IB Maths IA guide: how to plan, write and score well (AA and AI)

What is the IB Maths IA (the exploration)?

The Maths internal assessment (IA) is called the exploration. It is one individual piece of writing about an area of mathematics that you choose. You decide the question, do the mathematics, and explain what you did and what it tells you. Your teacher marks it out of 20 and the IB moderates a sample. It counts for 20% of your final grade, and if no exploration is submitted there is no grade for the course.

The task is the same at SL and HL, and the same in Analysis and approaches (AA) and Applications and interpretation (AI). What differs is the level of mathematics expected in one criterion, which we cover in the AA or AI section.

It is written in your own words, about 12 to 20 pages long, double-spaced. Diagrams and graphs count towards the pages, and the bibliography does not. The IB says the quality of the mathematical writing matters, not the length. The IB suggests about 10 to 15 hours of teaching time for the whole task, including explaining it, class work time and checking authenticity. Most students also work on it at home for weeks.

The rules that trip students up

• One exploration, one author. Talking ideas through with friends, choosing topics together and asking for feedback on your writing are all fine. Working together on a single exploration is not. The writing and the calculations must be yours.
• Your own data. If your data come from a group experiment, each student collects their own. Pooled data may be used in the analysis if you describe clearly what was pooled and what was yours.
• One draft. Your teacher may read and advise on one draft, orally or in writing. They may tell you that errors exist but should not correct them. The next version you hand in is the final one.
• Acknowledge everything. Ideas, pictures, diagrams, graphs and formulae that are not yours need a source, in a standard style used consistently. The bibliography is not marked, but a missing one can be treated as an academic honesty problem.
• No reuse. The exploration cannot be the same piece of work as your Extended Essay.
• Ethics. If you survey or test people, ethical practice applies from the planning stage.

What counts as a topic

Any area of mathematics can work: a model of something real, an investigation of a pattern, or an application. The IB offers a list of starting ideas to teachers, but you are not limited to it. The mathematics should be about the level of your course. You do not need to go beyond the syllabus, and you are not penalised for staying inside it. Technology is encouraged where it helps but is not required, and an analytic approach is not marked down for avoiding it.

You should be able to explain every stage of your own work, and a classmate on the same course should be able to follow it.

How the Maths IA is marked: the five criteria

The 20 marks are split across five criteria. Your teacher chooses, for each, the level that fits your work best, and gives whole marks. A top mark does not need a flawless piece, and the descriptors reward understanding more than difficulty.

Notice how the marks fall. Presentation and Mathematical communication together are 8 of the 20. Engagement and Reflection are another 6. So 14 marks reward how you write, think and decide, and only 6 reward the mathematics itself. A good piece of mathematics that is badly explained loses a lot.

Criterion A: Presentation (4 marks)

What it rewards. Three words: coherent, well organised and concise. Coherent means logical and easy to follow, and it meets its own aim. Well organised means a stated aim, an introduction, a conclusion, and graphs and tables placed where they are used rather than in an appendix. Concise means nothing repeated and nothing that the reader does not need.

Where marks go. A wide or unreachable aim; work that starts well and tapers off; explaining things your classmates already know, such as how to find a mean; describing calculator buttons; showing the same calculation six times instead of once with a table; a plan at the start that the rest of the piece does not follow.

Our tip. Write your aim in one sentence. Then, for every section, write down how it serves that sentence. A section with no answer is a candidate for cutting.

Criterion B: Mathematical communication (4 marks)

What it rewards. Correct mathematical language and notation, key terms and variables defined where they are needed, suitable representations (formulae, tables, graphs, diagrams), and logical working. Software notation is acceptable only when it is pasted software output. In your own writing, use proper notation. A single suitable form of representation can be enough if it suits the topic, and small slips that do not hurt clarity are not punished at the top.

Where marks go. Calculator syntax such as x^2 in your own text; undefined parameters; graphs without titles, labelled axes or units; tables split across pages; tiny graphs; a long run of trivial calculations; symbols added for show.

