GCSE Mathematics

• Algebraic expressions: expanding and factorising

  ≈ 3.5%1 h 15 min

  Simplifying expressions, expanding single, double and triple brackets, and factorising, including quadratics with $a \ne 1$ and the difference of two squares.

  • Simplify expressions by collecting like terms and using index laws, and know the difference between an expression, equation, formula and identity
  • Expand products of two or three binomials, e.g. $(x+2)(x-3)(2x+1)$, and simplify the result (Higher for triple brackets)
  • Factorise by taking out common factors, and factorise quadratics $x^2 + bx + c$ and $ax^2 + bx + c$
  • Factorise using the difference of two squares, e.g. $9x^2 - 25y^2$, and spot it inside harder expressions
  • Use and prove identities with the $\equiv$ sign, e.g. find $a$ and $b$ such that $3(x+a) + b \equiv 3x + 11$

• Formulae, rearranging and algebraic fractions

  ≈ 2.5%1 h 10 min

  Substituting into formulae, changing the subject (including when it appears twice), and simplifying and solving with algebraic fractions.

  • Substitute positive, negative and fractional values into formulae and expressions, including kinematics formulae such as $v^2 = u^2 + 2as$
  • Rearrange formulae to change the subject, including cases with powers, roots and fractions
  • Rearrange when the new subject appears twice by collecting terms and factorising, e.g. make $x$ the subject of $y = \frac{x + 3}{x - 2}$ (Higher)
  • Simplify algebraic fractions by factorising, and add, subtract, multiply and divide them (Higher)
  • Solve equations involving algebraic fractions that lead to linear or quadratic equations (Higher)

• Linear equations and inequalities

  ≈ 2.5%1 h

  Solving linear equations (including with brackets, fractions and unknowns on both sides), forming equations from contexts, and solving and representing linear inequalities.

  • Solve linear equations with brackets, fractions and unknowns on both sides
  • Form and solve equations from geometric and worded contexts (angles, perimeters, ages, costs), and interpret the solution
  • Solve linear inequalities, including double inequalities, and show solutions on a number line, as integer lists or in set notation
  • Represent inequalities in two variables as regions on a graph, and identify or shade the region satisfying several inequalities (Higher)

• Quadratic equations and inequalities

  ≈ 3.5%1 h 30 min

  Solving quadratics by factorising, the formula and completing the square, finding turning points, and solving quadratic inequalities.

  • Solve quadratic equations by factorising, including $ax^2 + bx + c = 0$ with $a \ne 1$
  • Solve quadratics using the quadratic formula, giving answers to a stated accuracy or in exact surd form
  • Complete the square, $x^2 + bx + c = (x + p)^2 + q$, and use it to solve equations and find the turning point of a quadratic graph (Higher)
  • Solve quadratic inequalities such as $x^2 - x - 12 < 0$ using a sketch and represent the solution set (Higher)
  • Form and solve quadratic equations from area, number and geometric contexts, and reject solutions that do not fit the context

• Simultaneous equations

  ≈ 2.5%1 h 10 min

  Solving two linear simultaneous equations algebraically and graphically, and a linear with a quadratic (including a circle) at Higher tier.

  • Solve two linear simultaneous equations by elimination or substitution, including when both equations must be scaled
  • Form and solve simultaneous equations from contexts (e.g. costs of tickets) and interpret the solution
  • Find approximate solutions graphically as the intersection of two graphs
  • Solve one linear and one quadratic equation algebraically, including a line and a circle $x^2 + y^2 = r^2$, and interpret the solutions as intersection points (Higher)

• Sequences and series

  ≈ 2.5%1 h

  Term-to-term and position-to-term rules, linear, quadratic and geometric sequences, and arithmetic series where the specification includes them.

  • Generate terms from term-to-term and position-to-term rules, and recognise triangular, square, cube, Fibonacci-type and simple geometric sequences
  • Find and use the $n$th term of a linear sequence, and decide whether a given number is a term
  • Find the $n$th term of a quadratic sequence using second differences (Higher)
  • Work with geometric sequences with a common ratio, including surd ratios such as $\sqrt{2}$ (Higher)
  • Use the formulae for the $n$th term and the sum of the first $n$ terms of an arithmetic series (Edexcel International GCSE content)

• Straight-line graphs and coordinate geometry

  ≈ 2.5%1 h

  Coordinates, gradients, $y = mx + c$, parallel and perpendicular lines, midpoints and lengths of line segments.

