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Advanced Subsidiary, 2013– Use of Mathematics and UOM4 Qualification Code 5351 3.2 Sequencing of units There is no pre-determined dependency of units, although in the case of the Applyin

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Version 1.0: Date 0511

General Certificate of Education

Use of Mathematics 5351

2013

Material accompanying this Specification

• Specimen and Past Papers and Mark Schemes

• Reports on the Examination

• A Teachers’ Guide

SPECIFICATION

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This specification will be published annually on the AQA Website (www.aqa.org.uk) If there are any changes to the specification centres will be notified in print as well as on the Website The version on the Website is the definitive version of the specification

Further copies of this specification booklet are available from:

AQA Logistics Centre, Unit 2, Wheel Forge Way, Ashburton Park, Trafford Park, Manchester, M17 1EH

Telephone: 0870 410 1036 Fax: 0161 953 1177

or

can be downloaded from the AQA Website: www.aqa.org.uk

COPYRIGHT

AQA retains the copyright on all its publications However, registered centres for AQA are permitted to copy material from this booklet for their own internal use, with the following important exception: AQA cannot give permission to centres to photocopy any material that is acknowledged to a third party even for internal use within the centre

Set and published by the Assessment and Qualifications Alliance

The Assessment and Qualifications Alliance (AQA) is a company limited by guarantee registered in England and Wales 3644723 and a registered charity number 1073334 Registered address AQA, Devas Street, Manchester, M15 6EX

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Advanced Subsidiary, 2013– Use of Mathematics

Subject Content

9 Content and Assessment – Applying Mathematics 13

Key Skills and Other Issues

11 Spiritual, Moral, Ethical, Social, Cultural and Other Issues 46

Awarding and Reporting

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Use of Mathematics – Advanced Subsidiary, 2013

Appendices

A FSMQ Using and Applying Statistics 6990 49

B FSMQ Working with Algebraic and Graphical

Techniques 6991 87

D FSMQ Using and Applying Decision Mathematics 6994 158

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Background Information

Free-Standing Mathematics Qualifications

1.1 Introduction to this

Qualification

It is the aim of each of the ten Free-Standing Mathematics Qualifications to encourage candidates to apply mathematical principles in their studies, work or interests

The new Advanced Subsidiary qualification in Use of Mathematics

has been developed to make a significant contribution towards the promotion of mathematical study beyond 16, especially among

“non-mathematicians” However, the qualification is suitable for and may be used by both pre-16 and post-16 candidates

It will prove particularly useful to those working in the areas of:

• Technology, including engineering

Mathematics Full specifications for these FSMQs can be found in the

Appendices of this document Applying Mathematics does not exist as a

separate Free-Standing Mathematics Qualification

Candidates embarking on AS level study in mathematics subjects are expected to have achieved at least Grade C in GCSE Mathematics or equivalent, and to have covered all the material in the Intermediate Tier In addition, candidates will be expected to be able to use the material listed in section 4 whenever it is required

Advanced Subsidiary is designed to provide an appropriate assessment of knowledge, understanding and skills expected of candidates who have completed the first half of a full Advanced Level qualification The level of demand of the AS examination is that expected of candidates half-way through a full A Level course of study

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Use of Mathematics 5351

at Advanced Subsidiary Level

This Advanced Subsidiary (AS) award comprises three assessment units of equal weighting

AS Use of Mathematics 5351

FSMQ Working with Algebraic and Graphical Techniques 6991

moderated by AQA written paper 1 hour 50% of assessment short and extended 30 minutes for this unit answer questions

coursework portfolio assessed by centre and moderated

by AQA

50% of assessment for this unit

coursework portfolio assessed by centre and moderated by AQA

50% of assessment for this unit

written paper short and extended answer questions

1 hour

30 minutes 50% of assessment for this unit

written paper short and extended answer questions

1 hour

30 minutes 50% of assessment for this unit

written paper 1 hour 30% of assessment

written paper 1 hour 70% of assessment short and extended 30 minutes for this unit

answer questions

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and UOM4

Qualification Code

5351

3.2 Sequencing of units There is no pre-determined dependency of units, although in the case

of the Applying Mathematics unit, candidates will build on the

mathematical knowledge, skills and understanding they develop in

working towards FSMQ Working with Algebraic and Graphical Techniques

3.3 Entry Codes The following unit entry codes should be used:

FSMQ Working with Algebraic and Graphical Techniques FSMQ Using and Applying Statistics

