Cambridge IGCSE Mathematics

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Number

Whole numbers

C1.1, C1.5, C1.6; E1.1, E1.5, E1.6 - Integers and the number lineA minus sign changes where a number belongs, how you compare it and what a calculation means. You will use a number line to recognise integers, order positive and negative numbers, add and subtract, and work out changes that cross zero.
C1.1, C1.6; E1.1, E1.6 - Place value and whole-number calculationsA zero can change a number enormously without adding anything of its own. You will read and write large whole numbers, explain the value of each digit, and use the four operations with checks that can catch a misplaced digit.
C1.1; E1.1 - Factors, multiples and primesSome numbers can be arranged into several different rectangles; others allow only one row. You will use exact division to find factor pairs, distinguish prime and composite numbers, and decide whether a number is a multiple of another.
C1.1; E1.1 - Prime factorisationDifferent factor pairs can lead to the same set of prime building blocks. You will break a positive integer into prime factors using a tree or repeated division, then write repeated factors compactly using powers.
C1.1; E1.1 - Highest common factors and lowest common multiplesSharing objects into equal groups and finding when two repeating events meet use the same number facts in different ways. You will find highest common factors and lowest common multiples by listing and prime factors, then choose the one a problem needs.
C1.1, C1.3; E1.1, E1.3 - Squares, cubes and rootsPowers count repeated equal factors; roots reverse that process. You will recognise square and cube numbers, recall the key values, and work out other whole-number powers and exact roots. The same methods apply to Core and Extended.
C1.6; E1.6 - Signed products and order of operationsMultiplication and division with negative numbers follow a consistent pattern. You will use that pattern in directed calculations, work through brackets in the correct order, and see why a minus sign outside a square changes its value. These skills are shared by Core and Extended.

Fractions and decimals

C1.4, C1.5; E1.4, E1.5 - Fractions as numbersA fraction is a number, even when it is written in several different ways. You will move between improper fractions and mixed numbers, simplify equivalent fractions, and compare their sizes using equal-sized parts. This lesson covers the shared Core and Extended fraction skills.
C1.6; E1.6 - Adding and subtracting fractionsAdding fractions means combining counts of equal-sized parts. You will use common denominators to add and subtract, simplify the result, and handle improper fractions, mixed numbers and negative values. The methods are the same for Core and Extended.
C1.1, C1.6; E1.1, E1.6 - Multiplying and dividing fractionsTaking a fraction of a quantity can make it smaller, while dividing by a fraction can make it larger. For Core and Extended, you will multiply fractions, use reciprocals to divide, and calculate with mixed numbers and negative fractions. C1.1/E1.1 supplies the reciprocal idea; C1.6/E1.6 puts it t...
C1.4, C1.5, C1.6; E1.4, E1.5, E1.6 - Decimal place value and calculationsA decimal point tells you the size of every digit, so a small change in its position can change an answer tenfold. For Core and Extended, C1.4/E1.4 develops decimal notation, C1.5/E1.5 compares values, and C1.6/E1.6 covers the four operations. You will use place value, signs and inverse operation...
C1.4, C1.5; E1.4, E1.5 - Fractions, decimals and percentagesA fraction, a decimal and a percentage can describe exactly the same amount. For Core and Extended, C1.4/E1.4 connects these representations and C1.5/E1.5 lets you compare their values. You will convert terminating decimals in both directions, interpret percentages above one hundred, and order a...
C1.1; E1.1 - Rational and irrational numbersSome exact numbers can be written as fractions of integers, while others cannot. For Core and Extended, C1.1/E1.1 uses this difference to distinguish rational and irrational numbers. You will recognise terminating and recurring decimals, classify exact roots, and separate a number from a rounded...

