To hit 100% in Grade 12 Higher Grade Mathematics, you need mastery across every topic, not just understanding. Below is a comprehensive list of 300 mathematics skills, grouped by key curriculum areas.
These align with what’s required in South Africa’s CAPS and IEB curricula for Mathematics Grade 12 (Higher Grade) — meaning they apply to both public and private school learners aiming for full marks.
Simplify algebraic expressions
Factorise trinomials
Factorise by grouping
Factorise the difference of squares
Factorise perfect square trinomials
Simplify rational algebraic fractions
Determine excluded values of rational expressions
Simplify complex algebraic fractions
Perform addition and subtraction of rational expressions
Multiply and divide rational expressions
Solve linear equations
Solve quadratic equations by factorisation
Solve quadratic equations using the quadratic formula
Solve quadratic equations by completing the square
Solve simultaneous linear equations
Solve simultaneous linear and quadratic equations
Solve equations with algebraic fractions
Solve equations involving exponents
Solve equations with radicals (surds)
Apply substitution to simplify equations
Understand and use inequalities
Solve quadratic inequalities
Solve rational inequalities
Represent inequalities graphically
Manipulate algebraic identities
Apply distributive and associative laws
Apply commutative laws
Simplify expressions with exponents
Apply laws of exponents to simplify algebraic terms
Convert between radical and exponential form
Simplify surd expressions
Rationalise denominators
Simplify binomial expansions (a + b)², (a - b)²
Use the binomial theorem for higher powers
Identify patterns in binomial expansions
Calculate specific terms in an expansion
Apply Pascal’s triangle
Use sigma notation in expansions
Substitute and evaluate algebraic expressions
Manipulate equations for subject change
Solve literal equations (formulae)
Use substitution in algebraic problem solving
Verify algebraic identities
Manipulate and simplify expressions with absolute values
Solve equations with absolute values
Translate word problems into algebraic equations
Model real situations using equations
Determine conditions for equations to have real roots
Find the discriminant of quadratic equations
Interpret discriminant results (two, one, or no real roots)
Identify function types (linear, quadratic, exponential, etc.)
Understand the definition of a function
Use function notation f(x), g(x)
Determine domain and range
Sketch linear functions
Sketch quadratic functions
Find the turning point of a parabola
Determine the axis of symmetry
Determine intercepts of functions
Sketch exponential functions
Sketch logarithmic functions
Understand asymptotes
Sketch hyperbolic functions
Apply transformations to functions (shifts, reflections, stretches)
Identify effects of a, q, and p in y = a(x + p)² + q
Determine inverse functions
Sketch inverse functions graphically
Identify where a function is increasing or decreasing
Interpret graphs in context
Solve equations graphically
Solve inequalities graphically
Determine points of intersection between graphs
Compare function shapes and transformations
Write equations of transformed functions
Model real-life problems using functions
Sketch piecewise-defined functions
Determine symmetry of graphs (even/odd)
Find average rate of change
Use the concept of gradient in function analysis
Interpret function tables and graphs
Evaluate composite functions (f∘g)(x)
Determine domains of composite functions
Interpret horizontal and vertical shifts
Identify turning points from given equations
Analyze the effect of a parameter on graph shape
Find inverse of exponential functions
Sketch log functions with transformations
Solve exponential equations
Solve logarithmic equations
Apply log laws to simplify expressions
Convert between exponential and log forms
Interpret function graphs for real-world meaning
Use calculators for function evaluations accurately
Interpret gradient as a rate of change
Analyze function symmetry
Identify intercepts and asymptotes
Write piecewise-defined functions
Compare growth rates of exponential vs. polynomial functions
Determine intervals of positivity or negativity of a function
Identify concavity visually from graphs
Understand definitions of sine, cosine, and tangent
Use the unit circle to find trig ratios
Use degrees and radians interchangeably
Simplify trig expressions using identities
Apply reciprocal identities
Apply quotient identities
Use Pythagorean identities
Solve basic trig equations
Solve for unknown angles in triangles
Solve for unknown sides in triangles
Use sine rule
Use cosine rule
Calculate area of triangle using ½ab sin C
Determine general solutions of trig equations
Find reference angles
Identify principal angles
Use reduction formulas
Simplify trig expressions using compound angle formulas
Use double-angle formulas
Use sum and difference formulas
Solve equations involving multiple angles
Interpret graphs of sine, cosine, tangent
Determine amplitude, period, and phase shift
Sketch sine and cosine graphs
Sketch tangent graphs
Determine equation from a given trig graph
Analyze transformations in trig graphs
Apply trig to height and distance problems
Use bearings in trig problems
Interpret periodic motion graphs
Simplify trig expressions to a single ratio
Prove trig identities
Solve mixed trig equations
Evaluate trig ratios without a calculator
Use co-function identities
Convert between radians and degrees
Solve real-world problems with angles of elevation and depression
Apply trig in navigation and surveying contexts
Model periodic phenomena using trig functions
Use inverse trig functions
Restrict domains for inverse functions
Sketch inverse trig graphs
Determine solutions in given intervals
Identify symmetry in trig functions
Use exact trig values for special angles
Verify trig identities algebraically
