AQA A LEVEL MATHS TEACHING GUIDANCE, Study Guides, Projects, Research of Mathematics

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AS and A-level Mathematics

Teaching Guidance

AS 7356 and A-level 7357

For teaching from September 2017

For AS and A-level exams from June 2018

Version 1.0, May 2017

Our specification is published on our website (aqa.org.uk). We will let centres know in writing about

any changes to the specification. We will also publish changes on our website. The definitive version of our specification will always be the one on our website and may differ from printed versions.

You can download a copy of this teaching guidance from our All About Maths website (allaboutmaths.aqa.org.uk/). This is where you will find the most up-to-date version, as well as information on version control.

AS and A-LEVEL MATHEMATICS TEACHING GUIDANCE

General information - disclaimer

This AS and A-level Mathematics teaching guidance will help you plan your teaching by further explaining how we have interpreted content of the specification and providing examples of how the content of the specification may be assessed. The teaching guidance notes do not always cover the whole content statement.

The examples included in this guidance have been chosen to illustrate the level at which this content will be assessed. The wording and format used in this guidance do not always represent how questions would appear in a question paper. Not all questions in this guidance have been

through the same rigorous checking process as the ones used in our question papers.

Several questions have been taken from legacy specifications and therefore represent higher levels of AO1 than will be found in a suite of exam papers for this A-level Mathematics specification.

This guidance is not, in any way, intended to restrict what can be assessed in the question papers

based on the specification. Questions will be set in a variety of formats including both familiar and unfamiliar contexts.

All knowledge from the GCSE Mathematics specification is assumed.

Subject content

The subject content for AS and A-level Mathematics is set out by the Department for Education (DfE) and is common across all exam boards.

This document is designed to illustrate the detail within the content defined by the DfE.

Content in bold type is contained within the AS Mathematics qualification as well as the A-level Mathematics qualification. Content in standard type is contained only within the A-level Mathematics qualification.

AS and A-LEVEL MATHEMATICS TEACHING GUIDANCE

Only assessed at A-level

Examples

1 Assuming 2 is a rational number we can write 2 = a

b

, where a and b are positive whole

numbers with no common factors.

(a) Show that a must be even.

(b) Show that b must be even.

(c) Using parts (a) and (b), explain why there is a contradiction and state what conclusion can be made about 2 as a result.

B (^) Algebra and functions

B1 Understand and use the laws of indices for all rational

exponents.

Assessed at AS and A-level

Teaching guidance

Students should be able to:

• understand and use the following laws:

x a^ × x b^ = x a + b

x a = a x

x

a

÷ x

b

= x

a − b

( )

a (^) a x b= b^ x a^ = bx

a b ab x x

• apply these laws when solving problems in other contexts, for example simplification of expressions before integrating/differentiating, solving equations or transforming graphs.

Examples

1 (a) Write down the values of p , q and r , given that:

(i) 64 = 8 p

(ii)

= 8 q

(iii) 8 = 8 r

(b) Find the value of x for which

8

x

2 Find^2 d

 +^ + 

∫ x^ x x

B2 Use and manipulate surds, including rationalising the

denominator.

Assessed at AS and A-level

Teaching guidance

Students should be able to:

• demonstrate they understand how to manipulate surds and rationalise denominators.
• show a result.
• answer algebraic questions.

Note: many calculators will perform simplifications but students should understand that the command words ‘show that’ require detailed mathematical reasoning using precise notation.

Examples

1 Show that

is an integer and find its value.

(^2) Rationalise the denominator of the fraction 3 2

a

a

, where a is a positive integer.

3 A rectangle has length^ ( 9 +^5 3 )cm and area^ (^15 +^7 3 )cm

2

Show that the width of the rectangle is (^) ( m + n (^3) ), where m and n are integers.

4 (a) Expand^ ( )

2 x − 1

(b) Hence find^ ( ) d

2 ∫ x^ −^1 x

5 (a) Write x^5 in the form xk where k is a fraction.

(b) (^) Find ( ) d

∫ x^ x

AS and A-LEVEL MATHEMATICS TEACHING GUIDANCE

B3 Work with quadratic functions and their graphs; the

discriminant of a quadratic function, including the

conditions for real and repeated roots; completing the

square; solution of quadratic equations including solving

quadratic equations in a function of the unknown.

Assessed at AS and A-level

Teaching guidance

Students should:

• be able to sketch graphs of quadratics, ie of y = ax^2 + bx + c
• be able to identify features of the graph such as points where the graph crosses the axes, lines of symmetry or the vertex of the g raph.
• know and use the following:

b

2

− 4 ac > 0 Distinct real roots

b

2

− 4 ac = 0 Equal roots

b

2

− 4 ac < 0 No real roots

Note: a quadratic described as having real roots will be such that (^) b^2^ − 4 ac ≥ 0

• be able to complete the square and use the resulting expression to make deductions, such as the maximum/minimum value of a quadratic or the number of roots. Note: unless specified in a question, any correct method for solving a quadratic will be acceptable. Students should become familiar with using calculators to solve quadratic equations. Use of appropriate technology should be encouraged throughout the teaching of this specification. It is a requirement from the DfE that ‘the use of technology…must permeate the study of AS and A-level Mathematics’.

