Edexcel GCSE Mathematics A Linear Higher · Mathematics A Linear Higher Teacher Guide Edexcel GCSE...

Mathematics ALinearHigherTeacher Guide

Edexcel GCSE

Series director: Keith Pledger

Series editor: Graham Cumming

Authors:Chris BastonJulie BolterGareth ColeGill DyerAndrew EdmondsonMichael FlowersKeren HughesPeter JollyJoan KnottJean LinskyGraham NewmanRob PepperJoe PetranKeith PledgerRob SummersonKevin TannerBrian Western

A01_MSAF_TG_GCSE_0877_FM.indd 1 12/05/2010 12:16

Published by Pearson Education Limited, a company incorporated in England and Wales, having its registered office at Edinburgh Gate, Harlow, Essex, CM20 2JE. Registered company number: 872828

Edexcel is a registered trademark of Edexcel Limited

Text © Chris Baston, Julie Bolter, Gareth Cole, Gill Dyer, Michael Flowers, Karen Hughes, Peter Jolly, Joan Knott, Jean Linsky, Graham Newman, Rob Pepper, Joe Petran, Keith Pledger, Rob Summerson, Kevin Tanner, Brian Western and Pearson Education Limited 2010

The rights of Chris Baston, Julie Bolter, Gareth Cole, Gill Dyer, Michael Flowers, Karen Hughes, Peter Jolly, Joan Knott, Jean Linsky, Graham Newman, Rob Pepper, Joe Petran, Keith Pledger, Rob Summerson, Kevin Tanner and Brian Western to be identified as the authors of this Work have been asserted by them in accordance with the Copyright, Designs and Patent Act, 1988.

First published 201013 12 11 1010 9 8 7 6 5 4 3 2 1

British Library Cataloguing in Publication DataA catalogue record for this book is available from the British LibraryISBN 978 1 84690 082 2

Copyright noticeAll rights reserved. No part of this publication may be reproduced in any form or by any means (including photocopying or storing it in any medium by electronic means and whether or not transiently or incidentally to some other use of this publication) without the written permission of the copyright owner, except in accordance with the provisions of the Copyright, Designs and Patents Act 1988 or under the terms of a licence issued by the Copyright Licensing Agency, Saffron House, 6 –10 Kirby Street, London EC1N 8T (www.cla.co.uk). Applications for the copyright owner’s written permission should be addressed to the publisher.

Typeset by Pantek Arts, Maidstone, Kent.

Printed in Great Britain at Ashford Colour

AcknowledgementsThe publisher would like to thank the following for their kind permission to reproduce their photographs:We are grateful to the following for permission to reproduce copyright material:Every effort has been made to trace the copyright holders and we apologise in advance for any unintentional omissions. We would be pleased to insert the appropriate acknowledgement in any subsequent edition of this publication.

DisclaimerThis material has been published on behalf of Edexcel and offers high-quality support for the delivery of Edexcel qualifications. This does not mean that the material is essential to achieve any Edexcel qualification, nor does it mean that it is the only suitable material available to support any Edexcel qualification. Edexcel material will not be used verbatim in setting any Edexcel examination or assessment. Any resource lists produced by Edexcel shall include this and other appropriate resources. Copies of official specifications for all Edexcel qualifications may be found on the Edexcel website: www.edexcel.com


iii

Contents

1 Number 1.1 Understanding prime factors, LCM and HCF 21.2 Understanding squares and cubes 41.3 Understanding order of operations 61.4 Using a calculator 81.5 Understanding the index laws 10

2 Expressions and sequences 2.1 Collecting like terms 122.2 Using substitution 142.3 Using the index laws 162.4 Fractional and negative powers 182.5 Term-to-term and position-to-term definitions 202.6 The nth term of an arithmetic sequence 22

3 Fractions 3.1 Adding and subtracting fractions and mixed numbers 243.2 Multiplying fractions and mixed numbers 263.3 Dividing fractions and mixed numbers 283.4 Fraction problems 30

4 Decimals and estimation 4.1 Conversion between fractions and decimals 324.2 Carrying out arithmetic using decimals 344.3 Decimal places 364.4 Significant figures 384.5 Estimating calculations by rounding 404.6 Estimating calculations involving decimals 424.7 Manipulating decimals 444.8 Converting recurring decimals to fractions 464.9 Upper and lower bounds of accuracy 484.10 Calculating the bounds of an expression 50

5 Angles and polygons 5.1 Angle properties of parallel lines 525.2 Proving the angle properties of triangles and quadrilaterals 545.3 Using the angle properties of triangles and quadrilaterals 565.4 Angles of elevation and depression 585.5 Bearings 605.6 Using angle properties to solve problems 625.7 Polygons 64

