Exploring maths Tier 2 teacher's book - Pearson Schools

Teacher’s B

ook

Anita Straker, Tony Fisher, Rosalyn Hyde, Sue Jennings and Jonathan Longstaff e 2

ii | Exploring maths Tier 2 Introduction

Exploring mathematics: Tier 2 NC levels 3 and 4 (3a, 4c, 4b)

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Exploring maths Tier 2 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 2 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 2 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 2 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 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 2 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 2 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 2 Introduction | ix

Where can I fi nd…?

Historical and cultural referencesN2.1 The use of negative numbers in 7th century India Class book p.7

S2.1 The origins of Venn and Carroll diagrams Class book p.16

N2.2 Babylonian numerals Class book p.24

N2.2 Counting boards in the Middle Ages Class book p.29

N2.2 Magic squares Class book p.32

N2.2 Sudoku Class book p.33

N2.2 Babylonian multiplication tables Class book p.36

S2.2 Zero Home book p.13

S2.2 The number 2520 Home book p.14

A2.1 Eratosthenes’ sieve Class book p.53

A2.1 Pythagoras and number patterns Class book p.54

N2.3 Egyptian fractions Class book p.65

G2.2 The fi rst use of 360° in Mesapotamia Class book p.112

G2.2 The use of metal protractors on ships Class book p.116

N2.4 Decimal numbers and alternatives to the decimal point Class book p.121

N2.4 Decimal currency Class book p.129

A2.2 Pythagoras’ study of the property of numbers Class book p.140

A2.2 Leonhard Euler and the development of the idea of mappings Class book p.147

A2.2 Descartes and his naming of Cartesian coordinates Class book p.149

A2.2 The fi rst graphing calculator made by Casio in 1985 Class book p.152

A2.2 Ptolemy’s use of geometry to show functions Home book p.49

G2.3 M.C. Escher and patterns from tessellations Class book p.162

N2.5 Harriet Quimby – the fi rst woman to fl y alone across the English channel Class book p.169

N2.5 Taking measurements to solve real problems Class book p.182

N2.5 Use of metric and imperial units Home book p.58

S2.3 William the Conqueror and the fi rst survey of England in 1086 Class book p.189

G2.4 The introduction of the metric system in France in 1790 Class book p.203

G2.4 The world’s tallest buildings Class book p.209

G2.4 The use of sundials to tell the time Class book p.210

G2.4 The world’s fi rst train timetables published by George Bradshaw in 1839 Class book p.213

N2.6 Nine Chapters on the Mathematical Art – written over 2000 years ago in China Class book p.234

S2.4 Roman dice Class book p.243

G2.5 The tangram – a very old Chinese puzzle Class book p.261

G2.5 The fi ve solids named after Plato Class book p.263

A2.3 Al-Khwarizmi and the origins of algebra Class book p.273

A2.3 Johann Bernoulli’s study of functions in depth Class book p.282

A2.3 Niccolo Tartaglia and the invention of round brackets Home book p.95

A2.3 The Enigma machine – used to send coded messages in WW2 Home book p.98

S2.5 William Playfair and the use of the fi rst ever pie chart Class book p.292

N2.7 Mathematical language Class book p.306

x | Exploring maths Tier 2 Introduction

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

N2.2 Lesson 1: Adding on a calculator Teacher’s book p. 30

A2.1 Lesson 4: Using number grids to explore multiples Teacher’s book p. 60

N2.3 Lesson 3: Calculating fractions of numbers using a calculator Teacher’s book p.76

Lesson 4: Decimal place value on a calculator Teacher’s book p. 79

N2.4 Lesson 4: Make money calculations using a calculator Teacher’s book p. 132

Lesson 6: Using a calculator Teacher’s book p. 136

A2.2 Lesson 5: Using interactive graphing software to explore coordinates Teacher’s book p. 154

N2.5 Lesson 2: Converting units of measurement using a calculator Teacher’s book p. 184

Lesson 6: Using a calculator as part of solving a word problem Teacher’s book p. 192

S2.3 Lesson 3: Using an interactive number line to explore intervals Teacher’s book p. 206

G2.4 Lesson 1: Using a simulation to explore decimal place value Teacher’s book p. 220

