Edexcel IGCSE Mathematics A
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Number
Numbers and the number system
1.1 - Integers and the order of operationsRead whole numbers accurately, compare numbers below zero and calculate changes such as temperature rises. You will also use signed multiplication and division, then decide which operation to do first when a calculation has several steps.
1.1 - Factors and multiplesUse number properties to decide which integers divide a total exactly and which values two counting patterns share. You will identify primes, list factors and multiples, and use common values to solve grouping and timing problems.
1.2 - Equivalent fractions and orderingFractions let you describe equal parts, compare shares and calculate part of a quantity. You will simplify fractions, change between mixed and improper forms, put fractions in order and express one quantity as a fraction of another.
1.2 - Calculating with fractionsKeep fraction calculations exact as you add, subtract, multiply and divide. You will use common denominators when combining shares, convert mixed numbers before multiplying, and use reciprocals to understand why division by a small fraction can make an answer larger.
1.2, 1.3 - Fractions, decimals and percentagesThe same amount can wear three different labels. In 1.3 you will read and order decimals, then turn a terminating decimal into an exact fraction. In 1.2 you will convert fractions to decimals and percentages, so you can compare quantities written in different forms.
1.3 - Recurring decimals as fractionsAn endless decimal can still represent an exact fraction. You will use place value and algebra to cancel repeating tails, including decimals with nonrecurring digits or a whole-number part. The key is to align the tails before subtracting.
1.4 - Powers, roots and index lawsPowers record repeated multiplication, and roots reverse it. You will recognise square and cube numbers, calculate roots and use integer index laws, including zero and negative indices. The final part adds fractional indices for positive numerical bases.
1.4 - Prime factorisation, HCF and LCMPrime factors reveal how whole numbers are built. You will express integers as products of prime powers, then use those powers to find the highest common factor and lowest common multiple. You will also decide which quantity answers a grouping or repeating-events problem.
1.4 - Surds and rationalising denominatorsSurds keep roots exact, so you can calculate without losing information to rounding. You will simplify, combine and multiply surds, then remove roots from denominators. You will also learn how to check that a different-looking answer still has the same value.
1.5 - Sets and Venn diagramsSets give you a precise way to sort objects into groups. You will describe membership, select unions, intersections and complements, and place elements in Venn diagrams. The same language works for numbers, letters and people.
1.5 - Set notation and counting elementsYou will use algebra to define sets, distinguish elements from subsets, and count overlapping groups without counting anyone twice. Set sizes help you solve practical problems even when no list of names is available. Start with membership, complements and Venn regions, then add a careful counting...
1.6 - Percentages and percentage changeA percentage compares an amount with a chosen whole using parts per hundred. You will change between percentage, fraction and decimal forms, compare amounts fairly, and calculate percentage parts and changes. Choosing the right whole is the key to every method here.
1.6 - Reverse percentagesA sale label tells you today's price. Can you recover yesterday's? You will undo percentage increases and decreases, choose the correct base, and check that your answer recreates the given amount.
1.6 - Compound interest and depreciationInterest can earn interest of its own. You will use repeated multipliers for compound growth and falling values. The Higher sections combine different changes and find unknown deposits, rates and whole numbers of years.
1.7 - Ratios and sharing quantitiesRatios describe how quantities fit together. You will simplify ratios, connect them to fractions, and share amounts when you know a total or a difference. The key is to work out what one equal part represents.
1.7 - Proportional reasoningA known price, recipe or scale can help you find an unknown amount. You will find a value for one unit, recognise direct proportion, and use ratio parts and map scales. You will also check when a constant-rate model is reasonable.
1.8 - Rounding and estimationA quick estimate can catch a misplaced decimal point before it costs you marks. You will round integers, decimals and significant figures, then use simple rounded values to check a calculation's size. You will also see why rounding too early can change an answer.
1.8 - Bounds and accuracyA rounded measurement hides a range of possible values. Find that range, read its endpoints and, at Higher tier, use it to decide whether a calculation or promise is reliable.
1.9 - Standard formStandard form makes very large and very small numbers manageable. Convert and calculate with powers of ten, read calculator displays and, at Higher tier, build a model for a problem with large quantities.
1.10 - Everyday number and metric unitsConvert metric measurements without changing what they represent. Use compatible units for calculations, distinguish length from area and volume, then decide how many whole containers an event needs.
