EDUCIFLY BLOG
IB Math IA Examples: 4 Annotated Explorations and What Each One Scored
Most students searching for IB Math IA examples want one thing. They want to see a finished exploration, know what it scored, and know why it scored that. Lists of topic ideas do not answer that. So this page does the opposite of a topic list. Below are four full explorations — one for each course route — broken down page by page, with a mark for every criterion and the exact change that would push each one higher.
Quick answer: The IB Maths exploration is worth 20% of your final grade in all four routes: Analysis and Approaches (AA) SL and HL, and Applications and Interpretation (AI) SL and HL. It is marked out of 20 across five criteria: Presentation (4), Mathematical communication (4), Personal engagement (3), Reflection (3) and Use of mathematics (6). The IB suggests 12 to 20 pages. Truly marked example IAs live in your teacher's Teacher Support Material; everything else online is unmarked by the IB, so treat it as a model of shape, not of score.
A quick note on who wrote this. Educifly's IB Math specialists have coached explorations since 2018. The four examples below are worked models we built to show the criteria in action. They are not copies of student work, and the marks are our reading of how the published criteria would apply. Use them the way a chef uses a recipe photo — to see what "done" looks like.
If you have not picked a subject yet, start with our list of IB Math IA topics. If you have a topic and need the shape of the write-up, you are in the right place.
What "IB Math IA examples" actually means
An IB Math IA example is a complete mathematical exploration, written by a student, that you can read end to end. The useful ones come with marks and examiner comments attached.
The IA is not an essay about maths. It is a piece of mathematical writing with one clear aim, where you do the maths yourself and explain every decision. The IB calls it "the exploration". Your school may call it the IA. They are the same thing.
Three things make an example worth your time:
It has a stated aim, not just a topic. "Trigonometry" is a topic. "Can a single sine function model the tide at Marina Beach across a whole month?" is an aim.
It shows the working, not the summary. You should be able to follow the algebra.
It comes with marks per criterion. Without that, you cannot tell a 19 from a 13.
Most free "examples" online fail the third test. Keep reading and you will get all three.
The 20 marks you are actually judged on
Every exploration on the current course is marked out of 20 using five criteria. Here is what each one is really asking for, in plain English.
Criterion | Max marks | What the examiner is asking | Where students lose marks |
|---|---|---|---|
A: Presentation | 4 | Is it organised, easy to follow, and concise? | Padding. Long introductions. Pages that do not serve the aim. |
B: Mathematical communication | 4 | Is the notation correct and consistent? | Calculator syntax on the page. Undefined variables. Unlabelled graphs. |
C: Personal engagement | 3 | Did you make this your own? | Copying a standard proof and adding nothing. |
D: Reflection | 3 | Do you question your own results as you go? | A "limitations and extensions" list bolted on at the end. |
E: Use of mathematics | 6 | Is the maths right, and at the level of your course? | Formula-plugging with no understanding shown. |
Two details matter enormously and almost nobody mentions them.
First, criterion E is the only one that differs between SL and HL. The IB says plainly that "the descriptors for criterion E are different for SL and HL". At SL, maths that is correct with thorough understanding earns the full 6. At HL, that same work earns 4. To reach 6 at HL your maths must be precise and show both sophistication and rigour. This is the single biggest reason strong HL students are surprised by their IA mark.
Second, there is a hard cap. If the level of maths is not right for your course, criterion E is capped at 2 marks out of 6. An exploration built only on prior learning — the maths you knew before the DP started — hits that cap no matter how neatly it is written.
You can read how this fits with every other IA in the diploma in our guide to what an internal assessment is.
Example 1 — Math AA SL: the hanging chain
Aim: Does a parabola or a catenary better describe a chain hanging between two hooks, and why does the difference show up at the ends?
Why it works as an AA SL exploration. The student started from a function they already knew. That is what IB subject reports keep asking for. They did not learn a new field of maths to impress anyone.
