Lesson 6 of 8
Objective: SWBAT sketch exponential functions on a semi-logarithmic plane, and to begin to understand the inverse relationship between exponents and logarithms.
Today's opener is on the second slide of the lesson notes, and it helps to set the tone of this review lesson. Students will apply a lot of what they know and what they have seen over the last few weeks, as they also learn something new about the use and purpose of logarithmic scale.
For this first review topic, students have the opportunity to practice writing rules for arithmetic and geometric sequences. I use the word "Quick" on this opener to show students that we're going to move quickly through a lot of material today. It's not that there isn't urgency on other days, but today's lesson is in contrast to a lot what we've done lately, in our deep, "slow-thinking" investigations of specific situations.
I give students three or four minutes to try writing these rules on their own, then I post the solutions (on slide #3) for students to check their work. If anyone has specific questions, I do slow down to answer them. If students want to go over recursive definitions, we will, but there's no need to force that today. After a quick vibe-check, we're ready to move on.
By the end of today's lesson, my goal is for all students to have reconsidered everything they already know about graphing functions on the coordinate plane. To get ready for that, here's a quick review of a few things that should feel very familiar.
I call these "Quick Sketches", and the instructions are laid out on slide #4 of the lesson notes, then followed by three rounds of exercises on slides #5-7. In each round, students will have just a minute to sketch three graphs on the same axes. I specifically instruct students not to use graph paper. The goal here is to see how much kids know and can apply in a short amount of time. After students get a minute to sketch what they can, I give them another minute or two to compare what they've got with a neighbor, and we'll troubleshoot as a class if necessary.
There's a little review embedded in each set. In the first one, I'll be happy as long as kids recognize that these are three parallel lines, and I'll be especially happy if they pay attention the relative distance of each line from the origin. In the second set, all students should notice that these lines have the same y-intercept, and I hope that most students recognize that two of the lines are perpendicular.
The third set is where this lesson starts to transition toward new material. Students will recognize that these are exponential functions, and most of my kids will be able to come up with a rough sketch of an exponential curve. Most will also know that the greater the base, the more quickly the graph will shoot upward, but precisely how that happens is something we haven't looked at too deeply yet.
Transition: So...What's Hard About That?
Even if we had already studied the behaviors of exponential curves in depth, however, how much would that knowledge help us? Because of the nature of exponential functions, their graphs are hard to read. I use slides #8-10 to show students what I mean - and often, I'm just clarifying ideas that kids brought up already. On slide #9 and 10, we see all four graphs from Quick Sketches #3. In the first illustration, we see what happens when we count by 1's on both axes. All four graphs hit the point (0,1), and then they quickly "accelerate" out of the frame. In the second illustration, we see that if we scale the axes to accommodate the largest outputs on our domain, we get more information about the rate at which each graph "takes off" but the early behavior of each graph is harder to assess.
I try to elicit student ideas here, and to build up the kind of informal vocabulary that I've used in the preceding paragraph. Then it's time to transition to the main part of today's lesson, where we answer the question: ok, so what can we do about that?
My goal is for students to leave today's class with a basic understanding of how logarithmic scale works and why it exists. We're already moving toward that goal. When we get to this part of the lesson, students have already acknowledged some of the difficulties of graphing exponential functions on the traditional linear-linear plane. Now we're going to look back at some of the examples we've seen previously.
To begin, we'll go back to a linear model of the cost of air travel from the previous unit. On slide #12 of the lesson notes, I take students back to the model we created that day. I post the slide, and say, "Do you remember this? Who can explain what we're looking at?" A few students take turns interpreting this model. Next, I show a similar, and slightly simplified graph on slide #13. I ask students to interpret the coefficients in the equation, and then to predict the cost of a few flights with different lengths. Finally on slide #14, I post two lines, without labels, and ask students which airline is more expensive. The answer is that it depends on the length of the flight. For flights shorter than 600 miles, the blue line is lower, and beyond that, the red. The point is simple, and I make sure that everyone is with me on this: lines make it pretty easy to compare.
"That's why it's nice to see a straight line," I say, transitioning to slide #15, "on a graph like this." Clearly, this line shows the steady increase of processing power predicted by Moore's Law, which we studied last week. I wait a few beats. If they're with me, alarm bells should start going off in the heads of my students, for at least one of the following reasons: (a) students will remember graphing this data, and seeing that the graph was definitely not a straight line, (b) the mid-graph caption notes that this "curve" indicates "transistor count doubling every two years," and (c) what the heck is up with that y-axis?
I flip to slide #16, where it's a similar story. The data looks straight, but that's not how we recall graphing it, and the y-axis appears to breaking a cardinal rule about maintaining consistent scale in our labeling of an axis. For one more example, we revisit the headlining graph from Gapminder on slides #17 and 18: first with a graph that uses a log scale, and then with another that doesn't. I ask the class, "Which graph is easier to read?"
I tell everyone that these graphs employ "logarithmic scales," and that they don't break that cardinal rule about setting up your axes. It turns out that you can use a geometric sequence just as well as you can employ an arithmetic one when you set up the axes of a graph. Next, we'll investigate how this is done and the results of doing so.
We're going to graph the same four exponential functions from Quick Graphs #3, but this time on a "semi-log" plane that has a linear-scaled x-axis and a log-scaled y-axis.
I prepare copies of this semi-log graph paper by printing on both sides of the page, and distribute one to each student. (You can find more options for printing all sorts of graph paper at this terrific utility of a site: http://www.peregraph.com/documents/graphpaper.)
