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Skills available for Northern Ireland key stage 5 maths curriculum

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KS5.6 Algebra and functions

KS5.6.1 Laws of indices for all rational exponents
KS5.6.2 Use and manipulation of surds
KS5.6.3 Quadratic functions and their graphs; the discriminant of a quadratic function; completing the square; solution of quadratic equations
KS5.6.4 Simultaneous equations: analytical solution by substitution, e.g. of one linear and one quadratic equation
KS5.6.5 Solution of linear and quadratic inequalities
KS5.6.6 Algebraic manipulation of polynomials, including expanding brackets and collecting like terms, factorisation and simple algebraic division; use of the Factor Theorem and the Remainder Theorem
KS5.6.7 Graphs of functions; sketching curves defined by simple equations; geometrical interpretation of algebraic solution of equations; use of intersection points of graphs of functions to solve equations
KS5.6.10 Knowledge of the effect of simple transformations on the graph of y = f(x) as represented by y = af(x), y = f(x) + a, y = f(x + a), y = f(ax)

KS5.7 Coordinate geometry in the (x,y) plane

KS5.7.1 Equation of a straight line, including the forms y - y₁ = m(x - x₁) and ax + by + c = 0; conditions for two straight lines to be parallel or perpendicular to each other
KS5.7.2 Coordinate geometry of the circle using the equation of a circle in the form (x - a)² + (y - b)² = r², and including use of the following circle properties:
- KS5.7.2.a the angle in a semicircle is a right angle;
  - Angles in inscribed right triangles (12-T.5)
- KS5.7.2.b the perpendicular from the centre to a chord bisects the chord;
- KS5.7.2.c the perpendicularity of radius and tangent
  - Tangent lines (12-T.3)

KS5.11 Differentiation

KS5.KS5.1 The derivative of f(x) as the gradient of the tangent to the graph of y = f (x) at a point; the gradient of the tangent as a limit; interpretation as a rate of change; second order derivatives
KS5.KS5.2 Differentiation of xⁿ, and related sums and differences
KS5.KS5.3 Applications of differentiation to gradients, tangents and normals, maxima and minima and stationary points, increasing and decreasing functions

KS5.12 Integration

KS5.KS5.1 Indefinite integration as the reverse of differentiation
KS5.KS5.2 Integration of xⁿ
KS5.KS5.3 Approximation of area under a curve using the trapezium rule; interpretation of the definite integral as the area under a curve; evaluation of definite integrals

Northern Ireland

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KS5.6 Algebra and functions

KS5.6.1 Laws of indices for all rational exponents

KS5.6.2 Use and manipulation of surds

KS5.6.3 Quadratic functions and their graphs; the discriminant of a quadratic function; completing the square; solution of quadratic equations

KS5.6.4 Simultaneous equations: analytical solution by substitution, e.g. of one linear and one quadratic equation

KS5.6.5 Solution of linear and quadratic inequalities

KS5.6.6 Algebraic manipulation of polynomials, including expanding brackets and collecting like terms, factorisation and simple algebraic division; use of the Factor Theorem and the Remainder Theorem

KS5.6.7 Graphs of functions; sketching curves defined by simple equations; geometrical interpretation of algebraic solution of equations; use of intersection points of graphs of functions to solve equations

KS5.6.10 Knowledge of the effect of simple transformations on the graph of y = f(x) as represented by y = af(x), y = f(x) + a, y = f(x + a), y = f(ax)

KS5.7 Coordinate geometry in the (x,y) plane

KS5.7.1 Equation of a straight line, including the forms y - y1 = m(x - x1) and ax + by + c = 0; conditions for two straight lines to be parallel or perpendicular to each other

KS5.7.2 Coordinate geometry of the circle using the equation of a circle in the form (x - a)² + (y - b)² = r², and including use of the following circle properties:

KS5.7.2.a the angle in a semicircle is a right angle;

KS5.7.2.b the perpendicular from the centre to a chord bisects the chord;

KS5.7.2.c the perpendicularity of radius and tangent

KS5.8 Sequences and series

KS5.8.1 Sequences, including those given by a formula for the nth term and those generated by a simple relation of the form x(n+1) = f(xn)

KS5.8.2 Arithmetic series, including the formula for the sum of the first n natural numbers

KS5.8.3 The sum of a finite geometric series; the sum to infinity of a convergent geometric series, including the use of r < 1

KS5.8.4 Binomial expansion of (1+ x)n for positive integer n; the notations n! and (n r)

KS5.9 Trigonometry

KS5.9.1 The sine and cosine rules, and the area of a triangle in the form 1/2absinC

KS5.9.2 Radian measure, including use for arc length and area of sector

KS5.9.3 Sine, cosine and tangent functions; their graphs, symmetries and periodicity

KS5.9.5 Knowledge and use of tanθ = sinθ/, Cosθ and sin²θ + cos²θ = 1

KS5.9.7 Solution of simple trigonometric equations in a given interval

KS5.1 Exponentials and logarithms

KS5.10.1 y = a to the x power and its graph

KS5.10.2 Laws of logarithms: logax + logay = loga(xy) logax -logay = loga(x/y) klogax = logax to the k power

KS5.10.3 The solution of equations of the form a to the x power = b

KS5.11 Differentiation

KS5.KS5.1 The derivative of f(x) as the gradient of the tangent to the graph of y = f (x) at a point; the gradient of the tangent as a limit; interpretation as a rate of change; second order derivatives

KS5.KS5.2 Differentiation of xn, and related sums and differences

KS5.KS5.3 Applications of differentiation to gradients, tangents and normals, maxima and minima and stationary points, increasing and decreasing functions

KS5.12 Integration

KS5.KS5.1 Indefinite integration as the reverse of differentiation

KS5.KS5.2 Integration of xn

KS5.KS5.3 Approximation of area under a curve using the trapezium rule; interpretation of the definite integral as the area under a curve; evaluation of definite integrals

KS5.7.1 Equation of a straight line, including the forms y - y₁ = m(x - x₁) and ax + by + c = 0; conditions for two straight lines to be parallel or perpendicular to each other

KS5.8.4 Binomial expansion of (1+ x)ⁿ for positive integer n; the notations n! and (n r)

KS5.10.2 Laws of logarithms: log_ax + log_ay = log_a(xy) log_ax -log_ay = log_a(x/y) klog_ax = log_ax to the k power

KS5.KS5.2 Differentiation of xⁿ, and related sums and differences

KS5.KS5.2 Integration of xⁿ