KS3 & GCSE Maths

Learn to form and solve equations from area and perimeter at GCSE: write expressions for sides in terms of x, set up an equation, and solve for the unknown.

Calculate the gradient between two coordinate points at GCSE using (y₂ − y₁) ÷ (x₂ − x₁): positive, negative, zero and undefined gradients with examples.

Decide which method to use for GCSE quadratic equations: compare factorising, the quadratic formula, completing the square, and graphical methods.

Learn how to calculate gradient from a table of values at KS3, work through a linear and a non-linear example, and understand why gradient changes in a curve.

Master the second-differences method to find the nth term of any quadratic sequence at GCSE, with a full worked example and a clear table of differences.

Learn the formula Sn = a(1 − rⁿ) ÷ (1 − r) for the sum of a geometric series at GCSE, with a worked example and a comparison of finite and infinite series.

Learn to expand and simplify algebraic expressions at KS3: remove brackets, collect like terms, and write the result in its simplest form.

How to solve ratio problems involving algebra at GCSE — set up an equation from the ratio, solve for the unknown, then find each quantity. Worked examples included.

How to classify arithmetic and geometric sequences at KS3. Spot the common difference or ratio, identify other sequence types, and avoid mixing up the two main kinds.

How to factorise ax² + bx + c when a ≠ 1 at GCSE using the ac method. Find a factor pair, split the middle term, then group and factorise — fully worked examples.

How to read the gradient m and y-intercept c directly from y = mx + c at KS3. Identify parallel lines, sketch graphs quickly, and rearrange equations into the right form.

How to solve equations graphically at GCSE: draw the curve and a suitable line, read off the intersections, and use one graph to solve multiple different equations.

How to solve GCSE equations that require expanding double brackets — recognise the quadratic, rearrange to ax² + bx + c = 0, then solve. Fully worked examples.

How to solve linear inequalities at GCSE: two-step examples, the negative-number flip rule, number-line representation, and writing answers in set notation.

The perfect square identities (a+b)² and (a−b)² let you expand or evaluate squared expressions instantly. Learn the patterns, the proof, and GCSE uses.

The sum formula for arithmetic series lets you add long sequences without listing every term. Learn both forms with step-by-step GCSE maths worked examples.

The factor theorem states that if f(a) = 0 then (x − a) is a factor. Learn how to use it to factorise cubic expressions in GCSE Higher maths with examples.

Understand average and instantaneous rate of change at GCSE — use chord gradients for average rates and tangent gradients for the instantaneous rate at a point.

Learn to divide an algebraic expression by a single term at KS3 — split each term in the numerator over the same denominator and cancel common factors.

Learn the substitution method for GCSE simultaneous equations — rearrange one equation for a variable, substitute into the second and solve step by step.

Use algebra to find missing angles at KS3 — set up equations from angle-sum facts in triangles, on straight lines and around a point, then solve for x.

Write and simplify algebraic expressions for the perimeter and area of shapes at KS3 — form expressions from side lengths, collect like terms and evaluate.

Divide a polynomial by a linear expression at GCSE Higher using algebraic long division: step-by-step method with fully worked examples and guidance on spotting factors.

Understand exponential growth and decay at GCSE: use the model y = abˣ to represent bacteria, population and radioactive decay, and read or sketch the exponential graph.

Set up and solve GCSE quadratic equations from geometry: write sides as expressions in x, use area or Pythagoras, rearrange to ax² + bx + c = 0 and solve.

Compare the GCSE proof methods — deductive algebraic proof, proof by exhaustion and disproof by counter-example — and learn when to apply each, with worked examples.

Understand domain and range for GCSE Higher maths: what inputs a function accepts, what outputs it produces, with mapping diagrams and worked examples.

Master proof by exhaustion in GCSE maths: cover every possible case to prove a statement, with worked examples on last digits, remainders, and parity.

Recognise the six key graph types at GCSE — linear, quadratic, cubic, reciprocal, exponential, and trigonometric — from their equations and characteristic shapes.

Form and solve simultaneous equations from GCSE word problems: translate sentences into algebra, choose an efficient method, and interpret the solution in context.

Solve equations graphically at KS3 by drawing two functions and finding where they intersect, with worked examples for linear, quadratic, and cubic equations.

Understand and use position vectors at GCSE: describe points relative to an origin, find the vector between two points, and calculate midpoints using column vectors.

Solve simultaneous linear equations at GCSE using elimination and substitution with step-by-step worked examples, including equations that need scaling first.

Learn how to write algebraic formulae from word descriptions at KS3 maths — turning real-life rules into equations with variables, operations and constants, step by step.

Learn how to interpret the gradient and y-intercept of a straight-line graph in real-life contexts at GCSE maths — rates of change, starting values, and units explained.

Learn how to disprove mathematical conjectures at GCSE using a single counter-example — what counts as valid, how to find one efficiently, and exam question types.

Find the turning point, axis of symmetry and minimum or maximum of a quadratic graph at GCSE maths — using completing the square or the vertex formula.

How to collect like terms and simplify algebraic expressions at KS3 maths — group terms with the same letter and power, then add or subtract the coefficients.

Understand the three index laws for GCSE maths — multiplying, dividing and raising powers — with worked examples and exam tips for AQA and Edexcel.

How to multiply algebraic terms at KS3 maths: multiply coefficients together and add indices on matching letters — with worked examples from simple to harder expressions.

How to plot and read quadratic graphs at KS3 maths — complete a table of values for y = x², identify the vertex and line of symmetry, with worked examples.

How to rearrange harder formulae for GCSE maths: make the subject appear twice, collect and factorise, or deal with squares and square roots — step by step.

Learn to read velocity-time graphs at GCSE maths: find acceleration from gradient, calculate distance from area under the graph, with worked examples.

Use the discriminant b² − 4ac to find how many real roots a quadratic has at GCSE higher tier, with worked examples, equal-roots questions and tangent problems.

Factorising by grouping at GCSE — split a four-term expression into two pairs, factorise each separately, then extract the common bracket. Fully worked examples.

