The IB Mathematics Internal Assessment, called the Exploration, carries 20% of your final mark in IB DP Mathematics — across all four routes: AA HL, AA SL, AI HL, AI SL. Unlike the science IAs, the Maths Exploration is a piece of independent mathematical writing — typically 12–20 pages — exploring a topic of your choice using mathematical techniques.

This guide goes deeper than our overview of IB IA topic ideas across all subjects to focus specifically on Mathematics: 25 topic ideas with research questions split across AA and AI, the criteria broken down with what each one rewards, common pitfalls and what a high-scoring Exploration looks like.

Free resource: Use Tutopiya’s revision checklists for IB Maths AA and IB Maths AI to confirm your topic uses techniques you understand at depth.

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How the IB Mathematics Exploration is marked

The Exploration is assessed against five criteria for a total of 20 marks:

Criterion	Marks	What it rewards
Presentation	4	Logical structure, clarity, coherent narrative
Mathematical communication	4	Notation, terminology, well-defined symbols
Personal engagement	3	Genuine curiosity, original choices, your voice
Reflection	3	Critical thinking about results and process
Use of mathematics	6	Mathematical sophistication and accuracy at HL or SL level

A grade 7 Exploration typically sits at 17–19/20. Use of mathematics carries 6 of the 20 marks — and is where HL and SL diverge sharply (HL Explorations need genuinely HL-level techniques; SL Explorations rewarded for SL-level mathematics applied well).

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The five criteria, broken down

Presentation (4 marks)

Examiners want a coherent narrative with introduction, body and conclusion. Section headings help. The Exploration should read as a journey through a mathematical question, not a textbook chapter.

Mathematical communication (4 marks)

Examiners want correct notation, defined symbols and consistent terminology. Common mark-loss points: using = when ≈ is correct, mixing radians and degrees, undefined variables.

Personal engagement (3 marks)

Genuine curiosity — why this topic. Original design choices. Your voice. Generic intros lose marks here.

Reflection (3 marks)

Examiners want critical thinking about your results — not just stating them. “My model fits the data with R² = 0.94, but the residuals show systematic deviation at low values, suggesting…” earns full reflection marks. “My model fits the data well” earns one.

Use of mathematics (6 marks)

The most heavily weighted criterion. HL Explorations need HL-level mathematics — calculus, complex numbers, vectors at full HL depth, statistical inference, proof. SL Explorations are rewarded for SL-level mathematics applied accurately and meaningfully.

A common HL pitfall: choosing a topic that uses only SL maths. Even a beautifully written Exploration on, say, geometric probability, will lose 2–3 of the 6 use-of-maths marks at HL because the techniques don’t extend to HL level.

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25 IB Mathematics Exploration topic ideas

These are deeper and more specific than the 10 in our overview article. Topics are tagged for AA and AI suitability.

Calculus and modelling (AA HL/SL, some AI HL)

1. Optimisation — Designing the most material-efficient soft-drink can: optimising surface area for fixed volume. (AA HL/SL)
2. Volume of revolution — Modelling the volume of a wine glass using a function and the disc method. (AA HL)
3. Newton’s law of cooling — Modelling the cooling of a hot drink and using calculus to predict ambient time. (AA HL/SL, AI HL)
4. Population growth — Comparing exponential and logistic growth models for a real population. (AA HL/SL, AI HL/SL)
5. Cycloids and curves — Investigating the path traced by a point on a rolling wheel using parametric equations. (AA HL)

Statistics and probability (AA, AI all routes)

6. Hypothesis testing — Is there a significant difference between resting heart rates of athletes and non-athletes? (Two-sample t-test.) (AA HL, AI HL/SL)
7. Chi-squared independence — Is preference for music genre independent of gender at school? (Chi-squared test.) (AA HL/SL, AI all routes)
8. Correlation and regression — Does Big Mac price correlate with GDP per capita across countries? (AI HL/SL, AA SL)
9. Monte Carlo simulation — Estimating π using random sampling in a unit square. (AI HL, AA HL)
10. Binomial vs Poisson — Modelling the number of cars passing a junction per minute. (AI HL/SL)

Number theory and algebra (AA HL especially)

11. Modular arithmetic — Investigating divisibility tests using modular arithmetic. (AA HL)
12. Continued fractions — Approximating irrational numbers using continued fractions. (AA HL)
13. Pythagorean triples — Generating Pythagorean triples and proving the parameterisation. (AA HL/SL)
14. Complex numbers — Investigating the geometric meaning of complex multiplication using Argand diagrams. (AA HL)

Geometry and trigonometry

15. Spherical geometry — Calculating great-circle distances between cities and comparing with flat-map distances. (AA HL, AI HL)
16. Tessellations — Investigating which regular polygons tessellate the plane and proving why. (AA HL/SL)
17. Fourier series — Approximating a square wave using Fourier series. (AA HL — extension)

Real-world applications (AI focused)

18. Loan amortisation — Modelling a mortgage payment schedule and the effect of overpayment. (AI HL/SL)
19. Population pyramids — Comparing demographic structure across countries using statistical measures. (AI HL/SL)
20. Sports analytics — Predicting football match outcomes using Poisson goal modelling. (AI HL)

