OCR GCSE Mathematics

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Number

Factors and powers

1.02a - Factors, multiples and primesNumber names tell you how numbers are built and how they relate. You will use place value, classify odd, even and prime numbers, distinguish factors from multiples, and connect squares and cubes with their roots.
3.01a, 3.01b - Integer powers and rootsPowers compress repeated multiplication; roots recover equal factors. In 3.01a you will use positive and negative integer indices, and in 3.01b you will calculate powers and roots and recognise familiar powers of two, three, four and five.
1.02b - Prime factorisationPrime numbers are the indivisible building blocks of whole-number multiplication. You will recognise primes beyond twenty, split composite numbers into prime factors, and use powers to record the unique collection of prime factors in a positive integer greater than one.
1.02c - Highest factors and lowest multiplesCommon factors help you share or cut exactly; common multiples help you match repeating quantities. You will find the highest common factor and lowest common multiple using lists and prime factorisations, then choose between them in context.
1.03a - Order of operationsWhy does 6+2×5 mean 16, rather than 40? Learn the agreed order for brackets, powers, roots and the four operations, so that a calculation has one clear meaning.
3.01c - Laws of indicesCan you simplify a product of powers without working out the large numbers? Learn why multiplying powers adds indices, dividing subtracts them and raising a power multiplies them, including zero and negative integer indices.

Fractions

2.01a - Equivalent fractions and mixed numbersCan two differently written fractions represent exactly the same amount? Learn to change the size of the parts without changing the value, simplify fractions and move between mixed numbers and improper fractions.
2.01b - Adding and subtracting fractionsHow much is two thirds of a metre plus one quarter of a metre? Learn to count equal-sized fractional units, then use the same idea with subtraction, negative fractions and mixed numbers.
2.01b - Multiplying and dividing fractionsCan dividing make a positive number larger? Learn to multiply fractional amounts, divide by using reciprocals, and combine fraction operations with brackets, mixed numbers and negative signs.
2.01c, 3.03a - Fractions of quantitiesCan taking a fraction of an amount give more than you started with? Use 2.01c to calculate fractional amounts and compare two quantities, then use 3.03a to keep fraction calculations exact without a calculator.

Decimals and percentages

2.02a - Decimals and fractionsWhy can one fraction fit into a short decimal while another seems to go on for ever? Learn to read decimal place value and move between fractions and decimals, using equivalent fractions and division to explain the digits.
2.02b, 2.02c - Calculating with decimalsCan dividing by a number make your answer larger? Use place value to add, subtract, multiply and divide positive and negative decimals without a calculator. You will see why decimal division changes both numbers together, and use inverse operations to check your answers.
2.03a, 2.03b - Percentages of quantitiesWhich is larger: 25% of 80, or 80% of 25? Understand percentages as parts per hundred, convert between number forms, find percentages of quantities and express one quantity as a percentage of another.
2.04a, 2.04b - Comparing numbersWhich is larger: 3/8 or 38%? Compare signed numbers on a number line, put fractions, decimals and percentages into a common form, and use equality and inequality symbols precisely.
2.03c - Percentage change and original valuesA price has been reduced by 12.5%. Can you recover the old price by adding 12.5% to the sale price? Learn to increase and decrease amounts, represent changes with multipliers and reverse a percentage change by finding the original whole.

Accuracy and estimation

4.01a - Rounding to useful accuracyHow can the same measurement round to 6.8 or 6.80, and why might the extra zero matter? Learn to round to a place value, decimal places and significant figures, then choose sensible accuracy for money, measurements and practical decisions.
4.01b - Estimating calculationsAn estimate can expose a misplaced decimal point before you trust a calculator answer. You will choose manageable approximations, estimate expressions with several operations and roots, and explain what an estimate can tell you.
4.01c - Error intervalsA displayed measurement does not usually tell you the exact size of an object. You will recover the interval of possible values from rounding or truncation, use the correct inequality signs and decide what a measurement guarantees.
4.01c - Bounds in calculationsRounded inputs create a range of possible answers. You will choose endpoints for sums, differences, products and quotients, then distinguish the largest possible count from an excluded limit for a continuous measurement.

