So I decided to try a new beginning to (non-AP) calculus this year. Instead of doing an algebra bootcamp and diving into limits, I decided to teach kids a new kind of function transformation. I’d say this is something that makes my classroom uniquely mine (this is my contribution to Mission 1 of Explore the MTBoS). I don’t think anyone else I know does something like this.
You see, I was talking with a fellow calculus teacher, and we had a big realization. Yes, calculus is hard for kids because of all the algebra. But also, calculus involves something that students have never seen before.
It involves transformations that morph one graph into another graph. And not just standard up, down, left, right, stretch, shrink, reflect transformations. Although they do transform functions, they don’t make them look too different from the original. Given a function and a basic up, left, reflect, shrink transformation of it, you’d be able to pair them up and say they were related… But in calculus, students start grappling with seriously weird and abstract transformations. For example: if you hold an f(x) graph and an f'(x) graph next to each other — they don’t look alike at all. You would never pair them up and say “oh, these are related.”
So I wanted to start out with a unit on abstract and weird function transformations. Turns out, even though the other teacher and I had brainstormed 5 different abstract function transformations, I got so much mileage out of one of them that I didn’t have to do anything else. You see: I introduced my kids to integrals, without ever saying the word integrals. Well, to be fair, I introduced them to something called the area transformation and the only difference between this and integrals is that we can’t have negative area. 
You can look at this geogebra page to see what I mean by area functions.
Here’s the packet I created (.docx)
That packer was just the bare backbones of what we did. There was a lot of groupwork in class, a lot of conceptual questions posed to them, and more supplemental documents that were created as I started to realize this was going to morph into a much larger unit because I was getting so much out of it. (I personally was finding so much richness in it! A perfect blend of the concrete and the abstract!)
Here are other supplemental documents:
The benefits I’m already reaping:
- It’s conceptual, so those kids who aren’t strong with the algebraic stuff gain confidence at the start of the year
- Kids start to understand the idea of integration as accumulation (though they don’t know that’s what they are doing!)
- Kids understand that something can be increasing at a decreasing rate, increasing at a constant rate, or increasing at an increasing rate. They discovered those terms, and realized what that looks like graphically.
- Kids already know why the integral of a constant function is a linear function, and why the integral of a linear function is a quadratic function.
- Kids are talking about steepness and flatness of a function, and giving the steepness and flatness meaning… They are making statements like “because the original graph is close to the x-axis near x=2, not much area is being added as we inch forward on the original graph, so the area function will remain pretty flat, slightly increasing… but over near x=4, since the original function is far from the x-axis, a lot of area is being added as we inch forward on the original graph, so the area function shoots up, thus it is pretty steep”
- Once we finish investigating the concept of “instantaneous rate of change” (which is soon), kids will have encountered and explored the conceptual side of both major ideas of calculus: derivatives and integrals. All without me having used the terms. I’m being a sneaky teacher… having kids do secret learning.
I mean… I worked these kids hard. Here is a copy of my assessment so you can see what was expected of them.
I love it.
I’m going to put a picture gallery below of some things from my smartboards.
 To be super technical, I am having kids relate and