FSMQ Modelling with Calculus FSMQ Using and Applying Decision Mathematics

The Subject Code for entry to the Use of Mathematics AS award is

5351

3.4 Classification Codes Every specification is assigned to a national classification code

indicating the subject area to which it belongs

Centres should be aware that candidates who enter for more than one GCE qualification with the same classification code will have only one grade (the highest) counted for the purpose of the School and College Performance Tables

The classification code for this specification is 2200

3.5 Private Candidates This specification is not available to private candidates

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3.6 Access Arrangements and

Special Consideration

We have taken note of equality and discrimination legislation and the interests of minority groups in developing and administering this specification

We follow the guidelines in the Joint Council for Qualifications (JCQ)

document: Access Arrangements, Reasonable Adjustments and Special

Consideration: General and Vocational Qualifications This is published on

the JCQ Website (http://www.jcq.org.uk) or you can follow the link from our website (http://www.aqa.org.uk)

3.7 Language of Examination Assessments in this subject are provided in English only

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9

Scheme of Assessment

Mathematics

Before you start this unit Candidates must be able to: This includes:

plot by hand accurate graphs of data pairs and linear and simple quadratic functions in all four quadrants

quadratics of the type

c bx ax

y= 2 + +

recognise and predict the general shapes of graphs of direct proportion, other linear and quadratic functions

quadratics of the type y=kx2 +c

find linear functions to model data pairs calculating gradients and intercepts of linear graphs rearrange basic algebraic

use index notation indices which are positive and

negative, integers and fractions solve quadratic equations • factorising

• using the formula

a

ac b b x

2 4

2 −

±

−

=

understand basic ideas of

between theoretical probabilities and those determined by survey or experimentation

• understanding that if the probability of an event A

occurring is p( )A then the probability of it not occurring

is 1 − p( )A

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A course based on this specification aims to promote:

• application of mathematical principles

• the application of mathematics to candidates’ studies, work or interests

• the development of a mathematics curriculum that is integrated with other areas of candidates’ studies, work or interests

• development of skills that enable candidates to communicate, use and interpret their mathematics

• the solution of substantial and realistic problems encountered by adults

• the ability to solve open-ended problems

• the development of mathematical modelling skills

• the development of mathematical reasoning skills

• appropriate use of ICT

• enjoyment of mathematics and the development of confidence in using mathematics

The Applying Mathematics unit will assess a candidate’s ability to apply

mathematical principles to analyse and make sense of situations, to solve problems and to draw conclusions Candidates should be able

to develop and use mathematics as a model of reality and have an awareness of any limitations this may introduce into their analysis of a situation

In particular the assessment components will concentrate on:

• the processes involved when mathematics is used to solve problems

• developing clarity in the communication of mathematics

• some new mathematical techniques associated with simulating random events and using recurrence relations

• reading and making sense of the mathematics of other people

In their studies of this unit candidates will build on the mathematical knowledge, skills and understanding they develop in working towards

FSMQ Working with Algebraic and Graphical Techniques The two units

can be studied alongside each other

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The Advanced Subsidiary (AS) award comprises three assessment units of equal weighting

6991

FSMQ Working with Algebraic and Graphical Techniques

The written paper comprises short and extended answer questions All questions are compulsory A graphics calculator may be used The coursework portfolio is assessed

by the centre and moderated by AQA

EITHER

6990

FSMQ Using and Applying Statistics

OR

6992

FSMQ Modelling with Calculus

OR

6994

Written Paper 60 marks and coursework

1 hour 30 minutes

FSMQ Using and Applying Decision Mathematics

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4

Applying Mathematics

8.1 Introduction Applying Mathematics is assessed through two written examinations

(externally set and marked): a mathematical comprehension paper lasting one hour; and a further paper lasting one and a half hours

The grading of the Applying Mathematics unit will be reported on the

uniform mark scale A candidate’s uniform mark is calculated from his/her raw mark for the unit by using the grade boundaries set by the awarding committee

8.2 External Assessment The comprehension paper will be one hour in duration It

encourages candidates to communicate their results, working and reasoning clearly by allowing time for them to present their solutions with care

The questions will be based in contexts using data from a Data Sheet that will be made available to candidates up to 14 days, and at least 7 days, prior to the examination Candidates will be expected to familiarise themselves with the contexts outlined in the Data Sheet and should discuss these with others, including teachers

The Data Sheet issued in advance of the examination must not be taken into the examination A clean copy of this will be supplied with the question paper Candidates will be required to answer all

questions

The second written paper will be one and a half hours in duration It will include questions that test the mathematical principles set out in section 9 The accuracy of the candidates’ work will be credited in the usual way and, in addition, marks will be awarded for use of correct notation and clarity of mathematical argument

