## Maths

Unit A1 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 ½abSinC.
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)

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 acos θ + bsin θ in the equivalent forms of Rcos(θ ± a) or rsin(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)

E5 Understand and use tan θ≡ cos θ/sin θ
Understand and use sin2θ +cos2θ ≡1; sec2θ ≡ 1 + tan2θ and cosec2θ ≡ 1 + cot2θ

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)

Unit A2 Modulus

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)

Unit A3 Functions

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)

Unit A4 Graphs and Transformations

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) and y = f(ax) and combinations of these transformations.

TLMaths – B9: Graph Transformations (google.com)
Graph Transformations | Revision | MME (mmerevise.co.uk)

Unit A5 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 2 and the infinity of primes, and application to unfamiliar proofs)

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

Unit A6 Vectors

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)

Unit A7 Kinematics

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

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)

Unit A8 Forces and Newtons Laws

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 µR 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)

Unit A9 Moments

S1 Understand and use moments in simple static contexts.

TLMaths – S1: Moments
Moments | Revision | MME (mmerevise.co.uk)

Unit B1 Numerical Methods

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 xn + 1= g(xn) .
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

Unit B2 Sequences and Series

D2 Work with sequences including those given by a formula for the nth term and those generated by a simple relation of the form xn+1 = f(xn); increasing sequences; decreasing sequences; periodic sequences.

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 nth 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 nth 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

Unit B3 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.
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)

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 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 xn, for rational values of n, and related constant multiples, sums and differences.
Differentiate ekx and akx, sin 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.

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)

Unit B5 Parametric Equations

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)

Unit B6 Integration

H2  Integrate xn (excluding n= 1), and related sums, differences, and constant multiples.
Integrate ekx,1/x, sin 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

Unit B7 Differential Equations

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)

Unit B8 Probability

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)

Unit B9 Statistical Distributions

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)

Unit B10 Statistical Distributions

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)