I love composing functions. I always have. It's one of the topics in algebra I get most excited about. I don't really know why. It's cool that functions have this operation that the real numbers don't have. It's cool that this operation shows up repeatedly in calculus. But, ultimately, I just think it's really fun.
Every time I go to teach function compositions, I think I finally have it nailed. I use all the right colors to differentiate between the inside function and outside function; I start out with an application problem to motivate the use; and I'm pretty sure my enthusiasm during this lesson is off the charts.
Still, something was missing.
In the past, I've started with composing at a general expression. We would start by finding f(g(x)), for example. I guess I did it that way because that's how the textbooks always did it.
This year, I started with composing at a number. We started by finding f(g(3)), for example. I'm not sure why I never thought of this before. I feel like a total idiot that it took me this long, but it is what it is. I did 2-3 examples of evaluating at a real number, and the kids took off from there. As I walked around the room, checking their work, it was clear they nailed it (at least for the day, right?).
Now onto the general case. But first, a bit of a digression. I showed them this slide:
We worked these together. The magic bullet? f(a+1). In the past, I would skip from f(a) to f(x+1). This is where I lost the kids. x+1 was too much too fast because the original function is in terms of x. But adding that one little line made all the difference in the world. From here, we could find f(g(x)) where g(x)=x+1. No problemo. Here is my notebook file.
Now if they can just remember that (x+1)^2 isn't x^2+1 I will be a very happy lady.
Kate Nowak wrote this in one of her posts that I've always remembered:
I just find it stunning that you can plan out a lesson 95% correctly and it will miss most of your kids. And you can change one little thing - add three rows to a table - and now all the kids basically get [it]...and tell you why they don't get why this is such a big deal. I feel a little like I have super powers.