Programme of Study: Key Stage 4, Year 11
Skills available for England Key Stage 4, Year 11 maths objectives
Dimension
Print skill plan
Working mathematically
Develop fluency
Consolidate their numerical and mathematical capability from key stage 3 and extend their understanding of the number system to include powers, roots {and fractional indices}.
Rational numbers
Roots
Powers
Standard form
Consolidate their numerical and mathematical capability from key stage 3 and extend their understanding of the number system to include powers, roots {and fractional indices}.
Rational numbers
Roots
Powers
Standard form
Select and use appropriate calculation strategies to solve increasingly complex problems, including exact calculations involving multiples of π {and surds}, use of standard form and application and interpretation of limits of accuracy.
Rational numbers
Standard form
Indices
Radical expressions
Select and use appropriate calculation strategies to solve increasingly complex problems, including exact calculations involving multiples of π {and surds}, use of standard form and application and interpretation of limits of accuracy.
Rational numbers
Standard form
Indices
Radical expressions
Consolidate their algebraic capability from key stage 3 and extend their understanding of algebraic simplification and manipulation to include quadratic expressions, {and expressions involving surds and algebraic fractions}.
Quadratic expressions
Rational expressions
Consolidate their algebraic capability from key stage 3 and extend their understanding of algebraic simplification and manipulation to include quadratic expressions, {and expressions involving surds and algebraic fractions}.
Quadratic expressions
Rational expressions
Extend fluency with expressions and equations from key stage 3, to include quadratic equations, simultaneous equations and inequalities.
Linear inequalities
Simultaneous linear equations
Quadratic inequalities
Quadratic equations
Simultaneous quadratic and linear equations
Extend fluency with expressions and equations from key stage 3, to include quadratic equations, simultaneous equations and inequalities.
Linear inequalities
Simultaneous linear equations
Quadratic inequalities
Quadratic equations
Simultaneous quadratic and linear equations
Move freely between different numerical, algebraic, graphical and diagrammatic representations, including of linear, quadratic, reciprocal, {exponential and trigonometric} functions.
Equations of linear functions
Graphs of linear functions
Quadratic functions
Exponential functions
Linear, quadratic and exponential functions
Trigonometric functions
Move freely between different numerical, algebraic, graphical and diagrammatic representations, including of linear, quadratic, reciprocal, {exponential and trigonometric} functions.
Equations of linear functions
Graphs of linear functions
Quadratic functions
Exponential functions
Linear, quadratic and exponential functions
Trigonometric functions
Use mathematical language and properties precisely.
Properties of operations and equality
Direct and inverse variation
Functions
Linear functions
Quadratic functions
Polynomial expressions
Congruence
Similarity
Transformations
Statistics
Use mathematical language and properties precisely.
Properties of operations and equality
Direct and inverse variation
Functions
Linear functions
Quadratic functions
Polynomial expressions
Congruence
Similarity
Transformations
Statistics
Reason mathematically
Extend and formalise their knowledge of ratio and proportion, including trigonometric ratios, in working with measures and geometry, and in working with proportional relations algebraically and graphically.
Proportional relationships
Direct and inverse variation
Similar figures
Similar solids
Trigonometric ratios
Extend and formalise their knowledge of ratio and proportion, including trigonometric ratios, in working with measures and geometry, and in working with proportional relations algebraically and graphically.
Proportional relationships
Direct and inverse variation
Similar figures
Similar solids
Trigonometric ratios
Extend their ability to identify variables and express relations between variables algebraically and graphically.
Equations
Linear functions
Quadratic functions
Exponential functions
Linear, quadratic and exponential functions
Extend their ability to identify variables and express relations between variables algebraically and graphically.
Equations
Linear functions
Quadratic functions
Exponential functions
Linear, quadratic and exponential functions
Make and test conjectures about the generalisations that underlie patterns and relationships; look for proofs or counter-examples; begin to use algebra to support and construct arguments {and proofs}.
Sequences
Properties of operations
Linear functions
Quadratic expressions
Make and test conjectures about the generalisations that underlie patterns and relationships; look for proofs or counter-examples; begin to use algebra to support and construct arguments {and proofs}.
Sequences
Properties of operations
Linear functions
Quadratic expressions
Reason deductively in geometry, number and algebra, including using geometrical constructions.
Congruence
Similarity
Reason deductively in geometry, number and algebra, including using geometrical constructions.
Congruence
Similarity
Interpret when the structure of a numerical problem requires additive, multiplicative or proportional reasoning.
