How to write the IB Maths AA Exploration

The Maths AA Exploration is the one piece of coursework your Mathematics: Analysis & Approaches grade is marked on internally — worth about 20% of your final grade at both SL and HL. Most students lose marks not because they cannot do the mathematics, but because they choose a flat, impersonal topic or never learn what each marking criterion actually rewards. This guide walks you through the whole thing: what the Exploration is, how it is marked, exactly how to write each part, and what separates a top-band investigation from an average one. The Exploration is sometimes called the maths IA, but its proper name is the Exploration, and that word matters: you are exploring an idea, not writing up an experiment.

The distinction is worth dwelling on, because it shapes every decision you make. A science practical has a procedure, a result and an evaluation; an Exploration has an idea you pursue wherever the mathematics leads. There is no fixed method to follow and no single right answer waiting at the end. That freedom is exactly what trips students up: faced with a blank page, many reach for the safest, dullest topic they can think of, reproduce a derivation from a textbook, and hand in something competent but lifeless. An examiner can spot that in the first paragraph, and it caps your marks before you have written a line of real mathematics. The students who do well treat the Exploration as a small piece of genuine mathematical writing — they have a question they actually want answered, they chase it through the algebra and calculus themselves, and they stop to think about what their answers mean. This guide is built around that mindset, criterion by criterion, so that by the end you know not just what to write but why each part earns the marks it does.

One more framing point before the detail. The five criteria are not weighted equally, and that should shape where you spend your effort. Use of mathematics alone carries six of the twenty marks — almost a third — and Presentation, communication, engagement and reflection share the rest. A common error is to pour hours into a beautifully typeset document with elementary mathematics inside it; the presentation marks are capped at four, so a polished exploration of trivial maths can never reach the top band. The reverse error — dense, sophisticated mathematics presented in an impenetrable wall of symbols with no reflection — fails just as surely. The best Explorations are balanced: ambitious mathematics, clearly communicated, genuinely owned, and honestly reflected upon.

The IB Maths AA Exploration at a glance

The Maths AA Exploration is a written investigation of a mathematical topic of genuine personal interest. There is no strict word count; instead it is expected to run to roughly 12–20 pages, with the length governed by how much mathematics the question genuinely needs rather than by padding. It is marked out of 20 across five criteria, and it is worth about 20% of the final grade for both Standard and Higher Level. Because Analysis & Approaches is the more abstract of the two Mathematics courses, the strongest Explorations lean on rigorous, analytic work — calculus, algebra, functions, trigonometry and proof — pursued in depth rather than across a sprawling list of techniques.

The page guidance is a guide, not a target, and it is easy to misread. Twelve to twenty pages does not mean you should stretch a thin idea until it fills twelve, nor compress a rich one to fit twenty. It means a focused piece of mathematics usually needs about that much room to be developed properly: enough space to define your functions, work through the analysis in full, draw the graphs that matter and reflect as you go, but not so much that the argument drowns. If your Exploration is running short, the honest fix is almost never to add words — it is to deepen the mathematics or sharpen the question. If it is running long, the fix is to cut the passages that do not move the investigation forward. Length is a symptom, not a goal.

It is also worth knowing what counts as an AA-appropriate topic. Analysis & Approaches rewards the analytic strand of the syllabus, so calculus (optimisation, areas, rates of change), algebra (sequences, series, complex numbers at HL), functions (transformations, modelling with families of curves), trigonometry and formal proof all sit comfortably at the centre of a strong Exploration. Pure data-gathering with little underlying analysis sits better in the Applications & Interpretation course; if your idea is essentially "collect data and fit a line", it may be in the wrong subject. For AA, aim for an idea where the interesting part is the mathematics itself — a result you derive, a function you analyse, a proof you construct — rather than the data behind it.

