Maths study guides
79 free maths guides across every exam we teach, from GCSE to AP, A level and university admissions tests.
AP Calculus
- AP Calculus AB vs BC: which should you take?AP Calculus AB vs BC compared: what each course covers, how the exams work, how hard BC is and how to choose.
- AP Calculus AB units, explainedAll eight AP Calculus AB units explained in plain English, plus the two extra units in AP Calculus BC and how to study each one.
- AP Calculus formula sheet: what to memoriseThere is no formula sheet in the AP Calculus AB or BC exam. Here are the derivatives, integrals, theorems and BC formulas to memorise.
- AP Calculus AB review and study planAn AP Calculus AB review plan: what to revise in each unit, free-response tips, calculator skills and a plan for the final weeks.
- AP Calculus BC units, explainedAll ten AP Calculus BC units explained, including the BC-only topics, what to focus on and how to prepare for the exam.
- AP Calculus BC series, made clearAP Calculus BC series made clear: sequences, convergence tests, Taylor and Maclaurin series, radius of convergence and error bounds.
- Related rates in AP CalculusRelated rates in AP Calculus made clear: a step-by-step method, worked examples with ladders and cones, and the mistakes to avoid.
- AP Calculus derivative rulesEvery derivative rule you need for AP Calculus AB and BC: power, product, quotient and chain rules, trig, exponential, log and inverse functions.
- Arc length, explained with examplesArc length explained for AP Calculus BC: the formula for y = f(x) and parametric curves, speed and total distance, worked examples and practice with answers.
- The chain rule, explained with examplesThe chain rule explained step by step: the formula, spotting the inner and outer function, examples with powers, trig, exponentials and logs, plus practice.
- Euler's method, explained with examplesEuler's method explained for AP Calculus BC: the formula, a step-by-step table, over- or underestimates from concavity, and step size, with practice.
- Derivatives of exponentials and logarithmsDerivatives of e^x, ln x, a^x and log base b explained: proofs, chain rule cases like ln(f(x)) and logarithmic differentiation, with practice and answers.
- The fundamental theorem of calculus, explainedThe fundamental theorem of calculus explained: Part 1 and Part 2, derivatives of accumulation functions, net change and AP-style questions, with practice.
- Implicit differentiation, explained with examplesImplicit differentiation explained: why it works, a four-step method, examples with circles, products and trig, tangent lines and second derivatives.
- Improper integrals, explained with examplesImproper integrals explained for AP Calculus BC: infinite limits and discontinuities, convergence and divergence, the p-integral rule, with worked examples.
- Critical points, inflection points and the derivative testsInflection points, concavity and the first and second derivative tests explained for AP Calculus, with absolute extrema, justifications and practice.
- Integration by parts, explained with examplesIntegration by parts explained: the formula, choosing u with LIATE, worked examples with ln x and e^x sin x, definite integrals and the tabular method.
- The intermediate value theorem, explainedThe intermediate value theorem explained for AP Calculus: what it says, why continuity matters, showing a root exists, and writing a full-credit justification.
- L'Hôpital's rule, explained with examplesL'Hôpital's rule explained: the conditions, 0/0 and ∞/∞ limits, other indeterminate forms, and how to justify it on AP free-response, with practice.
- Limits in calculus, explained with examplesLimits explained for AP Calculus: one-sided limits, substitution, factoring, rationalising, limits at infinity, the squeeze theorem and continuity.
- Linear approximation, explained with examplesLinear approximation explained for AP Calculus: local linearity, the tangent line formula, estimates, over- or underestimates and differentials, with practice.
- Logistic growth, explained with examplesLogistic growth explained for AP Calculus BC: the logistic differential equation, carrying capacity, fastest growth at L/2 and the solution, with practice.
- The mean value theorem, explained with examplesThe mean value theorem explained: what it says, the conditions, Rolle's theorem, how it compares with the IVT and EVT, and AP-style examples with answers.
- Polar coordinates, explained for AP Calculus BCPolar coordinates explained for AP Calculus BC: converting to Cartesian, common polar curves, dy/dx, and area inside and between polar curves, with practice.
- The product rule and quotient rule, explainedThe product rule and quotient rule explained: the formulas, where they come from, when to rewrite instead, combining with the chain rule, with practice.
- Riemann sums, explained with examplesRiemann sums explained for AP Calculus: left, right and midpoint sums, the trapezoidal rule, over- and underestimates and sums from tables, with practice.