6 Collecting and recording data 6.1 Introduction to statistics 666.2 Sampling methods 686.3 Stratified sampling 706.4 Collecting data by observation and experiment 726.5 Questionnaires 746.6 Two-way tables 76


Contents

iv

6.7 Sources of bias 786.8 Secondary data 80

7 Measure 7.1 Converting between units of measure 827.2-4 Compound measures; Speed; Density 84

8 Congruence, symmetry and similarity 8.1 Congruent triangles 868.2-3 Symmetry in 2D shapes; Symmetry of special shapes 888.4 Recognising similar shapes 908.5 Similar triangles 92

9 Expanding brackets and factorising 9.1 Expanding brackets 949.2 Factorising by taking out common factors 969.3 Expanding the product of two brackets 989.4 Factorising quadratic expressions 100

10 Area and volume 1 10.1 Perimeter of shapes 10210.2-3 Area of rectangles, triangles, parallelograms and trapeziums;

Problems involving perimeter and area 10410.4 Circles 10610.5 Circumference and area of a circle 10810.6 Drawing 3D shapes 11010.7 Elevations and plans 11210.8-10 Volume of a cuboid; Volume of a prism; Volume of a cylinder 114

11 Averages and range 11.1-3 Finding the mode and median; Calculating the mean; Using the three types

of average 11611.4 Using frequency tables to find averages 11811.5-6 Modal class and median of grouped data; Estimating the mean of grouped data 12011.7 Range, quartiles and inter-quartile range 122

12 Constructions and loci 12.1 Perpendicular lines 12412.2 Constructing and bisecting angles 12612.3 Loci 12812.4 Regions 13012.5 Constructing triangles 13212.6 Scale drawings and maps 134

13 Linear equations 13.1 Solving simple equations 13613.2 Solving linear equations containing brackets 13813.3 Solving linear equations with the unknown on both sides 14013.4 Solving linear equations containing fractions 14213.5 Solving simple linear equations 14413.6 Distinguishing between ‘equation’, ‘formula’, ‘identity’ and ‘expression’ 146

14 Percentages 14.1 Working out a percentage of a quantity 14814.2 Finding the new amount after a percentage increase or decrease 150


Contents

v

14.3 Working out a percentage increase or decrease 15214.4 Working out compound interest 15414.5 Calculating reverse percentages 156

15 Graphs 15.1 Drawing straight line graphs by plotting points 15815.2 Finding the midpoint of a line segment 16015.3 The gradient and y-intercept of a straight line 162 The gradient of a straight line 16415.4 The equation y = mx + c 166 The equation of a straight line y = mx + c 16815.5 Parallel and perpendicular lines 17015.6 Real-life graphs 172

16 Ratio and proportion 16.1 Introducing ratio 17416.2 Solving ratio problems 17616.3 Sharing a quantity in a given ratio 17816.4 Using direct proportion 18016.5 Using inverse proportion 182

17 Transformations 17.1 Understanding transformations 18417.2 Using translations 18617.3 Transforming shapes using reflections 18817.4 Transforming shapes using rotations 19017.5 Enlargements and scale factors 19217.6 Centre of enlargement and the scale factor 19417.7 Combinations of transformations 196

18 Processing, representing and interpreting data 18.1-2 Producing pie charts; Interpreting pie charts 19818.3 Representing and interpreting data in a stem and leaf diagram 20018.4 Interpreting comparative and composite bar charts 20218.5-7 Drawing and interpreting frequency diagrams and histograms; Drawing and

using frequency polygons; Drawing and using histograms with unequal class intervals 204

18.8-9 Drawing and using cumulative frequency graphs; Finding quartiles from a cumulative frequency graph 206

18.10 Drawing and interpreting box plots 208

19 Inequalities and formulae 19.1 Representing inequalities on a number line 21019.2 Solving simple linear inequalities in one variable 21219.3 Finding integer solutions to inequalities in one variable 21419.4 Solving graphically several linear inequalities in two variables 21619.5 Using formulae 21819.6 Deriving an algebraic formula 22019.7 Changing the subject of a formula 22219.8 Changing the subject in complex formula 224

20 Pythagoras’ theorem and trigonometry 1 20.1-2 Pythagoras’s Theorem; Applying Pythagoras’ Theorem 22620.3 Finding the length of a line segment 228


Contents

vi

20.4-5 Trigonometry in right-angled triangles; Working out lengths of sides using trigonometry 230

21 More graphs and equations 21.1 Graphs of quadratic functions 23221.2 Graphs of cubic functions 23421.3 Graphs of reciprocal functions 23621.4 Graphs of exponential functions 23821.5 Solving equations by the trial and improvement method 240

