Mathematics

205 guides

Number

• Compound interest formula for GCSE Maths, with examplesThe compound interest formula for GCSE Maths, A = P(1 + r/100)ⁿ: multipliers, compound vs simple interest, depreciation, finding the rate or years.
• Compound measures: speed, density, pressure and ratesWhat compound measures are, the formulas for speed, density and pressure, converting compound units, rates and best buys, with worked GCSE examples.
• Exponent calculator: powers, fraction exponents and rootsFree exponent calculator: work out any power exactly, including negative and fraction exponents, simplify roots and see the exponent rules applied.
• GCF calculator: greatest common factor with stepsFind the GCF (greatest common factor, also called the HCF) of up to six numbers by prime factors or Euclid's algorithm, with full working and examples.
• How to multiply fractions: 3 steps, examples and practiceHow to multiply fractions in 3 steps, with whole numbers, mixed numbers, cancelling first, word problems and questions you can check on the page.
• Integers: definition, examples and rules for operationsIntegers are whole numbers and their negatives. See examples, integers on a number line, and the rules for adding, subtracting, multiplying and dividing.
• LCM calculator: least common multiple with stepsFind the LCM (least or lowest common multiple) of up to six numbers with prime factor steps, a Venn diagram, worked examples and a quick check.
• Log calculator: any base, ln, log10, antilog, with stepsWork out logs in any base, including ln and log base 10, find a number from its log or the base, and see the log rules at work, with examples.
• Long division calculator: step by step with remaindersLong division set out step by step: remainders, decimals and recurring decimals, with the written layout built up one digit at a time and examples.
• Math notepad: type equations in real notation, get answersFree online math notepad: type fractions, powers and roots the way they look on paper, and see each line worked out as a number as you go.
• Metric system: units, prefixes and how to convertThe metric system explained: base units, the prefixes from kilo to milli, how to convert between metric units, and metric to US customary conversions.
• Number line: how to use one, with negatives and examplesWhat a number line is and how to use it: plot negatives, fractions and decimals, add and subtract, compare numbers and graph inequalities, with examples.
• PEMDAS: order of operations rules with examplesWhat PEMDAS stands for, the order of operations rules step by step, worked examples from easy to hard, the mistakes to avoid and practice problems.
• Percent error calculator: formula, steps and examplesWork out percent error from a measured value and a true value, with absolute and relative error, percent difference, tolerance ranges and examples.
• Perfect squares: list from 1 to 900, patterns and testsA perfect square is a whole number times itself. See the list of perfect squares to 900, how to tell if a number is one, and perfect square trinomials.
• Prime factor decomposition: method, HCF and LCMHow to write a number as a product of prime factors with a factor tree or division, in index form, and use it for HCF, LCM, square roots and cube roots.
• Prime factorization calculator with a factor treeFree prime factorization calculator: break any whole number up to a trillion into primes, with a factor tree, index form and the number of factors.
• Product rule for counting: GCSE method and examplesThe product rule for counting: multiply the number of choices at each step. GCSE examples with codes, menus and restrictions, plus practice questions.
• Rational numbers: definition, examples and decimalsA rational number is any number you can write as a fraction of two integers. See examples, rational vs irrational numbers, and repeating decimals.
• Reciprocal: what it is, how to find it, with examplesThe reciprocal of a number is 1 divided by it. Find the reciprocal of whole numbers, fractions, decimals and negatives, and see the graph of y = 1/x.
• Reverse percentage: how to find the original valueHow to do reverse percentages: find the original value after a percentage increase or decrease using a multiplier, with worked examples and questions.
• Scientific notation calculator: convert and calculateFree scientific notation calculator: convert numbers to and from scientific notation exactly, and add, subtract, multiply or divide them with every step.
• Scientific notation: how to write, convert and calculateWhat scientific notation is, how to convert large and small numbers into it and back, and how to multiply, divide, add and subtract with it, with examples.
• Significant figures calculator: count and round sig figsCount the significant figures in any number and round to as many as you need, with each digit labelled by its rule and worked examples to check.
• Square root: meaning, table, simplifying and examplesWhat a square root is, a table of square roots, how to find and estimate them, how to simplify square roots like √72, and the square root graph.
• Surds in maths: how to simplify surds, with examplesWhat surds are, how to simplify surds, add, multiply and rationalise them, with GCSE worked examples, common mistakes and practice questions.
• Types of numbers: integers, rational, irrational and moreThe types of numbers explained: natural numbers, integers, rational, irrational, real and complex, plus primes, squares, cubes, factors and multiples.

