Further Mathematics

Converting Fractions to Decimals

Converting Decimals to Fractions

Simplifying Fractions

Adding and Subtracting Fractions

Multiplying Fractions

Dividing Fractions

Understanding Decimals

Rounding Decimals

Adding and Subtracting Decimals

Multiplying Decimals

Dividing Decimals

Understanding Percentages

Converting Percentages to Fractions

Converting Percentages to Decimals

Percentage Increase and Decrease

Solving Percentage Problems

Understanding Ratios

Simplifying Ratios

Using Ratios in Problems

Understanding Proportion

Direct Proportion Problems

Inverse Proportion Problems

Order of Operations (BIDMAS)

Applying BIDMAS in Calculations

Common Errors in BIDMAS

The Product Rule for Counting

Worked Examples of Product Rule

Simplifying Surds

Adding and Subtracting Surds

Multiplying Surds

Dividing Surds

Rationalising the Denominator

Exact Calculations Using Surds

Common Errors in Surds Manipulation

Mixed Problems on Surds

Understanding Function Notation

Domain of a Function

Range of a Function

Composite Functions

Inverse Functions

Expanding Single Brackets

Expanding Double Brackets

Expanding Triple Brackets

Using Pascal’s Triangle in Expansions

Factorising Simple Expressions

Factorising Quadratic Expressions

Factorising Higher Degree Polynomials

Manipulating Algebraic Fractions

Adding and Subtracting Rational Expressions

Multiplying and Dividing Rational Expressions

Simplifying Complex Rational Expressions

Rearranging Formulae to Change the Subject

Using the Factor Theorem

Factorising Using the Factor Theorem

Completing the Square for Quadratics

Finding Turning Points via Completing the Square

Sketching Quadratic Graphs

Sketching Exponential Graphs

Interpreting Graphs of Functions

Solving Linear Equations

Solving Quadratic Equations by Factorisation

Solving Quadratic Equations Using the Formula

Solving Quadratic Equations by Completing the Square

Graphical Solutions to Quadratic Equations

Solving Simultaneous Linear Equations

Solving Simultaneous Equations with One Quadratic

Solving Simultaneous Equations Graphically

Solving Linear Equations in Three Variables

Solving Linear Inequalities

Solving Quadratic Inequalities

Simplifying Expressions with Index Laws

Solving Equations with Fractional Indices

Solving Equations with Negative Indices

Proving Divisibility Using Algebraic Proof

Proving Identities Using Algebraic Proof

Finding the nth Term of Linear Sequences

Worked Examples: nth Term of Linear Sequences

Identifying Patterns in Linear Sequences

Using the nth Term to Find Specific Terms

Solving Problems with Linear Sequences

Understanding Quadratic Sequences

Finding the nth Term of Quadratic Sequences

Worked Examples: nth Term of Quadratic Sequences

Using the nth Term for Quadratic Sequences

Identifying Patterns in Quadratic Sequences

Solving Problems with Quadratic Sequences

Comparing Linear and Quadratic Sequences

Introduction to Limiting Values

Finding Limiting Values of Sequences

Worked Examples: Limiting Values of Sequences

Understanding Converging Sequences

Understanding Diverging Sequences

Using Limiting Values in Problem Solving

Arithmetic Sequences: Basics and Formulas

Geometric Sequences: Basics and Formulas

Solving Problems with Arithmetic Sequences

Solving Problems with Geometric Sequences

Special Sequences: Fibonacci and Others

Finding Differences Between Sequence Terms

Finding Sums of Sequence Terms

Worked Examples: Summing Sequence Terms

Applications of Sequences in Real-Life Problems

Common Exam Traps in Sequence Questions

Practice Problems: Linear Sequences

Practice Problems: Quadratic Sequences

Practice Problems: Limiting Values

Practice Problems: Arithmetic Sequences

Practice Problems: Geometric Sequences

Exam Techniques for Sequence Questions

Calculating Gradients of Lines

Gradients of Parallel Lines

Gradients of Perpendicular Lines

Using Pythagoras to Find Distance

Distance Between Two Points Formula

Definition of Midpoint

Finding Midpoint Between Two Points

Using Ratios to Divide a Line Segment

Equation of a Straight Line: y = mx + c

Equation of a Straight Line: Point-Slope Form

Interpreting Gradient and Y-Intercept

Drawing Straight Lines from Equations

Equation of a Circle with Centre (0, 0)

Writing Circle Equation from Radius

Circle Geometry: Angle in a Semi-Circle

Circle Geometry: Perpendicular Bisector of a Chord

Circle Geometry: Tangent-Radius Angle

Circle Geometry: Equal Length Tangents

Equation of a Circle with Centre (a, b)