Lesson 2: Using a simulation of a set of scales to explore weight Teacher’s book p. 222

N2.6 Lesson 4: Calculating fractions of numbers using a calculator Teacher’s book p. 245

S2.4 Lesson 1: Using an interactive number line to explore probability Teacher’s book p. 258

G2.5 Lesson 3: Using a simulation to explore Venn diagrams Teacher’s book p. 275

Lesson 5: Using interactive geometry software to explore drawing 2D shapes on grids

Teacher’s book p. 278

Lesson 6: Using interactive geometry software to explore matching equal sides of 2D shapes

Teacher’s book p. 280

A2.3 Lesson 1: Ordering operations on a calculator Teacher’s book p. 294

R2.1 Lesson 1: Using a simulation of a set of scales to explore weight Teacher’s book p. 352

Lesson 5: Using a simulation to explore Venn diagrams Teacher’s book p. 360

Exploring maths Tier 2 Introduction | xi

Related unitsUnits from diff erent tiers can be aligned if necessary.

For example, Unit N2.1 Properties of numbers in Tier 2 can be used alongside the Tier 1 Unit N1.1 Properties of numbers and the Tier 3 Unit N3.1 Properties of numbers.

Tier 1 Tier 2 Tier 3

N1.1 Properties of numbers N2.1 Properties of numbers N3.1 Properties of numbers

N1.2 Adding and subtracting

N1.3 Multiplying and dividing

N1.4 Mental calculations

N2.2 Whole numbers N3.2 Whole numbers and decimals

N1.5 Fractions

N1.6 Money and decimals

N1.7 Number and measures

N2.3 Fractions, decimals and percentages

N2.4 Decimals

N2.5 Decimals and measures

N3.3 Fractions and percentages


N1.8 Multiplying and dividing 2 N2.6 Fractions, percentages and direct proportion

N3.5 Percentages, ratio and proportion

N1.9 Solving number problems N2.7 Solving number problems N3.6 Solving number problems

A1.1 Patterns and sequences A2.1 Patterns and sequences

A2.2 Sequences, functions and graphs

A3.1 Patterns and sequences

A3.3 Functions and graphs

A3.4 Using algebra

A2.3 Expressions and equations A3.2 Equations and formulae

G1.1 Properties of shapes

G1.5 More properties of shapes

G2.5 Properties of shapes G3.4 Properties of shapes

G1.2 Angles and symmetry G2.2 Angles G3.2 Angles

G3.5 Constructions

G1.2 Angles and symmetry G2.3 Symmetry and refl ection G3.3 Transformations

G1.3 Measures 1

G1.4 Measures 2

G1.6 Measures 3

G2.1 Length, perimeter and area

G2.4 Measures


G3.1 Area and perimeter


S1.1 Graphs and charts 1 S2.1 Graphs, charts and tables S3.1 Grouped data and simple statistics

S1.2 Graphs and charts 2 S2.3 Enquiry 1 S3.3 Enquiry 1

S1.3 Graphs and charts 3 S2.5 Enquiry 2 S3.4 Enquiry 2

S2.2 Probability 1 S3.2 Probability 1

S2.4 Probability 2 S3.5 Probability 2

R2.1 Revision unit 1 R2.1 Revision unit 1 R3.1 Revision unit 1

R2.2 Revision unit 2 R2.2 Revision unit 2 R3.2 Revision unit 2

xii | Exploring maths Tier 2 Introduction

Contents

N2.1 Properties of numbers 21 Square numbers 42 Multiples and divisibility 63 Positive and negative integers 8Mental test 10Check up 11Answers 12

S2.1 Graphs, charts and tables 141 Tally charts, bar charts and pictograms 162 Venn and Carroll diagrams 183 Mode and range 20Check up and resource sheets 22Answers 24

N2.2 Whole numbers 281 Place value, ordering and rounding 302 Mental addition and subtraction 323 Written methods 344 Problem solving 365 Multiplying and dividing by 10, 100 and 1000 386 Multiplication tables 407 Multiplication 428 Division 44Mental test 46Check up and resource sheets 47Answers 48

A2.1 Patterns and sequences 521 Continuing sequences 542 Sequences from rules 563 Multiples 584 Factors 605 Investigating patterns 626 Making general statements 64Check up and resource sheets 66Answers 67

