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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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Page 1: Edexcel GCSE in Mathematics (Specification B 1388 )1223023076.web1.sitemove.co.uk/_files/maths/edexcel gcse...UG009825 – Edexcel GCSE Mathematics 1388 – Teachers’ Guide: Content

EDEXCEL GCSE

Teachers’ Guide – Content Exemplification (Modular)

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

July 2003

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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

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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

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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 Edexcel’s 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.

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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 data

project and an investigational task ✔

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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 data

project 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 data

project 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.

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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.

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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 , Σx, Σfx, 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.)

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

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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.

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

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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 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

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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

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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’ means‘number 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

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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

,, 41

21 ,,,,, 3

231

1001

101

51

43

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

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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)

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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

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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

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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

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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°

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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

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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: 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

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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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Contents Examples

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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GCSE Mathematics (Modular) 1388 – Foundation Tier: Stage 3

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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 1¾ 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 ½ 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 π ≈ 3.14 or use the πbutton 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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GCSE Mathematics (Modular) 1388 – Intermediate Tier: Stage 1

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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’ means‘number 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’ and‘expression’

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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Contents Examples

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 type‘If ... 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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Contents Examples

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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Contents Examples

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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Contents Examples

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 1¾ 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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Contents Examples

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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Contents Examples

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

kìãÄÉê=çéÉê~íáçåë=~åÇ=íÜÉ=êÉä~íáçåëÜáéëÄÉíïÉÉå=íÜÉã

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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Contents Examples

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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Contents Examples

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 drawings

compare angles in two similar shapes

use of non-integer scale factors to find themissing length of a missing side in each of twosimilar triangles that have a pair ofcorresponding sides given

read and construct scale drawings

draw lines and shapes to scale

estimate lengths using a scale diagram

Measures and construction

4. Pupils should be taught to:

Measures

a) recognise that measurements given tothe nearest whole unit may be inaccurate byup to one half in either direction

understand and use compound measures,including speed and density

length of a page of a book can normally bemeasured to the nearest mm

Mensuration

d) find the surface area of simple shapesby using the formulae for the areas of trianglesand rectangles

find volumes of cuboids, recalling the formulaand understanding the connection to countingcubes and how it extends this approach

find the area of a compound shape made fromrectangles and triangles by counting squares

find the surface area of a cuboid of dimensions7 cm × 8 cm × 9cm

find the volume of a compound solidconstructed from cubes and cuboids

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Contents Examples

calculate volumes of right prisms and ofshapes made from cubes and cuboids

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

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

volumes of cylinders

candidates need to know π ≈ 3.14 or use the πbutton on their calculator

includes simple fractions of circle, e.g.semicircle

answers in terms of π may be required

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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

b) identify key questions that can beaddressed by statistical methods

c) discuss how data relate to a problem

identify possible sources of bias and plan tominimise it

occupancy of cars may vary through the day

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 primary and secondary data touse

consider fairness

Processing and representing data

4. Pupils should be taught to:

a) draw and produce, using paper andICT, pie charts for categorical data, anddiagrams for continuous data, including linegraphs (time series), scatter graphs, frequencydiagrams, stem-and-leaf diagrams, cumulativefrequency tables and diagrams, box plots

scatter diagram weight against height

frequency polygons for grouped continuousdata

box and whisker diagrams

b) understand and use estimates ormeasures of probability from theoreticalmodels, or from relative frequency

an estimate of the number of times an eventwith a probability of 7

2 happening over 300tries is 300 × 7

2

what is the probability of throwing a 3 or 4from a die, or a 6 twice?

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Contents Examples

e) find the median, quartiles andinterquartile range for large data sets andcalculate the mean for large data sets withgrouped data

calculate the mean and range for a set of data

the expression ‘estimate’ will be used whenappropriate

mean of grouped data using halfway values

i) 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;

pie charts

c) look at data to find patterns andexceptions

e) consider and check results, and modifytheir approaches if necessary

f) appreciate that correlation is ameasure of the strength of the associationbetween two variables

distinguish between positive, negative andzero correlation using lines of best fit

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

‘there is some evidence from this samplethat ...’

probabilities must be written as fractions,decimals or percentages

h) compare experimental data andtheoretical probabilities

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 parameters.