Algebra

Expressions and formulae

6.01a, 6.01b - Writing and simplifying expressionsHow can letters make a calculation shorter? Learn to recognise algebraic expressions and statements, identify terms and factors, and collect like terms without changing their meaning.
6.01c - Multiplying and dividing algebraic termsWhy is x+x different from x× x? Multiply and divide algebraic terms by treating coefficients and variable factors carefully, then combine powers without losing the restrictions on division.
6.02b, 6.02e - Substituting into formulaeCan the same formula work with positive and negative inputs? Substitute values into expressions and formulae, keeping signs, powers, roots and fractions in the right order, then use supplied formulae for motion with constant acceleration.
6.02a - Building expressions from situationsHow can one rule describe many different purchases or shapes? Choose variables with clear meanings, translate operations in the right order, and build expressions and formulae from fixed amounts, changing amounts and geometric measurements.

Brackets and factorisation

6.01d, 6.01e - Expanding and factorising a bracketWhy must a number outside brackets multiply every term inside? Use distribution to expand brackets, combine expanded expressions, and reverse the process by taking out the greatest common factor.
6.01d, 6.01e - Quadratic expressionsWhat happens when both factors contain a letter? Multiply two binomials, reverse the expansion to factorise monic quadratics, and use the cancellation in a difference of two squares, including exact square roots.
6.01c–6.01e - More demanding algebraic productsFractional powers, extra brackets and a coefficient before a squared term all make algebra look less familiar. You will extend index laws in 6.01c, expand several binomials in 6.01d and factorise general quadratics in 6.01e. The same habit helps throughout: recognise the structure before doing th...
6.01f - Completing the squareThe same quadratic can be written as an expanded expression or as a shifted square plus a constant. You will complete the square for monic and non-monic quadratics, keep fractional values exact and check each result by expansion. This form will later make important features of quadratic graphs ea...
3.03b - Rationalising denominatorsAn exact answer can be written in several equivalent ways. You will remove surds from denominators, first for a single surd and then for a sum or difference, while keeping the value unchanged. Simplifying the surds before choosing a method often saves work.
6.01g - Algebraic fractionsAlgebraic fractions follow the same arithmetic rules as numerical fractions, but their denominators can rule out particular values of the variable. You will factorise before cancelling, multiply and divide fractions, and combine sums and differences over a common denominator. Keeping the original...

Equations and rearranging

6.03a - Solving linear equationsAn equation says that two expressions have the same value. You will keep that balance while undoing operations, collecting unknown terms, and handling brackets and fractions. You will also turn a word problem into an equation and decide what its solution means in the original situation.
2.02a - Recurring decimals as fractionsA recurring decimal has an infinite tail, but it can still represent an exact fraction. You will use division to identify repeating blocks, then align and subtract shifted copies to convert recurring decimals back into fractions. The method works when the repeating block starts immediately or fol...
6.02c - Changing the subjectA formula can answer a different question when you rearrange it. You will make a chosen letter the subject, including when it is squared, in a denominator or appears twice, and check which values the rearrangement permits.
6.03b - Solving factorised quadraticsA quadratic equation can have two answers, even when the question looks like one calculation. You will use factorisation to find both roots of monic quadratics, then decide which roots fit a problem's conditions.
6.02d, 6.03b - Solving general quadraticsWhat can you do when a quadratic has no convenient integer factors? Under 6.03b, you will rearrange and solve general quadratics by factorisation, completing the square or the quadratic formula; 6.02d adds recognising and using that formula. You will keep exact roots when needed and understand wh...
6.03c - Simultaneous linear equationsOne equation in two unknowns can leave many possibilities. A second equation can pin down the same pair of values. You will understand simultaneous solutions, find them by elimination or substitution, and build pairs of equations from information about two unknown quantities.
6.02c, 6.03a - Equations with algebraic fractionsMultiplying away a denominator can make an equation much easier, but it can also conceal a forbidden value. Under 6.03a, you will solve equations containing algebraic fractions; under 6.02c, you will rearrange fractional formulae. You will keep the original restrictions and check the resulting an...
6.03c - Simultaneous equations with quadraticsA line and a curve can share two points, so simultaneous equations may have two different solution pairs. You will substitute to obtain a quadratic, keep each coordinate with its matching partner, and form equations for pairs of unknown numbers.

