Mathematics · Year 10 · Number Systems and Proportionality · Autumn Term

Growth and Decay: Depreciation and Population

Modelling real-world situations involving percentage decrease (depreciation) and population changes.

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

About This Topic

Growth and decay models use exponential functions to represent real-world changes, such as asset depreciation and population fluctuations. In Year 10, students calculate percentage decreases to track how car values drop by 20% annually or how populations decline due to factors like disease. They compare linear approximations with true exponential curves, revealing how small percentage changes compound over time. Key skills include setting up iterative calculations and interpreting graphs of y = a(1 - r)^n for decay.

This topic aligns with GCSE standards in Ratio, Proportion and Rates of Change, linking number systems to practical applications in finance and biology. Students design problems, like modelling rabbit populations with a 5% annual decrease, fostering problem-solving and model comparison. Understanding compound effects prepares them for advanced topics like compound interest and logistic growth.

Active learning suits this topic well. When students simulate depreciation by passing objects around groups and updating values each round, or plot real census data on shared graphs, they grasp compounding intuitively. Collaborative design of population scenarios encourages peer critique, making abstract percentages concrete and relevant.

Key Questions

1. Compare the mathematical models for growth and decay scenarios.
2. Explain how depreciation affects the value of assets over time.
3. Design a problem involving population change that requires an exponential model.

Learning Objectives

• Calculate the value of an asset after a specified period, given an annual depreciation rate.
• Compare the long-term effects of linear versus exponential decay models on asset value.
• Analyze population data to identify trends and predict future sizes using exponential decay models.
• Design a word problem involving population decline that requires the use of an exponential decay formula.
• Critique the suitability of exponential decay models for different real-world depreciation scenarios.

Before You Start

Percentages and Percentage Change

Why: Students need a solid understanding of how to calculate percentage increases and decreases to model depreciation and population changes.

Introduction to Exponential Functions

Why: Familiarity with the basic form and behavior of exponential functions, y = ab^x, is necessary for understanding decay models.

Key Vocabulary

Depreciation	The decrease in the value of an asset over time, often due to wear and tear, obsolescence, or market factors.
Exponential Decay	A process where a quantity decreases at a rate proportional to its current value, resulting in a curve that gets progressively flatter.
Depreciation Rate	The percentage by which the value of an asset decreases each period, typically annually.
Model	A mathematical representation used to describe or predict real-world phenomena, such as population changes or asset value.

Watch Out for These Misconceptions

Common Misconception

Decay is linear, like subtracting a fixed amount each year.

What to Teach Instead

Exponential decay multiplies by a factor less than 1, so decreases accelerate relative to current value. Pair discussions of car value tables reveal the curve's shape, helping students contrast it with straight-line models through shared sketches.

Common Misconception

Percentage decrease applies to the original amount every time.

What to Teach Instead

Each decrease is from the updated value, compounding the effect. Group simulations with physical tokens passed and reduced show this buildup clearly, as students track and debate totals aloud.

Common Misconception

Growth and decay use the same formula structure.

What to Teach Instead

Growth uses (1 + r), decay (1 - r); signs differ but principles align. Whole-class matching activities expose this symmetry, with students articulating differences via think-pair-share.

Active Learning Ideas

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.

30 min·Pairs

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.

45 min·Small Groups

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.

25 min·Whole Class

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.

35 min·Individual

Real-World Connections

• Automotive industry professionals use depreciation models to determine the resale value of vehicles, impacting pricing strategies for dealerships and financing options for consumers.
• Economists and financial analysts track the depreciation of capital assets like machinery and buildings to inform business investment decisions and calculate tax liabilities.
• Conservation biologists use population decay models to assess the risk of extinction for endangered species, guiding conservation efforts and habitat management strategies.

Assessment Ideas

Quick Check

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.

Discussion Prompt

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.

Exit Ticket

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.

Frequently Asked Questions

How do you explain depreciation to Year 10 students?

Start with familiar assets like phones or bikes, showing tables of value after successive 10-20% drops. Use calculators for iterations, then graphs to visualise the curve. Connect to insurance quotes for relevance, emphasising compound effects build financial awareness over time.

What real-world examples illustrate population decay models?

Use endangered species data, like tiger populations declining 4% yearly, or historical events such as Black Death impacts scaled to percentages. Students input census figures into spreadsheets, projecting futures and debating conservation implications. This grounds maths in biology and ethics.

How can active learning help teach growth and decay?

Activities like relay depreciations or card-based simulations let students physically manipulate values, revealing compounding without rote formulas. Group plotting of real data fosters collaboration, while designing problems builds ownership. These approaches make exponents tangible, boosting retention and application skills.

What is the difference between growth and decay mathematical models?

Both are exponential: growth y = a(1 + r)^n expands from a base, decay y = a(1 - r)^n contracts toward zero. Compare via side-by-side tables and graphs in pairs; students spot mirrored shapes. This comparison clarifies when to use each in scenarios like investments versus asset loss.