Fractions explained: add, multiply, divide and simplify
Learn how fractions work, with step-by-step examples for adding, multiplying, dividing and simplifying them. Ideal for KS3 and middle school.
KS3Middle School➗ Mathematics · 46 guides
Maths study guides for GCSE, KS3 and high school
Revise GCSE and KS3 maths, or work through US Algebra 1, Algebra 2 and Geometry. Each guide explains the idea in plain English, walks through worked examples step by step, and finishes with five practice questions where every answer is explained.
Solving linear equations step by step (with examples)
Learn the balance method for solving linear equations, including brackets and x on both sides, with worked examples for GCSE and Algebra 1.
KS3GCSEMiddle SchoolHigh SchoolPercentages: how to find, increase and decrease them
Work out a percentage of an amount, increase or decrease by a percentage and calculate percentage change, with worked examples for KS3 and GCSE.
KS3GCSEMiddle SchoolPythagoras' theorem: how to find a missing side
Use a² + b² = c² to find any side of a right-angled triangle. Worked examples for GCSE, and the Pythagorean theorem in US Geometry.
KS3GCSEMiddle SchoolHigh SchoolMean, median, mode and range: how to work them out
Find the mean, median, mode and range of a data set with clear worked examples, and learn which average to use when there is an outlier.
KS3GCSEMiddle SchoolHigh SchoolSolving quadratic equations: factorising and the formula
Solve quadratics by factorising and with the quadratic formula, using worked examples. Covers GCSE quadratic equations and Algebra 1 and 2 methods.
GCSEA-LevelHigh SchoolExpanding and factorising brackets (factoring) explained
Multiply out single and double brackets, take out common factors and use the difference of two squares. GCSE factorising and US factoring.
GCSEHigh SchoolSOH CAH TOA: how to find sides and angles in a triangle
Use sine, cosine and tangent to find missing sides and angles in right-angled triangles, with step-by-step examples for GCSE and high school Geometry.
GCSEHigh SchoolArea and circumference of a circle: formulas and examples
Learn the area and circumference formulas for circles, how to use radius and diameter, and how to work with semicircles, with worked examples.
KS3GCSEMiddle SchoolHigh SchoolSimultaneous equations (systems of equations) step by step
Solve two equations with two unknowns using elimination and substitution. Worked examples for GCSE simultaneous equations and Algebra 1 systems.
GCSEHigh SchoolProbability basics: how to calculate the chance of an event
Learn how to calculate probability, use 'not' events and find the probability of two events together, with worked examples for KS3 and GCSE.
KS3GCSEMiddle SchoolHigh SchoolBIDMAS and PEMDAS: the order of operations explained
Learn the correct order for working out a calculation, with worked examples and common traps. BIDMAS in the UK is the same as PEMDAS in the US.
KS3GCSEMiddle SchoolHigh SchoolNegative numbers: how to add, subtract, multiply and divide
Learn the rules for working with negative numbers, including subtracting a negative and multiplying signs, with number-line examples.
KS3GCSEMiddle SchoolHigh SchoolFactors, multiples and prime numbers explained
Learn what factors, multiples and prime numbers are, how to list them and how to test whether a number is prime, with worked examples.
KS3GCSEMiddle SchoolHigh SchoolHCF and LCM using prime factorisation (GCF and LCM)
Write numbers as products of primes, then find the highest common factor (greatest common factor in the US) and the lowest common multiple.
GCSEMiddle SchoolHigh SchoolMixed numbers and improper fractions: convert and calculate
Convert between mixed numbers and improper fractions, then multiply and compare them, with step-by-step examples and practice questions.
KS3GCSEMiddle SchoolFractions, decimals and percentages: how to convert them
Switch between fractions, decimals and percentages, including recurring decimals, with worked examples and practice questions to check your answers.
KS3GCSEMiddle SchoolHigh SchoolRatio and proportion: how to simplify, share and scale
Simplify ratios, share an amount in a ratio and solve scaling problems with the unit method, with worked examples for GCSE and middle school.
KS3GCSEMiddle SchoolHigh SchoolIndices and exponent rules: the laws of indices explained
Learn the index laws for multiplying, dividing and raising powers, plus zero and negative indices. UK indices and US exponent rules, with examples.
GCSEHigh SchoolStandard form (scientific notation): write and calculate
Write very large and very small numbers in standard form, convert them back and multiply them, with worked examples for GCSE and high school.
GCSEHigh SchoolSimplifying algebraic expressions: collecting like terms
Collect like terms, multiply and divide algebraic terms and simplify expressions with brackets, with step-by-step examples and practice questions.
KS3GCSEMiddle SchoolHigh SchoolRearranging formulae: how to change the subject
Make any letter the subject of a formula using inverse operations, with worked examples including squares and square roots.
