IGCSE Mathematics Syllabus: Structure, Topics and Exam Format

The IGCSE Mathematics syllabus is organized into five content areas: Number, Algebra and Graphs, Coordinate Geometry, Geometry, and Statistics and Probability. Cambridge Assessment International Education offers two tiers—Core (grades C to G) and Extended (grades A* to E)—with Extended covering all Core content plus additional topics at greater depth. The assessment consists of two written papers, with specific calculator and formula sheet rules depending on which examination board syllabus you follow.

What Is in the IGCSE Mathematics Syllabus?

The Cambridge IGCSE Mathematics syllabus (0580/0980) divides mathematical content into five major strands. Each strand contains multiple subtopics that build progressively in difficulty.

Number includes arithmetic operations, fractions, decimals, percentages, ratio and proportion, indices and standard form, and basic number theory concepts like factors and multiples. Students work with directed numbers (positive and negative), order of operations, and numerical estimation.

Algebra and Graphs covers algebraic notation, manipulation of expressions, solving equations and inequalities (linear and quadratic), sequences, functions, and graphical representation. Extended tier students additionally study quadratic functions in depth, exponential growth and decay, and more complex simultaneous equations.

Coordinate Geometry focuses on the Cartesian plane, straight-line graphs including gradient and intercept, distance between points, midpoint of a line segment, and the relationship between parallel and perpendicular lines. Extended students work with more sophisticated coordinate problems involving circles and transformations.

Geometry encompasses properties of angles, triangles, quadrilaterals, polygons, circles, and three-dimensional shapes. Students learn geometric reasoning, similarity and congruence, Pythagoras' theorem, trigonometry in right-angled triangles, and mensuration (area, volume, surface area). Extended tier extends trigonometry to non-right triangles using the sine and cosine rules.

Statistics and Probability includes data collection and representation (tables, charts, histograms, cumulative frequency), measures of central tendency and spread, and probability calculations including combined events. Extended students study more complex statistical diagrams and probability tree diagrams.

IGCSE Mathematics Topics by Section

Number

The number strand begins with natural numbers, integers, rational and irrational numbers, establishing the number system students will use throughout. Key topics include:

• Four operations with whole numbers, decimals, and fractions
• Converting between fractions, decimals, and percentages
• Percentage calculations including percentage change, reverse percentages, and compound interest
• Ratio and proportion, including direct and inverse proportion
• Indices (powers) and roots, index laws, and standard form (scientific notation)
• Upper and lower bounds in calculations
• Approximation and estimation to check reasonableness of answers

Extended tier adds set notation and Venn diagrams for three sets, more complex proportion problems, and surds (expressions involving roots like \(\sqrt{2}\)) including rationalization of denominators.

Algebra and Graphs

Algebraic skills develop from basic simplification to solving complex equations. Core topics include:

• Algebraic notation and vocabulary (terms, expressions, equations, formulae)
• Simplifying expressions by collecting like terms
• Expanding brackets, including products like \((x + 3)(x - 2)\)
• Factorization of expressions, including common factors and simple quadratics
• Solving linear equations in one unknown
• Solving simultaneous linear equations in two unknowns
• Solving quadratic equations by factorization
• Linear inequalities and representing solutions on a number line
• Sequences including finding the \(n\)th term of arithmetic sequences
• Drawing and interpreting graphs of linear and simple quadratic functions

Extended tier requires solving quadratic equations using the formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) and by completing the square, solving simultaneous equations where one is linear and one quadratic, manipulating algebraic fractions, solving harder inequalities, and working with functions including composite and inverse functions.

Coordinate Geometry

Students learn to work systematically with points, lines, and shapes in the coordinate plane:

• Plotting and reading coordinates in all four quadrants
• Finding the gradient of a line from two points using \(m = \frac{y_2 - y_1}{x_2 - x_1}\)
• Finding the equation of a straight line in forms \(y = mx + c\) and \(ax + by = c\)
• Parallel lines (equal gradients) and perpendicular lines (product of gradients equals \(-1\))
• Finding the length of a line segment and the midpoint of a line segment

Extended students apply coordinate geometry to prove geometric properties, work with the equation of a circle \(x^2 + y^2 = r^2\), and solve more complex problems involving transformations in the coordinate plane.

