Pearson Edexcel International GCSE in Mathematics (Specification A) · 4MA1 · Free · no account needed

Incandio Mathematics

53 modules across 16 strands of Pearson Edexcel International GCSE Mathematics (Specification A), completely free. Every module explains what the idea is, why it works, when a question is asking for it, the misconceptions that actually cost marks and worked examples laid out the way an examiner wants working shown — then five questions that climb Grade 6 → 9+, marked on your own device with an authored diagnosis of the mistake you made.

Also known as Edexcel IGCSE Maths — Pearson’s own name for the qualification is the International GCSE, and that is the name every Incandio page uses.

Foundation Tier targets grades 5–1 and Higher Tier grades 9–4. Modules marked Higher do not appear on Foundation papers. The five practice questions in every module climb Grade 6 → 9+, so they are pitched at Higher Tier throughout.

Then the part a textbook cannot do: you teach the method back to Ember, an AI pupil who asks the awkward question, and defend it against the mathematician with the strongest claim on it.

The strands

Number — 8 modules

Place value, fractions, decimals, indices, standard form, bounds, surds and sets — the arithmetic bedrock of both papers, and nearly a fifth of all marks by itself.

• Integers, Place Value & Ordering — Negative numbers, the order of operations, and putting any mix of numbers in order — the arithmetic everything else stands on.
• Fractions & Mixed Numbers — Adding, multiplying and dividing fractions with certainty — and the bar-model picture that makes worded fraction problems fall open.
• Decimals & Recurring Decimals — Multiplying and dividing decimals without a wobble, and the Higher-tier trick that turns any recurring decimal into an exact fraction.
• Powers, Roots & Index Laws — What indices really mean — including the zero, negative and fractional ones that look strange until you see the pattern they complete.
• Standard Form — Writing enormous and tiny numbers as a×10ⁿ, and calculating with them confidently — the physicist's notation, on both papers.
• Rounding, Estimation & Bounds — Significant figures, one-significant-figure estimates that sanity-check everything, and the upper and lower bounds of a rounded measurement.
• Surds — Exact answers with square roots: simplifying √48, expanding surd brackets, and rationalising denominators. Higher Tier. (Higher Tier)
• Sets & Venn Diagrams — Set notation, intersections and unions, and the counting arguments that solve 'how many study both?' problems in two lines.

Percentages — 3 modules

The most-asked number topic: multipliers, percentage change, reverse percentages and compound interest — appearing on almost every paper, fumbled on almost every paper.

• Percentage Basics — What 'per cent' actually means, finding a percentage of an amount, and writing one number as a percentage of another.
• Multipliers & Percentage Change — The multiplier method — one number that does the whole job — for increases, decreases, repeated change and 'what was the overall change?'
• Reverse Percentages & Compound Interest — Finding the original price before a change (the question everyone gets backwards), and money growing multiplier-on-multiplier.

Ratio & Proportion — 3 modules

Sharing in ratios, direct and inverse proportion, and the compound measures — speed, density, pressure — that dress proportion in real-world clothes.

• Ratio — Sharing in a ratio, finding totals from parts, and combining two ratios into one — all powered by the value of ONE part.
• Direct & Inverse Proportion — Quantities that scale together, quantities that trade off, and the algebraic y = kx² machinery of Higher Tier proportion.
• Speed, Density & Compound Measures — Speed–distance–time, density–mass–volume and the unit conversions between them — including the average-speed trap everyone falls into once.

Algebra — 8 modules

Expressions, equations, quadratics, simultaneous equations, inequalities and algebraic fractions. Number and algebra together carry about 60% of the qualification.

• Expressions & Index Laws — Collecting like terms and running the index laws on letters — the fluency layer every other algebra module assumes.
• Expanding & Factorising — Brackets out, brackets back in — including the double brackets and quadratic factorising that unlock every quadratic equation.
• Linear Equations — Solving any equation with one unknown — both sides, brackets, fractions — by doing the same thing to both sides, every time.
• Formulae & Rearranging — Substituting into formulae and changing the subject — including the Higher-tier case where the new subject appears twice.
• Quadratic Equations — Four ways to solve a quadratic — factorising, the formula, completing the square, and reading a graph — why each one works, how to choose between them, and how to reason about how many solutions there are.
• Simultaneous Equations — Two unknowns, two equations — elimination, substitution, and the Higher-tier pairing of a line with a curve.
• Inequalities — Solving inequalities like equations — with the one extra rule about negatives — and reading off the integers they allow.
• Algebraic Fractions — Simplifying, combining and solving with fractions whose parts are algebra — factorise first, and they behave exactly like number fractions. Higher Tier. (Higher Tier)

Sequences — 2 modules

nth terms of linear sequences, and the second-difference machinery of quadratic sequences at Higher Tier.

• Linear Sequences & the nth Term — Finding the rule that generates a sequence — and using it to predict, test and bound terms without listing hundreds of them.
• Quadratic Sequences — Sequences whose differences themselves change — spotting the n² engine underneath and building the exact rule. Higher Tier. (Higher Tier)

Coordinate Geometry — 2 modules

y = mx + c, gradients, midpoints and lengths — then parallel and perpendicular lines at Higher Tier.

• Straight Lines: y = mx + c — Gradient as a rate of climb, intercept as the starting height — and building a line's equation from any two of its points.
• Parallel & Perpendicular Lines — Same gradient means parallel; gradients multiplying to −1 mean perpendicular — and the constructions that follow. Higher Tier. (Higher Tier)

Graphs — 2 modules

Travel graphs and real-life graphs where gradients mean speeds and areas mean distances; quadratic, cubic and reciprocal curves.