Algebra

• Absolute value: meaning, equations, inequalities and graphAbsolute value is a number's distance from zero, so it is never negative. Learn the notation, solve equations and inequalities, and graph y = |x|.
• Argand diagram: plot complex numbers, modulus and argumentHow to plot complex numbers on an Argand diagram, find the modulus and argument, add and multiply geometrically, and draw loci, with to-scale diagrams.
• Arithmetic sequence calculator: nth term and sumFree arithmetic sequence calculator: the nth term rule, any term and the sum of n terms, from a and d or from the first few terms, with full working.
• Arithmetic series formula: Sₙ = n/2(2a + (n − 1)d), proofThe arithmetic series formula Sₙ = n/2[2a + (n − 1)d] and Sₙ = n/2(a + l): the proof, sigma notation, finding n, two-condition problems and examples.
• Binomial expansion calculator: any power with stepsExpand (a + b)ⁿ with the binomial theorem and Pascal's triangle, find any term or coefficient, and expand negative and fractional powers as series.
• Binomial expansion formula: (a + b)ⁿ with examplesThe binomial expansion formula for (a + b)ⁿ and (1 + x)ⁿ, Pascal's triangle, finding one coefficient, fractional and negative powers, with examples.
• Completing the square: method, examples and calculatorHow to complete the square in 4 steps, with worked examples, solving quadratics, finding the turning point and the a ≠ 1 case, plus a free calculator.
• Complex number calculator: Argand diagram and polar formAdd, multiply and divide complex numbers exactly, find the modulus, argument and polar form, and use De Moivre for powers and roots, with examples.
• Composite functions: how to find fg(x), with examplesWhat a composite function is, how to work out fg(x) and gf(x), why the order matters, solving equations, and domain and range at A-level.
• Determinant calculator: 2x2, 3x3 and up with stepsFind the determinant of a matrix up to 5 by 5 exactly, by the 2 by 2 formula, cofactor expansion or row reduction, with worked examples to check.
• Expanding double brackets: FOIL, grid method and examplesHow to expand double brackets with FOIL and the grid method, from (x + 2)(x + 6) to squared brackets, difference of two squares and triple brackets.
• Expanding single brackets: method and examplesHow to expand single brackets: multiply the term outside by every term inside, with negatives, letters, expand and simplify, and solving equations.
• Exponent rules: all 7 laws of exponents with examplesAll the exponent rules in one chart: multiplying, dividing, powers of powers, zero, negative and fractional exponents, with worked examples and practice.
• Factoring calculator: factor polynomials step by stepFree factoring calculator: factor trinomials, polynomials, differences of squares and cubes completely, with each method named and every step shown.
• Geometric series calculator: sum, nth term and S∞Free geometric series calculator: the sum of n terms, the sum to infinity and the nth term from a and r, as exact fractions, or the rule from a few terms.
• How to factor a quadratic equation: 6 methods, examplesHow to factor a quadratic equation step by step: common factors, x² + bx + c, the ac method, difference of squares, perfect squares and solving.
• Inequality calculator: solve and graph on a number lineFree inequality calculator: solve linear, quadratic and rational inequalities with steps, and get the answer on a number line and in interval notation.
• Interval notation: brackets, infinity and examplesInterval notation explained: when to use square or round brackets, how to write infinity and unions, and how to turn inequalities into intervals.
• Inverse matrix calculator: 2x2, 3x3 and up with stepsFind the inverse of a matrix up to 5 by 5 with exact fractions: the 2 by 2 formula, Gauss-Jordan steps for 3 by 3 and bigger, and examples to check.
• Iteration in maths: solving equations by iterationHow to solve equations by iteration: change of sign, rearranging to x = g(x), using x(n+1) = g(x(n)), cobweb diagrams and Newton-Raphson, with examples.
• Log rules: the laws of logarithms with examplesAll the log rules in one place: product, quotient and power laws, change of base and ln rules, with worked examples, solving equations and practice.
• Matrix calculator: multiply, inverse, determinant, RREFAdd, multiply, invert and row reduce matrices up to 5 by 5 with exact fractions, plus determinants, eigenvalues and worked examples to check.
• Matrix multiplication calculator: steps and rulesMultiply matrices up to 5 by 5 with every entry worked out as a row times a column, the size rule, and examples you can check in the calculator.
• Partial fraction decomposition calculator with stepsFree partial fraction decomposition calculator: split a fraction into partial fractions, with linear, repeated and quadratic factors and full working.
• Partial fractions questions with worked answersPartial fractions questions with full answers: the two methods for finding constants, repeated factors, integrating with partial fractions, and practice.
• Point-slope form: formula, examples and how to use itThe point-slope form formula, how to write a line from a point and slope or two points, and how to convert it to slope-intercept and standard form.
• Polynomial: definition, degree, operations and graphsWhat a polynomial is, how to find its degree, how to add, multiply, divide and factor polynomials, and what their graphs look like, with worked examples.
• Proof by contradiction: method and examplesHow proof by contradiction works, how to write one, the proofs that √2 is irrational and that there are infinitely many primes, and more examples.
• Quadratic equation: what it is and 4 ways to solve itWhat a quadratic equation is, standard, vertex and factored form, and how to solve one by factoring, square roots, completing the square or the formula.
• Quadratic formula calculator: roots, graph and workingSolve any quadratic equation with the quadratic formula: enter a, b and c to see the roots and the parabola, then follow the full working by hand.
• Quadratic graphs: plot, sketch and read them (GCSE)How to plot and sketch quadratic graphs: the parabola shape, roots, turning point, line of symmetry and solving equations from a graph, with live graphs.
• Quadratic sequences: how to find the nth termHow to spot a quadratic sequence and find its nth term an² + bn + c, step by step, with worked GCSE examples, a check method and practice questions.
• Quadratic simultaneous equations: method and examplesHow to solve simultaneous equations where one is quadratic: substitute, solve the quadratic, find both pairs. Circles, parabolas, tangents and practice.
• Remainder theorem: statement, proof and worked examplesThe remainder theorem explained: why f(a) is the remainder, dividing by ax − b, finding unknown coefficients, and its link to the factor theorem.
• Sequences in maths: nth term, types and examplesSequences explained: term-to-term and nth term rules, arithmetic, geometric, quadratic and Fibonacci sequences, recurrence relations and series.
• Set notation in A-level maths: symbols and inequalitiesThe set notation symbols you need for A-level maths, how to write the solution of an inequality in set notation, and how sets are used in probability.
• Sigma notation: how to read, write and evaluate sumsSigma notation explained: what Σ means, how to evaluate a sum, rules for splitting sums, arithmetic and geometric series, and Σ in statistics.
• Simultaneous equations: how to solve them, with examplesHow to solve simultaneous equations by elimination, substitution and graphs, with worked GCSE examples, word problems and practice questions with answers.
• Standard form of a linear equation: Ax + By = CStandard form of a linear equation is Ax + By = C. Convert to and from y = mx + b, graph with intercepts and write equations, with worked examples.
• Synthetic division calculator with steps and remainderFree synthetic division calculator: divide a polynomial by x − c or ax − b, with every step of the table, the quotient, remainder and factor theorem.
• System of equations calculator: 2 or 3 variables, with stepsFree system of equations calculator: solve 2 or 3 linear equations by elimination or substitution, or a line with a curve, with exact answers and a graph.
• Vertex form calculator: vertex, steps and graphConvert y = ax² + bx + c to vertex form y = a(x − h)² + k by completing the square or with h = −b/2a, and check the vertex on a graph.
• What is a logarithm? Logs explained with examplesWhat a logarithm is, how to read and work out logs, log base 10 and natural logs, the log rules and why they work, solving equations and real uses.

