Teach students to compare simple, compound, and continuous interest using formulas, graphs, real-world savings scenarios, activities, assessments, and financial decision-making.

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Money Time Machine: Comparing Simple, Compound, and Continuously Compounded Interest

Materials Needed

Calculator or spreadsheet app
Paper, pencil, and colored pencils
Graph paper or graphing tool
Three labeled containers, envelopes, or jars: Simple, Compound, and Continuous
Coins, play money, or small counters
Interest comparison cards, prepared by the instructor
Optional: computer or tablet for a spreadsheet or graphing calculator

Lesson Overview

Grade/Standard: MA.912.FL.3.1

Length: Two days, 50 minutes per day

Essential Question: How does the way interest is added affect how money grows over time?

Real-World Connection: Banks, savings accounts, credit cards, investments, and loans use different interest models. Understanding these models helps people compare financial choices.

Learning Objectives

By the end of the two-day lesson, the learner will be able to:

Identify and explain the differences among simple, compound, and continuously compounded interest.
Use the correct formula to calculate the future value of an account.
Compare how the three types of interest grow over the same period.
Represent interest growth using tables, graphs, and written explanations.
Use evidence from calculations to recommend the best option for a realistic financial situation.

Success Criteria

The learner is successful when they can:

Correctly identify the variables in each formula.
Substitute values accurately and use appropriate calculator settings.
Calculate at least four out of five comparison problems correctly.
Explain why compound interest eventually grows faster than simple interest.
Explain why continuous compounding produces slightly more growth than periodic compounding at the same stated rate.
Support a financial recommendation with numbers and clear reasoning.

Key Formulas and Vocabulary

Type of Interest	Formula	Meaning
Simple interest	A = P(1 + rt)	Interest is calculated only on the original principal.
Compound interest	A = P(1 + r/n)^nt	Interest is added at regular intervals and earns additional interest.
Continuous compounding	A = Pe^rt	Interest is modeled as being added constantly.

A: final amount
P: principal, or starting amount
r: annual interest rate written as a decimal
t: time in years
n: number of compounding periods per year
e: Euler’s number, approximately 2.71828

Day 1: The Three Ways Money Grows

Introduction: Hook and Objectives — 8 minutes

Show the learner three pretend savings accounts. Place the same starting amount of play money in each container.

Hook: “Imagine that each account starts with $1,000 and earns 6% interest for 10 years. Will all three accounts have the same amount of money? Which account would you choose?”

Ask the learner to make a prediction and briefly explain their reasoning. Record the prediction for review at the end of Day 2.

Tell the learner:

“Today we will learn three ways interest can grow. We will calculate each type, organize the results in a table, and begin comparing which method creates more money over time.”

I Do: Direct Instruction and Modeling — 12 minutes

Step 1: Model Simple Interest

Use the example:

Principal: $1,000 Rate: 6% Time: 10 years

Convert the percentage to a decimal:

6% = 0.06

Substitute into the formula:

A = P(1 + rt)

A = 1,000(1 + 0.06 × 10)

A = 1,000(1.6) = $1,600

Explain that simple interest adds the same amount each year:

$1,000 × 0.06 = $60 per year

Step 2: Model Compound Interest

Suppose the account compounds monthly, so n = 12.

A = P(1 + r/n)^nt

A = 1,000(1 + 0.06/12)¹²⁽¹⁰⁾

A ≈ $1,819.40

Explain that each month’s interest is added to the account, so future interest is calculated on both the original money and previously earned interest.

Step 3: Model Continuous Compounding

A = Pe^rt

A = 1,000e^0.06(10)

A ≈ $1,822.12

Emphasize that continuous compounding does not mean the difference will always be large. It represents the mathematical limit of compounding more and more frequently.

Quick Check

Ask the learner:

Which formula uses the number of compounding periods, n?
Which method adds the same dollar amount each year?
Why is the continuous result slightly larger than the monthly compound result?

We Do: Build a Comparison Table — 15 minutes

Work together to complete the table. Use a calculator and round money to the nearest cent.

Scenario: An account starts with $1,000, earns 6% annually, and grows for 1, 5, 10, and 20 years. Compound interest is compounded monthly.

Time	Simple Interest	Monthly Compound	Continuous Compound
1 year	$1,060.00	$1,061.68	$1,061.84
5 years	$1,300.00	$1,348.85	$1,349.86
10 years	$1,600.00	$1,819.40	$1,822.12
20 years	$2,200.00	$3,309.14	$3,320.12

Discuss the following questions:

Which method grows the least after 20 years?
When does the difference between simple and compound interest become noticeable?
Why is the difference between monthly and continuous compounding relatively small?
What happens when the account is left untouched for a longer period?

You Do: Interest Detective Challenge — 10 minutes

Give the learner the following three account descriptions. The learner should identify the type of interest, write the formula, and calculate the final amount.

$500 earns 4% simple interest for 6 years.
$500 earns 4% interest compounded quarterly for 6 years.
$500 earns 4% interest compounded continuously for 6 years.

Answer check:

Simple: $620.00
Quarterly compound: $635.03, approximately
Continuous: $635.62, approximately

Day 1 Closure — 5 minutes

Ask the learner to complete this exit reflection:

“Simple interest is different from compound interest because…”
“Continuous compounding means…”
“My prediction about the $1,000 account was accurate/inaccurate because…”

Day 2: Compare, Graph, and Make a Financial Recommendation

Introduction and Review — 5 minutes

Review the three formulas using a verbal memory strategy:

Simple: “Principal times one plus rate times time.”
Compound: “Principal times one plus a fraction of the rate, raised to the number of periods.”
Continuous: “Principal times e raised to rate times time.”