AS and A-LEVEL MATHEMATICS TEACHING GUIDANCE

4 (a) Using the substitution Y = 3 x , show that the equation

9 x^ − 3 x^ +^1 + 2 = 0

can be written as

( Y − 1)( Y − 2) = 0

(b) Hence show that the equation 9 x^ − 3 x +^1 + 2 = 0 has a solution

x = 0

and, by using logarithms, find the other solution, giving your answer to four decimal places.

5 Given that

3 sin^2 3cos cos 2

show that 1 cos 2

6 Solve the equation

3 2 x^ − 3 x^ +^1 − 4 = 0

giving your answer in an exact form.

B4 Solve simultaneous equations in two variables by

elimination and by substitution, including one linear and

one quadratic equation.

Assessed at AS and A-level

Teaching guidance

Students should be able to:

• understand the relationship between the algebraic solution of simultaneous equations and the points of intersection on the corresponding graphs.
• in the case of one linear and one quadratic equation, recognise the geometrical significance of the discriminant of the resulting quadratic.

Note: simultaneous equations could arise from problems set on a variety of topics including mechanics and statistics.

Examples

1 The straight line L has equation

y = 3 x − 1

The curve C has equation

y = ( x + 3)( x − 1)

(a) (^) Sketch on the same axes the line L and the curve C , showing the values of the intercepts on the

x -axis and y -axis.

(b) Show that the x -coordinates of the points of intersection L and C satisfy the equation

x^2 − x − 2 = 0

(c) Hence find the coordinates of the points of intersection of L and C.

2 The curve^ C^ has equation

y = k ( x^2 + 3)

where k is a constant.

The line L has equation

y = 2 x + 2

Show that the x -coordinates of any points of intersection of the curve C with the line L satisfy

the equation

kx^2 − 2 x + 3 k − 2 = 0

B5 Solve linear and quadratic inequalities in a single

variable and interpret such inequalities graphically,

including inequalities with brackets and fractions.

Express solutions through correct use of 'and' and 'or',

or through set notation.

Represent linear and quadratic inequalities such as

y > x + 1 and ax

+ bx + c graphically.

Assessed at AS and A-level

Teaching guidance

Students should be able to:

• give the range of values which satisfy more than one inequality.
• illustrate regions on sketched graphs, defined by inequalities.
• define algebraically inequalities that are given graphically.

Notes: Dotted/dashed lines or curves will be used to indicate strict inequalities.

Overarching theme 1.3 is of particular relevance here. Students are required to demonstrate an understanding of and use the notation (language and symbols) associated with set theory (as set out in Appendix A of the specification). Students are required to apply this notation to the solutions of inequalities.

AS and A-LEVEL MATHEMATICS TEACHING GUIDANCE

Examples

1 Find the possible values of k which satisfy the inequality 3 k^2 − 2 k − 1 < 0

2 The diagram shows the graphs of y = x + 5 and y = x^2 − 4 x + 5.

State which pair of inequalities defines the shaded region. Circle your answer.

y < x + 5

and

y < x^2 − 4 x + 5

y ≤ x + 5

or

y > x^2^ − 4 x + 5

y ≤ x + 5

and

y ≥ x^2^ − 4 x + 5

y ≥ x + 5

or

y < x^2^ − 4 x + 5

3 Find the values for x which satisfy both inequalities x^2^ + 2 x > 8 and 3 2( x + 1 ) ≤ 15.

AS and A-LEVEL MATHEMATICS TEACHING GUIDANCE

Only assessed at A-level

Teaching guidance

Students should be able to:

• understand the Factor Theorem where the divisor is of the form ( ax + b ).
• simplify rational expressions.
• carry out algebraic division where the divisor is of the form ( ax + b ).

Note: any correct method will be accepted, eg by inspection, by equating coefficients or by formal division.

Examples

1 Express^

x x x x

in the form ax^2 + bx +

c x

, where a , b and c are integers.

2 The polynomial f( x ) is defined by f( x ) = 4 x^3 − 7 x − 3

(a) Find f(−1)

(b) Use the Factor Theorem to show that 2 x + 1 is a factor of f( x )

(c) Simplify the algebraic fraction

3 2

x x

x x

B7 Understand and use graphs of functions; sketch curves

defined by simple equations including polynomials, the

modulus of a linear function, y =

a

x

and y =

a

x^2

(including their vertical and horizontal asymptotes);

interpret algebraic solution of equations graphically; use

intersection points of graphs to solve equations.

Understand and use proportional relationships and their

graphs.

Assessed at AS and A-level

Teaching guidance

Students should be able to:

• understand, use and sketch straight-line graphs (including vertical and horizontal).
• understand and use polynomials up to cubic (including sketching curves).
• understand and use cubic polynomials with at least one linear factor.
• distinguish between the various possibilities for graphs of cubic polynomials indicating where graphs meet coordinate axes.
• understand and use graphs of the functions

a

y

x

= and = 2

a

y

x

as well as simple transformations of these graphs (including sketching curves).

• use the following:

Proportionality ( ∝ ) Equation

y is proportional to x y = kx

y is proportional to x n^ y = kx n

y is inversely proportional to x n n

k

y

x