AQA GCSE Mathematics

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Number

Structure and calculation

N1, N2 - Place value, ordering and inequality symbolsEvery digit in a number is worth an amount that depends on the column it sits in. In this lesson you will read place value, multiply and divide by 10, 100 and 1000, use one known calculation to find others, and put decimals and negative numbers in order using inequality symbols. These ideas sit u...
N2 - Written methods for the four operationsWritten methods turn a large calculation into small steps you can trust. You will add, subtract, multiply and divide integers and decimals, handle negative numbers, and use these skills for money and account balances. The methods work on both tiers, including Paper 1, where you cannot use a calcu...
N3 - Inverse operations and order of operationsThe same numbers can give different answers when you change the order of the operations. You will learn which operations come first, how brackets, powers, roots and fraction bars group a calculation, and how inverse operations undo a process. You will also use cancellation to make a calculation s...
N4 - Factors, multiples, primes and prime factorisationFactors tell you how a number can be shared exactly; multiples tell you what happens when equal groups are repeated. You will recognise primes, break whole numbers into prime factors, and use the highest common factor and lowest common multiple to solve grouping and timing problems. You will also...
N1, N2 - Equivalent fractions, mixed numbers and orderingFractions describe equal parts of a whole, but the same amount can have many different fraction names. You will use N2 to make equivalent fractions, simplify them and change between improper fractions and mixed numbers. You will then use N1 to compare and order fractions, including negative fract...
N2 - Calculating with fractions and mixed numbersFractions let you calculate with equal parts exactly. You will add, subtract, multiply and divide positive and negative fractions, including improper fractions and mixed numbers. Each operation has a reason behind its rule, so you can choose a method rather than guess which numbers to combine.
N5 - Systematic listing and the product ruleA list is useful only if you can trust that nothing is missing. You will organise choices in lists, tables and branching diagrams, decide when order matters, and count possibilities without repeats. The final part introduces the product rule for Higher tier.
N6 - Powers and rootsPowers compress repeated multiplication, while roots undo it. You will work with squares, cubes and higher powers, recognise useful number facts, and distinguish a root from the power that produced it. The final part estimates unfamiliar powers and roots for Higher tier.
N7 - Calculating with roots and indicesYou will calculate with roots, combine powers efficiently and work out what zero and negative indices mean. The laws come from counting and cancelling equal factors, so you can reconstruct them when memory lets you down. These methods keep large calculations manageable on a non-calculator paper.
N7 - Fractional indices (Higher)A fraction in an index combines a root with a power. You will read each part of the index, choose a short order of calculation and handle negative fractional indices without a calculator. The same index laws connect all these forms.
N8 - Exact answers with pi and surdsAn exact answer keeps the full value, even when no short decimal can do that. You will preserve fractions and multiples of π through calculations, then work with surds in the Higher parts. Knowing when to stop is as useful as knowing how to calculate.
N9 - Standard formStandard form writes a very large or very small number with a manageable coefficient and a power of ten. You will convert, compare and calculate in this form, both by hand and with a calculator. You will also learn to read a calculator display without losing the exponent.

Measures and accuracy

N13 - Metric units and conversionsA measurement needs a number and a unit. In N13 you will choose sensible units, convert length, mass, time and money, and handle square and cubic units. You will also connect litres to cubic centimetres and use an imperial conversion when it is supplied.
N14, N15 - Rounding, significant figures and estimationAn answer can be accurate without needing every digit. N15 shows you how to round to decimal places or significant figures and choose sensible accuracy in a context. N14 uses these simpler numbers to estimate calculations and check whether an answer, including a calculator answer, makes sense.
N15, N16 - Error intervals and limits of accuracyA rounded measurement describes a range of possible values. In N15 you will write this range using inequality symbols, for both rounding and truncation. In N16 you will use the limits of accuracy to decide which measurements and claims are possible.
N16 - Upper and lower bounds (Higher)When measured inputs are rounded, the calculated answer also has uncertainty. In N16 you will choose upper and lower bounds for sums, differences, products and quotients. You will then decide which digits of a final answer are actually guaranteed by the measurements.

