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ii | Exploring maths Tier 5 Introduction

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Exploring maths Tier 5 Introduction | iii

IntroductionThe materialsThe Exploring maths scheme has seven tiers, indicated by the seven colours in the table below. Each tier has:

a class book for pupils;

a home book for pupils;

a teacher’s book, organised in units, with lesson notes, mental tests (for number units), facsimiles of resource sheets, and answers to the exercises in the class book and home book;

a CD with interactive books for display, either when lessons are being prepared or in class, and ICT resources for use in lessons.

Content, structure and diff erentiationThe tiers are linked to National Curriculum levels so that they have the maximum fl exibility. They take full account of the 2007 Programme of Study for Key Stage 3, the Secondary Strategy’s renewed Framework for teaching mathematics in Years 7 to 11, published in 2008, and the possibility of taking the statutory Key Stage 3 test before the end of Year 9. Standards for functional skills for Entry Level 3 and Level 1 are embedded in Tiers 1 and 2. Tier 3 begins to lay the groundwork for level 2.

Labels such as ‘Year 7’ do not appear on the covers of books but are used in the table below to explain how the materials might be used.

The Exploring maths scheme as a whole off ers an exceptional degree of diff erentiation, so that the mathematics curriculum can be tailored to the needs of individual schools, classes and pupils.

Year 7 Year 8 Year 9

Extra supportFor pupils who achieved level 2 or a weak level 3 at KS2, who will enter the level 3–5 test at KS3 and who are likely to achieve Grade F–G at GCSE.

Tier 1NC levels 2–3(mainly level 3)


Tier 3NC levels 4–5(both levels 4 and 5)

SupportFor pupils who achieved a good level 3 or weak level 4 at KS2, who will enter the level 4–6 test at KS3 and who are likely to achieve Grade D–E at GCSE.



Tier 4NC level 5–6(mainly level 5)

CoreFor pupils who achieved a secure level 4 at KS2, who will enter the level 5–7 test at KS3 and who are likely to achieve B–C at GCSE.




ExtensionFor pupils who achieved level 5 at KS2, who will enter the level 6–8 test at KS3 and who are likely to achieve A or A* at GCSE.




Gifted and talentedFor gifted pupils who achieved a strong level 5 at KS2, who may be entered early for the level 6–8 test for KS3 and who are likely to achieve A* at GCSE.




iv | Exploring maths Tier 5 Introduction

There are at least fi ve tiers available for each of the year groups 7, 8 and 9. The range of tiers to be used in Year 7 can be chosen by the school to match the attainment of their incoming pupils and their class organisation. Teachers of mixed-ability classes can align units from diff erent tiers covering related topics (see Related units, p. xi).

The Results Plus Progress entry test, published separately, guides teachers on placing pupils in an appropriate tier at the start of Year 7. The test analysis indicates which topics in that tier may need special emphasis. Similar computer assessments are available for other years (see Computer-mediated assessments, p. viii).

Pupils can progress to the next tier as soon as they are ready, since the books are not labelled Year 7, Year 8 or Year 9. Similarly, work on any tier could take more than a year where pupils need longer to consolidate their learning.

Pupils in any year group who have completed Tier 4 or above successfully could be entered early for the Key Stage 3 test if the school wishes. Single-level tests for pupils working at particular national curriculum levels, which pupils can take in the winter or summer of any calendar year, are currently being piloted in ten local authorities as part of the Making good progress project. The tiered structure of Exploring maths is ideally suited to any extension of this pilot.

Each exercise in the class book off ers diff erentiated questions, so that teachers can direct individual pupils to particular sections of the exercises. Each exercise starts with easier questions and moves on to harder questions, identifi ed by underscored question numbers. More able pupils can tackle the extension problems.

If teachers feel that pupils need extra support, one or more lessons in a unit can be replaced with or supplemented by lessons from revision units.

Organisation of the unitsEach tier is based on 100 lessons of 50 to 60 minutes, plus 10 extra lessons to use for revision or further support, either instead of or in addition to the main lessons.

Lessons are grouped into units, varying in length from three to ten lessons. The number of lessons in a unit increases slightly through the tiers so that there are fewer but slightly longer units for the higher tiers.

Each unit is identifi ed by a code: N for number, A for algebra, G for geometry and measures, S for statistics and R for revision. For example, Unit N4.2 is the second number unit for Tier 4, while Unit G6.3 is the third geometry and measures unit for Tier 6. Mathematical processes and applications are integrated throughout.

The units are shown in a fl owchart giving an overview for the year (see p. ii). Some units need to be taught before others but schools can determine the precise order.

Schools with mixed-ability classes can align units from diff erent tiers covering related topics. For example, Unit G4.2 Measures and mensuration in Tier 4 can be aligned with the Tier 3 Unit G3.1 Mensuration and the Tier 5 Unit G5.2 Measures and mensuration. For more information on where to fi nd related units, see p. xi.

Revision unitsEach optional revision unit consists of fi ve stand-alone lessons on diff erent topics. These lessons include national test questions to help pupils prepare for tests.

Revision lessons can be taught in any order whenever they would be useful. They could be used with a whole class or part of a class. Schools that are entering pupils for national tests may wish to use, say, fi ve of the revision lessons at diff erent points of the spring term and fi ve in the early summer term.

Exploring maths Tier 5 Introduction | v

The revision lessons can either replace or be taught in addition to lessons in the main units. Units where the indicative number of lessons is given as, say, 5/6 lessons, are units where a lesson could be replaced by a revision lesson if teachers wish.

Balance between aspects of mathematicsIn the early tiers there is a strong emphasis on number and measures. The time dedicated to number then decreases steadily, with a corresponding increase in the time for algebra, geometry and statistics. Mathematical processes and applications, or using and applying mathematics, are integrated into the content strands in each tier.

The lessons for each tier are distributed as follows.

