Mathematics

Assumptions, Logical Steps and Conclusions

Proof by Deduction: Structuring a Clear Argument

Proving Statements by Completing the Square

Proof by Exhaustion: Checking a Finite Set of Cases

Choosing Cases Efficiently in Exhaustion Proofs

Disproof by Counterexample

Writing a Counterexample Clearly

Proof by Contradiction: The Core Idea

Proving √2 Is Irrational

Proving There Are Infinitely Many Primes

Applying Contradiction to Unfamiliar Statements

Proofs Involving Series Results

Proof Presentation: Symbols, Connectives and Clarity

Converting Between Fractional and Surd Index Form

Simplifying Surds

Rationalising Denominators

Quadratic Functions: Key Features and Sketching

Discriminant and the Number of Real Roots

Completing the Square for Quadratics

Solving Quadratics by Factorising

Solving Quadratics Using the Formula

Solving Quadratic Equations in a Function of the Unknown

Simultaneous Equations: One Linear and One Quadratic

Linear Inequalities in One Variable

Quadratic Inequalities in One Variable

Inequalities with Brackets and Fractions

Writing Inequality Solutions Using Set Notation

Shading Linear and Quadratic Regions

Polynomial Expansion and Factorising (Including Cubics)

Algebraic Division by a Linear Factor

The Factor Theorem in Practice

Simplifying Rational Expressions by Factorising

Sketching Polynomial Graphs (Including Cubics and Quartics)

Modulus Graphs of Linear Functions

Reciprocal Graphs and Asymptotes

Solving Equations by Graph Intersection

Proportion and Modelling with “∝”

Composite Functions

Inverse Functions and Reflection in y = x

Domain and Range for Composite and Inverse Functions

Graph Transformations: y = af(x) and y = f(x) + a

Graph Transformations: y = f(x + a) and y = f(ax)

Combining Multiple Transformations

Transformations Involving Modulus

Partial Fractions (Up to 3 Terms, Linear/Repeated Linear)

Using Partial Fractions in Integration and Differentiation

Using Partial Fractions in Series Expansion

Functions in Modelling: Assumptions, Limits and Refinements

Using y − y₁ = m(x − x₁)

Using ax + by + c = 0

Equation of a Line Through Two Points

Perpendicular Line Conditions

Building and Using Straight-Line Models in Context

Circle Equation in (x − a)² + (y − b)² = r²

Finding Centre and Radius from a Circle Equation

Expanding and Rearranging Circle Equations

Completing the Square for Circle Form

Angle in a Semicircle: Using the Theorem

Perpendicular from Centre to a Chord

Radius–Tangent Perpendicular Property

Finding the Equation of a Tangent to a Circle

Finding the Circumcircle of a Triangle

Parametric Curves: Understanding x(t), y(t)

Converting Between Parametric and Cartesian Forms

Recognising Standard Curves from Parametrics

Restricting Parameter Domains to Sections of Curves

Parametric Modelling of Shapes

Parametric Modelling of Motion

Using Factorials and Binomial Coefficients

Pascal’s Triangle and Binomial Coefficients

Linking Binomial Expansion to Binomial Probabilities

Binomial Expansion for Rational Powers

Validity Condition for Rational Binomial Expansion

Using Binomial Expansion for Approximation

Modelling Error and Range of Validity in Approximations

Defining Sequences by an nth Term Formula

Defining Sequences by a Recurrence Relation

Identifying Increasing, Decreasing and Periodic Sequences

Sigma Notation Basics

Standard Sigma Results (Including ∑n)

Arithmetic Sequences: nth Term

Arithmetic Series: Sum to n Terms

Proving the Arithmetic Series Sum Formula

Geometric Sequences: nth Term

Finite Geometric Series Sums

Infinite Geometric Series and |r| < 1

Proving the Geometric Series Sum Formula

Using Logs to Find n in Geometric Contexts

Modelling with Arithmetic Sequences and Series

Modelling with Geometric Sequences and Series

Sine Rule and When to Use It

Cosine Rule and When to Use It

Area of a Triangle Using ½ab sin C

The Ambiguous Case in the Sine Rule

Radians: Converting Between Degrees and Radians

Arc Length s = rθ

Sector Area A = ½r²θ

Small Angle Approximations (in Radians)

Graphs of sin x, cos x and tan x

Periodicity and Symmetry of Trig Graphs

Exact Values for Standard Angles

Sec, cosec, cot: Definitions and Links to sin/cos/tan

Inverse Trig Functions: Meaning, Range and Domain

Core Identities: sin²θ + cos²θ = 1

Core Identities: sec²θ and cosec²θ Forms

Double Angle Formulae

Compound Angle Formulae (A ± B)

Geometric Proofs of Compound/Double Angle Results

Writing a cosθ + b sinθ as r cos(θ ± α) or r sin(θ ± α)

Solving Trig Equations in an Interval

Quadratic Trig Equations (in sin, cos or tan)

Trig Equations with Multiple Angles

Proving Trig Identities

Trig Modelling in Context (Waves, Circular Motion, etc.)