N2.3 Fractions, decimals and percentages 701 Fractions of shapes 722 Equivalent fractions 743 Fractions of quantities 764 Decimal place value 785 Tenths and hundredths 806 Percentages of quantities 82Mental test 84Check up and resource sheets 85Answers 86

G2.1 Length, perimeter and area 901 Length 922 Perimeter 943 Finding areas by counting squares 964 Area of rectangles 98Check up and resource sheets 100Answers 101

S2.2 Probability 1 1041 Probability scale 1062 Probability experiments 108Check up and resource sheets 110Answers 111

G2.2 Angles 1141 Amounts of turn 1162 Measuring angles 1183 Drawing angles 120Check up and resource sheets 122Answers 123

N2.4 Decimals 1241 Decimals and the number line 1262 Ordering decimals 1283 Rounding 1304 Decimals and money 1325 Adding and subtracting decimals 1346 Using a calculator 136Mental test 138Check up and resource sheets 139Answers 140

A2.2 Sequences, functions and graphs 1441 Sequences 1462 Function machines 1483 Finding the rule 1504 Mapping diagrams 1525 Coordinates 1546 Graphs 156Check up and resource sheets 158Answers 160

G2.3 Symmetry and refl ection 1661 Refl ection 1682 Symmetry 1703 Translation 172Check up and resource sheets 174Answers 176

Tier2

Exploring maths Tier 2 Introduction | xiii

N2.5 Decimals and measures 1801 Metres, centimetres and millimetres 1822 Converting units of measurements 1843 Reading scales 1864 Time 1885 Remainders 1906 Word problems 192Mental test 194Check up and resource sheets 195Answers 196

S2.3 Enquiry 1 2001 Collecting and organising data 2022 Venn and Carroll diagrams 2043 Bar charts and pie charts 2064 Bar-line graphs 2085 Mode and range 210Check up and resource sheets 212Answers 214

G2.4 Measures 2181 Converting units 2202 Reading scales 2223 Estimating and measuring 2244 Time intervals 2265 Timetables 228Check up and resource sheets 230Answers 233

N2.6 Fractions, percentages and direct proportion 236

1 Comparing fractions 2382 Fractions, decimals and percentages 2403 Percentages of quantities 2424 Working with fractions 2445 Introducing ratio and proportion 2466 Scaling up and down 248Mental test 250Check up and resource sheets 251Answers 252

S2.4 Probability 2 2561 Probability scale 2582 Probability games 2603 Equally likely outcomes 262Check up and resource sheets 264Answers 266

G2.5 Properties of shapes 2681 Parallel and perpendicular lines 2702 Properties of shapes 2723 Classifying shapes 2744 Angles 2765 Drawing 2D shapes on grids 2786 Making shapes and solids 2807 Nets 282Check up and resource sheets 284Answers 286

A2.3 Expressions and equations 2921 Order of operations 2942 Using brackets 2963 Letters for numbers 2984 Collecting like terms 3005 Substitution 3026 Inverse operations 3047 Solving simple equations 306Check up and resource sheets 308Answers 309

S2.5 Enquiry 2 3121 Collecting data 3142 Representing data 3163 Interpreting data 3184 Drawing conclusions and writing a report 3205 Line graphs 3226 More line graphs 324Check up and resource sheets 326Answers 328

N2.7 Solving number problems 3341 Solving problems with whole numbers 3362 Solving problems with number sequences 3383 Solving problems with decimals 3404 Solving problems with fractions 342Mental test 344Check up and resource sheets 345Answers 346

R2.1 Revision unit 1 3501 Measures and measuring scales 3522 Solving number problems 3543 Sequences and patterns 3564 Perimeter and area 3585 Drawing and interpretinggraphs and charts 360Mental test 362Answers 363

R2.2 Revision unit 2 3661 Inverse operations 3682 Equivalent fractions, decimals and percentages 3703 Expressions and equations 3724 Symmetry and refl ection 3745 Probability 376Mental test 378Check up and resources 379Answers 381

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 purple provide useful consolidation and enrichment opportunities. These should be included wherever possible.

Properties of numbers

Previous learningBefore they start, pupils should be able to:

recall addition and subtraction facts for each number to 20

recognise odd and even numbers

recognise multiples of 2, 5 and 10.

