Mathematics
CAPS
Grade
Teacher’s Guide
Zonia C Jooste • Karen Press • Moeneba Slamang
Lindi van Deventer
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Study & Master
Mathematics
Grade 4
Teacher’s Guide
Zonia C Jooste • Karen Press • Moeneba Slamang • Lindi van Deventer
SM_Maths_G4_TG_TP_Eng.indd
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Cambridge University Press
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© Cambridge University Press 2012
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First published 2012
ISBN 978-1-107-28400-5
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Contents
1 Introduction v
2Planning and organising
your Mathematics teaching xi
3 Lesson plans xv
Term 1
Unit 1 Count, order and compare
numbers 2
Unit 2 Addition and subtraction
facts up to 20 7
Unit 3 Add and subtract multiples
of 10 8
Unit 4 Multiplication and division 11
Unit 5 Problem-solving 14
Unit 6 Write and solve number
sentences 16
Unit 7 Reverse operations in number
sentences 18
Unit 8 Use number sentences to
describe and solve problems 21
Unit 9 Solve number sentences 23
Unit 10 Patterns in number sentences 25
Assessment task 1 Number sentences 28
Assessment task 1 Number sentences:
solutions 29
Unit 11 Count, order and compare
numbers (2) 30
Unit 12 Place value and representing
numbers 33
Unit 13 Place value 35
Unit 14 Represent and compare
numbers 37
Unit 15 Swap and regroup numbers 38
Assessment task 2 Numbers and place
value 41
Assessment task 2 Numbers and place
value: solutions 42
Unit 16 Revision: find connections 43
Unit 17 Number sentences and
problem-solving 44
Unit 18 Strategies for adding and
subtracting 46
Unit 19 Use different methods and
operations 48
Unit 20 More strategies for adding
and subtracting 49
Unit 21 Add and subtract with 3-digit
numbers 51
Unit 22 More strategies for adding
and subtracting (2) 52
Unit 23 Calculations with 4-digit
numbers 54
Assessment task 3 Addition and
subtraction 56
Assessment task 3 Addition and
subtraction: solutions 57
Unit 24 Patterns in counting sequences 59
Unit 25 Number grids and patterns 61
Unit 26 Number groups and patterns 63
Assessment task 4 Numeric patterns 67
Assessment task 4 Numeric patterns:
solutions 68
Unit 27 Multiplication by grouping
and repeated addition 69
Math G4 TG.indb 3
Unit 28 Know the multiplication
tables 73
Unit 29 Round off and estimate in
real life 75
Unit 30 Use grouping and sharing 77
Unit 31 Division facts and rules 80
Assessment task 5 Multiplication and
division 84
Assessment task 5 Multiplication and
division: solutions 85
Term 2
Unit 1 Count and order 132
Unit 2 Compare and represent
numbers 136
Unit 3 Place value 138
Unit 4 Estimate and round off 139
Assessment task 1 Counting, place value
and estimation 142
Assessment task 1 Counting, place value
and estimation: solutions 144
Unit 32 Revision of Grade 3 work 86
Unit 33 24-hour time 87
Unit 34 Read time in 5-minute
intervals 89
Revision and consolidation 90
Unit 35 Read calendars 90
Unit 36 Read timetables 92
Unit 37 History of time 93
Revision 93
Unit 5 Add and subtract multiples
of 10 145
Unit 6 Strategies for adding and
subtracting 146
Unit 7 More strategies for adding and
subtracting 148
Unit 8 Add and subtract with 3- and
4-digit numbers 150
Use tally marks 95
Draw up a tally table 96
Show data in pictographs 98
Show data in bar graphs 99
Project 100
Unit 42 Explain data 100
Unit 43 Data from pictographs 101
Unit 44 Data from pie charts 102
Unit 45 Data from bar graphs 103
Unit 46 Draw your own bar graph 103
Revision 104
Remedial activities 104
Extension activities 105
Self-assessment 105
Unit 47 Different shapes 106
Unit 48 Triangles and quadrilaterals 108
Unit 49 Pentagons and hexagons 110
Unit 50 Put shapes together 111
Revision 111
Revision activity 112
Remedial activities 112
Extension activities 112
Additional class activity 113
Assessment task 7 Properties of
2-D shapes 113
Assignment 113
Assessment task 7 Properties of
2-D shapes: solutions 114
Assignment 114
Self-assessment 114
Unit 51 Equal sharing and multiples 115
Unit 52 Multiplication and division
strategies 116
Unit 53 Basic multiplication and
division facts 117
Unit 54 Multiplication and division
flow diagrams 119
Unit 55 Number rules for multiplication
and division 120
Unit 55 Ratio and rate 123
Assessment task 8 Multiplication and
division 126
Assessment task 8 Multiplication and
division: solutions 128
Order and compare fractions 156
Represent fractions 157
Equal sharing 160
Calculations with fractions 164
Equivalent fractions 166
Count and calculate fractions 168
Assessment task 3 Common fractions 172
Assessment task 3 Common fractions:
solutions 173
Unit 15 Revision of Grade 3 work 174
Unit 16 Work with centimetres (cm) and
millimetres (mm) 175
Unit 17 Tricky measurements 176
Unit 18 Understand units of
measurement 176
Unit 19 Convert between kilometres,
metres and millimetres 177
Unit 20 Convert between centimetres
and metres 179
Unit 21 Convert between millimetres
and centimetres 180
Unit 22 Round off measurements 180
Unit 23 Problem-solving with distance
and length 181
Revision and consolidation 182
Revision 183
Assessment task 4 Common
fractions 184
Assessment task 4 Common fractions:
solutions 185
Unit 24 Basic multiplication facts 186
Unit 25 Multiplication strategies 187
Unit 26 Problem-solving with
multiplication 189
Unit 27 Multiplication and estimation 190
Unit 28 Patterns in multiplication 192
Unit 29 More multiplication methods 192
Assessment task 5 Multiplication 194
Assessment task 5 Multiplication:
solutions 196
Unit 30 Flat and curved surfaces 197
Assessment task 6 Time 94
Assessment task 6 Time: solutions 95
Unit 38
Unit 39
Unit 40
Unit 41
Assessment task 2 Addition and
subtraction 152
Assessment task 2 Addition and
subtraction: solutions 154
Unit 9
Unit 10
Unit 11
Unit 12
Unit 13
Unit 14
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Unit 31 Shapes and faces of
3-D objects 199
Unit 32 Straight, flat faces: polyhedra 200
Investigation 202
Revision 202
Remedial activities 202
Extension activity 203
Project 203
Unit 33 Explore geometric patterns 204
Unit 34 Identify and extend patterns 206
Unit 35 Extend patterns 207
Unit 36 Input and output numbers
(values) 209
Assessment task 6 Patterns 211
Assessment task 6 Patterns:
solutions 212
Unit 37 What is symmetry? 213
Investigation 214
Revision 215
Remedial activities 215
Extension activities 215
Unit 38 Round off to add and
to subtract 216
Unit 39 Different ways to add 216
Unit 40 Different ways to subtract 218
Unit 41 Check addition and subtraction
calculations 220
Unit 42 Solve story problems 222
Unit 43 Division with and without
remainders 223
Unit 44 Division with remainders 225
Unit 45 Division with 3-digit numbers
and remainders 226
Unit 46 Problem-solving with
division 227
Assessment task 7 Whole numbers
(division) 230
Assessment task 7 Whole numbers
(division): solutions 232
Term 3
Unit 1 What do you remember? 236
Unit 2 Measure capacity and
volume 237
Unit 3 Understand volume and
capacity 238
Unit 4 Estimate and round off 239
Unit 5 Calculations with litres
and millilitres 241
Unit 6 Calculate capacity with
fractions 242
Revision 242
Assessment task 1 Measurement 243
Assessment task 1 Measurement:
solutions 244
Recognise fraction parts 245
Fractions of whole numbers 246
Equivalent fractions 248
Equal sharing and
problem-solving 250
Unit 11 Count, order and calculate
with fractions 252
Assessment task 2 Common
fractions 254
Assessment task 2 Common fractions:
solutions 256
Unit 7
Unit 8
Unit 9
Unit 10
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Unit 12 Rules for operations 257
Unit 13 Count, compare, represent
numbers and place values 260
Assessment task 3 Counting and
place value 264
Assessment task 3 Counting and
place value: solutions 266
Unit 14 Addition and subtraction
facts 267
Investigation 269
Unit 15 Problem-solving 269
Unit 16 Double, halve and round off for
estimations and calculations 272
Unit 17 Different strategies for
calculations 274
Unit 18 Side views and top views 277
Investigation 278
Unit 19 Side views and plan views 279
Remedial activities 280
Extension activities 280
Project 280
Unit 20 Sort 2-D shapes 282
Unit 21 Investigate circles 283
Unit 22 Investigate polygons 284
Investigation 285
Unit 23 Patterns and pictures with
2-D shapes 285
Remedial activities 287
Extension activities 287
Unit 24 Use tally marks 288
Unit 25 Show data on graphs 289
Unit 26 Explain data 290
Unit 27 More graphs 291
Project 292
Remedial activities 293
Extension activities 293
Unit 28 Patterns in number grids 294
Unit 29 Finding rules 298
Unit 30 Rules for number patterns 299
Assessment task 4 Number patterns 301
Assessment task 4 Number patterns:
solutions 302
Unit 31 Quick calculations 303
Unit 32 Count, order and compare
numbers and place value 305
Unit 33 Problem-solve with
whole numbers 306
Assessment task 4 Number patterns 307
Assessment task 4 Number patterns:
solutions 308
Unit 34 Multiplication strategies 310
Unit 35 Basic multiplication facts 312
Unit 36 Round off and solve
problems 312
Unit 37 Write number sentences 315
Unit 38 Balance and inspect
number sentences 316
Unit 39 Equations and problemsolving 318
Unit 40 Make new shapes 320
Unit 41 Tangrams 321
Revision 322
Remedial activities 323
Extension activities 323
Assignment 323
Project 323
Term 4
Unit 1 Revise rules for working with
numbers 326
Unit 2 Represent numbers and place
value 328
Unit 3 Problem-solving 331
Unit 4 Inverse operations 332
Unit 5 More calculations 334
Unit 6 Use estimating and
problem-solving 336
Assessment task 1 Whole number addition
and subtraction 340
Assessment task 1 Whole number addition
and subtraction: solutions 242
Revision 343
Estimate 344
More addition and subtraction 345
More multiplication
and division 346
Unit 11 Problem-solving 349
Unit 12 Recognise and compare
3-D objects 352
Unit 13 Faces and models of
3-D objects 352
Unit 14 Statements about 3-D objects 353
Remedial activities 354
Extension activities 354
Unit 15 Order and compare fractions 355
Unit 16 Calculate with fractions 356
Unit 17 Fractions of whole numbers 358
Unit 18 Problem-solving with
fractions 359
Revision 362
Unit 19 Basic division facts 364
Unit 20 Divide by 10 and 100 365
Unit 21 Strategies for division 367
Assessment task 2 Division 369
Assessment task 2 Division: solutions 370
Unit 22 Perimeter 371
Unit 23 Area 374
Unit 24 Volume 375
Revision and consolidation 376
Project 376
Rubric to assess the project 377
Unit 25 Work with grids 378
Unit 26 Grids on maps 379
Remedial activities 379
Extension activities 379
Unit 27 Tessellations 380
Unit 28 Describe patterns 381
Remedial activities 381
Extension activities 382
Unit 29 Geometric patterns 382
Unit 30 Growing patterns 385
Unit 31 Use place value to add
and subtract 387
Unit 32 Use 10-strips to add
and subtract 389
Unit 33 Probability 391
Unit 34 Experiments and actual
outcomes 392
Revision 394
Remedial activities 395
Extension activities 395
Unit 7
Unit 8
Unit 9
Unit 10
5
6
Resources 397
Documents 435
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1. Introduction
The amended National Curriculum and Assessment Policy Statements for
Grades R–12 came into effect in January 2012. They replaced the National
Curriculum Statements Grades R–9 (2002) and the National Curriculum
Statements Grades 10–12 (2004). The National Curriculum and Assessment
Policy Statement (CAPS) for Intermediate Phase Mathematics (Grades 4–6)
replaces the Subject Statements, Learning Programme Guidelines and Subject
Assessment Guidelines that were used before then.
The instructional time for subjects in the Intermediate Phase is given in the table
below.
Table 1 Instructional time for Intermediate Phase subjects
Subject
Time allocation
per week (hours)
Home Language
6
First Additional Language
5
Mathematics
6
Science and Technology
3, 5
Social Sciences
3
Life Skills:
4
Creative Arts
1, 5
Physical Education
1, 5
Religion Studies
1
The Mathematics curriculum: aims and skills
The aims of the National Curriculum for Mathematics, as set out in the CAPS,
are to develop the following qualities in learners:
• a critical awareness of how mathematical relationships are used in social,
environmental, cultural and economic relations
• confidence and competence to deal with any mathematical situation without
being hindered by a fear of mathematics
• a spirit of curiosity and a love for mathematics
• an appreciation of the beauty and elegance of mathematics
• recognition that mathematics is a creative part of human activity
• deep understanding of concepts needed to make sense of mathematics
• acquisition of specific knowledge and skills necessary for:
–the application of mathematics to physical, social and mathematical
problems
–the study of related subject matter (e.g. other subjects)
–the further study of mathematics.
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The CAPS lists the following specific skills that learners must acquire to develop
their essential mathematical skills:
• correct use of the language of mathematics
• ability to understand and use number vocabulary, number concept and
calculation and application skills
• ability to listen, communicate, think, reason logically and apply the
mathematical knowledge gained
• ability to investigate, analyse, represent and interpret information
• ability to pose and solve problems
• awareness of the important role that mathematics plays in real-life situations,
including the personal development of the learner.
Problem-solving and mathematics
This Mathematics course is designed to encourage learner-centred and activitybased learning through problem-solving, an approach that should be applied
throughout the course.
Problem-solving is one of the unique features of learning and teaching
mathematics. Learners should be able to:
• make sense of problems
• analyse, synthesise (create), determine and execute solution strategies
• estimate, confirm (validate) and interpret the solutions appropriate to the
context.
Problem-solving does not necessarily imply solving word problems. Word
problems could be examples of extending problems that test learners’
mathematical knowledge. These problems involve the use and validation of
techniques learnt in all the content areas of Intermediate Phase Mathematics.
In a problem-solving situation, it may be highly unlikely that learners have had
previous instruction on how to tackle the problems they are facing. Learners
should invent their own solution strategies using different problem-solving
procedures. There are no ready-made recipes or blueprints for searching for and
finding problem-solving solutions.
Solutions and strategies are not as obvious in problem-solving situations as they
are in word problems. In word problems, it is easy to identify which operations
to apply to solve the problem. Problem-solving is not a topic that can be learnt.
It is a process in which learners can explore situations by applying different
skills. Learners construct new meaning by building on previous knowledge and
experiences in an active, cooperative environment.
Learners do not learn problem-solving techniques by memorising rules or
consulting checklists. You should raise consistent awareness of the different
techniques suitable for different problem-solving situations. You could give the
problem as a homework task, group activity or introduction to new concepts
(knowledge), or deal with it in an oral or written situation that applies to all
learners without gender or culture bias. Throughout this course, learners are
presented with different possible strategies for solving problems, and are
encouraged to choose or develop strategies that work most effectively in given
contexts.
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Mathematics Grade 4 Teacher’s Guide
introduction
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Keep in mind that it is important to acknowledge that people are fundamentally
different, and experience problem situations differently. Expect learners to apply
a wide range of different methods and ideas in the problem-solving process.
Monitor learner groups carefully and encourage discussions and arguments while
questioning learners about their progress. An important aspect of the learning
of mathematics involves creative initiatives by learners to use the strategies
and methods they know when they are confronted with new problems, and to
experiment with different approaches to solving the problems.
Lead a class discussion on making mistakes, working well together, useful
steps to keep in mind during a lesson, and enjoyment as an important part of
mathematics activity. Discuss each aspect and ask learners questions such as:
How do you feel when you have made a mistake? Why do you feel this way?
Explain, for example, what it means to work towards a common goal. Take note
of learners who seem reluctant to attempt problems that they find difficult, and
help them to use their existing knowledge to solve new problems.
Inclusivity in the Mathematics classroom
The ultimate aim of an inclusive school is to contribute towards the development
of an inclusive society, where diversity is respected and used as a tool for
building a stronger community.
Inclusive education is a process in which barriers to successful learning are
identified and then removed for every learner. This starts at the school level,
where the physical environment should be designed to accommodate learners
who are challenged, where the school principal, the staff and the parents/
guardians work together to create a good school ethos and where specialised
equipment and/or personnel are provided for these learners.
You should highlight daily the aspects of Mathematics that encourage
cooperative learning and respect for diversity. Plan activities on an individual,
pair or group basis so that you can meet the different needs of learners.
Homogeneous groups or pairs (in which all the learners have more or less
the same level of skill and knowledge) are appropriate when the purpose of
the group is to assist learners who have a common special educational need.
Use homogeneous groups to cope with differentiated learning. For example,
learners who have completed a class activity can be given an individual or group
extension activity while you work with the rest of the class or with a group that
needs more intensive input from you to help them understand and complete an
activity. The intention is not for these groups to be fixed groups, but that learners
move to different groups according to their needs and progress.
Heterogeneous groups have a number of advantages. These groups consist
of learners with diverse backgrounds, gender, languages and abilities.
Heterogeneous groups expose learners to new ideas, generate more discussion,
and allow explanations to be given and received more frequently – this helps to
increase understanding. Peer-tutoring, where two learners with different skills
are paired, can be a mutually enriching experience.
Mathematics Grade 4 Teacher’s Guide
Math G4 TG.indb 7
introduction
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Content areas in Intermediate Phase Mathematics
Mathematics in the Intermediate Phase covers five content areas:
• Numbers, operations and relationships
• Patterns, functions and algebra
• Space and shape (geometry)
• Measurement
• Data handling.
Each content area contributes towards the acquisition of specific skills. The
table, Mathematics content knowledge, on page 9 in the CAPS document shows
the general focus of the Mathematics content areas in curriculum as a whole, and
the specific content focus in the Intermediate Phase.
Each content area is divided into topics. All the content areas must be taught
every term. The tables for time allocation per topic on pages 24 (Grade 4),
25 (Grade 5) and 26 (Grade 6) of the CAPS document set out the sequences of
topics per term for each grade. This Mathematics course is structured to follow
the sequences of topics set out in the CAPS table for each grade, term by term.
The full descriptions of concepts and skills for each content area, as well as
additional teaching guidelines, are given in the detailed tables that follow these
overview tables in the CAPS document. The Learner’s Book and Teacher’s
Guide for this course provide cross references to the relevant sections of these
tables; this will help you to check that you are covering the required concepts
and skills as you work through the units in the course.
The units in each term of this Mathematics course are clearly structured
according to these content areas. At the same time, you will find that
opportunities are provided in each content area to use concepts and skills
relating to other content areas. For example, learners use concepts and contexts
from Measurement, and Space and shape to solve problems in the Numbers,
operations and relationships, and Patterns, functions and algebra content
areas. In this way, learners are able to integrate the concepts, techniques and
problem-solving strategies they learn across all content areas, and increase their
awareness of mathematics as a coherent body of knowledge that covers a wide
range of contexts and concepts.
Mental mathematics
Mental mathematics is a central part of the Intermediate Phase curriculum
content. It should be part of the daily mathematics activity in the classroom
throughout the year. In this Mathematics course most content units start with
Mental maths activities. These activities are designed to relate to the content that
follows in the main unit, and also to revise skills and problem-solving strategies
that learners have used earlier in the year. They are a vital part of the course, as
they serve to keep learners actively thinking and talking about mathematics with
you and with their peers, on a daily basis.
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Mathematics Grade 4 Teacher’s Guide
introduction
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Weighting of content areas
Mathematics content areas are weighted for two purposes: firstly the weighting
gives guidance on the amount of time needed to adequately cover the content
in each content area; secondly the weighting gives teachers guidance on the
spread of content in the examination (especially in the end-of-year summative
assessment).
The weighting of the content is the same for each grade in the Intermediate
Phase. The table on the next page shows the weightings, per grade.
Table 2 Weighting of content areas in Intermediate Phase Mathematics
Weighting of content areas
Content area
Grade 4
Grade 5
Grade 6
Numbers, operations and relationships*
50%
50%
50%
Patterns, functions and algebra
10%
10%
10%
Space and shape (geometry)
15%
15%
15%
Measurement
15%
15%
15%
Data handling
10%
10%
10%
100%
100%
100%
Total
*The weighting of the Numbers, operations and relationships content area has been
increased to 50% for all three grades, in order to ensure that learners are sufficiently
numerate when they enter the Senior Phase.
