There are two purposes to today's lesson. One is for students to see how the parameters of linear and exponential functions affect the shapes of their graphs. The other is to give students a chance to play with Desmos, a free online graphing calculator.
In order to achieve these goals, I teach a pretty loosely structured lesson today in the computer lab. There is no product today, and no handout on which kids have to record their work. I don't even insist that they take notes. I really just want kids to interact with these graphs and to gain some fluency using this powerful tool. Despite the common perception that high school students are hyper-connected and tech-savvy, many of my students often lack the opportunity to use a computer. I believe that showing them something cool, and then allowing time to play with it, is a great use of today's class. So a tool they've seen before, but on which they haven't done their own work from scratch.
As students arrive in the computer lab, I instruct them to:
Of course, my own use of parentheses in that second instruction indicates that it's not always so easy to write a mathematical expression on a computer. On the board I might simply write:
On a computer, however, it's not as obvious how to write a fraction. In fact, if you try to enter that equation on Desmos by pressing the keys "y, =, 1, /, 2, x," your result will not be a straight line:
My classes are split about 50/50 on who initially gets each result. What's fun about this is that kids are curious to know what's going on here. That inverse function makes a pretty compelling shape, so just as they're wondering why they didn't get a nice straight line, students are also curious about the graph that shows up in its place. I encourage students to help each other out if they see a neighbor who didn't get the straight line, and I circulate to assist anyone who is completely stuck. The simplest usage tip is to use the right arrow key to move out of the fraction - and this is necessary whether we use parentheses or not.
It's also useful to note that this is an explicit example of how an expression changes based on whether a variable is in the numerator or denominator of a fraction. Some kids can need a good bit cajoling to believe that it matters, but this evidence is pretty convincing.
The Common Core Standard F-LE.5 says that students should be able to "Interpret the parameters in a linear or exponential function in terms of a context." For the previous two days of class, our focus has been on sketching and interpreting graphs in context. Today, on the other hand, our focus will turn to the parameters of linear and exponential functions. I'm going to ignore context temporarily, and really try to get kids to understand what a parameter is and how different parameters can affect the graph of a function.
In this narrative video, I give an overview of how we get started. I try to share a lot of what I say to kids in this video, but of course, in the actual classroom there's wait time: giving kids a chance to explore, to question, and to explain themselves.
As I describe in this narrative video, my next move is to show kids how to set up a table of values on Desmos, and then to think back to the two payment plans that I introduced to kick off this unit. For "Plan A," the daily salary grows at a constant rate, so it can be modeled with a linear function. For "Plan B," which we discuss here, the pay doubles each day. Now we have the chance to see what the graph of an exponential function looks like.
Finally, we investigate the parameters of an exponential function. At the very least, I want kids to see what happens as the base changes, and that may be all we have time for. If there's more time, there are a few ways the conversation can go. Check out this narrative video to see how I lead kids through the final part of today's lesson.