📐 Mathematics – Key Stage 4

Pupils should be taught to:

• MA-KS4-N-01apply systematic listing strategies, {including use of the product rule for counting}
• MA-KS4-N-02{estimate powers and roots of any given positive number}
• MA-KS4-N-03calculate with roots, and with integer {and fractional} indices
• MA-KS4-N-04calculate 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}
• MA-KS4-N-05calculate with numbers in standard form A × 10n, where 1 ≤⃒ A < 10 and n is an integer
• MA-KS4-N-06{change recurring decimals into their corresponding fractions and vice versa}
• MA-KS4-N-07identify and work with fractions in ratio problems
• MA-KS4-N-08apply and interpret limits of accuracy when rounding or truncating, {including upper and lower bounds}

Pupils should be taught to:

• MA-KS4-A-01simplify 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 2 squares; {factorising quadratic expressions of the form ax2 + bx + c}; simplifying expressions involving sums, products and powers, including the laws of indices
• MA-KS4-A-02know 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}
• MA-KS4-A-03where appropriate, interpret simple expressions as functions with inputs and outputs; {interpret the reverse process as the ‘inverse function’; interpret the succession of 2 functions as a ‘composite function’}
• MA-KS4-A-04use the form y = mx + c to identify parallel {and perpendicular} lines; find the equation of the line through 2 given points, or through 1 point with a given gradient
• MA-KS4-A-05identify and interpret roots, intercepts and turning points of quadratic functions graphically; deduce roots algebraically {and turning points by completing the square}
• MA-KS4-A-06recognise, sketch and interpret graphs of linear functions, quadratic functions, simple cubic functions, the reciprocal function y = [Fraction: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}
• MA-KS4-A-07{sketch translations and reflections of the graph of a given function}
• MA-KS4-A-08plot 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
• MA-KS4-A-09{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}
• MA-KS4-A-10{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}
• MA-KS4-A-11solve 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
• MA-KS4-A-12solve 2 simultaneous equations in 2 variables (linear/linear {or linear/quadratic}) algebraically; find approximate solutions using a graph
• MA-KS4-A-13{find approximate solutions to equations numerically using iteration}
• MA-KS4-A-14translate simple situations or procedures into algebraic expressions or formulae; derive an equation (or 2 simultaneous equations), solve the equation(s) and interpret the solution
• MA-KS4-A-15solve linear inequalities in 1 {or 2} variable {s}, {and quadratic inequalities in 1 variable}; represent the solution set on a number line, {using set notation and on a graph}
• MA-KS4-A-16recognise 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}
• MA-KS4-A-17deduce expressions to calculate the nth term of linear {and quadratic} sequences.

Pupils should be taught to:

• MA-KS4-R-01compare lengths, areas and volumes using ratio notation and/or scale factors; make links to similarity (including trigonometric ratios)
• MA-KS4-R-02convert between related compound units (speed, rates of pay, prices, density, pressure) in numerical and algebraic contexts
• MA-KS4-R-03understand that X is inversely proportional to Y is equivalent to X is proportional to [Fraction:1/Y]; {construct and} interpret equations that describe direct and inverse proportion
• MA-KS4-R-04interpret the gradient of a straight line graph as a rate of change; recognise and interpret graphs that illustrate direct and inverse proportion
• MA-KS4-R-05{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}
• MA-KS4-R-06set up, solve and interpret the answers in growth and decay problems, including compound interest {and work with general iterative processes}

Pupils should be taught to:

• MA-KS4-GM-01interpret and use fractional {and negative} scale factors for enlargements
• MA-KS4-GM-02{describe the changes and invariance achieved by combinations of rotations, reflections and translations}
• MA-KS4-GM-03identify and apply circle definitions and properties, including: centre, radius, chord, diameter, circumference, tangent, arc, sector and segment
• MA-KS4-GM-04{apply and prove the standard circle theorems concerning angles, radii, tangents and chords, and use them to prove related results}
• MA-KS4-GM-05construct and interpret plans and elevations of 3D shapes
• MA-KS4-GM-06interpret and use bearings
• MA-KS4-GM-07calculate arc lengths, angles and areas of sectors of circles
• MA-KS4-GM-08calculate surface areas and volumes of spheres, pyramids, cones and composite solids
• MA-KS4-GM-09apply the concepts of congruence and similarity, including the relationships between lengths, {areas and volumes} in similar figures
• MA-KS4-GM-10apply Pythagoras’ Theorem and trigonometric ratios to find angles and lengths in right-angled triangles {and, where possible, general triangles} in 2 {and 3} dimensional figures
• MA-KS4-GM-11know 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°, 60°
• MA-KS4-GM-12{know and apply the sine rule, [Fraction:a/sinA] = [Fraction:b/sinB] = [Fraction:c/sinC] , and cosine rule, a2 = b2 + c2 - 2bc cos A, to find unknown lengths and angles}
• MA-KS4-GM-13{know and apply Area = [Fraction:1/2] ab sin C to calculate the area, sides or angles of any triangle}
• MA-KS4-GM-14describe translations as 2D vectors
• MA-KS4-GM-15apply 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}

Pupils should be taught to:

• MA-KS4-P-01apply the property that the probabilities of an exhaustive set of mutually exclusive events sum to 1
• MA-KS4-P-02use a probability model to predict the outcomes of future experiments; understand that empirical unbiased samples tend towards theoretical probability distributions, with increasing sample size
• MA-KS4-P-03calculate the probability of independent and dependent combined events, including using tree diagrams and other representations, and know the underlying assumptions
• MA-KS4-P-04{calculate and interpret conditional probabilities through representation using expected frequencies with two-way tables, tree diagrams and Venn diagrams}

Pupils should be taught to:

• MA-KS4-S-01infer properties of populations or distributions from a sample, whilst knowing the limitations of sampling
• MA-KS4-S-02interpret and construct tables and line graphs for time series data
• MA-KS4-S-03{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}
• MA-KS4-S-04interpret, 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 measures of central tendency (including modal class) and spread {including quartiles and inter-quartile range}
• MA-KS4-S-05apply statistics to describe a population
• MA-KS4-S-06use 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.