GCSE graphical inequalities guide — drawing solid and dashed boundary lines, shading the correct region, and identifying the solution region for multiple inequalities.

KS3 guide to writing algebraic expressions from words — using letters for unknowns, forming perimeter expressions, and translating word problems into algebra.

Estimate the area under a graph at GCSE using the trapezium rule — split into strips, apply the formula, and judge whether your answer over- or under-estimates.

Master composite and inverse functions at GCSE — learn to calculate fg(x), distinguish it from gf(x), and find f⁻¹(x) step by step.

Explore Fibonacci, triangular and other special number sequences at KS3 — spot the pattern, find any term, and know how to recognise each sequence type.

Estimate the gradient of a curve at GCSE by drawing a tangent at a point — learn to calculate rise over run and interpret rate of change in context.

Find the midpoint and distance between two coordinate pairs at GCSE — apply the midpoint formula and Pythagoras with step-by-step worked examples.

Solve equations containing fractions at GCSE — multiply through by the LCM to clear denominators, then solve step by step with worked examples.

Use the quadratic formula at GCSE to solve any quadratic equation — master the discriminant, avoid calculator errors, and check answers efficiently.

How to translate word problems into equations and solve them at KS3 maths. Step-by-step strategy with worked examples for Year 7, 8, and 9 students.

What number machines and function machines are, how to work forwards and backwards, and how they link to algebra in KS3 maths. With worked examples.

Difference of two squares GCSE explained: spot a² − b² instantly and factorise it into (a + b)(a − b), with worked examples and common exam traps.

Master expanding triple brackets GCSE style: multiply three linear brackets into a cubic expression, with fully worked examples and a checking method.

Function notation GCSE maths made clear: what f(x) means, how function machines work, and how to evaluate and combine functions with worked examples.

Learn geometric sequences GCSE maths: spot the common ratio, apply the nth term formula ar^(n-1), and compare geometric with arithmetic sequences.

Master parallel and perpendicular lines GCSE maths: same gradient for parallel lines, negative reciprocal for perpendicular ones, with worked steps.

Solve simultaneous equations linear and quadratic GCSE style using substitution, with fully worked examples, a circle case, and a check on solution count.

Master algebraic proof for GCSE: use algebra to prove results about integers, consecutive numbers, odd and even expressions, with worked examples and common techniques.

Understand cubic graphs at KS3: plot y = x³ and related curves, identify key features like the point of inflection, and interpret cubic graphs in context.

Explore exponential graphs at GCSE: sketch y = aˣ curves, recognise growth and decay, find key features, and interpret them in real-life modelling contexts.

Master graph transformations for GCSE: translate, reflect and stretch any curve using f(x) notation, with step-by-step worked examples and key rules.

Explore inverse proportion at GCSE: recognise y ∝ 1/x and y ∝ 1/x² relationships, find the constant of proportionality, and solve problems with worked examples.

Learn iteration for GCSE maths: use iterative formulas to find approximate solutions to equations that cannot be solved algebraically, with step-by-step examples.

Form and use proportion equations at GCSE: write direct and inverse proportion using the constant k, find k from a table of values, and solve problems with full working.

Solve quadratic inequalities at GCSE: find where a quadratic expression is positive or negative, write the solution on a number line, and use set notation correctly.

Learn about reciprocal graphs at KS3: understand the shape of y = k/x, plot a hyperbola, identify asymptotes, and recognise inverse proportion from a graph.

Solve simultaneous equations graphically at GCSE: draw two lines or curves, find their intersection, and check your answer algebraically — with worked examples.

Factorise and solve quadratic equations at GCSE — from simple expressions like x² + 5x + 6 to the difference of two squares, with step-by-step worked examples.

Go beyond basic plotting — master gradient and y = mx + c at GCSE, find equations from two points, and interpret parallel and perpendicular lines with examples.

Find the gradient and equation of a straight line at GCSE: use y = mx + c, calculate gradient from two points, and write equations in different forms.

Solve quadratic equations at GCSE using factorisation, the quadratic formula, and completing the square, with worked examples and common exam mistakes explained.

Find the nth term of a quadratic sequence at GCSE: identify second differences, use the standard method, and build fluency with fully worked examples.

Interpret real-life graphs at KS3 maths — from conversion graphs to water depth and cooling curves — learning what gradient and shape tell you about a situation.

Learn the trial and improvement method at KS3 maths: narrow down a solution to an equation by systematic substitution, with worked examples and a clear table.

Master algebraic fractions at GCSE: simplify by cancelling common factors, add and subtract by finding a common denominator, and multiply and divide with clear examples.

Master completing the square at GCSE maths: write a quadratic in the form (x + p)² + q, find the vertex of a parabola, and solve quadratic equations with clear examples.

Learn to find the gradient of a straight line at KS3 maths: use rise over run from a graph or two coordinates, and interpret gradient as a real-world rate of change.

Learn to solve linear inequalities and show solutions on a number line at KS3 maths: use open and closed circles, solve multi-step inequalities, and handle negatives.

Learn to plot quadratic graphs at GCSE maths: build a table of values, identify roots, vertex, and line of symmetry, and read solutions to equations from the graph.

Learn to solve quadratic equations by factorising at GCSE maths: factor pairs, double brackets, difference of two squares — with clear step-by-step worked examples.

Learn how to factorise algebraic expressions at KS3 maths: finding common factors, factorising into single brackets, and checking answers with worked examples.

Learn how to rearrange formulae and change the subject at KS3 — inverse operations, step-by-step method and worked examples including multi-step rearrangements.

Step-by-step guide to solving simultaneous equations by elimination and substitution — with worked examples for KS3 and GCSE maths students.

Step-by-step guide to solving linear equations with unknowns on both sides at KS3 maths, with worked examples, bracket equations, and tips for Year 8 and 9.

Learn how to expand double brackets in KS3 maths using FOIL — First, Outer, Inner, Last — with worked examples, negative terms and how to simplify.

