## Maths

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 ½abSinC.

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)

E4 Understand and use the definitions of secant, cosecant and cotangent and of arcsin, arccos and arctan; their relationships to sine, cosine and tangent; understanding of their graphs; their ranges and domains.

TLMaths – E4: Further Trigonometry (google.com)

E2 Understand and use the standard small angle approximations of sine , cosine and

tangent . Sin θ ≈ θ, Cos θ ≈ 1 − θ²/2, Tan θ ≈ θ where θ is in radians

TLMaths – E2: Small Angle Approximation (google.com)

E6 Understand and use double angle formulae; use of formulae for sin (*A *± *B) *, cos (*A *± *B) *and tan (*A *± *B) *; understand geometrical proofs of these formulae.

Understand and use expressions for *a*cos θ + *b*sin θ in the equivalent forms of *R*cos(θ ± *a*) or *r*sin(θ ± *a)*

TLMaths – E6: Compound Angles & Equivalent Forms (google.com)

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 0, π/6, π/4, π/3, π/2, π and multiples thereof and exact values of tan for 0, π/6, π/4, π/3, π/2, π 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 tan θ≡ cos θ/sin θ

Understand and use sin^{2}θ +cos^{2}θ ≡1; sec^{2}θ ≡ 1 + tan^{2}θ and cosec^{2}θ ≡ 1 + cot^{2}θ

TLMaths – E5: Trigonometric Identities (google.com)

Basic Trig Identities | MME (mathsmadeeasy.co.uk)

E8 Construct proofs involving trigonometric functions and identities.

TLMaths – E8: Proving Trigonometric Identities (google.com)

B7 Understand and sketch the modulus of linear functions.

Interpret algebraic solutions of equations graphically; use intersection points of graphs to solve equations.

TLMaths – B7: Graphs & Proportion (google.com)

The Modulus Function | Revision | MME (mmerevise.co.uk)

B8 Understand and use composite functions, inverse functions, and their graphs.

TLMaths – B8: Functions

Composite and Inverse Functions | Revision | MME (mmerevise.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 & Rational Expressions

Polynomials | Revision | MME (mmerevise.co.uk)

B10 Decompose rational functions into partial fractions (denominators not more complicated than squared linear terms and with no more than three terms, numerators constant or linear).

TLMaths – B10: Algebraic Fractions

Partial Fractions | Revision | MME (mmerevise.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*) and *y *= f(*ax*) and combinations of these transformations.

TLMaths – B9: Graph Transformations (google.com)

Graph Transformations | Revision | MME (mmerevise.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 2 and the infinity of primes, and application to unfamiliar proofs)

TLMaths – A1: Proof (google.com)

Proof | Revision | MME (mmerevise.co.uk)

Proof by Contradiction | Revision | MME (mmerevise.co.uk)

J1 Use vectors in two dimensions and in three dimensions.

TLMaths – J1: Introducing Vectors (google.com)

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

TLMaths – J5: Vector Problems (google.com)

3D Vectors | Revision | MME (mmerevise.co.uk)

Q3 Understand, use and derive the formulae for constant acceleration for motion in a straight line; extend to 2 dimensions using vectors.

TLMaths – Q3: SUVAT (google.com)

SUVAT Equations | Revision | MME (mmerevise.co.uk)

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

v = dr/dt, a = dv/dt = d²r/dt², r = ʃ v dt, v = ʃa dt

Extend to 2 dimensions using vectors

TLMaths – Q4: Calculus in Kinematics

Non-Uniform Acceleration | Revision | MME (mmerevise.co.uk)

Vectors in Mechanics | Revision | MME (mmerevise.co.uk)

Q5 Model motion under gravity in a vertical plane using vectors; projectiles.

TLMaths – Q5: Projectiles

Projectiles | Revision | MME (mmerevise.co.uk)

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

TLMaths – R1: Introducing Forces & Newton’s First Law

Forces | Revision | MME (mmerevise.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); extend to situations where forces need to be resolved (restricted to 2 dimensions).

TLMaths – R2: Newton’s Second Law

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; resolving forces in 2 dimensions; equilibrium of a particle under coplanar forces.