Graphs

• Coordinate geometry: lines, circles and worked examplesCoordinate geometry for GCSE and A-level: gradient, midpoint and distance, equations of lines, parallel and perpendicular lines, circles and tangents.
• Cubic graph calculator: plot y = ax³ + bx² + cx + dFree cubic graph calculator: plot y = ax³ + bx² + cx + d, read the roots and y-intercept, and find the turning points, with practice questions.
• Cubic graphs: shapes, roots and how to sketch themWhat cubic graphs look like, positive and negative cubics, how to sketch one from its roots and y-intercept, repeated roots and turning points.
• Distance formula calculator: distance between two pointsFree distance formula calculator: type two points to get the exact distance as a surd and a decimal, with the right triangle drawn and every step.
• Exponential graphs: y = aˣ, eˣ, growth and decayHow exponential graphs work: the shape of y = aˣ, the asymptote, the point (0, 1), growth and decay, y = eˣ, transformations and log graphs.
• Graph plotter: plot any equation and find its key pointsPlot any equation y = f(x) online, then find its intercepts, turning points, derivative and the area under it. How to type each function, with examples.
• Graphing linear equations: slope, intercepts and examplesHow to graph linear equations three ways: from slope and y-intercept, from a table of values and from the intercepts, with worked examples and live graphs.
• How to find the centre of a circle: 3 methodsHow to find the centre of a circle: by construction with two chords, from the equation by completing the square, and from three points, with examples.
• Linear graph calculator: plot y = mx + b, find interceptsFree linear graph calculator: plot y = mx + b, change the slope and intercept, and read where the line crosses each axis, with practice questions.
• Logarithmic graph: shape, asymptote and transformationsWhat a logarithmic graph looks like, its asymptote and x-intercept, and how a and c stretch and shift it, with a graph you can change and worked examples.
• Polar coordinates calculator: convert and plot curvesFree polar coordinates calculator: convert between polar and rectangular coordinates exactly, in radians or degrees, and plot polar curves with their area.
• Reciprocal graphs: y = 1/x, y = k/x and asymptotesReciprocal graphs explained: the shape of y = 1/x and y = k/x, asymptotes, plotting from a table, transformations and inverse proportion, with live graphs.
• Slope calculator: find the slope of a line from two pointsFree slope calculator: type two points to get the slope as an exact fraction, the line in y = mx + b form and every step, with worked examples.
• Transformations of graphs: shifts, stretches, reflectionsTransformations of graphs explained: f(x) + a, f(x + a), af(x), f(ax), −f(x) and f(−x), with a summary table, point mapping and worked examples.

