## Maths

Unit A1 Linear Graphs

C1
Understand and use the equation of a straight line, including the forms y y1 = m(x x1) and ax + by + c = 0; gradient conditions for two straight lines to be parallel or perpendicular.
Use straight line models in a variety of contexts.

TLMaths – C1: Coordinate Geometry (google.com)

B4
Solve simultaneous equations in two variables by elimination and by substitution.

TLMaths – B4: Simultaneous Equations (google.com)
Simultaneous Equations | Revision | MME (mathsmadeeasy.co.uk)

B3
Work with quadratic functions and their graphs; the discriminant of a quadratic function, including the conditions for real and repeated roots; completing the square; solution of quadratic equations including solving quadratic equations in a function of the unknown.

B4
Solve simultaneous equations in two variables by elimination and by substitution focusing on one linear and one quadratic equation.

TLMaths – B4: Simultaneous Equations (google.com)

B5
Solve linear and quadratic inequalities in a single variable and interpret such inequalities graphically, including inequalities with brackets and fractions.
Express solutions through correct use of ‘and’ and ‘or’, or through set notation.
Represent linear and quadratic inequalities such as y > x + 1 and ax2 + bx + c graphically.

Unit A3 Polynomials

B6
Manipulate polynomials algebraically, including expanding brackets and collecting like terms,
factorisation and simple algebraic division; use of the factor theorem.
Simplify rational expressions including by factorising and cancelling, and algebraic division (by linear expressions only).

Unit A4 Binomial Expansion

D1
Understand and use the binomial expansion of (a + bx)n for positive integer n; the notations n! and nCr; link to binomial probabilities.

TLMaths – D1: Binomial Expansion (google.com)

Unit A5 Graphs Transformation

B7
Understand and use graphs of functions; sketch curves defined by simple equations including polynomials, the modulus of a linear function,

and (including their vertical and horizontal asymptotes); interpret algebraic solution of equations graphically; use intersection points of graphs to solve equations.
Understand and use proportional relationships and their graphs.

TLMaths – B7: Graphs & Proportion (google.com)
The Modulus Function | MME (mathsmadeeasy.co.uk)

B9
Understand the effect of simple transformations on the graph of y = f(x) including sketching associated graphs:
y = af(x), y = f(x) + a, y = f(x + a), y = f(ax), y = -f(x) and y = f(-x)

TLMaths – B9: Graph Transformations (google.com)

Unit A6 Circles

C2 Understand and use the coordinate geometry of the circle including using the equation of a circle in the form (x – a)² + (y – b)² = r² ; completing the square to find the centre and radius of a circle; use the following properties:
• the angle in a semicircle is a right angle
• the perpendicular from the centre to a chord bisects the chord
• the radius of a circle at a given point on its circumference is perpendicular to the tangent to the circle at that point.

Unit A7 Trigonometry

E1
Understand and use the definitions of sine, cosine and tangent for all arguments; the sine and cosine rules; the area of a triangle in the form

.
Work with radian measure, including use for arc length and area of sector.

Sine & Cosine Rules | MME (mathsmadeeasy.co.uk)
Arc Length & Area of a Sector | MME (mathsmadeeasy.co.uk)

E3
Understand and use the sine, cosine and tangent functions, their graphs, symmetries and periodicity.
Know and use exact values of sin and cos for  and multiples thereof and exact values of tan for and multiples thereof.

TLMaths – E3: Trig Graphs (google.com)

E5
Understand and use
Understand and use sin2θ +cos2θ ≡1

Introduction to trigonometric identities

TLMaths – E5: Trigonometric Identities (google.com)
Basic Trig Identities | MME (mathsmadeeasy.co.uk)

E7
Solve simple trigonometric equations in a given interval, including quadratic equations in sin, cos and tan and
equations involving multiples of the unknown angle.

TLMaths – E7: Trig Equations (google.com)

Unit A8 Vectors

J1
Use vectors in two dimensions

TLMaths – J1: Introducing Vectors (google.com)

J2
Calculate the magnitude and direction of a vector and convert between component form and magnitude/direction form.

TLMaths – J2: Magnitude & Direction of a Vector (google.com)

J3
Add vectors diagrammatically and perform the algebraic operations of vector addition and multiplication by scalars, and understand their geometrical interpretations.

TLMaths – J3: Resultant & Parallel Vectors (google.com)

J4
Understand and use position vectors; calculate the distance between two points represented by position vectors.