Powers and accuracy

C1.7; E1.7 - Integer index lawsPowers can make a calculation much shorter, even when an index is zero or negative. You will use repeated factors to explain the index laws, then combine them to find exact numerical values. This lesson is for Core and Extended.
E1.7 - Fractional indicesAn index can tell you to take a root as well as a power. You will interpret fractional and negative fractional indices, choose a sensible order for the calculation, and find exact numerical values. This lesson is Extended only.
C1.8; E1.8 - Writing numbers in standard formStandard form separates a number's digits from its size. You will write large and small positive numbers using powers of ten, convert them back, and compare their magnitudes. This lesson is for Core and Extended, including non-calculator work.
C1.8; E1.8 - Calculating in standard formYou can multiply tiny and huge numbers without writing out all their zeros. You will use index laws for products and quotients, match scales for sums and differences, and check calculator displays. Core calculates with standard form on Paper 3 only; Extended also calculates with standard form on...
C1.9; E1.9 - Rounding and estimationA rounded answer should keep the information the problem needs. You will round to decimal places, significant figures and named place values, make quick estimates, and decide whether a practical answer needs rounding up or down. This lesson is for Core and Extended.
C1.10; E1.10 - Limits of rounded measurementsA rounded measurement stands for a range of possible values. You will find that range, identify its lower and upper bounds, and decide what the recorded measurement does and does not tell you.
E1.18 - Simplifying and combining surdsSome square roots cannot be written as terminating or recurring decimals. In this Extended lesson, you will keep them exact, simplify them by extracting square factors, combine matching surds and multiply surds without losing accuracy.

Proportional reasoning

C1.11; E1.11 - Simplifying and sharing ratiosA ratio compares quantities by equal-sized parts. You will put quantities into common units, simplify ratios with two or three terms, and divide a total fairly according to a stated ratio.
C1.11; E1.11 - Scaling quantities and comparing valueProportional reasoning connects a known quantity to a new one by keeping the same relative amounts. You will scale through one unit, adapt recipes, use map scales and compare prices fairly, while checking when proportional reasoning is justified.
C1.13; E1.13 - Percentages of quantitiesPercentages describe amounts relative to a chosen whole. You will calculate percentages of quantities, express one amount as a percentage of another, and choose the correct base for deposits, discounts, earnings, profit and loss.
C1.13; E1.13 - Percentage change and interestPercentages can change a price, measure a rise in earnings or describe interest on savings. You will use multipliers for increases and decreases, calculate percentage change, and distinguish simple interest from compound interest.
E1.13, E1.17 - Reverse percentages and repeated changeA percentage change uses the value at the start of that change. You will recover an original amount, then model repeated growth and decay, including population change and depreciation. These Extended methods connect reverse percentages in E1.13 with exponential change in E1.17.

Measures and rates

C1.15; E1.15 - Time, clocks and timetablesA journey can finish on the next day even when it lasts less than two hours. You will convert time units, read clocks and timetables, calculate durations and compare local times. C1.15 and E1.15 share these skills, including the important difference between decimal hours and hours with minutes.
C1.16, C1.12; E1.16, E1.12 - Money and exchange ratesMoney calculations need both accurate arithmetic and the correct currency. You will calculate costs and change, convert currencies in both directions and decide how to round a payment. C1.16 and E1.16 provide the money skills; C1.12 and E1.12 connect exchange rates to proportional reasoning.
C1.12; E1.12 - Rates and average speedA rate tells you how much of one quantity corresponds to one unit of another. You will use pay, flow and fuel rates, calculate average speed and apply supplied formulas for density, pressure and population density. C1.12 and E1.12 share these methods, with units helping you choose each calculation.
C1.14; E1.14 - Using a scientific calculator wellA calculator follows the expression you enter, including every bracket and sign. You will enter calculations accurately, retain intermediate precision and interpret answers involving time and money. These C1.14 and E1.14 skills are for calculator Papers 3 and 4, using a scientific calculator with...
E1.10 - Calculating with boundsRounded measurements describe intervals of possible values, so calculations using them also have limits. You will choose the inputs that make a sum, difference, area or speed as small or large as possible. This Extended E1.10 lesson keeps the rounding intervals and the calculation separate so tha...