Rates
Proportions
Metric and imperial units
Interpret when the structure of a numerical problem requires additive, multiplicative or proportional reasoning.
Rates
Proportions
Metric and imperial units
Explore what can and cannot be inferred in statistical and probabilistic settings, and express their arguments formally.
Statistics
Probability
Explore what can and cannot be inferred in statistical and probabilistic settings, and express their arguments formally.
Statistics
Probability
Assess the validity of an argument and the accuracy of a given way of presenting information.
Assess the validity of an argument and the accuracy of a given way of presenting information.
Solve problems
Develop their mathematical knowledge, in part through solving problems and evaluating the outcomes, including multi-step problems.
Develop their mathematical knowledge, in part through solving problems and evaluating the outcomes, including multi-step problems.
Develop their use of formal mathematical knowledge to interpret and solve problems, including in financial contexts.
Financial problems
Triangles
Circles
Probability
Develop their use of formal mathematical knowledge to interpret and solve problems, including in financial contexts.
Financial problems
Triangles
Circles
Probability
Make and use connections between different parts of mathematics to solve problems.
Make and use connections between different parts of mathematics to solve problems.
Model situations mathematically and express the results using a range of formal mathematical representations, reflecting on how their solutions may have been affected by any modelling assumptions.
Equations
Linear functions
Linear inequalities
Simultaneous equations
Exponential growth and decay
Pythagoras' theorem
Model situations mathematically and express the results using a range of formal mathematical representations, reflecting on how their solutions may have been affected by any modelling assumptions.
Equations
Linear functions
Linear inequalities
Simultaneous equations
Exponential growth and decay
Pythagoras' theorem
Select appropriate concepts, methods and techniques to apply to unfamiliar and non-routine problems; interpret their solution in the context of the given problem.
Select appropriate concepts, methods and techniques to apply to unfamiliar and non-routine problems; interpret their solution in the context of the given problem.
Number
Apply systematic listing strategies, {including use of the product rule for counting}.
Apply systematic listing strategies, {including use of the product rule for counting}.
{Estimate powers and roots of any given positive number.}
{Estimate powers and roots of any given positive number.}
Calculate with roots, and with integer {and fractional} indices.
Roots
Integer indices
Fractional indices
Calculate with roots, and with integer {and fractional} indices.
Roots
Integer indices
Fractional indices
Calculate exactly with fractions, {surds} and multiples of π {simplify surd expressions involving squares [for example √12 = √(4 × 3) = √4 × √3 = 2√3] and rationalise denominators}.
Rational numbers
Complex fractions
Radical expressions
Calculate exactly with fractions, {surds} and multiples of π {simplify surd expressions involving squares [for example √12 = √(4 × 3) = √4 × √3 = 2√3] and rationalise denominators}.
Rational numbers
Complex fractions
Radical expressions
Calculate with numbers in standard form A × 10n, where 1 ≤ A < 10 and n is an integer.
Calculate with numbers in standard form A × 10n, where 1 ≤ A < 10 and n is an integer.
{Change recurring decimals into their corresponding fractions and vice versa.}
{Change recurring decimals into their corresponding fractions and vice versa.}
Identify and work with fractions in ratio problems.
Ratios
Rates
Proportions
Identify and work with fractions in ratio problems.
Ratios
Rates
Proportions
Apply and interpret limits of accuracy when rounding or truncating, {including upper and lower bounds}.
Apply and interpret limits of accuracy when rounding or truncating, {including upper and lower bounds}.
Algebra
Simplify and manipulate algebraic expressions (including those involving surds {and algebraic fractions}) by:
Simplify and manipulate algebraic expressions (including those involving surds {and algebraic fractions}) by:
factorising quadratic expressions of the form x2 + bx + c, including the difference of two squares; {factorising quadratic expressions of the form ax2 + bx + c}
factorising quadratic expressions of the form x2 + bx + c, including the difference of two squares; {factorising quadratic expressions of the form ax2 + bx + c}
simplifying expressions involving sums, products and powers, including the laws of indices
Integer indices
Fractional indices
Monomials
Polynomials
Rational expressions
Radical expressions
simplifying expressions involving sums, products and powers, including the laws of indices
Integer indices
Fractional indices
Monomials
Polynomials
Rational expressions
Radical expressions
Know the difference between an equation and an identity; argue mathematically to show algebraic expressions are equivalent, and use algebra to support and construct arguments {and proofs}.
Properties of addition and multiplication
Expressions
Equations
Know the difference between an equation and an identity; argue mathematically to show algebraic expressions are equivalent, and use algebra to support and construct arguments {and proofs}.