  • Plot and interpret straight-line graphs, and identify the gradient and $y$-intercept from $y = mx + c$ or after rearranging
  • Find the equation of a line through two points, or through one point with a given gradient
  • Use the gradient conditions for parallel lines ($m_1 = m_2$) and perpendicular lines ($m_1 m_2 = -1$) to find equations (Higher)
  • Find the midpoint and length of a line segment from coordinates
  • Interpret gradients and intercepts of real-life linear graphs (e.g. conversion graphs, taxi fares) as rates and fixed charges

• Quadratic, cubic, reciprocal, exponential, trigonometric and circle graphs

  ≈ 2.5%1 h 15 min

  Recognising, plotting, sketching and interpreting non-linear graphs, including circles centred at the origin and tangents to circles.

  • Plot and recognise the shapes of quadratic, cubic, reciprocal ($y = \frac{k}{x}$) and exponential ($y = k^x$) graphs
  • Sketch quadratic graphs showing roots, $y$-intercept and turning point, and use graphs to solve equations approximately
  • Recognise and sketch $y = \sin x$, $y = \cos x$ and $y = \tan x$ for $0^\circ \le x \le 360^\circ$ and use symmetry to find solutions (Higher)
  • Recognise the equation of a circle centred at the origin, $x^2 + y^2 = r^2$, and find the equation of a tangent at a given point (Higher)
  • Use graphs to find approximate solutions of equations such as $x^3 - 3x = 1$ by drawing a suitable line

• Graph transformations, gradients of curves and areas under graphs

  ≈ 2%1 h

  Translations and reflections of $y = f(x)$, estimating gradients and areas under curves, and interpreting distance-time and velocity-time graphs.

  • Identify and sketch the translations $y = f(x) + a$ and $y = f(x + a)$ and the reflections $y = -f(x)$ and $y = f(-x)$, and state the effect on key points (Higher)
  • Estimate the gradient of a curve at a point by drawing a tangent, and interpret it as an instantaneous rate of change (Higher)
  • Estimate the area under a curve using trapezia or strips, interpret it (e.g. distance travelled), and say whether it is an over- or underestimate (Higher)
  • Interpret distance-time and velocity-time graphs: speed as gradient, acceleration as gradient, distance as area

• Functions and iteration

  ≈ 2%1 h

  Function notation, composite and inverse functions, domain and range, and finding approximate roots by iteration.

  • Use function notation $f(x)$: evaluate $f(a)$ and solve $f(x) = k$ (Higher)
  • Find and interpret composite functions $fg(x)$ and inverse functions $f^{-1}(x)$ (Higher)
  • State the domain and range of simple functions and values that must be excluded from a domain (International GCSE and Cambridge Extended content)
  • Show that an equation has a root between two values by a change of sign (England GCSE boards, Higher)
  • Rearrange an equation into an iterative formula $x_{n+1} = g(x_n)$ and use iteration to find a root to a given accuracy (England GCSE boards, Higher)

• Algebraic argument and proof

  ≈ 1.5%45 min

  Using algebra to prove statements, disprove them by counterexample, and complete “show that” questions rigorously.

  • Represent even, odd and consecutive integers and multiples algebraically ($2n$, $2n + 1$, $n$, $n + 1$)
  • Construct algebraic proofs of number facts, e.g. prove that the difference between the squares of two consecutive odd numbers is a multiple of 8 (Higher)
  • Disprove a statement with a counterexample and explain why an argument is invalid
  • Present “show that” answers as a clear chain of steps that reaches the given result with no gaps

• Differentiation (International Specifications)

  ≈ 1.5%1 h 10 min

  Edexcel International GCSE Mathematics A onlyEdexcel International GCSE Mathematics B onlyCambridge IGCSE Mathematics onlyCambridge IGCSE (9-1) Mathematics only

  Differentiating polynomials (and simple negative powers), gradients of curves, tangents, turning points and kinematics. This is required for Edexcel 4MA1 Higher, 4MB1 and Cambridge 0580/0980 Extended.

