IB Math IA examples
The IB Mathematics Internal Assessment is worth 20% of your final grade, and it is the one component where you control the topic, the approach, and the depth of exploration. That freedom is also what makes it difficult. Marks are often lost where the mathematics is correct: the exploration shows little personal engagement, stays below the level of the course, or never states its aim clearly. This guide follows one example exploration through its key passages, then covers topics that work for each course and a structure that suits most explorations.
What the criteria look for
The IB Math IA is marked out of 20 on five criteria. A and B are worth 4 marks each, C and D 3 marks each, and E 6 marks:
- Criterion A: Presentation (4 marks). Is the exploration coherent, well organised and concise, with an introduction, a clearly described aim and a conclusion, and graphs and tables placed where they are discussed?
- Criterion B: Mathematical communication (4 marks). Are notation, symbols and terminology correct, consistent and defined, and do you use more than one form of representation (formulae, diagrams, tables, graphs) where it helps?
- Criterion C: Personal engagement (3 marks). Does the work show you thinking independently or creatively, presenting ideas in your own way or testing your own predictions, rather than reproducing a textbook treatment?
- Criterion D: Reflection (3 marks). Do you discuss limitations, surprises, and what you would do differently? Is your thinking visible throughout?
- Criterion E: Use of mathematics (6 marks). Is the mathematics relevant to the aim? Is it commensurate with the level of the course? Is it applied correctly? Do you show that you understand it, not only that you can carry it out?
Take an SL exploration that scores 4, 4, 3, 3 and 5 on criteria A to E: 19 out of 20. If its mathematics stays at the level of the course but shows only limited understanding (3 on Criterion E) and its reflection only describes the results (1 on Criterion D), the same exploration scores 15. This guide is about closing that gap.
An annotated IB Math IA example
Complete explorations with their marks and the moderator's reasoning are rarely public (the reasons are further down this page). Instead, here is one exploration followed through its key passages, each written twice: the way drafts often read, and the way the upper levels of the criteria describe it. We made up the exploration, the measurements and both versions to show the difference. The numbers are realistic but were not measured. They do not come from a student's IA, and no mark is claimed for either version.
The exploration: the paper cone cups at a school water dispenser. For a fixed volume, which cone shape uses the least paper, and do the real cups use it? The mathematics is optimisation with derivatives, which is in the Analysis and Approaches SL syllabus, so read this as an SL exploration; at HL the same aim would need mathematics that shows the sophistication the HL descriptors ask for. The five criteria are the same for Applications and Interpretation.
The central result, h = r√2, is a standard textbook exercise, so deriving it would show little personal engagement on its own. What makes the exploration the student's own is testing the result against real cups and following up what the test showed. The passages appear in the order they come in the exploration, so the criteria run A, B, E, C, D.
The aim (Criterion A: Presentation)
In this exploration I will be looking at cones and how they are used in real life, because cones are everywhere, from ice cream to traffic cones.
The paper cups at our school's water dispenser are cones, and the bin beside it fills with them every day, which made me wonder whether their shape wastes paper. Aim: for a cone that holds a fixed volume, find the ratio of height to radius for which the cone uses the least paper, and test whether these cups use it. I measured the cups, modelled the paper as the curved surface of a cone, and minimised that area using differentiation.
Criterion A rewards an exploration that is coherent, well organised and concise, and the guide describes a well-organised exploration as one that has an introduction, describes its aim and has a conclusion. The weaker opening never states an aim: cones in real life is a topic, not a question. The stronger one gives a reason for the choice, which helps a reader follow it, then says what will be found, about which object and by what method, in three sentences a reader can hold on to for the rest of the exploration.
Setting up the model (Criterion B: Mathematical communication)
A = pi*r*sqrt(r^2+h^2) and V = 1/3pi r^2 h. I put h into A and differentiated on my calculator and got h = 1.41r.
Let r be the radius of the rim, h the height of the cone and l its slant height, all in cm (Figure 1), and let V be its volume in cm³, which is fixed; for the comparison later, V is the volume of the real cups. The paper forms the curved surface, of area A cm², where A = πrl and l = √(r² + h²), so
A = πr√(r² + h²)(1)
The volume V = ⅓πr²h gives h = 3V/(πr²). Because A is positive, A is smallest where A² is smallest, and minimising A² avoids differentiating a square root. Substituting for h in (1) and squaring:
A² = π²r⁴ + 9V²/r²(2)
Criterion B looks for relevant, appropriate and consistent notation and terminology, with key terms and variables defined, and it accepts calculator notation such as ^ and * only in software output. The stronger version defines every symbol with its units, labels the lengths on a diagram, numbers the equations it will refer back to, and explains the one step (minimising A² instead of A) that a reader might otherwise stop at.
