Lesson 17: Different Compounding Intervals

Different Compounding Intervals

5 min

Narrative

The Warm-up prompts students to think about equivalent ways to express a compounded interest calculation. The given expressions describe the same initial investment for the same period of time, but one expression counts the number of months while the other counts the number of years. This work prepares students to look at different compounding intervals in this lesson and upcoming ones.

If time is limited, focus only on the first question.

Launch

Arrange students in groups of 2. This situation was originally mentioned in an earlier lesson. Remind students that they worked with this situation in a previous activity.

Student Task

$1,000 is deposited into a bank account that earns 1% interest each month. Each month, the interest is added to the account, and no other deposits or withdrawals are made.

To calculate the account balance in dollars after 3 years, Elena wrote: $1,000 \boldcdot (1.01)^{36}$ and Tyler wrote: $1,000 \boldcdot \left((1.01)^{12}\right)^3$ .

Discuss with a partner:

Why do Elena's expression and Tyler's expression both represent the account balance correctly?
Kiran said, "The account balance is about $1,000 \boldcdot (1.1268)^3$ ." Do you agree? Why or why not?

Sample Response

Sample explanation: There are 36 months in 3 years. Since interest is compounded monthly at 1%, we can multiply the initial balance by 1.01 thirty-six times. This is what Elena did by writing $1,000 \boldcdot (1.01)^{36}$ . Tyler's expression is also correct because a property of exponents tells us that when raising a power to a power we can multiply the exponents, so $\left((1.01)^{12}\right)^3 =(1.01)^{12 \boldcdot 3} =(1.01)^{36}$ .
Agree. Sample reasoning: $1.01^{12} \approx 1.1268250$ , so the effective interest rate for one year is approximately 12.68%. The balance after 3 years is therefore about $1,000 \boldcdot (1.1268)^3$ .

Activity Synthesis (Teacher Notes)

Invite groups to share their explanations for why Tyler's and Elena's expressions both represent the account balance after 3 years. If not already mentioned in students explanations, highlight the following:

In Elena's expression, $(1.01)^{36}$ correctly represents the 1% interest applied or compounded every month for 36 months, which is 3 years.
In Tyler's expression, $(1.01)^{12}$ represents the 1% interest compounded every month for 12 months or a year. So for 3 years, we need to multiply the initial balance by that yearly rate 3 times, or $(1.01)^{12} \boldcdot(1.01)^{12} \boldcdot(1.01)^{12}$ , which is equivalent to $\left((1.01)^{12}\right)^3$ .
By the power of powers property, we also know that $\left((1.01)^{12}\right)^3 =(1.01)^{36}$ .
(If time permits:) Evaluating $(1.01)^{12}$ gives approximately 1.1268. This means that 12.68 is the effective annual rate, which is the rate that Kiran used in his expression.

Standards

Addressing

A-SSE.A·Interpret the structure of expressions
HSA-SSE.A·Interpret the structure of expressions.

15 min

Knowledge Components

All skills for this lesson

No KCs tagged for this lesson

Different Compounding Intervals

5 min

Narrative

If time is limited, focus only on the first question.

Launch

Arrange students in groups of 2. This situation was originally mentioned in an earlier lesson. Remind students that they worked with this situation in a previous activity.

Student Task

$1,000 is deposited into a bank account that earns 1% interest each month. Each month, the interest is added to the account, and no other deposits or withdrawals are made.

To calculate the account balance in dollars after 3 years, Elena wrote: $1,000 \boldcdot (1.01)^{36}$ and Tyler wrote: $1,000 \boldcdot \left((1.01)^{12}\right)^3$ .

Discuss with a partner:

Why do Elena's expression and Tyler's expression both represent the account balance correctly?
Kiran said, "The account balance is about $1,000 \boldcdot (1.1268)^3$ ." Do you agree? Why or why not?

Sample Response

Sample explanation: There are 36 months in 3 years. Since interest is compounded monthly at 1%, we can multiply the initial balance by 1.01 thirty-six times. This is what Elena did by writing $1,000 \boldcdot (1.01)^{36}$ . Tyler's expression is also correct because a property of exponents tells us that when raising a power to a power we can multiply the exponents, so $\left((1.01)^{12}\right)^3 =(1.01)^{12 \boldcdot 3} =(1.01)^{36}$ .
Agree. Sample reasoning: $1.01^{12} \approx 1.1268250$ , so the effective interest rate for one year is approximately 12.68%. The balance after 3 years is therefore about $1,000 \boldcdot (1.1268)^3$ .

Activity Synthesis (Teacher Notes)

In Elena's expression, $(1.01)^{36}$ correctly represents the 1% interest applied or compounded every month for 36 months, which is 3 years.
In Tyler's expression, $(1.01)^{12}$ represents the 1% interest compounded every month for 12 months or a year. So for 3 years, we need to multiply the initial balance by that yearly rate 3 times, or $(1.01)^{12} \boldcdot(1.01)^{12} \boldcdot(1.01)^{12}$ , which is equivalent to $\left((1.01)^{12}\right)^3$ .
By the power of powers property, we also know that $\left((1.01)^{12}\right)^3 =(1.01)^{36}$ .
(If time permits:) Evaluating $(1.01)^{12}$ gives approximately 1.1268. This means that 12.68 is the effective annual rate, which is the rate that Kiran used in his expression.

Standards

Addressing

A-SSE.A·Interpret the structure of expressions
HSA-SSE.A·Interpret the structure of expressions.

Different Compounding Intervals

Warm-upReturns over Three Years5 minKCs

Narrative

Launch

Student Task

Sample Response

ActivityContemplating Credit Cards15 minKCs

ActivityWhich One Would You Choose?15 minKCs

Knowledge Components

Different Compounding Intervals

Warm-upReturns over Three Years5 minKCs

Narrative

Launch

Student Task

Sample Response

ActivityContemplating Credit Cards15 minKCs

ActivityWhich One Would You Choose?15 minKCs

5 min

15 min

15 min

5 min

15 min

15 min