Objectives based on NC levels 3 and 4 (mainly level 4)In this unit, pupils learn to:

understand practical problems in familiar and unfamiliar contexts

identify and obtain necessary information to tackle a problem

represent problems using words or diagrams

look for and visualise patterns

manipulate numbers

represent, explain, compare and discuss methods and results

begin to generalise

communicate solutions to practical problems, giving explanations

and to:

identify squares of numbers to 12 � 12

recognise multiples and use simple tests of divisibility

recognise negative numbers in practical contexts

solve problems requiring calculation with temperature.

Objectives in colour lay the groundwork for Functional Skills at level 1.

Lessons 1 Square numbers

2 Multiples and divisibility

3 Positive and negative integers

About this unit Pupils’ confi dence in responding to numbers in everyday situations is strengthened by having a good ‘feel for number’. This means being aware of signifi cant relationships between numbers and knowing at a glance which properties they possess and which they do not.

In this unit pupils learn to use patterns to help them to recall number facts and recognise number properties.

Assessment This unit includes:

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

a self-assessment section (N2.1 How well are you doing? class book p. 11);

a set of questions to replace or supplement questions in the exercises or homework tasks, or to use as an informal test (N2.1 Check up, CD-ROM).

Common errors and misconceptions

Look out for pupils who:

have diffi culty in remembering number facts, such as addition and subtraction facts to 20, or multiplication facts to 10 � 10;

confuse squaring and doubling;

lack confi dence in working in the negative part of the number line, and who think that �3 � 5 � 8, or that �3 � 5 � 8.

N2.1

2 | N2.1 Properties of numbers

Key terms and notation problem, solution, method, pattern, relationship, order, solve, explain, represent

calculate, calculation, calculator, add, subtract, multiply, divide, divide exactly, divisible, sum, total, diff erence, product, greater than (�), less than (�), value

positive, negative, integer, odd, even, multiple, square, perfect square, square root, digit sum

temperature, degrees Celsius (°C)

Practical resources calculators for pupils individual whiteboards

Exploring maths Tier 2 teacher’s bookN2.1 Mental test, p. 10Answers for Unit N2.1, pp. 12–13

Tier 2 CD-ROMPowerPoint fi les N2.1 Slides for lessons 1 to 3Tools and prepared toolsheets Calculator tool Toolsheets 3.1 and 3.2 Number jumps toolTier 2 programs Number grids Number sorter

Tier 2 class bookN2.1, pp. 1–12N2.1 How well are you doing? p. 11

Tier 2 home bookN2.1, pp. 1–3

Tier 2 CD-ROMN2.1 Check up

Useful websites Multiples (an NRICH package of problems and puzzles)nrich.maths.org/public/viewer.php?obj_id�5530

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

Multi sequencerwww.amblesideprimary.com/ambleweb/numeracy.html

N2.1 Properties of numbers | 3

Learning points

When a number is multiplied by itself the result is a square number.

A square number can be represented by dots arranged in the shape of a square.

81 is the square of 9. It can be written as 92.

1 Square numbers

Starter

Main activity

Say that this unit is about special properties of numbers. This lesson is about square numbers.

Show slide 1.1. Point to the 3 by 3 pattern of dots.

How many dots are there? How are they arranged?

Write 9 in the box below the dots and 3 � 3 in the box below. Continue up to 6 � 6.

Point to the row of numbers 1, 4, 9, 16, 25, 36. Say that these are called square numbers � each is the result of multiplying a number by itself and can be represented by dots arranged in a square shape.

Say that there is a special way of reading and writing square numbers. Point to 12 and 22 saying ‘one squared, two squared’. Ask pupils to write on their whiteboards 32, 42, 52 and 62. Enter these in the table.

Launch Number grids. Choose a multiplication grid with ten rows and columns, a start number of 0 and a step of 1. Say that you will highlight the fi rst six numbers in the sequence of square numbers. Click to highlight 1, 4, 9, 16, 25, 36. Point out that 16 or 4 � 4 lies in the fourth column and the fourth row.

What are the next numbers in the sequence? [49, 64, 81, 100]

How would we write seven squared?

What number do you square to get 81?