The use of a graphics calculator will be expected

Candidates will be expected to remember all appropriate formulae as

no formulae sheet will be provided

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Subject Content

Applying Mathematics

9.1 About this unit

Guidance for Students

In your study of this unit you will:

• extend your use of mathematics in the areas of algebra, functions and graphs when modelling different situations

• learn some new mathematics to develop models of situations using simulations and recurrence relations

• learn to think critically about how you and others use mathematics including developing the ability to comprehend short

mathematical texts

You may study this unit alongside the other two units of the qualification This will prove particularly useful when you are developing and extending algebraic and graphical techniques as well as when you are exploring the processes involved in applying

mathematical models to analyse different situations

You should refer to the sample assessment material for exemplification of the ways in which you might be expected to apply the knowledge, skills and understanding you will develop Your teacher has further examples in Guidance for Teachers, section 9.4 of this document

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9.2 What you need to Learn This unit emphasises ‘process’ skills – the skills you need to develop

to understand mathematical models of real situations

In your studies for this unit you should use a graphics calculator - you will be expected to have access to such a calculator in examinations Graph plotting facilities on a computer may be used as an alternative, particularly if you want a printout of graphical work

You should learn to use a

plot graphs of data pairsplot graphs of functionsuse function facilities ex , sin x, etc

trace graphs finding intersections of functions

with other functions and axes use zoom facilities (if possible) finding significant features of

functions such as turning points

Whenever you use a calculator you should record your working as well as the result You should be able to sketch graphs you have plotted with your graphics calculator showing clearly all significant features such as intercepts with axes, turning points and asymptotes.

9.3 Learning Objectives The six Learning Objectives define the skills, techniques and

understanding that you need to develop during your study of this unit You will learn to:

LO1 understand how mathematics can be used to model different

situations

LO2 use simulation techniques to develop mathematical models for

random events

LO3 extend, develop and use a range of algebraic, graphical and

numerical techniques when modelling situations

LO4 appreciate that general mathematical principles may be

applicable in a range of different contexts

LO5 make sense of mathematics LO6 work accurately, structure mathematical arguments carefully and

communicate mathematics clearly

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LO1 Understanding how

mathematics can be used to

model different situations

Throughout the course leading to this qualification you will use mathematical models of a range of real life contexts It is important that you appreciate how these models relate to the situation and are aware of their limitations

Note that the ‘Links’ column indicates the appropriate sub-section

of the coursework portfolio requirements for FSMQ Working with

Algebraic and Graphical Techniques

LO1a appreciate the main

stages involved in developing a mathematical model of

a real situation

you may find a diagram useful in assisting your understanding of the stages of the process

for example:

LO1b understand that

simplifying assumptions will be made when a mathematical model is developed and that these may introduce limitations into the usefulness of the model

for example:

when modelling the growth of savings in a bank account a simplifying assumption may be that interest is added twice a year In fact interest is often calculated daily, which can have the effect of increasing or decreasing the amount

of interest added to the account over

a period

1a

* References 1a, 1b, etc refer to the appropriate coursework portfolio requirement of

the FSMQ Working with Algebraic and Graphical Techniques.

Define problem

observe

validate

Analyse Set up a model

Interpret Predict

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LO1c be able to interpret the

main features of mathematical models

in terms of the real situations which they model and use these models to predict what will happen in situations for which you have no data

for example:

• when using direct proportion to model the relationship between two variables understand that when one variable is zero so is the other

• when using a function to model

a situation you may use it to predict long term behaviour - perhaps by using linear extrapolation

1a, b

of mathematical models

for example:

by considering the predicted behaviour in extreme (t→ ∞) or simple (t= 0) cases

1a

general mathematical model allows you to solve a variety of related problems

for example:

the model a=B×c t gives the rate

of radioactive decay of all radioactive substances where the coefficients B and c depend on the substance

1a

LO2 Using simulation techniques

to develop mathematical models

for random events

Mathematics can be used to model many different and varied

situations In FSMQ Working with Algebraic and Graphical Techniques,

algebra, functions and graphs were used to develop models Here you will learn new techniques so that you can model situations involving random events such as people arriving at a shop or cars leaving a motorway junction You will then be able to explore the situations and answer questions such as “How many payment points should a shop have?” or “How long will a queue of vehicles be at traffic lights?”