22 Quadratic and simultaneous equations 22.1 Solving simultaneous equations 24222.2 Setting up equations in two unknowns 24422.3 Using graphs to solve simultaneous equations 24622.4 Solving quadratic equations by factorisation 24822.5 Completing the square 25022.6 Solving quadratic equations by completing the square 25222.7 Solving quadratic equations using the formula 25422.8 Solving algebraic fraction equations leading to quadratic equations 25622.9 Setting up and solving quadratic equations 25822.10 Constructing graphs of simple loci 26022.11 Solving simultaneous equations when one is linear and the other is quadratic 26222.12 Solving simultaneous equations when one is linear and one is a circle 264

23 Area and volume 2 23.1-2 Sectors of circles; Problems involving circles in terms of _ 26623.3 Units of area 26823.4-6 Volume of a pyramid and a cone; Volume of a sphere; Volume of harder shapes 27023.7 Units of volume 27223.8-9 Surface area of a prism; Surface area of harder shapes 27423.10 Coordinates in three dimensions 276

24 Line diagrams and scatter graphs 24.1-4 Drawing and using line graphs; Drawing and using scatter graphs; Recognising

correlation; Drawing lines of best fit 27824.5 Using lines of best fit to make predictions 280

25 Indices, standard form and surds 25.1 Using zero and negative powers 28225.2 Using standard form 28425.3 Working with fractional indices 28625.4 Using surds 288

26 Similar shapes 26.1 Areas of similar shapes 29026.2 Volumes of similar shapes 29226.3 Lengths, areas and volumes of similar shapes 294

27 Proportion 27.1 Direct proportion 27.2 Further direct proportion 29827.3 Writing statements of proportionality and formulae 30027.4 Problems involving square and cubic proportionality 30227.5 Problems involving inverse proportion 304


Contents

vii

28 Probability 28.1 Writing probabilities as numbers 30628.2 Mutually exclusive outcomes 30828.3 Estimating probability from relative frequency 31028.4 Finding the expected number of outcomes 31228.5 Independent events 31428.6 Probability tree diagrams 31628.7 Conditional probability 318

29 Pythagoras’ Theorem and trigonometry 2 29.1-2 Pythagoras’ theorem and trigonometry in three dimensions;

Angle between a line and a plane 32029.3 Trigonometric ratios for any angle 32229.4 Finding the area of a triangle using ½absinC 32429.5-6 The sine rule; Using the sine rule to calculate an angle 32629.7-8 The cosine rule; Using the cosine rule to calculate an angle 32829.9 Using trigonometry to solve problems 330

30 Transformations of functions 30.1 Using functional notation 33230.2 Translation of a curve parallel to the axes 33430.3 Stretching a curve parallel to the axes 33630.4 Rotation about the origin and reflection in the axes 338

31 Circle geometry 31.1 Isosceles triangle in a circle 34031.2 Tangents to a circle 34231.3 Circle theorems 34431.4 More circle theorems 346

32 Algebraic fractions and algebraic proof 32.1 Simplifying algebraic fractions 34832.2 Adding and subtracting algebraic fractions 35032.3 Multiplying and dividing algebraic fractions 35232.4 Algebraic proof 354

33 Vectors 33.1 Vectors and vector notation 35633.2 The magnitude of vectors 35833.3 Addition of vectors 36033.4 Parallel vectors 36233.5 Solving geometric problems in two dimensions 364

Functional skills Climbing Snowdon 366All at sea 367Energy efficiency 368Communication 369Music sales 370Going on holiday

Answers 371


viii

Introduction

Edexcel’s GCSE Mathematics materialsThis GCSE Mathematics course has been developed by Edexcel to support you in teaching our new GCSE Mathematics specifi cations. All the materials have been fully referenced to the specifi cations. The course offers the following components for each of the Foundation and Higher Tiers:

• Student Book with graded questions, lots of support for the new Assessment Objectives and our unique Examiner insight from Results Plus.

• ActiveTeach CD-ROM to support you in your use of ICT for whole-class teaching, and in your lesson planning and management.

• Teacher’s Guide, providing lesson objectives, topic grades, ideas for activities including the use of ICT in ActiveTeach and resource sheets to support students completing exercises in the Student Book. Word and Pdf fi les of all material available on the CD-ROM which is included and which will integrate with the ActiveTeach.

• Practice Books: with one-to-one matching of Student book exercises. Pdfs of the material for upload to the school VLE or network are available separately and will integrate with ActiveTeach.

• Targeted Practice Books: providing support for G to F students; extension material for A to A* students and Booster C material for those all important borderline D/C students. Pdfs of the material for upload to the school VLE or network are available separately and will integrate with ActiveTeach.