What is the square of ten?

What is the next square number after 100? And after that?

Ask the class to chant the sequence of square numbers. Point to the numbers as pupils say them. Click on ‘Hide products’ and chant again.

If you prefer, use slide 1.2 instead of Number grids. You can white out the slide for the last chant by pressing W on the keyboard, or by right-clicking then choosing Screen � White screen. Press W or right-click again to restore the slide.

Use the Calculator tool to show pupils how to use the x2 key on their calculators.

What is six squared? What is two squared?

What is the sum of six squared and two squared? [40]

Demonstrate the key sequence 6 x2 + 2 x2

= .

Slide 1.1

Slide 1.2

SIM

TO



Homework

Review

Explain that 40 is the sum of two square numbers, and that so is 13. Ask pupils to discuss in pairs what the two squares that sum to 13 might be [2 and 3].

Which other numbers up to 30 are the sum of two diff erent squares?

Ask the pairs to investigate this. Discuss how to work systematically, e.g. add 12 to each of 22, 32, 42 and 52. Since 12 added to 62 is too big, now add 22 to each of 32, 42 and 52, and so on.

Why don’t we need to try adding two squared to one squared?[same as 12 � 22]

Leave the pairs to work for a few more minutes, then gather the complete set of results: 5, 10, 13, 17, 20, 25, 26, 29.

How do we know that we have found all the possibilities?

Establish that because they have worked systematically through all the possible pairs they must have checked them all.

Say that a mystery number, when squared, has the answer 225.

Is the mystery number less than 10? [no – its square would be less than 100]

Is the mystery number more than 20?

Confi rm with the Calculator tool that 202 � 400, so the number is less than 20.

Could the mystery number be even?

Establish that the product of two even numbers is always even, so the mystery number is odd (and lies between 10 and 20). Write 11, 13, 15, 17 and 19 on the board.

Which of these numbers could it be? Explain why.

Draw out that 15 is the only number which, when multiplied by itself, will result in a number with a units digit of 5. Confi rm using the Calculator tool.

Sum up with the points on slide 1.3.

Ask pupils to do N2.1 Task 1 in the home book (p. 1).

Select individual work from N2.1 Exercise 1 in the class book (p. 2).

TO

Slide 1.3


Learning points

A multiple of 5 is a number that divides exactly by 5.

A number is a multiple of 2 if its last digit is even.

A number is a multiple of 3 if its digit sum is a multiple of 3.

A number is a multiple of 4 if half of it is even.

A number is a multiple of 5 if its last digit is 0 or 5.

A number is a multiple of 10 if its last digit is 0.

2 Multiples and divisibility

Starter

Main activity

Say that this lesson is about multiples. Remind pupils that a multiple of a number divides exactly by the number.

Draw a large square box on the board. Ask pupils to suggest some numbers below 60. If they are multiples of 5, write them in the box. If not, write them outside.

Once there are at least three numbers in the box, ask:

What is my rule for putting numbers in the box?

Continue asking for numbers until pupils recognise the rule. Repeat with multiples of 7, multiples of 2 and multiples of 11. Invite pupils who think that they know the rule to the board to write another number in the box.

Write 96 on the board.

Is this number odd or even? How do you know?

Point out that the last digit is even, so the whole number is even. An even number is divisible by 2, i.e. divides exactly by 2 with no remainder. It is also a multiple of 2. Say a few numbers and ask pupils to say if the number is divisible by 2.

Return to 96.

Is this number divisible by 3?

Tell the class that there is a quick way to fi nd out by adding up all the digits, i.e. 9 � 6 � 15. Explain that 15 is called the digit sum. Because it is a multiple of 3, the number 96 is a multiple of 3. Test a few more numbers for divisibility by 3.

Return to 96.

Is this number divisible by 4?

Say that there is an easy way to fi nd out. We know that 96 is even so it divides exactly by 2, so fi nd half of 96. Point out that because 48 is even, 96 can be divided exactly by 2 and then exactly by 2 again. So 96 is divisible by 4. Test a few more numbers for divisibility by 4.

5527

1 23

59

20

35



Review

Slide 2.1

Slide 2.2

Homework

SIM

Return to 96.

Is this number divisible by 5 or by 10?

Confi rm that it is not, since it does not end in 5 or 0.

