TOK lesson starters using mathematics

Is this a proof? 10 min

Task. Show the first ten cases of a pattern that holds (for example, sums of the first n odd numbers are squares). Ask: how many cases would convince you? Then show a pattern that fails late (circle regions from points: 1, 2, 4, 8, 16, then 31).

Debrief. Why does mathematics demand proof when science accepts evidence? Was anyone 'right' to be convinced?

Follow-up reading: Why can one counterexample overturn a mathematical claim that thousands of examples could not establish?

Monty Hall vote 15 min

Task. Play the three-door game with the class. Take a vote on whether switching helps, then run 30 trials in pairs and pool the results.

Debrief. What changed minds: the argument or the data? What does that say about how we come to know?

Follow-up reading: Why do some correct mathematical results feel wrong, even to experts?

Rank the maps 10 min

Task. Show the same region on two or three projections. Ask groups to rank them for 'accuracy', then reveal what each preserves.

Debrief. Can a representation be accurate and misleading? Who chooses the maps we grow up with?

Follow-up reading: Can a representation be accurate and misleading at the same time?

Headline surgery 10 min

Task. Give a headline that a risk 'doubles'. Ask students to find the absolute risk and rewrite the headline twice: once to alarm, once to reassure, both true.

Debrief. If both headlines are true, what is the knowledge? What responsibility does the writer have?

Follow-up reading: How should we judge a statistical claim reported in the news?

Invented or discovered: the line-up 15 min

Task. Students stand on a line from 'invented' to 'discovered' for each of: the number 7, the equals sign, π, the axioms of geometry, the Mandelbrot set. Ask a few to justify their spot.

Debrief. Did people move between items? Perhaps the answer differs for notation, definitions and results.

Follow-up reading: Is mathematics discovered or invented?

Four-colour challenge 10 min

Task. Students try to draw a map that needs five colours. After a few minutes explain that it cannot be done, and how the proof was found.

Debrief. Do we know the theorem is true? Who checked it? Would you accept a proof you could never read?

Follow-up reading: Can we know a result is true if no human can check every step of its proof?

The 0.999… debate 10 min

Task. Ask for a show of hands: is 0.999… equal to 1? Present two short arguments that it is (thirds, and the difference between them), then poll again.

Debrief. Why do proofs fail to remove the feeling of doubt? Is knowing the same as being convinced?

Follow-up reading: Did the mathematical idea of a limit resolve Zeno's paradoxes, or only give us a way to calculate?

Which average? 10 min

Task. Give a small set of incomes with one very high value. Ask half the class to argue for the mean and half for the median as 'the average income'.

Debrief. Both are correct calculations. Who decides which one represents the group?

Follow-up reading: Can a single average represent a group fairly?

Same votes, different winners 15 min

Task. Use a class ranked vote on four options (for example, a trip). Count it by first-past-the-post, by elimination rounds and by points. Compare the winners.

Debrief. Is there a fair system? What can mathematics settle, and what is left to values?

Follow-up reading: Can mathematics tell us what a fair voting system is?

Trust your calculator? 10 min

Task. Ask students to compute something where rounding bites (for example, (1 + 10⁻¹⁵) − 1 on a calculator or spreadsheet, or a long chain of rounded steps). Compare answers.

Debrief. When do we check a tool, and how? What do we know when we only trust it?

Follow-up reading: How far should we trust tools whose workings we cannot check?

Base twelve for a day 10 min

Task. Ask students to write 'one hundred and forty-four' in base 12 and add two numbers in base 12. Then show a clock and ask where base 60 survives.

Debrief. Which parts of mathematics are conventions and which would be the same anywhere?

Follow-up reading: Is mathematics the same in every culture?

Who is responsible? 15 min

Task. Describe the 2020 exam-grade algorithms in a few neutral sentences. Assign roles: student, statistician, regulator, school. Each says who was responsible and why.

Debrief. Can mathematics be blamed? Where did the human choices sit?

Follow-up reading: Can an algorithm be biased if it only follows mathematics?

Beautiful or not? 10 min

Task. Show four short proofs or equations, one of them messy. Students rank them for beauty and explain their top choice in one sentence.

Debrief. Did people agree? Is beauty evidence of truth or only a reaction of the knower?

Follow-up reading: Is beauty a reliable guide to mathematical truth?

Two models, one data set 15 min

Task. Give early data from a growth situation. Half the class fit an exponential model, half a logistic one, using a GDC. Compare their predictions a long way ahead.

Debrief. The data was the same; the conclusions were not. Where did the difference come from?

Follow-up reading: How do the assumptions built into a model shape the conclusions it can reach?

Using these in maths lessons

Many IB maths topic pages on this site now carry a one-line "TOK link" to a matching question, for example Proof by induction (AA HL) and Correlation, regression, and Pearson's r (AI SL). Students can follow it after the lesson.

For classes doing the IA or EE, the IA hub and EE hub have their own guides.