Finding the optimum (Criterion E: Use of mathematics)
dA/dr = 0 so h = 1.41r, which is the minimum.
Differentiating (2) with respect to r:
d(A²)/dr = 4π²r³ − 18V²/r³
Setting this equal to zero gives r⁶ = 9V²/(2π²), which has exactly one positive solution. Substituting V = ⅓πr²h back in gives r⁶ = r⁴h²/2, so h² = 2r² and h = r√2 ≈ 1.41r. The second derivative, d²(A²)/dr² = 12π²r² + 54V²/r⁴, is positive for every r > 0, so d(A²)/dr is increasing: it is negative before this point and positive after it. A² therefore decreases and then increases, and this stationary point gives the smallest curved surface area of any cone with this volume.
Criterion E asks for relevant mathematics commensurate with the level of the course, used correctly and with understanding demonstrated. The weaker line asserts both the result and that it is a minimum without showing either. The stronger one derives the result, shows that it is a minimum rather than a maximum, and explains why it is the minimum over every cone of this volume. At HL this derivation still counts as commensurate, because SL content is part of the HL course, but it shows no sophistication, which the guide defines as HL mathematics or SL mathematics used in a way beyond what an SL student could reasonably be expected to do: level 5 asks for sophistication or rigour, and level 6 for precise mathematics with both.
Testing it against the real cups (Criterion C: Personal engagement)
I found online that cone cups are about 7 cm wide and 9 cm tall, so they are not the optimal shape. I found this very interesting, because I have always been interested in maths and in how things are designed.
I expected a manufacturer to minimise paper, so I predicted that the cups would be close to h = 1.41r. The cup in my hand already disagreed: h = 1.41r gives a cone about 1.4 times as wide as it is deep, and these cups are deeper than they are wide. To find out how far apart they are, I measured ten cups with vernier callipers. The paper gives slightly under the jaws, so I read each length only to the nearest millimetre: the rim diameters ranged from 6.9 cm to 7.1 cm and the heights from 8.8 cm to 9.1 cm, with means of 7.00 cm and 9.00 cm; I took r = 3.50 cm, h = 9.00 cm and V = 115.5 cm³. The real cup has h/r = 2.57, far from 1.41. For this volume the optimum is r = 4.27 cm and h = 6.04 cm, which would use 99.3 cm² of paper instead of the real cup's 106.2 cm², 6.5% less. To see what that saving would cost, I cut out and built the optimal cone. Its net is a sector of 208°, against 130° for the real cup, and the cone is 8.5 cm across but only 6.0 cm deep: a bowl rather than a cup.
Saying you are interested earns nothing on Criterion C: the guide looks for engagement shown in the work. The stronger passage checks the result against the object in the student's hand, measures to find out how far apart they are, and builds the optimal cone to see what the saving would cost.
What the result means (Criterion D: Reflection)
My model was accurate and I found the optimal cup. If I did this again, I would measure more cups.
The model answers a narrower question than the cup's designers faced. It counts only the curved surface, but a real cup uses paper in two more places. The glued overlap along the seam runs the length of the slant height, 9.66 cm on the real cup against 7.40 cm on the optimal one, so it favours the optimal shape and would make the saving larger. The rolled rim runs round the circumference, 22.0 cm on the real cup against 26.8 cm on the optimal one, so it favours the real cup and would make the saving smaller. I have not measured the width of either yet. Even ignoring the seam, the rim would cancel the 6.9 cm² saving only if its strip were at least 1.2 cm wide (a strip of width w extends the cone past its rim and adds πrw(2l + w)/l of paper). The rolled rim looks only a few millimetres across, but rolling hides the paper inside it, so this is the first thing to measure, by unrolling a cup. More importantly, the optimum ignores how the cup is used: a wide, shallow cone is harder to hold and spills more easily, and a saving of 6.5% of the paper is probably not worth that, which suggests the designers optimised for use rather than material. A next step would be to add the seam and the rim to the model as strips of measured width and see where the optimum moves.
Describing results is limited reflection. The stronger passage evaluates the model's assumptions, says which way each would move the answer, weighs the result against the real problem, and proposes a specific next step: critical reflection in the sense of Criterion D. In a full exploration, reflection like this runs through the work instead of waiting for the last page.