  • Differentiate $ax^n$ for integer $n$ and sums of such terms, e.g. $y = 2x^3 - \frac{4}{x}$
  • Find the gradient of a curve at a point and the equation of the tangent at that point
  • Find turning points by solving $\frac{dy}{dx} = 0$ and decide whether each is a maximum or a minimum
  • Apply differentiation to kinematics: velocity $v = \frac{ds}{dt}$ and acceleration $a = \frac{dv}{dt}$ (Edexcel International GCSE)

• Functions, asymptotes, logarithms and GDC graph work (0607)

  ≈ 1%1 h 10 min

  Cambridge IGCSE International Mathematics only

  Cambridge 0607 function content: sketching and analysing graphs on a graphic display calculator, asymptotes, transformations of functions, and the logarithmic function as the inverse of the exponential.

  • Use a graphic display calculator to sketch graphs, produce tables of values, and find zeros, local maxima and minima, intersections and the vertex of a quadratic
  • Identify vertical and horizontal asymptotes of functions such as $f(x) = \frac{1}{x-2} + 3$ and $f(x) = \tan x$ from their graphs
  • Describe and identify the translations $y = f(x) + k$ and $y = f(x + k)$, and find a quadratic function from its vertex, its $x$-intercepts or a given point, using $y = a(x - h)^2 + k$
  • Use the logarithmic function as the inverse of the exponential function ($y = a^x$ is equivalent to $x = \log_a y$) and solve $a^x = b$ as $x = \frac{\log b}{\log a}$, e.g. in compound interest and growth problems (logs are base 10 unless stated)

About 14 h 25 min of study, lessons and core practice

• Ratio and its applications

  ≈ 3.5%1 h

  Simplifying ratios, sharing in a ratio, linking ratios with fractions, combining ratios, and ratio problems with algebra.

  • Simplify ratios (including with different units) and write them in the form $1 : n$
  • Share a quantity in a given ratio and solve problems where the difference or one part is known
  • Relate ratios to fractions and to linear functions, e.g. if $a : b = 3 : 5$ then $a = \frac{3}{5}b$
  • Combine two ratios into a three-part ratio, and solve ratio problems that need algebra, e.g. ratios that change after an amount is added (Higher)
  • Use ratio in scale drawings, maps, recipes and best-buy comparisons

• Percentages, growth and decay, and financial mathematics

  ≈ 4%1 h 20 min

  Percentage change, reverse percentages, simple and compound interest, depreciation, repeated percentage change, and everyday financial numeracy.

  • Use decimal multipliers for percentage increase and decrease, and calculate percentage change and percentage profit or loss
  • Solve reverse-percentage problems to find the original amount, e.g. a price after a 15% discount is £68
  • Calculate simple interest, compound interest and depreciation, and use $A = P(1 + \frac{r}{100})^n$ for repeated percentage change, including finding $n$ by trial or iteration
  • Model exponential growth and decay (populations, radioactive decay, bacteria) and interpret the multiplier
  • Solve financial problems in context: wages, overtime, tax and National Insurance deductions, bills, VAT, budgets, loans and savings, and currency exchange (a major focus of WJEC Unit 1)

• Direct and inverse proportion

  ≈ 3%1 h

  Recognising proportional relationships, forming equations with a constant of proportionality, and interpreting proportion graphs.

  • Solve direct and inverse proportion problems using unitary and multiplier methods (e.g. recipes, workers and time)
  • Form and use equations of the form $y = kx^n$ and $y = \frac{k}{x^n}$, including square, cube and square-root relationships (Higher)
  • Recognise and sketch graphs of direct and inverse proportion, and read the constant of proportionality from them
  • Explain the effect on $y$ of scaling $x$, e.g. if $y \propto x^2$ and $x$ is tripled, $y$ is multiplied by 9

• Compound measures, units and time

  ≈ 3%1 h

  Speed, density, pressure and other rates, converting units (including area and volume units), and time and money calculations.