TLMaths – J4: Position Vectors (google.com)

J5
Use vectors to solve problems in pure mathematics and in context, including forces

TLMaths – J5: Vector Problems (google.com)
Mechanics 1 – M1 – Vectors (1) Introduction – Full tutorial-(Edexcel, OCR, MEI & AQA) – YouTube

Unit A9 Kinematics

Q1
Understand and use the language of kinematics: position; displacement; distance travelled; velocity; speed; acceleration.

TLMaths – P1: Quantities & Units in Mechanics (google.com)
TLMaths – Q1: Displacement, Velocity & Acceleration (google.com)

Q2
Understand, use and interpret graphs in kinematics for motion in a straight line: displacement against time and interpretation of gradient; velocity against time and interpretation of gradient and area under the graph.

TLMaths – Q2: Graphs of Motion (google.com)

Q3
Understand, use and derive the formulae for constant acceleration for motion in a straight line.

Vectors in Mechanics | MME (mathsmadeeasy.co.uk)

Q4
Use calculus in kinematics for motion in a straight line:

Unit A10 Forces

R1
Understand the concept of a force; understand and use Newton’s first law.

TLMaths – R1: Introducing Forces & Newton’s First Law (google.com)

R2
Understand and use Newton’s second law for motion in a straight line (restricted to forces in two perpendicular directions or simple cases of forces given as 2-D vectors)

TLMaths – R2: Newton’s Second Law (google.com)

R3
Understand and use weight and motion in a straight line under gravity; gravitational acceleration, g, and its
value in SI units to varying degrees of accuracy. (The inverse square law for gravitation is not required
and g may be assumed to be constant, but students should be aware that g is not a universal constant but
depends on location).

TLMaths – R3: Weight & Tension (google.com)

R4
Understand and use Newton’s third law; equilibrium of forces on a particle and motion in a straight line (restricted to forces in two perpendicular directions or simple cases of forces given as 2-D vectors); application to problems involving smooth pulleys and connected particles.

TLMaths – R4: Newton’s Third Law and Pulleys (google.com)

Unit B2 Exponentials and Logarithms

F3 Understanding the Exponential and Logarithm Functions
Exponentials and Logarithms | MME (mathsmadeeasy.co.uk)
The Exponential Function | MME (mathsmadeeasy.co.uk)

F1 Graphs of Exponentials and Logarithms
The Exponential Function | MME (mathsmadeeasy.co.uk)
Modelling Exponential Growth and Decay | MME (mathsmadeeasy.co.uk)

F2 Finding and using the Gradient of an Exponential Function
TLMaths – F2: Exponential Models (google.com)
Differentiating Exponential Functions | MME (mathsmadeeasy.co.uk)

F4 Understand and use the laws of logarithms:
loga x + loga y ≡ loga (xy)

loga x – loga y ≡ loga (x/y)

kloga x ≡ loga xk

TLMaths – F4: Laws of Logarithms (google.com)
Exponentials and Logarithms | MME (mathsmadeeasy.co.uk)
Equations Involving Exponentials | MME (mathsmadeeasy.co.uk)
The Exponential Function | MME (mathsmadeeasy.co.uk)

F5
Solve equations of the form ax= b.

TLMaths – F5: Exponential & Logarithmic Equations (google.com)
Equations Involving Exponentials | MME (mathsmadeeasy.co.uk)
The Exponential Function | MME (mathsmadeeasy.co.uk)

F6
Use logarithmic graphs to estimate parameters in relationships of the form y = axn and y = kbx , given data for x and y.
TLMaths – F6: Reduction to Linear Form (google.com)
Reduction to Linear Form | MME (mathsmadeeasy.co.uk)

F7
Understand and use exponential growth and decay; use in  modelling (examples may include the use of e in continuous compound interest, radioactive decay, drug concentration decay, exponential growth as a model for population growth); consideration of limitations and refinements of exponential models.
TLMaths – F7: Exponential Growth & Decay (google.com)
Modelling Exponential Growth and Decay | MME (mathsmadeeasy.co.uk)
Equations Involving Exponentials | MME (mathsmadeeasy.co.uk)

Unit B3 Proof

A1
Understand and use the structure of mathematical proof, proceeding from given assumptions through a series of logical steps to a conclusion; use methods of proof, including proof by deduction, proof by exhaustion.
Disproof by counter example.
Proof by contradiction (including proof of the irrationality of

and the infinity of primes, and application to unfamiliar proofs).

