IGCSE Maths Lesson Notes
The full Cambridge IGCSE Mathematics (0580) syllabus, from number and algebra to trigonometry and statistics — worked the way it is taught, with the method marks students throw away.
All 47 topics across 10 units
Start anywhere — each topic stands on its own.
Number
10 topics-
Fractions, decimals and percentages
Fractions, decimals and percentages: the four operations with fractions, mixed numbers, converting between all three forms, recurring decimals to fractions, and ordering.
7 min read Read notes -
Indices and standard form
Indices and standard form: all the index laws, zero, negative and fractional indices, converting to and from standard form, and calculating with standard form — with the mistakes that cost marks.
12 min read Read notes -
Money, exchange rates, time and speed
Money and measures: currency conversion, the 24-hour clock and timetables, converting time units, speed distance time, average speed and travel graphs.
6 min read Read notes -
Percentage change, reverse percentage and compound interest
Percentages: percentage increase and decrease, multipliers, percentage change, profit and loss, reverse percentages, and simple vs compound interest with the growth and decay formula.
9 min read Read notes -
Ratio and proportion
Ratio and proportion: simplifying ratios, sharing in a given ratio, finding a total from one part, difference questions, scales and maps, and direct and inverse proportion.
8 min read Read notes -
Rounding, estimation and limits of accuracy
Rounding and estimation: decimal places, significant figures, the difference between them, estimating by rounding to 1 significant figure, and choosing the right accuracy for your answer.
6 min read Read notes -
Set notation and Venn diagrams
Sets: union, intersection, complement, subsets and the universal set, reading and shading Venn diagrams, n(A) notation, and solving Venn diagram problems with algebra.
8 min read Read notes - Extended
Surds
Surds: what a surd is, simplifying using square factors, adding and subtracting like surds, multiplying and expanding brackets, and rationalising the denominator with conjugates.
8 min read Read notes -
Types of number, primes, HCF and LCM
Types of number: primes, factors and multiples, square and cube numbers, rational and irrational numbers, prime factorisation, and finding HCF and LCM including word problems.
8 min read Read notes - Extended
Upper and lower bounds
Upper and lower bounds: finding bounds from rounded values, the half-a-unit rule, and calculating maximum and minimum values for sums, differences, products and quotients.
7 min read Read notes
Algebra and graphs
11 topics- Extended
Algebraic fractions
Algebraic fractions: simplifying by factorising, adding and subtracting with common denominators, multiplying and dividing, and solving equations containing algebraic fractions.
6 min read Read notes - Extended
Differentiation and gradients of curves
Differentiation: the power rule, differentiating constants, finding the gradient at a point, stationary points and turning points, and distinguishing maxima from minima.
7 min read Read notes - Extended
Direct and inverse proportion
Direct and inverse variation: writing proportion statements, finding the constant k, proportion to a square, cube or square root, and inverse proportion problems.
6 min read Read notes -
Expanding, factorising and algebraic manipulation
Algebraic manipulation: collecting like terms, expanding single, double and triple brackets, factorising with common factors and grouping, index laws in algebra, and simplifying expressions.
8 min read Read notes - Extended
Functions, composite and inverse
Functions: function notation, evaluating f(x), solving f(x) = k, composite functions fg(x), inverse functions, and finding domain and range.
7 min read Read notes -
Graphs of functions and graphical solutions
Graphs: tables of values, recognising linear, quadratic, cubic, reciprocal and exponential graphs, asymptotes, solving equations graphically, and finding the gradient of a curve with a tangent.
8 min read Read notes -
Linear equations and inequalities
Solving linear equations and inequalities: balancing, brackets and fractions, unknowns on both sides, flipping the inequality sign, number lines, integer solutions and shading regions on a graph.
8 min read Read notes - Extended
Quadratic equations and expressions
Quadratic equations: expanding and factorising, solving by factorisation, the quadratic formula, completing the square, turning points and word problems — with the mistakes that cost marks.
14 min read Read notes -
Rearranging formulae and changing the subject
Changing the subject of a formula: reversing operations in the right order, dealing with squares and square roots, brackets, fractions, and when the subject appears twice.
7 min read Read notes -
Sequences
Sequences: term-to-term rules, the nth term of linear and quadratic sequences, second differences, geometric and exponential sequences, and checking whether a number is in a sequence.
8 min read Read notes -
Simultaneous equations
Simultaneous equations: the elimination method, substitution, when to add or subtract, solving graphically, quadratic simultaneous equations, and forming equations from word problems.
8 min read Read notes
Coordinate geometry
1 topic-
Straight line graphs, gradient and midpoint
Coordinate geometry: gradient, midpoint and length of a line segment, y = mx + c, finding the equation of a line, parallel and perpendicular gradients, and perpendicular bisectors.
10 min read Read notes
Geometry
5 topics-
Angles, parallel lines and polygons
Angle rules: angles on a line and around a point, alternate, corresponding and co-interior angles, triangle and quadrilateral properties, and interior and exterior angles of polygons.
8 min read Read notes - Extended
Circle theorems
Circle theorems: angle in a semicircle, angle at the centre, angles in the same segment, cyclic quadrilaterals, tangent and radius, equal tangents and the alternate segment theorem — with the reasons that earn marks.
9 min read Read notes -
Congruence and similarity
Congruence and similarity: the difference between them, the congruence conditions SSS, SAS, ASA and RHS, proving triangles similar, corresponding sides, and finding missing lengths.