Each passage is shown under one criterion, but every criterion is judged across the whole exploration, and most drafts sit somewhere between the two versions. The useful question is which passages in yours still read like the weaker one: the Math IA grader marks your own draft criterion by criterion and points to them.
IB Math IA topic ideas for AA and AI
Analysis and Approaches (AA), SL and HL: topics that work
1. Modelling the spread of a rumour using differential equations
This classic exploration uses a logistic differential equation to model how information spreads through a population. It suits AA HL: solving the logistic differential equation by separating variables is additional higher level content (AHL 5.18). A student who chooses a real data source, such as post counts from a news story that went viral, gives Personal engagement something to reward: the context is self-chosen and the data set is one they assembled. The mathematics involves solving the logistic equation, fitting parameters to data, and comparing the model against observed values. At HL, level 5 of Criterion E asks for correct mathematics that shows sophistication or rigour, and level 6 for precise mathematics that shows sophistication and rigour, both with thorough knowledge and understanding. Going beyond the syllabus, for example with a model that adds a time delay, helps only if it serves the aim and is fully understood; the guide does not require mathematics beyond the syllabus for the highest levels.
Common mistake: reproducing the textbook logistic model step by step tends to score low on Criterion C. Engagement shows in what you do with the model: testing a prediction against real figures, questioning an assumption, or trying a second approach and comparing them.
2. Investigating the golden ratio in architecture or music
A perennially popular topic, but one that frequently scores low because students state that golden ratio connections "prove" aesthetic preference without measuring anything systematically. A stronger version measures a defined set of buildings or compositions, chosen before looking for φ, computes the ratios and compares their mean with φ ≈ 1.618, using their standard deviation to judge whether the gap is large, then reflects honestly on whether the data supports the claim. Correlation does not answer this question: two dimensions can be almost perfectly correlated while their ratio is nowhere near φ. Ratios, a mean and a standard deviation are the most elementary statistics in AA and are unlikely to carry Criterion E on their own, so an AA version needs more, for example the regression line of the longer dimension on the shorter: if every ratio were φ, the line would have gradient φ and pass through the origin, so compare the gradient with φ and check that the intercept is close to zero, because a gradient near φ with a large intercept means the ratios are not φ. At HL, a proof by induction of Binet's formula for the Fibonacci numbers could add the rigour the upper levels look for. Hypothesis tests are not in the AA syllabus, so an AA student who uses one must show they understand it. Reflection such as "the ratios cluster around 1.5, not 1.618, which suggests…" is the kind of thinking Criterion D rewards.
3. Optimisation of a packaging design
Calculus-based optimisation is reliable AA SL content. A student who chooses a product they actually use (a protein bar wrapper, a tea tin, a particular shoe box) and measures its real dimensions before comparing them with the theoretical optimum gives Criterion C real evidence. The mathematics should justify the optimum, for example with a second derivative test, rather than only find it. At HL a single optimisation like this uses only SL techniques: it is part of the HL course, but on its own it shows none of the sophistication that level 6 of Criterion E requires, so an HL version needs more, for example a container whose volume has to be found as a volume of revolution (AHL 5.17).
Applications and Interpretation (AI), SL and HL: topics that work
4. Regression analysis of Premier League goal data
An AI exploration built around regression is appropriate if the variables have a plausible relationship and the student goes beyond a single regression line. A strong version compares linear, quadratic, and exponential models, uses residual analysis to evaluate fit, and discusses which model is most appropriate and why. Non-linear regression, the sum of squared residuals and R² are HL content in Applications and Interpretation (AHL 4.13), so at SL use them only if you can show you understand them. A chi-squared test for independence between venue (home or away) and result (win or no win) for one club over several seasons can add a second strand of analysis; it is in the syllabus at both SL and HL. Keep to one club: a table built from both teams in every match pairs each home win with an away loss, so its observations are not independent.
5. Using Voronoi diagrams to optimise emergency service locations
Voronoi diagrams are in the AI syllabus at both SL and HL. An exploration that applies them to a real map, such as the nearest ambulance station for each district of a city or the nearest recycling point in a neighbourhood, works well because the application is practical and the mathematics is used rather than described. The student should take the stations' positions from a real map, construct the Voronoi cells from perpendicular bisectors, find the point furthest from every existing station (the syllabus's toxic waste dump problem, solved at a Voronoi vertex or on the boundary) as the site for a new one, and reflect on what straight-line distance ignores (traffic, road layout, capacity constraints).
