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EDEXCEL GCSE Teachers’ Guide – Content Exemplification (Modular) Edexcel GCSE in Mathematics (Specification B 1388 ) First examination 2003 July 2003

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  • EDEXCEL GCSE

    Teachers Guide Content Exemplification (Modular)

    Edexcel GCSE in Mathematics(Specification B 1388 )First examination 2003

    July 2003

  • Edexcel Foundation is an innovative force in education and training, combining the skills andexperience of two internationally recognised awarding and assessment bodies BTEC, a leadingprovider of applied and vocational qualifications, and London Examinations, one of the major GCSEand GCE examining boards in the UK.

    Acknowledgements

    This document has been produced by Edexcel on the basis of consultation with teachers, examiners,consultants and other interested parties. Edexcel recognises and values all those who contributed theirtime and expertise to the development of the qualification.

    Authorised by Peter Goff

    Publications Code UG009825

    All the material in this publication is copyright Edexcel Foundation July 2003

  • Contents

    Introduction 1Scheme of Assessment 2External Assessment 4Foundation Tier 6 Stage 1 6 Stage 2 20 Stage 3 29Intermediate Tier 37 Stage 1 37 Stage 2 52 Stage 3 60Higher Tier 69 Stage 1 69 Stage 2 83 Stage 3 91

  • UG009825 Edexcel GCSE Mathematics 1388 Teachers Guide: Content Exemplification (Modular) Issue 2 July 2003 1

    Introduction

    This publication has been produced with the intention of providing additional assistance for teacherswho are preparing to teach Edexcels modular GCSE in Mathematics.

    The material contained in this publication is divided into two columns. The first column identifies thesubject content of the specification, and the second column offers examples to illustrate the nature ofthe content item.

    These examples in conjunction with the specimen papers are intended as guidance only for theinterpretation of the subject content.

  • 2 UG009825 Edexcel GCSE Mathematics 1388 Teachers Guide: Content Exemplification (Modular) Issue 2 July 2003

    Scheme of assessment

    Entry tiers

    Candidates for this qualification must be entered for one of three tiers.The grades available for each tier are as follows:

    Tier Grades available

    Foundation G to D

    Intermediate E to B

    Higher C to A*

    Candidates achieving a mark below the minimum for the award of the lowest grade in each tier willbe ungraded.

    Centres are advised to enter candidates for each tier as follows:

    Tier Estimated Grade

    Foundation G, F, E

    Intermediate D, C

    Higher B, A, A*

    Assessment of the specification consists of:

    For foundation tier candidates:

    Paper Weighting Time calculator

    Section A (25 mins) Paper 8 (module test 1) 15% 50 minutes

    Section B (25 mins)

    Section A (25 mins) Paper 11 (module test 2) 15% 50 minutes

    Section B (25 mins)

    Paper 14 25% 1hour

    Paper 15 25% 1hour

    Coursework

    Paper 7a or 7b(coursework) 20%

    Coursework consists of a handling dataproject and an investigational task

  • UG009825 Edexcel GCSE Mathematics 1388 Teachers Guide: Content Exemplification (Modular) Issue 2 July 2003 3

    For intermediate tier candidates:

    Paper Weighting Time calculator

    Section A (25 mins) Paper 9 (module test 1) 15% 50 minutes

    Section B (25 mins)

    Section A (25 mins) Paper 12 (module test 2) 15% 50 minutes

    Section B (25 mins)

    Paper 16 25% 1 hour 15 mins

    Paper 17 25% 1 hour 15 mins

    Coursework

    Paper 7a or 7b(coursework) 20%

    Coursework consists of a handling dataproject and an investigational task

    For higher tier candidates:

    Paper Weighting Time calculator

    Section A (25 mins) Paper 10 (module test 1) 15% 50 minutes

    Section B (25 mins)

    Section A (25 mins) Paper 13 (module test 2) 15% 50 minutes

    Section B (25 mins)

    Paper 18 25% 1 hour 15 mins

    Paper 19 25% 1 hour 15 mins

    Coursework

    Paper 7a or 7b(coursework) 20%

    Coursework consists of a handling dataproject and an investigational task

    Candidates may retake modules at a different entry tier. Candidates final tier of entry is determinedby the written papers the candidate takes.

  • 4 UG009825 Edexcel GCSE Mathematics 1388 Teachers Guide: Content Exemplification (Modular) Issue 2 July 2003

    External AssessmentModule Tests (papers 8 13)

    ResitsStudents may resit any module test once only and the better result will count towards the final award.The shelf-life of individual modules is limited only by the shelf-life of the specification. The fullqualification may be retaken more than once.The papersModule tests will be combined question/answer books containing both shorter and longer questions.

    Module 1 (examination papers 8, 9 and 10) will be timetabled in one session and module 2(examination papers 11, 12 and 13) in another.

    The focus of assessment for module 1 in each tier will be the material identified in the stage 1 contentfor that tier.

    The focus of assessment for module 2 in each tier will be the material identified in the stage 2 contentfor that tier, but knowledge of the content in the associated stage 1 will be assumed.

    Section A in each module test will be a non-calculator section. In this section calculators, slide rules,logarithm tables and all other calculating aids are forbidden.

    Module test section A may test any topic in the subject content appropriate to the stage and tier ofentry, except those that expressly require the use of a calculator.

    Module test section B may test any topic in the subject content appropriate to the stage and tier ofentry, except those that expressly prohibit the use of a calculator.

    Each module test will carry a maximum mark of 38 (equally divided between sections A and B).

    Whilst each module test will assess the full range of grades at each tier, the emphasis will be on thelower grades available for the tier of entry.

    Examination papers 14 19

    Examination papers 14 19 will be combined question/answer books containing both shorter andlonger questions.

    Examination papers 14, 16 and 18 will be timetabled in one session and examination papers 15, 17and 19 in another.

    Examination papers 14, 16 and 18 will be non-calculator papers. In these papers calculators, sliderules, logarithm tables and all other calculating aids are forbidden.

    The non-calculator examination papers may test any topic in the subject content appropriate to the tierof entry, except those that expressly require the use of a calculator.

    The with-calculator examination papers may test any topic in the subject content appropriate to thetier of entry, except those that expressly prohibit the use of a calculator.

    Each examination paper will carry a maximum mark of 62.

    There will be two parallel examination papers for each tier.

    Whilst each examination paper will assess the full range of grades at each tier, there will be anemphasis on the higher grades available for the tier of entry.

  • UG009825 Edexcel GCSE Mathematics 1388 Teachers Guide: Content Exemplification (Modular) Issue 2 July 2003 5

    The examination papers for each tier will assess the full range of content given in the specification forthat tier. However there will be a slight emphasis on the material identified in stage 3 of the relevanttier.

    There will be common questions across examination papers to aid standardisation and comparabilityof awards between tiers.

    General issues relating to both module tests and examination papers

    There will be a number of questions demanding the unprompted solution of multi-step problems.

    There will be a number of questions requiring the use of manipulative algebra.

    Taken together the module tests and examination papers in the higher and intermediate tiers will offera balanced assessment across the grades available in the tier.

    In the foundation tier, about one third of the marks awarded across the module tests and examinationpapers will be allocated to grade G material and the remaining marks will be balanced across the othergrades.

    Questions on the intermediate tier module tests and examination papers will assume knowledge fromthe foundation tier. However, material related to grades below the range of the tier will not be thefocus of assessment.

    Questions on the higher tier module tests and examination papers will assume knowledge from theintermediate tier. However, material related to grades below the range of the tier will not be the focusof assessment.

    Diagrams will not necessarily be drawn to scale and measurements should not be taken from diagramsunless instructions to this effect are given.

    Each candidate may be required to use mathematical instruments e.g. pair of compasses, ruler,protractor.

    Formulae sheets will be provided for foundation, intermediate and higher tiers.

    Calculators

    Candidates will be expected to have access to a suitable electronic calculator for all the modular testsand examination papers 15, 17 and 19.

    The electronic calculator to be used by candidates attempting foundation tier module tests andexamination paper 15 should have, as a minimum, the following functions:

    =================+, , , , x 2, x, memory, brackets, xy, yx1

    The electronic calculator to be used by candidates attempting intermediate and higher tier moduletests and examination papers 17 and 19 should have, as a minimum, the following functions:

    +, , , , x 2, x, memory, constant function, brackets, xy, yx1

    , x , x, fx, standard form,sine, cosine, tangent and their inverses.

