Free IB Maths AI SL lesson: Quadratic models (fitting to data)
A ready-to-teach IB Maths AI SL lesson on Quadratic models (fitting to data) (Unit 2 · Functions): 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, 19 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
- Graphing Parabolas with the GDC
- Modelling Projectile Motion & Optimisation Contexts
- Fitting a Quadratic to Data
- Quadratic Models & Optimising with Technology
Key ideas on the slides
- In a model h(t) = at² + bt + c, c is the initial value, the vertex gives the maximum (or minimum) and the roots give when the quantity is zero.
- The maximum height of a projectile is the y-coordinate of the vertex; the time it happens is the t-coordinate, t = −b/2a.
- Fit a quadratic to data with quadratic regression on the GDC; check it with a plotted graph.
- Reject roots outside the model's domain (a negative time) and give answers in context with units.
By the end of the lesson, students should be able to answer a question like: Find the quadratic regression equation for the points (1,3), (2,5), and (3,9).
Key vocabulary
- model: A mathematical description of a real situation.
- domain of the model: The input values for which the model makes sense.
- regression: Fitting a curve of a chosen type to data.
Common mistakes this lesson tackles
- Interpreting in context: parameters, derivatives and justifications. Name the quantity and its units: "0.35 is the cost in dollars per extra kilometre". Base every justification on a calculated value from the question.
- Giving the maximum value when the question asks when it happens (or the reverse).
- Keeping a negative time as an answer.
- Using the model far outside the data range.
The worked example
Using h(t) = −5t² + 20t + 2, use your GDC to find the time at which the ball hits the ground.
- Set h(t) = 0: −5t² + 20t + 2 = 0.
- Open the polynomial solver on the GDC.
- Enter a = −5, b = 20, c = 2.
- Read: t = −0.0976… or t = 4.0976…
- Reject the negative time.
- State: ball lands at t = 4.10 s (3 s.f.).
Tip: Quadratic physical models often produce one positive and one negative root — reject the negative one because negative time isn't physical.
Calculator tips
- Use the graph's maximum and zero tools; quadratic regression (QuadReg) fits a model to data.
Lesson plan
| 0–5 min | Do now: students write down what they remember about Parallel & perpendicular lines and Straight-line graphs (m and c) |
| 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 SL Unit 2: Quadratic Modelling 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 AI SL Unit 2: Quadratic Modelling practice questions on ibmathrevision.com, where the free questions are open and the full bank comes with a plan.