# Changing compounding intervals, continuous compounding, and the base e

## • Understand functions of the form $f(t) = P(1 + r/n)^{nt}$ which use a given compounding frequency, $n$ (A-SSE.A.1$^\star$).  • Solve problems given different compounding intervals (F-LE.A.2$^\star$).  • Understand (informally) how the base $e$ arises in the context of compounding intervals as $n$ becomes arbitrarily large (F-BF.A.1a$^\star$).

The base $e$ is very commonly used in scientific and other modeling applications. The fundamental reason that $e$ is a useful base for an exponential function is beyond the scope of this course. However, an approach using compound interest that shows $e$ arising as the natural base for a quantity being compounded continuously can serve as a way to develop understanding at this level. First, students must understand functions of the form $f(t) = P(1 + r/n)^{nt}$ which show a given compounding interval, $n$. This section examines that approach.

1 The Bank Account

WHAT: This task describes the basics of an interest-bearing savings account and presents the equation, $B = 500(1+.05/12)^{12t}$. Students are asked to interpret parameters and explain their meaning in context (MP.2).

WHY: The sample activity Loan Ranger carefully develops the idea of different compounding intervals. This task requires students to think in the other direction: here’s a function – interpret it.

2 Compounding with a 5% Interest Rate

WHAT: In this task students study the effect of compounding more and more frequently. They see that the effect tails off and see how an exponential function base $e$ emerges as the limiting case.

WHY: The purpose of this task is to provide a motivation for using the base $e$ in exponential functions that can be understood at the high school level.

## External Resources

1 Loan Ranger

#### Description

WHAT: In this activity students study compound interest with different compounding intervals in the context of credit card debt. They compare the consequences of not paying balances on two cards with different interest rates, observe the effect of adding monthly compounding to the model, and compare the effect of paying the minimum versus paying more, both with regard to how much interest is paid and with regard to how long it takes to pay off the debt.

WHY: In the previous section students saw the form of an expression that describes compounding monthly. In this activity have the opportunity to build exponential functions that describe compounding at any period (F-BF.1ab$^\star$, F-LE.A.2$^\star$). The activity also provides and opportunity to learn more about credit card debt, an important topic for them to know about (MP.4). Finally, the activity prepares them to study continuous compounding and learn about the base e in a following activity.

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