IB Maths IA Ideas: 40 Topics with a Starting Question

Searching for Maths IA ideas usually returns long lists of topic names, which is not much help. What helps is a starting question you can test and the maths that carries it. Below are 40 original starting points grouped by area. Each one has a concrete question, the mathematics involved, which course it fits, and a plain note on difficulty. They are starting points only: you will need to make the question your own, collect or find your own data, and decide your own route through it.

How to use these ideas

Look for a sparkPick two or three that make you want to find out. Interest carries you through the 12 to 20 pages, and it shows in personal engagement.
Check the levelCriterion E rewards maths at the right level for your course (6 of the 20 marks). At HL it expects HL-level maths, so an idea marked for AA HL is not a shortcut for an SL course, and a gentle idea needs deepening to earn a high mark.
Make a changeChange the case, the data or the question. A copy of an idea will not show your own choices. Add your own measurement, your own comparison or your own extension.
Run a one-page trialSpend an hour on a rough version to see if the maths holds up before you commit.
Talk to your teacherYour teacher can tell you whether the topic is realistic for your course and whether your school has a preference.

The Fit labels are our guide to where the maths naturally sits. Courses differ between AA and AI, and SL and HL, and your teacher knows what your class has covered. These criteria are for the current courses (exams up to November 2028); new courses from 2029 change them.

Algebra and functions (8 ideas)

1. How many ways can a fixed sum be made from a set of coin values, and how does the count grow? Maths: generating functions, recurrences, sequences. Fit: AA SL and HL. Difficulty: moderate; needs care with counting.
2. Which closed form fits the pattern of the number of regions made by lines crossing in a circle? Maths: sequences, induction, polynomial fitting. Fit: AA SL and HL. Difficulty: moderate; good for proof.
3. How fast does Newton's method converge to different roots, and when does it fail? Maths: iteration, derivatives, convergence. Fit: AA SL and HL. Difficulty: moderate; computer help needed.
4. Can a cubic or quartic sound like a real shape, such as a bridge arch, if fitted to measured points? Maths: polynomial regression, residuals, modelling. Fit: AA and AI, SL and HL. Difficulty: gentle; the evaluation is the real work.
5. What patterns appear in the digits of fractions, and how are they linked to number theory? Maths: recurring decimals, modular arithmetic, series. Fit: AA HL. Difficulty: stretching; abstract.
6. How do the roots of unity build regular polygons, and what does that say about drawing a pentagon with a ruler and compass? Maths: complex numbers, polynomials, Euler form. Fit: AA HL. Difficulty: stretching; HL complex numbers.
7. How does the choice of base change how long a number is, and what is the best base for storing information? Maths: logarithms, optimisation of cost. Fit: AA and AI, SL and HL. Difficulty: gentle; an interesting question about efficiency.
8. Which function best describes the pitch of a sound as a string's tension changes, using a tuner app? Maths: power functions, log scales, regression. Fit: AI SL and HL. Difficulty: gentle; measurement quality decides the mark.

Calculus (8 ideas)

9. What shape of can uses the least metal for a given volume, and why do real cans differ from the ideal? Maths: optimisation, derivatives, volume and area. Fit: AA SL and HL. Difficulty: gentle; the surprise is in the evaluation.
10. How does the best angle to throw a ball change if the landing height is different from the launch height? Maths: parametric equations, derivatives, trigonometry. Fit: AA SL and HL. Difficulty: moderate; needs a clean model.
11. What is the volume of a vase or bowl, and how close can integration get to measured capacity? Maths: volumes of revolution, numerical integration. Fit: AA HL. Difficulty: moderate; measure carefully.
12. How does the area under a curve compare with the trapezium and Simpson's methods for real data? Maths: integration, numerical methods, error. Fit: AA and AI, SL and HL. Difficulty: gentle to moderate; error analysis adds depth.
13. How does the depth of water in a tapering tank change as it drains, and can you predict the time to empty? Maths: differential equations, separable variables. Fit: AA HL. Difficulty: stretching; check the physics assumptions.
14. How well do Maclaurin series approximate a function of your choice, and how many terms are enough? Maths: series, remainder, convergence. Fit: AA HL. Difficulty: moderate; the criteria want use of the maths, not copying.
15. When is the rate of change of a stock price or sports score greatest, using real data and a smoothed curve? Maths: differentiation, smoothing, interpretation. Fit: AA and AI, SL and HL. Difficulty: gentle; be careful with overfitting.
16. What path does a pursuer take when chasing a target, and does it ever catch it? Maths: differential equations, parametric curves. Fit: AA HL. Difficulty: stretching; solves a classic problem in an original setting.