Algebra

Expressions

C2.1, C2.5; E2.1, E2.5 - Letters, expressions and substitutionLetters let you describe a calculation before you know all its numbers. You will read algebraic notation, turn words into expressions and formulas, and find their values by substitution, including negative numbers and fractions.
C2.2; E2.2 - Collecting like termsSome expressions look longer than the quantities they describe. You will identify terms and coefficients, combine matching terms without changing their value, and distinguish powers and products that must stay separate.
C2.2; E2.2 - Expanding and taking out common factorsMultiplying a whole sum and multiplying its separate terms give the same result. You will use that fact to expand brackets, handle negative signs, and reverse the process by taking out the highest common factor.
C2.2; E2.2 - Expanding two bracketsTwo brackets multiply two whole quantities, so every term in one must multiply every term in the other. You will organise those products, collect the resulting terms and check an expansion without confusing a successful numerical check with a proof.
E2.2 - Several brackets and grouped factorsThis Extended lesson takes the distributive law beyond two brackets in one variable. You will expand products with several letters or three brackets, then reverse expansion to factorise grouped terms and cubic expressions with a common factor.
E2.2 - Factorising quadratic expressionsFactorising reverses expansion: a sum becomes a product whose structure is easier to see. You will factorise monic and non-monic quadratics, recognise square patterns and check that every factor has been found. This lesson is for the Extended route.
E2.2 - Completing the squareA quadratic can be written as a square plus a correction, even when it has no simple integer factors. You will build that form, handle coefficients other than one and verify that your new expression is identical to the original. This lesson is for the Extended route.

Indices and fractions

C2.4; E2.4 - Index laws with lettersLetters obey the same integer index laws as numbers. You will simplify products, quotients and powers, state when zero is excluded and recognise an unknown exponent by writing both sides with a common base. This lesson covers the shared Core and Extended content.
E2.4 - Fractional powers with lettersFractional powers connect algebraic indices to roots. You will simplify fractional and negative powers with their real-number conditions, combine index laws and solve common-base equations whose exponents contain linear expressions. This lesson is for the Extended route.
E2.3 - Calculating with algebraic fractionsAlgebraic fractions follow the same four operations as numerical fractions, with an extra responsibility: some variable values may make the original calculation undefined. You will factorise before cancelling, use common denominators for sums and differences, and retain every original restriction...

Equations and formulas

C2.5; E2.5 - Solving linear equationsAn equation turns an unknown number into something you can find. You will keep equality true while undoing operations, solve equations with brackets and numerical fractions, and build an equation from a situation before checking the answer.
C2.5; E2.5 - Simultaneous linear equationsOne equation can leave many possible pairs of numbers. You will find a pair that satisfies two equations together, use elimination or substitution, and construct equations for two unknown quantities in a situation.
C2.5; E2.5 - Changing the subjectA formula can describe the same relationship with a different quantity on its own. You will reverse operations to change the subject, handle numerical and algebraic fractions, and check that a rearranged formula still represents the original relationship.
E2.5 - Rearranging powers and repeated subjectsSome formulas hide the target inside a power, a root or more than one term. In this Extended lesson, you will isolate the relevant expression, choose the correct inverse operation, and collect repeated targets before factorising and dividing.
E2.5 - Solving quadratic equationsQuadratic equations can have two, one or no real solutions. In this Extended lesson, you will solve by factorisation, completing the square and the supplied quadratic formula, keep exact roots when needed, and decide which solutions fit a situation.
E2.5 - Equations with algebraic denominatorsAn equation can appear to offer an answer that makes its original fractions impossible. You will identify excluded values, clear numerical and linear algebraic denominators, and check every candidate solution in the original equation.
E2.5 - A line and a nonlinear equationTwo simultaneous equations can have more than one solution pair. You will substitute a linear relationship into a nonlinear equation, find every resulting pair, and decide which pairs fit a stated situation.