The teacher’s book, class book and home book

Teacher’s bookEach unit starts with a two-page overview of the unit. This includes:

the necessary previous learning and the objectives for the unit, with the process skills and applications listed fi rst for greater emphasis;

the titles of the lessons in the unit;

a brief statement on the key ideas in the unit and why they are important;

brief details of the assessments integrated into the unit;

common errors and misconceptions for teachers to look out for;

the key mathematical terms and notation used in the unit;

the practical resources required (equipment, materials, paper, and so on);

the linked resources: relevant pages in the class book and home book, resource sheets, assessment resources, ICT resources, and so on;

references to useful websites (these were checked at the time of writing but the changing nature of the Internet means that some may alter at a later date).

The overview is followed by lesson notes. Each lesson is described on a two-page spread. There is enough detail so that non-specialist teachers could follow the notes as they stand whereas specialist mathematics teachers will probably adapt them or use them as a source of ideas for teaching.

Number AlgebraGeometry and

measuresStatistics

Tier 1 54 1 30 15

Tier 2 39 19 23 19

Tier 3 34 23 24 19

Tier 4 26 28 27 19

Tier 5 20 29 29 22

Tier 6 19 28 30 23

Tier 7 17 29 29 25

TOTAL 209 157 192 142

30% 23% 27% 20%

vi | Exploring maths Tier 5 Introduction

Each lesson identifi es the main learning points for the lesson. A warm-up starter is followed by the main teaching activity and a plenary review.

The lesson notes refer to work with the whole class, unless stated otherwise. For example, where pupils are to work in pairs, the notes make this clear.

All the number units include an optional mental test for teachers to read out to the class, with answers on the same sheet.

All units in the teacher’s book include answers to questions in the class book, home book, check ups and resource sheets. The answers are repeated in the answer section at the back of the teacher’s book.

Class bookThe class book parallels the teacher’s book and is organised in units. The overall objectives for the unit, in pupil-friendly language, are shown at the start of the unit, and the main objective for each individual lesson is identifi ed.

Interesting information to stimulate discussion on the cultural and historical roots of mathematics is shown throughout the units in panels headed ‘Did you know that…?’

The exercises include activities, games or investigations for groups or individuals, practice questions and problems to solve. Questions are diff erentiated, with easier questions at the beginning of each exercise. Harder questions are shown by underlining of the question number. Challenging problems are identifi ed as extension problems. The exercises for each lesson conclude with a summary of the learning points for pupils to remember.

Answers to exercises and functional skills activities in the class book are given in the teacher’s book.

Each unit ends with a self-assessment section for pupils called ‘How well are you doing?’ to help them to judge for themselves their grasp of the work. Answers to these self-assessment questions are at the back of the class book for pupils to refer to.

Home bookEach lesson has an optional corresponding homework task. Homework tasks are designed to take most pupils about 15 to 20 minutes for Tiers 1 and 2, 25 minutes for Tiers 3, 4 and 5, and 30 minutes for Tiers 6 and 7.

Homework is normally consolidation of class work. It is assumed that teachers will select from the homework tasks and will set, mark and follow up homework in accordance with the school’s timetable. Because each school’s arrangements for homework are diff erent, feedback and follow-up to homework is not included in the lesson notes. It is assumed that teachers will add this as appropriate.

If the homework is other than consolidation (e.g. Internet research, collecting data for use in class), the lesson notes state that it is essential for pupils to do the homework. The next lesson refers to the homework and explains how it is to be used.

Answers to the homework tasks are given in the teacher’s book.

The ActiveTeach CD-ROMThe ActiveTeach contains interactive versions of the Teacher’s Book, Class Book, Home Book and a variety of ICT resources. Full notes on how to use the ActiveTeach are included on the CD-ROM in the Help tab.

Teachers can use the interactive version of the Teacher’s Book when they are planning or teaching lessons.

From the contents page of the Teacher’s Book, teachers can navigate to the lesson notes for the relevant unit, which are then displayed in a series of double page spreads.

Exploring maths Tier 5 Introduction | vii

Clicking on the thumbnail of the PowerPoint slide or the triangular icon shown on the edges of the pages allows teachers to view ICT resources, resource sheets, and other Microsoft Offi ce program fi les. All these resources, as well as exercises in the Class Book and tasks in the Home Book, can be accessed by clicking on the reference to the resource in the main text.

There is also an option for teachers to use a resource palette to put together their own set of resources ready for a particular lesson, choosing from any of the Exploring maths resources in any tier, and adding their own if they wish. This option will be especially useful for teachers of mixed ability classes.

Interactive versions of the Class Book and Home Book can be displayed in class. From the contents page, teachers can go to the relevant unit, which is then shown in a series of double page spreads. It is possible to zoom in and enlarge particular worked examples, diagrams or photographs, points to remember, homework tasks, and so on. Just as in the Teacher’s Book, clicking on the triangular icon launches the relevant resource.

ICT resourcesEach tier has a full range of ICT resources, including: a custom-built toolkit with over 60 tools, Flash animations, games and quizzes, spreadsheets and slides.

The diff erent resources are coded as follows.

Check ups (CU)

Each unit is supplemented by an optional check-up for pupils in the form of a PDF fi le to print and copy (see also the section on Assessment for learning).

Resource sheets (RS)

Some units have PDF fi les of resource sheets to print and copy for pupils to write on in class.

Tools (TO)

These general purpose teaching tools can be used in many diff erent lessons. Examples are:

– an interactive calculator, similar to an OHP calculator (in most cases, the scientifi c calculator will be needed);

– number lines and grids; – a graph plotter; – simulated dice and spinners; – squared paper and dotty paper; – drawing tools such as a protractor, ruler and compasses.

Simulations (SIM)

Some of these are animations to play and pause like a video fi lm. Others are interactive and are designed to generate discussion; for example, the teacher may ask pupils to predict an outcome on the screen.

Quizzes (QZ)

These are quizzes of short questions for pupils to answer, e.g. on their individual whiteboards, usually at the start or end of a lesson.

Interactive teaching programs (ITP)

These were produced by the Primary Strategy and are included on the CD-ROM with permission from the DCSF.

PowerPoint presentations (thumbnails)

These are slides to show in lessons. Projected slides can be annotated, either with a whiteboard pen or with the pen tool on an interactive whiteboard. Teachers without access to computer and data projector in their classrooms can print the slides as overhead projector transparencies and annotate them with an OHP pen.

viii | Exploring maths Tier 5 Introduction

Excel fi les (XL)

These are spreadsheets for optional use in particular lessons.