Learn how to expand single brackets in KS3 maths step by step, using the distributive law — with worked examples, negative terms and common mistakes.

Learn how to solve equations with brackets at KS3 maths. Expand, simplify, and balance equations with clear worked examples for Year 8 and Year 9 pupils.

Learn how to solve two-step equations at KS3 using inverse operations and the balance method, with worked examples, negative numbers and checking answers.

Learn how to substitute values into formulae at KS3. Step-by-step worked examples covering positive numbers, negatives, and squared terms for Year 7, 8 and 9 pupils.

A clear KS3 guide to inequalities: reading inequality symbols, solving linear inequalities, representing solutions on a number line, and common exam mistakes.

Learn how to find the nth term of a linear sequence at KS3, continue sequences, and use the formula to find any term, with worked examples for Year 8 and 9.

A KS3 guide to solving one-step linear equations using inverse operations — with worked examples for addition, subtraction, multiplication and division.

Learn what like terms are in KS3 algebra, how to identify them, and how to collect them to simplify expressions — with worked examples and practice.

What is algebra and why does it matter at KS3? This Year 7 guide explains variables, expressions, equations and terms with clear examples and no assumed knowledge.

Master the area of an annulus (ring shape) for GCSE maths: use π(R² − r²) to find the area between two concentric circles, with clear worked examples and exam tips.

Tackle multi-step angle problems at GCSE by combining parallel-line rules, triangle and polygon angle sums, and circle theorems in a single diagram with full working.

Learn two methods to find the area of a regular polygon at GCSE — splitting into triangles and using the apothem formula — with a worked hexagon example.

Master GCSE mixed volume and surface area problems with a step-by-step strategy, a complete formulae table, a compound-shapes worked example, and the shared-face trap.

Use the section formula to find coordinates that divide a line segment in a given ratio at GCSE: internal division, external division and worked examples.

Master describing transformations at KS3: give full descriptions of reflections, rotations, translations, and enlargements with all required details.

Find the equation of a perpendicular bisector at GCSE: locate the midpoint, find the negative reciprocal gradient, then use y = mx + c.

Understand similar triangles at GCSE: use the AA, SAS and SSS similarity conditions to prove triangles are similar and find missing side lengths.

Learn about vertically opposite angles at KS3: what they are, why they are always equal, and how to use them to find missing angles in a diagram.

How to construct a perpendicular bisector at KS3 using compasses and a ruler. Step-by-step method, how to check accuracy, and where the construction is used in loci.

Find the area and perimeter of a semicircle in GCSE maths using half-circle formulas, with worked examples and questions involving composite shapes.

Find the area of a kite using the diagonal formula — half the product of the two diagonals. This KS3 guide covers the formula, the proof, and worked examples.

Learn how to construct a regular hexagon and equilateral triangle using only compasses and a ruler in KS3 maths, with clear step-by-step instructions.

Learn how to bisect an angle accurately with a compass and ruler. This KS3 construction guide gives clear steps, explains the locus reasoning, and exam tips.

Three proofs of Pythagoras' theorem for GCSE — the four-triangle dissection, similar triangles, and an area rearrangement — explained with clear written working.

Calculate the surface area of compound 3D shapes in GCSE maths by identifying exposed faces. Step-by-step examples combine cylinders, cones and hemispheres.

Calculate area and perimeter of GCSE compound shapes involving circles — combine rectangles, triangles and sectors using exact answers in terms of π.

Learn how to perform combined transformations at GCSE — apply two successive reflections, rotations or translations and find the single equivalent transformation.

Master GCSE loci and constructions — draw perpendicular bisectors, angle bisectors and solve harder region problems that satisfy two or more conditions at once.

Solve GCSE bearings problems that require trigonometry: find back bearings, form non-right-angled triangles from bearing data and apply the sine rule or cosine rule.

Test whether a triangle is right-angled at GCSE using the converse of Pythagoras: if a² + b² = c², the triangle has a right angle opposite the longest side.

Calculate the magnitude of a vector and find resultant vectors at GCSE: use |a| = √(x² + y²), add and subtract column vectors, and apply scalar multiplication.

Find the midpoint of a line segment at KS3 by averaging the x- and y-coordinates: M = ((x₁+x₂)/2, (y₁+y₂)/2) — with examples and finding a missing endpoint.

Find all solutions of sinθ = k, cosθ = k and tanθ = k in a given range at GCSE: use inverse trig to find the first solution, then apply symmetry to find all others.

Learn the two key GCSE trigonometric identities — sin²θ + cos²θ = 1 and tanθ = sinθ/cosθ — with proofs, worked simplification examples and exam tips.

Calculate the area of a circular segment at GCSE by subtracting the triangle area from the sector area, using ½r² sin θ for the triangle with worked examples.

Understand and apply the four congruence criteria — SSS, SAS, ASA, and RHS — to decide whether two triangles are congruent at KS3 maths.

Learn to construct accurate triangles using a ruler and compasses at KS3, covering SSS, SAS, and ASA methods with step-by-step instructions and diagrams.

Use coordinates, gradients, midpoints, and distances to prove geometric facts at GCSE Higher — squares, rectangles, and parallel lines with worked examples.

Identify cross-sections of 3D solids at GCSE — prisms, pyramids, cylinders, and cones — and understand how slicing affects shape, area, and volume reasoning.

Learn how to fully describe any transformation in GCSE maths — translation, reflection, rotation, and enlargement — with every defining feature included.

Calculate the surface area of square-based and triangular pyramids at GCSE by summing the base and slant-face triangles, with net diagrams and worked examples.

Use position vectors at GCSE Higher to prove geometric facts: collinearity, parallel lines, and midpoints — with step-by-step vector algebra proofs.

Calculate angles of elevation and depression at GCSE using trigonometry — with labelled diagrams, worked examples, and tips for setting up the right-angled triangle.

Calculate interior and exterior angles of regular polygons at GCSE using the formulas for n-sided shapes — with worked examples for hexagons, octagons, and beyond.