Inequalities and reasoning

6.04a - Linear inequalitiesAn inequality describes a range of possible values rather than one answer. Learn to solve linear inequalities, keep their direction correct and show exactly which values are included on a number line.
6.04a - Quadratic inequalitiesA quadratic can be positive for two separate ranges of inputs. Find its zero boundaries, determine its sign between and beyond them, then state the complete solution with the correct endpoints and set notation.
6.01a - Identities and algebraic argumentsTwo expressions can look different yet give the same result for every input. Learn to distinguish an identity from an equation to solve, show equivalence using algebra and build arguments that explain why a result holds.
6.01a - Proof with algebraAlgebra can explain patterns for every integer, including ones you could never finish checking. Represent integers precisely, prove divisibility and parity claims, and judge whether a chain of reasoning really establishes its conclusion.

Functions and sequences

6.05a - Inputs and outputsA function turns an allowed input into one output using a fixed rule. Read and write those rules as expressions or machines, follow their operation order and work backwards to recover a missing input.
6.05a - Inverse and composite functionsAn inverse function reverses a rule, while a composite function applies rules in succession. Use words, expressions and machine diagrams to build these processes, keep their order straight and decide when a reverse rule gives one unambiguous answer.
6.06a, 6.06b - Rules for sequencesHow could you find the 100th number in a pattern without listing the first 99? Learn the difference between a term and its position, generate sequences from rules, find arithmetic nth terms and recognise triangular, square and cube numbers.
6.06b - Fibonacci and geometric sequencesHow can a pattern grow when the amount added keeps changing? Learn to generate Fibonacci sequences, recognise quadratic patterns and identify geometric progressions by a constant positive rational multiplier.
6.06a, 6.06b - Quadratic and other sequence rulesCan the differences between terms reveal a formula for a whole sequence? In this Higher lesson, learn subscript and recurrence notation, explain the second-difference method for quadratic nth terms, and build other rules from fractions and repeated surd multipliers.

Ratio and proportion

Ratio calculations

5.01a, 5.01c - Ratios and fractionsA drink contains two measures of concentrate for every three measures of water. What fraction is concentrate? Learn to simplify ratios in common units, write them as one to n and distinguish a comparison between parts from a fraction of the whole.
5.01b - Sharing in a ratioIf two people share £84 in the ratio two to five, why should you divide by seven? Learn to share a total, reconstruct missing amounts from one share or a difference, and divide a quantity between three or more recipients.
5.01d - Using ratios in problemsA recipe for four people must feed six: should you add the same amount to every ingredient or multiply them all? Learn to scale recipes, connect ratios through a shared quantity and turn a fixed comparison into a linear input-output rule.

Geometry and measures

Shapes and angles

8.01a–8.01c, 8.01e, 8.01f - Geometrical language and measurementA diagram can communicate a whole set of instructions, provided its labels mean the same thing to everyone. Learn the language for lines, angles and polygons, then use rulers, protractors and compasses to turn a written description into a clear figure.
8.03a–8.03c - Angles on linesA single marked angle can determine several others without a protractor. Use turns, straight lines and parallel lines to calculate missing angles, explaining the geometrical reason for each step.
8.03d, 8.04a–8.04c - Angles and properties of polygonsWhy can a regular octagon have equal angles without any of them being right angles? Build the angle rules for triangles and larger polygons, use quadrilateral properties with reasons, and identify reflection and rotational symmetry.
9.02a, 9.02b - Congruent trianglesTwo triangles can match exactly even when one is turned over or points in a different direction. Match their vertices correctly, use the four congruence tests, and transfer equal lengths and angles only after you have justified the match.
8.03a–8.03d, 8.04a, 8.04b - Proof with angles and shapesA measured diagram can suggest a result, but a proof must establish it for every figure satisfying the given conditions. Build formal chains from angle facts, triangle properties and quadrilateral properties, keeping each reason beside the statement it supports.

