Eduqas GCSE Mathematics

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Number

Number sense

FN1–FN2, HN1–HN2 - Place value and number orderA number's digits tell you its value only when you know where they sit. Learn to read decimal place values, compare positive and negative numbers, and write comparisons with mathematical symbols. This begins FN1 and HN1 on number order, and FN2 and HN2 on place value; comparing fractions and perf...
FN2–FN3, HN2–HN3 - Calculating with integersCalculate accurately with positive and negative integers, including using written methods. You will use place value to explain carrying and exchanging, choose what a remainder means, and check an answer by reversing the operation.
FN3, FN6, HN3, HN6 - Powers and rootsRead powers as repeated multiplication and use roots to reverse that process. You will recognise useful powers of small numbers, distinguish a square root from the solutions of a square equation, and calculate expressions in the correct order.

Factors and fractions

FN4, HN4 - Factors and prime numbersFind factors and multiples systematically, and use factors to distinguish prime and composite numbers. You will break a whole number into prime factors and see why different routes lead to the same prime product.
FN4, HN4 - Highest common factors and lowest common multiplesFind the highest common factor and lowest common multiple using lists and prime factors. You will explain why the two methods use different prime powers, and decide which one solves a grouping or repeating-cycle problem.
FN1–FN3, FN10, HN1–HN3, HN10 - Equivalent fractions and number orderUse equal scaling to recognise and simplify equivalent fractions. You will convert between improper fractions and mixed numbers, locate signed fractions on a number line, and compare fractions using common denominators or exact decimals.
FN2, FN8, HN2, HN8 - Adding and subtracting fractionsLearn to add and subtract fractions by counting equal-sized parts of the same whole. You will calculate exactly with proper fractions, improper fractions and mixed numbers, then apply the same ideas to negative values. This develops the fraction arithmetic in FN2 and HN2 and the exact calculation...
FN2–FN3, FN8, FN12, HN2–HN3, HN8, HN12 - Multiplying and dividing fractionsCan multiplying make a number smaller, and dividing make it larger? Learn to use fractions as operators, multiply signed fractions and mixed numbers, and explain division using reciprocals. This develops FN12 and HN12 on fraction operators, with the fraction operations, cancellation and exact cal...

Decimals and accuracy

FN2–FN3, HN2–HN3 - Calculating with decimalsWhy is 4.62÷0.06 larger than 4.62÷0.6? Learn to calculate with decimal place values, scale multiplication and division correctly, and check signed answers using their size and inverse operations. This develops decimal arithmetic in FN2 and HN2, with the inverse-operation reasoning i...
FN10, HN10 - Fractions and terminating decimalsFractions and decimals can name exactly the same number. Learn to convert decimal place values into simplified fractions, use written division to find terminating decimals, and choose a convenient exact form for comparisons and calculations. This covers the terminating-decimal content shared by F...
FN14–FN15, HN14–HN15 - Rounding and estimationLearn to round numbers without losing their place value, and to estimate calculations before trusting a calculator display. FN15 and HN15 cover decimal places and significant figures; FN14 and HN14 use those ideas to judge whether an answer has a sensible size.
FN15–FN16, HN15–HN16 - Accuracy and error intervalsA rounded measurement describes a range of possible values, rather than one exact value. Learn to write those ranges with inequalities, distinguish rounding from truncation, and report accuracy honestly. FN15 and HN15 supply the interval notation; FN16 and HN16 explain what the limits mean for a...
FN13, FR1, FG12, HN13, HR1, HG14 - Units and conversionsConvert everyday measures while keeping the quantity unchanged, calculate time across clock boundaries, and explain why area and volume need different conversion factors from length. FN13, HN13, FG12 and HG14 cover the standard measures; FR1 and HR1 connect their units, including squared and cube...

Powers and exact values

FN3, FN7, HN3, HN7 - Laws of indicesPowers can make a calculation look enormous even when its answer is small. Learn to combine integer powers, explain zero and negative indices, and simplify powers of powers before calculating. FN7 and HN7 supply the index laws; FN3 and HN3 connect negative powers to reciprocals and division.
FN9, HN9 - Standard formStandard form separates a number's size from its leading digits. Learn to write and compare very large or small values, convert quantities to matching units, and calculate without losing powers of ten. These are the interpretation and calculation skills required by FN9 and HN9.
FN5, HN5 - Listing possibilitiesFind every possible outcome without missing one or counting one twice. You will organise lists, use tables and trees, and decide how restrictions and the order of choices affect the answer.

