LEARN · EXPLAIN · REVISE
Build the foundations
Read the idea, work independently, then explain what changed.
Start from the missing skill
- Try six starting checks
- Return to the concepts and foundation examples of the relevant lesson.
- Attempt the four foundation exercises, then the standard and transfer tasks.
- Revisit your latest checked wrong answers and write a correction.
Concept of a set
Apply concept of a set with explicit conditions.
Basic relations between sets
Apply basic relations between sets with explicit conditions.
Basic operations on sets
Apply basic operations on sets with explicit conditions.
Sufficient and necessary conditions
Apply sufficient and necessary conditions with explicit conditions.
Universal and existential quantifiers
Apply universal and existential quantifiers with explicit conditions.
Properties of equalities and inequalities
Apply properties of equalities and inequalities with explicit conditions.
Basic inequality: AM–GM
Apply basic inequality: am–gm with explicit conditions.
Quadratic functions, equations and inequalities
Apply quadratic functions, equations and inequalities with explicit conditions.
Function concept and representations
Interpret and use function concept and representations with domain checks.
Basic properties of functions
Interpret and use basic properties of functions with domain checks.
Power functions
Interpret and use power functions with domain checks.
Applications of functions I
Interpret and use applications of functions i with domain checks.
Exponents
Interpret and use exponents with domain checks.
Exponential functions
Interpret and use exponential functions with domain checks.
Logarithms
Interpret and use logarithms with domain checks.
Logarithmic functions
Interpret and use logarithmic functions with domain checks.
Applications of functions II
Interpret and use applications of functions ii with domain checks.
General angles and radian measure
Use general angles and radian measure with explicit angle units and domains.
Concept of trigonometric functions
Use concept of trigonometric functions with explicit angle units and domains.
Reduction formulas
Use reduction formulas with explicit angle units and domains.
Graphs and properties of trigonometric functions
Use graphs and properties of trigonometric functions with explicit angle units and domains.
Trigonometric identities and transformations
Use trigonometric identities and transformations with explicit angle units and domains.
The function y=A sin(ωx+φ)
Use the function y=a sin(ωx+φ) with explicit angle units and domains.
Applications of trigonometric functions
Use applications of trigonometric functions with explicit angle units and domains.
Concept of plane vectors
Connect geometric and algebraic forms of concept of plane vectors.
Operations on plane vectors
Connect geometric and algebraic forms of operations on plane vectors.
Basis theorem and coordinate representation
Connect geometric and algebraic forms of basis theorem and coordinate representation.
Applications of plane vectors
Connect geometric and algebraic forms of applications of plane vectors.
Concept of complex numbers
Connect geometric and algebraic forms of concept of complex numbers.
Arithmetic of complex numbers
Connect geometric and algebraic forms of arithmetic of complex numbers.
Trigonometric representation of complex numbers (starred)
Connect geometric and algebraic forms of trigonometric representation of complex numbers (starred).
Basic solid figures
Use definitions, valid spatial reasoning and metric checks for basic solid figures.
Oblique drawings of solids
Use definitions, valid spatial reasoning and metric checks for oblique drawings of solids.
Surface areas and volumes of simple solids
Use definitions, valid spatial reasoning and metric checks for surface areas and volumes of simple solids.
Positions of points, lines and planes in space
Use definitions, valid spatial reasoning and metric checks for positions of points, lines and planes in space.
Parallel lines and planes in space
Use definitions, valid spatial reasoning and metric checks for parallel lines and planes in space.
Perpendicular lines and planes in space
Use definitions, valid spatial reasoning and metric checks for perpendicular lines and planes in space.
Random sampling
Use random sampling to state a justified conclusion.
Estimating a population from a sample
Use estimating a population from a sample to state a justified conclusion.
Statistical case study
Use statistical case study to state a justified conclusion.
Random events and probability
Use random events and probability to state a justified conclusion.
Independence of events
Use independence of events to state a justified conclusion.
Frequency and probability
Use frequency and probability to state a justified conclusion.
Space vectors and their operations
Add, scale and multiply space vectors; distinguish scalar and vector results.