Whilst working with these new modelling tools you can continue to work towards other learning objectives of the unit, particularly LO1

- understanding how mathematics can be used to model different situations

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meant by

- random event

- discrete probability distribution LO2b be able to use tables

and graphics calculators to find random numbers

you should be aware that graphics calculators generate pseudo- random numbers (Graphics calculators use algorithms to generate random numbers and a particular model may start with the same ‘seed’ when you switch on

unless you change this.)

LO2c be able to use

random numbers, together with a probability distribution to simulate discrete random events

for example:

if the probability that a train is:

on time is 0.1, 0-5 minutes late is 0.2, 5-10 minutes late is 0.2, 10-30 minutes late is 0.4, over 30 minutes late is 0.1, you can assign the random numbers 0-9 generated by your graphics calculator to these events as follows:

0: on time

1, 2: 0-5 minutes late

3, 4: 5-10 minutes late

5, 6, 7, 8: 10-30 minutes late 9: over 30 minutes late you should be able to build a table for such situations that gives the status of significant factors at specific instants using your simulation

LO2d be able to interpret

models that involve the simulation of discrete random events

in particular be aware of limitations due to simplifying assumptions and simulations of small numbers of occurrences

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LO3 Extending, developing and

using a range of numerical,

algebraic and graphical

techniques when modelling

situations

You should learn how developing algebraic, graphical and numerical understanding of mathematics can assist you with your analysis of situations and your communication of mathematical ideas

You will learn some new algebraic, graphical and numerical

techniques building on those used in FSMQ Working with Algebraic

and Graphical Techniques

• appreciate when

algebraic, graphical

or numerical methods (or combinations of these) are most appropriate, when they are

inappropriate and when possibly unsound

• establish links between these different ways of understanding mathematical ideas

t 3 − t +1=0

you will not be able to solve this algebraically and should be aware that any solution of this equation found using a graphics calculator is not exact

• you should avoid looking for solutions to equations by

substituting numerical values

• substitution is a valid method of checking the validity of a solution

1a, c

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LO3b • find solutions to

equations

• show that particular

solutions to equations are valid

• know when each of these methods is appropriate

you should be able to:

• use algebraic methods to

find solutions

• substitute numerical values

into expressions to show or verify that solutions are

• two linear equations

• one linear and one quadratic equation

you should:

• be aware that in general a

system of n equations is needed to find n unknowns

• have a graphical understanding of when systems of equations have:

- one or more solutions

- no unique solution

- no solution

• understand that every equation in a system of equations is satisfied simultaneously by each solution

• substitute solutions you find into all equations to check their validity

1c

LO3d be able to solve linear

inequalities you should be able to: • substitute numerical values

into linear expressions to determine whether or not inequalities hold

• solve equations algebraically

to assist in the solution of inequalities

• present your solutions graphically:

- using dashed lines to indicate boundaries not included

- using full lines to indicate boundaries included

- using shading to indicate regions not included

1c

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LO3e use algebraic and graphical

techniques to investigate continuous models in a wide variety of contexts such as population modelling, drug absorption and traffic flow

This includes the use of any of the functions specified in the

FSMQ Working with Algebraic and

Graphical Techniques

1a,b,2

LO3f investigate discrete models

using recurrence relations in applications such as

population growth and investment

This includes:

• having an understanding of key differences between discrete and continuous models

• using subscript notation, e.g u n

• understanding that to find a sequence of values you need

an initial value, x0, and a relation between x n+1 and the previous term x n

• understanding the difference between recurrence relations such as a n+1 =ka n +b and closed forms such as

kn a

If you are to appreciate the power of mathematics as a tool that can be used

to analyse a wide range of different situations, you need to be able to extract and appreciate general mathematical principles that underpin the work that you have been doing

particular ideas in mathematics across a range

of situations or contexts

you should aim to be able to transfer your use of mathematics into your work in other areas,

for example you should be:

• familiar with using a range of different notations for common mathematical models (e.g s=t2 + 2t+ 1 ,

) 1 ( , ) 1 ( ) (

g c = c+ 2 N =m m+

are all quadratic functions and have the usual properties of

such functions)

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• able to use algebraic and graphical models of direct proportion in different situations such as problems involving motion at constant speed (uniform motion) and those involving scaling lengths

in plans and maps and be able

to recognise the principles that underpin and are applicable in both situations

• able to identify the use of quadratic models in a range of situations using different notations such as those above and then be able to use mathematical methods to assist your analysis of these situations

