Explain the relationship between simple interest and linear growth. Explain the relationship between compound interest and exponential growth and the relationship between continuously compounded interest and exponential growth.

### Clarifications

*Clarification 1*: Within the Algebra 1 course, exponential growth is limited to compound interest.

General Information

**Subject Area:**Mathematics (B.E.S.T.)

**Grade:**912

**Strand:**Financial Literacy

**Date Adopted or Revised:**08/20

**Status:**State Board Approved

## Benchmark Instructional Guide

### Connecting Benchmarks/Horizontal Alignment

### Terms from the K-12 Glossary

- Simple Interest

### Vertical Alignment

Previous Benchmarks

Next Benchmarks

### Purpose and Instructional Strategies

In grade 7, students solved problems involving simple interest. In Algebra I, students explain the relationship between simple interest and linear growth and the relationship between compound interest and exponential growth. In later courses, students will extend this to include continuously compounded interest.- In MA.912.FL.3.2, students became familiar with simple and compound interest and how to use the formulas for each to solve real-world problems. In this benchmark, students will make connections between simple interest and linear growth and between compound interest and exponential growth. To help students discover this relationship, consider guiding them to form a table.
- For example, Kianna and Samantha both receive $1,000 cash from graduation gifts from family and friends. They each decide to invest their money in an investment account. Kianna’s investment earns 10% in simple interest. Samantha’s investment earns 10% in compound interest annually. Guide students to create the interest formulas below and use them to create the table below to compare the growth of their investments over time.
- Kianna’s Interest Earned would be represented by $A$ = 1000 · 0.1 · $t$.
- Kianna’s Total Value would be represented by $A$ = 1000(1 + 0.1$t$).
- Samantha’s Interest Earned would be represented by
$A$ = 1000(1 + 0.1)
^{$t$}− 1000. - Samantha’s Total Value would be represented by $A$ = 1000(1 + 0.1)
^{$t$}

- For example, Kianna and Samantha both receive $1,000 cash from graduation gifts from family and friends. They each decide to invest their money in an investment account. Kianna’s investment earns 10% in simple interest. Samantha’s investment earns 10% in compound interest annually. Guide students to create the interest formulas below and use them to create the table below to compare the growth of their investments over time.

- Once completed, ask students what relationships they observe in the behavior of Kianna’s versus Samantha’s investment. Students should quickly discover Kianna’s investment exhibits linear growth while Samantha’s shows exponential growth.
- Solidify this understanding by having students graph the two functions that represent the total value of the two investments.
- Once students make this discovery, begin a conversation with them about which type of interest would be more advantageous for long-term investments. Take this opportunity to make connection to MA.912.F.1.6 (i.e., What if Kianna received $10,000 in gifts? Would the simple interest account be a better investing tool?).
- Remember the expectation for this benchmark is for students to explain why these relationships occur. Be sure to discuss the equations formed and that the variation of years is used as a factor in the simple interest formula and as an exponent in the compound interest formula. .

### Common Misconceptions or Errors

- When forming compound interest equations, students sometimes forget to convert the interest rate from a percent value to a decimal value before substituting it into the formula.

### Strategies to Support Tiered Instruction

- Instruction includes making the connection to determining linear and exponential functions (MA.912.F.1.8) from a financial context.
- For students who need extra support in converting a percentage to a decimal, instruction includes students thinking about percent as “per one-hundred.”
- For example, when writing 8% as a decimal, ask “8% is how many per 100?” Then write 8% = 8/100 which is equivalent to 0.08.

### Instructional Tasks

*Instructional Task 1 (MTR.3.1, MTR.4.1, MTR.5.1, MTR.7.1)*

- Phoenix invests in a savings account that applies simple interest.
- Part A. How will her investment grow, linearly or exponentially? Justify your answer.
- Part B. If Phoenix invests $725 and earns an annual rate of 4.2%, write an equation that would represent the total amount she would have at the end of each year.
- Part C. How long will it take for her initial investment to double?
- Part D. If instead the savings account had interest at the same rate but was compounded annually, how much money would she have after the amount of time found in Part C?

### Instructional Items

*Instructional Item 1*

- Trevarius invests in a savings account that applies compound interest annually. How will his investment grow, linearly or exponentially? Justify your answer.

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

## Related Courses

This benchmark is part of these courses.

1200310: Algebra 1 (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))

1200320: Algebra 1 Honors (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))

1200330: Algebra 2 (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))

1200340: Algebra 2 Honors (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))

1200370: Algebra 1-A (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))

1200380: Algebra 1-B (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))

1200400: Foundational Skills in Mathematics 9-12 (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))

7912070: Access Mathematics for Liberal Arts (Specifically in versions: 2014 - 2015, 2015 - 2018, 2018 - 2019, 2019 - 2022, 2022 - 2023, 2023 and beyond (current))

7912080: Access Algebra 1A (Specifically in versions: 2014 - 2015, 2015 - 2018, 2018 - 2019, 2019 - 2022, 2022 and beyond (current))

7912090: Access Algebra 1B (Specifically in versions: 2014 - 2015, 2015 - 2018, 2018 - 2019, 2019 - 2022, 2022 and beyond (current))

1200315: Algebra 1 for Credit Recovery (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))

1200375: Algebra 1-A for Credit Recovery (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))

1200385: Algebra 1-B for Credit Recovery (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))

7912075: Access Algebra 1 (Specifically in versions: 2014 - 2015, 2015 - 2018, 2018 - 2019, 2019 - 2022, 2022 and beyond (current))

7912095: Access Algebra 2 (Specifically in versions: 2016 - 2018, 2018 - 2019, 2019 - 2022, 2022 and beyond (current))

1207350: Mathematics for College Liberal Arts (Specifically in versions: 2022 and beyond (current))

## Related Access Points

Alternate version of this benchmark for students with significant cognitive disabilities.

MA.912.FL.3.AP.4: Identify the relationship between simple interest and linear growth. Identify the relationship between compound interest and exponential growth.

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