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 (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:

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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 years. Discuss questions such as:

• “Are all of these expressions equivalent? How do you know?” (Make sure students understand that and are equivalent, but that  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.)

Math Community

Invite 2–3 students to share the norm they chose and how it supported the work of the group or a realization they had about a norm that would have worked better in this situation. Provide these sentence frames to help students organize their thoughts in a clear, precise way:

• “I picked the norm ‘.’ It really helped me/my group because .”
• “I picked the norm ‘.’ During the activity, I realized that the norm ‘’ would be a better focus because .”

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 because 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 pays only at the end of the compounding interval. In that event, the option that compounds interest every 4 months would pay interest only twice.

Notice how students deal with the length of the investment. Some students may recognize the structural similarity between the situation here and that in the previous activity (that a 1% monthly rate leads to a higher interest than a 12% annual rate does) 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, because they are chosen to enable students to find and compare effective annual rates.

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. Students 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?" (Because the interest is given on only certain months, the balances jump up at certain times rather than changing smoothly. This means that if I’m planning to take out the money at a certain time, I need to choose the better option based on how long the money has been invested.)
• "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 , and so every two years it will grow by a factor of , and every ten years it will grow by a factor of . In this context, 10-year periods are convenient because . 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 , or (because 20 years is 2 decades) . Because  , 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.