GCSE Maths worked examples and methods
146 topics, each with the method written out in order, the key facts to learn and the lesson explanation in full. Pick a topic, read the method, then try the questions on its practice page.
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Number
Grade 1 Place value
Method: value of a digitGrade 1 Addition & subtraction
Method: column additionGrade 1 Multiplication & division
Method: grid method for multiplicationGrade 1 Negative numbers
Method: adding and subtracting on a number lineGrade 1 Rounding
Method: rounding to a column, d.p. or s.f.Grade 1 BIDMAS
Method: one operation per lineGrade 1 Powers & roots
Method: working out a powerGrade 1 Factors & multiples
Method: listing factors in pairsGrade 1 Writing, simplifying & ordering fractions
Method: simplifying a fractionGrade 1 Time
Method: counting on to find a durationGrade 1 Metric conversions
Method: multiply or divide?Grade 2 Calculation problems
Method: label every stepGrade 2 Using a calculator
Method: type it as one calculationGrade 2 Systematic listing
Method: fix the first, cycle the restGrade 2 Fractions of an amount
Method: divide then multiplyGrade 2 Fractions, decimals & percentages (FDP)
Method: fraction to decimal by divisionGrade 3 Error intervals
Method: rounded numbersGrade 3 Fractions
Method: add or subtract with a lowest common denominatorGrade 3 Estimating
Method: round to 1 s.f. and calculateGrade 3 Percentages
Method: build-up (non-calculator)Grade 3 Percentage change
Method: multiplierGrade 3 Non-calculator arithmetic tricks
Method: decimal shuffle from a known factGrade 4 Compound interest & depreciation
Method: single multiplier with a powerGrade 4 Indices
Method: simplify a product or quotientGrade 4 Prime factors, HCF & LCM
Method: factor treeGrade 5 Reverse percentages
Method: divide by the multiplierGrade 5 Standard form
Method: multiply or divideGrade 6 Recurring decimals to fractions
Method: Shift and subtractGrade 6 Fractional & negative indices
Method: Root, power, flipGrade 6 The product rule for counting
Method: Slots and choicesGrade 6 Repeated percentage change
Method: Multiplier to a powerGrade 7 Surds
Method: Simplify a surdGrade 7 Bounds
Method: Half a unit either side
Method: value of a digitGrade 1 Addition & subtraction
Method: column additionGrade 1 Multiplication & division
Method: grid method for multiplicationGrade 1 Negative numbers
Method: adding and subtracting on a number lineGrade 1 Rounding
Method: rounding to a column, d.p. or s.f.Grade 1 BIDMAS
Method: one operation per lineGrade 1 Powers & roots
Method: working out a powerGrade 1 Factors & multiples
Method: listing factors in pairsGrade 1 Writing, simplifying & ordering fractions
Method: simplifying a fractionGrade 1 Time
Method: counting on to find a durationGrade 1 Metric conversions
Method: multiply or divide?Grade 2 Calculation problems
Method: label every stepGrade 2 Using a calculator
Method: type it as one calculationGrade 2 Systematic listing
Method: fix the first, cycle the restGrade 2 Fractions of an amount
Method: divide then multiplyGrade 2 Fractions, decimals & percentages (FDP)
Method: fraction to decimal by divisionGrade 3 Error intervals
Method: rounded numbersGrade 3 Fractions
Method: add or subtract with a lowest common denominatorGrade 3 Estimating
Method: round to 1 s.f. and calculateGrade 3 Percentages
Method: build-up (non-calculator)Grade 3 Percentage change
Method: multiplierGrade 3 Non-calculator arithmetic tricks
Method: decimal shuffle from a known factGrade 4 Compound interest & depreciation
Method: single multiplier with a powerGrade 4 Indices
Method: simplify a product or quotientGrade 4 Prime factors, HCF & LCM
Method: factor treeGrade 5 Reverse percentages
Method: divide by the multiplierGrade 5 Standard form
Method: multiply or divideGrade 6 Recurring decimals to fractions
Method: Shift and subtractGrade 6 Fractional & negative indices
Method: Root, power, flipGrade 6 The product rule for counting
Method: Slots and choicesGrade 6 Repeated percentage change
Method: Multiplier to a powerGrade 7 Surds
Method: Simplify a surdGrade 7 Bounds
Method: Half a unit either side
Algebra
Grade 1 Coordinates
Method: plotting and reading a pointGrade 2 Simplifying algebra
Method: collecting like termsGrade 2 Writing an expression
Method: build the expression one phrase at a timeGrade 2 Function machines