- U-substitution, explained with examplesU-substitution explained for AP Calculus and A level Maths: when to use it, choosing u, the step-by-step method and changing limits, with practice.
- Volumes of revolution, explained with examplesVolumes of revolution explained for AP Calculus: disc and washer methods, rotating about any axis, known cross-sections and setting up the integral.
GCSE Maths
- GCSE Maths revision: a plan that worksA practical GCSE Maths revision plan: what to revise first, how to use past papers, and a week-by-week plan for Foundation and Higher tier.
- Circle theorems, explainedAll eight GCSE circle theorems explained in plain English, with how to spot them in exam questions and how to give reasons for full marks.
- GCSE Maths topics: the full listThe full list of GCSE Maths topics for Foundation and Higher tier, grouped into the six content areas, with the Higher-only topics marked.
- GCSE Maths vectorsGCSE Maths vectors explained: column vectors, adding and multiplying vectors, and how to answer Higher tier vector proof questions.
- GCSE Maths trigonometryGCSE Maths trigonometry explained: SOHCAHTOA, exact values, the sine rule, the cosine rule and the area of a triangle, with when to use each.
- GCSE Maths surdsSurds for GCSE Maths Higher explained: what a surd is, simplifying, adding and multiplying surds, expanding brackets and rationalising the denominator.
- Mean, median, mode and range, explainedMean, median, mode and range explained for GCSE Maths: averages from frequency tables and grouped data, missing values, and practice with answers.
- Bearings, explained with examplesBearings explained for GCSE Maths: the three rules, measuring and drawing, back bearings, and bearings with trigonometry, with practice questions and answers.
- Upper and lower bounds, explained with examplesUpper and lower bounds explained for GCSE Maths: bounds from rounding, error intervals, and bounds in calculations, with practice questions and answers.
- Box plots, explained with examplesBox plots explained for GCSE Maths: the five-number summary, quartiles, drawing a box plot, IQR, outliers and comparing distributions, with practice.
- Area of a circle, explained with examplesArea of a circle explained for GCSE Maths: A = πr², using the diameter, answers in terms of π, finding the radius, semicircles and sectors, with practice.
- Compound interest, explained with examplesCompound interest explained for GCSE Maths: the multiplier method, the formula, compound vs simple interest, depreciation and reverse problems, with practice.
- Completing the square, explained with examplesCompleting the square explained for GCSE and A level Maths: the method, when a is not 1, solving equations and finding turning points, with practice.
- Cumulative frequency, explained with examplesCumulative frequency explained for GCSE Maths: the table, plotting the graph, reading the median, quartiles and IQR, with practice questions and answers.
- Volume of a cylinder, explained with examplesVolume of a cylinder explained for GCSE Maths: V = πr²h, using the diameter, finding the height or radius, capacity in litres and surface area, with practice.
- Factorising quadratics, explained with examplesFactorising quadratics explained for GCSE Maths: common factors, x² + bx + c, difference of two squares, and the ac method when a is not 1, with practice.
- Histograms, explained with examplesHistograms explained for GCSE Maths: frequency density, unequal class widths, reading frequencies and estimating within a class, with practice and answers.
- Index laws, explained with examplesLaws of indices explained for GCSE Maths: multiplying and dividing powers, zero, negative and fractional indices, and solving index equations, with practice.
- Inequalities, explained with examplesInequalities explained for GCSE Maths: number lines, solving linear inequalities, flipping the sign, integer solutions, quadratic inequalities and regions.
- Straight line graphs, explained with examplesStraight line graphs explained for GCSE Maths: gradient, y = mx + c, the equation from two points, parallel and perpendicular lines, with practice and answers.
- The nth term, explained with examplesThe nth term explained for GCSE Maths: linear sequences, checking if a number is in a sequence, quadratic and geometric sequences, with practice and answers.
- Percentage change, explained with examplesPercentage change explained for GCSE Maths: the formula, increase and decrease, profit and loss, multipliers, reverse percentages and repeated changes.
- Pie charts, explained with examplesPie charts explained for GCSE Maths: calculating angles, drawing accurately, reading frequencies and percentages, and comparing two pie charts, with practice.
- Prime factors, HCF and LCM, explainedPrime factors, HCF and LCM explained for GCSE Maths: factor trees, products of primes, and HCF and LCM with Venn diagrams and word problems, with practice.
