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# Standard #: MA.912.FL.3.1

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Compare simple, compound and continuously compounded interest over time.

### Clarifications

Clarification 1: Instruction includes taking into consideration the annual percentage rate (APR) when comparing simple and compound interest.

### General Information

Subject Area: Mathematics (B.E.S.T.)
Strand: Financial Literacy
Status: State Board Approved

### Terms from the K-12 Glossary

• Exponential function
• Simple interest

### Vertical Alignment

Previous Benchmarks

Next Benchmarks

### Purpose and Instructional Strategies

In Algebra I, students solved problems involving simple and compound interest, using arithmetic operations and graphing. In Math for College Liberal Arts, students will compare simple, compound and continuously compounded interest over time in a variety of real-world contexts. In later courses, students will describe the advantages and disadvantages of financial and investment plans.
• In this benchmark, students will describe the difference between simple and compound interest. Instruction should feature a variety of real-world contexts related to money and business.
• The simple interest formula, $I$ = $p$$r$$t$, calculates only the interest earned over time. Each year’s interest is calculated from the initial principal, not the total value of the investment of that point in time. Lead students to generate the formula by exploring patterns (MTR.5.1). Considering the following:
• Jalen invests \$100 in a bank account that pays 5% simple interest each year. How much interest does Jalen earn in 15 years? What is the total value of his investment in 15 years?
• Have students begin exploration by creating a table like the one below.

• Be sure to highlight the difference between the simple interest formula and those for compound interest. The formula $I$ = $p$$r$$t$ only calculates the interest, whereas the compound interest formulas calculate the worth of the total investment
• The simple interest amount formula, $A$ = $P$(1 + $r$$t$), calculates the total value of an investment over time.
• The compound interest formula, $A$ = $P$(1 + $\frac{\text{r}}{\text{n}}$)$n$$t$also calculates the total value of the investment over time. Each month/year’s interest is calculated from the total value of the investment of that point in time. Lead students to generate the formula by exploring patterns (MTR.5.1).
Consider the following:
• Jalen invests \$100 in a bank account that pays 5% interest compounded quarterly. How much interest does Jalen earn in 15 years? What is the total value of his investment in 15 years?
• Guide students through an exploration beginning with the simple interest formula.
• The interest for one quarter would be
$I$ = $p$$r$$t$
$I$ = (100)(0.05)($14$)
$I$ = 100($0.054$)
• The total amount of the investment after one quarter would then be
$A$= 100 + 100($0.054$)
$A$1 = 100(1 + $0.054$)
• The value of the investment after 2 quarters, $A$2, would be
$A$$A$1 + $A$1($0.054$)
$A$$A$1(1 + $0.054$)
$A$= 100(1 + $\frac{\text{0.05}}{\text{4}}$)(1 + $\frac{\text{0.05}}{\text{4}}$)
$A$100(1 + $\frac{\text{0.05}}{\text{4}}$)2
• The value of the investment after 3 quarters, $A$3, would be
$A$$A$2 + $A$2($0.054$)
$A$= $A$2(1 + $0.054$)
$A$= 100(1 + $\frac{\text{0.05}}{\text{4}}$)(1 + $\frac{\text{0.05}}{\text{4}}$)(1 + $\frac{\text{0.05}}{\text{4}}$)
$A$100(1 + $\frac{\text{0.05}}{\text{4}}$)3
• The value of the investment after 4 quarters, $A$4, would be
$A$$A$3 + $A$3($0.054$)
$A$$A$3(1 + $0.054$)
$A$= 100(1 + $\frac{\text{0.05}}{\text{4}}$)(1 + $\frac{\text{0.05}}{\text{4}}$)(1 + $\frac{\text{0.05}}{\text{4}}$)(1 + $\frac{\text{0.05}}{\text{4}}$)
$A$100(1 + $\frac{\text{0.05}}{\text{4}}$)4
• At this point, student should see the connection between the exponent 4 and the number of quarters the investment has accumulated interest. Speed the exploration up by moving to 8, 12 and 16 quarters to build the idea of the exponent being the product of the number of years and the number of times the investment compounds each year.
• After 8 quarters (2 years), $A$8, would be
$A$100(1 + $\frac{\text{0.05}}{\text{4}}$)8
$A$8 100(1 + $\frac{\text{0.05}}{\text{4}}$)4(2)
• After 12 quarters (3 years), $A$12, would be
$A$12 100(1 + $\frac{\text{0.05}}{\text{4}}$)12
$A$12 100(1 + $\frac{\text{0.05}}{\text{4}}$)4(3)
• Students should now see the pattern emerge for the compound interest formula. Have them replace the numbers in the equation to generate $A$ = $P$(1 + $\frac{\text{r}}{\text{n}}$)$n$$t$
• Ask students how they would describe the difference between the simple and compound interest to someone who does not know (MTR.4.1)
• Instruction guides students to consider the concept of continuous compounding by examining investment situations where compounding is done increasingly more often (MTR.5.1)
• The continuously compounded interest formula, $A$ = $P$$e$$r$$t$, calculates the total value of an investment over time. Instead of calculating interest on a finite number of compounding periods (e.g., monthly, annually, daily), continuous compounding calculates interest assuming a constant compounding over an infinite number of compounding periods. Continuously compounded interest assumes interest is compounded and added to the amount accrued an infinite number of times.
• Instruction includes defining continuous compounding. This includes, but is not limited to daily, weekly, monthly, quarterly, semiannually and annually.
• Instruction includes comparison of simple, compound and continuously compounded interest using various representations, such as graphs, tables, equations and written description. Students will be guided to interpret key features of linear and exponential relationships in terms of context to connect to simple and compound interest problems.
• Instruction includes the Annual Percentage Yield (AYP), $Y$ = (1 + $\frac{\text{r}}{\text{n}}$)$n$ − 1. The Annual Percentage Yield allows you to compare compound interest rates with differing compounding periods.
• For example, a student may be asked to determine which a better investment is: 1.28% compounded weekly or 1.35% compounded monthly.
The better investment would be compounded monthly at 1.35%
• Instruction includes the discussion of Annual Percentage Rate (APR) – the total interest that will be paid in the year. If the interest is paid in smaller time increments, the APR will be divided up.

