AQA GCSE Specification

Version 2.0

General Certificate of

Secondary Education

Mathematics (Modular) 2005Specification B

This specification should be read in conjunction with:Specimen and Past Papers and Mark Schemes

Examiners’ Reports

Teachers’ Guide

AQA GCSE 3302

This specification will be published annually on the AQA Website (www.aqa.org.uk). If thereare 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.

In the Spring Term before the start of the course, details of any year-specific information, suchas set tests, theme/topics, will be notified to centres in print and on the Website.

Vertical black lines indicate a significant change or addition to the specification.

Copyright © 2003 AQA and its licensors. All rights reserved.

COPYRIGHTAQA retains the copyright on all its publications, including the specimen units and markschemes/teachers guides. However, the registered centres for AQA are permitted to copy materialfrom this booklet for their own internal use, with the following exception: AQA cannot givepermission to centres to photocopy any material that is acknowledged to a third party even for internaluse within the centre.

Set and published by the Assessment and Qualifications Alliance.

The Assessment and Qualifications Alliance 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.Dr Michael Cresswell, Director General.

General Certificate of Secondary Education, 2005 examination - Mathematics B (Modular)

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Contents

Background Information

1 The Revised General Certificate of Secondary Education 7

2 Specification at a Glance 9

3 Availability of Assessment Units and Entry Details 10

Scheme of Assessment

4 Introduction 13

5 Aims 15

6 Assessment Objectives 16

7 Scheme of Assessment 17

Subject Content

8 Summary of Subject Content 21

9 Module 1 25

10 Module 2 31

11 Module 3 33

12 Module 4 44

13 Module 5 48

Mathematics B (Modular) - General Certificate of Secondary Education, 2005 examination

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Key Skills and Other Issues

14 Key Skills – Teaching, Developing and Providing

Opportunities for Generating Evidence 67

15 Spiritual, Moral, Ethical, Social, Cultural and Other Issues 72

Internal Assessment (Coursework)

16 Nature of the Coursework Modules 74

17 Assessment Criteria for the Coursework Modules 76

Option T – Centre-Assessed Modules 2 and 4

18 Guidance on Setting the Centre-Assessed Modules 80

19 Supervision and Authentication 81

20 Standardisation 82

21 Administrative Procedures 83

22 Moderation 84

Option X – AQA-Assessed Modules 2 and 4

23 Guidance on Setting the AQA-Assessed Modules 85

24 Supervision and Authentication 86

25 Administrative Procedures 87

Awarding and Reporting

26 Grading, Shelf-Life and Re-Sits 88


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Appendices

A Grade Descriptions 91

B Formulae Sheets 94

C AQA-set Coursework Tasks for Module 2 97

D AQA-set Coursework Tasks for Module 4 102

E Record Forms 109

F Overlaps with Other Qualifications 119


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Background Information

1 The Revised General Certificate

of Secondary Education

Following a review of the National Curriculum requirements, and theestablishment of the National Qualifications Framework, all the unitaryawarding bodies revised their GCSE syllabuses for examination in 2003.

1.1 National QualificationsFramework

GCSE has the following broad equivalence to General NationalVocational Qualifications (GNVQ).

GCSE GCSE GNVQ

Two GCSE Grades D-G One (Double Award) DD-GG One 3-Unit GNVQ Foundation†

Grades A*-C One (Double Award) A*A*-CC Intermediate††

Four GCSE Grades D-G Two (Double Award) DD-GG One 6-Unit GNVQ Foundation

Grades A*-C Two (Double Award) A*A*-CC Intermediate

† only available until 2003†† only available until 2005

1.2 Changes at GCSE

Key Skills All GCSE specifications must identify, as appropriate, opportunities forgenerating evidence on which candidates may be assessed in the “main”Key Skills of Communication, Application of Number and InformationTechnology at the appropriate level(s). Also, where appropriate, they mustidentify opportunities for developing and generating evidence foraddressing the “wider” Key Skills of Improving own Learning and

Performance, Working with Others and Problem Solving.

Spiritual, moral, ethical,social, cultural,environmental, health andsafety and European Issues

All specifications must identify ways in which the study of the subjectcan contribute to an awareness and understanding of these issues.

ICT The National Curriculum requires that students should be givenopportunities to apply and develop their ICT capacity through the useof ICT tools to support their learning. In each specification candidateswill be required to make effective use of ICT in ways appropriate to theneeds of the subject.


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Tiering In GCSE Mathematics the scheme of assessment must includequestion papers targeted at three tiers of grades, i.e. A* - C (Higher),B – E (Intermediate) and D – G (Foundation).

Candidates should be entered at the tier appropriate to theirattainment. In GCSE Mathematics (Modular) each candidate mayenter for each individual module at a different tier of entry. However,the final range of grades available to a candidate is determined by thetier of entry for Module 5. Candidates who fail to achieve the markfor the lowest grade available at each tier of Module 5 will be recordedas unclassified (U).

Citizenship Students in England are required to study Citizenship as a NationalCurriculum subject. Each GCSE specification must signpost, whereappropriate, opportunities for developing citizenship knowledge, skillsand understanding.

1.3 Changes to the MathematicsCriteria

• Internal assessment (coursework) is now compulsory.

• Internal assessment comprises two tasks:

� the AO4 task – a handling data task which counts as half ofthe AO4 weighting;

� the AO1 task – an investigative task which assesses AO1 in thecontext of AO2 and/or AO3 and counts as half of the AO1weighting.

• The other half of the AO1 and AO4 weightings are assessed in thewritten papers.

• New subject content has been added to the Programme of Study,particularly in AO4, whilst other subject content has been deleted.

• Some questions demanding the unprompted solution of multi-stepproblems are required.

• The proportion of marks allocated to grade G on Foundation tierhas been increased to about one third, leaving the remaining marksbalanced across grades D, E and F.

• Grade descriptors have been modified to reflect the newProgramme of Study.


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2 Specification at a Glance

Mathematics B (Modular)

Option T and Option X• This is one of two specifications offered by AQA. Specification A

is a traditional linear scheme; Specification B is modular and issuitable for both pre-16 and post-16 candidates.

• There are three tiers of assessment, Foundation (D-G),Intermediate (B-E) and Higher (A*-C).

• Centres in Northern Ireland/Wales must refer to the Statement inSection 8.1 of this specification.

Foundation Tier GCSE 3302

Intermediate TierHigher Tier

Modules 1, 3 and 5are available in all

Module 1

Written Paper 11% of the total assessment2 × 25 minutes (All tiers)

Section A – CalculatorSection B – Non-calculator

Module 2

Coursework (AO4 task) 10% of the total assessment

three tiersSee entry codeinformation in

section 3.2 Either

OPTION TCentre-Set or AQA-Set taskCentre-Marked

Or

OPTION XAQA-Set taskAQA-Marked

Module 3

Written Paper 19% of the total assessment2 × 40 minutes (All tiers)

Section A – CalculatorSection B – Non-calculator

Module 4

Coursework (AO1 task) 10% of the total assessment

Either

OPTION TCentre-Set or AQA-Set taskCentre-Marked

Or

OPTION XAQA-Set taskAQA-Marked

Module 5

Written papers 50% of the total assessment

Non-calculatorFoundation tier 1 hourIntermediate and Higher tiers 1 hour 15 minutes

CalculatorFoundation tier 1 hourIntermediate and Higher tiers 1 hour 15 minutes


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3 Availability of Assessment Units

and Entry Details

3.1 Availability of AssessmentUnits

Specification B is a modular assessment of GCSE Mathematicsdesigned to be taken over a one or two year course of study. To offermaximum flexibility to centres and to suit different teachingprogrammes, Modules 1 to 4 can be taken in any order and candidatescan enter at different tiers for the different modules. Module 5 is thecertificating module and must be taken in the final examination series.This is to meet the QCA requirement that at least 50% of thequalification is externally examined at the end of the course.

Examinations based on this specification will be available as follows:

SeriesAvailability of Modules

Availability ofCertification

Module 1 Module 2 Module 3 Module 4 Module 5

March All tiers All tiers All tiers All tiers_ _

June All tiers All tiers All tiers All tiers All tiers All tiers

November All tiers All tiers All tiers All tiersIntermediate

tier onlyIntermediate

tier only

3.2 Entry Codes Normal entry requirements apply, but the following informationshould be noted.

A separate entry is needed for each of the five modules. In addition,an entry for the overall subject award, 3302, must be submitted by 21February for the June examination or 7 October for the Novemberexamination.

More detailed information, including component codes, will be issuedto examination centres in a separate document.


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3.3 Prohibited Combinations Candidates entering for Module 5 of this Specification are prohibitedfrom entering for any other GCSE Mathematics specification that willbe certificated in the same examination series.

Candidates may enter only for a single tier in each module, in aparticular examination series.

Each specification is assigned a national classification code, indicatingthe subject area to which it belongs.

Centres should be aware that candidates who enter for more than oneGCSE qualification with the same classification code, will have onlyone grade (the highest) counted for the purpose of the School andCollege Performance Tables.

The classification code for this specification is 2210.

3.4 Private Candidates Private candidates should normally enter for Specification B OptionX. Specification B Option T is only available for private candidateswhere:

• the candidate attends an AQA centre which will supervise thecoursework, or,

• the candidate has a coursework module mark that can be carriedforward (see Section 26.5).

Private candidates should write to AQA for a copy of Supplementary

Guidance for Private Candidates.


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3.5 Special Consideration Special consideration may be requested for candidates whose work hasbeen affected by illness or other exceptional circumstances. Theappropriate form and all relevant information should be forwarded tothe AQA office which deals with such matters for the centreconcerned. Special arrangements may be provided for candidates withspecial needs.

Details are available from AQA and centres should ask for a copy ofCandidates with Special Assessment Needs, Special Arrangements and Special

Conditions.

3.6 Language of Examinations All assessment will be through the medium of English. Assessmentmaterials will not be available in Welsh or Gaeilge.


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Scheme of Assessment

4 Introduction

4.1 National Criteria This AQA GCSE (modular) in Mathematics: (B) complies with thefollowing:

• the GCSE Subject Criteria for Mathematics;

• the GCSE and GCE A/AS Code of Practice;

• the GCSE Qualification Specific Criteria;

• the Arrangements for the Statutory Regulation of ExternalQualifications in England, Wales and Northern Ireland: CommonCriteria.

4.2 Rationale AQA offers a suite of qualifications for GCSE Mathematics.Specification A is a traditional scheme and is a development of theformer NEAB GCSE Mathematics syllabus A and SEG GCSEMathematics syllabus 2510T and 2510X. Specification B is a modularscheme suitable for both pre-16 and post-16 candidates; it is adevelopment of the former SEG GCSE Modular Mathematicssyllabus 2540.

Specification A and Specification B have common coursework tasks;this allows candidates the flexibility to move from one scheme ofassessment to the other.

4.3 Specification B There are two options within Specification B, allowing alternativeapproaches for the Internal Assessment (coursework) Modules 2 and4. In Option T centres may choose from the bank of courseworktasks provided by AQA or they may set their own coursework tasks;centres mark their own coursework tasks with moderation ofcandidates’ coursework by AQA. In Option X centres must choosefrom the bank of coursework tasks provided by AQA (AQA-Set tasks)and candidates’ coursework is marked by AQA (see appendices C andD).

Specification B used in pre-16 centres.Mathematics is essentially a holistic subject, and as such should betaught in this way with appropriate connections being made betweenthe sections on Number and algebra, Shape, space and measures, andHandling data, as required in the National Curriculum. For exampleNumber underpins the whole of Mathematics. Modular MathematicsSpecification B is designed to be more reflective of the way in whichcandidates are likely to revise for examinations when they tend tocover just one area of Mathematics at a time. Specification B allowscandidates to take modules early in the course based on Handling data(AO4) and the mainly number part of Number and algebra (AO2). The


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final module has to comprise 50% of the external written assessmentand this concentrates on the mainly algebraic part of Number and algebra

(AO2) and the whole of Space, shape and measures (AO3). Thecoursework has been separated into two further modules to allow forincreased flexibility as to when the tasks are submitted.

Division into discrete topic areas gives candidates much more insightinto their strengths and weaknesses. Specification B provides a naturallink between KS3/KS4 (which are taught holistically) and A-levelwhere Mathematics is examined in discrete topic areas, but notnecessarily taught as such. The modular nature of the specificationcan allow candidates who fail to obtain a GCSE Grade C at KS4 tocarry forward some of their module results into post-16 education.

Specification B used in a post-16 centre gives links to Free-Standing Mathematics Qualifications (FSMQs) and the Key Skill ofApplication of Number, and in some cases this could lead to co-teachingopportunities.

4.4 Prior level of attainment andrecommended prior learning

There is progression of material through all levels at which the subjectis studied. This specification therefore builds on the Key Stage 3Programme of Study.

It is also expected that candidates will have reached the required levelof literacy through study at Key Stage 3.

4.5 Progression This qualification is a recognised part of the National QualificationsFramework. As such, GCSE Mathematics provides progression fromKey Stage 3 to GCE A/AS Mathematics or further study at Advancedor Advanced Subsidiary level in other subjects or further study atGNVQ level, or directly into employment.


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5 Aims

The aims set out below are consistent with the 1999 NationalCurriculum Order for Mathematics and the GCSE Criteria forMathematics. Most of the aims are reflected in the AssessmentObjectives; others are not because they cannot be readily translatedinto assessment objectives.

This specification encourages candidates to:

a. consolidate their understanding of mathematics;

b. be confident in their use of mathematics;

c. extend their use of mathematical vocabulary, definitions and formalreasoning;

d. develop the confidence to use mathematics to tackle problems in thework place and everyday life;

e. take increasing responsibility for the planning and execution of theirwork;

f. develop an ability to think and reason mathematically;

g. learn the importance of precision and rigour in mathematics;

h. make connections between different areas of mathematics;

i. realise the application of mathematics in the world around them;

j. use ICT appropriately;

k. develop a firm foundation for appropriate further study.


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6 Assessment Objectives

6.1 Assessment Objectives A course based on this specification requires candidates to demonstratetheir knowledge, understanding and skills in the following assessmentobjectives. These relate to the knowledge, skills and understanding inthe Programme of Study.

AO1 Using and applying mathematics

AO2 Number and algebra

AO3 Shape, space and measures

AO4 Handling data

The Assessment Objective AO1, Using and applying mathematics, will beassessed in contexts provided by the other assessment objectives.

6.2 Quality of WrittenCommunication

This specification does not formally assess quality of writtencommunication.


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7 Scheme of Assessment

7.1 Assessment UnitsOption T and Option X

The Scheme of Assessment has a modular structure. The subjectcontent of the specification is assessed by five separate modules whichcomprise the following components.

Module 1 Written Paper(Section A – Calculator)(Section B – Non-Calculator)

Foundation Tier 2 x 25 minutesIntermediate Tier 2 x 25 minutesHigher Tier 2 x 25 minutes

11 % of the total assessment 2 x 20 marks

This written paper is the same for Option T or Option X.Assesses AO4 (Handling data).All questions are compulsory. Question and answer booklet.

Module 2 Internal Assessment 1 – AO4 task

10 % of the total assessment 24 marks

EITHER Option Tone task set and marked by thecentre

OR Option Xone task, selected from a bank oftasks provided by AQA, andmarked by AQA (Appendix C)

Coursework task set in the context of AO4 (Handling data).

Module 3 Written Paper(Section A – Calculator)(Section B – Non-Calculator)

Foundation Tier 2 x 40 minutesIntermediate Tier 2 x 40 minutesHigher Tier 2 x 40 minutes

19 % of the total assessment 2 x 32 marks

This written paper is the same for Option T or Option X.Assesses mainly the number part of AO2 (Number and algebra).All questions are compulsory. Question and answer booklet.


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Module 4 Internal Assessment 2 – AO1 task

10 % of the total assessment 24 marks

EITHER Option TOne task set and marked by thecentre

OR Option Xone task, selected from a bank oftasks provided by AQA, andmarked by AQA (Appendix D)

Coursework task set in the context of AO2 and/or AO3.

Module 5(Terminal Module)

Written PaperPaper 1 (Non-Calculator)

Foundation Tier 60 marks 1 hourIntermediate Tier 70 marks 1 hour 15 minsHigher Tier 70 marks 1 hour 15 mins

25 % of the total assessment

Written PaperPaper 2 (Calculator)

Foundation Tier 60 marks 1 hourIntermediate Tier 70 marks 1 hour 15 minsHigher Tier 70 marks 1 hour 15 mins

25 % of the total assessment

Both written papers are the same for Option T or Option X.Both papers assess the mainly algebra part of AO2 (Number andalgebra) and the whole of AO3 (Shape, space and measures).All questions are compulsory.Question and answer booklet.


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7.2 Weighting of AssessmentObjectives

The approximate relationship between the relative percentageweighting of the Assessment Objectives and the overall Scheme ofAssessment is shown in the following table.

Module Weightings (%)Assessment

Objectives Module 1

(Written)

Module 2

(Coursework)

Module 3

(Written)

Module 4

(Coursework)

Module 5

(Written)

Overall Weighting of

Assessment

Objectives (%)

AO1 Using andapplying mathematics 1* 2* 10 7* 20

AO2 Number andalgebra 17 23 40

AO3 Shape, space and

measures 20 20

AO4 Handling data 10 10 20

Overall Weighting of

Modules (%) 11 10 19 10 50 100

* On the written papers the assessment of AO1 is subsumed withinthe other Assessment Objectives covered by the Module. Thus 10%of the total written paper assessment will also assess Using and ApplyingMathematics within the contexts of the questions.

Candidates’ marks for each module are scaled to achieve the correctweightings.

7.3 Written papers The written papers at the Intermediate and Higher tiers offer balancedassessment across the grades available at those tiers. At Foundationtier about one third of the marks are allocated to grade G and theremaining marks are balanced across grades D, E and F.

Common questions will be set on papers at adjacent tiers. Somequestions will be designed to assess the unprompted solution of multi-step problems.

