Power Sprints 9 & 10 today. These are the final sprints of this type, so it's important to collect them and take stock of the students' progress. Report on the gains they've made and point out the increased difficulty of the sprints compared to the first ones. All of this practice with simple powers will be very useful as we begin our study of logarithms!
Today, we're going to begin our first set of "Exponent Sprints". There are 8 of these sprints total; today we do numbers 1 & 2. Explain that now students will be given the base and the power, but they'll have to solve for the exponent. It's obvious to you, the teacher, that these are logarithms, but I won't introduce that term yet. I want my students to already think of "solving for the exponent" as something easy before I complicate matters with an unfamiliar term.
The first exponent sprint may throw some of them for a loop, but I would expect a big improvement on exponent sprint 2. These are the same powers they've been practicing for some time now; they're just being asked to view them from a different angle.
It's time to discuss the solutions to the Basketball Rebounds problem from the previous lesson. We've already discussed the data table and the equations, so today's focus is on the graph. I'll begin this conversation by using a document camera to share one student's work. The class will discuss the accuracy of the graph, as well as the details like scaling and labeling. It's also important to point out the restricted domain (non-negative integers) and how this is expressed graphically. (MP 6)
When it's clear that everyone is comfortable with the model, I like to end by asking when the basketball will stop bouncing. Our model implies that it will never stop! So, as simple and useful as this model is, it clearly falls short of reality since we've made some simplifying assumptions along the way. (MP 4)
Students will have an easier time understanding exponential growth & decay if they can compare it to something familiar. That means we need to take a brief look back at linear functions. (Please see this quick video for more on the rationale behind this lesson.)
I will pose the question this way:
"What we've just seen is an example of an exponential function. To better understand what makes it exponential, let's compare it to something more familiar. In general, what makes linear functions linear?"
Then I'll point out that I'd like the students to give me an explanation with reference to the graph, the data, and the equation. (See the whiteboard.)
After giving everyone a minute to reflect, I'll ask them to discuss it with their neighbors for a few minutes. Finally, I'll ask for a volunteer to share her thoughts. The rest of the class will then comment and critique. Together, we should arrive at the following points.
Linear functions are linear because:
1. The graph is a straight line.
2. The graph is straight because the slope is constant.
3. The slope is constant because successive y-values have a constant difference. (Illustrate this with a data table.)
4. That constant difference arises from or leads to an equation of the form y = mx + b. (There are other forms, but this is the most familiar and "reveals" the constant slope.)
Next, I'll ask the question, "What makes an exponential function exponential?" Again, I'll give the students some time to think individually and then to discuss with their neighbors before beginning a class conversation. (See the whiteboard after this conversation.)
My goal is to link the expression of exponential growth in the form of data, an equation, and a graph. (MP 7) In this case, however, it is probably simplest to begin with the equation. (For an interesting misconception, see my reflection on this section.)
The conversation will begin when I call for a volunteer to share his thoughts on the answer to my question. The rest of the class will then be invited to comment on that answer, to critique it, to add to it, etc. We'll begin by contrasting exponential growth with linear growth, and then move on to the unique characteristics inherent in exponential growth. My aim is for something like the following:
1. The equation contains a constant base raised to a variable power. (From the basketball problem, we have a familiar example of the form y = a(b^x).)
2. This leads to data which does not have a constant difference, but a constant ratio. In other words, we don't add the same amount from one y-value to the next, but rather we multiply by the same amount each time.
3. This always leads to a curved graph that approaches zero (or some other finite number) in one direction, but approaches infinity in the other.
With only a few minutes left, I'll hand out this worksheet. Based on our discussion today, everyone should be prepared for the first two tasks (these will be homework).
So, for the remainder of the class period, I would like everyone to work individually on creating the linear and exponential equations. Meanwhile, I will move around the room checking in with students individually until it's time to go.