Our tip. Read a page aloud as if to someone who cannot see it. Wherever you stumble, a symbol or a label is missing.

Criterion C: Personal engagement (3 marks)

What it rewards. How far you make the mathematics your own. The IB is clear that this is not a measure of effort, and not a measure of how interested you say you are. It looks for evidence in the work: thinking independently or creatively, presenting ideas your way, looking from different angles, making predictions and testing them. A teacher’s comment that you were engaged does not count. Textbook-style or reproduced mathematics rarely reaches the top.

Where marks go. Interest stated and then not visible in the maths; a routine task with your name on it; following a source’s method with new numbers; a topic chosen to fit a technique you wanted to use; a second method bolted on to look thorough.

Our tip. List three choices you made that another student on the same topic might not have made. If you cannot, look for the places where your data, your context or your constraint should have changed the mathematics.

Criterion D: Reflection (3 marks)

What it rewards. How you review, analyse and evaluate your own exploration. It can sit in the conclusion but is best found all the way through. Describing results is limited. Meaningful reflection links back to the aim, says what you learned, and weighs a limitation or a different approach. Critical reflection goes further: it changes what you do next, and it considers strengths, weaknesses and what the result means.

Where marks go. A list of limitations and extensions at the end that repeats what should have been done earlier; “sources of error” named and never investigated; reflection on how the project felt rather than on the mathematics; self-praise in place of testing.

Our tip. After each section, write one sentence that begins “so what?” and one that says what you will do next because of it.

Criterion E: Use of mathematics (6 marks)

What it rewards. Mathematics that is relevant to the aim, at the level of your course, correct, and understood. “Understood” means you show it by reasoning, not that you reach a right answer. A correct answer alone does not show understanding, even at the lower levels. You do not need many ideas, and even one topic is enough. The IB says plainly that a few things done properly beat many done badly.

Where marks go. Following a procedure like an exam question without saying why; many techniques, some wrong; a model chosen because the software offered it, with a high R² as the only argument; a test used without checking its conditions; complicated mathematics that does not help your aim; hand calculation of something technology does in a second, shown as if it proved understanding.

Our tip. List every piece of mathematics you use. Next to each, write why it is the right tool here and what else you could have used.

Three habits run through all five criteria: give reasons, not lists; give numbers, not adjectives; and link the criteria, so that a prediction you make is tested in E and what you learn feeds D.

AA or AI, SL or HL: what changes in the Maths IA

Less than most students expect. Criteria A to D are worded the same for AA and AI, and the same at SL and HL. You are marked on presentation, communication, engagement and reflection in exactly the same way whichever course you take.

Criterion E is the one that moves, in two ways. First, the mathematics is judged against your own course. An AA student is judged against AA content, an AI student against AI content, and SL against SL, HL against HL. Everything is read as “commensurate with the level of the course”. Second, HL has a different set of descriptors, which add sophistication and rigour at the top.

The last point of the table is worth repeating. You never have to go beyond your syllabus. If you do, you must explain it for a reader who has not met it, and you must show that it beats what is inside the syllabus. Maths wholly from before the IB cannot earn the higher levels in E at either level.

Topics that belong to one course

Some topics sit inside AI and outside AA. Graph theory, Voronoi diagrams, Markov chains, matrices and eigenvalues, the Poisson distribution, chi-squared and t-tests, slope fields and phase portraits are AI content, mostly at HL. An AA student who builds an exploration on them is working beyond the syllabus. That is allowed, but it asks for more explanation and a clear reason, and it does not buy extra marks. Check any topic against your own course before you commit.

Criterion E at SL and at HL

The figure below is our plain reading of the two sets of descriptors. It is not the official wording, so read the real descriptors in your subject guide before you rely on a detail.

Three HL words need unpacking. Sophistication means mathematics at HL level, or SL mathematics used in a complex way that an SL student could not be expected to manage. It can come from challenging ideas, a different view of the problem, or a link between areas of the course. Rigour means clear logic and language, with the claims that matter justified or proved rather than shown by one example. Precise means free of errors, with suitable accuracy throughout.