The fundamental theorem of space vectors
Recognise a basis and use unique vector coordinates and affine combinations.
Coordinate representation of space vectors
Calculate spatial lengths, angles, projections and perpendicular directions from coordinates.
Applications of space vectors
Use direction and normal vectors to establish angles, distances and spatial relations.
Inclination and slope of a line
Connect slope, inclination and parallel/perpendicular directions, including vertical lines.
Equations of a line
Choose point-slope, two-point, intercept or general form without losing exceptional lines.
Intersections and distance formulas
Solve intersections and calculate point-line and parallel-line distances with geometric checks.
Equations of a circle
Construct circle equations from centers, radii, diameters and point constraints.
Relations between lines and circles
Classify intersection and tangency using distances, discriminants and common chords.
Ellipse
Use the focal definition, standard equation and geometric properties of an ellipse.
Hyperbola
Use the distance-difference definition, asymptotes and branch restrictions.
Parabola
Relate focus, directrix and vertex, and retain conditions in line-parabola intersections.
Concept of a sequence
Connect general terms, recurrence relations and partial sums.
Arithmetic sequences
Derive terms and sums and solve indexed arithmetic models.
Geometric sequences
Connect ratios, indexed powers and finite geometric sums.
Mathematical induction
Build proofs with a verified base, an explicit induction hypothesis and a valid step.
Derivative: concept and meaning
Distinguish average and instantaneous rates and derive tangents from a limit.
Derivative operations
Apply power, product, quotient and chain rules with elementary derivatives.
Applications of derivatives
Use derivative signs, boundary values and models to justify extrema and optimization.
Addition and multiplication counting principles
Count disjoint alternatives and sequential choices without overlap or omission.
Permutations and combinations
Distinguish ordered selections, unordered sets and symmetry in arrangements.
Binomial theorem
Use the general term to locate coefficients, constants and coefficient sums.
Conditional and total probability
Use conditional denominators, disjoint partitions and Bayes inversion.
Discrete random variables and distributions
Construct a distribution and compute event probabilities from its support.
Numerical characteristics of random variables
Compute expectation and variance and distinguish transformation from random independence.
Binomial and hypergeometric distributions
Choose the correct model for repeated independent trials or sampling without replacement.
Normal distribution
Standardise a normal variable and interpret supplied table probabilities and symmetry.
Correlation of paired data
Interpret scatter patterns and compute a correlation from centered paired summaries.
Linear regression and applications
Fit a least-squares line and distinguish prediction, residual and extrapolation.
Contingency tables and independence tests
Compute expected counts and a chi-square statistic and state a bounded conclusion.
Polynomial division and partial fractions
Divide polynomials and choose decomposition numerators matched to each denominator factor.
Variation and financial models
Identify direct, inverse, joint and partial variation and apply stated interest or depreciation models.
Euclidean figures and circle theorems
Use similarity, angle and circle theorems with labeled points and compatible arcs.
Linear programming and discrete feasibility
Draw a feasible region and optimise a linear objective, then check integer restrictions when required.
Matrices, determinants and linear systems
Add and multiply matrices, use determinants and solve systems without assuming invertibility.
Polar coordinates
Convert representations and eliminate polar variables without losing the geometric range.
Integration, area and school extensions
Find antiderivatives, evaluate definite integrals and distinguish signed accumulation from geometric area.
Spatial lines, planes, spheres and cross products
Construct spatial equations and use normals, cross products and direction tests.
Parametric curves and preserved domains
Eliminate parameters while retaining ranges, missing points and the geometry of the locus.
Infinite geometric series and error control
Check convergence before summing and interpret recurring decimals and truncation errors.
Radicals, rational equations and simultaneous equations
Choose an equivalent method and explain exclusions before calculating.
Absolute values and solution sets
Choose an equivalent method and explain exclusions before calculating.
Inverse functions and domain restrictions
Choose an equivalent method and explain exclusions before calculating.
Exponential and logarithmic equations
Choose an equivalent method and explain exclusions before calculating.