Mathematics · 9th Grade · Exponential Functions and Finance · Weeks 28-36

Exponential Growth and Decay Models

Identifying the constant percent rate of change in exponential relationships.

• Common Core State Standards
• CCSS.Math.Content.HSF.LE.A.1
• CCSS.Math.Content.HSF.LE.A.1c

About This Topic

Exponential growth and decay involve relationships where a quantity changes by a constant percentage rate over equal intervals of time. Unlike linear growth, which adds the same amount each time, exponential growth multiplies by the same factor. This is a fundamental Common Core standard that models real-world phenomena like population growth, radioactive decay, and viral spread.

Students learn to write equations in the form f(t) = a(1 + r)^t, where 'a' is the initial amount and 'r' is the rate of change. This topic comes alive when students can engage in 'simulation games', like modeling the spread of a rumor or the decay of 'radioactive' dice. Collaborative investigations help students see how small percentage changes can lead to massive differences over time.

Key Questions

1. Differentiate how a constant growth rate differs from a constant growth amount.
2. Explain what determines if an exponential function will grow or decay.
3. Construct how we represent half-life and doubling time mathematically.

Learning Objectives

• Analyze exponential growth and decay scenarios to identify the constant percent rate of change.
• Compare and contrast linear growth (constant amount) with exponential growth (constant percent rate).
• Explain the mathematical conditions that determine whether an exponential function represents growth or decay.
• Construct exponential models representing real-world phenomena involving half-life and doubling time.
• Calculate the future value of an investment or the remaining amount of a substance given an initial value and a constant percent rate of change.

Before You Start

Introduction to Functions

Why: Students need a foundational understanding of what a function is and how to evaluate it before working with exponential function models.

Linear Growth and Decay

Why: Understanding how to identify and model constant additive rates of change provides a necessary contrast for grasping constant multiplicative rates of change.

Percentage Calculations

Why: Students must be proficient in calculating percentages, including finding a percentage of a number and determining percentage increase or decrease, to work with rates of change.

Key Vocabulary

Exponential Growth	A process where a quantity increases by a constant percentage over equal time intervals. The growth factor is greater than 1.
Exponential Decay	A process where a quantity decreases by a constant percentage over equal time intervals. The growth factor is between 0 and 1.
Growth Factor	The constant multiplier applied to a quantity in each time period for exponential growth or decay. It is represented as (1 + r) for growth and (1 - r) for decay.
Rate of Change (r)	The constant percentage by which a quantity increases or decreases over a specific time interval, expressed as a decimal in the exponential model.
Doubling Time	The fixed amount of time it takes for a quantity undergoing exponential growth to double in size.
Half-Life	The fixed amount of time it takes for a quantity undergoing exponential decay to reduce to half of its initial amount.

Watch Out for These Misconceptions

Common Misconception

Students often think that a rate of 5% growth means the 'b' value in the equation is 0.05.

What to Teach Instead

Use the 'Rumor Mill' activity. Peer discussion helps students realize that if you only multiply by 0.05, you are losing 95% of your value. They must use (1 + 0.05) or 1.05 to keep the original amount and add the growth.

Common Misconception

Confusing the 'initial value' (a) with the 'growth factor' (b).

What to Teach Instead

Use the 'M&M Decay' activity. Collaborative analysis helps students see that the number of candies they started with is 'a,' while the percentage that survives each round is 'b,' keeping the roles of the two numbers distinct.

Active Learning Ideas

Simulation Game: The Rumor Mill

One student 'starts' a rumor. Every 30 seconds, everyone who 'knows' the rumor tells two more people. Students track the number of people who know the rumor at each interval, create a table, and discover the exponential growth pattern as the whole class is quickly involved.

25 min·Whole Class

Inquiry Circle: M&M Decay

Groups start with a cup of M&Ms. They shake them and pour them out; any candy with the 'm' facing down is 'decayed' and removed. They repeat this multiple times, recording the remaining candy to model exponential decay and find the 'half-life' of their sample.

40 min·Small Groups

Think-Pair-Share: Growth or Decay?

Give students several equations (e.g., y = 500(1.05)^x and y = 200(0.85)^x). Pairs must identify the starting value and the percentage rate of change for each, and then explain how they know if the function is growing or shrinking.

20 min·Pairs

Real-World Connections

• Biologists use exponential decay models to track the rate at which a medication breaks down in the body, informing dosage recommendations for patients.
• Financial analysts project the future value of investments using exponential growth formulas, considering compound interest rates to advise clients on retirement planning.
• Environmental scientists model the spread of invasive species or the decline of endangered populations using exponential growth and decay principles to inform conservation strategies.

Assessment Ideas

Quick Check

Present students with two scenarios: one describing linear growth (e.g., saving $50 each week) and one describing exponential growth (e.g., a savings account earning 5% interest annually). Ask students to write the formula for each and explain in one sentence why one represents a constant amount of change and the other a constant percent rate of change.

Exit Ticket

Provide students with a data set showing the population of a city over several years. Ask them to: 1. Calculate the approximate annual percent growth rate. 2. Write the exponential function modeling this growth. 3. Predict the population in 5 more years.

Discussion Prompt

Pose the question: 'Imagine you have two options: Option A doubles your money every day for 30 days. Option B gives you $1 million dollars on day 30. Which option would you choose and why?' Facilitate a discussion where students must justify their choice using concepts of exponential growth and doubling time.

Frequently Asked Questions

What is the difference between growth and decay?

In exponential growth, the quantity increases over time because the growth factor is greater than 1. In exponential decay, the quantity decreases because the growth factor is between 0 and 1.

How can active learning help students understand exponential functions?

Active learning strategies like 'M&M Decay' turn an abstract formula into a physical event. When students see their pile of candy shrink by roughly half each time, the concept of a 'constant ratio' becomes a visible reality. This hands-on experience makes the math feel like a description of a natural process, which helps them remember the structure of the equation much better than just memorizing it.

What is 'half-life'?

Half-life is the amount of time it takes for a quantity to decay to exactly half of its original value. It is a common way to describe the rate of decay in things like medicine or radioactive materials.

Why does exponential growth start slow but then speed up?

Because you are multiplying the current amount, not the starting amount. As the total grows, the 'percentage' being added becomes a much larger number, leading to the characteristic 'explosion' of values seen on an exponential graph.