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They include: • route maps to allow you to plan how to deliver the specification in the way that will best suit youand your students • teaching guidance to outline clearly the possible s

MATHEMATICS

DRAFT 7366

Specification

For teaching from September 2017 onwards

For AS exams in 2018 onwards

Version 0.1 9 June 2016

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DRAFT SPECIFICA

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1.1 Why choose AQA for AS Further Mathematics 5

1.2 Support and resources to help you teach 5

3.3 Optional application 1 – mechanics 16

3.4 Optional application 2 – statistics 17

3.5 Optional application 3 – discrete 19

5.2 Overlaps with other qualifications 27

5.3 Awarding grades and reporting results 27

5.5 Previous learning and prerequisites 28

5.6 Access to assessment: diversity and inclusion 28

5.7 Working with AQA for the first time 28

Are you using the latest version of this specification?

• You will always find the most up-to-date version of this specification on our website at

1 Introduction

1.1 Why choose AQA for AS Further Mathematics

Maths is essential for many higher education courses and careers We’ve worked closely with

higher education to ensure this qualification gives your students the best possible chance to

progress and realise their potential

A specification with freedom – assessment design that rewards

understanding

We want students to see the links between different areas of maths and to apply their maths skills

across all areas That’s why our assessment structure gives you the freedom to teach further

maths your way

Consistent assessments are essential, which is why we’ve worked hard to ensure our papers are

clear and reward your students for their mathematical skills and knowledge

You can find out about all our Further Mathematics qualifications at aqa.org.uk/maths

1.2 Support and resources to help you teach

We’ve worked with experienced teachers to provide you with a range of resources that will help

you confidently plan, teach and prepare for exams

Teaching resources

Visit aqa.org.uk/7366 to see all our teaching resources They include:

• route maps to allow you to plan how to deliver the specification in the way that will best suit youand your students

• teaching guidance to outline clearly the possible scope of teaching and learning

• lesson plans and homework sheets tailored to this specification

• tests and assessments that will allow you to measure the development of your students as theywork through the content

• textbooks that are approved by AQA

• training courses to help you deliver AQA Mathematics qualifications

• subject expertise courses for all teachers, from newly-qualified teachers who are just getting

started, to experienced teachers looking for fresh inspiration

• training courses to help you deliver AQA Further Mathematics qualifications

• subject expertise courses for all teachers, from newly qualified teachers who are just getting

started to experienced teachers looking for fresh inspiration

Preparing for exams

Visit aqa.org.uk/7366 for everything you need to prepare for our exams, including:

• past papers, mark schemes and examiners’ reports

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• specimen papers and mark schemes for new courses

• Exampro: a searchable bank of past AQA exam questions

• example student answers with examiner commentaries

Analyse your students' results with Enhanced Results Analysis (ERA)

Find out which questions were the most challenging, how the results compare to previous years

and where your students need to improve ERA, our free online results analysis tool, will help you

see where to focus your teaching Register at aqa.org.uk/era

For information about results, including maintaining standards over time, grade boundaries and our

post-results services, visit aqa.org.uk/results

Keep your skills up-to-date with professional development

Wherever you are in your career, there’s always something new to learn As well as subject

specific training, we offer a range of courses to help boost your skills

• Improve your teaching skills in areas including differentiation, teaching literacy and meeting

Ofsted requirements

• Prepare for a new role with our leadership and management courses

You can attend a course at venues around the country, in your school or online – whatever suits

your needs and availability Find out more at coursesandevents.aqa.org.uk

Help and support

Visit our website for information, guidance, support and resources at aqa.org.uk/7366

If you'd like us to share news and information about this qualification, sign up for emails and

This draft qualification has not yet been accredited by Ofqual It is published to enable teachers to

have early sight of our proposed approach to AS Further Mathematics Further changes may be

required and no assurance can be given that this proposed qualification will be made available in

its current form, or that it will be accredited in time for first teaching in September 2017 and first

award in August 2018

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All students must study this content.

• Compulsory content (page 12)

• OT1: Mathematical argument, language and proof (page 11)

• OT2: Mathematical problem solving (page 11)

• OT3: Mathematical modelling (page 12)

Options

Students must study two of these options

• Optional application 1 – mechanics (page 16)

• Optional application 2 – statistics (page 17)

• Optional application 3 – discrete (page 19)

How it's assessed

• Written exam: 1 hour 30 minutes

Paper 2

What's assessed

One question paper answer booklet on Discrete and One question paper answer booklet on

Statistics

How it's assessed

• Written exam: 1 hour 30 minutes

How it's assessed

• Written exam: 1 hour 30 minutes

Paper 2

What's assessed

One question paper answer booklet on Mechanics and one question paper answer bookelt on

Discrete

How it's assessed

• Written exam: 1 hour 30 minutes

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3 Subject content

The subject content in sections A to L is compulsory for all students Students must study two of

the optional applications The optional applications are mechanics (MA1 to MA6), statistics (SA1 toSG1) and discrete (DA1 to DG9)