6. Comparing body mass index between two age groups
A statistics-heavy AI exploration using publicly available health data. The student needs individual-level data, not published averages, for two age groups, applies the course's pooled two-sample t-test to compare their mean BMI, and checks what that test assumes: independent samples, similar variances in the two groups, and roughly normal data, which BMI often is not, although with large samples the sample means are close to normal anyway. The key to Criterion D here is recognising that statistical significance is not the same as practical significance: with a large sample, a difference too small to matter can still be significant.
Structure of a high-scoring IB Math IA
Your teacher marks the exploration and a moderator may read it cold. A clear structure shows organisation (Criterion A) and makes the mathematics easier to follow (Criterion B). This structure works for most explorations:
- Introduction: why this topic, and what the aim is. State the aim explicitly. Do not start with "Mathematics is everywhere."
- Background mathematics (optional): only the theory a reader needs to follow your exploration, not textbook definitions of things your reader already knows.
- Exploration (most of the work): your calculations, models, graphs and reasoning. Show working, label every figure, and explain what each step means, not just what it is.
- Reflection: what you found, what surprised you, the limitations of your model, and what you would do differently. Reflection that runs through the exploration counts, not only a closing section.
- Conclusion: what you found in relation to your aim, with no new material.
- Bibliography: cite every data source, textbook and website you used.
Length: the guide suggests approximately 12 to 20 pages with double line spacing, including diagrams and graphs but not the bibliography, and adds that the quality of the mathematical writing matters, not the length.
The most common reasons IB Math IAs score below expectations
Choosing a topic that is too broad
"The mathematics of climate change" cannot be explored in 20 pages. A focused question can, with a specific dataset: does a linear or an exponential model better fit the decline in September Arctic sea ice extent since 1979? Narrow your aim to something you can answer with the mathematics you know.
Listing results without explaining them
A student who writes "the derivative is 2x, therefore the minimum is at x=0" without explaining why this matters for the aim is losing marks on Criteria E and D: Use of mathematics asks for understanding to be demonstrated, and Reflection for results linked to the aim. Every result should be connected back to the research question.
Copying a well-known example
The "SIR model for disease spread" and the "mathematics of music and Fourier series" are familiar to anyone who marks IAs, and teachers and moderators recognise them quickly. If you choose a familiar topic, Criterion C depends on showing your own thinking: an unusual angle, predictions you test yourself, or a question the standard treatment does not ask.
Weak or absent reflection
A 1 on Criterion D is one of the most preventable mark losses in the Math IA. Students who write one paragraph at the end saying "in conclusion, my model was reasonably accurate" are describing, not reflecting. Reflection means asking: what did I assume? What could go wrong? How does this connect to real-world constraints? What mathematics could extend this exploration?
Why full Math IA examples are hard to find, and what to use instead
Complete, high-scoring Math IAs are rarely published in full, and for good reason: the work belongs to the student who wrote it, and the marked examples the IB annotates go to teachers in support material rather than onto the open web. What does circulate online usually lacks the two things that would help you: the mark it received and the comments explaining that mark. A PDF with no score attached cannot tell you why it scored what it did.
So what most people want from an example is the answer to this: what separates a top-band Math IA from an average one? The difference lies in specific things each criterion rewards:
| Criterion | What an average one does | What a top-band one does |
|---|---|---|
| A: Presentation | Sections exist, but the reader has to reconstruct what the exploration is trying to find out. | The aim is visible from the first page and the exploration stays coherent and concise around it. |
| B: Mathematical communication | Notation drifts, symbols appear undefined, graphs are unlabelled. | Notation and terminology are correct throughout; every graph and table is labelled and referred to in the text. |
| C: Personal engagement | The introduction asserts that the topic is interesting to the student. | Engagement is shown, not claimed: own data, an extension nobody assigned, an approach chosen independently. |
| D: Reflection | The ending summarises what was done. | The reflection evaluates the mathematics itself, its limits, and what the result does not establish. |
| E: Use of mathematics | Routine procedures, correctly executed, only just at the level of the course. | Mathematics commensurate with the level, and clearly understood rather than merely performed. |
See a full marked report
If you want to see what marking against published descriptors looks like on a whole piece of work, we publish three complete IBLens reports on the same TOK-style title at three quality levels, with the commentary explaining each band placement: Sample IBLens Reports. They are our own demonstration essays, so we can show them in full.
The faster route: check your own draft
Reading someone else's work tells you what good looks like in general. It cannot tell you what is missing from yours. IBLens marks your draft against these criteria. Without an account you get a free range for your total and, for most drafts, your weakest criterion with its feedback and the top risks. Ask your teacher first whether your school allows outside feedback on this work.
Check my Math IA against the criteria →