    Calculators with any of the following facilities are prohibited from any module test or examination:

    Databanks; retrieval of text or formulae; QWERTY keyboards; built-in symbolic algebramanipulations; symbolic differentiation or integration.

    Calculators which are not permitted in any paper include calculators such as Texas TI-89, TI-92,Casio cfx-9970G, Hewlett Packard HP 48G, Casio C-300. (NB There are almost certainly others thatare not permitted.)

  • Edexcel GCSE Mathematics (Modular) 1388 Foundation Tier: Stage 1

    6 UG009825 Edexcel GCSE Mathematics 1388 Teachers Guide: Content Exemplification (Modular) Issue 2 July 2003

    Foundation Tier: Stage 1Pupils should be taught the knowledge, skills and understanding contained in this specificationthrough:

    a) extending mental and written calculation strategies and using efficient procedures confidently tocalculate with integers, fractions, decimals, percentages, ratio and proportion

    b) solving a range of familiar and unfamiliar problems, including those drawn from real-lifecontexts and other areas of the curriculum

    c) activities that provide frequent opportunities to discuss their work, to develop reasoning andunderstanding and to explain their reasoning and strategies

    d) activities focused on developing short chains of deductive reasoning and correct use of the =sign

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

    f) practical work in which they draw inferences from data, consider how statistics are used in reallife to make informed decisions, and recognise the difference between meaningful andmisleading representations of data

    g) activities focused on the major ideas of statistics, including using appropriate populations andrepresentative samples, using different measurement scales, using probability as a measure ofuncertainty, using randomness and variability, reducing bias in sampling and measuring, andusing inference to make decisions

    h) substantial use of tasks focused on using appropriate ICT (for example, spreadsheets, databases,geometry or graphic packages), using calculators correctly and efficiently, and knowing whennot to use a calculator.

  • GCSE Mathematics (Modular) 1388 Foundation Tier: Stage 1

    UG009825 Edexcel GCSE Mathematics 1388 Teachers Guide: Content Exemplification (Modular) Issue 2 July 2003 7

    Ma2 Number and algebra

    Contents Examples

    Using and applying number and algebra Using and applying statements apply to allthree stages; thus examples given must be seenin the context of the relevant stage.

    1. Pupils should be taught to:

    Problem solving

    a) select and use suitable problem-solvingstrategies and efficient techniques to solvenumerical and algebraic problems

    b) break down a complex calculation intosimpler steps before attempting to solve it

    c) use algebra to formulate and solve asimple problem identifying the variable,setting up an equation, solving the equation andinterpreting the solution in the context of theproblem

    d) make mental estimates of the answersto calculations

    use checking procedures, including use ofinverse operations

    work to stated levels of accuracy

    Communicating

    e) interpret and discuss numerical andalgebraic information presented in a variety offorms

    f) use notation and symbols correctly andconsistently within a given problem

    g) use a range of strategies to createnumerical, algebraic or graphicalrepresentations of a problem and its solution

    Hire Purchase calculations with deposit andmonthly payments, comparison of creditmethods, loan or hire rates

    for angles in a triangle derive and solvex + 2x + 3x = 180

    set up a formula for the cost of photocopying nsheets at 3p per sheet and a standard charge of20p

    estimate the answer to 97 6.2, 78

    20332

    348 23 approximates to 300 20 (i.e. to onesignificant figure)

    96 8 = 12 is also 8 12 = 96

    give an answer to 2 decimal places, or to 1significant figure

    charts, timetables, shopping details,advertisements, information in newspapers, TVprogramme lists, holiday costs lists

    money notation, correctly given algebraicexpressions, mathematical notation

    cost of hiring car: 20 plus 50p per mile or 10plus 75p per mile find which is cheaper

  • Edexcel GCSE Mathematics (Modular) 1388 Foundation Tier: Stage 1

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    Contents Examples

    h) present and interpret solutions in thecontext of the original problem

    correct calculations using foreign exchangerates, units of measure; interpretation ofnegative money as a deficit or loss

    maximum number of stamps bought for 10must be a whole number; rounding up is notappropriate

    Reasoning

    j) explore, identify, and use pattern andsymmetry in algebraic contexts, investigatingwhether particular cases can be generalisedfurther, and understanding the importance of acounter-example

    k) show step-by-step deduction in solvinga problem

    using simple codes that substitute numbers forletters

    clear working, line by line

    Numbers and the number system

    2. Pupils should be taught to:

    Integers

    a) use their previous understanding ofintegers and place value to deal with arbitrarilylarge positive numbers and round them to agiven power of 10

    understand and use positive numbers, both aspositions and translations on a number line

    order integers

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

    Write 50 million in figures, write 5739 to thenearest 10, 100 or 1000

    temperature changes, changes in heightinvolving positive and negative numbers

    find the warmest temperature from a list, placea series of positive and negative numbers inascending or descending order

    pick the multiples of 4 from a list of numbers

    write out all the common factors of 18 and 32

    NB: does not include prime numbers

  • GCSE Mathematics (Modular) 1388 Foundation Tier: Stage 1

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    Contents Examples

    Fractions

    c) understand equivalent fractions,simplifying a fraction by cancelling allcommon factors

    order fractions by rewriting them with acommon denominator

    compare shaded fractions of shapes, givinganswers in simplified form

    express 3528 in its simplest form

    write 83 , 4

    1 , 103 in descending order (in order of

    decreasing size)

    write two fractions between 41 and 2

    1

    explain why 43 is the same as 8

    6

    Decimals

    d) use decimal notation and recognise thateach terminating decimal is a fraction

    0.137 = 1000137

    order decimals write in ascending order: 0.32, 2.3, 0.23, 3.2

    Percentages

    e) understand that percentage meansnumber of parts per 100 and use this tocompare proportions

    interpret percentage as the operator so manyhundredths of

    10% means 10 parts per 100

    find what percentage of a shape is shaded

    15% of Y means 10015 Y

    demonstrate how to work out 40% of 250

    Ratio

    f) use ratio notation, including reductionto its simplest form and its various links tofraction notation.

    in maps and scale drawings, paper sizes andgears

    1:5 is 61 and 6

    5

    51 of a whole is 1:4

    writing in the form 1: n

    the simplest form of 4:6 is 2:3

  • Edexcel GCSE Mathematics (Modular) 1388 Foundation Tier: Stage 1

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    Contents Examples

    Calculations

    3. Pupils should be taught to:

    Number operations and the relationshipsbetween them

    a) add, subtract, multiply and divideintegers and then any number

    851 37 or 777 37, 27.6 23

    non-calculator methods such as 365 54,753 36, temperature changes

    multiply or divide any number by powers of 10,and any positive number by a number between0 and 1

    3 0.4, 0.7 1000, 8.8 10

    b) use brackets and the hierarchy ofoperations

    insert brackets so that 20 3 2 = 34

    BIDMAS

    work out 5 (2 + 3) and 5 2 + 3

    c) calculate a given fraction of a givenquantity, expressing the answer as a fraction

    perform short division to convert a simplefraction to a decimal

    for scale drawings and construction of models,down payments, discounts

    find 43 of 48, 4

    3 of 6

    write 83 as a decimal

    d) understand and use unit fractions asmultiplicative inverses;

    multiply a fraction by an integer

    multiply a fraction by a unit fraction

    by thinking of multiplication by 51 as division

    by 5, or multiplication by 76 as multiplication

    by 6 followed by division by 7 (or vice versa)

    103 4,

    31

    54

    Mental methods

    g) recall all positive integer complementsto 100

    recall all multiplication facts to 10 10, anduse them to derive quickly the correspondingdivision facts;

    recall the fraction-to-decimal conversion offamiliar simple fractions

    37 + 63 = 100 , (100 37) = 63

    8 7 = 56, 56 7 = 8 and 56 8 = 7

    ,, 4121 ,,,,, 3231100110151 43

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    Contents Examples

    h) round to the nearest integer in interest calculations, or in carrying ourestimates of complex calculations

    i) develop a range of strategies for mentalcalculation

    derive unknown facts from those they know

    add and subtract mentally numbers with up totwo decimal places

    13.76 5.21, 20.08 + 12.4

    multiply and divide numbers with no more thanone decimal digit, using the commutative,associative, and distributive laws andfactorisation where possible, or place valueadjustments

    14.3 4, 56.7 7, 36.2 2

    23.2 3 = 69.6

    what is 2.32 3, 23.2 30, 2.32 30

    Written methods

    j) use standard column procedures foraddition and subtraction of integers anddecimals

    calculating shopping bills, energy bills, oraddition of lengths given as decimals of a unit