• Assessment Pack containing End of chapter tests; extra AO2 and AO3 practice questions; and a set of Exam Practice Papers with mark schemes. Word and Pdf fi les of all material are available on the accompanying CD-ROM which will integrate with ActiveTeach.

• ResultsPlus Booster C, designed to boost the grades of your D/C borderline students. It provides web-delivered homeworks and tests for individual formative assessment with detailed teacher and pupil feedback.

• ResultsPlus Progress tests provide web-delivered individual summative assessment matched to the new specifi cation.

• SupportPlus website contains information about the specifi cations, training events, support and sample materials. An Edexcel Maths-users-only area gives further detailed support for teaching the specifi cation.

Support for teaching the new Assessment Objectives Assessment Objective

What it is What this means

Approx % of marks in the exam

AO1 Recall and use knowledge of the prescribed content.

Questions testing your knowledge of each topic.

45-55

AO2 Select and apply mathematical methods in a range of contexts.

Questions asking you to decide what method you need to use to get to the correct answer.

25-35

AO3 Interpret and analyse problems and generate strategies to solve them.

Problem solving: Deciding how and explaining why

15-25

The new assessment objectives means that question styles within the exam are changing, with more problem-solving, open-style questions being set. These new question types are clearly marked in the Student books and also have dedicated spreads for further practice. Yet more questions are available in the Practice Books and in the Assessment Pack. Further interactive support is offered in the ActiveTeach with interactive examples and in the Assessment pack

The examination and the courseWritten by examiners who thoroughly understand the new specifi cation, all the material you need to prepare students for the examination is available from this course and has been carefully developed and reviewed.• All questions show targeted grades.• ResultsPlus examiner tips help students to gain those

extra few marks in the examination.• Past exam questions and exam style questions can

be found in the Chapter Review at the end of each chapter. These have been chosen or specifi cally written to ensure they are a true refl ection of the style of questions that might appear in the examination.

• Written by examiners. the Assessment Pack contains: • editable end-of-chapter tests for you to use or

adapt for student assessment. All answers are provided.

• a bank of AO2 and AO3 questions for extra practice of these assessment objectives

• examination practice papers, to help your students become familiar with the types of questions they will be asked in the exam.

• Formative and summative electronic tests are available through our online ResultsPlus Booster C and Results Plus Progress tests.

• References to the specifi cation are given in student friendly language in the Student books and in full in the Teacher’s Guide.

• The Teacher Guide CD contains electronic copies of the scheme of work and the specifi cation both of which contain references to the relevant sections of the Student book, Practice books and Teacher Guide.

• A blank self-assessment sheet is available on the Teacher Guide CD to enable students to refl ect on a topic. Cross referencing on the self-assessment sheet points students back to the book if they need to revise.

prime factor highest common factor (HCF) lowest common multiple (LCM)

Chapter 1 Number

4

GCSE 2010Nc Use the concepts and vocabulary of factor (divisor), multiple, common factor, Highest Common Factor (HCF), Least Common Multiple (LCM), prime number and prime factor decomposition.

FS Process skillsExamine patterns and relationships

FS PerformanceLevel 1 Use appropriate checking procedures at each stage

Specifi cation 1.2 Understanding prime factors, LCM and HCF

Concepts and skills

• Find the prime factor decomposition of positive integers.

• Find the highest common factor (HCF) and the least common multiple (LCM) of two or three numbers.

Functional skills

• L1 Add, subtract, multiply and divide whole numbers using a range of mental methods.

Prior key knowledge, skills and concepts

• Students should already know their multiplication tables up to 10 × 10.

• Students should be able to fi nd factors, multiples and prime numbers.

Starter

• Check that students understand the terms prime number, factor and multiple. List the factors of 12. (1, 2, 3, 4, 6, 12) List the multiples of 6 between 10 and 40. (12, 18, 24, 30, 36) List the fi rst ten prime numbers. (2, 3, 5, 7, 11, 13, 17, 19, 23, 29)

• Introduce the word ‘common’ into some questions.Find two common factors of 12 and 18. (1, 2, 3, 6) Find two common multiples of 3 and 4. (12, 24, 36 etc)

Main teaching and learning

• Tell students that they are going to fi nd out how to write any positive whole number as a product of its prime factors. Check that students understand the meaning of the word product.

• Explain that this can be done by using a factor tree (or repeated division). Draw a factor tree to show how 120 can be broken down into its prime factors (see Example 4).

• Discuss the fact that you can start with any two numbers that multiply to give 120. Draw a second factor tree for 120 starting with a different factor pair to show that the same result is reached.

• Tell students that they are going to fi nd the HCF and LCM of two numbers.

• Explain that there are different methods that can be used to do this depending on the size of the numbers involved.