1.10 - Time, money and currency conversionsRead clock intervals, convert durations to decimal hours and calculate payments. Use a stated exchange rate in either direction, keeping track of the currency and the order of any fees.
1.11 - Using a scientific calculatorMake the calculator evaluate the expression you intend. Group fractions, powers and roots correctly, retain full precision and check whether the final result makes mathematical sense.
Algebra
Equations, formulae and identities
2.1 - Algebraic symbols and indicesAlgebra lets you describe many calculations with one statement. You will read letters as numbers, use the usual rules of arithmetic, and simplify integer powers. The final part extends powers to roots for Higher learners.
2.2, 2.3 - Expressions, substitution and like termsIn 2.3 you will read conventional algebraic notation and replace letters by given values. In 2.2 you will evaluate those expressions and collect like terms. Careful brackets and signs let you change the form without changing its meaning.
2.2 - Expanding brackets and common factorsExpanding turns a product into a sum; factorising reverses that change. You will multiply through brackets, find common factors, and expand two simple linear factors. The final part extends the method to more general products at Higher.
2.2 - Factorising quadratic expressionsYou will reverse bracket expansion by choosing factors with the right coefficients and signs. Both tiers factorise quadratics with leading coefficient 1, including missing terms. Higher learners also factorise quadratics with other leading coefficients and extract common factors first.
2.2 - Algebraic fractionsYou will simplify, multiply, divide, add and subtract fractions containing algebra. This Higher lesson uses numerical, linear and quadratic numerators and denominators. You will also keep track of the inputs for which the original calculation is defined.
2.2 - Completing the squareRewrite a quadratic as a multiple of one squared bracket plus a constant. This form reveals values that are hidden in expanded algebra. You will handle positive, negative and fractional numbers inside the bracket and check that every coefficient stays the same.
2.2 - Constructing algebraic proofsUse algebra to show that a claim holds for every permitted value. You will represent whole families of numbers, reveal factors and prove identities. You will also see why one failed case can settle a question that a thousand successful checks cannot.
2.3 - Forming expressions and formulaeTurn verbal rules, labelled dimensions and practical situations into algebra. You will define variables, choose operations and derive expressions and formulae. The same rule can then answer many questions, from a rectangle's area to which printing service costs less.
2.3 - Changing the subject of a formulaRewrite a formula so the required letter stands alone. Start with a letter that appears once, including a squared quantity such as a radius. The Higher extensions develop cubic powers, roots inside formulae and letters appearing in more than one term.
2.4 - Forming and solving linear equationsFind an unknown while keeping both sides of an equation equal. You will handle brackets, unknowns on both sides and fractional coefficients. Then use the same reasoning to turn a situation into an equation and interpret its solution.
2.5 - Direct and inverse proportionLearn to turn a proportion statement into an equation, find its constant and solve for either variable. You will connect eight power, root and reciprocal models to their graphs, so you can check whether an answer fits the relationship.
2.6 - Simultaneous linear equationsLearn to find two unknown values that satisfy two conditions at once. You will solve exactly by elimination and substitution, build equations from paired data and, at Higher tier, connect the answer to the intersection of two lines.
2.7 - Solving quadratics by factorisationFactorisation turns a quadratic equation into a product of simpler factors. Learn to find every root, including zero and repeated roots, and check that none has been lost. Foundation uses monic quadratics; Higher also uses general factorisable quadratics.
2.7 - Quadratic formula and completing the squareLearn two reliable ways to solve quadratics when factorisation is inconvenient. You will keep both roots, recognise when real roots exist and give exact or rounded answers as requested.
2.7 - Forming quadratic equations from contextsTurn a situation into a quadratic equation, solve it and decide which roots fit the original conditions. You will use areas, number relationships, squared lengths and journey times, then check each answer in the situation that produced it.
2.7 - Linear and quadratic simultaneous equationsFind every pair of values that satisfies a line and a quadratic equation together. Learn how substitution finds exact intersections, why brackets matter when squaring, and how to recognise one solution or no real solutions.
2.8 - Linear inequalities and number linesDescribe sets of possible values with inequalities, solve linear inequalities and show the answers on number lines. Learn how endpoint circles and negative division change what your answer means.
2.8 - Regions defined by linear inequalitiesUse coordinate graphs to show where several linear inequalities hold together. Learn to draw boundaries, choose the correct side and identify the common region, then extend the method to harder sloping constraints.
