Arithmetic sequences and series — students discover the nth term, derive the Gauss sum formula from first principles, and decode sigma notation through a concert hall design scenario.

Geometric sequences and series — students build a branching fractal, derive the sum formula via telescoping cancellation, discover convergence to a finite total, and confront the paradox of infinitely many pieces with finite area.

Geometry of nth roots of complex numbers — students discover the equal-spacing formula from polygon geometry, then reverse-engineer a mystery tile using De Moivre's theorem.

Descriptive statistics review — sampling methods, cumulative frequency, mean, standard deviation, box plots, and linear transformations.

Bivariate statistics — scatter diagrams, Pearson's correlation coefficient, regression lines, and interpretation of results in context.

Bayes' theorem — prior and posterior probabilities, sequential updating with two detection signals, and sensitivity analysis of a decision threshold through an AI-detection scenario.

Binomial distribution — students discover P(X = k) by tilting a Plinko simulator, derive the formula via tree diagrams and Pascal's triangle, then find E(X) and Var(X) through hands-on exploration.

Evaluating limits involving trigonometric and rational terms — Maclaurin series, L'Hôpital's rule, and conditions for finite limits.

Predator-prey ecological modelling — separable ODEs, equilibrium analysis, Euler's method for systems, phase portrait interpretation, and model critique.