Geometer’s Sketchpad fi les (GSP)

These are dynamic geometry fi les for optional use in particular lessons.

Other ICT resources, such as calculators, are referred to throughout the units.

The table on p. x identifi es those lessons where pupils have an opportunity to use ICT for themselves.

Assessment for learningThere is a strong emphasis on assessment for learning throughout Exploring maths.

Learning objectives for units as a whole and for individual lessons are shown on slides and in the class book for discussion with pupils.

Potential misconceptions are listed for teachers in the overview pages of each unit.

Key questions for teachers to ask informally are identifi ed in the lesson notes.

The review that concludes every lesson allows the teacher to judge the eff ectiveness of the learning and to stress the learning points that pupils should remember.

The points to remember are repeated in the class book and home book.

A self-assessment section for pupils, ‘How well are you doing?’, is included in each unit in the class book to help pupils to judge for themselves their grasp of the work.

Optional revision lessons provide extra support in those areas where pupils commonly have diffi culty.

Each unit on the CD-ROM includes an optional check-up of written questions.

Each number unit of the teacher’s book includes an optional mental test of 12 questions for teachers to read to the class.

The mental test could be used as an alternative to part of the last lesson of the unit. About 20 minutes of lesson time is needed to give the test and for pupils to mark it. Answers are on the same sheet.

The written check-ups include occasional questions from national tests. Teachers could use some or all of the questions, not necessarily on the same occasion, and pupils could complete them in class, at home, or as part of an informal test. For example, some written questions could be substituted for the fi nal homework of a unit and the mental test could be used as an alternative to part of the last lesson. Answers to the written check-ups are given in the teacher’s book.

Computer-mediated assessmentsExploring maths is complemented by Results Plus Progress, a series of stimulating on-line computer-mediated assessments supporting Key Stage 3 mathematics, available separately.

There is an entry test for Year 7 to guide teachers on placing pupils in an appropriate tier at the start of the course. For each of Years 7, 8 and 9, there are two end-of-term assessments for the autumn and spring terms, and an end-of-year assessment.

Each product off ers sets of interactive test questions that pupils answer on computers, either in school or on home computers with internet access. Because the tests are taken electronically, the products off er instant marking and analysis tools to identify strengths and weaknesses of individuals or groups of pupils. Future units from Exploring maths that are dependent on the same skills are identifi ed so that teachers are aware of the units that they may need to adapt, perhaps by adding in extra revision or support lessons.

Results Plus Progress has been developed by the Test Development Team at Edexcel, who have considerable experience in producing the statutory national end-of-key-stage tests and the optional tests for Years 7 and 8.

Exploring maths Tier 5 Introduction | ix

Where can I fi nd…?

Historical and cultural references

N5.1 Euclid’s Fundamental Theorem of Arithmetic Class book p.6

A5.1 Oresme and Descartes and the invention of a coordinate system Class book p.20

A5.1 Graphing calculators Class book p.24

G5.1 The history of π Class book p.37

N5.2 The Ancient Egyptians’ use of fractions Class book p.56

N5.2 Use of the ratio system in the seventeenth century Class book p.65

N5.2 Exchange rates Class book p.71

A5.2 Fahrenheit and Celsius Class book p.100

A5.2 Euler’s formula Class book p.101

A5.2 The history of solving equations Class book p.107

G5.2 Regular polygons in man-made and natural environments Class book p.119

G5.2 The building of the Hagia Sophia Class book p.124

G5.2 How emperor penguins huddle to keep warm Class book p.129

N5.3 A googol and googolplex Class book p.137

S5.2 Abraham de Moivre and the development of the theory of probability Class book p.155

G5.3 Pyramids and plane symmetry Class book p.187

G5.3 Islamic patterns Class book p.194Home book p.61

A5.4 The Rhind Mathematical Papyrus Class book p.214

S5.3 William Farr – the fi rst chief statistician Class book p.224

S5.3 The coordinate system developed by Rene Descartes Class book p.227

G5.4 Carl Friedrich Gauss and the construction of a heptadecagon Class book p.242

G5.4 Logo and turtle graphics Class book p.246

A5.5 The origins of algebra Class book p.272

S5.4 Girolamo Cardano’s work on probability Class book p.300

S5.4 The Monty Hall Game Class book p.312

N5.4 Mathematicians who helped to develop our number system Class book p.316Home book p.98

N5.4 Ramanujan and the sum of two cubes Class book p.322

R5.1 The Whetstone of Witte – the fi rst algebra book written in English Class book p.338

R5.2 The Moscow Papyrus Class book p.355

x | Exploring maths Tier 5 Introduction

ICT lessons: hands-on for pupilsPupils have many opportunities for hands on use of ICT.

N5.1 Lesson 1: Using the xy key on a calculator Teacher’s book p.4

Lesson 2: Using Excel to construct spreadsheets Teacher’s book p.6

A5.1 Lesson 3: Generating sequences using spreadsheets Teacher’s book p.20

G5.1 Lesson 2: Using dynamic geometry software to fi nd � Teacher’s book p.38

N5.2 Lesson 1: Using a calculator to simplify fractions Teacher’s book p.54

Lesson 2: Using a calculator to multiply fractions Teacher’s book p.56

Lesson 3: Using a calculator to divide fractions Teacher’s book p.58

A5.2 Lesson 6: Using graph-plotting software for trial and improvement Teacher’s book p.108

G5.2 Lesson 1: Exploring angles and lines with dynamic geometry software

Teacher’s book p.116

Lesson 7: Using Excel to create spreadsheets and graphs Teacher’s book p.128

N5.3 Lesson 5: Consolidating and extending calculator skills Teacher’s book p.152

S5.2 Lesson 4: Using spreadsheets in probability experiments Teacher’s book p.170

A5.3 Lesson 1: Using graph-plotting software to generate graphs Teacher’s book p.182

Lessons 2 and 5: Using graph-plotting software to sketch and transform linear graphs


Lesson 6: Using graph-plotting software to generate quadratic graphs


G5.3 Lesson 2: Creating transformations using dynamic geometry software


Lesson 6: Using dynamic geometry software to explore area and perimeter


A5.4 Lesson 2: Using graph-plotting software to solve geometry problems


Lesson 3: Using the interactive algebra program Frogs (optional) Teacher’s book p.226

S5.3 Lesson 5: Producing graphs using Excel Teacher’s book p.242

G5.4 Lesson 2: Drawing regular polygons with Logo Teacher’s book p.260

A5.5 Lesson 8: Using graph-plotting software to solve problems Teacher’s book p.302

N5.4 Lesson 1: Internet research on the history of our number system Teacher’s book p.328

Functional skillsStandards for functional skills for Entry Level 3 and Level 1 are embedded in Tiers 1 and 2.