6 min read Read notes -
Constructions, scale drawings and loci
Constructions and loci: constructing triangles with compasses, perpendicular bisectors and angle bisectors, the four standard loci, and scale drawings.
6 min read Read notes -
Symmetry
Symmetry: lines of symmetry, order of rotational symmetry, symmetry of common quadrilaterals and regular polygons, and planes of symmetry in 3D solids.
6 min read Read notes
Mensuration
4 topics-
Area and perimeter
Area and perimeter: formulae for rectangles, triangles, parallelograms and trapezia, compound shapes, the perpendicular height rule, shaded regions and unit conversion.
6 min read Read notes -
Circles, arcs and sectors
Circles: circumference and area, arc length and sector area using the angle over 360, segment area, perimeter of a sector, and answers in terms of pi.
6 min read Read notes - Extended
Similar shapes: area and volume scale factors
Similar shapes: the length, area and volume scale factors k, k² and k³, working forwards and backwards with square and cube roots, and similar solids.
6 min read Read notes -
Surface area and volume
Surface area and volume: prisms, cylinders, cones, spheres and pyramids, composite solids, working backwards from a volume, unit conversion and the mistakes that cost marks.
8 min read Read notes
Trigonometry
6 topics- Extended
3D trigonometry
3D trigonometry: finding the diagonal of a cuboid, identifying the right triangle inside a solid, the angle between a line and a plane, and two-stage Pythagoras.
6 min read Read notes -
Bearings
Bearings: the three-figure rule, measuring clockwise from north, back bearings, using parallel north lines and co-interior angles, and combining bearings with trigonometry.
6 min read Read notes -
Pythagoras' theorem
Pythagoras' theorem: finding the hypotenuse and shorter sides, when to add and when to subtract, composite shapes, isosceles triangles and 3D applications.
6 min read Read notes -
Right-angled trigonometry (SOHCAHTOA)
SOHCAHTOA: labelling opposite, adjacent and hypotenuse, finding missing sides and angles, inverse trig functions, angles of elevation and depression, and exact trig values.
9 min read Read notes - Extended
Sine rule, cosine rule and area of a triangle
The sine rule, cosine rule and the ½ab sin C area formula: how to choose the right rule, finding sides and angles, the obtuse angle case, and the mistakes that cost marks.
9 min read Read notes - Extended
Trigonometric graphs and equations
Trig graphs: the shapes of sin, cos and tan, amplitude and period, using symmetry to find all solutions in a range, and the CAST/quadrant rule.
7 min read Read notes
Transformations and vectors
2 topics-
Transformations
Transformations: reflection, rotation, translation and enlargement, how to describe each one fully for full marks, negative and fractional scale factors, and finding the centre of rotation or enlargement.
8 min read Read notes -
Vectors
Vectors: column vectors, magnitude, adding and subtracting, scalar multiples, vector paths in geometry, ratios and midpoints, parallel vectors and proving points are collinear.
7 min read Read notes
Probability
2 topics-
Basic probability
Probability: the probability scale, complement events, adding vs multiplying, with and without replacement, 'at least one' questions, relative frequency and expected frequency.
7 min read Read notes -
Tree diagrams and combined events
Probability tree diagrams: drawing the branches, multiplying along and adding across, with and without replacement, conditional probability and 'at least one' questions.
7 min read Read notes
Statistics
5 topics-
Averages, range and data from tables
Mean, median, mode and range: finding averages from frequency tables, estimating the mean of grouped data with midpoints, the modal class, and working backwards from a given mean.
9 min read Read notes -
Bar charts, pie charts and pictograms
Statistical diagrams: drawing and reading bar charts, calculating pie chart angles, working backwards from a pie chart, pictogram keys, and criticising misleading graphs.
6 min read Read notes - Extended
Cumulative frequency and box plots
Cumulative frequency graphs and box-and-whisker plots: building the table, plotting at upper bounds, reading the median and quartiles, interquartile range, and comparing distributions.
8 min read Read notes - Extended
Histograms
Histograms with unequal class widths: frequency density, class width, why area represents frequency, drawing and reading histograms, and finding frequencies from a graph.
6 min read Read notes -
Scatter diagrams and correlation
Scatter diagrams: plotting points, types of correlation, drawing and using a line of best fit, interpolation and extrapolation, outliers, and why correlation is not causation.
6 min read Read notes
Exam skills
1 topic-
Exam technique, calculator use and past paper strategy
How to earn more marks in IGCSE Cambridge Maths: showing working for method marks, calculator settings, accuracy and units, reading command words, timing, and how to use past papers effectively.
8 min read Read notes
Notes get you started. A tutor gets you the grade.
These notes came out of one-to-one Maths lessons. If a topic still will not click, sit in one — the first lesson is free.
About these IGCSE Maths notes
Do these notes cover the whole Cambridge IGCSE Mathematics syllabus?
Yes — all 47 topics across the 10 units of the undefined syllabus are covered, grouped in the order the syllabus lists them. Nothing is held back behind a sign-up.
Who wrote them?
Tutopiya Maths tutors, from their own one-to-one teaching. The emphasis in each topic reflects where students actually struggle rather than how much space a textbook gives it.
How should I use them alongside past papers?
Read the topic first, then attempt questions on it, then come back to the mistakes section at the end of the topic. Most lost marks in a past paper map onto something already listed there.
Are these notes up to date with the 0580 specification?
They follow the current undefined specification. Exam boards do revise syllabuses, so confirm the objectives for your exam session against the official Cambridge specification.
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