Year 12 skills
Algebra
Skills
- Simplification and expansion of algebraic expressions
- Simplification of algebraic expressions involving integer indices
- Simplification of algebraic expressions involving rational indices
- Simultaneous linear equations in three unknowns
- Simultaneous linear equations in four unknowns
- Simultaneous nonlinear equations
- Simplify nested surds
- Partial sums of arithmetic series
- Algebra of absolute-value functions
- Level 1: Solving inequalities using sign charts
- Level 2: Solving inequalities using sign charts
- Long division of polynomials
- Rationalise a binomial surd denominator
- Simplification of rational algebraic expressions
Exponentials and logarithms
Skills
- Evaluation of logarithms
- Simplification of logarithmic expressions
- Equations involving exponents
- Level 1: Logarithm laws
- Level 1: Logarithmic equations
- Level 2: Logarithm laws
- Level 2: Logarithmic equations
- Level 2: Logarithmic equations and domains
- Interpreting and using logarithmic scales
- Graph features of translated logarithmic functions
- Inverse relationship of exponential and logarithmic functions
Functions and relations
Tests
Skills
- Function transformations
- Composite functions
- Level 1: Composite functions
- Domain and range of composite functions
- Level 1: Composite function equations
- Level 2: Composite function equations
- Level 2: Composite functions
- Level 1: Continuity and discontinuity of functions
- Level 2: Continuity and discontinuity of functions
- Sketching polynomial functions
- Identifying transformed polynomials
- Sketching power functions
- Sketching sum, different and product of two functions
- Intersection of two graphs
- Absolute value function
- Level 1: Restricting a function to make it one-to-one
- Domain of an inverse function
- Range of an inverse function
- Level 2: Restricting a function to make it one-to-one
- Level 2: Solving inverse function equations
- Level 3: Solving inverse function equations
- Level 4: Solving inverse function equations
- Level 1: Transformations of functions
- Level 2: Transformations of functions
- Level 3: Transformations of functions
Coordinate geometry
Tests
Coordinate geometry: Find coordinates: points, midpoints, planar shapesPremium- Coordinate geometry: Find coordinates: points, midpointsPremium
Skills
Geometry
Skills
Trigonometry
Skills
- Pythagoras theorem
- Conversions between radians and degrees
- Evaluate the sine and cosine functions
- Find amplitude, period and range of trigonometric functions
- Sketch trigonometric functions
- Find the unknown angle using trigonometric functions
- The sine and cosine laws
- Co-function identities
- Angle sum and difference identities
- Evaluate trigonometric functions
- General solution of trigonometric equations
- Trigonometric relations involving circles
- Trigonometric relations involving complex geometric shapes
Calculus
Skills
- The number e as the base giving a limit of 1
- Limit of the exponential difference quotient
- Level 1: Average and instantaneous rates of change
- Identify a differentiable function
- Level 1: Differentiability of piecewise functions
- Level 2: Differentiability of piecewise functions
- Evaluation of a derivative at a point
- Level 1: Differentiate using the chain rule
- Level 2: Differentiate using the chain rule
- Level 1: Apply the chain rule to composite functions
- Level 2: Apply the chain rule to composite functions
- Level 3: Apply the chain rule to composite functions
- Level 1: Apply the product rule
- Level 2: Apply the product rule
- Differentiation of fractions
- Level 1: Differentiation of logarithmic functions
- Level 2: Differentiation of logarithmic functions
- Level 1: Differentiation of trigonometric functions
- Level 2: Differentiation of trigonometric functions
- Level 3: Differentiation of trigonometric functions
- Level 4: Differentiation of trigonometric functions
- Level 2: Average and instantaneous rates of change
- Level 1: Indirect differentiation
- Level 2: Indirect differentiation
- Differentiation problems involving the cubic function
- Level 1: Differentiation of mixed functions
- Level 2: Differentiation of mixed functions
- Level 3: Differentiation of mixed functions
- Level 1: Differentiation of indirectly defined relations
- Level 2: Differentiation of indirectly defined relations
- Level 1: Gradient and tangent of a curve at a point
- Level 2: Gradient and tangent of a curve at a point
- Level 1: Application of differentiation to graphs
- Level 2: Application of differentiation to graphs
- Level 1: Local minimum and local maximum values
- Level 2: Local minimum and local maximum values
- Differentiation problems involving stationary points
- Level 1: Stationary points, inflection points and intervals of increase or decrease
- Level 2: Stationary points, inflection points and intervals of increase or decrease
- Level 1: Differentiation word problems
- Level 2: Differentiation word problems
- Level 3: Differentiation word problems
- Increments formula - linear approximation
- Indefinite integration of polynomials
- Indefinite integration of surds
- Indefinite integration of trigonometric functions
- Level 1: Indefinite integration and antiderivatives
- Level 2: Indefinite integration and antiderivatives
