GCSE Maths topics

Simultaneous equations

Two unknowns, two equations, one consistent story. Read the full explainer →

Written against: AQA 8300 A19 · Foundation and Higher

Builds on: Solving two-step linear equations — “Two-step fluency before two equations at once.” · Linear graphs y = mx + c

Calculating in standard form

Multiply and divide numbers in standard form using the index laws.

Written against: AQA 8300 N9 · Foundation and Higher

Builds on: Laws of indices — “Multiplying the powers of ten uses the index laws.” · Standard form: large numbers — “You can't calculate in a form you can't read.”

Recurring decimals

Some fractions become decimals that repeat forever — recognise them and write them with dot notation.

Written against: AQA 8300 N10 · Higher only

Builds on: Fractions ↔ decimals ↔ % — “You need the fraction–decimal bridge before asking which decimals repeat.”

Estimating roots

Trap a square root between two whole numbers — √50 lies between 7 and 8.

Written against: AQA 8300 N6 · Higher only

Builds on: Squares, cubes & roots — “You can't trap √50 between squares you don't know.”

Parallel lines on graphs

Lines with the same gradient never meet — recognise parallel lines from their equations.

Written against: AQA 8300 A9 · Foundation and Higher

Builds on: Linear graphs y = mx + c — “Parallel is a claim about gradients — read them first.”

Fractional enlargements

Enlarge by a factor like ½ — the image shrinks, but it's still called an enlargement.

Written against: AQA 8300 G7 · Foundation and Higher

Builds on: Scale factors as multipliers — “A factor of ½ only surprises you until multipliers don't.” · Enlarging shapes — “Whole-number factors first; fractions bend the intuition.”

Plans & elevations

Draw a solid from the front, the side and above — three flat views that pin down one 3D shape.

Written against: AQA 8300 G13 · Foundation and Higher

Builds on: Nets of 3D shapes — “Nets flatten surfaces; elevations flatten views.” · Naming 3D solids

Independent events

One event happening tells you nothing about the other — coins have no memory.

Written against: AQA 8300 P8 · Foundation and Higher

Builds on: Probabilities sum to 1 — “Independence talk needs sure-footed single-event probability.” · Theory versus experiment

Line of best fit

Draw one straight line through the middle of a scatter cloud, then use it to predict — carefully.

Written against: AQA 8300 S6 · Foundation and Higher

Builds on: Scatter graphs & correlation — “The line summarises a cloud you can already read.”

Correlation is not causation

Ice cream sales and drownings rise together — the correlation is real, the cause is the summer.

Written against: AQA 8300 S6 · Foundation and Higher

Builds on: Scatter graphs & correlation — “First see the pattern, then question its story.”

Quadratic expressions

Expand and factorise with x² — the next algebra.

Written against: AQA 8300 A4 · Foundation and Higher

Builds on: Collecting like terms — “Like terms keep x² tidy.” · Multiplying & dividing negatives — “Sign slips are where double brackets die.” · Expanding and simplifying — “Expand-and-collect is the whole of double brackets.” · Graphs of quadratics

Trigonometry first steps

Sine, cosine and tangent as ratios in right-angled triangles.

Written against: AQA 8300 R12 · AQA 8300 G20 · Foundation and Higher

Builds on: Pythagoras: the hypotenuse · Ratio & proportion problems · Angles in polygons — “Trig lives inside triangles’ angle facts.”

Error intervals

A rounded number hides a range — write the interval of values it could have been.

Written against: AQA 8300 N15 · AQA 8300 N16 · Foundation and Higher

Builds on: Significant figures — “The interval's width is set by the rounding you applied.”

Fractional indices

A power of ½ means square root — indices and roots turn out to be one system.

Written against: AQA 8300 N7 · Higher only

Builds on: Squares, cubes & roots · Negative & zero indices — “Half-powers extend the same pattern past zero and the negatives.”

Function notation f(x)

Write rules as f(x) = 3x + 2 and read f(5) as 'feed in 5' — tidier packaging for machines you know.

Written against: AQA 8300 A7 · Higher only

Builds on: Substituting into expressions — “f(3) is substitution with tidier packaging.” · Function machines

Rearranging multi-step formulae

Change the subject when it appears in several steps — brackets, fractions and all.

Written against: AQA 8300 A5 · Foundation and Higher

Builds on: Unknowns on both sides · Rearranging simple formulae — “Two-step subjects before tangled ones.”

