The purpose of this lesson is for students to uncover and understand the formulas for exponential growth and decay using their prior knowledge of exponential functions. I like this task because first students use multiple representations to represent exponential growth and then they are asked to connect their equations with a given formula for compound interest. Next, students try to write their own formulas for exponential decay.
I start class by reading through the first paragraph of Up a Little, Down a Little together. I spend some time talking with students about compound interest so they understand how it works. Students are often surprised to learn that they can make interest on their interest!
Students get to work on modeling the amount of money in Mama Bigbuck's account after t years. Students should be ready to make tables, graphs, and equations to represent this function. They can work in small groups.
I start the discussion by having students share out their representations. My goal for this discussion is for students to be able to connect their work with the formula for exponential growth and to see that the interest they are earning is in the 1 + r part of the formula. I want them to understand why it's 1 plus the decimal of the interest rate. I try to surface the idea in the discussion that they are keeping 100 percent of their money AND making an additional 3%. This idea can be a tricky one for students to grasp so I want to hear lots of different students express this idea in their own words.
Next, we look at the graph of the equation they created on desmos and play around with the interest rate. I ask students how to predict how the curve would change if they interest rate was 5% or 10% compounded annually.
Next I ask students what the opposite of this kind of increase might be. I ask them what would happen if something went down by a certain percent each year. We talk about the word depreciation and students get to work on the next part of the task.
I expect many students to approach the depreciation problem by finding 3% of 20,000 and subtracting from $20,000 and then using the new amount and repeating those same steps. I want to push their thinking here about how much value remains on the car and I ask them if there's a way they can compute that number without thinking about how much money was lost. I am pushing for students to recognize that the car is keeping 97% of its value if it is losing 3%. Students often struggle with this idea, so I might have them make two tables and compare their values.
The final part of the task asks students to create a general formula for depreciation. Here they may need guidance to look back at the growth formula and see how it might be different for depreciation. Again the 1 + r part of the formula can be a useful jumping off point for students. I might ask them what should stay the same and what should change in the exponential growth formula to make it reflect decay instead.
I begin our second discussion by having students share out their formulas. We spend some time talking about how they developed them and what each part represents. We may work through some disagreement and/or confusion about which one correctly represents Mama's car's loss of value. Then we look at a graph of their equation. I put the appreciation and the depreciation graphs side by side and ask students to talk about similarities and differences. I record their comments on a chart. I want students to talk about how the bank account first grows slowly but then grows more quickly and the car first depreciates quickly and then slows down.
It's a fun question to ask students when the car will be worth nothing. I like to let them debate this question a bit before talking about the difference between the mathematical model and what would be a likely scenario in real life.
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