Properties of addition and multiplication
Expressions
Equations
Where appropriate, interpret simple expressions as functions with inputs and outputs; {interpret the reverse process as the 'inverse function'; interpret the succession of two functions as a 'composite function'}.
Identify functions
Evaluate functions
Composite functions
Inverse functions
Where appropriate, interpret simple expressions as functions with inputs and outputs; {interpret the reverse process as the 'inverse function'; interpret the succession of two functions as a 'composite function'}.
Identify functions
Evaluate functions
Composite functions
Inverse functions
Use the form y = mx + c to identify parallel {and perpendicular} lines; find the equation of the line through two given points, or through one point with a given gradient.
Gradient
Equations
Parallel and perpendicular lines
Use the form y = mx + c to identify parallel {and perpendicular} lines; find the equation of the line through two given points, or through one point with a given gradient.
Gradient
Equations
Parallel and perpendicular lines
Identify and interpret roots, intercepts and turning points of quadratic functions graphically; deduce roots algebraically {and turning points by completing the square}.
Identify and interpret roots, intercepts and turning points of quadratic functions graphically; deduce roots algebraically {and turning points by completing the square}.
Recognise, sketch and interpret graphs of linear functions, quadratic functions, simple cubic functions, the reciprocal function y = 1/x with x ≠ 0, {the exponential function y = kx for positive values of k, and the trigonometric functions (with arguments in degrees) y = sin x, y = cos x and y = tan x for angles of any size}.
Linear functions
Quadratic functions
Exponential functions
Trigonometric functions
Function families
Recognise, sketch and interpret graphs of linear functions, quadratic functions, simple cubic functions, the reciprocal function y = 1/x with x ≠ 0, {the exponential function y = kx for positive values of k, and the trigonometric functions (with arguments in degrees) y = sin x, y = cos x and y = tan x for angles of any size}.
Linear functions
Quadratic functions
Exponential functions
Trigonometric functions
Function families
{Sketch translations and reflections of the graph of a given function.}
{Sketch translations and reflections of the graph of a given function.}
Plot and interpret graphs (including reciprocal graphs {and exponential graphs}) and graphs of non-standard functions in real contexts, to find approximate solutions to problems such as simple kinematic problems involving distance, speed and acceleration.
Plot and interpret graphs (including reciprocal graphs {and exponential graphs}) and graphs of non-standard functions in real contexts, to find approximate solutions to problems such as simple kinematic problems involving distance, speed and acceleration.
{Calculate or estimate gradients of graphs and areas under graphs (including quadratic and other non-linear graphs), and interpret results in cases such as distance-time graphs, velocity-time graphs and graphs in financial contexts.}
{Calculate or estimate gradients of graphs and areas under graphs (including quadratic and other non-linear graphs), and interpret results in cases such as distance-time graphs, velocity-time graphs and graphs in financial contexts.}
{Recognise and use the equation of a circle with centre at the origin; find the equation of a tangent to a circle at a given point.}
{Recognise and use the equation of a circle with centre at the origin; find the equation of a tangent to a circle at a given point.}
Solve quadratic equations {including those that require rearrangement} algebraically by factorising, {by completing the square and by using the quadratic formula}; find approximate solutions using a graph.
Square roots
Factorising
Completing the square
Quadratic formula
Solve quadratic equations {including those that require rearrangement} algebraically by factorising, {by completing the square and by using the quadratic formula}; find approximate solutions using a graph.
Square roots
Factorising
Completing the square
Quadratic formula
Solve two simultaneous equations in two variables (linear/linear {or linear/quadratic}) algebraically; find approximate solutions using a graph.
Number of solutions
Linear equations
Linear and quadratic equations
Solve two simultaneous equations in two variables (linear/linear {or linear/quadratic}) algebraically; find approximate solutions using a graph.
Number of solutions
Linear equations
Linear and quadratic equations
{Find approximate solutions to equations numerically using iteration.}
{Find approximate solutions to equations numerically using iteration.}
Translate simple situations or procedures into algebraic expressions or formulae; derive an equation (or two simultaneous equations), solve the equation(s) and interpret the solution.
Equations
Simultaneous equations
Translate simple situations or procedures into algebraic expressions or formulae; derive an equation (or two simultaneous equations), solve the equation(s) and interpret the solution.
Equations
Simultaneous equations
Solve linear inequalities in one {or two} variable{s}, {and quadratic inequalities in one variable}; represent the solution set on a number line, {using set notation and on a graph}.