2.8 - Quadratic inequalitiesFind where a quadratic expression is positive, negative or zero, then give its complete solution set. Use factor signs, graphs and number lines to choose the intervals and include the correct endpoints.
Sequences, functions and graphs
3.1 - Generating and describing sequencesLearn to build a sequence one term at a time or jump straight to a chosen position. You will compare familiar integer patterns and describe a rule clearly enough for someone else to continue it.
3.1 - Arithmetic sequences and nth termsLearn to turn a constant-step pattern into an nth-term rule and decide whether a number belongs to it. Higher students will also use first-term notation and recover an arithmetic sequence from two separated terms.
3.1 - Sums of arithmetic seriesLearn to add an arithmetic sequence efficiently and explain why its sum formula works. You will use totals to solve a saving-budget problem. This lesson is for Higher students.
3.2 - Functions, domain and rangeLearn to read a function as a rule connecting inputs and outputs. You will use function notation, identify forbidden inputs and find the outputs actually reached from a given domain. This lesson is for Higher students.
3.2 - Composite and inverse functionsLearn to combine function rules in the right order and find a function that reverses a mapping. You will keep track of allowed inputs and choose the correct square-root branch when an inverse needs a domain restriction. This lesson is for Higher students.
3.3 - Coordinates and midpointsA coordinate pair turns a position into two numbers. You will plot and read points in all four quadrants, use geometric information to locate vertices and find midpoints or missing endpoints. These skills let you solve a map problem exactly, even when the picture is not drawn accurately.
3.3 - Gradients and equations of straight linesA straight line has one constant steepness, even if you measure it using different-sized steps. You will find that gradient and connect it to the equation of the line. The Higher parts use two coordinates and find equations of parallel and perpendicular lines.
3.3 - Plotting linear and quadratic graphsAn equation is a compact way to describe many points. You will calculate those points and turn them into straight lines or smooth quadratic curves, including equations that fix one coordinate. A well-chosen table and a few checks help you spot a wrong point before it spoils the graph.
3.3 - Interpreting real-life and conversion graphsThe same shape can tell different stories when the axis labels change. You will interpret journeys, changing speeds and curved graphs, then draw and use a straight conversion graph. Reading the quantities, units and scales comes before doing any calculation.
3.3 - Cubic and reciprocal graphsCubic and reciprocal graphs can bend, turn or split into separate branches. You will use tables, signs and excluded inputs to recognise and draw their actual shapes, including equations with different variable letters. Checking the equation first prevents a familiar-looking curve from leading you...
3.3 - Sine, cosine and tangent graphsSine, cosine and tangent keep working after an angle passes a full turn. You will draw their graphs in degrees, recognise their repeating patterns and find all the angles that give a chosen output. Keep your calculator in degree mode throughout.
3.3 - Transformations of function graphsA small change to a formula can move a whole graph or change its size. You will apply f(x)+a, f(x+a), af(x) and f(ax) to linear, quadratic, sine and cosine functions. You will also work backwards from a graph's features to its equation, keeping trigonometric angles in degrees.
3.3 - Tangents and graph intersectionsA curve's gradient changes from place to place, and a line can meet a curve more than once. You will estimate local gradients with tangents, read all relevant intersections and turn equality of graph outputs into an equation. Drawing and algebra will help you check each other.
3.4 - Differentiation and rates of changeDifferentiation finds how quickly a function changes at one input. You will connect this rate to a tangent, then differentiate integer powers, including negative powers. The original function gives a graph's height; its derivative gives the gradient.
3.4 - Gradients and turning pointsDifferentiation can locate a curve's horizontal tangents and points with any chosen gradient. You will find their coordinates, then use the original graph's shape to identify maxima and minima. A zero gradient does not always mean that the graph turns.
3.4 - Calculus in motion and practical modelsA changing position gives a velocity, and a changing velocity gives an acceleration. You will use differentiation for motion along a straight line, track distance after reversals and find rates or greatest values in simple practical models. Every model has its own units and valid interval.
Shape, space and measure
Geometry and trigonometry
4.1 - Angles, parallel lines and trianglesYou will identify angle types and find unknown angles in lines and triangles. You will learn which angles are equal, which add to a fixed total, and how to explain your choice. These ideas let you calculate angles even when a drawing cannot be measured accurately.