Geometry

• Acute angle: definition, examples and acute trianglesWhat an acute angle is, how it compares with right, obtuse and reflex angles, how to measure one, and how to tell if a triangle is acute, with examples.
• Adjacent angles: definition, examples and finding xAdjacent angles share a vertex and a side without overlapping. See examples and non-examples, linear pairs, angles around a point, and how to solve for x.
• Alternate interior angles: theorem, converse and examplesAlternate interior angles lie between two lines on opposite sides of a transversal, and are equal when the lines are parallel. Proofs and examples.
• Angles: types, angle facts and how to find themWhat angles are, the types of angles from acute to reflex, how to measure them, every angle fact you need, and how to find missing angles, with examples.
• Bearings GCSE maths: the 3 rules, back bearings, trigHow to measure and calculate bearings for GCSE Maths: the three rules, back bearings, using angle facts, and Pythagoras and trigonometry with bearings.
• Circle theorems GCSE: all 8 rules with diagramsAll the GCSE circle theorems with a diagram for each, the reasons to write in the exam, and worked examples you can follow step by step.
• Complementary angles: definition, examples and problemsWhat complementary angles are, how to find the complement of an angle, how they differ from supplementary angles, and how to solve problems with algebra.
• Concave and convex: meaning, shapes, lenses and curvesConcave vs convex explained: what each word means, polygons, lenses and mirrors in physics, and convex and concave curves in A-level maths.
• Congruent: meaning in math, symbol and congruent trianglesWhat congruent means, the ≅ symbol, how to show figures are congruent with rigid motions, and the SSS, SAS, ASA, AAS and HL rules for triangles.
• Corresponding angles: definition, theorem and examplesCorresponding angles sit in matching positions where a line crosses two others, and they are equal when the lines are parallel. Examples and practice.
• Diameter: definition, formulas and how to find itThe diameter is the distance across a circle through its centre, twice the radius. Find it from the radius, circumference or area, with worked examples.
• How to find the area of a triangle: every methodHow to find the area of a triangle: ½ × base × height, right-angled and obtuse triangles, ½ab sin C, Heron's formula and coordinates, with examples.
• Hypotenuse calculator: from two sides or a side and angleFind the hypotenuse of a right triangle from the two legs or from one side and an angle, with the formulas, special triangles and worked examples.
• Line segment: definition, notation, length and midpointWhat a line segment is, how it differs from a line and a ray, how to name it, and how to find its length and midpoint on a number line or a graph.
• Loci maths: the 4 GCSE loci and how to construct themLoci maths for GCSE: the four standard loci, ruler and compass constructions step by step, shading regions, and worked examples with diagrams.
• Midpoint calculator: find the midpoint of two pointsFind the midpoint of a line segment from two points, with exact fractions, the distance and gradient, a coordinate plot and worked examples.
• Obtuse angle: definition, examples and obtuse trianglesWhat an obtuse angle is, how it compares with acute, right and reflex angles, how to measure one, and how to tell if a triangle is obtuse, with examples.
• Parallel lines: definition, angles, slopes and examplesWhat parallel lines are, the ∥ symbol, the angles made by a transversal, how to prove lines are parallel, and parallel lines on a graph, with examples.
• Pentagon shape: sides, angles, types and areaA pentagon is a shape with 5 sides and 5 angles adding to 540°. See the types of pentagon, the angles of a regular pentagon and how to find its area.
• Perimeter: formulas for every shape, with examplesWhat perimeter is and how to find it: formulas for squares, rectangles, triangles, polygons and circles, plus compound shapes, missing sides and examples.
• Perpendicular lines: meaning, slopes and equationsWhat perpendicular means, how to spot perpendicular lines from their slopes, and how to find the equation of a perpendicular line, with worked examples.
• Polyhedron: definition, types, Euler's formula and examplesWhat a polyhedron is, how to count its faces, edges and vertices, Euler's formula, the 5 Platonic solids, prisms, pyramids and nets, with examples.
• Pythagorean theorem calculator: find any side with stepsFind the hypotenuse or a missing leg of a right triangle with the Pythagorean theorem, worked examples you can check, and a right triangle test.
• Right angle: definition, symbol and right trianglesWhat a right angle is, its symbol, how to check for one with the 3-4-5 rule, perpendicular lines, and right triangles and the Pythagorean theorem.
• Scalene triangle: definition, types, area and examplesWhat a scalene triangle is, how it differs from isosceles and equilateral triangles, its types and properties, and how to find its area and angles.
• Supplementary angles: definition, examples and finding xSupplementary angles add up to 180°. Learn how to find a supplement, how they differ from complementary angles, and how to solve for x, with examples.
• Trapezoid: definition, properties, area and perimeterWhat a trapezoid is, its parts, types and properties, and the area and perimeter formulas, with worked examples, angle problems and practice.
• Vectors GCSE Maths: column vectors and vector proofsVectors for GCSE Maths: column vectors, adding, subtracting and multiplying by a scalar, vector diagrams, and how to write vector proofs, with examples.
• Vertical angles: definition, theorem and how to find xVertical angles are the opposite angles made when two lines cross, and they are always equal. See why, then solve for x with worked examples.
• What is a polygon? Definition, types, names and anglesA polygon is a flat, closed shape with straight sides. See polygon names from 3 to 20 sides, the types, interior and exterior angles, and examples.