Two cautions from the examiners’ side. Standard techniques used with no link to the context are not sophisticated, and neither is complicated out-of-syllabus mathematics used by substituting into a formula. Very few explorations reach the top levels. A well-chosen, correct, well-explained piece at your own level is a much better target.

We have studied the AA wording closely. The AI HL examiner reports quote the same HL wording about sophistication, so the structure carries across, but check your own course guide for the exact AI descriptors and your exam session.

How to choose a Maths IA topic

The examiners’ own words are that a topic’s suitability depends far more on your framing, clarity and engagement than on the topic itself. A fashionable topic with a weak question loses to an ordinary topic with a sharp one. Use these seven tests.

1. A real question. Start from something you want to know about the world around you, not from a technique you want to show off. A question first, then the mathematics it needs.
2. Mathematics you can explain. At your course level, or close to it. If you cannot explain a step without the page, it is too far.
3. Room to adapt. Choose a problem where the first method will probably push back, so you have to change something. That is where Criteria C and D get their evidence.
4. Data you can get. Ideally your own, collected or built by you. If the exploration needs data, check on day one that you can get it.
5. Not too big. Many variables and constraints tend to produce shallow work. A narrow question done thoroughly beats a wide one.
6. Mostly mathematics. If it needs lots of physics, biology or game rules, it becomes an explainer of that subject.
7. Depth for HL. If you take HL, the topic should leave room to prove something, link two areas, generalise or test.

Topics that need a twist

Examiners have named some topics as routine over several years: the volume of an egg, the Gini coefficient, the correlation between two arbitrary variables, projectiles, cooling objects, the travelling salesman problem, Voronoi diagrams combined with it, the surface area of a bottle, and solids of revolution. This is not a ban. The same reports say framing matters more than topic. It does mean these topics need your own data, an unusual constraint, or a real test of the model, and a route that a textbook does not already give.

Be careful with templates. Work that is too like another student’s, or like a script posted online, puts you at risk of a finding of malpractice.

Where to look for ideas

Our list of 97 Maths IA ideas for AA and AI is sorted by topic, with the SL and HL fit for each. Our free Maths IA topic finder on photonresources.com has a full guide for each of the same 97 topics, and filters by course, level and syllabus area. Each guide gives you the checks a good question must pass and the traps to avoid, not a question to copy.

Use both as starting points only. Start from your own interests and course, look at two to four topics that fit your level, and write your own aim.

How to write a Maths IA aim or research question

The IB asks for an aim. A formal research question is not required. We still recommend writing one strong question, because it forces you to be specific, and a specific question is far easier to answer, test and reflect on than a topic title. If your teacher wants a stated aim, write the question and then add one sentence that says what you will do to answer it.

A strong Maths question has four parts, shown below. The example is our own, invented to show the pattern.

Weak, better, strong

These three examples are our own. They are not topics to copy.

What changed each time? The strong versions name the situation and the quantities, state the mathematics, give a specific comparison or test, and leave a place for the answer to surprise you. They also include a decision you must make (which model, which cost, which interval). That decision is where evidence for Criteria C and D comes from. And none of them can be answered with a single formula.

The 5-point check

Before you commit, score your question on five points: focus, variables, feasibility, data, and fit with the criteria. This is the check our Coach applies to a research question.

A good question also allows a surprise. “The model fails” and “there is no effect” are acceptable findings, and often more interesting than a neat success.

Modelling, data and technology in the Maths IA

Many strong explorations are modelling explorations: you build a mathematical description of something real, work with it, and test it. Not every exploration has to be one. A pure investigation of a pattern is allowed, and so is an application. But the modelling habit of “build, use, test, revise” is a good way to produce the reflection that Criterion D wants, so it is worth knowing.

