Lesson 18

In this lesson, students examine different ways to express repeated percent increase. Using rules of exponents, it is possible to group exponential expressions in different ways to highlight different aspects of a situation. For example, if an investment grows by 1% each month, then the effective annual percentage rate earned is about 12.68%, since \((1.01)^{12} \approx 1.1268\). After 6 years, the total interest earned is about 2.05, since \((1.01)^{72} \approx 2.05\). This means that the account balance will double (roughly) every 6 years.

If \$1,000 were invested at this rate, with no further deposits or withdrawals, here are some different ways to see what the balance, in dollars, would be after 20 years:

• \(1,\!000 \boldcdot (1.01)^{240}\)
• \(1,\!000 \boldcdot \left((1.01)^{12}\right)^{20}\)
• about \(1,\!000 \boldcdot (1.268)^{20}\)
• about \(1,\!000 \boldcdot 2^4\)

The expressions highlight different growth rates for different time periods (a month, a year, 5 years).

In this lesson, we distinguish between growth rate (previously known as percent change or interest rate) and growth factor (defined earlier in this unit). In functions of the form \(a \boldcdot (1+r)^x\), the growth rate is \(r\), and the growth factor is \(1+r\).

As students write and work across different expressions, they practice using structure and repeated reasoning (MP7 and MP8). In choosing an expression strategically to highlight a particular aspect of the situation, they reason abstractly and concretely (MP2). 

Technology isn't required for this lesson, but there are opportunities for students to choose to use appropriate technology to solve problems. We recommend making technology available.

Devices that can run Desmos (recommended) or other graphing technology should be avalable as an optional tool for students.