Higher number

HN6–HN7 - Fractional indicesA fractional power describes a root followed by a power. Learn to calculate these values exactly, combine them with negative indices and estimate roots between nearby known values. HN7 supplies fractional-index calculations; HN6 adds estimates of powers and roots of positive numbers.
HN8 - Working with surdsKeeping roots exact can turn an awkward decimal calculation into a short one. Learn to simplify square roots, calculate with surd expressions and remove roots from denominators. This develops the surd calculation, square-factor and rationalisation requirements of HN8.
HN10 - Recurring decimals as fractionsLearn to read recurring-decimal notation, recognise repetition in division, and convert recurring decimals into exact fractions. This develops HN10 in both directions, including decimals with a non-recurring start before their repeating block.
HN16 - Calculating with boundsLearn how rounding uncertainty passes through sums, differences, products and quotients, and decide how accurately an answer can be reported. This develops HN16 by choosing the right limiting values, distinguishing possible values from unattainable endpoints, and interpreting answers in context.
HN5 - Counting with the product ruleCount large sets of choices without writing every outcome. You will connect multiplication to systematic lists, adjust counts for restrictions, and check whether a calculation counts ordered arrangements or unordered selections.

Algebra

Expressions and formulae

FA1, FA3, FA17, HA1, HA3, HA21 - Writing algebraic expressionsAlgebra describes a calculation before you know the numbers. Learn to read its compact notation, distinguish expressions from mathematical statements, and write expressions and formulae from words. FA1 and HA1 supply the notation, FA3 and HA3 its vocabulary, and FA17 and HA21 connect it to situat...
FA2, FA5, HA2, HA5 - Substituting into formulaeOne formula can describe many different rectangles, journeys or experiments. Learn to replace its letters with values, keep negative signs and powers in the right places, and interpret the result with suitable units. FA2 and HA2 cover substitution, including scientific formulae; FA5 and HA5 contr...
FA4, HA4 - Collecting terms and expanding bracketsDifferent expressions can describe exactly the same calculation. Learn to collect like terms, multiply a single term over brackets, and combine these methods while keeping every sign. These are the collecting and single-bracket skills in FA4 and HA4.
FA4, HA4 - Taking out common factorsExpanding turns a product into a sum; factorising reverses that process. Learn to find common numerical and algebraic factors, take out the greatest common factor, and check whether your factorisation is complete. This teaches the common-factor method in FA4 and HA4.
FA4, HA4 - Exact expressions with rootsA square root can stay exact while you rearrange an expression around it. Learn to collect matching root terms, expand brackets containing roots, and take out common factors. These FA4 and HA4 methods use roots already given in the question, just as earlier algebra used letters.
FA4, HA4 - Simplifying powers in algebraThe index laws count equal factors whether the base is a number or a letter. Learn to simplify products, quotients and powers of algebraic terms, then distinguish those calculations from adding like terms. This develops the sums, products, powers and index-law scope within FA4 and HA4.

Equations and inequalities

FA14, HA17 - Solving linear equationsAn equation asks which value makes two expressions equal. Learn to undo operations, deal with brackets and fractions, and solve equations with the unknown on both sides. FA14 and HA17 also include using a supplied graph to estimate a solution, so you will connect equality with the crossing of two...
FA17, HA21 - Forming equations from problemsAn equation becomes useful when you can build it from a situation. Learn to choose an unknown, express the relationships in words as algebra, and decide whether a solution makes sense in its context. FA17 and HA21 also cover forming formulae and linked equations; here you will prepare those relat...
FA5, HA5 - Changing the subjectA formula may give the wrong quantity for the question you want to answer. Learn to rearrange it while keeping an equivalent relationship, including brackets and squares, and check a proposed rearrangement. FA5 and HA5 cover using formulae and changing their subject.
FA18, HA22 - Linear inequalitiesAn inequality can allow a whole range of values rather than one answer. Learn to read and solve inequalities in one variable, list integer solutions when requested, and represent solution sets on number lines. This teaches FA18 and the shared one-variable linear work within HA22.