4.2 - Polygons and quadrilateralsYou will recognise polygons from their properties and calculate their angles. You will see why the angle sums work and use an exterior angle to find the number of sides. A name such as “hexagon” counts sides; it does not promise that the shape is regular.
4.2, 4.3 - Congruence and symmetryYou will recognise congruent polygons using the same-shape-and-size idea from 4.2. You will then use 4.3 to identify reflection axes and count matching positions during a rotation. A shape can have one kind of symmetry without the other, so you will test them separately.
4.4 - Measuring, estimating and bearingsYou will read scales, calculate elapsed times and choose sensible estimates. You will also measure angles to the nearest degree and describe directions using three-figure bearings. Each skill starts by choosing the correct unit, starting point or zero.
4.4 - Speed, density and pressureYou will calculate average speed, density and pressure using quantities with matching units. You will also rearrange the relationships and explain what happens when one quantity changes. These compound measures describe an amount per unit of something else; the pressure formula is supplied in que...
4.5 - Drawing and constructing shapesLearn to measure and draw accurately, then use given lengths and angles to construct triangles and other shapes. Work on paper with a sharp pencil, a ruler marked in millimetres, a protractor and compasses.
4.5, 4.11 - Maps and scale drawingsA map or plan shrinks real lengths by one consistent scale. Use construction skills from 4.5 to make accurate plans, and the map skills in 4.11 to convert distances and compare routes. You will work with ratio scales, scale bars and labelled lengths.
4.5 - Perpendicular and angle bisectorsLearn to construct two kinds of bisector with a straight edge and compasses, and understand why their arcs locate the correct line. Make the constructions on paper and leave the arcs visible.
4.6 - Circles, chords and tangentsCircle vocabulary helps you choose the right geometrical property. Foundation and Higher share the terms, chord bisection and tangent properties; the final two sections add Higher intersecting chord products. You will calculate lengths and angles and explain which features make each property valid.
4.6 - Circle angle theoremsThis Higher lesson teaches the five circle angle theorems and the meaning of a cyclic quadrilateral. You will identify the relevant arc or chord, calculate unknown angles and give a geometrical reason for each step. The diagrams help you trace the angles; formal proofs of these theorems are not r...
4.7 - Chains of geometrical reasoningAn angle answer becomes convincing when each step has a reason. You will connect facts about lines, triangles and polygons to find missing values. You will also use chord and tangent facts, with circle-angle reasons in the Higher section.
4.8 - Pythagoras in two dimensionsYou will use a right angle to connect three side lengths, find a missing length and solve problems in a flat diagram. The same method helps you calculate a diagonal or a sloping support. You will also decide whether a final answer needs ordinary rounding or rounding up.
4.8 - Right-angled trigonometry and bearingsYou will connect an acute angle to the side ratios of a right triangle, calculate missing lengths and angles, and use these methods with bearings. Bearings turn a journey into perpendicular north–south and east–west distances. Keep your calculator in degree mode throughout.
4.8 - Obtuse angles, elevation and depressionYou will extend trigonometry to obtuse angles and use angles of elevation and depression to calculate heights and distances. Clear angle geometry matters as much as calculator work. Use degree mode and keep full calculator precision until the final answer.
4.8 - Sine rule, cosine rule and triangle areaYou will choose rules for triangles without a right angle, calculate missing sides and angles, and check whether more than one triangle is possible. You will also find area from two sides and their included angle. Use acute and obtuse trigonometry in degree mode.
4.8 - Three-dimensional Pythagoras and trigonometryFind lengths and angles inside solids by bringing the right triangle into view. You will use a perpendicular projection to find a line's angle with a plane, then choose between different triangles in a pyramid.
4.9 - Perimeter and area of polygonsMeasure the boundary and the region inside a shape, then use the right units for each. You will convert measurements, choose areas for triangles and quadrilaterals, and work out how much material a compound shape needs.
4.9 - Circumference, circle area and semicirclesUse radius and diameter to find circular lengths and areas. You will distinguish the curved arc from a semicircle's complete boundary, then measure shapes made by joining semicircles to rectangles.
4.9 - Arc lengths and sector areasUse a central angle as a fraction of a full turn to find an arc length or sector area. You will include the straight edges in a sector's perimeter and work backwards to find an unknown angle or radius, using degrees throughout.