Progression in content areas across the Intermediate Phase
The Intermediate Phase Mathematics curriculum is structured to enable
learners to develop their skills and knowledge in each content area in a careful
progression from Grade 4 to Grade 6. A summary of this progression is provided
on pages 11–22 of the CAPS document.
Mathematics Grade 4 Teacher’s Guide
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introduction
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Math G4 TG.indb 10
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2. P
lanning and organising your
Mathematics teaching
This Teacher’s Guide is an essential component of this series. It gives clear
guidelines about how to teach the concepts the learners need to master and how
to organise activities in the classroom. It contains a collection of photocopiable
resources that are required for some of the activities in the Learner’s Book. You
can also use these resources to repeat activities at different times during the year,
if you want to revise particular concepts and methods with learners.
The Learner’s Book is structured according to the term-by-term sequence of
topics for the grade set out in the CAPS document. Most units are preceded by
a mental mathematics section that is integrated with the content to be covered in
the unit that follows. You may want to do the mental mathematics activity at one
time in the day and then proceed with the next unit later in the day. It is essential
to keep to the rhythm of daily mental mathematics activities, so that learners
continue to develop and consolidate their mathematical skills.
Resources in the classroom
In the Intermediate Phase learners move from work with concrete apparatus
to focus more on written and oral work based on the content in the Learner’s
Book. However, it is still important that they use concrete apparatus such as flard
cards, Dienes blocks and geometric shapes and objects to help them consolidate
their understanding of place value, shape, space, pattern, division (sharing)
and grouping, and other concepts that they will work with in this phase. The
Teacher’s Guide indicates what equipment will be useful for this purpose,
in relation to each unit of the course.
Learners will also need to have access to instruments and equipment for practical
activities in certain content areas, particularly the Measurement content area,
where they need to use analogue and digital watches, stopwatches, scales and
thermometers, as well as tape measures, trundle wheels, measuring jugs and
spoons and droppers. Since much of this equipment is used in Grades 4, 5 and
6, you could arrange with teachers across the phase to have a collection of such
equipment available for use by the learners in all three grades.
The photocopiable resources provided in this Teacher’s Guide can be used
throughout the year to repeat activities such as the mental mathematics games
learners play, revising number concepts such as place value.
A teaching strategy that builds conceptual and social skills
The learning experiences in this course are designed for group work, pair work,
individual work and for the whole class to do together. This cultivates an ethos
of cooperation and working together. Letting learners work together is a very
useful and successful teaching strategy. It helps them to develop social skills
such as cooperating in teams, taking turns, showing respect and responsibility,
as well as listening and communicating effectively through interactive learning.
Helping learners overcome barriers to learning Mathematics
Learners who experience barriers to learning Mathematics should be given many
opportunities for activity-based learning, to help them overcome their barriers
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at the pace that works for them. They should be given more time to do practical
examples, using concrete objects and practical experiences, than other learners.
Moving too soon to abstract work may make these learners feel frustrated, and
they may then lose mathematical understanding and skills they have developed.
When organising daily classroom activities, allow more time for these learners
to complete tasks, use their own strategies to develop their thinking skills, and
do assessment activities. You may also need to reduce the number of activities
you give to these learners, without leaving out any of the concepts and skills that
need to be introduced and consolidated.
Revision work
The term-by-term content schedule for each grade includes periods set aside for
revision work. During this period of the term you can repeat activities from units
throughout the term, let learners play again the games they played during the
Mental maths units, or design new revision activities using the notes provided
for each unit in this Teacher’s Guide. Use the revision periods as a way to assess
learners’ readiness to complete formal assessment tasks for the term.
Assessment
The purpose of assessment is to inform you, the learners and their parents or
guardians about their performance. Assessment also serves as a tool for you to
reflect on and analyse your own teaching practice, as this has an influence on the
learners’ performance. You can use your assessment to see whether you need to
provide more opportunities for some or all of the learners to develop a particular
skill or master a concept in a given topic.
You should develop a well-planned process to identify, record and interpret
the performance of your learners throughout the year, using both informal and
formal assessment methods. Keep a record of the learners’ performance on
assessment sheets, and summarise this information on a report form or card
to give the learners and their parents or guardians at certain times of the year.
You may photocopy the various assessment sheets provided in the back of this
Teacher’s Guide to use in your classroom.
Assessment methods
You can use various methods to assess the learners’ progress during the year.
Any assessment method involves four steps:
• generate and collect evidence of learners’ achievement
• evaluate this evidence
• record your findings
• use this information to understand learners’ development and help them
improve the process of learning, and also to improve your teaching.
Before you undertake any assessment of learners’ work, decide on a set of
criteria or standards for what they should be able to understand and do, and
base your assessment on these criteria. It is important that you give the learners
clear instructions about what you expect of them, so that they can complete
the assessment tasks correctly and honestly. Once an assessment task has been
completed, discuss your assessment with the learners and give them feedback to
help them increase their ability to do the task successfully.
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Mathematics Grade 4 Teacher’s Guide
PLANNING
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Term-by-term assessment
The term-by-term content guidelines in the CAPS document specify which
content areas are to be assessed in each term of the year. This Teacher’s Guide
includes assessment tasks for the content areas covered in each term. You may
choose to do the assessment of a particular content section straight after that
section has been completed, or to schedule the assessment at another time during
the term. The assessment tasks are resources that you can use as part of your
overall assessment plan for the year.
Self-assessment
Throughout the year the mental maths sections of this course include activities
that learners can complete on a Mental maths grid. This is a self-assessment tool
that will enable the learners monitor their own achievements, and indicate where
they feel they need help with a particular aspect of the content. You should use
the completed grids as part of your own assessment of each learner’s progress
throughout the year.
Formal assessment requirements for Intermediate Phase Mathematics
The table below sets out the formal assessment requirements for Intermediate
Phase Mathematics, as specified in the CAPS document.
Table 4 Minimum requirements for formal assessment: Intermediate Phase Mathematics
Minimum requirements per term
Forms of
Number of
Weighting
assessment tasks per year Term 1 Term 2 Term 3 Term 4
School-based
assessment
(SBA)
Tests
3
Examination
1
Assignment
2
Investigation
1
Project
1
Total
8
1
1
1
1
1
1
75%
1
1
2
2
2
2
To be completed before the final examination at the end of the year
Final
examination
1
End of the year
25%
Assessing learners who experience barriers to learning
Learners who experience barriers to learning should be given opportunities
to demonstrate their competence in ways that suit their needs. You may have
to consider using some or all of the following methods when assessing these
learners’ skills and knowledge:
• Allow these learners to use concrete apparatus for a longer time than other
learners in the class.
• Break up assessment tasks (especially written tasks) into smaller sections for
learners who have difficulty concentrating for long periods, or give them short
breaks during the task.
• Learners who are easily distracted may need to do their assessment tasks in a
separate venue.
Mathematics Grade 4 Teacher’s Guide
Math G4 TG.indb 13
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• Use a variety of assessment methods, as some learners may not be able to
demonstrate what they can do using certain types of assessment.
For example, a learner may be able to explain a concept orally but have
difficulty writing it down.
Reporting learners’ performance
Reporting is the process of communicating learners’ performance to the learners,
and to parents and guardians, schools and other stakeholders. You can use report
cards, parent meetings, school visitation days, parent–teacher conferences, phone
calls, letters and other appropriate methods to make your reports.
Records of learner performance should provide evidence of the learner’s
conceptual progression within a grade and her/his readiness to progress to the
next grade.
Formal assessment is reported in all grades using percentages. The table below
sets out the scale of achievement to be used for recording and reporting levels
of competence in the Intermediate Phase. You should also use comments to
describe learners’ performance, as appropriate.
Table 5 S
cale of achievement for the National Curriculum Statement
Grades 4 – 6
Rating code
xiv
Math G4 TG.indb 14
Description of competence
Percentage
7
Outstanding achievement
80 –100
6
Meritorious achievement
70 –79
5
Substantial achievement
60 – 69
4
Adequate achievement
50 –59
3
Moderate achievement
40 – 49
2
Elementary achievement
30 –39
1
Not achieved
0 –29
Mathematics Grade 4 Teacher’s Guide
PLANNING
2012/09/14 5:32 PM
3. Lesson plans
Note: For all terms, time for Mental maths activities is included in the time for a unit.
Term 1
Unit Title
LB pages Time
Content area: Number, operations and relationships
Topic: Whole numbers: revise Grade 3 work
5 hours
1
Count, order and compare numbers
1–5
1 hour
2
Addition and subtraction facts up to 20
5–6
1 hour
3
Add and subtract multiples of 10
6–7
1 hour
4
Multiplication and division
8–9
1 hour
5
Problem-solving
10
1 hour
Assessment task 1 Number sentences
Content area: Patterns, functions and algebra
Topic: Number sentences
3 hours
6
Write and solve number sentences
11–12
1 hour
7
Reverse operations in number sentences
13–14
8
Use number sentences to describe and solve problems
14–16
30 minutes
9
Solve number sentences
17–18
1 hour
10
Patterns in number sentences
18–19
30 minutes
Assessment task 2 Numbers and place value
Content area: Number, operations and relationships
Topic: Whole numbers: adding and subtracting
10 hours
11
Count, order and compare numbers (2)
20–21
1 hour
12
Place value and representing numbers
22–23
1 hour
13
Place value
23–25
1 hour
14
Represent and compare numbers
25–26
1 hour
15
Swap and regroup numbers
26–28
1 hour
16
Revision: find connections
29
17
Number sentences and problem-solving
30–31
1 hour
18
Strategies for adding and subtracting
31–32
1 hour
19
Use different methods and operations
32–33
1 hour
20
More strategies for adding and subtracting
34
21
Add and subtract with 3-digit numbers
35–36
22
More strategies for adding and subtracting (2)
36–37
23
Calculations with 4-digit numbers
37–38
1 hour
1 hour
Assessment task 3: Addition and subtraction
Content area: Patterns, functions and algebra
Topic: Numeric patterns
4 hours
24
Patterns in counting sequences
39–40
2 hours
25
Number grids and patterns
40–41
1 hour
26
Number groups and patterns
41–42
1 hour
Assessment task 4: Numeric patterns
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Unit Title
LB pages Time
Content area: Number, operations and relationships
Topic: Whole numbers: multiplication and division
4 hours
27
Multiplication by grouping and repeated addition
43–45
1 hour
28
Know multiplication tables
46–48
1 hour
29
Round off and estimate in real life
48–49
1 hour
30
Use grouping and sharing
50–51
1 hour
31
Division facts and rules
52–54
Assessment task 5: Multiplication and division
Content area: Measurement
Topic: Time
6 hours
32
Revision of Grade 3 work
55–56
1 hour
33
24-hour time
56–58
1 hour
34
Read time in 5-minute intervals
58–60
1 hour
35
Read calendars
61–62
1 hour
36
Read timetables
63–64
1 hour
37
History of time
66–66
1 hour
Assessment task 6: Time
Revision
66
Content area: Data handling
Topic: Collect, organise and represent data
5 hours
38
Use tally marks
67–68
1 hour
39
Draw up a tally table
68–69
1 hour
40
Show data in pictographs
70–72
1 hour
41
Show data in bar graphs
72–74
2 hours
Topic: Analyse, interpret and report data
4 hours
42
Explain data
74–76
1 hour
43
Data from pictographs
76–77
1 hour
44
Data from pie charts
77–79
1 hour
45
Data from bar graphs
79–80
1 hour
Topic: Represent data
46
1 hour
Draw your own bar graph
80–819
Revision
82
Content area: Space and shape
Topic: Properties of 2-D shapes
1 hour
5 hours
47
Different shapes
83–84
1 hour
48
Triangles and quadrilaterals
85–86
1 hour
49
Pentagons and hexagons
86–88
1 hour
50
Put shapes together
88–90
2 hours
Assessment task 7: Properties of 2-D shapes
Revision
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Mathematics Grade 4 Teacher’s Guide
87–88
lesson plans
2012/09/14 5:32 PM
Unit Title
LB pages Time
Content area: Numbers, operations and relationships
Topic: Whole numbers: multiplication and division
5 hours
51
Equal sharing and multiples
91
1 hour
52
Multiplication and division strategies
92–94
1 hour
53
Basic multiplication and division facts
94–95
1 hour
54
Multiplication and division flow diagrams
95–96
30 minutes
55
Number rules for multiplication and division
96–97
30 minutes
56
Ratio and rate
97–100
1 hour
Assessment task 8: Multiplication and division
Mathematics Grade 4 Teacher’s Guide
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Term 2
Unit Title
LB pages Time
Content area: Number, operations and relationships
Topic: Whole numbers
6 hours
1
Count and order
102–104
1 hour
2
Compare and represent numbers
104–106
3
Place value
106–105
1 hour
4
Estimate and round off
107–109
1 hour
1 hour
Assessment task 1 Counting, place value and estimation
Topic: Whole numbers: addition and subtraction
5
Add and subtract multiples of 10
110
6
Strategies for adding and subtracting
111
7
More strategies for adding and subtracting
112
1 hour
8
Add and subtract with 3- and 4-digit numbers
112–113
1 hour
Assessment task 2: Addition and subtraction
Content area: Number, operations and relationships
Topic: Common fractions
6 hours
9
Order and compare fractions
114–115
1 hour
10
Represent fractions
116–117
1 hour
11
Equal sharing
117
1 hour
12
Calculations with fractions
118–119
1 hour
13
Equivalent fractions
120–121
1 hour
14
Count and calculate fractions
121–122
1 hour
Assessment task 3: Common fractions
Content area: Measurement
Topic: Length
6 hours
15
Revision of Grade 3 work
123–126
1 hour
16
Working with centimetres (cm) and millimetres (mm)
126–128
1 hour
17
Tricky measurements
128–129
18
Understand units of measurement
129–132
1 hour
19
Convert between kilometres, metres and millimetres
133–134
1 hour
20
Convert between centimetres and metres
134–135
21
Convert between millimetres and centimetres
135–136
22
Round off measurements
136–137
23
Problem-solving with distance and length
138–139
Revision and consolidation
139–140
1 hour
1 hour
Assessment task 4: Length
Content area: Number, operations and relationships
Topic: Whole numbers: multiplication
6 hours
24
Basic multiplication facts
141–142
1 hour
25
Multiplication strategies
142–144
1 hour
26
Problem-solving with multiplication
144–145
1 hour
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lesson plans
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Unit Title
LB pages Time
27
Multiplication and estimation
145–146
1 hour
28
Patterns in multiplication
146–147
1 hour
29
More multiplication methods
148
1 hour
Assessment task 5: Multiplication
Content area: Space and shape
Topic: Properties of 3-D objects
5 hours
30
Flat and curved surfaces
149–150
1 hour
31
Shapes and faces of 3-D objects
150–151
1 hour
32
Straight, flat faces: polyhedra
151–154
3 hours
Revision
154
Content area: Patterns, functions and algebra
Topic: Geometric patterns
4 hours
33
Explore geometric patterns
155–156
1 hour
34
Identify and extend patterns
157–158
1 hour
35
Extend patterns
158–159
1 hour
36
Input and output numbers (values)
160–161
1 hour
Assessment task 6: Patterns
Content area: Space and shape
Topic: Symmetry
37
2 hours
What is symmetry?
162–165
Revision
163
Content area: Numbers, operations and relationships
Topic: Whole numbers: addition and subtraction
2 hours
4 hours
38
Round off to add and subtract
166
1 hour
39
Different ways to add
167
1 hour
40
Different ways to subtract
168–169
1 hour
41
Check addition and subtraction calculations
169–170
1 hour
Content area: Numbers, operations and relationships
Topic: Whole numbers: division
4 hours
42
Solve story sums
171–172
1 hour
43
Division with and without remainders
172–174
1 hour
44
Division with remainders
174–175
30 minutes
45
Division with 3-digit numbers and remainders
175–176
30 minutes
46
Problem-solving with division
176
1 hour
Assessment task 7: Whole numbers (division)
Mathematics Grade 4 Teacher’s Guide
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Term 3
Unit Title
LB pages Time
Content area: Measurement
Topic: Capacity and volume
6 hours
1
What do you remember?
178
1 hour
2
Measuring capacity and volume
179
1 hour
3
Understand volume and capacity
180–182
1 hour
4
Estimate and round off
182–184
5
Calculations with litres and millilitres
184–185
1 hour
6
Calculate capacity with fractions
186–187
1 hour
Revision
187
Assessment task 1: Measurement
Content area: Numbers, operations and relationships
Topic: Fractions of whole numbers
1 hour
7
Recognise fraction parts
188–189
30 minutes
8
Fractions of whole numbers
189–190
30 minutes
Topic: Whole numbers: number rules
1 hour 30 minutes
9
Equivalent fractions
191–192
30 minutes
10
Equal sharing and problem-solving
192–193
30 minutes
11
Count, order and calculate with fractions
194–195
30 minutes
Assessment task 2: Common fractions
Topic: Whole numbers: adding and subtracting
3 hours 30 minutes
12
Rules for operations
196–198
30 minutes
13
Count, compare, represent numbers and place value
199–200
30 minutes
Assessment task 3: Counting and place value
14
Addition and subtraction
201
30 minutes
15
Problem-solving
202–203
30 minutes
16
Double, halve and round off for estimations and calculations
203–204
30 minutes
17
Different strategies for calculating
204–205
1 hour
Content area: Space and shape
Topic: View objects
2 hours
18
Side views and top views
206–207
1 hour
19
Side views and plan views
208–209
1 hour
Revision
Content area: Space and shape
Topic: Properties of 2-D shapes
4 hours
20
Sort 2-D shapes
210
1 hour
21
Investigate circles
211
1 hour
22
Investigate polygons
212–213
1 hour
23
Patterns and pictures with 2-D shapes
214
1 hour
xx
Math G4 TG.indb 20
Mathematics Grade 4 Teacher’s Guide
lesson plans
2012/09/14 5:32 PM
Unit Title
LB pages Time
Content area: Data handling
Topic: Collect, organise and present data
3 hours
24
Use tally marks
215–216
1 hour
25
Show data on graphs
216–218
2 hours
Topic: Analyse, interpret and report data
4 hours
26
Explain data
218–220
2 hours
27
More graphs
221–222
2 hours
Content area: Patterns, functions and algebra
Topic: Numeric patterns
5 hours
28
Patterns in number grids
223–225
1 hour
29
Finding rules
225–226
2 hours
30
Rules for number patterns
226–227
2 hours
Assessment task 4: Number patterns
Content area: Numbers, operations and relationships
Topic: Whole numbers: addition and subtraction
4 hours
31
Quick calculations
228–229
1 hour
32
Count, order and compare numbers and place value
229–230
1 hour
33
Problem-solve with whole numbers
231–229
2 hours
Assessment task 5: Addition and subtraction
Topic: Whole numbers: multiplication
4 hours
34
Multiplication strategies
232–233
1 hour
35
Basic multiplication facts
234
1 hour
36
Round off and solving problems
235
2 hours
Content area: Patterns, functions and algebra
Topic: Number sentences
3 hours
37
Write number sentences
236–237
1 hour
38
Balance and inspect number sentences
238–239
1 hour
39
Equations and problem-solving
239–241
1 hour
Content area: Space and shape
Topic: Transformations
3 hours
40
Make new shapes
242–243
1 hour
41
Tangrams
243–244
2 hours
Revision
244
Mathematics Grade 4 Teacher’s Guide
Math G4TG M0.indd 21
lesson plans
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2012/09/17 2:49 PM
Term 4
Unit Title
LB pages Time
Content area: Numbers, operations and relationships
Topic: Whole numbers: working with whole numbers
2 hours
1
Revise rules for working with numbers
246–247
1 hour
2
Represent numbers and place value
247–249
1 hour
Topic: Whole numbers: addition and subtraction
4 hours
3
Problem-solving
250
1 hour
4
Inverse operations
251–252
1 hour
5
More calculations
252–253
1 hour
6
Use estimating and problem-solving
253
1 hour
Assessment task 1: Whole number addition and subtraction
Content area: Measurement
Topic: Mass
5 hours
7
Revision
254–255
1 hour
8
Estimate
255
1 hour
9
More addition and subtraction
255–256
1 hour
10
More multiplication and division
256–257
1 hour
11
Problem-solving
257–258
1 hour
Content area: Space and shape
Topic: Properties of 3-D objects
4 hours
12
Recognise and compare 3-D objects
259–260
1 hour
13
Faces and models of 3-D objects
261–262
2 hours
14
Statements about 3-D objects
262–263
1 hour
Self-assessment
Content area: Numbers, operations and relationships
Topic: Common fractions
5 hours
15
Order and compare fractions
264–265
1 hour
16
Calculate with fractions
265–266
1 hour
17
Fractions with whole numbers
267
1 hour
18
Problem-solving with fractions
268–270
2 hours
Revision
270
Content area: Numbers, operations and relationships
Topic: Whole numbers: division
3 hours
19
Basic division facts
271
1 hour
20
Divide by 10 and 100
272–273
1 hour
21
Strategies for division
273–274
1 hour
Assessment task 2: Division
Content area: Measurement
Topic: Perimeter, area and volume
7 hours
22
Perimeter
275–277
3 hours
23
Area
278–280
2 hours
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Mathematics Grade 4 Teacher’s Guide
lesson plans
2012/09/14 5:32 PM
Unit Title
24
LB pages Time
Volume
280
Revision and consolidation
281
2 hours
Content area: Space and shape
Topic: Position and movement
2 hours
25
Work with grids
282–283
1 hour
26
Grids on maps
284
1 hour
Content area: Space and shape
Topic: More transformations
3 hours
27
Tessellations
285–287
2 hours
28
Describe patterns
287–288
1 hour
Content area: Patterns, functions and algebra
Topic: Geometric patterns
2 hours
29
Geometric patterns
289–290
1 hour
30
Growing patterns
291–292
1 hour
Content area: Numbers, operations and relationships
Topic: Whole numbers: addition and subtraction
3 hours
31
Use place value to add and subtract
293–294
1 hour
32
Use 10-strips to add and subtract
294–295
2 hours
Content area: Data handling
Topic: Probability
2 hours
33
Probability
296–297
1 hour
34
Experiments and actual outcomes
297–298
1 hour
Revision
299
Mathematics Grade 4 Teacher’s Guide
Math G4 TG.indb 23
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Math G4 TG.indb 24
2012/09/14 5:32 PM
TERM
Revise whole numbers
Unit 1
Unit 2
Unit 3
Unit 4
Unit 5
Count, order and compare
numbers
Addition and subtraction
facts up to 20
Add and subtract multiples of 10
Multiplication and division
Problem-solving
Number sentences
Write and solve number sentences
Reverse operations in number
sentences
Unit 8 Use number sentences to
describe and solve problems
Unit 9 Solve number sentences
Unit 10 Patterns in number sentences
Unit 6
Unit 7
Whole numbers: adding and
subtracting
Unit 11 Count, order and compare
numbers (2)
Unit 12 Place value and representing
numbers
Unit 13 Place value
Unit 14 Represent and compare numbers
Unit 15 Swap and regroup numbers
Unit 16 Revision: find connections
Unit 17 Number sentences and
problem-solving
Unit 18 Strategies for adding and
subtracting
Unit 19 Use different methods and
operations
Unit 20 More strategies for adding
and subtracting
Unit 21 Add and subtract with 3-digit
numbers
Unit 22 More strategies for adding and
subtracting (2)
Unit 23 Calculations with 4-digit numbers
Numeric patterns
Unit 24 Patterns in counting sequences
Unit 25 Number grids and patterns
Unit 26 Number groups and patterns
Whole numbers: multiplication
and division
Unit 27 Multiplication by grouping
and repeated addition
Unit 28
Unit 29
Unit 30
Unit 31
1
Know the multiplication tables
Round off and estimate in real life
Use grouping and sharing
Division facts and rules
Time
Unit 32 Revision of Grade 3 work
Unit 33 24-hour time
Unit 34 Read time in 5-minute intervals
Revision and consolidation
Unit 35 Read calendars
Unit 36 Read timetables
Unit 37 History of time
Revision
Collecting and organising data
Unit 38
Unit 39
Unit 40
Unit 41
Unit 42
Unit 43
Unit 44
Unit 45
Unit 46
Use tally marks
Draw up a tally table
Show data in pictographs
Show data in bar graphs
Project
Explain data
Data from pictographs
Data from pie charts
Data from bar graphs
Draw your own bar graph
Revision
Properties of 2-D shapes
Unit 47
Unit 48
Unit 49
Unit 50
Different shapes
Triangles and quadrilaterals
Pentagons and hexagons
Put shapes together
Investigation
Revision
Revision activity
Whole numbers: multiplication
and division
Unit 51 Equal sharing and multiples
Unit 52 Multiplication and division
strategies
Unit 53 Basic multiplication and
division facts
Unit 54 Multiplication and division
flow diagrams
Unit 55 Number rules for multiplication
and division
Unit 55 Ratio and rate
1
Math G4 TG.indb 1
2012/09/14 5:32 PM
Revise whole numbers
Mental maths
In Term 1, eight hours is allocated to Mental maths, presented in
10 minutes per day. This means that most units start with a short
Mental maths activity, and some units have more than one. The
concepts dealt with in the mental maths activities always fit in with
the content of the main lessons.