Method: forwards for the outputGrade 2 Solving one-step equations
Method: balance both sidesGrade 2 Sequences: rules and special sequences
A list with a ruleGrade 3 Substitution
Method: bracket, replace, BIDMASGrade 3 Solving linear equations
Method: balance methodGrade 3 Drawing linear graphs
Method: table of valuesGrade 4 Real-life & distance–time graphs
Method: speed from a distance–time graphGrade 4 Inequalities
Method: solve like an equationGrade 4 Forming & solving equations
Method: form and solveGrade 4 Sequences (nth term)
Method: difference and zero termGrade 4 Expanding & factorising
Method: expand and simplifyGrade 5 Changing the subject of a formula
Method: reverse the operationsGrade 5 Expanding & factorising quadratics
Method: expand with a grid or FOILGrade 5 Solving quadratics by factorising
Method: factorise and solveGrade 5 Drawing quadratic graphs
Method: table, plot, curveGrade 5 Cubic and reciprocal graphs
Method: table and plot a cubicGrade 5 Simultaneous equations
Method: eliminationGrade 5 Simultaneous equations graphically
Method: draw both lines and read offGrade 5 Midpoint of a line segment
Method: average the coordinatesGrade 5 Gradient of a line
Method: two points formulaGrade 5 Equation of a line
Method: gradient and a pointGrade 5 Exam technique: “show that” questions
Method: forwards from the given informationGrade 6 Expanding triple brackets
Method: Two then oneGrade 6 Parallel & perpendicular lines
Method: Gradient then pointGrade 6 Inequalities on graphs
Method: Line, test, shadeGrade 7 The quadratic formula
Method: identify, discriminant, substituteGrade 7 Harder factorising (ax² + bx + c)
Method: Split the middle termGrade 7 Algebraic fractions
Method: Factorise then cancelGrade 7 Harder rearranging formulae
Method: Clear, expand, collect, factorise, divideGrade 7 Trigonometric & exponential graphs
Method: Second solution by symmetryGrade 7 Inverse & composite functions
Method: Inside out for compositesGrade 7 Iteration
Method: Sign changeGrade 7 Geometric sequences
The common ratioGrade 8 Completing the square
Method: Coefficient of $x^2$ is 1Grade 8 Quadratic simultaneous equations
Method: Substitute the linear into the quadraticGrade 8 Transforming graphs y = f(x)
Method: Track one point at a timeGrade 8 Proof
Method: Prove a statement for all integersGrade 8 The nth term of a quadratic sequence
Method: Second differences then subtract $an^2$Grade 8 Quadratic inequalities
Method: Sketch and shadeGrade 8 Velocity–time graphs
Method: Straight-line graph — gradient and areaGrade 8 Gradient of a curve
A curve keeps changingGrade 8 Equation of a circle
Pythagoras on a grid
Method: plotting and reading a pointGrade 2 Simplifying algebra
Method: collecting like termsGrade 2 Writing an expression
Method: build the expression one phrase at a timeGrade 2 Function machines
Method: forwards for the outputGrade 2 Solving one-step equations
Method: balance both sidesGrade 2 Sequences: rules and special sequences
A list with a ruleGrade 3 Substitution
Method: bracket, replace, BIDMASGrade 3 Solving linear equations
Method: balance methodGrade 3 Drawing linear graphs
Method: table of valuesGrade 4 Real-life & distance–time graphs
Method: speed from a distance–time graphGrade 4 Inequalities
Method: solve like an equationGrade 4 Forming & solving equations
Method: form and solveGrade 4 Sequences (nth term)
Method: difference and zero termGrade 4 Expanding & factorising
Method: expand and simplifyGrade 5 Changing the subject of a formula
Method: reverse the operationsGrade 5 Expanding & factorising quadratics
Method: expand with a grid or FOILGrade 5 Solving quadratics by factorising
Method: factorise and solveGrade 5 Drawing quadratic graphs
Method: table, plot, curveGrade 5 Cubic and reciprocal graphs
Method: table and plot a cubicGrade 5 Simultaneous equations
Method: eliminationGrade 5 Simultaneous equations graphically
Method: draw both lines and read offGrade 5 Midpoint of a line segment
Method: average the coordinatesGrade 5 Gradient of a line
Method: two points formulaGrade 5 Equation of a line
Method: gradient and a pointGrade 5 Exam technique: “show that” questions
Method: forwards from the given informationGrade 6 Expanding triple brackets
Method: Two then oneGrade 6 Parallel & perpendicular lines
Method: Gradient then pointGrade 6 Inequalities on graphs
