IB Math AI HL Internal Assessment — the complete guide
AI HL students have the richest toolkit of any IB Mathematics course for the exploration — Markov chains, graph theory, linear programming, non-linear regression, differential equations, hypothesis testing — and almost all of it is naturally applied. The challenge is the opposite of AA: not finding an application, but making sure the mathematics, not the technology, is what the examiner sees.
This guide sets out what Criterion E requires at AI HL, six topics built on HL content, and two complete worked explorations.
What Criterion E expects at AI HL
For AI HL, "commensurate with the level of the course" means content from the AI HL syllabus: matrices and eigenvalues, Markov chains, graph theory algorithms, linear programming, further differential equations, non-linear regression and the HL statistics extensions. A chi-squared test on a survey, however carefully done, is SL mathematics and caps Criterion E at 3/6 for an HL student.
| Clearly at AI HL level | Transition matrices and steady states; eigenvalues and eigenvectors in a real model; Dijkstra, Kruskal, Prim and the Chinese postman problem; linear programming with three or more constraints; logistic and other non-linear regression with transformation; Euler's method and coupled or second-order differential equations; Poisson and t-distribution tests; confidence intervals. |
|---|---|
| Borderline — add the HL layer | Linear regression, chi-squared tests, binomial/normal probability, compound interest and amortisation (all SL). Promote them: add a Markov model to the survey, an eigen-analysis to the matrix, an LP sensitivity analysis, a logistic alternative to the linear fit. |
| Not enough on its own | Descriptive statistics, Voronoi diagrams drawn by hand, percentage change, single-variable optimisation by reading a graph. |
For the full level descriptors of every criterion, see the IB Math IA criteria and rubric; for the 6-mark criterion in depth, see Criterion E — Use of Mathematics.
Six IA topic ideas at AI HL level
Each one has a research question you could use tomorrow, the mathematics that makes it course-level, and where the data comes from. None of them appears in the general topic list, so they are not the ones every examiner has seen.
Ranking football teams with eigenvectors
Can an eigenvector ranking of Premier League results predict the final table better than points alone?
Adjacency matrices, eigenvalues and eigenvectors, Perron–Frobenius idea, comparison with the official table using rank correlation.
Real results data; the same mathematics as Google PageRank, applied to something you follow.
The Chinese postman problem for a delivery route
What is the shortest route that covers every street in my neighbourhood at least once, and how much does it improve on the driver's current route?
Graph theory: Eulerian trails, odd vertices, minimum-weight pairing, Dijkstra for shortest pairings.
Map a real estate or campus; time the actual route for the comparison.
Queue modelling at a café or metro gate with the Poisson distribution
How well does a Poisson model predict arrivals per minute at a café counter, and what staffing level keeps the expected wait under two minutes?
Poisson distribution, goodness-of-fit test, expected values, simple queue reasoning, optimisation of a cost function.
Count arrivals yourself for an hour; primary data lifts Criterion C.
Predator–prey or two-population models with Euler's method
Can a coupled system of differential equations reproduce the observed cycles of two interacting populations, and how sensitive is the model to step size?
Coupled first-order ODEs, Euler's method, phase portraits, equilibrium points and their stability (eigenvalues of the Jacobian).
Published ecological data or your own simulation; the step-size analysis is the rigour.
Linear programming for a school timetable or diet plan
What combination of meals minimises cost while meeting daily nutrient targets, and which constraint is the most binding?
Linear programming with several constraints, simplex or technology solution, shadow prices and sensitivity analysis.
Use real supermarket prices and nutrition labels; the sensitivity analysis is HL-level thinking.
Markov chains and weather or traffic-light patterns
Using 90 days of observed weather, what is the long-run proportion of sunny, cloudy and rainy days predicted by a Markov chain, and how fast does the chain converge?
Transition matrices, matrix powers, steady-state vector by solving πP = π, eigenvalue interpretation of convergence rate.
Daily observation of anything with states — weather, a junction's lights, a bus being late or on time.
Two complete AI HL explorations, summarised
These are two of the 16 worked IA examples on this site, chosen because their core technique sits squarely at AI HL level. Here you get the idea, the mathematics and the reflection; the full page shows every calculation step.
Worked example 13: Smartphone Brand Loyalty: A Markov Chain Analysis
"Using a Markov chain model based on survey data from 200 students, what is the predicted long-run market share of Apple, Samsung and Huawei?"
Mathematical concepts used
- Transition matrix P (rows = current state, cols = next state)
- State vector multiplication: sn+1 = sn · P
- Steady-state distribution: solve πP = π with Σπᵢ = 1
- Eigenvalue approach: steady state is the eigenvector of PT for λ = 1
What the answer means — and how to reflect on it
The exploration's answer is a definite prediction — Apple settles at about 48% whatever the starting shares — and a timescale: roughly a decade. Test it: start from s₀ = (0.2, 0.2, 0.6) and show the chain converges to the same π. That independence from the start is the key property of a regular Markov chain, and demonstrating it with your own matrix is both rigorous and engaging.