In Modules 1 and 3, the written papers are divided into 2 separatesections. The first section is the calculator paper and this is issued tocandidates at the beginning of the examination. After this section hasbeen completed (after 25 minutes for Module 1 and 40 minutes forModule 3) candidates are instructed to place their calculators beneaththeir seat. The second section (the non-calculator paper) is thenissued. At the end of the examination, the two sections are taggedtogether and the papers are collected in.

Module 5 written papers are taken on two separate days, with the non-calculator paper on the first day and the calculator paper on thesecond day.

Formulae sheets for the Foundation, Intermediate and Higher tierpapers of Module 5 are provided in Appendix B.

On the non-calculator papers the use of a calculator, slide rules,logarithmic tables and all other aids is forbidden. On the calculatorpapers, candidates will be required to demonstrate the effective use ofa calculator.


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7.4 Calculators Candidates will be expected to have a suitable electronic calculator foruse with the calculator papers. The calculator should possess thefollowing as a minimum requirement:

• Foundation tier – four rules and a square, square root, brackets,reciprocal and power function and a memory facility;

• Intermediate and Higher tiers – as for Foundation tier togetherwith a constant function, standard form and appropriateexponential, trigonometric and statistical functions.

Further guidance on regulations relating to calculators can be obtainedfrom Instructions for the Conduct of Examinations.

7.5 Coursework modules Apart from the choice of coursework tasks and the method ofassessment, the nature of the Coursework Modules 2 and 4 is the samefor Option T and Option X. Information about the administrativearrangements for Option T Modules 2 and 4 can be found in Section21 and for Option X Modules 2 and 4 in Section 25. AQA set taskscan be found in Appendices C and D of this specification.

7.6 Entry policy Centres are encouraged to enter candidates aiming to achieve gradesE, F and G for the Foundation tier, grades C and D for theIntermediate tier and grades A*, A and B for the Higher tier.


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Subject Content

8 Summary of Subject Content

8.1 Introduction There are three tiers of entry for GCSE Mathematics candidates:Foundation, Intermediate and Higher. In the National Curriculum,published in 1999, the Key Stage 4 Programme of Study was directedinto two tiers. The division of the Programme of Study into threetiers in the subject content of this specification is common to allAwarding Bodies. Thus:

the subject content of the Foundation tier is based on the FoundationProgramme of Study but does not include the grade C material;

the subject content of the Intermediate tier is based on the HigherProgramme of Study but does not include the grade A and A*material;

the subject content of the Higher tier is based on the HigherProgramme of Study but does not include the grade D (or lower)material.

In general, the Intermediate tier content of the specification subsumesthe Foundation tier content. However, questions on the Intermediatetier do not focus directly on material that is outside the grade range ofthe tier. Similarly, the Higher tier content subsumes the Intermediateand Foundation tier content, but questions on the examination papersfor the Higher tier do not focus directly on material that is outside thegrade range of the tier.

This GCSE Specification has been written against the Key Stage 4Programme of Study for England. Candidates entering for this GCSEin Northern Ireland and Wales must be taught all the material requiredby the National Curriculum in their own country.

8.2 Assessment Objectives Within the modules of this specification the subject content ispresented under the following Assessment Objectives.

The Assessment Objective AO1 (Using and applying mathematics) isassessed in contexts provided by the other Assessment Objectives.

AO2 Number and algebra

1.

2.

3.

4.

5.

6.

Using and applying number and algebra

Numbers and the number system

Calculations

Solving numerical problems

Equations, formulae and identities

Sequences, functions and graphs


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AO3 Shape, space and measures

1.

2.

3.

4.

Using and applying shape, space and measures

Geometrical reasoning

Transformations and coordinates

Measures and construction

AO4 Handling data

1.

2.

3.

4.

5.

Using and applying handling data

Specifying the problem and planning

Collecting data

Processing and representing data

Interpreting and discussing results

8.3 Modules Module 1

This includes all of the subject content from AO4 (Handling data) ofthe National Curriculum for Mathematics, divided into three tiers ofentry.

Module 2

This is an internally assessed module assessing the using and applyingsection of AO4 (Handling data). The marking criteria are given inSection 17.5.

Module 3

This includes the mainly number subject content from AO2 (Number

and algebra) of the National Curriculum. At the Foundation andIntermediate tiers, only number topics are examined in this module.At the Higher tier some algebra topics are also examined.

Module 4

This is an internally assessed module assessing the using and applyingsections of AO2 (Number and algebra) and/or AO3 (Shape, space andmeasures). The marking criteria are given in Section 17.6.

Module 5

This includes the mainly algebra subject content from AO2 (Number

and Algebra) and all of the subject content from AO3 (Shape, Space andMeasures). At the Foundation and Intermediate tiers selected numbertopics from AO2 (Number and algebra) are also assessed. At the Highertier only the algebra topics from AO2 (Number and algebra) are assessedin this module.


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8.4 Breadth of Study In addition to the required knowledge, skills and understanding, theNational Curriculum Programme of Study also specifies the Breadthof Study expected.

Foundation Tier Pupils should be taught the knowledge, skills and understandingthrough:

a. extending mental and written calculation strategies and using efficientprocedures confidently to calculate with integers, fractions, decimals,percentages, ratio and proportion;

b. solving a range of familiar and unfamiliar problems, including thosedrawn from real-life contexts and other areas of the curriculum;

c. activities that provide frequent opportunities to discuss their work, todevelop reasoning and understanding and to explain their reasoningand strategies;

d. activities focused on developing short chains of deductive reasoningand correct use of the ‘=’ sign;

e. activities in which they do practical work with geometrical objects,visualise them and work with them mentally;

f. practical work in which they draw inferences from data, consider howstatistics are used in real life to make informed decisions, andrecognise the difference between meaningful and misleadingrepresentations of data;

g. activities focused on the major ideas of statistics, including usingappropriate populations and representative samples, using differentmeasurement scales, using probability as a measure of uncertainty,using randomness and variability, reducing bias in sampling andmeasuring, and using inference to make decisions;

h. substantial use of tasks focused on using appropriate ICT (forexample, spreadsheets, databases, geometry or graphic packages),using calculators correctly and efficiently, and knowing when not touse a calculator.

Intermediate/Higher Tiers Pupils should be taught the knowledge, skills and understandingthrough:

a. activities that ensure they become familiar with and confident usingstandard procedures for the range of calculations appropriate to thislevel of study;

b. solving familiar and unfamiliar problems in a range of numerical,algebraic and graphical contexts and in open-ended and closed form;

c. using standard notations for decimals, fractions, percentages, ratio andindices;

d. activities that show how algebra, as an extension of number usingsymbols, gives precise form to mathematical relationships andcalculations;

e. activities in which they progress from using definitions and shortchains of reasoning to understanding and formulating proofs inalgebra and geometry;


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f. a sequence of practical activities that address increasingly demandingstatistical problems in which they draw inferences from data andconsider the uses of statistics in society;

g. choosing appropriate ICT tools and using these to solve numerical andgraphical problems, to represent and manipulate geometricalconfigurations and to present and analyse data.

8.5 Subject Content Presentation The subject content for each module is shown in three columns,representing the Programmes of Study for Key Stage 4 divided intothree tiers of entry. The subject content is taken directly from theStatutory Orders for Mathematics.

To maintain the coherence of the topics, the statements have beengiven in full for each tier. Where the wording is almost the same asthe previous tier with just a small addition, the additional material is inbold type face. In the Module 3 Foundation and Intermediate tiers thestatements for some number topics are shown in Module 3 but areshaded to show that they are not examined until Module 5. Thestatements are then repeated in Module 5.

For each of the written paper modules, Modules 1, 3 and 5, the usingand applying statements are given at the beginning. These statementswill be mainly tested, and indeed some can only be tested, in thecoursework tasks. However, 10% of the total written paperassessment also has to assess using and applying mathematics withinthe contexts of questions appropriate to that paper.

Each statement is referenced to the appropriate statement in theFoundation or Higher Programme of Study.


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9 Module 1

AO4: Handling data

1. Using and applying handling data

Problem solving

Foundation tier Intermediate tier Higher tier

Pupils should be taught to:

4F1a carry out each of the four aspects of the handlingdata cycle to solve problems:

4H1a carry out each of the four aspects of the handlingdata cycle to solve problems:


(i) specify the problem and plan: formulate questionsin terms of the data needed, and consider whatinferences can be drawn from the data; decide whatdata to collect (including sample size and dataformat) and what statistical analysis is needed



(ii) collect data from a variety of suitable sources,including experiments and surveys, and primary andsecondary sources



(iii) process and represent the data: turn the raw datainto usable information that gives insight into theproblem



(iv) interpret and discuss: answer the initial question bydrawing conclusions from the data

(iv) interpret and discuss the data: answer the initialquestion by drawing conclusions from the data


4F1b identify what further information is needed to pursuea particular line of enquiry

4F1c select and organise the appropriate mathematics andresources to use for a task

4F1d review progress while working; check and evaluatesolutions

4H1b select the problem-solving strategies to use instatistical work, and monitor their effectiveness(these strategies should address the scale andmanageability of the tasks, and should considerwhether the mathematics and approach used aredelivering the most appropriate solutions)



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Communicating


4F1e interpret, discuss and synthesise informationpresented in a variety of forms

4F1f communicate mathematically, including using ICT,making use of diagrams and related explanatory text

4H1c communicate mathematically, with emphasis on theuse of an increasing range of diagrams and relatedexplanatory text, on the selection of their

mathematical presentation, explaining itspurpose and approach, and on the use ofsymbols to convey statistical meaning

4H1c communicate mathematically, with emphasis on theuse of an increasing range of diagrams and relatedexplanatory text, on the selection of theirmathematical presentation, explaining its purposeand approach, and on the use of symbols to conveystatistical meaning

Reasoning

4F1h apply mathematical reasoning, explaining inferencesand deductions

4H1d apply mathematical reasoning, explaining and

justifying inferences and deductions, justifyingarguments and solutions

4H1d apply mathematical reasoning, explaining andjustifying inferences and deductions, justifyingarguments and solutions

4H1e identify exceptional or unexpected cases whensolving statistical problems


4F1i explore connections in mathematics and look forcause and effect when analysing data

4H1f explore connections in mathematics and look forrelationships between variables when analysingdata

4H1f explore connections in mathematics and look forrelationships between variables when analysing data

4H1g recognise the limitations of any assumptions and theeffects that varying the assumptions could have onthe conclusions drawn from data analysis


2. Specifying the problem and planning


4F2a see that random processes are unpredictable 4H2a see that random processes are unpredictable

4F2b identify questions that can be addressed by statisticalmethods

4H2b identify key questions that can be addressed bystatistical methods

4F2c discuss how data relate to a problem 4H2c discuss how data relate to a problem; identifypossible sources of bias and plan to minimise it

4H2c identify possible sources of bias and plan tominimise it


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4F2d identify which primary data they need to collect andin what format, including grouped data, consideringappropriate equal class intervals

4H2d identify which primary data they need to collect andin what format, including grouped data, consideringappropriate equal class intervals

4H2d select and justify a sampling scheme and amethod to investigate a population, includingrandom and stratified sampling

4F2e design an experiment or survey; decide whatsecondary data to use

4H2e design an experiment or survey; decide whatprimary and secondary data to use

4H2e decide what primary and secondary data to use

3. Collecting data


4F3a design and use data-collection sheets for groupeddiscrete and continuous data; collect data usingvarious methods, including observation, controlledexperiment, data logging, questionnaires and surveys

4H3a collect data using various methods, includingobservation, controlled experiment, data logging,questionnaires and surveys

4F3b gather data from secondary sources, includingprinted tables and lists from ICT-based sources

4H3b gather data from secondary sources, includingprinted tables and lists from ICT-based sources

4F3c design and use two-way tables for discrete andgrouped data

4H3c design and use two-way tables for discrete andgrouped data

4H3d deal with practical problems such as non-response ormissing data

4H3d deal with practical problems such as non-response ormissing data

4. Processing and representing data


4F4a draw and produce, using paper and ICT, pie chartsfor categorical data, and diagrams for continuousdata, including line graphs for time series, scattergraphs, frequency diagrams and stem-and-leafdiagrams

4H4a draw and produce, using paper and ICT, pie chartsfor categorical data, and diagrams for continuousdata, including line graphs (time series), scattergraphs, frequency diagrams, stem-and-leaf diagrams,cumulative frequency tables and diagrams, box

plots

4H4a draw and produce, using paper and ICT, cumulativefrequency tables and diagrams, box plots andhistograms for grouped continuous data

4F4b calculate mean, range and median of small data setswith discrete and then continuous data; identify themodal class for grouped data

see 4H4e see 4H4e


28 ��


4F4c understand and use the probability scale

4F4d understand and use estimates or measures ofprobability from theoretical models (includingequally likely outcomes)

4H4b understand and use estimates or measures ofprobability from theoretical models, or from relativefrequency

4H4b understand and use estimates or measures ofprobability from theoretical models, or from relativefrequency

4F4e list all outcomes for single events, and for twosuccessive events, in a systematic way

4H4c list all outcomes for single events, and for twosuccessive events, in a systematic way

4F4f identify different mutually exclusive outcomes andknow that the sum of the probabilities of all theseoutcomes is 1

4H4d identify different mutually exclusive outcomes andknow that the sum of the probabilities of all theseoutcomes is 1

4H4e find the median, quartiles and interquartile range forlarge data sets and calculate the mean for large datasets with grouped data

4H4e find the median, quartiles and interquartile range forlarge data sets and calculate the mean for large datasets with grouped data

4H4f calculate an appropriate moving average 4H4f calculate an appropriate moving average

4H4g know when to add or multiply two probabilities:if A and B are mutually exclusive, then theprobability of A or B occurring is P(A) + P(B),whereas if A and B are independent events, theprobability of A and B occurring is P(A) × P(B)

4H4h use tree diagrams to represent outcomes ofcompound events, recognising when events areindependent

4H4h use tree diagrams to represent outcomes ofcompound events, recognising when events areindependent

4F4h draw lines of best fit by eye, understanding whatthese represent

4H4i draw lines of best fit by eye, understanding whatthese represent

4H4i draw lines of best fit by eye, understanding whatthese represent

4H4j use relevant statistical functions on a calculator orspreadsheet

4H4j use relevant statistical functions on a calculator orspreadsheet


�� 29

5. Interpreting and Discussing Results



4F5a relate summarised data to the initial questions 4H5a relate summarised data to the initial questions

4F5b interpret a wide range of graphs and diagrams anddraw conclusions

4H5b interpret a wide range of graphs and diagrams anddraw conclusions; identify seasonality and trendsin time series

4H5b identify seasonality and trends in time series

4F5c look at data to find patterns and exceptions 4H5c look at data to find patterns and exceptions

4F5d compare distributions and make inferences, using theshapes of distributions and measures of average andrange

4H5d compare distributions and make inferences, using theshapes of distributions and measures of average andspread, including median and quartiles

4H5d compare distributions and make inferences, using theshapes of distributions and measures of average andspread, including median and quartiles; understandfrequency density

4F5e consider and check results and modify theirapproach if necessary

4H5e consider and check results and modify theirapproach if necessary

4F5f have a basic understanding of correlation as ameasure of the strength of the association betweentwo variables; identify correlation or no correlationusing lines of best fit

4H5f appreciate that correlation is a measure of thestrength of the association between two variables;distinguish between positive, negative and zerocorrelation using lines of best fit; appreciatethat zero correlation does not necessarily imply‘no relationship’ but merely ‘no linearrelationship’

4H5f appreciate that correlation is a measure of thestrength of the association between two variables;distinguish between positive, negative and zerocorrelation using lines of best fit; appreciate thatzero correlation does not necessarily imply ‘norelationship’ but merely ‘no linear relationship’

4F5g use the vocabulary of probability to interpret resultsinvolving uncertainty and prediction

4H5g use the vocabulary of probability to interpret resultsinvolving uncertainty and prediction [for example,‘there is some evidence from this sample that …’]

4F5h compare experimental data and theoreticalprobabilities

4H5h compare experimental data and theoreticalprobabilities

4F5i understand that if they repeat an experiment, theymay – and usually will – get different outcomes, andthat increasing sample size generally leads to betterestimates of probability and populationcharacteristics

4H5i understand that if they repeat an experiment, theymay – and usually will – get different outcomes, andthat increasing sample size generally leads to betterestimates of probability and population parameters

4F5j discuss implications of findings in the context of theproblem


30 ��


4F5k interpret social statistics including index numbers[for example, the General Index of Retail Prices];time series [for example, population growth]; andsurvey data [for example, the National Census]


�� 31

10 Module 2

AO4: Handling data

1. Using and applying handling data

Problem solving



4F1a carry out each of the four aspects of the handlingdata cycle to solve problems:












(iv) interpret and discuss: answer the initial question bydrawing conclusions from the data



4F1b identify what further information is needed to pursuea particular line of enquiry

4F1c select and organise the appropriate mathematics andresources to use for a task

4F1d review progress while working; check and evaluatesolutions




32 ��

Communicating


4F1e interpret, discuss and synthesise informationpresented in a variety of forms

4F1f communicate mathematically, including using ICT,making use of diagrams and related explanatory text

4H1c communicate mathematically, with emphasis on theuse of an increasing range of diagrams and relatedexplanatory text, on the selection of their

mathematical presentation, explaining itspurpose and approach, and on the use ofsymbols to convey statistical meaning

4H1c communicate mathematically, with emphasis on theuse of an increasing range of diagrams and relatedexplanatory text, on the selection of theirmathematical presentation, explaining its purposeand approach, and on the use of symbols to conveystatistical meaning

Reasoning

4F1h apply mathematical reasoning, explaining inferencesand deductions

4H1d apply mathematical reasoning, explaining and

justifying inferences and deductions, justifyingarguments and solutions

4H1d apply mathematical reasoning, explaining andjustifying inferences and deductions, justifyingarguments and solutions



4F1i explore connections in mathematics and look forcause and effect when analysing data

4H1f explore connections in mathematics and look forrelationships between variables when analysingdata

4H1f explore connections in mathematics and look forrelationships between variables when analysing data




�� 33

11 Module 3

AO2: Number and Algebra

1. Using and applying number and algebra

Problem solving



2F1a select and use suitable problem-solving strategies andefficient techniques to solve numerical and algebraicproblems