Quadratic sequences

Sequences whose differences change steadily — the n² family, one peek ahead.

Written against: AQA 8300 A24 · Foundation and Higher

Builds on: Recognising special sequences · nth term of linear sequences — “Linear nth terms first — the difference of differences comes next.”

Proportion formulae

Write y = kx or y = k/x from a description, find k, and use the formula.

Written against: AQA 8300 R13 · Higher only

Builds on: Direct proportion y = kx — “y = kx must mean something before you write it down.” · Inverse proportion — “Half the formulae here are the inverse kind.” · Rearranging simple formulae

Arcs & sectors

A slice of circle: the angle tells you what fraction of the circumference and area you keep.

Written against: AQA 8300 G18 · Foundation and Higher

Builds on: Circumference of a circle — “An arc is a fraction of a circumference you can already find.” · Area of a circle — “A sector is a fraction of an area you can already find.” · Fractions of amounts

Column vectors

Write a slide as a column vector — across and up in one bracket.

Written against: AQA 8300 G24 · Foundation and Higher

Builds on: Translating shapes — “Column vectors formalise the slides you've been doing.” · Coordinates in four quadrants

Sampling & bias

You can't ask everyone — how you choose who to ask decides what your data is worth.

Written against: AQA 8300 S1 · Foundation and Higher

Builds on: Experimental probability · The data-handling cycle — “Sampling is the collect step, interrogated.”

Frequency polygons

Join the midpoints of grouped data — a line that lets two distributions share one grid. Read the full explainer →

Written against: AQA 8300 S4 · Foundation and Higher

Builds on: Line graphs over time · Grouped frequency tables — “No classes, no midpoints to join.”

Quartiles & interquartile range

Cut the ordered data into quarters — the middle half's width is a spread outliers can't wreck. Read the full explainer →

Written against: AQA 8300 S4 · Higher only

Builds on: Outliers & their effect · Averages & range — “Quartiles slice the ordered list the median already cut.”

The product rule for counting

Work out how many combinations there are by multiplying the number of choices at each stage instead of listing them.

Written against: AQA 8300 N5 · Higher only

Builds on: Listing outcomes systematically · Sample space diagrams

Equations and identities

Tell an equation (true for some values) from an identity (true for every value), and use the identity sign.

Written against: AQA 8300 A3 · AQA 8300 A6 · Foundation and Higher

Builds on: Term, expression, equation, formula · Expanding and simplifying

Showing expressions are equivalent

Use algebra to show that two expressions are the same and to back up a mathematical argument.

Written against: AQA 8300 A6 · Foundation and Higher

Builds on: Expanding and simplifying · Writing expressions from words

Finding the equation of a line

Find the equation of a line from two points, or from one point and the gradient.

Written against: AQA 8300 A9 · Foundation and Higher

Builds on: Finding a line's equation · Gradient as a rate

Reading a quadratic graph

Read the roots, the intercepts and the turning point from the graph of a quadratic.

Written against: AQA 8300 A11 · AQA 8300 A18 · Foundation and Higher

Builds on: Graphs of quadratics · Solving equations with graphs

Cubic and reciprocal graphs

Recognise, sketch and read simple cubic graphs and the graph of y = 1/x, including in real contexts.

Written against: AQA 8300 A12 · AQA 8300 A14 · Foundation and Higher

Builds on: Graphs of quadratics · Inverse proportion graphs · Substituting negatives & squares

Exponential graphs

Recognise, sketch and read graphs of the form y = k to the power x, including in real contexts.

Written against: AQA 8300 A12 · AQA 8300 A14 · Higher only

Builds on: Geometric sequences · Negative & zero indices · Percentage multipliers

Gradient of a curve

Estimate the gradient at a point on a curve by drawing a tangent, and say what it means, for example as a speed.

Written against: AQA 8300 A15 · AQA 8300 R15 · Higher only

Builds on: Gradient as a rate · Real-life graphs · Graphs of quadratics

Area under a graph

Estimate the area under a graph and say what it means, for example as a distance.

Written against: AQA 8300 A15 · Higher only

Builds on: Area of a trapezium · Real-life graphs · Speed, distance, time

Equation of a circle

Recognise and use the equation of a circle centred on the origin.

Written against: AQA 8300 A16 · Higher only

Builds on: Pythagoras: the hypotenuse · Coordinates in four quadrants

Iteration

Find an approximate solution to an equation by repeating a calculation, feeding each answer back in.