  • Use and rearrange speed = distance ÷ time, density = mass ÷ volume and pressure = force ÷ area, including average speed over multi-stage journeys
  • Convert between metric units, including area units (e.g. $\text{cm}^2$ to $\text{m}^2$) and volume units, and between compound units (e.g. km/h to m/s)
  • Solve rate problems such as flow rates, fuel consumption, unit pricing and exchange rates
  • Calculate with time (12- and 24-hour clock, timetables, time differences) and money

About 4 h 20 min of study, lessons and core practice

• Angles, parallel lines and polygons

  ≈ 2.5%1 h

  Angle facts, angles in parallel lines, properties of triangles and quadrilaterals, interior and exterior angles of polygons, and geometric reasoning.

  • Apply angle facts (on a line, at a point, vertically opposite, in a triangle and a quadrilateral) and give a reason for each step
  • Use alternate, corresponding and co-interior angles on parallel lines, naming the correct reason
  • Use the properties of special triangles and quadrilaterals, and the symmetry of 2D shapes
  • Calculate interior and exterior angles of regular and irregular polygons, and find the number of sides from an angle
  • Write multi-step geometric proofs with full reasons, e.g. show that a triangle formed in a diagram is isosceles

• Constructions, loci, bearings and scale drawings

  ≈ 2%1 h

  Ruler-and-compass constructions, loci and regions, three-figure bearings, and scale drawings and maps.

  • Construct perpendicular bisectors, angle bisectors, perpendiculars from a point and triangles from given information using ruler and compasses, leaving arcs visible
  • Draw loci (a fixed distance from a point or line, equidistant from two points or lines) and shade regions that satisfy several conditions
  • Measure and calculate three-figure bearings, including back bearings found with parallel-line angle facts
  • Use scales on maps and scale drawings, including ratios such as 1 : 25 000

• Transformations

  ≈ 2%55 min

  Reflections, rotations, translations and enlargements (including fractional and negative scale factors), combined transformations and invariant points.

  • Reflect in lines such as $x = a$, $y = b$, $y = x$ and $y = -x$, and rotate about a given centre through a given angle and direction
  • Translate using column vectors, and fully describe a single transformation that maps one shape to another
  • Enlarge shapes from a centre with positive, fractional and negative scale factors (Higher for negative), and find the centre and scale factor
  • Describe combined transformations as a single transformation and identify invariant points (Higher)

• Congruence and similarity

  ≈ 2.5%1 h

  Congruence criteria and proof, similar shapes, and the scale factors for length, area and volume.

  • Use the congruence conditions SSS, SAS, ASA and RHS to prove that two triangles are congruent, with a reason for each statement (Higher)
  • Find missing lengths in similar shapes, including overlapping and nested similar triangles
  • Use the scale factors $k$, $k^2$ and $k^3$ for length, area and volume of similar shapes and solids (Higher)
  • Solve problems that mix similarity with ratio, e.g. find the mass or cost of a similar solid

• Perimeter, area, circles, arcs and sectors

  ≈ 2.5%1 h

  Perimeter and area of 2D shapes (including compound shapes), circumference and area of circles, and arc lengths and sector areas.

  • Calculate the perimeter and area of rectangles, triangles, parallelograms, trapezia and compound shapes
  • Calculate the circumference and area of circles and semicircles, and give answers in terms of $\pi$ when asked
  • Calculate arc lengths and sector areas using $\frac{\theta}{360}$ of the circle, and reverse the process to find an angle or radius (Higher)
  • Solve multi-step area problems, e.g. costs of materials, shaded regions and segments

• Volume, surface area and 3D shapes

  ≈ 2.5%1 h

  Properties of 3D shapes, plans and elevations, and the volume and surface area of prisms, cylinders, pyramids, cones, spheres and frustums.

  • Identify faces, edges and vertices, and draw and interpret plans and elevations of 3D shapes
  • Calculate the volume and surface area of cuboids, prisms and cylinders
  • Calculate the volume and surface area of pyramids, cones and spheres (and hemispheres) using given formulae, and find the volume of a frustum (Higher)
  • Solve problems involving capacity, density, rates of filling and changes of shape, e.g. melting a sphere into cylinders

• Pythagoras’ theorem and right-angled trigonometry

  ≈ 3%1 h 10 min

  Pythagoras’ theorem in 2D and 3D, sine, cosine and tangent in right-angled triangles, exact trigonometric values, and angles of elevation and depression.

