SWBAT use spreadsheet software and graphs to compare linear, quadratic, and exponential growth.

Writing formulas in a spreadsheet lays foundations for writing recursive rules for arithmetic and geometric sequences.

5 minutes

During today's computer-based lesson, students will learn how to use spreadsheet formulas to recreate the table they worked on during as I've done here. I project my screen at the front of the computer lab for students to see.

The first bit of functionality students will need to learn about is how to resize columns to be wide enough to fit the column titles. Some students will recognize that this can be done by clicking and dragging, while others will wonder what happened to their text when they move their cursor to the right. I show everyone how to double-click in between the column headers to automatically resize each column.

18 minutes

Today's lesson is a continuation of yesterday's lesson, in which students compared the growth of two different payment plans. This lesson finds my classes half-done with that work (here is the double-sided handout that they will continue to use). If you haven't already seen that lesson, please check it out for some background on how we got here.

Today, these are my goals:

- To introduce the use of basic formulas in spreadsheet software.
- To build a vocabulary for thinking about the recursive definitions of sequences; in other words to see that the two steps for writing a spreadsheet formula are to enter the first term and then to define a rule for generating the next terms in a column. I'd like to be able to reference this example tomorrow when I introduce the idea of a recursive rule for an arithmetic or geometric sequence.
- To show students how quickly and efficiently they can finish the work from yesterday's lesson, given the right tool.

Take a look at this narrative video to see how I take my students through today's process. There is a such a great wow-factor when kids see how easy it is to fill in each column, so engagement is high. In between each step that I've shared in the video, I spend plenty of time circulating to make sure every kid has what they should on the screen. I help to troubleshoot, sure, but I also try to identify students who can serve to help out classmates in need. There are plenty of kids who are quick to embrace this task, and others for whom the use of software like this is completely new.

When the tables are complete on Excel, students can check their answers then move on to graphing of these two scenarios - there is space to do that on the back of the *How will your Salary Grow?* document.

20 minutes

Once students confirm that they've got an accurate table of values, they can move on to graphing these two scenarios. On the *How Will Your Salary Grow?* handout, I provide students with a 30x30 graphing area and positive axes. It's pretty easy to decide how to scale and label the x-axis, but scaling the y-axis is actually a wonderfully rich task. The first time I taught this lesson, that came as a surprise to me, but now I expect it. When kids have to figure out how they're going to represent a range that stretches from $1 to over $1 billion, all sorts of interesting conversations come naturally.

The first question kids will have is whether or not they should graph the "Daily Pay" or "Total Pay" for each payment plan. I tell them it's up to them, but I also suggest that between the four members of each group, it might make the most sense for two people to take on each option. It's also an opportunity to ask students to compare the Daily and Total columns for each payment plan - there's interesting stuff to notice in either case.

The big conversation is what comes next - and it's terrific how naturally kids have discuss this with each other: *What are we going to count by on this axis?* I really try to stay out of these conversations. Of course, when kids get to talking math, I'm going to have a lot to say, but it's so important to let them have their own thoughts. Scaling decisions are wrapped up in all sorts of politics - every time we see a graph in the media, someone has made a decision exactly what part to show us - and I want students to experience that process.

Once their scaling decisions are made, students begin plotting points, which yields another set of questions. If every grid square counts for, say, $40 million, then where does $1 go? What is $3 when compared to $3,000,000? (Answer: basically nothing.) What is $1 million compared to $40 million? (On this graph, it's barely above the x-axis!)

Enterprising students will ask if they can ignore a few points. I say that's fine and let them go for it, and I leave it up to them to see how incomplete the story can be when they leave out the highest points. After all, it's only at these especially high numbers that we can see what's really happening at the end of the month.

**Big Reveal: Excel Scatter Plots**

Finally, with a few minutes left in class, I show everyone how to make these plots on Excel. If kids weren't sold on this tool beforehand, this really captures their attention now. They can check their work, see how the software chose to scale the axes, and they get a good look at the shape of an exponential curve. It's a dramatic moment, and again the conversation can go off in any number of directions. I have fun with this, just seeing where kids go and what kinds of ideas we can have.