• Discuss the best method for fi nding the HCF and LCM for two small numbers (e.g. 4 and 6). Show students how these can be found by making a list of the factors and fi rst few multiples of 4 and 6.

• Discuss why this method would not be appropriate for large numbers (e.g. 240 and 280).

• Explain how writing large numbers as the product of prime factors can be used to fi nd the LCM and HCF.

Common misconceptions

• Remind students to include the multiplication signs when writing a number as a product of its prime factors. (These are often incorrectly replaced by addition signs or commas.)

Enrichment

• Suggest that students use the Venn diagram method to fi nd the HCF and LCM of three large numbers (e.g. 240, 300 and 420).

• Students might like to know that the HCF of two numbers must be a factor of the difference between them. So the HCF of 210 and 250 must be a factor of 40. They may like to explore this and consider why this is the case.

Plenary

• Ask for the HCF of pairs of small numbers e.g. 2 and 6 (2), 4 and 10 (2), 6 and 12 (6).

• Ask for the LCM of pairs of small numbers e.g. 2 and 6 (6), 4 and 10 (20), 6 and 12 (12).

Linkswww.heinemann.co.uk/hotlinks

ActiveTeach resourcesMultiples and factors quizLadder methodBBB Video: Supercross

Resources

M01_MSAH_TG_GCSE_0822_BLAD.indd 4 12/02/2010 15:05

Section 1.2 Understanding prime factors, LCM and HCF

5

Questions are targeted at the grades indicated

1 Write down all the factors of each of the following numbers.

a 12 ...................................................................... b 30 ......................................................................

c 36 ...................................................................... d 40 ......................................................................

2 Complete the following factor trees.

a b c

3 Use this factor tree to write 54 as a product of its prime factors.

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

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

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

4 Write each of the following numbers as the product of its prime factors.a 24 ............................................... b 40 ...............................................

c 50 ............................................... d 72 ...............................................

e 100 ...............................................

How can factor trees be drawn before working through to fi nd out how many factors there are?

5 a Write out all the factors of (i) 4 ............................................... (ii) 6 ............................................... (iii) 10 ............................................... (iv) 16 ............................................... (v) 20 ...............................................

b Use part a to help you write down the highest common factor (HCF) of (i) 4 and 10 ............................................... (ii) 6 and 16 ............................................... (iii) 4 and 20 ............................................... (iv) 16 and 20 ............................................... (v) 10 and 16 ...............................................

6 a Write out the fi rst ten multiples of (i) 4 ............................................... (ii) 6 ...............................................

b Write out the fi rst six multiples of (i) 10 ............................................... (ii) 16 ............................................... (iii) 20 ...............................................

c Use your answers to parts a and b to help you write down the lowest common multiple (LCM) of

(i) 4 and 6 ............................................... (ii) 6 and 10 ............................................... (iii) 4 and 16 ............................................... (iv) 16 and 20 ............................................... (v) 6 and 20 ...............................................

7 a Use your answers to question 3 to write 42, 70 and 84 as products of their prime factors......................................................................................................................................................................................................................................

b Find the HCF and LCM of 42 and 70......................................................................................................................................................................................................................................

c Find the HCF and LCM of 70 and 84......................................................................................................................................................................................................................................

8 Find the HCF and LCM of the following pairs of numbers.

a 60 and 84 ............................................................... b 70 and 105 ...............................................................

c 72 and 96 ............................................................... d 84 and 96 ...............................................................

Write down the factors in pairs.

2 33 3

6 9

54

21

42

10

70

21

84

Remember to draw factor trees fi rst.

Factors are numbers that go exactly into the given number.

Multiples are the numbers in the times table of the given number.

C

G

D


Specifi cation references for the 2010 specifi cation and for Functional skills.

Objectives that link directly to the specifi cation and are included in each section in the Student Book.

Prior knowledge, skills and concepts highlighted where applicable.

Key mathematical vocabulary pulled out; this is also available on the ActiveTeach with written and spoken defi nitions in English and a multilingual spoken glossary.

Sections detailing common misconceptions, and possible enrichment activities to challenge students and check their understanding are also included where appropriate.

Hints help students tackle the work on their own.

Photocopiable remediation exercise worksheets provide extra questions and help support students.

A topic from Edexcel’s GCSE Mathematics courseAs well as the concise Starters, Main teaching and learning points and Plenary, the lesson notes also contain:

Teaching and Learning Saving you time and guiding you through the new specifi cation, the Teacher’s guide contains concise, easy-to-read Lesson plans and extra Guided Practice Worksheets which are available as editable Word fi les and pdfs on the CD-ROM.

