IGCSE Mathematics: Study Guide & Free Lessons
Short lessons, worked examples, and practice that connects.
Exact number and place value
An exact fraction preserves a value without rounding. Factors and multiples describe different relationships between whole numbers.
Ratios and percentage changes
A ratio compares parts, whereas a percentage compares a quantity with a stated base. Reverse percentages recover the original base.
Equations and algebraic structure
An equation states equality; an inequality describes a set of permitted values. Equivalent operations preserve the relevant relation.
Coordinates, gradients and lines
Gradient compares vertical change with horizontal change. A midpoint averages coordinates, while a line equation links all its points.
Quadratics and number patterns
Quadratic expressions can be written in expanded, factored or completed-square form. A sequence formula must match its index convention.
Angles and similar shapes
Angles in a straight line total 180 degrees. Similar shapes have equal corresponding angles and a constant length scale factor.
Area, volume and circular measures
Area counts square units and volume counts cubic units. A formula becomes meaningful only when the correct dimensions and units are inserted.
Right triangles and trigonometric ratios
Pythagoras relates the side lengths of a right triangle. Trigonometric ratios connect an acute angle with ratios of those sides.
Vectors and transformations
A translation adds a fixed displacement to every point. Reflections and rotations preserve distance but change position or orientation.
Probability with and without replacement
A probability compares favourable outcomes with the relevant possible outcomes. Conditional information can change the denominator.
Summaries from frequency data
A mean uses every value, a median locates the ordered centre and a range compares the extremes. A frequency tells how often a value occurs.
Sets and function notation
A union includes either set, while an intersection includes their shared members. A function assigns exactly one output to each allowed input.
Carry measurement bounds through a calculation
Find the tight interval consistent with rounded measurements and use it in a derived quantity.
Preserve exact roots and use fractional indices
Simplify a surd, rationalise a denominator and interpret a fractional power without rounding.
Connect standard form, recurring decimals and compound factors
Carry out operations in different number representations while preserving their values.
Combine and simplify algebraic fractions on their domains
Use common denominators and factor cancellation without losing excluded inputs.
Use curves, domains and intersections to solve problems
Interpret sign intervals, choose an inverse branch and identify intersections without confusing outputs with inputs.
Use derivatives for gradients and stationary points
Differentiate simple polynomials, build a tangent and distinguish a minimum from a maximum.
Chain circle theorems and state the reasons
Recognize which circle theorem applies and combine it with triangle or right-angle facts.
Choose sine, cosine or a spatial right triangle
Handle a non-right triangle or a three-dimensional diagonal using the correct geometric relationship.
Combine solid volumes and distinguish scale powers
Use cone, sphere and composite-solid formulas while matching length, area and volume scale factors.
Connect direction, scale and vector geometry
Use a consistent reference direction, convert a map scale and interpret displacement magnitudes and ratios.
Read histogram areas and estimate grouped summaries
Use frequency density when class widths differ and distinguish exact counts from grouped estimates.
Read percentiles and distinguish association from prediction
Use cumulative totals for intervals and percentiles, and read a fitted line without claiming causation.
0580 Extended foundations for2025–2027; Core is not a separately implemented route.
A clear route through the material
Follow the topics in order or use the prerequisites to choose a starting point. Read the worked example before focused practice, then use review cards to revisit the idea.
Continue your preparation
Scope: Extended foundations · exam years 2025–2027