Inequalities and sequences

C2.6; E2.6 - Inequalities on a number lineOne small line under an inequality sign decides whether a boundary value is allowed. You will read strict and inclusive inequalities, represent intervals with the correct circles, and list integer values that satisfy both ends of an interval.
E2.6 - Solving linear inequalitiesSolving an inequality finds a whole range of values rather than one required value. You will preserve or reverse its direction correctly, solve compound inequalities, and interpret the result when a quantity must be a whole-number count.
C2.7; E2.7 - Patterns and linear sequencesA pattern can tell you what comes next or let you jump straight to a distant term. You will connect number and shape patterns, distinguish a term's position from its value, and derive rules for sequences with a constant difference.
C2.7; E2.7 - Square and cube sequencesSome sequences grow by an ever larger amount. You will recognise square and cube patterns, find simple nth-term rules, and relate a new sequence to one you already know. A rule lets you reach a distant term without listing everything before it.
E2.7 - Quadratic, cubic and exponential sequencesA sequence can hide a square, a cube or a repeated multiplier behind extra terms. You will use differences and comparisons with familiar sequences to uncover nth-term rules, then use those rules at any position. This lesson develops the Extended sequence methods.
E2.8 - Direct and inverse proportionThe same change in an input can double, quadruple or halve an output, depending on the relationship. You will translate proportion statements into equations, find their constants and calculate unknown quantities. This Extended lesson includes proportion involving squares, cubes and roots.

Geometry

Shapes and construction

C4.1; E4.1 - Lines, angles and polygonsGeometry begins with descriptions you can trust. You will name lines, angles, triangles, quadrilaterals and polygons, and use their properties to decide what a shape really is even when it has been turned round.
C4.1, C4.2; E4.1, E4.2 - Solids and their netsA flat drawing can describe a three-dimensional object, or even fold into one. You will recognise solids and their faces, edges and vertices, then draw and interpret nets with the matching lengths that make them fit together.
C4.1; E4.1 - Parts of a circleA circle has several different kinds of lines, curves and regions. You will distinguish them precisely, use the relationship between radius and diameter, and explain the difference between an arc, a sector and a segment.
C4.2; E4.2 - Measuring and constructing trianglesA careful drawing can solve a geometry problem without guessing its shape. You will use a ruler and protractor accurately, construct triangles from three side lengths with compasses, and combine two triangles to construct a rhombus.
C4.1, C4.3; E4.1, E4.3 - Scale drawings and bearingsA plan can put a long journey on one page, but distance alone does not locate the destination. You will draw and interpret lengths to scale, use three-figure bearings to give directions, and find the bearing for a return journey.
C4.5; E4.5 - Line and rotational symmetryA shape can look unchanged after a fold or a turn, but these are different kinds of symmetry. You will recognise both, find rotational order, and use symmetry to explain equal lengths and angles in familiar shapes.
E4.1, E4.5 - Planes and symmetry in solidsA solid needs more than a mirror line to describe its symmetry. E4.1 introduces planes, perpendicular bisectors, hemispheres and frustums; E4.5 uses planes and rotation axes to describe how three-dimensional shapes match themselves.

Angle reasoning

C4.6; E4.6 - Angles at points and in trianglesAn angle can be forced by the lines around it without being measured. You will calculate angles at points, at crossing lines, and inside triangles and quadrilaterals, giving a geometric reason for each step.
C4.6; E4.6 - Angles in parallel linesParallel lines let an angle at one crossing determine an angle at another. You will distinguish corresponding, alternate and co-interior angles, then combine these facts with straight-line and triangle sums in reasoned chains.
C4.6; E4.6 - Angles in polygonsA regular polygon's angles follow from its number of sides, so you can calculate them without measuring. You will build interior-angle sums from triangles, understand exterior angles as turns, and work backwards from an angle to the number of sides.
E4.6 - Reasoning with irregular polygonsIrregular polygons keep the angle-sum rules without having equal angles. You will calculate missing interior and exterior angles, handle an inward-pointing corner carefully, and combine a polygon sum with angles in a smaller triangle.

Similarity and circles

C4.1, C4.4; E4.1, E4.4 - Similar shapes and length scale factorsA photograph can change size while keeping its shape. You will recognise similar and congruent shapes, match corresponding sides and angles, and use a length scale factor to find missing lengths.
E4.4 - Explaining triangle similarityTwo triangles can have the same shape even when they overlap or face different ways. You will match their vertices, show similarity using geometric reasons, and use the resulting correspondence to calculate lengths.
C4.7; E4.7 - Right angles in circlesA circle can guarantee a right angle without a protractor. You will use a diameter to identify an angle in a semicircle, use the right angle between a tangent and its radius, and combine these facts with triangle angles while giving precise reasons.
E4.7 - Angles in the same circleThe same chord can connect several angles in a circle. You will identify the arc or segment each angle uses, calculate centre and circumference angles, use cyclic quadrilaterals, and select the correct angle between a tangent and a chord.
E4.8 - Chords and tangentsCircle symmetry controls lengths as well as angles. You will use perpendicular distances to equal chords, locate the centre using chord bisectors, and set equal the two tangent segments drawn from one external point.

