In today's task, students are asked to compare rates of growth between a linear function and an exponential function given a real world example about net income. Because the task uses some large number and because I know my students will struggle with the percent increase for the exponential function, I use a Warm Up for today's class that will get them ready for the more complex task to follow.
The upside to the Warm Up is that I think my students really need it and it will help scaffold the central task for today. The downside is that sometimes it eats up a lot of time and the lesson gets stretched to two days. In a different class setting, I might skip the Warm Up and get right to the main event.
I start class by projecting the Potato Chips Warm Up on the Smartboard and handing each student a copy. My focus for the Warm Up is for students to see that rather than finding 10% of the price of the potato chips and adding it to the previous price, they can instead multiply by 1.10 each time. This repeated multiplication should be familiar to them from our previous work on Geometric Sequences. I try to explain this 1.10 factor as the chips keeping 100% of their value and increasing by 10%. If students struggle with understanding, I sometimes let them do it out the long way (finding 10% and adding) and then show them the two resulting tables. We can also use the table they generated the long way and look at the ratios in the out column to find the common factor.
Now we're ready for the main task of the lesson. We read through Getting Down to Business together. Students may need some clarification about what "net income" means so we'll have a short discussion about that. We also need to discuss how we are going to work with these large numbers. I like to let students decide, but I might push them toward doing everything in millions to make the numbers a little easier to work with and to help them limit small math mistakes (miscounting zeros). They may need reminding that if they do work in millions this will change how they write their equations and they'll have to keep that in mind as they get answers.
Next, I let students get to work in pairs or small groups on parts a, b, and c. In part a they will write a recursive formula for each company and this should be pretty straightforward for students at this point. I anticipate that it will be harder for them to write the explicit equations and I'll be supporting them through this part of the work (and encouraging them to work together). Once students are down with Questions a through c, we will have a mini discussion as a whole group before they tackle Question d.
To start the group discussion I will have different students share out the various recursive and explicit equations they have written. The discussion will follow the Getting Down to Business. Our work here will focus on how students can tell from the two kinds of equations which one is linear and which is exponential? I will be asking them questions like "Where do you see the linear rate of change in this equation?" We will compare and contrast both sets of equations and spend some time talking about whether or not they should be discrete or continuous. I will try to get students to make arguments for both kinds of functions.
Next we'll preview and think aloud about Question d. Question d. asks students in what year the net income of both companies will be the same. I want students to think together about how they might answer that problem. I'm hoping to elicit ideas about making tables and graphs (the other two representations we have been working with regularly).
This is really the meat of the lesson where students are exploring the functions to see how they grow, when they cross, and how much each company make at x years. I let students work together in paris and small groups. I watch out for the following issues:
I anticipate that this work will take us right to the end of today's class. We will have to discuss their findings at the start of the next class. I will want to spend time working with them on plugging in years to the explicit functions to predict how much money each company will have after a certain number of years.
Depending on where they end up, I might ask them a closing question to give them something to think about. If they have not quite found where the two companies make the same amount of net income, I might ask them to make a prediction. If they have already found that point, I might ask them how big the difference in net income between the companies will be in 20 years.
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