Use the formula Area = ½ab sin C to find the area of any triangle at GCSE — including obtuse-angled triangles — with worked examples and exam tips.

Write geometric proofs using angle facts at GCSE: chain angle reasons step by step, prove angles are equal or supplementary, and present working that earns full marks.

Learn to draw 3D shapes on isometric dot paper at KS3: position the grid correctly, represent cuboids and compound solids, and interpret existing isometric drawings.

Find the equation of a tangent to a circle at any point — a GCSE Higher skill using the perpendicular radius rule and the straight-line equation, with worked examples.

Understand the exterior angle theorem at KS3 maths: the exterior angle equals the sum of the two non-adjacent interior angles, with a short proof and worked examples.

Understand 3D coordinates for KS3 maths: how to plot and read points in three dimensions using (x, y, z) notation, with worked examples and grid diagrams.

Learn the key parts of a circle for KS3 maths — radius, diameter, chord, arc, tangent, sector and segment — with clear diagrams and definitions.

How to find the perimeter of composite shapes with curved edges at GCSE maths — combine arc lengths with straight sides using the formula arc = (θ/360) × 2πr.

How to find the volume of composite solids at GCSE maths — break compound 3D shapes into simpler parts, calculate each volume and add or subtract them.

What is a column vector? KS3 guide to column notation, adding and subtracting vectors, and using column vectors to describe translations. Worked examples included.

KS3 guide to six quadrilateral types — square, rectangle, rhombus, parallelogram, trapezium, and kite — with parallel sides, diagonals, and symmetry properties.

GCSE guide to the surface area of a cylinder — the formula 2πr² + 2πrh explained step by step, with worked examples for both closed and open cylinders.

KS3 guide to polygon names from triangle to dodecagon, regular vs irregular, convex vs concave polygons, and interior angle sums — with a quick-reference table.

KS3 guide to the four types of triangle — equilateral, isosceles, scalene, and right-angled — with properties, angle facts, and worked examples.

KS3 guide to the volume of any prism — triangular prisms, trapezoidal prisms, and L-shaped prisms — using volume = cross-sectional area × length. Worked examples.

Enlarge shapes using fractional and negative scale factors at GCSE — find object images, describe enlargements fully, and locate the centre of enlargement.

Find the volume of a frustum at GCSE Higher — reconstruct the full cone, remove the smaller cone using similar triangles, and subtract the volumes.

Calculate the volume of a sphere at GCSE using V = 4/3πr³ — identify the radius, answer in terms of π, and find the volume of hemispheres too.

How to count vertices, edges, and faces of common 3D shapes at KS3. Includes Euler's formula, a complete reference table, and exam-style examples.

How to calculate the area and circumference of a circle at KS3 maths. Step-by-step worked examples, formulae using pi, and exam tips for Year 7–9.

How to find the area of a trapezium and a parallelogram at KS3 maths. Formulae, step-by-step worked examples, and common mistakes for Year 7–9 pupils.

What tessellations are and how to identify regular, semi-regular, and irregular tessellations at KS3 maths. Includes why some shapes tile and others do not.

How to calculate the volume of cones and pyramids at GCSE maths. Formulae, step-by-step worked examples, and common exam mistakes for Year 10–11.

Congruent triangle proofs GCSE explained: use SSS, SAS, ASA or RHS to prove two triangles congruent, with a worked geometric proof and exam-style layout.

Nets of 3D shapes KS3 guide: what a net is, how to identify valid nets of cubes, cuboids, cylinders and prisms, with a worked surface area example.

Learn similar shapes area and volume scale factor GCSE rules: square the linear factor for area, cube it for volume, with worked examples and a summary table.

Sine rule and cosine rule GCSE explained: how to solve non-right-angled triangles, find sides and angles, and calculate area, with worked examples.

Trigonometric graphs GCSE maths guide: the shape, period and key points of y = sin x, y = cos x and y = tan x, with a value table and worked examples.

Extend Pythagoras' theorem and SOHCAHTOA into three dimensions at GCSE: find diagonals of cuboids and angles in 3D solids using right-angled triangles step by step.

Learn to write and use the equation of a circle at GCSE: x² + y² = r² and (x − a)² + (y − b)² = r², with worked examples and exam tips.

Learn to draw and interpret plans and elevations at KS3: front, side, and top views of 3D shapes, with a step-by-step method and common shape examples.

Understand rotation, reflection, and translation at KS3: how to describe and perform each transformation fully, with worked examples on coordinate grids.

Calculate the surface area of spheres and cones at GCSE: use the correct formulae, understand curved and total surface area, with fully worked examples.

Master arc length and sector area at GCSE maths: use the angle as a fraction of 360° to find arc lengths and sector areas, with clear worked examples and formula cards.

Master all 8 circle theorems for GCSE maths: angle at the centre, cyclic quadrilateral, tangent-radius, angles in the same segment, and more — with worked examples.

Understand congruent and similar shapes at KS3 maths: congruent shapes are identical in size and shape; similar shapes have equal angles and proportional sides.

Learn compass constructions and loci at KS3 maths: perpendicular bisector, angle bisector, and drawing the locus of points with clear step-by-step examples.

Understand vectors at GCSE maths: column vector notation, adding and subtracting vectors, multiplying by a scalar, and finding vector paths in geometry questions.

Learn how to find interior and exterior angles of any polygon using simple formulas — with worked examples for KS3 maths students.

Learn how to calculate the volume and surface area of a cylinder using π — with step-by-step worked examples and common mistakes for KS3 maths students.

Understand enlargement and scale factor at KS3: how to enlarge shapes, find missing lengths, and use centres of enlargement with worked examples for Year 7–9.

A step-by-step KS3 guide to working out and measuring bearings — three-figure notation, how to draw them, and common bearing problems explained clearly.

How to find the surface area of prisms and cuboids at KS3 — unfolding into nets, adding face areas, with full worked examples for triangular and rectangular prisms.

Understand trigonometry SOHCAHTOA at KS3 and GCSE — labelling sides, choosing the right ratio, and finding missing sides and angles with worked examples.