Sequences and functions

FA19–FA21, HA23–HA25 - Linear sequencesFind distant terms without writing out an entire sequence. FA19 and HA23 introduce rules for generating terms; FA20 and HA24 identify arithmetic progressions; FA21 and HA25 let you describe a linear sequence using its position. You will also decide whether a given number can occur in a sequence.
FA19–FA20, HA23–HA24 - Recognising number patternsSome sequences grow by changing amounts, while others multiply or combine earlier terms. FA20 and HA24 cover triangular, square, cube, quadratic, Fibonacci-type and simple geometric patterns. FA19 and HA23 also let you generate terms from stated rules. Learn to recognise the pattern and justify e...
FA7, HA7 - Inputs and outputsTurn an expression into a rule that acts on an input and produces an output. FA7 and this part of HA7 connect algebra to number machines: read the operations, calculate outputs in the correct order, and work backwards to recover a missing input.

Straight-line graphs

FA8–FA9, HA8–HA9 - Coordinates and plottingA coordinate pair locates a point; an equation can describe a whole line of points. FA8 and HA8 cover coordinates in all four quadrants. This part of FA9 and HA9 teaches you to calculate a table of values and draw an accurate straight-line graph, ready to study its gradient and equation later.
FA9–FA10, HA9–HA10 - Gradient and interceptCan two lines cross the vertical axis at different heights yet stay the same distance apart? FA9 and HA9 connect straight-line equations to parallel lines. FA10 and HA10 explain how gradient and intercept describe a line's direction and starting value.
FA9–FA10, HA9–HA10 - Finding a line equationFind a straight line's equation from a gradient and point, or from two points, using FA9 and HA9. Interpret the gradient and intercept in models using FA10 and HA10, and check that the equation makes sense for the quantities involved.
FA16–FA17, HA19, HA21 - Simultaneous linear equationsOne equation can allow many pairs of values; two equations can identify the pair that works in both. FA16 and the linear part of HA19 cover elimination, substitution and graphical solutions. FA17 and HA21 turn linked real information into equations and interpret the answers.

Quadratics and curves

FA4, HA4 - Expanding two bracketsLearn to multiply two brackets, explain why a middle term sometimes disappears, and check an expansion before trusting it. This develops the two-binomial expansion skills in FA4 and HA4 and prepares you to reverse the process when factorising.
FA4, HA4 - Factorising quadratic expressionsLearn to turn a quadratic expression into a product of brackets by reversing expansion. This develops the monic quadratic and difference-of-squares factorisation in FA4 and HA4, including choosing signs and taking a common factor first.
FA15, HA18 - Solving quadratics by factorisingLearn why a product equal to zero gives separate equations, how to find every solution of a factorisable quadratic, and how to check the roots. This develops FA15 and HA18 using monic quadratics already written equal to zero; the factorisation makes their solutions visible.
FA11–FA12, FA15, HA11–HA12, HA18 - Quadratic graphs and rootsHow can a curve reveal two answers to one equation? FA12 and HA12 cover drawing quadratic graphs; FA11 and HA11 connect their roots, intercepts and turning points. Use the graphical parts of FA15 and HA18 to estimate solutions from crossings.
FA12, HA12 - Cubic and reciprocal graphsWhy does one curve pass through the origin while another approaches both axes without touching them? FA12 and HA12 cover recognising, sketching and interpreting simple cubic functions and the reciprocal function. Use calculated points, signs and allowed inputs to distinguish their shapes.
FA13, HA14 - Graphs in real situationsA horizontal travel graph can represent a long stop, while a steep graph can represent fast movement. FA13 and HA14 connect graphs, including reciprocal and non-standard models, to quantities such as distance, time and speed. Read approximate solutions with units and use the supplied constant-acc...