Statistics and probability (9 ideas)

17. Is a sports player's run of successes streaky or just luck, according to a hypothesis test? Maths: binomial model, runs tests, p-values. Fit: AI SL and HL. Difficulty: moderate; the data must be genuinely independent.
18. How do waiting times at a canteen or bus stop compare with a Poisson or exponential model? Maths: Poisson distribution, goodness of fit. Fit: AI HL, AA HL. Difficulty: moderate; collect your own observations.
19. How accurate are the odds in a card or dice game, and what strategy is best? Maths: conditional probability, expected value. Fit: AA and AI, SL and HL. Difficulty: gentle; good first project.
20. Does the number of hours of sleep relate to test scores in a class, and how strong is the evidence? Maths: correlation, regression, confidence. Fit: AI SL and HL. Difficulty: gentle; consent and anonymity first.
21. How large must a sample be before a poll's margin of error becomes small? Maths: sampling distributions, confidence intervals. Fit: AI HL, AA HL. Difficulty: moderate; simulation helps.
22. Does the Benford first-digit pattern appear in data sets from different fields? Maths: logarithms, chi-squared tests. Fit: AI SL and HL. Difficulty: moderate; the data choice matters.
23. What does a Markov chain say about the long-run weather or board-game positions? Maths: matrices, transition probabilities, steady states. Fit: AA HL, AI HL. Difficulty: stretching; HL matrices.
24. How reliable are screening tests when the condition is rare? Maths: Bayes' theorem, tree diagrams. Fit: AA and AI, SL and HL. Difficulty: gentle; the conclusion is counter-intuitive.
25. How do different tie-break systems change the winner in a small league? Maths: probability, simulation, combinatorics. Fit: AA and AI, SL and HL. Difficulty: moderate; simulate many seasons.

Geometry and trigonometry (7 ideas)

26. What is the most efficient shape for a honeycomb or packing, and why do bees use hexagons? Maths: tessellations, area and perimeter, optimisation. Fit: AA and AI, SL and HL. Difficulty: gentle; the maths must be deeper than the shapes.
27. How can the height of a tall building be found from shadows or angles, and how accurate is the method? Maths: trigonometry, error propagation. Fit: AA and AI, SL and HL. Difficulty: gentle; the maths needs raising with error work.
28. How do planetary orbits relate to ellipses, and how well can orbital data be fitted? Maths: conics, polar forms, parametric equations. Fit: AA HL. Difficulty: stretching; needs real data.
29. What does a Fourier series say about the shape of a sound wave from a musical instrument? Maths: periodic functions, integration, sums of sines. Fit: AA HL. Difficulty: stretching; computer help needed.
30. How does a map projection distort areas and distances on a sphere? Maths: spherical geometry, trigonometry, scale factors. Fit: AA HL. Difficulty: stretching; clear definitions needed.
31. What is the shortest route that visits several points on a map, and how does a heuristic compare with the best answer? Maths: graph theory, optimisation, distance. Fit: AI SL and HL. Difficulty: moderate; reduce to a small case.
32. What are the proportions behind a favourite painting or building, and do golden-ratio claims hold up? Maths: ratios, measurement, tests of fit. Fit: AA and AI, SL and HL. Difficulty: gentle; avoid an assertion, test it.

Modelling (8 ideas)

33. How does a drink cool in different cups, and how well does Newton's law of cooling fit? Maths: exponential decay, regression, residuals. Fit: AA and AI, SL and HL. Difficulty: gentle; temperature readings need care.
34. How do two models of population growth compare with published data for one country? Maths: logistic and exponential models, fitting. Fit: AA and AI, SL and HL. Difficulty: moderate; avoid just plotting a curve.
35. How does the speed of a spreading rumour or disease depend on contact rate, in a simple simulation? Maths: recurrence relations, differential equations, simulation. Fit: AA HL, AI HL. Difficulty: stretching; explain every assumption.
36. How does loan or savings interest compound, and when does a repayment plan break even? Maths: geometric series, logarithms, financial formulas. Fit: AA and AI, SL and HL. Difficulty: gentle; add comparison of plans.
37. How should a business set a price to maximise profit, using a demand curve of measured data? Maths: functions, derivatives, regression. Fit: AA and AI, SL and HL. Difficulty: moderate; use real data.
38. How does the height of a bouncing ball shrink on each bounce, and what does the model predict for the total distance? Maths: geometric series, regression, convergence. Fit: AA and AI, SL and HL. Difficulty: gentle; strong for data and graphs.
39. How can the queue length at a shop be predicted from arrivals and service times? Maths: probability models, simulation, averages. Fit: AI HL, AA HL. Difficulty: stretching; collect observations.
40. What is the best allocation of a fixed budget across several options, and how does the answer change if costs change? Maths: linear programming, sensitivity analysis. Fit: AI SL and HL. Difficulty: moderate; test the sensitivity.

Common mistakes with IA ideas

Choosing a topic because it sounds advanced, then finding the maths is only copying a formula.
Picking an idea with no data you can get, so the investigation stalls.
Starting with a broad topic such as "probability" instead of a question.
Skipping the evaluation: every idea above needs you to say how reliable the model is and what you would improve.

Once you have a shortlist, read how the work is judged in Maths IA criteria explained and see what a plan looks like in annotated outlines. The main Maths IA guide has the overview.

IB Maths IA ideas: 40 topics with a starting question