3.1 Overarching themes

A-level specifications in mathematics must require students to demonstrate the overarching

knowledge and skills contained in sections OT1, OT2 and OT3 These must be applied, along with

associated mathematical thinking and understanding, across the whole of the detailed content set

OT1.1 Construct and present mathematical arguments through appropriate use of

diagrams; sketching graphs; logical deduction; precise statements involvingcorrect use of symbols and connecting language, including: constant, coefficient,expression, equation, function, identity, index, term, variable

OT1.2 Understand and use mathematical language and syntax as set out in the content

OT1.3 Understand and use language and symbols associated with set theory, as set out

in the content

OT1.5 Comprehend and critique mathematical arguments, proofs and justifications of

methods and formulae, including those relating to applications of mathematics

3.1.2 OT2: Mathematical problem solving

Content

OT2.1 Recognise the underlying mathematical structure in a situation and simplify and

abstract appropriately to enable problems to be solved

OT2.2 Construct extended arguments to solve problems presented in an unstructured

form, including problems in context

OT2.3 Interpret and communicate solutions in the context of the original problem

OT2.6 Understand the concept of a mathematical problem solving cycle, including

specifying the problem, collecting information, processing and representinginformation and interpreting results, which may identify the need to repeat thecycle

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OT2.7 Understand, interpret and extract information from diagrams and construct

mathematical diagrams to solve problems, including in mechanics

3.1.3 OT3: Mathematical modelling

Content

OT3.1 Translate a situation in context into a mathematical model, making simplifying

assumptions

OT3.2 Use a mathematical model with suitable inputs to engage with and explore

situations (for a given model or a model constructed or selected by the student)

OT3.3 Interpret the outputs of a mathematical model in the context of the original

situation (for a given model or a model constructed or selected by the student)

OT3.4 Understand that a mathematical model can be refined by considering its outputs

and simplifying assumptions; evaluate whether the model is appropriate

OT3.5 Understand and use modelling assumptions

3.2 Compulsory content

3.2.1 B: Complex numbers

Content

B1 Solve any quadratic equation with real coefficients; solve cubic or quartic

equations with real coefficients given sufficient information to deduce at least oneroot for cubics or at least one complex root or quadratic factor for quartics

Content

B2 Add, subtract, multiply and divide complex numbers in the form x + iy with x and y

real; understand and use the terms ‘real part’ and ‘imaginary part’

Content

B3 Understand and use the complex conjugate; know that non-real roots of

polynomial equations with real coefficients occur in conjugate pairs

Knowledge/skill

B4 Use and interpret argand diagrams

Content

B5 Convert between the Cartesian form and the modulus-argument form of a

complex number (knowledge of radians is assumed)

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B6 Multiply and divide complex numbers in modulus-argument form (knowledge of

radians and compound angle formulae is assumed)

Content

B7 Construct and interpret simple loci in the argand diagram such as |z-a| > r and arg

( z − a ) = θ (knowledge of radians is assumed)

C3 Use matrices to represent linear transformations in 2D; successive

transformations; single transformations in 3D (3D transformations confined toreflection in one of x = 0, y = 0, z = 0 or rotation about one of the coordinate axes)(knowledge of 3D vectors is assumed)

• Calculate and use the inverse of non-singular 2 x 2 matrices

3.2.3 D: Further algebra and functions

Content

D1 Understand and use the relationship between roots and coefficients of polynomial

equations up to quartic equations

Content

D2 Form a polynomial equation whose roots are a linear transformation of the roots of

a given polynomial equation of at least cubic degree

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D3 Understand and use formulae for the sums of integers, squares and cubes and

use these to sum other series

Content

D4 Understand and use the method of differences for summation of series

Content

D6 Recognise and use the Maclaurin series for ex , ln 1 +x , sinx , cosx , and

1 +xn , and be aware of the range of values of x for which they are valid (proofnot required)

D13 Graphs of rational functions of form ax+b

cx+d ; asymptotes, points of intersection withcoordinate axes or other straight lines; associated inequalities

Content

D14 Graphs of rational functions of form ax2 +bx+c

dx2 +ex+f

Content

D15 Using quadratic theory (not calculus) to find the possible values of the function

and coordinates of the stationary points of the graph for rational functions of form

F5 • Calculate and understand the properties of the vector product

• Understand and use the equation of a straight line in the form (r – a) × b = 0

H1 Understand the definitions of hyperbolic functions sinh x, cosh x and tanh x,

including their domains and ranges, and be able to sketch their graphs

3.2.8 L: Coordinate geometry

Content

L1 Simple loci

3.3 Optional application 1 – mechanics

3.3.1 MA: Dimensional analysis

Content

MA1 Finding dimensions of quantities

Content

MA2 Prediction of formulae

3.3.2 MB: Momentum and collisions

Content

MB1 Conservation of momentum for linear motion and cases where velocities are given

as simple one or two dimensional vectors (no resolving of forces)

