Free IB Maths AI HL lesson: Coupled differential equations & phase portraits
A ready-to-teach IB Maths AI HL lesson on Coupled differential equations & phase portraits (Unit 5 · Calculus): 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.
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, 28 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
- Setting Up & Interpreting Coupled Linear Systems
- Euler's Method for Coupled Systems
- Solving Coupled Systems with Eigenvalues & Eigenvectors
- Sketching Trajectories & Classifying Equilibrium Points
- Equilibrium Points & Long-Term Behaviour
- Real, Complex & Imaginary Eigenvalues
- Second-Order Differential Equations as Coupled Systems & with Euler's Method
- Differential Equations in Context: Population, Predator–Prey & Combat Models
Key vocabulary
- equilibrium point: A point where all the rates are zero.
- phase portrait: A diagram of trajectories in the x–y plane.
- saddle point: An equilibrium that attracts along one direction and repels along another.
Common mistakes this lesson tackles
- Coupled systems and phase portraits: equilibria, eigenvectors, trajectories. Equilibrium: dx/dt = 0 and dy/dt = 0. Write the system as a matrix, find eigenvalues and eigenvectors (in the form (1, k)), and use their signs: both negative → stable node, opposite signs → saddle, complex → spiral.
- Classifying with eigenvector directions instead of the eigenvalues' signs.
- Drawing trajectories that cross.
Lesson plan
| 0–5 min | Do now: students write down what they remember about Reading trajectories on a slope field and Sketching slope fields |
| 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 AI HL Unit 5: Differential Equations practice questions.
What schools get
- Set this lesson as homework, marked by AI against the mark scheme, with every result in your markbook (school Pro 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 AI HL Unit 5: Differential Equations practice questions on ibmathrevision.com, where the free questions are open and the full bank comes with a plan.