Lesson 17

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.

Arrange students in groups of 2.

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.

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 \(1,\!000 \boldcdot (1.268)^3\)

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.

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.

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.

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.

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.

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.

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.

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.

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.