1. Advanced Number Theory and Proof
- Properties of integers, divisibility, primes, GCD, and LCM in advanced contexts
- Prime factorization and structure of integers
- Euclidean algorithm and applications
- Modular arithmetic with congruences, residues, and divisibility arguments
- Rational and irrational number classification
- Proofs of irrationality and number-classification arguments
- Direct proof, contradiction, counterexample, and structured case analysis
- Parity, divisibility, remainders, and constraints
- Diophantine equations and integer-solution reasoning
- Olympiad-style number puzzles involving remainders, parity, divisibility, and integer constraints
- Generalization of numerical patterns with proof

2. Advanced Algebra, Expressions, and Equations
- Polynomial identities and advanced factorization techniques
- Polynomial operations and algebraic equivalence
- Rational expressions, complex fractions, and restrictions
- Rearranging formulae in more complex applied contexts
- Quadratic equations and expressions: factorizing, completing the square, formula, and discriminant reasoning where appropriate
- Systems of equations, including linear systems and introductory linear-nonlinear systems
- Radical equations and analysis of extraneous roots
- Exponential equations in structured cases
- Algebraic modelling from complex contexts
- Validity of algebraic transformations and error analysis
- Algebraic proof and identity-based reasoning

3. Functions and Graphical Analysis
- Linear, quadratic, exponential, and piecewise functions
- Function notation, domain, range, and restrictions
- Transformations of functions: shifts, stretches, reflections, and combinations
- Inverse functions, one-to-one conditions, and basic inverse construction
- Key features of graphs: intercepts, roots, vertex, turning points, asymptotes, and introductory end behaviour
- Graphical solution of equations and systems
- Comparing functions across equations, tables, graphs, and verbal descriptions
- Modelling real-life data with suitable function families
- Evaluating model fit, limitations, and contextual meaning
- Graph interpretation in abstract and applied contexts

4. Inequalities and Optimization
- Linear, compound, and quadratic inequalities
- Absolute value inequalities
- Sign charts, interval reasoning, and solution-set notation
- Systems of inequalities and feasible regions
- Constraint-based word problems
- Maximum and minimum reasoning without calculus
- Introductory optimization and linear programming
- Interpretation of feasible and optimal solutions
- Checking validity and context constraints in optimization problems
- Error analysis in inequality transformations

5. Coordinate Geometry and Vectors
- Distance, midpoint, gradient, and equations of lines
- Parallelism and perpendicularity in coordinate settings
- Circle equations, tangent ideas, and intersections
- Parabola equations and links to quadratic functions
- Coordinate proofs involving lines, shapes, and relationships
- Transformations in the coordinate plane
- Vectors in the plane: representation, magnitude, direction, addition, subtraction, and scalar multiplication
- Dot product and geometric interpretation
- Vector applications to geometry, including parallelism, perpendicularity, and angle reasoning
- Connections between coordinate, vector, algebraic, and geometric methods

6. Geometry, Constructions, and Trigonometry
- Triangle similarity, congruence, and geometric proof
- Advanced angle chasing and structured geometric reasoning
- Circle geometry: chords, tangents, angles, and cyclic quadrilaterals
- Compass-straightedge constructions and justification
- Transformational geometry and scale reasoning
- Perimeter, area, surface area, and volume in appropriate contexts
- Pythagorean theorem and geometric applications
- Trigonometric ratios and right-triangle applications
- Trigonometric identities and equations at an appropriate level
- Sine rule, cosine rule, and area of triangles
- Bearings, elevation/depression, and indirect measurement
- Geometric modelling and optimization problems

7. Sequences, Series, and Recursion
- Arithmetic and geometric sequences and sums
- Sigma notation and nth-term formulas
- Recursive and explicit definitions
- Recursive patterns and iterative processes
- Links between sequences and linear or exponential functions
- Real-life modelling using growth, decay, repeated change, and finance contexts
- Pattern generalization in olympiad-style problems
- Proofs of patterns and introductory induction-style reasoning where appropriate
- Summation reasoning and formula interpretation
- Recursive modelling and repeated operations

8. Probability and Combinatory
- Counting principles, including addition and multiplication rules
- Systematic listing and case organization
- Permutations and combinations
- Pascal’s Triangle and binomial coefficients
- Probability of simple and compound events
- Conditional probability and independence
- Tree diagrams, two-way tables, and structured sample spaces
- Introductory Bayes’ Theorem in controlled contexts
- Inclusion-exclusion principle
- Experimental versus theoretical probability
- Olympiad-style probability reasoning in non-routine setups
- Counting with restrictions, overlapping cases, or cases requiring careful organization

9. Data Analysis, Statistics, and Modelling
- Interpreting large datasets using measures of center and spread
- Mean, median, mode, range, variability, and standard deviation
- Comparing distributions and identifying outliers
- Sampling methods, bias, and limitations of data collection
- Correlation, association, and regression line interpretation
- Correlation versus causation and evaluation of claims
- Conceptual confidence intervals and uncertainty
- Statistical fallacies and misleading graphs
- Using data in modelling, decisions, and evidence-based conclusions
- Evaluating assumptions and limitations in statistical models

10. Critical Thinking, Mathematical Modelling, and Investigations
- Logical reasoning and paradox-style problems
- Statements, conditions, implications, and negation
- Always/sometimes/never reasoning
- Counterexamples and proof-based claim testing
- Functional equations and creative constraints
- Mathematical modelling of real-world and abstract problems
- Assumptions, constraints, limitations, and evaluation of models
- Multi-step problem solving and strategy selection
- Quantitative and financial reasoning where relevant
- Inquiry-style scenarios, problem creation, and mathematical explorations
- Proofs of patterns and generalizations
- Error analysis and critique of mathematical arguments