t use it in precal; I just want it to simmer in their backbrain for a few months until we need it in August.

For homework, they repeat the process on two more polynomials. They get to choose which two from a list I give them. Then they have to write about what they notice, what they wonder, and how they could check those wonderings. I’m hoping some of them will notice the  Power Rule; the list they can choose from leads them their nicely. If they don’t get it, well, we repeat this with a few tweaks in Calculus, so they’ll get it then. My goal for the overview of Calculus is primarily to get them thinking about a few big ideas. We have all of next year to do stuff with it. We’ll probably also do a class discussion next time about what they noticed and wondered, and I might get them to the Power Rule that way.

Their exit ticket was to tell me why we could find the slope by zooming in. Most of them, with a bit of a nudge, could tell me that we zoomed in, saw a line, and from there we could find the slope. No one used the phrase locally linear, but I’m ok with that for this level.

I really, really liked the way this worked today. Their reactions when we zoomed out made my day. Also, in the past I’ve done this with the graphing calculator instead of Desmos (this is the first year we’re 1-1), and it was a pain. Desmos was perfect for this lesson – it’s easy to use, they’re familiar with it, and the sliders are a huge help.

I also liked that this only took about 25-30 minutes; we fit this in after a warmup and a quiz. With a whole hour, we could have done more with the theory, or I could have given them more time to practice. Things to think about….