Internal Assessment · Maths AA · Maths AI · 12 min read
Maths IA topics: 36 ideas for AA and AI, each with an aim and the maths it needs
Thirty-six exploration ideas sorted by course, each turned into a question you could actually answer in 12–20 pages.
On this page
- How do you know if a Maths IA topic will work?
- AA-leaning Maths IA topic ideas
- AI-leaning Maths IA topic ideas
- Maths IA ideas that work for either course
- Which Maths IA topics are overdone, and how do you make them fresh?
- How do you turn a topic into a Maths IA aim?
- Does your topic have to match AA or AI?
- How do you test your shortlist before committing?
A good Maths IA topic is a specific question you can answer with maths you understand, using data you can actually get, with enough depth to fill 12–20 pages without padding. Below are 36 ideas sorted into AA-leaning, AI-leaning and either-course topics, each with a concrete aim, the maths it would use and an honest note on difficulty.
Treat each as a starting point to change until the question is yours. For what a finished exploration looks like, see the annotated Maths IA example.
How do you know if a Maths IA topic will work?
Run every candidate through four tests before you commit weeks to it.
- Is there a real link to you? Not "I have always loved maths", but a reason the question matters to you: your sport, your commute, a game you play. Personal engagement (criterion C) is easiest to show when the question would exist even without the IA.
- Can you get the data? Statistics topics need data you can collect or download this month. If it depends on someone else replying to an email, have a plan B.
- Is the maths at your level? Use of mathematics (criterion E) is judged against your course and level. Going a little beyond is fine if you learn it properly; three pages of maths you cannot explain is not.
- Can you go deep in 12–20 pages? A good topic has one clear aim and a second layer: a model to refine, a comparison to make, a generalisation to prove. If the whole exploration fits in one paragraph, it is too thin; if it needs three branches of maths, it is too wide.
Then check you can write the aim as one sentence with a mathematical verb: model, optimise, compare, test, prove. If not, you have an area, not a topic; our guide to sharpening a research question shows how to narrow it.
AA-leaning Maths IA topic ideas
Calculus, functions, trigonometry, series and proof. Where a topic needs HL content or maths beyond the syllabus, the table says so; syllabus placements refer to the current course, so check your own subject guide.
| Idea and aim | Maths it uses | Difficulty and feasibility |
|---|---|---|
| Packaging with thicker ends. What dimensions minimise the metal in a 330 ml can when the lid is thicker than the wall, and does that explain real can shapes? | Optimisation with a constraint, first and second derivatives, generalising to a ratio k | SL-friendly. The classic version is overdone; the thickness ratio makes it yours (the optimum height-to-diameter ratio equals k). |
| Rugby conversions. After a try scored a distance a from the near post, how far back should the kicker go to maximise the angle between the posts? | Trigonometry, compound angle formula for tan, differentiation | SL or HL. Neat result: the best distance is the geometric mean of the distances to the two posts. Extend to a kicker with limited range. |
| The lifeguard problem. Where should I enter the water to reach a swimmer fastest, given my running and swimming speeds? | Distance–time functions, optimisation, links to Snell's law | SL-friendly. Measure your own running and swimming speeds for real parameters. |
| Milk first or milk later? How well does Newton's law of cooling model my tea, and when should milk go in to drink it soonest? | Exponential models, linearising with ln, separable differential equations (HL) | Very feasible with a kitchen thermometer; evaporation and lids give reflection material. |
| Chain or arch? Is a hanging chain better modelled by a parabola or by a catenary, and how big is the difference? | Exponential functions (cosh defined from ex), least-squares fitting, arc length (beyond syllabus) | HL-ambitious. Photograph a chain against a grid. |
| Volume from a photo. How accurately can a volume of revolution predict how much my water bottle holds? | Fitting functions to a profile, volumes of revolution (HL) | Overdone (see below). Fresh if you compare modelling choices and measure the true volume. |
| Daylight where I live. How well does a sinusoidal model predict daylight hours, and when is the day length changing fastest? | Sinusoidal functions, derivatives, comparing latitudes | SL-friendly, with free sunrise-table data. Compare several latitudes. |
| The lift ride. Can I reconstruct the height of each floor from my phone's accelerometer during a lift ride? | Kinematics, numerical integration, error accumulation | Feasible with a free sensor app. Drift in the integrated data is good reflection material. |