Revision of Grade 3 work
During the first week in Term 1, learners revise Grade 3 work. The
first Mathematics lesson will be spent entirely on Mental maths.
Note that rote counting is not very meaningful to some learners.
Relate counting to other mathematical concepts and to real-life
contexts. For example, if they count in 10s to 100, they should
say how many 10s they have counted and then link it to repeated
addition and multiplication. If they count to 150, they should realise
that 15 × 10 = 150 because they have counted 10 fifteen times.
If you are working with time, for example, they could count in 5
(school days in a week), 7 (days in a week), 12 (months in a year),
60 (minutes in an hour) or 24 (hours in a day). Learners should
experience Mathematics concepts holistically, and understand the
relationships between various concepts and contexts.
Keep in mind that the outcomes to be achieved as stated in the
curriculum are the minimum that learners should know. If some
concepts are not stated, this does not imply that you should not deal
with them. Your learners will perform at the level that you allow
them to perform.
Unit 1
Count, order and compare numbers
Mental Maths
Learner’s Book page 2
This activity could serve as an icebreaker on the first day of
school. Take the class outside if the weather permits. The activity
can also help you learn the learners’ names.
1. Prepare A5-size cards. Attach string to the cards so that
learners can hang them around their necks. Ask the learners
to write their names at the top on both sides of the cards. You
can use one side for single-digit numbers, and the other side
for 3- and 4-digit numbers. Let each learner write a number
on the card according to the number of learners in the class,
for example, each one will have a number in the range 0–30 or
0–40, and so on.
2
Math G4 TG.indb 2
Mathematics Grade 4 Teacher’s Guide
TERM 1
2012/09/14 5:32 PM
Use the instructions in the Learner’s Book to find out what
they know about basic number facts. You might ask:
• All those with even numbers step forward.
• All those with odd numbers sit down.
• What number comes straight after your number?
• What is 5 more than your number? (… 10 more? … 100
more? 500 more?) and so on.
Break up the circle. Ask those with numbers that are
multiples of 3, 4 or 5 to line up. Ask them why, for example,
the number 4 or 8 is not in the group when they count from
3 in 3s (multiples of 3). Learners have fun working to form
number bonds of 20, 17 and 40. They should realise that
double a number is the same as multiplying by 2, and halving
is the same as dividing by 2. Let them count on and back
from different numbers. Then let them use two numbers to
make their own calculations.
Examples
3 and 7
3 + 7 = 10
7–3=4
1
7 × 3 = 21
7 ÷ 3 = 22 or 2 remainder 1
2. Set up a number pin board: New maths words. Choose a
number for the day. Give each group six strips of paper to
write down number facts for each number selected and pin
their strips to the wall or number board.
This gives learners opportunities to practise concepts such as
basic operations, doubling and halving. Stretch their thinking
by suggesting facts such as half of 48 = 24, 8 + 8 + 8 = 24,
100 – 86 = 24, and so on.
3. Work as a class. Learners make two sets of numbers from the
page numbers. They order the numbers and write them on the
board as they name the numbers.
Ask what they notice about the numbers (one set is in 2s, the
other in 1s). The numbers in the first set are called natural
numbers. The numbers in the second set are called even
numbers.
Let learners name the numbers that are missing in the second
set. Find out if they know these numbers are called uneven or
odd numbers.
Help learners realise that, when you count in consecutive
even numbers, the units are always in the order 2; 4; 6; 8;
0 or 0; 2; 4; 6; 8 and counting in consecutive odd numbers
always results in 1; 3; 5; 7; 9 as units.
Solutions
95; 96; 97; 98; 99; 100; 101; 102
238; 240; 242; 244; 246; 248; 250
Mathematics Grade 4 Teacher’s Guide
Math G4 TG.indb 3
TERM 1
3
2012/09/14 5:32 PM
4. Learners work on their own. They use the Mental maths
grid to record their solutions (see below). They will use this
template to record their achievements on a regular basis, to
monitor their own progress and to serve as motivation to
improve their results.
a) 100
b) 100
c) 100
d) 248
e) 250
f) 238
g) 250
h) 250
i) 10 003
j) 997
Mental calculations
Name:
Task 1
Task 2
Task 3
Task 4
Task 5
Task 6
Task 7
Date:
a)
100
b)
100
c)
100
d)
248
e)
250
f)
238
g)
250
h)
250
i)
99 7
j)
258 7
Shade the blocks below to show your progress.
10
9
8
7
6
5
4
3
2
1
Reflection
What do I do well?
..............................................................................................................................................
..............................................................................................................................................
..............................................................................................................................................
What can I do better next time?
..............................................................................................................................................
..............................................................................................................................................
..............................................................................................................................................
4
Math G4 TG.indb 4
Mathematics Grade 4 Teacher’s Guide
TERM 1
2012/09/14 5:32 PM
For learners who struggle, use copies of the 200-grid template (see
resources) to help them compare sizes of 2-digit numbers. Learners
may find it difficult to determine numbers less than or more than
when multiples of 10 are involved. Address this concept consistently
in Mental maths. More activities on this concept are provided later
in this term.
If learners give incorrect solutions, let them check using a number
line, 100-grid, a calculator or the numbers in the previous activity.
Do not only tell them the solution is incorrect and move on.
If learners struggle with calculations that are larger than multiples
of 10, work with a smaller number range so that they could see the
jumps by looking for a pattern. Use a number line to consolidate
this understanding. Also use the following list for further mental
calculations and allowing learners to look for relationships.
4 + 6 = n 10 – 4 = n
5 + 6 = n 11 – 5 = n
5+5=n
4 + 7 = n 11 – 4 = n
5 + 7 = n 12 – 5 = n 85 + 5 = n
4 + 8 = n 12 – 4 = n
5 + 8 = n 13 – 5 = n 795 + 5 = n
4 + 9 = n 13 – 4 = n
5 + 9 = n 14 – 5 = n 20 – 5 = n
94 + 6 = n 50 – 4 = n 85 + 6 = n 41 – 5 = n 100 – 5 = n
344 + 7 = n 111 – 4 = n 495 + 9 = n 102 – 5 = n 900 – 5 = n
Activity 1.1
1.
Learner’s Book page 4
Explain that in real life the names of people in a telephone
directory and words in a dictionary are ordered alphabetically.
We also order names of days of the week and months of the
year. Ask them to think of other places we list numbers in
order (for example, house numbers, numbers on clocks and
telephones.)
Explain that they should arrange the house numbers in order.
In (a) the numbers are arranged in ascending order (from
smallest to largest) and in (b) in descending order (from largest
to smallest) order. Learners work on their own. Assess their
ability to order numbers. You can write the numbers in the two
sets on cards to stick on the board for arrangement during the
class feedback, or use them to assist learners who struggle.
Make sets of number cards to address the learners’ different
levels of development. Let slow learners work with a lower
number range and advanced learners with higher number
ranges in different intervals, for example:
65
70
1 000
997
1 001
69
67
1 002
996
998
68
60
999
Check that learners can recognise the sets of numbers as odd
and even numbers.
Mathematics Grade 4 Teacher’s Guide
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2, 3. Ask the learners where and why we compare numbers in real
life. We compare numbers when we look at the prices of food,
clothes, populations in different countries, our ages, and so on.
Learners work on their own to determine numbers that are 1, 2,
4, 5 and 10 more or less than others. They should realise that
the numbers are linked to the numbers in the previous question.
Let them explain their strategies.
Encourage learners to use place value. For example, 10 more
than 230 means that the tens digit becomes 1 more, so 30
becomes 40, which is 10 more. You could give learners a list of
these problems to practise for homework.
Learners should explain their solutions. In comparing the
numbers, they will use knowledge of place value, for example,
900 is more than 100. Lead a class discussion about learners’
methods to solve these problems. Ask them if they observe any
patterns in the sets of numbers. Learners should realise that
they first have to look at the hundreds to compare the numbers.
Introduce the learners to the more than (>) and less than (<)
relationship signs if they do not know these signs yet.
4. The learners place the numbers in descending order.
Solutions
1. a) 129; 131; 133; 135; 137; 139; 141; 143; 145
b) 234; 232; 230; 228; 226; 224; 222; 220; 218
2. a) 2 more than 129 = 131
b) 10 more than 230 = 240
c) 10 less than 133 = 123
d) 5 more than 145 = 150
e) 2 less than 230 = 228
f) 4 more than 226 = 230
g) 10 more than 141 = 151
h) 1 more than 139 = 140
3. a) 931 > 139
b) 143 < 413
c) 220 > 202
d) 541 > 145
e) 218 < 228
f) 145 < 154
g) 133 < 331
h) 224 < 242
4. a) 96; 94; 69; 64; 49; 46
b) 385; 375; 365; 355; 345; 335; 325
c) 982; 928; 892; 829; 298; 289
d) 440; 404; 220; 202; 44; 22
5. 95; 96; 97; 98; 99; 100; 101; 102; 103
6. Learners’ examples
6
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Mathematics Grade 4 Teacher’s Guide
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Unit 2
Addition and subtraction facts up to 20
From their Grade 3 work, learners should already know addition and
subtraction facts (bonds) of various numbers up to 20.
Mental Maths Learner’s Book page 5
Let the learners work together as a class but name individuals to
give the remaining addition and subtraction bonds of 15. They
complete the addition and subtraction combinations for 20 for
homework. Ask different groups to do combinations for different
numbers, such as 13, 14 and 16. Groups can record their work
on sheets of paper and display these in the classroom. Encourage
them to work systematically.
You should focus learners’ attention on the commutative
property of addition – that is, that the numbers can be added in
different orders.
For example, ask the learners what they observe in the
calculations 0 + 15 = 15 and 15 + 0 = 15 or 2 + 13 = 15 and
13 + 2 = 15.
Ask the learners to check if this also works with the subtraction
bonds. For example, 15 – 5 = 10; is 5 – 10 = 15? Use true and
false statements such as: Are the following statements true or
false? 20 – 7 = 7 – 20, 18 – 9 = 9 – 18. They should just be aware
that you can change the order of the numbers in addition and
still get the same answer, but not in subtraction. Ask the learners
to investigate the relationship between addition and subtraction
(inverse operations), for example, 15 = 11 + 4; 15 – 11 = 4 or
15 – 4 = 11.
Use the following list for a mental test:
1. 5 + 7 = n
2. 7 + 5 = n
3. 12 – 5 = n
4. 12 – 7 = n
5. 8 + 7 = n
6. 7 + 8 = n
7. 15 – 7 = n
8. 15 – 8 = n
9. 16 – 7 = n
10. 16 – 9 = n
Encourage the learners to build up or break down to 10 or use
doubling before they perform the test, for example:
5 + 9 = 5 + 5 + 4 or 16 – 7 = 16 – 6 – 1.
Activity 2.1
Learner’s Book page 5
Encourage learners to look for relationships or links or connections
between numbers and within calculations. Tell them, for example
that people who are related have something in common. Let them
give examples of their own relatives and how they are related.
Mathematics is the science of pattern. Looking for patterns or
relationships is an important skill. It could help to solve problems
much more easily and quickly.
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1, 2. They should realise that the numbers are swapped or switched
but the answer stays the same. Allow learners to explain these
observations. They should realise that it is easier to calculate
13 + 7 than 7 + 13. They do not have to know the term
commutative to describe this property.
Some learners may not see the connection that, for example,
17 – 9 = 8 is one less than 16 – 9 = 7, but may rather reason
that 16 – 9 = 16 – 8 + 1 = 9 because they understand the
concept of halving.
3. Learners use logical thinking and should realise that 6 + 9 = 15
is 1 more than 5 + 9 = 14 because 6 is 1 more than 5.
4. Learners use the commutative property for addition. They
switch subtractors and differences in subtraction, and look at
the relationship between addition and subtraction.
Solutions
Unit 3
2. a) 6 + 7 = 13 and 7 + 6 = 13
c) 3 + 8 = 11 and 8 + 3 = 11
e) 7 + 5 = 12 and 5 + 7 = 12
b) 4 + 9 = 13 and 9 + 4 = 13
d) 4 + 7 = 11 and 7 + 4 = 11
f) 3 + 9 = 12 and 9 + 3 = 12
3. a) 6 + 9 = 15
7 + 9 = 16
c) 16 – 9 = 7
15 – 7 = 8
b) 7 + 8 = 15
9 + 8 = 17
d) 8 + 5 = 13
9 + 4 = 13
4. a) 7 + 9 = 16
9 + 7 = 16
16 – 9 = 7
16 – 7 = 9
b) 8 + 6 = 14
6 + 8 = 14
14 – 6 = 8
14 – 8 = 6
c) 7 + 5 = 12
5 + 7 = 12
12 – 5 = 7
12 – 7 = 5
d) 11 + 9 = 20
9 + 11 = 20
20 – 9 = 11
20 – 11 = 9
Add and subtract multiples of 10
Mental Maths Learner’s Book page 6
Many learners struggle to calculate problems such as 70 + 40, 600 + 700,
130 – 40 and 1 000 – 10, which should be done by quick mental recall. If they
know the number bonds, such as 7 + 4, 6 + 7, 13 – 4 and 10 – 1, as covered
during this week, they should be able to connect this knowledge to mental
calculations with operations involving multiples of 10. Demonstrate that 70 + 40
is 7 + 4 and a zero added to get 110 ((7 + 4) × 10), or show how to build or break
down to the nearest multiple of 10, for example, 130 – 30 – 10. Tell the learners
that children often find it easy to calculate with 5 and 10. Ask them to suggest
reasons. They may guess it is because we have five fingers and five toes on one
hand or foot and ten on two hands or feet.
8
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People have used finger counting since the earliest times, many thousands of
years ago.
●
●
●
●
●
●
●
●
●
●
●
0
10
20
30
40
50
60
70
80
90
100
Do short counting exercises in 2s, 3s, 5s, 10s, 50s and 100s. The numbers the
learners say in the different intervals are called multiples – more of the same
number.
1. a) Write down the numbers as the learners name the missing multiples of 10
on the number line.
Demonstrate the connection between these counting numbers. Point out
that these are all whole numbers. Counting numbers are whole numbers
that include zero. Natural numbers start from 1; 0 (zero) is not regarded
as a natural number.
Use the New maths words board, and write new words on cards as they
arise during lessons. Include the multiples of 100 as given below, and
ask them to look at the position of the digits. This will be preparation for
developing place value concept.
0
1
2
3
4
5
6
7
8
9
10
0
10
20
30
40
50
60
70
80
90
100
0 100 200 300 400 500 600 700 800 900 1 000
b) Draw an empty or blank number line on the board. Let the learners plot
the points, starting from the right because they have to count back. This
could help them estimate and plot equal spaces, which is required in
measurement. They fill in the multiples of 10 from 150 to 90.
90
100
110
120
130
150
150
c) The learners should find out that each strip has 10 dots. They should
realise that counting the dots in ones will take too long. Some might
realise that they could count the number of strips and multiply by 10
instead of doing repeated addition. There are 14 strips. The total number
of dots is 140. Extend the activity by asking how many dots there will be
in other numbers of strips, for example, in 8, 9, 11, 12 or 13 strips. At this
stage learners should have discovered a rule for multiplication by 10.
2. Write down the numbers as the learners say them. Ask them if the numbers
43; 53; 63; 73; 83; 93 and 57; 47; 37; 27; 17; 7 are multiples of 10. Although
+ 10
+ 10
+ 10
+ 10
+ 10
they add or subtract 10 as with the multiples of 10, you should explain that
here they counted
in intervals
of 1063and not in 73
multiples of8310.
43
53
93
+ 10
a)
43
+ 10
53
– 10
b)
7
+ 10
63
– 10
17
+ 10
73
– 10
27
+ 10
83
– 10
37
93
– 10
47
57
– 10 line gives
– 10multiples
– 10
10 30; ...),–but
10 the second
c) The first number
of 10 (10;–20;
number line
shows adding
10 to27each term37(or intervals
of 10) – it57does
17
47
7
not show multiples of 10 (7; 17; 27; ...).
