IB Maths IA Topic Idea Bank
Every idea below is tagged with the exact IB courses and levels it suits, plus the specific maths it draws on, so you can filter down to something that's genuinely a good fit for you rather than scrolling a generic list. Frequently submitted topics are flagged with a note on how to make them stand out instead of reading as a rehash. Filter by theme, level, or course, or search by keyword, then jump straight into practice questions on the matching technique for any idea that catches your eye.
Optimising a football free kick trajectory
Model the optimal angle and speed for a free kick to clear the defensive wall and score, using projectile motion equations and calculus to find maximum ranges.
Modelling basketball shooting percentage vs. distance
Collect or use real data on field-goal percentage by distance, fit a regression model, and use it to determine the optimal shot location on the court.
Is there home advantage in football? A statistical analysis
Use real match data from a league to test whether home advantage is statistically significant using chi-squared or t-tests.
Predicting marathon finishing times
Use data on age, training mileage and half-marathon times to build a regression model predicting marathon finish times.
Modelling the swing of a golf ball
Use aerodynamic equations to model how spin affects the curved path of a golf ball, and optimise spin rate for maximum distance.
The mathematics of Fibonacci in music
Explore how Fibonacci numbers and the golden ratio appear in musical composition, instrument design, and tuning systems.
Fourier analysis of musical tones
Use Fourier series to decompose a musical note into its harmonic components and explore how timbre is created mathematically.
Does tempo affect emotional response? A statistical study
Survey participants about emotional responses to music at different tempos and test whether the correlation is statistically significant.
Modelling population growth using logistic equations
Compare exponential and logistic growth models for a real animal population, identifying the carrying capacity and inflection point.
The fractal dimension of coastlines
Use the box-counting method to estimate the fractal dimension of different coastlines and compare your results.
Modelling the spread of a disease (SIR model)
Build and analyse an SIR model for disease spread using a real epidemic dataset.
The golden ratio in plant spirals (phyllotaxis)
Count and analyse the spiral patterns in sunflower seeds or pine cones, exploring their connection to Fibonacci numbers.
Optimising a loan repayment schedule
Model different mortgage repayment strategies, calculating total interest paid and optimal strategies.
Modelling cryptocurrency price volatility
Analyse the statistical distribution of daily returns for Bitcoin and compare its volatility to traditional assets.
Supply, demand, and equilibrium price modelling
Use mathematical functions to model supply and demand curves, find equilibrium, and investigate the effect of taxes or subsidies.
The mathematics of compound interest and wealth inequality
Model how compound interest accelerates wealth accumulation over time and explore its link to wealth inequality statistics.
Optimising the shape of an arch bridge
Compare parabolic and catenary arch shapes, deriving the optimal geometry to maximise load-bearing efficiency for a given span.
Minimising material in packaging design
Use calculus to find the dimensions of a cylindrical can or rectangular box that minimise surface area for a given volume.
The mathematics of domes: Brunelleschi vs. geodesic
Compare the structural mathematics of traditional masonry domes with geodesic structures, analysing surface area and load distribution.
Drug concentration models in the bloodstream
Model how a drug's concentration changes over time after a dose, exploring dosing intervals and steady-state levels.
BMI and health outcomes: a statistical investigation
Analyse a real health dataset to test the correlation between BMI and a health outcome, assessing statistical significance.
Modelling the growth of a tumour
Use and compare exponential and Gompertz growth models for tumour size data, discussing clinical implications.
The golden ratio in art and aesthetics
Investigate whether the golden ratio genuinely appears in famous artworks and whether audiences perceive golden-ratio rectangles as more aesthetic.
Perspective and vanishing points in Renaissance painting
Use projective geometry to analyse how Renaissance painters achieved realistic perspective, and verify using real paintings.
Fractals in Islamic geometric art
Explore the self-similar patterns in Islamic tiling and calculate their approximate fractal dimensions.
Encryption and the mathematics of RSA
Explain and demonstrate the mathematical basis of RSA encryption - prime factorisation, Euler's theorem - and implement a simple version.
PageRank: the maths behind Google's algorithm
Use Markov chains and matrices to model how PageRank assigns scores to webpages and simulate it on a small graph.
Modelling Wi-Fi signal strength with distance
Measure real Wi-Fi RSSI data at varying distances and fit a logarithmic model, comparing it to the theoretical path loss formula.