Learn the angle sum rules for triangles and quadrilaterals at KS3, with worked examples for finding missing angles in Year 7, 8, and 9 maths.

Learn the main angle types and key angle rules at KS3, including angles on a straight line, around a point, and vertically opposite angles, with worked examples.

How to find the area of rectangles and triangles at KS3 maths. Clear formulae, step-by-step worked examples, and compound shape problems for Year 7 and 8 pupils.

Understand coordinates and the four quadrants for KS3 maths. Covers plotting points, negative coordinates, midpoints and common mistakes for Year 7 and Year 8.

A clear KS3 introduction to Pythagoras' theorem: what a²+b²=c² means, finding the hypotenuse or a shorter side, and common exam mistakes explained with worked steps.

How to find the volume of a cuboid at KS3. Covers the formula, worked examples with whole numbers and decimals, unit conversion, and exam tips for Year 7 and 8.

What is pi, and how do you use it to find the circumference and area of a circle? KS3 guide with worked examples, the key formulas and common mistakes.

Master the grid method for multiplication at KS3: partition numbers into hundreds, tens and units, multiply each part separately, then add the partial products.

Understand hire purchase and credit agreements for GCSE maths: calculate total repayment, deposit, monthly instalments, and compare the true cost of borrowing.

Build confidence with money and everyday maths at KS3: calculate change, work out totals on receipts, understand VAT, and plan a simple budget using percentages.

Learn to work with negative fractions at KS3: place them on a number line, add and subtract negative fractions, and apply the sign rules for multiplication and division.

Explore negative numbers in real-life contexts at KS3: temperature differences, bank balances, sea level, floors in a building, and directed distances on a number line.

Learn the step-by-step proof that √2 is irrational at GCSE Higher: assume √2 = p/q in lowest terms and reach a contradiction using even-number reasoning.

Understand modular arithmetic at KS3 using clock and calendar examples, with worked examples for mod 7 and mod 12 and a clear table of remainders.

Understand equivalent fractions at KS3: what they are, how to make them by multiplying or dividing, and how to use them to compare and simplify fractions.

Learn to express one quantity as a fraction of another at KS3 — write the part over the whole, use the same units, and simplify the result fully.

Master zero and negative indices at KS3: understand why any number to the power 0 equals 1, and how a negative index means a reciprocal.

How to estimate non-perfect square roots at KS3 without a calculator. Find the bounding perfect squares, then narrow down to one decimal place step by step.

Learn how measurement error and accuracy work at KS3. Covers reading scales, upper and lower bounds from rounding, and why all measurements carry uncertainty.

Key metric–imperial conversions for GCSE maths — length, mass and capacity. Learn which facts to memorise, how to convert in both directions, and exam-style examples.

How to multiply and divide mixed numbers at KS3 — convert to improper fractions first, then use standard fraction methods. Worked examples and errors to avoid.

How to order fractions, decimals and percentages together at KS3. Convert everything to decimals first, compare, then write the final list in the original forms.

How to solve ratio word problems at KS3 — translate words into a ratio, simplify, then share or scale the quantities using the unitary method or the parts method.

Multi-step speed–distance–time problems at GCSE: rearrange the formula, convert units, find average speed for a whole journey, and read distance–time graphs.

Master adding and subtracting mixed numbers at KS3 — convert to improper fractions, find a common denominator, then convert back and simplify clearly.

Multiplying a fraction by an integer is a key KS3 skill. Learn to cancel before multiplying, use worked examples, and solve word problems confidently.

Percentage error measures how accurate an estimate is compared to the true value. Learn the formula with clear KS3 worked examples and common mistakes avoided.

Learn what truncation means in GCSE maths, how to truncate to decimal places and significant figures, and find the correct error interval every time.

Learn to add and subtract hours and minutes without a calculator at KS3 — convert between time units, bridge through clock boundaries and check with estimation.

Build speed and confidence with key mental arithmetic strategies for KS3 — learn doubling, halving, near-number adjustments and partitioning step by step.

Practise reading scales accurately at KS3 — work out the interval between divisions on rulers, weighing scales, thermometers and measuring cylinders.

Understand the different types of numbers at KS3 — natural numbers, integers, rational numbers and square numbers, and how each set contains the previous.

Master written column addition and subtraction at KS3 — align digits by place value, carry into the next column and always check your answer by estimation.

Solve GCSE mortgage and loan problems: calculate total repayment, total interest paid and monthly payments, and compare two loan deals with worked examples.

Find the result of two or more different percentage changes in sequence at GCSE: multiply the decimal multipliers together to find the equivalent single change.

Understand gross pay, net pay, income tax bands, and National Insurance at GCSE maths, with step-by-step worked examples for all personal finance questions.

Know when and how to leave GCSE maths answers in exact form — as surds, fractions, or multiples of pi — rather than rounding to a decimal, with worked examples.

Multiply and divide surds at GCSE using the product and quotient laws, expand surd brackets, and simplify results to exact form — with fully worked examples.

Multiply, divide, and add numbers in standard form at GCSE — with step-by-step worked examples, common pitfalls, and non-calculator tips.

Master your scientific calculator for GCSE maths: enter fractions, powers, roots, trigonometry, and standard form correctly — with Casio fx tips and common mistake fixes.

Learn how to add and subtract surds at GCSE by collecting like surd terms — including cases where you must simplify first — with clear worked examples and exam tips.

Use upper and lower bounds correctly in GCSE maths — learn which bounds to combine for the maximum or minimum value of a sum, difference, product or quotient.

Learn the divisibility rules for 2, 3, 4, 5, 6, 8, 9, and 10 at KS3 maths — quick tests to tell whether a number divides exactly, without doing the full division.

Learn how to express one quantity as a percentage of another at KS3 — divide the part by the whole, multiply by 100, and use the skill for test scores and comparisons.

Master adding, subtracting, multiplying, and dividing with negative numbers at KS3 maths — clear sign rules, a worked example for each operation, and common mistakes.