Structure, page by page:
Pages | Content |
|---|---|
1 | Title page: title and page count only. No name, no school. |
2 | The question, in one paragraph, plus why a chain and not a bridge cable. |
3–4 | Getting 11 coordinates from a photograph using free geometry software. Explains the scaling. |
5–6 | Fitting y = ax² + c by hand using three points, then checking with regression. |
7–8 | Introducing cosh x as (eˣ + e⁻ˣ)/2, since exponentials are on the SL syllabus. Fitting a·cosh(x/a). |
9–10 | Comparing both models with a residual table and a residual plot. |
11 | Why the parabola fails near the ends: the gradient grows too slowly. Short calculus check. |
12 | Conclusion and one honest limitation about the photograph's lens distortion. |
Our mark: 17 out of 20.
Criterion | Mark | Reason |
|---|---|---|
A: Presentation | 4 | Twelve pages, no padding, every section serves the aim. |
B: Mathematical communication | 3 | Variables defined well, but two graphs are missing axis units. |
C: Personal engagement | 3 | Own photograph, own data, own decision to test the ends separately. |
D: Reflection | 2 | Reflection is real but sits mostly in the conclusion. |
E: Use of mathematics | 5 | Correct and well understood. Slightly thin: the residual comparison is descriptive. |
The one change that moves it up. Take the reflection out of the conclusion and put it where it happened. When the parabola fits well in the middle, say so there, and ask on the spot whether the middle is where you should be judging fit at all. Reflection that answers a reader's question at the moment the question arises reads as critical. Reflection saved for the end reads as a checklist. That single edit is worth a mark, taking this to 18.
Students on this route often want a second pair of eyes on the modelling choice. That is exactly what our IB Math AA tutors do in a normal session.
Example 2 — Math AA HL: how many terms of a series do you need?
Aim: Find the smallest number of terms of the Maclaurin series for cos x that approximates cos x to within 0.0005 on the interval 0 to π/2, and check whether the theoretical error bound is tight.
Why it works at HL. Maclaurin series are HL-only content. The student did not stop at "here is the series". They asked a question the textbook does not answer: is the error bound close to the real error, or wildly cautious?
Structure, page by page:
Pages | Content |
|---|---|
1–2 | The question and why approximation error is worth caring about. |
3–4 | Deriving the series from repeated differentiation. Short proof of the four-step derivative pattern. |
5–7 | Building polynomials with 2, 3 and 4 terms. Table of actual error at nine points. |
8–10 | The Lagrange error bound. Applying it, then comparing bound with actual error. |
11–12 | A graph of bound minus actual error. Explaining why the gap widens near π/2. |
13–14 | Testing the same method on sin x to see if the pattern holds. Conclusion. |
Our mark: 18 out of 20.
Criterion | Mark | Reason |
|---|---|---|
A: Presentation | 3 | Strong, but pages 5 to 7 repeat similar calculations that should have been summarised. |
B: Mathematical communication | 4 | Notation is precise throughout. Every inequality is set up properly. |
C: Personal engagement | 2 | Reads like a very good textbook chapter. The maths is not made personal. |
D: Reflection | 3 | Questions the bound's usefulness at each stage, not just at the end. |
E: Use of mathematics | 6 | Precise, sophisticated and rigorous. The extension to sin x shows structure. |
The one change that moves it up. Personal engagement is not about effort or about liking maths. It is about interacting with the mathematics in your own way. Here, the fix is small: before doing the calculation, write down a prediction. "I expect three terms will be enough, because the fourth term is already smaller than 0.0005 at x = π/2." Then test it and say whether you were right. Making and testing your own conjecture is engagement. That takes this to 19.
Example 3 — Math AI SL: how fast does a cup of chai cool?
Aim: Model the temperature of a cup of chai over 40 minutes, and test whether a lid changes the rate in the way the model predicts.
Why it works at AI SL. AI rewards modelling real situations with technology. A kitchen thermometer and 40 minutes is enough data for a full exploration.
Structure, page by page:
Pages | Content |
|---|---|
1–2 | The question and the set-up: two identical cups, one lidded, readings every two minutes. |
3 | The raw data table. Room temperature measured, not assumed. |
4–5 | Why a linear model cannot be right: it predicts the chai freezes. Plotting the failure. |
6–8 | Fitting an exponential model to the temperature gap above room temperature. |
9–10 | Finding the time for the gap to halve. Interpreting the constant in words. |
11–12 | The lidded cup: predicted versus observed, with a residual table. |
13 | Conclusion and limitations. |
Our mark: 16 out of 20.