Algebraic forms

HA4 - More demanding factorisationHow can you factorise a quadratic when its first term is 6x² rather than x²? Extend the HA4 algebra methods to nonmonic quadratics, products of three or more brackets and surd coefficients. Use expansion to check that every rewritten expression is identical to the original.
HA4, HA11, HA18 - Completing the squareCan you find a quadratic's lowest value without trying a long list of inputs? HA4 rewrites quadratics as a square plus a constant. HA11 uses that form to locate turning points, and HA18 uses it to solve quadratic equations exactly or approximately.
HA18 - Solving general quadratic equationsA quadratic may have awkward coefficients, irrational roots or a root that makes no sense in its context. HA18 covers rearranging before solving, using the supplied quadratic formula, and choosing between factorisation, completing the square and the formula. Keep exact roots until a decimal answe...
HA4 - Algebraic fractionsCan cancelling a factor make a forbidden input allowed? HA4 uses factorisation and fraction arithmetic to simplify algebraic fractions, multiply or divide them, and combine sums or differences. Keep the original restrictions alongside the simplified expression so that it represents the same allow...
FA6, HA6 - Identities and algebraic argumentsLearn the difference between an equation and an identity, and use equivalent expressions to explain why a calculation works for every input. This develops FA6 and HA6 through expansion, reverse expansion and short algebraic arguments about proposed rules.

Higher functions and equations

HA6 - Proof with algebraLearn to represent arbitrary integers and construct algebraic proofs about odd, even and divisible numbers. This develops the general proofs in HA6, including stating why a factor proves divisibility and using a counterexample to reject a false universal claim.
HA7 - Composite and inverse functionsDoes “square, then subtract” give the same result as “subtract, then square”? HA7 names input-output rules, combines them in a stated order, and reverses rules when each output identifies exactly one allowed input.
HA19 - Linear and quadratic simultaneous equationsCan a line and a curve have more than one common point? HA19 uses substitution to find all coordinate pairs that satisfy a linear and a quadratic equation together, then explains what those pairs mean on a graph.
HA20 - Numerical solutions and iterationHow can you find a root accurately when an equation has no convenient factorisation? HA20 uses systematic trials, decimal searches, bisection and iterative rules to obtain numerical solutions. Keep enough precision, check the equation and distinguish evidence of convergence from a guarantee.
HA22 - Quadratic and graphical inequalitiesAn equation may give just two boundary values. An inequality can ask for every value between them, outside them, or throughout a region. HA22 uses signs and graphs to identify the complete solution set, including whether its boundaries belong.
HA23–HA25 - Quadratic and geometric sequence rulesThe gaps in a sequence may change regularly even when the terms do not. How can that reveal a formula for a distant term? HA23–HA25 connects second differences to quadratic rules, checks every coefficient, and extends geometric patterns to exact surd ratios.

Ratio and proportion

Percentages and growth

FN12, FR10, HN12, HR10 - Percentages as operatorsRead percentages as parts per hundred, convert them to fractions and decimals, and use them to calculate amounts. FN12 and HN12 explain percentage operators; FR10 and HR10 connect those operators with percentages above one hundred and comparisons between quantities.
FR10, HR10 - Percentage change and original valuesUse a multiplier to increase or decrease an amount, measure a change against its original value, and work backwards from a changed amount. FR10 and HR10 connect these three uses of percentages so that you can choose the correct reference amount.
FR10, FR16, HR10, HR17 - Interest, growth and decayDistinguish simple interest from compound interest, then use repeated percentage multipliers to model growth and depreciation. FR10 and HR10 cover simple interest; FR16 and HR17 connect repeated changes with equations, time periods and sensible interpretation of answers.

Rates and proportionality

FR1, FR12, HR1, HR12 - Unit rates and compound measuresCompare prices and pay using amounts per unit, relate distance to time, and convert rates by changing both units correctly. FR1/HR1 cover standard and compound-unit conversions; FR12/HR12 cover interpreting and using those measures.
FR2, FR12, HR2, HR12 - Density and pressureInterpret mass per unit volume and force per unit area, use their relationships in either direction, and compare measures in consistent units. FR2/HR2 establish the concepts; FR12/HR12 apply their compound units.
FR8, FR11, FR14–FR15, HR8, HR11, HR14–HR15 - Direct and inverse proportionUse equal ratios for direct proportion and a constant product for inverse proportion, then interpret their equations and graphs. This is shared Foundation and Higher content; constructing more general power models is treated separately.