Carbon emissions and global temperature: a regression analysis
Use historical CO₂ and global temperature data to fit a regression model and discuss the strength and limitations of the correlation.
Modelling plastic waste accumulation in the ocean
Build a mathematical model for ocean plastic accumulation rates using real data, and project future levels under different scenarios.
Optimal placement of wind turbines
Use power output equations and geometric optimisation to determine the spacing and layout of a wind farm that maximises energy output.
Sugar content in food and health correlations
Analyse nutritional data from a sample of supermarket products to test hypotheses about sugar content across food categories.
The mathematics of bread fermentation (yeast growth)
Model the growth of yeast in bread dough using exponential and logistic models, measuring real CO₂ production over time.
Optimising coffee cooling: Newton's Law
Collect temperature data for cooling coffee and fit Newton's Law of Cooling, exploring optimal timing for adding milk.
Statistical analysis of menu pricing strategies
Compare menu pricing across restaurant categories using statistical tests to assess whether pricing follows particular distributions.
Exploring the Limiting Ratio of the Fibonacci Sequence
Prove how the ratio of consecutive Fibonacci numbers converges to the Golden Ratio and explore its geometric appearance in nature.
Modeling Complex Impedance in AC Circuits
Analyze how resistors, capacitors, and inductors affect an AC circuit by representing voltage and current as complex numbers.
Fractals and Mandelbrot Set Boundary Dimensions
Explore how iterative functions on complex numbers form the Mandelbrot set and investigate the mathematics of fractal dimensions.
Using Matrices to Solve Leontief Input-Output Economic Models
Model an economy of interdependent industries to find the production vectors required to meet external consumer demands.
The Mathematics of Musical Temperament
Investigate the logarithmic relationship between musical notes in Equal Temperament compared to Just Intonation, analyzing the geometric progression of frequencies.
Benford's Law and Detecting Financial Fraud
Analyze the frequency distribution of leading digits in legitimate financial data vs. fabricated datasets to test the statistical effectiveness of Benford's Law.
Evaluating Investment Strategies with Compound Interest and Annuities
Model the growth of a savings fund over 30 years, comparing regular monthly contributions against lump-sum investments with varying compounding periods.
Finding Roots via the Newton-Raphson Method
Investigate conditions under which Newton-Raphson successfully converges to find real roots of quintic equations, tracking error rates across iterations.
Investigating Properties of Magic Squares
Explore the algebraic constraints required to form 3×3 and 4×4 magic squares, deriving formulas for the magic constant using arithmetic progressions.
Continued Fractions and Rational Approximations of π
Analyze how infinite continued fractions generate increasingly accurate rational approximations of irrational numbers like π.
Population Genetics with the Hardy-Weinberg Principle and Matrices
Model changes in allele and genotype frequencies over generations using transition matrices to determine if a population reaches genetic equilibrium.
Properties of Pythagorean Triples and Generating Formulas
Prove Euclid's formula for generating primitive Pythagorean triples and investigate patterns in the perimeters and areas of these right-angled triangles.
Calculating Pi via the Archimedean Polygon Method
Recreate Archimedes' method of bounding π by calculating perimeters of inscribed and circumscribed polygons as the number of sides approaches infinity.
Logarithmic Scale Design: Decibels vs. Richter Scale
Contrast how the base-10 logarithmic scale is applied in acoustics versus seismology, examining the dramatically different growth curves.
Modeling the Trajectory of a Basketball Shot
Use video tracking to capture coordinates of a basketball shot, fit a quadratic equation, and find the maximum height and ideal launch angle.
The Mathematics of Architecture: Modeling the Gateway Arch
Investigate the equation of the St. Louis Gateway Arch to determine if it is a parabola or catenary curve, comparing their structural differences mathematically.
Modeling Tidal Movements using Sinusoidal Functions
Extract 48 hours of coastal tide height data and fit a sine or cosine wave to predict high and low tide times.
Optimizing the Shape of an Egg using Parametric Functions
Photograph a bird's egg, map its coordinates, and establish parametric equations that overlay its profile to calculate its exact volume.
Analyzing the Wing Profile of an Aircraft with Joukowsky Transformations
Explore how the Joukowsky transform maps a circle in the complex plane into an aerodynamic airfoil shape, evaluating how circle parameters affect wing shape.
