# Lesson 17

Different Compounding Intervals

## 17.1: Returns Over Three Years (5 minutes)

### Warm-up

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.

### Student Facing

Earlier, you learned about a bank account that had an initial balance of \$1,000 and earned 1% monthly interest. Each month, the interest was added to the account and no other deposits or withdrawals were 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?

### Student Response

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### Activity Synthesis

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.

## 17.2: Contemplating Credit Cards (15 minutes)

### Activity

Building on their work in the warm-up, students make several compound-interest calculations in a credit card context. They revisit and explore how nominal and effective interest rates are used by credit institutions.

In a borrowing context, the nominal *annual percentage rate, *or the nominal APR, is 12 times the monthly interest rate. The *effective annual percentage rate *is the compounded interest rate if no payments are made over a year. As in other compounding situations, the nominal APR is lower than the effective annual rate (i.e., the actual rate cardholders pay). For this reason, credit card companies usually report the nominal APR rather than the effective annual rate to make the card more appealing.

Students also practice writing different expressions to represent the same quantity in an exponential situation. Look for these variations in students' work and ask them to share later:

- \(1,\!000 \boldcdot \left((1.02)^{12}\right)^t\)
- \(1,\!000 \boldcdot (1.02)^{12t}\)
- about \(1,\!000 \boldcdot (1.268)^3\)

### Launch

Give students an overview of credit cards, in case students are unfamiliar. Consider using the explanation: Credit card companies allow their clients (the cardholders) to borrow money. In return, the companies charge interest, a percentage of the borrowed amount, until the debt is paid. The percentage charged every year is called the annual percentage rate (APR). The companies allow their card holders to pay incrementally, by making a certain minimum amount of payment every month. Interest is charged on the remaining owed amount (the balance). Many companies charge late fees if the minimum payment is not made.

Ask: A credit card company lists an annual percentage rate of 24%. A cardholder makes \$1,000 of purchases on his credit card. How much interest could he expect to owe if he doesn't pay the company for a year? Poll the class.

Students will have seen the terms *nominal interest rate *and* effective interest rate *from a previous lesson. Remind them as needed.

*Reading, Conversing, Writing: MLR5 Co-Craft Questions.*To help students make sense of the language of mathematical comparisons, start by displaying only the context of the situation without the questions. Give students 1–2 minutes to write their own mathematical questions. Invite students to share their questions with the class, then reveal the activity’s questions. Revoice questions that use unfamiliar or new terms such as “nominal or effective APR” as well as those that seek to generalize for an unspecified time period. This will build student understanding of mathematical language used in a credit card context.

*Design Principle(s): Maximize meta-awareness; Cultivate conversation*

### Student Facing

A credit card company lists a nominal APR (annual percentage rate) of 24% but compounds interest monthly, so it calculates 2% per month.

Suppose a cardholder made \$1,000 worth of purchases using his credit card and made no payments or other purchases. Assume the credit card company does not charge any additional fees other than the interest.

- Write expressions for the balance on the card after 1 month, 2 months, 6 months, and 1 year.
- Write an expression for the balance on the card, in dollars, after \(m\) months without payment.
- How much does the cardholder owe after 1 year without payment? What is the
*effective*APR of this credit card? - Write an expression for the balance on the card, in dollars, after \(t\) years without payment. Be prepared to explain your expression.

### Student Response

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### Student Facing

#### Are you ready for more?

A bank account has an annual interest rate of 12% and an initial balance of \$800. Any earned interest is added to the account, but no other deposits or withdrawals are made. Write an expression for the account balance:

- After 5 years, if interest is compounded \(n\) times per year.
- After \(t\) years, if interest is compounded \(n\) times per year.
- After \(t\) years, with an initial deposit of \(P\) dollars and an annual interest percentage rate of \(r\), compounded \(n\) times per year.

### Student Response

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### Anticipated Misconceptions

For students struggling to work with the expressions in this activity, refer them to the warm-up and the multiple options used to represent the values seen there.

### Activity Synthesis

Select previously identified students to share their expressions for the account balance after \(t\) years. Discuss questions such as:

- “Are all of these expressions equivalent? How do you know?” (Make sure students understand that \(1,\!000 \boldcdot \left(1.02^{12}\right)^t\) and \(1,\!000 \boldcdot \left(1.02^{12t}\right)\) are equivalent, but \(1,\!000 \boldcdot (1.268)^t\) is a convenient approximation for the other two.)
- “Why do you think the credit card company advertises the nominal annual percentage rate rather than the effective annual percentage rate?” (The nominal APR of 24% sounds better because it is less than the effective annual rate of 26.8%. Credit card companies choose the percentage rate so that it works to their advantage. Banks advertise the effective annual percentage for their interest-bearing accounts for the same reason.)

If time permits, elaborate on context: Though they advertise an annual percentage rate (APR), credit card companies (and banks) do not calculate interest on an annual basis. Credit card statements are sent out once a month and calculations of any interest due can be done more frequently, with daily calculations being the shortest time increment that would realistically occur.