Labeling the Axes
I post slide #19 of the lesson notes, which displays the same line-log axes students just received. We take a moment to look at it together, and I open the floor to questions and observations. Some students might take a stab at explaining how to label that y-axis. Others will just wonder what we're doing here. Soon we get started with labeling the axes.
There are no surprises on the x-axis, but we do have a chance to distinguish between "major" and "minor" grid lines. I say that for now we're only going to concern ourselves with the scale marks that extend vertically across the entire graph, the "major" ones.
Then we get to why we're here: the y-axis. I show students how to count that there are "six decades" on this graph, and I identify the major grid lines as those that border those decades. "We will label these major grid lines as powers of 10," I say. "But what's 10^0?" When students recall that it's 1, I say, "That's where we start. From there, you can consider this axis to be labeled with the geometric sequence 1, 10, 100, ..." Students should follow along, and at this point their paper should look like this.
Next we get to the minor grid lines. I say, "We're using what's called logarithmic scale, but what is a logarithm? Let's see." I invite students to type log(1), log(10) and log(100) on your calculators and ask, "What do you these results represent?" From there, we develop a working definition that logarithms tell us what power we'll have to raise 10 to. It's important to note that I'm not teaching a deep lesson on logs here. If this were a Pre-Calculus class, I would use this activity to start a whole unit on logarithms. In the context of the class I'm teaching, my goal is just to expose students to this rich idea, and to get to the resulting graphs.
I ask students to make some predictions: what about log(2) through log(9)? Or log(20) through log(99)? Slides #20 and 21 summarize what students will see on their calculators, and there are a couple important observations to make. First, we see that the difference between log(1) and log(2) is greater than the difference between log(9) and log(10), with decreasing differences in between. That explains why the minor grid lines are not equally spaced, but getting closer together in each decade. Secondly, we see that log(20) is the same as log(2) + 1, or more generally that log(a) + 1 = log(10a), and that explain why each decade is the same. I show students how to finish labeling the axis, at which point their papers should look like this.
As students finish labeling the y-axis, they gain new insights about place value, and making comparisons between numbers. I like to ask kids questions like, "Who would be happier to get $10: someone with a dollar in their pocket or someone who has a million dollars in their bank account?" Putting aside the answers that a sociologist or psychology student might provide, it's a percentage thing. For whom does $10 make a bigger impact? Now, we can make relative comparisons. If I have $1000, look at where $2000 is. It's the same as $10 to $20 or $100,000 to $200,000, percentage-wise. In both cases, the numbers are doubling. This graph gives us a way to talk about that. With the axes set up, it's time for my favorite part: "Ok, so what does a graph look like on here?" I ask.
Now students will plot points for those four exponential functions from the start of today's lesson. I encourage everyone to use a different color for each one (I always keep a giant box of colored pencils handy), and to carefully plot each point for each function. Students don't need to much guidance to get started, and I just love watching as they start to realize what's happening: on these axes, an exponential function will make a straight line. Here are the results of graphing y=2^x in blue, y=3^x in orange, y=4^x in purple, and y=10^x in red.
So how can interpret what we see here? I tell students that on linear-logarithmic axis, a straight line represents a function changing by a constant factor or constant percentage, just like we see in geometric sequences. And just like we can compare the slopes of linear functions to assess the pace of growth, we can do the same with these lines: a steeper slope represents a greater common ratio. Rather than these exponential functions quickly getting "out of hand" as they do on traditional axes, we can chart exponential growth toward much greater numbers.
"Correlation" implies a linear relationship. Here, we're allowed to expand on that idea. As a group, we take another look at the examples. Think about total change vs. percent change - and how "exciting" each advance is - in the context of each example. Back in the 1980's, if you could squeeze another 100,000 transistors onto a chip, that was a huge deal! These days, an increase of 100,000 transistors would "only" take a chip from 2.0 Ghz to 2.01 Ghz.
To summarize, my main goal is for students to approach the understanding that a straight line on a logarithmic graph means that a quantity is growing by the percent or factor over time. That's what a geometric sequence does - this way of scaling allows us to see that. We might take one more look at those Gapminder graphs: Why is it sometimes necessary to use linear scales, and other times to use logarithmic scale on Gapminder? For which variables? That's a rich question!
Depending on the context in which you teach this lesson, there are all sorts of rich questions and ideas to explore - I hope that this lesson provides a starting place for that! Here's one more idea that I'll get to if I have time, but it's always been fascinating to me: for any two quantities on a log scale, their geometric mean is halfway between those two numbers. See what I mean?
Sometimes there's plenty of time left in today's class, and other times we only get as far as what I've already shared. On the last slide of the lesson notes, I include some extension questions. In my highest-achieving classes, I might tell students to choose one task and submit it for an assignment. In other classes, we might just look at one or two of these.
- Try sketching a few similar exponential functions that don’t start at 1, like y = 3*2x and y = 5*2x
- Try sketching a geometric sequence whose common ratio is not a whole number, like 80, 120, 180, 270, 405, …
- What does exponential decay look like on our linear-logarithmic axes?
- Is the Fibonacci Sequence more like a linear function or more like an exponential function?
Other extensions depend on what else my students are studying in other classes. It's worth it to touch base with their science teacher: maybe a lesson on the Richter Scale, measurements of sound, acidity vs. alkalinity, or astronomical measurements will make sense here.
There's so much here, and like I mentioned above, if this was precalc course, I would explore this idea in greater depth. As a stand-alone lesson in the context of this course, my students really enjoy it.