Learn how multiplying or dividing by 10, 100, and 1000 shifts digits in the place-value system at KS3 maths, with worked examples for whole numbers and decimals.

How to find percentage change in KS3 maths: use the formula (change ÷ original) × 100 to calculate the percentage increase or decrease between two values.

How to calculate simple interest in KS3 maths: use the formula I = PRT ÷ 100 to find interest on savings or loans — with worked examples and exam tips.

Understand time problems for KS3 maths: convert between 12-hour and 24-hour clock, calculate time intervals and read bus or train timetables — with worked examples.

GCSE percentage multipliers explained — use a single decimal to increase or decrease in one step, combine successive changes, and reverse a percentage change.

Learn what cube numbers are and how to find cube roots at KS3 — recognise perfect cubes, use the cube root symbol, and solve exam problems.

Convert between currencies using exchange rates at KS3 — multiply to go from pounds, divide to return, and compare best-value holiday money deals.

Use the pressure, force and area formula triangle at GCSE — calculate any one of the three quantities given the other two, with compound measure examples.

Rationalise surd denominators at GCSE — use multiplication by the surd or its conjugate to produce a rational denominator, with worked examples.

How to add and subtract decimal numbers at KS3 maths using the column method. Includes lining up decimal points, worked examples, and common mistakes.

How to compare and order fractions at KS3 maths using common denominators. Covers mixed numbers, negative fractions, and worked examples for Year 7–9.

How to find a percentage of an amount and apply it to real-life VAT and profit problems at KS3 maths. Worked examples with money for Year 7–9.

How to calculate percentage profit and loss at KS3 maths. Step-by-step worked examples with money contexts, formulae, and exam tips for Year 8 and 9.

Learn converting recurring decimals to fractions GCSE style, using the algebraic method with fully worked examples for single, double and mixed repeats.

Mixed numbers and improper fractions KS3 explained, with clear steps and worked examples for converting between the two forms in either direction.

A step-by-step guide to negative and fractional indices GCSE questions, covering reciprocal and root rules with fully worked higher-tier examples.

Rational and irrational numbers GCSE explained: clear definitions, how to tell them apart, and worked examples involving fractions, decimals and surds.

Understand bounds and error intervals for GCSE: find upper and lower bounds, write error intervals with inequality notation, and tackle compound measure problems.

Learn long division and long multiplication at KS3: follow the column methods step by step, avoid common errors, and build confidence with large-number calculations.

Master place value and decimals at KS3: understand digit values in whole numbers and decimals, compare and order numbers, and multiply or divide by powers of 10.

Learn prime factor decomposition at KS3 maths: use factor trees or repeated division to write any integer as a product of prime factors, then find HCF and LCM.

Master compound interest and depreciation for GCSE maths — learn the multiplier method with step-by-step worked examples and exam tips.

A clear GCSE guide to simplifying surds — find the largest square factor, rationalise the denominator, and add or subtract like surds with worked examples.

Understand the difference between significant figures and decimal places — with clear rules for rounding and worked examples for KS3 maths students.

Master compound measures at KS3 maths: speed, density, and pressure formulas with worked examples, unit conversions, and formula triangles for Year 8 and 9.

Learn how to find the original amount before a percentage change using reverse percentages — a GCSE maths method with clear worked examples included.

Master converting between fractions, decimals and percentages at KS3. Step-by-step methods with worked examples, a conversion table, and common mistakes to avoid.

Learn how to add and subtract fractions at KS3 with clear step-by-step worked examples, common mistakes, and tips for finding common denominators in Year 7 and 8.

Step-by-step guide to multiplying and dividing fractions at KS3: worked examples, the keep-change-flip rule for division, and common mistakes to avoid.

Master rounding to decimal places and significant figures at KS3, plus estimation strategies for checking answers in Year 7, 8, and 9 maths.

Learn how to simplify fractions at KS3 with clear worked examples, the highest common factor method, and tips for Year 7 and 8 maths.

Negative numbers explained for KS3: how to add, subtract, multiply and divide with negatives, number line tips, and common sign-error mistakes to avoid.

What are prime numbers, factors and multiples? KS3 guide with worked examples, factor trees, and HCF and LCM methods for Year 7 and Year 8 maths pupils.

Understand standard form at KS3: how to convert large and small numbers, multiply and divide in standard form, with worked examples for Year 8 and 9 maths.

Decide which statistical diagram to use for GCSE: compare bar charts, histograms, pie charts, scatter graphs, box plots, and frequency polygons with a decision guide.

Learn to draw and interpret frequency polygons at GCSE: plot midpoints against frequency for grouped data and compare two distributions on the same diagram.

Learn to read and draw line graphs at KS3: plot points over time, choose appropriate scales, identify trends, and read off values between plotted data points.

Understand positive skew, negative skew, and symmetric distributions for GCSE: compare mean and median positions, and describe shape from a histogram or box plot.

Learn a step-by-step method for comparing two distributions using box plots at GCSE, with a sample comparison table and typical exam question wording.

Learn the standard deviation formula for GCSE statistics, work through a five-step example, and understand what a low or high result tells you about spread.

Learn to draw and read frequency diagrams for grouped data at KS3: set up equal class intervals, draw bars correctly, and interpret the results.

How to draw a line of best fit and use it to make predictions at GCSE. Covers interpolation, extrapolation, when predictions are reliable, and common exam errors.

Back-to-back stem and leaf diagrams compare two data sets side by side. Learn to draw them, find medians and ranges, and compare distributions in GCSE maths.

Understand the difference between a population and a sample in KS3 statistics — why we sample, what makes a sample representative, and how bias is avoided.

The complement rule P(not A) = 1 − P(A) is an essential KS3 probability shortcut. Learn when to use it, see worked examples, and avoid common mistakes.

Understand how to collect primary and secondary data at KS3 — design data collection sheets, choose an appropriate sampling method and identify sources of bias.

Spot misleading statistics and graphs at GCSE — truncated axes, biased samples and cherry-picked averages explained with real examples and exam question tips.