Mensuration

Area and perimeter

C5.1; E5.1 - Metric units and conversionsA metre contains one hundred centimetres, but a square metre contains many more than one hundred square centimetres. You will convert metric lengths and masses, explain square and cubic conversion factors, and connect the volume of a container to its capacity in litres.
C5.2; E5.2 - Perimeters and areas of polygonsTwo shapes can reach the same height and stretch across the same base yet cover different amounts of space. You will find areas and perimeters of rectangles, triangles, parallelograms and trapezia, choose perpendicular heights, and work backwards to missing dimensions.
C5.3; E5.3 - Circle area and circumferenceDoubling a circular mat's radius doubles the distance round it, but uses four times as much material. You will calculate circumference and area, distinguish radius from diameter, keep exact answers in terms of π, and recover a circle's size from either measure.
C5.3; E5.3 - Arcs and sectorsA slice of a circle has both a curved edge and two straight edges. You will use equal shares of a full turn to find arc lengths and sector areas, then decide which lengths belong in the sector's total perimeter.
E4.1, E5.3 - Major and minor sectorsTwo points on a circle give you two routes round its edge. You will distinguish major and minor arcs and sectors, use angles that do not divide a full turn evenly, and work backwards from a sector's area or arc length.
C5.5; E5.5 - Compound areas and perimetersA shape made by joining or cutting familiar shapes still has an area and one outside boundary to follow. You will split compound shapes into useful pieces, find missing lengths, and combine straight edges with circular arcs while keeping exact answers.

Surface area and volume

C5.4; E5.4 - Prisms and cylindersThe amount a container holds and the material needed to make it are different measurements. In C5.4 and E5.4, you will calculate volumes using uniform cross-sections, build surface areas from nets, and distinguish a cylinder's curved area from its whole surface.
C5.4; E5.4 - Pyramids, cones and spheresPointed and rounded solids need different formulas from prisms. In C5.4 and E5.4, you will use the supplied volume and surface formulas, distinguish perpendicular height from slant height, and calculate both the curved and flat surfaces of half a sphere.
C5.5; E5.5 - Compound solids and exposed surfacesA joined solid can hide faces, and a cut can create new ones. In C5.5 and E5.5, you will add or subtract volumes, identify the exposed boundary of compound solids, and keep exact answers when parts of circles appear.
E4.4 - Area and volume scale factorsDoubling every length does more than double the material or capacity. In this Extended lesson, E4.4 connects corresponding lengths to areas, surface areas and volumes, and shows how to recover a length ratio from an area or volume ratio.
E5.5 - Frustums and similar solidsA frustum is the part left when a parallel cut removes the top of a cone or pyramid. In this Extended E5.5 lesson, you will reconstruct the missing full solid by similarity, subtract volumes, and identify the curved or flat surfaces exposed by the cut.

Trigonometry

Right-angled triangles

C6.1; E6.1 - Pythagoras and missing lengthsA diagonal can be shorter than a journey round two sides, but its length is not a guess. You will use the right-angle relationship between three sides, find either missing side, and check whether lengths can belong to a right-angled triangle.
C6.2; E6.2 - Using sine, cosine and tangentAn angle can tell you a missing length even when Pythagoras has only one known side to work with. You will name sides relative to an acute angle, choose sine, cosine or tangent, and rearrange a ratio when the unknown is above or below the fraction line.
C6.2; E6.2 - Finding angles in right trianglesSide lengths can reveal an angle when you reverse a trigonometric ratio. You will choose an inverse sine, cosine or tangent calculation, find the acute angle in degrees, and keep enough precision to round the final angle correctly.
C6.2; E6.2 - Bearings and linked right trianglesA map or a larger shape may hide several right triangles. You will choose a useful triangle, turn bearings into the angle a ratio needs, and carry an unrounded shared length from one triangle into the next.