TLMaths – R4: Newton’s Third Law and Pulleys

Connected Particles | Revision | MME (mmerevise.co.uk)

R5 Understand and use addition of forces; resultant forces; dynamics for motion in a plane.

TLMaths – R5: F=ma & Differential Equations

R6 Understand and use the **F **≤ *µ***R **model for friction; coefficient of friction; motion of a body on a rough surface; limiting friction and statics.

TLMaths – R6: The Coefficient of Friction

Friction | Revision | MME (mmerevise.co.uk)

S1 Understand and use moments in simple static contexts.

TLMaths – S1: Moments

Moments | Revision | MME (mmerevise.co.uk)

I1 Locate roots of f(*x*) = 0 by considering changes of sign of f(*x*) in an interval of *x *on which f(*x*) is sufficiently well-behaved. Understand how change of sign methods can fail.

TLMaths – I1: The Change of Sign Method

I2 Solve equations approximately using simple iterative methods; be able to draw associated cobweb and staircase diagrams.

Solve equations using the Newton-Raphson method and other recurrence relations of the form *x _{n }*

_{+ 1}= g(

*x*) .

_{n}Understand how such methods can fail.

TLMaths – I2: The x=g(x) Method & The Newton-Raphson Method

Newton-Raphson Method | Revision | MME (mmerevise.co.uk)

I3 Understand and use numerical integration of functions, including the use of the trapezium rule and estimating the approximate area under a curve and limits that it must lie between.

TLMaths – I3: Numerical Integration

I4 Use numerical methods to solve problems in context.

TLMaths – I4: Numerical Methods in Context

Locating Roots | Revision | MME (mmerevise.co.uk)

H4 Understand and use integration as the limit of a sum.

TLMaths – H4: Integration as the Limit of a Sum

D2 Work with sequences including those given by a formula for the *n*th term and those generated by a simple relation of the form *x _{n}*

_{+1 }= f(

*x*); increasing sequences; decreasing sequences; periodic sequences.

_{n}TLMaths – D2: Sequences

D3 Understand and use sigma notation for sums of series.

TLMaths – D3: Sigma Notation

D4 Understand and work with arithmetic sequences and series, including the formulae for

*n*th term and the sum to

*n*terms.

TLMaths – D4: Arithmetic Sequences

Arithmetic Series | Revision | MME (mmerevise.co.uk)

D5 Understand and work with geometric sequences and series including the formulae for the

*n*th term and the sum of a finite geometric series; the sum to infinity of a convergent geometric series, including the use of |

*r*| < 1; modulus notation.

TLMaths – D5: Geometric Sequences

Geometric Series | Revision | MME (mmerevise.co.uk)

D6 Use sequences and series in modelling.

TLMaths – D6: Modelling with Sequences

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.

Extend to any rational

*n*, including its use for approximation; be aware that the expansion is valid for ǀ bx/a ǀ < 1 (Proof not required.)

TLMaths – D1: Binomial Expansion

Binomial Expansion | Revision | MME (mmerevise.co.uk)

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 *and for sin *x *and cos *x.*

Understand and use the second derivative as the rate of change of gradient; connection to convex and concave sections of curves and points of inflection.

TLMaths – G1: Differentiation from First Principles

Differentiation from First Principles | Revision | MME (mmerevise.co.uk)

G2 Differentiate *x ^{n}*, for rational values of

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

Differentiate e

*and*

^{kx}*a*, sin

^{kx}*kx*, cos

*kx*and tan

*kx*related sums, differences and constant multiples.

Understand and use the derivative of ln

*x*.

TLMaths – G2: Differentiation

More Differentiation | Revision | MME (mmerevise.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

G4 Differentiate using the product rule, the quotient rule and the chain rule

TLMaths – G4: Further Differentiation

Product Rule | Revision | MME (mmerevise.co.uk)

Quotient Rule | Revision | MME (mmerevise.co.uk)

Chain Rule | Revision | MME (mmerevise.co.uk)

G5 Differentiate simple functions and relations defined implicitly for first derivative only.

