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GCE Further Mathematics (2017) Poll 1 Which boards do you have experience of teaching Further Maths with?

GCE Further Mathematics (2017) Poll 2 Which Edexcel Further Maths modules have you previously taught?

GCE Further Mathematics (2017) Poll 3 How many people are you viewing this presentation with?

GCE Further Mathematics (2017) Poll 4 What is the structure of your course?

GCE Further Mathematics (2017) A Level Reforms

The A level reforms All new AS and A levels will be assessed at the same standard as they are currently All new AS and A levels will be fully linear AS levels will be stand-alone qualifications The content of the AS level can be a sub-set of the A level content to allow co-teachability, but marks achieved in the AS will not count towards the A level

The A level reforms A level Further Mathematics 50% core (all pure mathematics) 50% optional and can include pure mathematics mechanics statistics decision mathematics any other

The A level reforms - content AS level Further Mathematics 20% core (all pure mathematics) 10% compulsory (selected from the A level core) 70% optional (same options as A level) We have made an additional 20% of the content compulsory (taken from A level core)

The A level reforms Requirement for the assessment of problem solving, communication, proof, modelling, application of techniques Requirement that candidates have a calculator with the ability to compute summary statistics and access probabilities from standard statistical distributions an iterative function the ability to perform calculations with matrices up to at least order 3 3 (further mathematics only)

Overview of the specification A Level Further mathematics Paper 1: Core Pure Mathematics 1 25% 1 hour 30 minutes 75 marks Compulsory content any content on either paper Paper 2: Core Pure Mathematics 2 25% 1 hour 30 minutes 75 marks Paper 3: Further Mathematics Option 1 25% 1 hour 30 minutes 75 marks Paper 4: Further Mathematics Option 2 25% 1 hour 30 minutes 75 marks Students take two optional papers with options available in Further Pure Mathematics Further Statistics Further Mechanics Decision Mathematics

A level Further Mathematics options For papers 3 and 4 students choose a pair of options, either any two from column A, or a matching pair from columns A and B

Overview of the specification AS Further Mathematics

GCE Further Mathematics (2017) Overview of content

AS Level Content Unit Title Estimated hours 1 Complex numbers (part 1) a Introduction of complex numbers, basic manipulation FP1 Ch 1 b Argand diagrams FP1 Ch 1 c Modulus and argument FP1 Ch 1 d Loci FP2 Ch 3 2 Matrices a Matrix addition, subtraction and multiplication FP1 Ch 4 b Inverse of 2 2 and 3 3 matrices FP3 Ch 6 c Simultaneous equations New d Linear transformations 3 Complex numbers (part 2) Complex conjugate, division and solving polynomial a equations FP1 Ch 4 FP3 Ch 6 FP1 Ch1

AS Level Content Unit Title Estimated hours 4 Series a Sums of series FP2 Ch 2 5 Algebra and functions a Roots of polynomial equations New b Formation of polynomial equations New 6 Proof a Proof by mathematical induction FP1 Ch 6 7 Vectors a Vector and Cartesian equations of a line and a plane FP3 Ch 5 b Scalar product C4 Ch5 FP3 Ch 5 c Problems involving points, lines and planes FP3 Ch 5 8 Calculus a Volumes of revolution C4 Ch 6

A Level Content Unit Title Estimated hours 1 Complex numbers A Know and use z = re iθ = r(cos θ + i sin θ) FP2 Ch 3 b De Moivre s theorem FP2 Ch 3 c The nth roots of z = re iθ and complex roots of unity FP2 Ch 3 2 Hyperbolic functions a sinh x, cosh x, tanh x and their inverses FP3 Ch 1 b FP3 Ch 4 3 Polar coordinates a Convert between Cartesian and polar and sketch r(θ) FP2 Ch 7 b Area enclosed by a polar curve FP2 Ch 7

A Level Content Unit Title Estimated hours 4 Further algebra and functions (series) a Method of differences FP2 Ch 2 b Maclaurin series FP2 Ch 6 5 Further calculus a Improper integrals New b Mean value of a function New c Integrate using partial fractions FP3 Ch 4 d Differentiate inverse trigonometric functions and integrate using trigonometric substitutions FP3 Ch 4 e Further volumes of revolutions FP3 Ch 4 6 Differential equations a Integrating factors to solve first order differential equations FP2 Ch 4 b Second order differential equations of the form y + ay + by = f(x) FP2 Ch 5 c Modelling FP2 Ch 5

Schemes of Work

Complex Numbers Radians Compound angle formula Series Sigma notation AS Level Key Prerequisites

AS Level: What s New? Complex Number: Knowledge of arg(z 1 z 2 ) = arg(z 1 )+arg(z 2 ) was specifically not required in FP1 but is in new specification. Matrices: Understand and use zero and identity matrices. Find invariant points and lines for a linear transformation. Understand and use singular and non-singular matrices. Properties of inverse matrices. Solve three linear simultaneous equations in three variables by use of the inverse matrix. Interpret geometrically the solution and failure of solution of three simultaneous linear equations. Further Algebra: Understand and use the relationship between roots and coefficients of polynomial equations up to quartic equations. Form a polynomial equation whose roots are a linear transformation of the roots of a given polynomial equation (of at least cubic degree).

