A Program of The Actuarial Foundation. Aligned with Common Core State and NCTM Standards.

What is an actuary? An actuary is an expert in statistics who works with businesses, governments, and organizations to help them plan for the future. Actuarial science is the discipline that applies math and statistical methods to assess risk.

4-8

DURATION
30

COLLECTION
Solving the Unknown with Algebra

# Lesson 2: Analyzing Change/Growth and Decay Formula

In this lesson, students will be able to identify what interest is as it pertains to saving and investing, as well as calculate simple and compound interest.

STANDARDS (CCSS and NCTM):

• Grade 6: Percent, formula, financial literacy, rates, exponents (CCSS 6.RP.A.3.c, 6.EE.A.1 & 2.c)
• Grade 7: Proportional relationships, formula, financial literacy, simple interest, compound interest (CCSS 7.RP.A.3, 7.EE.B.3)
• Grades 6–8: (CCSS MP1, 2, 4, 6 & 8); NCTM Algebra

TIME REQUIRED: 30–40 minutes, depending on class review needs; additional time for worksheets

OBJECTIVE
Students will be able to

• Define percent, interest, interest rate, and principal.
• Find interest using a percent of a quantity, or rate per 100.
• Calculate simple and compound interest by applying formulas, including those with exponents.

MATERIALS

• Ads from financial institutions indicating interest rates paid on savings and investment accounts
• Student calculators (for Worksheet 2; can also be done without calculators)
• Worksheet 2 (PDF) — Content Connections: Percentages, Exponents
• Bonus Worksheet 2 (PDF) — Content Connections: Conversion of Percentages to Decimals, Exponents
• Take-Home Activity 2 (PDF) — Content Connections: Percentages, Exponents
• Resource: Mini-Poster (PDF)

DIRECTIONS

INTRODUCTION TO NEW MATERIAL

1. Show a savings or investment ad to the class. Ask students to explain what the interest rate (e.g., 1.75%) means. Explain that the interest rate tells how much money the financial institution gives to the depositor in return for depositing money with the institution.

2. Define financial-literacy vocabulary for students:

• percent: a rate per 100
• interest: money paid in return for lending or investing money
• interest rate: how much money, per \$100 loaned or invested, will be paid in return for lending or investing money
• principal: the original amount of money loaned or invested
3. Revisit the savings or investment ad displayed at the beginning of the lesson. Ask: If 1.75% was paid to a lender for investing \$400, how much interest would the lender get? How do you know? (\$7; to find the percent of an amount, multiply the amount by the percent written as a decimal. To convert a percent into a decimal, divide the percent by 100, or simply move the decimal point two places to the left.) Ask students to calculate the amount of interest that would be paid to a lender if the lender invested \$600, \$900, or \$2,000. (\$10.50, \$15.75, or \$35)

4. Tell students that interest isn’t collected just one time. Instead, it can grow over many years. The formula I = p • r • t, where I = interest, p = principal, r = interest rate (in decimal form), and t = time (in years), determines how much interest is collected over a number of years. Demonstrate how to substitute a given value in the formula for a variable. For instance, if a person deposits \$1,000, you can substitute 1,000 into the formula for p: I = 1,000 • r • t. You can substitute given values for any variables in the formula.

5. Using the example of a \$1,000 CD deposited for one year with a 2% interest rate, show the calculation \$1,000 • 0.02 • 1 = \$20. Ask: If the depositor earns \$20 in interest, how much money does the depositor now have? (\$1,020) What general formula might show the total amount a depositor has after earning interest? (Principal + Interest, or Principal + Principal • Rate • Time) Tell students that interest also comes into play when a financial institution loans money. In these cases, the borrower pays interest to the financial institution. When people use credit cards and pay them back over time, they need to pay back the amount they spent along with interest.

6. Explain to the class that the interest rate is in effect the entire term of the CD. Ask: What would happen if the bank offered a two-year CD? (The bank will pay the interest¬, \$20, at the end of each year. So, a total of \$40 in interest will be paid.) Demonstrate for students that this follows the formula by using 2 for t in the formula: I = prtI = \$1,000 • 0.02 • 2 = \$40. To show how much the depositor has in total, add the interest to the principal: \$1,000 + \$40 = \$1,040.

7. Have students calculate the amount of interest the depositor earns after 3, 4, 5, and 6 years. (\$60, \$80, \$100, \$120) Ask: Is the amount of interest earned proportional to the number of years it is invested? How do you know? (The amount of interest is proportional. Students might suggest a few ways to determine this, such as demonstrating the growth in a table or graph or noting that the equation for interest is proportional. If your students would benefit from a further discussion about this, you may want to take the time to draw the corresponding table or graph, or discuss how to find the constant of proportionality in the interest equation.)

8. Ask what would happen if the depositor kept the first year’s interest in the account to “grow.” Show students how the principal for a second year would turn into \$1,020 instead of \$1,000. Ask: How can you use the formula to determine how much interest would the depositor earn in the second year? (I = prtI = 1,020 • 0.02 • 1 = \$20.40; the depositor would earn an extra 40¢.) Explain that this is an example of compound interest.

9. Indicate that there is a formula that can be used to calculate how money grows with compound interest: y = a(1+r)n where y = ending value, a = principal, r = interest rate (in decimal form), and n = the number of time periods, which at this stage of development can be considered time in years. The amount of interest can be determined by subtracting the principal from the ending principal amount. Provide students with a number of combinations of a, r, and n to work with to practice the skill.

GUIDED PRACTICE:

10. Have pairs of students come up with examples of principals, interest rates, and numbers of years to work with. Have them calculate and compare the simple interest versus the compound interest earned in these situations. Have them describe the patterns they see. For example, what makes the difference between simple interest and compound interest greatest, a large principal, a large interest rate, or a large amount of time?

INDEPENDENT PRACTICE

11. Distribute Worksheet 2: A Case of Interest (PDF) and classroom calculators. Read the introduction with the class and review the key facts before students complete the worksheet. Review answers as a class. Worksheets Answer Key (PDF)

12. Students will build on these skills in Bonus Worksheet 2: The Case of the Smelly Sandwich (PDF) and Take-Home Activity 2: The Case of the Decaying Car (PDF). If students require additional support, review worksheets as a class. Worksheets Answer Key (PDF)

EMAIL THIS