What the mathematical exploration is, how long it should be, how much it counts, how the work is spread over the two years, how SL and HL differ, and what the strongest explorations have in common.
The internal assessment in Mathematics: Analysis and Approaches is a single piece of written work called the mathematical exploration. Each student chooses an area of mathematics, poses a focused question about it, and investigates that question in a report of about 12 to 20 pages. The exploration counts for 20% of the final grade at both SL and HL, so it carries as much weight as a full written paper in some subjects.
The exploration is different from every other part of the course. There is no time limit in a hall, no question set by someone else and no single correct answer. The student decides what to investigate, which mathematics to use and how to present it. This freedom is what makes the exploration rewarding, and it is also the main reason weak explorations go wrong: without a clear aim, the work drifts.
The exploration is meant to show what a student can do with mathematics when given time to think. It rewards understanding, independent thinking and clear explanation, rather than speed or recall. A good exploration reads like a short article written for a classmate: someone who has studied the same course, but who has not thought about this particular question, should be able to follow every step.
There is no fixed list of topics. Most explorations fall into one of a few broad kinds, and many combine two of them.
Kind | What the student does | Example of a focused aim |
|---|---|---|
Modelling | Builds a mathematical model of a real situation, fits it to data and tests how well it works | Find whether a quadratic or a cosine model better describes the outline of an arch, using points measured from a photograph |
Statistical investigation | Collects or finds data, analyses it and draws a conclusion with stated confidence | Test whether a player's free-throw success depends on whether the previous shot went in |
Optimisation | Sets up a quantity to maximise or minimise and finds the best value | Find the seat in a cinema from which the screen appears largest, measured by the angle it subtends at the eye |
Pure mathematics | Investigates a pattern, conjectures a general result and proves it | Determine for which the sum of the first odd cubes is a perfect square |
Simulation | Uses technology to estimate a quantity that is hard to calculate exactly, then compares the estimate with theory | Estimate the expected number of packs needed to complete a sticker collection and compare with the exact result |
Every kind can earn full marks. What matters is that the mathematics is at the level of the course, used correctly and understood, and that the student's own thinking shapes the work.
The exploration is not a research report about mathematics. A history of , a biography of a mathematician, or a description of how a famous proof works, retold from a book or website, shows reading but little use of mathematics. It is also not a set of textbook exercises with a story wrapped around them. If every calculation could be copied from a worked example in a textbook, there is no room for personal engagement or genuine reflection.
The exploration should be roughly 12 to 20 pages long with double line spacing. That count includes graphs, tables and diagrams, but not the bibliography or any appendices. A typical text font at 12 point with normal margins is expected. The page range is a guide to the scale of the work: fewer than 12 pages usually means too little mathematics or too little explanation, and more than 20 pages usually means the work is not concise.
The exploration is typed, and mathematics is written with an equation editor so that it looks as it would in a textbook: rather than x^2, rather than sqrt(x), rather than 3E5. Each graph and table is placed in the text where it is discussed, with a number and a short caption. Writing and notation are covered in detail in "Writing, technology and referencing".
The exploration is marked out of 20 using five criteria. The same criteria are used at SL and HL.
Criterion | What it assesses | Marks |
|---|---|---|
A Presentation | Whether the work is coherent, well organised and concise | 0–4 |
B Mathematical communication | Whether notation, terminology and representations are appropriate and consistent | 0–4 |
C Personal engagement | Whether the student's own thinking and ownership drive the work | 0–3 |
D Reflection | Whether the student evaluates the results, methods and limitations critically | 0–3 |
E Use of mathematics | Whether the mathematics is relevant, at the level of the course, correct and understood | 0–6 |
Total | 20 |
The mark out of 20 is scaled to 20% of the final grade. Because 6 of the 20 marks come from criterion E alone, the quality of the mathematics matters most, but criteria A to D together are worth 14 marks. A mathematically strong exploration that is badly organised, impersonal and unreflective can still lose almost half the available marks.
The teacher marks the final exploration using the criteria. The IB then asks for a sample of explorations from each school and an external moderator marks them again. If the moderator's marks differ consistently from the teacher's, the marks of every student at that school are adjusted. A student's final IA mark can therefore go up or down after it leaves the school.
Moderation explains why teachers often mark drafts cautiously. A teacher who gives generous marks to every exploration does not help students, because the moderator's adjustment applies to all of them.