Reflect on the assumptions with evidence. Transition probabilities are not constant — a new model launch changes them — so the chain is really time-dependent; discuss what happens if P changes every few years. The ‘memoryless’ assumption fails if people who switched once are more likely to switch again; propose a second-order chain with six states (brand now, brand before) as an extension. And your sample is one year group in one school — say what a national survey might change. Each of these is specific, mathematically framed and testable: that is what ‘critical reflection’ means.
See every calculation step for this exploration, explained line by line →Worked example 14: Route Optimisation Using Dijkstra's Algorithm
"How can Dijkstra's shortest-path algorithm minimise travel time between Dubai Metro stations, and how does the optimal route change under congestion?"
Mathematical concepts used
- Weighted directed graph: vertices (stations), edges (journey time)
- Dijkstra's algorithm: greedy approach, priority queue, O(V²) complexity
- Adjacency matrix representation and path reconstruction
- Sensitivity: recompute with ×1.5 weight on congested edges
What the answer means — and how to reflect on it
The exploration answers its question — 10 minutes via Union, ADCB and Deira — and shows the answer is robust to congestion on one link. Push further: find the weight at which A→B must rise before the direct A→C route becomes optimal (solve 3 + 2 > 6 ⇒ AB > 4). That threshold analysis turns a single answer into an understanding of the network's sensitivity.
Explain why Dijkstra is correct: when a vertex is chosen as the smallest unvisited, every other route to it must pass through some unvisited vertex whose label is already larger, so it cannot be shorter — provided no edge has negative weight. State that condition and note that travel times are always positive, so the algorithm applies. Compare its efficiency with checking every path (there are only a few here, but the number of paths grows exponentially with network size, which is why real apps use Dijkstra or its descendants). Suggesting an extension to the full 50-station network using a spreadsheet implementation, or to time-dependent weights, shows you understand the algorithm's scope.
See every calculation step for this exploration, explained line by line →How the AI HL IA is marked — the five criteria
20 marks in total, identical for every course except the level descriptors of Criterion E. Your teacher marks it; the IB moderates a sample.
| Criterion | Marks | What full marks look like |
|---|---|---|
| A — Presentation | 4 | Coherent, well organised and concise. Clear sections, numbered figures, 12–20 pages. |
| B — Mathematical communication | 4 | Correct notation, variables defined with units, graphs and tables labelled, consistent throughout. |
| C — Personal engagement | 3 | Your own question, your own data or decisions, independent thinking that is visible in every section. |
| D — Reflection | 3 | Critical reflection on results, assumptions, limitations and extensions — throughout, not just at the end. |
| E — Use of mathematics | 6 | Relevant mathematics at HL level, correct, precise, thoroughly understood, with sophistication and rigour. |
| Total | 20 | 20% of the final grade |
How to start your AI HL IA this week
AI HL IA — word count, format and other questions
How is the AI HL IA different from the AI SL IA?
Same criteria, same 20 marks, same page limit. The only difference is Criterion E: HL explorations are marked against the HL syllabus and the 5th and 6th marks require sophistication and rigour. In practice that means an HL IA must contain at least one HL technique at its core — a Markov chain, graph algorithm, LP, eigen-analysis or HL differential equation.
Can I let technology do the heavy lifting in an AI HL IA?
Technology is expected in AI, but it has to be visible mathematics that the examiner can follow. Show the transition matrix and one multiplication by hand, then let the GDC compute P¹⁰; show the Dijkstra table for the first three iterations, then the software result. "Technology checks; it does not replace working."
Is there a word count for the IB Math IA?
No — the IB sets a page limit, not a word count. The exploration should be 12–20 pages including graphs and tables but excluding the bibliography and appendices. Most strong IAs land at 14–18 pages and roughly 2,500–4,000 words of prose; if you are over 20 pages you lose the "concise" descriptor in Criterion A, so the maximum there drops to 3/4.
What format should the IA be in?
A typed document (Word, Google Docs or LaTeX) exported to PDF, with page numbers, a title that states the research question, headed sections, numbered figures and tables, and an equation editor for every piece of mathematics. Double spacing is not required; 11–12 pt text is normal. Raw data and repetitive calculations go in an appendix.
How much of my grade is the IA worth?
The exploration is worth 20% of the final grade for every IB Mathematics course (AA SL, AA HL, AI SL and AI HL). The remaining 80% comes from the written papers. It is marked by your teacher out of 20 and then moderated by the IB.
Can I use technology — a GDC, Excel, Desmos, Python?
Yes, and for most explorations you should. Technology is expected for data fitting, graphs and repetitive calculation, but Criterion E rewards the mathematics you show: formula → substitution → result, with technology used to check or to extend. "My calculator gave r = 0.81" with no formula earns little; showing the formula, one worked calculation and then the technology output earns full credit.
How many drafts can my teacher look at?
The IB allows the teacher to give written feedback on one draft. They may discuss ideas and answer questions throughout, but only one full draft receives written comments, so make that draft as complete as possible — including the bibliography and appendix.
Want a second pair of eyes on your AI HL IA?
Mr Ahmad has guided IB Mathematics: Applications and Interpretation HL students in Dubai and online from topic choice to final draft — including the honest "is this at course level?" conversation that saves marks in Criterion E.
IA tutoring — how it works →