Written against: AQA 8300 A20 · AQA 8300 R16 · Higher only

Builds on: Substituting into expressions · Term-to-term sequences · Calculator fluency

Inequalities on a graph

Show the region of a graph that satisfies one or more inequalities in two variables.

Written against: AQA 8300 A22 · Higher only

Builds on: Solving linear inequalities · Plotting lines from tables · Horizontal & vertical lines

Fibonacci-type sequences

Recognise and use sequences where each term is the sum of the two before it.

Written against: AQA 8300 A24 · Foundation and Higher

Builds on: Term-to-term sequences · Building equations from problems

Simple interest and money maths

Work out simple interest and handle everyday financial calculations, with the vocabulary from N2b.

Written against: AQA 8300 N2 · AQA 8300 R9 · Foundation and Higher

Builds on: Percentages of amounts · Percentage increase & decrease

Density and pressure

Use density and pressure as compound measures, and convert their units.

Written against: AQA 8300 R1 · AQA 8300 R11 · Foundation and Higher

Builds on: Density as a rate · Speed, distance, time · Converting area & volume units · Calculating density (PHYS)

More constructions

Construct a perpendicular to a line from or at a point and an angle of 60 degrees, and know that the perpendicular is the shortest route from a point to a line.

Written against: AQA 8300 G2 · Foundation and Higher

Builds on: Constructing bisectors · Constructing triangles

Loci

Find and draw the set of points that obey a rule, such as being the same distance from two points.

Written against: AQA 8300 G2 · Foundation and Higher

Builds on: Constructing bisectors · Parts of a circle · Simple scale drawings

Geometric proof

Use angle facts, congruence, similarity and quadrilateral properties to prove results about angles and sides.

Written against: AQA 8300 G6 · Foundation and Higher

Builds on: Geometric reasoning chains · Congruence criteria · Similar shapes

Circle theorems: angles

Use the theorems about angles at the centre, in a semicircle, in the same segment and in a cyclic quadrilateral.

Written against: AQA 8300 G10 · Higher only

Builds on: Parts of a circle · Special triangles · Geometric reasoning chains

Volume of pyramids, cones and spheres

Find the volume of a pyramid, a cone and a sphere.

Written against: AQA 8300 G17 · Foundation and Higher

Builds on: Volume of cylinders · Volume of prisms · Substituting into formulae

Surface area of pyramids, cones and spheres

Find the surface area of a pyramid, a cone and a sphere.

Written against: AQA 8300 G17 · Foundation and Higher

Builds on: Surface area of cylinders · Surface area of prisms · Pythagoras: the hypotenuse

Similar shapes: areas and volumes

Use the fact that areas scale by the square, and volumes by the cube, of the length scale factor.

Written against: AQA 8300 G19 · Higher only

Builds on: Similar shapes · Converting area & volume units · Squares, cubes & roots

Histograms

Draw and read histograms, including ones with bars of different widths.

Written against: AQA 8300 S3 · Higher only

Builds on: Grouped frequency tables · Bar charts · Density as a rate

Cumulative frequency graphs

Draw a running-total graph for grouped data and read values from it.

Written against: AQA 8300 S3 · Higher only

Builds on: Grouped frequency tables · Averages from frequency tables · Line graphs over time

Repeated percentage change

Apply a percentage change several times — and see why two 10% rises are not 20%.

Written against: AQA 8300 R16 · Foundation and Higher

Builds on: Percentage multipliers — “Repeated change is repeated multiplication — multipliers keep it honest.” · Percentage increase & decrease — “Chaining changes assumes single changes are safe.” · Index notation

Combined events

Two dice, two spinners — multiplying chances.

Written against: AQA 8300 P8 · Foundation and Higher

Builds on: Sample space diagrams — “Two dice need the grid before the multiplication.” · Independent events — “Multiplying chances assumes the events ignore each other.” · Probability of single events — “Two dice are two single events, multiplied.”

Surds: exact roots

Keep √2 as √2 — an exact answer instead of a rounded one.

Written against: AQA 8300 N8 · Higher only

Builds on: Squares, cubes & roots — “Surds are roots that refuse to simplify — know roots first.” · Estimating roots — “Exact roots matter once you've felt approximations wobble.”

Recurring decimals to fractions

Turn 0.363636… into 4/11 with an equation trick — multiply, subtract, solve.

Written against: AQA 8300 N10 · Higher only

Builds on: Solving two-step linear equations — “The trick is solving a two-step equation.” · Recurring decimals — “You can only convert a repeat you can notate.”

