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Key Stage 4 Mathematics

At KS4 we follow topics sequentially:


N1 Integers

N1.1 Classifying numbers

N1.2 Calculating with integers

N1.3 Multiples, factors and primes

N1.4 Prime factor decomposition

N1.5 LCM and HCF


N2 Powers, roots and standard form


N2.1 Powers and roots

N2.2 Index laws

N2.3 Negative indices and reciprocals

N2.4 Fractional indices

N2.5 Surds

N2.6 Standard form


N3 Fractions


N3.1 Equivalent fractions

N3.2 Finding fractions of quantities

N3.3 Comparing and ordering fractions

N3.4 Adding and subtracting fractions

N3.5 Multiplying and dividing fractions


N4 Decimals and rounding


N4.1 Decimals and place value

N4.2 Terminating and recurring decimals

N4.3 Calculating with decimals

N4.4 Rounding

N4.5 Upper and lower bounds


N5 Percentages


N5.1 Fractions, decimals and percentages

N5.2 Percentages of quantities

N5.3 Finding a percentage change

N5.4 Increasing and decreasing by a percentage

N5.5 Reverse percentages

N5.6 Compound percentages


N6 Ratio and proportion


N6.1 Ratio

N6.2 Dividing in a given ratio

N6.3 Direct proportion

N6.4 Inverse proportion

N6.5 Proportionality to powers

N6.6 Graphs of proportional relationships






A1 Algebraic manipulation


A1.1 Using index laws

A1.2 Multiplying out brackets

A1.3 Factorization

A1.4 Factorizing quadratic expressions

A1.5 Algebraic fractions


A2 Linear equations


A2.1 Equations, formulae and identities

A2.2 Balancing equations

A2.3 Equations with brackets 

A2.4 Equations with fractions

A2.5 Using equations to solve problems


A3 Formulae


A3.1 Substituting into formulae

A3.2 Problems that lead to equations to solve

A3.3 Changing the subject of a formula

A3.4 Manipulating more difficult formulae

A3.5 Generating formulae


A4 Inequalities


A4.1 Representing inequalities on number lines

A4.2 Solving linear inequalities

A4.3 Inequalities and regions

A4.4 Inequalities in two variables

A4.5 Quadratic inequalities


A5 Simultaneous equations


A5.1 Solving simultaneous equations graphically

A5.2 The elimination method

A5.3 The substitution method

A5.4 Simultaneous linear and quadratic equations

A5.5 Problems leading to simultaneous equations


A6 Quadratic equations


A6.1 Solving quadratic equations by factorization

A6.2 Completing the square

A6.3 Using the quadratic formula

A6.4 Equations involving algebraic fractions

A6.5 Problems leading to quadratic equations


A7 Sequences


A7.1 Generating sequences from rules

A7.2 Linear sequences

A7.3 Quadratic sequences

A7.4 Geometric sequences

A7.5 Other types of sequence


A8 Linear and real-life graphs


A8.1 Linear graphs

A8.2 Gradients and intercepts

A8.3 Parallel and perpendicular lines

A8.4 Interpreting real-life graphs

A8.5 Distance-time graphs

A8.6 Speed-time graphs


A9 Graphs of non-linear functions


A9.1 Plotting curved graphs

A9.2 Graphs of important non-linear functions

A9.3 Using graphs to solve equations

A9.4 Solving equations by trial and improvement

A9.5 Function notation

A9.6 Transforming graphs





S1 Lines, angles and polygons


S1.1 Parallel lines and angles

S1.2 Triangles

S1.3 Quadrilaterals

S1.4 Angles in polygons

S1.5 Congruence

S1.6 Similarity


S2 Pythagoras’ theorem


S2.1 Introducing Pythagoras’ theorem

S2.2 Identifying right-angled triangles

S2.3 Pythagorean triples

S2.4 Finding unknown lengths

S2.5 Applying Pythagoras’ theorem in 2-D

S2.6 Applying Pythagoras’ theorem in 3-D


S3 Trigonometry


S3.1 Right-angled triangles

S3.2 The three trigonometric ratios

S3.3 Finding side lengths

S3.4 Finding angles

S3.5 Angles of elevation and depression

S3.6 Trigonometry in 3-D


S4  Further trigonometry


S4.1 Sin, cos and tan of any angle

S4.2 Sin, cos and tan of 30°, 45° and 60°

S4.3 Graphs of trigonometric functions

S3.4 Area of a triangle using ½ab sin C

S3.5 The sine rule

S4.6 The cosine rule


S5 Circles


S5.1 Naming circle parts

S5.2 Angles in a circle

S5.3 Tangents and chords

S5.4 Circumference and arc length

S5.5 Areas of circles and sectors


S6 Transformations


S6.1 Symmetry

S6.2 Reflection

S6.3 Rotation

S6.4 Translation

S6.5 Enlargement

S6.6 Combining transformations


S7 Vectors


S7.1 Vector notation

S7.2 Multiplying vectors by scalars

S7.3 Adding and subtracting vectors

S7.4 Vector arithmetic

S7.5 Finding the magnitude of a vector

S7.6 Using vectors to solve problems


S8 Measures


S8.1 Converting units

S8.2 Accuracy in measurement

S8.3 Calculations involving bounds

S8.4 Compound measures 

S8.5 Bearings

S9 Construction and loci


S9.1 Constructing triangles

S9.2 Geometrical constructions

S9.3 Imagining paths and regions

S9.4 Loci

S9.5 Combining loci


S10 Length, area and volume


S10.1 Dimensions of length, area and volume

S10.2 Polygons

S10.3 Cubes and cuboids

S10.4 Prisms and pyramids

S10.5 Cylinders, cones and spheres

S10.6 Lengths, areas and volumes of similar shapes






D1 Planning and collecting data

D1.1 Specifying the problem and planning

D1.2 Types of data

D1.3 Collecting data

D1.4 Sampling

D1.5 The stages of research


D2 Averages and range

D2.1 The mode

D2.2 The mean

D2.3 Calculating the mean from frequency tables

D2.4 The median

D2.5 Comparing data


D3 Presenting data


D3.1 Bar charts

D3.2 Line graphs

D3.3 Pie charts

D3.4 Stem-and-leaf diagrams

D3.5 Scatter graphs


D4 Moving averages and cumulative frequency


D4.1 Moving averages

D4.2 Plotting moving averages

D4.3 Cumulative frequency

D4.4 Using cumulative frequency graphs

D4.5 Box-and-whisker diagrams


D5 Frequency diagrams for continuous data


D5.1 Grouping continuous data

D5.2 Frequency diagrams

D5.3 Frequency polygons

D5.4 Histograms

D5.5 Frequency density


D6 Probability

D6.1 The language of probability

D6.2 The probabilities of single events

D6.3 The probabilities of combined events

D6.4 Tree diagrams

D6.5 Experimental probability