Free IB Maths AA HL lesson: Euler's method for differential equations
A ready-to-teach IB Maths AA HL lesson on Euler's method for differential equations (Unit 5 · Calculus): editable slides and teacher notes, free; a printable worksheet in 3 levels with answers for school plans. About 55 minutes.
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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, 27 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
- Euler's Method & Modelling Applications
- Homogeneous Equations & the Substitution y = vx
- Modelling with Differential Equations
- Choosing the Method
- Euler's Method vs the Exact Solution
Key ideas on the slides
- For dy/dx = f(x, y) with step h: xₙ₊₁ = xₙ + h, yₙ₊₁ = yₙ + hf(xₙ, yₙ).
- Each step follows the tangent from the current point, so errors build up; a smaller h usually gives a better approximation.
- For coupled systems, update each variable with its own rate using the old values: xₙ₊₁ = xₙ + h(dx/dt), yₙ₊₁ = yₙ + h(dy/dt).
- Set out a table (or use the GDC's sequence/spreadsheet) and give the answer to the accuracy asked.
By the end of the lesson, students should be able to answer a question like: dy/dx = x + y, y(0) = 1. Use Euler's method with h = 0.1 to estimate y(0.3).
Key vocabulary
- Euler's method: A step-by-step numerical method for approximating the solution of a DE.
- step length: h, the increase in x (or t) at each step.
- approximation: An estimate of the true value.
- differential equation: An equation involving a derivative.
- general solution: The solution with an arbitrary constant.
Common mistakes this lesson tackles
- Differential equations: integrating factor and the constant of integration. For dy/dx + P(x)y = Q(x): integrating factor e^(∫P dx), then (IF × y)′ = IF × Q. Add + C at the integration step and use the condition to find C before rearranging.
- Using the new y in f(xₙ, yₙ) before it has been calculated.
- Taking the wrong number of steps for the interval.
- Copying the formula without applying it.
Calculator tips
- Use the recursive sequence mode or a spreadsheet: u(n+1) = u(n) + 0.1(n·0.1 + u(n)).
- Check a particular solution with a slope-field or Euler plot if available.
Lesson plan
| 0–5 min | Do now: students write down what they remember about Derivatives of trig, inverse trig, exponential & log functions and Volumes of revolution |
| 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 the IB Maths AA HL Unit 5: Differential Equations and Maclaurin 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: Differential Equations and Maclaurin practice questions on ibmathrevision.com, where the free questions are open and the full bank comes with a plan.