SWBAT observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly or quadratically.

One key word in standard F-LE.3 is "eventually"! Students explore a situation that reveals how exponential growth might take a little while to catch up with linear growth, but that when it does...

5 minutes

**It's the New Marking Period, and We're Starting a New Unit**

Today is the first day of the third quarter. It's also the first day of a new unit called "Linear and Exponential Functions". I make a big deal of the fact that the year is half-over. On the agenda, I write "Week 21 of 40," and this is usually enough to kick off a useful little conversation about how fast time can go, and therefore how we can thoughtfully figure out how to use it well.

One concrete way that I signify the change in marking period is to assign new seats. For this week, I've arranged everyone in "Alphabet Groups" by placing everyone in groups of 4, in alphabetical order. This ensures that everyone is sitting with someone new, that groups are heterogenous, and that there's no need for students to read too deeply into why I asked each student to sit where they are.

**Opener**

So as students arrive, they see the seating arrangement projected on the front board, they find their new group, and with that new group they have to come up with an answer to the question:

**Would you rather make $25 per hour or $50,000 per year?**

Of course, it depends on how many hours you work in a year. I wait for students to get this conversation going, and the best case scenario is for me to be able to do nothing as students talk about the prompt. I circulate, and if I have to, I'll guide students by asking how many hours they'd like to work each day, how many days they'd like to work each week, and how many weeks they'd like to work each year. Sometimes students will know that someone can earn "time-and-a-half" for working overtime, and that adds an interesting twist to their answers.

The two goals of this opener are to get students talking to their new group members and to prime them for the discussion of a more preposterous choice of two ways to get paid. Five minutes is enough for us to achieve both of these goals.

5 minutes

With the end of the second marking period come progress reports, so for the fourth time this year I distribute mastery-based progress reports to each student. At this point, students have a pretty good grasp on how they're graded in this class, and they get better at talking in terms of * mastery* all the time. Please see my lesson,

While the progress reports allow us to look back at everything that has happened so far, we also want to look ahead to the new unit. I take this opportunity to introduce the four Student Learning Targets (SLTs) that will be our focus for this unit. They are posted on the back wall of the classroom, and there is space to add work as it happens.

"So we've got four weeks to master four learning targets," I say. "Is everyone ready?" The new unit is called "Linear and Exponential Functions". I point to the word *Linear*, and I ask if anyone has heard of linear functions. They have; that's what the last unit was about. Now, we're going to continue to work with linear functions, as we move on to comparing them with exponential functions. I point to the word *Exponential*, and ask if anyone knows this word.

30 minutes

**First Task: Listen to this Scenario**

As students take a moment to look over their progress reports, I distribute the How Will Your Salary Grow? handout that we'll use today. I say that this is a group assignment. Then, as I describe in this narrative video, I start by telling a story and giving a verbal, informal description of the two payment plans. Starting from this informal description of a relationship, students will construct a linear and exponential function (F-LE.2).

Of course, this is based on a pretty famous scenario. There are some nice twists to the way things are set up here. First of all, even though the the *daily pay *for Plan A makes an arithmetic sequence, the *total pay* for that plan is a finite series. It's not the focus of today's lesson, so I don't make explicit reference to the mathematical idea of a series or to the quadratic function that can be used to model the growth of its sum for successive days. I like to build exposure to other, upcoming ideas into my lessons, and we'll be able to reference this when we get to the quadratics unit later in the year.

In terms of a solution to this problem, I love that in this particular framing of the problem, it takes until Day 29 for the total earnings from Plan B to surpass - dramatically - the total earnings from Plan A. But we'll get to that later.

**Second Task: Group Discussion**

I take clarifying questions about each plan, then students start on the next task, which is to discuss with each member of their group which "Payment Plan" they'd choose. "You should write a quote from each person in your group," I say, "that explains why they would choose their payment plan." Most students are quick to pick Plan A, which is perfect for the purposes of this lesson. Some students will perceive that something fishy is going on, and choose Plan B just because they're accustomed to expecting the unexpected in a math class. A few students will choose Plan B out of defiance or to prove that they don't need the money, which is awesome, because they're in for a big payday surprise.

As these discussions happen, I circulate to listen in and to take a peek at how students recorded the details of each payment plan. By looking at what students have written, I can instantly assess how well they can generalize from my verbal description.

**Third Task: Make a Table**

As each group concludes their discussion of which plan they would choose, I direct them to the next task: "On a separate sheet of paper, make a table showing how much you'd get paid every day for each of these two payment plans."

I emphasize once again that this is a group assignment, and that students should share the work to make sure it gets done efficiently and accurately. I wait for everyone to get started, and soon students notice that there are two things they have to track for each plan: the amount they'd be paid daily, and the total amount of pay they'd receive. I give students a little while to plan and get their own ideas going. After seeing what they can do on their own, I set up my own table on the board. In some classes, I'm copying this idea from my students, and I give credit where its due. In others, I'm providing this example as an answer to questions about how exactly to accomplish this task. Either way, I want to make students feel like this set up is their idea, so my questioning reflects that.

For the remainder of class, students have time to work together to finish these tables. As they do, I visit each group to ask if they've noticed any patterns or developed any shortcuts. Even though it's hard not to say too much or give anything away, I work hard to do that, because I want kids to have the chance to make this their own.

**Fourth Task: Make a Graph**

The final task on this assignment is to create a graph of the two payment plans. Most of my students won't quite get to making the graph today. Please see tomorrow's lesson for a description of how I introduce the use of Microsoft Excel as a tool for making the table before setting students to the task of graphing these two scenarios.

3 minutes

With a few minutes left in class I ask if anyone has changed their mind about which plan they would choose, or if there have been any surprises yet. In most classes, students still need a little more time to complete their tables, so they may not yet have seen that Plan B will surpass Plan A. I say that students should complete their tables tonight, then, if they can, move on to creating graphs.

To close, I acknowledge that making these tables is kind of busy work. "Was it difficult to fill out this table?" I ask, and students are quick to say that nothing was too *hard* about this task, but that it was time-consuming, and that's a little annoying. To frame tomorrow's lesson, which takes place in the computer lab, I say that we're going to use computers to do this work more efficiently, and to check today's work.

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