Mathematics Grade 4 Teacher’s Guide
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Use the 200-grid to help learners practise adding and subtracting 10 and
multiples of 10.
a) 70 + 10 = 80
b) 0 + 10 = 10
c) 10 + 80 = 90
d) 90 + 10 = 100
e) 10 – 0 = 10
f) 100 – 10 = 90
4. a) 110 – 10 = 100
b) 50 – 10 = 40
c) 22 – 10 = 12
d) 53 – 10 = 43
e) 96 – 10 = 86
f) 103 – 10 = 93
g) 58 + 10 = 48
h) 94 + 10 = 104
i) 94 + 10 = 84
j) 106 + 10 = 96
5. Learners identify multiples of 10 in the solutions.
Activity 3.1
Learner’s Book page 7
1. The number cards above each set of calculations give learners
clues to help them solve the problems. Learners solve each
calculation by looking for a connection or relationship with the
number card. They should realise that, for example, 50 + 25 is 5
more than 50 + 20, and 60 – 39 is 1 more than 60 – 40.
a)
c)
50 + 20 = 70
b)
50 + 25 = 75
52 + 20 = 72
20 + 59 = 79
51 + 20 = 71
60 + 20 = 80
d)
60 + 23 = 83
24 + 60 = 84
66 + 20 = 86
29 + 60 = 89
60 – 40 = 20
60 – 39 = 21
60 – 38 = 22
60 – 35 = 25
60 – 34 = 26
90 – 50 = 40
90 – 49 = 41
90 – 45 = 45
90 – 47 = 43
90 – 48 = 42
2. Ask the learners what the answers to 7 – 2 and 1 + 9 are. Then
ask them the solutions to 70 – 20 and 10 + 90. Allow them to
look for relationships or patterns and explain their thinking
and strategies. This should be a consistent practice that allows
learners to develop effective mental calculation strategies that
they can use when working with larger numbers, for example,
800 – 500 or 8 000 – 50.
a) 70 – 20 = 50
b) 100 – 40 = 60
c) 60 – 20 = 40
d) 80 – 50 = 30
e) 90 – 20 = 70
f) 100 – 70 = 30
g) 80 – 30 = 50
h) 100 – 80 = 20
i) 10 + 90 = 100
j) 20 + 80 = 100
k) 30 + 70 = 100
l) 40 + 60 = 100
m) 50 + 50 = 100
n) 90 + 10 = 100
o) 80 + 20 = 100
p) 70 + 30 = 100
3. Learners explain connections they notice.
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Unit 4
Multiplication and division
Mental Maths Learner’s Book page 8
1. The arrangement of the cans forces the learners to count in
groups of 10. Discourage counting in 1s. There are 110 cans
altogether.
2. Some learners may count the cans by adding 10 repeatedly
while others may multiply the number of stacks by 10.
Ask them to look at the picture and let them compare their
methods. Encourage them to use the shorter method.
Be sensitive when you compare learners’ strategies and
always suggest that, for example, Let’s look at Sipho and
Peter’s methods. Encourage both learners but ask the two
learners whose strategy they think is more effective.
a) 4 stacks = 40 cans
b) 7 stacks = 70 cans
c) 5 stacks = 50 cans
d) 9 stacks = 90 cans
3. Learners should notice the relationship between repeated
addition and multiplication.
a) 10 + 10 + 10 = 3 × 10 = 30
b) 10 + 10 = 2 × 10 = 20
c) 10 + 10 + 10 + 10 + 10 + 10 + 10 + 10 = 8 × 10 = 80
d) 10 + 10 + 10 + 10 + 10 + 10 + 10 = 7 × 10 = 70
e) 10 + 10 + 10 + 10 + 10 + 10 = 6 × 10 = 60
f) 10 + 10 + 10 + 10 + 10 = 5 × 10 = 50
g) 10 + 10 + 10 + 10 = 4 × 10 = 40
h) 10 + 10 + 10 + 10 + 10 + 10 + 10 + 10 + 10 + 10
= 10 × 10 = 100
4. Learners may find it easy to apply equal grouping for
division. They may be more competent in explaining division
than multiplication, because they often share objects at home.
Allow them to use the method they are comfortable with,
but compare their methods to encourage the use of more
effective strategies. You can explain the relationship between
multiplication and division through repeated addition and
repeated subtraction.
Three bags = 10 + 10 + 10 = 30 oranges and 30 oranges
= 30 – 10 – 10 – 10 = 3 bags. The learners should understand
that the number of bags is the number of groups of 10
subtracted. They share out the groups until 0 (zero) oranges
remain.
a) 10 = 1 bag (1 group of 10 or 10 – 10 = 0)
b) 30 = 3 bags (3 groups of 10 or 30 – 10 – 10 – 10 = 0)
c) 50 = 5 bags
(5 groups of 10 or 50 – 10 – 10 – 10 – 10 – 10 = 0)
Mathematics Grade 4 Teacher’s Guide
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5. This question allows learners to observe the relationship
between equal grouping, repeated subtraction and division.
They should realise that division is a short method for equal
grouping and repeated subtraction.
a) 40 – 10 – 10 – 10 – 10 = 0
40 ÷ 10 = 4
b) 30 – 10 – 10 – 10 = 0
30 ÷ 10 = 3
c) 10 – 10 = 0
10 ÷ 10 = 1
d) 20 – 10 – 10 = 0
20 ÷ 10 = 2
Activity 4.1
Learner’s Book page 9
1. Learners count the number of fingers of all the learners in the
class. If some of them count in 5s this is acceptable, but ask
them which is the quickest way to count the fingers. Let them
make up multiplication problems to calculate the number of
fingers of 5, 10, 20 and 30 learners. Then ask them how many
learners show 10, 20 or 30 fingers and let them make up division
problems for these. They should realise that 0 (zero) learners
show 0 fingers, and 0 fingers means there are 0 learners.
2. Learners write multiplication problems to calculate the number
of fingers of:
a) 0 learners: 0 × 10 = 0 fingers
b) 1 learner: 1 × 10 = 10 fingers
c) 2 learners: 2 × 10 = 20 fingers
d) 3 learners: 3 × 10 = 30 fingers
e) 4 learners: 4 × 10 = 40 fingers
f) 5 learners: 5 × 10 = 50 fingers
g) 6 learners: 6 × 10 = 60 fingers
h) 7 learners: 7 × 10 = 70 fingers
i) 8 learners: 8 × 10 = 80 fingers
j) 9 learners: 9 × 10 = 90 fingers
k) 10 learners: 10 × 10 = 100 fingers
l) 11 learners: 11 × 10 = 110 fingers
They should realise that they have created the 10 times table
using their fingers. Except for 10 × 0, they could observe that
you add a zero to the end of the number when you multiply by
10. You can now extend the activity to include for example:
25 × 10 = n
30 × 10 = n
80 × 10 = n
99 × 10 = n
100 × 10 = n
3. 20 ÷ 10 = 2 learners
4. a)
b)
c)
d)
12
Math G4 TG.indb 12
0 fingers = 0 ÷ 10 = 0 learners
10 fingers = 10 ÷ 10 = 1 learner
20 fingers = 20 ÷ 10 = 2 learners
30 fingers = 30 ÷ 10 =3 learners
Mathematics Grade 4 Teacher’s Guide
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e) 40 fingers = 40 ÷ 10 = 4 learners
f) 50 fingers = 50 ÷ 10 = 5 learners
g) 60 fingers = 60 ÷ 10 = 6 learners
h) 70 fingers = 70 ÷ 10 = 7 learners
i) 80 fingers = 80 ÷ 10 = 8 learners
j) 90 fingers = 90 ÷ 10 = 9 learners
k) 100 fingers = 100 ÷ 10 = 10 learners
l) 110 fingers = 110 ÷ 10 = 11 learners
Allow learners to observe that they created the division table
for 10. Ask them to look for the connections and patterns in
the multiplication and division tables. They should realise that
division is the opposite of multiplication, so, multiplication and
division are inverse operations and for example, 0 × 10 = 0 and
0 ÷ 10 = 0 or 3 × 10 = 30 and 30 ÷ 3 = 10.
5. The learners practise the above relationship. Tell them they will
learn more about multiplication and division by 0 later.
a) 3 × 10 = 30
30 ÷ 10 = 3
b) 8 × 10 = 80
80 ÷ 10 = 8
c) 6 × 10 = 60
60 ÷ 10 = 6
d) 90 ÷ 10 = 9
9 × 10 = 90
e) 50 ÷ 10 = 5
5 × 10 = 50
f) 10 ÷ 10 = 1
1 × 10 = 10
g) 4 × 10 = 40
40 ÷ 10 = 4
h) 7 × 10 = 70
70 ÷ 10 = 7
i) 2 × 10 = 20
20 ÷ 10 = 2
j) 120 ÷ 10 = 12
12 × 10 = 120
k) 0 × 10 = 0
0 ÷ 10 = 0
6. Multiplication and divsion are inverse operations.
7. Learners list the multiplication and division tables for 10 in the
correct order. They must learn this by heart once they have a
conceptual understanding of multiplication and division by 10.
8. Listen carefully to the learners’ explanations for multiplying and
dividing 10 by 0. Teaching the concept of 0 is often neglected
because 0 is regarded as nothing and not a number. Zero (0) is
a number in its own right, just like the other numbers, because
it answers the question, How many? If we include calculations
with 0 as in this exercise, learners will be able to accept 0 as
a number and develop understanding of this abstract concept.
Some learners in higher grades and even teachers are often not
able to explain their understanding of the concept of 0 because
they have simply learnt rules for working with this number,
for example, Any number multiplied or divided by 0 is 0. This
rule is true, but learners should discover the rule for themselves
by engaging in activities like the ones in this lesson. Learners
should be able to reason that 0 × 10 = 0 and 0 ÷ 10 = 0 if there
are 0 fingers and 0 learners.
Mathematics Grade 4 Teacher’s Guide
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Unit 5
Problem-solving
Mental Maths Learner’s Book page 10
Learners solve word problems mentally. The problems involve
the four basic operations. Read the problems to the learners. The
numbers they will work with mostly involve multiples of 10.
They record their solutions on their Mental maths grids. Allow
learners to discuss their solutions and strategies afterwards.
Question 1, 5 and 6 are repeated addition problems. At this
stage your learners should, however, solve the problem using
multiplication.
Learners should know that 8 × 2 = 16, insert 0 and understand
the problem as 20 groups of 8:
8 × 20 → 8 × 2 → 16 × 10 = 160 slices of pizza
1. 20 × 8 = 2 × 8 × 10 = 160
2. 120 ÷ 10 = 12 packets
This is a grouping problem that learners should solve by
dividing. They should know the rule by now that you take
away the 0 when you divide a multiple of 0 by 10, but
understand that 120 is 12 groups of 10.
3. 90 ÷ 10 = 9 sweets each
This is a sharing problem that requires understanding that 90
is shared out equally by subtracting 10 until 0 remains:
90 – 10 – 10 – 10 – 10 – 10 – 10 – 10 – 10 – 10 = 0.
4. 44 ÷ 10 = 4 boxes
This is a grouping problem in which the remainder is rejected
because it does not form a group of 10. Learners could also
count in tens to the closest multiple to find that 10 + 10 + 10
+ 10 = 40 so that 4 boxes can be filled.
5. 15 × 10 = 150 colouring pencils
Check which learners add up 10 fifteen times. Ask them
to compare their method to those of learners who use
multiplication.
6. 25 × 10 = R250 saved
7. 50 + 50 = 100
8. 110 – 30 = 80 learners left in the hall.
Some learners might count back from the larger number
or they might break up the smaller number to reason that
110 – 10 – 20 = 80.
9. 100 – 40 = 60 marbles
Some learners might use a counting on or counting back
strategy.
10. 31 – 21 = 10 counters
This is a change problem with the unknown in the beginning:
n + 21 = 31. Some learners might count on from 21 to 31 and
others might recognise that 21 needs ten more to make 31.
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Activity 5.1
Learner’s Book page 10
Ask the learners where in real life we solve problems involving
numbers. Talk to them about number problems that people have to
solve every day, such as shopping, calculating time and filling up a
car with petrol. Learners can work in groups to solve these problems.
1. Jabulani gets 1 part and Leon 2 parts of R750. So dividing R750
into 3 parts results in R250 + R250 + R250. Jabulani gets R250
and Leon gets R500. This is a proportional sharing problem.
Some learners might start with a trial and improvement strategy.
They might count in 25s and 50s until the sums of the two
sequences add up to 750, for example:
25
50
75
100 = 250
50
100
150
200 = 500
If learners are stuck, use smaller numbers, such as R2 and R4
and after some weeks they have, for example R24. This means
that one must get one part and the other two parts of R24, so:
R8 + R8 + R8 so that one gets R8 and the other R16.
2. 300 – n = 159 300 – 159 = 141
This is a change problem with the unknown in the middle.
If learners know that 150 is half of 300 they can reason that
300 – 159 = 300 – 150 – 9 = 141. Learners may use different
strategies. Examine and compare the strategies they use.
3. a) Adding consecutive odd numbers results in the tens and
units in the answers forming the multiples of 4. The numbers
that are added form consecutive odd numbers in the
columns.
b) Possible answers:
51 + 53 = 104
53 + 55 = 108
55 + 57 = 112
57 + 59 = 116
c) Possible answers:
101 + 103 = 204
103 + 105 = 208
105 + 107 = 212
107 + 109 = 216
d) Zodwa is right. If you add two consecutive odd numbers, the
answer will always be an even number. All the answers are
even numbers.
Make sure that the learners understand what consecutive numbers
mean. Tell them when they count in whole or counting numbers
(0; 1; 2; 3; 4; ... 99; 100; 101; ...) or natural numbers (1; 2; 3; ... 48;
49; 50; ...) the numbers follow on each other in an orderly way and
are called consecutive numbers.
This activity builds on knowledge developed of addition of
multiples of 10, even and uneven numbers. The learners should also
look at the pattern in the solutions, for example: the tens and units
form multiples of 4. Ask the learners to investigate what would
happen if they add three consecutive odd numbers. Will the answers
still be even numbers?
Mathematics Grade 4 Teacher’s Guide
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number sentences
unit 6
Learner’s Book page 11
Write and solve number sentences
Writing and solving number sentences prepare learners for work
that involves algebraic expressions and equations. First allow
learners to work as a class and in groups before they work on their
own. Learners with reading problems should be supported; explain
methods and problem situations in the learners’ home languages,
and let them describe their ideas about how to solve the problems
using the home language or another language in which they can
express themselves freely.
Mental Maths
Learner’s Book page 11
Copy the I have ... whole numbers: basic operations template
on cardboard and cut out the cards. You could have the cards
laminated and use them regularly to allow learners to practise the
basic operations but also to read number sentences. Each card
has a statement and a question.
I have 20. Who has
double this plus 1?
I have 41.
Who has 2 less?
I have 39.
Who has ...?
Play the game with the learners. You can start reading the first
card. The learner who has the answer to your statement and
question answers and poses the next statement and question. The
game involves a number chain so that you will answer the last
question.
Use some of the cards after the game and ask learners to create
number sentences, for example (20 + 20) + 1 = n, 41 – 2 = n.
Double 20 plus 1: (2 × 20) + 1 = 41
Number sentences consist of numbers and symbols, which can
be operations (+, –, ÷ and ×) and relation signs (=, <, > and ≠).
Example of a closed number sentence: 4 + 5 = 9.
Activity 6.1
Learner’s Book page 11
The learners solve word problems. Ensure that they understand
the context and structure of the problem. Explain how the flow
diagram links to the problem. Let the class work together to
describe the problems with number sentences or flow diagrams.
Ask them to work in groups to solve the problems. They should
not struggle with the number concept because they have worked
with basic calculations before. Let them work in groups to solve the
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investigations. Work with them to help them understand the context
of the problems. You could also ask advanced learners to do these
problems as enrichment or let the learners do them for homework.
In question 1 the learners should realise that × 3 × 2 is the same as
× 6 and 12 ÷ 2 ÷ 3 is the same as 12 ÷ 6. Let the learners solve the
number sentences they created in question 2. Assist the learners in
noticing the relationship between 4 × 8 or 8 × 4 and 800 × 4.
In question 3 you could show the learners the short cut to multiply
by 11 (they will develop this skill later in this term). They use the
inverse operation, count on, doubling or breaking up numbers to
solve question 4. You should expect various strategies offered by the
learners.
Solutions
1. a) 1 × 3 × 2 = n
=3×2
or 1 × 6 = R6
= R6
b) 12 ÷ 2 ÷ 3 = n
=6÷3
or 12 ÷ 6 = R2
= R2
2. a) 4 × 2 = n
4×2=8
b) 4 × 8 = n
4 × 8 = 32
c) 100 × 8 × 4 = n
800 × 4 = 3 200
3. 11 × 12 = n
11 × 12 = 132
11 × 12 = 132
ABC
1+2
4. n + 350 = 725
725 – 325 = 400
or
350 + 350 = 700
400 – 25 = R375
700 + 25 = R725
or
350 + 25 = 375
or
350 + 50 = 400
375 + 25 = 400
725 – 50 = 675
400 + 325 = 725
400 + 275 = 675
25 + 25 + 325 = R375 50 + 50 + 275 = R375
Activity 6.2
Learner’s Book page 12
In question 1 the learners have to realise that they have to subtract the
mother’s age from the granny’s age first (52 – 25). They could do this:
(50 – 25) + 2 = 27. The mother’s age is 3 times the daughter’s, so they
divide by 3 so that Zonia is 9 years old. You could demonstrate at this
stage that we solve calculations in brackets first.
If the learners struggle with the problem in question 2, encourage
them to make a sketch. They first have to find the distance between
A and B.
Solutions
1. (52 – 25) ÷ 3 = n
27 ÷ 3 = 9
Zonia is 9 years old.
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2. From A to B: 15 – 11 = 4 km
From B to C: 12 – 4 = 8 km
? km
A
11 km
B
C
D
12 km
15 km
Unit 7
Reverse operations in number sentences
Mental Maths Learner’s Book page 13
Help the learners understand the structure of the I think of a
number problems that they have to solve mentally. They need
to work backwards using inverse operations to solve these
problems. Write the first number sentence on the board and ask
learners to give number sentences for the rest of the problems.
They should realise that the phrase I think of a number represents
the unknown number in the calculation.
Allow the learners to check solutions by substituting the place
holder (n) with the answer. They should understand that
doubling is multiplying by 2 and halving is dividing by 2.
1. a) n – 9 = 11
b) n × 6 ÷ 3 = 8
11 + 9 = 208 × 3 ÷ 6 = 24 ÷ 6
20 – 9 = 11 42 ÷ 7 = 6
4 × 6 ÷ 3 = 24 ÷ 3
=8
c) n ÷ 7 = 6
d)Double n – 5 = 25
7 × 6 = 42 2 × n – 5 = 25
42 ÷ 7 = 6 25 + 5 ÷ 2 = 30 ÷ 2
= 15
Double 15 – 5 = 25
e) Half of n + 8 = 20
f)
n × 9 – 10 = 71
n ÷ 2 + 8 = 20 71 + 10 ÷ 9 = 81 ÷ 9
20 – 8 × 2 = 12 × 2 =9
= 24 9 × 9 – 10 = 81 – 10
Half of 24 + 8 = 20
= 71
g) n + 25 ÷ 4 = 25
25 × 4 – 25 = 100 – 25
= 75
75 + 25 ÷ 4 = 25
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h)
n + 250 × 2 = 1 000
1 000 ÷ 2 – 250 = 500 – 250
= 250
(250 + 250) × 2 = 1 000
i)
n – 100 + 50 = 150
j)
n ÷ 7 × 4 = 28
150 – 50 + 100 = 100 + 10028 ÷ 4 × 7 = 7 × 7
= 200
= 49
200 – 100 + 50 = 150 49 ÷ 7 × 4 = 28
2. Answers will differ.
Activity 7.1
Learner’s Book page 13
1. Learners need to reverse operations to solve the problems
involving bigger numbers based on the problems they worked
with in the Mental maths activity. They use breaking up or
building up numbers to solve the problems.
a) n – 435 = 686
b) n + 567 – 100 = 525
686 + 435 = n525 + 100 – 567 = n
625 – 567 = 620 – 520
680 + 420 = 1 100
6 + 15 = 21
= 100
n = 1 121 100 – (5 + 47) = 100 – 52
n = 48
2. The learners give number sentences and use inverse operations
to check solutions. They should use brackets to show which
calculations they perform first, for example (1 × 6) – 2 = 6 – 2 = 4.
Where they have to give input numbers, they work with the
inverse operations so that – 2 becomes + 2 and × 6 becomes ÷ 6,
for example, (1 × 6) – 2 = 4 becomes (4 + 2) ÷ 6 = 1. Ask the
learners to give the 10th, 20th and 100th terms in the sequence.
