Cambridge IGCSE Mathematics (0580) Extended Syllabus Notes (2025)

Number

• Types of Numbers:

  • Natural Numbers: Positive counting whole numbers starting from $1$ (or $0$ depending on convention).
  • Integers: Whole positive numbers, negative numbers, and zero.
  • Prime Numbers: Numbers greater than $1$ that have exactly two distinct positive factors: $1$ and themselves.
  • Squares and Cubes: Numbers resulting from multiplying an integer by itself twice (square) or three times (cube).
  • Reciprocals: The multiplicative inverse of a given number $x$, expressed as $\frac{1}{x}$.
  • Rational Numbers: Any number that can be expressed as a quotient or fraction $\frac{a}{b}$ of two integers, where $a$ is the numerator and $b$ is a non-zero denominator ($b \neq 0$).
  • Irrational Numbers: Real numbers that cannot be written as a simple fraction $\frac{a}{b}$; their decimal representations are non-terminating and non-repeating.

• Indices and Surds:

  • Indices: Rules governing powers, exponents, fractional indices, and negative indices.
  • Surds: Operations involving irrational roots, specifically simplifying surds and rationalising the denominator (Extended syllabus requirement).
  • Recall: Memorization and direct recall of specific square numbers, cube numbers, and their corresponding roots.

• Applied Number Concepts:

  • Standard Form: Representing numbers in scientific notation $a \times 10^n$, where $1 \le a < 10$ and $n$ is an integer.
  • Fractions, Decimals, and Percentages: Converting between, ordering, and calculating operations across all three forms.
  • Money: Financial mathematics involving transactions, interest rates, profit, loss, and currency exchange.
  • Rates: Quantifying change over standard intervals, such as speed or unit costs.
  • Growth and Decay: Exponential increases or decreases over designated time intervals.
  • Estimation: Rounding figures to specified decimal places or significant figures to approximate mathematical values.
  • Time: Calculations involving timestamps, intervals, time zones, 12-hour/24-hour clocks, and timetables.

Algebra and Graphs

• Algebraic Manipulation:

  • Expanding: Multiplying out single, double, or complex brackets.
  • Factorising: Reversing expansion to express algebraic expressions as products of their factors.
  • Simplifying: Combining like terms and reducing algebraic fractions to their simplest terms.
  • Solving Equations: Computing exact solutions for linear equations as well as simultaneous equations.

• Functions (Extended):

  • Notation: Utilizing standard functional notation such as $f(x)$, $g(x)$, or $f: x \mapsto ...$
  • Domain and Range: Defining the domain (set of all valid input values for $x$) and range (set of all possible output values produced by the function).
  • Composite Functions: Combining two or more functions to form a single function, written as $f(g(x))$
  • Inverse Functions: Determining the inverse function $f^{-1}(x)$ to reverse the operation of $f(x)$.

• Graphs and Curve Sketching:

  • Linear Graphs: Plotting and analyzing straight-line graphs.
  • Quadratic Functions: Plotting and analyzing parabolic curves.
  • Reciprocal Functions: Plotting hyperbolic curves of reciprocal forms.
  • Curve Sketching: Identifying key features such as roots, intercepts, and turning points to draw qualitative sketches.
  • Fractional-Power Functions (Extended): New syllabus additions involving functions with fractional powers, such as $x^{(\frac{1}{2})}$ and $x^{(-\frac{1}{2})}$.

Coordinate Geometry

• Cartesian Coordinates and Lines:

  • Cartesian Coordinates: Locating points in two-dimensional space defined by $(x, y)$ coordinates.
  • Straight-Line Graphs: Constructing linear equations and graphing them on coordinate axes.
  • Gradients: Determining the steepness of a line using the formula $m = \frac{y_2 - y_1}{x_2 - x_1}$.
  • Intercepts: Finding where a graph crosses the $x$-axis ($x$-intercept) and $y$-axis ($y$-intercept) using standard equation forms such as $y = mx + c$.

• Perpendicular Lines (Extended):

  • Perpendicular Gradient Relationship: Applying the rule that two perpendicular lines with gradients $m_1$ and $m_2$ satisfy $m_1 \times m_2 = -1$
  • Equations of Perpendicular Lines: Finding the exact linear equation of a line that runs perpendicular to a given line through a specific point.

Geometry

• Geometric Shapes and Vocabulary:

  • Triangles: Identifying properties of equilateral, isosceles, scalene, and right-angled triangles.
  • Quadrilaterals: Identifying properties of squares, rectangles, parallelograms, rhombuses, trapeziums, and kites.
  • Polygons: Calculating interior and exterior angles of regular and irregular polygons.
  • Circles: Understanding terminology such as radius, diameter, circumference, tangent, chord, secant, arc, sector, and segment.