    Equations, formulae and identities

    5. Pupils should be taught to:

    Use of symbols

    a) distinguish the different roles played byletter symbols in algebra, knowing that lettersymbols represent definite unknown numbersin equations

    5x + 1 = 16 is an equation

    b) understand that the transformation ofalgebraic expressions obeys and generalises therules of arithmetic

    manipulate algebraic expressions by collectinglike terms, by multiplying a single term over abracket

    ab = a b = b a = baa + a + a = 3a,

    x + 5 2x 1 = 4 x5(2x + 3) = 10x + 15

    simplify: 3a + 2c a 3c + 2,a b 2 (= 2ab)

  • Edexcel GCSE Mathematics (Modular) 1388 Foundation Tier: Stage 1

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    Contents Examples

    Sequences, functions and graphs

    6. Pupils should be taught to:

    Graphs of linear functions

    b) use the conventions for coordinates inthe plane

    plot points in all four quadrants

  • GCSE Mathematics (Modular) 1388 Foundation Tier: Stage 1

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    Ma3 Shape space and measures

    Contents Examples

    Using and applying shape, space andmeasures

    Using and applying statements apply to allthree stages; thus examples given must be seenin the context of the relevant stage.

    1. Pupils should be taught to:

    Problem solving

    a) select problem-solving strategies andresources, including ICT tools, to use ingeometrical work, and monitor theireffectiveness

    b) select and combine known facts andproblem-solving strategies to solvecomplex problems

    c) identify what further information is neededto solve a geometrical problem; breakcomplex problems down into a series oftasks

    Communicating

    d) interpret, discuss and synthesisegeometrical information presented in avariety of forms

    e) communicate mathematically, bypresenting and organising results andexplaining geometrical diagrams

    f) use geometrical language appropriately

    Reasoning

    h) distinguish between practicaldemonstrations and proofs

    i) apply mathematical reasoning, explainingand justifying inferences and deductions

    j) show step-by-step deduction in solving a geometrical problem

    use spreadsheets to find maximum areas ofrectangle

    use geometry to draw regular slopes or slopesof given dimension

    measure sides of a rectangle to work out area

    draw and construct diagrams from giveninformation

    use of letters to identify points, lines and angles

    explain why regular hexagons tessellate andwhy regular pentagons do not

    name shapes and solids

    give explanations showing application ofgeometrical properties

    deduce lengths in a diagram

    find the angles in an isosceles triangle givenone of the angles

  • Edexcel GCSE Mathematics (Modular) 1388 Foundation Tier: Stage 1

    14 UG009825 Edexcel GCSE Mathematics 1388 Teachers Guide: Content Exemplification (Modular) Issue 2 July 2003

    Contents Examples

    Geometrical reasoning

    2. Pupils should be taught to:

    Angles

    b) distinguish between acute, obtuse,reflex and right angles

    estimate the size of an angle in degrees

    identify the acute, obtuse, reflex and right-angles in a diagram

    Properties of triangles and other rectilinearshapes

    f) recall the essential properties of specialtypes of quadrilateral, including square,rectangle, parallelogram, trapezium andrhombus

    classify quadrilaterals by their geometricproperties

    list and compare the properties of each

    draw sketches of shapes

    kites and arrowheads

    name all quadrilaterals that have a pair ofopposite sides that are parallel

    Properties of circles

    i) recall the definition of a circle and themeaning of related terms, including centre,radius, diameter, circumference

    Transformations and coordinates

    3. Pupils should be taught to:

    Specifying transformations

    a) understand that rotations are specifiedby a centre and an (anticlockwise) angle

    questions will clearly specify clockwise or anti-clockwise when required

    rotate a shape about the origin

    measure the angle of rotation using rightangles, simple fractions of a turn

    centre confined to (0, 0)

    quarter, half, three-quarters clockwise and anti-clockwise

    understand that reflections are specified by amirror line, at first using a line parallel to anaxis

    reflect simple shapes in a mirror line

  • GCSE Mathematics (Modular) 1388 Foundation Tier: Stage 1

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    Contents Examples

    Properties of transformations

    b) recognise and visualise rotations andreflections including reflection symmetry androtation symmetry of 2-D shapes

    transform triangles and other 2-D shapes byrotation and reflection, recognising that thesetransformations preserve length and angle, sothat any figure is congruent to its image underany of these transformations

    state the order of rotational symmetry

    draw in lines of symmetry

    rotation of simple shapes by a simple fractionof a turn about the origin, (anti)clockwise

    Coordinates

    e) understand that one coordinateidentifies a point on a number line, twocoordinates identify a point in a plane and threecoordinates identify a point in space, using theterms 1-D, 2-D and 3-D

    use axes and coordinates to specify points in allfour quadrants

    locate points with given coordinates

    given a 3-D coordinate on axes or on adiagram, write down a similar coordinaterelative to it

    for 2-D

    identify and plot coordinates in 2D

    Measures and construction

    4. Pupils should be taught to:

    Measures

    a) interpret scales on a range of measuringinstruments, including those for time and mass

    convert measurements from one unit to another

    mm, cm, m, km, ml, l, mg, g, kg, tonnes,seconds, minutes, hours, days, weeks, monthsand years

    use correct notation for time, differentiatingbetween 2.30 for 2 2

    1 h, and 2.3 as in 2 h 30 m

    imperial conversions: conversion betweenimperial units will be given

    make sensible estimates of a range of measuresin everyday settings

    metric equivalents should be known

    change 7.4 kg to grams, mm to cm, inches tofeet

    b) understand angle measure using theassociated language

    use (three-figure) bearings to specify directione.g. 072, 314

  • Edexcel GCSE Mathematics (Modular) 1388 Foundation Tier: Stage 1

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    Contents Examples

    Construction

    d) measure and draw lines to the nearestmillimetre, and angles to the nearest degree

    draw triangles and other 2-D shapes using aruler and protractor, given information abouttheir side lengths and angles

    a ruler of length 30 cm may be needed

  • GCSE Mathematics (Modular) 1388 Foundation Tier: Stage 1

    UG009825 Edexcel GCSE Mathematics 1388 Teachers Guide: Content Exemplification (Modular) Issue 2 July 2003 17

    Ma4 Handling Data

    Contents Examples

    Using and applying handling data Using and applying statements apply to allthree stages; thus examples given must be seenin the context of the relevant stage.

    1. Pupils should be taught to:

    Problem solving

    a) carry out each of the four aspects of thehandling data cycle to solve problems:

    (i) specify the problem and plan: formulatequestions in terms of the data needed, andconsider what inferences can be drawnfrom the data; decide what data to collect(including sample size and data format)and what statistical analysis is needed

    (ii) collect data from a variety of suitablesources, including experiments andsurveys, and primary and secondarysources

    (iii) process and represent the data: turn theraw data into usable information that givesinsight into the problem

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

    b) identify what further information isneeded to pursue a particular line of enquiry

    c) select and organise the appropriatemathematics and resources to use for a task

    d) review progress while working; checkand evaluate solutions

    Communicating

    e) interpret, discuss and synthesiseinformation presented in a variety of forms

    f) communicate mathematically, includingusing ICT, making use of diagrams andrelated explanatory text

    improve questions; design data collection sheet

    draw up a suitable data collection table for asurvey

    refine a vague question into a better format

    use suitable data collection techniques

    draw appropriate and different types of graphsto show data in a suitable format so thatdecisions can be made

    calculate total frequency or the fx column froma discrete frequency table

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    Contents Examples

    explain deficiencies in questions on datacollection techniques

    Reasoning

    h) apply mathematical reasoning, explaininginferences and deductions

    i) explore connections in mathematics andlook for cause and effect when analysingdata

    compare two data sets with paired variables ona scatter graph to check for correlation

    increased temperature results in increasedconsumption of ice-cream but not vice versa