| Racing to π. How many terms do the Leibniz, Nilakantha and Machin-type series need for a given accuracy, and why? | Series, partial sums, error bounds, Maclaurin series (HL) | Overdone as a history tour. Works if you model the error against the number of terms. |
| Pick's theorem. Why does area = I + B/2 − 1 hold for lattice polygons, and what changes for a polygon with holes? | Proof, triangulation, induction-style arguments | Pure and demanding. Engagement comes from your own test cases and the extension. |
| Polite numbers. Which whole numbers can be written as a sum of two or more consecutive positive integers, and in how many ways? | Arithmetic series, factorisation, proof (powers of 2 are the exceptions) | Proof topic, good at HL. Build your own table of examples first. |
| Map distortion. How much does the Mercator projection inflate the area of the places I have lived? | Trigonometry, scale factors (area scale is sec2 of latitude), integrating sec x (HL techniques) | SL or HL. At latitude 72° areas are inflated about tenfold. |
| Newton's method in the complex plane. Which root of z3 = 1 does each starting point lead to? | Complex numbers (HL), iteration, Newton's method (beyond syllabus) | HL-ambitious. Pretty pictures are easy; analysis is hard. |
AI-leaning Maths IA topic ideas
Modelling, statistics, networks and technology. Graph theory, Markov chains, the Poisson distribution and logistic models are AI HL topics.
| Idea and aim | Maths it uses | Difficulty and feasibility |
|---|---|---|
| Where should the next defibrillator go? Using the locations of existing defibrillators in my town, where is the point furthest from any of them? | Voronoi diagrams, perpendicular bisectors, largest empty circle | SL-friendly; many towns publish location maps. Walking versus straight-line distance gives reflection. |
| My paper round. What is the shortest route that covers every street on my delivery round? | Graph theory, Chinese postman problem (AI HL), shortest paths (Dijkstra's algorithm is beyond the syllabus) | AI HL, very feasible. Compare the optimal route with the one you walk. |
| Visiting every campus building. What is the shortest tour of the buildings on an open day, and how close do the bounds get? | Travelling salesman upper and lower bounds, minimum spanning trees (AI HL) | AI HL. Measure distances yourself on a map. |
| My sleep as a Markov chain. If tonight is a good, average or bad night, what happens next, and what is my long-run distribution? | Transition matrices, matrix powers, steady states (AI HL) | AI HL. Needs about two months of tracker data. Test whether the chain predicts the next month. |
| The canteen queue. How many tills does my school canteen need so that the expected wait stays under five minutes? | Poisson distribution (AI HL), simulation, expected values | AI HL. Count arrivals per minute yourself; check whether Poisson actually fits. |
| Who studies how? Is preferred revision method independent of year group at my school? | χ2 test for independence, expected frequencies, degrees of freedom | SL. Overdone as a bare survey; needs enough responses for every expected frequency to reach 5. Get consent. |
| Engine size and fuel use. Is a linear or a power model better for predicting fuel consumption from engine size? | Linear and power regression, residuals, log–log linearisation (AI HL) | SL with technology for the regressions; HL if you linearise. Government vehicle datasets are free. The comparison of models carries it. |
| The life of a hit song. Does a logistic model describe the cumulative streams of a song after release, and can it predict the plateau? | Logistic models (AI HL), regression, comparing with exponential | HL. Public chart data. Make predictions early in a song's life, then check them. |
| Does money buy points? How strongly does wage bill rank with league position across several seasons? | Spearman's rank correlation, outliers, Pearson's r for comparison | SL. Common; fresher across seasons or leagues. |
| Lease or buy? For a car my family is considering, which deal costs less over five years once interest and resale value are included? | Annuities, amortisation, the finance solver, inflation | SL. Real quotes make it personal; state every assumption. |
| Breaking a Hill cipher. How many letters of known plaintext do I need to recover a 2 × 2 Hill cipher key? | Matrices and inverses (AI HL), arithmetic modulo 26 (beyond syllabus) | HL. Good structure; make sure the modular arithmetic is explained, not just used. |
| Is 500 ml really 500 ml? Do the bottles of water at my supermarket contain the volume on the label? | Normal distribution, t-test (check which version your level covers) | SL or HL. Measure carefully: measurement error is the reflection. |
| The area of a lake. How close does the trapezoidal rule get to the official area of a lake I swim in, and how many strips do I need? | Trapezoidal rule, percentage error, Simpson's rule as an extension (beyond syllabus) | SL. Use a scaled map; vary the strip count systematically. |
Maths IA ideas that work for either course
Each can go in an AA direction (derive, prove, use calculus) or an AI direction (model, simulate, test).