• At a glance specifi cation references and detail.• Starter ideas to check that students have the required

prior knowledge.• Main teaching and learning points to help you teach

the topic itself.• Plenary questions to test understanding and

application of the mathematics.


ix

Introduction

Edexcel’s GCSE Mathematics materialsThis GCSE Mathematics course has been developed by Edexcel to support you in teaching our new GCSE Mathematics specifi cations. All the materials have been fully referenced to the specifi cations. The course offers the following components for each of the Foundation and Higher Tiers:

• Student Book with graded questions, lots of support for the new Assessment Objectives and our unique Examiner insight from Results Plus.

• ActiveTeach CD-ROM to support you in your use of ICT for whole-class teaching, and in your lesson planning and management.

• Teacher’s Guide, providing lesson objectives, topic grades, ideas for activities including the use of ICT in ActiveTeach and resource sheets to support students completing exercises in the Student Book. Word and Pdf fi les of all material available on the CD-ROM which is included and which will integrate with the ActiveTeach.

• Practice Books: with one-to-one matching of Student book exercises. Pdfs of the material for upload to the school VLE or network are available separately and will integrate with ActiveTeach.

• Targeted Practice Books: providing support for G to F students; extension material for A to A* students and Booster C material for those all important borderline D/C students. Pdfs of the material for upload to the school VLE or network are available separately and will integrate with ActiveTeach.

• Assessment Pack containing End of chapter tests; extra AO2 and AO3 practice questions; and a set of Exam Practice Papers with mark schemes. Word and Pdf fi les of all material are available on the accompanying CD-ROM which will integrate with ActiveTeach.

• ResultsPlus Booster C, designed to boost the grades of your D/C borderline students. It provides web-delivered homeworks and tests for individual formative assessment with detailed teacher and pupil feedback.

• ResultsPlus Progress tests provide web-delivered individual summative assessment matched to the new specifi cation.

• SupportPlus website contains information about the specifi cations, training events, support and sample materials. An Edexcel Maths-users-only area gives further detailed support for teaching the specifi cation.

Support for teaching the new Assessment Objectives Assessment Objective

What it is What this means

Approx % of marks in the exam

AO1 Recall and use knowledge of the prescribed content.

Questions testing your knowledge of each topic.

45-55

AO2 Select and apply mathematical methods in a range of contexts.

Questions asking you to decide what method you need to use to get to the correct answer.

25-35

AO3 Interpret and analyse problems and generate strategies to solve them.

Problem solving: Deciding how and explaining why

15-25

The new assessment objectives means that question styles within the exam are changing, with more problem-solving, open-style questions being set. These new question types are clearly marked in the Student books and also have dedicated spreads for further practice. Yet more questions are available in the Practice Books and in the Assessment Pack. Further interactive support is offered in the ActiveTeach with interactive examples and in the Assessment pack

The examination and the courseWritten by examiners who thoroughly understand the new specifi cation, all the material you need to prepare students for the examination is available from this course and has been carefully developed and reviewed.• All questions show targeted grades.• ResultsPlus examiner tips help students to gain those

extra few marks in the examination.• Past exam questions and exam style questions can

be found in the Chapter Review at the end of each chapter. These have been chosen or specifi cally written to ensure they are a true refl ection of the style of questions that might appear in the examination.

• Written by examiners. the Assessment Pack contains: • editable end-of-chapter tests for you to use or

adapt for student assessment. All answers are provided.

• a bank of AO2 and AO3 questions for extra practice of these assessment objectives

• examination practice papers, to help your students become familiar with the types of questions they will be asked in the exam.

• Formative and summative electronic tests are available through our online ResultsPlus Booster C and Results Plus Progress tests.

• References to the specifi cation are given in student friendly language in the Student books and in full in the Teacher’s Guide.

• The Teacher Guide CD contains electronic copies of the scheme of work and the specifi cation both of which contain references to the relevant sections of the Student book, Practice books and Teacher Guide.

• A blank self-assessment sheet is available on the Teacher Guide CD to enable students to refl ect on a topic. Cross referencing on the self-assessment sheet points students back to the book if they need to revise.

prime factor highest common factor (HCF) lowest common multiple (LCM)

Chapter 1 Number

4

GCSE 2010Nc Use the concepts and vocabulary of factor (divisor), multiple, common factor, Highest Common Factor (HCF), Least Common Multiple (LCM), prime number and prime factor decomposition.

FS Process skillsExamine patterns and relationships

FS PerformanceLevel 1 Use appropriate checking procedures at each stage

Specifi cation 1.2 Understanding prime factors, LCM and HCF

Concepts and skills

• Find the prime factor decomposition of positive integers.

• Find the highest common factor (HCF) and the least common multiple (LCM) of two or three numbers.

Functional skills

• L1 Add, subtract, multiply and divide whole numbers using a range of mental methods.