*Engagement: Develop Effort and Persistence.*Encourage and support opportunities for peer interactions. Invite students to talk about their ideas with a partner before sharing with the whole class. Display sentence frames to support students when they discuss: “Are all these expressions equivalent?” For example, “ _____ and _____ are alike because….” or “_____ is different because….”

*Supports accessibility for: Language; Social-emotional skills*

## 17.3: Which One Would You Choose? (15 minutes)

### Activity

This task prompts students to build mathematical models to compare two different interest options. Along the way, they need to make assumptions, most notably about the length of the investment as the better investment option may depend on what they assume to be true.

The given rates and compounding intervals are chosen so that it is not clear how the investments would pay for an investment time that is not a multiple of the compounding interval. For example, if the investment is left for 9 months, the option that compounds interest every 3 months will make 3 payouts, but what about the option that compounds interest every 4 months? Students might assume that each method only pays at the *end* of the compounding interval. In that event, the option that compounds interest every 4 month would only pay interest twice.

Notice how students deal with the length of the investment. Invite those who take this variable into account to share during the discussion. Some students may recognize the structural similarity between the situation here and that in the previous activity (i.e. that a 1% monthly rate leads to a higher interest than a 12% annual rate) and reason accordingly.

In asking "What if?" questions, stating their assumptions, and applying what they know about exponential growth to solve a real-world problem, students are engaging in aspects of mathematical modeling (MP4). While the investment context is authentic and practical, the rates in this task are theoretical, as they are chosen to enable students to find and compare effective annual rates.

### Launch

Arrange students in groups of 2-4. If time is limited, consider asking half of each group to analyze the first investment option and the other half to analyze the second option and then discuss their findings.

*Representing, Conversing: MLR7 Compare and Connect.*Ask students to prepare a visual display that shows their mathematical model for each investment option. Students should consider how to display their calculations so that another student can interpret them. Invite students to quietly circulate and read at least 2 other visual displays in the room. Give students quiet think time to consider what is the same and what is different about the displays. Next, ask students to return to their group to discuss what they noticed. Listen for and amplify observations of different time intervals of when one option would be better than the other. This will help students communicate their choice and rationale of their preferred option by using mathematical evidence to support their decision.

*Design Principle(s): Cultivate conversation*

*Action and Expression: Provide Access for Physical Action.*Provide access to tools or assistive technologies, such as a blank table of values, graphing calculator, or graphing software. Some students may benefit from calculating the total balance for each option using multiple time frames.

*Supports accessibility for: Organization; Conceptual processing; Attention*

### Student Facing

Suppose you have $500 to invest and can choose between two investment options.

- Option 1: every 3 months 3% interest is applied to the balance
- Option 2: every 4 months 4% interest is applied to the balance

Which option would you choose? Build a mathematical model for each investment option and use them to support your investment decision. Remember to state your assumptions about the situation.

### Student Response

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### Student Facing

#### Are you ready for more?

Is there a period of time during which the first option (3% interest rate, compounded quarterly) will *always* be the better option? If so, when might it be? If not, why might that be?

### Student Response

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### Anticipated Misconceptions

Students may be unsure about what to do because the length of time of the investment is not given. Encourage them to try out different time frames, or prompt them to choose different time frames for different cases. They should still support any decisions with mathematical reasoning.

### Activity Synthesis

Invite students to share their choice and their rationales. Discuss questions such as:

- What is the nominal annual interest rate for each option? (12%. There are four quarters or four 3-month periods in a year, so the nominal annual rate is 4 times 3% or 12%. There are three 4-month periods in a year, so the nominal annual rate is 3 times 4% or 12%).
- Do they have the same effective annual rate? Why or why not? (No, because the periodic interest is not calculated with the same frequency.)
- How does the length of investments influence your calculations and investment choice?
- Will the 4% investment option ever be the better option? When? (It would be the preferred option if the investment option is, say, 4 months or 8 months.)

Option 2 will be favorable for a length of investment that is a multiple of 4 months but not a multiple of 3 months, up until a certain time, at which point the shorter compounding period in Option 1 will always present an advantage over the higher rate in Option 2. The extension problem prompts students to think about whether the exact length of investment would continue to matter, or whether one option would always outperform the other for some domain.

## 17.4: Changes Over the Years (20 minutes)

### Optional activity

In this activity, students interpret an exponential equation in context. The equation describes college tuition cost as a function of time in years. Students are invited to examine how the tuition changes each decade.

Along the way, students observe that the rate of change is the *same* for any decade and that the time it takes tuition cost to double remains consistent. (If desired, this may be an opportunity to introduce the Rule of 72, a method for estimating the doubling time of a quantity that grows exponentially. In an earlier lesson, when comparing the graphs of the owed amounts at 12%, 24%, and 30% interest rates, students made estimates of when each loan would double. If the Rule of 72 is introduced, consider referring back to that activity to test the rule.) Students may also notice that the current rate of growth for tuition is likely unsustainable.

For the second question, students may reason inductively (by evaluating the expressions for certain 10-year periods) or deductively (by using the structure of the expressions and properties of exponents). Identify students who use each approach so they can share later.