2H1a select and use appropriate and efficient techniquesand strategies to solve problems of increasingcomplexity, involving numerical and algebraicmanipulation


2H1b identify what further information may be required inorder to pursue a particular line of enquiry and givereasons for following or rejecting particularapproaches


2F1b break down a complex calculation into simpler stepsbefore attempting to solve it

2H1c break down a complex calculation into simpler stepsbefore attempting a solution and justify their

choice of methods

2F1c use algebra to formulate and solve a simple problem– identifying the variable, setting up an equation,solving the equation and interpreting the solution inthe context of the problem

2F1d make mental estimates of the answers to calculations;use checking procedures, including use of inverseoperations; work to stated levels of accuracy

2H1d make mental estimates of the answers to calculations;present answers to sensible levels of accuracy;understand how errors are compounded incertain calculations

Assessed in Module 5


34 ��

Communicating


2F1e interpret and discuss numerical and algebraicinformation presented in a variety of forms

2H1e discuss their work and explain their reasoning usingan increasing range of mathematical language andnotation


2F1g use a range of strategies to create numerical,algebraic or graphical representations of a problemand its solution

2H1f use a variety of strategies and diagrams forestablishing algebraic or graphical representations ofa problem and its solution; move from one form ofrepresentation to another to get differentperspectives on the problem

2H1f move from one form of representation to another toget different perspectives on the problem

2F1h present and interpret solutions in the context of theoriginal problem

2H1g present and interpret solutions in the context of theoriginal problem

2F1f use notation and symbols correctly and consistentlywithin a given problem

2H1h use notation and symbols correctly and consistentlywithin a given problem

2H1i examine critically, improve, then justify their choiceof mathematical presentation

2H1i examine critically, improve, then justify their choiceof mathematical presentation; present a concise,reasoned argument

Reasoning

2F1j explore, identify, and use pattern and symmetry inalgebraic contexts [for example, using simple codesthat substitute numbers for letters], investigatingwhether particular cases can be generalised further,and understanding the importance of a counter-example

2H1j explore, identify, and use pattern and symmetry inalgebraic contexts, investigating whether a particularcase may be generalised further and understand theimportance of a counter-example; identify

exceptional cases when solving problems

2H1j understand the importance of a counter-example;identify exceptional cases when solving problems

2H1k understand the difference between a practicaldemonstration and a proof


2F1k show step-by-step deduction in solving a problem 2H1l show step-by-step deduction in solving a problem 2H1l derive proofs using short chains of deductivereasoning



�� 35


2H1m recognise the significance of stating constraints andassumptions when deducing results; recognise thelimitations of any assumptions that are made and theeffect that varying the assumptions may have on thesolution to a problem


2. Numbers and the number system

Integers


2F2a use their previous understanding of integers andplace value to deal with arbitrarily large positivenumbers and round them to a given power of 10;understand and use positive numbers, both aspositions and translations on a number line; orderintegers; use the concepts and vocabulary of factor(divisor), multiple and common factor

2H2a use their previous understanding of integers andplace value to deal with arbitrarily large positivenumbers and round them to a given power of 10;understand and use negative integers both aspositions and translations on a number line; orderintegers; use the concepts and vocabulary of factor(divisor), multiple, common factor, highestcommon factor, least common multiple, primenumber and prime factor decomposition

2H2a use the concepts and vocabulary of highest commonfactor, least common multiple, prime number andprime factor decomposition

Powers and roots

2F2b use the terms square, positive square root, cube; useindex notation for squares, cubes and powers of 10

2H2b use the terms square, positive square root, negativesquare root, cube and cube root; use index notationand index laws for multiplication and division ofinteger powers; use standard index form,expressed in conventional notation and on acalculator display

2H2b use index laws for multiplication and division ofinteger powers; use standard index form, expressedin conventional notation and on a calculator display



36 ��

Fractions


2F2c understand equivalent fractions, simplifying afraction by cancelling all common factors; orderfractions by rewriting them with a commondenominator

2H2c understand equivalent fractions, simplifying afraction by cancelling all common factors; orderfractions by rewriting them with a commondenominator

Decimals

2F2d use decimal notation and recognise that eachterminating decimal is a fraction [for example,

0.137 = 1000

137 ]; order decimals

2H2d recognise that each terminating decimal is a fraction

[for example, 0.137 = 1000

137 ]; recognise that

recurring decimals are exact fractions, and thatsome exact fractions are recurring decimals [for

example, 7

1 = 0.142857142857…]; order decimals

2H2d recognise that recurring decimals are exact fractions,and that some exact fractions are recurring decimals

[for example, 7

1 = 0.142857142857…]

Percentages

2F2e understand that ‘percentage’ means ‘number of partsper 100’ and use this to compare proportions;interpret percentage as the operator ‘so manyhundredths of’ [for example, 10% means 10 parts

per 100 and 15% of Y means 100

15 × Y]; use

percentage in real-life situations [for example,commerce and business, including rate of inflation,VAT and interest rates]

2H2e understand that ‘percentage’ means ‘number of partsper 100’, and interpret percentage as the operator ‘somany hundredths of’ [for example, 10% means 10

parts per 100 and 15% of Y means 100

15 × Y]

Ratio

2F2f use ratio notation, including reduction to its simplestform and its various links to fraction notation [forexample, in maps and scale drawings, paper sizes andgears]

2H2f use ratio notation, including reduction to its simplestform and its various links to fraction notation

2H2f use ratio notation, including reduction to its simplestform and its various links to fraction notation



�� 37

3. Calculations

Number operations and the relationships between them



2F3a add, subtract, multiply and divide integers and thenany number; multiply or divide any number bypowers of 10, and any positive number by a numberbetween 0 and 1

2H3a multiply or divide any number by powers of 10, andany positive number by a number between 0 and 1;find the prime factor decomposition of positiveintegers; understand ‘reciprocal’ asmultiplicative inverse, knowing that any non-zero number multiplied by its reciprocal is 1(and that zero has no reciprocal, because

division by zero is not defined); multiply anddivide by a negative number; use index laws tosimplify and calculate the value of numericalexpressions involving multiplication anddivision of integer powers; use inverseoperations

2H3a multiply or divide any number by a number between0 and 1; find the prime factor decomposition ofpositive integers; understand ‘reciprocal’ asmultiplicative inverse, knowing that any non-zeronumber multiplied by its reciprocal is 1 (and thatzero has no reciprocal, because division by zero isnot defined); multiply and divide by a negativenumber; use index laws to simplify and calculate thevalue of numerical expressions involvingmultiplication and division of integer, fractional andnegative powers; use inverse operations,understanding that the inverse operation ofraising a positive number to power n is raising

the result of this operation to power n

1

2F3b use brackets and the hierarchy of operations 2H3b use brackets and the hierarchy of operations

2F3c calculate a given fraction of a given quantity [forexample, for scale drawings and construction ofmodels, down payments, discounts], expressing theanswer as a fraction; express a given number as afraction of another; add and subtract fractions bywriting them with a common denominator; performshort division to convert a simple fraction to adecimal

2H3c calculate a given fraction of a given quantity,expressing the answer as a fraction; express a givennumber as a fraction of another; add and subtractfractions by writing them with a commondenominator; perform short division to convert asimple fraction to a decimal; distinguish betweenfractions with denominators that have only

prime factors of 2 and 5 (which are representedby terminating decimals), and other fractions(which are represented by recurring decimals)

2H3c distinguish between fractions with denominators thathave only prime factors of 2 and 5 (which arerepresented by terminating decimals), and otherfractions (which are represented by recurringdecimals); convert a recurring decimal to a fraction

[for example, 0.142857142857… = 7

1 ]

2F3d understand and use unit fractions as multiplicativeinverses [for example, by thinking of multiplication

by 5

1 as division by 5]; multiply and divide a fraction

by an integer, and multiply a fraction by a unitfraction

2H3d understand and use unit fractions as multiplicativeinverses [for example, by thinking of multiplication

by 5

1 as division by 5; or multiplication by 7

6 as

multiplication by 6 followed by division by 7 (or viceversa)], multiply and divide a given fraction by aninteger, by a unit fraction and by a general fraction

2H3d understand and use unit fractions as multiplicativeinverses [for example, by thinking of multiplication

by 7

6 as multiplication by 6 followed by division by 7

(or vice versa)]; multiply and divide a given fractionby a unit fraction and by a general fraction


38 ��


2F3e convert simple fractions of a whole to percentages ofthe whole and vice versa [for example, analysingdiets, budgets or the costs of running, maintainingand owning a car]

2H3e convert simple fractions of a whole to percentages ofthe whole and vice versa; then understand themultiplicative nature of percentages as operators

[for example, a 15% increase in value Y, followed bya 15% decrease is calculated as 1.15 × 0.85 × Y];calculate an original amount when given thetransformed amount after a percentage change;reverse percentage problems [for example, giventhat a meal in a restaurant costs £36 with VAT at

17.5%, its price before VAT is calculated as £175.1

36 ]

2H3e understand the multiplicative nature of percentagesas operators [for example, a 15% increase in value Y,followed by a 15% decrease is calculated as1.15 × 0.85 × Y]; calculate an original amountwhen given the transformed amount after apercentage change; reverse percentage problems [forexample, given that a meal in a restaurant costs £36with VAT at 17.5%, its price before VAT is

calculated as £175.1

36 ]

2F3f divide a quantity in a given ratio [for example, share£15 in the ratio of 1:2]

2H3f divide a quantity in a given ratio 2H3f divide a quantity in a given ratio

Mental methods

2F3g recall all positive integer complements to 100[for example, 37 + 63 = 100]; recall all multiplicationfacts to 10 × 10, and use them to derive quickly thecorresponding division facts; recall the cubes of 2, 3,4, 5 and 10, and the fraction-to-decimal conversionof familiar simple fractions [for example,

81

,32

,31

,1001

,101

,51

,41

,21

]

2H3g recall integer squares from 2 × 2 to 15 × 15 and thecorresponding square roots, the cubes of 2, 3, 4, 5and 10

2H3g recall integer squares from 2 × 2 to 15 × 15 and thecorresponding square roots, the cubes of 2, 3, 4, 5

and 10, the fact that n0 = 1 and n-1 =

n

1 for

positive integers n [for example,100 = 1; 9-1 =

9

1 ],

the corresponding rule for negative numbers

[for example, 25

1

5

12

2 ==−5 ], nn =2

1

and

33

1

nn = for any positive number n [for example,

525 2

1

= and 4643

1

= ]



�� 39


2F3h round to the nearest integer and to one significantfigure; estimate answers to problems involvingdecimals

2H3h round to a given number of significant figures;develop a range of strategies for mental calculation;derive unknown facts from those they know; convert

2H3h round to a given number of significant figures;convert between ordinary and standard index formrepresentations [for example,

2F3i develop a range of strategies for mental calculation;derive unknown facts from those they know [for

example, estimate 85 ]; add and subtract numbers

mentally with up to two decimal places [for example,13.76 – 5.21, 20.08 + 12.4]; multiply and dividenumbers with no more than one decimal digit,[for example, 14.3 × 4, 56.7 ÷ 7] using thecommutative, associative, and distributive laws andfactorisation where possible, or place valueadjustments

between ordinary and standard index formrepresentations [for example,0.1234 = 1.234 × 10-1], converting to standard indexform to make sensible estimates for calculationsinvolving multiplication and/or division

0.1234 = 1.234 × 10-1], converting to standard indexform to make sensible estimates for calculationsinvolving multiplication and/or division

Written methods

2F3j use standard column procedures for addition andsubtraction of integers and decimals

2F3k use standard column procedures for multiplicationof integers and decimals, understanding where toposition the decimal point by considering whathappens if they multiply equivalent fractions

2F3l use efficient methods to calculate with fractions,including cancelling common factors before carryingout the calculation, recognising that, in many cases,only a fraction can express the exact answer

2H3i use efficient methods to calculate with fractions,including cancelling common factors before carryingout the calculation, recognising that, in many cases,only a fraction can express the exact answer

2F3m solve simple percentage problems, including increaseand decrease [for example, VAT, annual rate ofinflation, income tax, discounts]

2H3j solve percentage problems, including increase anddecrease [for example, simple interest, VAT, annualrate of inflation]; and reverse percentages

2H3j solve percentage problems, [for example, simpleinterest, VAT, annual rate of inflation]; and reversepercentages


40 ��


2F3n solve word problems about ratio and proportion,including using informal strategies and the unitarymethod of solution [for example, given that m

identical items cost £y, then one item costs £m

y and

n items cost £(n × m

y ), the number of items that

can be bought for £z is z × y

m ]

2H3k represent repeated proportional change using amultiplier raised to a power [for example, compoundinterest]

2H3k represent repeated proportional change using amultiplier raised to a power [for example, compoundinterest]

2H3l calculate an unknown quantity from quantities thatvary in direct proportion

2H3l calculate an unknown quantity from quantities thatvary in direct or inverse proportion

2H3m calculate with standard index form [for example,2.4 × 107 × 5 × 103 = 12 × 1010

= 1.2 × 1011,

(2.4 × 107) ÷ (5 × 103) = 4.8 × 103]

2H3m calculate with standard index form [for example,2.4 × 107 × 5 × 103 = 12 × 1010

= 1.2 × 1011,

(2.4 × 107) ÷ (5 × 103) = 4.8 × 103]

2H3n use surds and π in exact calculations, without acalculator

2H3n use surds and π in exact calculations, without acalculator; rationalise a denominator such as

3

3

3

1=

Calculator methods

2F3o use calculators effectively; know how to entercomplex calculations and use function keys forreciprocals, squares and powers

2H3o use calculators effectively and efficiently; know howto enter complex calculations; use an extendedrange of function keys, includingtrigonometrical and statistical functions relevantacross this programme of study

2H3o use calculators effectively and efficiently, know howto enter complex calculations; use an extended rangeof function keys, including trigonometrical andstatistical functions relevant across this programmeof study

2F3p enter a range of calculations, including thoseinvolving measures [for example, time calculations inwhich fractions of an hour must be entered asfractions or as decimals]



�� 41


2F3q understand the calculator display, interpreting itcorrectly [for example, in money calculations, orwhen the display has been rounded by thecalculator], and knowing not to round during theintermediate steps of a calculation

2H3p understand the calculator display, knowing when tointerpret the display, when the display has beenrounded by the calculator, and knowing not to roundduring the intermediate steps of a calculation

2H3q use calculators, or written methods, to calculate theupper and lower bounds of calculations, particularlywhen working with measurements

2H3r use standard index form display and know how toenter numbers in standard index form

2H3r use standard index form display and know how toenter numbers in standard index form

2H3s use calculators for reverse percentage calculations bydoing an appropriate division

2H3s use calculators for reverse percentage calculations bydoing an appropriate division

2H3t use calculators to explore exponential growth anddecay [for example, in science or geography], using amultiplier and the power key

4. Solving numerical problems


2F4a draw on their knowledge of the operations and therelationships between them, and of simple integerpowers and their corresponding roots, to solveproblems involving ratio and proportion, a range ofmeasures including speed, metric units, andconversion between metric and common imperialunits, set in a variety of contexts

2F4b select appropriate operations, methods and strategiesto solve number problems, including trial andimprovement where a more efficient method to findthe solution is not obvious

2H4a draw on their knowledge of operations and inverseoperations (including powers and roots), and ofmethods of simplification (including factorisationand the use of the commutative, associative anddistributive laws of addition, multiplication andfactorisation) in order to select and use suitablestrategies and techniques to solve problems andword problems, including those involving ratio andproportion, repeated proportional change, fractions,percentages and reverse percentages, surds, measuresand conversion between measures, and compoundmeasures defined within a particular situation

2H4a draw on their knowledge of operations and inverseoperations (including powers and roots), and ofmethods of simplification (including factorisationand the use of the commutative, associative anddistributive laws of addition, multiplication andfactorisation) in order to select and use suitablestrategies and techniques to solve problems andword problems, including those involving ratio andproportion, repeated proportional change, fractions,percentages and reverse percentages, inverseproportion, surds, measures and conversionbetween measures, and compound measures definedwithin a particular situation



42 ��


2F4c use a variety of checking procedures, includingworking the problem backwards, and consideringwhether a result is of the right order of magnitude

2F4d give solutions in the context of the problem to anappropriate degree of accuracy, interpreting thesolution shown on a calculator display, andrecognising limitations on the accuracy of data andmeasurements

2H4b check and estimate answers to problems; select andjustify appropriate degrees of accuracy for answers toproblems; recognise limitations on the accuracy ofdata and measurements

2H4b check and estimate answers to problems; select andjustify appropriate degrees of accuracy for answers toproblems; recognise limitations on the accuracy ofdata and measurements

5. Equations, formulae and identities

Use of symbols


2F5a assessed in Module 5 2H5a assessed in Module 5 2H5a distinguish the different roles played by lettersymbols in algebra, using the correct notationalconventions for multiplying or dividing by a givennumber, and knowing that letter symbols representdefinite unknown numbers in equations [forexample, x2 + 1 = 82], defined quantities or variablesin formula [for example, V = IR], general,unspecified and independent numbers in identities[for example, (x + 1)2 = x2 + 2x + 1, for all x] and infunctions they define new expressions or quantitiesby referring to known quantities [for example,

y = 2 – 7x, f(x) = x3; y = x

1 with x ≠ 0]

2F5b assessed in Module 5 2H5b assessed in Module 5 2H5b understand that the transformation of algebraicentities obeys and generalises the well-defined rulesof generalised arithmetic [for example,

a(b + c) = ab + ac]; manipulate algebraic expressionsby collecting like terms

2H5c assessed in Module 5 2H5c know the meaning of and use the words ‘equation’and ‘expression’


�� 43

Direct and inverse proportion


2H5h set up and use equations to solve word and otherproblems involving direct proportion or inverseproportion

[for example, y∝ x, y∝ x2, y∝

x

1 , y∝ 2

1

x

], and relate

algebraic solutions to graphical representation of theequations

6. Sequences, functions and graphs

Quadratic functions


2H6e assessed in Module 5 2H6e generate points and plot graphs of simple quadraticfunctions [for example, y = x2; y = 3x

2 + 4], thenmore general quadratic functions [for example,x

2 – 2x + 1]; find approximate solutions of aquadratic equation from the graph of thecorresponding quadratic function; find theintersection points of the graphs of a linear andquadratic function, knowing that these are theapproximate solutions of the correspondingsimultaneous equations representing the linear andquadratic functions


44 ��

12 Module 4

AO2: Number and algebra


Problem solving









2H1c break down a complex calculation into simpler stepsbefore attempting a solution and justify theirchoice of methods





�� 45

Communicating














Reasoning


2H1j explore, identify, and use pattern and symmetry inalgebraic contexts, investigating whether a particularcase may be generalised further and understand theimportance of a counter-example; identify









46 ��

AO3: Shape, space and measures

1. Using and applying shape, space and measures

Problem solving



3F1a select problem-solving strategies and resources,including ICT tools, to use in geometrical work, andmonitor their effectiveness

3H1a select the problem-solving strategies to use ingeometrical work, and consider and explain theextent to which the selections they made wereappropriate


3F1b select and combine known facts and problem-solving strategies to solve complex problems

3H1b select and combine known facts and problem-solving strategies to solve more complexgeometrical problems

3H1b select and combine known facts and problem-solving strategies to solve more complex geometricalproblems

3F1c identify what further information is needed to solvea geometrical problem; break complex problemsdown into a series of tasks

3H1c develop and follow alternative lines of enquiry 3H1c develop and follow alternative lines of enquiry,justifying their decisions to follow or rejectparticular approaches

Communicating

3F1d interpret, discuss and synthesise geometricalinformation presented in a variety of forms

3F1e communicate mathematically, by presenting andorganising results and explaining geometricaldiagrams

3H1d communicate mathematically, with emphasis on acritical examination of the presentation andorganisation of results, and on effective use ofsymbols and geometrical diagrams


3F1f use geometrical language appropriately 3H1e use precise formal language and exact methods foranalysing geometrical configurations

3F1g review and justify their choices of mathematicalpresentation



�� 47

Reasoning


3F1h distinguish between practical demonstrations andproofs


3F1i apply mathematical reasoning, explaining andjustifying inferences and deductions

3H1f apply mathematical reasoning, progressing frombrief mathematical explanations towards fulljustifications in more complex contexts


3H1g explore connections in geometry; pose conditionalconstraints of the type ‘If … then …’, and askquestions ‘What if …?’ or ‘Why?’