Master GCSE tree diagram probability — draw two-stage and three-stage trees, multiply along branches and add outcomes to solve combined event problems.

How to draw and interpret scatter graphs at GCSE: identify correlation type and strength, draw a line of best fit, interpolate, extrapolate and spot causation errors.

Learn how to estimate population size using the mark-recapture method at GCSE, with the Lincoln-Petersen formula, worked examples, and exam tips.

Learn how to calculate and plot three-point moving averages at GCSE, smooth out seasonal variation in time series data, and identify underlying trends.

Compare statistical distributions at GCSE using median and IQR — interpret box plots and cumulative frequency curves and write full exam-style comparison answers.

Read and draw conversion graphs at KS3 — use straight-line graphs to convert between units such as miles and kilometres, degrees Celsius and Fahrenheit, and currencies.

Identify and fix bias in GCSE statistics questionnaires: leading questions, overlapping options, sampling bias, and how to run a pilot study — with worked examples.

Learn how to compare two data sets at KS3 maths using the mean or median for average and the range for spread, then write a conclusion that links back to the context.

Learn the three main sampling methods at KS3 maths — random, systematic, and convenience — what bias is, and how to choose an appropriate sample for your enquiry.

List all possible outcomes systematically at KS3 maths using organised tables and ordered lists — essential for probability, combinations, and counting problems.

Understand the statistical enquiry cycle at KS3 maths: how to pose a question, plan data collection, process results, draw conclusions, and evaluate your investigation.

How to find the interquartile range at KS3 maths: locate the lower and upper quartiles from ordered data, then subtract to measure the spread of the middle half.

Understand when to use the mean, median or mode as an average in KS3 maths — compare each measure's strengths and weaknesses with real-life examples and exam tips.

Work backwards from an average at KS3: multiply the mean by how many values there are to get the total, then find the missing number, combined means and removed values.

How to find the estimated mean from grouped frequency tables at GCSE — use class midpoints, multiply by frequency, divide by total. Worked example included.

Learn to draw a pie chart at KS3: convert frequencies to angles using 360° ÷ total, then draw each sector with a protractor. Includes a full worked example.

KS3 guide to tally charts and pictograms — recording data in groups of five, reading pictogram keys, drawing your own diagrams, and choosing the right chart.

Apply the AND and OR rules of probability at GCSE — know when to multiply or add, handle dependent events, and avoid the most common exam errors.

Understand the difference between discrete, continuous and categorical data at KS3 — and learn how data type determines which chart you should draw.

How to read, complete, and analyse grouped frequency tables at KS3 maths. Includes modal class, estimated mean, and step-by-step worked examples.

How to draw and interpret stem and leaf diagrams at KS3 maths. Includes back-to-back diagrams, finding the median, and worked examples for Year 7–9.

Frequency trees GCSE maths explained: how to draw, label and read a frequency tree from raw data, with a fully worked example and common exam pitfalls.

Learn stratified sampling GCSE maths with the formula, step-by-step method and a fully worked example, so every group in a population is fairly represented.

Understand time series graphs GCSE maths: how to plot data over time, spot the trend, calculate moving averages and interpret patterns with a worked example.

Understand conditional probability at GCSE: find P(A|B), use tree diagrams for events without replacement, and apply the multiplication rule with worked examples.

Understand frequency density, draw histograms with unequal class widths at GCSE, and interpret histogram area as frequency with worked exam examples.

Learn to draw cumulative frequency graphs and box plots at GCSE maths: find median, quartiles, and interquartile range from grouped data with step-by-step examples.

Learn to draw and interpret frequency polygons at KS3 maths: plot midpoints from a grouped frequency table, join them in order, and compare two distributions.

Master histograms at GCSE maths: calculate frequency density using frequency ÷ class width, draw and read histograms with unequal class widths, and find frequencies.

Learn Venn diagrams and set notation at KS3 maths: understand union, intersection, and complement, sort data into sets, and use Venn diagrams to calculate probabilities.

Step-by-step guide to finding the mean, median, and mode from a frequency table — with worked examples and common mistakes to avoid for KS3 maths.

A clear KS3 guide to reading and completing two-way tables — find missing values, calculate totals, and use tables to find fractions and probabilities.

Learn how to read and draw bar charts at KS3, including frequency bar charts and comparative bar charts, with step-by-step examples for Year 7 and 8 maths.

Understand mean, median, mode, and range at KS3 with clear definitions, worked examples, and tips for choosing the right average in Year 7, 8, and 9 maths.

Common GCSE Maths exam mistakes examiners see every year — from rounding errors to skipped working — and practical steps to avoid losing marks.

How to revise A-level Maths: a subject-specific technique covering Pure, Statistics and Mechanics, past-paper strategy, and how to sequence revision by topic weighting.

Angles on parallel lines for KS3 maths: alternate, co-interior, and corresponding angle rules explained with worked examples and geometric reasons for Year 8 and 9.

Understand direct proportion and the unitary method for KS3 maths: what direct proportion means, how to use the unitary method, and worked examples for Year 8 and Year 9.

Learn how to find fractions of an amount at KS3 with two clear methods, worked examples, and common mistakes to avoid in Year 7, 8, and 9 maths.

Clear explanation of highest common factor (HCF) and lowest common multiple (LCM) for KS3 maths, with prime factor trees, Venn diagrams, and worked examples.

Step-by-step guide to converting metric units at KS3: length, mass, and capacity conversions with worked examples for Year 7, Year 8, and Year 9 maths pupils.

How to draw and interpret scatter graphs at KS3 maths: correlation types, lines of best fit, and making predictions — worked examples for Year 8 and Year 9 pupils.

How to find a percentage of an amount at KS3: two reliable methods with worked examples, percentage increase and decrease, and reverse percentage explained.

How to read and interpret pie charts at KS3 maths: calculate sector angles, find frequencies from sectors, and draw your own — with worked examples for Year 7 to 9.

Step-by-step guide to plotting straight line graphs at KS3: table of values, gradient, y-intercept, and reading equations in y = mx + c form for Year 8 and 9.