### Student Facing

- The function \(f\) defined by \(f(x)= 15 \boldcdot (1.07)^x\) models the cost of tuition, in thousands of dollars, at a local college \(x\) years since 2017.
- What is the cost of tuition at the college in 2017?
- At what annual percentage rate does the tuition grow?
- Assume that before 2017 the tuition had also been growing at the same rate as after 2017. What was the tuition in 2000? Show your reasoning.
- What was the tuition in 2010?
- What will the tuition be when you graduate from high school?

- Between 2000 and 2010 the tuition nearly doubled.
- By what factor will the tuition grow between 2017 and 2027? Show your reasoning.
- Choose another 10-year period and find the factor by which the tuition grows. Show your reasoning.
- What can you say about how the tuition changes over any 10-year period (assuming the function \(f\) continues to be an accurate model)? Explain or show how you know that this will
*always*be the case.

### Student Response

### Activity Synthesis

Highlight the fact that *every* year, the tuition grows by a factor of \(1.07\) and so every two years it will grow by a factor of \((1.07)^2\), and every ten years it will grow by a factor of \((1.07)^{10}\). In this context, 10-year periods are convenient because \(1.07^{10} \approx 1.97\). So approximately every 10 years, the tuition doubles.

Ask students by what factor the tuition will grow between 2017 and 2037, assuming this trend continues. In thousands of dollars, it would be \(15 \boldcdot (1.07)^{20}\), or (because 20 years is 2 decades) \(15 \boldcdot \left((1.07)^{10}\right)^2\). Since \((1.07)^{10} \approx 2\) this means that in 2037 the tuition would be about $60,000! Another way to see this is that from 2017 to 2027 the tuition will double to $30,000, and then from 2027 to 2037 it doubles again to $60,000.

*Writing, Conversing: MLR1 Stronger and Clearer Each Time.*Use this routine to help students improve their writing by providing them with multiple opportunities to clarify their explanations through conversation. Give students time to meet with 2–3 partners to share their response to the question “Assuming this trend continues, by what factor will the tuition grow between 2017 and 2037? Explain or show your reasoning.” Provide listeners with prompts for feedback that will help their partner add detail to strengthen and clarify their ideas. For example, students can ask their partner, “Can you write that expression in a different way that will give the same result?” or “Will this be true for any interval of 20 years?” Provide students with 3–4 minutes to edit and revise their response. This will help students interpret an exponential equation in the context of college tuition.

*Design Principle(s): Optimize output (for explanation)*

## Lesson Synthesis

### Lesson Synthesis

We looked at how to calculate account balances when the interest is calculated at different time intervals. Review how compounding intervals affect the quantity being studied, and how to represent the different patterns of increase using expressions.

Let's say you have \$1,000 in a bank account that pays 8% annual interest and you make no other deposits or withdrawals.

- Which of these options for calculating interest would you prefer: 8% calculated at the end of one year, 4% calculated every half a year, 2% calculated every quarter (or 3 months), or 1% calculated every month? Why?
- For each option, what expression would you write to represent the balance in the account after one year?
- If the bank calculates 1/12 of the interest every month, will you have 8% of \$1,000 (or \$80) after one year? Why or why not?

## 17.5: Cool-down - How Often Is It Calculated? (5 minutes)

### Cool-Down

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## Student Lesson Summary

### Student Facing

In many situations, interest is calculated more frequently than once a year. How often the interest is compounded and calculated and added to the previous amount affects the overall amount of interest earned (or owed) over time.

Suppose a bank account has a balance of \$1,000 and a nominal annual interest rate of 6% per year. No additional deposits or withdrawals are made.

If the bank compounds interest annually, the account will have one interest calculation in one year, at a 6% rate. If it compounds interest every 6 months, the account will see two interest calculations in one year, at a 3% rate each time (because \(6 \div 2 = 3\)). If it is compounded every 3 months, there will be 4 calculations at 1.5% each time, and so on.

This table shows the nominal interest rates used for different compounding intervals, as well as the corresponding expressions for the account balance in one year.

compounding interval |
compounding frequency per year |
nominal interest rate | account balance in one year |
---|---|---|---|

annually (12 months) | 1 time | 6% | \(1,\!000 \boldcdot (1+0.06)\) |

semi-annually (6 months) | 2 times | 3% | \(1,\!000 \boldcdot (1+0.03)^2\) |

quarterly (3 months) | 4 times | 1.5% | \(1,\!000 \boldcdot (1+0.015)^4\) |

monthly (1 month) | 12 times | 0.5% | \(1,\!000 \boldcdot (1+0.005)^{12}\) |

If we evaluate the expressions, we find these account balances:

- annually: \(1,\!000 \boldcdot (1.06) = 1,\!060\)
- semi-annually: \(1,\!000 \boldcdot (1.03)^2 = 1,\!060.90\)
- quarterly: \(1,\!000 \boldcdot (1.015)^4 \approx 1,\!061.36\)
- monthly: \(1,\!000 \boldcdot (1.005)^{12} \approx 1,\!061.68\)

Notice that the more frequently interest is calculated, the greater the balance is.