Criterion | Mark | Reason |
|---|---|---|
A: Presentation | 3 | Clear, but the raw data table belongs in an appendix. |
B: Mathematical communication | 3 | Good, though "r²" is used before it is defined. |
C: Personal engagement | 3 | Own experiment, own control, own second cup. |
D: Reflection | 3 | Notices mid-exploration that the room warmed up, and adjusts. |
E: Use of mathematics | 4 | Correct, but the model choice is justified by r² alone. |
The one change that moves it up. This is the most common way marks leak out of an AI exploration, and IB subject reports say so directly: students pick a model, quote an r² value, and stop. A high r² does not make a model right for the purpose.
The fix is to justify the model before fitting it. One paragraph: "If the chai loses heat in proportion to how much hotter it is than the room, then the gap should fall by the same fraction every minute. That is exactly what an exponential does." Now the maths follows from an assumption you can state and test, rather than from a button on a calculator. That takes criterion E to 6 and the total to 18.
Our IB Math AI tutors spend most of their IA time on this one habit, because it is worth two marks in almost every modelling exploration.
Example 4 — Math AI HL: which café queue should I join?
Aim: Model the queue at the school café as a Markov chain with four states, find the long-run distribution, and test it against three weeks of observations.
Why it works at HL. Transition matrices and steady states are AI HL content. The student also chose something checkable — they could go and count.
Structure, page by page:
Pages | Content |
|---|---|
1–2 | The question, and the four states defined precisely (0, 1–3, 4–6, 7+ people). |
3–4 | Three weeks of observations, summarised. Full table moved to the appendix. |
5–6 | Building the transition matrix from observed frequencies. Every assumption listed. |
7–9 | Raising the matrix to higher powers. Watching the rows converge. |
10–11 | Solving for the steady-state vector directly, and showing both methods agree. |
12–13 | A chi-squared goodness of fit test comparing predicted and observed proportions. |
14–15 | Where the memoryless assumption breaks: the 10:45 bell. Conclusion. |
Our mark: 20 out of 20.
Criterion | Mark | Reason |
|---|---|---|
A: Presentation | 4 | Fifteen pages, appendix used properly, no repetition. |
B: Mathematical communication | 4 | Matrices, vectors and test statistics all notated consistently. |
C: Personal engagement | 3 | Three weeks of self-collected data and an original state definition. |
D: Reflection | 3 | The bell problem is spotted, tested and honestly reported as a failure of the model. |
E: Use of mathematics | 6 | Two independent routes to the same steady state. That is rigour. |
Why this one hits full marks. It is not because the maths is hardest. It is because the student solved the same problem two ways and got the same answer. Verifying your own result is the cheapest sophistication mark in the whole rubric, and almost nobody does it.
The same topic at 19 out of 20 and at 11 out of 20
Topic choice matters less than most students think. Execution matters more. Here is one topic — modelling a Ferris wheel with a trigonometric function — written two ways.
The 11-mark version | The 19-mark version | |
|---|---|---|
Aim | "To explore trigonometry using a Ferris wheel." | "Can one sine function predict my height on the London Eye at any minute, and how wrong is it in the first 30 seconds?" |
Data | Numbers taken from a website. | Wheel diameter and rotation time measured from a timed video. |
Maths | Substitutes into y = a·sin(b(t − c)) + d. | Derives each parameter from the physical set-up, then checks against the video. |
Reflection | "A limitation is that I assumed constant speed." | Tests constant speed against the video, finds the first 30 seconds are slower, and quantifies the error. |
Presentation | 22 pages, 6 of them screenshots. | 14 pages, 3 figures, every one referenced in the text. |
Communication | Writes "y=a*sin(b*(t-c))+d". | Uses proper notation and defines every parameter with units. |
Same topic. Eight marks apart. That is 8% of the final grade in the subject.
15 real IB Maths IA titles students have submitted
The IB publishes a list of previously submitted exploration titles in its teacher support material. Here is a selection, with what each one would actually demand of you. Read the IB's own warning first: these titles "attained a variety of marks". A published title is not a good title. Some of these scored badly.
