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IB Math IA examples: what top marks look like

Searching for Math IA examples usually turns up real student work with no marks attached and no explanation. So we wrote our own: original example explorations, annotated criterion by criterion, with weak and strong versions side by side. They are teaching examples, not real student IAs.

By Ejaz Ahmad · 22 October 2026 · 10 min read

Two graphs of the same data: one unlabelled and one with labelled axes, units and a marked minimum
  • Original examples
  • Weak vs strong
  • Criteria A to E
  • AA and AI

A high-scoring exploration is hard to picture from a mark scheme. The words "clear", "relevant" and "insightful" mean little until you see a page that has them and a page that does not. That is what this post gives you.

Read this first. Everything below was written by us at Photon Academy. These are not real student explorations, they carry no IB marks, and the sentences are short illustrations, not full pages. Use them to see the difference between levels, then do your own with your own question and data. If you want the overview first, read the IB Math IA guide; if you need ideas, browse 40 Math IA topic guides.

How to read these IB Math IA examples

The Math IA has five criteria out of 20: A Presentation (4), B Mathematical communication (4), C Personal engagement (3), D Reflection (3) and E Use of mathematics (6). For each example we show where each criterion is earned, and a pair of short excerpts: a weaker version and a stronger one. The weak excerpts are the slips I see most often in my classes; none comes from a real student.

Where the 20 marks sitWhere the 20 marks sitA Presentation4B Communication4C Personal engagement3D Reflection3E Use of mathematics6Total: 20 marks
The same five criteria in every example below. A and B reward organisation and clear writing, C and D reward ownership and thinking, E rewards the mathematics.

Example 1: the best can (calculus, AA SL with HL extensions)

Aim: to find the shape of a cylinder that uses least material for a fixed volume, then explain why real cans differ. Topic guide: Cans: the least metal for a given volume.

Page plan for example 1 (14 pages)Page plan for example 1 (14 pages)1.53.53.53.52Introduction and aim (1.5)Model and derivation (3.5)Real cans, measured (3.5)Extension: seam and rim costs (3.5)Reflection and conclusion (2)
A 14-page plan. Reflection also appears inside the three middle sections, not only at the end.
Two graphs of the same data for a cylinder of volume 330 cubic centimetres: an unlabelled version and a labelled version with the minimum markedxyWeaker: what is this?minimum: r = 3.74 cm, S = 264 cm²Radius r (cm)Surface area S (cm²)246300350400Stronger: reads on its ownAxes titled "x" and "y", no units, no scale numbers, no caption.Titled axes with units, scale numbers and the key result marked.
The same data twice. Criterion B asks that a reader needs no help to understand a graph. The right-hand version also gives the reader the key result without hunting for it.

Excerpts for example 1, criterion by criterion

CriterionWeaker excerpt (ours)Stronger excerpt (ours)
A PresentationPage 1 is a history of the cylinder, then 3 pages of definitions of derivative. The aim is never stated."Aim: to find the shape of a 330 cm³ can that uses least metal, and to explain why shop cans differ from it." Everything after serves that sentence.
B Communication"S = 2*pi*r^2 + 2*V/r so S' = 4*pi*r - 2V/r^2 ... answer = 3.7". The symbol V is never defined."Let V = 330 cm³ be the fixed volume and r the radius in cm. Then S(r) = 2πr² + 2V/r, and S′(r) = 4πr − 2V/r²." The graph is labelled and referred to by number.
C Personal engagement"I have always liked cans." Then the standard textbook result is copied."I measured 12 cans from my kitchen and found most are taller than the ideal. I decided to test whether seam cost explains this." The cans and the decision appear later in the maths.
D ReflectionFinal page: "The model worked well. If I had more time I would use better data."After the cans table: "Only 3 of 12 cans were within 5% of the ideal ratio, so volume alone cannot be the driver. I will add a cost term." Later: "Taking the rim cost as constant is a strong assumption; I test 2 values."
E Use of mathematicsThe derivative is found with a calculator command and the answer is read off.The derivative is set to zero by hand, the second derivative confirms a minimum, and the answer is sense-checked against the sample cans. For HL, a justification that the minimum is global.

Notice what separates the columns. It is rarely cleverness. It is a stated aim, defined symbols, a visible decision and a reflection that changes what happens next.

Example 2: how does a drink cool? (modelling, AA or AI)

Aim: to test how well Newton's law of cooling describes a drink in cups of different materials. Topic guide: Cooling drinks: does Newton's law of cooling fit?.

This is a classic that is easy to make dull and hard to make excellent. Most weak versions fit a curve, quote R² and stop. The strong ones ask where the model fails.