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Intermediate: Stage 3Ma2 Number and algebra

Contents Examples

Numbers and the number system

2. Pupils should be taught to:

Powers and roots

b) use index notation and index laws formultiplication and division of integer powers

82, 22 × 23, 35 ÷ 32

Calculations

3. Pupils should be taught to:

Number operations and the relationshipsbetween them

a) understand ‘reciprocal’ as multiplicativeinverse, knowing that any non-zero numbermultiplied by its reciprocal is 1 (and thatzero has no reciprocal, because division byzero is not defined)

use index laws to simplify and calculate thevalue of numerical expressions involvingmultiplication and division of integer powers

use inverse operations

find the reciprocal of 0.2, giving your answer inits simplest form

evaluate 4

53

222 × , work out 38 ÷ 36

e) understand the multiplicative natureof percentages as operators

calculate an original amount when given thetransformed amount after a percentagechange

reverse percentage problems

a 15% increase in value Y, followed by a 15%decrease is calculated as 1.15 × 0.85 × Y

given that a meal in a restaurant costs £36 withVAT at 17.5%, its price before VAT iscalculated as £ 1751

36.

calculate the original price given the sale price

Mental methods

g) recall integer squares from 2 × 2 to15 × 15 and the corresponding square roots, thecubes of 2, 3, 4, 5 and 10

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Contents Examples

h) develop a range of strategies formental calculation

derive unknown facts from those they know

convert between ordinary and standardindex form representations, converting tostandard index form to make sensibleestimates for calculations involvingmultiplication and/or division

estimate 7.16.52.208.4

−× using 1 significant figure

estimate 3√85

0.1234 = 1.234 × 10−1

019406382.

= 2

3

10941103826

−××

.

. ≅ 2

3

102106

−×× = 3 × 105

têáííÉå=ãÉíÜçÇë

j) solve percentage problems includingreverse percentages

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

sale price, discounts, percentage profit/loss

find the original price given the sale price

k) represent repeated proportionalchange using a multiplier raised to a power

compound interest, repeated calculations, e.g.bouncing ball reaching 70% of previous heighton each bounce

l) calculate an unknown quantity fromquantities that vary in direct proportion

m) calculate with standard index form 2.4 × 107 × 5 × 103 = 12 × 1010 = 1.2 × 1011 ,(2.4 × 107) ÷ (5 × 103) = 4.8 × 103

3.1 × 10−3 × 4.5 × 10−4

n) use surds and π in exact calculations,without a calculator

use of Pythagoras leading to an answer of √13,or the area of a circle given as 100π

Calculator methods

r) use standard index form display andhow to enter numbers in standard indexform

use of calculators in performing calculationswith numbers given in standard index form

s) use calculators for reversepercentage calculations by doing anappropriate division

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Contents Examples

Solving numerical problems

4. Pupils should be taught to:

a) draw on their knowledge ofoperations and inverse operations (includingpowers and roots), and of methods ofsimplification (including factorisation and theuse 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 ratioand proportion, repeated proportionalchange, fractions, percentages and reversepercentages, surds, measures and conversionbetween measures, and compound measuresdefined within a particular situation

recipes, sale prices, foreign currencies

non-calculator methods such as 365 × 54,753 ÷ 36

6 pencils cost 78p, what will 8 cost?

speed calculations and density

b) recognise limitations on the accuracyof data and measurements.

reading that a calculator result of 1.9999999can be 2

know that a distance of 10.0 km lies between9.5 km and 10.5 km

Equations, formulae and identities

5. Pupils should be taught to:

Use of symbols

b) manipulate algebraic expressions usingthe difference of two squares and bycancelling common factors in rationalexpressions

x2 – 9 = (x + 3)(x – 3)

2(x + 1)2/(x + 1) = 2(x + 1)

Index notation

d) use index notation for simple integerpowers, and simple instances of index laws

x3 × x2 = x5; x2/x3 = x−1; (x2)3 = x6

Formulae

g) change the subject of a formula,including cases where the subject occurstwice, or where a power of the subjectappears

3(w + y )= 5y + 7 to y =

p = rq + s to q =

find r given that A = π r2

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Contents Examples

Simultaneous linear equations

i) find the exact solution of twosimultaneous equations in two unknowns byeliminating a variable, and interpret theequations as lines and their common solutionas the point of intersection

solve algebraically:2x + y = 7, 3x – y = 13;4y + 5x = 3, 2y – x = −2

draw graphs and solve for x and y:y = 3x – 4, y = 2 – x

j) solve several linear inequalities intwo variables and find the solution set

shade the region defined byy > 3, y ≤ 7 – x, x > 0

Quadratic equations

k) solve quadratic equations byfactorisation

solve by factorising:x2 – 8x + 15 = 0ax2 + bx + c = 0 with a = 1

Sequences, functions and graphs

6. Pupils should be taught to:

Graphs of linear functions

b) recognise (when values are given form and c) that equations of the formy = mx + c correspond to straight-line graphsin the coordinate plane

match equations with simple sketch graphs

sketch the graphs of equations of the formy = mx + c

c) find the gradient of lines given byequations of the form y = mx + c (whenvalues are given for m and c)

understand that the form y = mx + crepresents a straight line and that m is thegradient of the line, and c is the value of they-intercept

explore the gradients of parallel lines

state the gradient of any line y = mx + c

find the equation of a line from a drawn line

give the equation of a line parallel to a drawnline

know that the lines represented by theequations y = −5x and y = 3 – 5x are parallel,each having gradient (−5)