Geometry and measures

Angles and shapes

FG1, FG4, HG1, HG4 - Describing shapes and symmetryA square can be turned so that it looks like a diamond, but its properties stay the same. Learn the language and markings used to describe geometry in FG1 and HG1, then use the triangle and quadrilateral properties of FG4 and HG4 to recognise shapes, describe their symmetries and sketch them from...
FG3, HG3 - Angle facts and parallel linesFind missing angles without measuring the drawing. FG3 and HG3 supply the facts about turns, intersecting lines and parallel lines; use those facts to build equations and explain each step.
FG3–FG4, HG3–HG4 - Angles in triangles and polygonsWhy does adding a side to a polygon add 180° to its interior angle total? Derive triangle and polygon angle sums using FG3 and HG3, then use the special-triangle properties in FG4 and HG4 to find missing angles. Connect the angles of regular polygons to one full turn.
FG5–FG6, HG5–HG6 - Congruence and geometric proofWhen does matching a few sides or angles guarantee that two whole triangles match? FG5 and HG5 give four congruence tests. Use them with angle and quadrilateral facts in FG6 and HG6 to justify results and distinguish a proof from a convincing-looking sketch.

Constructions and scale

FG2, HG2 - Ruler and compass constructionsHow can compass arcs locate a midpoint without measuring half the length? Use the constructions in FG2 and HG2 to draw perpendiculars, bisect angles and build accurate figures. The equal distances left by the compass explain why each construction works.
FG2, HG2 - Loci and regionsWhere could a point be if it must stay near one place and equally far from two others? FG2 and HG2 use compass constructions to describe all positions satisfying a distance rule. Learn to distinguish a boundary from a region and combine several conditions accurately.
FR3, FG13, HR3, HG15 - Scale drawings and bearingsLearn to turn map lengths into real distances, construct a consistent scale drawing, and describe directions using three-figure bearings. FR3 and HR3 supply the scale-factor reasoning; FG13 and HG15 combine accurate measurement with maps, angles and bearings.

Transformations and vectors

FG7, FG9, FG20, HG7, HG11, HG24 - Reflections, rotations and translationsLearn to reflect, rotate and translate shapes, and give the information that makes each transformation precise. FG7, FG9, HG7 and HG11 develop these movements on coordinates; FG20 and HG24 describe translations with two-dimensional vectors.
FG7, FG17, FR13, HG7, HG19, HR13 - Enlargement and similarityLearn to enlarge a shape from a centre using positive integer and fractional scale factors, and use corresponding lengths in similar figures. FG7 and HG7 supply the transformation; FG17, HG19, FR13 and HR13 connect congruence, length ratios and similarity.
FG21, HG25 - Calculating with vectorsLearn to represent directed movements with arrows and column vectors, combine routes, and multiply vectors by numbers. This develops the addition, subtraction, scalar and representation skills shared by FG21 and HG25.

Area and circles

FG14–FG15, HG16–HG17 - Perimeter and polygon areasMeasure the distance around a shape and the space it covers. FG14 and HG16 supply area methods for rectangles, triangles, parallelograms and trapezia; FG15 and HG17 extend these to perimeters and composite figures. You will choose useful lengths, explain the formulae and keep length units separat...
FN8, FG8, FG15, HN8, HG9, HG17 - Circles, circumference and areaDescribe a circle, measure its boundary and calculate its area. FG8 and HG9 supply the vocabulary; FG15 and HG17 supply circumference, area and composite-shape calculations. FN8 and HN8 let you keep multiples of π exact, so you can choose between an exact answer and a requested decimal.
FG16, HG18 - Arc lengths and sector areasUse a sector's central angle to find its share of a circle. FG16 and HG18 connect that fraction to arc length, sector perimeter and sector area. You will also work backwards from an area to its central angle.

Solids and volume

FG10–FG11, HG12–HG13 - Three-dimensional shapes and viewsA drawing can show a solid's front while hiding what lies behind it. Use FG10 and HG12 to identify solids from their faces, surfaces, edges and vertices, then use FG11 and HG13 to construct and interpret views from above, in front and from the side. Nets and cube arrangements help you decide what...
FG14, HG16 - Volumes of prisms and cylindersCalculate how much space a solid occupies from its cross-section and length. FG14 and HG16 cover cuboids, other right prisms and cylinders. You will also reverse volume calculations to find a length and connect cubic units to litres and millilitres.
FG15, HG17 - Surface area and curved solidsCalculate the area exposed on a solid and distinguish it from the volume inside. FG15 and HG17 cover surface areas of prisms and cylinders, formulae for cones, spheres and pyramids, and solids formed by joining or removing these shapes. You will track which faces remain exposed and choose perpend...