1 – MME – A Level Maths – Pure – Proof – YouTube

Unit B4 Differentiation

G1
Understand and use the derivative of f(x) as the gradient of the tangent to the graph of y = f(x) at a general point (x, y); the gradient of the tangent as a limit; interpretation as a rate of change; sketching the gradient function for a given curve; second derivatives; differentiation from first principles for small positive
integer powers of x.
Understand and use the second derivative as the rate of change of gradient.

TLMaths – G1: Differentiation from First Principles (google.com)
Differentiation from First Principles | MME (mathsmadeeasy.co.uk)

G2
Differentiate xn, for rational values of n, and related constant multiples, sums and differences.

G3
Apply differentiation to find gradients, tangents and normals, maxima and minima and stationary points. Identify where functions are increasing or decreasing.

Convex and Concave Curves | MME (mathsmadeeasy.co.uk)
Applications of Differentiation | MME (mathsmadeeasy.co.uk)
Differentiating Exponential Functions | MME (mathsmadeeasy.co.uk)

Unit B5 Integration

H1
Know and use the Fundamental Theorem of Calculus.

TLMaths – H1: Fundamental Theorem of Calculus (google.com)

H2
Integrate xn (excluding n= –1), and related sums, differences and constant multiples.

TLMaths – H2: Indefinite Integrals (google.com)

H3 Evaluate definite integrals; use a definite integral to find the area under a curve

TLMaths – H3: Definite Integrals (google.com)

Unit B6 Data

L1 Interpret diagrams for single-variable data, including understanding that area in a histogram represents frequency.
Connect to probability distributions.
TLMaths – L1: Box Plots, Cumulative Frequency & Histograms (google.com)
Cumulative Frequency and Boxplots | MME (mathsmadeeasy.co.uk)

L2
Interpret scatter diagrams and regression lines for bivariate data, including recognition of scatter diagrams which include distinct sections of the population (calculations involving regression lines are excluded).
Understand informal interpretation of correlation.
Understand that correlation does not imply causation.

TLMaths – L2: Scatter Graphs (google.com)
Correlation and Regression | MME (mathsmadeeasy.co.uk)

L3
Interpret measures of central tendency and variation, extending to standard deviation.
Be able to calculate standard deviation, including from
summary statistics.

TLMaths – L3: Central Tendency & Variation (google.com)
Mean and Standard Deviation | MME (mathsmadeeasy.co.uk)

L4
Recognise and interpret possible outliers in data sets and statistical diagrams.
Select or critique data presentation techniques in the context of a statistical problem.
Be able to clean data, including dealing with missing data, errors and outliers.

TLMaths – L4: Outliers & Cleaning Data (google.com)
Outliers and Interquartile Range | MME (mathsmadeeasy.co.uk)

Unit B7 Probability and Distribution

M1
Understand and use mutually exclusive and independent events when calculating probabilities.
Link to discrete and continuous distributions.

TLMaths – M1: Venn Diagrams, Tree Diagrams & Two-Way Tables (google.com)
Probability and Venn Diagrams | MME (mathsmadeeasy.co.uk)
Mutually Exclusive Events and Independence | MME (mathsmadeeasy.co.uk)
Conditional Probability and Tree Diagrams | MME (mathsmadeeasy.co.uk)

N1
Understand and use simple, discrete probability distributions (calculation of mean and variance of discrete random variables is excluded), including the binomial distribution, as a model; calculate probabilities using the binomial distribution.

TLMaths – N1: Discrete Random Variables & The Binomial Distribution (google.com)
The Binomial Distribution | MME (mathsmadeeasy.co.uk)

Unit B8 Sampling

K1
Understand the terms ‘population’ and ‘sample’.
Use samples to make informal inferences about the population.
Understand and use sampling techniques, including simple random sampling and opportunity sampling.
Select or critique sampling techniques in the context of solving a statistical problem, including understanding that different samples can lead to different conclusions about the population.

TLMaths – K1: The Large Data Set & Sampling Methods (google.com)

Unit B9 Hypothesis Testing

O1
Understand and apply the language of statistical hypothesis testing, developed through a binomial model: null hypothesis; alternative hypothesis, significance level, test statistic, 1-tail test, 2-tail test, critical value, critical region, acceptance region, p-value;

TLMaths – O1: Introducing Hypothesis Testing (google.com)