Algebra

Notation, vocabulary and manipulation

A1–A3 - Algebraic notation, substitution and vocabularyAlgebra lets one calculation stand for many different numbers. You will read its shorthand in A1, put values into expressions and scientific formulae in A2, and use the vocabulary in A3 to say exactly what each line means. A small bracket can change an answer completely, so reading comes before c...
A4 - Simplifying, expanding single brackets and factorisingYou can change the appearance of an expression without changing its value. Learn to collect matching terms, simplify products and powers, expand a single bracket, and reverse that process by taking out a common factor. These are the tools that make longer algebra manageable.
A4 - Expanding double brackets and factorising quadraticsMultiplying two brackets creates a quadratic expression. Learn to account for every product, then work backwards to find the brackets of a quadratic. You will also use the difference of two squares, including simple exact square-root factors.
A4 - Three brackets and harder factorising (Higher)Longer expressions become manageable when you keep one stage visible at a time. Expand three brackets, factorise quadratics whose leading coefficient is not one, and work exactly with surd coefficients. The same distributive rule still does the work; the extra challenge is choosing a useful order.
A4 - Algebraic fractions (Higher)Algebraic fractions follow the fraction rules you already use with numbers. You will simplify them by factorising, multiply and divide them, and add and subtract them using common denominators. The useful habit is to see whole factors before doing any cancelling.
A5 - Using and rearranging formulaeA formula describes a relationship between quantities. You will read formulae in words and symbols, use them with values, and rearrange them to make a different letter the subject. Doing the same thing to both sides lets one relationship answer several different questions.
A6 - Algebraic arguments and proofAlgebra lets you explain why a pattern works, rather than just check a few examples. You will distinguish equations from identities, show that expressions are equivalent, and use algebra to test claims. The final Higher part builds complete proofs about whole numbers.
A7 - Functions, composites and inversesA function turns an input into one output by following a rule. You will interpret simple expressions as function machines. The Higher parts introduce function notation, combine functions in a chosen order, and find an inverse rule that reverses a process.

Graphs

A8–A10 - Coordinates and straight-line graphsA graph turns an equation into a picture of all its solutions. You will locate coordinates in A8, plot straight-line equations in A9, and connect their steepness and axis crossings to their equations in A10. These links let you predict a line before drawing it.
A9 - Equations of straight linesA gradient tells you a line's direction, but one more piece of information fixes its position. You will use parallel lines, a point and gradient, or two points to work out a straight-line equation. The final part uses perpendicular gradients at Higher tier.
A11, A12 - Quadratic graphs, roots and turning pointsQuadratic graphs curve, turn, and often cross the horizontal axis twice. You will recognise and sketch their shapes in A12, and use roots, intercepts, symmetry and turning points to describe them in A11. Factorised algebra gives exact axis crossings that a drawing can only estimate.
A11 - Turning points by completing the square (Higher)Completing the square rewrites a quadratic so that its turning point becomes visible. You will connect a squared bracket to a minimum or maximum, create that form from an expanded expression, and handle coefficients other than one. The result is exact, even when a plotted graph would only give an...
A12 - Cubic, reciprocal and exponential graphsSome curves pass through the origin; others get close to an axis without ever reaching it. In A12 you will recognise, sketch and interpret simple cubic and reciprocal graphs. The Higher part adds exponential graphs, whose equal steps multiply the height instead of adding to it.
A13 - Translating and reflecting graphs (Higher)A small change to a function's equation can move its whole graph. In A13 you will translate graphs horizontally and vertically, reflect them in either coordinate axis, and sketch the result using a few key points. Tracking coordinates explains the signs, so you do not have to guess which way a gr...
A14 - Real-life graphsA graph can turn a rule into a useful answer, even when solving its equation is awkward. In A14 you will plot and interpret real-life graphs, follow journeys, connect speed with acceleration, and use reciprocal models. The Higher part uses exponential graphs to estimate times in a decay model.
A15, R15 - Gradients and areas under graphs (Higher)A curve records more than its plotted heights. In A15 you will calculate or estimate gradients and areas, and interpret them in journeys and financial models. R15 connects a chord's gradient to an average rate and a tangent's gradient to the rate at one instant.

