IB Maths IA Ideas Guide

How to Use This List

This list is a starting point, not a blueprint. The best IAs come from genuine curiosity — use these ideas to spark your own angle, not to copy an existing approach. An idea that's been investigated a hundred times is not a problem if your specific question, data, or analysis is original.

For each idea, ask: "Can I find data or construct a mathematical model that goes beyond what we've covered in class?" If yes, it has potential. If the exploration leads straight to a textbook result with no extension, consider a different direction.

AA Ideas — Calculus and Analysis

• SIR epidemic modelling: Use differential equations to model disease spread. Explore how varying infection rate and recovery rate affects the epidemic peak. (Difficulty: HL)
• Taylor series convergence: Investigate how many terms of the Maclaurin series for sin x or ln(1+x) are needed for a given precision. Visualise convergence vs divergence for different x values. (Difficulty: HL)
• Packaging optimisation: Use calculus to minimise surface area for a fixed volume — then extend to non-standard shapes like cylinders with hemispherical ends. (Difficulty: SL/HL)
• Euler's method accuracy: Compare numerical solutions of ODEs via Euler's method to exact solutions. Investigate how step size affects error. (Difficulty: HL)
• Newton's law of cooling: Collect temperature-time data and fit an exponential decay model. Explore how starting temperature and ambient conditions affect the model. (Difficulty: SL/HL)
• Brachistochrone problem: Investigate which curve minimises travel time under gravity between two points. Introduction to calculus of variations. (Difficulty: HL+)

AA Ideas — Statistics and Probability

• Central Limit Theorem demonstration: Collect samples of varying size from a skewed real-world distribution. Show empirically how sample means approach normality. (Difficulty: SL/HL)
• Poisson modelling: Model a real-world rare-event process (goals per football match, calls per hour) and test the fit to a Poisson distribution. (Difficulty: SL)
• Bayes' theorem and medical testing: Investigate false positive rates for a real diagnostic test. How does prevalence affect predictive value? (Difficulty: SL/HL)
• Normal distribution verification: Collect real measurement data and test normality using a chi-squared goodness of fit test. (Difficulty: HL)

AA Ideas — Geometry, Number Theory, and Algebra

• Modular arithmetic and RSA cryptography: Explore the mathematics of public-key encryption at a conceptual level using modular arithmetic and Euler's theorem. (Difficulty: HL)
• Continued fractions: Investigate the convergents of continued fraction representations of irrational numbers like √2 and π. What patterns emerge? (Difficulty: HL)
• Inscribed polygon areas approaching π: Use limits of n-gon area formulae as n→∞ to derive π geometrically. (Difficulty: SL/HL)
• Conic sections and their parametric forms: Investigate how varying parameters in parametric equations traces different conic sections. Explore reflective properties. (Difficulty: HL)

AI Ideas — Statistics and Modelling

• Multiple regression analysis: Collect real data on a measurable outcome (e.g., house prices, exam results) and model it using multiple regression. Assess the significance of each predictor. (Difficulty: HL)
• Logistic population growth: Fit a logistic model to real population data (country, species, or microbial colony). Compare to exponential fit. (Difficulty: SL/HL)
• Chi-squared independence testing: Investigate whether two categorical variables are independent using real survey or census data. (Difficulty: SL)
• Time series analysis: Investigate seasonal trends in a real dataset using moving averages and trend lines. (Difficulty: SL)
• Spotify or Netflix data modelling: Use streaming data to model listening/viewing patterns. Apply appropriate statistical tests. (Difficulty: SL/HL)

AI Ideas — Graph Theory and Networks

• Dijkstra's algorithm on a real network: Apply shortest path algorithms to a real road or transit network. Compare theoretical efficiency to actual route choices. (Difficulty: HL)
• Minimum spanning tree: Model a real logistics or infrastructure problem (cable laying, road network) as a minimum spanning tree problem. (Difficulty: HL)
• Chinese Postman Problem: Apply route inspection algorithms to find the most efficient route for a real-world service (mail delivery, street cleaning). (Difficulty: HL)

What Makes a Strong Research Question

From the ideas above, a strong research question:

• Is specific: "How does infection rate β affect the peak of an SIR epidemic?" not "How does maths apply to disease?"
• Requires mathematics at or beyond your course level to answer
• Allows genuine exploration — you should be able to change a variable or approach and see what happens
• Leads naturally to reflection on assumptions, limitations, and what would improve the model