IB Extended Essay in Mathematics — Complete Guide

1. What is the Mathematics Extended Essay?

The Extended Essay (EE) is a 4000-word independent research essay submitted as part of the IB Diploma. It is an opportunity to investigate a topic of personal interest in depth, demonstrating academic writing, research, and analytical skills.

A Mathematics EE is an independent mathematical investigation. Unlike a science EE (which requires experiments), a Maths EE is built on mathematical reasoning, proof, or modelling. The investigation should go beyond IB syllabus content while remaining accessible.

Three Types of Mathematics EE

Pure Mathematics Investigation

Explore a theorem, proof technique, number-theoretic property, or algebraic structure in depth. Example: properties of Fibonacci numbers, number bases, proof methods in combinatorics.

Applied Mathematics / Mathematical Modelling

Apply mathematics (calculus, statistics, linear algebra) to a real-world context. Fit models to data, evaluate model accuracy, discuss limitations. Example: modelling epidemic spread using the SIR model.

Historical Mathematical Analysis

Investigate the development of a mathematical idea, the historical significance of a result, or contributions of a mathematician. Must include substantial mathematical content — not just biography.

Key point: The EE is assessed on the quality of your mathematical investigation and argumentation, not the difficulty of the mathematics alone. A well-argued investigation on an accessible topic can score higher than a confused treatment of advanced material.

2. Choosing a Research Question

The research question (RQ) is the most important decision in your EE. A good RQ is specific, answerable, and genuinely investigable within 4000 words.

Characteristics of a Strong Research Question

Narrow scope — not "prove Fermat's Last Theorem" (impossible at this level)
Original angle — an interesting twist on known mathematics, not a textbook proof
Accessible to IB level but clearly going beyond the syllabus
Clear methodology — you can describe how you will investigate it
Answerable in words — "To what extent..." or "How does...?" frames allow nuanced conclusions

Example Research Questions by Category

Category	Example Research Question
Pure / Number Theory	"How does the Euclidean algorithm's efficiency compare to Bézout's identity method for finding GCD of large integers?"
Pure / Number Theory	"What are the properties of Mersenne primes, and what patterns emerge from their distribution?"
Pure / Geometry	"Which geometric constructions are possible using origami, compared to compass-and-straightedge methods?"
Applied / Statistics	"To what extent does the birthday paradox hold in real populations of IB school cohorts?"
Applied / Modelling	"How accurately can a logistic regression model predict outcomes in a binary classification dataset?"
Applied / Physics	"How effectively does the SIR model describe the spread of a contagious disease in a confined school population?"
Game Theory	"To what extent does Nash equilibrium predict observed behaviour in repeated Prisoner's Dilemma games?"
Historical	"How did Euler's approach to the Königsberg bridge problem establish the foundational concepts of graph theory?"
Historical	"How did the development of non-Euclidean geometry challenge the philosophical assumptions of 18th-century mathematics?"
Combinatorics	"How many distinct colouring patterns exist for a $3 \times 3$ grid using $k$ colours, accounting for rotational symmetry?"
Probability	"How does the Gambler's Ruin problem illuminate the limitations of the martingale betting strategy?"
Calculus	"How accurately can numerical integration methods (Trapezoid Rule, Simpson's Rule) approximate integrals with known exact values?"

Topics to avoid: "The history of calculus" (too broad, no clear investigation), "Why is $e$ irrational?" (single result, no room for development), "Applications of mathematics in everyday life" (not a mathematical investigation). Always ask: what mathematical argument am I making?

3. Mathematical Investigation Methodology

Step-by-Step Investigation Framework

The Five-Stage Investigation Process

Define your research question precisely

Frame a specific, answerable question. State clearly what you will investigate and what you expect to find (your conjecture or hypothesis). Define all terms in the RQ mathematically.

Example: "To what extent does the logistic growth model accurately describe population growth in isolated environments? I will test this by comparing logistic curve fits to historical population data from island populations, evaluating fit using $R^2$ and residual analysis."

Background mathematics

Present the necessary theory, definitions, and theorems needed to understand your investigation. Show understanding beyond the IB syllabus. Cite sources for theorems you use but do not prove from scratch.

Be selective — include only what is relevant. Do not copy large sections of textbooks. Demonstrate your own understanding by explaining concepts in your own words.

Investigation and exploration

This is the core of your essay (approximately 60% of the total length). Present examples, data, computations, or proofs systematically. Look for patterns, make conjectures, test them with further examples.

For pure math: develop your argument step by step, prove sub-results before using them
For applied math: present data clearly, show calculations, compare model to reality
For historical: analyse primary/secondary sources critically; do not just narrate

Proof or verification

Mathematical claims must be justified. For pure math: attempt rigorous proof (by induction, contradiction, or direct argument). For applied math: test your model against data, compute error measures, discuss when the model breaks down. For historical analysis: evaluate the significance and correctness of historical arguments.

