GCSE Mathematics Topics
77 topics with worked solutions.
3D Pythagoras and Trigonometry
Applying Pythagoras and trigonometry inside 3D shapes.
Algebraic Fractions
Simplifying, adding, subtracting, multiplying and dividing algebraic fractions.
Algebraic Proof
Constructing algebraic proofs about odd, even and consecutive numbers and divisibility.
Angle Basics
Angles on a line, around a point and vertically opposite angles.
Angles in Parallel Lines
Alternate, corresponding and co-interior angles with reasons.
Angles in Polygons
Interior and exterior angles of regular and irregular polygons.
Averages and Range
Mean, median, mode and range from lists and frequency tables.
Basic Probability
Probability scales, calculating simple probabilities and mutually exclusive events.
Bearings
Measuring, drawing and calculating three-figure bearings, often with trigonometry.
Best Buys and Value for Money
Comparing prices and pack sizes to decide the best value option.
BIDMAS and Order of Operations
Applying the correct order of operations in calculations.
Box Plots
Drawing and comparing box plots using medians and quartiles.
Charts and Graphs
Bar charts, pictograms, line graphs and frequency polygons.
Circle Theorems
Applying circle theorems with correct reasons to find missing angles.
Circles, Sectors and Arcs
Circumference and area of circles, arc lengths and sector areas.
Completing the Square
Writing quadratics in completed square form and using it to solve equations and find turning points.
Composite and Inverse Functions
Forming composite functions and finding inverse functions.
Compound Interest
Calculating compound interest and repeated percentage change using multipliers.
Compound Measures
Working with rates such as speed, density and pressure and converting units.
Conditional Probability
Calculating probabilities that depend on previous outcomes.
Congruent Triangles
Conditions for congruence and simple congruence proofs.
Constructions and Loci
Ruler and compass constructions and locus problems.
Cubic and Reciprocal Graphs
Recognising and sketching cubic, reciprocal and exponential graphs.
Cumulative Frequency
Cumulative frequency curves, medians and interquartile range.
Decimals
Calculating with decimals and converting between decimals and fractions, including recurring decimals.
Density, Mass and Volume
Using the density formula and solving problems involving mass and volume.
Direct and Inverse Proportion
Setting up and using equations for quantities in direct or inverse proportion.
Error Intervals
Finding upper and lower bounds of rounded values and writing error intervals with inequalities.
Exact Trigonometric Values
Knowing and using the exact trig values for 0, 30, 45, 60 and 90 degrees.
Exchange Rates
Converting between currencies and solving money conversion problems.
Expanding Brackets
Expanding single, double and triple brackets.
Factorising
Factorising into single brackets, quadratics and the difference of two squares.
Factors, Multiples and Primes
Prime factorisation, factors and multiples, and using product of primes.
Fractions
Simplifying, ordering and calculating with fractions, including mixed numbers.
Fractions, Decimals and Percentages
Converting fluently between fractions, decimals and percentages and comparing proportions.
Functions
Function notation, evaluating functions and solving equations involving functions.
Growth and Decay
Exponential growth and decay problems including depreciation.
HCF and LCM
Finding highest common factors and lowest common multiples, including from prime factorisations.
Histograms
Frequency density, drawing histograms and interpreting unequal class widths.
Indices
Laws of indices including negative and fractional powers.
Inequalities
Solving linear and quadratic inequalities and representing solutions on number lines and graphs.
Iteration
Using iterative formulae to find approximate solutions to equations.
Linear Equations
Solving linear equations including unknowns on both sides and equations with brackets or fractions.
Negative Numbers
Adding, subtracting, multiplying and dividing with negative numbers.
Nth Term
Finding the nth term of linear and quadratic sequences.
Percentage Change
Calculating percentage increase, decrease, profit and loss.
Percentages
Finding percentages of amounts, percentage increase and decrease, and reverse percentages.
Perimeter and Area
Perimeter and area of rectangles, triangles, parallelograms, trapezia and compound shapes.
Pie Charts
Drawing and interpreting pie charts using angles and proportions.
Place Value
Understanding the value of digits in integers and decimals, and ordering numbers by size.
Plans and Elevations
Drawing and interpreting plans and elevations of 3D solids.
Probability Trees
Tree diagrams for independent and dependent events, with and without replacement.
Proportion
Solving problems involving proportional relationships, recipes and scaling.
Pythagoras' Theorem
Finding missing sides in right-angled triangles, including in 2D and 3D problems.
Quadratic Equations
Solving quadratics by factorising and interpreting the solutions.
Quadratic Graphs
Plotting and interpreting quadratic graphs, roots, intercepts and turning points.
Ratio
Simplifying ratios, sharing amounts in a given ratio and solving ratio problems.
Rearranging Formulae
Changing the subject of a formula, including where the subject appears twice.
Relative Frequency
Estimating probability from experiments and expected outcomes.
Rounding and Estimation
Rounding to decimal places and significant figures, and using estimates to check calculations.
Sampling
Random and stratified sampling and identifying bias.
Scatter Graphs
Correlation, lines of best fit and using them to estimate values.
Sequences
Recognising arithmetic, geometric, quadratic and Fibonacci-type sequences.
Similar Shapes
Length, area and volume scale factors for similar shapes.
Simplifying Expressions
Collecting like terms and simplifying algebraic expressions.
Simultaneous Equations
Solving linear and linear-quadratic simultaneous equations by elimination and substitution.
Sine and Cosine Rules
Using the sine rule, cosine rule and the area formula for non-right-angled triangles.
Speed, Distance and Time
Calculating speed, distance and time, including average speed and unit conversion.
Standard Form
Writing very large and very small numbers in standard form and calculating with them.
Straight Line Graphs
Gradient, intercepts and the equation y = mx + c, including parallel and perpendicular lines.
Surds
Simplifying surds, expanding brackets with surds and rationalising denominators.
The Quadratic Formula
Using the quadratic formula to solve equations that do not factorise.
Transformations
Reflections, rotations, translations and enlargements, including negative scale factors.
Trigonometry (SOHCAHTOA)
Using sine, cosine and tangent to find missing sides and angles in right-angled triangles.
Vectors
Vector notation, vector arithmetic and geometric vector proofs.
Venn Diagrams
Drawing Venn diagrams and using set notation to find probabilities.
Volume and Surface Area
Volume and surface area of prisms, cylinders, cones, pyramids and spheres.