Tiers 3 and 4 begin to lay the groundwork for the content and process skills for functional skills at level 2. This continues in Tier 5.

Activities to encourage the development of functional skills are integrated throughout the Tier 5 class book.

In addition, there are four specifi c activities. These can be tackled at any point in the year, including the beginnings and ends of terms. They are all group activities which lend themselves to further development and follow-up. Many of the questions are open ended.

Exploring maths Tier 5 Introduction | xi

Related units

Tier 4 Tier 5 Tier 6

N4.1 Properties of numbers N5.1 Powers and roots N6.1 Powers and roots

N4.2 Whole numbers, decimals and fractions

N4.3 Fractions, decimals and percentages

N5.3 Calculations and calculators N6.3 Decimals and accuracy

N4.4 Proportional reasoning N5.2 Proportional reasoning N6.2 Proportional reasoning

N4.5 Solving problems N5.4 Solving problems N6.4 Using and applying maths

A4.1 Linear sequences A5.1 Sequences and graphs A6.2 Linear functions and graphs

A4.3 Functions and graphs A5.3 Functions and graphs A6.3 Quadratic functions and graphs

A4.5 Using algebra A5.4 Using algebra A6.4 Using algebra

A4.2 Expressions and formulaeA4.4 Equations and formulae

A5.2 Equations and formulaeA5.5 Equations, formulae and graphs

A6.1 Expressions and formulae

G4.1 Angles and shapes G5.2 2D and 3D shapes

G4.4 Constructions G5.4 Angles and constructions G6.1 Geometrical reasoning

G4.3 Transformations G5.3 Transformations G6.3 Transformations and loci

G4.2 Measures and mensuration G5.1 Measures and mensuration G6.4 Measures and mensuration

G6.2 Trigonometry 1G6.5 Trigonometry 2

S4.2 Enquiry 1 S5.1 Enquiry 1 S6.1 Enquiry 1

S4.3 Enquiry 2 S5.3 Enquiry 2 S6.3 Enquiry 2

S4.1 Probability S5.2 Probability 1 S6.2 Probability 1

S5.4 Probability 2 S6.4 Probability 2

R4.1 Revision unit 1 R5.1 Revision unit 1 R6.1 Revision unit 1

R4.2 Revision unit 2 R5.2 Revision unit 2 R6.2 Revision unit 2

The activities focus on these process skills:

identifying the mathematics in a situation and mathematical questions to ask;

recognising that a situation can be represented using mathematics;

selecting the information, methods, operations and tools to use, including ICT;

making an initial model of a situation using suitable forms of representation;

changing values in the model to see the eff ects on answers;

examining patterns and relationships;

interpreting results and drawing conclusions;

considering how appropriate and accurate results and conclusions are;

choosing appropriate language and forms of presentation to communicate results and solutions.

FS1 Should you buy or rent a TV set? Class book p.54

FS2 Where is the mathematics? Class book p.112

FS3 Cutting it up Class book p.220

FS4 Shoe sizes Class book p.298

xii | Exploring maths Tier 5 Introduction

Contents

N5.1 Powers and roots 21 Integer powers of numbers 42 Estimating square roots 63 Prime factor decomposition 8Mental test 10Check up 11Answers 12

A5.1 Sequences and graphs 141 Generating sequences 162 Making generalisations 183 Using computers 204 Sketching linear graphs 225 Rearranging linear equations 246 Graphs representing real-life contexts 26Check up 28Answers 29

G5.1 Measures and mensuration 341 Perimeter and area 362 Finding � 383 Area of a circle 404 Solving circle problems and using � 425 Volume of prisms 446 Surface area of prisms 46Check up 48Answers 49

N5.2 Proportional reasoning 521 Adding and subtracting fractions 542 Multiplying fractions 563 Dividing fractions 584 Percentage change 605 Ratio 626 Direct proportion 64Mental test 66Check up and resource sheets 67Answers 68

S5.1 Enquiry 1 721 Stem-and-leaf diagrams 742 Starting a statistical investigation 1 763 Completing a statistical investigation 1 784 Data collection sheets 805 Starting a statistical investigation 2 826 Completing a statistical investigation 2 84Check up and resource sheets 86Answers 88

A5.2 Equations and formulae 961 Multiplying out brackets 982 Factorising expressions 1003 Substituting into formulae 1024 Changing the subject of a formula 1045 Solving linear equations 1066 Trial and improvement 108Check up 110Answers 111

G5.2 2D and 3D shapes 1141 Exploring angles and lines 1162 Solving problems 1183 Solving longer problems 1204 Drawing 3D objects 1225 Drawing plans and elevations 1246 More plans and elevations 1267 Solving problems using surface area and volume 1288 Surface area and volume of prisms 130Check up and resource sheets 132Answers 134

N5.3 Calculations and calculators 1421 Powers of 10 1442 Rounding and approximation 1463 Mental calculations with decimals 1484 Written calculations with decimals 1505 Using a calculator 1526 Problems involving measures 154Mental test 156Check up and resource sheets 157Answers 158

S5.2 Probability 1 1621 Simple probability 1642 Equally likely outcomes with two events 1663 Mutually exclusive events 1684 Practical probability experiments 1705 Simulating probability experiments 172Check up and resource sheets 174Answers 176

A5.3 Functions and graphs 1801 Generating linear graphs using ICT 1822 Sketching graphs 1843 Drawing accurate graphs 1864 Direct proportion 1885 Refl ecting graphs in y � x 190

