GCSE Further Maths Guide

Curriculum Guide

The further maths GCSE is an exciting opportunity to study maths at a deeper level. It is not for the faint hearted – most schools reserve this course for only the very best gifted mathematicians. To complete two GCSE qualifications (the regular GCSE Maths and the Further Maths GCSE) within years 10 and 11 is intense. Work hard and revise well, this curriculum guide is designed to support your hard work. Thanks to Corbett Maths, Transom, Dr Frost and 1st Class Maths for supplying the resources linked below.

Before you begin, you should check that you have understood everything from Year 9, Year 10 and Year 11.

You may also like the Casio ClassWizz guide, it’s super helpful to know how to use your calculator quickly and efficiently in exams.

AQA Further Maths Formula Sheet is available here.

GCSE Further Maths Practice Questions: This sheet has hundreds of practice questions for GCSE Further Maths.

Topic 1: Number

Key Concept	Revision Video / Resource	Practice	Extension
Understand the Product Rule for Counting	Product Rule for Counting – Corbett Maths	Transum – Product Rule for Counting (use the tabs at the top to gradually increase difficulty) Corbett Maths Textbook: Product Rule for Counting	Exam Style Questions Product Rule for Counting – 1st Class Maths
Manipulate Surds:	Surds Revision Video	Transum – Surds Corbett Maths -Surds	Dr Frost Full Coverage – Surds
Add, Subtract, Multiply and Divide Surds.	See above	Surds Practice Worksheet Transum Surds Practice	See above
Expand Brackets with Surds.	See above	See above	See above
Be able to Rationalise a Denominator.	See above	See above	_{See above}

Topic 2: Algebra

Key Concept	Revision Video / Resource	Practice	Extension
Algebra Prerequisite Knowledge	Algebra Basics Revision Playlist
Functions:
1. Understand and use Composite Functions	Composite Functions Video	Transum – Functions Corbett Maths – Functions
2. Be able to identify Domain and Range of a Function.	Domain and Range Video Desmos Activity – Domain and Range		1st Class Maths: Domain and Range
3. Be able to find the inverse of a function.	Inverse Functions Video Desmos Activity – Inverse Functions	Corbett Maths Functions Practice Questions	Dr Frost Maths – Full Coverage of Functions
Expanding Brackets:	Expanding Brackets Video	Expanding Brackets Practice
1. Be able to expand triple brackets.	Expanding Triple Brackets Video	Transum – Triple Brackets	Expanding Triple Brackets Practice
2. Be able to expand (a+b)ⁿ for any positive integer n.	Binomial Expansion Video	Transum – Pascals Triangle Transum – Binomial Expansion	Corbett Maths – Binomial Expansion with Pascals Triangle
Algebraic Fractions: Be able to simplify algebraic fractions and solve algebraic fraction equations.	Simplifying Algebraic Fractions	Transum – Algebraic Fractions Use the tabs at the top of the page to increase difficultly.	Dr Frost Full Coverage – Algebraic Fractions
Understand the Factor Theorem for Polynomials	The Factor Theorem – Corbett Maths Video	Corbett Maths – The Factor Theorem Exam Style Questions	1st Class Maths: The Factor Theorem
Be able to complete the square for any quadratic.	Completing the Square Video	Transum – Completing the Square	Dr Frost Full Coverage – Completing the Square
Be able to Draw and Sketch Graphs	Drawing and Sketching Functions – Corbett Maths	Drawing and Sketching Functions Practice Questions Markscheme for above
Be able to solve Quadratic Equations.	Solving Quadratics Video Solving Quadratic Problems	Solving Quadratic Problems Practice Questions	Dr Frost Solving Quadratics Full Coverage
Be able to solve Quadratic Simultaneous Equations	Solving Quadratic Simultaneous Equations	Transum: Quadratic Simultaneous Equations	MathsGenie: Quadratic Simultaneous Equations Markscheme for above
Be able to solve linear equations with three unknown variables.	Solving Simultaneous Equations with 3 Variables.	Simultaneous Equations with Three Unknowns – Practice Questions, Corbett Maths	1st Class Maths: Simultaneous Equations with 3 Variables
Be able to solve linear inequalities.	Linear Inequalities Video	Linear Inequalities Practice – Transum	Inequalities Exam Style Questions
Be able to solve quadratic inequalities.	Quadratic Inequalities Video	Quadratic Inequalities – Transum	Quadratic Inequalities Exam Style Questions – Corbett Maths
Be able to simplify expressions with indices.	Laws of Indices Video	Transum – Laws of Indices Corbett Maths – Laws of Indices	Dr Frost Full Coverage: Laws of Indices
Understand Algebraic Proof	Algebraic Proof Video	Algebraic Proof – Corbett Maths Practice Questions
Linear n^th term	n^th term video	Transum – nth Term Practice
Quadratic n^th term	Quadratic n^th term – MathsGenie	Quadratic nth Term Practice Questions