Trigonometry

• Compound angle formula: sin, cos, tan (A ± B) with examplesThe compound angle formulae for sin(A ± B), cos(A ± B) and tan(A ± B): a proof, exact values, solving equations and the R cos(θ − α) form, with examples.
• Cosine rule: formula, finding angles and worked examplesThe cosine rule for a side or an angle, when to use it instead of the sine rule, 6 worked examples, practice questions and what each exam gives you.
• Double angle formula for sin, cos and tan, with examplesThe double angle formulae for sin 2A, cos 2A and tan 2A, how to derive them, the three forms of cos 2A, solving equations, proofs and integration.
• Exact trig values: table, triangles and how to rememberThe exact trig values of sin, cos and tan for 0°, 30°, 45°, 60° and 90°, the two exact value triangles, a memory trick and worked GCSE examples.
• Harmonic form: a sin x + b cos x as R sin(x + α)How to write a sin x + b cos x in harmonic form R sin(x ± α) or R cos(x ± α): finding R and α, maximum and minimum values, and solving equations.
• Law of cosines calculator: SAS and SSS trianglesFree law of cosines calculator: find the third side from two sides and the angle between them, or every angle from three sides, with worked examples.
• Law of sines calculator: solve triangles, ambiguous caseFree law of sines calculator: solve any triangle from two angles and a side, or two sides and an angle, with both answers in the ambiguous case.
• Right triangle calculator: every side, angle and the areaFree right triangle calculator: type any two values, one of them a side, to get every side and angle, the area and the perimeter, with worked examples.
• Sine rule: formula, examples and the ambiguous caseThe sine rule formula for sides and angles, when to use it instead of the cosine rule, the ambiguous case, area of a triangle, examples and practice.
• Small angle approximation: sin, cos and tan with examplesThe small angle approximations for sin, cos and tan in A-level Maths, why they work, how accurate they are, and worked examples of exam-style questions.
• SOHCAHTOA calculator: find a missing side or angleFree SOHCAHTOA calculator: find a missing side or angle of a right triangle with sin, cos or tan, learn which ratio to pick, and check worked examples.
• SOHCAHTOA: what it means and how to use it, with examplesWhat SOHCAHTOA stands for, how to label opposite, adjacent and hypotenuse, how to find missing sides and angles, and angle of elevation problems.
• Trig graphs: sin, cos and tan graphs, plus sec and cosecThe sin, cos and tan graphs with their key points, periods and asymptotes, how to solve equations and transform them, and the sec, cosec and cot graphs.
• Trig identities: full list, proofs and worked examplesEvery trig identity you need: reciprocal, quotient, Pythagorean, sum and difference, double-angle and half-angle, with proofs and worked examples.
• Unit circle: chart, radians, sin, cos, tan and how to use itThe unit circle explained: a full chart in degrees and radians, how it gives sin, cos and tan, quadrant signs, reference angles and worked examples.