Constructions and scale

8.02a–8.02c - Ruler and compass constructionsHow can you find an exact middle without measuring the length? Use equal compass radii to construct perpendicular bisectors (8.02a), angle bisectors (8.02b) and perpendicular lines (8.02c). Understand why the perpendicular gives the shortest distance to a line.
8.02d - Loci and equidistanceWhere could a lamp stand if it must be near one doorway and equally far from two walls? Turn distance conditions into circles, bisectors and shaded regions. Combine them carefully, including only the boundaries the wording allows.
10.01c - Scale drawings and bearingsA small map can locate a place kilometres away, but only if length and direction are read correctly. Convert map scales, measure clockwise bearings from north and build scale drawings that preserve both. Find the bearing for the return journey too.

Measurement and area

10.01a - Converting unitsDoes changing metres to centimetres multiply an area by 100 too? Convert length, mass, area, volume, capacity, time and money without changing the quantity. Use the same reasoning when measurements contain letters, and see why square and cubic units need different factors.
10.02a, 10.03a–10.03c - Perimeter and polygon areaDoes cutting a notch out of a shape always increase its perimeter? Trace rectilinear boundaries (10.02a), then build the area rules for triangles (10.03a), rectangles and parallelograms (10.03b) and trapezia (10.03c). Choose perpendicular heights, use square units and work backwards to missing di...
8.05a - Parts of a circleWhich part of a wheel does a spoke follow, and which part does a tyre follow? Learn the names of the straight lines, curved lines and regions in a circle, so that a diagram tells you exactly what to measure.
3.03a, 6.02d, 10.02b - Circumference and arc lengthHow much edging goes round a circular flower bed, and how much goes round just part of it? Use the circumference formula from 6.02d and 10.02b, preserve exact multiples of pi from 3.03a, and calculate arc lengths as fractions of a full circle.
6.02d, 10.03d - Circle and sector areaA sprinkler covers a circular garden, but sometimes turns through only part of a full revolution. Use the circle-area formula from 6.02d and 10.03d to find how much ground it waters, calculate sector areas, and work backwards to an unknown radius or angle.
10.02c, 10.03e - Composite perimeters and areasTwo gardens begin as identical rectangles. One gains a semicircle, and the other loses a semicircle: could their perimeters still be equal? Use 10.02c to trace composite boundaries and 10.03e to add or subtract the areas of simpler regions.

Solids and volume

8.01d, 8.06a, 8.06b - Solids, plans and elevationsCould two different buildings have the same front view? Use the solid vocabulary in 8.01d and 8.06a to describe three-dimensional shapes, then use 8.06b to connect solids with plans, elevations and isometric drawings.
10.04a - Volume and surface area of prismsCan two differently shaped containers hold the same amount? Learn to calculate the volume of cuboids, right prisms and cylinders, then unfold their surfaces to work out how much material a container needs.
10.04b, 10.04c - Cones, spheres and pyramidsWhy does joining two solids remove some of their surface area? Use the supplied cone and sphere formulae in 10.04b, build pyramid calculations in 10.04c, and distinguish the space inside a composite solid from the surfaces you can actually touch.
10.01b - Speed, density and pressureIs travelling at 36 km/h faster than travelling at 12 m/s? Understand what “per” means, use speed, pay, unit prices, density and pressure, and convert compound units correctly even when the quantities contain letters.