    Collecting data

    3. Pupils should be taught to:

    a) design and use data-collection sheetsfor grouped discrete data

    collect data using various methods, includingobservation, controlled experiment, datalogging, questionnaires and surveys

    understand and use tallying methods

    sort, classify and tabulate (categorical orqualitative) data and discrete or continuousquantitative data

    grouping of discrete and continuous data intoclass intervals of equal width

    design and criticise questions for aquestionnaire

    b) gather discrete data from secondarysources, including printed tables and lists fromICT-based sources

    abstract data from lists and tables

    Processing and representing data

    4. Pupils should be taught to:

    a) draw and produce, using paper andICT, pie charts for categorical data andfrequency diagrams

    pictograms and bar charts

    b) calculate mean, range and median ofsmall data sets with discrete data

    write down the range from a bar chart

    find the mode

    find mode, mean and median from a list

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    Contents Examples

    identify the modal class for grouped discretedata

    state the modal class from a grouped frequencytable

    c) understand and use the probabilityscale

    mark events and/or probabilities on aprobability scale 0 1

    e) list all outcomes for single events, andfor two successive events, in a systematic way

    three coins, two dice

    list all outcomes of coin tossing and dicethrowing, e.g. H1, H2, H3, H4, H5, H6, T1, T2,T3, T4, T5, T6

    Interpreting and discussing results

    5. Pupils should be taught to:

    b) interpret a wide range of graphs anddiagrams and draw conclusions

    g) use the vocabulary of probability tointerpret results involving uncertainty andprediction

    probabilities must be written as fractions,decimals or percentages

    j) discuss implications of findings in thecontext of the problem

  • Edexcel GCSE Mathematics (Modular) 1388 Foundation Tier: Stage 2

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    Foundation Tier: Stage 2Ma2 Number and Algebra

    Contents Examples

    Numbers and the number system

    2. Pupils should be taught to:

    Powers and roots

    b) use the terms square, positive squareroot, cube

    use index notation for squares and cubes andpowers of 10

    pick out square, cube numbers from a list

    find the square of 4, 64; estimate 85

    find the value of 32, 23, etc, up to 106

    3 102

    Calculations

    3. Pupils should be taught to:

    Number operations and the relationshipsbetween them

    c) express a given number as a fraction ofanother

    if 750 of 1000 is given to Penelope, whatfraction is this?

    e) convert simple fractions of a whole topercentages of the whole and vice versa

    analysing diets, budgets or the costs of running,maintaining and owning a car

    change 52 and 0.65 into a percentage

    70 out of 200 as a percentage

    Mental methods

    g) recall the cubes of 2, 3, 4, 5 and 10

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    Contents Examples

    Calculator methods

    o) use calculators effectively: know howto enter complex calculations and use functionkeys for reciprocals, squares and powers

    find 3.32 + 4.35.7 (4.2 + 0.5)2

    3.33 + 4.3

    calculator functions include +, , , , x2,x,memory, brackets, xy, x1/y

    32.53.2 +

    Solving numerical problems

    4. Pupils should be taught to:

    a) draw on their knowledge of theoperations and the relationships between them,and of simple integer powers and theircorresponding roots, to solve problemsinvolving ratio and proportion, a range ofmeasures including speed, metric units, andconversion between metric and commonimperial units, set in a variety of contexts

    recipes, sale prices, foreign exchange rates

    average speed in kph, metres per second

    Equations, formulae and identities

    5. Pupils should be taught to:

    Index notation

    c) use index notation for simple integerpowers

    2 2 2 2 = 24, 23 32 = 8 9 = 72

    Linear equations

    e) solve linear equations, with integercoefficients, in which the unknown appears oneither side or both sides of the equation

    5x + 5 = 7 + x, x + 4 = 11, 5x 3 = 7

    Formulae

    f) use formulae from mathematics andother subjects expressed initially in words andthen using letters and symbols

    formulae for the area of a triangle, the areaenclosed by a circle,wage earned = hours worked rate per hour

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    substitute numbers into a formula convert temperatures between degreesFahrenheit and degrees Celsius

    find x given y in y = 8x 1

    Sequences, functions and graphs

    6. Pupils should be taught to:

    Sequences

    a) generate terms of a sequence usingterm-to-term and position-to-term definitions ofthe sequence

    find the next term after 15 if the rule is add 4

    find the next term (or the 10th term) in1,2,4,8,..; 5,9,13,17,..; 1,4,5,8,9,12,13,

    express the rule for finding the nth term inwords for simple sequences, e.g. goes up inthrees

    draw the next diagrammatic term in a sequenceof diagrams

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    Ma3 Shape, Space and measures

    Contents Examples

    Geometrical reasoning

    2. Pupils should be taught to:

    Angles

    a) recall and use properties of angles at apoint, angles on a straight line (including rightangles), perpendicular lines, and oppositeangles at a vertex

    sum of 3 angles on a straight line

    sum of 4 angles at a point, of which one is 90

    Properties of triangles and other rectilinearshapes

    c) understand that the angle sum of a triangleis 180 degrees

    find the missing angle in a triangle, given twoangles

    d) use angle properties of equilateral,isosceles and right-angled triangles

    understand congruence

    explain why the angle sum of any quadrilateralis 360 degrees

    find missing angles in shapes, given one ormore of these angles

    identify shapes which are congruent

    as two triangles

    e) use their knowledge of rectangles,parallelograms and triangles to deduceformulae for the area of a parallelogram, and atriangle, from the formula for the area of arectangle

    g) calculate and use sums of the interiorand exterior angles of quadrilaterals, pentagonsand hexagons

    calculate and use angles of regular polygons

    a tessellation of at least 6 kites

    irregular polygons of sides 4, 5, 6 only

    complete a tessellation using a regular polygonwhere possible

    interior angle + exterior angle = 180

    given the size of each exterior angle of aregular polygon, find the number of sides

    sum of exterior angles of a polygon sum to360

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    Contents Examples

    Properties of circles

    i) understand that inscribed regular polygonscan be constructed by equal division of a circle

    construct an accurate drawing of a regularhexagon inside a circle

    Transformations and coordinates

    3. Pupils should be taught to:

    Specifying transformations

    a) understand that translations arespecified by a distance and direction, andenlargements by a centre and positive scalefactor

    describe transformations fully

    Properties of transformations

    b) recognise and visualise translations

    transform triangles and other 2-D shapes bytranslation, recognising that thesetransformations preserve length and angle, sothat any figure is congruent to its image underthis transformation

    c) recognise, visualise and constructenlargements of objects using positive scalefactors greater than one

    from a given centre such as the origin, or anenlargement of a simple shape on a squaredgrid

    enlargement scale factor 2 21

    d) recognise that enlargements preserveangle but not length

    identify the scale factor of an enlargement asthe ratio of the lengths of any twocorresponding line segments and apply this totriangles

    understand the implications of enlargement forperimeter

    compare angles in two similar shapes

    simple examples of similar triangles with apositive scale factor greater than one

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    Contents Examples

    Measures and construction

    4. Pupils should be taught to:

    Construction

    e) use straight edge and compasses to dostandard constructions, including an equilateraltriangle with a given side

    construct an equilateral triangle with side oflength 4 cm

    construction lines need to be shown

    regular hexagon inside a circle

    the pair of compasses should have a span of atleast 6 cm

    draw circles and arcs to a given radius

    triangles with only the side lengths given willrequire accurate construction

    construct parallel lines

    Mensuration

    f) find areas of rectangles, recalling theformula, understanding the connection tocounting squares and how it extends thisapproach

    recall and use the formulae for the area of aparallelogram and a triangle

    find the surface area of simple shapes using thearea formulae for triangles and rectangles;

    calculate perimeters and areas of shapes madefrom triangles and rectangles

    find the area of a shape by counting squares

    find the area of a compound shape made fromrectangles and triangles by counting squaresand using formulae

    Find the surface area of a cuboid of dimensions7 cm 8 cm 9 cm

    compound shapes, e.g.

