The complete guide to planning, writing, and submitting a top-scoring Mathematical Exploration. Free tools for every stage — for AA & AI, SL & HL.
The Internal Assessment — officially called the Mathematical Exploration — is a piece of independent written work worth 20% of your final IB Maths grade. It is marked by your teacher and then moderated externally by the IB, so the same mark scheme applies to every student in the world.
Unlike your exams, the IA is completed during the course with no time pressure. You choose your own topic, investigate it at your own pace, and submit a polished written exploration. That makes it one of the most controllable parts of the IB — and with the right approach, one of the easiest places to score well.
The exploration is not a history-of-maths essay, a biography of a mathematician, or a summary of a textbook chapter. It is an active mathematical investigation: you do calculations, build models, test hypotheses, and reflect on what you find. The IB wants to see you doing mathematics, not describing it.
There is no single compulsory format, but examiners consistently reward explorations that follow a clear, logical structure. A strong IA typically has these seven components:
A precise, specific title that tells the reader exactly what you investigated — not just "The Mathematics of Music" but "Modelling the Relationship Between Tempo and Emotional Response Using Regression Analysis."
Explain the context of your topic and — crucially — your personal reason for choosing it. State your aim clearly: what question are you trying to answer? This is where personal engagement marks are earned or lost. Keep this section to 1–2 pages; don't pad it with general background.
Briefly introduce any theory the reader needs to follow your work. Define all variables and notation here. Don't overdo this — examiners don't need you to re-derive the quadratic formula. Include only what is genuinely necessary to understand your exploration.
The largest section — where you actually do the mathematics. Present your calculations, models, graphs, and tables clearly. Show all working. Connect each result back to your aim: don't just calculate — interpret. Weave in reflection as you go rather than saving it all for the end. This section earns most of your marks on Criteria B, D, and E.
Answer your original question directly. Summarise the key findings without simply restating everything you've already written. Discuss whether the results matched your expectations and what they mean in the real-world context of your topic.
Acknowledge the limitations of your approach — the assumptions you made, the data you couldn't obtain, the models that oversimplify. Suggest meaningful extensions: what would you do differently? What could be explored further? This earns Criterion D marks and is often where students leave marks on the table.
List every source you used. Use a consistent format (MLA or APA) and cite within the text wherever you quote or use an idea from elsewhere. Academic honesty is taken seriously — a missing bibliography can trigger a formal review.
Every IB Maths IA is marked against the same five criteria, totalling 20 marks. Understanding exactly what each criterion rewards — and what it penalises — is the single most valuable thing you can do before you start writing.
Is the exploration well-organised, coherent, and easy to follow? Every graph labelled, every table titled, logical flow from start to finish.
Is notation used correctly and consistently? Are variables defined? Do you use multiple forms of representation (equations, graphs, tables)?
Does the exploration feel genuinely yours? Is there evidence of independent thinking, creativity, or a personal connection to the topic?
Do you critically evaluate your results, acknowledge limitations, and suggest meaningful extensions — throughout the exploration, not just at the end?
Is the mathematics relevant, accurate, and sufficiently sophisticated? SL: go beyond routine procedures. HL: demonstrate genuine HL-level content. Examiners penalise maths that is too simple for the level.
Students who score 18–20 out of 20 consistently share the same qualities. Here's what distinguishes a Grade 7 IA from an average one:
The best IAs come from genuine curiosity. "I fractured my wrist and wanted to understand the biomechanics of bone stress" will always outscore "I chose statistics because I like sport." The more specific the question, the deeper the mathematics.
Top explorations use mathematics at or above the course level — differential equations, regression modelling, hypothesis testing, calculus optimisation — applied correctly and interpreted meaningfully. Simple arithmetic or basic averages will not score above 4/6 on Criterion E.
Don't save all your reflection for the conclusion. After each result, ask: What does this mean? Does it make sense? What assumptions did I make? What would change if I modified the model? Examiners reward reflection that is embedded in the work.
Every variable defined. Every graph labelled. Notation consistent from page 1 to page 18. Examiners should never have to guess what a symbol means. Use an equation editor for all mathematical expressions — never write equations by hand or paste them as images.
If your exploration uses data, collecting it yourself — measuring, surveying, recording — scores far better on personal engagement than downloading a generic dataset. If you must use secondary data, justify your specific choice and show what you added to the analysis.
No model is perfect, and examiners know that. Identifying your own limitations shows mathematical maturity. "My model assumes a constant carrying capacity, which may not hold seasonally — a worthwhile extension would be to model K as a periodic function" is exactly the kind of reflection that earns full marks on Criterion D.
The IA counts for 20% of your final grade regardless of whether you are SL or HL, and the same five criteria apply. The key difference is the expected level of mathematical sophistication in Criterion E.
SL students are expected to go beyond routine procedures — simple statistics or basic algebra will not score highly. Regression analysis, modelling with functions, or inferential statistics (hypothesis testing, chi-squared) are appropriate SL-level mathematics.
HL students must demonstrate genuine HL content. Differential equations, complex numbers, advanced calculus, eigenvalues, or the HL statistics topics are expected. An HL student who writes an SL-level exploration will be marked down on Criterion E regardless of how well they perform on the other four criteria.
Use the free tools in this guide to plan, write, and polish your IA — step by step.
💡 Topic Idea Bank
Browse 35+ topic ideas. Filter by theme, level, or course — or search by keyword.
📅 IA Planner & Timeline
Follow the 7 stages of a strong IA, and generate a personal timeline from your submission date.
📊 Criteria Guide
The IB Maths IA is marked out of 20 across 5 criteria. Learn exactly what examiners look for.
✅ Pre-Submission Checklist
Work through every item before handing in your IA. Aim for 100%.
⚠️ Common Pitfalls
The most frequent mistakes IB examiners see — and exactly how to avoid them.