TLMaths – G5: Implicit Differentiation & Parametric Differentiation

Implicit Differentiation | Revision | MME (mmerevise.co.uk)

C3 Understand and use the parametric equations of curves and conversion between Cartesian and parametric forms.

TLMaths – C3: Parametric Equations

C4 Use parametric equations in modelling in a variety of contexts.

TLMaths – C4: Parametric Equation Modelling

Parametric Equations | Revision | MME (mmerevise.co.uk)

G4 Using differentiation to solve problems involving connected rates of change and inverse functions.

TLMaths – G4: Further Differentiation

G5 Differentiate simple functions and relations defined parametrically, for first derivative only.

TLMaths – G5: Implicit Differentiation & Parametric Differentiation

Differentiating Parametric Equations | Revision | MME (mmerevise.co.uk)

H2 Integrate ** x^{n} **(excluding

**–**

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

Integrate e

*, sin*

^{kx},1/x*kx*, cos

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

TLMaths – H2: Indefinite Integrals

Integrating Trig Functions | Revision | MME (mmerevise.co.uk)

H3 Evaluate definite integrals; use a definite integral to find the area under a curve and the area between two curves.

TLMaths – H3: Definite Integrals & Parametric Integration

H5 Carry out simple cases of integration by substitution and integration by parts; understand these methods as the inverse processes of the chain and product rules respectively.

(Integration by substitution includes finding a suitable substitution and is limited to cases where one substitution will lead to a function which can be integrated; integration by parts includes more than one application of the method but excludes reduction formulae.)

TLMaths – H5: Further Integration

Integration by Substitution | Revision | MME (mmerevise.co.uk)

Integration by Parts | Revision | MME (mmerevise.co.uk)

H6 Integrate using partial fractions that are linear in the denominator.

TLMaths – H6: Integration with Partial Fractions

G6 Construct simple differential equations in pure mathematics and in context, (contexts may include kinematics, population growth and modelling the relationship between price and demand).

TLMaths – G6: Forming Differential Equations

H7 Evaluate the analytical solution of simple first order differential equations with separable variables, including finding particular solutions. (Separation of variables may require factorisation involving a common factor.)

TLMaths – H7: Differential Equations

H8 Interpret the solution of a differential equation in the context of solving a problem, including identifying limitations of the solution; includes links to kinematics.

TLMaths – H8: Differential Equations in Context

Differential Equations | Revision | MME (mmerevise.co.uk)

M2 Understand and use conditional probability, including the use of tree diagrams, Venn diagrams, two-way tables.

Understand and use the conditional probability formula P(AǀB) = P(AՈB)/P(B)

TLMaths – M2: Conditional Probability

Conditional Probability and Tree Diagrams | Revision | MME (mmerevise.co.uk)

M3 Modelling with probability, including critiquing assumptions made and the likely effect of more realistic assumptions.

TLMaths – M3: Modelling with Probability

Probability and Venn Diagrams | Revision | MME (mmerevise.co.uk)

N2 Understand and use the Normal distribution as a model; find probabilities using the Normal distribution. Link to histograms, mean, standard deviation, points of inflection and the binomial distribution.

TLMaths – N2: The Normal Distribution

The Normal Distribution | Revision | MME (mmerevise.co.uk)

The Standard Normal Distribution | Revision | MME (mmerevise.co.uk)

N3 Select an appropriate probability distribution for a context, with appropriate reasoning, including recognising when the binomial or Normal model may not be appropriate.

TLMaths – N3: Appropriate Distributions

Selecting a Distribution | Revision | MME (mmerevise.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; extend to correlation coefficients as measures of how close data points lie to a straight line and be able to interpret a given correlation coefficient using a given

*p*-value or critical value (calculation of correlation coefficients is excluded).

TLMaths – O1: Introducing Hypothesis Testing

Hypothesis Testing | Revision | MME (mmerevise.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

Binomial Distribution Hypothesis Tests | Revision | MME (mmerevise.co.uk)

O3 Conduct a statistical hypothesis test for the mean of a Normal distribution with known, given or assumed variance and interpret the results in context.

TLMaths – O3: Sample Means Hypothesis Testing

Normal Distribution Hypothesis Tests | Revision | MME (mmerevise.co.uk)