• Circle Theorems and Constructions (Extended):

  • Circle Theorems: Proving and calculating angles using circle rules:
  • The angle subtended at the centre is twice the angle subtended at the circumference.
  • Angles in the same segment are equal.
  • The angle in a semicircle is a right angle ($90^\circ$).
  • Opposite angles in a cyclic quadrilateral sum to $180^\circ$.
  • The angle between a tangent and a chord equals the angle subtended in the alternate segment.
  • A tangent is perpendicular to the radius at the point of contact ($90^\circ$).
  • Constructions: Constructing net layouts for 3D solids and precise geometric triangle constructions using geometric tools.

Mensuration

• Perimeters and Areas:

  • 2D Shapes: Formulas and techniques for calculating perimeters and areas of plane figures.
  • Circle Arcs and Sectors: Calculating arc length and sector area based on the central angle fraction of a full circle.

• Surface Areas and Volumes:

  • Standard Solids: Determining surface area and volume for basic 3D solids.
  • Specific 3D Geometry: Applying exact volume and surface area formulas for spheres, cones, and pyramids.

• Compound Shapes and Solids:

  • Dividing multi-part two-dimensional shapes into simpler standard shapes to solve for total area or perimeter.
  • Combining or subtracting volume and surface area expressions for multi-part three-dimensional solids.

Trigonometry

• Right-Angled Triangle Trigonometry:

  • Pythagoras' Theorem: Solving side lengths using $a^2 + b^2 = c^2$
  • Trigonometric Ratios: Using basic ratios defined for right-angled triangles:
  • Sine: $\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}$
  • Cosine: $\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}$
  • Tangent: $\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}$

• Non-Right Triangles and 3D Trigonometry (Extended):

  • Sine Rule: Applying $\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}$ to find unknown sides or angles.
  • Cosine Rule: Applying $a^2 = b^2 + c^2 - 2bc\cos(A)$ to find unknown sides or angles.
  • Area of Non-Right Triangles: Computing area via $\text{Area} = \frac{1}{2}ab\sin(C)$
  • 3D Trigonometry: Applying right-angled and non-right-angled trigonometric methods to solve problems in three-dimensional space.

• Exact Trigonometric Values (Extended):

  • Recalling exact trigonometric ratios for key non-calculator angles without a calculator:
  • $0^\circ$
  • $30^\circ$
  • $45^\circ$
  • $60^\circ$
  • $90^\circ$

Transformations and Vectors

• Transformations:

  • Reflection: Flipping a shape across a specified line of reflection.
  • Rotation: Rotating a shape around a fixed centre point, specifically including $90^\circ$ rotations.
  • Enlargement: Scaling a shape relative to a centre of enlargement using positive and fractional scale factors.
  • Translation: Moving a shape positionally across a plane using translation vectors.

• Vector Syllabus Requirement:

  • Vector operations and vector arithmetic are restricted strictly to the Core curriculum.
  • The Extended syllabus no longer tests vector arithmetic.

Probability

• Probability Scale and Events:

  • Scale ranging from $0$ (impossible) to $1$ (certain).
  • Computing probabilities for simple events and combined events.

• Probability Representations:

  • Outcome Tables: Representing combined sample spaces in tabular format.
  • Venn Diagrams: Organizing set-based outcomes and probabilities.
  • Tree Diagrams: Tracking sequential independent or dependent events.

• Frequency Measures:

  • Expected Frequency: Calculating the expected occurrence count of an event over multiple trials.
  • Relative Frequency: Determining experimental probability estimates based on empirical outcome data.

Statistics

• Data Classification and Tabulation:

  • Classification of data (discrete vs continuous, qualitative vs quantitative).
  • Tally tables and frequency distribution tables.

• Graphical Data Representation:

  • Histograms: Constructing and reading histograms with unequal class widths using frequency density.
  • Frequency Density Formula: Calculated as $\text{Frequency Density} = \frac{\text{Frequency}}{\text{Class Width}}$
  • Scatter Diagrams: Plotting bivariate data to analyze correlation and draw lines of best fit.

Syllabus Changes: 2024 to 2025

• Calculator Policies:

  • Papers 1 and 2: Transitioned to strict non-calculator examinations.
  • Papers 3 and 4: Calculator permitted examinations.

• New Topics Introduced to Extended Syllabus:

  • Surds (simplifying and rationalising denominators).
  • Fractional-power graphs (such as $x^{(\frac{1}{2})}$ and $x^{(-\frac{1}{2})}$).
  • Domain and range of functions.
  • Exact trigonometric values ($0^\circ$, $30^\circ$, $45^\circ$, $60^\circ$, $90^\circ$).

• Topics Removed from Extended Syllabus:

  • Vector arithmetic (vectors are retained in Core only).
  • Box plots.
  • Linear programming.
  • Data collection procedures.