Mathematics · Grade 11

Active learning ideas

Modeling Exponential Growth and Decay

Exponential functions come alive when students collect real data and watch patterns emerge, rather than relying on abstract formulas. Students build intuition by seeing how a small change today can produce dramatic differences later, which is impossible to grasp from a lecture alone. Active modeling turns the abstract into the concrete through touchable simulations and calculable outcomes.

• Ontario Curriculum Expectations
• HSF.LE.A.1
• HSF.LE.A.2

• 30–45 min
• Pairs → Whole Class
• 4 activities

Activity 01

• Simulation Game
• 35 min · Pairs

Simulation Game: Penny Doubling Growth

Pairs start with 1 penny and double it each round on paper grids to represent generations. They record totals every 10 doublings, graph results, and predict when pennies cover a desk. Discuss proportional rate observations.

How do exponential models differ from linear models in their long-term predictions?

Facilitation Tip

During Penny Doubling Growth, circulate and ask each group to predict the number of pennies at step 15 before they calculate it, forcing them to confront the steepness of exponential growth.

What to look for

Present students with two scenarios: one linear growth (e.g., saving $50 per week) and one exponential growth (e.g., doubling $50 each week). Ask them to calculate the amount after 10 weeks for both and write one sentence comparing the outcomes.

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

Activity 02

• Problem-Based Learning
• 40 min · Small Groups

Dice Roll: Radioactive Decay

Small groups roll 50 dice per 'half-life,' removing those showing 4 or higher as decayed. Record survivors each round, plot on class graph. Compare to exponential formula predictions.

Why is the rate of change in an exponential function proportional to its current value?

Facilitation Tip

For Dice Roll: Radioactive Decay, limit students to one die per group to slow the process and allow time for plotting between rolls, making the half-life pattern visible.

What to look for

Pose the question: 'Imagine a country's population is growing exponentially. What are two potential challenges this growth might create for the environment and two potential challenges for the economy?' Facilitate a class discussion where students justify their answers using concepts of proportionality.

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

Activity 03

• Problem-Based Learning
• 45 min · Small Groups

Spreadsheet: Compound Interest Race

Small groups input principal, rates, and times into shared spreadsheets to compare account growth. Adjust variables, create graphs, and debate best savings strategy based on results.

Evaluate the ethical implications of using exponential models to predict population growth or resource depletion.

Facilitation Tip

In the Spreadsheet: Compound Interest Race, assign different interest rates to each group so their graphs diverge visibly, sparking immediate comparisons during the wrap-up.

What to look for

Give students a radioactive decay problem involving half-life (e.g., Carbon-14 dating). Ask them to calculate the remaining amount of the substance after a specific time and briefly explain the meaning of 'half-life' in their own words.

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

Activity 04

• Problem-Based Learning
• 30 min · Whole Class

Whole Class: Population Debate

Project exponential models for city growth. Students vote on predictions at intervals, then reveal actual graphs. Discuss ethical limits like resource strain.

How do exponential models differ from linear models in their long-term predictions?

Facilitation Tip

During the Population Debate, assign roles (economist, environmentalist, policy maker) so students must defend their positions using the exponential model they just built.

What to look for

Present students with two scenarios: one linear growth (e.g., saving $50 per week) and one exponential growth (e.g., doubling $50 each week). Ask them to calculate the amount after 10 weeks for both and write one sentence comparing the outcomes.

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

A few notes on teaching this unit

Start with a concrete simulation that students can physically manipulate, like doubling pennies, before introducing any formulas. Avoid teaching the general form f(t) = a•b^t until students have grappled with the concept of repeated multiplication through hands-on trials. Research shows this sequence reduces reliance on memorized procedures and builds durable understanding of proportional change. Always link the math to ethical questions so students see why accuracy matters beyond the classroom.

Students will accurately distinguish exponential from linear change by tracing values over time in multiple contexts. They will use proportional reasoning to predict future amounts and connect these calculations to real-world consequences. Classroom discourse will reveal their growing confidence in explaining why exponential curves bend and decay approaches zero without ever reaching it.

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Watch Out for These Misconceptions

• During Penny Doubling Growth, watch for students who sketch straight lines rising from their data points.

  Have each group plot their points on graph paper and connect them with a smooth curve. Ask them to measure the vertical distance between consecutive points to show how the gap widens, then prompt them to redraw their line through the points to reveal the bend.

• During Dice Roll: Radioactive Decay, watch for groups that add or subtract a fixed number of dice each round instead of removing half.

  Ask students to count the survivors after each roll and record the fraction remaining. Have them compare their data to a half-life chart on the board, then re-run the trial if their counts don’t align with the expected halving pattern.

• During Spreadsheet: Compound Interest Race, watch for students who assume decay reaches zero after a few periods.

  Ask students to zoom into the last rows of their spreadsheet and add a column calculating the remaining fraction. Challenge them to find when the amount drops below 0.01% of the original, then discuss why the graph never touches zero even if the values become negligible.

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Methods used in this brief

• Simulation Game
• Problem-Based Learning