Method: Line, test, shadeGrade 7 The quadratic formula
Method: identify, discriminant, substituteGrade 7 Harder factorising (ax² + bx + c)
Method: Split the middle termGrade 7 Algebraic fractions
Method: Factorise then cancelGrade 7 Harder rearranging formulae
Method: Clear, expand, collect, factorise, divideGrade 7 Trigonometric & exponential graphs
Method: Second solution by symmetryGrade 7 Inverse & composite functions
Method: Inside out for compositesGrade 7 Iteration
Method: Sign changeGrade 7 Geometric sequences
The common ratioGrade 8 Completing the square
Method: Coefficient of $x^2$ is 1Grade 8 Quadratic simultaneous equations
Method: Substitute the linear into the quadraticGrade 8 Transforming graphs y = f(x)
Method: Track one point at a timeGrade 8 Proof
Method: Prove a statement for all integersGrade 8 The nth term of a quadratic sequence
Method: Second differences then subtract $an^2$Grade 8 Quadratic inequalities
Method: Sketch and shadeGrade 8 Velocity–time graphs
Method: Straight-line graph — gradient and areaGrade 8 Gradient of a curve
A curve keeps changingGrade 8 Equation of a circle
Pythagoras on a grid
Statistics
Grade 1 Pictograms
Method: reading a pictogramGrade 2 Averages
Method: order first, then find everythingGrade 2 Pie charts
Method: angle per personGrade 2 Bar charts
Method: reading a bar chartGrade 2 Stem and leaf
Method: drawing an ordered diagramGrade 2 Frequency polygons
Method: drawing a frequency polygonGrade 3 Frequency trees
Method: complete the treeGrade 3 Two-way tables
Method: fill in the gapsGrade 3 Line graphs and time series
Data over timeGrade 4 Averages from frequency tables
Method: mean from a frequency tableGrade 4 Scatter graphs
Method: describe the correlationGrade 4 Sampling & bias
Method: criticise a sampling methodGrade 6 Cumulative frequency
Method: Table to curveGrade 6 Box plots
Method: Five values to a box plotGrade 6 Capture recapture
Method: estimate the populationGrade 7 Histograms
Method: Table to histogram
Method: reading a pictogramGrade 2 Averages
Method: order first, then find everythingGrade 2 Pie charts
Method: angle per personGrade 2 Bar charts
Method: reading a bar chartGrade 2 Stem and leaf
Method: drawing an ordered diagramGrade 2 Frequency polygons
Method: drawing a frequency polygonGrade 3 Frequency trees
Method: complete the treeGrade 3 Two-way tables
Method: fill in the gapsGrade 3 Line graphs and time series
Data over timeGrade 4 Averages from frequency tables
Method: mean from a frequency tableGrade 4 Scatter graphs
Method: describe the correlationGrade 4 Sampling & bias
Method: criticise a sampling methodGrade 6 Cumulative frequency
Method: Table to curveGrade 6 Box plots
Method: Five values to a box plotGrade 6 Capture recapture
Method: estimate the populationGrade 7 Histograms
Method: Table to histogram
Geometry & measures
Grade 2 Angles
Method: fact, subtraction, reasonGrade 2 Area & perimeter
Method: formula, substitute, calculateGrade 3 Conversions & units
Method: area and volume conversionsGrade 3 Scale drawings
Method: map to realGrade 3 Area and circumference of circles
Method: circumference or area of a full circleGrade 3 Transformations
Method: describe fullyGrade 3 Area of compound shapes
Method: split and addGrade 4 Pythagoras
Method: finding the hypotenuseGrade 4 Angles in parallel lines
Method: find one angle with a reasonGrade 4 Angles in polygons
Method: interior angle sumGrade 4 Surface area
Method: face listGrade 4 Volume of a prism
Method: cross-section × lengthGrade 4 Cylinders
Method: volumeGrade 4 Loci and constructions
Method: perpendicular bisector of ABGrade 4 Bearings
Method: back bearingGrade 4 Plans & elevations
Method: draw the three views of a cube solidGrade 5 Spheres & cones
Method: substitute and roundGrade 5 Sectors and arcs
Method: fraction of the circleGrade 5 Similar shapes: lengths
Method: find the scale factor, then scaleGrade 5 SOHCAHTOA (trigonometry)
Method: missing side (unknown on top)Grade 5 Exact trig values
Method: rebuild from the special trianglesGrade 5 Vectors
Method: combine vectorsGrade 6 Similar shapes: area & volume
Method: Length factor firstGrade 6 Enlarging with negative scale factors
Method: Vector from the centreGrade 6 Circle theorems
Method: Spot, state, solveGrade 7 Area of any triangle
Method: Two sides and the angle betweenGrade 7 The sine rule
Method: Find a sideGrade 7 The cosine rule
