Free IB Maths AI SL lesson: Nearest-point problems from Voronoi
A ready-to-teach IB Maths AI SL lesson on Nearest-point problems from Voronoi (Unit 3 · Geometry & Trigonometry): 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, 29 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
- Piecewise Linear Models
- Nearest-Neighbour Interpolation in Context
Key ideas on the slides
- A Voronoi diagram splits the plane into cells: each cell contains the points closer to its site than to any other site.
- Cell edges lie on perpendicular bisectors of pairs of sites: through the midpoint, with gradient the negative reciprocal of the gradient joining the sites.
- A vertex is equidistant from three (or more) sites: solve two perpendicular-bisector equations simultaneously.
- Adding a site: draw the perpendicular bisectors between the new site and its neighbours and remove the edges that are now closer to the new site.
By the end of the lesson, students should be able to answer a question like: Use the Voronoi diagram to find the closest hospital to point (3,7).
Key vocabulary
- site: One of the given points that generate the diagram.
- cell: The region of points closest to one site.
- vertex: A point where three or more cells meet, equidistant from their sites.
- perpendicular bisector: The line at right angles through the midpoint of a segment.
Common mistakes this lesson tackles
- Voronoi diagrams and bisectors: midpoint, negative reciprocal, vertices. Perpendicular bisector: midpoint of the two sites, gradient −1/m, then y − y₁ = m(x − x₁). A vertex is where two bisectors meet: solve them together. Vertical and horizontal bisectors are x = … and y = ….
- Using the gradient between the sites instead of its negative reciprocal.
- Finding the midpoint with a sign error.
- Placing a vertex by eye instead of solving the bisector equations.
The worked example
Voronoi diagram of hospitals H₁, H₂, H₃. Accident at point P. Which hospital dispatches?
- Locate P on the diagram.
- Observe which cell contains P.
- State: dispatch from the hospital whose cell contains P (guaranteed nearest by construction).
Tip: No distance calc needed — cell boundaries already guarantee the closest site.
Calculator tips
- Graph the bisectors and use intersect to check the vertex.
Lesson plan
| 0–5 min | Do now: students write down what they remember about Constructing a Voronoi diagram and Area = ½ ab sin 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 3: Voronoi Diagrams 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 3: Voronoi Diagrams practice questions on ibmathrevision.com, where the free questions are open and the full bank comes with a plan.