L04b understand when

mathematical methods will lead to solutions

for example, when you have three

unknowns that you want to find you need to have a system of three equations involving these

unknowns

general features of graphs

of functions, including intercepts with axes and asymptotes

• the x-coordinates of the points

of intersection of the graphs

( )x

y= f and y= g( )x are solutions of the equation

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• the nature of discontinuities

in functions of the form

( )x =k x

f , ( )x x k a

−

= g

• limiting values of functions

LO4d develop your understanding

of how geometric transformations can be applied to the graphs of basic functions and be able

to use these when working with graphs that model real situations

including:

• the transformations of graphs

of y= f( )x as defined in

FSMQ Working with Algebraic

and Graphical Techniques

• having an appreciation of the symmetries of functions such

as quadratic and trigonometric functions and being able to use such properties when working with models based on such

mathematical working by developing sub-steps if necessary

1c

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new situations to mathematics in situations with which you are familiar

for example, when calculating

the possible percentage error due to rounding to the nearest year in a population model such as P=Aekt, you may relate this to earlier experience of calculating percentage errors in finding the area of a rectangle when you rounded lengths to the nearest centimetre

1c

you in making sense of mathematics

for example, this could involve

you in considering:

• extremes (e.g in modelling a population using P= 100 − 30 × 0 8t, considering what

happens as t→ ∞)

• boundary conditions (e.g

when t = 0 the initial population predicted by the model

t

P= 100 − 30 × 0 8 is

70 30

turnover annual

debt g outstandin

representations / understanding (algebraic, graphical or numerical) to explain and make sense of mathematics

for example, when considering

the recurrence relation,

n

x +1= 1 − , you may calculate successive numerical values x1, x2etc

and then plot a graph of x n

against n

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working by:

• questioning whether your solutions are reasonable and / or valid

• using checking techniques wherever possible

for example:

• you could consider the magnitude of solutions (e.g when modelling population growth of a country if you calculate that the population will increase ten-fold in ten years consider whether this is reasonable)

• you could consider the dimensions of values you are calculating (e.g

h l

k× × has the dimensions of area if kis

a dimensionless constant and both l and h are lengths)

• if you have found solutions to simultaneous equations check that these are correct by substituting the values you have found back into the equations

1a

all

degree of accuracy of values you calculate

for example:

• when using a calculator to find the area of a circle of radius 10 cm (correct to the nearest cm) your calculator will use

π 3.14159265 = You should not quote your final answer to the accuracy of the value on

your calculator display

All

• when using values read from graphs think carefully about the accuracy of any values that you subsequently calculate

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• write clear and unambiguous mathematical statements

• use notation correctly

this includes correct use of brackets

your mathematics should read correctly (as an English sentence), e.g when solving

0 5

10t − t2 = write:

(2 ) 0 5

5

10t− t2 = t −t =

2

or =

=

⇒t t

NOT:

(2 ) 0 , 2 5

for example:

when working with a population that may be modelled by the function

t

P= 100 − 30 × 0 8 you could:

• most effectively show how this varies with time by plotting or sketching a graph of P against t

• explore the long term behaviour predicted by this population model by substituting a large value for t and note that P is

approximately 100

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9.4 Guidance for Teachers This unit has been designed so that although its assessment will be by

examination, it can be studied alongside the compulsory FSMQ

Working with Algebraic and Graphical Techniques and one of the optional

units FSMQ Modelling with Calculus, FSMQ Using and Applying Statistics and FSMQ Using and Applying Decision Mathematics This is the

preferred method of study, as it will allow candidates to develop the process skills of mathematical modelling, comprehension and communication that are at the heart of this unit as they apply mathematical principles in their work for the other two units If possible, study for this unit should be ‘ramped’ so that as progress is made in the other two units the proportion of time spent for this unit gradually increases As candidates develop the skills required by the first two units they will find it increasingly possible to demonstrate the more advanced skills demanded by this unit

It is important that candidates meet a wide range of different contexts in which they apply their mathematics as they study this unit These should span Science, Technology, Economics, Geography and other disciplines

Although the unit has been presented by defining six Learning Objectives

it is not intended that candidates should develop these in isolation from each other In particular it is essential that those objectives which target specific process skills are addressed whilst candidates are engaged with meaningful activity; this may be whilst they are studying the two Free-Standing Mathematics Qualifications or whilst working with the new content of this unit

Learning objectives LO1, LO4, LO5 and LO6 are particularly, although not exclusively, aimed at developing important skills in the areas of mathematical modelling, comprehension and communication

Learning objectives LO2 and LO3 introduce new mathematical content including modelling random events and using recurrence relations This section exemplifies the type of work this new content requires