Solving equations and inequalities

A17 - Solving linear equationsAn equation asks which value makes two expressions equal. You will solve linear equations using inverse operations, brackets and unknowns on both sides, then read approximate solutions from a graph. Checking your answer in the starting equation turns a guess into a result you can trust.
A18 - Solving quadratic equations by factorisingA quadratic equation can have two solutions, one repeated solution, or no real solution. You will use factorised expressions to find exact solutions and graphs to read approximate ones. The key is to turn one quadratic equation into simpler linear equations.
A18 - Quadratic formula and completing the square (Higher)Some quadratics do not split into brackets with whole-number coefficients. You will solve them using the quadratic formula or completing the square, keep exact surd answers, and give rounded answers when asked. You will also put equations into a suitable form before choosing a method.
A19 - Simultaneous linear equationsTwo unknowns usually need two pieces of information. You will solve pairs of linear equations by elimination or substitution, and read approximate solutions from intersecting graphs. Each answer is a pair of values that must satisfy both equations at the same time.
A19 - Linear and quadratic simultaneous equations (Higher)When a line meets a quadratic curve, there can be two intersection points. You will use substitution to find both pairs of coordinates, including exact surd answers, and use graphs for approximate solutions. Each pair must satisfy both of the original equations.
A20 - Approximate solutions by iteration (Higher)Some equations do not have a convenient exact solution. You will use a starting value and a repeated calculation to get closer to a solution, read suffix notation correctly, and decide how many digits your answer needs. A calculator does the arithmetic; you still choose what to feed back into it.
A21 - Forming and solving equationsAn equation starts with a relationship: two costs are equal, the sides add to a perimeter, or two purchases have known totals. You will turn words and procedures into algebra, form one or two equations, and interpret the answers in the original problem. Choosing what your letters mean is the firs...
A22 - Solving linear inequalitiesAn inequality can have a whole range of answers. You will solve linear inequalities, show their solutions on number lines, and decide which whole-number values are allowed. Most steps work like solving equations, with one important change when you multiply or divide by a negative number.
A22 - Quadratic inequalities, set notation and regions (Higher)An inequality may select an interval, two separate ranges, or a region of a coordinate grid. You will use quadratic roots to find intervals, write solution sets clearly, and combine linear inequalities in two variables. Graphs make the difference between a boundary and a whole solution set visible.

Sequences

A23, A24 - Sequence rules and special sequencesA sequence is an ordered list with a rule behind it. In A23 you will generate terms from rules and picture patterns. In A24 you will recognise number families, from triangular numbers to sequences that grow by multiplication, and use their patterns to continue them correctly.
A25 - The nth term of a linear sequenceA term-to-term rule gets you to the next term. An nth-term rule lets you reach any position in one calculation. In A25 you will build linear nth-term rules, including decreasing sequences and sequences with missing terms, then use your rule to decide whether a number appears.
A24, A25 - Quadratic nth term and harder sequences (Higher)Changing differences can hide a simple algebraic rule. In A25 you will deduce quadratic nth terms and use their structure to explain properties of a whole sequence. In A24 you will work with surd ratios and other given rules, keeping exact values and checking which type of sequence you actually h...

Ratio, proportion and rates of change

Units, scale and ratio

R1 - Converting standard and compound unitsA unit change can turn a small number into a large one without changing the quantity. In R1 you will use standard conversions in calculations, then convert speed, pay, prices, density and pressure. You will also write conversions using letters, so the same reasoning works for any value.
R3, R4 - Comparing quantities as fractions and ratiosA fraction and a ratio can compare the same quantities in different forms. R3 shows how to express one quantity as a fraction of another, including answers greater than one. R4 adds ratio notation, order and simplest form, so you can make fair comparisons even when the units or numbers look diffe...
R2 - Scale diagrams and mapsA scale drawing shrinks a real place while keeping its lengths in the same proportions. In R2 you will turn map lengths into real distances, work backwards to drawing lengths, and choose dimensions for a scale plan. You will then use a plan to solve a geometrical problem.
R5 - Sharing in a ratioA ratio tells you how many equal parts each person receives, rather than the value of a part. This part of R5 shows how to share a total using a part : part or part : whole ratio. You will also work backwards from two amounts to describe their division as a ratio.
R5 - Ratio problems and best buysRatios help with recipes, mixtures, conversions and shopping decisions. This part of R5 shows how to choose the quantity that corresponds to one part, scale it, and compare like with like. You will also use a known share or difference to recover missing amounts.
R6–R8, N11 - Ratios, fractions and proportionA ratio, a fraction and a straight-line equation can all describe the same relationship. R6 connects ratios with multipliers, R7 treats proportion as equality of ratios, and R8 links ratios to fractions and linear functions. N11 lets you use fractions inside ratio problems, including a fraction o...

Proportion, rates and growth

R10 - Direct and inverse proportionSolve direct and inverse proportion problems by deciding what stays fixed. You will scale quantities, use tables and given equations, and read the same relationships from graphs.
R11 - Speed, density and other compound unitsUse speed, rates of pay, unit prices, density and pressure to connect two measurements. You will choose a calculation from the meaning of “per”, keep units consistent and compare quantities fairly.
R13, R14 - Proportion equations and graphsR13 connects direct and inverse proportion equations with fixed ratios, products and reciprocals. R14 connects these relationships to graphs and interprets a straight-line gradient as a rate of change. You will use both the shape and the scale of a graph to explain what it represents.
R13 - Forming proportion equations (Higher)Turn a proportion statement into an equation, use a known pair of values to find its constant, then solve the completed model. You will work with direct and inverse proportion, including powers and roots, and explain how changing one quantity affects another.
R16 - Compound interest, growth and decayModel repeated percentage changes using multipliers, powers and year-by-year calculations. You will calculate compound interest, follow growth and decay, recover missing values and interpret when a target is reached. The final Higher part adds other operations to a repeated process.