Cryptography and game theory

21. RSA cryptography — How does RSA encryption work and why does the maths protect the message? (AA HL)
22. Game theory — Investigating Nash equilibria in simple two-player games. (AA HL, AI HL)

Sequences and series

23. Fibonacci and the golden ratio — Investigating the Fibonacci sequence and its convergence to the golden ratio. (AA HL/SL)
24. Compound interest — Comparing the long-run effect of monthly vs annual compounding. (AI HL/SL, AA SL)

Open-ended investigation

25. Dimensional analysis — Using dimensional analysis to predict scaling laws in nature (e.g. why elephants don’t gallop). (AA HL, AI HL — physics-adjacent, ambitious)

For broader topic ideas across the IB subjects, see our overview of IA topic ideas by subject. For broader maths preparation, our guide on tips to score 45 points in IB Maths covers complementary strategies.

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What a high-scoring Mathematics Exploration looks like

A 18/20 Exploration shares five common features:

1. A focused mathematical question that can be addressed in 12–20 pages.
2. Mathematics at the appropriate level for your route (HL Explorations using HL techniques).
3. A clear narrative with introduction, mathematical development, results and reflection.
4. Critical reflection on results — limitations, alternative approaches, extensions.
5. Correct mathematical notation and communication throughout.

Length: typically 12–18 pages including diagrams.

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Common Mathematics Exploration pitfalls

Five errors come up consistently:

1. Topic uses only SL-level maths at HL. This is the single biggest reason HL candidates lose Use-of-Mathematics marks. Choose a topic that demands HL techniques.
2. Excessive narrative, insufficient maths. Ten pages of text and two pages of equations is a low-scoring pattern. Aim for at least 30–40% mathematical content.
3. Generic introduction. “I have always been interested in maths” loses Personal Engagement marks. Be specific about why this topic chose you.
4. No reflection. Stating results without discussing them costs Reflection marks.
5. Notation errors. Mixing radians and degrees, undefined variables, = where ≈ is correct.

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How to plan your Mathematics Exploration timeline

A useful Exploration timeline runs over 3–4 months:

• Month 1: Topic selection, draft research question, initial reading.
• Month 2: Mathematical investigation, data collection if applicable, first full draft.
• Month 3: Teacher feedback, revisions, second draft.
• Month 4: Final revisions, mathematical-notation review, submission.

The candidates who score 18+ are the ones who finish a complete first draft by month 2 and spend two months refining the mathematics. Candidates who finish a first draft in month 4 rarely break 14/20.

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Frequently asked questions

How long should an IB Maths Exploration be?

The IB recommends 12–20 pages. Most high-scoring Explorations sit at 14–18 pages.

Does my topic have to use HL-level maths if I’m sitting AA HL?

Yes — to score full marks in Use of Mathematics at HL, the techniques used must be at HL level. Topics that use only SL techniques typically cap at 4/6 in this criterion.

Can I do an Exploration on a topic outside the syllabus?

Yes — but you must justify the topic and demonstrate the mathematics rigorously. Out-of-syllabus topics often score well on Personal Engagement but require careful Use of Mathematics treatment.

How does AA differ from AI in terms of Exploration topics?

AA Explorations typically lean toward pure mathematics (calculus, complex numbers, proof). AI Explorations typically lean toward applied mathematics (statistics, modelling, real-world data). Both routes can score full marks; the topic must match the route.

What’s the most common reason Mathematics Explorations lose Personal Engagement marks?

Generic introductions and an absence of original choices. Examiners want to see your voice — specific reasons, original parameters, personal context.

Can I include code or simulations in my Exploration?

Yes — code (Python, MATLAB, Geogebra) is welcome. Include the code in an appendix and explain the underlying maths in the body.

How does the Exploration mark connect to my final IB Maths grade?

The Exploration contributes 20% of your subject grade across all four IB Maths routes. See our IB DP Mathematics grade boundaries guide for how the Exploration mark is combined with written papers to award a 1–7 grade.

Should I include proofs in my Exploration?

For AA HL especially, proofs significantly enhance Use of Mathematics marks. Even a short rigorous proof of a key result demonstrates mathematical sophistication.

Can I use a textbook example as the basis for my Exploration?

You can use a textbook idea as a starting point, but the Exploration must extend or apply it in a way that demonstrates your mathematical thinking. Reproducing a textbook proof verbatim costs Personal Engagement marks.

What’s the difference between a 14/20 and an 18/20 Exploration?

The 18 has level-appropriate mathematics, a clear narrative, critical reflection on results and correct notation throughout. The 14 typically misses one of those — most often the level-appropriate maths at HL or the critical reflection.

How many hours should I spend on the Exploration?

The IB allocates 10 hours of class time plus your independent work. High-scoring Explorations typically involve 30–50 hours total.

Is a single research question enough, or should I have multiple questions?

A single focused question is usually best. Multiple questions risk fragmenting the narrative; keep one central question and use the body to investigate it from different angles.

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Last reviewed: 29 April 2026. The IB Mathematics Exploration is moderated against published criteria. Always work from the official IB Mathematics subject guide and your teacher’s feedback. For broader subject-IA ideas, see our IB IA topic ideas by subject overview.

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