    ,

    g) find volumes of cuboids, recalling theformula and understanding the connection tocounting cubes and how it extends thisapproach

    calculate volumes of shapes made from cubesand cuboids

    find the volume of a compound solidconstructed from cubes, by counting thosecubes

    how many smaller boxes can fit in a larger box,dimensions given

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    Ma4 Handling Data

    Contents Examples

    Specifying the problem and planning

    2. Pupils should be taught to:

    a) see that random processes areunpredictable

    practical applications of probability

    roll a dice 60 times and read the results

    b) identify questions that can be addressedby statistical methods

    construct a survey of meals eaten in a schoolcanteen

    car occupancy throughout the day

    c) discuss how data relate to a problem

    d) identify which primary data they needto collect and in what format, includinggrouped data, considering appropriate equalclass intervals

    e) design an experiment or survey

    decide what secondary data to use

    consider fairness

    Collecting data

    3. Pupils should be taught to:

    a) design and use data-collection sheetsfor continuous data

    understand and use tallying methods

    Processing and representing data

    4. Pupils should be taught to:

    draw and produce, using paper and ICT,diagrams for continuous data, including scattergraphs and stem-and-leaf diagrams

    scatter diagram (graph) weight against height

    stem and leaf for height of 30 pupils from aschool

    b) calculate mean, range and median ofsmall data sets with continuous data

    identify the modal class for grouped continuousdata

    find the mode, mean or median of six heightsmeasured to the nearest cm

    state the modal class from a grouped frequencytable

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    Contents Examples

    d) understand and use estimates ormeasures of probability from theoreticalmodels (including equally likely outcomes)

    probabilities must be written as fractions,decimals or percentages

    dice , spinners, coins

    least likely, most likely

    estimate a probability as n1 or n

    a (or anequivalent decimal or percentage)

    f) identify different mutually exclusiveoutcomes and know that the sum of theprobabilities of all these outcomes is 1

    probability of NOT is 1 n1

    find a missing probability from a list or table

    h) draw lines of best fit by eye,understanding what these represent

    use line of best fit

    Interpreting and discussing results

    5. Pupils should be taught to:

    a) relate summarised data to the initialquestions

    b) interpret a wide range of graphs anddiagrams and draw conclusions

    c) look at data to find patterns andexceptions

    an isolated point on a scatter diagram

    use search criteria in lists and tables

    d) compare distributions and makeinferences, using the shapes of distributions andmeasures of average and range

    comparison of mean and range for twodistributions, one set given

    e) consider and check results and modifytheir approach if necessary

    f) have a basic understanding ofcorrelation as a measure of the strength of theassociation between two variables

    identify correlation or no correlation using linesof best fit

    describe correlation in words, e.g. as []goes up, [] goes down or positivecorrelation, negative correlation

    h) compare experimental data andtheoretical probabilities

    understand that rolling a dice 60 times shouldgive about 10 occurrences of each of 1, 2, 3, 4,5 and 6

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    Contents Examples

    i) understand that if they repeat anexperiment, they may and usually will getdifferent outcomes, and that increasing samplesize generally leads to better estimates ofprobability and population characteristics

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    Foundation tier: Stage 3Ma2 Number and algebra

    Contents Examples

    Numbers and the number system

    2. Pupils should be taught to:

    Percentages

    e) use percentage in real-life situations commerce and business, including rate ofinflation (or depreciation), VAT and interestrates

    Calculations

    3. Pupils should be taught to:

    Number operations and the relationshipsbetween them

    c) add and subtract fractions by writingthem with a common denominator 3

    2 + 43

    d) divide a fraction by an integer43 5

    f) divide a quantity in a given ratio share 15 in the ratio of 1:2

    Mental methods

    h) round to one significant figure

    estimate answers to problems involvingdecimals

    in interest calculations, or in carrying outestimates of complex calculations

    estimate 7.16.52.208.4

    as 25

    26205 =

    Written methods

    k) use standard column procedures formultiplication of integers and decimals,understanding where to position the decimalpoint by considering what happens if theymultiply equivalent fractions

    42.7 14, 3.27 12

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    Contents Examples

    l) use efficient methods to calculate withfractions, including cancelling common factorsbefore carrying out the calculation, recognisingthat, in many cases, only a fraction can expressthe exact answer

    100 3, 43 3

    2 , 32 7

    3

    m) solve simple percentage problems,including increase and decrease

    VAT, annual rate of inflation (or depreciation),income tax, discounts, sale prices

    n) solve word problems about ratio andproportion, including using informal strategiesand the unitary method of solution

    given that m identical items cost y, then oneitem costs y/m and n items cost (n y/m), thenumber of items that can be bought for z isz m/y

    6 pencils cost 78p, what will 8 cost?

    Calculator methods

    p) enter a range of calculations, includingthose involving measures

    time calculations in which fractions of an hourmust be entered as fractions or as decimals

    realising that that 2 hours 35 minutes is 2 6035

    q) understand the calculator display,interpreting it correctly, and knowing not toround during the intermediate steps of acalculation

    in money calculations, or when the display hasbeen rounded by the calculator; 5 2 = 2.50,not 2.5

    Solving numerical problems

    4. Pupils should be taught to:

    b) select appropriate operations, methodsand strategies to solve number problems,including trial and improvement where a moreefficient method to find the solution is notobvious

    reverse rate problems: from the total cost findhow many adult/child tickets bought

    given the area, find the perimeter of a square

    c) use a variety of checking procedures,including working the problem backwards, andconsidering whether a result is of the rightorder of magnitude

    d) give solutions in the context of theproblem to an appropriate degree of accuracy,interpreting the solution shown on a calculatordisplay, and recognising limitations on theaccuracy of data and measurements

    appropriate rounding up or down incalculations when remainders occur in realcontexts

    383 23 approximates to 400 20

    bill/rate calculations using 0.35p/unit; also withpercentage calculations

    interpret 3.6 pounds from a calculator displayas 3.60

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    Contents Examples

    Equations, formulae and identities

    5. Pupils should be taught to:

    Use of symbols

    a) distinguish the different roles played byletter symbols in algebra, knowing that lettersymbols represent defined quantities orvariables in formulae, general, unspecified andindependent numbers in identities and infunctions they define new expressions orquantities by referring to known quantities

    V = IR is a formula3x + 2x = 5x for all values of x is an identity

    y = 2x

    b) manipulate algebraic expressions bytaking out single common term factors

    x2 + 3x = x(x + 3)factorise: 6x 4

    Index notation

    c) substitute positive and negative numbersinto expressions such as 3x2 + 4 and 2x3

    C = 9

    )1( +yx . Find C when x = 30 and y = 7

    Linear equations

    e) solve linear equations that require priorsimplification of brackets, including those thathave negative signs occurring anywhere in theequation, and those with a negative solution

    3(x 4) = 30, 8 + x = 5(4 + x), 3x + 5 = 1

    Formulae

    f) derive a formula find the perimeter of a rectangle given its areaA and the length l of one side

    find the inverse of simple rules derivingexpressions

    Sequences, functions and graphs

    6. Pupils should be taught to:

    Graphs of linear functions

    b) plot graphs of functions in which y isgiven explicitly in terms of x, or implicitly

    y = 2x + 3, x + y = 7

    y = ax + b where b is an integer, a is a fractionor an integer, including negative values of x

    candidates may be required to draw, label andscale axes

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    Contents Examples

    c) construct linear functions from real-lifeproblems and plot their corresponding graphs

    discuss and interpret graphs arising from realsituations

    electricity bills, fixed charge and cost per unit,car hire, converging graphs

    bath filling

    Interpret graphical information

    e) interpret information presented in arange of linear and non-linear graphs

    graphs describing trends, conversion graphs,distance-time graphs, graphs of height orweight against age, graphs of quantities thatvary against time, such as employment

    travel graphs (as distinct from distance-timegraphs)

    graphs of water filling different-sizedcontainers

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    Ma3 Shape, space and measures

    Contents Examples

    Geometrical reasoning

    2. Pupils should be taught to:

    Properties of triangles and other rectilinearshapes

    c) use parallel lines, alternate angles andcorresponding angles

    understand the properties of parallelograms anda proof that the angle sum of a triangle is 180degrees

    understand a proof that the exterior angle of atriangle is equal to the sum of the interiorangles at the other two vertices

    corresponding or F anglesalternate or Z angles

    find the three missing angles in a parallelogramwhen one angle is given

    Properties of circles

    i) recall the meaning of the terms chord,tangent and arc as they relate to the circle

    3-D shapes

    j) explore the geometry of cuboids(including cubes), and shapes made fromcuboids

    know the terms face, edge, vertex

    k) use 2-D representations of 3-D shapesand analyse 3-D shapes through 2-Dprojections and cross-sections, including planand elevation.

    nets

    draw common 2-D and 3-D (solids or shapes)in different orientations on grids, including anisometric grid

    front and side elevations and plans of shapesmade from simple solids

    Transformations and coordinates

    3. Pupils should be taught to:

    Properties of transformations

    b) recognise and visualise rotations,reflections and translations, including reflectionsymmetry of 3-D shapes

    identify and draw in planes of symmetry

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    Contents Examples

    c) understand from this that any two circlesand any two squares are mathematicallysimilar, while, in general, two rectangles arenot

    d) use and interpret maps and scale drawings draw lines and shapes to scale

    estimate lengths using a scaled diagram

    read and construct scale drawings

    Coordinates

    e) find the coordinates of points identifiedby geometrical information

    find the coordinates of the midpoint of the linesegment AB, given points A and B

    find the coordinates of the fourth vertex of aparallelogram with vertices at (2, 1) (7, 3) and(5, 6)

    given endpoints (1, 2), (5, 3) deduce themidpoint is (3, 2 2

    1 )