Right-angled triangles

FG6, FG18, HG6, HG20 - Pythagoras' theoremFind an unknown side of a right-angled triangle from its other two sides. FG6 and HG6 explain why the square-area relationship works; FG18 and HG20 apply it to lengths, diagonals and distances in two dimensions. You will identify the longest side first and choose whether to add or subtract squares.
FG18, HG20 - Right-angled trigonometryUse an acute angle and a side ratio to solve a right-angled triangle. FG18 and HG20 define sine, cosine and tangent and apply them to missing lengths and angles in two dimensions. You will label sides relative to the chosen angle, choose a suitable ratio and use a calculator in degree mode.
FG19, HG21 - Exact trigonometric valuesCalculate with special angles without replacing their ratios by rounded decimals. FG19 and HG21 require exact sine and cosine values at 0°,30°,45°,60°,90°, and exact tangent values at 0°,30°,45°,60°. You will derive the acute-angle values from...

Higher geometry

HG7–HG8 - Negative enlargements and combined transformationsLearn how a negative scale factor places an image beyond the centre, and how the order of transformations affects the result. HG7 develops negative enlargement; HG8 concerns combined movements, their invariants and equivalent single transformations.
HG19, HR13 - Area and volume scale factorsLearn why similar figures have different scale factors for length, area and volume, and how to work backwards from an area or volume ratio. HG19 supplies the similarity relationships; HR13 expresses them as ratios and uses them in context.
HG10 - Angles in circlesLearn to prove and use angle theorems in a circle, then connect them in reasoned angle chains. HG10 requires the geometry that justifies an answer as well as its numerical value.
HG10 - Tangents and chordsLearn to prove and use the relationships between radii, tangents and chords, including the alternate-segment theorem. HG10 connects equal radii, triangle congruence, Pythagoras and the angle theorems already proved.

Higher trigonometry

HG22 - Sine and cosine rulesFind sides and angles in triangles which need not contain a right angle. The sine rule links opposite side–angle pairs; the cosine rule uses two sides and their included angle, or all three sides. This is Higher content.
HG23 - Area of any triangleCalculate a triangle's area from two sides and their included angle, even when its perpendicular height is unknown. Rearrange the same formula for a side or angle, and use a justified triangle split in larger shapes. This is Higher content.
HG20 - Three-dimensional trigonometryTurn a problem in a solid into a flat triangle with justified lengths and right angles. Use successive Pythagoras or trigonometry calculations, then identify the exact angle requested. This is the Higher three-dimensional contribution to HG20.
HG25 - Geometric arguments with vectorsUse directed routes and position vectors to prove geometric relationships. Convert a division ratio into a fraction of a whole segment, then justify parallelism or collinearity with the correct point and nonzero-vector conditions. This is the Higher proof contribution to HG25.

Higher graphs

Probability

Statistics

Comparing distributions

FS4, HS5 - Averages and rangeCalculate mean, median, mode and range from raw data, use summary conditions to recover missing information, and compare distributions in context. This is the shared central-tendency and range content of FS4 and HS5.
FS4–FS5, HS5–HS6 - Averages from frequency tablesCalculate exact summaries from discrete frequencies, estimate a mean from grouped measurements, and distinguish an exact median or mode from a median or modal class. FS4–FS5 and HS5–HS6 connect these summaries to meaningful descriptions of distributions.
FS6, HS7 - Scatter graphs and predictionA line of best fit predicts a typical outcome. Does its prediction apply exactly to every individual? FS6 and HS7 use paired observations to describe relationships, estimate within a trend, and judge when a prediction goes beyond the evidence.

Higher statistics

HS4 - Histograms and frequency densityCan a shorter histogram bar contain more observations than a taller one? HS4 makes frequency proportional to area, so class width matters as much as height. Use that relationship to construct fair diagrams, recover frequencies and estimate portions of groups.
HS4–HS5 - Cumulative frequency and quartilesHow can a grouped table tell you roughly how long half the journeys take, or what proportion exceed a time limit? HS4–HS5 accumulates frequencies, reads quartiles from a curve, and uses the resulting estimates to compare distributions.
HS5–HS6 - Box plots and comparing distributionsTwo services can have similar typical waits but very different consistency. HS5–HS6 summarises distributions with five numbers, represents them on box plots, and uses centre and spread together while considering outliers and the population sampled.