IGCSE Maths 0580 notes, quizzes & practice

The words that sort every number into its box — natural, integer, prime, square, cube, rational, irrational — plus the tools built on them: prime factorisation, HCF and LCM, set notation with Venn diagrams, powers and roots, reciprocals, and the inequality symbols used to put numbers in order.

One idea in three outfits: a fraction, a decimal and a percentage are the same amount in different clothing. Convert fluently between them (including recurring decimals at Extended), add and multiply mixed numbers, work out percentage change and reverse percentages, grow money with simple and compound interest, and handle time, timetables and currency without slipping.

Ratio splits amounts fairly by a recipe — simplify it, divide in it, scale it for a map or a menu. Proportion lets one quantity drive another, from doubling a recipe to inverse square laws at Extended. Rates bundle distance, money and flow per unit of time, with speed, density and pressure as the examined headline acts.

Three tools for writing very big and very small numbers exactly: the index laws that collapse products into single powers, standard form A × 10ⁿ that keeps astronomy and microbes on the same page, and surds that keep square roots exact. Non-calculator papers reward every one of these.

How to say “about how much” convincingly: round to decimal places or significant figures, estimate hard calculations by simplifying every number to one significant figure, and — at Extended — trap a rounded measurement inside upper and lower bounds, then push those bounds through perimeters, areas and speeds.

The grammar of algebra: swap letters for numbers, multiply brackets out cleanly, then run the process backwards — pulling common factors out, spotting the difference of two squares, factorising quadratics, completing the square, and cancelling rational expressions the honest way.

Getting from “what is true” to “what is the number”: undo linear equations in tidy steps, trap two unknowns with simultaneous pairs, crack quadratics by factorising or the formula, rearrange formulas for any letter, and solve inequalities without flipping a sign at the wrong moment.

Reading the rhythm of a list of numbers: describe it term by term, then compress it into an nth term you can evaluate anywhere — differences for linear runs, second differences for quadratics, pattern-hunting for cubics and for the doubling-and-halving geometric families.

Pictures that calculate: read rates straight off travel and conversion graphs, treat area under a speed–time graph as distance, build tables of values and plot them accurately, harvest roots and intersections from a curve, and sketch the classic families — parabolas, cubics, reciprocals and exponentials.

The Extended-only toolkit: feed numbers through function machines forwards and backwards, compose two functions in the right order, then switch to calculus — differentiate powers of x, hunt stationary points, classify them as maxima or minima, and pin tangents onto curves.

Algebra meets the x–y plane: measure a line’s slope from any two points, halve a segment to its midpoint, stretch Pythagoras across it for its length, then write equations — including the parallel clone and the perpendicular twin, all the way up to bisectors (Extended).

Angle-chasing with receipts: every fact worth a mark — straight lines, full turns, triangles, parallel-line pairs — plus the polygon machinery of interior and exterior angles, the area and volume tricks of similar shapes, and bearings measured the sailor’s way.

Six theorems and a pile of isosceles triangles: double angles from centre to rim, harvest equal angles from the same segment, close cyclic quadrilaterals to 180°, bounce tangents off radii, and let the alternate segment hand you angles nobody drew yet.

Move whole shapes with four machines — flip, spin, slide, stretch — and describe each move precisely; then upgrade points to column vectors and push them around with addition, scalars and magnitudes, finishing with vector proofs on parallelograms (Extended).

Measure everything: convert units without losing zeros, find perimeters and areas of the standard plane shapes, slice circles into arcs and sectors, and box, roll and round shapes into prisms, cylinders, cones and spheres.

From ladder-on-a-wall Pythagoras to full sine-rule navigation: pick the right ratio, keep the calculator in degrees, and climb from right-angled triangles to exact values, trig equations and 3-D diagonals (Extended).

Put chance on the 0-to-1 line, estimate it from experiments, then combine events with sample spaces, Venn diagrams and tree diagrams — finishing with conditional probabilities read straight off tables (Extended).

Turn raw numbers into answers: sort data into two-way tables, summarise with mean, median, mode and range, draw pie charts and stem-and-leaf diagrams, then predict with lines of best fit and read off quartiles (Extended).