Geometry

Geometric content spans plane geometry and three-dimensional shapes:

• Angle properties: angles on a straight line, angles at a point, vertically opposite angles
• Parallel lines and transversals: corresponding, alternate, and co-interior angles
• Angle properties of triangles, quadrilaterals, and polygons
• Properties of special quadrilaterals (parallelogram, rhombus, trapezium, kite)
• Circle theorems (Extended tier has more theorems than Core)
• Symmetry: line symmetry and rotational symmetry
• Similarity and congruence of triangles
• Pythagoras' theorem in two and three dimensions
• Trigonometric ratios: sine, cosine, and tangent in right-angled triangles
• Mensuration: perimeter, area, volume, and surface area of standard shapes
• Geometric constructions using ruler and compass
• Transformations: reflection, rotation, translation, enlargement
• Vectors in two dimensions (notation, addition, scalar multiplication)

Extended tier includes the sine rule \(\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\), the cosine rule \(a^2 = b^2 + c^2 - 2bc \cos A\), the area formula \(\frac{1}{2}ab\sin C\) for any triangle, and additional circle theorems with formal proofs.

Statistics and Probability

Data handling and probability develop from basic representation to statistical inference:

• Reading and interpreting tables, bar charts, pie charts, pictograms, and line graphs
• Calculating mean, median, mode, and range from lists and frequency tables
• Drawing and interpreting stem-and-leaf diagrams, scatter diagrams, and histograms
• Cumulative frequency diagrams and using them to find median and quartiles
• Basic probability including mutually exclusive events and combined events
• Probability from two-way tables and Venn diagrams

Extended tier adds box-and-whisker plots, histograms with unequal class intervals (where frequency density is required), more sophisticated use of cumulative frequency for interquartile range, and probability tree diagrams for dependent and independent events.

Core vs Extended: What Changes?

The Core curriculum targets grades C to G and covers fundamental mathematical concepts that every student should master. The Extended curriculum targets grades A* to E, includes all Core content, and adds approximately 30% more material at higher conceptual difficulty.

Students taking Core papers work with simpler numbers, more structured questions, and less algebraic manipulation. For example, Core students solve quadratic equations only by factorization, while Extended students must also use the quadratic formula and completing the square.

In geometry, Core students learn basic circle properties (such as the angle in a semicircle is a right angle), while Extended students prove and apply the full set of circle theorems including alternate segment theorem and theorems involving tangents and chords.

Extended tier introduces topics absent from Core entirely: set notation with Venn diagrams for three sets, surds, function notation including composite and inverse functions, solving simultaneous equations where one is quadratic, sine and cosine rules for any triangle, vectors in geometric proofs, and more sophisticated statistical techniques.

The difficulty gap between tiers is substantial. A student comfortable with Extended content at grade E level demonstrates mathematical maturity beyond a Core grade C student, even though both receive a C on their certificate. Schools typically decide which tier suits a student based on performance in earlier years and projected grades.

Exam Papers and Calculator Rules

Cambridge IGCSE Mathematics assessment consists of two written papers taken in the same examination session. The structure differs between Core and Extended:

Core tier: Paper 1 (1 hour, short-answer questions, calculator not allowed) and Paper 3 (2 hours, short-answer and structured questions, calculator allowed).

Extended tier: Paper 2 (1 hour 30 minutes, short-answer questions, calculator not allowed) and Paper 4 (2 hours 30 minutes, short-answer and structured questions, calculator allowed).

The non-calculator papers test numerical fluency, algebraic manipulation, and geometric reasoning without computational aids. Students must perform arithmetic with fractions and decimals by hand, simplify surds, and solve equations without technological support. These papers typically contain shorter questions worth 1-4 marks each.

Calculator papers include longer structured questions worth 5-8 marks that may involve multiple steps, real-world contexts, and more complex calculations. The calculator enables students to focus on problem-solving strategy rather than arithmetic, but questions are designed so that excessive calculator dependence leads to errors—understanding remains essential.

Calculators must be silent, non-programmable, and without symbolic algebra capability. Scientific calculators are standard; graphic display calculators are permitted if they meet the non-programmable criterion. Calculators with QWERTY keyboards or text-retrieval functions are prohibited.

Formula Sheet and Calculator Use

Cambridge provides a formula sheet with each examination paper listing key formulae. Students should familiarize themselves with what appears on this sheet during revision—not to memorize it, but to understand when each formula applies.