Extended trigonometry

E6.2 - Elevation, depression and shortest distanceA sight line can tell you the height of an inaccessible object, provided you measure its angle from the right reference. You will interpret elevation and depression, connect right-angled triangles, and find the shortest distance from a point to a straight line. This lesson is for Extended.
E6.3 - Exact trigonometric valuesA calculator gives decimals for trigonometric ratios, but a carefully chosen triangle gives their exact values. You will derive the values for special angles, use surds in exact calculations, and explain what happens at zero and ninety degrees. This lesson is for Extended.
E6.5 - Sine and cosine rulesA triangle need not contain a right angle for its sides and angles to be connected. You will choose between the sine and cosine rules, calculate with obtuse angles, and decide when the same given information describes two different triangles. This lesson is for Extended.
E6.5 - Triangle area using sineTwo sides and the angle between them can determine a triangle's area even when its height is hidden. You will derive the sine-area formula, preserve exact answers, and work backwards from an area while checking possible angle branches. This lesson is for Extended.
E6.6 - Lengths and angles in three dimensionsA space diagonal belongs to a flat triangle even though the surrounding object is three-dimensional. You will locate useful right triangles in solids, calculate lengths in stages, and identify the angle between a line and a plane using a perpendicular projection. This lesson is for Extended.

Coordinate geometry

Lines and coordinates

C3.1; E3.1 - Coordinates in four quadrantsA pair of numbers can locate one point precisely, but their order matters. You will use Cartesian coordinates to plot and read positions on scaled axes and to complete shapes, developing the two-dimensional coordinates in C3.1 and E3.1.
C3.2, C3.3; E3.2, E3.3 - Drawing lines and finding gradientsA straight-line equation describes a whole set of points. You will make tables and draw linear graphs for C3.2 and E3.2, then measure their gradients from a grid for C3.3 and E3.3, including horizontal and vertical lines.
C3.5; E3.5 - Equations of straight linesA line's equation records its gradient and where it crosses an axis. You will interpret these features and find a fully simplified equation from a graph, including horizontal and vertical lines, for C3.5 and E3.5.
C3.6; E3.6 - Parallel line equationsParallel lines share a direction but occupy different positions. You will recognise equal gradients, find the equation of a parallel line through a stated point and check that it meets both conditions, as required by C3.6 and E3.6.

Extended line geometry

E3.3, E3.4 - Gradient, midpoint and distanceTwo coordinates can tell you a line's direction, a segment's centre and its length without a drawn grid. You will calculate gradient for E3.3 and use averages and Pythagoras to find midpoint and distance for E3.4. These methods also work when points lie in different quadrants.
E2.11, E3.2, E3.5 - Linear equations in different formsDifferent-looking equations can describe the same straight line. You will rearrange and obtain line equations for E3.5, draw their graphs for E3.2, and recognise and sketch the linear family in E2.11. Choosing a useful form makes gradients, intercepts and special cases easier to see.
E3.7 - Perpendicular lines and bisectorsA right angle has an algebraic signature: perpendicular sloping lines have related gradients. You will explain that relationship, find a perpendicular line through a point and combine it with midpoint calculations to find perpendicular bisectors, as required by E3.7.