A strip of sixteen pages marking where reflection appears in a weaker and a stronger explorationWeaker: one reflection, on the last pageStronger: reflection where it is earnedPages 1 to 16 shown as cells; each dot marks a place where the student questions their own work.The dots at pages 4, 7, 9 and 12 follow a poor fit, a surprise, a change of method and a comparison.
Criterion D rewards reflection built into the piece. A single paragraph at the end is the weaker pattern.
CriterionWeaker excerpt (ours)Stronger excerpt (ours)
A PresentationEight near-identical worked cooling curves, one per minute.One worked fit shown in full, then a table of cooling constants for all three cups.
B CommunicationScreenshot of regression output with no comment, constants written as "k=0.0421E0".Output explained: "The fitted constant k = 0.042 min⁻¹ (3 s.f.) means the excess temperature falls by about 4% per minute."
C Personal engagementData copied from a website for "a cup of tea".Own readings every minute for three cups, with a note on why the room was kept at one temperature.
D Reflection"The model fits very well (R² = 0.99).""R² is high, but the residual plot curves, so the model underestimates early heat loss. Evaporation is the likely cause, so I repeat with lids on."
E Use of mathematicsExponential fit chosen because the software offered it.The model is derived from the assumption that the cooling rate is proportional to the excess temperature; at HL the differential equation is solved and the constant is interpreted.
Three steps of increasing mathematical quality from plug-in to understood to rigorousPlug-inNumbers into a commandanswer read offUnderstoodMethod justified, steps shown, result checkedRigorous (HL)Claims justifiedlinks across topicsCriterion E: the same topic can sit on any step. Climbing is about reasons, checks and understanding.
Criterion E in three steps. SL top marks need the second step done throughout; at HL the third step is also expected.

Example 3: is a run of successes luck? (statistics, AI)

Aim: to decide whether a sports player's streaks differ from what chance predicts. Topic guide: Hot streaks: is a player streaky or just luck?.

CriterionWeaker excerpt (ours)Stronger excerpt (ours)
A PresentationA 3-page description of the sport's rules.One paragraph of context, then the question: is the number of runs unusual for a 62% shooter?
B Communication"p-value = 0.0034" with no hypothesis written."H₀: shots are independent with p = 0.62. H₁: runs are fewer than expected." The test, the statistic and the p-value are in that order.
C Personal engagementUses the most famous player in the world with the first dataset found.Chooses a local league player, explains why, and records how the dataset was cleaned.
D Reflection"The result is significant, so the player is streaky.""Significance does not prove streakiness: shots are not independent trials. I test a simulation of a luck-only player to see how often such a p-value arises."
E Use of mathematicsRuns test used as a black box.The binomial model is checked, the runs distribution explained, and a simulation compared with theory.

What top marks look like, in one table

Across all three examples the pattern repeats:

CriterionWeaker workStronger work
A PresentationNo aim, padding, repeated workingStated aim; every section earns its place; one example plus a table
B CommunicationCalculator syntax, unlabelled graphsDefined symbols, proper notation, graphs that stand alone
C Personal engagementA personal intro that goes nowhereOwn data and decisions shown in the maths
D ReflectionOne closing paragraphReflection after results, with a judgement and consequence
E Use of mathematicsRoutine, copied or mostly pre-courseReasoned, checked and understood; rigorous at HL

From my classes: students often ask whether a harder topic scores higher. In my experience it rarely does. A modest topic done with a clear aim, careful working and real reflection beats an ambitious topic that is only half understood. If a marker asks "why this method?" and you cannot answer, the difficulty is working against you.

If you want a deeper HL example, the roots of unity and the pentagon guide shows how AA HL content can drive a whole exploration. Your own writing must stay your own: the aim of these examples is to show the standard, not to be copied.

How to use these examples well

  • Put the stronger excerpt next to your own paragraph and ask what is different: a defined symbol, a visible decision, a sense-check?
  • Read the weaker excerpt as a checklist of slips to avoid.
  • Pick one criterion per day and revise it in your draft, rather than rereading everything.
  • Ask the IA & EE Coach for feedback: it reads your draft against the criteria and asks questions, but never writes or edits any of it.

Where to go next

Start with the IB Math IA guide, then pick a question in the Math IA topic explorer. The IA and EE guide shows how the Math IA compares with the others, and the GDC guide helps with data and regression. For reference, Desmos is useful for drawing graphs, Khan Academy statistics explains tests and the IB Diploma Programme page is the official starting point. Not sure whether AA or AI suits your topic? Read AA vs AI, and see the AA and AI formula booklets. Other subjects: Physics, Chemistry, Biology.

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