Walking round the cycle

1. Ask a real question. This is your aim. It should be narrow enough to answer.
2. Simplify. Decide which quantities matter and what you will assume. Write the assumptions down, because you will test them later.
3. Build the model. Choose a function or method and say why. “The software offered it” is not a reason. “The loss is proportional to what is left” is.
4. Work with it. Solve, compute and graph. Use technology for routine calculation and show one worked case.
5. Test it on data it has not seen. Keep some data back. Quantify the error rather than listing possible errors.
6. Reflect. Does it answer your aim? If not, change an assumption or the model and go round again.

Technology: what to use and how to write it up

Technology is encouraged, because it lets you spend your pages on understanding rather than arithmetic. The rule of thumb is simple: let the tool do the calculation, and use your pages to say why you asked it and what the result means. Do not describe the buttons you pressed.

If you paste software output, the software’s notation may stay, because the IB accepts it as output. If you write it yourself, switch to proper notation. Cite any code, template or tutorial you followed, and make clear which mathematics is yours.

When you need data

Data you collect yourself are credited. Data simply taken off the internet generally are not, unless you explain why you chose them and what you did to them.

Using mathematics well: Criterion E in practice

Here is a small worked example of the habit that earns marks in E: choose a model for a reason, test it properly, and quantify the error. The numbers are invented and the example is far smaller than a real exploration. Do not copy it. It is there to show how understanding becomes visible on the page.

Suppose you pump up a bicycle tyre that has a slow puncture and record the pressure above atmospheric pressure every two days. You keep the day 12 and day 14 readings back.

Step 1: the tempting straight line

The points curve, but a straight line is the obvious first try. Least squares on the six build points gives p = 211.3 − 13.96t, with R² = 0.977. That looks excellent. A student who stops there has found a high R² and nothing else.

The residuals tell a different story. They are +9.7, −1.4, −9.5, −6.5, −1.6 and +9.3 kPa. They are large at both ends and negative in the middle, a U-shape. A U-shaped residual pattern means the straight line is the wrong shape, however high R² is.

Step 2: choose a model for a reason

Air leaks out faster when the pressure is higher, so take the rate of loss to be proportional to the pressure above atmospheric: dp/dt = −kp. That gives p = ae^−kt, an exponential decay. Taking logarithms gives ln p = ln a − kt, a straight line in t, which we can fit.

Least squares on the points (t, ln p) gives a slope of −0.1009 and an intercept of 5.398, with R² = 0.9996. So k = 0.1009 per day and a = e^5.398 = 221.0 kPa. The half-life of the excess pressure is ln 2 ÷ 0.1009 = 6.87 days.

Step 3: test it on data it has not seen

The exponential model predicts p(12) = 221.0 e^{−0.1009×12} = 65.8 kPa. The measured value is 65 kPa, so the error is +1.3%. For day 14 it predicts 53.8 kPa against 54 kPa, an error of −0.4%.

The straight line predicts 43.8 kPa on day 12 and 15.9 kPa on day 14. Those are errors of −32.6% and −70.6%. This is how you replace “the linear model is not very good” with a number.

What the example shows

• Relevant. The model comes from a reason about the situation, and it answers the question.
• Understanding. The student explains why the model has this form and why a logarithm turns it into a line.
• Technology used sensibly. The fit is a calculator or spreadsheet step. The page shows one worked case, not six.
• Tested. The prediction was checked against held-back data, with errors as percentages.
• Reflection. The residual pattern, not the R², is what changed the decision.

At HL, the same example would earn more by adding rigour: derive the model from the differential equation rather than stating it, and test the assumption behind it. Is the loss really proportional to the pressure, or does something change at low pressure? The mathematics stays at course level. The depth is in the justification.

Notice what the example does not do. It does not add a third model “for completeness”, and it does not claim the exponential holds for ever, since it never reaches zero. That limit belongs in your reflection.

Mathematical communication and presentation

Criteria A and B are the easiest marks to protect, because they come from habits rather than cleverness. Most lost marks here are small and avoidable with an hour of careful reading.

