IA Planning Guide

IB Maths IA Topic Ideas: How to Choose a Topic That Actually Works

Written by Mr Raj, an IB Mathematics examiner, IA moderator and workshop leader.  •  Originally published March 2026  •  Updated September 2026

Most students who search for “IB Maths IA topic ideas” are really asking two questions at once: what could I write about, and will it actually work. The second question matters more, and it is the one most lists ignore.

There is no topic that guarantees a high mark, and no topic that guarantees a low one. I have moderated explorations on the Golden Ratio that were genuinely excellent, and explorations on original, ambitious ideas that fell apart within two pages. What separates them is never the topic name on the cover page. It is what the student actually does with the mathematics once the topic is chosen.

This guide is informed by recurring themes highlighted in recent Mathematics: Analysis and Approaches subject reports, the documents the IB publishes each session summarising how explorations actually performed, and where they tend to go wrong. It gives you topic ideas, but more importantly it gives you a way of judging whether an idea of your own is mathematically sound before you spend twenty hours on it.

Quick note on terms: the Mathematics exploration does not need a formal “research question” in the way an Extended Essay does. It needs a clear mathematical aim: the focus of what you are investigating. I use “aim” and “focus” throughout this guide rather than “research question” for that reason.

What actually makes a good IB Maths IA topic?

A strong topic is not defined by subject matter. It is defined by whether it gives you room to do the following things, all of which the criteria reward directly:

• A clear mathematical aim. You can state in one or two sentences what you are investigating and why it needs mathematics to answer.
• Appropriate scope. Your exploration should be as concise as the mathematics allows: every section contributes to the mathematical development or interpretation, not compressed into a summary or stretched into padding.
• Mathematics you understand. Every method you use, you could explain to a classmate without notes.
• Room for mathematical choices. You decide which model, which method, which variables to include, and can justify the decision mathematically.
• Space to interpret and reflect. The results actually mean something, so you have something genuine to say about them.
• Real ownership. The exploration develops your own reasoning rather than reproducing a worked method from a textbook or website.

Notice that “sounds impressive” and “sounds original” are not on that list. A topic that ticks every box above using nothing more advanced than quadratics or a chi-squared test can score very highly. A topic that name-drops university-level mathematics but gives you none of the above usually cannot.

Good, risky and overused IA topics

Green means a strong starting point. Amber means it can work, but is often mishandled. Red means high-risk, usually worth reconsidering. Nothing in this table is banned, and none of it is a rule set by the IB. Red does not mean forbidden, it means “handled badly more often than not”. A red-flagged idea developed with real mathematical judgement can still outscore a green-flagged idea developed lazily.

The same topic can produce a weak or a strong IA

This is usually the most useful exercise a student can do before starting. Take the vague version of an idea and ask what it would actually require you to calculate.

Is your idea actually better suited to a Maths Extended Essay?

This is not a formal IB distinction, but it is a genuinely useful practical test. Subject reports have repeatedly noted explorations that read like research reports or Extended Essays rather than explorations: heavy on background, theory and literature, light on the student’s own mathematical work.

A rough way to tell the two apart:

• IA-shaped: aim → mathematics → your own result → a mathematical decision or comparison → refinement → reflection.
• EE-shaped: background → theory → more theory → literature review → explanation of a famous result someone else derived.

A useful early warning sign: if you find yourself needing several pages of background theory before you can do anything of your own, the idea may simply be too large for the IA’s scope. A related sign is a bibliography that dwarfs the mathematics: a long list of sources is not a substitute for using mathematics yourself, and a large bibliography is not something an exploration needs in the first place. The exploration should be written for a peer studying the same course, not for a general academic audience, so it does not need a formal literature review, a methodology section written in the style of a science report, or the structural apparatus of an EE.

This cuts both ways. An ambitious, original idea can produce an outstanding exploration, but only if you understand and control the mathematics well enough to make it your own, rather than presenting it as a summary of someone else’s work.

Criterion-by-criterion advice

The exploration is assessed against five criteria: A Presentation (4 marks), B Mathematical communication (4 marks), C Personal engagement (3 marks), D Reflection (3 marks) and E Use of mathematics (6 marks), for 20 marks in total. Here is what tends to separate strong and weak work on each.

Criterion A: Presentation (4 marks)

• State the aim clearly, early, and stick to it.
• Build a logical mathematical narrative: each section should follow from the last.
• Be concise. A shorter exploration that says everything once is stronger than a longer one that repeats itself.
• Skip the methodology section written like a science report. Say what you are doing and do it.
• Do not repeat near-identical calculations five times when one worked example and a summary table would do.
• Do not use appendices to hide mathematics that should be in the main body, or to pad out the exploration artificially. Essential working belongs in the exploration itself.
• Make sure the conclusion actually concludes: it should answer the aim, not just restate what was done.