  • Use Pythagoras’ theorem to find missing sides in right-angled triangles, and test whether a triangle is right-angled
  • Use $\sin$, $\cos$ and $\tan$ to find sides and angles in right-angled triangles, including angles of elevation and depression and bearings
  • Know the exact values of $\sin\theta$ and $\cos\theta$ for $\theta = 0^\circ, 30^\circ, 45^\circ, 60^\circ, 90^\circ$ and $\tan\theta$ for $\theta = 0^\circ, 30^\circ, 45^\circ, 60^\circ$, and use them without a calculator
  • Apply Pythagoras and trigonometry in 3D, e.g. the angle between a line and a plane in a cuboid or pyramid (Higher)

• Sine rule, cosine rule and area of a triangle

  ≈ 2.5%1 h 10 min

  Solving any triangle with the sine and cosine rules, the area formula $\frac{1}{2}ab\sin C$, and multi-step problems with bearings.

  • Use the sine rule to find a missing side or angle, and recognise the ambiguous case for an angle (Higher)
  • Use the cosine rule to find a missing side or angle (Higher)
  • Calculate the area of a triangle using $\frac{1}{2}ab\sin C$, and use it in reverse to find a side or angle (Higher)
  • Choose the correct rule in multi-step problems involving bearings, quadrilaterals split into triangles and 3D shapes (Higher)

• Circle theorems

  ≈ 2%1 h

  Circle vocabulary and the standard circle theorems, used to find angles and write geometric proofs.

  • Name the parts of a circle: radius, diameter, chord, tangent, arc, sector and segment
  • Apply the circle theorems: angle at the centre is twice the angle at the circumference, angle in a semicircle is 90°, angles in the same segment are equal, and opposite angles of a cyclic quadrilateral add to 180° (Higher)
  • Apply the tangent theorems (tangent perpendicular to radius, equal tangents from an external point), the perpendicular from the centre bisects a chord, and the alternate segment theorem (Higher)
  • Give the correct theorem as the reason for each step, and prove circle theorems or related results (Higher)

• Vectors

  ≈ 2%1 h

  Column vectors, vector arithmetic, and using vectors to prove geometric results such as collinearity and parallel lines.

  • Add, subtract and multiply column vectors by scalars, and represent them as directed line segments
  • Express vectors in a diagram in terms of $\mathbf{a}$ and $\mathbf{b}$, including points that divide a line in a given ratio (Higher)
  • Use vectors to prove that lines are parallel or that points are collinear (Higher)
  • Find the magnitude of a vector $\sqrt{x^2 + y^2}$ (International GCSE and Cambridge Extended content)

• Matrices and matrix transformations (4MB1)

  ≈ <1%1 h

  Edexcel International GCSE Mathematics B only

  Matrix operations, determinants and inverses of 2 × 2 matrices, and using matrices to represent and combine transformations. This is Edexcel International GCSE Mathematics B content.

  • Add, subtract and multiply matrices of compatible orders and multiply by a scalar, stating when a product is defined
  • Find the determinant and inverse of a 2 × 2 matrix and use the inverse to solve simultaneous equations
  • Use 2 × 2 matrices to represent reflections, rotations and enlargements, find the image of a shape, and identify the transformation a matrix represents
  • Combine transformations by multiplying matrices, and use the determinant to find the area scale factor

About 11 h 15 min of study, lessons and core practice

• Probability basics, relative frequency and sample spaces

  ≈ 2.5%50 min

  The probability scale, mutually exclusive and exhaustive outcomes, relative frequency and expected outcomes, and sample space diagrams.

  • Use the 0–1 probability scale, and use the fact that probabilities of mutually exclusive, exhaustive outcomes sum to 1
  • Calculate relative frequencies from experiments, explain why more trials give better estimates, and judge whether a dice or spinner is biased
  • Calculate expected outcomes, e.g. the expected number of sixes in 300 rolls
  • List outcomes in sample space diagrams, tables and grids, and use them to find probabilities of combined events

• Tree diagrams and combined events

  ≈ 2.5%1 h

  Independent and dependent events, the AND and OR rules, and tree diagrams including selections without replacement.

  • Use the addition rule for mutually exclusive events and the multiplication rule for independent events
  • Draw and use tree diagrams for independent and dependent events, including selections without replacement (Higher for dependent events)
  • Solve problems where the probability is given algebraically or the answer requires forming and solving an equation, e.g. a bag with $n$ red counters (Higher)
  • Find “at least one” probabilities using $1 - P(\text{none})$

• Venn diagrams, set notation and conditional probability

  ≈ 2.5%1 h

  Sets and set notation, Venn diagrams and two-way tables for probabilities, and conditional probability.