Vectors and matrices

• Angle between two vectors calculator: 2D and 3DFind the angle between two vectors in 2D or 3D from the dot product, cos θ = a·b ÷ (|a||b|), with the magnitudes, worked examples and a quick check.
• Cross product calculator: 3D vectors with formula and stepsFind the cross product of two 3D vectors with the formula, the right-hand rule, its magnitude and area, and worked examples you can check.
• Dot product calculator: 2D and 3D, with the angleFree dot product calculator: find the dot product of two vectors in 2D or 3D, and the angle between them, with the formula worked through.
• Eigenvalue calculator: eigenvalues and eigenvectorsFree eigenvalue calculator: find the eigenvalues and eigenvectors of a 2 by 2 to 5 by 5 matrix exactly, with the characteristic polynomial and every step.
• How to find the inverse of a matrix: 2x2 and 3x3 stepsHow to find the inverse of a matrix: the 2x2 formula, the 3x3 method with minors, cofactors and the adjugate, how to check it, and solving equations.
• Unit vector: definition, formula and how to find oneWhat a unit vector is, how to find the unit vector in the direction of any vector in 2D or 3D, the i, j, k unit vectors, and worked examples.
• Vector calculator: magnitude, direction, add, dot and crossFree vector calculator: magnitude and direction, adding two vectors, the dot product, cross product and angle between them, in 2D or 3D.
• What is a matrix in maths? Definition, types and examplesA matrix is a rectangular grid of numbers in rows and columns. Learn matrix types, how to add, multiply and invert matrices, and what they are used for.

Calculus

• Area under curve calculator: shaded, with stepsFind the area under a curve between two limits with a definite integral, with working, a shaded graph and the total area when part of it is below the axis.
• Chain rule: formula, examples and how to use itThe chain rule formula, how to differentiate a function of a function step by step, 8 worked examples, practice questions with answers and exam tips.
• Derivative calculator with steps: every rule shownFree derivative calculator with steps: each rule named, second and higher derivatives, the slope and tangent at any point, and partial derivatives in x.
• Differential equation calculator with steps and slope fieldSolve first order and second order differential equations with an initial condition: general and particular solutions, full working and a slope field.
• Differentiation from first principles: formula and examplesThe first principles formula f'(x) = lim (f(x + h) − f(x))/h, why it works, step by step proofs for x², x³ and sin x, worked examples and practice.
• How to find the gradient of a curve: tangents and dy/dxHow to find the gradient of a curve at a point: drawing a tangent, using chords, and differentiating, with GCSE and A-level worked examples.
• Integral calculator with steps: definite and indefiniteFree integral calculator with steps: indefinite integrals, definite integrals with exact values, improper integrals and the area under the curve shaded.
• Integration A-level Maths: methods, formulas and examplesA-level Maths integration in one place: standard integrals, definite integrals, areas, substitution, parts, partial fractions and differential equations.
• Integration by parts: formula, LIATE and worked examplesThe integration by parts formula, how to choose u with LIATE, the tabular method, and worked examples from x e^x to e^x sin x, plus definite integrals.
• Integration by substitution: questions with full answersIntegration by substitution step by step, with worked examples, changing the limits of definite integrals, and practice questions with answers.
• Limit calculator with steps: at a point and at infinityFree limit calculator with steps: limits at a point or at infinity, one-sided limits, L'Hôpital's rule and a table of values closing in from each side.
• Newton-Raphson method: formula, examples and when it failsThe Newton-Raphson method for solving f(x) = 0: the formula, why it works, worked examples, proving a root to a given accuracy and when the method fails.
• Parametric differentiation: dy/dx, tangents and normalsHow to differentiate parametric equations: dy/dx = (dy/dt) ÷ (dx/dt), with worked examples for gradients, tangents, normals and stationary points.
• Parametric equations: to Cartesian form, dy/dx, examplesParametric equations explained: what a parameter is, converting to a Cartesian equation, circles, trig identities, domains, dy/dx, tangents and modelling.
• Parametric integration: area under a parametric curveHow to find the area under a curve given by parametric equations using ∫ y (dx/dt) dt, changing the limits to t, worked examples and practice questions.
• Point of inflection: how to find one, with examplesWhat a point of inflection is, how to find one with the second derivative, stationary and non-stationary points of inflection, and convex and concave.
• Product rule: formula, examples and how to use itThe product rule formula, how to differentiate a product in four steps, 6 worked examples, the proof, practice questions and what the booklet gives.
• Quotient rule differentiation: formula, steps and examplesQuotient rule differentiation explained: d/dx(u/v) = (v du/dx − u dv/dx) ÷ v², how to remember it, step-by-step examples, tan x, and stationary points.
• Rate of change calculator: average and instantaneousFind the average and instantaneous rate of change of a function, watch the secant become the tangent, and follow worked examples you can check.
• Reverse chain rule: integration by inspection, with examplesHow the reverse chain rule integrates functions like x(x² + 1)⁴ and 4x/(x² + 5) by inspection: the patterns, the adjust-the-constant method and examples.
• Taylor series calculator: Taylor and Maclaurin, with stepsFree Taylor series calculator: expand a function about any centre to order 12, with every derivative and coefficient, plotted against the curve.
• Trapezium rule: formula, examples and over or underThe trapezium rule formula for estimating an integral, how to use it step by step, when it overestimates or underestimates, worked examples and practice.
• What is a derivative? Meaning, definition and examplesWhat a derivative is, in plain words and in symbols: the slope of a tangent, the limit definition, the power rule, worked examples and real uses.
• What is an integral? Meaning, area and worked examplesWhat an integral is: area under a curve, adding up tiny pieces, and the reverse of differentiation. Definite and indefinite integrals with worked examples.