Criterion B: Mathematical communication (4 marks)

• Use correct, consistent notation, and define variables when you introduce them.
• Label every graph and table (axes, units, legend), and only include a graph or table if it does real work.
• Keep units, approximation signs and significant figures/decimal places consistent throughout.
• Write equations properly rather than in calculator or software syntax (for example, write the closed form rather than pasting a GDC command).
• Ask of every visual: does this help the reader, or is it here to fill space?

If you need a refresher on standard IB notation and formulae for the areas you plan to use, the IB Maths formula booklet guide is a useful reference while you write.

Criterion C: Personal engagement (3 marks)

This is the most misunderstood criterion. “I chose football because I love football” is not personal engagement: it is a comment about the context, not the mathematics. Personal engagement is about your relationship with the mathematics itself, and it shows up through what you actually did, not through what you claim in the introduction:

• Making and justifying your own mathematical decisions, rather than following a template.
• Collecting or selecting your own data, and being deliberate about why.
• Adapting a standard model to fit your own situation rather than using it unmodified.
• Comparing two methods and forming a view on which suits your question better.
• Testing an assumption rather than simply stating it.
• Following up on an unexpected or messy result instead of smoothing over it.

There is no set of phrases that manufactures this. It comes from the choices you actually make while doing the mathematics.

Criterion D: Reflection (3 marks)

Strong reflection happens throughout the exploration, not only in a closing paragraph titled “limitations”. A single generic sentence at the end (“my model was not perfect”) earns very little. Reflection that engages with the mathematical process earns considerably more. Questions worth asking as you go include:

• What does this result actually mean in context?
• Is the answer realistic, and how would I know if it were not?
• Which assumption matters most to the outcome, and what happens if it fails?
• Why did one model perform differently from another?
• What changed when I varied a parameter, and why?
• Is it reasonable to extrapolate beyond the data I have?
• What is this model fundamentally unable to capture?

Criterion E: Use of mathematics (6 marks)

The single most important point on this page: more advanced mathematics does not automatically mean more marks. This criterion rewards mathematics that is correct, relevant to the aim, and demonstrably understood: shown through explanation, justification and (where appropriate) rigour, not through difficulty. Mathematics copied from a paper and used mechanically, without the student showing they understand why it works, consistently underperforms compared to simpler mathematics used with genuine command.

AA SL versus AA HL: choosing a topic at the right level

SL students do not need university-level mathematics to score well. Functions, statistics, probability, sequences and series, geometry, trigonometry, and accessible calculus (differentiation and integration of standard functions) are all entirely suitable for a high-scoring SL exploration, provided they are used with real understanding and applied to a genuine question.

HL students have access to more sophisticated tools (further calculus, complex numbers, matrices, more advanced statistics) and can push further with them. But the same rule applies at HL as at SL: the mathematics still has to be understood and communicated properly, not just deployed. Most topics below can be developed at either level; what changes is the depth and rigour of the mathematics you bring to it, not the topic itself. Treat “SL” and “HL” labels on any topic list, including this one, as a starting suggestion rather than a rule.

Topic ideas by category

These are starting points, not finished IAs. Use them to spark a direction, then narrow the aim yourself using the sections above. Each includes the main risk to watch for.

Sport

• Free-throw or penalty-kick angle optimisation. Quadratic/trigonometric modelling of trajectory against a self-collected sample of attempts. Main danger: using generic online data instead of your own, which removes the personal engagement. SL/HL: both, HL can extend to optimisation with calculus.
• Comparing two methods of ranking players or teams. Matrix or weighted-scoring methods compared against real results. Main danger: description without comparison. You need to justify which method is mathematically more defensible, not just present both.

Finance and economics

• Comparing loan or mortgage repayment structures. Geometric series and recurrence relations to compare amortisation under different interest assumptions. Main danger: a single formula plugged into one scenario with no comparison or sensitivity analysis. SL/HL: SL-friendly.
• Testing a simple trading or investment rule against real data. A defined recurrence-based rule tested against a specific index over a set period. Main danger: scope creep. Pick one rule, one index, one period, and go deep rather than wide.

Geometry

• Stability and shape of arches or bridges. Parabolic versus catenary forms, compared using functions and curvature. Main danger: description of shapes without a genuine comparison or optimisation question. SL/HL: HL suits a calculus-based curvature comparison.
• Symmetry and transformations in a specific pattern or artwork. Matrix transformations or symmetry-group reasoning applied to a real, chosen design. Main danger: cataloguing symmetries rather than analysing them mathematically.