    Measures and construction

    4. Pupils should be taught to:

    Measures

    a) know rough metric equivalents ofpounds, feet, miles, pints and gallons

    metric/imperial approximate conversions:Metric Imperial1 kg 2.2 pounds1 l 1 pints4.5 l 1 gallon8 km 5 miles30cm 1 foot

    c) understand and use speed simple calculations: find the average speed for200 miles travelled in 2 hours

    Construction

    d) understand, from their experience ofconstructing them, that triangles satisfying SSS,SAS, ASA and RHS are unique, but SSAtriangles are not

    construct cubes, regular tetrahedra, square-based pyramids and other 3-D shapes fromgiven information

    draw two different triangles with the same SSA

    nets and models

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    Contents Examples

    Mensuration

    h) find circumferences of circles and areasenclosed by circles, recalling relevant formulae

    candidates need to know 3.14 or use the button on their calculator

    i) convert between area measures,including square centimetres and square metres,and volume measures, including cubiccentimetres and cubic metres

    5 m2 = 5 1002 cm2 = 50 000 cm2 and5 m3 = 5 1003 cm3 = 50 000 000 cm3

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    Ma4 Handling Data

    Contents Examples

    Collecting data

    3. Pupils should be taught to:

    c) design and use two-way tables fordiscrete and grouped data.

    Processing and representing data

    4. Pupils should be taught to:

    a) draw and produce, using paper andICT, line graphs for time series

    Interpreting and discussing results

    5. Pupils should be taught to:

    k) interpret social statistics includingindex numbers

    time series

    and survey data

    the General Index of Retail Prices

    population growth

    the National Census

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    Intermediate tier: Stage 1Pupils should be taught the Knowledge, skills and understanding contained in this specificationthrough:

    a) activities that ensure they become familiar with and confident using standard procedures forthe range of calculations appropriate to this level of study

    b) solving familiar and unfamiliar problems in a range of numerical, algebraic and graphicalcontexts and in open-ended and closed form

    c) using standard notations for decimals, fractions, percentages, ratio and indices

    d) activities that show how algebra, as an extension of number using symbols, gives preciseform to mathematical relationships and calculations

    e) activities in which they progress from using definitions and short chains of reasoning tounderstanding and formulating proofs in algebra and geometry

    f) a sequence of practical activities that address increasingly demanding statistical problems inwhich they draw inferences from data and consider the uses of statistics in society

    g) choosing appropriate ICT tools and using these to solve numerical and graphical problems, torepresent and manipulate geometrical configurations and to present and analyse data.

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    Ma2 Number and algebra

    Contents Examples

    Using and applying number and algebra Using and applying statements apply to allthree stages; thus examples given must be seenin the context of the relevant stage.

    1. Pupils should be taught to:

    Problem solving

    a) select and use appropriate andefficient techniques and strategies to solveproblems of increasing complexity, involvingnumerical and algebraic manipulation

    b) identify what further informationmay be required in order to pursue aparticular line of enquiry and give reasonsfor following or rejecting particularapproaches

    c) break down a complex calculation intosimpler steps before attempting a solution andjustify their choice of methods

    d) make mental estimates of the answersto calculations

    present answers to sensible levels of accuracy

    understand how errors are compounded incertain calculations

    Communicating

    e) discuss their work and explain theirreasoning using an increasing range ofmathematical language and notation

    f) use a variety of strategies and diagramsfor establishing algebraic or graphicalrepresentations of a problem and its solution;

    move from one form of representation toanother to get different perspectives on theproblem

    compound interest

    Credit and Finance plan calculations withdeposit and monthly payments, comparison ofcredit methods, loan or hire rates

    7820342 =

    8020040 ,

    231.097.4610 =

    2.05600

    round practical measurements correct to 1mm ifappropriate

    by premature approximation; the effect ofsquaring on errors

    derive equations from problems

    derivation of equations such as linear andsimultaneous equations

    by generating a problem through use of algebra

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    Contents Examples

    g) present and interpret solutions in thecontext of the original problem

    h) use notation and symbols correctly andconsistently within a given problem

    i) examine critically, improve, thenjustify their choice of mathematicalpresentation

    Reasoning

    j) explore, identify, and use pattern andsymmetry in algebraic contexts, investigatingwhether a particular case may be generalisedfurther and understand the importance of acounter example

    identify exceptional cases when solvingproblems

    k) understand the difference between apractical demonstration and a proof

    l) show step by step deduction in solving aproblem

    m) recognise the significance of statingconstraints and assumptions when deducingresults

    recognise the limitations of any assumptionsthat are made and the effect that varying theassumptions may have on the solution to aproblem

    money notation, correctly given algebraicexpressions, mathematical notation

    simplification of fractions, percentages, ratio

    clear presentation of working out

    the limitations of the substitution of numericalvalues as demonstrating the validity of a proof,whereas one value which does not work issufficient to demonstrate that a hypothesis isfalse

    clear working, line by line

    e.g. the number of articles bought must be aninteger

    Numbers and the number system

    2. Pupils should be taught to:

    Integers

    a) use their previous understanding ofintegers and place value to deal with arbitrarilylarge positive numbers and round them to agiven power of 10

    understand and use negative integers both aspositions and translations on a number line

    write a quarter of a million in figures

    write 125 739 to the nearest thousand

    given 87 132 = 11484 find 87 0.132

    temperature calculations, changes in height;3 2, 7 + 3

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    Contents Examples

    order integers

    use the concepts and vocabulary of factor(divisor), multiple, common factor, highestcommon factor, least common multiple,prime number and prime factordecomposition

    find the warmest temperature from a list, placea series of file references in ascending ordescending order

    find LCM and HCF

    pick from a list the prime numbers, and primefactors of a number

    Powers and roots

    b) use the terms square, positive squareroot, negative square root, cube and cube root

    4 = 2 but x2 = 4 x = 2cube root of 125

    Fractions

    c) understand equivalent fractions,simplifying a fraction by cancelling allcommon factors

    order fractions by rewriting them with acommon denominator

    cancelling

    express 3528 in its simplest form (lowest terms)

    write 83 , 4

    1 , 103 in descending order

    write down two fractions between 41 and 2

    1

    Decimals

    d) recognise that each terminating decimalis a fraction

    recognise that recurring decimals are exactfractions, and that some exact fractions arerecurring decimals

    order decimals

    0.137 = 1000137

    71 == 0.142857142857...

    the notation 30. , 540. will be used to indicaterecurring decimals

    write in ascending order: 0.32, 2.3, 0.23, 3.2

    Percentages

    e) understand that percentage meansnumber of parts per 100, and interpretpercentage as the operator so many hundredthsof

    10% means 10 parts per 100 and 15% of Ymeans 100

    15 Y

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    Contents Examples

    Calculations

    3. Pupils should be taught to:

    Number operations and the relationshipsbetween them

    a) multiply or divide any number bypowers of 10, and any positive number by anumber between 0 and 1

    find the prime factor decomposition ofpositive integers

    multiply and divide by a negative number

    3 0.4,0.7 1000, 8.8 10,5.436 0.12, 9 30

    calculate shopping bills, energy bills

    write 252 as 22 32 7

    3 4, 5 2, 12 +3

    b) use brackets and the hierarchy ofoperations

    insert brackets so that 20 3 2 = 34

    BIDMAS

    work out 5 (2 + 3) and 5 2 + 3

    c) calculate a given fraction of a givenquantity, expressing the answer as a fraction 4

    3 of 48; 32 of 4

    3

    if 750 of 1000 is given to Penelope, whatfraction is this?

    express a given number as a fraction of another

    add and subtract fractions by writing them witha common denominator

    perform short division to convert a simplefraction to a decimal

    distinguish between fractions withdenominators that have only prime factorsof 2 and 5 (which are represented byterminating decimals), and other fractions(which are represented by recurringdecimals);

    write 60 out of 200 as a fraction in its simplestform

    32 + 4

    3 , 2 21 1 5

    4

    write 83 as a decimal

    d) understand and use unit fractions asmultiplicative inverses

    by thinking of multiplication by 51 as division

    by 5, or multiplication by 76 as multiplication

    by 6 followed by division by 7 (or vice versa)