• Travel & Real-Life Graphs — Distance–time and speed–time graphs read as stories: gradients are speeds, flat lines are stops, and area under a speed–time graph is distance.
• Quadratic & Other Curves — Parabolas, cubics and reciprocal curves — reading intercepts, turning points and symmetry, and solving equations where curves meet lines.

Functions — 1 module

Function notation, composite functions and inverses — the machine picture that makes them all one idea. Higher Tier.

• Function Notation, Composite & Inverse — f(x) as a machine: feeding it values, chaining two machines together, and running a machine backwards. Higher Tier. (Higher Tier)

Calculus — 2 modules

Yes, really — 4MA1 Higher Tier includes differentiation: gradients of curves, turning points, and velocity and acceleration in kinematics.

• Differentiation — The gradient of a curve at a single point — the power rule, and why it works. Higher Tier, and genuinely on this specification. (Higher Tier)
• Calculus in Motion: Kinematics — Displacement, velocity and acceleration as a chain of derivatives — the exam's favourite way to dress calculus in a story. Higher Tier. (Higher Tier)

Geometry — 4 modules

Angle facts with named reasons, the special shapes, polygons, and bearings — where the reasons earn as many marks as the arithmetic.

• Angles & Parallel Lines — Angle facts on lines, at points and between parallels — and the exam habit that doubles your marks: naming the reason for every step.
• Triangles & Quadrilaterals — The special triangles and quadrilaterals, their angle totals, symmetries and diagonal properties — the shape vocabulary of every geometry question.
• Polygons — Interior and exterior angles of any polygon — driven by the one unbreakable fact that exterior angles always total 360°.
• Bearings & Scale Drawings — Three-figure bearings measured clockwise from north, back-bearings, and map scales — the navigation mathematics on both papers.

Circles & Circle Theorems — 2 modules

Tangents, chords and the five classical circle theorems — the Higher paper's favourite angle-chase territory.

• Circle Properties & Tangents — Radii, chords and tangents, and the right angles they secretly generate — the foundation the circle theorems build on.
• Circle Theorems — The five classical theorems — centre and circumference, semicircle, same segment, cyclic quadrilateral, alternate segment — and how to chain them. Higher Tier. (Higher Tier)

Mensuration — 4 modules

Perimeter, area, arcs and sectors, volume and surface area, and the k, k², k³ ladder of similar shapes.

• Perimeter & Area — Rectangles, triangles, parallelograms, trapezia and compound shapes — with the height rule that separates area marks from area guesses.
• Circles, Arcs & Sectors — Circumference and area, then the fraction-of-a-circle thinking that turns arcs and sectors into one idea instead of four formulas.
• Volume & Surface Area — Prisms, cylinders, spheres and cones — filling solids (volume) and wrapping them (surface area), including the formulas the sheet won't give you.
• Similar Shapes & Scale Factors — When one shape is an enlargement of another: lengths scale by k, areas by k², volumes by k³ — and how to run those factors in any direction.

Trigonometry — 4 modules

Pythagoras and SOHCAHTOA on both tiers; the sine and cosine rules and full 3-D work at Higher.

• Pythagoras' Theorem — a² + b² = c² — finding missing sides in right-angled triangles, and spotting the hidden right triangles inside other shapes.
• Right-Angled Trigonometry — Sine, cosine and tangent: finding sides and angles in right triangles, from ladders to angles of elevation — plus the exact values worth knowing.
• The Sine & Cosine Rules — Trigonometry escapes the right angle: solving ANY triangle, plus the ½ab sin C area formula. Higher Tier. (Higher Tier)
• 3-D Pythagoras & Trigonometry — Space diagonals, angles between lines and planes — finding the flat right-angled triangle hiding inside every solid. Higher Tier. (Higher Tier)

Vectors & Transformations — 2 modules

The four transformations described completely, and the a-and-b vector algebra that ends most Higher papers.

• Transformations — Translation, reflection, rotation and enlargement — performing each, and describing them COMPLETELY, because a partial description scores nothing.
• Vectors — Column vectors, magnitude, and the a-and-b algebra of geometric proof — the topic that ends most Higher papers. Higher Tier. (Higher Tier)

Statistics — 3 modules

Averages and spread, pie charts and histograms, cumulative frequency and box plots — the strand students under-revise because it looks easy.

• Averages & Spread — Mean, median, mode and range — from lists, tables and grouped data — and the combined-mean and reverse-mean questions that catch everyone once.
• Charts, Pie Charts & Histograms — Pie chart angles, scatter-graph correlation, and the Higher-tier histogram rule that area — not height — is frequency.
• Cumulative Frequency & Box Plots — Running totals, medians and quartiles read from the curve, box plots, and comparing distributions properly. Higher Tier. (Higher Tier)

Probability — 3 modules

Fair counting, tree diagrams, and conditional probability — multiply along the branches, add between them.

• Probability Basics — Probability as fair counting: single events, 'not' events, expected frequency and relative frequency — the foundations every tree diagram stands on.
• Combined Events & Tree Diagrams — Multiply along branches, add between them — with and without replacement, and the 'at least one' shortcut that saves whole minutes.
• Conditional Probability — Probability when you already know something — shrinking worlds, 'given that' questions, and combining branches over changing conditions. Higher Tier. (Higher Tier)

The specification, and exam practice

• The 4MA1 specification — papers, assessment objectives and every specification-mapped topic.
• Exam Ready — exam-style practice for every module, marked on Incandio.
• Incandio — learn every idea, teach it, then defend it.