Content

MB2 Coefficient of Restitution and Newton’s Experimental Law Use in direct collisions

and impacts with a fixed smooth surface

Content

MB3 Impulse and its relation to momentum (in one- or two-dimensions) (no resolving of

forces) Use of Ft =mv−mu

Content

MB4 Impulse for variable forces One dimension only Use of I =∫Fdt

3.3.3 MC: Work, energy and power

MC6 Elastic Potential Energy using modulus of elasticity Use of EPE= kx22 and

EPE= λx2l2 Use in conservation of energy problems

3.4 Optional application 2 – statistics

3.4.1 SA: Discrete random variables (DRV) and expectation

Content

SA1 Understand probability tables and distributions defined by a probability density

function (pdf) with finite possibilities

Content

SA2 Evaluate probabilities from a DRV

Content

SA3 Evaluate measures of average and spread for a DRV to include mean, variance,

standard deviation, mode or median

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SA4 Understand expectation and know the formulae: E(X) = Σxipi; E(X²) = Σxi²pi; Var(X)

= E(X²) -(E(X))²

Content

SA5 Understand expectation of simple linear functions of DRVs and know the

formulae: E(aX+b) = aE(X)+b and Var (aX+b)= a² Var (X)

Content

SA6 Know discrete uniform distribution and its pdf Understand when the discrete

uniform distribution can be used as a model

SB2 Know Poisson formula and calculate Poisson probabilities using the formula,

tables or equivalent calculator function

Content

SB3 Know mean, variance and standard deviation of a Poisson distribution Use the

result that, if X ~ Po(λ) then the mean and variance of X are equal

Content

SB4 Understand the distribution of the sum of independent Poisson distributions

Content

SB5 Formulate hypotheses and carry out a hypothesis test of a population mean from

a single observation from a Poisson distribution using direct evaluation of Poissonprobabilities

3.4.3 SC: Type I and Type II errors

Content

SC1 Understand Type I and Type II errors and define in context Calculate probability

of making Type I error from tests based on a Poisson distribution

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3.4.4 SD: Continuous random variables (CRV)

Content

SD1 Understand and use a probability density function, f(x), for a continuous

distribution and understand the differences between discrete and continuousdistributions

SD4 Find mean, variance and standard deviation for given CRV function, f(x) Know

the formulae E(X)= ∫xf(x)dx, E(X2)= ∫x² f(x)dx, Var(X)=E(X² )-(E(X) )²

Content

SD5 Understand expectation of simple linear functions of CRVs and know the formulae

E(aX+b)=aE(X)+b and Var (aX+b)= a² Var (X) ]

3.4.5 SE: Chi tests for association

SE4 Identification of sources of association in the context of a question

3.5 Optional application 3 – discrete

DA4 Planar graphs, Euler’s formula V–E+F=2, Kuratowski’s Theorem, complete

graphs, the notation Kn bipartite graphs, the notation Km,n

DB3 Travelling salesperson problem, upper bounds and the nearest neighbour method,

upper and lower bounds by use of minimum spanning trees

3.5.4 DD: Linear programming

Content

DD1 Formulation of constrained optimisation problems

Content

DD2 Graphical solution of two-variable problems, including those with integer solutions

3.5.5 DE: Critical path analysis

DF1 Pay-off matrix, play-safe strategies, stable solutions, dominance and pay-off

matrix reduction, saddle points, value of the game

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4 Scheme of assessment

Find past papers and mark schemes, and specimen papers for new courses, on our website at

aqa.org.uk/pastpapers

This specification is designed to be taken over one or two years

This is a linear qualification In order to achieve the award, students must complete all

assessments at the end of the course and in the same series

AS exams and certification for this specification are available for the first time in May/June 2018

and then every May/June for the life of the specification

All materials are available in English only

Our AS exams in Further Mathematics include questions that allow students to demonstrate their

ability to:

• recall information

• draw together information from different areas of the specification

• apply their knowledge and understanding in practical and theoretical contexts

4.1 Aims

Courses based on this specification should encourage students to:

• understand mathematics and mathematical processes in ways that promote confidence, foster

enjoyment and provide a strong foundation for progress to further study

• extend their range of mathematical skills and techniques

• understand coherence and progression in mathematics and how different areas of mathematicsare connected

• apply mathematics in other fields of study and be aware of the relevance of mathematics to the

world of work and to situations in society in general

• use their mathematical knowledge to make logical and reasoned decisions in solving problems

both within pure mathematics and in a variety of contexts, and communicate the mathematical

rationale for these decisions clearly

• reason logically and recognise incorrect reasoning

• generalise mathematically

• construct mathematical proofs

• use their mathematical skills and techniques to solve challenging problems which require them

to decide on the solution strategy

• recognise when mathematics can be used to analyse and solve a problem in context

• represent situations mathematically and understand the relationship between problems in

context and mathematical models that may be applied to solve them

• draw diagrams and sketch graphs to help explore mathematical situations and interpret

solutions

• make deductions and inferences and draw conclusions by using mathematical reasoning

• interpret solutions and communicate their interpretation effectively in the context of the problem

• read and comprehend mathematical arguments, including justifications of methods and

formulae, and communicate their understanding

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