GCSEHigh SchoolSolving inequalities and showing them on a number line
Solve linear inequalities, including double inequalities, and remember to reverse the sign when dividing by a negative, with worked examples.
GCSEHigh SchoolStraight line graphs: gradient and y = mx + c
Plot straight lines, work out the gradient and read the equation from a graph. Covers y = mx + c (slope-intercept form), with worked examples.
GCSEHigh SchoolFind the equation of a line from two points
Work out the gradient (slope), find the y-intercept and write the equation of a line, plus parallel and perpendicular lines and midpoints.
GCSEA-LevelHigh Schoolnth term of a sequence: how to find it (with examples)
Find the nth term of arithmetic sequences, test whether a number is in a sequence and meet geometric and quadratic sequences, with worked examples.
GCSEHigh SchoolQuadratic graphs: roots, turning points and symmetry
Sketch parabolas and find the roots, y-intercept, line of symmetry and turning point (vertex) of a quadratic, with step-by-step worked examples.
GCSEA-LevelHigh SchoolCompleting the square: method, examples and solutions
Rewrite quadratics in completed-square form to solve equations and find the turning point, with step-by-step examples for GCSE Higher and Algebra 2.
GCSEA-LevelHigh SchoolFunction notation: f(x), composite and inverse functions
Understand f(x) notation, work out composite functions such as fg(x) and find inverse functions, with worked examples for GCSE Higher and Algebra 2.
GCSEA-LevelHigh SchoolExponential growth and decay: formulas and worked examples
Model repeated percentage change with y = a × bˣ, including compound growth, depreciation and doubling, with step-by-step examples.
GCSEA-LevelHigh SchoolDirect and inverse proportion: find k and solve problems
Use y = kx and y = k/x to solve direct and inverse proportion problems, including y proportional to x squared, with worked examples.
GCSEHigh SchoolAngles in parallel lines and polygons: rules and reasons
Learn the angle rules for straight lines, triangles, parallel lines and polygons, and how to give a reason for each step, with worked examples.
KS3GCSEMiddle SchoolHigh SchoolArea of 2D shapes: rectangle, triangle, trapezium and more
Calculate the perimeter and area of rectangles, triangles, parallelograms, trapeziums (trapezoids) and compound shapes, with worked examples.
KS3GCSEMiddle SchoolHigh SchoolVolume and surface area of 3D shapes: formulas and examples
Find the volume and surface area of cuboids, prisms, cylinders, cones and spheres, with worked examples and a quick formula list.
GCSEHigh SchoolSimilar shapes: scale factors, area and volume
Use scale factors to find missing lengths, and learn how area and volume scale with k squared and k cubed, with worked examples.
GCSEHigh SchoolTransformations: reflection, rotation, translation, enlargement
Describe and carry out the four transformations on a coordinate grid, including the details examiners expect, with worked examples.
GCSEHigh SchoolCongruent triangles: SSS, SAS, ASA and RHS explained
Learn the congruence conditions for triangles, why AAA does not prove congruence and how to use them in simple proofs, with worked examples.
GCSEHigh SchoolSine rule and cosine rule: formulas and worked examples
Solve non-right-angled triangles with the sine and cosine rules (law of sines and law of cosines) and find a triangle's area, with examples.
GCSEA-LevelHigh SchoolSurds (radicals): simplifying and rationalising
Simplify surds, calculate with them and rationalise denominators, with step-by-step examples. UK surds and US radicals, for GCSE Higher and Algebra 2.
GCSEA-LevelHigh SchoolScatter graphs: correlation and the line of best fit
Plot scatter graphs, describe correlation, draw a line of best fit and make estimates, and learn why correlation does not mean causation.
KS3GCSEMiddle SchoolHigh SchoolTree diagrams and conditional probability explained
Draw tree diagrams for independent and dependent events, including picks without replacement, with worked examples and common mistakes.
GCSEHigh SchoolCumulative frequency graphs and box plots explained
Find the median, quartiles and interquartile range, draw box plots (box and whisker plots) and compare data sets, with worked examples.
GCSEHigh SchoolReverse percentages and compound interest: step by step
Work backwards from a final price to the original amount, and calculate compound interest, with worked examples and practice questions.
GCSEHigh SchoolCircle theorems explained: rules, examples and reasons
Learn the main circle theorems and use them to find angles with a reason for every step. GCSE Higher and high school Geometry, with worked examples.
GCSEA-LevelHigh SchoolLogarithms explained: laws, examples and solving equations
Understand logarithms as the inverse of powers, use the log laws and solve simple exponential equations. Algebra 2 and A-Level, with examples.
High SchoolA-LevelAPVenn diagrams and set notation: how to read and use them
Use sets, Venn diagrams and the symbols for union, intersection and complement to solve problems and find probabilities, with examples.
KS3GCSEMiddle SchoolHigh School