Shape Changes for 0 < a < 1 vs a > 1

The Natural Exponential y = e^x and Its Graph

Graphs of y = e^t

Differentiating e^(kx) and Why It Models Proportional Change

When to Choose an Exponential Model

Logs as Inverses of Exponentials

The Natural Log ln x and Its Graph

Solving Equations like e^(ax+b) = p

Solving Equations like ln(ax+b) = q

Log Laws: Product, Quotient and Power

Using log_a a = 1 and Related Facts

Change of Base and Calculator Log Use

Solving a^x = b Using Logs

Log-Linearising Power Models y = ax^n

Log-Linearising Exponential Models y = kb^x

Estimating Parameters from Log Graphs

Exponential Growth Models in Context

Exponential Decay Models in Context

Continuous Compound Interest with e

Refining Models and Discussing Limitations

Derivative as a Limit (First Principles Idea)

Derivatives as Rates of Change in Context

Sketching a Gradient Function from a Given Curve

Differentiation from First Principles for x^n (Small Integers)

Differentiation from First Principles for sin x

Differentiation from First Principles for cos x

Second Derivative as Rate of Change of Gradient

Concavity and Convexity from fʺ(x)

Points of Inflection and Sign Change of fʺ(x)

Differentiating x^n for Rational n

Differentiating e^(kx) and a^(kx)

Differentiating sin(kx), cos(kx), tan(kx)

Differentiating ln x

Tangents and Normals to Curves

Stationary Points: Finding and Classifying

Increasing/Decreasing Intervals Using f′(x)

Product Rule

Quotient Rule

Chain Rule

Differentiating sec x, cosec x and cot x

Connected Rates of Change Problems

Differentiating Inverse Functions Using dy/dx = 1/(dx/dy)

Implicit Differentiation (First Derivative Only)

Parametric Differentiation (First Derivative Only)

Tangents and Normals for Implicit/Parametric Curves

Setting Up Simple Differential Equations from Context

The Constant of Integration and Families of Curves

Fundamental Theorem of Calculus (What It Connects)

Finding a Curve Given f′(x) and a Point

Integrating e^(kx)

Integrating 1/x to ln|x|

Integrating sin(kx) and cos(kx)

Using Trig Identities to Integrate (sin², cos², tan²)

Definite Integrals: Evaluating Exactly

Area Under a Curve Using a Definite Integral

Area Between Two Curves

Areas with Parametric Curves

Integration as the Limit of a Sum

Substitution: Choosing a Suitable Substitution

Substitution as Reverse Chain Rule

Recognising ∫ f′(x)/f(x) dx = ln|f(x)| + c

The Integral of ln x

Integration by Parts (One Application)

Integration by Parts (Multiple Applications)

Integration by Parts as Reverse Product Rule

Integrating from Partial Fractions (Linear Denominators)

Separable Differential Equations: Separation Step

Solving Separable Differential Equations Analytically

Finding Particular Solutions from Initial Conditions

Interpreting Differential Equation Solutions in Context

Validity and Limitations of DE Models for Large Values

When the Change of Sign Method Fails

Fixed-Point Iteration: Rearranging to x = f(x)

Performing Iterations Accurately

Divergence in Iteration

Cobweb Diagrams for Fixed-Point Iteration

Staircase Diagrams for Fixed-Point Iteration

Newton–Raphson Method: Formula and Setup

Newton–Raphson as a Tangent Process (Geometric Meaning)

Failure of Newton–Raphson Near Small Gradients

Other Recurrence Relations xₙ₊₁ = g(xₙ)

Numerical Integration with the Trapezium Rule

Trapezium Rule: Using Equal Step Sizes

Overestimates and Underestimates from Curve Shape

Bounding the True Area Using a Sketch

Choosing a Method for Problems in Context

Unit Vectors i, j (and k) Notation

Magnitude of a Vector

Direction of a Vector

Converting Between Component and Magnitude–Direction Form

Vectors in the Direction of a Given Vector

Vector Addition Algebraically

Scalar Multiplication and Geometric Meaning

Triangle Law of Vector Addition

Parallelogram Law of Vector Addition

Parallel Vectors and Scalar Multiples

Position Vectors and Points in Space

Vector Between Two Points (AB = b − a)

Distance Between Two Points in 2D Using Coordinates

Distance Between Two Points in 3D Using Coordinates

Solving Geometry Problems with Vectors

Finding Unknown Position Vectors (e.g. Parallelograms)

Using Vectors in Context (Velocity, Displacement, Forces)

Census vs Sample: Pros and Cons

Why Sampling Variability Happens

Simple Random Sampling

Opportunity (Convenience) Sampling

Systematic Sampling

Quota Sampling

Choosing a Sampling Method for a Scenario

Critiquing a Sampling Method for Bias

How Different Samples Can Give Different Conclusions

Frequency Polygons: Reading and Comparing Distributions

Box Plots: Median, Quartiles and Spread

Identifying Outliers from Box Plots

Cumulative Frequency Diagrams: Reading Percentiles

Scatter Diagrams: Plotting and Interpreting

Explanatory vs Response Variables

Correlation: Direction and Strength

Why Correlation Does Not Imply Causation

Interpolation from a Scatter Plot

The Dangers of Extrapolation

Distinct Sections in Scatter Data (Sub-populations)

Mean, Median and Mode in Context

Range and Interpercentile Range

Variance and Standard Deviation Meaning

Calculating Standard Deviation from Raw Data

Calculating Standard Deviation from Summary Statistics

Coding Data and Its Effect on Mean/SD

Linear Interpolation for Percentiles (Grouped Data)

Spotting and Handling Missing Data

Correcting Errors in Data Sets

Cleaning Data with Outliers (Given a Rule)

Choosing a Suitable Diagram for a Data Type

Independent Events

Using Venn Diagrams for Probability

Using Tree Diagrams for Probability

Conditional Probability as “Given”