Prior key knowledge, skills and concepts

• Students should already know their multiplication tables up to 10 × 10.

• Students should be able to fi nd factors, multiples and prime numbers.

Starter

• Check that students understand the terms prime number, factor and multiple. List the factors of 12. (1, 2, 3, 4, 6, 12) List the multiples of 6 between 10 and 40. (12, 18, 24, 30, 36) List the fi rst ten prime numbers. (2, 3, 5, 7, 11, 13, 17, 19, 23, 29)

• Introduce the word ‘common’ into some questions.Find two common factors of 12 and 18. (1, 2, 3, 6) Find two common multiples of 3 and 4. (12, 24, 36 etc)

Main teaching and learning

• Tell students that they are going to fi nd out how to write any positive whole number as a product of its prime factors. Check that students understand the meaning of the word product.

• Explain that this can be done by using a factor tree (or repeated division). Draw a factor tree to show how 120 can be broken down into its prime factors (see Example 4).

• Discuss the fact that you can start with any two numbers that multiply to give 120. Draw a second factor tree for 120 starting with a different factor pair to show that the same result is reached.

• Tell students that they are going to fi nd the HCF and LCM of two numbers.

• Explain that there are different methods that can be used to do this depending on the size of the numbers involved.

• Discuss the best method for fi nding the HCF and LCM for two small numbers (e.g. 4 and 6). Show students how these can be found by making a list of the factors and fi rst few multiples of 4 and 6.

• Discuss why this method would not be appropriate for large numbers (e.g. 240 and 280).

• Explain how writing large numbers as the product of prime factors can be used to fi nd the LCM and HCF.

Common misconceptions

• Remind students to include the multiplication signs when writing a number as a product of its prime factors. (These are often incorrectly replaced by addition signs or commas.)

Enrichment

• Suggest that students use the Venn diagram method to fi nd the HCF and LCM of three large numbers (e.g. 240, 300 and 420).

• Students might like to know that the HCF of two numbers must be a factor of the difference between them. So the HCF of 210 and 250 must be a factor of 40. They may like to explore this and consider why this is the case.

Plenary

• Ask for the HCF of pairs of small numbers e.g. 2 and 6 (2), 4 and 10 (2), 6 and 12 (6).

• Ask for the LCM of pairs of small numbers e.g. 2 and 6 (6), 4 and 10 (20), 6 and 12 (12).

Linkswww.heinemann.co.uk/hotlinks

ActiveTeach resourcesMultiples and factors quizLadder methodBBB Video: Supercross

Resources


Section 1.2 Understanding prime factors, LCM and HCF

5

Questions are targeted at the grades indicated

1 Write down all the factors of each of the following numbers.

a 12 ...................................................................... b 30 ......................................................................

c 36 ...................................................................... d 40 ......................................................................

2 Complete the following factor trees.

a b c

3 Use this factor tree to write 54 as a product of its prime factors.

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

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

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

4 Write each of the following numbers as the product of its prime factors.a 24 ............................................... b 40 ...............................................

c 50 ............................................... d 72 ...............................................

e 100 ...............................................

How can factor trees be drawn before working through to fi nd out how many factors there are?

5 a Write out all the factors of (i) 4 ............................................... (ii) 6 ............................................... (iii) 10 ............................................... (iv) 16 ............................................... (v) 20 ...............................................

b Use part a to help you write down the highest common factor (HCF) of (i) 4 and 10 ............................................... (ii) 6 and 16 ............................................... (iii) 4 and 20 ............................................... (iv) 16 and 20 ............................................... (v) 10 and 16 ...............................................

6 a Write out the fi rst ten multiples of (i) 4 ............................................... (ii) 6 ...............................................

b Write out the fi rst six multiples of (i) 10 ............................................... (ii) 16 ............................................... (iii) 20 ...............................................

c Use your answers to parts a and b to help you write down the lowest common multiple (LCM) of

(i) 4 and 6 ............................................... (ii) 6 and 10 ............................................... (iii) 4 and 16 ............................................... (iv) 16 and 20 ............................................... (v) 6 and 20 ...............................................

7 a Use your answers to question 3 to write 42, 70 and 84 as products of their prime factors......................................................................................................................................................................................................................................

b Find the HCF and LCM of 42 and 70......................................................................................................................................................................................................................................

c Find the HCF and LCM of 70 and 84......................................................................................................................................................................................................................................

8 Find the HCF and LCM of the following pairs of numbers.

a 60 and 84 ............................................................... b 70 and 105 ...............................................................

c 72 and 96 ............................................................... d 84 and 96 ...............................................................

Write down the factors in pairs.

2 33 3

6 9

54

21

42

10

70

21

84

Remember to draw factor trees fi rst.

Factors are numbers that go exactly into the given number.