Using and Applying Decision Mathematics

Working with Algebraic and Graphical Techniques

Modelling with Calculus

Using and Applying

Statistics

Applying Mathematics

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9.5 Learning Objectives The Learning Objectives are stated in detail in section 9.3

Note that the ‘Links’ column indicates where you may develop the required objectives as candidates produce their coursework portfolio

evidence for FSMQ Working with Algebraic and Graphical Techniques The

column indicates the appropriate sub-section of the coursework portfolio requirements

LO1 Understanding how

mathematics can be used

to model different

situations

This learning objective is central to all the work that candidates will do for each unit of this qualification Following study of this unit, candidates should be aware of the different stages of the modelling cycle in which the real world is simplified and analysed using mathematics They should understand the different stages that are necessary in a complete cycle (LO1a), and it is suggested that a diagram such as that given on the next page, will assist candidates in appreciating these different stages In particular emphasis should be placed on understanding and considering the transition activities of:

• being aware of likely limitations to models due to simplifying assumptions being made as the model is developed (LO1b) Note that candidates will not be expected to develop mathematical models

of their own from scratch in the assessment of this unit They will, however, be expected to work with given models and be aware of how such models may be limited

• being able to interpret the outcome of mathematical analysis in real terms (LO1c) and, in doing so, being able to consider the likely validity of a model (LO1d)

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It is important that candidates come to appreciate the power of general mathematical models and how they may be adapted to analyse closely related situations (LO1e)

There will be many opportunities throughout the course for candidates to consider the process of mathematical modelling For example:

• in producing work for the portfolio for FSMQ Working with

Algebraic and Graphical Techniques

• in LO2 of this unit (developing mathematical models for random events)

• in LO3 of this unit (investigating discrete models using recurrence relations)

In discussion of the other Learning Objectives of the unit, guidance is given as to how features of mathematical modelling might be

Candidates will perhaps find it useful to use a graphics calculator or spreadsheet software to develop the tables and graphical displays required here

Define problem

observe

validate

Analyse Set up a model

Interpret Predict

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Example: Comments:

The receptionist at a doctor’s surgery carries out a survey to find how long each patient spends with the doctor He finds that times of 8, 9, 10, 11 and 12 minutes have the following probabilities associated with them

Time (minutes) Probability Candidates should be aware of

the limitations likely to have been introduced into this model due to the simplified probabilities being used

• a patient arrives to see the doctor every 8 minutes

• the first patient arrives at the surgery at 10:00 and is immediately seen by the doctor

Candidates should be able to use the randomly generated integers from 0-9 so that they can be used

to assign the waiting times of 8, 9,

10, 11 and 12 minutes, e.g

0 → 8 minutes 1,2 → 9 minutes 3,4 → 10 minutes 5,6,7 → 11 minutes 8,9 → 12 minutes

A table is used to organise the simulation:

patient no arrival

time random no length of

consult

time start time end time wait length at queue

arrival

Candidates should be able to develop tables

of their own - in examinations they should be able to either organise information into tabular form themselves or work with given tables

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Results may be presented graphically: Candidates should be able to develop appropriate graphical

displays to illustrate their findings Candidates should be aware of the limitations of this model

For example,

• it is unlikely that patients will arrive to see a doctor every 8 minutes - perhaps the arrival

of patients could also be modelled as random events

• the length of time spent by patients with the doctor is modelled by a simplified probability distribution A more detailed probability distribution could be used

• the table gives information for only 10 patients

• the simulation could be run a number of times to

investigate what effect this will have

Other situations that candidates could investigate include:

• the length of queues at traffic lights

• queuing at supermarket check-outs

• radioactive decay

LO3 Extending, developing

and using a range of

numerical, algebraic and

graphical techniques when

modelling situations

Much of the work towards this Learning Objective can be developed as

candidates study the FSMQ Working with Algebraic and Graphical

Techniques

Developing an ability to use alternative techniques as appropriate when carrying out mathematical analysis is to be encouraged In particular candidates should be able to use algebraic, graphical and numerical techniques in their mathematics and their communication of this Candidates should be able to decide when it is most appropriate to use each of these techniques Work for this particular aspect of the Learning

Objective can be carried out as candidates study FSMQ Working with

Algebraic and Graphical Techniques or the additional content of this

Learning Objective

Patient wait time

0 5 10 15 20 25 30

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The following examples illustrate the type of work that will be required

of candidates when using continuous and discrete models

Example: ( a continuous function as a model) Comments:

In the 17th Century Galileo discovered that all objects fall to the earth with the same acceleration and that this is constant This might imply that the speed of a falling object would always be increasing