Notation and symbols

Use proper mathematical notation in your own writing. Use an equation editor so that variables are italic, subscripts sit properly and an equation does not split across two lines. Define every symbol where you first use it, and let each letter mean one thing only. If k is a rate on page 4, it should not be a constant on page 9.

Write a multiplication as a multiplication and a vector as a vector. Choose either Greek letters or Latin names for a quantity and stay with your choice. If you paste a formula from a PDF, check that it did not break when it was pasted.

Graphs and tables

Every graph needs a title, labelled axes with units, and a scale that fits the data. It must be big enough to read. Pick the right type: a scatter plot for two measured variables, not a histogram. Every table needs a caption and units, and should sit on one page. If a large table would break the flow, show a sample in the body and put the rest in an appendix. The marker does not need to read an appendix to follow your argument.

Place each graph or table next to the text that uses it, and refer to it by number. The IB asks for relevant graphs and tables to be in the body, with appendices kept for bulk material such as large data sets.

Setting out a test, and concision

If you use a hypothesis test, write it the way a mathematician would: the hypotheses, the test statistic, the p-value and a conclusion in context. State the conditions and check them. A result that is not significant does not prove the null hypothesis.

Put your aim on page 1 or 2, introduce each section by saying why it is needed, and return to the aim in the conclusion. Show a calculation once, with units, and summarise the rest in a table. Do not teach the syllabus. Finally, check the upload: missing equations and pages after conversion have cost students marks.

Personal engagement and reflection that earn marks

These two criteria are 6 of the 20 marks, and they are the ones students find most abstract. Both are easier once you see what they look like in the work.

Engagement is visible in your decisions

Marks go to evidence, not to enthusiasm. The IB describes three rough steps. A little evidence earns the first mark. Real engagement on a few occasions, which moves the exploration forward and shows what you intended, earns the second. Many high-quality moments, which steer the work in a creative way and show that you understand the context, earn the third.

Ask yourself: what is one decision here that someone else doing this topic might have made differently? Where does my own data change what I do next? Which part could I still defend if my source were taken away? If the answers are thin, the problem is usually the topic, not the writing. The examiners say the level available for engagement is often limited by the choice of task, and that routine tasks with a personal touch, such as modelling “my” vase with the textbook method, rarely score.

Researching new mathematics and describing it shows interest but not much more. It has to be applied in a way that is yours, or lead to a “what if” question that you then answer. Bolting on a second method to look thorough is noticed.

Reflection is a habit, not a final section

Reflection earns its marks when it changes what you do. After each section, say what the result means and what it makes you do next. A long list of limitations and extensions at the end adds little, because by then it is too late to act on them.

Look back at the tyre example. A student who reports R² = 0.977 for the straight line has described a result. A student who plots the residuals, sees the U-shape, asks why and changes the model has reflected. The change of direction is the evidence.

• Test the model on data it was not built from. Testing it on the same data it was fitted to cannot earn full credit.
• Quantify errors. Give bounds, or say which source of error matters most and by how much, rather than listing them.
• Look beyond R². Examine residuals, and discuss outliers or influential points in context before you keep or remove them.
• Check test conditions and interpret the result correctly.
• Ask “how reliable is this?” and “what would make it more reliable?” about your own result.

Two things do not count as critical reflection: a deliberately poor first method that you later improve, and a limitations list that is generic enough to appear in any exploration. The IB’s examiner reports also say that generic limitation and extension lists, some of them probably written by AI tools, add little. Write yours from your own work.

Maths IA structure and page count

The IB asks for about 12 to 20 pages, double-spaced, with graphs and diagrams included and the bibliography left out. It does not set section headings. The aim must appear, there must be an introduction and a conclusion, and the pages should flow. You do not need a contents page, a methodology section or a plan.

The figure below is how we would split a 16-page exploration. It is our habit, not an IB rule. Notice that reflection is not a section of its own: it is woven through the mathematics, and the “testing and reflection” block is where the biggest tests and their consequences live.

