﻿ KS4 List
Number Algebra Ratio Proportion & Rates of Change Geometry & Measure Probability Statistics

KS4 - Number

Apply systematic listing strategies, {including use of the product rule for counting} {estimate powers and roots of any given positive number}

Calculate with roots, with integer {and 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}

Change recurring decimals into their corresponding fractions and vice versa

Identify and work with fractions in ratio problems

Apply and interpret limits of accuracy when rounding or truncating, {including upper and lower bounds}

Key Stage 4 Topics

The topics required for the GCSE are those listed in the KS3 and KS4 Programmes of Study. Below are the KS4 topics grouped under the titles NUMBER, ALGEBRA, RATIO PROPORTION & RATES OF CHANGE, GEOMETRY & MEASURE, PROBABILITY, AND STATISTICS. These may be viewed by scrolling down the page. To access the resources available for a topic, click on the required topic’s descriptor below or use the above buttons to locate and browse the required group of topics.

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KS4 - Algebra

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 2 squares; {factorising quadratic expressions of the form ax2 + bx + c}
• simplifying expressions involving sums, products and powers, including the laws of indices

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}

Where 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’}

Use 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

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 = 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}

{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

{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}

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

Solve 2 simultaneous equations in 2 variables (linear/linear {or linear/quadratic}) algebraically; find approximate solutions using a graph

{Find approximate solutions to equations numerically using iteration}

Translate simple situations or procedures into algebraic expressions or formulae; derive an equation (or 2 simultaneous equations), solve the equation(s) and interpret the solution

Solve 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}

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}

Deduce expressions to calculate the nth term of linear {and quadratic} sequences.

KS4 - 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)

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

Interpret the gradient of a straight line graph as a rate of change; recognise and interpret graphs that illustrate direct and inverse proportion

{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}

KS4 - Geometry and Measure

Interpret and use fractional {and negative} scale factors for enlargements

{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

{Apply and prove the standard circle theorems concerning angles, radii, tangents and chords, and use them to prove related results}

Construct and interpret plans and elevations of 3D shapes

Interpret and use bearings

Calculate arc lengths, angles and areas of sectors of circles

Calculate surface areas and volumes of spheres, pyramids, cones and composite solids

Apply the concepts of congruence and similarity, including the relationships between lengths, {areas and volumes} in similar figures

Apply 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

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°, 60°

{Know and apply the sine rule, a/sinA = b/sinB = c/sinC , and cosine rule, a2 = b2 + c2 - 2bc cos A, to find unknown lengths and angles}

{Know and apply Area = ab sin C to calculate the area, sides or angles of any triangle}

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}

KS4 - Probability

Apply the property that the probabilities of an exhaustive set of mutually exclusive events sum to 1

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 and interpret conditional probabilities through representation using expected frequencies with two-way tables, tree diagrams and Venn diagrams}

KS4 - Statistics

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

{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:

• 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}

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.