Ask the learner to explain, without looking at notes, which method uses a constant increase and which methods allow interest to earn interest.

I Do: Model How to Compare Accounts — 10 minutes

Present this scenario:

“You have $2,000 to save for 15 years. Bank A offers 5% simple interest. Bank B offers 5% compounded monthly. Bank C offers 5% compounded continuously. Which account produces the greatest final balance?”

Model the comparison:

Bank A: Simple interest

A = 2,000(1 + 0.05 × 15) = $3,500.00

Bank B: Monthly compound interest

A = 2,000(1 + 0.05/12)¹²⁽¹⁵⁾ ≈ $4,226.49

Bank C: Continuous compound interest

A = 2,000e^0.05(15) ≈ $4,234.68

Model the written conclusion:

“Bank C produces the greatest balance, approximately $4,234.68. It earns about $734.68 more than the simple-interest account. However, Bank B is very close because monthly compounding approximates continuous compounding.”

We Do: Graph the Growth — 15 minutes

Use the Day 1 comparison table for $1,000 at 6% interest. The learner may create the graph by hand or use a spreadsheet.

Instructions:

Label the horizontal axis “Time in Years.”
Label the vertical axis “Account Value.”
Plot the values for 1, 5, 10, and 20 years.
Use a different color for each interest type.
Connect each set of points with a smooth line.
Write one observation about the shape of each graph.

Guide the learner toward these observations:

Simple interest creates a straight-line pattern.
Compound interest creates an upward-curving pattern.
Continuous compounding also creates an upward-curving pattern and is slightly above periodic compounding.
The longer the time, the more important compounding becomes.

You Do: Financial Choice Project — 15 minutes

The learner becomes a financial advisor. They may choose either scenario.

Choice A: Saving for a Goal

You have $1,500 to save for 12 years. Compare:

4.5% simple interest
4.5% compounded quarterly
4.5% compounded continuously

Choice B: Comparing Investment Offers

You have $3,000 to invest for 8 years. Compare:

3.75% simple interest
3.75% compounded monthly
3.75% compounded continuously

The learner must submit:

The formula used for each option.
The final amount for each option.
The total interest earned for each option.
A table or graph comparing the results.
A recommendation supported by at least two numerical facts.

Formative Assessment During the Project

Pause and ask the learner to answer:

Did you convert the interest rate from a percent to a decimal?
Did you use the correct value for n?
Is your time measured in years?
Does your answer make sense compared with the starting amount?
Which result should be largest, and why?

Conclusion and Summative Assessment — 5 minutes

Ask the learner to explain the lesson in a short “financial news report” or written paragraph:

“Today I compared simple, compound, and continuously compounded interest. Simple interest grows by adding interest only to the original principal. Compound interest adds interest to the account at regular intervals, so the account earns interest on interest. Continuous compounding models interest being added constantly. Over longer periods, compound and continuous interest usually grow faster than simple interest.”

Then administer the following exit assessment.

Exit Assessment

Write the formula for simple interest.
Calculate the final amount for $800 at 5% simple interest for 4 years.
Calculate the final amount for $800 at 5% compounded annually for 4 years.
Calculate the final amount for $800 at 5% compounded continuously for 4 years.
In two or three sentences, explain why the compound and continuous results are different from the simple-interest result.

Exit Assessment Answer Key

A = P(1 + rt)
$960.00
$972.41, approximately
$977.04, approximately
Compound and continuous interest allow previously earned interest to earn additional interest. Over time, this produces more growth than simple interest, which calculates interest only on the original principal.

Assessment Plan

Formative Assessments

Prediction during the opening hook
Formula and vocabulary quick checks
Guided comparison table
Interest Detective Challenge
Graph observations
Teacher or parent questioning during independent work

Summative Assessment

Evaluate the Financial Choice Project and exit assessment.

Category	Points
Correct formulas and substitutions	4
Accurate calculations	4
Clear table or graph	3
Evidence-based recommendation	3
Explanation of the three interest types	3
Total	17

Differentiation and Adaptations

Support for Learners Who Need More Scaffolding

Provide a formula card with each variable labeled.
Use a calculator or spreadsheet to reduce computational demands.
Complete the first substitution step together before the learner continues.
Use whole-number rates and shorter time periods before introducing decimals.
Provide sentence frames such as: “The account with the greatest final amount is ___ because ___.”
Allow the learner to explain comparisons verbally instead of writing a full paragraph.

Extension for Advanced Learners

Compare annual, quarterly, monthly, daily, and continuous compounding at the same rate.
Investigate how long it takes for an account to double using different interest models.
Use logarithms to solve for time in a compound-interest equation.
Explore the effect of changing the interest rate while keeping the principal and time fixed.
Research the difference between nominal annual interest rates and effective annual rates.

Flexible Learning Formats

Hands-on: Use containers and play money to represent interest being added.
Digital: Build a spreadsheet with columns for time, simple interest, compound interest, and continuous interest.
Auditory: Explain each formula aloud using the “financial news report” format.
Visual: Color-code formulas and create a comparison graph.
Kinesthetic: Move counters into the three labeled containers as interest is calculated.

Key Takeaways

Simple interest is calculated only on the original principal.
Compound interest earns interest on the principal and previously earned interest.
Continuous compounding models interest being added constantly.
At the same principal, rate, and time, continuous compounding generally produces slightly more than periodic compounding.
The effects of compounding become more significant as time increases.
Financial decisions should be based on both calculations and the terms of the account.

Simple vs. Compound Interest Lesson Plan