Solving quadratics by factorising

If two brackets multiply to zero, one of them is zero — the first quadratic solving method.

Written against: AQA 8300 A11 · AQA 8300 A18 · Foundation and Higher, with some Higher-only content

Builds on: One-step equations · Quadratic expressions — “A factorised quadratic hands you its roots.” · Unknowns on both sides · Equations with fractions

Compound interest

Money growing by the same percentage each year — repeated multipliers, with savings attached.

Written against: AQA 8300 R16 · Foundation and Higher

Builds on: Percentage multipliers — “No multipliers, no compounding.” · Repeated percentage change — “Compound interest is repeated percentage change with money attached.”

Trigonometry: finding angles

Use inverse sin, cos and tan to recover an angle from two sides.

Written against: AQA 8300 G20 · Foundation and Higher

Builds on: Trigonometry first steps — “Finding angles inverts the ratios — meet them forwards first.”

Tree diagrams

Draw branching chances for two-stage events — multiply along the branches, add down the ends.

Written against: AQA 8300 P6 · Foundation and Higher

Builds on: Independent events — “Branch probabilities multiply only when events are independent.” · Multiplying fractions · Combined events — “Trees draw the multiplication of chances.”

Calculating with surds and rationalising denominators

Calculate with surds, and rewrite a fraction so that no root is left on the bottom.

Written against: AQA 8300 N8 · Higher only

Builds on: Expanding and simplifying · Equivalent fractions · Surds: exact roots

Upper and lower bounds in calculations

Use upper and lower bounds, including to find the largest and smallest possible result of a calculation.

Written against: AQA 8300 N16 · Higher only

Builds on: Estimating before calculating · Error intervals

Difference of two squares

Recognise and factorise an expression that is one square minus another.

Written against: AQA 8300 A4 · Foundation and Higher

Builds on: Squares, cubes & roots · Quadratic expressions

Expanding three or more brackets

Multiply out three or more brackets and simplify the result.

Written against: AQA 8300 A4 · Higher only

Builds on: Index laws in algebra · Quadratic expressions

Factorising harder quadratics

Factorise a quadratic that has a number in front of the x squared.

Written against: AQA 8300 A4 · Higher only

Builds on: Factorising into a bracket · Quadratic expressions

Algebraic fractions

Simplify and manipulate fractions that contain algebra.

Written against: AQA 8300 A4 · Higher only

Builds on: Adding fractions, unlike denominators · Factorising into a bracket · Quadratic expressions

Algebraic proof

Write a complete proof of a general statement using algebra.

Written against: AQA 8300 A6 · Higher only

Builds on: Showing expressions are equivalent · Quadratic expressions

Inverse functions

Find the function that reverses another one.

Written against: AQA 8300 A7 · Higher only

Builds on: Inverse operations · Function notation f(x) · Rearranging multi-step formulae

Composite functions

Apply one function and then another, and write the result as a single rule.

Written against: AQA 8300 A7 · Higher only

Builds on: Function machines · Function notation f(x)

Perpendicular lines

Use y = mx + c to tell when two lines meet at right angles.

Written against: AQA 8300 A9 · Higher only

Builds on: Parallel lines on graphs · Gradient as a rate · Dividing by a fraction

Turning points by completing the square

Find the turning point of a quadratic from its completed-square form.

Written against: AQA 8300 A11 · Higher only

Builds on: Completing the square · Reading a quadratic graph

Trigonometric graphs

Recognise and sketch the graphs of sine, cosine and tangent for angles of any size.

Written against: AQA 8300 A12 · Higher only

Builds on: Graphs of quadratics · Trigonometry first steps

Transforming graphs

Sketch what happens to a graph when its function is translated or reflected.

Written against: AQA 8300 A13 · Higher only

Builds on: Graphs of quadratics · Translating shapes · Reflecting shapes · Function notation f(x)

Tangent to a circle

Find the equation of the line that just touches a circle at a given point.

Written against: AQA 8300 A16 · Higher only

Builds on: Finding a line's equation · Equation of a circle · Perpendicular lines · Circle theorems: tangents and chords

Completing the square

Rewrite a quadratic as a squared bracket plus a number, and use that to solve it.

Written against: AQA 8300 A18 · Higher only

Builds on: Squares, cubes & roots · Quadratic expressions

The quadratic formula

Solve any quadratic equation with the quadratic formula.