### Common Misconceptions or Errors

• Many institutions advertise investments by using the annual percentage yield (APY) rather than the annual percentage rate (APR) since it is greater for investments that compound multiple times annually. Students should be careful when analyzing investments to see which percentage is being advertised.
• For simple interest scenarios, students may forget to add the principal to the interest they calculate using the simple interest formula when calculating the total amount of an investment over time.
• For compound interest scenarios, students may forget to subtract the principal from the total amount they calculate using a compound interest formula when calculating the total interest earned over time.
• Students may confuse the compounding period variable with time variable.
• Students may assume continuously compounding accounts will have a substantially larger returns, when in fact, the difference in the total interest earned through continuous compounding is not very high when compared to traditional compounding periods.
• Students may incorrectly use the rate.
• For example, students may be given the rate of 4.2% and they use 4.2 instead of 0.042. To address this misconception, students should use multiple representations of percentages.

• A principal investment of \$5,000 had an interest rate of 3%.
• Part A. Complete the table below.

• Part B. After 5 years, what is the best way to invest the money?
• Part C. Which investment will have the greatest value in 15 years? Explain why.
• Part D. Which investment will have the least value in 15 years? Explain why.

### Instructional Items

Instructional Item 1
• To the nearest cent, how much more does an investment of \$3500 earn over 10 years, compounded continuously at 3.5%, than a \$3500 investment over 10 years, compounded quarterly at 3.5%?

Instructional Item 2
• Which is the better financial investment: 3.85% compounded weekly or 3.74% compounded monthly?

*The strategies, tasks and items included in the B1G-M are examples and should not be considered comprehensive.

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#### Related Access Points

 Access Point Number Access Point Title MA.912.FL.3.AP.1 Compare simple and compound interest over time.

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