Step-by-step guide to using a protractor to measure and draw angles at KS3. Covers acute, obtuse, and reflex angles with common errors for Year 7 and Year 8.

Plain-English guide to indices and powers for KS3 maths: index notation, the laws of indices, zero and negative powers, with worked examples for Year 8 and Year 9.

A year-by-year breakdown of KS3 maths topics for Years 7, 8 and 9 — number, algebra, geometry, ratio and statistics — mapped to the national curriculum.

How to multiply and divide decimals at KS3 using integer methods and powers of ten. Worked examples for Year 7, 8, and 9 with common mistakes to avoid.

What is BIDMAS and why does order of operations matter? KS3 guide with worked examples, common mistakes, and why 2 + 3 × 4 equals 14, not 20.

How to calculate percentage increase and decrease at KS3: two methods, multiplier shortcut, worked examples for Year 7, 8, and 9, plus reverse percentage.

How to find the perimeter and area of compound shapes at KS3 by splitting into rectangles and triangles. Worked examples for Year 7, 8, and 9 maths pupils.

How to draw probability tree diagrams at KS3 maths: multiply along branches, add between branches, with and without replacement — worked examples for Year 8 and 9.

Clear guide to square numbers and square roots for KS3: what they are, how to calculate them, the first 15 perfect squares, and worked examples for Year 7 and 8 pupils.

Clear guide to symmetry and transformations at KS3: reflections, rotations, translations, and enlargements explained with worked examples for Year 7, 8, and 9.

What is ratio and proportion in KS3 maths? Clear definitions, worked examples, simplifying ratios, sharing in a ratio, and direct proportion explained.

Understand the probability scale from 0 to 1 at KS3: learn the vocabulary — impossible, unlikely, even chance, likely, certain — and place events on the scale.

Understand mutually exclusive events at KS3: what they are, how to spot them, and how to use the addition rule P(A or B) = P(A) + P(B).

Understand relative frequency at KS3: how to calculate it from experiments, use it to estimate probability, and explain why more trials give better estimates.

Calculate probabilities for events with and without replacement at GCSE using tree diagrams — understand how removing an item changes the sample space and fractions.

Use Venn diagrams to calculate GCSE probability — fill in regions from given information, find union and intersection probabilities, and use conditional probability.

How to find expected frequency in KS3 maths probability: multiply the theoretical probability of an event by the number of trials to predict how often it should occur.

How to use the product rule for counting at GCSE maths: multiply the choices at each stage, handle repeats and restrictions, with worked PIN and menu examples.

Relative frequency and experimental probability GCSE explained: the formula, how it estimates true probability from trials, and a worked dice-bias example.

Sample space diagrams KS3 maths, explained with a two-dice grid: how to list every possible outcome of combined events and use the grid to find probabilities.

A complete KS3 guide to probability basics: the probability scale, calculating simple probabilities, listing outcomes, and complementary events for Year 7 to Year 9.

Solve GCSE density, mass and volume problems using D = M ÷ V: rearrange the formula triangle, work with g/cm³ and kg/m³, and apply it with cylinders, spheres and prisms.

Solve power proportion problems at GCSE: if y ∝ xⁿ, write y = kxⁿ, find k from a known pair, then calculate unknowns. Covers y ∝ x², y ∝ x³, y ∝ √x and y ∝ 1/x².

Divide a quantity into three parts using ratio a:b:c at KS3: add the parts, find the value of one part, then multiply to find each share — with worked examples.

GCSE ratio problems guide: combining ratios, finding unknowns from one given share, ratio with algebra, and ratio in real-life contexts. Worked examples throughout.

How to read and create scale drawings and interpret map scales at KS3 maths. Worked examples, unit conversions, and exam tips for Year 7–9 students.

How to use the speed, distance, and time formulae at KS3 maths. The SDT triangle, unit conversions, and step-by-step worked examples for Year 7–9.

A step-by-step KS3 guide to simplifying ratios — find the HCF, divide both parts, and handle ratios with different units or three parts.

Step-by-step guide to sharing amounts in a given ratio at KS3 — find the total parts, find the value of one part, then multiply. Includes worked examples.

Explore direct proportion graphs at KS3: how to recognise y = kx as a straight line through the origin, find k, and use the graph to solve proportion problems.

Learn inverse proportion at KS3: what it means, how to recognise it, and how to solve problems where one quantity increases as the other decreases.

Learn to write a ratio in the form 1:n at GCSE: divide both parts by the first term to find n, and use this form to compare ratios easily.

Solve best buy problems GCSE maths sets using unit pricing, with a clear method and worked examples comparing pack sizes to find the best value option.

GCSE Maths Foundation vs Higher tier: which to choose based on grade cap, current attainment, and how to know if switching tier is the right move.

A gcse maths problem solving question technique for multi-step, wordy exam questions: decode the question, choose a method, show working, and check the answer.

Using the GCSE maths formula sheet: which formulas are given in the exam, which ones students must memorise, and how to use the sheet efficiently.

Five reliable strategies for checking GCSE maths answers under exam conditions: estimation, inverse operations, substitution, unit checks and digit verification.

Learn to convert compound units at GCSE: change m/s to km/h, g/cm³ to kg/m³, and other rates using a step-by-step method with no formula to memorise.

GCSE guide to converting units of area and volume — why cm² to m² needs a factor of 10 000, and cm³ to m³ needs 1 000 000 — with worked examples and common mistakes.

KS3 guide to converting between metric and imperial units — miles to km, kilograms to pounds, litres to pints — with key conversion facts and worked examples.

The exact values of sin, cos, and tan for 0°, 30°, 45°, 60°, and 90° at GCSE maths. Derivations, a memory table, and exam-style worked examples.

What is A-level Maths really like? A guide to the topics covered, the jump in difficulty from GCSE, and how to decide if it's right for you.

Understand distance-time graphs at KS3: what the gradient means, how to read speed, and how to spot stationary periods — with a worked interpretation.