Real submitted title | Maths it needs | Best fit |
|---|---|---|
The volume of an egg | Volumes of revolution, curve fitting | AA HL |
Optimum dimensions of an aluminium drink can | Optimisation, differentiation | AA SL |
Modelling the cooling of a cup of tea | Exponential models, regression | AI SL |
The Ferris wheel | Trigonometric modelling | AA SL / AI SL |
Graph theory — finding the shortest path | Graphs and networks, algorithms | AI HL |
Origami applications to mathematics | Geometry, angle construction, proof | AA SL |
The open Knight's Tour on a chessboard | Combinatorics, graph theory | AA HL |
Approximation of pi | Series, limits, geometry | AA HL |
Euler's totient theorem | Number theory, modular arithmetic | AA HL |
Exploring card counting in blackjack using probability | Conditional probability, expected value | AA SL / AI SL |
The Monty Hall problem | Conditional probability, simulation | AI SL |
The Tower of Hanoi puzzle | Recurrence relations, proof by induction | AA HL |
How many bicycles are there in Amsterdam? | Estimation, sampling, error bounds | AI SL |
Will female swimmers ever overtake male swimmers? | Regression, extrapolation, residuals | AI SL / AI HL |
Modelling Arctic sea ice cover | Trigonometric and exponential models | AI HL |
Notice the pattern. Every one of these is a question or a measurable object. None of them is a chapter heading.
24 more example research questions, sorted by course
These are Educifly's own suggestions, written as aims rather than topics. Each is narrow enough to answer in 14 pages.
Math AA SL
Which of three functions best models the arch of my local railway bridge, judged by residuals?
How does the volume of a paper cone change as I cut a larger sector from the circle, and where is the maximum?
Can I predict the number of visible spirals on a pine cone from its length using a linear model?
How accurate is the small-angle approximation sin x ≈ x, and where exactly does it break down?
What shape should a rain gutter be bent into to carry the most water?
How many possible distinct seating plans does my class have if two students refuse to sit together?
Math AA HL
Does Newton's method always converge for cubic functions, and what happens near a turning point?
How well does the Maclaurin series for eˣ approximate compound interest at very short intervals?
Can I find the exact volume of a rugby ball by revolving an ellipse, and how close is the ball I own?
Why does the harmonic series diverge so slowly, and how many terms are needed to pass 10?
What is the shortest closed path through the five buildings on my campus, and can I prove it is shortest?
How does the number of solutions to a trigonometric equation change as I scale its argument?
Math AI SL
Does the number of daily steps in my family follow a normal distribution, and what does that mean for a step goal?
How well does a logistic model describe the number of students who joined a school club over one year?
Is there a real correlation between a football team's possession share and goals scored, or is it noise?
How much does a phone battery's charge time depend on starting percentage, and is the relationship linear?
Can I model the queue time at the school canteen well enough to choose when to go?
Does the exchange rate between two currencies follow a trend I can model over 12 months?
Math AI HL
Can a Markov chain predict weather patterns in my city better than assuming tomorrow equals today?
Does a chi-squared test show that dice from three brands are equally fair?
How do the eigenvalues of a population matrix predict the long-run age structure of a herd?
Can a Voronoi diagram place a new fire station in my district better than the current one?
How sensitive is a loan repayment model to small changes in the interest rate?
Does a differential equation model of a spreading rumour match what happened in my school group chat?
Stuck between two of these? Pick the one where you can already explain the mathematics to a classmate without notes. That is the IB's own test for a workable topic.
Where to find real, marked IB Math IA examples
There is one source of genuinely IB-marked explorations, and it is not a website you can sign up for.
1. Your teacher's Teacher Support Material. The IB publishes a set of explorations assessed by experienced teachers, with the marks and the reasoning attached. Your teacher can open these on the programme resource centre. This is the only place where the marks are authoritative. Ask for them. Most teachers are happy to share and simply are not asked.
2. Your school's own archive. Many schools keep past explorations from earlier cohorts, with permission. These come with your own teacher's marks, which is useful, because your teacher marks first and the IB moderates a sample afterwards.
