SWBAT distinguish between exponential growth and decay, as they continue to create mathematical models of real phenomena.

Once students understand a process of gathering, analyzing, presenting, and interpreting data, they'll be able to explore all sorts of new questions.

10 minutes

Today's opener is on the second slide of the lesson notes. I use this opportunity to provide a "textbook example" of what students have been studying so far, and to review function notation with my students. I like this problem because it's about population growth, which students can count on seeing at some point, and because kids are encouraged to talk to each other about what they know so far.

More importantly, I really want kids to feel comfortable interpreting the parameters of an exponential function. My use of the word *parameters* is a little different than it's been in the last few lessons. Previously, I've used this word to mean all of the factors that might affect a mathematical model that we develop from researching data. Today, I'm using a little more formally, to mean the coefficients in an algebraic rule.

We'll spend more time on this in the next few lessons. Today, I have a chance to check in on how much my students already know, and to prepare for the next few lessons.

5 minutes

To set the stage for today's lesson, it's story time. I put up the third slide of the lesson notes, and I ask students if anyone has heard of the Nintendo Entertainment System. Responses are always mixed, but the fact remains that even for kids born in the late '90s this system is a cultural icon, and kids love hearing a few stories about their teacher's own childhood. I tell my class about how I used to play all these games, how I used to trade cartridges with friends, and just like they do now, how I'd eagerly anticipate a new release or a conversation about how to beat a game. "Back then, I had to ask a friend, call my uncle or buy a magazine to figure out how to beat a game," I tell students, "there was no Internet to teach me how to do it!"

Then comes the purpose of this story. I point to the Zelda cartridge on the slide. "Do you know how many hours I spent playing this game?" I ask the class. They don't. I don't either. Countless. "For all those countless hours of entertainment," I say, "do you know how much data was on these cartridges? In other words, if you put this file on a computer, how much space would it take up?"

So, did you know: the amount of data on a typical NES cartridge was 128-384 kilobytes. What does that mean? Well, that's 0.128 - 0.384 megabytes (note again that these number are based on powers of 2), or just a fraction of the size of many digital folders, or as we'll see today single MP3 files.

If you're not so into talk about video games, you might also relay the genesis of the "Save" icon in many programs. It's a floppy disk. Most kids have truly never seen one. And why would they? A 3.5" floppy disk would not even hold an MP3 file!

60 minutes

Next, we get to the meat of today's lesson. This investigation follows a similar flow to the previous lesson on Moore's law. Students will gather data from an online source, analyze that data by running both linear and exponential regressions, present that data graphically, and then interpret their models. They will also briefly review function notation.

What's new here is that students will be able to compare exponential growth to exponential decay. They will see that, while computing power could, theoretically, increase infinitely -- has it stopped yet?? -- the cost of storing data cannot decrease below $0.00. The Moore's Law investigation was about exponential growth. Today's lesson shows students that the capacity of a hard drive has increased in a similar way to the number of transistors on a computer chip, and that's mirrored by a dramatic decrease in the cost of storing data.

**Research**

Students start by doing some quick research on the typical size of an MP3 file. Anything from 3 megabytes (MB) to 10 MB will do, and it's important to make it clear that settling on a particular number is a decision that will affect our models. "It's up to you what number you decide on," I say. "In fact, you might learn more if you choose a different value from some of your classmates."

On Storing MP3s investigation handout, I provide this link: http://www.mkomo.com/cost-per-gigabyte. I start by giving kids a slightly older version of the article, from 2009. Linked from there is an update we'll use in the interpretation.

The handout says "pick whichever you want from each year," but I say, "Try to have some reason or method for selecting the data that you do." For example, on the Desmos graphs linked below, I've chosen the biggest hard drive for each year. Students could also choose to use the most or least expensive, or a midrange size for a given year. Again: decisions, all of these.

In the data, you'll see that there are fields missing from some rows. That's a good thing. Data is really like that. We use what we can get our hands on, and we try to at least be aware of what we don't know.

**Analysis**

From here, the structure of the lesson is familiar to students. When we run one regression for hard drive size over time, and another for the cost of storing each MP3 over time, we're presented with the difference between exponential growth and decay. The first case should look very similar to our work with Moore's Law. In fact, the common ratio is similar enough (depending on data choices) to the one we got in Moore's law.

Students might notice that the exponential decay model is similar to our Auto Depreciation investigation from earlier in the unit. With values that start high, and in which we repeatedly multiply by a number between 0 and 1.

As I note above, there are many decisions students make that will influence their results: which subset of hard drives have they chosen? What mp3 file size have they assumed? And then, going a level deeper, as long as the mp3 size is consistent throughout a model, how much does that really change the model? Where/how does this value show up in our models? If you want to focus on how different coefficients influence the graph of an exponential function, this moves us toward scaling our models without changing the insight they offer.

Over the course of this unit, my goal is for students deeply understand a process that can be applied far and wide. I discuss that idea in this narrative video. This exercise gives students an opportunity to practice the process, and I hope to see students getting comfortable with that. As they further develop these skills, we can talk more about extensions, next steps, and my favorite: how we might address new research questions! Today, my role is to circulate, troubleshoot steps with individual students, and push the thinking of those who "already get it". Also in the video, I discuss the growing role that I hope the interpretation step will take today. The better students understand *what *they're doing, the more room we have for discussing where this knowledge might take us.

**Other Notes**

- Here are links to example graphs I made on Desmos: the decreasing cost of storing MP3's over time and increasing size of hard drives over time - keep in mind the decisions I've made when making these models: to assume that an MP3 takes up 5 MB of space, and to choose only the highest-capacity hard drive for each year.
- In the original article, we also notice the graph on the linked article is a straight line. How did that happen? That's what tomorrow's lesson is about.
- The exponential model seems to fit the data better, but it also seems to be less effective after 2009. See Matt Komorowski's linked update from 2014. I try to engage students in conversations about some of the points raised there.
- It doesn't take a whole lesson, but I love including a brief mention of Xeno's Paradox when get to talking about exponential decay. I'll tell a student to stand on one side of the room, and then repeatedly walk halfway to the opposite wall. It's another way to illustrate the idea of the limit that storage costs will approach.