3F1j show step-by-step deduction in solving a geometricalproblem

3H1h show step-by-step deduction in solving a geometricalproblem

3H1i state constraints and give starting points whenmaking deductions


3H1j understand the necessary and sufficient conditionsunder which generalisations, inferences and solutionsto geometrical problems remain valid


48 ��

13 Module 5

AO2: Number and Algebra


Problem solving









2H1c break down a complex calculation into simpler stepsbefore attempting a solution and justify theirchoice of methods






�� 49

Communicating














Reasoning


2H1j explore, identify, and use pattern and symmetry inalgebraic contexts, investigating whether a particularcase can be generalised further and understand theimportance of a counter-example; identify









50 ��

2. Numbers and the number system

Integers



2F2a use the concepts and vocabulary of factor (divisor),multiple and common factor

2H2a assessed in Module 3 2H2a assessed in Module 3

Powers and roots

2F2b use the terms square, positive square root, cube; useindex notation for squares, cubes and powers of 10

2H2b use the terms square, positive square root, negativesquare root, cube and cube root; use index notationand index laws for multiplication and division ofinteger powers; use standard index form,expressed in conventional notation and on acalculator display

2H2b assessed in Module 3

Fractions

2F2c understand equivalent fractions, simplifying afraction by cancelling all common factors; orderfractions by rewriting them with a commondenominator

2H2c understand equivalent fractions, simplifying afraction by cancelling all common factors; orderfractions by rewriting them with a commondenominator

Decimals

2F2d use decimal notation 2H2d assessed in Module 3 2H2d assessed in Module 3

Percentages

2F2e understand that ‘percentage’ means ‘number of partsper 100’ and use this to compare proportions;interpret percentage as the operator ‘so manyhundredths of’ [for example, 10% means 10 parts

per 100 and 15% of Y means 100

15 × Y]

2H2e understand that ‘percentage’ means ‘number of partsper 100’, and interpret percentage as the operator ‘somany hundredths of’ [for example, 10% means 10

parts per 100 and 15% of Y means 100

15 × Y]


�� 51

3. Calculations

Number operations and the relationships between them



2F3a add, subtract, multiply and divide integers and thenany number; multiply or divide any number bypowers of 10, and any positive number by a numberbetween 0 and 1


2F3b use brackets and the hierarchy of operations 2H3b assessed in Module 3

Mental methods

2F3g recall all positive integer complements to 100[for example, 37 + 63 = 100]; recall all multiplicationfacts to 10 × 10, and use them to derive quickly thecorresponding division facts; recall the cubes of 2, 3,4, 5 and 10

2H3g assessed in Module 3 2H3g assessed in Module 3

Calculator methods

2F3o use calculators effectively; use function keys forreciprocals, squares and powers

2H3o assessed in Module 3 2H3o assessed in Module 3

4. Solving numerical problems


2F4a draw on their knowledge of simple integer powersand their corresponding roots, to solve problems



52 ��

5. Equations, formulae and identities

Use of symbols



2F5a distinguish the different roles played by lettersymbols in algebra, knowing that letter symbolsrepresent definite unknown numbers in equations[for example, 5x + 1 = 16], defined quantities orvariables in formulae [for example, V = IR], general,unspecified and independent numbers in identities[for example, 3x + 2x = 5x, for all values of x] and infunctions they define new expressions or quantitiesby referring to known quantities [for example,y = 2x]

2H5a distinguish the different roles played by lettersymbols in algebra, using the correct notationalconventions for multiplying or dividing by agiven number, and knowing that letter symbolsrepresent definite unknown numbers in equations[for example, x2 + 1 = 82], defined quantities orvariables in formulae [for example, V = IR], general,unspecified and independent numbers in identities[for example, (x + 1)2 = x2

+ 2x + 1, for all x], and infunctions they define new expressions or quantitiesby referring to known quantities [for example,

y = 2 – 7x; f(x) = x3; y =

x

1 with x ≠ 0]

2H5a distinguish the different roles played by lettersymbols in algebra, using the correct notationalconventions for multiplying or dividing by a givennumber, and knowing that letter symbols representdefinite unknown numbers in equations [forexample, x2 + 1 = 82], defined quantities or variablesin formulae [for example, V = IR], general,unspecified and independent numbers in identities[for example, (x + 1)2 = x2

+ 2x + 1, for all x], and infunctions they define new expressions or quantitiesby referring to known quantities [for example,

y = 2 – 7x; f(x) = x3; y =

x

1 with x ≠ 0]

2F5b understand that the transformation of algebraicexpressions obeys and generalises the rules ofarithmetic; manipulate algebraic expressions bycollecting like terms, by multiplying a single termover a bracket, and by taking out single termcommon factors [for example,x + 5 – 2x – 1 = 4 – x; 5(2x + 3) = 10x + 15;

)3(32 +=+ xxxx ]

2H5b understand that the transformation of algebraicentities obeys and generalises the well-defined rulesof generalised arithmetic [for example,a(b + c) = ab + ac]; expand the product of twolinear expressions [for example,(x + 1)(x + 2) = x2 + 3x + 2]; manipulate algebraicexpressions by collecting like terms, multiplying asingle term over a bracket, taking out commonfactors [for example, 9x – 3 = 3(3x – 1)], factorisingquadratic expressions, including the difference

of two squares [for example, x2 – 9 = (x + 3)(x – 3)],and cancelling common factors in rationalexpressions [for example,2(x + 1)2/(x + 1) = 2(x + 1)]

2H5b understand that the transformation of algebraicentities obeys and generalises the well-defined rulesof generalised arithmetic [for example,

a(b + c) = ab + ac]; expand the product of two linearexpressions [for example,(x + 1)(x + 2) = x2

+ 3x + 2]; manipulate algebraicexpressions by collecting like terms, multiplying asingle term over a bracket, taking out commonfactors [for example, 9x – 3 = 3(3x – 1)], factorisingquadratic expressions, including the difference oftwo squares [for example, x2 – 9 = (x + 3)(x – 3)], andcancelling common factors in rational expressions[for example, 2(x + 1)2/(x + 1) = 2(x + 1)]

2H5c know the meaning of and use the words ‘equation’,‘formula’, ‘identity’ and ‘expression’

2H5c know the meaning of and use the words ‘equation’,‘formula’, ‘identity’ and ‘expression’


�� 53

Index notation


2F5c use index notation for simple integer powers;substitute positive and negative numbers intoexpressions such as 3x

2 + 4 and 2x3

2H5d use index notation for simple integer powers, andsimple instances of index laws [for example,

x3 × x2 = x5;

3

2

x

x = x-1; (x2)3 = x6]; substitute positive

and negative numbers into expressions such as3x

2 + 4 and 2x3

2H5d use simple instances of index laws [for example,

x3 × x2 = x5;

3

2

x

x = x-1; (x2)3 = x6]

Equations

2H5e set up simple equations [for example, find the anglea in a triangle with angles a, a + 10, a + 20]; solvesimple equations [for example, 5x = 7; 11 – 4x = 2;

3(2x + 1) = 8; 2(1 – x) = 6(2 + x); 4x2 = 49; 3 =

x

12 ]

by using inverse operations or by transforming bothsides in the same way

2H5e set up simple equations [for example, find the angle ain a triangle with angles a, a + 10, a + 20]; solvesimple equations [for example, 5x = 7; 11 – 4x = 2;

3(2x + 1) = 8; 2(1 – x) = 6(2 + x); 4x2 = 49; 3 =

x

12 ]

by using inverse operations or by transforming bothsides in the same way

Linear Equations

2F5e solve linear equations, with integer coefficients, inwhich the unknown appears on either side or onboth sides of the equation; solve linear equationsthat require prior simplification of brackets,including those that have negative signs occurringanywhere in the equation, and those with a negativesolution

2H5f solve linear equations in one unknown, with integeror fractional coefficients, in which the unknownappears on either side or on both sides of theequation; solve linear equations that require priorsimplification of brackets, including those that havenegative signs occurring anywhere in the equation,and those with a negative solution

2H5f solve linear equations in one unknown, with integeror fractional coefficients, in which the unknownappears on either side or on both sides of theequation


54 ��

Formulae


2F5f use formulae from mathematics and other subjectsexpressed initially in words and then using letters andsymbols [for example, formulae for the area of atriangle, the area enclosed by a circle,wage earned = hours worked × rate per hour];substitute numbers into a formula; derive a formula[for example, convert temperatures between degreesFahrenheit and degrees Celsius, find the perimeter ofa rectangle given its area A and the length l of oneside]

2H5g use formulae from mathematics and other subjects[for example, formulae for the area of a triangle or aparallelogram, area enclosed by a circle, volume ofa prism, volume of a cone]; substitute numbersinto a formula; change the subject of a formula,including cases where the subject occurs twice,

or where a power of the subject appears [for

example, find r, given that A = 2rπ , find x given

y = mx + c]; generate a formula [for example, findthe perimeter of a rectangle given its area A and thelength l of one side]

2H5g use formulae from mathematics and other subjects[for example, formulae for the area of a triangle or aparallelogram, area enclosed by a circle, volume of aprism, volume of a cone]; substitute numbers into aformula; change the subject of a formula, includingcases where the subject occurs twice, or where apower of the subject appears [for example, find r,

given that A = 2rπ , find x given y = mx + c];

generate a formula [for example, find the perimeterof a rectangle given its area A and the length l of oneside]

Simultaneous linear equations

2H5i find the exact solution of two simultaneousequations in two unknowns by eliminating a variable,and interpret the equations as lines and theircommon solution as the point of intersection

2H5i find the exact solution of two simultaneousequations in two unknowns by eliminating a variable,and interpret the equations as lines and theircommon solution as the point of intersection

2H5j solve simple linear inequalities in one variable, andrepresent the solution set on a number line; solveseveral linear inequalities in two variables and findthe solution set

2H5j solve simple linear inequalities in one variable, andrepresent the solution set on a number line; solveseveral linear inequalities in two variables and findthe solution set

Quadratic equations

2H5k solve quadratic equations by factorisation 2H5k solve quadratic equations by factorisation,completing the square and using the quadraticformula


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Simultaneous linear and quadratic equations


2H5l solve exactly, by elimination of an unknown, twosimultaneous equations in two unknowns, one ofwhich is linear in each unknown, and the other islinear in one unknown and quadratic in the other[for example, solve the simultaneous equationsy = 11x – 2 and y = 5x

2], or where the second is ofthe form x2

+ y2 = r2

Numerical methods

2H5m use systematic trial and improvement to findapproximate solutions of equations where there is nosimple analytical method of solving them [forexample, x3 – x = 900]

2H5m use systematic trial and improvement to findapproximate solutions of equations where there is nosimple analytical method of solving them [forexample, x3 – x = 900]

6. Sequences, functions and graphs

Sequences


2F6a generate terms of a sequence using term-to-term andposition-to-term definitions of the sequence

2H6a generate common integer sequences (includingsequences of odd or even integers, squared

integers, powers of 2, powers of 10, triangularnumbers); generate terms of a sequence using term-to-term and position-to-term definitions of thesequence; use linear expressions to describe thenth term of an arithmetic sequence, justifying itsform by reference to the activity or context fromwhich it was generated

2H6a generate common integer sequences (includingsequences of odd or even integers, squared integers,powers of 2, powers of 10, triangular numbers); uselinear expressions to describe the nth term of anarithmetic sequence, justifying its form by referenceto the activity or context from which it wasgenerated


56 ��

Graphs of linear functions


2F6b use the conventions for coordinates in the plane;plot points in all four quadrants; plot graphs offunctions in which y is given explicitly in terms of x[for example, y = 2x + 3], or implicitly [for example,x + y = 7]

2H6b use conventions for coordinates in the plane; plotpoints in all four quadrants; recognise (whenvalues are given for m and c) that equations ofthe form y = mx + c correspond to straight-linegraphs in the coordinate plane; plot graphs offunctions in which y is given explicitly in terms of x(as in y = 2x + 3), or implicitly (as in x + y = 7)

2H6b recognise (when values are given for m and c) thatequations of the form y = mx + c correspond tostraight-line graphs in the coordinate plane

2F6c construct linear functions from real-life problemsand plot their corresponding graphs; discuss andinterpret graphs arising from real situations

2H6c find the gradient of lines given by equations of theform y = mx + c (when values are given for m and c);understand that the form y = mx + c represents astraight line and that m is the gradient of the line, andc is the value of the y-intercept; explore the gradientsof parallel lines [for example, know that the linesrepresented by the equations y = –5x and y = 3 – 5x

are parallel, each having gradient (–5)]

2H6c find the gradient of lines given by equations of theform y = mx + c (when values are given for m and c);understand that the form y = mx + c represents astraight line and that m is the gradient of the line, andc is the value of the y-intercept; explore the gradientsof parallel lines and lines perpendicular to theselines [for example, know that the lines representedby the equations y = –5x and y = 3 – 5x are parallel,each having gradient (–5) and that the line with

equation 5

xy = is perpendicular to these lines

and has gradient 5

1 ]

Interpret graphical information

2F6e interpret information presented in a range of linearand non-linear graphs [for example, graphsdescribing trends, conversion graphs, distance-timegraphs, graphs of height or weight against age,graphs of quantities that vary against time, such asemployment]

2H6d construct linear functions and plot thecorresponding graphs arising from real-lifeproblems; discuss and interpret graphs modellingreal situations [for example, distance-time graph for aparticle moving with constant speed, the depth ofwater in a container as it empties, the velocity-timegraph for a particle moving with constantacceleration]

2H6d construct linear functions and plot thecorresponding graphs arising from real-lifeproblems; discuss and interpret graphs modellingreal situations [for example, distance-time graph for aparticle moving with constant speed, the depth ofwater in a container as it empties, the velocity-timegraph for a particle moving with constantacceleration]


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Quadratic functions


2H6e generate points and plot graphs of simple quadraticfunctions [for example, y = x2; y = 3x

2 + 4], thenmore general quadratic functions [for example,y = x2 – 2x + 1]; find approximate solutions of aquadratic equation from the graph of thecorresponding quadratic function

2H6e generate points and plot graphs of simple quadraticfunctions [for example, y = x2; y = 3x

2 + 4], thenmore general quadratic functions [for example,y = x2 – 2x + 1]; find approximate solutions of aquadratic equation from the graph of thecorresponding quadratic function

Other functions

2H6f plot graphs of simple cubic functions [for example,

y = x3], the reciprocal function y = x

1 with x ≠ 0,

using a spreadsheet or graph plotter as well as penciland paper; recognise the characteristic shapes of allthese functions

2H6f plot graphs of simple cubic functions [for example,

y = x3], the reciprocal function y = x

1 with x ≠ 0, the

exponential function y = kx

for integer values of

x and simple positive values of k [for example,

y = x

2 ; y = ( )x2

1 ], the circular functions y = sinx

and y = cosx, using a spreadsheet or graph plotter aswell as pencil and paper; recognise the characteristicshapes of all these functions

Transformation of functions

2H6g apply to the graph of y = f(x) the transformationsy = f(x) + a, y = f(ax), y = f(x + a), y = af(x) forlinear, quadratic, sine and cosine functions f(x)

Loci

2H6h construct the graphs of simple loci 2H6h construct the graphs of simple loci, including the

circle x2 + y

2 = r

2 for a circle of radius r centred

at the origin of coordinates; find graphically theintersection points of a given straight line with

this circle and know that this corresponds tosolving the two simultaneous equationsrepresenting the line and the circle


58 ��

AO3: Shape, space and measures

1. Using and applying shape, space and measures

Problem solving



3F1a select problem-solving strategies and resources,including ICT tools, to use in geometrical work, andmonitor their effectiveness