4.10 - Solids, nets and surface areaRecognise common solids and describe their faces, edges and vertices, including hidden features. You will unfold solids into nets and add rectangle and triangle areas to find how much material covers their surfaces.
4.10 - Prisms, cylinders and volume unitsFind the space inside prisms and cylinders, choose which cylinder surfaces to count, and convert volumes into capacities. These methods help you decide how much a container holds and how much material covers it.
4.10 - Cones and spheresUse the supplied cone and sphere formulae, distinguish curved from total surface area, and find the exposed surfaces of joined solids. Choosing the right height and counting the right surfaces matter as much as substitution.
4.11 - Similar figures and length scalesMatch the sides and angles of similar figures, use a common length scale, and test whether an enlargement keeps the same shape. The method works even when a figure is turned or reflected.
4.11 - Area and volume scales of similar figuresUse length scales to compare areas and volumes of similar figures. Learn why the powers differ, work backwards from an area or volume ratio, and find unknown volumes from their difference.
Vectors and transformation geometry
5.1 - Vectors, resultants and magnitudeDescribe movements using vectors, combine directed journeys and calculate the length of a direct displacement. You will connect arrows, bold letters and column notation, then use signed components to keep track of where a journey ends.
5.1 - Vector geometry and proofsUse directed routes and scalar multiples to prove relationships between points and lines. You will find vectors to points dividing sides, prove parallelism and collinearity, and use a multiplier to recover a length ratio.
5.2 - Rotations, reflections and translationsMove shapes by translation, reflection and rotation, construct their images on paper and give complete descriptions of the movements. You will use corresponding vertices to check both the position and the unchanged shape and size.
5.2 - Enlargement and transformation descriptionsEnlarge shapes using positive scale factors, including fractions, and describe an enlargement completely. You will measure from a centre, construct images with or without a given centre, and distinguish the lengths that change from the angles that stay equal.
Handling data
Statistics and probability
6.1 - Tables, pictograms, bar charts and pie chartsTurn a collection of answers into a clear table or chart. You will construct and read pictograms, bar charts and pie charts, and decide what each display can tell you.
6.1 - Histograms and frequency densityLearn to construct and interpret histograms for continuous measurements with unequal class widths. You will use bar areas to recover frequencies and explain when a count is only an estimate.
6.1, 6.2 - Cumulative frequency and estimated quartilesBuild a cumulative-frequency diagram from grouped data, then use it to estimate a median, quartiles and counts above or between limits. You will distinguish recorded class totals from estimates inside classes.
6.2 - Mean, median, mode and rangeUse mean, median and mode to describe a typical value, and range to describe spread. Calculate from lists and frequency tables, recover missing information and choose a summary suited to the question.
6.2 - Grouped mean estimates and modal classGrouping measurements makes a table shorter but hides their precise values. Identify a modal class and estimate a mean with class midpoints, while explaining what information has been lost.
6.2 - Spread, quartiles and interquartile rangeAn average describes a centre but does not tell you how much observations vary. Calculate quartiles and interquartile range from discrete data, then compare spread and explain the effect of an extreme value.
6.3 - Probability, relative frequency and expectationLearn to describe chance, calculate probabilities from a model and estimate them from observations. Use probabilities to predict counts while allowing for uncertainty in what actually happens. These ideas apply at Foundation and Higher.
6.3 - Sample spaces and systematic outcomesLearn to record every possible outcome without omissions or double counting, then select the outcomes belonging to an event. Ordered lists and tables help you calculate probabilities and explain why some results are more likely than others. This lesson applies at Foundation and Higher.
6.3 - Venn probabilities and mutually exclusive eventsLearn to read probabilities from Venn regions, reconstruct missing counts and decide when event probabilities can be added. You will distinguish a whole circle from its “only” region and count each person once. This lesson applies at Foundation and Higher; its addition rule is for mutually exclus...
6.3 - Tree diagrams and independent eventsLearn to construct probability trees, multiply along paths and combine the routes that meet a target. You will use independence to decide which probabilities repeat and complements to shorten calculations. These tree methods are Higher content.
6.3 - Conditional probabilityLearn how new information changes the group used in a probability calculation. Update a bag after each draw without replacement, combine qualifying routes and restrict those routes when a condition is given. These are Higher methods.