Free IB Maths AA HL and AI HL lesson: Related rates of change
A ready-to-teach IB Maths AA HL and AI HL lesson on Related rates of change: editable slides and teacher notes, free; a printable worksheet in 3 levels with answers for school plans. About 55 minutes.
Free with a teacher account. Sign in with Google, Microsoft or email and download the slides and teacher notes.
One lesson for AA HL and AI HL
Related rates of change is taught in IB Maths AA HL and IB Maths AI HL, with the same key ideas. Choose your course above the download button: each version's slides use that course's own worked example, questions and practice link.
- AA HL (Unit 5 · Calculus)
- AI HL (Unit 5 · Calculus)
What's inside
- Slides (PowerPoint, editable) · freeabout 12 slides: do now, objectives, key ideas, a worked example step by step, “your turn”, practice QR code, exit ticket
- Teacher notes (PDF) · freelesson plan with timings, common mistakes, homework links
- Worksheet (PDF) · school plans6 questions, 32 marks: Fluency, Reasoning, Exam-style. Included with school plans and the teacher trial.
- Answers (PDF) · school plansthe answer to every worksheet question, from its mark scheme. Included with school plans and the teacher trial.
Learning objectives
- Related (connected) rates of change
Key ideas on the slides
- Link the rates with the chain rule: dV/dt = dV/dr × dr/dt.
- Write a formula connecting the quantities (e.g. V = (4/3)π r³), differentiate it, then substitute the values at the instant asked.
- A decreasing quantity has a negative rate.
- Give units: if V is in cm³ and t in seconds, dV/dt is in cm³/s.
By the end of the lesson, students should be able to answer a question like: A sphere's volume grows at 15 cm³ s⁻¹. Find dr/dt when r = 3 cm.
Key vocabulary
- rate of change: A derivative with respect to time, e.g. dr/dt.
- chain rule: dy/dt = (dy/dx)(dx/dt).
- instant: The particular moment at which the values are substituted.
Common mistakes this lesson tackles
- Substituting the values before differentiating.
- Losing the negative sign for a decreasing quantity.
- Writing the rates the wrong way up in the chain rule.
The worked example
A spherical balloon is being inflated such that its volume V is increasing at a constant rate of 10 cm³s⁻¹. Find the exact rate of change of its surface area A when the radius is exactly 5 cm.
- Identify the given rate dV/dt = 10, and state the formulas for a sphere: V = (4/3)π r³ and A = 4π r².
- Differentiate the volume with respect to radius: dV/dr = 4π r².
- Apply the chain rule to link volume and radius rates: dV/dt = dV/dr × dr/dt ⇒ 10 = 4π(5²) × dr/dt.
- Solve for the rate of change of the radius: dr/dt = 10/100π = 1/10π.
- Differentiate the surface area with respect to radius: dA/dr = 8π r.
- Apply the chain rule again to find the requested rate: dA/dt = dA/dr × dr/dt = 8π(5) × 1/10π = 40π × 1/10π = 4 cm²s⁻¹.
Tip: Set up your related rates chain rules explicitly (e.g., dA/dt = dA/dr × dr/dt) before substituting any numbers. Substituting constants into geometric formulas before differentiating will erroneously yield a derivative of zero!
Calculator tips
- No GDC shortcut: write the chain rule with the units.
Lesson plan
| 0–5 min | Do now: students write down what they remember about Implicit differentiation and Product & quotient rules |
| 5–8 min | Share the learning objective and success criteria |
| 8–20 min | Teach: the worked example, step by step; students copy the method |
| 20–25 min | Your turn: 1 mini-whiteboard question; check every board before moving on |
| 25–42 min | Practise: the worksheet (6 questions in 3 levels, with school plans and the teacher trial) or the topic's practice questions online; move up a level after two right in a row |
| 42–47 min | Plenary: 1-question exit ticket; collect it to plan the next lesson |
For students: practise this topic
The worksheet's QR code opens your course's practice questions:
- IB Maths AA HL Unit 5: Advanced Differentiation practice questions
- IB Maths AI HL Unit 5: Advanced Differentiation and Related Rates practice questions
What schools get
- Set this lesson as homework, marked by AI against the mark scheme, with every result in your markbook (with a School plan).
- This lesson's worksheet in three levels, with its answers, ready to print.
- Support, Core and Stretch versions and A/B copies of the worksheet from the full question bank, in the worksheet builder.
- Create lesson: this lesson rebuilt for your class's next scheme-of-work lesson, from the full bank.
School licences from €490 a year. Try everything free for 30 days as a teacher: no card.
Questions teachers ask
Is this lesson free?
The slides and the teacher notes are free to download with a free IB Math Revision teacher account (sign in with Google, Microsoft or email); no card. The printable worksheet in three levels and its answers are included with school plans and the free 30-day teacher trial.
Can I edit the slides?
Yes. The slides are an ordinary PowerPoint file: change, add or delete anything. They also open in Google Slides and Keynote. The logo and web address sit on the slide master, so your edits do not move them.
What do students use for homework?
The practice slide's QR code and web address take students to the IB Maths AA HL Unit 5: Advanced Differentiation practice questions on ibmathrevision.com, where the free questions are open and the full bank comes with a plan.