Simplify products and quotients of trig functions
Analyze phase difference between functions
Solve word problems using trig graphs
Interpret sine and cosine models in data
Understand the concept of limits
Determine limits from graphs
Calculate limits algebraically
Understand the definition of a derivative
Differentiate power functions
Differentiate exponential functions
Differentiate logarithmic functions
Differentiate trigonometric functions
Apply the chain rule
Apply the product rule
Apply the quotient rule
Differentiate implicit functions
Find the gradient of a tangent
Find the equation of a tangent
Find stationary points
Determine the nature of stationary points (max/min)
Use first derivative to find increasing/decreasing intervals
Use second derivative to test concavity
Sketch graphs using calculus
Apply differentiation to motion problems
Apply differentiation to optimization problems
Solve rates of change problems
Differentiate functions involving roots
Differentiate functions with constants
Apply higher-order derivatives
Differentiate piecewise functions where applicable
Understand integration as reverse differentiation
Integrate power functions
Integrate exponential functions
Integrate trigonometric functions
Apply u-substitution in integration
Solve area under curve problems
Determine area between two curves
Use definite integrals for accumulated quantities
Interpret integral as net change
Apply integration to motion problems
Use integration to find volume (if required)
Interpret average value of function via integration
Evaluate improper integrals (if limits infinite)
Solve real-world problems using integration
Sketch curves using first and second derivatives
Identify inflection points
Link differentiation and continuity
Apply chain rule in composite differentiation
Model growth and decay using calculus
Relate rate of change to graph features
Solve problems with given derivative functions
Integrate to reconstruct functions
Check integration using differentiation
Interpret graphical meaning of area and slope
Calculate gradient between two points
Find midpoint of a line segment
Determine distance between two points
Write equation of a line given point and gradient
Write equation of a line given two points
Determine parallel and perpendicular line equations
Calculate angles between lines
Determine intersection of two lines
Calculate length of perpendicular from a point to a line
Find equation of perpendicular bisector
Determine equation of a circle
Find center and radius from circle equation
Determine if a point lies on a circle
Find tangent to a circle
Determine gradient of tangent to circle
Find points of intersection between line and circle
Determine length of chord cut by line on circle
Find condition for tangency between line and circle
Solve geometric problems involving circles
Find equation of circle from geometric conditions
Derive circle equation from general form
Convert general circle equation to standard form
Determine circle and line intersections algebraically
Interpret circle geometry graphically
Determine radius from diameter endpoints
Use geometry to verify coordinates satisfy circle equation
Apply perpendicular radius-tangent theorem
Prove properties of chords and tangents algebraically
Solve coordinate geometry problems in context
Calculate points of contact for tangents
Calculate mean, median, mode
Determine range, variance, standard deviation
Interpret box-and-whisker plots
Identify outliers
Compare data sets statistically
Interpret histograms and frequency polygons
Construct cumulative frequency curves
Find percentiles and quartiles
Use scatter plots to identify correlation
Calculate correlation coefficient
Interpret line of best fit
Use least squares regression line conceptually
Interpret slope and intercept in regression context
Use regression line for prediction
Distinguish between causation and correlation
Understand sample vs population
Apply basic counting principles
Calculate factorial expressions
Compute permutations (nPr)
Compute combinations (nCr)
Solve problems involving nCr notation
Apply counting principles to probability
Understand definition of probability
Determine sample spaces
Calculate theoretical probabilities
Apply complementary probabilities
Use Venn diagrams
Determine union and intersection probabilities
Apply conditional probability
Use tree diagrams for dependent events
Apply multiplication rule
Apply addition rule
Interpret probability tables
Determine independent and dependent events
Apply probability to real-life problems
Use relative frequency to estimate probability
Compute probabilities using combinations
Solve binomial probability problems
Interpret probability distributions
Calculate expected value of discrete distributions
Identify arithmetic sequences
Determine common difference
Find nth term of arithmetic sequence
Calculate sum of n terms of arithmetic series
Identify geometric sequences
Determine common ratio
Find nth term of geometric sequence
Calculate sum of finite geometric series
Apply formula for infinite geometric series
Solve real-world problems using sequences
Apply sigma notation for series
Derive formulas for nth term and sum
Determine pattern rules from data
Calculate compound interest
Calculate simple interest
Apply future value and present value formulas
Use nominal and effective interest rate formulas
Compare interest rate periods (annual vs monthly)
Apply loan amortization formulas
Calculate annuity payments
Determine time required for investment to double
Solve for rate or time in interest formulas
Use logs to solve interest equations
Interpret financial growth graphs
Apply geometric sequences to finance
Model inflation or depreciation
Apply formulas for increasing or decreasing value
Solve present vs future value comparison problems
Determine break-even points in finance problems
Interpret compound growth problems graphically