Quadratic functions

e) find approximate solutions of aquadratic equation from the graph of thecorresponding quadratic function

solve x2 – 2x + 1 = 0solve x2 – 2x – 3 = 0 using the graph ofy = x2 – 2x + 1

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Contents Examples

Other functions

f) plot graphs of: simple cubic functions,the reciprocal function y = 1/x with x ≠ 0,using a spreadsheet or graph plotter as wellas pencil and paper

recognise the characteristic shapes of allthese functions

y = x3

match equations with their graphs

sketch graphs from given equations

Loci

h) construct the graphs of simple loci loci as defined on a coordinate grid

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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) investigate the geometry of cuboidsincluding cubes, and shapes made from cuboids

know the terms face, edge, vertex.

g) understand similarity of triangles andof other plane figures, and use this to makegeometric inferences

use of non-integer scale factors to find themissing length of a missing side in each of twosimilar triangles that have a pair ofcorresponding sides given

Properties of circles

h) explain why the perpendicular fromthe centre to a chord bisects the chord;

use the facts that the angle subtended by anarc at the centre of a circle is twice the anglesubtended at any point on thecircumference, the angle subtended at thecircumference by a semicircle is a rightangle, that angles in the same segment areequal, and that opposite angles of a cyclicquadrilateral sum to 180 degrees

3-D shapes

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

solve problems involving surface areas andvolumes of prisms and cylinders

draw common 2-D and 3-D shapes in differentorientations on grids, including an isometricgrid

total surface area = 2πrh + 2πr2

volumes of cylinders

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Contents Examples

Transformations and coordinates

3. Pupils should be taught to:

Properties of transformations

b) transform triangles and other 2-D shapesby translation, rotation and reflection andcombinations of these transformations

distinguish properties that are preservedunder particular transformations

reflection in any line

candidates may be asked to perform successivegeometrical transformations

e.g. angle is preserved

d) understand the difference betweenformulae for perimeter, area and volume byconsidering dimensions

recognise that 34 πr2 cannot represent the

volume of a sphere

Coordinates

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

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

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

given endpoints (–2, 4), (6, –5) deduce themidpoint is (2, – 2

1 )

use Pythagoras to calculate the length as√(82 + 92)

2 dimensions only

Vectors

f) understand and use vector notation the form ���

����

qp

is also required for translations

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Contents Examples

Measures and construction

4. Pupils should be taught to:

Construction

c) use straight edge and compasses to dostandard constructions including an equilateraltriangle with a given side, the midpoint andperpendicular bisector of a line segment, theperpendicular from a point to a line, theperpendicular from a point on a line, and thebisector of an angle

construct an equilateral triangle with side oflength 4 cm

construction lines and arcs are required to beshown

the pair of compasses should have a span up to6 cm

draw circles and arcs to a given radius.

Mensuration

d) convert between volume measuresincluding cm3 and m3

5 m2 = 5 × 1002 cm2 = 50 000 cm2

5 m3 = 5 × 1003 cm3 = 5 000 000 cm3

Loci

e) find loci, both by reasoning and byusing ICT to produce shapes and paths

a region bounded by a circle and an intersectingline

restricted to two dimensions

perpendicular bisector

angle bisector

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Ma4 Handling data

Contents Examples

Processing and representing data

4. Pupils should be taught to:

f) calculate an appropriate movingaverage

usually for 3 or 4 moving sets of data

3 or 4 point moving average

h) use tree diagrams to representoutcomes of compound events, recognisingwhen events are independent

enter probabilities to make a probability tree

draw a probability tree based on informationgiven

j) use relevant statistical functions on acalculator or spreadsheet

use of x key

Interpreting and discussing results

5. Pupils should be taught to:

b) identify seasonality and trends intime series

d) compare distributions and makeinferences, using shapes of distributions andmeasures of average and spread, includingmedian and quartiles

comparison of mean and range for twodistributions, one set given

f) appreciate that zero correlation doesnot necessarily imply ‘no relationship’ butmerely ‘no linear relationship’

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Higher 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 and efficienttechniques and strategies to solve problems ofincreasing complexity, involving numerical andalgebraic manipulation

b) identify what further information maybe required in order to pursue a particular lineof enquiry and give reasons for following orrejecting particular approaches

Communicating

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

f) move from one form of representationto another to get different perspectives on theproblem

compound interest

presenting answers appropriately

by generating a problem through use of algebra

i) examine critically, improve, then justifytheir choice of mathematical presentation

present a concise, reasoned argument

Reasoning

j) understand the importance of a counter-example

identify exceptional cases when solvingproblems

k) understand the difference between apractical demonstration and a proof

l) derive proofs using short chains ofdeductive reasoning

simplification of fractions, percentages andalgebraic expressions

the limitations of the substitution of numericalvalues as demonstrating the validity of a proof

based on algebra

based on geometry

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Contents Examples

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.