Statistics

• Bar graph: what it is, how to draw and read oneWhat a bar graph is, its types, how to draw and read one step by step, and how a bar graph differs from a histogram, with worked examples.
• Binomial distribution calculator: exactly, at most, at leastFind binomial probabilities: exactly, at most, at least or between, with the mean and standard deviation, a bar chart and worked examples.
• Binomial distribution: conditions, formula, mean, varianceWhat the binomial distribution is, when to use it, its formula, mean and variance, cumulative probabilities, the normal approximation and exam questions.
• Box plot calculator: five-number summary and outliersFree box plot calculator: draw box and whisker plots for up to four data sets, with quartiles, the IQR, outliers by the 1.5 × IQR rule and a comparison.
• Confidence interval calculator: means and proportionsFree confidence interval calculator for a mean (t or z) or a proportion, from summary values or raw data, with the margin of error and every step.
• Cumulative frequency graph: draw it, find median and IQRHow to draw a cumulative frequency graph step by step, then use it to estimate the median, quartiles, interquartile range and percentiles. GCSE example.
• Expected value calculator: E(X), variance and SDFree expected value calculator: enter outcomes and probabilities, as decimals or fractions, to get E(X), the variance and standard deviation, with steps.
• Frequency density formula: how to work it outThe frequency density formula, how to work it out for a histogram with unequal class widths, reading frequencies back, estimating from bars, and practice.
• Frequency polygons: how to draw and compare themHow to draw a frequency polygon from grouped data, compare two on one set of axes, and tell them apart from histograms and other frequency diagrams.
• Geometric distribution: formula, mean and examplesWhat the geometric distribution is, its formula, cumulative probabilities, mean and variance, how it differs from the binomial, and worked examples.
• Histogram maker: free, with frequency densityFree histogram maker: turn raw data or a grouped table into a correctly scaled histogram, with frequency density, cumulative frequency and the median.
• Hypothesis testing A-level Maths: steps and examplesHypothesis testing for A-level Maths: the language, the steps, binomial tests and critical regions, normal mean tests and correlation, with examples.
• Line graph: what it is, how to read and make oneWhat a line graph is, its parts, how to read trends and values from one, how to draw your own, and how it compares with bar graphs, with examples.
• Linear interpolation: formula, examples and grouped dataLinear interpolation explained: the interpolation formula, step-by-step examples, the median and percentiles of grouped data, and when it goes wrong.
• Linear regression calculator: line of best fit and rFit a least-squares line of best fit to paired data, with the correlation coefficient r, r squared, residuals and predictions, plus examples.
• Mean median mode calculator, with range and quartilesFree mean median mode calculator: get the mean, median, mode, range and quartiles of a list, a frequency table or grouped data, with every step shown.
• Normal distribution calculator: probabilities and curveFind normal distribution probabilities below, above, between or outside values for any mean and standard deviation, with the shaded curve and examples.
• Normal distribution: bell curve, z-scores and examplesThe normal distribution explained: the bell curve, the 68-95-99.7 rule, z-scores and the standard normal, probabilities and the normal approximation.
• P-value calculator: z, t and chi-square testsFree p-value calculator: run a z, t, chi-square or F test from your data or summary values and get the p-value, the critical value and a plain conclusion.
• PMCC: product moment correlation coefficient explainedThe product moment correlation coefficient (PMCC) explained: what r means, the formula, interpreting values, and the A-level test for correlation.
• Poisson distribution calculator: exact, at most, at leastFree Poisson distribution calculator: find P(X = k), P(X ≤ k), P(X ≥ k) or a range for any mean rate λ, with the graph shaded and worked examples.
• Probability calculator: normal, binomial and PoissonFree probability calculator: shade P(X < a), P(X > a) or a range on a normal, binomial or Poisson distribution and read the probability, with examples.
• Standard deviation calculator: sample and populationFind the sample or population standard deviation of raw data, a frequency table or grouped data, with the full working table and worked examples.
• Standard deviation formula for A-level Maths, with examplesThe standard deviation formula A-level Maths uses: σ from raw data, Sxx and summary statistics, frequency tables, coding and outliers, with examples.
• Stem and leaf diagram: how to draw and read oneHow to draw a stem and leaf diagram with a key, find the median, mode, range and quartiles from it, and compare data with a back-to-back diagram.
• T test calculator: one-sample, two-sample and pairedRun one-sample, two-sample and paired t tests from summary values or raw data, with the t statistic, p-value, critical value and a plain conclusion.
• Uniform distribution: discrete and continuous, with examplesThe uniform distribution explained: discrete and continuous, the pdf 1/(b − a), mean and variance formulas with derivations, and worked examples.
• Variance calculator: sample and population varianceFind the sample or population variance of a data set, frequency table or grouped data, with each squared deviation shown and worked examples.
• Z score calculator: z = (x − μ)/σ and probabilitiesFind the z-score of a value from the mean and standard deviation, with the probability to the left or right, a shaded normal curve and examples.