Tier5

Exploring maths Tier 5 Introduction | xiii

6 Simple quadratic graphs using ICT 192Check up 194Answers 195

G5.3 Transformations 1981 Planes of symmetry 2002 Combined transformations 2023 Islamic patterns 2044 Enlargements 2065 Enlargements in real-life applications 2086 Length, area and volume 210Check up and resource sheets 212Answers 214

A5.4 Using algebra 2201 Using graphs to solve problems 2222 Using algebra in geometry problems 2243 Using algebra in investigations 226Check up 228Answers 229

S5.3 Enquiry 2 2321 Calculating statistics 2342 Line graphs for time series 2363 Scatter graphs 2384 Collecting and organising data 2405 Analysing and representing data 2426 Interpreting data 2447 Reporting and evaluating 246Check up and resource sheets 248Answers 251

G5.4 Angles and constructions 2561 Angles in polygons 2582 Regular polygons 2603 Regular polygons and the circle 2624 Angle problems and polygons 2645 Polygons and parallel lines 2666 Constructions 2687 Constructing triangles 2708 Loci 2729 More loci 274Check up and resource sheets 276Answers 278

A5.5 Equations, formulae and graphs 2861 Factorising 2882 Working with algebraic fractions 2903 Working with formulae 2924 Forming equations 294

5 Visualising graphs 2966 Interpreting graphs 2987 Matching graphs to real-life situations 3008 Using graphs to solve problems 302Check up 304Answers 305

S5.4 Probability 2 3121 Theoretical and experimental probability 3142 Mutually exclusive events 3163 Using experimental probability 3184 Choice or chance? 320Check up and resource sheets 322Answers 323

N5.4 Solving problems 3261 History of our number system and zero 3282 Number puzzles based on 3 by 3 grids 3303 Exploring fractions 3324 Problems involving properties of numbers 3345 Using algebra and counter-examples 336Slide commentary 338Mental test 341Check up and resource sheets 342Answers 343

R5.1 Revision unit 1 3461 Using a calculator 3482 Using percentages to compare proportions 3503 Sequences, equations and graphs 3524 Angles and polygons 3545 Charts and diagrams 356Mental test 358Resource sheet 359Answers 360

R5.2 Revision unit 2 3641 Ratio and proportion 3662 Solving number problems 3683 Expressions, equations and formulae 3704 Circles and enlargements 3725 Probability 374Mental test 376Answers 377

Schools planning a shortened two-year programme for Key Stage 3 may not have time to teach all the lessons. The lessons in black cover the essential material for pupils taking this route. The lessons in blue provide useful consolidation and enrichment opportunities. These should be included wherever possible.

2 | N5.1 Powers and roots

Powers and roots

Previous learningBefore they start, pupils should be able to:

recognise and use multiples, factors (divisors), common factor, highest common factor, lowest common multiple and primes

use squares, positive and negative square roots, cubes and cube roots, and index notation for small positive integer powers.

Objectives based on NC levels 5 and 6 (mainly level 6)In this unit, pupils learn to:

identify problems and the methods needed to tackle them

select and apply mathematics to fi nd solutions

represent problems and synthesise information in diff erent forms

use accurate notation

calculate accurately, selecting mental methods or a calculator as appropriate

use appropriate checking procedures

record methods, solutions and conclusions

make convincing arguments to justify generalisations or solutions

interpret and communicate solutions to problems

and to:

use index notation for integer powers

know and use the index laws for multiplication and division of positive integer powers

extend mental methods of calculation with factors, powers and roots

use the power and root keys of a calculator

use ICT to estimate square roots and cube roots

use the prime factor decomposition of a number.

Objectives in colour lay the groundwork for Functional Skills.

Lessons 1 Integer powers of numbers

2 Estimating square roots

3 Prime factor decomposition

About this unit Sound understanding of powers and roots of numbers helps pupils to generalise the principles in their work in algebra and provides a foundation for later work on numbers in standard form and surds.

It also helps pupils to be aware of the relationships between numbers and to know at a glance which properties they possess and which they do not.

Calculators vary in the ways that powers and roots are entered. You may need to point out these diff erences to pupils and explain how to adapt the information in the class book or home book for their own calculators.

Assessment This unit includes:

an optional mental test which could replace part of a lesson (p. 10);

a self-assessment section (N5.1 How well are you doing? class book p. 9);

a set of questions to replace or supplement questions in the exercises or homework tasks, or to use as an informal test (N5.1 Check up, CD-ROM).

N5.1

N5.1 Powers and roots | 3

Common errors and misconceptions

Look out for pupils who:

think that n2 means n � 2, or that √__

n means n __ 2 ;

wrongly apply the index laws, e.g. 103 � 104 � 107, or 103 � 104 � 1012;

think that 1 is a prime number;

include 1 in the prime factor decomposition of a number;

confuse the highest common factor (HCF) and lowest common multiple (LCM);

assume that the lowest common multiple of a and b is always a � b.

Key terms and notation problem, solution, method, pattern, relationship, expression, solve, explain, systematic, trial and improvement

calculate, calculation, calculator, operation, multiply, divide, divisible, product, quotient

positive, negative, integer

factor, factor pair, prime, prime factor decomposition, power, root, square, cube, square root, cube root, notation n2 and √

__ n , n3 and 3 √

__ n

Practical resources scientifi c calculators for pupilsindividual whiteboards

computers with spreadsheet software, e.g. Microsoft Excel, or graphics calculators

Exploring maths Tier 5 teacher’s bookN5.1 Mental test, p. 10Answers for Unit N5.1, pp. 12–13

Tier 5 CD-ROMPowerPoint fi les N5.1 Slides for lessons 1 to 3Excel fi le N5.1 Square rootTools and prepared toolsheets Calculator toolTier 5 programs Multiples and factors quiz Ladder method HCF and LCM

Tier 5 class bookN5.1, pp. 1–10

Tier 5 home bookN5.1, pp. 1–3

Tier 5 CD-ROMN5.1 Check up

Useful websites Topic B: Indices: simplifyingwww.mathsnet.net/algebra/index.html

Factor treenlvm.usu.edu/en/nav/category_g_3_t_1.html

Grid gamewww.bbc.co.uk/education/mathsfi le/gameswheel.html


Learning points

A number a raised to the power 4 is a4 or a � a � a � a.