Modeling the Spread of a Viral Meme via Bass Diffusion Model
Chart cumulative views of a viral social media video and fit a Bass Diffusion curve to differentiate between innovators and imitators.
Braking Distance and Vehicle Velocity: A Polynomial Relationship
Explore how stopping distance scales non-linearly with velocity by fitting a model accounting for linear reaction time and quadratic braking distance.
Bézier Curves and Their Application in Computer Vector Graphics
Investigate how linear, quadratic, and cubic Bézier curves are calculated using control points to produce smooth paths in graphic design software.
Modeling the Depletion Rate of a Smartphone Battery
Record battery percentage drop under different tasks to construct a piecewise model of daily battery depletion.
Sound Wave Interference and Beats Phenomena
Prove algebraically how superposition of two sine waves with slightly different frequencies creates a wave with an oscillating amplitude envelope.
Modeling the Path of a Roller Coaster Loop
Design a piecewise function for a roller coaster path, ensuring smooth transitions by enforcing equal function values and first derivatives at transition points.
Carbon-14 Dating: Quantifying Archeological Timelines
Map the radioactive decay curve of Carbon-14 and solve for the age of ancient organic artifacts based on remaining isotope percentages.
Using Spherical Trigonometry to Map Global Flight Paths
Calculate shortest flight distances between major cities using the Haversine formula and compare to straight lines on a flat Mercator projection.
Finding the Optimal Tilt Angle for Solar Panels
Model the angle of incidence of solar rays on a roof at a specific latitude across seasons to calculate the optimal permanent tilt for a solar panel.
Voronoi Diagrams and Optimizing Urban Emergency Services
Construct a Voronoi diagram over a city map to find the nearest fire station for every neighborhood, identifying underserved regions.
Calculating the Surface Area and Volume of an Unconventional Vase
Divide an asymmetrical vase into thin cylinders and frustums, summing their dimensions to approximate the total capacity and surface area.
Exploring the Geometry and Structural Stability of Trusses
Calculate internal tensile and compressive forces acting on a simple triangular bridge truss under a localized load using vector components.
The Geometry of Perspectives: Anamorphic Art Creation
Investigate the geometric transformations needed to distort a 2D image so it appears correctly proportioned when viewed from a precise oblique angle.
Finding Shortest Network Paths: Minimum Spanning Trees in Logistics
Map delivery depots as vertices and travel times as weighted edges, applying graph algorithms to find the most cost-effective connecting transport network.
The Packing Problem: Optimizing Spheres in a Cylindrical Container
Investigate the maximum packing density of uniform spheres within cylindrical containers, finding the percentage of wasted volume.
Calculating Geodesics on Curved Surfaces
Determine mathematical properties of a geodesic (shortest path) constrained to the surface of a cone or paraboloid rather than a flat plane.
Elliptical Orbits and Kepler's Laws of Planetary Motion
Track coordinates of a planet around the sun to model its elliptical path, calculating semi-major axis, eccentricity, and orbital period.
Trigonometric Triangulation: Measuring Inaccessible Heights
Use a homemade clinometer to measure angles of elevation from multiple baseline points to calculate the height of a tall local landmark.
The Geometry of the Cycloid: The Brachistochrone Problem
Explore why a bead slides fastest from A to B along a cycloid path rather than a straight line, evaluating the underlying parametric properties.
Investigating the Properties of Regular Tilings and Tessellations
Prove mathematically why only triangles, squares, and hexagons can create a monohedral tessellation across a flat plane.
Finding the Center of Mass of Irregular Laminas
Find the balance point of a flat irregularly shaped polygon by treating it as a collection of sub-triangles and applying vector coordinate formulas.
Using Trigonometry to Optimize Stadium Seating Field of View
Model a movie theater screen or stadium pitch from a side profile to calculate the optimal seating distance that maximizes the vertical viewing angle.
Using Logistic Regression to Predict Football Match Outcomes
Gather historic league data to construct a logistic regression model predicting the probability of a team winning an upcoming match.
Chi-Square Test on Consumer Spending Habits
Survey a community to test whether there is a statistically significant relationship between age group and preferred streaming service.
Modeling Casino Roulette Strategies with Markov Chains
Simulate a betting strategy on a roulette wheel to find the probability of a player going bankrupt versus doubling their cash over long cycles.