Optimisation

• Packaging design for minimum material and fixed volume. Differentiation-based optimisation with a real, measured product. Main danger: this is a well-known template. The strength comes from choosing a genuinely original object and constraints, not a generic can or bottle.
• Optimal seating, queuing or layout under real constraints. Linear programming or constrained optimisation applied to a situation you can actually measure. Main danger: an unrealistic constraint set that makes the “optimal” answer trivial.

Probability and games

• Fairness of a specific game’s reward or scoring system. Expected value and variance analysis of a real game (board game, card game, or sport scoring system). Main danger: spending too long explaining the rules and too little on the mathematics. Keep rule explanation brief and get to the analysis quickly.
• Markov chains for a simple real process (weather, a board game, a queueing system). Transition matrices used to model and predict long-run behaviour. Main danger: setting up the matrix correctly but never interpreting what the steady state actually means.

Modelling

• Population growth in a small, controlled or local system. Comparing exponential and logistic models, estimating parameters from real data, evaluating fit through residuals. Main danger: fitting a model and stopping. The mathematics is in evaluating why one model fits better and where it breaks down.
• Cooling, heating or diffusion under varied real conditions. See the coffee example above. The strength is in comparing conditions and estimating your own constants, not reciting the model.

Music and sound

• Comparing tuning systems using frequency ratios. Logarithms and ratio analysis comparing equal temperament with just intonation. Main danger: presenting the ratios without a genuine mathematical comparison of why the systems diverge.
• Modelling a sound wave or chord using trigonometric functions. Superposition of sine waves fitted to a recorded sample. Main danger: relying entirely on software output without showing or understanding the underlying mathematics.

Architecture and design

• Proportional systems in a specific building or design tradition. Ratio and geometric analysis of a real, measured or well-documented structure. Main danger: turning into a descriptive history of the building rather than a mathematical analysis of it.

Personal data

• Analysis of your own tracked data (sleep, training, screen time, spending). Descriptive and inferential statistics, or time-series style modelling, applied to a genuinely personal dataset. Main danger: a dataset that is too small or too noisy to support the conclusions drawn from it. Be honest about this in your reflection.

Transport and routes

• Shortest-path or route-efficiency comparison for a real network. Graph theory (Dijkstra’s algorithm, minimum spanning trees) applied to a genuine local network: a bus route, a delivery problem, a school layout. Main danger: using a toy network with no real constraints, which removes the need for genuine decision-making.

Functions

• Comparing candidate functions for a real, measured curve (a bridge cable, a projectile path, a growth curve). Fitting and testing several function families against your own data, and justifying the best fit mathematically rather than by eye. Main danger: choosing the “best” function purely by highest R² without justifying why that family makes mathematical sense for the situation.

Calculus

• Related rates or optimisation in a real, physical situation you can measure. Differentiation applied to a self-designed scenario (a ladder, a shadow, a filling container). Main danger: this is one of the most templated areas online. Originality in the specific setup matters more than the calculus technique itself.

A short guide to regression, correlation and modelling

Regression and correlation are consistently among the most common approaches in Maths explorations, and there is nothing wrong with that. What separates a strong regression-based IA from a weak one is almost never the technique: it is whether the student treats the model as something to interrogate rather than something to produce. Before you finish, make sure you have genuinely asked:

• Why should this model fit? Is there a mathematical or physical reason to expect a linear, exponential, quadratic or logistic relationship, or did you just try several and pick the highest R²?
• What do the parameters mean? A gradient, an intercept, a growth rate: each should have an interpretation in context, not just a numerical value.
• What do the residuals show? A residual plot often reveals more mathematical insight than the R² value itself.
• Is the model sensible outside your observed data? Extrapolation is one of the easiest places to demonstrate (or fail to demonstrate) genuine understanding.
• Could a different model be justified, and why did you not use it?
• What assumptions are you making about the data (independence, measurement error, sample size)?
• Does correlation imply causation here? Almost never. Saying so, with a specific reason relevant to your data, is worth more than the correlation coefficient itself.

Testing five different regression models and reporting whichever has the highest R² is not, on its own, a strong use of mathematics. Choosing a model because you can justify it, then testing whether the data supports that choice, is.

Topics I would think twice about

None of these are prohibited. They appear repeatedly in subject reports as ideas that tend to go wrong, usually for the same handful of reasons, and it is worth knowing those reasons before you commit to one.