The formula sheet typically includes:

• Area formulae for circle, triangle, parallelogram, and trapezium
• Circumference of a circle
• Volume and surface area formulae for cuboid, prism, cylinder, pyramid, cone, and sphere
• Pythagoras' theorem
• Trigonometric ratios (though students should know \(\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}\) etc.)
• Sine rule, cosine rule, and area of triangle (Extended only)
• Quadratic formula (Extended only)

Formulae not provided include: percentage calculations, compound interest, gradient formula, midpoint formula, equation of a straight line, distance between two points, mean and range, and probability addition and multiplication rules. Students must recall these independently.

Effective calculator use means knowing when not to use it. In the calculator paper, simple calculations like \(3 \times 4\) or \(\frac{1}{2} + \frac{1}{3}\) should be done mentally or on paper—calculator errors and miskeying cause unnecessary mark loss. Use calculators for: evaluating trigonometric functions, computing with awkward decimals, finding powers and roots beyond squares and cubes, and checking arithmetic in multi-step problems.

How to Revise Each Topic

Effective revision requires active problem-solving, not passive reading. For each syllabus topic, follow this approach:

Identify the skill. Write down exactly what you should be able to do. For example, "solve quadratic equations by factorization" or "find the equation of a line parallel to a given line through a given point." If you cannot state the skill precisely, you do not yet understand the topic boundary.

Attempt past paper questions. Work through questions from previous examination papers on that specific topic. Time yourself under exam conditions for full questions. Mark your work honestly, identifying not just wrong answers but unclear reasoning or inefficient methods.

Analyze errors. When you make a mistake, classify it. Did you misread the question? Forget a formula? Make an arithmetic error? Misunderstand the concept? Each error type requires a different fix. Conceptual errors need re-learning; careless errors need better checking habits.

Build connections. Mathematics is a web of linked ideas. When revising coordinate geometry, notice how gradient connects to parallel lines, perpendicular lines, and rates of change. When studying probability, see how it links to fractions, percentages, and algebraic manipulation. Strong students revise by strengthening connections, not memorizing isolated procedures.

Practice without notes. Eventually, close your textbook and formula sheet (except the official exam one) and solve problems as you will in the exam. If you cannot proceed without looking up a method, you have found a gap. Return to worked examples, understand the reasoning, then try again from memory.

For topics you find difficult, work more examples at slightly easier levels before attempting exam-standard questions. If Extended tier quadratic equations frustrate you, first ensure you can factorize confidently, then expand brackets accurately, then solve by factorization before approaching the quadratic formula. Mathematical skill builds hierarchically—gaps in foundations cause struggles at higher levels.

Common Questions Students Ask

How much of the syllabus appears in each exam? Both papers cover the full syllabus breadth. You cannot predict which topics will dominate a particular paper. Examination papers balance content areas, but the specific questions vary. Revision must cover everything.

Can I use a formula even if I don't understand it? Yes, in the exam—examiners award marks for correct application. But understanding why a formula works makes it easier to remember, apply correctly, and adapt when questions present unfamiliar contexts. Rote memorization is fragile; understanding is robust.

Should I memorize the formula sheet? No. The formula sheet is provided to reduce memory burden. However, you must know which formula to apply in which situation, and you must memorize formulae not on the sheet. During revision, practice using the official formula sheet so you know what to look for during the exam.

How much working should I show? Show every step that is not trivial. Examiners award method marks even when the final answer is wrong. If you write only the answer and it is incorrect, you score zero. If you show clear reasoning with a minor arithmetic error, you may lose only one mark. Write enough that someone else could follow your logic.

What if I finish early? Check your work systematically. Recalculate numerical answers by a different method if possible. Verify that you answered the question asked—students often solve for \(x\) when the question asked for \(2x + 1\). Read each question again to confirm you completed all parts. Resist the temptation to leave early; checking catches errors.

Is Extended tier much harder than Core? Extended tier contains more content and more sophisticated reasoning, but it is designed for students who have engaged with that material throughout their course. A well-prepared Extended student finds their papers appropriately challenging, not impossibly hard. The tier decision should reflect your mathematical confidence and performance over time, not a last-minute choice.

The IGCSE Mathematics syllabus provides a structured pathway through essential mathematical concepts. Success comes from understanding the content organization, practicing systematically across all topics, and developing fluency in both calculator and non-calculator techniques. Whether taking Core or Extended tier, students who engage actively with problem-solving—rather than passively reading notes—build the skills and confidence the examination demands.