Reflection and limitations

Critically discuss: What are the limitations of your approach? What assumptions did you make? What could be improved? What further investigations are possible? What surprised you? Personal reflection on the mathematical journey is valued in Criterion E (Engagement).

4. Structure of the Essay

A well-structured EE is easy for the examiner to follow. The standard structure is:

Recommended Structure

Title page — title, research question, subject, word count, candidate name/number
Table of contents — with page numbers
Introduction (300–500 words) — context, personal motivation, the research question, overview of methodology
Background mathematics (400–600 words) — definitions, key theorems, relevant theory
Investigation / Main body (2000–2500 words) — the core mathematical argument, examples, data, proofs, analysis
Conclusion (400–600 words) — direct answer to RQ, evaluation of findings, limitations, extensions
Bibliography — all sources cited, in consistent format (MLA or APA)
Appendix — raw data, lengthy calculations, code (not counted in word count)

Word count rules: The IB counts words in the main body (introduction through conclusion). Equations, footnotes, bibliography, and appendices are excluded. Tables and figure captions may or may not be counted depending on how they are formatted — check the current IB guidelines.

5. Mathematical Writing Standards

A key assessment criterion is the quality of mathematical communication. The examiner assesses whether your mathematical writing is clear, precise, and follows accepted conventions.

Core Writing Principles

Define all variables and notation before first use — never leave a symbol undefined
Justify each step in proofs and derivations — do not skip steps and say "it follows that..."
Separate claim from proof — state the result first, then prove it
Use proper mathematical notation, not calculator notation (write $x^2$, not x^2)
Number all equations that you reference later in the text
Include sufficient detail for a reader of similar mathematical background to follow your reasoning
Integrate mathematics with text — don't just present calculations without explanation

Example of Good vs. Poor Mathematical Writing

Presenting a Proof

Poor: "The sum of the first $n$ odd numbers equals $n^2$. Proof: You can check this for small cases."
(No proof, no notation, no rigour.)

Better:

Theorem: For all positive integers $n$, $\displaystyle\sum_{k=1}^n (2k-1) = n^2$.

Proof by induction: Base case: $n=1$: $\sum_{k=1}^1 (2k-1) = 1 = 1^2$. ✓

Inductive step: Assume true for $n=m$, i.e., $\sum_{k=1}^m (2k-1) = m^2$. Then for $n=m+1$: $$\sum_{k=1}^{m+1}(2k-1) = m^2 + (2(m+1)-1) = m^2 + 2m + 1 = (m+1)^2. \quad \square$$

6. Assessment Criteria Breakdown

The EE is marked out of 34 points across five criteria. Understanding these criteria helps you target your effort effectively.

Focus and Method 6 marks

Is the research question clearly stated and appropriately scoped?
Is the methodology appropriate and clearly explained?
Is the investigation carried out in a thorough and systematic manner?

Top marks require: precise RQ, well-defined method that you actually follow, comprehensive investigation that addresses the RQ directly.

Knowledge and Understanding 12 marks

Is the mathematical knowledge used correct and appropriate?
Are mathematical language and notation used correctly and consistently?
Is technology (GDC, software) used appropriately where relevant?
Does the investigation demonstrate understanding beyond the IB syllabus?

This is the highest-weighted criterion. Errors in mathematics, undefined notation, or staying entirely within the syllabus all cost marks here.

Critical Thinking 12 marks

Are results correctly analysed and interpreted?
Are patterns identified and used to form conjectures?
Are proofs or arguments logically sound?
Is the significance of results discussed? Are limitations acknowledged?

Also 12 marks. Don't just compute — interpret, connect, prove. Ask "so what?" after every result.

Presentation 4 marks

Is the essay well-structured with clear introduction, body, and conclusion?
Are layout, notation, and presentation clear and consistent?
Is the bibliography complete and correctly formatted?

Engagement 6 marks — via RPPF

Assessed through the Reflections on Planning and Progress Form (RPPF)
Demonstrates intellectual curiosity, ownership of the investigation, and personal learning
Discuss what motivated you, what obstacles you encountered, and how your understanding evolved

The RPPF has three entries (submitted at the start, midpoint, and end of the process). Be honest and specific about your intellectual journey.

7. Common Mistakes to Avoid

Scope too broad: "A history of calculus" or "applications of mathematics" is not a specific mathematical investigation. You need a precise, answerable question.

Descriptive rather than analytical: Simply summarising what Wikipedia says about a topic earns zero marks for critical thinking. You must analyse, not just describe.

No clear mathematical progression: The investigation must build — each section should follow from and extend what came before. An essay that just presents disconnected examples is not an investigation.

Plagiarism: Copying proofs from textbooks without citation is academic misconduct. Always cite your sources, even for well-known results. You can use standard results, but you must acknowledge them.

Poor notation: Mixing LaTeX notation, calculator syntax, and informal language in the same essay creates confusion. Choose a notation system and use it consistently.

