Mathematics · Year 13

Active learning ideas

Rates of Change and Related Rates

Active learning works for rates of change because students must see how variables interact in real time, not just on paper. Working through geometry-based problems like a ladder or a cone forces students to connect abstract derivatives to physical motion, making the chain rule feel necessary rather than theoretical.

• National Curriculum Attainment Targets
• A-Level: Mathematics - Differentiation

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

Activity 01

• Problem-Based Learning
• 45 min · Small Groups

Demo Lab: Sliding Ladder

Provide ladders or poles against walls; students mark positions, pull bases outward at constant speed, and measure heights over time. Derive the related rates equation using Pythagoras, differentiate, and compare predicted top descent rate to measurements. Discuss discrepancies as a class.

Explain how the chain rule is fundamental to solving related rates problems.

Facilitation Tip

During the Sliding Ladder demo, stand at the back of the room and call out the base’s speed so students must listen carefully to track both variables without visual distractions.

What to look for

Provide students with a scenario: 'A spherical balloon is being inflated at a rate of 10 cm³/s. How fast is the radius changing when the radius is 5 cm?' Ask students to write down the equation they would differentiate, the differentiated equation, and the final answer.

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

Activity 02

• Problem-Based Learning
• 35 min · Small Groups

Relay Race: Cone Filling Problems

Divide class into teams; each station has a cone-filling scenario card with partial solutions. Teams complete one step, like setting up volume equation or applying chain rule, then rotate. Final team assembles full solutions and presents.

Analyze the relationship between different rates of change in a dynamic system.

Facilitation Tip

In the Cone Filling Relay, rotate groups every 10 minutes so students hear multiple approaches to the same problem and adapt their reasoning.

What to look for

Present a diagram of a ladder sliding down a wall. Ask students: 'If the base of the ladder is moving away from the wall at 2 m/s, what is the rate of change of the height of the top of the ladder? What information is missing to solve this?'

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

Activity 03

• Problem-Based Learning
• 30 min · Pairs

Desmos Simulation: Shadow Lengths

Pairs load Desmos graphs of related rates scenarios, like a person walking past a lamp post. Adjust sliders for speeds, observe shadow rate changes, and derive equations to match graphs. Share sliders and predictions with class.

Construct a solution to a related rates problem involving geometric shapes.

Facilitation Tip

For the Desmos Shadow simulation, pause the animation at key points and ask students to estimate rates before running calculations to build intuition.

What to look for

Pose the question: 'How is the chain rule used to connect the rate at which a shadow lengthens to the rate at which the object casting the shadow moves?' Facilitate a class discussion where students explain the variables and their derivatives.

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

Activity 04

• Problem-Based Learning
• 25 min · Pairs

Card Sort: Rate Setups

Distribute cards with diagrams, variables, and equations. Individuals or pairs match them into complete related rates problems, then solve one as a group. Class votes on trickiest setups and verifies solutions together.

Explain how the chain rule is fundamental to solving related rates problems.

Facilitation Tip

Keep Card Sorts small—three to four students per group—to ensure every voice contributes to the rate setup or solution process.

What to look for

Provide students with a scenario: 'A spherical balloon is being inflated at a rate of 10 cm³/s. How fast is the radius changing when the radius is 5 cm?' Ask students to write down the equation they would differentiate, the differentiated equation, and the final answer.

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

A few notes on teaching this unit

Teach this topic by having students verbalize each derivative’s meaning before writing equations. Avoid giving away the chain rule too early; instead, let students discover why they need it when they try to relate rates like dV/dt to dh/dt. Research shows that students who explain their steps aloud in pairs retain more than those who work silently. Also, alternate between hands-on labs and abstract problems to reinforce both intuition and precision.

By the end of these activities, students should confidently set up related rates problems, correctly apply the chain rule to implicit differentiation, and solve for unknown rates with clear reasoning. They will explain their steps aloud and justify each variable’s role in the system.

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Watch Out for These Misconceptions

• During the Sliding Ladder demo, watch for students assuming the ladder’s height decreases at a constant rate because the base moves at a constant speed.

  Prompt students to measure the height and base distances at multiple time points during the demo, then compare the actual rates of change to show they are not constant. Have them calculate dh/dt at two different positions to see the variation.

• During the Cone Filling Relay, watch for students forgetting to apply the chain rule when differentiating volume with respect to time.

  Circulate during the relay and ask each group to verbally explain how dV/dt connects to dh/dt before they write anything. If they omit the chain rule, ask them to explain the role of h(t) in the volume formula.

• During the Card Sort activity, watch for students confusing which given rate corresponds to which variable in the problem.

  Have groups physically label each rate card with its variable (e.g., dh/dt = 2 m/s) and place it next to the corresponding part of the diagram. Require them to justify their choices aloud before proceeding to the solution.

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Methods used in this brief

• Problem-Based Learning