take into account the size and appropriatenessof answers

possible error of half a unit in practicalmeasurements

Numbers and the number system

2. Pupils should be taught to:

Integers

a) use the concepts and vocabulary ofhighest common factor, least common multiple,prime number and prime factor decomposition

find LCM and HCF

Powers and roots

b) use index laws for multiplication anddivision of integer powers

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

22 × 23, 35 ÷ 32

write 35 000 as 3.5 × 104, 0.000 34 as 3.4 × 10−4

Decimals

d) recognise that recurring decimals areexact fractions, and that some exact fractionsare recurring decimals

71 == 0.142857142857...

the notation 3.0 � and 54.0 �� will be used toindicate recurring decimals

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

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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 by anumber between 0 and 1

find the prime factor decomposition of positiveintegers

multiply and divide by a negative number

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

write 252 as 22 × 32 × 7

3 × −4, −5 × −2, −12 ÷ +3

c) distinguish between fractions withdenominators that have only prime factors of 2and 5 (which are represented by terminatingdecimals), and other fractions (which arerepresented by recurring decimals)

convert a recurring decimal to a fraction 0.142857142857... = 71

d) multiply and divide a given fraction by aunit fraction and by a general fraction

by thinking of multiplication by 76 as

multiplication by 6 followed by division by 7(or vice versa);3 4

3 × 7, 2 21 ÷ 5, 2 2

1 × 3 75 , 1 3

2 ÷ 4 54

e) understand the multiplicative nature ofpercentages as operators

calculate an original amount when given thetransformed amount after a percentage change

reverse percentage problems

a 15% increase in value Y, followed by a 15%decrease is calculated as 1.15 × 0.85 × Y

given that a meal in a restaurant costs £36 withVAT at 17.5%, its price before VAT iscalculated as £ 175.1

36

calculate the original price given the sale price

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

Mental methods

g) recall integer squares from 2 × 2 to15 × 15 and the corresponding square roots, thecubes of 2, 3, 4, 5 and 10

h) round to a given number of significantfigures

round to a given number of decimal places

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Contents Examples

convert between ordinary and standard indexform representations

0.1234 = 1.234 × 10−1

Written methods

j) solve percentage problems

reverse percentages

percentage of quantities, simple interest, VAT,annual rate of inflation

sale price, discounts, percentage profit/loss

k) represent repeated proportional changeusing a multiplier raised to a power

compound interest, repeated calculations, e.g.bouncing ball reaching 70% of previous heighton each bounce

Calculator methods

r) use standard index form display andhow to enter numbers in standard index form

use of calculators in performing calculationswith number given in standard index form

s) use calculators for reverse percentagecalculations by doing an appropriate division

Solving numerical problems

4. Pupils should be taught to:

a) draw on their knowledge of operationsand inverse operations and of methods ofsimplification (including factorisation and theuse 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 ratioand proportion, repeated proportional change,fractions, percentages and reverse percentages,inverse proportion, measures and conversionbetween measures, and compound measuresdefined within a particular situation

recipes, sale prices, foreign currencies

answers in surd form

6 pencils cost 78p, what will 8 cost?

speed calculations and density

b) check and estimate answers toproblems

appropriate rounding up or down whenremainders occur in real contexts

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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, 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

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, taking out common factors

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

simplify 3x4y2 × x2y3, 3a2 ÷ 6a3b3, (2xy3)4,(a−2)−3

simplify 3(x + 2) + 5(2x – 1),expand x(2x2 – 5)9x – 3 = 3(3x – 1), 4x2 + 6xy = 2x(2x + 3y)

c) know the meaning of and use the words‘equation’, ‘formula’, ‘identity’ and‘expression’

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

Index notation

d) use index notation for simple instancesof index laws

x3 × x2 = x5; x2/x3 = x−1; (x2)3 = x6

Equations

e) set up simple equations

solve simple equations by using inverseoperations or by transforming both sides in thesame 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

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Contents Examples

Linear equations

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

632 −x +

32+x = 5,

417 x− = 2 – x

Formulae

g) use formulae from mathematics andother subjects

substitute numbers into a formula

generate a formula

for area of a triangle or a parallelogram, areaenclosed by a circle, volume of a prism, volumeof a cone

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

find the perimeter of a rectangle given itsarea A and the length l of one side

find the inverse of simple rules derivingexpressions

Numerical methods

m) use systematic trial and improvementto find approximate solutions of equationswhere there is no simple analytical method ofsolving them

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

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)

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

give the nth term of a linear sequence as analgebraic expression

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Contents Examples

Graphs of linear functions

b) recognise (when values are given for mand c) that equations of the form y = mx + ccorrespond to straight-line graphs in thecoordinate plane

match equations with simple sketch graphs

sketch the graph of equations of the formy = mx + c

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

the depth of water in a container as it empties

Quadratic functions

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

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

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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 selections theymade were appropriate

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

c) develop and follow alternative lines ofenquiry, justifying their decisions to follow orreject particular approaches

Communicating

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

e) use precise formal language andexact methods for analysing geometricalconfigurations

Reasoning

f) apply mathematical reasoning,progressing from brief mathematicalexplanations towards full justifications in morecomplex contexts

g) explore connections in geometry

pose conditional constraints of the type‘If ... then ...’