How the Maths AA Exploration is marked: the five criteria

Every mark comes from one of these five criteria. Write your Exploration with the criteria beside you and check what each one rewards:

How to write a Maths AA Exploration, step by step

1. Choose a topic of genuine interest. Start from mathematics you actually find interesting — a sport, a piece of music, a puzzle, a question you have always wondered about — so your personal engagement is real, and fix a clear aim.
2. Frame a focused research question. Sharpen the aim into one narrow, answerable question. "I like football" is not a question; "modelling the optimal angle for a free kick using calculus" is.
3. Plan AA-level mathematics. Map out the calculus, algebra, functions, trigonometry or proof you will use, and sanity-check that it is sophisticated enough for the course but still within your reach.
4. Carry out the mathematics with correct notation. Do the derivations yourself, define every variable, and write in proper mathematical notation rather than calculator shorthand.
5. Present clearly across 12–20 pages. Organise the exploration so it reads coherently and concisely, with clear, labelled graphs and tables that carry the argument forward.
6. Reflect critically as you go. Question your assumptions, weigh your results, and note the limitations of your model or method as they appear — not as an afterthought at the end.
7. Self-check against the five criteria. Read the draft against Presentation, Mathematical communication, Personal engagement, Reflection and Use of mathematics, and rewrite the weakest one.

A word on each of these steps in practice. The hardest is almost always the second — narrowing a broad enthusiasm into a single answerable question — and it is worth spending real time on it before you write anything else, because a vague question poisons everything downstream. A good test is whether you can state, in one sentence, what mathematical object you will produce: a maximum, a closed form, a proof, a model. If you cannot, the question is still too broad. The third step, planning the mathematics, is where you check ambition against ability: pick work that stretches you but that you can genuinely carry out and explain, because borrowed sophistication you cannot defend is worse than honest, slightly simpler mathematics that is fully yours. Steps six and seven are where most marks are quietly won or lost — reflection that runs throughout reads as genuine, whereas a single reflective paragraph bolted on at the end reads as exactly that.

Maths AA Exploration structure: what goes in each section

There is no single mandated layout, but the clearest structure that maps onto the criteria is:

• Introduction & rationale — why this topic interests you, and the aim of the exploration.
• Research question — one focused, answerable question stated explicitly.
• Background mathematics — the functions, theorems or techniques you will need, defined and introduced.
• Mathematical development — the calculus, algebra, trigonometry or proof, carried out step by step in correct notation.
• Results — outcomes presented in clearly labelled graphs and tables.
• Reflection — critical evaluation of results, assumptions and limitations, woven through rather than bolted on.
• Conclusion — a direct answer to the research question, with what you found and what it means.
• Sources — a consistent citation style throughout.

Notice that this structure is a skeleton, not a script. The sections need not appear as rigid headings, and the best Explorations let one flow into the next so that background mathematics introduces exactly the tool the development then uses, and reflection picks up the thread the results leave hanging. What matters is that an examiner can always tell where they are: what the question is, what you are doing, why you are doing it, and what you have concluded. If a reader has to flip back to remind themselves of your aim, the presentation is working against you. Keep the research question visible — restate it where it helps — and make sure every section is doing a job that the criteria reward.

What a strong vs weak Maths AA Exploration looks like

The fastest way to lift your marks is to see the difference. The three pairs below isolate the moves that most often separate a middling Exploration from a top-band one: the quality of the question, the cleanliness of the notation, and the depth of the reflection. In each pair the underlying topic is identical — what changes is the execution, and it is the execution that the criteria reward.

The topic and question

Mathematical notation

Reflection

Common mistakes that cost marks

• A theme instead of a question. "Something about music" is not a research question — narrow it until there is something definite to work out.
• No personal angle. A generic textbook topic caps Personal engagement; show why you chose it and how you made it your own.
• Calculator notation. Writing x^2 or 3*x throughout caps Mathematical communication — use proper mathematical notation.
• Trivial mathematics. Maths that is too easy for the AA course cannot reach the top band of Use of mathematics.
• Maths beyond you, copied. Borrowing an advanced result you do not understand is obvious to an examiner and risks academic-honesty issues too.
• Description instead of reflection. Narrating the steps is not reflection; evaluate your assumptions, results and limitations.
• Padding to fill pages. Length should come from the mathematics, not from filler — a concise exploration scores better than a bloated one.

If you take one thing from this list, let it be the connection between the question and everything else. Almost every mistake above traces back to a question that was too broad, too borrowed or too thin. A sharp, personal, genuinely AA-level question makes the rest of the Exploration easier to write: it tells you which mathematics to develop, it gives your reflection something concrete to evaluate, and it keeps your presentation focused because every section is visibly serving the same end. Spend disproportionate effort there, get a teacher to challenge it before you commit, and the remaining marks become far easier to earn.

Maths AA Exploration — frequently asked questions

Guidance written by experienced IB examiners and aligned to the current Mathematics: Analysis & Approaches guide. Not affiliated with or endorsed by the International Baccalaureate Organization.