Written against: AQA 8300 A18 · Higher only

Builds on: Substituting negatives & squares · Calculator fluency · Solving quadratics by factorising · Surds: exact roots

Simultaneous equations: one linear, one quadratic

Solve a pair of equations where one is linear and the other is quadratic.

Written against: AQA 8300 A19 · Higher only

Builds on: Simultaneous equations · Solving quadratics by factorising

Quadratic inequalities and set notation

Solve an inequality that contains a quadratic, and write solution sets in set notation.

Written against: AQA 8300 A22 · Higher only

Builds on: Inequalities on a number line · Solving quadratics by factorising · Reading a quadratic graph

Geometric sequences with surds, and other sequences

Work with geometric sequences whose multiplier is a surd, and with unfamiliar sequences defined in the question.

Written against: AQA 8300 A24 · Higher only

Builds on: Geometric sequences · Surds: exact roots

nth term of a quadratic sequence

Find the formula for the nth term of a quadratic sequence.

Written against: AQA 8300 A25 · Higher only

Builds on: nth term of linear sequences · Quadratic sequences

Negative scale factors

Enlarge a shape by a negative scale factor, which puts the image on the other side of the centre.

Written against: AQA 8300 G7 · Higher only

Builds on: Fractional enlargements · Enlarging shapes · Multiplying & dividing negatives

Combining transformations

Describe the result of one transformation followed by another, and say what stays unchanged.

Written against: AQA 8300 G8 · Higher only

Builds on: Describing transformations · Column vectors

Circle theorems: tangents and chords

Use the theorems about tangents, chords and the alternate segment.

Written against: AQA 8300 G10 · Higher only

Builds on: Parts of a circle · Congruence criteria · Circle theorems: angles

Proving circle theorems

Prove the circle theorems and use them to prove other results.

Written against: AQA 8300 G10 · Higher only

Builds on: Circle theorems: angles · Circle theorems: tangents and chords · Geometric proof

Composite solids and frustums

Find the volume and surface area of solids built from simpler ones, including a cone with its top cut off.

Written against: AQA 8300 G17 · Foundation and Higher

Builds on: Similar shapes · Volume of pyramids, cones and spheres · Surface area of pyramids, cones and spheres

Pythagoras and trigonometry in 3D

Find lengths and angles inside three-dimensional shapes.

Written against: AQA 8300 G20 · Higher only

Builds on: Pythagoras: shorter sides · Plans & elevations · Trigonometry first steps · Trigonometry: finding angles

Exact trigonometric values

Know the exact sine and cosine of 0, 30, 45, 60 and 90 degrees, and the exact tangent of 0, 30, 45 and 60 degrees.

Written against: AQA 8300 G21 · Foundation and Higher

Builds on: Special triangles · Pythagoras: the hypotenuse · Trigonometry first steps

The sine rule

Find a missing side or angle in any triangle with the sine rule.

Written against: AQA 8300 G22 · Higher only

Builds on: Equations with fractions · Trigonometry first steps · Trigonometry: finding angles

The cosine rule

Find a missing side or angle in any triangle with the cosine rule.

Written against: AQA 8300 G22 · Higher only

Builds on: Pythagoras: the hypotenuse · Substituting negatives & squares · Trigonometry first steps · Trigonometry: finding angles

Area of any triangle

Find the area of a triangle from two sides and the angle between them, or work back from the area to a side or an angle.

Written against: AQA 8300 G23 · Higher only

Builds on: Perimeter & area · Trigonometry first steps

Adding, subtracting and scaling vectors

Add and subtract vectors and multiply a vector by a number, on a diagram and in column form.

Written against: AQA 8300 G25 · Foundation and Higher

Builds on: Negative numbers · Column vectors

Vector proofs

Use vectors to build geometric arguments and proofs.

Written against: AQA 8300 G25 · Higher only

Builds on: Simplifying expressions · Ratio notation · Adding, subtracting and scaling vectors

Dependent events

Find probabilities when the first event changes the chances for the second, such as picking without replacement.

Written against: AQA 8300 P8 · Foundation and Higher

Builds on: Independent events · Tree diagrams · Combined events

Conditional probability

Work out the probability of one thing given that another has happened, using two-way tables, trees and Venn diagrams.

Written against: AQA 8300 P9 · Higher only

Builds on: Two-way tables · Venn diagrams for probability · Dependent events

Box plots

Draw and compare box plots.

Written against: AQA 8300 S4 · Higher only

Builds on: Comparing two distributions · Quartiles & interquartile range · Cumulative frequency graphs