3. Public sample sites. Clastify, RevisionDojo, Nail IB and similar sites host large libraries of student IAs. They are genuinely useful for seeing structure, length and formatting. Two honest cautions. The scores shown are usually self-reported, not confirmed by the IB. And the temptation is real: reusing another student's work is malpractice, and you and your teacher both have to sign that the exploration is your own.
4. Not the IB store. Complete subject guides can be bought there, but the assessed exemplars are inside the school-access material, not sold separately.
Read examples for shape and for the feel of a well-argued page. Never read one on your own topic before you have written your own plan. It is very hard to unsee someone else's structure.
Five topics examiners have asked students to move on from
IB subject reports name specific topics that keep appearing and keep scoring poorly. The problem is never that the maths is bad. It is that the topic has been written up so many times that the student ends up transcribing rather than exploring.
The golden ratio. Almost always ends as a summary of existing work.
The birthday paradox. The maths runs out after three pages.
The SIR epidemic model. Very common since 2020, and usually copied wholesale.
Fourier transforms and partial differential equations. Far beyond the syllabus, so understanding cannot be shown.
Purely descriptive historical topics. The IB says directly that these are not appropriate, because the criteria cannot be applied to them.
There is also a quieter trap. Volume of revolution used only to produce a number from a formula will not reach the top of criterion E, however tidy it looks. The examiner wants to see you understand why the method works.
What changes for students starting the course in 2027
The IB is updating both mathematics courses. The new courses launch in February 2027, first teaching begins in August 2027, and the first assessment is May 2029. If you are sitting exams in May 2028 or earlier, nothing below applies to you.
The exploration survives. The criteria do not.
Current course (to May 2028) | New course (from May 2029) | |
|---|---|---|
Task | The mathematical exploration | The mathematical exploration |
Weighting | 20% | 20% |
Criteria | A Presentation (4), B Mathematical communication (4), C Personal engagement (3), D Reflection (3), E Use of mathematics (6) | A Problem specification (4), B Abstraction (6), C Computation (4), D Interpretation (6) |
SL and HL | Criterion E differs | Identical criteria |
Hours | 15 hours | 30 hours |
The new criteria follow the IB's four-stage inquiry process: specify the problem, abstract it into mathematics, compute, then interpret. Personal engagement disappears as a separate criterion. Assumptions and refinement become worth real marks.
Content is being cut too, and no content is being added. In AA, SL loses financial applications and bivariate data; HL also loses proof by counter example and Euler's method. In AI, SL loses logarithms and the trapezoidal rule; HL loses complex numbers, the vector product, the Poisson distribution and several hypothesis tests. That matters for IA planning, because "commensurate with the level of the course" will mean something slightly different.
A realistic IA timeline
The IB tells schools to hand samples to moderators in April for a May session, or October for a November session. Teachers usually collect final explorations six to eight weeks before that. Work backwards from your school's date, not from the IB's.
When | What you should be doing |
|---|---|
End of DP Year 1 | Keep a running list of ideas. The IB suggests starting exploration work before Year 1 ends. |
Summer between years | Pick two or three candidate aims. Test each for two hours: can you already explain the maths? |
Early Year 2 | Choose one aim. Write the aim in a single sentence. Collect or generate data. |
8–10 weeks before the school deadline | Full first draft. This is the one draft your teacher may formally comment on. |
4–6 weeks before | Rewrite from the feedback. Move reflection inline. Cut anything that does not serve the aim. |
2 weeks before | Check notation, label every figure, number every page, finish the bibliography. |
School deadline | Submit. No exploration means no grade in mathematics at all. |
That last line is not a scare tactic. The IB states that students who do not submit an exploration will not receive a grade for mathematics.
How Educifly helps with the Maths IA
We are a small practice, not a marketplace. Tutors are matched by subject, and you keep the same tutor every week. IA, EE and TOK coaching is a paid add-on to regular subject tutoring rather than something bundled into it, so you only pay for it when you need it.
For the exploration specifically, the useful work is nearly always the same three things: sharpening a vague topic into an answerable aim, justifying the model before you fit it, and pulling reflection out of the conclusion and back into the body. If that is where you are stuck, our IA support is built for exactly that, and you can start with a free trial class to see whether the fit is right.
Frequently asked questions
What is the IB Maths IA?
