Math: Analysis & Approaches · IA research questions
Written and cross-checked by IB Diploma graduates: Alexander (41), Alejandro (40), Martina (40), Victoria (38), Martin (35), Maria (34)
· Checked against official IB guidance · Editorial policy
A strong Maths AA exploration takes a question you can push a real mathematical technique through — calculus, series, modelling with differential equations, probability — and ends with a result you can check against reality or a known value. The mathematics should be a genuine tool, not decoration.
Scope: Pick one focused technique at roughly HL/SL level and go deep. Bring in real data or a real measurement where you can so there is something to compare your model against. 12–20 pages.
Approach · Model with projectile equations, differentiate range with respect to angle, then compare the optimum to frame-by-frame video of real free throws.
Approach · Compute partial sums, tabulate error vs number of terms, and analyse the rate of convergence of each series.
Approach · Fit dP/dt = kP(1 − P/L) to published data, solve analytically, and evaluate residuals against an exponential model.
Approach · Derive the Fourier coefficients, sum numerically, and measure the maximum error (including the Gibbs overshoot) as terms increase.
Approach · Build the transition matrix (including jail rules), find the steady-state vector, and compare to a simulation of many games.
Approach · Derive y = a cosh(x/a), extract (x, y) coordinates from a photograph, fit a by least squares, and quantify the fit.
Approach · Optimise surface area subject to the volume constraint using calculus, then measure several real cans and discuss why they deviate.
Approach · Fit cubic Bézier segments to sampled outline points, and measure deviation between the curve and the true outline.
Approach · Collect a large sample of outcomes, apply a chi-squared goodness-of-fit test, and discuss Type I/II error and sample size.
Approach · Set up and numerically solve the SIR differential equations, vary the contact rate, and compare the S-curve to exponential growth.
Approach · Construct packings for several circle counts, compute the packing fraction, and compare to the π/(2√3) hexagonal limit.
Approach · Model both with recurrence relations, run them on real historical index data, and compare mean return and volatility.
Primary-source references used to verify key assessment facts on this page. Last source check: September 22, 2026.
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