The number that expresses the power is its index, so 2, 5 and 7 are the indices of a2, a5 and a7.

To multiply two numbers in index form, add the indices, so am � an � am � n.

To divide two numbers in index form, subtract the indices, so am � an � am � n.

When a negative number is raised to an even power, the result is positive; when a negative number is raised to an odd power, the result is negative.

1 Integer powers of numbers

Starter

Main activity

Tell pupils that in this unit they will learn more about powers and roots of numbers. This fi rst lesson is about fi nding positive and negative integer powers.

Remind pupils that when a number is multiplied by itself the product is called a power of that number. So a � a, or a squared, is the second power of a and is written as a2; a � a � a, or a cubed, is the third power of a and is written as a3; and so on. For a4 we say a to the power of 4, and similarly with higher powers.

The number that expresses the power is its index. So 5 and 7 are the indices of a5 and a7. When the index is 1, it is usually omitted: we write a, rather than a1.

Ask pupils to calculate mentally some powers of integers, e.g.

82, (�8)2, 25, (�2)5, 34, (�3)4, 53, (�5)3

Record answers on the board, and ask pupils what they notice. Draw out that for negative numbers even powers are positive and odd powers are negative.

What is (�1)123? What is (�1)124?

As an optional extension, include some decimals, e.g.

(0.1)3, (0.7)2, (�0.2)4

If appropriate, use the Calculator tool to show pupils how to use the xy keys of their calculators.

Explore raising a number to the power 0 to show that this always results in 1.

Discuss negative indices. Ask pupils to consider the pattern on the right.

10 000 � 104

1000 � 103

100 � 102

10 � 101

1 � 100

How is each number found from the one above it?

What is the pattern of the indices?

What are the next few lines of the pattern?

Establish that:

0.1 � 1 __ 10 � 10–1

0.01 � 1 ___ 100 � 10–2

0.001 � 1 ___ 1000 � 10–3

Similarly 1 _ 2 � 2–1, 1 _ 4 � 2–2 and 1 _ 8 � 2–3.

Show the table of powers of 2 on slide 1.1. Ask pupils to work in pairs to make up and record some multiplications using the table. Ask questions to help pupils to discover for themselves the rules for calculations with indices.Slide 1.1

TO


Homework

Review

Slide 1.2

Select individual work from N5.1 Exercise 1 in the class book (p. 1).

Slide 1.3

Slide 1.4

Homework

What do you notice about the indices in these calculations?

What is a quick way of multiplying powers of 2? Why does it work?

What is this calculation in index form?

32 � 256 � 8192 [25 � 28 � 213]

4096 � 512 [212 � 29 � 23]

16 384 � 64 [214 � 26 � 28]

What do you notice about the indices in these calculations?

What is a quick way of dividing one power of 2 by another?Why does it work?

Repeat with the powers of 4 on slide 1.2.

Now generalise. Write on the board m2 � m3.

What will this simplify to? Explain why.[m2 � m3 � (m � m) � (m � m � m) � m � m � m � m � m � m5]

Stress that the indices have been added, so that:

m2 � m3 � m2 � 3 � m5

Repeat with m5 � m2. Stress that for division the indices are subtracted, so that:

m5 � m2 � m5 � 2 � m3

Discuss negative indices, e.g.

m3 � m7 � m3 � 7 � m–4 � 1 ___ m4

m5 � m–3 � m5 � 3 � m2.

Show slide 1.3.

Point to two diff erent powers of 10. Ask pupils to multiply or divide them and to write the answer on their whiteboards.

Stress that the rules for multiplying and dividing numbers in index form apply to both positive and negative indices.

Ask pupils to remember the points on slide 1.4.

Ask pupils to do N5.1 Task 1 in the home book (p. 1).


Learning points

√__

n is the square root of n, e.g. √___

81 � �9.

3 √__

n is the cube root of n, e.g. 3 √____

125 � 5, 3 √____

�27 � �3.

Trial and improvement can be used to estimate square roots when a calculator is not available.

2 Estimating square roots

Starter

Main activity

Tell pupils that in this lesson they will be estimating the value of square roots.

Remind them that the square root of a is denoted by 2 √__

a , or more simply as √__

a , and that a square root of a positive number can be positive or negative, e.g. if a2 � 9, a � � √

__ 3 .

Show the grid on slide 2.1. Write on the board: x �1, z � 4.

Point to an expression on the grid. Ask pupils to work out its value mentally and to write the answer on their whiteboards. Ask someone to explain how they calculated it. After a while, change the values for x and z to x � 9 and z � 25.

Discuss how to estimate the positive square root of a number that is not a perfect square. For example, √

___ 70 must lie between √

___ 64 and √

___ 81 , so 8 � √

___ 70 � 9. Since

70 is closer to 64 than to 81, we expect √___

70 to be closer to 8 than to 9, perhaps about 8.4.

Show the class how they could fi nd √__

7 if they had only a basic calculator with no square-root key. Tell them that the process is called trial and improvement.

Explain that √__

7 must lie between 2 and 3, because 7 lies between 22 and 32.

Try 2.52 � 6.25 too smallTry 2.62 � 6.76 too smallTry 2.72 � 7.29 too bigTry 2.652 � 7.0225 very close but a little bit too bigTry 2.642 � 6.9696 very close but too smallTry 2.6452 � 6.986025 still too small

The answer lies between 2.645 and 2.65.All numbers between 2.645 and 2.65 round up to 2.65.So √

__ 7 � 2.65 correct to two decimal places.

Ask pupils to work in pairs and, using only the � key on their calculator, to fi nd √

___ 12 to two decimal places [answer: 3.46]. Establish fi rst that it must lie between

3 and 4.

Show the class how they could use a spreadsheet for this activity, without using the square-root function, e.g. use the Excel fi le N5.1 Square root.

