CORE MATHEMATICS 4 SCHEME OF WORK 2005-2006:

C4 UNIT ≈ 40 Lessons (the course should be started once C3 has been finished, i.e. November, aiming to complete by spring half-term). The first part of the summer term should be used to revise. The course leads to a June assessment of 1½hours (75 marks, ≈ 7 questions), QCA recommended calculators be permitted.

The text used is Heinemann P2 & Heinemann P3 (new edition). Other sources referenced include Core Maths 4(Staff copy) & Introducing Pure Mathematics (Smedley & Wiseman) (MAC & JLW have old copy, PJW, SAM and ACD new). Also remember to make use of the Heinemann “A-Level Exambank” CD-Rom containing past paper questions under topic headings (see MC for a copy). Homework of approximately 1 hour duration should be given after each lesson, including revision, past paper and review exercise questions whenever possible.

The formulae that candidates are expected to know are indicated in the appropriate sections in this scheme. Other formulae can be found in the booklet Mathematical Formulae including Statistical Formulae and Tables – please allow pupils to familiarise themselves with this document as early as possible within the course (new booklet for 2006 exams).

In order to assist pupils’ learning, give short timed tests as lesson starters, using examination style questions, as early on in the course as possible.

Knowledge of C1-C3 material is assumed and may be tested.

Time	Specification	Notes	Page and Exercise	Other Sources
2	Algebra Rational Functions Decomposition of rational functions into partial fractions (denominators not more complicated than repeated linear terms)	Quick recap on +/- of algebraic fractions from C2 as we now move into doing this in reverse NB Identities no longer covered in year 12, so a little more input on their use needed here. Denominators will include ones of the form (ax+b)(cx+d)(ex+f) and (ax+d)(cx+d)². It is possible that the degree of the numerator will exceed that of the denominator. Quadratic factors in the denominator such as (x²+a), a>0, are not required.	Heinemann C4 Ch. 1 p. 1
10	Integration Integration of e x , 1/x, sinx, cosx Integration using trigonometric identities Integration by parts (reverse of product rule) Integration by substitution (reverse of chain rule Integration of rational expressions, including those arising from partial fractions Evaluation of volume of revolution (leave out parametric ones) Numerical integration of functions	Examples will include standard functions such as sin3x, sec²2x and tanx, e 5x , 1/2x Recognition of functions of the form ∫ (f’(x)/f(x)dx=lnf(x)+c e.g. Integrate sin²x, tan²x, cos2xcos4x Include ∫ lnx dx. May need more than application of the by parts method e.g. ∫ x²e x dx Except in the simplest of cases, the substitution will be given Rational expressions e.g. 2/3x+5, 3/(x-1)², x/(x²+5), 2/(2x-1) 4 π∫ y 2 dx is required but not π∫ x 2 dy. Application of trapezium rule to functions covered in C3 &C4: use of increasing number of trapezia to improve accuracy and estimate error will be required. Questions will not require more than 3 iterations. Simpson’s rule will not be required.	Heinemann C4 Ch. 6 p.82 (leave out 6.10 and 6.11 – see differential equations later)	ACD has calculus sheet. PPT Integration (all methods) PPT Integration questions
3	Differentiation Implicit differentiation Differentiation of a x to give a x lna	Include examples of finding equations of tangents and normals to curves given implicitly So e x differentiates to e x lne which is equal to e x .	Heinemann C4 Ch. 4 p. 33 leave out 4.1, 4.4 and 4.5
5	Coordinate Geometry in the x-y plane Parametric equations of curves and conversion between parametric and Cartesian form. Parametric differentiation Area under a curve and volume of revolution	Candidates should be able to find the area under a curve given its parametric equations. They will not be expected to sketch a curve from its parametric equations. Include examples of finding equations of tangents and normals to curves given parametrically. π∫ y 2 dx is required but not π∫ x 2 dy.	Heinemann C4 Ch. 2 p. 9 Heinemann C4 Ch. 4.1 p. 33 Volume: Heinemann C4 Ch. 6.9 p. 106
6	Vectors Vectors in both two and three dimensions Magnitude of a vector i.e. familiarity with  a  The orthogonal unit vectors Position vectors The distance between two points Vector equations of a line The scalar product Use of the scalar product for calculating the angle between two lines	Algebraic operations of vector addition, subtraction and multiplication by a scalar, and their geometrical interpretations Including finding a unit vector in the direction of a Use of I , j , k . The definition of a vector in terms of its Cartesian components OB – OA = AB = b – a i.e. d² = (x 1 – x 2 )² + (y 1 – y 2 )² + (z 1 – z 2 )² To include the forms r = a + t b and r = c + t (d – c) Intersection, or otherwise, of two lines Candidates should know that if OA = a = a 1 i + a 2 j + a 3 k and OB = b = b 1 i + b 2 j + b 3 k then a . b = a 1 b 1 + a 2 b 2 + a 3 b 3 and cos ∠ AOB = a . b  a  b  Knowledge that if a . b = 0, and that a and b are non-zero vectors, then a and b are perpendicular	Heinemann C4 Ch. 5 p. 47	PPT Vectors
3	Binomial series The binomial expansion of (1+x) n and (ax+b) n where n is rational and  x  <1	Recap on C2 binomial where n is a positive integer Questions could include rational functions that require decompositions into partial fractions	Heinemann C4 Ch. 3 p. 21
3	Differential equations Formation of simple differential equations Analytical solution of simple first order differential equations with separable variables Exponential Growth and decay	This includes questions involving connected rates of change General and particular solutions will be required	Heinemann C4 Ch. 4 p. 38 (4.4 and 4.5) Heinemann C4 Ch. 6 p. 108 (6.10 and 6.11)