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    Contents Examples

    multiply and divide a given fraction by aninteger, by a unit fraction and by a generalfraction

    103 4, 4

    3 5, how many 32 oz portions in 7 oz,

    53 8

    4 , 32 2 4

    3 , 1 21 2 5

    4

    e) convert simple fractions of a whole topercentages of the whole and vice versa

    analysing diets, budgets or the cost of running,maintaining and owning a car

    change 52 and 0.6 into a percentage

    70 out of 200 as a percentage

    Written methods

    i) use efficient methods to calculate withfractions, including cancelling common factorsbefore carrying out the calculation, recognisingthat in many cases only a fraction can expressthe exact answer

    100 3, 32

    43 , 7

    332

    j) solve percentage problems, includingpercentage increase and decrease;

    percentages of quantities, simple interest, VAT,annual rate of inflation or depreciation

    sale price, discounts, percentage profit/loss

    Calculator methods

    p) understand the calculator display,knowing when to interpret the display, whenthe display has been rounded by the calculator,and not to round during the intermediate stepsof a calculation

    understand why 32 is given as 0.666667 by

    some (but not all) calculators

    problems associated with premature roundingwhilst undertaking calculations

    Equations, formulae and identities

    5. Pupils should be taught to:

    Use of symbols

    a) distinguish the different roles played byletter symbols in algebra, using the correctnotational conventions for multiplying ordividing by a given number, and knowing thatletter symbols represent definite unknownnumbers in equations, defined quantities orvariables in formula, general, unspecified andindependent numbers in identities, and infunctions they define new expressions orquantities by referring to known quantities

    y = 2 7x; f(x) = x3; y = 1/x with x 0

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    Contents Examples

    b) understand that the transformation ofalgebraic entities obeys and generalises thewell-defined rules of generalised arithmetic

    manipulate algebraic expressions by collectinglike terms, multiplying a single term over abracket and taking out common factors

    a(b + c) = ab + acab = a b = b a = baa + a + a = 3a, a a = a2

    9x 3 = 3(3x 1) , ax 2x = x(a 2)

    c) know the meaning of and use thewords equation, formula, identity andexpression

    x2 + 1 = 82 is an equationV = IR is a formulax(x + 2) = x2 + 2x for all x is an identity

    Equations

    e) set up simple equations;

    solve simple equations by using inverseoperations or by transforming both sides inthe same way

    find the angle a in a triangle with angles a,a + 10, a + 20

    5x = 7; 11 4x = 2; 3(2x + 1) = 8;2(1 x) = 6(2 + x); 4x2 = 49; 3 = 12/x

    Linear equations

    f) solve linear equations in one unknown,with integer or fractional coefficients, inwhich the unknown appears on either side or onboth sides of the equation

    61 x + 3

    1 x = 5

    solve linear equations that require priorsimplification of brackets, including those thathave negative signs occurring anywhere in theequation, and those with a negative solution

    5x + 17 = 3(x + 6), 8 + x = 5(4 + x)3(x 4) = 30

    Formulae

    g) use formulae from mathematics andother subjects

    substitute numbers into a formula

    for area of a triangle or a parallelogramarea enclosed by a circle, volume of a prism,volume of a cone

    knowledge of inverse operations, such asfinding x given y in y = 8x 1

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    Sequences, functions and graphs

    6. Pupils should be taught to:

    Sequences

    a) generate common integer sequences(including sequences of odd or even integers,squared integers, powers of 2, powers of 10,triangular numbers)

    generate terms of a sequence using term-to-term and position-to-term definitions of thesequence

    use linear expressions to describe the nth termof an arithmetic sequence, justifying its form byreference to the activity or context from whichit was generated

    find the next term (or the 10th term) in1,2,4,8,...; 5,9,13,17,; 1, 4,5,8,9,12,13,

    express the rule for finding the nth term inwords

    draw the next complex diagrammatic term in asequence of diagrams

    give the nth term of a linear sequence as analgebraic expression

    Graphs of linear functions

    b) use conventions for coordinates in theplane

    plot points in all four quadrants

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    Ma3 Shape, space and measures

    Contents Examples

    Using and applying shape, space andmeasures

    Using and applying statements apply to allthree stages; thus examples given must be seenin the context of the relevant stage.

    1. Pupils should be taught to:

    Problem solving

    a) select the problem-solving strategies touse in geometrical work, and consider andexplain the extent to which the selectionsthey made were appropriate

    b) select and combine known facts andproblem-solving strategies to solve morecomplex geometrical problems

    c) develop and follow alternative linesof enquiry

    Communicating

    review and justify their choice ofmathematical presentation

    d) communicate mathematically, withemphasis on a critical examination of thepresentation and organisation of results, andon effective use of symbols and geometricaldiagrams

    Reasoning

    distinguish between practicaldemonstrations and proofs

    f) apply mathematical reasoning,progressing from brief mathematicalexplanations towards full justifications inmore complex contexts

    g) explore connections in geometry

    pose conditional constraints of the typeIf ... then ...

    and ask questions What if ...? or Why?

    find the location of a ship, given its bearingfrom two lighthouses

    recall and use basic properties of angles (anglesat a point, angles on a straight line, oppositeangles) in more complex problems

    use 3 letter notation for an angle ABC

    use 2 letter notation AB for the line between Aand B

    give explanations showing application ofgeometrical properties

    determine whether a tessellation of a givenshape is possible

    e.g. what if we allow 2 regular polygons?

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    show step-by-step deduction in solving ageometrical problem

    i) state constraints and give startingpoints when making deductions

    Geometrical reasoning

    2. Pupils should be taught to:

    Properties of triangles and other rectilinearshapes

    a) distinguish between lines and linesegments

    use parallel lines, alternate angles andcorresponding angles

    understand the consequent properties ofparallelograms and a proof that the angle sumof a triangle is 180 degrees

    understand a proof that the exterior angle of atriangle is equal to the sum of the interiorangles at the other two vertices

    line segments are denoted by AB

    corresponding or F angles, alternate or Z angles

    find the three missing angles in a parallelogramwhen one angle is given

    b) use angle properties of equilateral,isosceles and right-angled triangles

    explain why the angle sum of a quadrilateral is360 degrees

    find missing angles

    as the angle sum of two triangles

    c) recall the definitions of special types ofquadrilateral, including square, rectangle,parallelogram, trapezium and rhombus

    classify quadrilaterals by their geometricproperties

    list and compare the properties of each

    name all quadrilaterals that have a pair ofopposite sides that are parallel.

    d) calculate and use the sums of theinterior and exterior angles of quadrilaterals,pentagons, hexagons

    calculate and use the angles of regular polygons

    a tessellation of at least 6 kites

    irregular polygons with 4, 5 or 6 sides only

    complete a tessellation using a regular polygon

    sum of the interior angles of an n-sided polygonis (n 2) 180 or 2n 4 right angles

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    Properties of circles

    h) recall the definition of a circle and themeaning of related terms, including centre,radius, chord, diameter, circumference, tangent,arc, sector and segment

    Transformations and coordinates

    3. Pupils should be taught to:

    Specifying transformations

    a) understand that rotations are specifiedby a centre and an (anticlockwise) angle

    use any point as the centre of rotation

    measure the angle of rotation, using rightangles, fractions of a turn or degrees

    understand that reflections are specified by a(mirror) line

    identify the point from which a shape has beenrotated

    rotate a triangle T by 60 anti-clockwise aboutthe point (1, 2) to give triangle U

    Properties of transformations

    b) recognise and visualise rotations,reflections and translations including reflectionsymmetry of 2-D and 3-D shapes, and rotationsymmetry of 2-D shapes

    transform triangles and other 2-D shapes bytranslation, rotation and reflection

    identify & draw in planes of symmetry

    state the order of rotational symmetry

    identify x = 0 or y = 0 as the line of symmetry

    reflection in any line

    Coordinates

    e) use axes and coordinates to specify points inall four quadrants

    locate points with given coordinates

    find the coordinates of points identified bygeometrical information

    for 2-D

    identify and plot points with given coordinatesin 2-D

    find the points which comprise a parallelogram

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    Measures and construction

    4. Pupils should be taught to:

    Measures

    a) use angle measure

    know that measurements using real numbersdepend on the choice of unit

    use bearings to specify direction, e.g. 072,314

    convert measurements from one unit to another conversions between Imperial measures will begiven

    metric/imperial approximate conversions:Metric Imperial1 kg 2.2 pounds1 l 1 pints4.5 l 1 gallon8 km 5 miles

    Construction

    b) draw approximate constructions oftriangles and other 2-D shapes, using a rulerand protractor, given information about sidelengths and angles

    construct specified cubes, regular tetrahedra,square-based pyramids and other 3-D shapes

    a ruler of length 30 cm may be needed

    nets and models

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    Ma4 Handling data

    Contents Examples

    Using and applying handling data Using and applying statements apply to allthree stages; thus examples given must be seenin the context of the relevant stage.