Multiples are the numbers in the times table of the given number.

C

G

D


Specifi cation references for the 2010 specifi cation and for Functional skills.

Objectives that link directly to the specifi cation and are included in each section in the Student Book.

Prior knowledge, skills and concepts highlighted where applicable.

Key mathematical vocabulary pulled out; this is also available on the ActiveTeach with written and spoken defi nitions in English and a multilingual spoken glossary.

Sections detailing common misconceptions, and possible enrichment activities to challenge students and check their understanding are also included where appropriate.

Hints help students tackle the work on their own.

Photocopiable remediation exercise worksheets provide extra questions and help support students.

A topic from Edexcel’s GCSE Mathematics courseAs well as the concise Starters, Main teaching and learning points and Plenary, the lesson notes also contain:

Teaching and Learning Saving you time and guiding you through the new specifi cation, the Teacher’s guide contains concise, easy-to-read Lesson plans and extra Guided Practice Worksheets which are available as editable Word fi les and pdfs on the CD-ROM.

• At a glance specifi cation references and detail.• Starter ideas to check that students have the required

prior knowledge.• Main teaching and learning points to help you teach

the topic itself.• Plenary questions to test understanding and

application of the mathematics.


Chapter 1 Number

x

Introduction (Continued)

• Common misconceptions, and possible enrichment activities are also included where appropriate.

• Editable scheme of work available on the CD-ROM.• Guided practice worksheets with remediation

questions. As these are support worksheets, some of the questions may fall below grade levels.

• Integrates fully with ActiveTeach if installed.

Digital products ActiveTeachICT is seamlessly incorporated into mathematics lessons by using the unique, networkable, VLE compatible ActiveTeach.

ActiveTeach is a front-of-class teaching tool allowing you to display the Student Books on your whiteboard or through your VLE, while giving access to a wealth of activities, video clips, quizzes and other activities.

• BBC Active clips bring maths to life. Accompanying each clip are teacher mediated questions, and a worksheet for students to complete.

• ResultsPlus interactive problem-solving activities provide whole class practice of the new AO2 and AO3 style questions with our unique three-part tool.

• ResultsPlus knowledge checkers test AO1 recall with a multiple choice test at the end of each chapter.

• High-quality interactive content integrates seamlessly with the Student Book.

• Multi-lingual glossary gives audio translations for common maths terms in fi ve languages.

• My lessons area allows you to personalise content by adding your own links, interacting directly with the text and saving your annotations, enabling you to reapply your thinking the next time you deliver the lesson.

ResultsPlus Booster CEasy to adopt, set up and administer, ResultsPlus Booster C is an online service that takes borderline D/C students through highly targeted practice to boost their performance and help them get that all-important C grade.• Dynamically generated guided practice questions,

labelled by grade, give students a variety of practice to meet their needs exactly.

• Edexcel exam-style questions onscreen, give the benefi ts of instant examiner feedback, and familiarity with the new GCSE question types and requirements.

• Online delivery ensures total currency of questions for the new specifi cation.

• Links to other course components, give students and teachers a clear, consistent learning experience.

• Advanced reporting tools give unmatched insight into student performance, enabling teachers to pinpoint exactly where individuals are going wrong.

• Works alongside ResultsPlus Progress to allow you to address the weak areas that ResultsPlus Progress diagnoses.

ResultsPlus Progress testsOur online diagnostic assessment service helps you improve your students’ performance before it’s too late. Great for embedding Assessment for Learning into your course, it gives you access to exactly the information you need, to help tackle areas of weakness for each student.• Ten topic-based tests, all with 25 questions that are

perfect for both linear and modular courses.• Each topic test can be taken individually or linked with

others to create more comprehensive unit, termly or mock-style assessments.

SupportPlus websitewww.edexcelmaths.com/supportplusOur dedicated website with information about the specifi cations, training events, support and sample materials. An Edexcel Maths users-only area gives detailed support including • Interactive Schemes of Work• Teaching Resources – Lesson Plans and Practice

Worksheets• Exam Question Editor• Updates from Subject Leader Graham Cumming• ICT Blog • Community Area• Answers to questions in printed materials not included

with the book

Icons used in the Student books• A02 A03 Assessment objective questions are classifi ed

as AO2 and/or 3. These questions follow the more open structure demanded by QCDA for the new specifi cation and are not available in earlier GCSE publishing schemes.

• Functional skills indicates questions that cover functional elements of GCSE maths.

• * Quality of Written Communication (QWC) identifi es questions that follow the style of QWC questions in the exam.

• Non-calculator indicates questions where students must not use a calculator to fi nd the answer. It does NOT indicate that the subject area covered by the question will only appear in the non-calculator paper of the exam.


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