Graphs and functions

Curves and equations

C2.10; E2.10 - Plotting quadratic and reciprocal graphsA table of values can reveal a curve, but the way you join its points matters. You will calculate and plot quadratic and reciprocal graphs, handle an excluded input and read roots and intersections to solve equations graphically for C2.10 and E2.10.
C2.11; E2.11 - Sketching lines and parabolasA sketch keeps a graph's essential features without calculating a long table. You will recognise and sketch straight lines and parabolas, marking intercepts, roots and symmetry for C2.11 and E2.11. The aim here is a clear shape and correct labelled features.
E2.10, E2.11 - Cubic, reciprocal and exponential curvesDifferent rules produce recognisable curve families, with different allowed inputs and long-term behaviour. You will build tables and draw the power and exponential graphs in E2.10, then recognise and sketch cubic, reciprocal and exponential forms for E2.11, using roots, symmetry and asymptotes.
E2.10, E2.11 - Turning points and growth modelsAn algebraic form can reveal a graph feature that a long table only estimates. You will find quadratic turning points by completing the square for E2.11, then draw and interpret exponential growth and decay and solve model equations by intersections for E2.10.
E2.6 - Regions defined by inequalitiesAn equation draws a boundary; an inequality chooses a side of it. You will draw the correct boundary, shade unwanted points and recover the inequalities defining an unshaded region. This lesson is for Extended.
E6.4 - Trigonometric graphs and equationsA turning radius connects triangle ratios to repeating graphs. You will recognise sine, cosine and tangent through a full turn and use their signs and symmetries to find every solution of a simple trigonometric equation. This lesson is for Extended; all angles are in degrees.

Functions and calculus

E2.13 - Functions, domain and rangeA function is a rule with specified allowed inputs, not just an expression on its own. You will use function notation and mappings, decide which inputs are permitted and find the outputs a function can actually produce. This lesson is for Extended.
E2.13 - Inverse and composite functionsTwo function machines can be connected, or one machine can be reversed. You will find inverse rules, choose restrictions that make reversing possible, and form and simplify composite functions in the correct order. This lesson is for Extended.
E2.12 - Gradients of curves and differentiationA curve can have a different gradient at every point. You will estimate a local gradient with a tangent, use differentiation to obtain an exact gradient rule for simple polynomials, and apply it at a specified point. This lesson is for Extended.
E2.12 - Stationary points and their natureA horizontal tangent identifies a stationary point, but its gradient alone does not tell you whether the point is a maximum or minimum. You will find stationary coordinates and justify their nature using nearby gradient signs, a second derivative or an accurate sketch. This lesson is for Extended.

Transformations and vectors

Transforming shapes

C7.1; E7.1 - Reflections and rotationsA shape can move without changing its size. You will draw reflections and rotations on a grid, then describe each movement precisely enough for someone else to reproduce it. C7.1 and E7.1 use the mirror line or the centre and turn to fix the image.
C7.1; E7.1 - Translations and enlargementsA translation moves every point by the same amount; an enlargement scales every distance from one centre. You will use column vectors, draw positive and fractional enlargements, and recover the movement from corresponding shapes. These are the translation and enlargement requirements of C7.1 and...
E7.1 - Negative enlargements and combined transformationsChanging a centre, the sign of a scale factor or the order of two movements can change an image completely. You will extend reflection to sloping lines, rotate about any centre, draw negative enlargements and carry out transformations in order. These are Extended applications of E7.1.

Vector reasoning

E7.2, E7.3 - Calculating with vectorsA vector records both the size and direction of a displacement. You will connect column, arrow and bold notation, combine movements and calculate their magnitudes. E7.2 supplies the arithmetic; E7.3 connects the two components to Pythagoras' theorem.
E7.4 - Position vectors and geometric pathsVector paths turn a diagram into algebra without needing to measure its angles. You will express directed journeys using two vectors, distinguish position vectors from displacements, and find points dividing a segment. These E7.4 methods prepare you to justify geometric relationships.
E7.4 - Parallel vectors and collinear pointsVector algebra can prove a geometric relationship even when a drawing is not to scale. You will recognise scalar multiples, justify collinearity using a common point, and connect ratio paths to similar triangles. These are geometric reasoning applications of E7.4.

Probability

Sets and chance

C1.2; E1.2 - Sets and two-set Venn diagramsA person can belong to two clubs without becoming two people. For Core and Extended, C1.2/E1.2 uses sets to organise such overlaps. You will describe sets, read union, intersection and complement notation, and fill two-set Venn diagrams without counting anyone twice.
E1.2 - Three sets and set relationshipsSet notation can describe one element, a whole collection, or a relationship between collections. This Extended lesson develops E1.2 through membership and subset symbols, three-set Venn regions and sets defined by conditions, including coordinate pairs. You will distinguish these meanings before...
C8.1; E8.1 - Single-event probabilityProbability measures chance on a scale from impossible to certain. For Core and Extended, C8.1/E8.1 connects that scale to equally likely outcomes and information in tables, graphs and two-set Venn diagrams. You will calculate single-event probabilities and use complements to find the chance that...
E8.1, E8.3 - Probability notation and Venn regionsProbability notation makes a long event description compact without changing its meaning. In this Extended lesson, E8.1 introduces event and complement notation, while E8.3 combines it with union and intersection in Venn diagrams. You will translate the symbols into regions and calculate probabil...