Skydivers know that this is not the case and that they approach a steady speed known as ‘terminal velocity’

Here a general model is introduced This serves to illustrate the potential power of continuous functions as

mathematical models The coefficients a and b will be different in various situations but here we have the general form of

a mathematical model

The vertical speed, v metres per second, of a skydiver t seconds after leaving an aircraft may be modelled by a function of the form

a

v= 1 − e− For a particular skydiver

The screen-shot of a graphics calculator below has a window:

xmin = −1 xmax = 20 ymin = −10 ymax = 70

Candidates should be able to use their graphics calculators to plot graphs of functions On

occasions they will also need to

be able to plot data alongside these functions to investigate the validity of using such

mathematical models They should be able to relate their graphs to the situation they are investigating

For the general model v=a(1 − e−bt)the effect of varying a is shown

methodically (e.g varying one

coefficient at a time) and be able

to draw appropriate conclusions

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For the general model v=a(1 − e−bt)the effect of varying b is shown below (a= 50)

In this case:

a gives the terminal velocity;

b affects how quickly the terminal velocity is reached

Example: (recurrence relations used to model a situation) Comments:

A car hire company is going to set

up a depot for hire cars in London and another in Bristol Customers will be able to return cars that they hire to either depot The company does some market research and finds:

• 4

1

of customers who hire cars in London each week will return them to Bristol The remainder will return them to London

• 2

1

of customers who hire cars in Bristol each week will return them to London The remainder will return them to Bristol

The linked recurrence relations used as an example here are quite complex for this unit It is not expected that candidates would

be able to develop them for themselves but they should be able to interpret their meaning in terms of the information given and be able to use the recurrence relations

This situation can be modelled using the recurrence relations:

n n

L

2

1 4

3

1 = +

+

n n

B

2

1 4

1

1 = +

+where L n and B n are the number

of cars in London and Bristol

respectively at the start of the nth

week

To assist in making sense of the recurrence relations candidates may substitute specific values (in this case the initial values

200

The model can be used to investigate what happens to the distribution of hire cars

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33

The table shows what happens if there are initially 200 cars in each

of London and Bristol

n London Bristol Candidates should be able to

organise their working into tabular form and be able to communicate their findings using graphs that they should be able to draw using graphics calculators

In many cases graphics calculators can be used easily to carry out the iterative calculations required when using recurrence relations Candidates will find it helpful to use spreadsheet software when carrying out work in this topic in the classroom

Extension opportunities - candidates may:

• investigate using different initial values of cars at each location

• use algebra to determine the steady state values, e.g in the steady state let

and ,

3 +

=

2

1 4

1

B L

B= +

0 50 100 150 200 250 300

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Rearrangement of these gives

400 and

2

1 4

1

= +

= B L B

400 3

2B+B= B=

133 3

LO4 Appreciating that

being able to transfer mathematics For example, being able to

recognise that the two models below are both of direct proportion and consequently be able to use and apply all of the underlying principles

of models of direct proportion (such as finding and understanding the meaning of the constant of direct proportion) is very powerful

Model B The supply of goods by manufacturers may be related to

price by the following model

Candidates should be particularly aware of general principles that apply to the other types of functions they meet throughout their studies for this qualification, i.e linear, quadratic, trigonometric and exponential functions

For example, candidates should be aware that they might meet linear models in a range of different situations In this case they should be aware that in each case a linear model is appropriate and be able to work with the different notation that is likely to occur in each, have a firm understanding of the general underlying features of linear models and be able to interpret these in the particular situation under

investigation To encourage understanding of this type you may ask candidates to look at a range of situations such as those highlighted below and ask them to draw out the common features of linear models and what these features tell them about each particular situation

Quantity supplied Price

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Immediately candidates should recognise from the graphical display that although a straight line model is possible some data points do not fit very well (e.g the data point for 1932)

Candidates should be aware of the limitations of such a model For example, the model predicts that eventually, in the year 2679, the Olympic record will reduce to zero

against T2 (Note that this graph does not show clearly the planets nearest the Sun because of the scale of the horizontal axis accommodating those planets which are much further away.)

In this case candidates should be aware that although these data are empirical the law T2α d3 has underlying physical principles (This law was first developed by Kepler in the 17th century.)