3F1b select and combine known facts and problem-solving strategies to solve complex problems

3H1b select and combine known facts and problem-solving strategies to solve more complexgeometrical problems

3H1b select and combine known facts and problem-solving strategies to solve more complex geometricalproblems

3F1c identify what further information is needed to solvea geometrical problem; break complex problemsdown into a series of tasks

3H1c develop and follow alternative lines of enquiry 3H1c develop and follow alternative lines of enquiry,justifying their decisions to follow or rejectparticular approaches

Communicating

3F1d interpret, discuss and synthesise geometricalinformation presented in a variety of forms

3F1e communicate mathematically, by presenting andorganising results and explaining geometricaldiagrams



3F1f use geometrical language appropriately 3H1e use precise formal language and exact methods foranalysing geometrical configurations




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Reasoning




3F1i apply mathematical reasoning, explaining andjustifying inferences and deductions





3F1j show step-by-step deduction in solving a geometricalproblem

3H1h show step-by-step deduction in solving a geometricalproblem



3H1j understand the necessary and sufficient conditionsunder which generalisations, inferences and solutionsto geometrical problems remain valid

2. Geometrical reasoning

Angles


3F2a recall and use properties of angles at a point, angleson a straight line (including right angles),perpendicular lines, and opposite angles at a vertex

3F2b distinguish between acute, obtuse, reflex and rightangles; estimate the size of an angle in degrees


60 ��

Properties of triangles and other rectilinear shapes


3F2c use parallel lines, alternate angles and correspondingangles, understand the properties of parallelogramsand a proof that the angle sum of a triangle is 180degrees; understand a proof that the exterior angle ofa triangle is equal to the sum of the interior angles atthe other two vertices

3H2a distinguish between lines and line segments; useparallel lines, alternate angles and correspondingangles, understand the consequent properties ofparallelograms and a proof that the angle sum of atriangle is 180 degrees; understand a proof that theexterior angle of a triangle is equal to the sum of theinterior angles at the other two vertices

3H2a distinguish between lines and line segments

3F2d use angle properties of equilateral, isosceles andright-angled triangles; understand congruence;explain why the angle sum of any quadrilateral is360 degrees

3H2b use angle properties of equilateral, isosceles andright-angled triangles; understand congruence;explain why the angle sum of any quadrilateral is 360degrees

3F2e use their knowledge of rectangles, parallelograms andtriangles to deduce formulae for the area of aparallelogram, and a triangle, from the formula forthe area of a rectangle

3F2f recall the essential properties of special types ofquadrilateral, including square, rectangle,parallelogram, trapezium and rhombus; classifyquadrilaterals by their geometric properties

3H2c recall the definitions of special types ofquadrilateral, including square, rectangle,parallelogram, trapezium and rhombus; classifyquadrilaterals by their geometric properties

3F2g calculate and use the sums of the interior andexterior angles of quadrilaterals, pentagons andhexagons; calculate and use the angles of regularpolygons

3H2d calculate and use the sums of the interior andexterior angles of quadrilaterals, pentagons andhexagons; calculate and use the angles of regularpolygons

3H2e understand and use SSS, SAS, ASA and RHSconditions to prove the congruence of trianglesusing formal arguments, and to verify standard rulerand compass constructions

3H2f understand, recall and use Pythagoras’ theorem in2-D problems; investigate the geometry of cuboidsincluding cubes, and shapes made from cuboids

3H2f understand, recall and use Pythagoras’ theorem in2-D , then 3-D problems; investigate the geometryof cuboids including cubes, and shapes made fromcuboids, including the use of Pythagoras’theorem to calculate lengths in three dimensions


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3H2g understand similarity of triangles and of other planefigures, and use this to make geometric inferences;understand, recall and use trigonometricalrelationships in right-angled triangles, and use theseto solve problems, including those involvingbearings

3H2g understand similarity of triangles and of other planefigures, and use this to make geometric inferences;understand, recall and use trigonometricalrelationships in right-angled triangles, and use theseto solve problems, including those involvingbearings, then use these relationships in 3-Dcontexts, including finding the angles between aline and a plane (but not the angle between twoplanes or between two skew lines); calculate the

area of a triangle using Cab sin 2

1 ; draw, sketch

and describe the graphs of trigonometricfunctions for angles of any size, includingtransformations involving scalings in either orboth the x and y directions; use the sine andcosine rules to solve 2-D and 3-D problems

Properties of circles

3F2i recall the definition of a circle and the meaning ofrelated terms, including centre, radius, chord,diameter, circumference, tangent and arc; understandthat inscribed regular polygons can be constructedby equal division of a circle

3H2h recall the definition of a circle and the meaning ofrelated terms, including centre, radius, chord,diameter, circumference, tangent, arc, sector andsegment; understand that the tangent at anypoint on a circle is perpendicular to the radius atthat point; understand and use the fact thattangents from an external point are equal in

length; explain why the perpendicular from thecentre to a chord bisects the chord; understandthat inscribed regular polygons can be constructedby equal division of a circle; use the facts that theangle subtended by an arc at the centre of acircle is twice the angle subtended at any pointon the circumference, the angle subtended atthe circumference by a semicircle is a right

angle, that angles in the same segment areequal, and that opposite angles of a cyclicquadrilateral sum to 180 degrees

3H2h recall the definition of a circle and the meaning ofrelated terms, including sector and segment;understand that the tangent at any point on a circle isperpendicular to the radius at that point; understandand use the fact that tangents from an external pointare equal in length; explain why the perpendicularfrom the centre to a chord bisects the chord; prove

and use the facts that the angle subtended by an arcat the centre of a circle is twice the angle subtendedat any point on the circumference, the anglesubtended at the circumference by a semicircle is aright angle, that angles in the same segment areequal, and that opposite angles of a cyclicquadrilateral sum to 180 degrees; prove and use thealternate segment theorem


62 ��

3-D shapes


3F2j explore the geometry of cuboids (including cubes),and shapes made from cuboids

3F2k use 2-D representations of 3-D shapes and analyse3-D shapes through 2-D projections and cross-sections, including plan and elevation

3H2i use 2-D representations of 3-D shapes and analyse3-D shapes through 2-D projections and cross-sections, including plan and elevation; solve

problems involving surface areas and volumes ofprisms and cylinders

3H2i solve problems involving surface areas and volumesof prisms, pyramids, cylinders, cones and spheres;solve problems involving more complex shapes

and solids, including segments of circles andfrustums of cones

3. Transformations and coordinates

Specifying transformations


3F3a understand that rotations are specified by a centreand an (anticlockwise) angle; rotate a shape about theorigin; measure the angle of rotation using rightangles or simple fractions of a turn; understand thatreflections are specified by a mirror line, at first usinga line parallel to an axis; understand that translationsare specified by a distance and direction, andenlargements by a centre and positive scale factor

3H3a understand that rotations are specified by a centreand an (anticlockwise) angle; use any point as thecentre of rotation; measure the angle of rotationusing right angles, fractions of a turn or degrees;understand that reflections are specified by a (mirror)line such as y = x or y = –x line; understand thattranslations are specified by giving a distance anddirection (or a vector), and enlargements by a centreand positive scale factor

3H3a use any point as the centre of rotation; measure theangle of rotation using fractions of a turn or degrees;understand that translations are specified by giving avector

Properties of transformations

3F3b recognise and visualise rotations, reflections andtranslations, including reflection symmetry of 2-Dand 3-D shapes, and rotation symmetry of 2-Dshapes; transform triangles and other 2-D shapes bytranslation, rotation and reflection, recognising thatthese transformations preserve length and angle, sothat any figure is congruent to its image under any ofthese transformations

3H3b recognise and visualise rotations, reflections andtranslations, including reflection symmetry of 2-Dand 3-D shapes, and rotation symmetry of 2-Dshapes; transform triangles and other 2-D shapes bytranslation, rotation and reflection andcombinations of these transformations;distinguish properties that are preserved underparticular transformations

3H3b transform triangles and other 2-D shapes bycombinations of translation, rotation and reflection;use congruence to show that translations,

rotations and reflections preserve length andangle, so that any figure is congruent to itsimage under any of these transformations;distinguish properties that are preserved underparticular transformations


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3F3c recognise, visualise and construct enlargements ofobjects using positive scale factors greater than one;understand from this that any two circles and anytwo squares are mathematically similar, while, ingeneral, two rectangles are not

3H3c recognise, visualise and construct enlargements ofobjects; understand from this that any two circlesand any two squares are mathematically similar,while, in general, two rectangles are not, then usepositive fractional scale factors

3H3c use positive fractional and negative scale factors

3F3d recognise that enlargements preserve angle but notlength; identify the scale factor of an enlargement asthe ratio of the lengths of any two correspondingline segments and apply this to triangles; understandthe implications of enlargement for perimeter; useand interpret maps and scale drawings

3H3d recognise that enlargements preserve angle but notlength; identify the scale factor of an enlargement asthe ratio of the lengths of any two correspondingline segments; understand the implications ofenlargement for perimeter; use and interpret mapsand scale drawings; understand the differencebetween formulae for perimeter, area andvolume by considering dimensions

3H3d understand the difference between formulae forperimeter, area and volume by consideringdimensions; understand and use the effect ofenlargement on areas and volumes of shapesand solids

Coordinates

3F3e understand that one coordinate identifies a point ona number line, two coordinates identify a point in aplane and three coordinates identify a point in space,using the terms '1-D', '2-D' and '3-D'; use axes andcoordinates to specify points in all four quadrants;locate points with given coordinates; find thecoordinates of points identified by geometricalinformation [for example, find the coordinates of thefourth vertex of a parallelogram with vertices at(2, 1) (–7, 3) and (5, 6)]; find the coordinates of themid-point of the line segment AB, given points Aand B

3H3e understand that one coordinate identifies a point ona number line, that two coordinates identify a pointin a plane and three coordinates identify a point inspace, using the terms '1-D', '2-D' and '3-D'; use axesand coordinates to specify points in all fourquadrants; locate points with given coordinates; findthe coordinates of points identified by geometricalinformation; find the coordinates of the midpoint ofthe line segment AB, given the points A and B, then

calculate the length AB

3H3e given the coordinates of the points A and B,calculate the length AB


64 ��

Vectors


3H3f understand and use vector notation 3H3f understand and use vector notation; calculate, andrepresent graphically the sum of two vectors, thedifference of two vectors and a scalar multiple ofa vector; calculate the resultant of two vectors;understand and use the commutative andassociative properties of vector addition; solve

simple geometrical problems in 2-D using vectormethods

4. Measures and construction

Measures


3F4a interpret scales on a range of measuring instruments,including those for time and mass; convertmeasurements from one unit to another; knowrough metric equivalents of pounds, feet, miles, pintsand gallons; make sensible estimates of a range ofmeasures in everyday settings

3F4b understand angle measure using the associatedlanguage [for example, use bearings to specifydirection]

3F4c understand and use speed

3H4a use angle measure [for example, use bearings tospecify direction]; know that measurements usingreal numbers depend on the choice of unit;recognise that measurements given to thenearest whole unit may be inaccurate by up toone half in either direction; convertmeasurements from one unit to another;

understand and use compound measures,including speed and density

3H4a know that measurements using real numbers dependon the choice of unit; recognise that measurementsgiven to the nearest whole unit may be inaccurate byup to one half in either direction; understand and usecompound measures, including speed and density


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Construction


3F4d measure and draw lines to the nearest millimetre, andangles to the nearest degree; draw triangles and other2-D shapes using a ruler and protractor, and giveninformation about their side lengths and angles;understand, from their experience of constructingthem, that triangles satisfying SSS, SAS, ASA andRHS are unique, but SSA triangles are not; constructcubes, regular tetrahedra, square-based pyramids andother 3-D shapes from given information

3H4b draw approximate constructions of triangles andother 2-D shapes, using a ruler and protractor, giveninformation about side lengths and angles; constructspecified cubes, regular tetrahedra, square-basedpyramids and other 3-D shapes

3F4e use straight edge and compasses to do standardconstructions, including an equilateral triangle with agiven side

3H4c use a straight edge and compasses to do standardconstructions, including an equilateral triangle with agiven side, the midpoint and perpendicularbisector of a line segment, the perpendicular

from a point to a line, the perpendicular from apoint on a line, and the bisector of an angle

3H4c use a straight edge and compasses to do standardconstructions, including an equilateral triangle with agiven side, the midpoint and perpendicular bisectorof a line segment, the perpendicular from a point toa line, the perpendicular from a point on a line, andthe bisector of an angle

Mensuration

3F4f find areas of rectangles, recalling the formula,understanding the connection to counting squaresand how it extends this approach; recall and use theformulae for the area of a parallelogram and atriangle; find the surface area of simple shapes usingthe area formulae for triangles and rectangles;calculate perimeters and areas of shapes made fromtriangles and rectangles

3F4g find volumes of cuboids, recalling the formula andunderstanding the connection to counting cubes andhow it extends this approach; calculate volumes ofshapes made from cubes and cuboids

3F4h find circumferences of circles and areas enclosed bycircles, recalling relevant formulae

3F4i convert between area measures, including cm2 andm2, and volume measures, including cm3 and m3

3H4d find the surface area of simple shapes by using theformulae for the areas of triangles and rectangles;find volumes of cuboids, recalling the formula andunderstanding the connection to counting cubes andhow it extends this approach; calculate volumes ofright prisms and of shapes made from cubes andcuboids; convert between volume measuresincluding cm3 and m3; find circumferences of circlesand areas enclosed by circles, recalling relevantformulae

3H4d find the surface area of simple shapes by using theformulae for the areas of triangles and rectangles;find volumes of cuboids, recalling the formula andunderstanding the connection to counting cubes andhow it extends this approach; calculate volumes ofright prisms; convert between volume measuresincluding cm3 and m3; calculate the lengths of arcsand the areas of sectors of circles


66 ��

Loci


3H4e find loci, both by reasoning and by using ICT toproduce shapes and paths [for example, a regionbounded by a circle and an intersecting line]

3H4e find loci, both by reasoning and by using ICT toproduce shapes and paths [for example, a regionbounded by a circle and an intersecting line]


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Key Skills and Other Issues

14 Key Skills – Teaching, Developing

and Providing Opportunities for

Generating Evidence

14.1 Introduction The Key Skills Qualification requires candidates to demonstrate levelsof achievement in the Key Skills of Communication, Application of Numberand Information Technology.

The units for the ‘wider’ Key Skills of Improving own Learning and

Performance, Working with Others and Problem Solving are also available.The acquisition and demonstration of ability in these ‘wider’ Key Skillsis deemed highly desirable for all candidates, but they do not form partof the Key Skills Qualification.

Copies of the Key Skills Units may be down loaded from the QCAweb site (www.qca.org.uk/keyskills).

The units for each Key Skill comprise three sections:

A What you need to know.

B What you must do.

C Guidance.

Candidates following a course of study based on this Specification forGCSE Mathematics (Modular) can be offered opportunities todevelop and generate evidence of attainment in aspects of the KeySkills of Communication, Application of Number, Information Technology,

Improving own Learning and Performance, Working with Others and ProblemSolving. Areas of study and learning that can be used to encourage theacquisition and use of Key Skills, and to provide opportunities togenerate evidence for Part B of the units, are signposted below.

14.2 Key Skills Opportunities inMathematics (Modular)

The signposting which follows indicates the opportunities to acquireand produce evidence of the Key Skills in AO2-4. AO1, Using and

applying mathematics which is assessed in the context of AO2-3, alsoprovides opportunities.


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Communication Level 1

What you must do … Signposting of Opportunities for GeneratingEvidence in Subject Content

AO2 AO3 AO4

C1.1 Take part in discussions � � �

C1.2 Read and obtain information � � �

C1.3 Write different types of documents

Communication Level 2


AO2 AO3 AO4

C2.1a Contribute to discussions � � �

C2.1b Give a short talk � � �

C2.2 Read and summarise information � � �

C2.3 Write different types of documents

Application of Number Level 1


AO2 AO3 AO4

N1.1 Interpret information from differentsources

� � �

N1.2 Carry out calculations � � �

N1.3 Interpret results and present findings � � �

Application of Number Level 2

What you must do … Signposting of Opportunities for Generating

Evidence in Subject Content

AO2 AO3 AO4

N2.1 Interpret information from differentsources

� � �

N2.2 Carry out calculations � � �

N2.3 Interpret results and present findings � � �


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Information Technology Level 1


AO2 AO3 AO4

IT1.1 Find, explore and developinformation

� � �

IT1.2 Present information, including text,numbers and images

� � �

Information Technology Level 2



AO2 AO3 AO4

IT2.1 Search for and select information � � �

IT2.2 Explore and develop informationand derive new information

� � �

IT2.3 Present combined information,including text, numbers and images

� � �

Improving own Learning and Performance Level 1



AO2 AO3 AO4

LP1.1 Confirm short-term targets and plan how these will be met

� � �

LP1.2 Follow plan to meet targets and improve performance

� � �

LP1.3 Review progress and achievements

� � �


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Improving own Learning and Performance Level 2


AO2 AO3 AO4

LP2.1 Help set short-term targets and planhow these will be met

� � �

LP2.2 Use plan and support from others,to meet targets

� � �

LP2.3 Review progress and identifyevidence of achievements

� � �

Working with Others Level 1

Signposting of Opportunities for Generating


What you must do …

AO2 AO3 AO4

WO1.1 Confirm what needs to be doneand who is to do it

� � �

WO1.2 Work towards agreed objectives � � �

WO1.3 Identify progress and suggestimprovements

� � �

Working with Others Level 2

Signposting of Opportunities for GeneratingEvidence in Subject Content

What you must do …

AO2 AO3 AO4

WO2.1 Plan work and confirm workingarrangements

� � �

WO2.2 Work cooperatively towardsachieving identified objectives

� � �

WO2.3 Exchange information on progressand agree ways of improving workwith others

� � �


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Problem Solving Level 1


AO2 AO3 AO4

PS1.1 Confirm understanding of givenproblems

� � �

PS1.2 Plan and try out ways of solvingproblems

� � �

PS1.3 Check if problems have beensolved and describe the results

� � �

Problem Solving Level 2



AO2 AO3 AO4

PS2.1 Identify problems and come upwith ways of solving them

� � �

PS2.2 Plan and try out options � � �

PS2.3 Apply given methods to check ifproblems have been solved anddescribe the results

� � �

The signposting in the twelve tables above represents the possibleopportunities to acquire and produce evidence of the Key Skillsthrough this specification. Such opportunities are dependent on thedetailed course of study delivered within centres.