Let them write down the number sequence created by the rule in
the diagrams. They work with inverse operations to complete the
flow diagrams. They should realise that they undo subtraction
when they add and they undo multiplication when they divide.
a) 1
2
3
4
7
5
6
×6
×6
×6
×6
×6
×6
×6
–2
–2
–2
–2
–2
–2
–2
4
10
16
22
28
34
40
a) (1 × 6) – 2 = n
(2 × 6) – 2 = n
(3 × 6) – 2 = n
(4 × 6) – 2 = n
(5 × 6) – 2 = n
(6 × 6) – 2 = n
(7 × 6) – 2 = n
6–2=4
12 – 2 = 10
18 – 2 = 16
24 – 2 = 22
30 – 2 = 28
36 – 2 = 34
42 – 2 = 40
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b) (4 + 2) ÷ 6 = 1
(10 + 2) ÷ 6 = 2
(16 + 2) ÷ 6 = 3
(22 + 2) ÷ 6 = 4
(28 + 2) ÷ 6 = 5
(34 + 2) ÷ 6 = 6
(40 + 2) ÷ 6 = 7
3. The learners work with inverse operations. They should realise
that they have to start where two numbers that are provided to
get the number in the centre, for example, 25 – 5 = 20 in (a)
and 54 – 9 = 45 in (b). This strategy now allows them to solve
the rest of the calculations, which are combinations of addition,
subtraction, multiplication and division of 5 and 9
a)
b)
4
×5
25
54
–5
–9
20
÷5
4
5
×9
45
+5
+9
25
54
4. a) 25 – 5 = 20
4 × 5 = 20
20 ÷ 5 = 4
25 – 5 = 20 or 20 + 5 = 25
÷9
5
b) 54 – 9 = 45
5 × 9 = 45
45 ÷ 9 = 5
45 + 9 = 54
5. a) Some learners might use repeated addition, for example:
(4 × 3) + (4 × 3) + (4 × 3) + (4 × 3) = 12 + 12 + 12 + 12
If they do this, check if they do this for every row. You could
encourage them to use doubling, for example:
24 + 24 = 48.
Tell the learners that you have another method:
(4 × 3) × 4
= 12 × 4
= (6 × 4) + (6 × 4)
= 24 + 24
= 48
The problems involve the strategies repeated addition and
doubling, and using the distributive property.
b) (4 × 4) + (4 × 4) or
(2 × 4) + (2 × 4) + (2 × 4) + (2 × 4)
= 16 + 16
=8+8+8+8
= 32
=4×8
= 32
c) (3 × 5) + (3 × 5) + (3 × 5)
= 15 + 15 + 15
= 30 + 15
= 45
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Unit 8
Use number sentences to describe and
solve problems
Mental Maths Learner’s Book page 14
1. Learners write number sentences on the board to show how
to calculate the number of strongmen, children and rings
in the picture. They give number sentences to show how to
calculate the number of cooldrink cans, eggs and biscuits
in the boxes. The focus is on the construction of number
sentences to assess whether the learners understand how to
solve the problems – not as much on the correct solutions.
The learners use multiplication and calculations with
brackets to solve the problems.
a) 3 × 1 = 3
b) 3 × 2 = 6
c) 6 × 6 = 36
2. (6 × 4) × 2
3. (5 × 30) × 2
= 24 × 2
= 150 × 2
= 48 cans of cooldrink
= 300 eggs
4. (8 × 10) × 2
= 80 × 2
= 160 biscuits
Activity 8.1
Learner’s Book page 15
1. The learners would probably use trial and improvement in
attempts to find the rule. They should find out, for example,
what they have to add to 1 and which number to multiply to get
9. Allow them to discover this on their own. Learners copy the
flow diagrams and fill in the missing numbers. Add 2 to each
input number and then multiply by 2.
1
2
3
4
5
6
7 8
144444424444443
+2
x3
144444424444443
9
12 15 18 21 24 27 30
2. Learners write number sentences. The answers are all multiples
of 3.
(1 + 2) × 3 = 9
(2 + 2) × 3 = 12
(3 + 2) × 3 = 15
(4 + 2) × 3 = 18
(5 + 2) × 3 = 21
(6 + 2) × 3 = 24
(7 + 2) × 3 = 27
(8 + 2) × 3 = 30
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3. The learners use the functions in the flow diagrams and give the
missing numbers.
a) 1 → 1
1×3–2=1
3 → 7
3×3–2=7
5 → 13
5 × 3 – 2 = 13
7 → 19
7 × 3 – 2 = 19
9 → 25
9 × 3 – 2 = 25
11 → 31
(31 + 2) ÷ 3 = 11
13 → 39
13 × 3 – 2 = 39
15 → 43
15 × 3 – 2 = 43
b) 2 → 4
2×3–2=4
4 → 10
4 × 3 – 2 = 10
6 → 16
6 × 3 – 2 = 16
8 → 22
8 × 3 – 2 = 22
10 → 28
10 × 3 – 2 = 28
14 → 40
(40 + 2) ÷ 3 = 14
16 → 46
16 × 3 – 2 = 46
18 → 52
18 × 3 – 2 = 52
In questions 4 to 6, learners write number sentences to show how they
would solve the problems. They write the number sentences to show
whether they make sense of the structure of the problems before they
solve the problems. They have to calculate the number of chocolates
on the strings and the total number of chocolate blocks displayed in
different slabs of chocolate. They then have to determine how many
learners could each get six blocks of chocolate and show how to
calculate the number of Easter eggs on display.
4. Answers will differ.
5. Some learners may use addition, while others may realise that
they can use multiplication and addition.
a) 8 + 4 + 2 = 8 + 2 + 4
b) 8 + 4 + 3 = n
= 14
= 15
c) 8 + 4 + 1 = n
d) 8 + 4 = n
= 13
= 12
e) 14 + 15 + 13 + 12
or
= 15 + 25 + 4 = 54
(8 × 4) + (4 × 4) + 2 + 3 + 1
= 32 + 16 + 4 + 2
= 54
6. The problem involves practising division, multiplication,
addition and using the associative property.
[(4 × 3) + (8 × 2) + (4 × 2) + (6 × 4)] ÷ 6
= (12 + 8 + 16 + 24) ÷ 6
= (20 + 40) ÷ 6
= 60 ÷ 6
= 10 children
Ten children can each get six blocks.
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7. The learners apply knowledge of multiplication, doubling
and the associative property. They identify doubles and group
numbers that add up to multiples of 10.
(4 × 3) + (4 × 3) + (4 × 4) + (3 × 6) + 12
= 12 + 12 + 16 + 18 + 12
= 24 + 16 + 30
= 40 + 30
= 70 Easter eggs
Unit 9
Solve number sentences
Mental Maths Learner’s Book page 17
Introduce the terms sum and product. Ask learners to write
number sentences on the board for each term. They should
demonstrate how to find the sum and the product of each pair
of numbers. Then ask them to show how to find the difference
between the numbers. Check that they subtract the smaller from
the bigger number. Ask what happens if you subtract the bigger
number from the smaller one.
Solutions
1. a) 5 + 9 = 14
2. a) 5 × 9 = 45
b) 7 + 8 = 15
b) 7 × 8 = 56
c) 9 + 8 = 17
c) 9 × 8 = 72
d) 6 + 9 = 15
d) 6 × 9 = 54
e) 7 + 9 = 16
e) 7 × 9 = 63
3. Ask the learners to look at the function that is applied to the
input number in the flow diagram. Let them give the number
sentence for each counting number or input value to get the
output value. Ask them to describe the patterns they observe.
Then ask them to give the inverse operations starting with
the output values.
4.
1
2
3
4
5
6
7
8
14444444244444443
×6
–4
14444444244444443
2
1×6–4=2
5 × 6 – 4 = 26
2×6–4=8
6 × 6 – 4 = 32
3 × 6 – 4 = 14
7 × 6 – 4 = 38
4 × 6 – 4 = 20
8 × 6 – 4 = 44
8 14 20 26 32 38 44
Activity 9.1
Learner’s Book page 17
The learners work with non-contextual problems to consolidate
basic calculation facts and apply inverse operations. They use the
basic knowledge they have developed to apply to calculations with
larger numbers. They have to solve calculations in brackets first.
Show them the following examples that illustrate the importance of
solving calculations in brackets first.
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Performing operations in brackets first:
7 × (3 + 7) = n
(20 – 10) ÷ 5 = n
7 × 10 = 70
10 ÷ 5 = 2
Ignoring the brackets to get incorrect answers:
7 × (3 + 7) = n
(20 – 10) ÷ 5 = n
21 + 7 = 28
20 – 2 = 10
Emphasise the consequences of ignoring the brackets by putting the
calculation in context.
For example, John drinks 3 cups of tea and 7 cups of coffee each
day. How many cups of tea and coffee does he drink altogether in a
week? Does he drink 70 cups or 52 cups?
(3 + 7) × 7 = 10 × 7
= 70 cups
3 + (7 × 7) = 3 + 49
= 52 cups
Also discuss the order of operations with reference to the above
examples.
1. a)
126 + 84 = n
b) 200 + 58 = n
126 + 4 + 80 = 130 + 80 200 + 50 + 8 = 258
n = 210
n = 258
c)
150 + 97 = n
d)
130 + 79 = n
150 + 50 + 47 = 247 130 + 70 + 90 = 209
n = 247
n = 209
e) 130 + 65 = n
f) 120 – 48 = n
n = 195 120 – 50 + 2 = 72
n = 72
g)
200 – 64 = n
h)
140 – 92 = n
200 – 60 – 40 = 136 140 – 40 – 50 – 2 = 48
n = 136
n = 48
i)
520 – 324 = n
520 – 320 – 4 = 196
n = 196
2. Learners must use inverse operations to work out answers and
not merely their knowledge of multiplication tables.
a) 45 ÷ 9 = 5
b) 450 ÷ 10 = 45
c) 500 ÷ 2 = 250
d) 39 ÷ 3 = 13
e) 36 ÷ 6 = 6
f) 35 ÷ 5 = 7
g) 3 × 6 = 18
h) 8 × 9 = 72
i) 56 × 8 = 448
3. a) (3 × 6) + 2 = n
b) (6 × 6) + 4 = n
18 + 2 = 20 36 + 4 = 40
c) (8 × 8) + 6 = n
d) (4 × 7) + 2 = n
64 + 6 = 70 28 + 2 = 30
e) (6 × 7) + 8 = n
f) 16 – (12 ÷ 2) = n
42 + 8 = 50
16 – 6 = 10
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g) 30 – (25 ÷ 5) = n
h) (16 – 9) × 7 = n
30 – 5 = 25
7 × 7 = 49
i) (40 – 8) ÷ 8 = n
j) (48 ÷ 8) × 9 = n
32 ÷ 8 = 4
6 × 9 = 54
4. The order of operations is not important for these problems. You
just calculate from left to right. Ask the learners to substitute the
place holder (n) with the solution to check each answer.
a) (36 ÷ n) + 2 = n
b) 9 + (n × 3) = 30
6 – 2 × n = 36 (30 – 9) ÷ n = 3
4 × n = 36
21 ÷ 7 = 3
4 × 9 = 36 9 + (7 × 3) = 30
(36 ÷ 9) + 2 = 6
c) 7 × (4 – n) = 21
d)(n – 4) × 2 = 32
21 ÷ 7 + n = 4 32 ÷ 2 + 4 = n
3 + 1 = 4
16 + 4 = 20
7 × (4 – 1) = 21 (20 – 4) × 2 = 32
Unit 10
Patterns in number sentences
There are some very interesting patterns in numbers. If you can
notice the patterns, this can make calculations very easy. Sometimes
you don’t even have to calculate – you can simply follow a pattern!
Mental Maths Learner’s Book page 18
The learners explore the patterns in number sentences and create
their own patterns. They should notice, for example, that the
difference between the numbers in the solutions to (a) and (b) is
8 and 9 respectively although the numbers are not multiples of 8
and 9. They could use this knowledge to check their solutions.
Solutions
1. a) 1 × 8 + 1 = 9
2 × 8 + 1 = 17
3 × 8 + 1 = 25
4 × 8 + 1 = 33
5 × 8 + 1 = 41
6 × 8 + 1 = 49
7 × 8 + 1 = 57
8 × 8 + 1 = 65
9 × 8 + 1 = 73
10 × 8 + 1 = 81
2. a) + 8
b) 1 × 9 + 2 = 11
2 × 9 + 2 = 20
3 × 9 + 2 = 29
4 × 9 + 2 = 38
5 × 9 + 2 = 47
6 × 9 + 2 = 56
7 × 9 + 2 = 65
8 × 9 + 2 = 74
9 × 9 + 2 = 83
10 × 9 + 2 = 92
b) + 9
3. Answers will differ.
Mathematics Grade 4 Teacher’s Guide
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Activity 10.1
Learner’s Book page 18
The following patterns develop appreciation for the beauty of
numbers. Young learners need to think about the predictability and
structure of patterns so that their ideas about patterns link to the
algebraic thinking and reasoning required in the higher grades. They
often find large numbers fascinating. Allow the learners to work
with calculators if possible to extend the patterns. Do not expect
them to read the large numbers. They look at the structure of the
numbers to fill in the next number sentence in each sequence. Lead
a class discussion about the different patterns they observe.
Solutions
1. The learners might notice that the sum of the 1s in (b) forms the
pattern 2; 3; 4; 5. In (c) the sum of the digits forms the multiples
of 8 (16; 24; 32; 40). In (d) the sum of the digits forms square
numbers (1; 4; 9; 16). Tell the learners they will learn about
square numbers later.
a) 12 345 × 8 + 5 = 98 765
b) 12 345 × 9 + 6 = 111 111
c) 98 765 × 9 + 3 = 888 888
d) 11 111 × 11 111 = 123 454 321
2. Learners check their answers.
3. Learners complete the number sentences and look for patterns.
In (a) and (b) they find short cuts to multiply by 11 and 99.
Remind them that they learnt a shortcut for multiplication by 11.
They should notice the following relationship in the numbers for
multiplying by 99:
9 × 3 = 27
99 × 13 = 1 287
9 + 9 = 18
In (c) the multiplicands increase by 1 and the multipliers
decrease by 1. Ask learners to look for patterns in the units and
tens digits and predict the solutions to the next three number
sentences in the series.
a) 11 × 11 = 121
b) 99 × 12 = 1 188 c) 22 × 22 = 484
11 × 12 = 132
99 × 13 = 1 287
23 × 21 = 483
11 × 13 = 143
99 × 14 = 1 386
24 × 20 = 480
11 × 14 = 154
99 × 15 = 1 485
25 × 19 = 475
11 × 15 = 165
99 × 16 = 1 584
26 × 18 = 468
11 × 16 = 176
99 × 17 = 1 683
27 × 17 = 459
11 × 17 = 187
99 × 18 = 1 782
28 × 16 = 448
11 × 18 = 198
99 × 19 = 1 881
29 × 15 = 435
11 × 19 = 209
99 × 20 = 1 980
30 × 14 = 420
4. The number series involve additing consecutive odd, natural
and even numbers. Learners should notice that the sum of
consecutive odd numbers is always an uneven or odd number
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while the sum of consecutive even numbers is always an even
number. The solutions in (a) are uneven multiples of 3 while
the solutions in (b) are consecutive multiples of 3. In (c) the
solutions are multiples of 6.
a)
1+3+5=9
b) 1 + 2 + 3 = 6
c) 0 + 2 + 4 = 6
3 + 5 + 7 = 15 2 + 3 + 4 = 9
2 + 4 + 6 = 12
5 + 7 + 9 = 21 3 + 4 + 5 = 12
4 + 6 + 8 = 18
7 + 9 + 11 = 27 4 + 5 + 6 = 15 6 + 8 + 10 = 24
9 + 11 + 13 = 33 5 + 6 + 7 = 18 8 + 10 + 12 = 30
11 + 13 + 15 = 39 6 + 7 + 8 = 21 10 + 12 + 14 = 36
13 + 15 + 17 = 45 7 + 8 + 9 = 24 12 + 14 + 16 = 42
5. Problem-solving. Learners may work in groups to carry out
the investigation. They will solve the problem by trial and
improvement. Encourage them to work systematically, starting
with 1.
a) 1 + 7 + 8 = 16
1; 7; 8
2 + 6 + 8 = 16
2; 6; 8
3 + 5 + 8 = 16
3; 5; 8
3 + 6 + 7 = 16
3; 6; 7
4 + 5 + 7 = 16
4; 5; 7
b) The learners have to find the father’s age first. They work
backwards and use inverse operations.
Dog’s age: 12 – 5 = n
12 – 5 = 7
Nosimphiwe’s age: n × 3 = 36
36 ÷ 3 = 12 years
Explain if necessary:
• Number sequence: 3; 5; 7; ...
• Number series:
1+2=2
3+2=5
5+2=7
Assessment task 1: number sentences
The assessment task requires of learners to display knowledge of
what they have learnt about patterns and number sentences. They
work with flow diagrams, solve and write number sentences and
solve word problems.
Mathematics Grade 4 Teacher’s Guide
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Assessment task 1 Number sentences
1. Horses are transported to a race course in six trucks. Each truck
transports three horses.
Complete these tables.
a)
Number of trucks
6
2
4
5
8
10
Number of horses
b)
Number of horses
Number of legs
4
3
7
6
5
9
(12)
2. Write a number sentence for each item in the
tables above.
3. Calculate.
a) (8 × 3) + 4 = __
b) (5 × 7) + 5 = __
c) 9 + (63 ÷ 7) = __
d) 70 – (6 × 9) = __
e) 100 – (25 × 3) = __
(12)
(5)
4. Write number sentences to show how you would solve
these problems.
a) How many groups of 8 in 72?
(1)
b) What is 9 groups of 6?
(1)
c) Erica has 27 marbles. Sipho has 24 more marbles
than Erica. How many marbles does Sipho have?
(3)
d) Zerick has some marbles. He won 57 more marbles.
He now has 120 marbles. How many marbles did he
have in the beginning?
(4)
e) Lee has 65 marbles. Simo has 67 marbles. How
many marbles do they have altogether? (2)
Total [40]
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Assessment task 1 Number sentences
1. Multiply by 3 and 4.
a) Number of Number of
trucks
6
2
4
5
8
10
horses
18
6
12
15
24
30
b)
Number of
horses
4
3
7
6
5
9
Solutions
Number of
legs
16
12
28
24
20
36
(12)
2. Write numbers sentences for the input and output numbers.
a) 6 × 3 = 18
b) 4 × 4 = 16
2 × 3 = 6
3 × 4 = 12
4 × 3 = 12
7 × 4 = 28
5 × 3 = 15
6 × 4 = 24
8 × 3 = 24
5 × 4 = 20
10 × 3 = 30
9 × 4 = 36
(12)
3. Solve number sentences that require knowledge of the use
of brackets and the order of operations.
a) (8 × 3) + 4 = n
b) (5 × 7) + 5 = n
= 24 + 4
= 35 + 5
= 28
= 40
c) 9 + (63 ÷ 7) = n
d) 70 – (6 × 9) = n
= 9 + 9
= 70 – 54
= 81
= 70 – 50 – 4
= 20 – 4
= 16
e) 100 – (25 × 3) = n
= 100 – 75
= 25
(5)
4. Write number sentences to show how to solve word problems.
a) 72 ÷ 8 = n(1)
=9
b) 9 × 6 = n(1)
c) 27 + 24 = n
or
27 + 24 = n
= 27 + 3 + 21
= 20 + 20 + 7 + 3 + 1
= 30 + 21
= 51
= 51
(3)
d) n + 57 = 120
120 – 57 = 120 – 50 – 7
(breaking down)
= 70 – 7
= 63
or n + 57 = 120
57 + 3 = 60
(counting on)
60 + 60 = 120
60 + 3 = 63
63 + 57 = 120
(4)
e)
65 + 67 = n
65 + 65 + 3 = 130 + 3
(near doubles)
= 133
(2)
Total [40]
Mathematics Grade 4 Teacher’s Guide
Math G4 TG.indb 29
TERM 1
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2012/09/14 5:32 PM
Whole numbers: adding and subtracting
Unit 11
Count, order and compare numbers (2)
Tell the learners that they will do different kinds of counting. They
will count in multiples of different numbers.
Mental Maths Learner’s Book page 20
1. Snakes and ladders
Learners work in pairs to count and figure out which
numbers they should roll with the imaginary dice. The
activity involves simple counting on and back but requires
logical thinking. They go up the ladders and go down the
snakes when they land on numbers 5, 6, 8 and 14.
a) 9
b) There are various combinations, for example, 3 and 4
and 2 and 5. They have to work out a strategy to get past
the snakes.
c) They get from Start to Home by rolling 4, 4 and 6.
d) They get Home in exactly five rolls by throwing 4, 3, 5,
1, 3
2. a) Learners estimate the number of squares. Write their
numbers on the board so that they can later check their
estimates against the actual number of squares. Ask them
to count the squares. Encourage them to use the most
effective method. If anyone starts counting in 1s, quickly
remind them to use groups. They count the squares in
groups of 2, 5 and 10. When they count in 3s, they will
realise that there are 2 left. Ask them why this is the case.
If they do not know, tell them that 3 is an uneven number
and 50 is not a multiple of 3. List the multiples of 2, 3, 5,
10 and 3 so that they understand this.
Let them check the accuracy of their estimates.
b) Counting in 2s: 25 groups
c) Counting in 5s: 10 groups
d) Counting in 10s: 5 groups
e) Counting in 3s: 16 groups remainder 2
f) Answers will differ.
g) Answers will differ.