In this case T2 =kd3

9.8 10 10.2 10.4 10.6 10.8 11 11.2

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Candidates should appreciate that this gives not only a linear relation but also one of direct proportion They should be able to develop and use an algebraic expression of such a law and have an awareness

of how the law applies in reality For example, candidates should understand that although all of the planets obey this law and consequently lie on the straight line, not every point on the line gives data for a planet

Within this learning objective candidates should also learn to appreciate general graphical features including asymptotes The following examples give contexts in which candidates may meet such asymptotes and give examples of the types of questions they might consider

Examples: Comments:

This graph shows how the concentration of a drug in the human body decays with time after a dose

This can be modelled by function

• explore the effect that altering the coefficients A and b has

This data and graph show the world population since 1870 The data given here clearly shows some form of exponential growth

such as P=Aekt where P is the population (in millions) and t is the number of years after some arbitrary zero Candidates should appreciate that functions of this type have a horizontal asymptote (in this case as t→ −∞ , P→ 0) Candidates should relate the constants A and k to reality (e.g A gives the population when

0

=

t )

Year Population (millions)

time, t Drug concentration

d 3 plotted against T2 for the planets of our

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37

The following example illustrates how candidates might be expected to work with mathematical principles concerning straight lines and quadratics to develop a model in an economic context This highlights how candidates should then be able to use the mathematical principles underpinning quadratics to make sense of the evolving model

The graph below models how the demand, D , for a particular product may vary with its price,

P

Candidates should be able to interpret this straight line model

in general terms such as,

• as the price of a product falls, demand for the product increases,

• there is a price, P0, at which demand for the product falls

to zero,

• when the price for the product falls to zero the demand reaches a maximum,

0

D Candidates should be able to find the equation of the function for this model,

D

P P P

0

0 −

= They should recognise that this is

a linear equation and should then

be able to work with the mathematical principles associated with this type of function

0 1000 2000 3000 4000 5000 6000

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product is found by multiplying the price, P, by the demand, D

Given that R=P×D candidates should be able to find R in terms

of D, i.e

2 0

0 0 0

0

D

P D P D D D

P P

.Candidates should recognise that

this is a quadratic function with

a maximum point They should then be able to work with this quadratic applying the

mathematical principles associated with this type of function For example, candidates should be able to use the factorised form

D D D

P P

The table below gives sunrise and sunset times on the first day of each month (January - July) in London, UK (all times GMT)

D R

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of the data that can be modelled by trigonometric functions

The sunrise time, T , can be

6 cos

• they should be able to predict

a function for the sunset times based on the function for the sunrise times and a graphical understanding of the data

LO5 Making sense of

mathematics

This Learning Objective is a key part of the unit It will be assessed in both assessment papers but in particular in the comprehension part of the assessment

Candidates should learn to read short pieces of text that not only include background information that gives the context, but also includes complete or partial mathematical working It will be useful if candidates are given such pieces of text at an early stage of the course and use them as resources on which they base their work towards other learning objects of this unit and in generating coursework portfolio evidence for the other units of the qualification

In particular, pieces of text of this type that include good examples of clear mathematical writing will prove useful to exemplify the type of work expected of candidates with LO6 Candidates should also meet a wide range of mathematical writing from a variety of contexts so that they experience how people working in many different fields may use mathematics Although at times the notation used may be unfamiliar (see the following examples) candidates should come to appreciate the underlying mathematical principles and that they are able to work with these to make sense of a wide range of different situations

00:00 04:00 08:00 12:00 16:00 20:00 00:00

Jan Feb Mar Apr May Jun Jul

Month Sunrise & sunset times

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Examples of strategies that candidates might adopt to make sense of algebraic equations and expressions are given below

Considering when factors of algebraic expressions are zero For example:

One possible formula used to calculate a wind chill temperature

( )wc

T in degrees Celsius for a wind speed of v kilometres per hour and still air temperature of T

degrees Celsius is:

045

0 v+ − v T−

is zero

Candidates should then be able to interpret this situation, i.e that this model predicts that when the air temperature is 33 degrees Celsius there is no wind chill effect

The profit a manufacturer makes may be modelled by

The term (P−AVC) is zero if

AVC

P= Candidates should be able to interpret this case, i.e

no matter what quantity is produced the manufacturer will

be unable to make a profit

Note that it is common for economists and others to use notation such as AVC to denote a single variable - it does not denote

C V

A× × Candidates should meet and be aware of a range of

to find the PDV of a future payment £F over n years at discount rate i, you use

( )i n

F PDV

05 0 ,

100 =

= i

F and n= 1 to find the amount of money that would need to be invested for one year

at an interest rate of 5% to ensure

a payment of £100 This gives

£95.24 Candidates could then verify by any familiar method that this investment would indeed give the desired future payment of

£100

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