Algebra – GCSE AQA Higher Maths revision notes

What this topic covers

Algebra has 4 subtopics:

• Notation, vocabulary and manipulation
• Graphs
• Solving equations and inequalities
• Sequences

Algebra lessons by subtopic

39 lessons, grouped by subtopic.

Notation, vocabulary and manipulation

• Substituting Values into Formulae and Expressions: Learn to substitute numerical values into algebraic formulae and expressions, including scientific formulae. Practice with various examples from mathematics and science contexts.
• Understanding Algebraic Language: Learn the key vocabulary of algebra including expressions, equations, formulae, inequalities, terms and factors. Understand what each term means and how to identify them in mathematical contexts.
• Expanding and Factorising Expressions: Master expanding products of binomials, factorising quadratic expressions including difference of two squares, and working with indices.
• Collecting Like Terms and Basic Algebraic Operations: Learn to collect like terms, multiply single terms over brackets, and take out common factors from algebraic expressions.
• Advanced Algebraic Manipulation: Develop skills in factorising quadratic expressions with coefficients and simplifying complex algebraic expressions through manipulation techniques.
• Substituting into and Interpreting Formulae: Substitute values into formulae correctly, interpret algebraic formulae in context, and avoid common notation errors.
• Changing the Subject of a Formula: Rearrange formulae to change the subject by applying inverse operations in the correct order; Identify and apply the correct sequence of steps when the subject appears once in a formula; Rearrange formulae where the subject appears more than once, requiring factorisation
• Equations vs Identities and Algebraic Arguments: Learn to distinguish between equations and identities, and develop skills in using algebra to construct mathematical arguments and show that expressions are equivalent.
• Algebraic Proofs: Explore how to construct formal algebraic proofs, building on your understanding of equivalent expressions and mathematical arguments to prove algebraic relationships.
• Interpreting Expressions as Functions: Learn to recognise when algebraic expressions can be interpreted as functions, identifying inputs and outputs in simple mathematical relationships.

Graphs

• Working with Coordinates in All Four Quadrants: Learn to identify, plot, and work with coordinates in all four quadrants of the coordinate plane, including positive and negative x and y values.
• Plotting Straight-Line Graphs: Learn how to plot graphs of equations that correspond to straight-line graphs in the coordinate plane, including identifying coordinates and drawing accurate linear graphs.
• Gradients and Intercepts of Linear Functions: Learn to identify and interpret gradients and intercepts of linear functions both graphically (from graphs) and algebraically (from equations). Understand what gradient and y-intercept represent in real-world contexts.
• Turning Points and Symmetry of Quadratics: Master finding turning points by completing the square and explore the symmetrical properties of quadratic functions, including more advanced techniques for analysing quadratic graphs.
• Quadratic Graphs - Roots and Intercepts: Learn to identify and interpret roots, intercepts and turning points of quadratic functions from their graphs, and discover how to find roots algebraically using factorisation and the quadratic formula.
• Recognising and Sketching Linear and Quadratic Graphs: Learn to recognise the key features of linear and quadratic functions, sketch their graphs accurately, and interpret the information they display including gradients, intercepts, and turning points.
• Straight-Line Graph Skills and Gradients: Build core graph skills by plotting coordinates, sketching straight-line graphs, finding gradients, and comparing lines using y = mx + c.
• Transformations of Linear and Quadratic Graphs: Interpret and sketch translations, reflections, and stretches of linear and quadratic graphs and compare how their shapes change.
• Interpreting Real-World Graphs: Learn to plot and interpret graphs of non-standard functions in real contexts, focusing on kinematic problems involving distance, speed and acceleration to find approximate solutions
• Interpreting Gradients and Areas Under Graphs: Learn to calculate and estimate gradients of linear and non-linear graphs, find areas under curves, and interpret these results in real-world contexts including distance-time graphs, velocity-time graphs, and financial situations.
• Circle Equations and Properties: Learn to recognise and work with the equation of a circle with centre at the origin, including understanding the relationship between the equation x² + y² = r² and the circle's radius.
• Tangents to Circles: Discover how to find the equation of a tangent to a circle at a given point, using coordinate geometry and the properties of perpendicular lines.

Solving equations and inequalities

• Solving Linear Equations Algebraically: Learn how to solve linear equations in one unknown using algebraic methods, including collecting like terms, using inverse operations, and checking solutions.
• Finding Approximate Solutions Using Graphs: Discover how to use graphs to find approximate solutions to equations by identifying intersection points and reading values from coordinate axes.
• Completing the Square and the Quadratic Formula: Solve quadratic equations by completing the square and using the quadratic formula, and choose the most efficient method.
• Solving Quadratic Equations by Factorising: Learn to solve quadratic equations by factorising, including equations that need rearranging into standard form first.
• Solving simultaneous equations algebraically: Learn to solve two simultaneous linear equations in two variables using algebraic methods including elimination and substitution
• Graphical solutions to simultaneous equations: Find approximate solutions to simultaneous equations using graphs, including linear/linear and linear/quadratic systems
• Solving Quadratics, Simultaneous Equations and Rearranging Formulae: Solve quadratic equations by factorising, completing the square and the quadratic formula, use the discriminant, solve linear–quadratic simultaneous equations and rearrange complex formulae.
• Numerical Solutions Using Iteration: Learn to find approximate solutions to equations using iterative methods, including recursive formulae with suffix notation.
• Solving Linear Equations with Brackets and Unknowns on Both Sides: Solve linear equations using the balance method, including equations with brackets and equations where the unknown appears on both sides.
• Creating Algebraic Expressions from Situations: Learn to translate real-world problems and situations into algebraic expressions and formulae. Practice identifying variables and writing mathematical representations.
• Solving Equations from Real Problems: Derive an equation or pair of simultaneous equations from a real-world or geometrical problem; Solve the derived equation(s) to find unknown values; Interpret the solution in context, including rejecting values that are not valid in the situation
• Representing Solution Sets: Master how to represent inequality solutions on number lines and graphs, including proper notation and conventions for boundaries.
• Advanced Inequalities: Solve quadratic inequalities and work with higher-level inequality problems, applying advanced techniques and representations.
• Solving Linear Inequalities: Learn to solve linear inequalities in one and two variables, understanding the basic principles and methods for finding solution sets.

Sequences

• Generating Sequence Terms: Learn how to generate terms of sequences using both term-to-term rules (where each term depends on the previous term) and position-to-term rules (where each term can be found directly from its position number). Practice finding specific terms and continuing sequences using both types of rules.
• Triangular, Square and Cube Numbers and Arithmetic Progressions: Learn to recognise and use sequences of triangular, square and cube numbers, and understand simple arithmetic progressions. Practice identifying patterns and finding terms in these important number sequences.
• Finding the nth Term of Linear Sequences: Learn to identify patterns in linear sequences and derive algebraic expressions for the nth term using the common difference and first term.

Revise Right Mocks '26

AQA GCSE Maths mock draft, 26–31 Oct. Draft coverage: Maths Paper 1 (Higher tier). Reviewed for an original-practice sitting; check student availability before registering. Marking and results will follow later, after separate approval. Eligible new accounts receive a five-day Pro trial when signup email verification completes. Previous trials keep their original expiry. After expiry, new papers need Pro unless you have an explicitly confirmed free sitting offer for this event. That offer adds no Pro time or AI learning. Saved attempts and released results remain readable. No card or automatic charge.

More in this course

Go back to the full GCSE AQA Higher Maths topic list, or practise this topic with past-paper questions.