What is the IB Mathematical Exploration?
The Mathematical Exploration is a 12–20 page investigation required for AA SL/HL and AI SL/HL. Students choose a question and use mathematics to investigate it.
Assessment note: University-level mathematics is not required. The IB rewards mathematics appropriate to the course, explained clearly, with personal engagement and critical reflection.
Assessment Criteria (20 marks)
Every IB Math exploration is assessed against five criteria. Use the rubric to see what each criterion requires.
Evaluates structure, coherence, organization, and conciseness.
- Well-Structured: Clear Introduction, Rationale, Aim, Analysis, and Conclusion.
- Coherent & Focused: Reader can follow the flow without confusion or unnecessary tangents.
- Concise: Aim for 12–20 pages. Avoid repetitive repetitive calculations; summarize large tables in appendices.
Evaluates mathematical language, notation, symbols, graphs, and terminology.
- Standard Notation: Use proper math notation (e.g. \( \int_{a}^{b} f(x)dx \), \( \chi^2 \)) instead of calculator syntax like
y=2x^2. - Graph Quality: Fully labeled axes, clear scales, titles, and units on every diagram.
- Defined Variables: Define every symbol upon first introduction (e.g. "Let \( r \) represent the rate in \(\text{m/s}\)").
Evaluates authentic interest, curiosity, independent thought, and creative thinking.
- Genuine Rationale: Avoid fake stories ("I love coffee so I modeled cooling"). Show real curiosity or collected primary data.
- Original Direction: Take a textbook model and adapt or customize it to your unique context.
- Independent Thought: Evidence of problem-solving when obstacles arise during modeling.
Evaluates critical evaluation, interpretation, and consideration of limitations.
- Ongoing Reflection: Do not save reflection only for the conclusion; reflect at each key step.
- Meaning in Context: What does an \( R^2 = 0.94 \) actually signify for your real-world system?
- Model Limitations: Analyze assumptions, percentage errors, boundary conditions, and valid domains.
Evaluates knowledge, understanding, sophistication (HL), and mathematical rigor.
Standard Level (SL)
- Demonstrates full, thorough understanding of SL syllabus topics.
- Completely correct mathematical execution without routine calculation errors.
- Rigorous explanations of why formulas work, not just plug-and-chug.
Higher Level (HL)
- Requires sophistication or mathematical depth commensurate with HL.
- Uses advanced concepts: differential equations, complex numbers, matrices, multivariable extensions.
- Thorough, complete mathematical justification with rigorous proof or error quantification.
Interactive IA Score Estimator
Select your estimated level for each criterion to calculate your total mark out of 20 and forecast your IB Grade.
50+ IA Topic Ideas
Filter topic ideas by course (AA or AI) and interest. Each topic suggests a mathematical question to investigate.
Modelling Damped Harmonic Motion in a Real Pendulum
Collect high-speed video data of a physical pendulum and formulate 2nd order differential equations to compare underdamped models with experimental decay.
Stochastic Markov Chains in Portfolio Credit Migration
Analyze transition matrices of corporate bond ratings over 10 years to determine stationary steady-state distributions and risk default probabilities.
Optimizing the Release Angle in Basketball Free Throws
Use projectile mechanics to derive the entry angle into the hoop, calculating clearance margins through trigonometric functions and quadratic envelopes.
Voronoi Diagrams in Football Territorial Dominance
Construct Voronoi cells from player coordinates in a match freeze-frame to measure pitch control and compare against successful pass completion rates.
Visualizing Complex Roots of Unity & Fractal Basins
Explore Newton-Raphson iteration in the complex plane \( z^3 - 1 = 0 \) to investigate fractal boundary behaviors and speed of convergence.
Route Optimization Using Travelling Salesperson Algorithms
Model school bus or delivery routes across your local area using graph theory, calculating bounds with nearest-neighbor and minimum spanning trees.
Surface Area & Volume Optimization of Coffee Cups
Model real coffee containers using solids of revolution. Determine if commercial cup dimensions minimize surface area (material cost) for a fixed volume.
Mortgage Amortization & Early Repayment Compounding
Derive the annuity formula using geometric series and simulate total interest savings under accelerated bi-weekly payment schedules.
Brachistochrone Curve: Deriving the Path of Quickest Descent
Formulate the calculus of variations equation for a sliding particle under gravity, solving the parametric cycloid equations and testing with 3D-printed tracks.
Coupled Lotka-Volterra Predator-Prey Simulations
Simulate deer and wolf population cycles using systems of first-order differential equations solved numerically via Euler's method.
The Mathematics of Musical Chords & Equal Temperament
Model pure harmonic ratios versus 12-tone equal temperament frequencies using geometric progressions and logarithmic cent scales.
Chi-Square Independence Test in Student Screen Time
Design a survey on student sleep quality versus evening screen time, testing hypothesis validity using contingency tables and \(\chi^2\) distributions.
Five-Stage Writing Roadmap
1. Introduction & Rationale
State your explicit mathematical aim. Explain why this question matters to you personally. Establish your scope and assumptions right away.
2. Methodology & Data
Explain how you collected data or set up your equations. Justify why chosen models fit the phenomena. Define all variables with correct units.
3. Mathematical Analysis
Show complete, rigorous working. Include diagrams, function graphs, and clear step-by-step logic. Don't hide steps; explain the 'why' behind each equation.
4. Ongoing Critical Reflection
Evaluate errors as they occur. Discuss goodness-of-fit, limitations of assumptions, and unexpected anomalies in your models.
5. Conclusion & Future Scope
Summarize answers to your original aim. Propose concrete ways to extend the mathematics. Include a full bibliography with formatted citations.
Pre-Submission Checklist
Check off each item before handing in your draft to your teacher or supervisor:
Frequently Asked Questions
Can I use the same topic as a classmate?
You can investigate a similar broad theme (e.g. projectile motion or epidemic models), but your specific research question, collected data, and mathematical angles must be completely independent. Collaboration is strictly prohibited under IB academic integrity rules.
Is an IA in Analysis & Approaches harder than Applications & Interpretation?
No, both are marked on the same 20-point rubric. AA IAs generally feature algebraic proof, calculus, or function analysis, while AI IAs feature data collection, mathematical modeling, technology, and statistical hypothesis testing.
What font, spacing, and formatting should I use?
Use a clean professional font (Arial, Calibri, Times New Roman, or Computer Modern if using LaTeX) at 11 or 12pt with 1.5 line spacing. Always write equations using Word Equation Editor, MathType, or LaTeX.