Geometry and measures

Properties and constructions

G1, G2 - Geometric terms and constructionsUse G1's geometric language to read, label and draw shapes accurately. Then use G2's ruler-and-compass constructions to make exact midpoints, right angles and angle bisectors, including an angle of 60°.
G2 - Loci and shortest distancesFind every possible position that fits a distance rule. Use circles, perpendicular bisectors and angle bisectors to construct loci, then combine conditions to locate a point or shade a region.
G3 - Angle facts and parallel linesUse angle sums and equal-angle relationships to work out missing angles. Learn the precise reasons for angles on straight lines, around points and between parallel lines, so you can explain each step without measuring a sketch.
G3 - Angles in triangles and polygonsDerive the angle sum of a triangle, then build angle sums for polygons from triangles. Use interior and exterior angles to work out missing angles and identify a regular polygon from its corners.
G4 - Properties of triangles and quadrilateralsIdentify triangles, special quadrilaterals and named polygons from their defining properties. Use equal sides, equal angles, parallel lines, diagonals and symmetry to work out missing information, including when a shape belongs to more than one family.
G5, G6 - Congruent triangles and geometric proofDecide when triangle measurements force exactly one shape. G5 supplies four congruence tests; G6 uses them, angle facts, quadrilateral properties and similarity to build proofs. You will finish by explaining why the squares on the two shorter sides of a right-angled triangle add to the square on...
G7 - Reflections, rotations and translationsMove a shape on a grid while keeping its exact size and shape. You will construct and recognise reflections, rotations and translations, then describe each movement precisely enough for someone else to reproduce it. These transformations make congruent images.
G7 - Enlargement and scale factorsConstruct a similar copy by scaling every position from one centre. You will enlarge shapes with whole-number and fractional scale factors and recover a centre and scale factor from a shape and its image. A separate Higher part extends the same idea to negative scale factors.
G8 - Combined transformations and invariant points (Higher)Track a shape through several movements and describe their combined effect. You will use column notation for translations, recognise when two movements equal one familiar transformation, and distinguish points that stay fixed from properties that stay unchanged.
G9, G10 - Circle parts and angle theoremsName the lines and regions in a circle accurately. Then use the angle theorems in the Higher parts, explain why they work and give clear reasons in geometric arguments.
G10 - Circle theorems about tangents and chords (Higher)Use and prove circle theorems involving tangents and chords. Build clear arguments from radii, right angles and congruent triangles, then connect a tangent angle to an angle at the circumference.
G11 - Geometry problems on coordinate axesUse coordinates to work out lengths along grid lines, locate midpoints and complete shapes. Check geometric claims with exact coordinate evidence, rather than relying on how a sketch looks.
G12, G13 - 3D shapes, plans and elevationsDescribe solid shapes using faces, surfaces, edges and vertices. Draw flat views from above, in front and from the side, then use those views to reason about arrangements of cubes.