Assessing Weather Forecast Accuracy Using Normal Distributions
Collect forecasted vs. actual temperatures over two months to analyze whether prediction errors are normally distributed around zero.
The Birthday Paradox: Probability Simulations vs. Empirical Reality
Derive the theoretical probability that shows how many people must be in a room for a shared birthday to exceed 50%, validating with real school data.
Modeling Queuing Times at a Supermarket via Poisson Distribution
Count customer arrivals per minute at a check-out to test whether frequency fits a Poisson distribution and calculate probability of severe bottlenecks.
Testing the Validity of the Central Limit Theorem
Use simulations or dice experiments to show how sample means become normally distributed as sample size increases, even from skewed distributions.
Analyzing the Fairness of Monopoly using Probability
Calculate the exact probabilities of landing on specific property colors in Monopoly, determining the mathematical return on investment for each property tier.
Examining the Gini Coefficient and Wealth Inequality
Model a nation's income distribution using a continuous function to plot its Lorenz curve, integrating the area underneath to compute its exact Gini coefficient.
Using Bayes' Theorem to Evaluate Medical Diagnostic Testing
Investigate how a medical test with 99% accuracy can still yield high false positive rates if the base prevalence of disease in the population is very low.
Evaluating Student Performance with Two-Way ANOVA Tests
Test whether differences in exam scores are significantly influenced by two factors: teaching methodology and time of day.
Statistical Modeling of Sleep Duration vs. Academic Stress
Collect sleep and stress-level survey responses to test whether a quadratic or linear regression model better fits the data trend.
The Monty Hall Problem: Mathematical Analysis and Extended Simulations
Prove the counterintuitive advantage of switching doors using tree diagrams, and extend the problem to setups involving N doors.
Estimating Wildlife Populations via Capture-Recapture Method
Simulate a closed ecosystem using colored beads to analyze the mathematical reliability and error margins of the capture-recapture estimation method.
Modeling the Flight Profile of a Water Rocket via Kinematics
Use sensor data from a water rocket launch to construct an acceleration function and integrate it to calculate peak velocity and altitude.
Finding the Surface Area of a Wine Glass via Solid of Revolution
Trace the silhouette of a glass goblet, fit a piecewise function to its curve, and apply the solid of revolution formula to calculate its volume.
Gabriel's Horn Paradox: Infinite Surface Area with Finite Volume
Prove that rotating y=1/x from x=1 to ∞ around the x-axis yields a shape with finite volume (π) but infinite surface area.
Optimizing Structural Cable Tension under Loading
Model how mechanical tension of a suspended wire changes as loading shifts, applying optimization to find the minimum sag under safe strain.
Drainage Rate of a Leaking Container using Torricelli's Law
Monitor falling water level of a draining cylinder and solve the differential equation from Torricelli's Law to compare theoretical vs. observed exit rates.
Optimizing Vehicle Fuel Efficiency via Speed Variation Curves
Gather fuel consumption data across different cruising speeds, fit an efficiency-vs-speed function, and differentiate to find the optimal cruising speed.
Modeling the Acceleration Profile of a Drag Racing Car
Analyze telemetry where acceleration drops as velocity rises, constructing an empirical function to calculate displacement via successive integration steps.
Evaluating Total Energy Consumption from Power Demand Graphs
Use smart-meter data tracking household wattage per minute over a day to estimate total kilowatt-hours consumed via numerical integration.
Mathematics of Skydiver Terminal Velocity with Air Resistance
Set up a differential equation for a falling body balancing gravity against velocity-dependent air resistance, solving for the velocity function and terminal velocity asymptote.
Optimizing a Protective Retaining Wall against Soil Pressure
Apply integration to calculate the total lateral force exerted by soil or water on a retaining wall, finding the height profile that minimizes structural stress.
Finding the Error Margin in Simpson's Rule Approximations
Investigate how spacing intervals affect Simpson's Rule accuracy when approximating non-integrable functions like e^(-x²), comparing against analytical bounds.
Maximizing Profit under Monopoly Tax Constraints
Model how a per-unit government tax shifts a producer's cost function, applying differentiation to find the new profit-maximizing price passed to consumers.
Graph Theory to Optimize Garbage Collection Routes
Map a neighborhood's streets as a graph to solve the Chinese Postman Problem, ensuring a vehicle traverses every street at least once with minimum distance.
