Like the opener of yesterday's lesson, today's opener gives students to a chance to practice evaluating functions. Yesterday, students were give one function and asked to evaluate it for a series of inputs. Today, they're given four different functions, and asked to evaluate each for the same pair of inputs. Like yesterday, I want to move pretty quickly through this opener, by giving kids a few minutes to try these exercises on their own, then leading them through a quick look at the first three (my notes look like this). It's important for students to complete these exercises, but even more importantly, I want students to use function notation in context, which is what they'll do when they engage in today's work.
Introducing an Exponential Function
What's most important here is exercise d, the exponential function, and I I use this problem to teach a brief lesson. One thing I love about this example is how it makes the 0 exponent rule so intuitive, but before getting to that, I focus on finding f(3). As you can see on these notes, I ask students to work through the multiplication one step at a time. I say, "what's 40 times 1.5 times 1.5 times 1.5?" and we work through it. I challenge students to learn to multiply by 1.5 in their heads, by adding half of a number to itself. When everyone is satisfied that
f(3) = 40 * (1.5)^3 = 40 * 1.5 * 1.5 * 1.5 = 135
we can then address f(0). I say, "What you get when you start with 40 and you don't multiply it by 1.5? In other words, when you multiply 40 by 1.5 zero times?" We discuss how this is different from "multiplying by zero." I know that some students will need to see this a few more times before they really get it, but I'm trying to pre-empt the confusion of multiplication versus exponents.
Without explicitly teaching the zero property of exponents, students have a basis for understanding why a number raised to the 0 power is 1. They're also ready to move on to the next part of the project, in which they'll work with exponential functions.
I don't spend any whole-class time discussing the challenge problem, but it's enough to get some kids thinking. It's not actually too challenging, but here's the little mind-trick: by tacking the word "challenge" onto something, it makes a bunch of kids want to figure it out. And because I didn't go over it, those students will want to show me their work on the problem or go over their solution, so it's another nice piece to talk about once work time commences.
Following the opener, the rest of today's class is a work period. I tell students that as soon as they finish Part 1a of the Spending & Saving Project, I'll give them Part 1b. I expect to have a roughly even split of kids finishing up Part 1a and kids moving on to Part 1b.
No matter which part of the project students are working on, my role is to circulate and to continuously urge them to interpret their work. If they're finding, say, f(2) for a particular function, I'll ask what that means. If they're solving for x from an output, I'll try to get them to say what they're figuring out in the context of a problem.
I love work days like this. When students have the time to do as much as they can, and their questions are driving the lesson, time is always well spent!
As I've noted above, not all of my students are ready for this part of the project today. When they are, here is what it looks like, and here is a video describing this part of the project.
What's most important to me is that I take any opportunity I get to encourage kids to interpret the mathematics in the context of each problem situation. Sure, they're getting practice using function notation, but more importantly, they should be thinking about the meaning of each exercise. Today, my role is to circulate and help kids as they need it. Whenever they don't immediately need me, I visit each group and ask them to interpret whatever problem they're working on.