IGCSE Mathematics Extended Study Guide & Review

Prepare for Cambridge IGCSE Mathematics 0580 Extended by connecting advanced techniques, solving multi-step problems under both calculator conditions and communicating methods clearly across all nine official groups.

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  1. 01Learn concepts
  2. 02Practise questions
  3. 03Review mistakes

How to study Cambridge IGCSE Mathematics 0580 Extended

Cambridge IGCSE Mathematics 0580 Extended rewards more than isolated formula recall: about half of each paper assesses analysis, interpretation and mathematical communication. Build advanced algebra through functions and differentiation alongside spatial reasoning in non-right-angled triangles and vector geometry. For every problem, translate the information into an equation, graph, diagram or statistical model before selecting a technique.

Rebuild an unfamiliar representation in Concept and retrieve formulas, conditions and notation in Mastery. Then apply them through the Extended Question Bank in both non-calculator and calculator modes. Compare the complete method with marking evidence, identify the first failure—representation, strategy, manipulation, calculator input, exact form, accuracy or communication—and repair that precise Topic before returning to mixed work.

Practise 0580 Extended by mathematical behaviour

Use different routines for advanced algebra, connected graphs and calculus, spatial reasoning, data models and the two official paper modes.

Control advanced algebra and functions

Algebraic fractions, quadratics, nonlinear simultaneous equations, inequalities, inverse and composite functions

Name restrictions and the target form before manipulating. Explain each transformation, verify solutions by substitution or graph and reject values outside the domain. When a solution fails, locate whether the error began in representation, legal manipulation, branch selection or interpretation rather than repeating the whole calculation.

Practise Functions

Connect graphs and differentiation

Function graphs, curve sketches, gradients, derivatives and stationary points

Move deliberately between equation, graph and derivative: predict key features, differentiate, solve the relevant condition and interpret the result on the original curve. Check coordinates and gradient signs against the sketch, and keep exact forms when requested instead of trusting an unsupported calculator display.

Practise Differentiation

Model trigonometry and vectors

Sine and cosine rules, ambiguous cases, 3D trigonometry, vector magnitude and vector proofs

Draw and label the geometry before choosing a relationship. Test whether the information creates an ambiguous case, preserve exact values until the end and justify collinearity or parallelism from vector relationships. Treat a correct number without the required geometrical or vector argument as incomplete communication.

Practise Vector Geometry

Build probability and statistics models

Conditional probability, combined events, cumulative frequency, grouped data and histograms

Construct the sample space, tree, table or frequency-density model before calculating. Distinguish conditional from independent events, label axes and intervals, retain grouped-data estimates as estimates and interpret conclusions in the stated context instead of reporting a detached value.

Practise Histograms

Transfer between Paper 2 and Paper 4

Exact arithmetic and surds without a calculator; scientific-calculator modelling, accuracy and interpretation

Rehearse shared Extended ideas under both tool conditions. On Paper 2 simplify and estimate without calculator dependence; on Paper 4 model before keying, check mode and magnitude, retain unrounded values and apply the stated accuracy only at the end. Keep necessary working visible on both papers.

Build an Extended test

Where to start

Begin from one known Extended Topic, or diagnose whether repeated losses come from strategy selection, algebra, representation, calculator control, accuracy or communication.

Choose your starting point

  1. I know the weak Topic

    Open its exact Extended route, identify the failed objective and practise the representation or decision that caused the first invalid step.

    Browse 72 Extended Topics

  2. I do not know what is weak

    Use a mixed Extended diagnostic to separate missing technique from strategy, representation, manipulation, calculator, accuracy and communication errors.

    Start an Extended diagnostic

Choose your Cambridge 0580 Extended starting point

  1. Explain and represent

    State what is known, choose an equation, graph, diagram or distribution, and justify why the selected Extended method fits the conditions.

    Review a Concept

  2. Retrieve and connect

    Reproduce the required formula, condition and notation without prompts, then connect it to a second representation or related Topic.

    Check Mastery

  3. Apply, classify and repair

    Solve a fresh Extended problem, compare every necessary stage with marking evidence and repair the first invalid decision before retrying.

    Practise Extended questions

Explore the 2028–2030 Cambridge 0580 Extended syllabus

The Extended route maps 72 Topics across nine official groups, from Number to Statistics. Use the map to locate one precise weakness, open its resource destination and connect it to related representations; Cambridge does not prescribe the published Topic order as a teaching sequence.