  • Use set notation: universal set $\xi$, union $A \cup B$, intersection $A \cap B$, complement $A'$, and $x \in A$
  • Complete and use Venn diagrams (two or three sets) and two-way tables to find probabilities, including with algebraic unknowns
  • Calculate conditional probabilities $P(A \mid B)$ from Venn diagrams, two-way tables and tree diagrams (Higher)
  • Use $n(A)$, subsets and the empty set, and shade regions such as $(A \cup B)' \cap C$ (International GCSE and Cambridge Extended content)

About 2 h 50 min of study, lessons and core practice

• Data collection, sampling, averages and spread

  ≈ 2.5%55 min

  Types of data, sampling methods and bias, mean, median, mode and range from lists and frequency tables (including grouped data), and comparing distributions.

  • Classify data as qualitative, discrete or continuous, understand populations and random samples, and identify sources of bias and ways to reduce it
  • Calculate mean, median, mode and range from lists and frequency tables, and estimate the mean of grouped data using midpoints
  • Identify the modal class and the class containing the median of grouped data, and explain why a grouped mean is only an estimate
  • Compare two distributions using one average and one measure of spread, written in context
  • Solve reverse-mean problems, e.g. find a missing value or the combined mean of two groups

• Statistical diagrams and scatter graphs

  ≈ 2%50 min

  Bar charts, pie charts, frequency polygons, time series, stem-and-leaf diagrams and scatter graphs with correlation and lines of best fit.

  • Construct and interpret bar charts, pie charts, pictograms, frequency polygons, stem-and-leaf diagrams and time-series graphs
  • Spot misleading features of graphs (broken axes, unequal scales, 3D effects) and criticise them
  • Plot scatter graphs, describe correlation, draw a line of best fit and use it for interpolation, with a warning about extrapolation
  • Recognise outliers and explain that correlation does not imply causation; find a linear regression line with a GDC (0607)

• Cumulative frequency, box plots and histograms

  ≈ 3%1 h 10 min

  Cumulative frequency graphs, quartiles and interquartile range, box plots, and histograms with unequal class widths.

  • Draw cumulative frequency graphs (plotting at upper class boundaries) and read off the median, quartiles, interquartile range and percentiles (Higher)
  • Draw and interpret box plots, and compare distributions using the median and interquartile range in context (Higher)
  • Draw and interpret histograms with unequal class widths using frequency density = frequency ÷ class width (Higher)
  • Estimate frequencies within part of a class interval from a histogram or cumulative frequency graph (Higher)

About 2 h 55 min of study, lessons and core practice

• Problem solving, mathematical communication and exam technique

  ≈ 3%1 h

  How to attack unstructured multi-step problems, communicate reasoning clearly, meet accuracy requirements and manage time on non-calculator and calculator papers.

  • Break unfamiliar multi-step problems into steps, choose methods from different topics, and link them (e.g. ratio with area, algebra with geometry)
  • Communicate reasoning clearly: set out working logically, state reasons and units, and write conclusions that answer the question asked
  • Evaluate and critique given solutions, assumptions and statistical claims, and identify errors in worked reasoning
  • Meet accuracy conventions (exact answers, significant figures, rounding only at the end) and use the calculator efficiently and non-calculator methods reliably
  • Plan exam time (about 1 mark per minute), secure accessible marks first, and check with estimation and inverse operations

• Investigation and modelling (0607 papers 5 and 6)

  ≈ <1%1 h

  Cambridge IGCSE International Mathematics only

  Tackling the Cambridge 0607 investigation (spot, generalise and justify patterns) and modelling (build, use and refine models of real situations) questions.

  • Organise results systematically in tables, spot patterns and write general rules in words and algebra
  • Test a conjecture with further cases and justify or prove a general result
  • Build a mathematical model of a real situation (linear, quadratic, exponential or piecewise), state its assumptions, and use a GDC to analyse it
  • Evaluate a model’s limitations and refine it in response to new information

About 2 h of study, lessons and core practice