- Level 2: Finding a curve from a stationary point
- Level 3: Finding a curve from a stationary point
- Riemann sums - estimating area under a curve
- Definite integration
- Level 1: Definite integration
- Level 2: Definite integration
- Displacement given velocity
- Position given acceleration and initial conditions
- General solutions of separable differential equations
- Particular solutions of separable differential equations
- Substitutions in differential equations
- Formulating differential-equation models
- Interpreting differential-equation models
- Standard Maclaurin series
- Combining standard Maclaurin series
- Using Maclaurin series for approximations
- Small-angle approximations
- Deriving Maclaurin expansions
- Convergence of Maclaurin series
Complex numbers
Skills
Discrete mathematics
Skills
- Vector magnitude
- Vector direction and components
- Vector scalar multiplication and addition
- Vector arithmetic
- Vector dot product
- Distance between position vectors
- Equations of lines in three dimensions
- Equations of planes in three dimensions
- Relationships between lines and planes in three dimensions
- Perpendicular feet and distances in three dimensions
- Angles between lines and planes
- Recursive sequences
- Rules for matrix elements
- Level 1: Transition matrices
- Level 2: Transition matrices
- Level 1: Leslie matrices
- Level 2: Leslie matrices
- Level 1: Permutation matrices
- Level 2: Permutation matrices
- Level 1: Communication matrices
- Level 2: Communication matrices
- Level 1: Dominance matrices
- Level 2: Dominance matrices
- Level 1: Counting problems (permutations and combinations)
- Level 2: Counting problems (permutations and combinations)
- Level 3: Counting problems (permutations and combinations)
Graph theory
Skills
Numbers
Skills
Time and date
Financial mathematics
Skills
- Present value of ordinary annuities
- Future value of ordinary annuities
- Perpetuities
- Depreciation - flat rate vs reducing balance crossover
- Depreciation - unit cost method
- Perpetuities - cumulative payments
- Compound interest - regular deposits, find the rate
- Compound interest - effective annual rate and interest earned
- Annuities - two-phase withdrawals, total years
- Making informed decisions - comparing investment options
- Making informed decisions - comparing loan offers
- Making informed decisions - break-even between payment plans
Set theory
Skills
Probability
Skills
- Level 1: Multi-stage probability experiments
- Level 2: Multi-stage probability experiments
- Conditional probability
- Identify discrete probability distributions
- Calculate probabilities given a discrete probability distribution
- Linear transformation of mean and standard deviation
- Bernoulli distribution - mean and variance
- Bernoulli distribution - modelling practical problems
- Conditions for a binomial distribution
- Binomial distribution probabilities
- Level 1: Conditional probability
- Level 2: Conditional probability
- Level 1: Discrete random variables
- Level 2: Discrete random variables
- Mean and variance of a binomial distribution
- Level 1: Probabilities from polynomial probability density functions
- Level 2: Probabilities from polynomial probability density functions
- Level 1: Probabilities from exponential probability density functions
- Level 2: Probabilities from exponential probability density functions
- Level 1: Statistics from polynomial probability density functions
- Level 2: Statistics from polynomial probability density functions
- Level 1: Statistics from exponential probability density functions
- Level 2: Statistics from exponential probability density functions
- Normal distribution - qualitative graph features
- Normal distribution - probabilities and quantiles
- Contexts suitable for a normal model
Challenge Winning the tic-tac-toe
Statistics
Skills
- Understanding frequency tables
- Understanding stem-and-leaf plots
- Understanding histograms
- Summary and spread of data
- Understanding box plots
- Understanding scatter plots
- Association and causation
- Least-squares regression lines
- Coefficient of determination
- Residuals, interpolation and extrapolation
- Time series plots and patterns
- Moving averages and smoothing
- Seasonal indices and time series models
- The concept of a random sample
- Sources of bias and ensuring randomness
- Sample proportion - mean and standard deviation
- Approximate normality of the sample proportion
- Simulated sampling distribution of the sample proportion
- Confidence interval and margin of error for a proportion
- Coverage of repeated confidence intervals
- Hypothesis testing concepts and terminology
- Formulating hypotheses and choosing tails
- One-tailed z-tests for a population mean
- Two-tailed z-tests for a population mean
- Interpreting hypothesis test conclusions
Mechanics
Skills
- Quantities and units in mechanics
- Resultant forces
- Forces and Newton's first law
- Weight and motion under gravity
- Resolving forces into components
- Friction on a rough surface
- Coefficient of friction
- Moments
- Newton's second law
- Newton's third law and connected bodies
- The language of kinematics
- Motion graphs
- Constant acceleration formulae
- Calculus in kinematics
- Projectiles