Exponential Growth and Decay Models

Exponential growth and decay come alive when students experience the multiplying process firsthand. Active learning helps students move beyond abstract formulas by letting them see how small rates compound over time. Through simulation and hands-on investigation, students build intuition that textbooks alone cannot provide.

• 9th Grade
• Mathematics
• 3 activities
• 20 min–40 min

Learning Objectives

1. Analyze exponential growth and decay scenarios to identify the constant percent rate of change.
2. Compare and contrast linear growth (constant amount) with exponential growth (constant percent rate).
3. Explain the mathematical conditions that determine whether an exponential function represents growth or decay.
4. Construct exponential models representing real-world phenomena involving half-life and doubling time.
5. 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.

Simulation Game

⏱ 25 min·Whole Class

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.

Prepare & details

Differentiate how a constant growth rate differs from a constant growth amount.

Facilitation Tip: During the Rumor Mill, walk around and listen for students to say, 'Each round the number multiplies by 1.05,' not 0.05.

Setup: Flexible space for group stations

Materials: Role cards with goals/resources, Game currency or tokens, Round tracker

• Apply
• Analyze
• Evaluate
• Create
• Social Awareness
• Decision-Making

Inquiry Circle

⏱ 40 min·Small Groups

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.

Prepare & details

Explain what determines if an exponential function will grow or decay.

Facilitation Tip: In the M&M Decay activity, ask groups to hold up their survival ratios before writing the formula to ensure they see a as the starting candies and b as the decay factor.

Setup: Groups at tables with access to source materials

Materials: Source material collection, Inquiry cycle worksheet, Question generation protocol, Findings presentation template

• Analyze
• Evaluate
• Create
• Self-Management
• Self-Awareness

Think-Pair-Share

⏱ 20 min·Pairs

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.

Prepare & details

Construct how we represent half-life and doubling time mathematically.

Facilitation Tip: For Growth or Decay Think-Pair-Share, provide a mix of scenarios and require each pair to justify their choice with both a formula and a verbal explanation.

Setup: Standard classroom seating; students turn to a neighbor

Materials: Discussion prompt (projected or printed), Optional: recording sheet for pairs

• Understand
• Apply
• Analyze
• Self-Awareness
• Relationship Skills

Teaching This Topic

Teach this topic by letting students grapple with the difference between adding and multiplying. Avoid starting with y = ab^x right away. Instead, begin with concrete simulations where students collect and analyze data. Research shows that when students discover the pattern themselves, they retain the concept longer. Emphasize the meaning of b: if b is 0.95, they are keeping 95%, not adding 5%. Use real-world contexts like interest, medicine absorption, or population to ground the abstract math.

What to Expect

By the end of these activities, students should confidently distinguish between linear and exponential change. They should write and interpret exponential equations with correct parameters and explain why constant percentage rates lead to rapid increases or decreases. Successful learning shows up when students naturally use growth and decay factors rather than fixed additions.

Watch Out for These Misconceptions

Common Misconception

During the Rumor Mill simulation, watch for students who write the growth factor as 0.05 instead of 1.05.

What to Teach Instead

In the Rumor Mill activity, redirect students by asking, 'If you start with 100 people and only multiply by 0.05, how many people remain?' Then guide them to see they must multiply by 1.05 to keep the original 100 and add 5%.

Common Misconception

During the M&M Decay collaborative investigation, watch for students confusing the initial number of candies with the decay factor.

What to Teach Instead

In the M&M Decay activity, have students label their data table with 'starting candies = a' and 'percentage surviving = b.' Ask them to calculate the survival factor (e.g., 80% becomes 0.8) and write it next to the formula.

Assessment Ideas

Quick Check

After the Rumor Mill simulation, present students with a linear and exponential scenario. Ask them to write the correct equation for each and explain in one sentence why the exponential model uses a constant percent rate.

Exit Ticket

After the M&M Decay activity, give students a data set showing the number of candies remaining over rounds. Ask them to calculate the decay factor (b), write the exponential equation, and predict how many candies will remain after one more round.

Discussion Prompt

During the Growth or Decay Think-Pair-Share, pose the doubling money question. Have students discuss their choice in pairs, then call on students to share how they used exponential reasoning to justify their selection.

Extensions & Scaffolding

• Challenge early finishers by giving them a scenario with a changing rate (e.g., a rumor spreading at 10% the first day, 15% the second) and ask them to model it with a piecewise exponential function.
• For students who struggle, provide a partially completed table in the M&M Decay activity showing the number of candies remaining but missing the decay factor column.
• Deeper exploration: Have students research a real-world exponential decay process (e.g., drug half-life) and create a presentation explaining how the exponential model applies, including the meaning of the parameters in context.

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.