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

solving angle problems (with reasons)

give explanations showing application ofgeometrical properties

determine whether a tessellation of a givenshape is possible

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Contents Examples

i) state constraints and give startingpoints when making deductions

number of sides of a polygon must be aninteger

j) understand the necessary andsufficient conditions under whichgeneralisations, inferences and solutions togeometrical problems remain valid.

Geometrical reasoning

2. Pupils should be taught to:

Properties of triangles and other rectilinearshapes

a) distinguish between lines and linesegments

f) understand, recall and use Pythagoras’theorem in 2-D problems

g) understand, recall and use trigonometricalrelationships in right angled triangles, and usethese to solve problems, including those usingbearings.

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

may involve angles of elevation and depression

Properties of circles

h) recall the definition of a circle and themeaning of related terms, including sector andsegment

understand that the tangent at any point on acircle is perpendicular to the radius at that point

understand and use the fact that tangents froman external point are equal in length

two tangents to a circle produce a situation thatis symmetrical

Transformations and coordinates

3. Pupils should be taught to:

Specifying transformations

a) use any point as the centre of rotation

measure the angle of rotation, using fractions ofa turn or degrees

identify the point from which a shape has beenrotated

could be a bearing

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Contents Examples

understand that translations are specified by avector

describe transformations

describe combinations of transformations

a translation by the vector ���

����

− 32

Properties of transformations

c) recognise, visualise and constructenlargements of objects using positive,fractional and negative scale factors

from a given centre

enlargement scale factor –1½

Coordinates

e) given the co ordinates of the points Aand B, calculate the length AB

given end-points (−2, 4), (6, −5), usePythagoras to calculate the length as √(82 + 92)

Measures and construction

4. Pupils should be taught to:

Measures

a) know that measurements using realnumbers depend on the choice of unit

length of a page of a book can normally bemeasured to the nearest mm.

jÉåëìê~íáçå

d) find the surface area of simple shapesby using the formulae for the areas of trianglesand rectangles

find volumes of cuboids, recalling the formulaand understanding the connection to countingcubes and how it extends this approach

calculate volumes of right prisms

find the surface area of a cuboid of dimensions7 cm × 8 cm × 9 cm

find the volume of a compound solidconstructed from cubes and cuboids

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

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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 beseen in 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 inferencescan be 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 ofsuitable sources, including experimentsand surveys, and primary and secondarysources

iii) process and represent the data:turn the raw data into usable informationthat gives insight into the problem

iv) interpret and discuss the data:answer the initial question by drawingconclusions from the data

b) select the problem-solving strategies touse in statistical work, and monitor theireffectiveness (these strategies should address thescale and manageability of the tasks, and shouldconsider whether the mathematics and approachused are delivering the most appropriatesolutions)

Communicating

c) communicate mathematically, withemphasis on the use of an increasing range ofdiagrams and related explanatory text, on theselection of their mathematical presentation,explaining its purpose and approach, and on theuse of symbols to convey statistical meaning

use suitable data collection techniques

calculate total frequency or the fx columnfrom a discrete frequency table

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Contents Examples

Reasoning

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

e) identify exceptional or unexpected caseswhen solving statistical problems

f) explore connections in mathematics andlook for relationships between variables whenanalysing data

g) recognise the limitations of any assumptionsand the effects that varying the assumptionscould have on the conclusions drawn from dataanalysis

explain deficiencies in questions on datacollection techniques

Specifying the problem and planning

2. Pupils should be taught to:

c) identify possible sources of bias andplan to minimise it

e) decide what primary and secondary datato use

Collecting data

3. Pupils should be taught to:

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

Processing and representing data

4. Pupils should be taught to:

a) draw and produce, using paper and ICT,cumulative frequency tables and diagrams, boxplots and histograms for grouped continuousdata

scatter diagram weight against height

emphasis on unequal class intervals,frequency density, and shapes of histograms

e) find the median, quartiles andinterquartile range for large data sets andcalculate the mean for large data sets withgrouped data

mean of grouped data using halfway values

calculate the mean and range for a set of data

the expression ‘estimate’ will be used whereappropriate

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Contents Examples

f) calculate an appropriate moving average usually for 3 or 4 moving sets of data

3 point or 4 point moving average

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

use line of best fit

j) use relevant statistical functions on acalculator or spreadsheet

Interpreting and discussing results

5. Pupils should be taught to:

b) identify seasonality and trends in timeseries

d) compare distributions and makeinferences, using shapes of distributions andmeasures of average and spread, includingmedian and quartiles

comparison of mean and range for twodistributions, one set given

from a set of data calculate the height of aperson on the 90th percentile

f) appreciate that correlation is a measureof the strength of the association between twovariables

distinguish between positive, negative and zerocorrelation using lines of best fit

appreciate that zero correlation does notnecessarily imply ‘no relationship’ but merely‘no linear relationship’