• Golden Ratio and Fibonacci in nature: becomes descriptive rather than analytical, almost every time.
• Birthday Paradox: the mathematics is short, well known, and rarely extended into something original.
• Standard SIR/SEIR epidemic models: fine when adapted to real, student-collected data; weak as a textbook reproduction.
• Generic bottle or can surface-area/volume optimisation: an extremely common template; needs a genuinely original object and constraint to stand out.
• Newton’s Law of Cooling with no individual data: valid mathematics, but adds little if it is not applied to your own measurements.
• Purely historical mathematics: tends to explain mathematics rather than use it.
• Very large datasets used for their own sake: size is not sophistication; a small, well-understood dataset usually produces a stronger exploration than a huge one skimmed superficially.
• Coding-dominated projects: code can support an exploration (simulation, computation, visualisation) but should never replace the mathematics. The exploration has to show mathematical reasoning, not just a working script.
• Advanced mathematics copied from a paper or website: this is the single strongest recurring warning in subject reports. If you cannot derive, explain and interpret it yourself, it will not score well regardless of how advanced it looks.
• Correlation studies across ten unrelated variables: breadth without depth; better to explore two or three variables thoroughly than ten superficially.

The 60-second IA topic test

Before you commit to an idea, answer these honestly:

• Can I explain my aim in one sentence?
• Could another student in my Maths class follow my mathematics without extra explanation?
• Will I actually be doing mathematics, rather than mostly explaining background?
• Do I have real mathematical choices to make, and can I justify them?
• Can I compare methods or test an assumption, rather than just presenting one path?
• Can I develop this fully without padding it out or compressing it into a summary?
• Can I explain why each method I plan to use is the right one for this question?
• Do I understand every piece of mathematics I intend to use: well enough to defend it in a viva-style conversation?

If you answered “no” to three or more of these, the topic is not necessarily wrong, but it needs narrowing before you start writing.

Frequently asked questions

Does my Maths IA topic need to be original?

No. Many strong explorations use familiar topic areas, and two students can write about the same theme and produce completely different, individually valid work. What needs to be original is your specific data, your specific modelling decisions, and your own reasoning, not the general subject.

Is regression a bad Maths IA topic?

No, it is one of the most common approaches and can score very highly. It becomes weak only when the model choice is not justified beyond “it had the best R²”, or when correlation is treated as if it proves causation. See the mini-guide above.

Does the Maths IA need a research question?

Not in the formal, Extended-Essay sense. What it needs is a clear mathematical aim: a focus that the reader can identify early and that the exploration answers by the end.

Can an SL student use HL mathematics, or get full marks without it?

An SL student can get full marks without HL-level mathematics. Use of mathematics is judged on whether the mathematics is correct, relevant and demonstrably understood at a level commensurate with the course, not on difficulty for its own sake. Bringing in HL material is not necessary, and it only helps if it is genuinely understood and used well.

What counts as personal engagement in a Maths IA?

Engagement with the mathematics, not with the topic. It shows through choices you make and justify, data you collect or select yourself, a model you adapt rather than use unmodified, and results you interpret rather than simply report. Liking the subject matter is not personal engagement on its own.

Can I use Desmos, Python or other coding tools?

Yes. Graphing and coding tools are a normal part of exploring, checking and generating results. They should support the mathematics, not replace it: you still need to present the mathematics properly in your own notation and explain the reasoning behind what the tool produces.

How advanced should the mathematics be?

As advanced as you can use with genuine understanding, and no further. More advanced mathematics does not automatically mean more marks. Simpler mathematics used correctly and with real command of it consistently outperforms advanced mathematics used mechanically.

How is a Maths IA different from a Mathematics Extended Essay?

This is not a formal IB distinction, but a useful practical one. An IA is built around your own mathematics: an aim, your working, your results and your reflection. If a topic needs substantial background theory or a literature review before you can do anything of your own, or if it mainly explains an existing famous result rather than investigating something yourself, it is likely better suited to an Extended Essay.

Are topics such as the Golden Ratio or Birthday Paradox banned?

No. Neither is prohibited. Both are simply overused and frequently handled in a descriptive, formulaic way, which is why they carry more risk than a less familiar topic. A student who develops either with genuine original analysis can still score well.

For general questions about lessons and booking, see the Turtle Maths FAQ.

This guide is informed by recurring themes highlighted in recent Mathematics: Analysis and Approaches subject reports, combined with Mr Raj’s experience as an IB examiner and IA moderator. It is independent guidance and does not represent an official position of the International Baccalaureate Organization. “IB” and “International Baccalaureate” are trademarks of the International Baccalaureate Organization. For ongoing, criterion-focused support once you have a direction in mind, see IB Maths tutoring.