The Exploration

📋What it actually is

A written report, roughly 12 to 20 pages with double line spacing, investigating an area of mathematics that you chose. Diagrams, graphs and tables count towards the length; the bibliography does not. Around 10 to 15 hours of class and personal time is the expectation.

The page guidance is guidance, not a rule with a penalty attached — the criteria reward quality of mathematical writing, not volume. A tight 13-page Exploration that does one thing properly outscores a sprawling 22-page one that does five things loosely, every time.

🎯The five criteria

The final mark is the sum of five independently applied criteria. Notice how the marks are distributed — it explains a great deal about where to spend your effort.

• A — Presentation (4 marks). Coherent, well organised and concise. A real introduction stating the aim, a body that develops logically, and a conclusion that answers the aim. Graphs and tables belong where they are discussed; appendices are for bulk data only. Concise is the word that costs AI students marks in particular: a full data table of 200 rows belongs in an appendix, and forty near-identical calculations should be summarised, not printed.
• B — Mathematical communication (4 marks). Correct notation, symbols and terminology throughout, key terms and variables defined, and multiple forms of representation where they help. Calculator and computer notation is only acceptable if it is genuine software-generated output; typing x^2 in your own prose pins you at the lower levels. Label every axis on every graph, with units.
• C — Personal engagement (3 marks). Evidence that you made the mathematics your own — thinking independently, presenting ideas in your own way, testing predictions, approaching the topic from more than one angle. It is explicitly not a measure of effort. In AI, collecting your own data is one of the most reliable routes to this criterion, because the choices you had to make are yours to describe.
• D — Reflection (3 marks). Reviewing, analysing and evaluating — and this is the criterion most often thrown away. Merely describing results is “limited reflection”, worth 1. Meaningful reflection links back to the aim, comments on what you learned, or weighs one approach against another. Critical reflection — discussing the implications of a result, the weaknesses of your method, what you would do next — is worth 3, and needs to appear throughout, not only in a closing paragraph.
• E — Use of mathematics (6 marks). The largest single criterion. The mathematics must be relevant, commensurate with the level of the course (not built entirely on prior learning), correct, and with understanding demonstrated. Getting the right answer is not enough — and this matters especially in AI, where the calculator does the arithmetic. Feeding numbers into a regression tool and reporting the output does not demonstrate understanding; explaining why that model, what the parameters mean and how good the fit is, does. At HL the top levels additionally require sophistication or rigour.

🧩Choosing a topic

Three tests a good AI topic passes.

1. It asks a question, not a subject. “Basketball statistics” is a subject and produces a report. “Does a player’s shooting accuracy depend on distance from the basket, and can it be modelled?” is a question and produces an investigation. Write your aim as a sentence with a question mark before you start.
2. The mathematics sits at course level. AI rewards modelling, statistical inference, optimisation and — at HL — graph theory, matrices and differential equations. A report whose entire mathematical content is a mean, a bar chart and a percentage is built on prior learning and will be capped on criterion E, however good the writing.
3. You can say something about it that is yours. If the whole report could have been assembled from three web pages, criterion C is already capped.

Directions that work well for AI specifically: collecting your own data and fitting a model to it, then testing the fit and criticising it; a hypothesis test on something you genuinely wanted to know about your school, your sport or your city; an optimisation problem with a real constraint you measured; comparing two models on the same data and arguing for one; a network or routing problem in a place you know.

Directions that reliably disappoint: a survey of 30 friends analysed with one \( \chi^{2} \) test and nothing else; a summary of a topic already on the syllabus; correlating two variables you found in a spreadsheet with no reason to expect a relationship; and anything where the data is so clean that there was nothing to decide.

📝A workable structure

• Introduction. The aim, in one clear sentence. Why you chose it — specifically and personally, in two or three sentences. What you plan to do.
• Background and data. Define your variables, state your units, and describe where the data came from. State your assumptions; you will want them later for criterion D.
• The mathematics. Developed in stages, each motivated by the last. Where you use technology, say what it did and why it was the sensible tool — and show that you understand the output rather than just possessing it.
• Testing and extension. Check your model against data you did not use to build it. Change a parameter. Try a second model and compare. This is where criteria C, D and E are earned together.
• Conclusion. Answer the aim explicitly. Then evaluate: the limitations, what you would do differently, the obvious next question.
• Bibliography. Not assessed for quality, but its absence can be flagged as an academic honesty issue. Cite everything, including images, formulae and data you did not generate.

⚠️Academic honesty

The Exploration must be your own work, and you will have to authenticate it. Your teacher may read and comment on one draft — they can tell you where something is wrong or unclear, but they may not correct it for you, and the next version you hand in is final. Collaboration is allowed and encouraged when generating ideas, sharing sources and giving each other feedback on writing; it is not allowed on a single shared piece of work. Group-collected data must be individually analysed, and the joint collection described in the report.

The same work cannot be submitted for both the Exploration and the Extended Essay.

📅A sensible timeline

• Well before the deadline: submit a topic and a short outline. Expect the first idea to be too broad; most good aims are the third version.
• Then: collect the data and do the mathematics before you write the report. Investigating and writing simultaneously is what produces reports that wander.
• Draft: the one draft your teacher may comment on. Hand in something complete, not a fragment.
• Final: read it once with only criteria C and D in mind, and once out loud for notation and axis labels. Both passes reliably find marks.