AS Level: What s New? Further Calculus: Derive formulae for and calculate volumes of revolution. Further Vectors Understand and use the vector and Cartesian forms of an equation of a straight line in 3-D. Understand and use the vector and Cartesian forms of the equation of a plane. Calculate the scalar product and use it to express the equation of a plane, and to calculate the angle between two lines, the angle between two planes and the angle between a line and a plane.

A Level: What s New?

GCE Further Mathematics (2017) Complex Numbers

Complex Numbers

Working in Exponential Form

De Moivre s Theorem

Finding Roots

Find the n distinct nth roots of re iθ for r 0 and know that they form the vertices of a regular n-gon in the Argand diagram. Finding Roots

Finding Roots: Model Working

Further DeMoivre s Work

As a Tool in Integration

Example Question

Loci

Transformations

GCE Further Mathematics (2017) 1 st Order Differential Equations

Separation of variables (from C4) Perfect Integrals 1 st Order Differential Equation Techniques Integrating Factors

Perfect Integrals

The Integrating Factor

Example

GCE Further Mathematics (2017) 2 nd Order Differential Equations

2 nd Order Differential Equations

Complementary Functions

Particular Integral

The General Solution

Predator Prey Models

Harmonic Motion

Damped Motion

Harmonic Motion

GCE Further Mathematics (2017) Matrices

Addition example 3x3 Matrices Subtraction example

3x3 Matrices Multiplication example

Example 3x3 Matrices Transpose matrix, A T Example. Find A T A

3x3 Matrices Determinants For the 2x2 matrix A = a c b d Similarly for the 3x3 matrix a b c d e f g h i

3x3 Matrices Example A matrix M is singular if det(m) =0 Show that A= is singular

Minors 3x3 Matrices Minors

3x3 Matrices Inverse Matrix, A -1

3x3 matrices Using a Calculator Casio 991ES

Matrix Transformations

Inverse Transformations

Simultaneous equations

3x3 Matrices Examination Question Further Mathematics Advanced Sample Assessment Materials Paper 2: Core Pure Mathematics.

3x3 Matrices Examination Question

GCE Further Mathematics (2017) Hyperbolic Functions

Hyperbolic Functions With circular trigonometric functions the point (sinx,cosx) describes the locus of a circle radius 1 For hyperbolic functions, the point (sinhx, coshx) gives the locus of poits on an (equilatoral) hyperbola. As an example, the graph of y=coshx will be a catenary (latin- catena, chain ) the curve taken by a chain suspended between two points.

Hyperbolic Functions - Definitions

Hyperbolic Functions Example 1

Hyperbolic Functions Example 2 Use the definition of sinh x to find, to 2 d.p, the value of x for which sinh x = 5.

Hyperbolic Functions Graphs

Hyperbolic Functions Identities Hyperbolic identities are very similar to trigonometric identities Osborn s rule: The corresponding hyperbolic identity can be formed from a trigonometric identity by changing the sign of the product (or implied product) of any 2 sine terms.

Hyperbolic Functions - Identities Example: Write down the hyperbolic identity corresponding to tan(a+b) The sign in the denominator has changed since tanatanb has the implied multiplication of 2 sines (sinasinb) hence a sign change by Osborn s rule.

Hyperbolic Functions - Identities Given that sinh x = ¾, find the exact value of (a) cosh x, (b) tanh x, (c) sinh 2x Unlike circular functions, the option of using a right-angled triangle with a cast diagram is not available and a hyperbolic identity must be used.

Hyperbolic Inverse Functions The function is reflected in the line y=x to find the inverse function. Note that coshx requires x 1 to be one to one and have an inverse.

Hyperbolic Functions -Equations Solve 2cosh 2 x 5sinh x = 5, giving your answer in natural logarithms

GCE Further Mathematics (2017) Integration Reduction Formulae

Itegration: Reduction Formulas.

Free Support Plan: schemes of work and course planners to help you deliver the qualifications in the best way for your centre content mapping documents Getting started guide Teach: topic-based resources to use in the classroom, particularly for the new and unfamiliar topics content exemplification Track and Assess: specimen papers secure mock papers practice papers assessment guide exemplar solutions Exam Wizard & Results Plus Develop: a full programme of launch and training events live and online pre-recorded getting ready to teach sessions our collaborative network events the famous Mathematics Emporium, led by Graham Cumming