Getting Ready to Teach Online course Core Pure

Transcription

1 Getting Ready to Teach Online course Core Pure

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

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

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

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

6 GCE Further Mathematics (2017) A Level Reforms

7 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

8 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

9 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)

10 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)

11 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

12 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

13 Overview of the specification AS Further Mathematics

14 GCE Further Mathematics (2017) Overview of content

15 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

16 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

17 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

18 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

19 Schemes of Work

20 Schemes of Work

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

22 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).

23 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.

24 A Level: What s New?

25 GCE Further Mathematics (2017) Complex Numbers

26 Complex Numbers

27 Complex Numbers

28 Working in Exponential Form

29 De Moivre s Theorem

30 De Moivre s Theorem

31 Finding Roots

32 Finding Roots

33 Finding Roots

34 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

35 Finding Roots: Model Working

36 Further DeMoivre s Work

37 As a Tool in Integration

38 As a Tool in Integration

39 Example Question

40

41 Loci

42

43 Transformations

44 GCE Further Mathematics (2017) 1 st Order Differential Equations

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

46 Perfect Integrals

47 The Integrating Factor

48 The Integrating Factor

49 Example

50 GCE Further Mathematics (2017) 2 nd Order Differential Equations

51 2 nd Order Differential Equations

52 2 nd Order Differential Equations

53 Complementary Functions

54 Particular Integral

55 Particular Integral

56 The General Solution

57 Predator Prey Models

58

59 Predator Prey Models

60 Predator Prey Models

61 Predator Prey Models

62 Harmonic Motion

63 Harmonic Motion

64 Damped Motion

65 Harmonic Motion

66 GCE Further Mathematics (2017) Matrices

67 Addition example 3x3 Matrices Subtraction example

68 3x3 Matrices Multiplication example

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

70 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

71 3x3 Matrices Example A matrix M is singular if det(m) =0 Show that A= is singular

72 Minors 3x3 Matrices Minors

73 3x3 Matrices Inverse Matrix, A -1

74 3x3 Matrices Inverse Matrix, A -1

75 3x3 matrices Using a Calculator Casio 991ES

76 Matrix Transformations

77 Inverse Transformations

78 Simultaneous equations

79 3x3 Matrices Examination Question Further Mathematics Advanced Sample Assessment Materials Paper 2: Core Pure Mathematics.

80 3x3 Matrices Examination Question

81 GCE Further Mathematics (2017) Hyperbolic Functions

82 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.

83 Hyperbolic Functions - Definitions

84 Hyperbolic Functions Example 1

85 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.

86 Hyperbolic Functions Graphs

87 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.

88 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.

89 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.

90 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.

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

92 GCE Further Mathematics (2017) Integration Reduction Formulae

93 Itegration: Reduction Formulas.

94 Itegration: Reduction Formulas.

95 Itegration: Reduction Formulas.

96 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