Launch Number sorter. Choose a two-way Carroll diagram and then ‘Is a multiple of 3 / Is a multiple of 4’. Involve pupils in dragging the numbers to the correct part of the diagram. Ask questions such as:

How do you know that 42 is a multiple of 3?

How do you know that 42 is not a multiple of 4?

How do you know that 24 is divisible by 3 and divisible by 4?

Write 96 on the board again.

Is 96 divisible by 6?

Explain that all multiples of 6 divide exactly by 2 and also by 3. We know that 96 divides exactly by 2 because it is even. We also know that it divides exactly by 3 because its digit sum of 15 is a multiple of 3. So 96 is divisible by 6.

Launch Number sorter again. This time choose a two-way Venn diagram, then ‘Is a multiple of 3 / Is a multiple of 6’. Involve pupils in dragging the numbers to the correct region. Ask, for example:

How do you know that 39 is a multiple of 3?

How do you know that 39 is not a multiple of 6?

Sum up with the points on slides 2.1 and 2.2.



Learning points

The negative number 6 is called ‘negative 6’ and written as �6.

Numbers get less as you count back along the number line beyond nought or zero, so �10 is less than �5.

Six degrees below zero is minus six degrees Celsius (�6 °C).

�10 °C is a lower temperature than �5 °C.

Always include the units when you write a temperature.

3 Positive and negative integers

Starter

Main activity

Say that this lesson is about positive and negative numbers.

Show the fi rst number line on Toolsheet 3.1. Point at random to divisions on the line and ask pupils to say the number. As they do so, click just below the division on the line to reveal the number.

Explain that the line shows the integers from �5 to 5. Integers are positive or negative whole numbers and zero. The negative integers are called ‘negative one’, ‘negative two’, and so on. Say that we usually don’t write �2 for ‘positive two’ but write 2.

Indicate the second line. Again, point at random to divisions on the line and ask pupils to say the number. Click to reveal the number. Ask questions like:

Tell me a number that is less than �20… that is more than �30…that lies between �20 and 10.

Record answers on the board using the � and � signs, e.g.

�60 � �20 �9 � �30 �20 � 0 � 10

Show the number line on Toolsheet 3.2. Explain that, just like a number line with only positive numbers, it is possible to add and subtract by counting steps along the line.

What is 5 more than �2? [record as �2 � 5 � 3]

If you wish, click on the Number jumps tool to show the jump from �2 to 3. Click on the line at the start point, drag the blue line in the direction of the jump, then click on the line at the end point.

What is 6 less than 4? [record as 4 � 6 � �2]

What is the diff erence between 5 and �3?

Stress that a diff erence is measured by the number of steps or the distance between the numbers, and can be recorded as 5 � (�3) � 8.

Which pairs of numbers have a diff erence of 4?

TO

TO

�3

3 5

0 5


Review

Slide 3.1

Slide 3.2

Slide 3.3

Slide 3.4

Homework

Show slide 3.1. Say that the thermometers show the temperatures in Leeds and Barcelona on the same day in winter. Point out the °C (degrees Celsius) abbreviation. Say that as you move down the scale the temperature is falling.

What is the temperature in Leeds? In Barcelona?

How much colder is it in Leeds than Barcelona?

Point out �5 °C on each scale. Invite a pupil to show �7 °C. Explain that this is read as ‘minus seven degrees Celsius’ not ‘negative seven degrees Celsius’ and that it means that the temperature is seven degrees Celsius below zero.

The temperature falls by 5 degrees in each city.What are the temperatures now?

On another day, the temperature in Leeds is 3 °C. Barcelona is 7 degrees colder. What is the temperature in Barcelona?

Remind pupils that they should include the units when they write a temperature.

Show slide 3.2 and work through the questions.

Show slide 3.3. Ask pupils to write the temperatures in order on their whiteboards from hottest to coldest. Use the temperatures to ask questions such as:

What is the diff erence between 12 °C and 37 °C?Between 12 °C and �12 °C?Between �12 °C and �2 °C?

The temperature is �2 °C.How many degrees must it rise to reach 12 °C?

The temperature falls from 37 °C to �12 °C.How many degrees has it fallen?

After each question, ask pupils how they worked out the answer.

