Mathematics · Year 10

Active learning ideas

Growth and Decay: Depreciation and Population

Active learning makes growth and decay tangible. When students manipulate depreciation tables or population tokens, they see how small percentages compound over time, turning abstract formulas into observable patterns. These hands-on experiences prevent misconceptions about linearity and reinforce the difference between iterative and fixed decreases.

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

• 25–45 min
• Pairs → Whole Class
• 4 activities

Activity 01

• Simulation Game
• 30 min · Pairs

Pairs: Depreciation Relay

Pairs start with a £10,000 car value and take turns applying a 15% annual depreciation over 10 years, recording values on a shared sheet. Switch roles midway and compare results. Discuss why values approach zero asymptotically.

Compare the mathematical models for growth and decay scenarios.

Facilitation Tip

During Depreciation Relay, circulate and ask each pair to verbalize their calculation step before moving to the next card.

What to look for

Present students with a scenario: 'A company buys a machine for $50,000. It depreciates by 15% each year. What is its value after 3 years?' Ask students to show their calculation steps and write the final value.

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

Activity 02

• Simulation Game
• 45 min · Small Groups

Small Groups: Population Simulation Cards

Provide cards with population events like birth rates or harvests causing percentage changes. Groups draw cards sequentially to model a village population over 20 years, plotting on mini-whiteboards. Groups present trends to class.

Explain how depreciation affects the value of assets over time.

Facilitation Tip

In Population Simulation Cards, pause after each round to ask groups to predict the next value aloud and challenge their reasoning.

What to look for

Pose the question: 'When might a linear model be a reasonable approximation for depreciation, and when is an exponential model essential?' Facilitate a class discussion where students justify their answers with examples.

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

Activity 03

• Simulation Game
• 25 min · Whole Class

Whole Class: Exponential Graph Match

Display graphs of growth and decay curves. Class votes on matching scenarios like phone battery drain or viral spread, then verifies with calculators. Adjust votes as evidence emerges.

Design a problem involving population change that requires an exponential model.

Facilitation Tip

During Exponential Graph Match, require students to annotate each matched graph with the formula they used and the r-value they chose.

What to look for

Give students a data set showing a declining population over several years. Ask them to calculate the approximate annual percentage decrease and write one sentence explaining whether this trend is likely to continue indefinitely.

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

Activity 04

• Simulation Game
• 35 min · Individual

Individual: Model Design Challenge

Students create a decay problem using real data, such as smartphone value loss, and solve it iteratively. Share one solution digitally for peer review.

Compare the mathematical models for growth and decay scenarios.

Facilitation Tip

In Model Design Challenge, ask students to include a short reflection on why their model fits the scenario better than a linear one.

What to look for

Present students with a scenario: 'A company buys a machine for $50,000. It depreciates by 15% each year. What is its value after 3 years?' Ask students to show their calculation steps and write the final value.

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

A few notes on teaching this unit

Teach this topic by starting with concrete manipulatives before moving to abstract formulas. Research shows that students grasp compound change better when they physically reduce tokens or build tables step-by-step. Avoid rushing to the formula—instead, let students discover the pattern through repeated calculation and comparison. Emphasize the meaning of r in context: whether it represents a loss of value or population decline, and how the sign changes alter the curve's behavior.

Successful learning looks like students fluently switching between recursive and explicit forms of exponential models, recognizing when an exponential approach is necessary, and explaining their reasoning with clear calculations and sketches. By the end, they should confidently compare linear approximations to exponential curves and justify their choices in context.

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Watch Out for These Misconceptions

• During Depreciation Relay, watch for students treating the 20% decrease as a fixed amount subtracted each year.

  Have students sketch a quick table on the back of their relay sheet after the first round and compare it to a linear decrease of $4,000 per year. Ask them to explain why their values diverge from a straight line.

• During Population Simulation Cards, watch for students applying the percentage decrease to the original population each round.

  Pass out a second set of tokens labeled ‘original amount’ and ask students to physically move tokens from the updated pile each time, forcing them to see the new base for the percentage.

• During Exponential Graph Match, watch for students assuming growth and decay formulas are interchangeable.

  Require students to write the formula they matched on an index card and place it under the correct header (growth or decay) on the board, then discuss why the sign in the formula must change with context.

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Methods used in this brief

• Simulation Game