14.3 Further Guidance More specific guidance and examples of tasks that can provideevidence of single Key Skills, or composite tasks that can provideevidence of more than one Key Skill, are given in the AQAspecification support material, particularly the Teachers’ Guide.

14.4 Exemptions for the Key SkillsQualification

GCSE A*- C examination performance on this specification providesexemptions for the external test in Application of Number at Level 2.


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15 Spiritual, Moral, Ethical, Social,

Cultural and Other Issues

15.1 Spiritual, Moral, Ethical,Social, Cultural and OtherIssues

Mathematics provides opportunities to promote:

• spiritual development, through explaining the underlying mathematicalprinciples behind some of the natural forms and patterns in theworld around us;

• moral development, helping pupils recognise how logical reasoningcan be used to consider the consequences of particular decisionsand choices helping them learn the value of mathematical truth;

• social development, through helping pupils work together productivelyon complex mathematical tasks and helping them see that theresult is often better than could be achieved separately;

• cultural development, through helping pupils appreciate thatmathematical thought contributes to the development of ourculture and is becoming increasingly central to our highlytechnological future, and through recognising that mathematiciansfrom many cultures have contributed to the development ofmodern day mathematics.

15.2 European Dimension AQA has taken account of the 1988 Resolution of the Council of theEuropean Community in preparing this specification and associatedspecimen papers.

15.3 Environmental Issues AQA has taken account of the 1988 Resolution of the Council of theEuropean Community and the Report Environmental Responsibility: An

Agenda for Further and Higher Education 1993 in preparing thisspecification and associated specimen papers.

15.4 Citizenship Coursework tasks, particularly those for AO4 Handling data, promotethe skills of enquiry and communication. They also encourage the skillof participation and responsible action in the educationalestablishment and/or communication.

15.5 Avoidance of Bias AQA has taken great care in the preparation of this specification andassociated specimen papers to avoid bias of any kind.

15.6 Health and Safety Coursework tasks, particularly those for AO4 Handling data provideopportunities to promote Health and Safety issues.


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15.7 ICT (a) Pupils should be given opportunities to apply and developtheir ICT capability through the use of ICT tools to supporttheir learning in mathematics.

(b) Pupils should be given opportunities to support their work by being taught to :

(i) find things out from a variety of sources, selecting and synthesising the information to meet their needs and developing an ability to question its accuracy, bias and plausibility;

(ii) develop their ideas using ICT tools to amend and refine their work and enhance its quality and accuracy;

(iii) exchange and share information, both directly and through electronic media;

(iv) review, modify and evaluate their work, reflecting critically on its quality, as it progresses.

15.8 Other issues Mathematics provides opportunities to promote:

• thinking skills, through developing pupils’ problem-solving skillsand deductive reasoning;

• financial capability, through applying mathematics to problems set infinancial contexts;

• enterprise and entrepreneurial skills, through developing pupils’ abilitiesto apply mathematics in science and technology, in economics andin risk assessment;

• work related learning, through developing pupils’ abilities to use andapply mathematics in workplace situations and in solving real-lifeproblems.


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Internal Assessment (Coursework)

16 Nature of the Coursework

Modules

16.1 Introduction There are two alternative approaches to the assessment of thecoursework modules:

• Option T centres may choose from a bank of coursework tasksprovided by AQA or they set their own coursework tasks; centresthen mark the coursework tasks with moderation of candidates’coursework by AQA;

• Option X centres choose from the bank of coursework tasksprovided by AQA in this specification and candidates’ courseworkis marked by AQA.

Apart from the choice of coursework tasks and the method ofassessment, the nature of the coursework is the same for Option Tand Option X. The following details apply to both Option T andOption X. It is not necessary to use the same option for both tasks.

The details for the coursework are also common to GCSEMathematics Specification A.

16.2 Module 2 Module 2 assesses the Handling data task (AO4 task) which must be setin the context of AO4. Candidates are expected to submit one taskonly. Tasks based on probability only, without data handling, areunlikely to score well on these criteria and should be avoided.Simulation activities are acceptable provided that they lead to statisticaltasks rather than probability tasks. Candidates may choose to usestatistical information from the Internet or other sources. TheAssessment Criteria for the AO4 task are given in section 17.5. TheAO4 task is marked out of a total of 24 marks. The coursework taskis expected to take approximately two weeks to complete, includinglesson and homework time. It is not permissible for the Handling data

project (AO4 task) to be re-used as the Module 4 coursework task.

16.3 Module 4 The Using and Applying Mathematics task (AO1 task) submitted forModule 4 must be set in the context of AO2 and/or AO3. One taskis expected, however, candidates may submit up to two tasks in orderto satisfy the assessment criteria for AO1. The Assessment Criteriafor the AO1 task are given in section 17.6. The AO1 task is markedout of a total of 24 marks and if two tasks are submitted, the bettermark in each strand should be used. The coursework task is expectedto take approximately two weeks to complete, including lesson andhomework time.


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16.4 General Module 2 and Module 4 will be offered three times a year in theMarch, June and November examination series. Centres may choosethe most appropriate examination series to submit the tasks forassessment. Centres may enter candidates for Module 2 and forModule 4 in different examination series. For example, centres mayenter candidates for Module 2 in the June of year 10 and for Module 4in the June of year 11.

16.5 Philosophy It is intended that coursework should be an integral part of theteaching and learning process. It must not be regarded as anadditional or separate part of this process. Therefore it is importantthat the scheme of work includes activities designed to develop thestrands that are assessed in Module 2 and Module 4. The Module 2AO4 coursework task provides an opportunity for candidates to carryout an extended piece of work using Handling data skills. The Module4, AO1 coursework task provides an opportunity for candidates toconduct an extended piece of work which enhances theirunderstanding of the mathematics of AO2 and/or AO3. Candidatesare expected to use appropriate mathematical skills to investigate andcarry out the tasks. These skills may involve the use of practicalequipment and computers where appropriate to the tasks. Tasksshould be chosen so that they are appropriate for the candidate and,by their nature, do not limit the mark that can be awarded.

Coursework also provides an appropriate method for generatingevidence for the six Key Skills: Communication, Application of Number,Information Technology, Improving own Learning and Performance, Working with

Others and Problem Solving.


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17 Assessment Criteria for the

Coursework Modules

17.1 Introduction There are two different sets of assessment criteria, one for each of thecoursework modules. For Module 2 (AO4 task) the assessmentcriteria for Handling data are used and for Module 4 (AO1 task) theassessment criteria for Using and applying mathematics are used.

17.2 Module 2Handling data (AO4 task)

Candidates will be assessed in terms of their attainment in each of thefollowing three strands which correspond to the Programme of Studyfor Handling data at National Curriculum Key Stages 3 and 4.

Strand Maximum mark

1 Specify the problem and plan 8

2 Collect, process and represent data 8

3 Interpret and discuss results 8

Maximum total mark 24

The score in each of the three strands should be that which reflectsthe best performance by the candidate in that strand. These marksshould be totalled to give a mark out of 24.

The criteria are to be used as best fit indicative descriptions and thestatements within them are not to be taken as hurdles. This meanscandidates’ work should be assessed in relation to the criteria taken asholistic descriptions of performance. The first consideration is whichof the descriptions in each strand best describes the work in acandidate’s project. Once that is established, the final step is to decidebetween the lower and the higher tier mark available for thatdescription; this decision may well involve looking again at the criteriaabove and below the selected best fitting criterion. It is notappropriate to take each statement in each description and regard it asa separate assessment criterion. Nor is it necessary to considerwhether the majority of the statements within a criterion have beenmet.

A mark of 0 should be awarded if a candidate’s work fails to satisfy therequirements for 1 mark.

Descriptions for higher marks subsume those for lower marks.

Where there are references to ‘at least the level detailed in the handlingdata paragraph of the grade description for grade X ’ , work which usesno technique beyond the specified grade is indicative of the lower ofthe two marks. To obtain the higher of the two marks requiresprocessing and analysis using techniques that best fit a moredemanding standard.

In these criteria, there is an intended approximate link between 7marks and grade A, 5 marks and grade C and 3 marks and grade F.


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17.3 Module 4Using and ApplyingMathematics (AO1 task)

Candidates will be assessed in terms of their attainment in each of thefollowing three strands which correspond to the three areas of theProgramme of Study for Using and applying mathematics at NationalCurriculum Key Stages 3 and 4.

Strand Maximum mark

1 Making and monitoring decisions tosolve problems

8

2 Communicating mathematically 8

3 Developing skills of mathematicalreasoning

8

Maximum total mark 24

The score in each of the three strands should be that which reflectsthe best performance by the candidate in that strand. These marksshould be totalled to give a mark out of 24.

The criteria are to be used as best fit indicative descriptions and thestatements within them are not to be taken as hurdles. It is necessary,however, for the majority of the statement to be met for the mark tobe awarded.

The mark descriptions within a strand are designed to be broadlyhierarchical. This means that, in general, a description at a particularmark subsumes those at lower marks. Therefore the mark awardedmay not be supported by direct evidence of achievement of lowermarks in each strand.

It is assumed that tasks which allow higher marks will involve a moresophisticated approach and/or treatment.

The AO1 coursework task must be set in the context of AO2 (Numberand algebra) and/or AO3 (Shape, space and measures).

In these criteria, there is an intended approximate link between 7marks and grade A, 5 marks and grade C and 3 marks and grade F.

17.4 Reporting of the CourseworkModules

The mark out of a total of 24 awarded for each Module is reported ona Uniform Mark Scale (see section 26.3). The rules for re-sitting andcarrying forward the coursework modules are also given in Sections26.5 and 26.6.


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17.5 Module 2 (AO4 task) – Assessment criteria for Handling dataStrand 1

Specify the problem and plan

Strand 2

Collect, process and represent data

Strand 3

Interpret and discuss results

1-2 Candidates choose a simple well-defined problem. Their aims

have some clarity. The appropriate data to collect are

reasonably obvious. An overall plan is discernible and some

attention is given to whether the plan will meet the aims. Thestructure of the report as a whole is loosely related to the

aims.

Candidates collect data with limited relevance to the problem and

plan. The data are collected or recorded with little thought given to

processing. Candidates use calculations of the simplest kind. The

results are frequently correct. Candidates present information andresults in a clear and organised way. The data presentation is

sometimes related to their overall plan.

Candidates comment on patterns in the data. They

summarise the results they have obtained but make little

attempt to relate the results to the initial problem.

3-4 Candidates choose a problem involving routine use of simple

statistical techniques and set out reasonably clear aims.Consideration is given to the collection of data. Candidates

describe an overall plan largely designed to meet the aims and

structure the project report so that results relating to some of

the aims are brought out. Where appropriate, they use asample of adequate size.

Candidates collect data with some relevance to the problem and

plan. The data are collected or recorded with some considerationgiven to efficient processing. Candidates use straightforward and

largely relevant calculations involving techniques of at least the level

detailed in the handling data paragraph of the grade description for

grade F. The results are generally correct. Candidates showunderstanding of situations by describing them using statistical

concepts, words and diagrams. They synthesise information

presented in a variety of forms. Their writing explains and informs

their use of diagrams, which are usually related to their overall plan.They present their diagrams correctly, with suitable scales and titles.

Candidates comment on patterns in the data and any

exceptions. They summarise and give a reasonably correctinterpretation of their graphs and calculations. They attempt

to relate the summarised data to the initial problem, though

some conclusions may be incorrect or irrelevant. They make

some attempt to evaluate their strategy.

5-6 Candidates consider a more complex problem. They choose

appropriate data to collect and state their aims in statistical

terms with the selection of an appropriate plan. Their plan is

designed to meet the aims and is well described. Candidatesconsider the practical problems of carrying out the survey or

experiment. Where appropriate, they give reasons for

choosing a particular sampling method. The project report is

well structured so that the project can be seen as a whole.

Candidates collect largely relevant and mainly reliable data. The

data are collected in a form designed to ensure that they can be

used. Candidates use a range of more demanding, largely relevant

calculations that include techniques of at least the level detailed inthe handling data paragraph of the grade description for grade C.

The results are generally correct and no obviously relevant

calculation is omitted. There is little redundancy in calculation or

presentation. Candidates convey statistical meaning through preciseand consistent use of statistical concepts that is sustained

throughout the work. They use appropriate diagrams for

representing data and give a reason for their choice of presentation,

explaining features they have selected.

Candidates comment on patterns in the data and suggest

reasons for exceptions. They summarise and correctly

interpret their graphs and calculations, relate the summarised

data to the initial problem and draw appropriate inferences.Candidates use summary statistics to make relevant

comparisons and show an informal appreciation that results

may not be statistically significant. Where relevant, they allow

for the nature of the sampling method in making inferencesabout the population. They evaluate the effectiveness of the

overall strategy and make a simple assessment of limitations.

7-8 Candidates work on a problem requiring creative thinking andcareful specification. They state their aims clearly in statistical

terms and select and develop an appropriate plan to meet

these aims giving reasons for their choice. They foresee and

plan for practical problems in carrying out the survey orexperiment. Where appropriate, they consider the nature and

size of sample to be used and take steps to avoid bias. Where

appropriate, they use techniques such as control groups, or

pre-tests of questionnaires or data sheets, and refine these toenhance the project. The project report is well structured and

the conclusions are related to the initial aims.

Candidates collect reliable data relevant to the problem underconsideration. They deal with practical problems such as non-

response, missing data or ensuring secondary data are appropriate.

Candidates use a range of relevant calculations that include

techniques of at least the level detailed in the handling dataparagraph of the grade description for grade A. These calculations

are correct and no obviously relevant calculation is omitted.

Numerical results are rounded appropriately. There is no

redundancy in calculation or presentation. Candidates use languageand statistical concepts effectively in presenting a convincing

reasoned argument. They use an appropriate range of diagrams to

summarise the data and show how variables are related.

Candidates comment on patterns and give plausible reasonsfor exceptions. They correctly summarise and interpret

graphs and calculations. They make correct and detailed

inferences from the data concerning the original problem

using the vocabulary of probability. Candidates appreciate thesignificance of results they obtain. Where relevant, they allow

for the nature and size of the sample and any possible bias in

making inferences about the population. They evaluate the

effectiveness of the overall strategy and recognise limitationsof the work done, making suggestions for improvement.

They comment constructively on the practical consequences

of the work.


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17.6 Module 4 (AO1 task) – Assessment criteria for Using and Applying Mathematics

Strand 1):Making and monitoring decisions to solve problems

Strand 2:Communicating mathematically

Strand 3:Developing skills of mathematical reasoning

1 Candidates try different approaches and find ways ofovercoming difficulties that arise when they are solvingproblems. They are beginning to organise their work andcheck results.

Candidates discuss their mathematical work and arebeginning to explain their thinking. They use andinterpret mathematical symbols and diagrams.

Candidates show that they understand a generalstatement by finding particular examples that match it.

2 Candidates are developing their own strategies forsolving problems and are using these strategies both inworking within mathematics and in applyingmathematics to practical contexts.

Candidates present information and results in a clear andorganised way, explaining the reasons for theirpresentation.

Candidates search for a pattern by trying out ideas oftheir own.

3 In order to carry through tasks and solve mathematicalproblems, candidates identify and obtain necessaryinformation; they check their results, consideringwhether these are sensible.

Candidates show understanding of situations bydescribing them mathematically using symbols, wordsand diagrams.

Candidates make general statements of their own,based on evidence they have produced, and give anexplanation of their reasoning.

4 Candidates carry through substantial tasks and solvequite complex problems by breaking them down intosmaller, more manageable tasks.

Candidates interpret, discuss and synthesise informationpresented in a variety of mathematical forms. Theirwriting explains and informs their use of diagrams.

Candidates are beginning to give a mathematicaljustification for their generalisations; they test them bychecking particular cases.

5 Starting from problems or contexts that have beenpresented to them, candidates introduce questions oftheir own, which generate fuller solutions.

Candidates examine critically and justify their choice ofmathematical presentation, considering alternativeapproaches and explaining improvements they havemade.

Candidates justify their generalisations or solutions,showing some insight into the mathematical structureof the situation being investigated. They appreciate thedifference between mathematical explanation andexperimental evidence.

6 Candidates develop and follow alternative approaches.They reflect on their own lines of enquiry whenexploring mathematical tasks; in doing so they introduceand use a range of mathematical techniques.

Candidates convey mathematical meaning throughconsistent use of symbols.

Candidates examine generalisations or solutionsreached in an activity, commenting constructively onthe reasoning and logic employed, and make furtherprogress in the activity as a result.

7 Candidates analyse alternative approaches to problemsinvolving a number of features or variables. They givedetailed reasons for following or rejecting particular linesof enquiry.

Candidates use mathematical language and symbolsaccurately in presenting a convincing reasoned argument.

Candidates' reports include mathematical justificationsexplaining their solutions to problems involving anumber of features or variables.

8 Candidates consider and evaluate a number ofapproaches to a substantial task. They exploreextensively a context or area of mathematics with whichthey are unfamiliar. They apply independently a range ofappropriate mathematical techniques.

Candidates use mathematical language and symbolsefficiently in presenting a concise reasoned argument.

Candidates provide a mathematically rigorousjustification or proof of their solution to a complexproblem, considering the conditions under which itremains valid.


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Option T – Centre-Assessed

Modules 2 and 4

18 Guidance on Setting the

Centre-Assessed Modules

18.1 Introduction Centres following Option T may choose from the AQA-set tasks ormay choose their own tasks based on the guidance provided in theTeachers’ Guide and coursework support materials.

The AQA-set tasks for submission in 2004 and 2005 for Module 2 aregiven in Appendix C and those for Module 4 in Appendix D.AQA-set tasks may be removed or added from year to year. It istherefore essential that candidates wishing to submit work underOption X use current versions.