Sum up the lesson with the points on slide 3.4.

Round off the unit by referring again to the objectives. Suggest that pupils fi nd time to try the self-assessment problems in N2.1 How well are you doing? in the class book (p. 11).




N2.1 Mental test

Read each question aloud twice.

Allow a suitable pause for pupils to write answers.

1 Write the next odd number after twenty-nine. 1998 KS2

2 Which is the lowest of these temperatures? 2005 KS2[Write on board: 2 °C �5 °C 5 °C 0 °C �1 °C]

3 What is three times three added to four times four? 2003 KS2

4 Write three even numbers that add to twenty. 2004 KS3

5 What temperature is ten degrees lower than seven degrees Celsius? 2006 KS2

6 What number is nine squared? 1997 KS3

7 Write down an even number that is a multiple of seven. 2005 KS3

8 What is the smallest whole number that is divisible by fi ve and by three? 2004 KS3

9 The temperature on Monday was minus eight degrees Celsius. 2005 PT[Write on the board �8 °C.]On Tuesday, it was ten degrees higher.What was the temperature on Tuesday?

10 What is the next square number after thirty-six? 2005 PT

11 Write a number that is a multiple of ten and also a multiple of twelve. 2006 PT

12 What number multiplied by eight equals forty-eight? 2005 KS3

Key:KS2 Key Stage 2 Mental Test PT Progress Test KS3 Key Stage 3 Mental Test Questions 1 to 3 are at level 3. Questions 4 to 12 are at level 4.

Answers 1 31 2 �5 °C 3 25 4 One of these sets of three numbers:

2, 2, 16; 2, 4, 14; 2, 6, 12; 2, 8, 10; 4, 4, 12; 4, 6, 10; 4, 8, 8; 6, 6, 8

5 �3 °C 6 81 7 e.g. 14, 28 8 15

9 2 °C 10 49 11 e.g. 60, 120 12 6


N2.1 Check up

1.2 | Tier 2 resource sheets | N2.1 Properties of numbers © Pearson Education 2008

N2.1 Check up [continued]

Properties of numbers (calculator allowed)

5 2003 Progress Test level 4

The 4th square number is 16.

What is the 5th square number?


Here is a grid with some numbers shaded. 1 2 3 4

5 6 7 8

9 10 11 12The grid continues. Will the number 35 be shaded?Write Yes or No.Explain your answer.


The diagram shows what is above and below sea level.

a What is about 50 m lower than the bird?

b An octopus is at about �40 m.About how many metres higher is the diver than the octopus?

30

20

10

0

�10

�20

�30

bird

kite

butterfly

boat

diver

fish

eel

Check up N2.1Write your answers in your book.

Properties of numbers (no calculator)

1 1995 level 3

Ali drew a picture to show what there is above and below the sea at Aber.

The anchor is at about �40 m.

a What is at about �10 m?

b What is at about �10 m?

c What is about 30 m higher than the chest?

2 2005 KS2 level 3

Which three of these numbers add to make a multiple of 10?

11 12 13 14 15 16 17 18 19

3 1997 KS2 level 3

One of these numbers when multiplied by itself gives the answer 49.Which number is it?

2 3 4 5 6 7 8 9

4 2002 KS2 level 4

Write all the multiples of 8 in this list of numbers.

18 32 56 68 72

© Pearson Education 2008 Tier 2 resource sheets | N2.1 Properties of numbers | 1.1