One-variable linear inequalities
One-variable quadratic inequalities
Two-variable inequalities
Solve linear inequalities in one {or two} variable{s}, {and quadratic inequalities in one variable}; represent the solution set on a number line, {using set notation and on a graph}.
One-variable linear inequalities
One-variable quadratic inequalities
Two-variable inequalities
Recognise and use sequences of triangular, square and cube numbers, simple arithmetic progressions, Fibonacci type sequences, quadratic sequences, and simple geometric progressions (rn where n is an integer, and r is a positive rational number {or a surd}) {and other sequences}.
Arithmetic and geometric sequences
Other sequences
Recognise and use sequences of triangular, square and cube numbers, simple arithmetic progressions, Fibonacci type sequences, quadratic sequences, and simple geometric progressions (rn where n is an integer, and r is a positive rational number {or a surd}) {and other sequences}.
Arithmetic and geometric sequences
Other sequences
Deduce expressions to calculate the nth term of linear {and quadratic} sequences.
Deduce expressions to calculate the nth term of linear {and quadratic} sequences.
Ratio, proportion and rates of change
Compare lengths, areas and volumes using ratio notation and/or scale factors; make links to similarity (including trigonometric ratios).
Similar figures
Similar solids
Trigonometry
Compare lengths, areas and volumes using ratio notation and/or scale factors; make links to similarity (including trigonometric ratios).
Similar figures
Similar solids
Trigonometry
Convert between related compound units (speed, rates of pay, prices, density, pressure) in numerical and algebraic contexts.
Convert between related compound units (speed, rates of pay, prices, density, pressure) in numerical and algebraic contexts.
Understand that X is inversely proportional to Y is equivalent to X is proportional to 1/Y; {construct and} interpret equations that describe direct and inverse proportion.
Understand that X is inversely proportional to Y is equivalent to X is proportional to 1/Y; {construct and} interpret equations that describe direct and inverse proportion.
Interpret the gradient of a straight line graph as a rate of change; recognise and interpret graphs that illustrate direct and inverse proportion.
Direct and inverse variation
Gradient
Interpret the gradient of a straight line graph as a rate of change; recognise and interpret graphs that illustrate direct and inverse proportion.
Direct and inverse variation
Gradient
{Interpret the gradient at a point on a curve as the instantaneous rate of change; apply the concepts of instantaneous and average rate of change (gradients of tangents and chords) in numerical, algebraic and graphical contexts.}
{Interpret the gradient at a point on a curve as the instantaneous rate of change; apply the concepts of instantaneous and average rate of change (gradients of tangents and chords) in numerical, algebraic and graphical contexts.}
Set up, solve and interpret the answers in growth and decay problems, including compound interest {and work with general iterative processes}.
Set up, solve and interpret the answers in growth and decay problems, including compound interest {and work with general iterative processes}.
Geometry and measures
Interpret and use fractional {and negative} scale factors for enlargements.
Interpret and use fractional {and negative} scale factors for enlargements.
{Describe the changes and invariance achieved by combinations of rotations, reflections and translations.}
{Describe the changes and invariance achieved by combinations of rotations, reflections and translations.}
Identify and apply circle definitions and properties, including: centre, radius, chord, diameter, circumference, tangent, arc, sector and segment.
Parts of a circle
Area and circumference
Arcs, chords, sectors and tangents
Identify and apply circle definitions and properties, including: centre, radius, chord, diameter, circumference, tangent, arc, sector and segment.
Parts of a circle
Area and circumference
Arcs, chords, sectors and tangents
{Apply and prove the standard circle theorems concerning angles, radii, tangents and chords, and use them to prove related results.}
Angles, tangents and chords
Inscribed angles
{Apply and prove the standard circle theorems concerning angles, radii, tangents and chords, and use them to prove related results.}
Angles, tangents and chords
Inscribed angles
Construct and interpret plans and elevations of 3D shapes.
Construct and interpret plans and elevations of 3D shapes.
Interpret and use bearings.
Interpret and use bearings.
Calculate arc lengths, angles and areas of sectors of circles.
Calculate arc lengths, angles and areas of sectors of circles.
Calculate surface areas and volumes of spheres, pyramids, cones and composite solids.
Surface area
Volume
Calculate surface areas and volumes of spheres, pyramids, cones and composite solids.
Surface area
Volume
Apply the concepts of congruence and similarity, including the relationships between lengths, {areas and volumes} in similar figures.
Congruent figures
Congruent triangles
Similar figures
Similar solids
Apply the concepts of congruence and similarity, including the relationships between lengths, {areas and volumes} in similar figures.