Transformations and similarity

7.01a, 8.01g - Coordinates in four quadrantsCan changing the order of two numbers move you to a different place? Use the four-quadrant coordinates in 7.01a to locate points, then apply 8.01g to solve shape problems and describe movements accurately.
9.01a–9.01c - Reflection, rotation and translationHow can a designer move a tile without changing its shape or size? Reflection in 9.01a flips it, rotation in 9.01b turns it, and translation in 9.01c slides it. You will find image coordinates and describe each move precisely enough for someone else to repeat it.
9.01d - Combining transformationsDoes “flip, then slide” give the same result as “slide, then flip”? In this Higher lesson you will follow sequences of isometries, test whether their order matters and describe an equivalent single move when one exists. You will also distinguish a property that stays unchanged from a point that s...
9.04b - EnlargementA drawing can be made twice as wide or half as wide without distorting it. Both moves are called enlargements in mathematics. You will use a centre and a positive scale factor to place the image, then work backwards to find those two details from corresponding shapes.
9.04b - Negative scale factorsCan an enlargement send a shape to the opposite side of its centre? In this Higher lesson a negative scale factor does exactly that. You will separate the factor's sign from its size, construct negative images and recover the centre and factor from corresponding shapes.
9.04a, 9.04c - Similar triangles and lengthsTwo triangular signs have the same shape but different sizes. Can one measured side tell you every other length? In 9.04a you will recognise and prove triangle similarity; in 9.04c you will match corresponding sides, calculate unknown lengths and compare supplied measurements using the appropriat...
9.04c - Area and volume scale factorsA model is twice as tall, twice as wide and twice as deep. Does it hold twice as much? This Higher lesson explains why similar areas use the square of the length factor and similar volumes use its cube. You will use those relationships forwards and backwards, and compare capacity with surface area.

Right-angled triangles

6.02d, 10.05a - Pythagoras' theoremA rectangular garden is 6 metres by 8 metres. How long is the straight path between opposite corners? Use squared areas to explain Pythagoras' theorem, then find missing lengths in right-angled triangles and simple two-dimensional shapes.
6.02d, 10.05b - Right-angled trigonometryA ramp rises at a fixed angle: how does its height change when its length doubles? Similar right-angled triangles turn that question into three useful side ratios. Learn to choose a ratio, find missing lengths and angles, and connect a ground-level direction to a bearing.
10.05c - Exact trigonometric valuesCan you find a triangle's height without rounding a calculator value? Two familiar shapes, a square and an equilateral triangle, give exact trigonometric ratios. Use them to build the standard values, understand the boundary angles and calculate exact lengths.
10.05a, 10.05b - Lengths and angles in three dimensionsHow long is a straight support running from one corner of a box to the opposite corner, and how steeply does it rise above the base? In this Higher tier lesson, find flat right-angled triangles inside solids, then apply Pythagoras and trigonometry in those triangles. The drawing represents three...

General triangles

6.02d, 10.05d - The sine ruleFind distances across a triangle without a right angle. The formula in 6.02d comes from a shared perpendicular height; 10.05d uses it to find sides and angles, including the cases where two different triangles fit the same measurements.
6.02d, 10.05e - The cosine ruleFind the third side when two sides and their included angle are known, or find an angle from three sides. Derive the 6.02d formula from Pythagoras, then use 10.05e to choose and combine triangle rules without losing accuracy.
6.02d, 10.03a - Triangle area using sineFind a triangle's area without being given its perpendicular height. Derive the 6.02d formula from base and height, then apply 10.03a to included angles, missing lengths and angles, and problems that first need a sine or cosine rule.

Circle theorems

8.05b–8.05d - Angles at the centre and circumferenceWhy can a camera move along a circular track while keeping the same viewing angle between two fixed posts? In 8.05b you will connect angles at the centre and circumference, in 8.05c explain the right angle made by a diameter, and in 8.05d prove why angles in the same segment stay equal.
8.05e–8.05g - Chords and tangentsHow can the centre of a circular design help you split a chord exactly or predict the angle where a straight line touches it? In 8.05e you will prove the chord-bisector relationship, in 8.05f connect a tangent to its radius, and in 8.05g link a tangent–chord angle to an angle across the circle.
8.05h - Cyclic quadrilateralsFour posts on a circular boundary form a quadrilateral. Why do an angle of 108° and its opposite angle have to make 180°? You will recognise cyclic quadrilaterals, prove their opposite-angle relationship and combine it with other circle theorems.