| Idea and aim | Maths it uses | Difficulty and feasibility |
|---|---|---|
| Snakes and ladders. What is the probability of finishing our family's board within 30 moves? | Transition matrices and powers (AI HL), or recursion and simulation (AA) | Either. Expected game length needs absorbing-chain ideas beyond the syllabus. |
| Benford's law. Do the first digits of my country's town populations follow P(d) = log10(1 + 1/d)? | Logarithms, χ2 goodness of fit (8 degrees of freedom), why the law holds | Either. AA students explain the logarithm; AI students test the fit. |
| Predicting race times. Does a power law predict my 10 km time from my 5 km time, and what exponent fits my own data? | Power functions, log–log regression, comparing with published exponents | Either. Your own running app history is ideal data. |
| Caffeine across a school day. How does the caffeine in my body build up with my usual drinks, and what dosing keeps it below a chosen level? | Exponential decay, repeated doses, geometric series for the long-run level | Either. Half-life varies between people: treat it as a parameter and test sensitivity. |
| Out of tune on purpose. How far are equal-tempered intervals from pure ratios, measured in cents? | Logarithms, geometric sequences (21/12), percentage error | Either. The fifth is off by about 2 cents, the major third by about 14. |
| Penalty kicks. What mixed strategy should a penalty taker use against a goalkeeper, and does my team's data match it? | Probability, expected value, simultaneous equations, game theory (beyond syllabus) | Either. Explain the equilibrium idea carefully. |
| Is the pity system worth it? In a game I play, what is the expected number of pulls to get a rare item, with and without a guaranteed drop? | Expected value, infinite geometric series, the geometric distribution (beyond syllabus) | Either. Use the drop rates the game publishes, and cite them. |
| Home advantage. Is there statistical evidence of home advantage in my local league this season? | Binomial or normal models, χ2 tests, confidence in the conclusion | Either. Choose a test you can justify; define "advantage" precisely. |
| Elo ratings. How would the final table of my club's league change if teams were ranked by Elo rating instead of points? | Logistic functions, iteration, rank correlation | Either. Clear structure; reflection on the choice of the K-factor. |
| Winning from deuce. If I win 55% of my points, what is my probability of winning a game, a set and a match? | Binomial probability, infinite geometric series (from deuce the probability is p2/(p2 + q2)) | Either. Check the model against your own match statistics. |
If your topic involves anything you can film (a throw, a jump, a swing, a bounce), the video-analysis guide shows how to collect motion data that the maths can actually use.
Which Maths IA topics are overdone, and how do you make them fresh?
No topic is banned, and an overdone topic done well still scores. The risk is that common topics invite the same write-up every year: a page of history, textbook maths and a predictable conclusion, which is weak on personal engagement and on use of mathematics.
| Overdone topic | Why it usually struggles | A fresher version |
|---|---|---|
| Golden ratio and Fibonacci in nature or faces | Mostly description; measurements that "find" φ by choosing what to measure | Test it: count the spirals on 30 pine cones you collected and check how often they are consecutive Fibonacci numbers |
| Monty Hall, birthday problem | Fully solved puzzles; the maths fits on one page | Generalise: how many people do you need for a 50% chance that two share a birthday within one day of each other? |
| Zombie or pandemic SIR models | Copied equations, invented parameters, simulation without analysis | Fit the model to real data from a school illness outbreak or a historical epidemic and judge where it fails |
| Volume of a wine glass or vase | One fitted curve, one integral, done | Compare several modelling choices (one polynomial, piecewise, more points) against the measured volume |
| The drinks can | The textbook answer, then "real cans are different" | Explain the difference: add thicker ends, a seam or a rim, and test the new optimum against measured cans |
| Height against shoe size | A correlation with no question behind it | Ask something that needs the answer: can footprint length estimate height well enough for a specific use? |
How do you turn a topic into a Maths IA aim?