�20 m

0 m

�20 m

�40 mchest

anchor

fish

diver

boatbird

hotel


N2.1 Answers

Class book

Exercise 11 a 100 b 225

c 400 d 1225

2 a 8 � 8 � 64 b 9 � 9 � 81

c 11 � 11 � 121 d 12 � 12 � 144

e 14 � 14 � 196 f 22 � 22 � 484

3 a 50 b 72 c 98

d 45 e 34 f 21

g 4 h 100

4 Other solutions may be possible.

a 25 � 16 � 9 b 50 � 25 � 25

c 17 � 16 � 1 d 29 � 25 � 4

e 85 � 49 � 36 f 52 � 36 � 16

g 61 � 25 � 36 h 125 � 100 � 25

i 3 � 4 � 1 j 16 � 25 � 9

k 20 � 36 � 16 l 15 � 16 � 1

m 36 � 100 � 64 n 21 � 25 � 4

o 64 � 100 � 36 p 77 � 81 � 4

5 1 � 12

1 � 3 � 22

1 � 3 � 5 � 32

1 � 3 � 5 � 7 � 42

1 � 3 � 5 � 7 � 9 � 52

1 � 3 � 5 � … � 19 � 102 � 100

Extension problem

6 There are 8 ways to write 150 as the sum of four squares:

144 � 4 � 1 � 1121 � 16 � 9 � 4100 � 25 � 16 � 981 � 64 � 4 � 181 � 49 � 16 � 464 � 49 � 36 � 164 � 36 � 25 � 2549 � 49 � 36 � 16

Exercise 21 a 10, 20, 30 b 3, 6, 9 c 6, 12, 18

d 9, 18, 27 e 21, 42, 63

2 a True b True c False

d True e False f True

g False h True

3 18, 56, 72

4 30, 45, 60

5 a No b Yes c No

d Yes e Yes f No

6 54 � 36 or 34 � 56

7 a 15 b 36 c 63

d 25 e 22 f 36 or 72

Extension problem

8 41 49 47 43 31 37

35 5545

3040

50 42

3648

5432 34

444652 38

5853 59

56

6033 51 57

39

Numbers from 30 to 60

Multiples of 5 Multiples of 3

Multiples of 2

Exercise 31 a �2 °C, �7 °C, �1 °C, �5 °C

b �1 °C

c �7 °C �5 °C �2 °C �1 °C 3 °C

2 The temperature rose by 10 degrees.

3 5 degrees colder

4City Temperature diff erence (degrees)

Belfast 9

Liverpool 10

Cardiff 10

Newcastle 9

London 10

Plymouth 11

York 9


5 11 cm

6 a 6 degrees b 6 degrees c 12 degrees

7 a �3 b 1 c 0 d �1

e 4 f �5 g �4 h �5

8 a �4 � 2 � �2 b �3 � 4 � 1

c 5 � 7 � �2 d �2 � 4 � �6

e 2 � 4 � �2 f �1 � 3 � 2

Extension problem

9�3 4 �1

2 0 �2

1 �4 3

How well are you doing?1 28 and 35

2 A �19 °C

B 16 degrees colder

C �22 °C

3 25

4 A 4, 16, 36 or 64

B 1, 9, 25, 49 or 81

C Any even number that is not a square number

D Any odd number that is not a square number

5 a 11 b 36

6 a 5 °C b �9 °C, �3 °C, 0 °C, 6 °C

7 a Any multiple of 10 that does not divide exactly by 20, e.g. 10, 30, 50, 70, 90, 110, …

b Any multiple of 20 must also be a multiple of 10, so it is not possible to put a number in section B.

Home book

Task 11

Numbers 3 6 7 5 4 9 11 15 12

Squares 9 36 49 25 16 81 121 225 144

2 a 32 � 42 � 52 b 62 � 82 � 102

c 92 � 122 � 152 d 52 � 122 � 132

3 For example: 150 � 1 � 49 � 100150 � 4 � 25 � 121150 � 25 � 25 � 100

Task 21 a 5, 10, 25 b 7, 14, 21

c 8, 16, 24 d 31, 62, 93

2 a 40, 50, 60 b 40, 45, 50, 60, 75

c 45, 63, 72, 81 d 16, 24, 32, 40, 72

e 24, 60, 72

3 41 43 46 47 49 50

31 34 35 37 38

30 39

45

36

48

44

32

4042

33

Numbers from 30 to 50

Multiples of 3 Multiples of 4

Task 3

1 �2 °C

2 A fall of 9 degrees

3 a �6 °C, �4 °C, 2 °C, 4 °C

b i 10 degrees ii 2 degreesiii 8 degrees iv 8 degrees

4 50 degrees

CD-ROM

Check up1 a Bird b Diver c Fish

2 Any three numbers from 11 to 19 inclusive that sum to a multiple of 10, e.g. 11, 12, 17

3 7

4 32, 56, 72

5 25

6 No. 35 will not be shaded because all the shaded numbers are even.

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Exploring maths Tier 2 teacher's book - Pearson Schools