CORE MATHEMATICS 3 SCHEME OF WORK 2005-2006:

C3 UNIT: ≈ 40 Lessons (the course should be completed by October half-term and it is likely the exam will be taken in January)

The texts used are Heinemann P2 and Heinemann P3 (new edition).  Other sources referenced include Heinemann C3 (staff copy), Introducing Pure Mathematics (Smedley & Wiseman) (MAC & JLW have old edition, 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.  Encourage pupils to use the likes of the website “AS Guru”.

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 version for 2005 onwards).

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.

It is assumed that pupils have a good knowledge of C1 &C2 material and their formulae.  Exam is 1½hr, approx 7 questions, with (QCA) calculators permitted.

Time

Specification

Notes

Page and Exercise

Other Sources
 

N.B. Background knowledge for the course is direct proof, proof by contradiction and disproof by counterexample.

Teach separately for a couple of lessons and/or try and include starters etc. in all topic areas where proof can readily be used.

    

2

• Algebra:

Simplification of rational expressions, including factorising and cancelling and algebraic division

Include +, -, x and ÷ of algebraic fractions

Denominators of rational expressions will be linear or quadratic e.g. 1/(ax + b); (ax + b)/(px² + qx +r);

(x³ + 1)/(x² – 1)

Heinemann C3 Ch. 1 p. 1

  

5

• Functions:

Definition of a function and the domain and range of a function.  

Composite functions

Inverse functions: if f -1 exists, then

f -1 f(x) = ff -1 (x) = x

A one-one or many-one mapping from (a subset of) R to R .  The notation f:x → … and f(x) will be used

fg will mean ‘do g first, then f’

Represent graphically too (reflection in y=x)

Heinemann C3 Ch.2 p. 11

  

3

• Exponentials and logarithms:

The function e x and its graph

The function lnx and its graph

Appreciation of lnx as the inverse of e x

Differentiation of ln x and e x .

To include y=e ax+b +c

Solution of equations of the form e ax+b +c=p and ln(ax+b)=q is expected

Heinemann C3 Ch. 3 p. 29

Heinemann C3 Ch. 8 p. 125, 127

PPT logs (true or false)

6

• Trigonometry: (use of degrees & radians)

Secant, cosecant and cotangent: their relation to sin, cos and tan and their graphs

Arcsin, arcos and arc tan: their relation to sin, cos and tan and their graphs

Trig. Identities and their use in solving equations and proving further identities:

Knowledge and use of 1 + tan 2 θ = sec 2 θ

                                     1 + cot 2 θ = cosec 2 θ

                                     cot θ = cos θ /sin θ

Compound Angle Formulae

Double & Half Angle Formulae

Harmonic form: expressions for acos θ + bsin θ in the form rcos( θ ± α ) or rsin( θ±α )

Restrict domains appropriately

Waldo “Trigonometric Functions” gives graphs

Restrict domains appropriately

Waldo “Trigonometric Functions” gives graphs

Use of all identities in this section as stated and in reverse to solve equations and prove identities e.g. Solve acosθ + bsinθ = c or cosx + cos3x =cos2x in a given interval; Prove cosxcos2x + sinxsin2x ≡ cosx

← Learn

← Learn

i.e. sin(A ± B), cos(A ± B) and tan(A ± B)

Learn sin2A ≡ 2sinAcosA; cos2A ≡ cos²A - sin²A;

Tan2A ≡ 2tanA/(1 - tan²A).  NB knowledge of the t(tan½ θ ) formula will not be required

Heinemann C3 Ch. 6 p. 73

Heinemann C3 Ch. 7 p. 95

PPT trig identities

GSP trig identities

ACD has trig identities sheet

3

• Transformations:

Modulus functions

Combinations of transformations of y = f(x)

Definition and notation.  The ability to sketch

 y = ax+b , y = f(x) and y = f (x) given the graph of y = f(x).  See Waldo “Modulus Functions”

Including y = af(x), y = f(x) + a, y = f(x + a), y = f(ax) and combinations of these transformations, but NOT y=f(ax+b).

e.g. sketch y=2f(3x), y=f(-x)+1 OR y=3+sin2x,

y=-cos(x+ π /4)

Waldo “Polynomial transformations” may be useful.

Heinemann C3 Ch. 5 p. 54

  

5

• Differentiation:

Differentiation of sinx, cosx, tanx and their sums and differences

Differentiation using the product rule, the quotient rule and the chain rule

The use of dy/dx = 1/(dx/dy)

Learn sinkx ⇒ kcoskx, coskx ⇒ -ksinkx

Learn f(x)+g(x) ⇒ f’(x)+g’(x), f(x)g(x) ⇒ f’(x)g(x)+f(x)g’(x) and f(g(x)) ⇒ f’(g(x))g’(x)

Differentiation of cosecx, cotx and secx are required.  Skill will be expected in the differentiation of functions generated from standard forms using products, quotients and composition, such as 2x 4 sinx, e 3x /x, cosx² and tan²2x.

E.G. For finding dy/dx for x=sin3y

Heinemann C3 Ch. 8 p. 120

PPT Chain rule

PPT Chain rule by inspection

PPT product and quotient rule

2

Numerical Methods:

Location of roots of f(x)=0 by considering changes of sign of f(x) in an interval of x in which f(x) is continuous

Approximate solutions of equations using simple iterative methods, including recurrence relations of the form x n+1 =f(x n )

Solution of equations by use of iterative procedures for which leads will be given

Heinemann C3 Ch.4 p. 42

PPT Numerical solutions

For information: movement from year 12 to year 13:

N/A

Omissions from work previously taught in year 13:

  1. Solving equations graphically (now C1)

  1. Remainder theorem (now in C2)

  1. The circle (now in C2)

  1. Binomial (now in C2)

  1. Logs and equations of the form a x =b (now in C2)

  1. Trapezium rule (now in C2)