30
Math G4 TG.indb 30
Mathematics Grade 4 Teacher’s Guide
TERM 1
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Activity 11.1
1. a)
Learner’s Book page 21
15
20
25
30
10
5
40
35
115
0
45
50
105 60
55
95 100 65
70
90
75
end start
110
85
80
Learners make copies of the grid.
Explain what is meant by one
pathway. They can move left, up,
right and down, but not backwards
while filling in the multiples of 5.
Let them work in pairs.
Show them a copy of the completed grid if they did not
succeed in completing it correctly. You could also choose
not to show it to them but give them copies of it and let them
complete it for homework.
b) 100; 105; 110; 115; 120; 125; 130; 135; 140; 145; 150
c) The units always end in 0 or 5. The hundreds and tens make
a pattern: 10, 10; 11, 11; 12, 12; 13, 13; 14, 14; 15. Let them
write down the 5 times table. They can now predict that
when you multiply by 5, the answer should always have a 5
or 0 as a unit.
2. a) Learners make equal spaces when they plot the points on
the number line and write 100 in the centre, and 50 and 150
halfway between 0 and 100, and 100 and 200, and then fill
in the other values.
0
25
50
75
100
125
150
175
200
As with the multiples of 5, the units are always 0 or 5. The
tens and units are always 0, 25, 50, 75.
b) There is a 0 or a 5 in the units place.
c) All multiples of 25 are also multiples of 5, but not all
multiples of 5 are multiples of 25.
3.
0
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19
20 21 22 23 24 25 26 27 28 29
30 31 32 33 34 35 36 37 38 39
40 41 42 43 44 45 46 47 48 49
50 51 52 53 54 55 56 57 58 59
60 61 62 63 64 65 66 67 68 69
70 71 72 73 74 75 76 77 78 79
80 81 82 83 84 85 86 87 88 89
90 91 92 93 94 95 96 97 98 99
100 101 102 103 104 105 106 107 108 109
Mathematics Grade 4 Teacher’s Guide
Math G4 TG.indb 31
TERM 1
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2012/09/14 5:32 PM
Let the learners take part in a class discussion about their
observations. You can guide them by asking questions such
as, Do you notice any patterns?; What do you notice about the
numbers that are not shaded?; What is different between the
shaded and unshaded numbers? The even numbers are also
multiples of 2. Learners should realise that 0 is an even number.
Many people struggle to make sense of this concept. They will
realise that 0 is included in the rows of even numbers and so it
is an even number. If they are not convinced, ask them to define
an even number. If you divide even numbers by 2, there are no
remainders: 0 ÷ 2 = 0; there is no remainder. When you count in
even numbers you skip the uneven numbers: 0; 2; 4; 6; ... 0 fits
this description.
There are various legitimate reasons for classifying 0 as an even
number. The numbers that are not shaded are the uneven or odd
numbers. Ask the learners to describe as many patterns as they
can observe in the sequences of even and uneven numbers.
a) They are all even numbers.
b) They are all odd numbers.
c) Zero (0) fits the definition of an even number; 0 is even.
4. a)
0
3
6
9
12
15
18
21
24
27
30
33
36
39
b) The numbers are multiples of 3.
c) • Multiples alternate between odd and even numbers.
• There are groupings of 3 or 4 multiples for each tens
digit.
5. Learners complete the number chain. They should notice that
it is only the hundreds digits that change. They may struggle
with working with numbers larger than 1 000. Write down the
number that they name for 995 + 100. If they are not able to
give the correct number, write down the correct number and
explain it:
Take 5 from 100 and add it to 995; it gives you 1 000 plus 95
is 1 095. Let learners read the number aloud and also other
examples such as 1 020 and 1 050.
a) 495; 595; 695; 795; 895; 995; 1 095
395
1 095
+ 100
+ 100
495
+ 100
595
+ 100
695
995
+ 100
895
+ 100
795
+ 100
b) No, they are not multiples of 100, because 100 cannot divide
into them without leaving a remainder.
32
Math G4 TG.indb 32
Mathematics Grade 4 Teacher’s Guide
TERM 1
2012/09/14 5:32 PM
6. People count various things in many contexts in real life, for
example, bank tellers count big amounts of money each day.
Ask them where else people use counting, for example, shops
take stock they have to count each item in the shop and farmers
count their cattle.
Unit 12
Place value and representing numbers
Mental Maths Learner’s Book page 22
Place value board
This activity helps learners develop an understanding of place
value. Let them work in groups. They must use all four counters
to make each number. Make copies of the board and give them
counters. Let them write down the numbers they create under
Th, H, T and U.
1. The smallest number you can make with 4 counters if you
put all four counters on the units: 4
2. The largest number: 400
3. Numbers less than 50: 40, 31, 22, 13 and 4
This is a good opportunity to help the learners understand 2-digit
and 3-digit numbers. Allow them to discover that the 2-digit and
3-digit numbers created always have digits with a sum of 4.
4. Numbers larger than 100: any number where you also put a
counter on the hundreds rod
5. The smallest even number: 4
6. The largest even number: 400
7. The smallest odd number: 13
8. The largest odd number: 301
9. If there is a thousands rod, the largest number: 4 000
Activity 12.1
Learner’s Book page 22
This activity is aimed at developing understanding of 0 as a place
holder that represents absent digits (empty places) in numbers.
Learners often struggle with calculations such as 4 005 – 2679
when they have to subtract digits from 0, because they incorrectly
reason that you cannot subtract a big number from a small number
or that any number subtracted from 0 is equal to 0. Learners
should understand that 0 is not nothing. It represents the empty
set and stands for no objects or no quantity. The activity could
assist learners to understand the importance of 0 as a place holder
– emphasise the implications of ignoring its importance, as shown
in the examples of different numbers that are written with the
same digits, where there is no 0 to separate them correctly. Tell
the learners that people have used their body parts to count since
Mathematics Grade 4 Teacher’s Guide
Math G4 TG.indb 33
TERM 1
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2012/09/14 5:32 PM
ancient times when they did not have numbers to count. The word
digit means finger, as fingers are the parts that were mostly used
for counting – as they are even today! Help them understand for
example that 4 050 is a 4-digit number. Also tell them that in big
numbers we leave spaces between the hundreds and thousands to
make the numbers easy to read.
Mathematicians from ancient times represented 0 in numbers with
an empty space. Representing numbers such as 404, 440 and 44 as
4 4, 44 and 44 could have been confusing. The numbers 440 and
44 looked the same and the spaces between digits were not always
represented equally.
Solutions
1. Mathematicians from long ago used an empty space to represent
0 (zero).
a) 530 → 53
b) 503 → 5 3
53 → 53
53 → 53
c) 5 003 → 5 3
d) 101 → 1 1
5 3 → 53
110 → 11
e) 2 020 → 2 2
f) 1 500 → 15
202 → 2 2
15 → 15
2. Give the learners an opportunity to discuss this way of
representing numbers. They should notice that there could be
confusion with numbers such as 530 and 53.
• 5
••
••
•
••
b) 303 → 3 5
d) 3 050 → 3 5
3. a) 350 → 35
c) 3 005 → 3
e) 3 500 → 35
4. Help learners realise that without the empty space, the numbers
would be read as 324.
3
2
4
0
3
0
2
4
3
2
0
4
3 240: three thousand two hundred and forty
3 024: three thousand and twenty four
3 204: three thousand two hundred and four
5.
43
34
Math G4 TG.indb 34
435
Mathematics Grade 4 Teacher’s Guide
1 206
4 520
TerM 1
2012/09/14 5:32 PM
6.
Unit 13
43 →
435 →
1 206 → 1 000
4 520 → 4 000
400
+ 200
+ 500
40 + 3
+ 30 + 5
+ 6
+ 20
Place value
Flard cards are widely used for developing understanding of place
value – to break up or expand and build up numbers.
Activity 13.1
Learner’s Book page 23
Some of the calculations involve carrying. Help the learners with
halving (e). The learners pack out the numbers:
600 + 60 + 4; 200 + 40 + 6 and 400 + 8.
The flard cards that show the newly created numbers are placed
below the original place values. Adding 4 to 200 + 40 + 6 results in:
200 + 40 + 6
+
4
200 + 50
200 + 50 = 250
1.
a) Add 4:
2 46
664
600 + 60 + 4
+ 4
600 + 60 + 6
+ 50
600 + 110 + 8
700 + 10 + 8
–200
c) Minus 200:
500 + 10 + 8
500 + 10 + 8
d) Double:
1 000
+ 20 + 16
1 000
+ 30 + 6
500 + 15 + 3
e) Halve:
500 + 10 + 8
b) Add 50:
200 + 40 + 6
+ 4
200 + 40 + 10
200 + 40
+ 50
200 + 100
300
–200
100
100
200
100
40 8
400
+ 8
+ 4
400
+ 12
400 + 10 + 2
+ 50
400 + 60 + 2
–200
200
200
400
500
250
200
+ 60
+ 60
+ 120
+ 20
+ 10
+ 60
+
+
+
+
+
+
2
2
4
4
2
2
2. Numbers in the calculations for 664:
600 + 60 + 8 = 668
700 + 10 + 8 = 718
500 + 10 + 8 = 518
1 000 + 30 + 6 = 1 036
500 + 10 + 8 = 518
Mathematics Grade 4 Teacher’s Guide
Math G4 TG.indb 35
TERM 1
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2012/09/14 5:32 PM
Numbers in the calculations for 246:
200 + 50 = 250
300
100
200
100
Numbers in the calculations for 408:
400 + 10 + 2 = 412
400 + 60 + 2 = 462
200 + 60 + 2 = 262
500 + 20 + 4 = 524
200 + 60 + 2 = 262
The numbers that were doubled and then halved again give the
same solution.
3. The learners write down the numbers represented by the Dienes
blocks (plastic or wooden blocks). Learners should understand
that they have to carry units and tens when there are 10 or more.
Use copies of the Dienes block cards (see resource section) to
help learners build up units more than 9 and tens more than 90.
Give each learner or pair of learners a set of cards. Ask the
learners to write the numbers in expanded notation.
a) 100 + 30 + 5 = 135
b) 300 + 10 + 10 = 320
c) 100 + 30 + 13 = 143
d) 200 + 110 + 2 = 312
e) 200 + 60 = 260
f) 100 + 8 = 108
4. Use the template in this guide to make copies of the flard cards
and copy the cards on hard card and have them laminated. Each
learner should have a set. If you cannot manage this, make
sets for pairs or groups of learners. They use the flard cards to
represent the numbers in expanded notation as in question 2.
a) 58: 50 + 8
b) 733: 700 + 30 + 3
c) 999: 900 + 90 + 9
d) 606: 600 + 6
e) 530: 500 + 30
f) 1 001: 1 000 + 1
g) 1 900: 1 000 + 900
h) 1 050: 1 000 + 50
Mental Maths Learner’s Book page 25
This activity further develops understanding of place value.
Learners should realise that they can only use the calculator
keys indicated – they cannot use the C or CE keys to erase the
numbers on the screen. Calculators are optional.
1. a) Make 510: enter 510 (or 500 + 10 =)
b) Make 643:
510 (on screen) + 100 + 10 + 10 + 10 + 1 + 1 + 1 =
c) Make 402:
643 (on screen) – 100 – 100 – 50 + 5 + 1 + 1 + 1 =
or 643 – 100 – 100 – 10 – 10 – 10 – 10 – 1 – 1 =
d) Double 402:
402 (on screen) + 100 + 100 + 100 + 100 +1 + 1 =
36
Math G4 TG.indb 36
Mathematics Grade 4 Teacher’s Guide
TERM 1
2012/09/14 5:32 PM
e) Make 884:
500 + 100 + 100 + 100 + 50 + 10 + 10 + 10 + 1 + 1 + 1 + 1 =
f) Halve 884: 884 (on screen) – 100 – 100 – 100 – 100 – 10
– 10 – 10 – 10 – 1 – 1 =
2. Learners should realise that in 674, for example, the place
value of the digits is 6 hundreds + 7 tens + 4 units. The value
of the digits is 600 + 70 + 4.
Some learners might not have worked with thousands in
Grade 3, but they will work with big numbers in Grade
4. First observe what they do and assist them afterwards.
You can remind them about the Indian mathematicians’
invention of place value boards and numbers based on 10.
If they wanted a number to have a bigger value than units,
they moved the number one space to the left to become tens,
another place to the left would indicate hundreds and another
space left would indicate thousands.
a) 674: 600
b) 857: 50
c) 560: 0
d) 912: 900
e) 410: 10
f) 1 795: 90
g) 2 001: 1
h) 3 500: 3 000
Unit 14
Represent and compare numbers
Mental Maths Learner’s Book page 25
1. The learners have to find out whether 100 will be included in
the number sequences if they count on. Some of them might
do the counting to find out while others might reason as
follows:
a) 0; 2; 4; 6; 8; 10; ... Yes. 100 is an even number or when
you count in 2s, it will be included.
b) 15; 20; 25; 30; ... Yes. 100 is a multiple of 5.
c) 24; 34; 44; 54; ... No. These numbers do not have 0 as a
unit or they are not multiples of 10.
d) 0; 3; 6; 9; 12; ... No. 100 is not a multiple of 100 or if you
count on in 3s from 90 it is 93, 96, 99, 102. You skip 100.
e) 1; 3; 5; 7; ... No. These are odd numbers. 100 is not an
odd number.
2. a) Counting back in 10’s. It will reach exactly 0.
b) Multiples of 11 less than 99. It will reach exactly 0,
because 11 can be subtracted each time.
c) Multiples of 25 less than 225. It will reach 0 as 25 can be
subtracted each time.
d) Multiples of 4 less than 36. It will reach 0 if 4 is
subtracted each time.
e) Multiples of 100 less than 1 000. It will reach 0 if 100 is
subtracted each time.
Mathematics Grade 4 Teacher’s Guide
Math G4 TG.indb 37
TERM 1
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2012/09/14 5:32 PM
Learners can count on and back to check their thinking for
questions 1 and 2.
Activity 14
Learner’s Book page 25
1. a) 450
c) 866
e) 2 109
b) 74
d) 1 327
2. Check whether learners include 0 for absent values in numbers.
a) 700 + 20 + 1 = 721
b) 1 000 + 400 + 7 = 1 407
c) 900 + 90 + 9 = 999
d) 4 000 + 10 = 4 010
e) 500 + 50 = 550
3. Tell learners that +, –; × and ÷ are operation signs or symbols
and >, = or < are relationship signs or symbols. You can use
your index finger and thumb of the left and right hand to
represent the signs to help learners remember the signs.
a) 305 < 350
b) 2 500 = 2 500
c) 111 > 110
d) 36 < 63
e) 880 > 808
Unit 15
Swap and regroup numbers
Mental Maths Learner’s Book page 26
1. The learners study the counter groupings to find that
swapping numbers when you add and multiply does not
influence the solutions. Let them explain what they observe
and understand.
2. The activity enforces knowledge of the commutative
property by explaining the addition and multiplication facts
represented by the counters. Ask a few learners to show their
understanding by writing number sentences on the board.
a) 4 + 5 = 5 + 4 = 9
b) 9 + 5 = 5 + 9 = 14
c) 5 × 1 = 1 × 5 = 5
d) 5 × 3 = 3 × 5 = 15
e) 3 × 6 = 6 × 3 = 18
f) 7 + 8 = 8 + 7 = 15
3. The learners display knowledge of the commutative property.
a) 8 + 9 = 17
b) 7 × 6 = 42
c) 10 × 1 = 10
d) 20 × 3 = 60
e) 12 + 13 = 25
f) 9 + 8 = 17
g) 6 × 7 = 42
h) 1 × 10 = 10
i) 3 × 20 = 60
j) 13 + 12 = 25
Activity 15.1
Learner’s Book page 27
The learners investigate using the commutative property.
1. Learners show that they understand that the arrangement of the
numbers in addition expressions does not influence the solutions.
38
Math G4 TG.indb 38
Mathematics Grade 4 Teacher’s Guide
TERM 1
2012/09/14 5:32 PM
The learners use the commutative property to multiplication.
They apply the commutative property to addition number
sentences that involve bigger numbers.
4, 5. Lead a class discussion about using the commutative property
for subtraction and division. Let the learners use calculators
to find out that commutativity is not applicable to these
operations. If you swap the numbers in these calculations,
the answers differ. Pose questions – some true and some false
– to help the learners to understand these relationships. For
example, ask them if the following statements are true or false.
They should explain their reasoning.
10 – 5 = 5 – 10
20 – 10 = 10 – 20
20 ÷ 2 = 2 ÷ 20
10 ÷ 5 = 5 ÷ 10
Learners can use calculators to check the solutions and explain
what they notice.
2.
3.
Solutions
1. Drawings will differ.
2. a) 4 × 8 = 32
b) 3 × 10 = 30
c) 5 × 7 = 35
d) 2 × 9 = 18
e) 21 = 3 × 7
f) 3 × 8 = 8 × 3
g) 4 × 10 = 10 × 4
h) 9 × 3 = 3 × 9
3. a) 6 + 9 = 15
b) 15 + 7 = 22
c) 25 + 10 = 35
d) 120 + 80 = 80 + 120
e) 350 + 50 = 50 + 350
4, 5. Explanations may differ.
Activity 15.2
8 × 4 = 32
10 × 3 = 30
7 × 5 = 35
9 × 2 = 18
7 × 3 = 21
6×8=8×6
9 + 6 = 15
7 + 15 = 22
10 + 25 = 35
Learner’s Book page 28
The learners develop understanding of the associative and
distributive properties.
1. They have to calculate the total number of dots on the flower
petals. The obvious way to do this is to multiply the number of
dots by the number of petals in each flower and calculate the
total number of dots. Encourage and help them to find an easier
way to count the dots. Let them compare their strategies. They
should recognise that they could use multiplication by 5, for
example, 6 × 4 = (1 × 4) + (5 × 4). They can now regroup the
numbers so that they have (5 × 4) + (5 × 4) + (5 × 4); one four
has been added to 4 × 4. It is now easy to add 20 + 20 + 20.
Mathematics Grade 4 Teacher’s Guide
Math G4 TG.indb 39
TERM 1
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2012/09/14 5:32 PM
Learners explore Peter’s strategy to calculate the number of dots.
They should realise that 4 dots are taken from (a) and added to (b)
– compensation. Adding 4 and then subtracting 4 again results in
0 – the additive property of 0: 60 + 4 – 4 = 60 + 0.
2. Learners should use effective strategies to calculate the number
of dots in each array. After they have illustrated their strategies,
use one of the diagrams to show how the sets of dots are
regrouped to calculate smarter using the distributive property.
a) (5 × 3) + (5 × 3) + (5 × 3) + (5 × 3)
= 4 × (5 × 3)
= 4 × 15
= (2 × 15) + (2 × 15)
= 30 + 30
= 60 dots
b) (7 × 3) × 3
c) 3 × (5 × 5) + (1 × 5) + (2 × 5)
= 21 × 3
= (3 × 25) + 5 + 10
= (20 × 3) + (1 × 3)
= 75 + 15
= 60 + 3
= 90 dots
= 63
d) (5 × 6) + (5 × 6) + (5 × 6)
= 30 + 30 + 30
= 90
3. Learners can regroup the numbers using the associative property
so that they make multiples of 10.
a) (32 + 8) + (46 + 4)
b) (25 × 4) × 3
= 40 + 50
= 100 × 3
= 90
= 300
c) (80 + 20) + (90 + 10) + (7 + 3)
= 100 + 100 + 10
= 210
d) (7 × 10) × (4 × 5)
e) (60 – 20) – (7 + 3)
= 70 × 20
= 40 – 10
= 1 400
= 30
Assessment task 2: numbers and place value
The learners have now reached the end of week 3. They complete
an assessment task on counting, ordering, comparing and
representing numbers and place value. They work on their own.
The marks will be used as informal assessment.
The following concepts and skills are being assessed:
• counting groups of 10 using repeated addition or multiplication
• multiples of 2, 3, 10, 25, and 50
• ordering numbers
• doubling and halving
• expanded notation
• number properties.