    1. Pupils should be taught to:

    Problem solving

    a) carry out each of the four aspects of thehandling data cycle to solve problems:

    (i) specify the problem and plan:formulate questions in terms of the dataneeded, and consider what inferences canbe drawn from the data;

    decide what data to collect (includingsample size and data format) and whatstatistical analysis is needed

    (ii) collect data from a variety of suitablesources, including experiments andsurveys, and primary and secondarysources

    (iii) process and represent the data: turnthe raw data into usable information thatgives insight into the problem

    (iv) interpret and discuss the data: answerthe initial question by drawing conclusionsfrom the data

    b) select the problem-solving strategiesto use in statistical work, and monitor theireffectiveness (these strategies should addressthe scale and manageability of the tasks, andshould consider whether the mathematicsand approach used are delivering the mostappropriate solutions)

    use suitable data collection techniques

    estimate how long it will take to carry out 50questionnaires

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    Communicating

    c) communicate mathematically, withemphasis on the use of an increasing rangeof diagrams and related explanatory text, onthe selection of their mathematicalpresentation, explaining its purpose andapproach, and on the use of symbols toconvey statistical meaning

    Reasoning

    d) apply mathematical reasoning,explaining and justifying inferences anddeductions, justifying arguments andsolutions

    e) identify exceptional or unexpectedcases when solving statistical problems

    f) explore connections in mathematics andlook for relationships between variableswhen analysing data

    g) recognise the limitations of anyassumptions and the effects that varying theassumptions could have on the conclusionsdrawn from data analysis

    calculate total frequency or the fx column froma discrete frequency table

    compare the mean height p for boys and girls toanswer the question are girls taller at 12 yearsold?

    the older the car, the lower its value

    explain deficiencies in questions on datacollection techniques

    Collecting data

    3. Pupils should be taught to:

    a) collect data using various methods,including observation, controlled experiment,data logging, questionnaires and surveys

    understand and use tallying methods

    sort, classify and tabulate (categorical) data anddiscrete or continuous quantitative data.

    grouping of discrete and continuous data intoclass intervals of equal width

    b) gather data from secondary sources,including printed tables and lists from ICT-based sources

    extract data from lists and tables

    c) design and use two-way tables fordiscrete and grouped data

    d) deal with practical problems such asnon-response or missing data

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    Processing and representing data

    4. Pupils should be taught to:

    c) list all outcomes for single events, andfor two successive events, in a systematic way

    three coins, two dice

    d) identify different mutually exclusiveoutcomes and know that the sum of theprobabilities of all these outcomes is 1

    if the probability of an event occurring is p,then the probability of the event not occurringis 1 p

    find a missing probability from lists and tables

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    Intermediate tier: Stage 2Ma2 Number and algebra

    Contents Examples

    Numbers and the number system

    2. Pupils should be taught to:

    Powers and roots

    b) use standard index form, expressed inconventional notation and on a calculatordisplay

    write 35 000 as 3.5 104, 0.00034 as 3.4 10-4

    work out 35000 0.007, giving your answer instandard form

    Ratio

    f) use ratio notation, including reductionto its simplest form and its various links tofraction notation

    if 41 of the class is boys, the ratio of boys to

    girls is 1:3, writing in the form 1: n

    Calculations

    k=~=~==~=

    3. Pupils should be taught to:

    f) divide a quantity in a given ratio 5000 in the ratio 10:8:7;

    Mental methods

    h) round to a given number of significantfigures;

    round to a given number of decimal places

    round 4567 to 3 significant figures, round0.0258 to 2 significant figures

    Calculator methods

    o) use calculators effectively andefficiently, knowing how to enter complexcalculations

    Find 3.32 + 4.3, 5.7 (4.2 + 0.5)2

    use an extended range of function keys,including trigonometrical and statisticalfunctions relevant across this programme ofstudy

    22

    22

    9.18.43.24.6

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    Contents Examples

    the electronic calculator to be used bycandidates should have, as a minimum, thefollowing functions: +, , , , x 2, x,memory, brackets, xy, yx , trigonometricfunctions

    Solving numerical problems

    4. Pupils should be taught to:

    b) check and estimate answers to problems;

    select and justify appropriate degrees ofaccuracy for answers to problems;

    appropriate rounding up or down in realcontexts when remainders occur

    bill/rate calculations involving a fraction of apenny per unit (e.g. 0.35p/unit)

    also with percentage calculations

    Equations, formulae and identities

    5. Pupils should be taught to:

    b) expand the product of two linearexpressions

    manipulate algebraic expressions byfactorising quadratic expressions

    (x + 1)(x + 2) = x2 + 3x + 2(2x + 5)(x 3) = 2x2 x 15

    4x2 + 6xy = 2x(2x + 3y), x2 + x = x(x + 1)x2 5x + 6 = (x 3)(x 2)ax2 +bx + c = 0, a = 1

    Index notation

    d) substitute positive and negativenumbers into expressions such as 3x2 + 4 and2x3

    Formulae

    g) change the subject of a formula find x given y = mx + c

    generate a formula find the perimeter of a rectangle given its areaA and the length l of one side

    find the inverse of simple rules, derivingexpressions

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    Simultaneous linear equations

    j) solve simple linear inequalities in onevariable, and represent the solution set on anumber line

    26 + n > 4n 7

    show 3 < x 4 on a number line

    Numerical methods

    m) use systematic trial andimprovement to find approximate solutionsof equations where there is no simpleanalytical method of solving them

    solve x3 x = 900 to 1 decimal placex3 + x = 12 to 2 decimal places

    Sequences, functions and graphs

    6. Pupils should be taught to:

    Graphs of linear functions

    b) plot graphs of functions in which y is givenexplicitly in terms of x (as in y = 2x + 3), orimplicitly (as in x + y = 7)

    y = ax + b where b is an integer, a is a fractionor an integer

    Interpreting graphical information

    d) construct linear functions and plot thecorresponding graphs arising from real-lifeproblems

    discuss and interpret graphs modelling realsituations

    distance-time graph for a particle moving withconstant speed, the velocity-time graph for aparticle moving with constant acceleration

    travel graphs, conversion graphs

    the depth of water in a container as it empties

    Quadratic functions

    e) generate points and plot graphs ofsimple quadratic functions, then moregeneral quadratic functions

    y = x2, y = 3x2 + 4y = x2 2x + 1y = ax2 + bx + c

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    Ma3 Shape, space and measures

    Contents Examples

    Geometrical reasoning

    2. Pupils should be taught to:

    Properties of triangles and other rectilinearshapes

    f) understand, recall and usePythagoras theorem in 2-D problems

    calculate the height (altitude) of an isoscelestriangle given the lengths of all three sides

    g) understand, recall and usetrigonometrical relationships in right-angledtriangles, and use these to solve problems,including those involving bearings

    may involve angles of elevation and depression

    right-angled triangles in 2-D only

    Properties of circles

    h) understand that the tangent at anypoint on a circle is perpendicular to theradius at that point

    understand and use the fact that tangentsfrom an external point are equal in length

    understand that inscribed regular polygons canbe constructed by equal division of a circle

    two tangents to a circle from an external pointproduce a situation that is symmetrical

    Transformations and coordinates

    3. Pupils should be taught to:

    Specifying transformations

    a) understand that translations arespecified by giving a distance and direction (ora vector), and enlargements by a centre and apositive scale factor

    a translation by the vector

    32

    describe transformations

    describe combinations of transformations

    Properties of transformations

    c) recognise, visualise and constructenlargements of objects

    from a given centre

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    understand from this that any two circles andany two squares are mathematically similar,while, in general, two rectangles are not, thenuse positive fractional scale factors

    enlargement scale factor 31 , 2 2

    1

    d) recognise that enlargements preserveangle but not length

    identify the scale factor of an enlargement asthe ratio of the lengths of any twocorresponding line segments

    understand the implications of enlargement forperimeter

    use and interpret maps and scale draw