The IB Maths IA is a written mathematical exploration that every Diploma Programme maths student must complete. You choose your own aim, do the mathematics yourself, and explain your reasoning. Your teacher marks it out of 20 and the IB moderates a sample from each school. It counts for 20% of your final grade in the subject.
How many marks is the IB Math IA worth?
The exploration is marked out of 20 and is worth 20% of your final mathematics grade at both SL and HL, in both AA and AI. Because the weighting is 20%, the arithmetic is simple: 18 out of 20 on the IA contributes 18 of your final 100 marks, while 12 out of 20 contributes 12. Six marks on a single piece of coursework can be the difference between a 5 and a 6.
How long should an IB Math IA be?
The IB says 12 to 20 pages is appropriate, and adds that an exploration may be shorter than 12 pages. There is no word count. Conciseness is marked directly under criterion A, so a focused 14-page exploration usually beats a padded 22-page one. Diagrams and graphs sit inside the page count; the bibliography does not.
Where can I find real IB Math IA examples?
The only IB-assessed examples are in the Teacher Support Material, which your teacher can open through the programme resource centre. Ask them for the "Examples of explorations" section. Public libraries like Clastify and RevisionDojo are useful for seeing structure and length, but their scores are self-reported rather than confirmed by the IB.
Is the Math IA the same for AA and AI?
Yes. The task, the five criteria, the 20 marks and the 20% weighting are identical across Analysis and Approaches and Applications and Interpretation. What differs is the mathematics that counts as appropriate. AA explorations lean towards proof, functions and calculus. AI explorations lean towards modelling, data and technology. AA explorations are usually judged on argument; AI explorations are usually judged on how well the model matches reality.
Is the IB Math IA harder at HL than SL?
Four of the five criteria are identical. Only criterion E, Use of mathematics, has different descriptors. The HL ladder is shifted up: work that is correct and shows thorough understanding earns 6 at SL but 4 at HL. To score 6 at HL, the mathematics must be precise and show both sophistication and rigour. So yes, the same exploration will usually score lower at HL.
Can two students in my class write about the same topic?
You can work in the same area of mathematics, and you may even end up with the same title. What you cannot do is submit explorations that are the same mathematically. The IB expects the final piece to be the individual work of one student. Sharing sources, brainstorming ideas and asking for peer feedback are all allowed and encouraged.
How many drafts can my teacher read?
One. Your teacher may give formal written or spoken feedback on a single draft, telling you how the work could be improved without editing it for you. After that, the next version you hand in must be the final one. Informal help — such as revising a topic you are struggling with — is allowed at any point, as long as it is not tied directly to the exploration.
Do I need to collect my own data for the Math IA?
No. There is no requirement to use external material or to gather data at all. Plenty of high-scoring AA explorations are entirely analytic. If you do use data or formulas from elsewhere, cite them and list them in the bibliography. Every source you consulted must appear there, whether or not you cited it in the text.
What counts as a good IB Math IA score?
Anything at or above 15 out of 20 is a strong result. Below about 12, the IA starts to drag your subject grade down noticeably. Most explorations that miss the top band lose marks in the same two places: criterion D, because reflection was written as an afterthought, and criterion E, because a model was used without being justified.
Which IB Math IA topics should I avoid?
Avoid topics that have been written up so often that you would be summarising rather than exploring. Subject reports repeatedly name the golden ratio, the birthday paradox and the SIR epidemic model. Also avoid mathematics far beyond your syllabus, such as Fourier transforms, because you cannot show understanding of it. And skip purely descriptive historical topics, which the IB says the criteria cannot be applied to.
Is the IB Math IA changing?
Yes, but not soon. The updated mathematics courses launch in February 2027, with first teaching in August 2027 and first assessment in May 2029. The exploration stays, and it stays at 20%. The criteria change to Problem specification (4), Abstraction (6), Computation (4) and Interpretation (6), and they will be the same at SL and HL. Anyone sitting exams in May 2028 or earlier uses the current five criteria.
Written by Educifly's IB Mathematics specialists. All course details verified against the IB's published subject briefs and curriculum updates for mathematics: analysis and approaches and mathematics: applications and interpretation, and the IB mathematics teacher support material.