Mathematics · Year 11

Active learning ideas

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.

• National Curriculum Attainment Targets
• GCSE: Mathematics - Ratio, Proportion and Rates of Change

• 20–35 min
• Pairs → Whole Class
• 4 activities

Activity 01

• 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.

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.

What to look for

Present students with two scenarios: Scenario A: A population of 1000 bacteria grows by 20% each hour. Scenario B: An investment of $1000 grows by 10% each year. Ask students to calculate the value after 3 periods for each scenario and identify which is growing faster initially and which will be larger after 10 years.

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

Activity 02

• 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.

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.

What to look for

Provide students with a scenario: 'A new car costs $25,000 and depreciates by 15% each year.' Ask them to write down the percentage multiplier for depreciation and calculate the car's value after 2 years.

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

Activity 03

• 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.

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.

What to look for

Pose the question: 'Imagine two towns, Town A with a population of 5000 growing at 50 people per year, and Town B with a population of 4000 growing at 10% per year. Which town's population will be larger in 10 years, and why? What type of growth does each represent?'

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

Activity 04

• 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.

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.

What to look for

Present students with two scenarios: Scenario A: A population of 1000 bacteria grows by 20% each hour. Scenario B: An investment of $1000 grows by 10% each year. Ask students to calculate the value after 3 periods for each scenario and identify which is growing faster initially and which will be larger after 10 years.

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

A few notes on teaching this unit

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.

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.

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Watch Out for These Misconceptions

• 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.

  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.

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

  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.

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

  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.

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Methods used in this brief

• Simulation Game