Slide 2.1

8 9

64 70 81

XL


Review

Homework

Point out that the strategy here is diff erent. We work systematically in tenths from 3 to 4, then in hundredths from 3.4 to 3.5, then in thousandths from 3.46 to 3.47.

You can use this fi le to estimate other square roots by overtyping 3, 3.4 and 3.46. If possible, pupils should develop similar spreadsheets, using either a computer or a graphics calculator.

Introduce root notation. Explain that if 729 is the cube of 9, then 9 is the cube root of 729, which is written as 3 √

____ 729 � 9. The cube root, fourth root, fi fth root, …

of a are denoted by 3 √ __

a , 4 √ __

a , 5 √ __

a , …

Use the Calculator tool to demonstrate how to fi nd roots. You may need to explain that some calculators have a cube root key as well as a general key like

x √ _ , or other variations, e.g.

To fi nd the value of 3 √ ____

216 , key in 3 x √ _ 2 1 6 .

[Answer: 6]

Ask pupils to work out 3 √ ___

64 and 3 √ _____

�125 . Explain that the cube root of a positive number is positive, and the cube root of a negative number is negative.

Show how to use a calculator with an example such as 5 √ ______

32 768 � 8.

Sum up the lesson by reminding pupils of the learning points.


TO



Learning points

Writing a number as the product of its prime factors is called the prime factor decomposition of the number.

You can use a tree method or a ladder method to fi nd a number’s prime factors.

To fi nd the highest common factor (HCF) of a pair of numbers, fi nd the product of all the prime factors common to both numbers.

To fi nd the lowest common multiple (LCM) of a pair of numbers, fi nd the smallest number that is a multiple of each of the numbers.

3 Prime factor decomposition

Starter

QZ

Main activity

Tell pupils that in this lesson they will be fi nding the prime factors of numbers and using them to fi nd common factors and multiples of a pair of numbers.

Remind them of the defi nitions of multiple, factor, factor pair and prime number.

Launch Multiples and factors quiz. Ask pupils to answer on their whiteboards. Use ‘Next’ and ‘Back’ to move through the questions at a suitable pace.

Write on the board three products such as:

11 � 5 � 3 2 � 2 � 13 2 � 3 � 5 � 5

What do you notice about the numbers in these products?

Establish that they are all prime numbers. Explain that when a number is expressed as the product of its prime factors it is called the prime factor decomposition of a number. Stress that because 1 is not a prime number it is not included in the decomposition.

How can we fi nd the prime factor decomposition of 80?

First explain the tree method, i.e. split 80 into a product such as 20 � 4, then continue factorising any non-prime number in the product. Repeat with 300.

Launch Ladder method. Use it to show the alternative method, where the number is repeatedly divided by any prime that will divide into it exactly.

Demonstrate with 63, dragging numbers from the grid to the 3 63 3 21 7 7 1

relevant positions. Drag 63 to the box contained in the sentence. Drag prime numbers to the circles on the right. Continue to divide by prime numbers until completed. The bottom square will become a circle. Express the answer as 63 � 3 � 3 � 7 � 32 � 7.

Repeat with 80, either on the board or using the simulation.

Show how to fi nd the highest common factor (HCF) and lowest common multiple (LCM) of a pair of numbers. Launch HCF and LCM.

SIM

804

2 5

10

2

300

30

2

15

3

52

20

5

4

2

2

SIM


Review

Homework

Select ‘Find the lowest common multiple’. Select 8 and 6 using the arrows by the numbers. Drag multiples of 8 and multiples of 6 from the 100-square to the answer boxes. (Numbers snap back to the 100-square if dragged from answer box.) Numbers common to both boxes change colour to blue.

Which numbers are both multiples of 8 and multiples of 6?

Which is the lowest number that is both a multiple of 8 and 6?

Drag the LCM into the box below the 100-square.

Repeat with diff erent numbers, then change to ‘highest common factor’, which works similarly. Select 24 and 18, then drag factors to the answer boxes.

Which numbers are both factors of 24 and factors of 18?

Which is the highest number that is both a factor of 24 and 18?

Repeat with diff erent numbers.

Show how to use a Venn diagram to fi nd the HCF and LCM of a pair of numbers such as 36 and 30. Explain that:

the overlapping prime factors give the HCF (2 � 3 � 2 � 3 � 6);

all the prime factors give the LCM (2 � 2 � 3 � 3 � 5 � 22 � 32 � 5 � 180).

Repeat with 18 and 24.

Sum up the lesson by stressing the points on slide 3.1.

Round off the unit by referring again to the objectives. Suggest that pupils fi nd time to try the self-assessment problems in N5.1 How well are you doing? in the class book (p. 9).


Slide 3.1

36 30

523

3

2



N5.1 Mental testRead each question aloud twice.

Allow a suitable pause for pupils to write answers.

1 What number is fi ve to the power three?

2 Write all the prime factors of forty-two.

3 Write down a factor of thirty-six that is greater than ten and less than twenty. 2005 KS3

4 What is the next number in the sequence of square numbers? 2004 KS3One, four, nine, sixteen, …

5 Look at the numbers. 1999 Y7Write down each number that is a factor of one hundred.

[Write on the board: 10 15 20 25 30 35 40 45 50]

6 Write two factors of twenty-four which add to make eleven. 2005 KS2

7 What is the square root of eighty-one? 2001 KS3

8 What number is fi ve cubed? 2003 KS3

9 The volume of a cube is sixty-four centimetres cubed. 2002 KS3What is the length of an edge of the cube?

10 What is the square of three thousand? 2001 KS3

11 To the nearest whole number, what is the square root of 2004 KS3eighty-three point nine?

12 I think of a number. I square my number and get the answer 2007 KS3one thousand six hundred. What could my number be?

Key:KS2 Key Stage 2 testY7 Year 7 optional test (1999)KS3 Key Stage 3 testQuestions 3 to 7 are at level 5; 8 to 11 are at level 6; 12 is at level 7

Answers

1 125 2 2, 3, 7 3 12 or 18 4 25

5 10, 20, 25, 50 6 3 and 8 7 �9 8 125

9 4 cm 10 9 000 000 11 9 12 �40


N5.1 Check up

© Pearson Education 2008 Tier 5 resource sheets | N5.1 Powers and roots | 1.1

Check up N5.1Write your answers in your book.

