Modeling Growth and Decay

Active learning helps students grasp how exponential functions model real-world change, because abstract formulas like A = A0 * (1/2)^(t/h) become concrete when students manipulate data and observe patterns. Through hands-on experiments and collaborative problem-solving, students see why growth slows, decay follows predictable timelines, and resource limits matter.

• Year 12
• Mathematics
• 4 activities
• 25 min–50 min

Learning Objectives

1. Calculate the half-life of a substance given two data points and an exponential decay model.
2. Analyze the limitations of exponential growth models when applied to real-world populations with finite resources.
3. Compare the steepness of exponential decay curves by altering the base of the exponential function.
4. Critique the validity of a given exponential model for a specific real-world scenario, such as radioactive decay or population growth.
5. Apply Newton's Law of Cooling to predict the temperature of an object over time given initial conditions and ambient temperature.

Problem-Based Learning

⏱ 35 min·Pairs

Pairs Graphing: Base Impact Challenge

In pairs, students use graphing calculators or Desmos to plot y = 10 * b^x for bases b = 1.05, 1.1, 1.2, and 1.5 over x from 0 to 50. They note doubling times and predict population scenarios. Pairs share findings in a class gallery walk.

Prepare & details

Explain how to determine the half-life of a substance using only two data points.

Facilitation Tip: During Pairs Graphing: Base Impact Challenge, circulate and ask each pair to defend why doubling the base from 2 to 3 will steepen the curve before they plot it.

Setup: Groups at tables with access to research materials

Materials: Problem scenario document, KWL chart or inquiry framework, Resource library, Solution presentation template

• Analyze
• Evaluate
• Create
• Decision-Making
• Self-Management
• Relationship Skills

Problem-Based Learning

⏱ 45 min·Small Groups

Small Groups: Half-Life Data Puzzle

Provide groups with two carbon dating data points, such as 100g remaining after 5730 years. Groups solve for half-life algebraically, then test with a third point. They extend to decay curves and discuss accuracy limits.

Prepare & details

Assess the limitations of an exponential growth model in a world with finite resources.

Facilitation Tip: In Small Groups: Half-Life Data Puzzle, provide stopwatches and identical samples so students can time each decay step together and compare notes.

Setup: Groups at tables with access to research materials

Materials: Problem scenario document, KWL chart or inquiry framework, Resource library, Solution presentation template

• Analyze
• Evaluate
• Create
• Decision-Making
• Self-Management
• Relationship Skills

Problem-Based Learning

⏱ 50 min·Whole Class

Whole Class: Cooling Law Experiment

Teacher prepares hot water thermometers at multiple stations. Class records temperatures every 2 minutes for 20 minutes. Together, they plot data on semi-log paper, fit an exponential model, and compute cooling constant.

Prepare & details

Analyze how changing the base of an exponential function affects the steepness of the curve.

Facilitation Tip: For Whole Class: Cooling Law Experiment, assign roles like timer, recorder, and grapher to ensure every student contributes to the shared dataset.

Setup: Groups at tables with access to research materials

Materials: Problem scenario document, KWL chart or inquiry framework, Resource library, Solution presentation template

• Analyze
• Evaluate
• Create
• Decision-Making
• Self-Management
• Relationship Skills

Problem-Based Learning

⏱ 25 min·Individual

Individual: Resource-Limited Simulation

Students model bacterial growth with discrete steps on spreadsheets, starting with 10 cells and a growth factor, but cap resources at 1000. They graph and identify when exponential assumption breaks, reflecting on logistic alternatives.

Prepare & details

Explain how to determine the half-life of a substance using only two data points.

Facilitation Tip: For Individual: Resource-Limited Simulation, set clear boundaries like fixed bead counts to force students to confront finite limits.

Setup: Groups at tables with access to research materials

Materials: Problem scenario document, KWL chart or inquiry framework, Resource library, Solution presentation template

• Analyze
• Evaluate
• Create
• Decision-Making
• Self-Management
• Relationship Skills

Teaching This Topic

Teach this topic by alternating between concrete experiments and abstract reasoning. Start with physical models to build intuition, then move to equations. Avoid rushing to formulas before students see the phenomena they model. Research shows students better understand decay when they plot real cooling data before calculating half-lives.

What to Expect

Successful learning looks like students using exponential models to make accurate predictions, explaining why growth curves plateau, and justifying how base size affects steepness. They should connect equations to graphs, data points, and real constraints without mixing up half-life with starting quantity.

Watch Out for These Misconceptions

Common Misconception

During Individual: Resource-Limited Simulation, watch for students assuming growth continues forever without noticing the fixed bead supply.

What to Teach Instead

Direct students to graph their population over time and label the point where growth stops, then ask them to explain what the plateau means in terms of resources.

Common Misconception

During Small Groups: Half-Life Data Puzzle, watch for groups believing half-life changes with sample size.

What to Teach Instead

Have groups overlay their decay curves on the same axes to see that all samples follow the same proportional decay rate, regardless of starting amount.

Common Misconception

During Whole Class: Cooling Law Experiment, watch for students interpreting the cooling curve as linear because it looks straight at first.

What to Teach Instead

Ask students to plot the same data on a logarithmic scale and compare it to the linear version to see the true exponential shape.

Assessment Ideas

Quick Check

After Small Groups: Half-Life Data Puzzle, give each group a new scenario with two data points and ask them to calculate the half-life using the formula, showing their steps on a whiteboard.

Discussion Prompt

During Individual: Resource-Limited Simulation, pause the activity and ask students to discuss in pairs what their graphs would look like if resources were unlimited, then share observations with the class.

Exit Ticket

After Pairs Graphing: Base Impact Challenge, have students sketch y = 3^x and y = 1.5^x on the same axes and write one sentence comparing how the bases affect the curves.

Extensions & Scaffolding

• Challenge early finishers to model a substance with a half-life of 7 minutes using a stopwatch and small candies, then predict the amount after 28 minutes.
• Scaffolding for struggling students: Provide pre-labeled graph paper with time intervals marked to help them plot data points accurately during Half-Life Data Puzzle.
• Deeper exploration: Have students research real-world half-life applications (e.g., medical isotopes) and present how the concept applies beyond the classroom.

Key Vocabulary

Half-life	The time required for a quantity to reduce to half its initial value, commonly used in radioactive decay and drug concentration.
Exponential Decay	A process where a quantity decreases at a rate proportional to its current value, resulting in a curve that gets progressively flatter.
Exponential Growth	A process where a quantity increases at a rate proportional to its current value, resulting in a curve that gets progressively steeper.
Newton's Law of Cooling	A law stating that the rate of heat loss of a body is directly proportional to the difference in temperatures between the body and its surroundings.
Base of an Exponential Function	The constant number that is raised to a variable exponent, influencing the rate of growth or decay of the function.