This is the first of several lessons that develop students' ability to use exponential expressions (and eventually functions) to model repeated interest calculations. Students graph the associated exponential functions for different interest rates. Though the term compounding is not yet used with students, the activities help students see that, in the context of borrowing money, high interest rates and compounding do not favor the borrower. In later lessons, students will learn that the same mechanisms can be favorable in the context of saving money.
To represent compounded interest with an exponential expression or an exponential function, students need to realize that the process of repeatedly applying, say, 5% interest \(n\) times, is the same as multiplying by \((1.05)^n\). This is an example of looking for and expressing regularity in repeated reasoning (MP8).
- Calculate the result of applying a percent increase repeatedly and write the result as a numerical value and expression.
- Use graphs to illustrate and compare situations in which different percent increases are applied to the same initial value.
- Write an expression of the form $(1+r)^n$ to represent a percent increase applied $n$ times.
Let's investigate what happens when we repeatedly apply a percent increase to a quantity.
Acquire devices that can run Desmos (recommended) or other graphing technology. It is ideal if each student has their own device. (Desmos is available under Math Tools.)
- I can use graphs to illustrate and compare different percent increases.
- I can write a numerical expression or an algebraic expression to represent the result of applying a percent increase repeatedly.