If you are at 22 pages, cut before you shrink the font: look for repeated calculations, syllabus explanations and software steps. If you are at 9, the question is usually too small for every criterion to have something to mark. Appendices are for bulk data and long tables, and everything the reader needs belongs in the body. Put the page count on the cover page, with no name or school.

A 10-week Maths IA timeline

Schools set their own deadlines, so treat this as a shape to work back from.

• Weeks 1 to 2. Shortlist two to four topics at your level. Draft a question, run the 5-point check, and write your aim in one sentence.
• Weeks 2 to 3. Plan the mathematics and get the data. Check early that your data exist, and keep some back to test the model.
• Weeks 4 to 6. Do the mathematics. Show a calculation once, graph as you go, and keep a log of the decisions you make and why.
• Weeks 5 to 7. Test the model on data it has not seen and write a sentence of reflection after each section.
• Weeks 6 to 7. Write the full first draft. A complete draft is more useful than a polished half.
• Week 8. Your teacher reads the one draft. Do not send a second.
• Weeks 9 to 10. Revise, check notation and references against the checklist below, and submit.

The mistakes we see most often, and the fix

These come from the drafts we read. They are patterns, not individuals.

1. The aim is too wide. “Explore the mathematics of sport” is a title, not an aim. Fix: state one question that names the quantities, the method and a test, and check that every section serves it.
2. The exploration teaches the syllabus. Pages go on defining a correlation coefficient or showing how to find a mean. Fix: assume your reader is on your course, and keep only what moves your argument.
3. Calculator syntax in the writing. We see ^, E and sqrt in the student’s own text. Fix: re-type in an equation editor. Only pasted output may keep software notation.
4. Graphs with no labels. Missing titles, axes without units, tiny images and split tables. Fix: look at each graph and ask “what do the axes say?”
5. Symbols that are undefined or reused. A letter stands for a constant on one page and a variable on another. Fix: list every symbol and where it is defined.
6. A high R² is the only argument. Often a high-order polynomial that fits everything and predicts nothing. Fix: choose a form the context supports, look at the residuals, and test on held-back data.
7. The model is tested on its own data. Reflection says “the model fits well” about the points it was built from. Fix: keep some data back and give the error as a number.
8. Chi-squared used on measurements. Or a test used with no check of its conditions. Fix: chi-squared works on frequencies only, and every test needs its conditions stated and checked.
9. Hand calculation shown as understanding. Long algebra, or a statistic worked by hand that technology gives in a second. Fix: let the tool calculate, and spend your words on why this method is right here.
10. A key derivation is announced but never done. Or the method in the text differs from what the code did. Fix: recompute your two or three headline numbers a second way and compare them with the draft.
11. Reflection is a list at the end. Limitations and extensions that could appear in anyone’s work. Fix: move each one to the section it belongs to, and say what you did about it.
12. A template topic. The method follows a video or a classmate and only the numbers are new. Fix: find the constraint, the data or the question that makes it yours, and cite anything you followed.

Maths IA checklist before you submit

Tick each box when you can point to it in your exploration. Your ticks are saved in this browser so you can come back to them.

How Photon Academy helps with the Maths IA

We give feedback, questions and teaching. We never write or rewrite a student’s IA, and we do not supply topics-as-answers, data, calculations or wording. The exploration and the marks are yours. There are two ways we can help.

In lessons at Photon Academy

Our Maths students, in both AA and AI, get IA support as part of tuition: choosing a topic, sharpening the aim, planning the mathematics, and feedback on the draft. Details are on our IB Maths tuition page. If you would like to talk it through first, message us on WhatsApp.

The IA & EE Coach, on photonresources.com

Our international study site has an IA & EE Coach for students anywhere. It gives questions and pointers, not rewrites.

There is one draft check per IA, in the spirit of the IB’s one-draft rule. Your school’s own rules come first, so ask your teacher which draft they have read.

Free: 97 full Maths IA topic guides on photonresources.com, with a finder that filters by course, level and syllabus area. Browse the 97 Maths IA topic guides.