Congruent figures
Congruent triangles
Similar figures
Similar solids
Apply Pythagoras' Theorem and trigonometric ratios to find angles and lengths in right-angled triangles {and, where possible, general triangles} in two {and three} dimensional figures.
Pythagoras' theorem
Trigonometric ratios
Apply Pythagoras' Theorem and trigonometric ratios to find angles and lengths in right-angled triangles {and, where possible, general triangles} in two {and three} dimensional figures.
Pythagoras' theorem
Trigonometric ratios
Know the exact values of sin θ and cos θ for θ = 0°, 30°, 45°, 60° and 90°; know the exact value of tan θ for θ = 0°, 30°, 45° and 60°.
Know the exact values of sin θ and cos θ for θ = 0°, 30°, 45°, 60° and 90°; know the exact value of tan θ for θ = 0°, 30°, 45° and 60°.
{Know and apply the sine rule, a/sin A = b/sin B = c/sin C, and cosine rule, a2 = b2 + c2 - 2bc cos A, to find unknown lengths and angles.}
{Know and apply the sine rule, a/sin A = b/sin B = c/sin C, and cosine rule, a2 = b2 + c2 - 2bc cos A, to find unknown lengths and angles.}
{Know and apply Area = 1/2 ab sin C to calculate the area, sides or angles of any triangle.}
{Know and apply Area = 1/2 ab sin C to calculate the area, sides or angles of any triangle.}
Describe translations as 2D vectors.
Describe translations as 2D vectors.
Apply addition and subtraction of vectors, multiplication of vectors by a scalar, and diagrammatic and column representations of vectors; {use vectors to construct geometric arguments and proofs}.
Addition and subtraction
Scalar multiplication
Apply addition and subtraction of vectors, multiplication of vectors by a scalar, and diagrammatic and column representations of vectors; {use vectors to construct geometric arguments and proofs}.
Addition and subtraction
Scalar multiplication
Probability
Apply the property that the probabilities of an exhaustive set of mutually exclusive events sum to one.
Apply the property that the probabilities of an exhaustive set of mutually exclusive events sum to one.
Use a probability model to predict the outcomes of future experiments; understand that empirical unbiased samples tend towards theoretical probability distributions, with increasing sample size.
Use a probability model to predict the outcomes of future experiments; understand that empirical unbiased samples tend towards theoretical probability distributions, with increasing sample size.
Calculate the probability of independent and dependent combined events, including using tree diagrams and other representations, and know the underlying assumptions.
Calculate the probability of independent and dependent combined events, including using tree diagrams and other representations, and know the underlying assumptions.
{Calculate and interpret conditional probabilities through representation using expected frequencies with two-way tables, tree diagrams and Venn diagrams.}
{Calculate and interpret conditional probabilities through representation using expected frequencies with two-way tables, tree diagrams and Venn diagrams.}
Statistics
Infer properties of populations or distributions from a sample, whilst knowing the limitations of sampling.
Infer properties of populations or distributions from a sample, whilst knowing the limitations of sampling.
Interpret and construct tables and line graphs for time series data.
Interpret and construct tables and line graphs for time series data.
{Construct and interpret diagrams for grouped discrete data and continuous data, ie, histograms with equal and unequal class intervals and cumulative frequency graphs, and know their appropriate use.}
{Construct and interpret diagrams for grouped discrete data and continuous data, ie, histograms with equal and unequal class intervals and cumulative frequency graphs, and know their appropriate use.}
Interpret, analyse and compare the distributions of data sets from univariate empirical distributions through:
Interpret, analyse and compare the distributions of data sets from univariate empirical distributions through:
appropriate graphical representation involving discrete, continuous and grouped data, {including box plots}
appropriate graphical representation involving discrete, continuous and grouped data, {including box plots}
appropriate measures of central tendency (including modal class) and spread {including quartiles and inter-quartile range}
Measures of central tendency and spread
Data displays
appropriate measures of central tendency (including modal class) and spread {including quartiles and inter-quartile range}
Measures of central tendency and spread
Data displays
Apply statistics to describe a population.
Apply statistics to describe a population.
Use and interpret scatter graphs of bivariate data; recognise correlation and know that it does not indicate causation; draw estimated lines of best fit; make predictions; interpolate and extrapolate apparent trends whilst knowing the dangers of so doing.
Use and interpret scatter graphs of bivariate data; recognise correlation and know that it does not indicate causation; draw estimated lines of best fit; make predictions; interpolate and extrapolate apparent trends whilst knowing the dangers of so doing.