9
syllabus groups
72
mapped topics

1. Number

Open unit questions
E1.1 Types of numberE1.2 SetsE1.3 Powers and rootsE1.4 Fractions, decimals and percentagesE1.5 OrderingE1.6 The four operationsE1.7 Indices IE1.8 Standard formE1.9 EstimationE1.10 Limits of accuracyE1.11 Ratio and proportionE1.12 RatesE1.13 PercentagesE1.14 Using a calculatorE1.15 TimeE1.16 MoneyE1.17 Exponential growth and decayE1.18 Surds

2. Algebra and graphs

Open unit questions
E2.1 Introduction to algebraE2.2 Algebraic manipulationE2.3 Algebraic fractionsE2.4 Indices IIE2.5 EquationsE2.6 InequalitiesE2.7 SequencesE2.8 ProportionE2.9 Graphs in practical situationsE2.10 Graphs of functionsE2.11 Sketching curvesE2.12 DifferentiationE2.13 Functions

3. Coordinate geometry

Open unit questions
E3.1 CoordinatesE3.2 Drawing linear graphsE3.3 Gradient of linear graphsE3.4 Length and midpointE3.5 Equations of linear graphsE3.6 Parallel linesE3.7 Perpendicular lines

4. Geometry

Open unit questions
E4.1 Geometrical termsE4.2 Geometrical constructionsE4.3 Scale drawingsE4.4 SimilarityE4.5 SymmetryE4.6 AnglesE4.7 Circle theorems IE4.8 Circle theorems II

5. Mensuration

Open unit questions
E5.1 Units of measureE5.2 Area and perimeterE5.3 Circles, arcs and sectorsE5.4 Surface area and volumeE5.5 Compound shapes and parts of shapes

6. Trigonometry

Open unit questions
E6.1 Pythagoras’ theoremE6.2 Right-angled trianglesE6.3 Exact trigonometric valuesE6.4 Trigonometric functionsE6.5 Non-right-angled trianglesE6.6 Pythagoras’ theorem and trigonometry

7. Transformations and vectors

Open unit questions
E7.1 TransformationsE7.2 Vectors in two dimensionsE7.3 Magnitude of a vectorE7.4 Vector geometry

8. Probability

Open unit questions
E8.1 Introduction to probabilityE8.2 Relative and expected frequenciesE8.3 Probability of combined eventsE8.4 Conditional probability

9. Statistics

Open unit questions
E9.1 Classifying statistical dataE9.2 Interpreting statistical dataE9.3 Averages and measures of spreadE9.4 Statistical charts and diagramsE9.5 Scatter diagramsE9.6 Cumulative frequency diagramsE9.7 Histograms

Cambridge IGCSE Mathematics 0580 Extended assessment

Extended candidates take two compulsory two-hour papers worth 100 marks and 50% each, with grades A*–E available. Around 50–60% of each paper assesses AO2 analysis, interpretation and communication, so revision must include strategy selection and connected multi-step work.

Paper / componentQuestionsTime% of grade
Paper 2: Non-calculator (Extended)Structured and unstructured compulsory questions; all Extended content except E1.14100 marksCalculators are prohibited. Candidates answer on the question paper, receive the formula list on page 2 and must show all necessary working; the list does not include every required formula.How to prepare: Develop exact arithmetic, surd manipulation, algebraic transformations, estimation, construction and diagram fluency without calculator dependence. Preserve exact values where required, communicate recoverable intermediate steps and check results through substitution, inverse operations, graph features or magnitude rather than relying on an unsupported answer.2 hours50%
Paper 4: Calculator (Extended)Structured and unstructured compulsory questions across all Extended content100 marksA permitted scientific calculator is required; algebraic and graphical calculators are not allowed. Candidates also receive the formula list, show necessary working and follow the stated accuracy conventions.How to prepare: Translate the context into mathematics before entering values, estimate the likely result and check calculator mode, sign and scale. Retain unrounded values for later parts; unless instructed otherwise, give non-exact numerical answers to three significant figures and angles in degrees to one decimal place, then interpret the result.2 hours50%

SourceCambridge International Education · Cambridge IGCSE Mathematics 0580 syllabus for exams in 2028, 2029 and 20300580 Extended · 2028–2030 · Version 1 · September 2025

Cambridge IGCSE Mathematics 0580 Extended questions

Extended candidates take Paper 2, a two-hour non-calculator paper, and Paper 4, a two-hour calculator paper. Each is worth 100 marks and 50% of the qualification. Both contain compulsory structured and unstructured questions; Paper 2 excludes only E1.14 Using a calculator, while Paper 4 may assess all Extended content.

No. A formula list is supplied on page 2 of both papers, but Cambridge states that it does not contain every formula candidates may need. Learn which relationships are supplied and retrieve the remaining required formulas, conditions and notation. The list also does not choose the method, justify a step or replace necessary working.

Extended uses Papers 2 and 4, offers grades A*–E and maps 72 E-coded Topics, compared with Core Papers 1 and 3 and grades C–G. It adds areas such as functions, differentiation, non-right-angled trigonometry, vector geometry, conditional probability and histograms, with about 50–60% AO2 demand on each paper.

Older questions can support individual Topic practice, but Cambridge warns that past papers may not reflect the current syllabus. For exact timed mocks, prioritise papers matching the current Paper 2 non-calculator and Paper 4 calculator format, then check every older question against the 2028–2030 Extended syllabus before treating it as relevant.

Practise the shared Extended content under both tool conditions instead of assigning whole Topics permanently to one paper. Build exact manipulation, estimation and construction without a calculator for Paper 2; for Paper 4, add scientific-calculator setup, mode and magnitude checks, retained precision and contextual interpretation. Keep complete, readable working in both modes.