Measurement and calculation

G14, G15 - Measuring, scale drawings and bearingsMeasurements need a number, a unit and a sensible way to read the instrument. In G14 you will use standard measures in geometrical problems. In G15 you will read rulers and protractors, connect measurements to scale drawings, and use compass directions and three-figure bearings to describe a jour...
G16, G17 - Perimeter and area of 2D shapesPerimeter measures the boundary; area measures the surface inside it. G16 gives the area methods for triangles, parallelograms and trapezia. G17 brings these together with rectangles to find perimeters and areas of composite shapes, including shapes with missing lengths.
G17 - Circumference and area of circlesA circle needs only one length to fix its size. In G17 you will use radius and diameter to work out circumference and area, keep exact answers in terms of π, and combine circles with straight edges in perimeter and area problems.
G16, G17 - Volume and surface area of prismsVolume measures the space inside a solid; surface area measures its exposed outside. In G16 you will calculate volumes of cuboids, other right prisms and cylinders. In G17 you will use familiar face areas to find their surface areas, including containers with an open end.
G18 - Arc length and sector areaAn arc and its sector take the same fraction of a circle's full turn. In G18 you will use that fraction to calculate arc lengths, sector areas and unknown angles. You will also distinguish a curved arc from the complete perimeter of a sector.
G19, R12 - Congruence and similar shapesDecide when shapes are exact copies and when one is a scaled copy of another. G19 gives you the geometry of congruence and similarity; R12 lets you express comparisons as ratios and use them to work out missing lengths.
G20 - Pythagoras' theoremWork out a missing length in a right-angled triangle, then use the same idea inside rectangles and on coordinate grids. You will learn when to add, when to subtract, and why the square root must come last.
G17 - Spheres, pyramids, cones and composite solidsWork out volumes and surface areas of curved and pointed solids, including shapes made by joining or removing pieces. You will distinguish a vertical height from a slant height, and count only the surfaces that are actually exposed.
G19 - Similar areas and volumes (Higher)Connect the length, area and volume ratios of similar shapes and solids. You will square or cube a length factor, reverse those steps with roots, and choose which kind of scale a context needs.
G20, G21 - Right-angled trigonometry and exact valuesAn angle can tell you a length, even when you cannot reach it with a tape measure. G20 gives you sine, cosine and tangent for right-angled triangles; G21 gives exact values for special angles. You will choose a ratio, find missing lengths and angles, and solve two-dimensional problems with a calc...
G20 - Pythagoras and trigonometry in 3D (Higher)A line through a solid can belong to a right triangle that is not on any outside face. You will find that triangle, use Pythagoras in stages, and work out lengths and angles in cuboids and pyramids. The arithmetic is familiar; choosing the correct cross-section is the new skill.
G22, G23 - Sine rule, cosine rule and triangle area (Higher)A triangle does not need a right angle for its sides and angles to be connected. G22 gives the sine and cosine rules for missing lengths and angles; G23 gives an area formula using two sides and the angle between them. You will choose a rule from the information available and recognise when an an...
A12 - Graphs of sin, cos and tan (Higher)Sine, cosine and tangent continue beyond the acute angles inside right triangles. You will recognise, sketch and interpret their graphs for angles in degrees, including negative angles and angles larger than a full turn. Their repeating patterns let a small set of exact values describe infinitely...
A16 - Equation of a circle and tangents (Higher)A circle centred at the origin has a short equation built from Pythagoras. You will recognise that equation, check whether points lie on the circle, and find tangent equations using perpendicular gradients. The coordinate rule links a curved shape to the straight line that just touches it.

Probability

P1–P5 - Probability, relative frequency and expected outcomesA probability describes a chance, not a promise. P1 and P2 connect experiments, fairness and expected outcomes; P3 and P4 connect the probability scale with complete sets of outcomes. P5 explains why a larger, unbiased experiment usually gives a more reliable estimate.
P6, P7 - Sample spaces and Venn diagramsA good picture makes it hard to miss an outcome or count it twice. P6 teaches you to organise sets using tables, grids, Venn diagrams and listing trees. P7 uses complete possibility spaces to turn equally likely outcomes into probabilities.
P8 - Tree diagrams and combined eventsA probability tree records both the possible paths and the chance of each step. You will calculate independent and dependent combined events, decide when to multiply and when to add, and use complements and overlapping events without double counting.
P9 - Conditional probability (Higher)New information changes which outcomes are still possible. You will calculate and interpret conditional probabilities using two-way tables, Venn diagrams, probability trees and expected frequencies. The key is choosing the right group as the whole.

Statistics

S1, S2 - Samples, tables and chartsUse a sample to learn about a larger group, while keeping its limits in view. Organise data in frequency and two-way tables, choose suitable charts, and construct and interpret bar charts, pictograms, vertical line charts, pie charts and time-series graphs.
S4 - Averages and rangeDescribe a data set using its mean, median, mode and range. Distinguish data types, find a missing value from a mean, and explain how an outlier can change the story an average tells.
S4, S5 - Frequency tables and comparing distributionsWork out averages from frequency tables and estimate a mean from grouped measurements. Compare distributions using a centre and a spread, interpret their displays, and use sample statistics to describe a population without claiming more than the data support.
S3 - Histograms (Higher)Construct and interpret histograms for continuous and grouped discrete data, including unequal class widths. Use frequency density to make area represent frequency, recover a missing scale, and estimate counts within a class while stating the assumption.
S3, S4 - Cumulative frequency and box plots (Higher)Construct cumulative frequency graphs, estimate medians and quartiles, and use five-number summaries to draw box plots. Compare distributions using medians and interquartile ranges, and distinguish what these displays reveal from what they hide.
S6 - Scatter graphs and correlationPlot paired numerical data, describe the direction and strength of correlation, and draw an estimated line of best fit. Use it for predictions while distinguishing interpolation from extrapolation and association from causation.