IB Internal Assessment Guide

The IB Mathematics Internal Assessment (IA), also called the Mathematical Exploration, is a written investigation where you use mathematics to answer a focused question. It typically counts for 20% of your final IB Math grade, so it is too important to treat as an afterthought. A strong IA does not try to impress with difficulty alone. It succeeds by being clear, mathematically sound, well-structured, and personal in its direction.

This guide explains what the IA is, how to choose a workable topic, how to structure the exploration, and how to align your writing with the assessment criteria.

What the IB Math IA is (and what it is not)

The IA is an individual investigation, usually 12 to 20 pages in length, where you:

• pose a mathematical question (or refine one into something precise),
• explore it using appropriate mathematics,
• interpret what your results mean in context,
• communicate the process clearly.

It is not a standard research essay, and it is not a simple collection of textbook exercises. The goal is to show mathematical thinking: making decisions, justifying steps, testing assumptions, and reflecting on limitations.

A useful way to think about it is: you are writing up the work of a small, careful mathematical project. The mathematics should be at an appropriate level for your course, but it should also be used purposefully rather than displayed for its own sake.

Understanding the assessment criteria

While the official wording can vary slightly between course levels and examination sessions, the IA is generally assessed through criteria that reward:

Mathematical communication

Your writing must be readable and mathematically correct. That includes:

• using correct notation consistently,
• defining variables and parameters,
• labeling graphs and tables,
• explaining steps in words, not just showing calculations,
• keeping the structure easy to follow.

A marker should not have to guess what a symbol represents or why a method was chosen.

Personal engagement

This does not mean adding opinion. It means your exploration shows ownership through choices you made, such as:

• selecting a question that connects to your interests,
• adapting a method to fit your data or constraints,
• comparing approaches and explaining why you kept one,
• reflecting on what surprised you or what you would improve.

Personal engagement often shows up in small, authentic decisions rather than dramatic storytelling.

Reflection

Reflection is more than a concluding paragraph. Throughout the IA, you should interpret results and evaluate your work. Examples include:

• commenting on whether results make sense in the real-world context,
• identifying limitations (measurement error, model assumptions, small sample size),
• considering how sensitive conclusions are to changes in parameters,
• explaining what the mathematics reveals and what it cannot reveal.

If you build a model, reflect on its fit and its purpose. If you prove a relationship, reflect on the meaning and potential generalizations.

Use of mathematics

This is the mathematical core. Examiners look for:

• appropriate level of mathematics for your course,
• correct application and reasoning,
• depth rather than breadth,
• connections between steps (not isolated techniques).

Using advanced methods poorly usually scores worse than using standard methods well and thoughtfully. The best IAs match the mathematics to the question and maintain accuracy from start to finish.

Choosing a strong topic: focused, feasible, and mathematical

Topic selection is where many IAs succeed or fail. A good topic is narrow enough to explore deeply, broad enough to sustain investigation, and clearly mathematical.

What works well

Strong topics typically fall into one of these patterns:

• Modeling a real situation: building a function or system that approximates behavior, then testing and refining it.
• Optimisation: minimizing cost, maximizing area, improving efficiency under constraints.
• Probability and statistics with purpose: analyzing data to answer a question, including justification of methods and interpretation.
• Pure mathematics exploration: investigating a conjecture, pattern, or property (sequences, graph theory, geometry, number theory) with proofs and examples.

What to avoid

Some topics tend to underperform:

• overly broad questions like “How is calculus used in physics?”
• topics that become descriptive rather than mathematical (history of a concept),
• investigations that rely on copying known results without your own reasoning,
• projects that are mostly technology output with minimal explanation.

A reliable check is to write your research question in one sentence. If it cannot be made precise, it is not ready.

Turning an interest into a research question

Start with an area you care about (sports, music, economics, architecture), then turn it into something measurable or provable. For example:

• Interest: basketball

Possible direction: “How does shot distance affect scoring probability?” Better research question: “Can a logistic model fit shot success probability as a function of distance, and how does model accuracy compare to a simple linear probability model within a chosen range?”

The improved version signals mathematics (model choice, comparison, accuracy criteria) and a manageable scope.

Planning your exploration before you write

A strong IA usually has a clear workflow:

1. Define the question and variables

Be explicit. If you use data, state source and units.

1. Choose methods that match the question

Decide early what mathematics you expect to use (regression, derivatives, binomial models, transformation geometry, proof by induction).

1. Collect or generate what you need

If you use data, ensure it is sufficient and relevant. If you simulate, explain assumptions.

1. Test an initial approach

Do a small trial. This often reveals whether the scope is realistic.

1. Refine

Adjust the question, improve the model, or narrow the domain.

This planning reduces the most common IA problem: a long write-up with no mathematical direction.

Recommended structure for a 12 to 20 page IA

A clear structure helps the reader see your reasoning and helps you meet the criteria.

Title page (optional but useful)

Include the title and research question. Avoid decorative formatting.

Introduction

• State the research question.
• Explain why it is interesting and what you aim to do.
• Define key terms and variables.
• Briefly outline your method.

Keep the introduction focused. It should set up the mathematics, not narrate your entire process.

Rationale and background (short)

Provide only the background needed to understand the exploration. If you reference a known theorem or model, summarize it in your own words and cite where appropriate.

Mathematics and investigation (main section)

This is where most marks are earned. Common best practices:

• Break work into labeled subsections with a logical sequence.
• Show derivations and reasoning, not just final formulas.
• Use graphs and tables to support arguments, and interpret them.
• When using technology (graphing calculators, spreadsheets, CAS), explain what you did and why. Do not outsource thinking to screenshots.

If you model something, make assumptions explicit. For example, if you assume a linear relationship, justify why it might be reasonable over your chosen interval.

Results and interpretation

Present results clearly and connect them back to the research question. If you computed an optimal value, explain what it means. If you fitted a curve, discuss fit quality and where the model fails.

Reflection and evaluation

Address:

• limitations of data or assumptions,
• sensitivity of results (what changes if inputs change?),
• alternative methods you considered,
• what you would do next with more time or better data.

Strong reflection is specific. “My model is not perfect” is weak. “Residuals increase for large values of $x$, suggesting the linear assumption breaks down beyond $x=12$” is meaningful.

Conclusion

Answer the research question directly. Summarize the main mathematical insight in a few sentences, without introducing new work.

References and appendices

Cite data sources, textbooks, articles, and tools. Put raw data or long computations in an appendix only if they are necessary for verification. The main narrative should remain readable.

Common pitfalls that lower IA scores

• Writing too much and saying too little: Pages of explanation without mathematics, or pages of calculations without interpretation.
• Unclear notation: Switching variables or using symbols without definition.
• Overreliance on technology: Graphs and regressions pasted in without discussion.
• Scope creep: Starting with one question, then drifting into several disconnected investigations.
• Minimal reflection: No critique of assumptions, no discussion of limitations.

Final checklist before submission

• Is the research question precise and answered by the end?
• Does every section contribute to the investigation?
• Is the mathematics correct, appropriate, and clearly explained?
• Are graphs labeled with units and interpreted in words?
• Have you shown personal decision-making and thoughtful reflection?
• Is the length within roughly 12 to 20 pages, with a clean layout and consistent formatting?

A high-scoring IB Math IA is not about finding the most complex topic available. It is about building a coherent mathematical argument that a reader can follow, verify, and learn from. Focus on clarity, sound mathematics, and genuine engagement with your question, and the marks tend to follow.