Schools set their own internal deadlines, but the stages are the same everywhere. A common plan over the two years of the course looks like this.
When (typical) | Stage | What the student produces |
|---|---|---|
First year, terms 1–2 | Introduction to the criteria; reading and marking past explorations; keeping a list of ideas | An idea log of possible questions |
First year, term 2–3 | Choosing a topic and agreeing a focused aim with the teacher | A one-page proposal: topic, aim, planned mathematics, data source |
End of first year | Research, data collection, the main mathematics | Notes, data, calculations, first graphs |
Start of second year | Writing a complete draft | One full draft |
Shortly after | Written feedback from the teacher (once only) | Feedback comments |
Second year, term 1 | Revising and submitting the final version | Final exploration and signed declaration |
Before the end of the course | Teacher marking and submission of a sample to the IB | (no further student work) |
The course allows class time for the exploration, typically around 10 to 15 hours, but most of the thinking and writing happens in a student's own time. Spreading the work over several months gives time to change direction if a first idea does not work.
A student may submit one draft, and the teacher may give written or oral feedback on it once. The feedback points out where the work falls short (for example, "criterion D: reflection appears only in the conclusion" or "the model is not tested against the data") but the teacher does not correct the mathematics or rewrite sections. After the final version is submitted, it cannot be changed.
The exploration must be the student's own work. The student and the teacher both confirm this when it is submitted. Ideas, data, images and formulas taken from anywhere else must be acknowledged where they are used. The same piece of work cannot be submitted both as the mathematics exploration and as an extended essay, although the two may be in related areas. Academic integrity, including the use of generative AI tools, is covered in "Writing, technology and referencing".
SL and HL students write the same kind of exploration, use the same length guidance and are marked with the same criteria A to D. The only difference is criterion E, Use of mathematics.
SL | HL | |
|---|---|---|
Length | 12–20 pages | 12–20 pages |
Weight | 20% | 20% |
Criteria A–D | Identical | Identical |
Criterion E: level of mathematics | Commensurate with SL | Commensurate with HL |
Criterion E: top band | Correct mathematics with thorough understanding | Precise mathematics showing sophistication and rigour, with thorough understanding |
"Commensurate with the level of the course" means the mathematics is at the same level as the course content or above it. An SL exploration that uses only mathematics from before the course, such as percentages, bar charts and the mean, cannot reach the higher levels of criterion E however well it is written. At HL the bar is higher: work that would earn 6 marks for criterion E at SL may earn 4 at HL if it uses only SL techniques.
An HL student does not have to use HL-only topics, but the mathematics as a whole must be at HL level. Using a familiar SL technique in a demanding way, such as setting up and solving an optimisation problem with several constraints, or proving a general result rather than checking cases, can meet that standard.
Explorations that score highly tend to share a set of features, whatever their topic.
A narrow, precise aim. The aim is a question that can be answered with mathematics, stated early and returned to in the conclusion. "To determine the launch angle that maximises the horizontal distance of my paper aeroplane, using a model fitted to 30 measured flights" is answerable; "to explore the mathematics of flight" is not.
Mathematics that is the student's own. The student chooses the method, applies it to a case no textbook has worked through, and explains why each step is valid. Formulas taken from a source are explained or derived, not just used.
The right level of mathematics. The methods come from the course (or are of a similar level), and are used because the question needs them, not to look impressive.
Correctness. Calculations are accurate, results are stated with appropriate rounding, and technology is used and explained properly.
Reflection throughout. Each result is followed by a short comment on what it means, whether it is reasonable and what its limitations are, not only in a final paragraph.
A clear story. An introduction gives the motivation and the aim, each section leads to the next, and the conclusion answers the aim. A reader never has to wonder why a calculation is there.
The difference between these two aims is the most common difference between a strong and a weak exploration. "Choosing a topic and an aim" shows how to narrow an interest down to an aim like Student Q's.
I can state the length, the weight and the five criteria of the exploration, with the marks for each.
I can explain how the teacher's marking and the IB's moderation work.
I can describe the stages of the exploration and when each typically happens.
I can explain that SL and HL differ only in criterion E, and what HL adds.
I can tell the difference between a topic and a focused aim.
I can list the features that strong explorations share.
With a free account, Augment remembers where you stopped in the notes and which questions you have marked.