Loses marksThe maths of music.
Earns marksHow far do the intervals of equal temperament differ from pure frequency ratios, and which intervals does that affect most?
Loses marksUsing statistics to analyse football.
Earns marksIs there evidence of home advantage in my club's league this season, and is it larger for goals scored or for goals conceded?
Loses marksAn investigation into calculus in sport.
Earns marksHow far back should a rugby kicker take a conversion to maximise the angle between the posts, and how does that change if the kicker can only reach 40 m?
Each strong version names one quantity to find or one claim to test, and has a second part that forces you to go beyond the first answer.
Does your topic have to match AA or AI?
No. Both courses use the same five criteria (20 marks, 20% of the grade at SL and HL), and neither restricts topics. What changes is what counts as maths "at your level" for criterion E: an AA student modelling cooling tea might solve the differential equation, while an AI student might compare models using residuals. The Maths IA criteria guide explains what each criterion rewards, and AA vs AI compared covers the rest of the course.
The new Maths curriculum (first exams May 2029) has new IA criteria: problem specification, abstraction, computation and interpretation. Everyone examined up to November 2028 is on the current course. The topic tests above still apply.
How do you test your shortlist before committing?
Spend half an hour on each of your best two or three ideas:
- Write the aim in one sentence.
- List the data you need and where it comes from. Get a small sample now.
- List the maths you expect to use, section by section. If it is all calculator output, find the step you will do by hand and explain.
- Sketch what a result would look like: a number, a graph, a proof, a decision.
- Write two limitations you can already see. If you cannot think of any, you do not understand the topic yet.
Take the best plan to your teacher, who knows your school's deadlines. Do the thinking and writing yourself: submitting AI-generated work as your own breaks the IB's academic integrity rules, and our guide to using AI honestly in the IB explains where the line is.
Questions students ask
Can I use a topic from a list of Maths IA ideas?
Yes, as a starting point. What earns marks is your own aim, your data and your decisions, and your teacher has to authenticate the work as yours. Two students with the same topic should end up with different explorations.
Does my Maths IA have to use maths beyond the syllabus?
No. Use of mathematics is judged against your course and level, so correct, well-understood maths from your own syllabus can score highly. Going beyond only helps if you clearly understand what you are doing; maths you cannot explain costs marks.
Is it better to collect my own data or use a published dataset?
Either can work. Your own data makes personal engagement easier to show; a published dataset (cited properly) gives you more data points. The marks come from what you decide to do with the data, not from where it came from.
Try one: Maths AA
A closed cylindrical can of radius r cm and height h cm holds 330 cm3. The metal for the top and bottom costs twice as much per cm2 as the metal for the curved side.
(a) Show that the cost of the metal is proportional to C = 660/r + 4πr2.
(b) Find the value of r that minimises C, justifying that it is a minimum, and find the ratio h : r for this can.
[7 marks]Show the answer and why
- Uses πr2h = 330 to write h = 330/(πr2)
- Cost ∝ 2πrh + 2 × 2πr2, and substituting for h gives 660/r + 4πr2
- dC/dr = −660/r2 + 8πr
- Sets the derivative equal to 0: r3 = 660/(8π)
- r = 2.97 cm (3 s.f.)
- d2C/dr2 = 1320/r3 + 8π > 0, so a minimum
- h = 11.9 cm, so h : r = 4 : 1
With equal costs the optimum is h = 2r (height equals diameter). Doubling the cost of the ends doubles the ratio: in general, if the ends cost k times as much, the optimum is h = 2kr. That is the kind of generalisation that turns a familiar textbook problem into an exploration.