Applications of e and Natural Logarithms

Active learning works for this topic because students often see the exponential function y = e^(kt) as just another formula to memorize. Hands-on modeling and comparison tasks help them recognize the underlying structure dy/dt = ky in real phenomena like cooling coffee or drug metabolism. When students build and interpret these models themselves, the parameters k and y_0 shift from abstract symbols to meaningful descriptors of growth or decay.

• 12th Grade
• Mathematics
• 3 activities
• 18 min–35 min

Learning Objectives

1. Design a mathematical model using the base of the natural logarithm, e, to represent continuous growth or decay scenarios.
2. Analyze the impact of parameter changes (growth rate k, initial value y_0) on the behavior of exponential growth and decay functions.
3. Evaluate the appropriateness of using continuous exponential models versus discrete models for specific real-world phenomena.
4. Calculate the time required for a quantity to double or halve given a continuous growth or decay rate.
5. Explain the relationship between a rate of change and its corresponding exponential growth or decay model.

Project-Based Learning

⏱ 35 min·Small Groups

Modeling Workshop: Build Your Own Decay Model

Groups choose a radioactive isotope (carbon-14, iodine-131, or uranium-238) from a reference card, look up its half-life, and derive the decay constant k. They then write the full model, graph it, and answer three specific questions: when is 10% remaining, what fraction remains after 5 half-lives, and what was the initial amount if 2g remains after a given time.

Prepare & details

Design a model using e and natural logarithms to represent a continuous process.

Facilitation Tip: During the Modeling Workshop, circulate and ask each group, 'What does the exponent kt represent physically in your scenario?' to ensure they connect k to a continuous rate.

Setup: Flexible workspace with access to materials and technology

Materials: Project brief with driving question, Planning template and timeline, Rubric with milestones, Presentation materials

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

Think-Pair-Share

⏱ 18 min·Pairs

Think-Pair-Share: Choosing the Right Model

Students are given five real-world scenarios (population growth, cooling coffee, bacterial culture, drug dosing, nuclear waste storage) and must decide with a partner whether continuous or discrete modeling is more appropriate and what sign k should have. Pairs share their reasoning and the class resolves any disagreements.

Prepare & details

Justify the choice of an exponential or logarithmic model for a given real-world scenario.

Facilitation Tip: For the Think-Pair-Share, assign each pair one correctly labeled scenario and one mislabeled scenario so they must defend their choice of model form.

Setup: Standard classroom seating; students turn to a neighbor

Materials: Discussion prompt (projected or printed), Optional: recording sheet for pairs

• Understand
• Apply
• Analyze
• Self-Awareness
• Relationship Skills

Project-Based Learning

⏱ 28 min·Small Groups

Parameter Investigation: What Does k Really Control?

Using a graphing calculator or Desmos, small groups graph y = 1000 * e^(kt) for k = 0.1, 0.5, 1.0, -0.1, -0.5, and -1.0. For each, they record the doubling or halving time, then derive the general formula t_double = ln(2)/k. Groups present their derivation and confirm the formula works across all cases.

Prepare & details

Evaluate the impact of changing parameters in a continuous growth model.

Facilitation Tip: In the Parameter Investigation, provide a graph of y = e^(kt) with k = 0.5 and k = –0.5 so students trace how k determines steepness and direction of change.

Setup: Flexible workspace with access to materials and technology

Materials: Project brief with driving question, Planning template and timeline, Rubric with milestones, Presentation materials

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

Teaching This Topic

Experienced teachers approach this topic by anchoring every activity in a concrete context students can visualize or relate to. Start with the differential equation dy/dt = ky to show the common structure across scenarios, then immediately move to modeling so students see how parameters change the curve. Avoid starting with the solution y = y_0 * e^(kt) as a black box. Research suggests that when students derive or at least justify the solution through activities like cooling experiments or population simulations, they retain the connection between the math and the phenomenon.

What to Expect

Students will recognize dy/dt = ky across contexts and write the correct e-based model with y_0 and k. They will interpret k as a continuous rate and distinguish it from discrete rates, use half-life correctly, and justify model choices with evidence from graphs and calculations. Successful learners will explain why the e-model is preferred for continuous processes and connect its parameters to real-world outcomes.

Watch Out for These Misconceptions

Common Misconception

During the Modeling Workshop, watch for students who default to y = a * b^t without considering whether the rate is continuous or discrete.

What to Teach Instead

Have them revisit the scenario’s language to identify if the rate is explicitly continuous (e.g., ‘grows continuously at 5% per hour’). Ask them to rewrite their model in the form y = y_0 * e^(kt) and explain how k relates to the 5% rate.

Common Misconception

During the Think-Pair-Share activity, watch for students who interpret the half-life as the time to reach zero.

What to Teach Instead

Direct them to the graph they sketched and ask them to mark each half-life on the curve, labeling the remaining quantity at each step to reinforce the asymptotic behavior. Then ask, 'Why does the curve never touch the horizontal axis?'

Assessment Ideas

Quick Check

After the Modeling Workshop, ask students to work individually on a scenario: 'A cup of tea cools from 90°C in a 20°C room with k = –0.1 per minute. Write the model and calculate the temperature after 5 minutes.' Collect responses to check if they use y = y_0 * e^(kt) correctly.

Discussion Prompt

During the Think-Pair-Share, ask each pair to present which model they chose for their assigned scenarios and justify their reasoning. Listen for statements that connect the model form to whether the rate is continuous or discrete.

Exit Ticket

After the Parameter Investigation, give students a half-life of 8 years and ask them to write the decay model and calculate the percentage remaining after 20 years. Use the results to see if they applied the half-life correctly and interpreted the model's parameters.

Extensions & Scaffolding

• Challenge students to create a model for a new context (e.g., drug concentration in the bloodstream) and predict the time when the concentration falls below a therapeutic threshold.
• Scaffolding: Provide a partially completed model with k given and ask students to derive the half-life formula from y = y_0 * e^(kt) using logarithms.
• Deeper exploration: Compare a discrete model y = y_0 * (1 + r/n)^(nt) to the continuous model and derive the relationship between the discrete rate r and the continuous rate k.

Key Vocabulary

Continuous Growth/Decay	A process where a quantity changes at a rate proportional to its current value, modeled by functions involving the base of the natural logarithm, e.
Exponential Growth Model	A function of the form y = y_0 * e^(kt), where y_0 is the initial amount, k is the continuous growth rate (k > 0), and t is time.
Exponential Decay Model	A function of the form y = y_0 * e^(kt), where y_0 is the initial amount, k is the continuous decay rate (k < 0), and t is time.
Half-life	The time it takes for a decaying substance to reduce to half of its initial amount, often modeled using exponential decay.