Teachers should note that in the AQA-set Handling Data tasks theword ‘hypothesis’ is used for a general prediction which is to betested.

It is important that teachers consider very carefully all types ofactivities which will provide valid evidence of achievement. Theactivities in which candidates are involved should be designed tomake reasonable demands and to enable positive achievement to bedemonstrated in relation to the assessment criteria. The tasks chosentherefore must be open to investigation by a variety of differentmethods, and open to investigations that permit candidates todemonstrate their best attainment in all three strands of the markingcriteria.

Teachers will find it helpful to refer to the assessment criteria whendesigning tasks. It is particularly important to ensure that the taskschosen do not limit the mark that can be achieved by the candidate.

18.2 Advice on group activities For the AO4 task it is permissible for candidates to collect data as agroup or class. It is important that teachers ensure that the analysisand writing up of this work is carried out individually by candidates,so that the requirements of the specification are met.

18.3 Coursework Advisers Coursework Advisers are available to assist centres with any mattersrelating to coursework.


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19 Supervision and Authentication

19.1 Supervision of Candidates’Work

Candidates’ work for assessment must be undertaken under conditionswhich allow the teacher to supervise the work and enable the work tobe authenticated. If it is necessary for some assessed work to be doneoutside the centre, sufficient work must take place under directsupervision to allow the teacher to authenticate each candidate’s wholework with confidence.

19.2 Guidance by the Teacher The work assessed must be solely that of the candidate concerned.Any assistance given to an individual candidate which is beyond thatgiven to the group as a whole must be recorded on the Candidate Record

Form.

19.3 Unfair Practice At the start of the course, the supervising teacher is responsible forinforming candidates of the AQA Regulations concerning malpractice.Candidates must not take part in any unfair practice in the preparationof coursework to be submitted for assessment, and must understandthat to present material copied directly from books or other sourceswithout acknowledgement will be regarded as deliberate deception.Centres must report suspected malpractice to AQA. The penalties formalpractice are set out in the AQA General Regulations.

19.4 Authentication of Candidates’Work

Both the candidate and the teacher are required to sign declarationsconfirming that the work submitted for assessment is the candidate'sown. The teacher declares that the work was conducted under thespecified conditions, and records details of any additional assistance.

Sample Candidate Record Forms for Option T are provided inAppendix E. Current Candidate Record Forms are available separately onthe AQA website under Administration/Procedures/CourseworkAdministration.


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20 Standardisation

20.1 Standardising Meetings Annual standardising meetings for both Specification A andSpecification B will usually be held in the autumn term. Centresentering candidates for the first time must send a representative to ameeting. Attendance is also mandatory in the following cases:

• where there has been a serious misinterpretation of thespecification requirements;

• where the nature of coursework tasks set by a centre has beeninappropriate;

• where a significant adjustment has been made to a centre’s marksin the previous year’s examination.

After the first year, attendance is at the discretion of centres. At thesemeetings support will be provided for centres in the development ofappropriate coursework tasks and assessment procedures.

20.2 Internal Standardisation ofMarking

The centre is required to standardise the assessments across differentteachers and teaching groups to ensure that all candidates at the centrehave been judged against the same standards. If two or more teachersare involved in marking a component, one teacher must be designatedas responsible for internal standardisation. Common pieces of workmust be marked on a trial basis and differences between assessmentsdiscussed at a training session in which all teachers involved mustparticipate. The teacher responsible for standardising the markingmust ensure that the training includes the use of reference and archivematerials such as work from a previous year or examples provided byAQA. The centre is required to send to the moderator the CentreDeclaration Sheet, duly signed, to confirm that the marking of centre-assessed work at the centre has been standardised. If only one teacherhas undertaken the marking, that person must sign this form.

A specimen Centre Declaration Sheet is provided in Appendix E.


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21 Administrative Procedures

21.1 Evidence to support theaward of marks

During the course teachers should keep records of their assessmentsin a form which facilitates the complete and accurate submission ofthe final assessments at the end of the course.

When the assessments are complete, the marks awarded under each ofthe assessment criteria must be entered on the Candidate Record Form,with supporting information given in the spaces provided. Aspecimen Candidate Record Form for Module 2 and for Module 4appears in Appendix E; the exact design may be modified before theoperational version is issued and the correct year’s Candidate RecordForms should always be used.

21.2 Recording AssessmentsThe candidates’ work must be marked according to the assessmentcriteria set out in Sections 17.5 and 17.6. The marks and supportinginformation must be recorded in accordance with the instructions inSection 21.3. The completed Candidate Record Form for each candidatemust be attached to the work and made available to AQA on request.

21.3 Submitting Marks and SampleWork for Moderation

The total component mark for each candidate must be submitted toAQA on the mark sheets provided or by Electronic Data Interchange(EDI) by the specified date and copies sent to the Moderator. Centreswill be informed which candidates’ work is required in the samples tobe submitted to the moderator.

21.4 Problems with IndividualCandidates

Teachers should be able to accommodate the occasional absence ofcandidates by ensuring that the opportunity is given for them to makeup missed assessments.

Special consideration should be requested for candidates whose workhas been affected by illness or other exceptional circumstances.Information about the procedure is issued separately.

If work is lost, AQA should be notified immediately of the date of theloss, how it occurred, and who was responsible for the loss. AQA willadvise on the procedures to be followed in such cases. Where specialhelp which goes beyond normal learning support is given, AQA mustbe informed so that such help can be taken into account whenassessment and moderation take place.

Candidates who move from one centre to another during the coursesometimes present a problem for a scheme of internal assessment.Possible courses of action depend on the stage at which the movetakes place. If the move occurs early in the course the new centreshould take responsibility for assessment. If it occurs late in thecourse it may be possible to accept the assessments made at theprevious centre. Centres should contact AQA at the earliest possiblestage for advice about appropriate arrangements in individual cases.


84 ��

21.5 Retaining Evidence The centre must retain the work of all candidates, with Candidate RecordForms attached, under secure conditions from the time it is assessed, toallow for the possibility of an enquiry upon results. The work may bereturned to candidates after the issue of results provided that noenquiry upon results is to be made which will include re-moderation ofthe coursework component. If an enquiry upon results is to be made,the work must remain under secure conditions until requested byAQA.

22 Moderation

22.1 Moderation Procedures Moderation of the coursework is by inspection of a sample ofcandidates' work, sent by post from the centre to a moderatorappointed by AQA. The centre marks must be submitted to AQAand the sample of work must reach the moderator by the specifieddate in the year in which the qualification is awarded.

The evidence must be presented in a clear and helpful way for themoderator. The candidates’ work must be annotated to identify, asprecisely as possible, where in the work the relevant criteria have beensatisfied so that the reasons why marks have been awarded are clear.Details must also be given of the context within which the work wasdone, to enable the moderator to judge the attainment inherent in thework.

Following the re-marking of the sample work, the moderator’s marksare compared with the centre marks to determine whether anyadjustment is needed to bring the centre’s assessments into line withstandards generally. In some cases it may be necessary for themoderator to call for the work of other candidates. In order to meetthis possible request, centres must have available the coursework andCandidate Record Form of every candidate entered for the examinationand be prepared to submit it on demand. Mark adjustments willnormally preserve the centre’s order of merit but, where majordiscrepancies are found, AQA reserves the right to alter the order ofmerit.

22.2 Post-Moderation Procedures On publication of the GCSE results, the centre is supplied with detailsof the final marks for the coursework component.

The candidates' work is returned to the centre after the examinationwith a report form from the moderator giving feedback to the centreon the appropriateness of the tasks set, the accuracy of theassessments made, and the reasons for any adjustments to the marks.

Some candidates' work may be retained by AQA for archive purposes.


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Option X - AQA-Assessed

Modules 2 and 4

23 Guidance on Setting the

AQA-Assessed Modules

23.1 Introduction Centres following Option X must select coursework tasks from thebank of AQA-set tasks provided in Appendix C for Module 2 orAppendix D for Module 4.

The AQA-set tasks and Mark Schemes will be published each year.Tasks may be removed or added from year to year. It is thereforeessential that the latest version is used each year.

Teachers should note that in the AQA-set Handling Data tasks theword ‘hypothesis’ is used for a general prediction which is to be tested.

23.2 Advice on group activities For the AO4 task it is permissible for candidates to collect data as agroup or class. It is important that teachers ensure that the analysisand writing up of this work is carried out individually by candidates, sothat the requirements of the specification are met.

23.3 Coursework Advisers Coursework Advisers are available to assist centres with any mattersrelating to coursework. Details will be provided when AQA knowswhich centres are following the specification.


86 ��

24 Supervision and Authentication

24.1 Supervision of Candidates’Work

Candidates’ work for assessment must be undertaken under conditionswhich allow the teacher to supervise the work and enable the work tobe authenticated. If it is necessary for some assessed work to be doneoutside the centre, sufficient work must take place under directsupervision to allow the teacher to authenticate each candidate’s wholework with confidence.

Private candidates who follow Option X and follow an open-learningcourse with a tutorial college, or attend a part-time course at a schoolor college, may have their work authenticated by their tutor.Candidates who do not have a tutor must make arrangements to carryout the tasks at their examination centre. In this case, the work shouldbe supervised and the examination officer must sign the declarationthat all the work has been carried out by the candidate.

24.2 Guidance by the Teacher The work assessed must be solely that of the candidate concerned.Any assistance given to an individual candidate which is beyond thatgiven to the group as a whole must be recorded on the Candidate RecordForm.

24.3 Unfair Practice At the start of the course, the supervising teacher is responsible forinforming candidates of the AQA Regulations concerning malpractice.Candidates must not take part in any unfair practice in the preparationof coursework to be submitted for assessment, and must understandthat to present material copied directly from books or other sourceswithout acknowledgement will be regarded as deliberate deception.Centres must report suspected malpractice to AQA. The penalties formalpractice are set out in the AQA General Regulations.

24.4 Authentication of Candidates’Work

Both the candidate and the teacher are required to sign declarationsconfirming that the work submitted for assessment is the candidate'sown. The teacher declares that the work was conducted under thespecified conditions, and records details of any additional assistance.

Sample Candidate Record Forms for Option X are provided inAppendix E. Current Candidate Record Forms are available separately onthe AQA website under Administration/Procedures/CourseworkAdministration.


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25 Administrative Procedures

25.1 Evidence of attainment Where there is ephemeral evidence of attainment, which does notform part of the candidate’s written record, brief notes of eachcandidate’s achievement in these skill areas should be supplied, withthe coursework, to AQA.

25.2 Problems with IndividualCandidates

Teachers should be able to accommodate the occasional absence ofcandidates by ensuring that the opportunity is given for them to makeup missed assessments.

Special consideration should be requested for candidates whose workhas been affected by illness or other exceptional circumstances.Information about the procedure is issued separately.

If work is lost, AQA should be notified immediately of the date of theloss, how it occurred, and who was responsible for the loss. AQA willadvise on the procedures to be followed in such cases. Where specialhelp which goes beyond normal learning support is given, AQA mustbe informed so that such help can be taken into account whenassessment and moderation take place.

Candidates who move from one centre to another during the coursesometimes present a problem for a scheme of internal assessment.Possible courses of action depend on the stage at which the movetakes place. If the move occurs early in the course the new centreshould take responsibility for assessment. If it occurs late in thecourse it may be possible to accept the assessments made at theprevious centre. Centres should contact AQA at the earliest possiblestage for advice about appropriate arrangements in individual cases.


88 ��

Awarding and Reporting

26 Grading, Shelf-Life and Re-Sits

26.1 Qualification Title The qualification based on this specification has the following title:AQA GCSE (modular) in Mathematics: (B).

26.2 Grading System The qualification will be graded on an 8 point grade scale A*, A, B, C,D, E, F, G.

The written paper modules are offered at three tiers of entry:Foundation tier, Intermediate tier and Higher tier. For candidatesentered for the Foundation tier, grades D-G are available. Forcandidates entered for the Intermediate tier, grades B-E are available.For candidates entered for the Higher tier, grades A*-C are available.Candidates may enter for each individual module at a different tier ofentry. However, the final range of grades available to a candidate isdetermined by the tier of entry of Module 5.

26.3 The determination ofcandidates’ final grades

For each module, candidates’ results are reported on a Uniform Mark

Scale which is related to grades by means of the followingcorrespondence.

Module 1 (Maximum uniform mark = 66)

Mark range Grade

59 - 6653 - 5846 - 5240 - 4533 - 3926 - 3220 - 2513 - 19 0 - 12

A*ABCDEFGU

Modules 2 and 4 (Maximum uniform mark = 60)

Mark range Grade

54 - 6048 - 5342 - 4736 - 4130 - 3524 - 2918 - 2312 - 17 0 - 11

A*ABCDEFGU


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Mark range Grade

103 - 114 91 - 102 80 - 90 68 - 79 57 - 67 46 - 56 34 - 45 23 - 33 0 - 22

A*ABCDEFGU


Mark range Grade

270 - 300240 - 269210 - 239180 - 209150 - 179120 - 149 90 - 119 60 - 89 0 - 59

A*ABCDEFGU

A candidate’s uniform mark is calculated from his/her raw mark forthe module by using the grade boundaries set by the awardingcommittee. For example, a candidate who achieved the minimum rawmark required for grade B on Module 1 receives a uniform mark of 46.(The marks required for each grade are published annually in the reporton the examination.)

A candidate cannot obtain a uniform mark corresponding to a gradewhich is above the range for the tier. For example, on Module 1 acandidate entered for the Foundation tier (grade range D-G) cannotobtain a uniform mark higher than 39, even if he/she achieves themaximum (raw) marks for the paper.

On individual modules there is a small ‘safety net’ for candidates whofail to reach the minimum mark required for the lowest grade availablein the tier. For example, on Module 1 a candidate entered for theIntermediate tier (grade range B-E) who just fails to reach the standardrequired for grade E does not obtain zero uniform marks. However,centres should note that such a candidate will normally be awardedfewer uniform marks than a Foundation tier candidate who reachesthe same standard.

A candidate’s overall uniform mark is obtained by adding together theuniform marks for the five modules. This overall mark is thenconverted to a grade by means of the following correspondence.


90 ��

Overall (Maximum uniform mark = 600)

Mark range Grade

540 - 600480 - 539420 - 479360 - 419300 - 359240 - 299180 - 239120 - 179

A*ABCDEFG

The final grade must be in a range which is available for thecandidate’s tier of entry for Module 5. For example, a candidateentering Module 5 at the Intermediate tier (grade range B - E), andwith uniform marks of 55, 48, 94, 50 and 234 for Modules 1, 2, 3, 4and 5 respectively, receives a total uniform mark of 481, whichcorresponds to a grade A, but the candidate is awarded grade B sincethis is the highest grade available on the Intermediate tier. Candidatesachieving less than the minimum uniform mark for the lowest gradeon the tier of entry for Module 5 will receive an Unclassified result.

26.4 Shelf-Life of Module Results The shelf-life of individual module results, prior to the award of thequalification, is limited only by the shelf-life of the specification.

26.5 Re-taking Modules andcarrying forward of ModuleResults

Modules 2 and 4, and each tier of Modules 1 and 3, may be re-takenonce before certification of the qualification. The best result for eachmodule will count towards the final award.

Candidates who wish to re-take the qualification after first certificationmay, on request, re-use results from Modules 1-4, but Module 5 mustbe taken again. For Modules 2 and 4 the two most recent results, andfor Modules 1 and 3 the two most recent results from each tier, will beconsidered, and the best of these results will count towards the finalaward. For example, if a candidate attempts Module 1 once at theHigher tier and twice at the Intermediate tier before first certification,then once more at the Intermediate tier before certificating again, theHigher tier attempt and the second and third Intermediate tierattempts are eligible to count towards the final award. In the case ofModule 5 the most recent attempt will always be the one that counts.

Candidates may take the whole qualification an unlimited number oftimes. There is no limit to the number of times a result for Modules1-4 may be re-used.

26.6 Minimum Requirements Candidates will be graded on the basis of work submitted forassessment.

26.7 Awarding and Reporting The regulatory authorities, in consultation with GCSE Awardingbodies, have developed a Code of Practice for GCSE qualificationsintroduced in September 2000. This specification complies with thegrading, awarding and certification requirements of the revised Codeof Practice.


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Appendices

A Grade Descriptions

Grade descriptions are provided to give a general indication of thestandards of achievement likely to have been shown by candidatesawarded particular grades. The descriptions must be interpreted inrelation to the content in the specification; they are not designed todefine that content. The grade awarded will depend in practice uponthe extent to which the candidate has met the assessment objectivesoverall. Shortcomings in some aspects of the candidates’ performancein the examination may be balanced by better performances in others.

Grade A Candidates give reasons for the choices they make when investigatingwithin mathematics itself or when using mathematics to analyse tasks:these reasons explain why particular lines of enquiry or procedures arefollowed and others rejected. Candidates apply the mathematics theyknow in familiar and unfamiliar contexts. Candidates usemathematical language and symbols effectively in presenting aconvincing reasoned argument. Their reports include mathematicaljustifications, explaining their solutions to problems involving anumber of features or variables.

Candidates understand and use rational and irrational numbers. Theydetermine the bounds of intervals. Candidates understand and usedirect and inverse proportion. They manipulate algebraic formulae,equations and expressions, finding common factors and multiplyingtwo linear expressions. In simplifying algebraic expressions, they userules of indices for negative and fractional values. In finding formulaethat approximately connect data, candidates express general laws insymbolic form. They solve problems using intersections and gradientsof graphs.

Candidates sketch the graphs of sine, cosine and tangent functions forany angle, and generate and interpret graphs based on these functions.Candidates use sine, cosine and tangent of angles of any size, andPythagoras’ theorem when solving problems in two and threedimensions. They use the conditions for congruent triangles in formalgeometric proofs. They calculate lengths of circular arcs and areas ofsectors, and calculate the surface area of cylinders and volumes ofcones and spheres.

Candidates interpret and construct histograms. They understand howdifferent methods of sampling and different sample sizes may affectthe reliability of conclusions drawn; they select and justify a sampleand method to investigate a population. They recognise when andhow to work with probabilities associated with independent andmutually exclusive events.