- Pythagoras' theorem, explained with examplesPythagoras' theorem explained for GCSE Maths: finding the hypotenuse and shorter sides, triples, distance between points and 3D problems, with practice.
- The quadratic formula, explained with examplesThe quadratic formula explained for GCSE Maths: when to use it, substituting without sign errors, the discriminant, surd answers and practice with answers.
- Ratio, explained with examplesRatio explained for GCSE Maths: simplifying, sharing in a ratio, finding totals from one part, combining ratios, best buys and scale, with practice and answers.
- Speed, distance and time, explained with examplesSpeed, distance and time explained for GCSE: the formula and triangle, unit conversions, average speed, distance-time graphs and km/h to m/s, with practice.
- Simultaneous equations, explained with examplesSimultaneous equations explained for GCSE Maths: elimination and substitution, linear and quadratic pairs, graphical solutions and word problems, with practice.
- The sine rule and cosine rule, explainedThe sine and cosine rules explained for GCSE and A level: finding sides and angles, which rule to use, the ambiguous case and ½ab sin C, with practice.
- Standard form, explained with examplesStandard form explained for GCSE Maths: writing large and small numbers, converting back, and multiplying, dividing, adding and subtracting, with practice.
- Area of a trapezium, explained with examplesArea of a trapezium explained for GCSE Maths: ½(a + b)h, why it works, finding a missing side or height, and compound shapes, with practice and answers.
- Probability tree diagrams, explained with examplesProbability tree diagrams explained for GCSE Maths: drawing them, multiplying and adding, with and without replacement, and conditional probability.
- Venn diagrams, explained with examplesVenn diagrams explained for GCSE Maths: two- and three-set diagrams, set notation, probabilities and conditional probability, with practice and answers.
A level Maths
- A level Maths topics: the full listEvery A level Maths topic in Pure, Statistics and Mechanics, how the exam papers are split by board, and how to revise them.
- SUVAT equations, explainedThe five SUVAT equations explained clearly: what s, u, v, a and t mean, how to choose the right equation and worked examples for GCSE and A level.
- A level Maths formula booklet: how to use itHow to use the A level Maths formula booklet for Edexcel, AQA and OCR, what is usually included, and the formulas you still need to memorise.
- A level Maths differentiationA level Maths differentiation explained: first principles, the standard rules, chain, product and quotient rules, implicit differentiation and applications.
- A level Maths integrationA level Maths integration explained: standard integrals, substitution, by parts, partial fractions, areas under curves and the trapezium rule.
- A level Maths mechanicsA level Maths mechanics explained: SUVAT, variable acceleration, Newton's laws, friction, connected particles, moments and projectiles.
- A level Maths statisticsA level Maths statistics explained: sampling, data, probability, binomial and normal distributions and hypothesis tests, with exam tips.
- A level Maths vectorsA level Maths vectors explained: notation, magnitude and direction, position vectors, unit vectors and solving geometric problems in 2D and 3D.
- Binomial expansion, explained with examplesBinomial expansion explained for A level Maths: Pascal's triangle, nCr, finding a coefficient, negative and fractional powers, and approximations.
- Differential equations, explained with examplesDifferential equations explained for A level Maths: forming equations, separating variables, general and particular solutions, growth and decay, with practice.
- Differentiation from first principles, explainedDifferentiation from first principles for A level Maths: the limit definition, proofs for x², x³ and 1/x, and how to set out a full-mark answer, with practice.
- Logarithms, explained with examplesLogarithms explained for A level Maths: the laws of logs, ln and e, solving exponential equations, and turning y = ax^n into a straight line, with practice.
- Parametric equations, explained with examplesParametric equations explained for A level Maths: converting to Cartesian form, dy/dx, tangents and normals, and areas under parametric curves, with practice.
- Partial fractions, explained with examplesPartial fractions explained for A level Maths: linear and repeated factors, improper fractions, the cover-up method, and use in integration, with practice.
- Trigonometric identities, explained with examplesTrig identities for A level Maths: core identities, sec, cosec and cot, compound and double angle formulae, R cos(x + a) and proofs, with practice.
TMUA and ESAT
- What is the TMUA?The TMUA explained: what it is, who needs it, the two papers, how it is scored and how to prepare. A guide from STEM-Tutor.
- The TMUA specification, topic by topicThe TMUA syllabus explained topic by topic: the maths behind Paper 1, logic and proof behind Paper 2, example questions and how to build a study plan.