1. Number Theory, Number Systems, and Proof Foundations
- Real numbers: rational and irrational numbers
- Ordering, comparing, and representing real numbers
- Divisibility, factors, multiples, primes, GCD, and LCM
- Prime factorization and numerical structure
- Euclidean algorithm at an introductory level
- Powers and exponent laws, including integer and rational exponents where appropriate
- Scientific notation and standard form
- Surds and radicals: simplifying, comparing, and operating
- Absolute value, intervals, and numerical bounds
- Estimation, approximation, significant figures, and reasonableness checks
- Introductory modular arithmetic: congruences, residues, and divisibility patterns
- Introductory proof involving parity, divisibility, and irrationality
- Direct proof, contradiction, and counterexample
- Olympiad-style number puzzles and introductory Diophantine equations

2. Algebra, Expressions, and Equations
- Algebraic notation, structure, and equivalent forms
- Expanding and factorizing expressions, including quadratics
- Polynomial operations and simple identities
- Algebraic identities and transformations
- Rational expressions, restrictions, and simplification
- Complex fractions and algebraic manipulation
- Rearranging formulae and changing the subject of a formula
- Linear equations and contextual equations
- Quadratic equations: factorizing, completing the square, and quadratic formula where appropriate
- Systems of linear equations: substitution, elimination, and graphical interpretation
- Introductory linear-nonlinear systems, especially line-parabola intersections
- Radical equations and possible extraneous solutions
- Introductory exponential equations in structured cases
- Validity of algebraic steps and checking solutions

3. Functions and Graphical Analysis
- Function concept: input-output, notation, domain, and range
- Linear functions: slope, intercepts, rate of change, and modelling
- Quadratic functions: roots, vertex, symmetry, and transformations
- Exponential functions: introductory growth and decay models
- Simple piecewise and contextual graphs
- Function evaluation and comparison
- Graphs from equations and equations from graphs
- Transformations of graphs: shifts, stretches, and reflections
- Inverse functions at an introductory level
- One-to-one functions and domain restrictions
- Key graph features: intercepts, roots, turning points, and introductory asymptote ideas
- Comparing functions using tables, graphs, equations, and descriptions
- Graphical problem solving and interpretation of intersections
- Modelling real-life data using basic function families

4. Inequalities and Optimization
- Linear inequalities and compound inequalities
- Interval notation and number-line representation
- Quadratic inequalities using sign charts and interval reasoning
- Absolute value inequalities at an introductory level
- Systems of inequalities in two variables
- Feasible regions and graphical constraints
- Contextual inequalities and word problems involving constraints
- Maximum and minimum reasoning in non-calculus contexts
- Introductory linear programming through small-scale graphical problems
- Reasonableness checks and interpretation of solution sets

5. Coordinate Geometry and Introductory Vectors
- Cartesian plane and coordinate representation
- Distance, midpoint, and gradient/slope
- Equations of lines in different forms
- Parallel and perpendicular lines
- Graphical and algebraic interpretation of line intersections
- Circle equations in basic forms
- Intersections involving lines and circles
- Parabola graphs and equations linked to quadratic functions
- Transformations in the coordinate plane: reflection, rotation, translation, and enlargement
- Vectors in the plane: representation, magnitude, direction, addition, and scalar multiplication
- Introductory dot product and basic vector geometry applications

6. Geometry, Constructions, and Trigonometry
- Properties of triangles, quadrilaterals, polygons, and composite figures
- Triangle congruence and similarity
- Scale factor reasoning and geometric proportionality
- Angle chasing and proof-style geometric reasoning
- Circle geometry: angles, chords, tangents, and cyclic quadrilaterals
- Compass-straightedge constructions at an appropriate level
- Perimeter, area, and composite figures
- Surface area and volume of prisms, cylinders, cones, pyramids, and spheres where appropriate
- Pythagorean theorem and distance applications
- Trigonometric ratios in right triangles: sine, cosine, and tangent
- Bearings, angle of elevation, and angle of depression
- Sine rule and cosine rule in core applications
- Area of triangles using trigonometry where appropriate
- Geometric modelling and introductory optimization

7. Sequences, Patterns, and Recursion
- Arithmetic sequences: term-to-term and nth-term rules
- Geometric sequences: common ratio and growth patterns
- Arithmetic and geometric sums
- Sigma notation at an introductory level
- Recursive and explicit definitions
- Sequence graphs and pattern interpretation
- Links between sequences and linear or exponential functions
- Repeated growth and decay in real-life contexts
- Iterative processes and recursive patterns
- Olympiad-style pattern generalization
- Introductory proof of patterns, including induction-style reasoning where appropriate

8. Probability and Combinatory
- Systematic listing and counting strategies
- Addition and multiplication counting principles
- Permutations and combinations
- Pascal’s Triangle and binomial coefficients at an introductory level
- Probability of simple and compound events
- Experimental and theoretical probability
- Tree diagrams and two-way tables
- Conditional probability and independence at an introductory level
- Introductory Bayes’ Theorem in controlled contexts
- Inclusion-exclusion principle at an introductory level
- Non-routine probability and counting problems

9. Data Analysis, Statistics, and Modelling
- Data collection, sampling methods, and bias
- Data displays: histograms, box plots, scatter plots, and frequency graphs
- Measures of center: mean, median, and mode
- Measures of spread: range, variability, and standard deviation concept and use
- Comparing distributions and identifying outliers
- Correlation and association at an introductory level
- Correlation versus causation
- Regression line interpretation at an introductory level
- Conceptual confidence ideas without heavy computation
- Misleading graphs, statistical fallacies, and limitations of claims
- Using data to build, evaluate, and communicate models and decisions

10. Mathematical Logic, Reasoning, Modelling, and Investigations
- Statements, conditions, and logical structure: if/then, and/or/not
- Always/sometimes/never reasoning
- Examples, counterexamples, and claim testing
- Validity of arguments and common reasoning errors
- Multi-step reasoning and elimination strategies
- Reading and critiquing mathematical arguments
- Functional equations at an introductory level
- Mathematical modelling of real-world problems with constraints
- Assumptions, limitations, and reasonableness checks in models
- Quantitative and financial literacy: percentages, tax, discounts, interest, comparisons, unit rates, density, speed, and cost efficiency
- Student-designed investigation-style scenarios and problem creation contexts