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Higher Tier: Stage 2

Ma2 Number and algebra

Contents Examples

Calculations

3. Pupils should be taught to:

Number operations and the relationshipsbetween them

a) understand ‘reciprocal’ asmultiplicative inverse, knowing that any non-zero number multiplied by its reciprocal is 1(and that zero has no reciprocal, becausedivision by zero is not defined)

use index laws to simplify and calculate thevalue of numerical expressions involvingmultiplication and division of integer,fractional and negative powers

find the reciprocal of 0.2

evaluate 4

53

222 × , (7 2

1)4

80, 4−2, 21

16 , 32

8 , 250.5

Mental methods

g) recall the fact that n0 = 1 and n−1 = n1 for

positive integers n and the correspondingrule for negative numbers

21

n = n and 31

n = 3 n for any positivenumber n

h) convert to standard index form to makesensible estimates for calculations involvingmultiplication and/or division

100 = 1; 9−1 = 91

5−2 = 251 = 25

1

251/2 = 5 and 641/3 = 4

0194.06382 =

2

3

1094.110382.6

−×× ≅

2

3

102106

−×× = 3 × 105

l) calculate an unknown quantity fromquantities that vary in direct or inverseproportion

m) calculate with standard index form 2.4 × 107 × 5 × 103 = 12 × 1010 = 1.2 × 1011 ,(2.4 × 107) ÷ (5 × 103) = 4.8 × 103

use surds and π in exact calculations, without acalculator

use of Pythagoras’ theorem leading to ananswer of √13, or the area of a circle given as100π

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Contents Examples

Calculator methods

q) use calculators, or written methods,to calculate the upper and lower bounds ofcalculations, particularly when working withmeasurements

find the maximum possible total length of154 mm and 87 mm, both to the nearest mm

bounds in area and volume calculations

t) use calculators to exploreexponential growth and decay, using amultiplier and the power key

in science or geography

population change

Solving numerical problems

4. Pupils should be taught to:

a) draw on their knowledge of operationsand inverse operations (including powers androots), and of methods of simplificationincluding surds, defined within a particularsituation

answers in surd form

b) select and justify appropriate degrees ofaccuracy for answers to problems

recognise limitations on the accuracy of dataand measurements

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

know that a distance of 10 km lies between 9.5km and 10.5 km

Equations, formulae and identities

5. Pupils should be taught to:

Use of symbols

b) expand the product of two linear expressions

manipulate algebraic expressions by factorisingquadratic expressions including the differenceof two squares and cancelling common factorsin rational expressions

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

x2 – 9 = (x + 3)(x – 3) , 4x2 + 6xy = 2x(2x + 3y)2(x + 1)2/(x + 1) = 2(x + 1)6x2 + x – 2 = (3x + 2)(2x – 1),

Formulae

g) change the subject of a formula,including cases where the subject occurs twice,or where a power of the subject appears

find r given that A = πr2,find x given y = mx + c

rqp = + s to q = ; 3(w + y ) = 5y + 7 to y =

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Contents Examples

Simultaneous linear equations

i) find the exact solution of twosimultaneous equations in two unknowns byeliminating a variable, and interpret theequations as lines and their common solution asthe point of intersection

solve algebraically:2x + y = 7, 3x – y = 134y + 5x = 3, 2y – x = − 2

draw graphs and solve for x and y: y = 3x – 4, y = 2 – x

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

solve several linear inequalities in two variablesand find the solution set

26 + n > 4n – 7

show n ≥ 2 on a number line

shade the region defined byy > 3, y ≤ 7 – x, x > 0

Quadratic equations

k) solve quadratic equations byfactorisation, completing the square andusing the quadratic formula

solve by factorising:x2 – 8x + 15 = 0, 2x2 − x – 1 = 0

solve x2 – 3x + 1 = 0 and give your answers to 1decimal place

solve the quadratic equation 3x2 – 9x + 2 = 0and leave your answer in surd form

Sequences, functions and graphs

6. Pupils should be taught to:

Graphs of linear functions

c) find the gradient of lines given byequations of the form y = mx + c (when valuesare given for m and c);

understand that the form y = mx + c representsa straight line and that m is the gradient of theline, and c is the value of the y-intercept

state the gradient of any liney = mx + c

find the gradient of any line ax + by = k byrearranging into the form y = mx + c

find the equation of a line

give the equation of a line parallel to a drawnline

Quadratic functions

e) find approximate solutions of aquadratic equation from the graph of thecorresponding quadratic function

solve x2 – 2x + 1 = 0solve x2 – 2x – 3 = 0 using the graph ofy = x2 – 2x +1

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Contents Examples

Other functions

f) plot graphs of: simple cubic functions,the reciprocal function y = 1/x with x ≠ 0, theexponential function y = kx for integer valuesof x and simple positive values of k, thecircular functions y = sin x and y = cos x,using a spreadsheet or graph plotter as well aspencil and paper

recognise the characteristic shapes of all thesefunctions

y = x3

y = 2x; y = ( 21 )x

given y = pqx, use the graph below to find thevalue of p and q.