Experiments and combined events

C8.2; E8.2 - Relative and expected frequencyAn experiment can suggest a probability without revealing its exact value. For Core and Extended, C8.2/E8.2 uses relative frequency to estimate chance and expected frequency to predict counts. You will also distinguish fair, biased and random processes, and explain why a prediction is not a guara...
C8.3; E8.3 - Sample spaces and combined eventsTwo results together can have more possibilities than the short list of their totals suggests. For Core and Extended, C8.3/E8.3 uses complete sample spaces and two-set Venn diagrams to calculate combined probabilities. You will keep ordered outcomes distinct, count events systematically and handl...
C8.3; E8.3 - Tree diagrams with replacementA tree keeps track of successive results, even when their chances are unequal. This shared Core and Extended lesson uses independent trials and draws with replacement. You will label branches, multiply probabilities along a complete path and add the paths that make an event happen.
E8.3 - Events without replacementRemoving an object changes what can happen next. This Extended lesson develops E8.3 by updating the remaining counts, building dependent probability trees and calculating events reached by several paths. You will also compare the same experiment with and without replacement.
E8.4 - Conditional probabilityNew information can change which outcomes are still possible. This Extended lesson uses E8.4 to calculate conditional probabilities from lists, tables, Venn diagrams and trees. You will interpret “given that”, keep only the relevant outcomes and choose the denominator for that restricted group.

Statistics

Collecting and displaying data

C9.1; E9.1 - Classifying and organising dataA list of observations becomes useful when you know what each value means and where it belongs. You will distinguish types of data, record frequencies without losing observations, and use two-way tables to keep two classifications together.
C9.4; E9.4 - Bar charts, pie charts and pictogramsThe same data can tell different stories when you change its display. You will draw and read ordinary, dual and stacked bar charts, turn frequencies into pie-chart angles, and use pictogram keys including parts of symbols.
C9.4; E9.4 - Stem-and-leaf and frequency distributionsA good display can organise observations while keeping their exact values. You will choose a clear stem-and-leaf key, order and retrieve the observations, and build simple frequency distributions without confusing a value with how often it occurs.

Summarising and comparing data

C9.3; E9.3 - Mean, median, mode and rangeOne number can summarise a set of data, but different summaries answer different questions. You will calculate mean, median, mode and range from lists and exact-value frequency tables, then choose a summary that suits the purpose and recognise the effect of an extreme observation.
C9.2; E9.2 - Interpreting and comparing dataA statistical conclusion needs more than a correct calculation. You will draw supported inferences from displays, compare both centre and spread in context, and explain what a sample or a few summary numbers leave uncertain.
E9.2, E9.3 - Quartiles and interquartile rangeThe range can be dominated by a single unusual observation. You will find quartiles and the interquartile range, then compare the middle and spread of data using summaries less sensitive to extreme values.
E9.3 - Grouped-data averagesA grouped table tells you which interval contains each observation, but it hides the exact values. You will use midpoints to estimate a mean, explain why it is an estimate, and identify the modal class without pretending to know an exact mode.

Relationships and distributions

C9.5; E9.5 - Scatter diagrams and correlationA scatter diagram keeps two measurements from the same person or object together. You will plot paired data, recognise correlation, and use a straight line of best fit for cautious estimates within the observed range.
E9.2, E9.6 - Cumulative frequency and percentilesGrouped counts can show approximately where the middle observations lie, even when their exact values are hidden. You will build cumulative frequency tables and smooth curves, read median and percentile estimates, and compare distributions using median and interquartile range.
E9.7 - Histograms and frequency densityThe tallest bar in a histogram need not contain the most observations. You will connect frequency to bar area, calculate frequency density, and draw or interpret histograms whose class widths differ.