40
Math G4 TG.indb 40
Mathematics Grade 4 Teacher’s Guide
TERM 1
2012/09/14 5:32 PM
Assessment task 2 Numbers and place value
1. Find a short way for counting the objects. Brackets show which
calculation should be performed first. Work out the totals.
a) 3 bunches with 10 bananas and 3 loose ones
b) 6 groups with 10 balloons and 2 loose ones
c) 4 bags with 10 apples and 6 loose ones
d) 5 pairs of hands with stretched fingers and 5 fingers on
one hand
e) 7 0bags with
(5)
+ 25 10 oranges each
+ 25 and 4 loose ones
+ 25
2. Complete the flow diagrams by counting in 25s and 50s.
a)
0
+ 25
+ 25
+ 25
+ 25
+ 25
b)
0
+ 50
+ 50
+ 25
+ 25
+ 50
+ 50
+ 50
(10)
0
+ 50
+ 50
+ 50
3. Fill in the missing numbers in the number sequences.
+ 50
a) 0; 3; 6; n; 12; 15; n; 21; 24; 27; +n50
b) n; 28; 26; 24; n; 20; 18; n; 14; 12; 10; 8; 6; 4; n; 0
c) 70; 80; n; 100; 110; 120; n; n; 150
(10)
4. Arrange numbers in descending order.
a) 885; 858; 824; 588; 482; 284; 248; 428
b) 605; 560; 506; 650; 602; 260; 620
(2)
5. Take the number 646 and follow the instructions.
a) Double 646.
b) Halve 646.
c) Add 300 to 646.
d) Subtract 346 from 646.
e) What must you add to 646 to get 700?
(5)
6. Write < or > to show which number is bigger.
a) 101 * 1 001
b) 240 * 204
c) 919 * 991
d) 727 * 772
e) 404 * 440
(5)
7. Write the numbers in expanded notation.
a) 555 = 500 + 50 + 5
b) 303 = 300 + 3
c) 330 = 300 + 30
(3)
8. Solve these problems. Show your calculations.
a) 4 + 7 = n 7 + 4 = n
b) 8 × 3 = n × 8
c) 12 + n = 8 + n
d) 14 + 19 + 16 + 11 = n
e) (4 × 5) + (4 × 5) + (4 × 5)
f) 25 + 3 + 5 + 7 + 5 = n
= n × (4 × 5)
= n
g) 3 × (6 + 10)
h) 9 × 4 = 36 36 = 4 × n
= (3 × n) + (3 × n)
= n + n
= n(15)
Total [55]
Mathematics Grade 4 Teacher’s Guide
Math G4 TG.indb 41
TERM 1
41
2012/09/14 5:32 PM
Assessment task 2 Numbers and place value
1. a) (3 × 10) + 3 = 33
c) (4 × 10) + 6 = 46
+ 25
e) (70 × 10)
+ 4 =2574
2 a) 0
25
+ 25
b)
0
+ 50
50
b) (6 × 10) + 2 = 62
d) (5 × 10) + 5 = 55
+ 25
50
+ 25
125
+ 25
+ 25
50
100
+ 25
+ 25
75
125
+ 25
100
+ 25
+ 50
100
+ 50
75
Solutions
(5)
150
250
200
+ 50
+ 50
(10)
100
150
50
+ 50
+ 50
+ 50
0
3. a) 0; 3; 6; 9; 12; 15; 18;
21; 24;
27; 30
250
200
+ 50
+ 50
b) 30; 28; 26; 24; 22; 20; 18; 16; 14; 12; 10; 8; 6; 2; 0
c) 70; 80; 90; 100; 110; 120; 130; 140; 150
(10)
4. a) 885; 858; 824; 588; 482; 428; 284; 248
b) 650; 620; 605; 602; 560; 506; 260
(2)
5. a) Double 646: 600 + 600 + 40 + 40 + 6 + 6
= 1 200 + 80 + 12
= 1 292
b) Halve 646: 300 + 20 + 3
= 323
c) Add 300 to 646: 600 + 300 + 40 + 6
900 + 40 + 6
= 946
d) Subtract 346 from 646:
646 – 346 = 600 – 300 + 40 – 40 + 6 – 6
= 300
c) 646 + n = 700
600 + 40 + 6 + 50 + 4 = 700
54 = n
(5)
6. a) 101 < 1 001
b) 240 > 204
c) 919 < 991
d) 727 < 772
e) 404 < 440
(5)
7. a) 555 = 500 + 50 + 5
b) 303 = 300 + 3
c) 330 = 300 + 30
(3)
8. a) 4 + 7 = 11 7 + 4 = 11 b) 8 × 3 = 3 × 8
c) 12 + 8 = 8 + 12
d) 14 + 19 + 16 + 11
= (14 + 16) + (19 + 11)
= 30 + 30
= 60
e) (4 × 5) + (4 × 5) + (4 × 5) f) 25 + 3 + 5 + 7 + 5
= 3 × (4 × 5) = (25 + 5) + (7 + 3) + 5
= 60 = 30 + 10 + 5
= 45
g) 3 × (6 + 10)
h) 9 × 4 = 36
36 = 4 × 9
= (3 × 10) + (3 × 6)
= 30 + 18
= 48
(15)
Total [55]
42
Math G4 TG.indb 42
Mathematics Grade 4 Teacher’s Guide
TERM 1
2012/09/14 5:32 PM
Unit 16
Learner’s Book page 29
Revision: find connections
In Term 1 this year the learners work with 3-digit numbers in
addition and subtraction. They break down numbers into place value
parts, compensate and fill up 10s as strategies to add and subtract
numbers in and out of context. During week 4, the Mental maths
sessions will involve the development of strategies to be applied
in addition and subtraction with 3-digit numbers. The learners will
fill up and break down multiples of 10, expand numbers and apply
grouping – the associative property – to calculate smarter.
Mental Maths Learner’s Book page 29
1. a) 26 + 4 = 30
b) 45 + 45 = 90
c) 38 + 12 = 50
d) 47 + 13 = 60
e) 91 + 0 = 91
f) 40 – 8 = 32
g) 35 – 0 = 35
h) 99 – 10 = 89
i) 67 – 25 = 42
j) 100 – 4 = 96
2. Learners work with calculations where the unknown is in
various positions – at the beginning, middle and end. They
also practise inverse operations.
a) 15 + 15 = 30
30 – 15 = 15
b) 18 + 12 = 30
30 – 12 = 18
c) 16 + 8 = 24
24 – 8 = 16
d) 19 + 11 = 30
30 – 11 = 19
e) 17 + 8 = 25
25 – 17 = 8
f) 25 – 10 = 15
15 + 10 = 25
g) 28 – 14 = 14
14 + 14 = 28
h) 27 – 15 = 12
15 + 12 = 27
i) 23 – 11 = 12
11 + 12 = 23
j) 100 – 4 = 96
96 + 4 = 100
3. Leaners explain what they notice about calculations.
Activity 16.1
Learner’s Book page 29
1. Learners add and subtract 3- and 2-digit numbers and apply
inverse operations. The calculations are simple so that the focus
is on identifying the inverse operations.
The answers are on the left and the checks on the right.
a) 150 + 150 300 – 150
= 100 + 100 + 50 + 50 = 300 – 100 – 50
= 300 = 150
b) 118 + 12 130 – 12
= 118 + 2 + 10 = 130 – 10 – 2
= 120 + 10 = 120 – 2
= 130 = 118
c) 216 + 8 224 – 8
= 216 + 4 + 4 = 224 – 10 + 2
= 220 + 4 = 214 + 2
= 224 = 216
Mathematics Grade 4 Teacher’s Guide
Math G4 TG.indb 43
TERM 1
43
2012/09/14 5:32 PM
d) 119 + 11 = 119 + 1 + 10
= 120 + 10
= 130
e) 317 + 8 = 317 + 3 + 5
= 320 + 5
= 325
f) 225 – 10 = 215
g) 128 – 14 = 114 (half of 28 is 14)
h) 527 – 15 = 527 – 20 – 5
= 507 + 5
= 512
i) 423 – 11 = 423 – 10 – 1 = 413 – 1 = 412
j) 900 – 4 = 900 – 10 + 6
= 890 + 6
= 896
130 – 11
= 130 – 10 – 1
= 120 – 1
= 119
325 – 8
= 325 – 10 + 2
= 315 + 2
= 317
215 + 10 = 225
114 + 14
= 128
15 + 512
= 512 + 8 + 7
= 520 + 7
= 527
11 + 412
= 412 + 10 + 1
= 422 + 1
= 423
896 + 4
= 800 + 96 + 4
= 800 + 100
= 900
2. Learners study the numbers in the grid. Ask them to look at the
2 × 2 squares to discover that the sum of the numbers is 20. Let
them add the numbers in the rows, columns and diagonals to
find out if the sum of the numbers is always 20. They explore
the numbers in more 2 × 2 squares to find more numbers with a
sum of 20. See examples below.
5 8
4 3
Unit 17
4 6
7 3
8 2
9 1
3 4
7 6
5 6
4 5
Number sentences and problem-solving
Mental Maths Learner’s Book page 30
1. Read the addition and subtraction problems to the learners
and help them understand the context of each problem.
Ask them to write open number sentences before doing
calculations. Encourage the learners to use effective mental
strategies and explain their thinking. Some learners might
apply advanced strategies and recall mental number facts
instantly but they are not always able to explain their
thinking and reasoning. They have to practise the skill
consistently. Other learners might need to practise strategies
that may appear simple as in the problems in question 1.
44
Math G4 TG.indb 44
Mathematics Grade 4 Teacher’s Guide
TERM 1
2012/09/14 5:32 PM
Understanding the contexts of these problems will help
learners solve the problems with larger numbers in the main
lesson.
2. The learners practise breaking up numbers in place value
parts again. Assist the learners in understanding and applying
the carrying and decomposition of numbers.
Solutions
1. a) R2,50 + R2,50 = R2 + R2 + 50c + 50c
= R5,00
b) 45 – 25 = 45 – 20 – 5
c) 50 – 35 = 50 – 25 – 10
= 25 – 5
= 25 – 10
= 20
= 15
d) 31 + 39 = 31 + 9 + 30
e) n + 12 = 28
= 40 + 30 28 – 12 = 28 – 10 – 2
= 70
= 18 – 2
= 16
2. a) 78 + 67 = 70 + 60 + 8 + 7
= 130 + 15
= 145
b) 294 + 189 = 200 + 100 + 90 + 80 + 4 + 9
= 300 + 170 + 13
= 400 + 80 + 3
= 483
c) 145 – 68 = (100 + 40 + 5) – (60 + 8)
= (130 + 15) – (60 + 8
= 70 – 7
= 77
d) 425 – 346 = (400 + 20 + 5) – (300 + 40 + 6)
= (300 + 110 + 15) – (300 + 40 + 6)
= 70 + 9
= 79
Activity 17.1
Learner’s Book page 30
Let the learners write number sentences before they solve the
problems. Let them work in groups. Allow learners to use strategies
they understand well.
Solutions
1. a) 357 + n = 475
475 – 357 = n
(inverse operation)
= 400 + 60 + 15
– 300 + 50 + 7 (decomposition)
100 + 10 + 8 = 118
118 litres of milk were sold on the second day.
Mathematics Grade 4 Teacher’s Guide
Math G4 TG.indb 45
TERM 1
45
2012/09/14 5:32 PM
b) 489 – 275 = n
= 400 + 80 + 9
– 200 + 70 + 5
200 + 10 + 4 = 214
214 people left early.
c) 1 000 – 859 = n
859 + 1 = 860 (counting on)
860 + 40 = 900
900 + 100 = 1 000
1 + 40 + 100 = 141
There are 141 chickens more than Farmer Brown has.
d) 546 + 454 = n
500 + 40 + 6
+ 400 + 50 + 4
= 900 + 90 + 10 = 900 + 100
1 000 people visited the fair.
e) n + 168 = 504
504 – 168 = n
500 – 100 = 400
400 – 60 = 340
340 – 8 = 332
332 + 4 = 336
336 guests arrived early.
(carrying)
(inverse operation)
(compensation)
2. Learners who have developed a good sense of number will realise
that they have to look for numbers in which the units give a sum
of 10 to get two numbers in the grid that add up to 1 000.
There are two possibilities:
386 + 614 = 1 000 and 462 + 538 = 1 000
Unit 18
Strategies for adding and subtracting
Mental Maths Learner’s Book page 31
Learners to record their answers on the Mental maths grid.
They should display knowledge of place value, addition and
subtraction.
1. (800 + 100) + 9 = 909
2. (700 + 100) + 46 = 846
3. (300 + 200) + 120 = 620
4. (800 + 100) + 29 = 929
5. (400 + 200) + 76 = 676
6. (700 – 300) + 87 = 487
7. 900 + 50 + 5 = 955
8. (500 – 200) + 20 + 1 = 321
9. (600 – 400) + 2 = 202
10. 100 – 10 = 90
46
Math G4 TG.indb 46
Mathematics Grade 4 Teacher’s Guide
TERM 1
2012/09/14 5:32 PM
Activity 18.1
Learner’s Book page 32
In this lesson, the learners investigate different addition and
subtraction strategies. Let them work in groups to check and discuss
how they understand the methods and how these strategies differ
from the ones that they use. Ask them to use inverse operations to
check the solutions in the examples.
Haadiya used breaking down numbers to solve the subtraction with
decomposition of tens to solve her problem. Roxy counted back on
the number line by subtracting the closest multiple of 100, 10 and 5.
Thabid used counting on to subtract. Anele used breaking down
numbers and Joshua broke down the second number to count on.
Solutions
1. Learners discuss different methods.
2. a) 118 + 357 = n
= 100 + 10 + 8
+ 300 + 50 + 7
400 + 60 + 15
400 + 70 + 5
= 475
475 – 357 = 118
(inverse operation)
(breaking down)
b) 214 + 275 = n
= 200 + 10 + 4
200 + 70 + 5
400 + 80 + 9
= 489
489 – 275 = 214
(inverse operation)
(breaking down)
(carrying)
Haadiya’s answer is correct.
Roxy’s answer is correct.
c) 141 + 859 = n
859 + 1 = 860
860 + 40 = 900
900 + 100 = 1 000
1 000 – 859 = 141
(inverse operation)
(counting on)
d) 1 000 – 454 = n
454 + 6 = 460
460 + 40 = 500
500 + 500 = 1 000
6 + 40 + 500 = 546
546 + 454 = 1 000
(inverse operation)
(counting on)
e) 336 + 168 = n
336 + 4 = 340
340 + 60 = 400
400 + 104 = 504
504 – 168 = 336
(inverse operation)
(building up)
Thabid’s answer is correct.
Anele’s answer is correct.
Joshua’s answer is correct.
Mathematics Grade 4 Teacher’s Guide
Math G4 TG.indb 47
TERM 1
47
2012/09/14 5:32 PM
Unit 19
Use different methods and operations
Mental Maths Learner’s Book page 32
Let the learners work in pairs to solve the problems.
Although they work with 1-digit numbers, it is not so easy
to decide which operation signs to use to get 5 as a solution.
a) 5 + 4 – 3 – 2 + 1 = 5
b) 2 + 9 – 3 – 4 + 1 = 5
c) 7 + 2 – 3 + 4 – 5 = 5
d) 7 + 5 + 4 – 3 – 8 = 5
e) 6 – 4 + 1 + 5 – 3 = 5
2, 3. There are eleven possible combinations, which include the
sum of 2, 3 and 4 numbers.
3+4=7
6 + 8 = 14
3+6=9
3 + 4 + 8 = 15
4 + 6 = 10
3 + 6 + 8 = 17
3 + 8 + 11
4 + 6 + 8 = 18
4 + 8 = 12
3 + 4 + 6 + 8 = 21
3 + 4 + 6 = 13
1.
Activity 19.1
Learner’s Book page 33
The learners use their own strategies to solve addition and
subtraction problems. Do not suggest a strategy at this stage. The
problems require carrying and decomposing. Observe how learners
deal with these problems. Below are strategies they could use.
1. a) 261 + 277 = n
250 + 250 = 500
(using near doubles)
10 + 20 = 30
1 + 7 = 8
261 + 277 = 538
b) 638 – 261 = n
638 = 600 + 30 + 8
261 = 200 + 60 + 1
500 + 130 + 8
– 200 + 60 + 1
300 + 70 + 7
638 – 261 = 377
c) 769 + 163 = n
769 = 700 + 60 + 9
163 = 100 + 60 + 3
800 + 120 + 12 = 900 + 30 + 2
769 + 163 = 932
d) 1 020 – 249 = n
1 020 – 250 = 750 + 20
= 770
770 + 1 = 771
1 020 – 249 = 771
48
Math G4 TG.indb 48
Mathematics Grade 4 Teacher’s Guide
(decomposition)
(carrying)
(compensation)
TERM 1
2012/09/14 5:32 PM
e) 900 – 567 = n
900 – 500 = 400
400 – 60 = 340
340 – 7 = 333
900 – 567 = 333
f) 1 007 – 498 = n
1 009 – 500 = 509
(break down and count back)
(add 2 to both numbers)
2. Learners check their solutions using inverse operations or
calculators.
3. a) 308 + 498 = 806
c) 808 – 587 = 219
e) 1 005 – 667 = 338
b) 700 – 497 = 203
d) 960 – 409 = 551
f) 1 500 – 878 = 622
4. a) 15 × 8 = n
= (15 × 4) + (15 × 4)
= 60 + 60
= 120
They paid R120.
b) 800 – 639 = n
800 – 600 = 200
200 – 30 = 170
170 – 4 = 161
161 seats were empty.
(breaking down)
Unit 20 More strategies for adding and
subtracting
Mental Maths Learner’s Book page 34
1. Help the learners make sense of the two strategies. One
learner uses adding the same number to both numbers. The
other counts on to make a multiple of 10, and breaking up.
Ask the learners to use the strategies to solve the problems.
a) 164 – 80 = n
= (164 + 20) – (80 + 20)
= 184 – 100
= 84
b) 257 – 70 = n
= (257 + 30) – (70 + 30)
= 287 – 100
= 187
c) 99 + 62 = n
= (99 + 1) + 61
= 100 + 61
= 161
Mathematics Grade 4 Teacher’s Guide
Math G4 TG.indb 49
TERM 1
49
2012/09/14 5:32 PM
d) 173 – 80 = n
= (173 + 20) – (80 + 20)
= 193 – 100
= 93
e) 139 + 42 = n
= (139 + 1) + 41
= 140 + 41
= 181
f) 199 + 51 = n
= (199 + 1) + 50
= 200 + 50
= 250
2. Answers may differ.
3. The learners use grouping by associating numbers that will
make the calculation easy. They make up groups of 10 or
multiples of 10.
a) 5 + 8 + 2 + 5 + 6 = (8 + 2) + (5 + 5) + 6 = 26
b) 7 + 4 + 1 + 3 + 6 = (7 + 3) + (6 + 4) + 1 = 21
c) 1 + 7 + 2 + 8 + 9 = (9 + 1) + (2 + 8) + 7 = 27
d) 16 + 3 + 4 + 17 = (16 + 4) + (17 + 3) = 40
e) 12 + 6 + 8 + 4 = (12 + 8) + (6 + 4) = 30
f) 12 + 4 + 8 + 6 + 9 = (12 + 8) + (4 + 6) + 9 = 39
g) 19 + 5 + 2 + 15 + 1 = (19 + 1) + (15 + 5) + 2 = 42
h) 27 + 11 + 9 + 3 + 6 = (27 + 3) + (11 + 9) + 6 = 56
i) 15 + 14 + 7 + 6 + 5 = (15 + 5) + (14 + 6) + 7 = 47
j) 18 + 13 + 12 + 17 = (18 + 12) + (13 + 17) = 60
Activity 20.1
Learner’s Book page 34
The learners perform calculations using blank number lines and
make reasonable estimates of how they should space the numbers
and arrows. You do not have to be too concerned about accurate
spacing. Learners break up the second number and add or subtract.
Let them use calculators to check their solutions.
1. a)
b)
c)
d)
e)
50
Math G4 TG.indb 50
–70
–8
129
59
–100
–40
356
–300
245
824
216
–45
–6
155
–50
324
285
–2
195
356
– 147
■=
209 = 206
545
369 = 149
■ =–149
824
558 = 266
■ =– 266
266
–90
Mathematics Grade 4 Teacher’s Guide
149
–8
274
–700
209
–45
200
–500
985
–7
256
545
129
78 = 51
■ =– 51
51
193
985
■ –=792
193= 193
TERM 1
2012/09/14 5:32 PM
f)
g)
h)
i)
j)
+70
+8
192
200
+100
+50
227
+9
327
+200
454
377
+50
654
636
+7
734
+50
936
+200
227
159 = 386
■ =+ 386
386
+30
704
+300
748
192
78 = 270
■ =+ 270
270
454
287 = 741
■ =+ 741
741
+6
986
■ =+ 992
636
356 = 992
992
–1
947
748
199 = 948
■ =+ 947
948
2. Learners check their calculations.
Unit 21
Add and subtract with 3-digit numbers
Mental Maths Learner’s Book page 35
1. Learners have to look for numbers with units that add up 10,
as a start.
67 + 133 = 200
126 + 74 = 200
108 + 92 = 200
101 + 99 = 200
149 + 51 = 200
2. Encourage the use of the words sum and difference.
75 is half of 150
73 + 27 = 100
86 is double 43
37 + 63 = 100
Activity 21.1
Learner’s Book page 35
Let learners work in pairs with a learner who is good at Maths to
help read the problems.