Graphs and functions

Curves and equations

6.03d, 7.01b, 7.01c - Quadratic graphsCan two different inputs give the same output, and can a curve cross an axis twice? Plot quadratic curves and use their symmetry, intercepts and turning points to connect a picture with an equation.
6.03d, 7.01c - Turning points and intersectionsCan an equation reveal a curve's lowest point before you draw it? Use completed-square form to identify turning points, then connect curve–line intersections with simultaneous equations. This lesson is for Higher tier.
7.01b, 7.01c - Cubic and reciprocal graphsCan a graph have a missing input and still contain infinitely many points? Plot cubic and reciprocal curves, recognise their basic shapes, and handle expressions that combine different terms.
7.01b, 7.01d - Exponential graphsA graph can keep falling without ever reaching zero. In 7.01b you will plot exponential values, then in 7.01d you will recognise growth, decay and the special case between them. These are Higher-tier graph skills.
6.03e - Approximating roots by sign changesYou can locate a solution without an exact formula for it. Use positive and negative outputs to trap a root, narrow its interval systematically and justify the number of decimal places you report. This is a Higher-tier method.

Further graphs

7.01f, 7.02b - Circles and perpendicular tangentsA circle equation describes every point the same distance from its centre. In 7.01f you will use an origin-centred circle, and in 7.02b you will find perpendicular lines and the tangent at a given point. These are Higher-tier coordinate-geometry skills.
7.01e - Sine, cosine and tangent graphsA rotating point can travel round a circle while its height rises and falls. Use that motion to understand the sine and cosine curves, then see why tangent has repeating breaks. All angles in this Higher-tier lesson are in degrees.
7.03a - Translating and reflecting graphsAn equation can move a whole graph without changing its shape. Track a few key coordinates to identify vertical and horizontal translations, then reflections in the coordinate axes. These are Higher-tier skills, and you can use them directly with equations without special function notation.
6.04b, 7.02a - Regions defined by inequalitiesAn inequality can describe a whole side of a line rather than one line. In 7.02a you will draw solid and dashed boundaries, then in 6.04b you will combine conditions to find a common region and its integer points. These are Higher-tier graph skills.

Graphs in context

7.04a, 7.04b - Real-world graphs and ratesHow can a falling line describe someone travelling forwards through time? In 7.04a, learn to read and construct graphs of journeys, money conversions and proportional relationships. In 7.04b, connect straight-line gradient to a rate, using the axis labels to explain what the number means.
7.04b - Average and instantaneous ratesA car can average 40 km/h over a journey while its speedometer shows many different speeds. Learn to distinguish an average over an interval from a rate at one instant, using chords and tangents. Then interpret the sign and units in journey, velocity and financial graphs.
7.04c - Areas under graphsCan an object travel twenty metres and finish where it started? Learn how areas under velocity-time graphs record motion, and why signed area and total distance can differ. Use rectangles, triangles and trapezia for straight segments, estimate curved areas, and work out what an area means from it...

Probability

Representing outcomes

11.02a, 11.02b - Systematic lists and sample spacesTwo fair dice can total seven in several ways, but total two in only one. In 11.02a, use outcome grids to calculate probabilities without treating different totals as equally likely. In 11.02b, list possibilities systematically, deciding first whether order matters and whether repetition is allowed.
11.02a, 11.02b - Counting and choosing representationsA menu can offer dozens of meals without listing each one. Under 11.02b you will count combined choices efficiently and decide when order creates a new outcome. Under 11.02a you will choose a representation that makes a probability problem manageable.
11.02c, 11.02e - Sets, Venn diagrams and additionIf 18 students play football and 14 make music, must there be 32 different students? Under 11.02c you will describe sets and use Venn diagrams to keep overlapping groups separate. Under 11.02e you will calculate probabilities of “not”, “or” and “both” without counting anybody twice.

Statistics

Data and sampling

12.01a - Populations, samples and biasCan asking more people make a survey less misleading? You will identify the population a study concerns, select a simple random sample and judge how far sample results support a population conclusion. The selection method matters as well as the number of responses.
12.02a - Tables and chartsCan two pie charts have identical sectors even when one survey contains twice as many people? You will classify and tabulate data, build suitable charts and compare groups without confusing a count with a proportion. A useful display makes the question easier to answer while preserving what the d...
12.02a, 12.03b - Time series and misleading graphsA graph can show the right numbers and still give the wrong impression. In 12.02a you will construct and interpret time-series graphs, separating long-term trends from seasonal changes. In 12.03b you will check whether scales, labels and presentation support the claims being made.