92 ��

Grade C Starting from problems or contexts that have been presented to them,candidates refine or extend the mathematics used to generate fullersolutions. They give a reason for their choice of mathematicalpresentation, explaining features they have selected. Candidates justifytheir generalisations, arguments or solutions, showing some insightinto the mathematical structure of the problem. They appreciate thedifference between mathematical explanation and experimentalevidence.

In making estimates candidates round to one significant figure andmultiply and divide mentally. They solve numerical problemsinvolving multiplication and division with numbers of any size using acalculator efficiently and appropriately. They understand the effects ofmultiplying and dividing by numbers between 0 and 1. Theyunderstand and use the equivalencies between fractions, decimals andpercentages and calculate using ratios in appropriate situations. Theyunderstand and use proportional changes. Candidates find anddescribe in symbols the next term or the nth term of a sequence,where the rule is quadratic; they multiply two expressions of the form(x + n); they simplify the corresponding quadratic expressions. Theysolve simple polynomial equations by trial and improvement andrepresent inequalities using a number line. They formulate and solvelinear equations with whole number coefficients. They manipulatesimple algebraic formulae, equations and expressions. Candidates usealgebraic and graphical methods to solve simultaneous linear equationsin two variables.

Candidates solve problems using angle and symmetry properties ofpolygons and properties of intersecting and parallel lines. Theyunderstand and apply Pythagoras’ theorem when solving problems intwo-dimensions. Candidates find areas and circumferences of circles.They calculate lengths, areas and volumes in plane shapes and rightprisms. Candidates enlarge shapes by a positive whole number orfractional scale factor. They appreciate the imprecision ofmeasurement and recognise that a measurement given to the nearestwhole number may be inaccurate by up to one half in either direction.They understand and use compound measures such as speed.

Candidates construct and interpret frequency diagrams. They specifyhypotheses and test them. They determine the modal class andestimate the mean, median and range of a set of grouped data,selecting the statistic most appropriate to their line of enquiry. Theyuse measures of average and range with associated frequencypolygons, as appropriate, to compare distributions and makeinferences. They draw a line of best fit on a scatter diagram byinspection. Candidates understand relative frequency as an estimate ofprobability and use this to compare outcomes of experiments.


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Grade F In order to carry through tasks and solve mathematical problems,candidates identify and obtain necessary information; they check theirresults, considering whether these are sensible. Candidates showunderstanding of situations by describing them mathematically usingsymbols, words and diagrams. They draw simple conclusions of theirown and give an explanation of their reasoning.

Candidates use their understanding of place value to multiply anddivide whole numbers and decimals by 10, 100 and 1000. They order,add and subtract negative numbers in context. They use all fouroperations with decimals to two places. They reduce a fraction to itssimplest form by cancelling common factors and solve simpleproblems involving ratio and direct proportion. They calculatefractional or percentage parts of quantities and measurements, using acalculator where necessary. Candidates understand and use anappropriate non-calculator method for solving problems involvingmultiplying and dividing any three-digit by any two-digit number. Insolving problems with or without a calculator, candidates check thereasonableness of their results by reference to their knowledge of thecontext or to the size of the numbers, by applying inverse operationsor by estimating using approximations. Candidates explore anddescribe number patterns and relationships including multiple, factorand square. They construct, express in symbolic form, and use simpleformulae involving one or two operations.

When constructing models and when drawing, or using shapes,candidates measure and draw angles as accurately as practicable anduse language associated with angle. They know the angle sum of atriangle and that of angles at a point. They identify all the symmetriesof 2-D shapes. They know the rough metric equivalents of imperialunits still in daily use and convert one metric unit to another. Theymake sensible estimates of a range of measures in relation to everydaysituations. Candidates calculate areas of rectangles and right-angledtriangles, and volumes of cuboids.

Candidates understand and use the mean of discrete data. Theycompare two simple distributions using the range and one of themode, median or mean. They interpret graphs and diagrams, includingpie charts, and draw conclusions. They understand and use theprobability scale from 0 to 1. Candidates make and justify estimates ofprobability by selecting and using a method based on equally likelyoutcomes or on experimental evidence as appropriate. Theyunderstand that different outcomes may result from repeating anexperiment.


94 ��

B Formulae Sheets for Module 5

Foundation Tier


�� 95

Intermediate Tier


96 ��

Higher Tier


�� 97

C AQA-Set Coursework Tasks for

Module 2 (2005)The following are the tasks for submission in 2004 and 2005.Details of the AQA-set tasks will be published annually.

AO4

Reaction Times

Grandad told Simon that some people have slower

reactions than other people.

Simon decided to test the reaction times of some of

his friends.

• Write down a hypothesis for him to test

• Design and carry out an investigation to find out

different ways in which reaction times can be

affected

Investigate further.

Context

This task is accessible to all candidates regardless of tier of entry.

It would normally follow on from work on setting up and testing

hypotheses, and statistical analysis.

1


98 ��

Sarah asked a sample of people to estimate

• the length of this line

• the size of this angle

Sarah then said that people estimate the length of lines

better than the size of angles.

• Write down a hypothesis to test how well people

estimate

• Design and carry out an investigation to test your

hypothesis

AO4 ContextThis task is accessible to all candidates regardless of tier of entry.It would normally follow on from work on setting up and testinghypotheses, and statistical analysis.

2 Guestimate



�� 99

AO4

Memory Game


Context




3

Ranjir collected 16 different objects. She put them on

a tray and covered them with a cloth.

She gathered some of her friends and sat them round

the tray. She removed the cloth for 30 seconds and let

them look at the objects.

After 30 seconds she covered the objects again and

asked her friends to write down as many objects as

they could remember.

• Write down a hypothesis to test in a memory game

like this


hypothesis


100 ��

AO4

Pulse Rate

Not everyone has the same pulse rate – and pulse rate can

be affected by a number of different things.

• Write a hypothesis about how someone’s pulse rate can

be affected

• Design and carry out an investigation to show different

ways in which pulse rate can be affected


Context


It would be most suitably done when candidates have covered

scatter graphs, plotting graphs of real experimental values, and

graphs of rates of change over time; in addition to work on setting

up and testing hypotheses, and statistical analysis.

4


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AO4

Read All About It

Suresh is comparing magazines and newspapers.

He chooses a passage from one newspaper and one

magazine. They each contain 100 words and he

counts the lengths of all the words.

Suresh then says that the magazine has the

shortest words.

• Write a hypothesis about the length of words in

newspapers and magazines


hypothesis


Context




5


102 ��

D AQA-Set Coursework Tasks for

Module 4 (2005)The following are the tasks for submission in 2004 and 2005.Details of the AQA-set tasks will be published annually.

AO1

Round and Round

Context

This task is most suitable for Foundation and/or Intermediate candidates.

It would normally follow on from work on sequences and algebraic equations.

Calculators will have to be used and this task offers a good opportunity to use

a spreadsheet.


1

6 Divide by 5Add 2

Write down

your result


�� 103

AO1

Trios

Three whole numbers, greater than zero, can be used

to form a trio.

For example:

(1, 2, 2) is a trio whose sum is 1 + 2 + 2 = 5

and

(2, 1, 2) is a different trio whose sum is also 5.

How many trios can you find with a sum of 5?


Context

This task is most suitable for Foundation and/or Intermediate candidates.

It could follow on from work on sequences.

2


104 ��

AO1

Fraction Differences

Context

This task is most suitable for Intermediate and/or Higher candidates.

It would normally follow on from work on sequences and fractions.


3

Ruth was investigating fraction differences.

She wrote down this sequence of fractions:

1 1 1 1 1 1 … …

Then she worked out the differences between the

consecutive fractions:

1 1 1 1 1 … …

Then she worked out the differences between the

fractions in her second series:

1 1 … …

1 2 3 4 5 6

2 6 12 20 30

3 12


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AO1

Tangled Triangles

Two students are discussing how to find the biggest

value of the area:perimeter ratio for triangles.

One of them suggests that this can be done with

measurements of 40, 60 and 80 – but forgets to say

what units were used, and whether they were angles

or sides.

Which triangle gives the biggest value for the

area:perimeter ratio?


Context

This task is most suitable for Intermediate and/or Higher candidates.

It would normally follow on from trigonometry work on the sine and

cosine rules. It provides an opportunity to use these in a practical situation.

4


106 ��

AO1

Equable Shapes

An equable shape is one in which:

• the perimeter

and

• the area

have the same numerical value.

Find out what you can about these shapes.


Context

This task is accessible to all candidates regardless of tier of entry. It

would normally follow work on mensuration of different shapes and it

provides an opportunity to use trigonometry and algebraic manipulation.

5


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AO1

Number Grid

Look at this number grid:

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48 49 50

51 52 53 54 55 56 57 58 59 60

61 62 63 64 65 66 67 68 69 70

71 72 73 74 75 76 77 78 79 80

81 82 83 84 85 86 87 88 89 90

91 92 93 94 95 96 97 98 99 100

• A box is drawn round four numbers

• Find the product of the top left number and the

bottom right number in this box

• Do the same with the top right and bottom left

numbers

• Calculate the difference between these products


Context


It can be completed by simple number manipulation or by algebraic methods.

6


108 ��

AO1

Trays

A shopkeeper asks a company to make some trays.

A net of a tray made from a piece of card measuring

18cm by 18cm is shown below:

Side

Side

Base

18 cm

[drawn to scale]

The shopkeeper says, “When the area of the base is

the same as the area of the four sides, the volume of

the tray will be a maximum”.

Investigate this claim.


Context


Candidates may tackle problems practically by making shapes or use

numerical or algebraic methods. It provides an opportunity for

candidates to use mensuration skills.

7


�� 109

E Record Forms

Samples of the Centre Declaration Sheet and Candidate Record Forms are given on the following pages.


110 ��

Centre-assessed work

Centre Declaration Sheet

Qualification: ✔ ELC GCSE GCE GNVQ VCE FSMQKey

Skills

Specification title: ……………………………………………………………………………… Unit code(s): ………………………

Centre name: ……………………………………………………………………… Centre no:

Authentication of candidates’ workThis is to certify that marks/assessments have been given in accordance with the requirements of thespecification and that every reasonable step has been taken to ensure that the work presented is that ofthe candidates named.

Any assistance given to candidates beyond that given to the class as a whole and beyond that described inthe specification has been recorded on the Candidate Record Form(s) and has been taken into account. The

marks/assessments given reflect accurately the unaided achievement of the candidates.

Signature(s) of teacher(s) responsible for assessment

Teacher 1:…………………………………………

Teacher 2: …………………………………………

Teacher 3:………………………………………….

Teacher 4: ………………………………………..

Teacher 5: ………………………………………..

Teacher 6: ………………………………………..

(continue overleaf if necessary)

Internal standardisation of marking

Each centre must standardise assessment across different teachers/assessors and teaching groups to ensurethat all candidates at the centre have been judged against the same standards.

If two or more teachers/assessors are involved in marking/assessing, one of them must be designated asresponsible for standardising the assessments of all teachers/assessors at the centre.

I confirm that [tick either (a) or (b)]

(a) the procedure described in the specification has been followed at this centre to ensure that the

assessments are of the same standard for all candidates; or

(b) I have marked/assessed the work of all candidates.

Signed: ……………………………………………………………………… Date: …………………………

Signature of Head of Centre: …………………………………………………………… Date: ………………………

This form should be completed and sent to the moderator with the sample of centre-assessed work.


�� 111


Candidate Record Form

2004

GCSE Mathematics B (Modular) Module 2 (Option T) 3302

Centre name: ......................................................................................... Centre no:

Candidate name: .................................................................................. Candidate no:

This side is to be completed by the candidate

Sources of advice and information

1. Have you received any help or information from anyone other than your subject teacher(s) in theproduction of this work? (Write YES or NO) ............................

2. If you have answered YES, give details below. Continue on a separate sheet if necessary.

................................................................................................................................................................................................................

.................................................................................................................................................................................................................3. If you have used any books, information leaflets or other materials (e.g. videos, software packages or

information from the Internet) to help you complete this work, you must list these below, unless they areclearly acknowledged in the work itself. To present material copied from books or other sources withoutacknowledgement will be regarded as deliberate deception.

…………………………………………………………………………………………………………………………………………………

…………………………………………………………………………………………………………………………………………………

…………………………………………………………………………………………………………………………………………………

NOTICE TO CANDIDATE

The work you submit for assessment must be your own.

If you copy from someone else or allow another candidate to copy from you, or if you

cheat in any other way, you may be disqualified from at least the subject concerned.

Declaration by candidate

I have read and understood the Notice to Candidate (above). I have produced the attached work without any helpapart from that which I have stated on this sheet.

Candidate’s signature: ....................................................................................................... Date: ..................................

This form should be completed and attached to the candidate’s work and retained at the Centre

or sent to the moderator as required.

PTO


112 ��


This side is to be completed by the teacher

Marks must be awarded in accordance with the instructions and criteria in section 87 of the specification.Supporting information to show how the marks have been awarded should be given in the form of annotationson the candidate’s work and in the spaces provided below.

Project title:

Module 2 – AO4 (one task only)

Strand Criteria for award of marksMax.

mark

Mark

awardedKey evidence

1 Specify the problem andplan

8

2 Collect, process andrepresent data

8

3 Interpret and discussresults

8

Total mark 24

Concluding comments

Details of additional assistance given (if any)

Record here details of any assistance given to this candidate which is beyond that given to the class as a wholeand beyond that described in the specification. Continue on a separate sheet if necessary.

Teacher’s signature: ……………………………………………………………………… Date: ………………………………


�� 113



2004

GCSE Mathematics B (Modular) Module 4 (Option T) 3302

Centre name: .......................................................................................... Centre no:

Candidate name: ................................................................................... Candidate no:



1. Have you received any help or information from anyone other than your subject teacher(s) inthe production of this work? (Write YES or NO) .............................


.................................................................................................................................................................................................................



…………………………………………………………………………………………………………………………………………………

…………………………………………………………………………………………………………………………………………………

NOTICE TO CANDIDATE





I have read and understood the Notice to Candidate (above). I have produced the attached work without anyhelp apart from that which I have stated on this sheet.

Candidate’s signature: ...................................................................................................... Date: ...................................

This form should be completed and attached to the candidate’s work and retained at the Centre

or sent to the moderator as required.

PTO


114 ��


This side is to be completed by the teacher

Marks must be awarded in accordance with the instructions and criteria in section 87 of the specification.Supporting information to show how the marks have been awarded should be given in the form of annotationson the candidate’s work and in the spaces provided below.

Project title:

Module 4 – AO1 task

Strand Criteria for award of marksMax.

mark

Mark

awardedKey evidence

1 Making and monitoringdecisions to solve problems

8

2 Communicatingmathematically

8

3 Developing skills ofmathematical reasoning

8

Total mark 24

Concluding comments


Record here details of any assistance given to this candidate which is beyond that given to the class as a whole and

beyond that described in the specification. Continue on a separate sheet if necessary.

Teacher’s signature: ……………………………………………………………………… Date: ………………………………


�� 115

AQA-assessed work

Candidate Record Form2004

GCSE Mathematics B (Modular) Module 2 (Option X) 3302





1. Have you received any help or information from anyone other than your subject teacher(s) inthe production of this work? (Write YES or NO) ............................


................................................................................................................................................................................................................

.................................................................................................................................................................................................................

3. If you have used any books, information leaflets or other materials (e.g. videos, software packages orinformation from the Internet) to help you complete this work, you must list these below, unless they areclearly acknowledged in the work itself. To present material copied from books or other sources withoutacknowledgement will be regarded as deliberate deception.

…………………………………………………………………………………………………………………………………………………

…………………………………………………………………………………………………………………………………………………

NOTICE TO CANDIDATE







This form should be completed and attached to the candidate’s work and sent to the examiner

PTO


116 ��


Teachers are strongly advised to provide comments as evidence of mathematical or statistical thinking

where this is not clearly communicated in the work. This may be done in the body of the script or on a

separate sheet.

Declaration by the teacher

Project title:


Record here details of any assistance given to this candidate which is beyond that given to the class as awhole and beyond that described in the specification. Continue on a separate sheet if necessary.

Teacher’s signature: ……………………………………………………………………… Date: ………………………………..

To be marked by the examiner


Strand Key evidenceFinal assessed

score (0–8)

1

2

3

Total score (max. 24)

Examiner’s initials


�� 117

AQA-assessed work


2004

GCSE Mathematics B (Modular) Module 4 (Option X) 3302



This side is to be completed by the candidate.


1. Have you received any help or information from anyone other than your subject teacher(s) inthe production of this work? (Write YES or NO) ............................


................................................................................................................................................................................................................



…………………………………………………………………………………………………………………………………………………

…………………………………………………………………………………………………………………………………………………

NOTICE TO CANDIDATE







This form should be completed and attached to the candidate’s work and sent to the examiner

PTO


118 ��


Teachers are strongly advised to provide comments as evidence of mathematical or statistical thinking

where this is not clearly communicated in the work. This may be done in the body of the script or on a

separate sheet.

Declaration by the teacher

Project title:


Record here details of any assistance given to this candidate which is beyond that given to the class as a wholeand beyond that described in the specification. Continue on a separate sheet if necessary.

Teacher’s signature: ……………………………………………………………………… Date: ………………………………..

To be marked by the examiner


Strand Key evidenceFinal assessed

score (0–8)

1

2

3

Total mark (max. 24)

Examiner’s initials


�� 119

F Overlaps with other

Qualifications

The subject content of this Specification is identical, though differentlystructured, to that of AQA GCSE Mathematics Specification A.

There is some overlap between Module 1 of this specification andGCSE Statistics.

There is a considerable overlap of skills and content between themodules of GCSE Mathematics (Modular) Specification B, Free-Standing Mathematics Qualifications (FSMQs) and the Key Skill ofApplication of Number. In some post-16 centres candidates on thedifferent courses may be grouped together.

Further information about the links between these subjects can beobtained from AQA (Guildford) as separate booklets.

AQA GCSE Specification

Documents

Transcript of AQA GCSE Specification