Ignoring limitations: All models and methods have limitations. If you don't discuss them, you lose critical thinking marks. "The model assumes..." is good; "The model is perfect" is not.

Conclusion doesn't answer the RQ: Your conclusion must directly address the research question you posed in the introduction. This is assessed under Criterion A.

8. Example Investigation Outline

Sample EE Outline

RQ: "To what extent does the logistic growth model accurately predict the spread of a contagious disease in a confined population?"

Introduction: Motivation (COVID-19, historical epidemics), context for mathematical modelling in epidemiology, statement of RQ, overview of method.

Background Mathematics:

The SIR model: $\dfrac{dS}{dt} = -\beta SI$, $\dfrac{dI}{dt} = \beta SI - \gamma I$, $\dfrac{dR}{dt} = \gamma I$
Logistic differential equation: $\dfrac{dP}{dt} = kP\left(1 - \dfrac{P}{L}\right)$
Solution: $P(t) = \dfrac{L}{1+Ae^{-kt}}$, derivation by separation of variables
Least-squares regression: definition, $R^2$ coefficient, residual analysis

Investigation:

Obtain anonymised infection count data from a historical outbreak (or use published data)
Use GeoGebra/Python to fit the logistic curve: estimate $k$, $L$, $A$ via least-squares
Compute $R^2$ and plot residuals — evaluate goodness of fit
Compare to linear and exponential models (show logistic is superior in S-curve phase)
Discuss when the logistic model breaks down (multiple waves, behaviour changes, data sparsity)

Conclusion: The logistic model provides a reasonable description of single-wave outbreak dynamics (e.g., $R^2 = 0.96$ in the example dataset), but systematically underestimates the timing of the inflection point. Limitations: the model assumes homogeneous mixing and constant transmission rate — real populations are heterogeneous. Extensions: SIR model with vaccination compartment.

9. Topics by Difficulty

The following table classifies EE topic areas by typical difficulty and suitability for different students.

Difficulty	Topic Area	Example Research Question
Accessible	Statistics / Probability	Birthday paradox verification in real school populations
Accessible	Number Theory	Properties of palindromic numbers in different bases
Accessible	Game Theory (intro)	Nash equilibrium analysis in simple 2×2 games
Accessible	Sequences & Series	Patterns in generalised Fibonacci sequences
Moderate	Calculus / Optimisation	Optimisation in economic models (Cobb–Douglas)
Moderate	Probability	Gambler's Ruin: analysis and simulation comparison
Moderate	Applied Modelling	Logistic model for population growth in island environments
Moderate	Combinatorics	Burnside's lemma applied to colouring problems
Challenging	Analysis	Convergence analysis of iterative root-finding methods
Challenging	Graph Theory	Chromatic polynomial and its properties for planar graphs
Challenging	Topology	Introduction to Euler characteristic and its invariance
Challenging	Differential Equations	Behaviour of solutions to the predator–prey (Lotka–Volterra) system

10. Recommended Resources

Essential References

IBO Mathematics EE Guide — read the criteria descriptions carefully; the official document defines exactly what is assessed
IB EE exemplars — the IBO releases scored exemplars with examiner commentary. Study these closely
Mathematical Gazette — accessible journal articles on recreational and applied mathematics
Plus Magazine (plus.maths.org) — excellent source of ideas for accessible mathematical investigations
NRICH (nrich.maths.org) — problem-solving resources that can inspire EE topics
Wolfram MathWorld — reference for mathematical definitions and theorems

Software Tools

GeoGebra — free, excellent for geometry, graphing, and interactive visualisations
Desmos — quick function plotting and regression fitting
Python with scipy, matplotlib, numpy — curve fitting, numerical methods, data visualisation
LaTeX / Overleaf — professional mathematical typesetting for the final document
GDC (TI-84, Casio) — regression functions, matrix computations

Citation Format

The IB recommends MLA or APA format, used consistently throughout. Use a citation manager (Zotero, Mendeley) or the IBO's online bibliography generator. Every source consulted — books, websites, journal articles — must be cited.

11. Timeline and Planning

Milestone	Approximate Timeline	Key Deliverable
Topic brainstorm	DP Year 1, Term 1	3–5 candidate research questions
Supervisor meeting 1	DP Year 1, Term 2	Confirmed RQ, initial reading list
RPPF Entry 1	After first supervisor meeting	Submitted to IBO coordinator
First draft	DP Year 1, Term 3	Complete draft (all sections present)
Supervisor meeting 2	DP Year 2, Term 1	Feedback on draft, RPPF Entry 2
Final submission	DP Year 2, early Term 2	Final essay + RPPF Entry 3

Start early: Students who start the EE in earnest in the first term of DP Year 1 consistently produce stronger work than those who leave it until Year 2. Mathematical investigations take time to develop — new ideas emerge slowly.

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