Modeling with the Natural Base, "e" (Lesson 4.6)
Unit 4 Day 8
CED Topic(s): 2.5
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Learning Targets

Describe the effects of compounding interest quarterly, monthly, weekly, daily, and continually and make use of structure to arrive at the compound interest formula.

Use an exponential model to make predictions about the dependent variable.

Understand "e" as the base rate of growth for all continually growing processes
Quick Lesson Plan
Experience First:
Today’s lesson is used to introduce the concept of “e”. Since this course has a heavy focus on modeling, an understanding of why and when the natural base is used in a modeling equation is valuable.
To motivate the introduction of “e”, students start by generating the compound interest formula and think about the effects of compounding interest multiple times a year. To introduce the topic, students are asked a ‘Would You Rather’ question to highlight the possibility of earning interest on your money and thus increasing the value of a bonus. The remainder of the activity focuses on a magical bank with 100% interest. This interest rate was chosen so that students could clearly see the structure of the formula and the limiting factor of “e” when compounding continuously. Students will recognize that 100% interest corresponds with doubling your money and will notice that earning interest multiple times a year actually allows you to more than double your money. When filling out the table in question 3b, many students began by writing $2000 as the final amount in the account after a year. Seeing that this turns out not to be the case is an important stepping stone of this lesson, so allow students to make this mistake and then revise their own thinking instead of telling them their answer is wrong.
To fill in the table for quarterly payouts, students will likely multiply their previous amount by 0.25 and then add this to their previous amount, whereas others might jump immediately to multiplying by 1.25. It is important that students see why these are equivalent in order to make sense of the 1+r portion of the equation.
In the last question, students will likely respond that yes, the money can triple if we simply compound more frequently. Allow students to explore the effects of compounding twice a day or even hourly. Our students did this without additional prompting and realized there was only a slight increase in money. Students realized that the amount earned at the end of a year would “level out” even though they could not pinpoint a specific value. The margin note for part b of question 6 will provide the support for their conjectures.
Monitoring Questions:

Describe how the amount in the account is changing each quarter. Is the amount earned each quarter constant? Growing linearly?

Why does the account not earn 100% interest at each payout?

Why, if the interest rate is being divided by 12, and the number of compoundings is being multiplied by 12, is the final amount not staying the same? Shouldn’t those two things cancel each other out?

What do you notice about how the amount in the account after one year changes as the number of compoundings increases? If the xaxis was the number of compoundings, and the yaxis was the amount in the account after one year, would the graph be increasing or decreasing? Concave up or concave down?

How could you determine the amount in the account after 2 years? After 5 years? After t years?

Can you write a general equation for the amount in the account after t years if interest is compounded n times a year and the interest rate is r?
Formalize Later:
Note that the context of compound interest is used as a vehicle for introducing “e”. The course framework for AP Precalculus does not explicitly mention any compound interest formulas but with the strong focus on modeling, compound interest is a popular contextual scenario for exponential growth. The main goal is not that students would be able to calculate compounded interest given any number of compoundings, but that they would be able to generate an exponential model based on a scenario, that they would be able to use structure to generalize a pattern, and that they would understand why an exponential model would have a natural base.
In the debrief, ask groups to articulate why one doesn’t earn 100% interest at each payout and how they knew what the interest rate per payout would be. The variables of r, n, and t will be defined in the debrief as well as in the Important Ideas. By thinking of the exponent as the number of total payments, students had no trouble generalizing to other values of t besides t=1.
Continue to emphasize that students came up with the compound interest formula on their own and could rely on this thinking process instead of having to memorize new formulas.
Encourage students to find the “e^x” button on their calculator instead of using 2.718 as the base. If students round too early their final answers will be off significantly.