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.

• Year 13
• Mathematics
• 4 activities
• 25 min–45 min

Learning Objectives

1. Calculate the rate of change of one variable given the rate of change of another related variable using implicit differentiation and the chain rule.
2. Analyze the interconnectedness of rates of change in a dynamic geometric system, such as a filling cone or a moving ladder.
3. Construct a mathematical model to solve a related rates problem, identifying all variables, constants, and their rates of change.
4. Explain the role of the chain rule in transforming a problem about rates of change in one variable into a problem about rates of change in another.
5. Critique the assumptions made when setting up a related rates problem, such as uniform rates or idealized shapes.

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.

Prepare & details

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.

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

⏱ 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.

Prepare & details

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.

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

⏱ 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.

Prepare & details

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.

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·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.

Prepare & details

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.

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 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.

What to Expect

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.

Watch Out for These Misconceptions

Common Misconception

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.

What to Teach Instead

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.

Common Misconception

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

What to Teach Instead

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.

Common Misconception

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

What to Teach Instead

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.

Assessment Ideas

Exit Ticket

After the Sliding Ladder demo, provide students with the exit ticket: 'A ladder is sliding down a wall. If the top is descending at 3 m/s when the base is 4 m from the wall, how fast is the base moving away when the ladder is 5 m long?' Ask them to write the equation they would differentiate, the differentiated equation, and the final answer.

Quick Check

During the Cone Filling Relay, after groups complete the first problem, ask them to hold up a whiteboard showing their setup before moving to the next cone size. Look for correct use of the chain rule and proper labeling of known and unknown rates.

Discussion Prompt

After the Desmos Shadow simulation, facilitate a class discussion where students explain how the chain rule connects the rate of the object’s movement to the rate of the shadow’s lengthening. Ask volunteers to sketch the variables on the board and label each derivative.

Extensions & Scaffolding

• Challenge students to design their own related rates problem using a real-world scenario, then solve it and exchange with peers.
• For students who struggle, provide partially completed rate setups with some derivatives already calculated to reduce cognitive load.
• Deeper exploration: Ask students to derive the general formula for the rate of change of volume in a cone filling with water, then test it with different dimensions in a spreadsheet.

Key Vocabulary

Related Rates	A problem in calculus where the rates of change of two or more related variables are considered simultaneously.
Implicit Differentiation	A technique used to differentiate equations where one variable cannot be easily expressed as a function of the other, differentiating with respect to a common variable, usually time.
Chain Rule	A rule in calculus for differentiating composite functions, essential for relating the rate of change of one variable to the rate of change of another through a common variable.
Rate of Change	The speed at which a variable changes over time, often represented by a derivative with respect to time (e.g., dy/dt).