Method: Find a side (SAS)Grade 7 Congruent triangles
Method: Three facts and a conditionGrade 7 Combined transformations
Order mattersGrade 7 3D Pythagoras & trigonometry
Method: Draw the 2D triangleGrade 8 Circle theorem proofs
Method: Numerical proof chainGrade 8 Equation of a tangent
Method: Radius gradient, then negative reciprocalGrade 8 Vector proof questions
Method: Route method for any vector
Method: fact, subtraction, reasonGrade 2 Area & perimeter
Method: formula, substitute, calculateGrade 3 Conversions & units
Method: area and volume conversionsGrade 3 Scale drawings
Method: map to realGrade 3 Area and circumference of circles
Method: circumference or area of a full circleGrade 3 Transformations
Method: describe fullyGrade 3 Area of compound shapes
Method: split and addGrade 4 Pythagoras
Method: finding the hypotenuseGrade 4 Angles in parallel lines
Method: find one angle with a reasonGrade 4 Angles in polygons
Method: interior angle sumGrade 4 Surface area
Method: face listGrade 4 Volume of a prism
Method: cross-section × lengthGrade 4 Cylinders
Method: volumeGrade 4 Loci and constructions
Method: perpendicular bisector of ABGrade 4 Bearings
Method: back bearingGrade 4 Plans & elevations
Method: draw the three views of a cube solidGrade 5 Spheres & cones
Method: substitute and roundGrade 5 Sectors and arcs
Method: fraction of the circleGrade 5 Similar shapes: lengths
Method: find the scale factor, then scaleGrade 5 SOHCAHTOA (trigonometry)
Method: missing side (unknown on top)Grade 5 Exact trig values
Method: rebuild from the special trianglesGrade 5 Vectors
Method: combine vectorsGrade 6 Similar shapes: area & volume
Method: Length factor firstGrade 6 Enlarging with negative scale factors
Method: Vector from the centreGrade 6 Circle theorems
Method: Spot, state, solveGrade 7 Area of any triangle
Method: Two sides and the angle betweenGrade 7 The sine rule
Method: Find a sideGrade 7 The cosine rule
Method: Find a side (SAS)Grade 7 Congruent triangles
Method: Three facts and a conditionGrade 7 Combined transformations
Order mattersGrade 7 3D Pythagoras & trigonometry
Method: Draw the 2D triangleGrade 8 Circle theorem proofs
Method: Numerical proof chainGrade 8 Equation of a tangent
Method: Radius gradient, then negative reciprocalGrade 8 Vector proof questions
Method: Route method for any vector
Probability
Grade 2 Probability
Method: count outcomesGrade 4 Combined events
Method: systematic listGrade 4 Relative frequency
What it isGrade 5 Probability trees
Method: draw and multiplyGrade 5 Venn diagrams
Method: overlap firstGrade 5 Sets and set notation
A setGrade 7 Conditional probability
Method: Tree without replacementGrade 8 Probability equation questions
Method: Build the equation from a tree
Method: count outcomesGrade 4 Combined events
Method: systematic listGrade 4 Relative frequency
What it isGrade 5 Probability trees
Method: draw and multiplyGrade 5 Venn diagrams
Method: overlap firstGrade 5 Sets and set notation
A setGrade 7 Conditional probability
Method: Tree without replacementGrade 8 Probability equation questions
Method: Build the equation from a tree
Ratio, proportion & rates of change
Grade 3 Writing & simplifying ratio
Method: simplify fullyGrade 3 Ratio
Method: total givenGrade 3 Proportion
Method: unitaryGrade 3 Exchange rates
Method: multiply or divide by the rateGrade 3 Best buy questions
Method: unit priceGrade 5 Ratio as a fraction or linear function
Method: ratio to fractionGrade 5 Direct & inverse proportion
Method: unitary method (direct)Grade 5 Speed, density and pressure
Method: formula triangleGrade 6 Problem-solving with ratio & algebra
Method: Let the parts be 3x and 5xGrade 7 Direct & inverse proportion (algebraic)
Method: Formula, k, formula
Method: simplify fullyGrade 3 Ratio
Method: total givenGrade 3 Proportion
Method: unitaryGrade 3 Exchange rates
Method: multiply or divide by the rateGrade 3 Best buy questions
Method: unit priceGrade 5 Ratio as a fraction or linear function
Method: ratio to fractionGrade 5 Direct & inverse proportion
Method: unitary method (direct)Grade 5 Speed, density and pressure
Method: formula triangleGrade 6 Problem-solving with ratio & algebra
Method: Let the parts be 3x and 5xGrade 7 Direct & inverse proportion (algebraic)
Method: Formula, k, formula
Prefer questions first? See GCSE Maths practice questions with answers or the key terms glossary.