Topic 3: Coordinate Geometry – Line Graphs

Key Concept	Revision Video / Resource	Practice	Extension
Be able to calculate the gradient between any two coordinates.	Gradient between 2 points. Desmos Activity – Graph Between Two Points	Gradient Practice Questions – Corbett Maths
Understand Parallel and Perpendicular Lines	Straight Line Graphs Revision Playlist Guess the Straight Line Activity – Desmos		Desmos: Interactive Graph of Parallel and Perpendicular Lines
Be able to find the expression for a perpendicular line through a specific coordinate.	Straight Line Graphs Exam Style Questions Video	Graph Match Activity – Transum Equation of a Line – Corbett Maths	Parallel and Perpendicular Lines MathsGenie Markscheme for Above
Use Pythagoras Theorem to calculate the distance between two coordinates.	Pythagoras Theorem Video Distance between two points – Corbett Maths Distance Between Two Points – Desmos	Transum: Distance between two points	Corbett Maths: Distance Between Two Points

Topic 3: Coordinate Geometry – Circle Graphs

Key Concept	Revision Video / Resource	Practice	Extension
Know the equations for a circle graph, and how the parameters link to the centre and radius.	Equation of a Circle – Addvance Maths Equation of a Circle On Desmos	Equation of a Circle Worksheet
Be able to find the equation of a tangent to a circle at a particular point.	See above		Circle Graphs Practice Questions 1st Class Maths – Equation of a Circle Questions

Topic 4: Calculus

Key Concept	Revision Video / Resource	Practice	Extension
Know that the derivative of a function gives gradient of a curve or the rate of change of a function.
Know that the gradient of a curve is equal to the gradient of a tangent at that point.
Be able to differentiate polynomials.	Differentiation Video	Differentiation Practice – Transum	Corbett Maths – Differentiation Practice Questions
Be able to find the equation of a tangent and normal to a curve.	Tangents Video Normals Video coming soon.		Corbettt Maths – Equation of a Tangent 1st Class Maths – Differentiation & Gradient
Use the derivative to find the regions where a function is increasing or decreasing.	Increasing and Decreasing Functions – Corbett Maths		Corbett Maths – Increasing/Decreasing Functions
Understand the second derivative, and that this measures the rate of change of the gradient function.	Second Order Derivatives – Corbett Maths Concavity – Desmos Activity		Corbett Maths – Increasing/Decreasing Functions
Use differentiation to find the maximum and minimum of functions.	Differentiation: Optimisation Video		Corbett Maths – Applications of the Derivative
Sketch a curve based on the maxima and minima of a function.	Stationary Points – Corbett Maths

Topic 5: Matrices & Matrix Transformations

Key Concept	Revision Video / Resource	Practice	Extension
Understand the concept of a matrix.
Be able to multiply a matrix by a scalar.	Multiplying by a scalar – Corbett Maths	See below
Be able to multiply a matrix by another matrix. (2 x 1 or 2 x 2 matrices)	Multiplying Matrices Video	Corbett Math Matrix Multiplication	1st Class Maths – Matrix Multiplication
Know the identity matrix and its properties.	The Identity Matrix – Corbett Maths		1st Class Maths – Identity Matrix
Transformations of a unit square in the xy plane. Including: Rotations and reflections.	Matrix Transformations – Corbett Maths	Corbett Maths – Metric Transformations Practice Questions	Dr Frost Full Coverage: Further Maths Matrix Algebra
Combine these transformations.	Matrix Transformations Interactive Graph	See above	1st Class Maths – Combing Transformations

Topic 6: Trigonometry

Key Concept	Revision Video / Resource	Practice	Extension
Prior Knowledge: See GCSE Guide	Notes: Area, Perimeter of rectangles and triangles. Circle Theorems Parallel lines rules Angles in Polygons Surface Area and Volume of 3D Shapes
Understand Geometric Proof: Be able to prove geometric properties.	Geometric Proof – Corbett Maths video	Corbett Maths – Geometric Proof Questions	MathsGenie: Proof of Circle Theorems Markscheme for above
Know and use the sine rule, cosine rule and area of a triangle using sine formula.	Sine Rule Video Cosine Rule Video Area of a Triangle using SineVideo	Advanced Trigonometry Questions: Corbett Maths
Be able to use Pythagoras Theorem in 3D.	3D Trigonometry Video	Corbett Maths: Further Maths 3D Trigonometry Questions.	Dr Frost Full Coverage: 3D Trig
Know the graphs of sin(x), cos(x) and tan(x).	Trig Graphs Video		Dr Frost Full Coverage: Trig Graphs and Equations
Know the tan identity and the pythagorean trig identity. Be able to use these to solve problems.	Trig Identities – Corbett Maths	See below
Be able to solve trigonometric equations for any value of theta/x.	Solving Trig Equations – Corbett Maths	Corbett Maths – Solving Trigonometric Equations	Dr Frost Full Coverage: Trig Equations 1st Class Maths – Trig Identities
Know the exact values of sine, cosine and tan for special angles (0, 30, 60, 90).	Special Trig Angles Video	Exact Trig Values Practice Exact Trig values answers