Growth and Decay Problems

Active learning helps students grasp exponential growth and decay because repetitive calculations and visual comparisons reveal patterns that abstract formulas alone cannot. By handling physical or graphical representations, students move from memorizing formulas to understanding why values double or shrink toward zero over time.

• Year 11
• Mathematics
• 4 activities
• 20 min–35 min

Learning Objectives

1. Calculate the final value of a quantity after multiple periods of exponential growth or decay using percentage multipliers.
2. Analyze how the initial value and the percentage multiplier influence the rate of exponential change in real-world scenarios.
3. Design a mathematical model to represent a given scenario of compound interest or radioactive decay.
4. Compare the long-term behavior of quantities undergoing different rates of exponential growth or decay.
5. Explain the relationship between percentage change and the corresponding multiplier for both growth and decay.

Simulation Game

⏱ 25 min·Pairs

Pairs: Multiplier Card Sort

Provide cards with scenarios, multipliers, and tables. Pairs match them, then calculate three iterations for each and plot points on mini-graphs. Partners swap sets to verify calculations and discuss patterns.

Prepare & details

Analyze the factors that influence the rate of exponential growth or decay.

Facilitation Tip: During Multiplier Card Sort, circulate and ask pairs to justify their matching choices using real-world examples like interest rates or decay rates.

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

Simulation Game

⏱ 35 min·Small Groups

Small Groups: Decay Simulation with Counters

Groups start with 100 counters representing atoms. Each round, remove a percentage based on the multiplier and record remaining. After five rounds, plot results and compare to calculated values.

Prepare & details

Predict the long-term behavior of a quantity undergoing exponential change.

Facilitation Tip: In Decay Simulation with Counters, ensure groups record each step’s count and percentage retained to make the gradual decay visible.

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

Simulation Game

⏱ 30 min·Whole Class

Whole Class: Prediction Relay

Divide class into teams. Teacher states a scenario and multiplier; first student calculates one step, tags next for the following step. Teams race to ten iterations, then graph and predict long-term trend.

Prepare & details

Design a mathematical model for a given real-world growth or decay scenario.

Facilitation Tip: In Prediction Relay, pause after each round to have students compare their predicted values with actual calculations before proceeding.

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

Simulation Game

⏱ 20 min·Individual

Individual: Real-World Model Builder

Students select a scenario like phone depreciation, choose a multiplier, and create a table or graph for five years. They write a short justification linking to rate factors.

Prepare & details

Analyze the factors that influence the rate of exponential growth or decay.

Facilitation Tip: For Real-World Model Builder, provide scaffolding tables so students focus on setting up equations rather than formatting data.

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

Teaching This Topic

Teach this topic by starting with concrete examples before introducing formulas. Use hands-on simulations to build intuition, then scaffold toward abstract reasoning. Avoid rushing to formulas; let students discover the power of repeated multiplication through guided discovery. Research shows that students better retain concepts when they experience the process physically before formalizing it.

What to Expect

Successful learning looks like students confidently selecting the correct multiplier for given percentage changes and articulating how repeated multiplication leads to non-linear patterns. They should connect real-world contexts to their calculations and explain why exponential change behaves differently from linear change.

Watch Out for These Misconceptions

Common Misconception

During Multiplier Card Sort, watch for students who group linear and exponential multipliers together, such as pairing 0.9 with a 10% decrease and 0.1 with a 90% decrease.

What to Teach Instead

Have pairs plot their matched multipliers on a number line and compare the resulting percentage changes. Ask them to explain why a multiplier of 0.9 corresponds to a 10% decrease but 0.1 would correspond to a 90% decrease, reinforcing the difference between iterative and fixed changes.

Common Misconception

During Decay Simulation with Counters, watch for students who assume the quantity halves each step when using a 50% decay rate.

What to Teach Instead

Ask groups to count the counters after each step and calculate the percentage retained. Direct them to notice that 50% decay means retaining 50% of the previous amount, not halving a fixed starting quantity.

Common Misconception

During Prediction Relay, watch for students who stop calculating after a few steps, assuming the pattern remains linear.

What to Teach Instead

After each round, have students graph their predictions and compare them to the exponential curve. Ask them to explain why the gap between their predictions and the actual values widens, reinforcing the concept of unbounded growth or asymptotic decay.

Assessment Ideas

Quick Check

After Multiplier Card Sort, present students with two scenarios and ask them to calculate values after 3 periods and compare growth rates. Collect their work to check for correct multiplier selection and accurate calculations.

Exit Ticket

After Decay Simulation with Counters, provide students with a scenario about depreciation and ask them to write the percentage multiplier and calculate the value after 2 years. Use their responses to assess understanding of percentage multipliers and iterative decay.

Discussion Prompt

During Prediction Relay, pose the question about Town A and Town B and ask students to justify their answers using calculations and reasoning about types of growth. Listen for explanations that distinguish linear from exponential growth and assess their ability to compare long-term behaviors.

Extensions & Scaffolding

• Challenge students who finish early to create a scenario where growth and decay interact, such as a population growing while resources decay exponentially.
• For students who struggle, provide pre-labeled graphs showing linear and exponential curves and ask them to match scenarios to the correct graph.
• Deeper exploration: Ask students to research and present on how compound interest or radioactive decay applies to careers like finance or nuclear medicine.

Key Vocabulary

Percentage Multiplier	A number used to increase or decrease a quantity by a fixed percentage in one step. For growth, it's greater than 1 (e.g., 1.05 for 5% growth); for decay, it's less than 1 (e.g., 0.95 for 5% decay).
Exponential Growth	A pattern where a quantity increases at a rate proportional to its current value, resulting in increasingly rapid increases over time.
Exponential Decay	A pattern where a quantity decreases at a rate proportional to its current value, resulting in increasingly slower decreases over time, often approaching zero.
Compound Interest	Interest calculated on the initial principal and also on the accumulated interest of previous periods, leading to exponential growth of the investment.
Depreciation	The decrease in the value of an asset over time, often modeled using exponential decay, such as the value of a car reducing each year.