y

(4, 8)

2

O x

match equations with their graphs

sketch graphs from given equations

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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

e) understand and use SSS, SAS, ASAand RHS conditions to prove the congruenceof triangles using formal arguments, and toverify standard ruler and compassconstructions

prove formally that that the base angles of anisosceles triangle are equal

g) calculate the area of a triangle using

21 ab sin C

Properties of circles

h) explain why the perpendicular from thecentre to a chord bisects the chord

3-D shapes

i) solve problems involving surface areasand volumes of prisms, pyramids, cylinders,cones and spheres

Transformations and coordinates

3. Pupils should be taught to:

Properties of transformations

b) transform triangles and other 2-Dshapes by combinations of transformations

reflection in any line

candidates may be asked to perform successivegeometrical transformations

describe fully a single transformation which isequivalent to a combination of twotransformations

recognise that these can be equivalent to asingle transformation and describe thistransformation fully

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Contents Examples

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

Vectors

f) understand and use vector notation

calculate, and represent graphically the sumof two vectors, the difference of two vectorsand a scalar multiple of a vector

calculate the resultant of two vectors

understand and use the commutative andassociative properties of vector addition

solve simple geometrical problems in 2-Dusing vector methods

the notation AB or a in bold type will be used

the form ���

����

qp

is also required

E.g. P (2, 3), Q (1, 7), ���

����

�−=

41

PQ

simple applications to geometry in twodimensions

the resultant of the vectors a and b is the vectora + b along the diagonal PR of theparallelogram PQRS

P Q

a + b b

S a R

the joining of the mid-points of the sides of anyquadrilateral form a parallelogram

Measures and construction

4. Pupils should be taught to:

Measures

a) recognise that measurements given tothe nearest whole unit may be inaccurate by upto one half in either direction

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Contents Examples

understand and use compound measures,including speed and density

Construction

c) use straight edge and compasses to dostandard constructions including an equilateraltriangle with a given side, the midpoint andperpendicular bisector of a line segment, theperpendicular from a point to a line, theperpendicular from a point on a line, and thebisector of an angle

a ruler of length 30 cm may be needed

construction lines and arcs are required to beshown

the pair of compasses should have a span up to6 cm

Mensuration

d) calculate the lengths of arcs and theareas of sectors of circles

the value of π will not be given

answers in terms of π may be required

Loci

e) find loci, both by reasoning and byusing ICT to produce shapes and paths

a region bounded by a circle and an intersectingline

restricted to two dimensions

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Ma4 Handling data

Contents Examples

Processing and representing data

4. Pupils should be taught to:

b) understand and use estimates ormeasures of probability from theoreticalmodels, or from relative frequency

an estimate of the number of times an eventwith a probability of 7

2 happening over 300tries is 300 × 7

2

g) know when to add or multiply twoprobabilities: if A and B are mutuallyexclusive, then the probability of A or Boccurring is P(A) + P(B), whereas if A and Bare independent events, the probability of Aand B occurring is P(A) × P(B)

what is the probability of throwing a 3 or a 4from a dice, or a 6 twice?

find the probability of obtaining balls of thesame colour one after another containing threered, four blue and seven green balls, withreplacement

when one ball is withdrawn and replaced and asecond ball withdrawn

h) use tree diagrams to representoutcomes of compound events, recognisingwhen events are independent

Interpreting and discussing results

5. Pupils should be taught to:

d) understand frequency density complete a frequency table from a histogram

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Higher Tier: Stage 3Ma2 Number and algebra

Contents Examples

Calculations

3. Pupils should be taught to:

Number operations and the relationshipsbetween them

a) use inverse operations, understandingthat the inverse operation of raising apositive number to power n is raising theresult of this operation to power n

1

Written methods

n) rationalise a denominator such as

33

31 =

simplify (3 – √2)2,

Calculator methods

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

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

find 3.32 + √4.3; 5.7 – (4.2 + 0.5)2

25.29.42.47.25 2

−−

calculator functions include +, −, ×, ÷, x2,√x,memory, brackets, xy, x1/y, trigonometric andstatistical functions

Equations, formulae and identities

5. Pupils should be taught to:

Direct and inverse proportion

h) set up and use equations to solveword and other problems involving directproportion or inverse proportion and relatealgebraic solutions to graphicalrepresentation of the equations

y ∝ x, y ∝ x2, y ∝ 1/x, y ∝ 1/x2

y = kx, y = kx2, y = xk , y = 2x

k

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Edexcel GCSE Mathematics (Modular) 1388 – Higher Tier: Stage 3

