IA & EE guides
IB Maths IA examples: annotated outlines
Four original outlines showing how a Maths IA is built: the aim, a page plan inside 12 to 20 pages and where each criterion is earned. They are teaching examples written by us, not real student explorations.
- 4 original outlines
- AA and AI
- SL and HL
- Not real student work
Searches for IB Maths IA examples often lead to student work that cannot be used as a model, because the marks are unclear and the topic may not suit you. So we have written four outlines from scratch. They are not real student explorations: no student wrote them and they carry no marks. Each shows an aim, a page plan inside the recommended 12 to 20 pages and where each criterion (A to E) is earned, so you can see how a good exploration is put together.
Please do not copy them. Your exploration needs your own question and data, and we never write or edit any part of a student's IA.
1. Packaging optimisation (AA, SL or HL)
Course fit: AA SL, with extensions for HL
Aim: To find the shape of a cylinder that uses least material for a fixed volume, and to explain why real cans differ.
A calculus exploration that moves from a classic result to the student's own measured data.
| Section | Pages | What happens here |
|---|---|---|
| Introduction and aim | 1 | Why cans, the aim and the plan. |
| Model and derivation | 3 | Surface area and volume, one-variable form, derivative, the ideal ratio. |
| Real cans | 3 | Measure cans of different sizes, compare their ratios with the ideal. |
| Extension: costs | 3 | Add seam and rim costs, re-optimise, compare. |
| Reflection and conclusion | 2 | What the model ignores, which assumptions mattered, what to try next. |
Where the criteria are earned.
- A: a clear aim and a sequence that follows it.
- B: defined variables, labelled graphs of cost against radius.
- C: own measurements and a cost model of the student's own.
- D: reflection after comparing the ideal with real cans.
- E: optimisation used and justified, with the second derivative or a test; extra depth in the cost extension.
2. Is a run of successes luck? (AI, SL or HL)
Course fit: AI SL and HL
Aim: To decide whether a sports player's streaks differ from what chance predicts.
A statistics exploration that uses publicly available match data and a test.
| Section | Pages | What happens here |
|---|---|---|
| Introduction and aim | 1 | The question, why it interests the student, and what data will be used. |
| Background | 2 | Binomial model, hypothesis tests, and what a run is. |
| Data and method | 3 | Which data, how it is cleaned, the test chosen and why. |
| Analysis | 4 | Calculations, graphs, p-values and a simulation of a luck-only player. |
| Reflection and conclusion | 2 | Independence assumptions, sample size, what the test cannot say. |
Where the criteria are earned.
- A: a short, clear plan in order.
- B: clear tables and a consistent notation for probabilities.
- C: own choice of player and data, with the reason stated.
- D: reflection on independence of shots and on sample size.
- E: hypothesis testing and a simulation checked against the theory.
3. How does a drink cool? (AA or AI, SL or HL)
Course fit: AA or AI, SL and HL
Aim: To test how well Newton's law of cooling describes a drink in cups of different materials.
A modelling exploration based on the student's own experiment.
| Section | Pages | What happens here |
|---|---|---|
| Introduction and aim | 1 | The question and what a good model would show. |
| Model | 2 | The exponential model, its differential equation form for HL, the room temperature term. |
| Data collection | 3 | Readings every minute for three cups, with notes on accuracy. |
| Fitting and analysis | 4 | Fitting, residual plots, comparing decay constants. |
| Reflection and conclusion | 3 | Where the model fails, other factors such as lids and evaporation, extension ideas. |
Where the criteria are earned.
- A: a clear aim and a logical order.
- B: labelled graphs with units, and defined constants.
- C: own experiment and a choice of materials.
- D: residuals interpreted and the model critiqued.
- E: exponential models fitted and checked; at HL the differential equation is solved and used.
4. Roots of unity and regular polygons (AA HL)
Course fit: AA HL
Aim: To use complex numbers to explain which regular polygons can be drawn with ruler and compass.
A pure-maths exploration that depends on HL content.
| Section | Pages | What happens here |
|---|---|---|
| Introduction and aim | 1 | The question and the route. |
| Complex numbers background | 3 | Euler form, De Moivre, roots of unity. |
| Constructing polygons | 4 | Triangle, square, pentagon: the coordinates and how to build each. |
| Which polygons are possible | 3 | Why some can and some cannot, with the key idea explained, not copied. |
| Reflection and conclusion | 2 | What was proved or only stated, limits of the explanation. |
Where the criteria are earned.
- A: a logical build-up from basics to the question.
- B: careful complex-number notation and clear diagrams.
- C: own choice of polygons and own diagrams.
- D: reflection on what was proved and what only assumed.
- E: HL-level maths used throughout, with reasoning.
What the four outlines share
- A question with an aim, not a topic name.
- Maths that does real work at the right level for the course.
- The student's own data, measurement or choices.
- Reflection placed where results appear, not saved for the end.
Each plan stays inside 12 to 20 pages, so there is room for graphs. For a topic of your own, browse 40 Maths IA ideas; to see how each criterion is judged, read the criteria explained; the Maths IA guide has the overview.
Our promise
We teach. We never write it for you.
This guide explains how the work is assessed and how to plan it. We never write, edit or rewrite any part of a student's IA or EE, and nor should anyone else: the IB treats work produced or heavily edited by someone else (a person or an AI tool) as not your own. If you use an AI tool for ideas or checking, say so and cite it, as your school will ask.
Keep reading
More on the Maths IA
Find an idea, then plan with the criteria in mind.
Written by Ejaz Ahmad, IB teacher since 2015. Checked against the IB's published guide, October 2026. Photon Resources is independent of the IB.