Unit 6, Lesson 16
Compounding Interest
Algebra 1
16.1
Warm-up
Standards Alignment
Building On
Addressing
HSF-BF.A.1.a
Determine an explicit expression, a recursive process, or steps for calculation from a context.
Building Toward
Instructional Routines
NoneMaterials
NoneActivity Narrative
In this Warm-Up, students practice writing an exponential expression for repeated interest calculations. To emphasize the exponential structure of compounded interest, they are prompted to write an expression before calculating a monetary value.
Launch
NoneActivity
NoneYou owe 12% interest each year on a \$500 loan. If you make no payments and take no additional loans, what will the loan balance be after 5 years?
Write an expression to represent the balance and evaluate it to find the answer in dollars.
Student Response
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Building on Student Thinking
Some students may find it difficult to write an expression because they are reasoning about the interest calculation in additive terms, for example, for the first year, for the second year, and so on. Encourage them to think about each year's calculation as a product and to refer to the work in a previous lesson, if needed.
Activity Synthesis
Invite students to share their expressions. If not already brought up in students' responses, bring up the following two expressions:
Highlight that the first expression with the 1.12 written out five times makes explicit the five separate years over which the interest is compounded. The second expression is shorthand for the first and is more succinct.
16.2
Activity
Instructional Routines
5 PracticesMaterials
NoneActivity Narrative
In this activity, students use a geometric context to investigate whether increasing an amount by 10% and then increasing the result by 10% again is the same as applying a 20% increase once. The work addresses a common misconception about successive percent increase. While the two increases (10% twice and 20% once) are in fact relatively close in value, they are not identical. The difference will become more pronounced the more times that the percent increase is applied.
The work prepares students to see how banks and credit card companies actually calculate or compound interest, which is explored in upcoming activities. Rather than charging customers with interest on an annual basis, they calculate and compound the interest on a daily or monthly basis, which typically results in a different actual rate for one year than the published annual interest rate.
The same principle applies for repeated percent decrease. For example, the result of decreasing by 10% three successive times is not the same as decreasing it by 30% once. Recall that earlier in the unit, students have seen that a car losing of its value in any given year (a varying amount) is not the same as losing of its original value every year (a fixed amount).
Monitor for students who:
- Choose a number for the height, apply each method of scaling, then compare the two resulting heights.
- Compare the scale factors for the two methods (think in terms of multiplying the height by for Andre's method and by 1.2 for Mai's method).
- Use a variable for the height, apply each method of scaling, and then compare the resulting expressions.
Plan to have students present in this order to support more abstract thinking.
16.3
Activity
Instructional Routines
5 PracticesMaterials
NoneActivity Narrative
Students now explore the idea of repeated percent change in a banking or savings context. This may be a good time to more formally introduce the term compounding. They examine the effect of compounding 1% interest each month for a year. From the previous activity, students should recognize that the result will be more than 12% interest on the initial balance. This is because for each month after the first one, the amount used to calculate the interest includes previously earned interest, so that amount is higher than the initial balance. In future lessons, students will continue to investigate the effects of compounding interest in the long term.
Monitor for students who use these different strategies to find the account balance at different times (the first question):
- Calculate the numerical values
- Write expressions
Plan to have students present in this order to support strategies that extend to larger numbers more easily.
Note that this activity simplifies what banks actually do to calculate interest rates. Financial institutions commonly calculate interest rates based on the number of days in the month or the year. The monthly interest rate for February, for instance, will be lower than that for March.
Lesson Synthesis
In this lesson, we compared the results of applying a percent increase, , two successive times versus applying twice the percent, , one time. Give students the following example:
Two years ago, a pound of snow crab was priced at \$25 at both Store A and Store B. Last year, Store A increased the price by 20%. This year, it increased the price by 20% again. Store B did not increase the price at all last year, but increased its price by 40% this year.
Discuss questions such as:
- “Are the current snow crab prices at the two stores equal? Why or why not?” (No, because at Store A the second 20% increase is applied to both the original price as well as the price increase from the year before.)
- “Write an expression to represent the current price at each store. Which parts of the expressions are the same? Which parts are different?” (Store A: . Store B: . Both include the original \$25 price, but the way the increase is calculated is different.)
- “In which of the two pricing situations do we see compounding? What does it mean?” (At Store A the increase is compounding because the second time the increase is applied it accounts for the original price as well as the price increase from the first year.)
- “Which option would result in more interest earned on a balance in a bank account: 1% interest calculated monthly or 6% calculated every six months? (Assume no money is added or removed from the account and the balance is left in the account for 6 months.) Why?” (The account that is calculated monthly would result in more interest earned because each month the increase is applied to the original deposit as well as the interest from the previous months. The account with 6 month interest only has the interest applied to the original amount after 6 months.)
Student Lesson Summary
Suppose a runner runs 4 miles a day this month. She is increasing her daily running distance by 25% next month, and then by 25% of that the month after. Will she be running 50% more than her current daily distance two months from now?
It is tempting to think that two months from now she will be running 6 miles, because twice 25% is 50%, and 50% more than her current daily distance is . But if we calculate the increase one month at a time, we can see that next month she will run or 5 miles. The month after that she will run or 6.25 miles.
So two months from now her daily distance will actually be: Two repeated 25% increases actually lead to an overall increase of 56.25% rather than of 50%, because . Applying a percent increase on an amount that has had a prior percent increase is called compounding.
Compounding happens when we calculate interest on money in a bank account or on a loan. An account that earns 2% interest every month does not actually earn 24% a year because the interest applies not only to the original amount, but also to any interest already included earlier. Let's say a savings account has \$300 and no other deposits or withdrawals are made. The account balances after some months are shown in this table.
| number of months | account balance in dollars |
|---|---|
| 1 | |
| 2 | |
| 3 | |
| 12 |
, so the account will grow by about 26.82% in one year. This rate is called the effective interest rate. It reflects how the account balance actually changes after one year.
The 24% is called the nominal interest rate. It is the stated or published rate and is usually used to determine the monthly, weekly, or daily rates (if interest were to be calculated at those intervals).