We start our new unit by reviewing where we left off at the end of Unit 2. I post the second slide of today's lesson notes and give students a few minutes to get started.
There are two tables of values here: one that represents a linear function, and another that doesn't. When students graph each set of points and run a linear regression for each data set, they'll see that one model hits all points perfectly, and the other does not. I take this opportunity to check in with kids and gauge their confidence about how to use their graphing calculators. I also pay attention to students as they react to the first model: I'm hoping that most kids are unsurprised by the fact that the line matches the points, and that many kids can explain why the equation of this first line is y = -3.5x + 24, independent of what they've learned about regression in this course.
After we collectively note that a line works for table (a) but not table (b), I show students that they already know almost everything they need to make a different sort of model. When they have data set (b) in their calculator, I show everyone that at the bottom of the STAT --> CALC menu, there's another type of regression labeled "ExpReg," which means "exponential regression". This yields the equation y = 48*0.5^2, the graph of which is an exponential curve, and which matches the second data set perfectly.
In the rest of today's lesson, we'll figure out when exponential models are useful!
This lesson is an introduction to distinguishing between linear and exponential functions, which will be one focus of Unit 3. This lesson is also about all the decisions we make when we collect data. There are many decisions to keep track of, and we must acknowledge the affect of each decision on the conclusions we draw. We can often use the word "parameters" to describe all of the inputs that might affect the outcome of an investigation. The word parameters has a formal definition in mathematics, and a more informal for general use. Because this word is new to many of students, I want them to be comfortable with its use in both contexts. A third objective of this lesson is about conducting research in the real world. How do we empower ourselves as consumers, by using tools that are available?
Students receive this Auto Depreciation Investigation handout, and I post an overview of first steps on slide #4 of the lesson notes. It's usually best to have students work in pairs for this assignment. I tell students to pick a popular car (rather than their "dream car" - we'll get to that tomorrow) that has been in production for at least 10 years.
In this narrative video, I describe the steps for collecting data on the Kelley Blue Book website, kbb.com. As I explain in the video, the key here is for students to notice that every time they click on an option on the site, they are setting the value of a variable - a parameter - that will affect the output from the site.
Students choose a make and model of a popular car, they identify options (it's best to keep these simple), and they create two "cases" that each suppose they drive the car a different number of miles each year. Then, they fill in the chart on the front of Auto Depreciation handout. As they do so, they're having the experience of conducting thoughtful consumer research that might come in handy someday. When they're done, they have data sets that we can use to run linear and exponential regressions, which we'll analyze on the back of the handout.
Here are two examples of completed handouts: one for a Toyota Camry that is driven 5,000 or 10,000 miles each year, and another for a Jeep Grand Cherokee driven 6,000 or 12,000 miles per year.
When the data collection is complete, students turn to the back of the Auto Depreciation handout. The first task is to plot a few sets of points. I give students the option of doing this by hand on graph paper, or on Desmos, but I definitely prefer Desmos in this case. An electronic version makes it easier to manipulate the graph and to quickly add different regression curves.
Students work through the steps, and soon enough, they can see that they data appears linear. The least-squares regression lines that our calculators give us seem to fit the data well enough. Keeping the data in their calculators, I remind students what they saw at the start of class: they can also run an exponential regression on the data. Then, they can graph both functions. Here's what the models look like for my Camry example with 10,000 miles per year, and here is what they look like for the Jeep Grand Cherokee with 12,000 miles per year.
When students try to describe which model fits their data better, there is some good-spirited debate, and you could make a good argument either way. The kicker happens when we move beyond our data. Question #5 asks which model will better predict the value of a 15-year-old car, and here, the exponential wins every time. For many data sets, the linear model is negative by the time year 15 rolls around, and as we look at the end behavior of each model, the advantages of the exponential model become clear. We talk about how even when a car barely runs, it's not quite worthless: you can usually find someone who will take it for parts.
If we get to it, we'll talk about interpreting our linear and exponential models, but there's no pressure to do that today. Over the course of this unit, students will become familiar with the idea that linear functions change by the same constant at each step, while exponential models change by a common factor at each step. So in the case of the Camry, we have a linear model that says the car loses $1193 in value each year, and an exponential model that says it loses 9.2% (or really, "keeps 90.8%") of its value each year. Which one feels more realistic?
To close today's class, I post today's Exit Slip prompt on slide #5 of the lesson notes. Students are prompted to explain which parameter has a greater influence on the value of a car: its age or its mileage (it's fun to recognize that word as "mile-age").
My goal is to get students to reference their research and to summarize what they've seen, and to give them one more chance to think about the many parameters they've seen today. When I read these exit slips, I'll have a nice snapshot of how well they got that idea, and the extent to which I'll need to emphasize it tomorrow.