Powers and roots (no calculator)

1 2001 level 6

Which two of the numbers below are not square numbers?

2 David says that 211 � 2048.

What is 210?

3 To the nearest whole number, what is the square root of 93.7?

4 If √ ___

81 � n � √ ____

144 , then n could be which of the following numbers?

9 11 12 13

5 Year 8 Optional Test level 6

Terry has 24 centimetre cubes.He uses them to make a cuboid that is one cube high.

Tina has 24 centimetre cubes.She uses them to make a solid cuboid that is two cubes high.

What could the dimensions of her cuboid be?

6 What is the biggest number that is a factor of both 105 and 135?

7 What is the smallest number that is a multiple of both 12 and 27?

Powers and roots (calculator allowed)

8 1996 level 6

Mary thinks of a number.

Which number did Mary think of?

24 25 26 27 28

1 cm

1 cm

1 cm

high

wide

long

First I subtract 3.76 Then I find the squareroot of what I get

My answer is 6.80


N5.1 Answers3 a 2 b 7 c 11 d 8

4 Each answer is correct to 1 d.p.

a 2.4 b 6.7 c 10.7 d 8.4

5 Between 700 and 750 slabs will be used. 26 � 26 is 678, which is too few, and 28 � 28 is 784, which is too many. So the exact number of slabs is 27 � 27 � 729.

6 a a � 9.7 to 1 d.p.

b a � 12.3 to 1 d.p.

c a � 20.4 to 1 d.p.

7 3.87 metres to 2 d.p.

Extension problem

8 2982 � 88 804888 is not a perfect square. There is no whole number between the square root of 8880 and the square root of 8889 but 298 lies between the √

______ 88 800 and √

______ 88 899 .

Exercise 31 a 3 � 22 b 3 � 5

c 3 � 7 d 23 � 3

e 33 f 2 � 33

2 a 1, 2, 5, 10, 25, 50 b 2 � 52

3 a 1, 3, 5, 9, 15, 45 b 5 � 32

4 e.g. 63 (with factors 1, 3, 7, 9, 21, 63)

5 a 72 and 30: HCF � 6, LCM � 360

b 50 and 80: HCF � 10, LCM � 400

c 48 and 84: HCF � 12, LCM � 336

6 HCF � 15, LCM � 600

7 HCF � 10, LCM � 360

Class book

Exercise 11 a 23 b 45

c 38 d (�1)4

e 5�2 f 6�1

2 a 64 b �243

c 256 d �128

e 1 f �1

g 1 __ 16 h 1 ___ 125

3 a 2401 b 15 625

c 1331 d 19 683

e 1024 f 11.39

g 32 157.43 h 11 272.96

4 a 28 b 35

c 104 d a8

e 54 f 124

g 80 h b3

5 11 6

21

31 2 1

54

2 7

6 a 19 � 32 � 32 � 12 b 41 � 62 � 22 � 12

c 50 � 52 � 42 � 32 d 65 � 62 � 52 � 22

e 75 � 72 � 52 � 12 f 94 � 72 � 62 � 32

or 92 � 32 � 22

7 Rachel and Hannah are 14 and 11 years old.

Extension problem

8 The smallest whole numbers are 6 and 10:102 � 62 � 100 � 36 � 64 � 43

103 � 63 � 1000 � 216 � 784 � 282

Exercise 21 a x � �3 b x � �7

c x � �12 d x � �1

2 a �1.41 to 2 d.p. b 2.15 to 2 d.p.

c �4 d �0.2

e �5 f �1.22 to 2 d.p.

g �1.73 to 2 d.p. h �1

22

2

5

3 5

75120

2

2

2

5 3

39040


8 a 2 and 3 b 6 c 378

9 a 28 and 40: HCF � 4, LCM � 280

b 200 and 175: HCF � 25, LCM � 1400

c 36 and 64: HCF � 4, LCM � 576

10 1050

Extension problems

11 a 22 � 4 (with factors 1, 2 and 4)

b 24 � 16 (with factors 1, 2, 4, 8, 16)

c 7 factors: 26 � 649 factors: 28 � 25611 factors: 210 � 102413 factors: 212 � 4096

12 420 days from now, since 420 is the lowest common multiple of 1, 2, 3, 4, 5, 6 and 7.

How well are you doing?1 a 32, 24, 52, 33 or 9, 16, 25, 27

b 57 � 55 � 52 � 3125 � 25 � 78 125

2 a 34 � 81 is the largest. 34 � 92

b 25 and 27 are not square numbers.

3 a 32 � 9 b 27 � 128

c 32 � 2 � 18

4 a a � 3 b b � 2

5 a HCF is 12 b LCM is 144

6 Suzy’s number is 4.9.

7 5 � 11 � 19 � 1045

Home book

Task 11 a 311 b 26 c 11�1 d x6

e 43 f 104 g 810 h z

2 a 28 � 33 � 13

b 72 � 43 � 23

c 1125 � 103 � 53

3 a (15)2 � 225(25)2 � 625

b (11)2 � 121(19)2 � 361(21)2 � 441(29)2 � 841(31)2 � 961

c (6)3 � 216

d (13)3 � 2197

Task 21 a 19 b 9

c 9 d 5

e 24 f 2

g 8.67 to 2 d.p. h 4.24 to 2 d.p.

2 18.8 cm to 1 d.p.

3 a 3 b 4

c 6 d 10

Task 31 a 23 � 3 � 7 b 35

2 a 2 � 32 � 52 b 5 � 7 � 17

3 45

4 936

5 7, 13 and 17

CD-ROM

Check up1 25 and 27 are not square numbers.

2 2048 � 2 � 1024

3 10

4 11

5 2 cm high by 1 cm wide by 12 cm long2 cm high by 2 cm wide by 6 cm long2 cm high by 3 cm wide by 4 cm long

6 15

7 108

8 50

Documents

Anita Straker, Tony Fisher, Rosalyn Hyde, Sue Jennings and ...assets.pearsonschool.com/asset_mgr/current/201214/...For example, Unit G4.2 Measures and mensuration in Tier 4 can be