Writing Circle Equation from Centre and Radius

Equation of Tangent to a Circle

Finding Gradient of Tangent at a Point

Finding Equation of Tangent to Circle

Worked Examples: Gradient and Distance

Worked Examples: Midpoint and Ratios

Worked Examples: Straight Line Equations

Worked Examples: Circle Equations

Exam Trap: Misinterpreting Gradient Signs

Exam Trap: Forgetting Units in Distance

Exam Trap: Incorrect Circle Equation Form

Rate of Change of y with Respect to x

Gradient of a Function at a Point

Differentiation of kx^n

Simplifying Expressions Before Differentiation

Finding dy/dx for Polynomial Functions

Equation of a Tangent to a Curve

Equation of a Normal to a Curve

Identifying Increasing Functions

Identifying Decreasing Functions

Second Derivative Notation (d²y/dx²)

Rate of Change of Gradient Function

Using Differentiation to Find Maxima

Using Differentiation to Find Minima

Determining Nature of Stationary Points

Using d²y/dx² to Classify Points

Maxima and Minima in Simple Problems

Sketching Curves with Maxima and Minima

Interpreting Curves with Stationary Points

Worked Example: Finding dy/dx Step-by-Step

Worked Example: Tangent Equation

Worked Example: Normal Equation

Worked Example: Increasing and Decreasing Functions

Worked Example: Maxima and Minima Using Differentiation

Exam Trap: Forgetting to Simplify Before Differentiation

Exam Trap: Misinterpreting d²y/dx² Results

Exam Trap: Incorrect Gradient Calculation for Tangents

Exam Trap: Confusing Tangents with Normals

Exam Trap: Missing Stationary Points on Curve Sketches

Application: Optimisation Problems Using Differentiation

Application: Real-Life Rate of Change Problems

Application: Curve Sketching in Context

Multiplying Two 2x2 Matrices

Multiplying a 2x2 Matrix by a 2x1 Matrix

Scalar Multiplication of Matrices

Understanding the Identity Matrix

Properties of the Identity Matrix

Matrix Representation of Transformations

Transformations of the Unit Square

Rotation by 90 Degrees About the Origin

Rotation by 180 Degrees About the Origin

Rotation by 270 Degrees About the Origin

Reflection in the Line x = 0

Reflection in the Line y = 0

Reflection in the Line y = x

Reflection in the Line y = -x

Enlargements Centered at the Origin

Understanding Combination of Transformations

Matrix Multiplication for Combined Transformations

Worked Examples: Combining Rotations

Worked Examples: Combining Reflections

Worked Examples: Combining Enlargements

Exam Trap: Order of Matrix Multiplications

Inverse Transformations Using Matrices

Checking Matrix Multiplication Results

Geometric Interpretation of Matrix Transformations

Identifying Transformation Types from Matrices

Zero Matrix and Its Role in Transformations

Diagonal Matrices in Transformations

Symmetry in Matrix Transformations

Using Determinants to Analyze Transformations

Effect of Determinant on Area Scaling

Worked Example: Area Scaling via Determinants

Transformations and Coordinate Systems

Exam Trap: Misinterpreting Transformation Directions

Matrix Transformations in Problem Contexts

Exam Strategy for Matrix Transformations

Calculating Area of Rectangles

Area of Triangles and Parallelograms

Area of Trapezia Step-by-Step

Surface Area of Prisms

Volume of Prisms Explained

Surface Area of Cylinders

Volume of Cylinders Formula

Surface Area of Spheres

Volume of Spheres Formula

Surface Area of Cones

Volume of Cones Calculation

Volume of Pyramids Formula

Circle Theorems: Angle at the Centre

Circle Theorems: Angles in Same Segment

Circle Theorems: Cyclic Quadrilateral

Circle Theorems: Alternate Segment

Circle Theorems: Tangent-Radius Angle

Geometric Proofs with Circle Theorems

Using Pythagoras' Theorem in 2D

Recognising Pythagorean Triples

Using Pythagoras' Theorem in 3D

Understanding Trigonometric Ratios

Solving Right-Angled Triangles

Sine Rule in Scalene Triangles

Cosine Rule in Scalene Triangles

Area of a Triangle Using Sine

Applying Trigonometry in 2D Problems

Applying Trigonometry in 3D Problems

Angle Between Line and Plane

Angle Between Two Planes

Graphs of y = sin x

Graphs of y = cos x

Graphs of y = tan x

Using Trigonometric Definitions

Special Triangles: 30-60-90

Special Triangles: 45-45-90

Using tan θ = sin θ / cos θ

Using sin²θ + cos²θ = 1

Solving Simple Trigonometric Equations

Common Errors in Trigonometric Problems

The Cosine Function: Definition and Basics

The Tangent Function: Definition and Basics

Sketching the Sine Graph

Sketching the Cosine Graph

Sketching the Tangent Graph

Key Features of Sine Graphs

Key Features of Cosine Graphs

Key Features of Tangent Graphs

Periodicity of Sine, Cosine, and Tangent

Amplitude and Frequency of Sine and Cosine

Phase Shift in Trigonometric Graphs

Transformations of Sine Graphs