## Maths

C1

Understand and use the equation of a straight line, including the forms *y *− *y*_{1} *= m*(*x *− *x*_{1}) 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)

Straight Lines | MME (mathsmadeeasy.co.uk)

Gradients, Tangents and Normals | MME (mathsmadeeasy.co.uk)

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.

TLMaths – B3: Quadratics (google.com)

Solving Quadratic Equations | MME (mathsmadeeasy.co.uk)

Quadratic Graphs | MME (mathsmadeeasy.co.uk)

The Quadratic Formula | MME (mathsmadeeasy.co.uk)

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)

Simultaneous Equations | MME (mathsmadeeasy.co.uk)

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 *ax*^{2} *+ bx + c *graphically.

TLMaths – B5: Inequalities (google.com)

Inequalities | MME (mathsmadeeasy.co.uk)

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).

TLMaths – B6: Polynomials (google.com)

Polynomials | MME (mathsmadeeasy.co.uk)

Algebraic Fractions | MME (mathsmadeeasy.co.uk)

Algebraic Division | MME (mathsmadeeasy.co.uk)

Cubics | MME (mathsmadeeasy.co.uk)

D1

Understand and use the binomial expansion of (*a + bx*)* ^{n} *for positive integer

*n*; the notations

*n*! and

*n*C

*r*; link to binomial probabilities.

TLMaths – D1: Binomial Expansion (google.com)

Binomial Expansion | MME (mathsmadeeasy.co.uk)

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)

Proportion | MME (mathsmadeeasy.co.uk)

The Modulus Function | MME (mathsmadeeasy.co.uk)

Cubics | MME (mathsmadeeasy.co.uk)

Function Graphs | MME (mathsmadeeasy.co.uk)

B9

Understand the effect of simple transformations on the graph of *y *= f(*x*) including sketching associated graphs:*y *= *a*f(*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)

Function Graphs | MME (mathsmadeeasy.co.uk)

Graph Transformations | MME (mathsmadeeasy.co.uk)

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.

TLMaths – C2: Circles (google.com)

Circle Geometry | MME (mathsmadeeasy.co.uk)

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.

TLMaths – E1: Trigonometry (google.com)

Trigonometry Basics | MME (mathsmadeeasy.co.uk)

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)

Trigonometry Basics | MME (mathsmadeeasy.co.uk)

Trig Graphs | MME (mathsmadeeasy.co.uk)

E5

Understand and use

Understand and use sin^{2}θ +cos^{2}θ ≡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)

Trig Equations | MME (mathsmadeeasy.co.uk)

J1

Use vectors in two dimensions

TLMaths – J1: Introducing Vectors (google.com)

Vector Basics | MME (mathsmadeeasy.co.uk)

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)

Vector Calculations | MME (mathsmadeeasy.co.uk)

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)

Vector Basics | MME (mathsmadeeasy.co.uk)

J4

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

TLMaths – J4: Position Vectors (google.com)

Position Vectors | MME (mathsmadeeasy.co.uk)

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

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)

Motion Graphs | MME (mathsmadeeasy.co.uk)

Velocity-Time Graphs – YouTube

Q3

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

TLMaths – Q3: SUVAT (google.com)

SUVAT Equations | MME (mathsmadeeasy.co.uk)

Motion Graphs | MME (mathsmadeeasy.co.uk)

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

Q4

Use calculus in kinematics for motion in a straight line:

TLMaths – Q4: Calculus in Kinematics (google.com)

Non-Uniform Acceleration | MME (mathsmadeeasy.co.uk)

https://youtu.be/gWkieXAqQZM?t=320

Applications of Differentiation | MME (mathsmadeeasy.co.uk)

R1

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

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

Forces | MME (mathsmadeeasy.co.uk)

Newton’s Laws | MME (mathsmadeeasy.co.uk)

Resolving Forces | MME (mathsmadeeasy.co.uk)

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)

Newton’s Laws | MME (mathsmadeeasy.co.uk)

Forces | MME (mathsmadeeasy.co.uk)

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)

Newton’s Laws | MME (mathsmadeeasy.co.uk)

Connected Particles | MME (mathsmadeeasy.co.uk)

Forces | MME (mathsmadeeasy.co.uk)

B1 Understand and use the laws of indices for all rational exponents.

Laws of Indices | MME (mathsmadeeasy.co.uk)

TLMaths – B1. Indices (google.com)

B2 Use and manipulate surds.

Surds | MME (mathsmadeeasy.co.uk)

TLMaths – B2. Surds (google.com)

F3 Understanding the Exponential and Logarithm Functions

TLMaths – F3: Logarithms (google.com)

Exponentials and Logarithms | MME (mathsmadeeasy.co.uk)

The Exponential Function | MME (mathsmadeeasy.co.uk)

F1 Graphs of Exponentials and Logarithms

TLMaths – F1: Exponentials (google.com)

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:*log _{a }x + log_{a }y ≡ log_{a }(xy)*

*log*

_{a }x – log_{a }y ≡ log_{a }(x/y)

*klog*

_{a }x ≡ log_{a }x^{k}

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 a

^{x}= 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 = ax

^{n}and y = kb

^{x}, 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)

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).

TLMaths – A1. Proof (google.com)

Proof | MME (mathsmadeeasy.co.uk)

Proof by Contradiction | MME (mathsmadeeasy.co.uk)

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

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 *x ^{n}*, for rational values of

*n*, and related constant multiples, sums and differences.

TLMaths – G2: Differentiation (google.com)

Differentiation | MME (mathsmadeeasy.co.uk)

G3

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

TLMaths – G3: Gradients (google.com)

Gradients, Tangents and Normals | MME (mathsmadeeasy.co.uk)

Convex and Concave Curves | MME (mathsmadeeasy.co.uk)

Stationary Points | MME (mathsmadeeasy.co.uk)

Applications of Differentiation | MME (mathsmadeeasy.co.uk)

Differentiating Exponential Functions | MME (mathsmadeeasy.co.uk)

H1

Know and use the Fundamental Theorem of Calculus.

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

Integration | MME (mathsmadeeasy.co.uk)

H2

Integrate *x ^{n} *(excluding

*n*= –1), and related sums, differences and constant multiples.

TLMaths – H2: Indefinite Integrals (google.com)

Integration | MME (mathsmadeeasy.co.uk)

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

TLMaths – H3: Definite Integrals (google.com)

Definite Integrals | MME (mathsmadeeasy.co.uk)

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)

Grouped Data | MME (mathsmadeeasy.co.uk)

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)

Grouped Data | 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)

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)

Probability Distributions | MME (mathsmadeeasy.co.uk)

The Binomial Distribution | MME (mathsmadeeasy.co.uk)

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)

Statistical Sampling | MME (mathsmadeeasy.co.uk)

Sampling Techniques | MME (mathsmadeeasy.co.uk)

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)

Hypothesis Testing | MME (mathsmadeeasy.co.uk)

O2

Conduct a statistical hypothesis test for the proportion in the binomial distribution and interpret the results in context.

Understand that a sample is being used to make an inference about the population and appreciate that the significance level is the probability of incorrectly rejecting the null hypothesis.

TLMaths – O2: Binomial Hypothesis Testing (google.com)

Binomial Distribution Hypothesis Tests | MME (mathsmadeeasy.co.uk)