The Mathematics of Flight Boarding Strategies
Compare the mathematical efficiency of airplane boarding methods by analyzing total aisle conflict points for strategies like back-to-front vs. window-middle-aisle.
Project Logistics using Critical Path Analysis (PERT)
Break a construction project into dependencies, constructing a network diagram to isolate the critical path that dictates the minimum completion timeline.
Game Theory: Optimal Strategy in Penalty Kicks
Analyze penalty kick data as a 2×2 zero-sum game, calculating the mixed-strategy Nash equilibrium for both the kicker and goalkeeper.
Assessing Traffic Flow and Congestion via Queue Theory
Collect arrival rate data at a traffic bottleneck, applying queue modeling equations to forecast how lane closures scale average driver delays.
Using the Leslie Matrix to Predict Endangered Species Populations
Research age-specific survival and birth rates of an endangered species to build a Leslie Matrix model, projecting population demographics 50 years forward.
Projectile Motion with Magnus Effect on Spinning Baseballs
Model the flight of a baseball pitch incorporating the Magnus force from angular spin, mapping deviations from a normal parabolic drop.
Modeling the Chaos of a Double Pendulum System
Investigate how microscopic changes to a double pendulum's release angle lead to wildly divergent paths, visualizing the breakdown of predictability.
Using Linear Programming to Optimize a Diet Plan on a Budget
Formulate linear inequalities for daily nutritional guidelines and use graphical optimization to find the lowest-cost grocery list satisfying all conditions.
The Mathematics of Map Coloring: The Four Color Theorem
Convert a geographical map into a vertex graph to explore the Four Color Theorem, verifying that four colors always suffice to color bordering regions distinctly.
Modeling a Stock Portfolio using Markowitz Efficiency
Select three stocks and use historical returns to calculate their covariance, finding the allocation that minimizes portfolio risk for a targeted return.
Analyzing Signal Noise using Fourier Series Approximations
Investigate how a complex periodic square or triangular wave can be constructed from an infinite sum of sine and cosine harmonics.
Evaluating the Efficiency of Different Keyboard Layouts
Run a text frequency count on English passages and use coordinate metrics to calculate total finger travel distances on QWERTY vs. Dvorak keyboard grids.
Modeling the Growth of a Financial Bubble via Hyper-Exponential Curves
Track asset price indices during historic financial bubbles to determine if market prices followed a hyper-exponential growth model before correction.
The Mathematics of Origami: Huzita-Hatori Axioms
Examine geometric transformations from precise paper folds, proving why origami can solve cubic equations like trisecting an angle that straightedge-and-compass cannot.
Analyzing the Complexity of Sorting Algorithms
Compare the mathematical execution steps required by Bubble Sort O(n²) versus Merge Sort O(n log n) as unsorted database size increases.
Designing an Optimal Maze Using Graph Spanning Trees
Analyze how spanning trees within grid graphs generate perfect mazes containing exactly one unique solution path between any two points.
Calculating the Surface Area of a Pringles Crisp (Hyperbolic Paraboloid)
Model the saddle shape of a snack crisp using a hyperbolic paraboloid equation, applying multivariable integration to estimate its exact surface area.
Using Affine Transformations to Generate Barnsley Fractal Ferns
Explore how applying four distinct coordinate transformations repeatedly over thousands of iterations creates a realistic natural fern shape.
The Shoelace Formula for Finding Irregular Plot Areas
Extract GPS coordinates of an irregularly shaped park on Google Maps, convert to a flat grid, and calculate the exact area using Gauss's Shoelace formula.
Analyzing Logic Gates and Boolean Algebra of Computer Processors
Investigate how complex computer operations are simplified by applying De Morgan's laws and Boolean simplification to networks of logic gates.
Evaluating Fair Division of Assets: The Adjusted Winner Procedure
Model an estate dispute mathematically using the Adjusted Winner procedure, proving how point allocation guarantees an envy-free, efficient distribution of assets.
Finding the Structural Center of Pressure on a Sailboat Keel
Model the underwater profile of a sailboat keel and calculate the exact center of hydrodynamic pressure using the ratio of first moment of area to total area.
The Mathematics of Voting Systems: Arrow's Impossibility Theorem
Analyze different voting systems to demonstrate Arrow's Theorem, which proves no voting system can satisfy all fairness criteria simultaneously.