Probability

• Combination calculator: nCr with and without repetitionWork out nCr, the number of ways to choose r items from n when order does not matter, with or without repetition, plus worked examples to check.
• Conditional probability questions with worked answersConditional probability questions from two-way tables, Venn diagrams and tree diagrams, with the formula, full worked answers and practice questions.
• How to find probability: formula, rules and examplesHow to find probability step by step: the basic formula, not, or and and rules, tree diagrams, experimental probability and worked examples.
• Permutation calculator: nPr with and without repetitionWork out nPr, the number of ordered arrangements of r items from n, with or without repetition, plus word arrangements and worked examples.
• Probability tree diagram questions with answersHow to draw and use probability tree diagrams, with worked questions on independent events, without replacement and algebra, plus questions to try.
• Sample space diagram: how to draw and use one (GCSE)How to draw a sample space diagram for two dice, a coin and a spinner, and use it to find probabilities and expected frequencies, with GCSE examples.
• Venn diagram rules: notation, filling in, probabilityVenn diagram rules for GCSE Maths: the universal set, intersection, union, complement, filling one in, probability, the addition rule, HCF and LCM.

Mechanics

• Displacement-time graph: gradient, velocity and examplesHow to read a displacement-time graph: gradient as velocity, flat, steep and curved lines, tangents, and how it differs from a distance-time graph.
• Velocity-time graph: gradient, area and examplesHow to read a velocity-time graph: the gradient gives acceleration and the area gives distance, with GCSE examples, curved graphs and common mistakes.

Courses

• Algebra course: from variables to the quadratic formulaAn interactive algebra course in 34 short sections: equations, inequalities, factoring, the quadratic formula, simultaneous equations and more.
• Calculus course: limits, derivatives and integralsAn interactive calculus course in 29 short sections: limits, differentiation rules, stationary points, integration, definite integrals and motion.
• Graphing functions course: lines, curves and transformationsAn interactive graphing functions course in 31 short sections: linear, quadratic, cubic, rational, exponential and root graphs, and transformations.
• Group theory course: symmetry, Cayley tables and LagrangeAn interactive group theory course in 24 short sections: symmetries, the group axioms, Cayley tables, modular arithmetic, cyclic groups and Lagrange.
• Probability course: from basics to binomial and normalAn interactive probability course in 36 short sections: sample spaces, Venn and tree diagrams, conditional probability, expected value and distributions.
• Trigonometry course: SOH CAH TOA to the unit circleAn interactive trigonometry course in 35 short sections: SOH CAH TOA, sides and angles, radians, the unit circle, trig graphs, sine and cosine rules.