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

Contents Examples

Simultaneous linear and quadraticequations

l) solve exactly, by elimination of anunknown, two simultaneous equations in twounknowns, one of which is linear in eachunknown, and the other is linear in oneunknown and quadratic in the other, orwhere the second is of the form x2 + y2 = r2

solve the simultaneous equations y = 11x – 2and y = 5x2

solve 4x – 3y = 24 x2 + y2 = 25

Sequences, functions and graphs

6. Pupils should be taught to:

Graphs of linear functions

c) explore the gradients of parallel linesand lines perpendicular to these lines

know that the lines represented by theequations y = −5x and y = 3 – 5x are parallel,each having gradient (−5) and that the line withequation y = 5

x is perpendicular to these linesand has gradient 5

1

Quadratic functions

e) find the intersection points of thegraphs of a linear and quadratic function,knowing that these are the approximatesolutions of the corresponding simultaneousequations representing the linear andquadratic functions

use the graph of y = x2 – 2x – 4 to solvex2 – 4x – 5 = 0 by drawing a suitable straightline

Transformation of functions

g) apply to the graph of y = f(x) thetransformations y = f(x) + a, y = f(ax),y = f(x + a), y = af(x) for linear, quadratic,sine and cosine functions f(x)

representing translations in the x and ydirection, reflections in the x and y-axis, andstretches parallel to the x and y axis

Loci

h) construct the graphs of simple loci,including the circle x2 + y2 = r2 for a circle ofradius r centred at the origin of coordinates

find graphically the intersection points of agiven straight line with this circle and knowthat this corresponds to solving the twosimultaneous equations representing the lineand the circle

loci as defined on a coordinate grid

perpendicular bisector of a line segment

using given graphs solve x + y = 7 x2 + y2 = 29

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Edexcel GCSE Mathematics (Modular) 1388 – Higher Tier: Stage 3

UG009825 – Edexcel GCSE Mathematics 1388 – Teachers’ Guide: Content Exemplification (Modular) – Issue 2 – July 2003 93

Ma3 Shape, space and measures

Contents Examples

Geometrical reasoning

2. Pupils should be taught to:

f) understand, recall and use Pythagoras’theorem in 3-D problems;

investigate the geometry of cuboids includingcubes, and shapes made from cuboids,including the use of Pythagoras’ theorem tocalculate lengths in three dimensions

g) understand similarity of triangles andof other plane figures, and use this to makegeometric inferences

use trigonometrical relationships in 3-Dcontexts, including finding the anglesbetween a line and a plane (but not the anglebetween two planes or between two skewlines)

draw, sketch and describe the graphs oftrigonometric functions for angles of anysize, including transformations involvingscalings in either or both the x and ydirections

use the sine and cosine rules to solve 2-D and3-D problems

calculate the diagonal through a cuboid, oracross the face of a cuboid

use of non-integer scale factors to find amissing side in each of two similar trianglesthat have a corresponding sides

may involve angles of elevation and depression

generate and interpret graphs based on thesefunctions

sketch the graph of y = 3 sin 2t°, y = tan x°

Properties of circles

h) prove and use the facts that the anglesubtended by an arc at the centre of a circle istwice the angle subtended at any point on thecircumference, the angle subtended at thecircumference by a semicircle is a right angle,that angles in the same segment are equal, andthat opposite angles of a cyclic quadrilateralsum to 180 degrees

prove and use the alternate segment theorem

any proof required will be in relation to adiagram, not purely by reference to a namedtheorem

prove other results dependent on these facts

3D shapes

i) solve problems involving morecomplex shapes and solids, includingsegments of circles and frustums of cones

answers in terms of π may be required

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Edexcel GCSE Mathematics (Modular) 1388 – Higher Tier: Stage 3

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

Contents Examples

Transformations and coordinates

3. Pupils should be taught to:

d) understand the difference betweenformulae for perimeter, area and volume byconsidering dimensions

understand and use the effect of enlargementon areas and volumes of shapes and solids

recognise that 234 rπ cannot represent the

volume of a sphere

know the relationships between linear, area andvolume scale factors of similar shapes

Measures and construction

4. Pupils should be taught to:

Mensuration

d) convert between volume measuresincluding cm3 and m3

5 m2 = 5 × 1002 cm2 = 50 000 cm2

5 m3 = 5 × 1003 cm3 = 5 000 000 cm3

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Edexcel GCSE Mathematics (Modular) 1388 – Higher Tier: Stage 3

UG009825 – Edexcel GCSE Mathematics 1388 – Teachers’ Guide: Content Exemplification (Modular) – Issue 2 – July 2003 95

Ma4 Handling data

Contents Examples

Specifying the problem and planning

2. Pupils should be taught to:

d) select and justify a sampling schemeand a method to investigate a population,including random and stratified sampling

use and understand how different sample sizesmay affect the reliability of conclusions drawn

consider fairness

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