Solutions
1. 234 + 256 + 187 = n
230 + 250 + 180 = 250 + 250 + 160
= 660 + 4 + 6 + 7
= 677 cubes
2. 156 + 95 + 105 = n
150 + 90 + 100 = 250 + 50 + 40
= 340 + 6 + 5 + 5
= 356 beads
(build up multiples of 10)
(break down)
(compensation)
Mathematics Grade 4 Teacher’s Guide
Math G4 TG.indb 51
TERM 1
51
2012/09/14 5:32 PM
3. 354 + n = 967
– 354
613 eggs were laid
4.
(column subtraction)
467 + n = 1 000
467 + 3 = 470
(count on)
470 + 30 = 500
500 + 500 = 1 000
3 + 30 + 500 = 533 more books received
5. 900 – 699 = n
900 – 700 = 200
(compensation)
200 + 1 = R201 less during the sale
6. 908 – 595 = n
913 – 600 = 313 more cellphones sold (add 5 to both numbers)
7.
395 + 99 = n
395 + 100 = 495
495 – 1 = R494 was the price before the decrease in price
8. 405 – 196 = n
409 – 200 = 209 samoosas were sold on Saturday
Unit 22 More strategies for adding and
subtracting (2)
Mental Maths Learner’s Book page 36
1. a) (26 + 24) + 25 = 50 + 25 b) 65 + 5 + 30 = 100
= 75 65 + 35 = 100
c) 80 – 37 = 80 – 30 – 7
d) 70 – 20 – 8 = 50 – 8
= 50 – 7
= 42
= 43
2. Answers will differ.
3. a) 100 – 2 = 98
b) 81 – 5 = 81 – 1 – 4
= 76
c) 79 + 9 = 79 + 1 + 8
d) 105 – 65 – 2 = 40 – 2
= 88
= 38
e) 200 – 75 = 125
f) 700 – 10 – 1 = 690 – 1
= 689
g) 56 – 26 = 50 – 20 – 6 + 6 h) 590 + 10 + 6 = 606
= 30 + 0 606 – 16 = 590
= 30
i) 1 000 – 10 + 1 = 991
j) 795 + 5 + 15 = 815
52
Math G4 TG.indb 52
Mathematics Grade 4 Teacher’s Guide
TERM 1
2012/09/14 5:32 PM
Activity 22.1
Learner’s Book page 37
1. Learners should discuss the strategies and ask whether they
prefer their own or these strategies. They practise using near
doubles to solve 3-digit number addition.
a) 257 + 259 = 250 + 250 + 7 + 9
= 500 + 16
= 516
b) 329 + 327 = 320 + 320 + 9 + 7
= 640 + 16
= 656
c) 448 + 457 = 440 + 440 + 8 + 17
= 880 + 25
= 905
d) 366 + 359 = 350 + 350 + 16 + 9
= 700 + 25
= 725
e) 458 + 459 = 450 + 450 + 8 + 9
= 900 + 17
= 917
2. The method also involves compensation – adding to build up
10s or 100s by taking away a number and adding it again.
a) 127 – 89 = 128 – 90
= 128 – 20 – 70
= 108 – 70
= 38
b) 254 – 59 = 255 – 60
= 250 – 50 – 10 + 5
= 200 – 10 + 5
= 195
c) 275 – 139 = 276 – 140
= 136
d) 466 – 278 = 468 – 280
= 460 – 260 – 20 + 8
= 200 – 20 + 8
= 188
e) 833 – 547 = 836 – 550
= 830 – 530 – 20 + 6
= 300 – 20 + 6
= 286
Mathematics Grade 4 Teacher’s Guide
Math G4 TG.indb 53
TERM 1
53
2012/09/14 5:32 PM
Unit 23 Calculations with 4-digit numbers
Mental Maths Learner’s Book page 37
Make sure that learners understand that the sum of the numbers
in each row on the lines should be the same. Circles with the
numbers 12, 14 and 3 are not on the same line. Let learners work
in groups.
1.
5
6
16
12
14
3
10
9
8
2. Answers will differ.
Activity 23.1
Learner’s Book page 38
Tell the learners that they are not expected to work with more
than 3-digit numbers during this term, but they are doing so well
that they can now start practising working with thousands (4-digit
numbers). This should not be difficult because they worked with
thousands during previous lessons. Let them work in pairs.
Assist the learners in noticing 0 as the place holder. They often
struggle with subtraction problems that involve 0s as digits.
Solutions
1. They break up numbers and work in columns. Encourage them
to write numbers in the right places. They can use the = sign,
which should always be arranged underneath each other.
a) 2 896 – 1 424 = n
2 000 – 1 000 → 1 000
800 – 400 → 400
90 – 20 →
70
6 – 4 → 2
2 896 – 1 424 = 1 472
b) 2 784 – 2 743 = n
2 000 – 2 000 →
700 – 700 →
80 – 40 →
4–3→
2 784 – 2 743 =
0
0
40
1
41
c) 2 059 – 1 019 = n
2 000 – 1 000 → 1 000
50 – 10 →
40
9 – 9 → 0
2 059 – 1 019 = 1 040
54
Math G4 TG.indb 54
Mathematics Grade 4 Teacher’s Guide
TERM 1
2012/09/14 5:32 PM
d) 2 650 – 1 140 = n
2 000 – 1 000 → 1 000
600 – 100 → 500
50 – 40 → 10
2 650 – 1 140 = 1 510
e) 2 999 – 2 576 = n
2 000 – 2 000 →
0
900 – 500 → 400
90 – 70 →
20
9 – 6 → 3
2 999 – 2 576 = 423
f) 2 987 – 1 940 = n
2 000 – 1 000 → 1 000
900 – 900 →
0
80 – 40 →
40
7 – 0 → 7
2 987 – 1 940 = 1 047
2. a) 545 – 325 = 220
c) 905 – 578 = 327
e) 2 478 – 1 343 = 1 135
b) 673 + 468 = 348
d) 557 – 509 = 48
f) 1 515 + 567 = 2 082
3. The learners should realise that they have to divide the sum of
the weights by two. Let them struggle with the problem. Only
intervene if you see that they are really stuck. The learners have
to calculate the total weight the donkey has to carry and divide it
by two to find how much each bag would take. If you help them
and they understand the context, ask them to find another way to
solve the problem.
Total weight: 202 kg
Two ways:
6 + 22 + 35 + 38 = 28 + 35 + 38
101 kg = 101 kg
6 + 9 + 17 + 22 + 47 = 38 + 28 + 35
101 kg = 101 kg
Assessment task 3: addition and subtraction
Tell the learners that they will be assessed on the work they
have done during the last two weeks. They will display
knowledge of addition and subtraction. The knowledge and skills
to be assessed are:
• building up and breaking down multiples of 10
• addition and subtraction using their own strategies
• addition on empty number lines
• word problems
Mathematics Grade 4 Teacher’s Guide
Math G4 TG.indb 55
TERM 1
55
2012/09/14 5:32 PM
Assessment task 3 Addition and subtraction
Work out the answers to questions 1 and 2.
1. a) 35 + 35 = n
b) 89 + 11 = n
c) 75 + 0 = n
d) 91 + 9 = n
e) 60 – 12 = n
f) 110 – 11 = n
g) 80 – 16 = n
h) 120 – 0 = n
i) 55 – 20 = n
j) n – 8 = 9
2. a)
b)
c)
d)
e)
(10)
n + 450 = 900
318 + 112 = n
400 – n = 375
900 – 15 = n
244 – 30 = n(5)
3. Break up the numbers to make it easy to work out the answers.
a) 256 + 367 = n
b) 550 – 367 = n
c) 606 + 228 = n
d) 487 – 209 = n
e) 326 + 426 = n(5)
4. Regroup the numbers and work out the answers.
a) 15 + 17 + 3 + 5 + 2 = n
b) 24 + 21 + 6 + 9 = n
c) 40 – 9 – 20 = n
d) 70 – 11 – 30 – 1 = n
e) 150 + 18 + 150 + 2 = n(5)
5. Draw empty number lines. Show how to work out
the answers.
a) 235 + 276
b) 456 + 358
(2)
6. Solve these problems.
a) Last week the learners collected cans for recycling.
This week they collected 359 cans. There are now
500 cans. How many cans did they collect last week?
(Use counting on.)
b) John has 134 marbles. Anele has 67 more marbles than
John. How many marbles does Anele have? (Break up
numbers.)
c) During the first week of May Spaza shop sold
346 loaves of bread. In the second week they sold
378 loaves. How many loaves of bread did they sell
altogether during the two weeks? (Break up numbers.) (3)
Total [30]
56
Math G4 TG.indb 56
Mathematics Grade 4 Teacher’s Guide
TERM 1
2012/09/14 5:32 PM
Assessment task 3 Addition and subtraction
Solutions
1. a) 35 + 35 = 30 + 30 + 5 + 5 b) 89 + 11 = 89 – 10 – 1
= 70
= 78
c) 75 + 0 = 75
d) 91 + 9 = 90 + 10
= 100
e) 60 – 12 = 60 – 10 – 2
f) 110 – 11 = 100 – 10 – 1
= 48
= 99
g) 80 – 16 = 80 – 10 – 6
h) 120 – 0 = 120
= 74
i) 55 – 20 = 50 – 20 + 5
j) n – 8 = 92
= 35 92 – 2 – 6 = 84
(10)
2. a)
n + 450 = 900
900 – 400 – 50 = 500 – 50
= 450 (half of 900)
450 + 450 = 900
b) 318 + 112 = 300 + 100 + 18 + 12
= 430
c)
400 – n = 375
400 – 300 – 75 = 25
400 – 25 = 375
d) 900 – 15 = 900 – 10 – 5
= 885
e) 244 – 30 = 210
(5)
3. The learners might use their own strategies but they should
be similar to the ones given below. You should check
whether they understand carrying in addition calculations and
decomposing in subtraction.
a) 256 + 367 = 200 + 300 + 50 + 60 + 6 + 7
= 500 + 110 + 13
= 600 + 20 + 3
= 623
b) 550 – 367 = 500
– 300
400
– 300
100
= 183
+
+
+
+
+
50
60
140
60
80
+
+
+
+
7
10
7
3
c) 606 + 228 = 600 + 200 + 20 + 6 + 8
= 820 + 10 + 4
= 834
d) 487 – 209 = (400 + 80 + 7) – (200 + 9)
= 200 + 70 + 17 – 9
= 270 + 8
= 278
Mathematics Grade 4 Teacher’s Guide
Math G4 TG.indb 57
TERM 1
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2012/09/14 5:32 PM
e) 326 + 426 = 300 + 400 + 20 + 20 + 6 + 6
= 740 + 10 + 2
= 752
(5)
4. Check if the learners can identify and group combinations of
numbers that add up to multiples of 10.
a) 15 + 17 + 3 + 5 + 2 = (15 + 5) + (17 + 3) + 2
= 20 + 20 + 2
= 42
b) 24 + 21 + 6 + 9 = (24 + 6) + (21 + 9)
= 30 + 30
= 60
c) 40 – 9 – 20 = 40 – 20 – 9
= 20 – 9
= 11
d) 70 – 11 – 30 – 1 = 70 – 30 – 11 – 1
= 40 – 10 – 2
= 38
e) 150 + 18 + 150 + 2 = (150 + 150) + (18 + 2)
= 320
(5)
5. Learners could count on and show the following calculations
on the number lines.
a) 235 + 276 = n
235 + 200 → 435 + 70 → 505 → + 5 → 510 + 1 → 511
b) 456 + 358 = n
450 + 350 → 800 + 10 → 810 + 4 → 814 (2)
6. If some learners struggle to read they could work together
in a group and you could read the problems to them without
helping them to solve the problems. Learners should write
number sentences to show their understanding of the
calculation process. In question (a), they should understand
that the unknown is at the start. They need to use the inverse
operation to solve the problem.
a) n + 359 = 500
500 – 359 = 500 – 300 – 50 – 9
= 150 – 9
= 141
141 cans were collected.
b) 134 + 67 = n
c) 346 + 378 = n
131 + 70 = 201 346 + 4 + 300 + 50 + 20 + 4
Anele has 201 marbles. 346 + 4 = 350
350 + 300 = 650
650 + 50 = 700
700 + 24 = 724
They sold 724 loaves. (3)
Total [30]
58
Math G4 TG.indb 58
Mathematics Grade 4 Teacher’s Guide
TERM 1
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Numeric patterns
Ask the learners where they observe patterns in real life. Explain
that we find patterns in our daily routines, the weather, designs and
in numbers. Let them give examples of these.
Unit 24 Patterns in counting sequences
Mental Maths Learner’s Book page 39
Ask the learners to describe the patterns in each row and between
rows. Let them look at the shapes to describe them according
to the number of sides. Relate this to the work with shapes that
learners do in their work with space and shape. You can write
these words on cards and paste on your New Maths words board.
Explain the meaning of the Latin prefixes: tri- 3; penta- 5; hexa6; hepta- 7; octa- 8; nona- 9; deca- 10. Gon means angle. You
should not spend much time on the terminology; rather focus
on the patterns in the number sequences. The activity allows
learners to practise multiples of numbers. The numbers in the
triangles are multiples of 3; in the squares there are multiples of
4, and so on. Ask the learners which multiples are common in the
sequences (important for Grade 5 work). Ask the learners to give
the next four numbers in each sequence.
Solutions
1. a)
c)
e)
g)
2. a)
d)
g)
3. a)
c)
e)
g)
15; 18; 21; 24
25; 30; 35; 40
35; 42; 56
45; 54; 63
+3
+6
+9
multiples of 3
multiples of 5
multiples of 7
multiples of 9
b)
d)
f)
h)
20; 24; 28; 32
30; 36; 42; 48
40; 48; 56
50; 60; 70
c) + 5
f) + 8
b)
d)
f)
h)
multiples of 4
multiples of 6
multiples of 8
multiples of 10
b) + 4
e) + 7
h) + 10
4. The number of sides of each shape gives the number to add
to each term to find the next term.
Activity 24.1
Learner’s Book page 40
In this activity, learners consolidate the multiplication tables. Allow
enough time for learners to explore the variety of patterns in the grid.
Ask them to identify multiples of different numbers. Let them look
for numbers that are common in different multiples. They should
Mathematics Grade 4 Teacher’s Guide
Math G4 TG.indb 59
TERM 1
59
2012/09/14 5:32 PM
explore the units in multiples to identify patterns. Multiples of 4, for
example, always end in 4, 8, 2, 6, 0. They can learn the multiples
by heart for homework. Our number system is based on a system
of patterns and predictability. Learners should therefore be able to
identify relationships in patterns and give reasons and evidence for
the existing patterns. When they explore sequences in numbers, they
should recognise the evidence for patterns and relationships.
You could give the learners copies of the grid. After they have
filled in the missing numbers, they should explain how they did
it. Let them describe the patterns they notice in the rows, columns
and diagonals. Ask them to create multiplication and division
calculations using the numbers in the grid, for example: 32 = 8 × 4
and 24 ÷ 6 = 4. Ask questions to consolidate of multiplication and
division facts, for example: How many 7s are there in 63? How
many 8s are there in 64? Let them name multiplication and division
facts for 30, for example:
30 = 5 × 6
6×5
3 × 10
10 × 3
60 ÷ 2
90 ÷ 3
(70 – 10) ÷ 3.
Solutions
1.
1
2
3
4
5
6
7
8
9
10
2
4
6
8
10
12
14
16
18
20
3
6
9
12
15
18
21
24
27
30
4
8
12
16
20
24
28
32
36
40
5
10
15
20
25
30
35
40
45
50
6
12
18
24
30
36
42
48
54
60
7
14
21
28
35
42
49
56
63
70
8
16
24
32
40
48
56
64
72
80
9
18
27
36
45
54
63
72
81
90
10
20
30
40
50
60
70
80
90
100
2, 3. Learners find relationships and describe patterns.
Activity 24.2
Learner’s Book page 40
Exploring patterns in calendars could help learners learn the facts of
the 7 times table. The learners should identify the multiples of 7 and
also intervals of 7. Let them explore the numbers and notice:
• The difference between the numbers in the columns is always 7.
• If you add the numbers below each other in the 1st two rows you
will get consecutive odd numbers as a sum:
0 + 7 = 7; 1 + 8 = 9; 2 + 9 = 11; 3 + 10 = 13, and so on.
• The difference between the numbers in the diagonals (bottom left
to right in row above) is always 6, for example:
11 – 5 = 6; 26 – 20 = 6; 29 – 23 = 6.
• Multiples of 7 appear in column 4.
60
Math G4 TG.indb 60
Mathematics Grade 4 Teacher’s Guide
TERM 1
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Remind the learners that when they investigate patterns, they should
ask questions such as: What would happen if I add the numbers?
and What would happen if I multiply the numbers? They use the
four basic operations. Ask them to explore the numbers in different
2 × 2 and 3 × 3 squares in the calendar. Below are relationships they
might notice.
3 + 11 = 14
4 + 10 = 14
3 + 10 = 13
4 + 11 = 15
10 – 3 = 7
11 – 4 = 7
10 – 4 = 6
11 – 3 = 8
4 + 12 + 20 = 36
6 + 12 + 18 = 36
13 – 6 = 7
20 – 13 = 7
20 – 6 = 14
4 + 11 = 15
5 + 12 = 17
6 + 13 = 19
3
4
10
11
4
5
6
11
12
13
18
19
20
unit 25 number grids and patterns
Mental Maths
Learner’s Book page 40
This activity allows learners to practise doubling, halving and
the four basic operations (+, –, ×, ÷). They can observe various
number patterns. Ask the learners what the numbers in the
grid are called. The first column contains natural or counting
numbers, the second column contains uneven numbers and the
third column contains even numbers. Let them describe the
patterns. The differences between the numbers in the first two
columns increase to form consecutive counting numbers:
0; 1; 2; 3; ... The difference between the numbers in the second
and third columns is 1. You double the numbers in the first
column to get the numbers in the third column. Ask the learners
to find the missing numbers in the extension of the table. They
must first double the numbers in the first column to fill in those
in the third column. They subtract 1 from the numbers in the
third column to fill in the numbers in the second column.
18 × 2 = 36 and 36 – 1 = 35
34 × 2 = 78 and 78 – 1 = 77
19 × 2 = 38 and 38 – 1 = 37
52 × 2 = 104 and 104 – 1 = 103
20 × 2 = 40 and 40 – 1 = 39
60 × 2 = 120 and 120 – 1 = 119
21 × 2 = 42 and 42 – 1 = 41
75 × 2 = 150 and 150 – 1 = 149
Mathematics Grade 4 Teacher’s Guide
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TerM 1
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2012/09/14 5:32 PM
Solutions
1. First column: natural numbers
Second column: odd numbers
Third column: even numbers
2.
even
18 × 2 = 36
19 × 2 = 38
20 × 2 = 40
21 × 2 = 42
34 × 2 = 68
52 × 2 = 104
60 × 2 = 120
75 × 2 = 150
18 35 36
19 37 38
20 39 40
21 41 42
34 67 68
52 103 104
60 119 120
75 149 150
odd
18 × 2 – 1 = 35
19 × 2 – 1 = 37
20 × 2 – 1 = 39
21 × 2 – 1 = 41
34 × 2 – 1 = 67
52 × 2 – 1 = 103
60 × 2 – 1 = 119
75 × 2 – 1 = 149
Activity 25.1
Learner’s Book page 40
Ask the learners to look at the grid they worked with in Mental
maths. They should realise that the numbers are created as follows:
• second column: double the number in the first column and
subtract 1 from the number in the third column
• third column: double first column
• first column: half of the number in the third column, or add 1 and
halve the number in the second column.
23
24
30
35
42
45
47
59
69
83
46
48
60
70
84
1
0
1
1 2
learners
4 3 3
21
2
0 125.20
Activity
1
0
1
1
2
3
0
1
2 1 2 2
4 3 3 4 3
5 6 4 5 6 4
7 8 5 7 8 5
9 10
9 10
2
2
8
15 16
17
51
55
61
67
81
52
56
62
68
82
Learner’s Book page 41
2
1. Ask the
to investigate
the numbers in the steps before
4 33 4 3
3
filling in the missing numbers in the sections from the steps.
5 6 4 5 6 45 6 4
They work
with natural, even and uneven numbers again. The
57 8 5
7 8 numbers
7 8and5 halve
learners double
and count on and back. They
9 10 the circles are double
9 10 below
9 10the numbers
should realise that
those in the circles. The numbers to the left and above are 1 less
or 1 more than those on the right and below.
b) 8 20
c)
a)
56
20 52 27
820
525627 52 27
8
56
26
30 54
57 58 3057 58 30
575458 53
1527 28 5315
53 54
15391640 15 16
39 27
40 2839154027 28
59 60
59 60
55 5629 30 55595660 55 56
1741 42 17 41 29
42 3041 4229 30
1438
14
15 16
17
14
26
28
31
34
41
14 38 26
38
26
e) 56
d) 52 27
26
52 5627
38 20
26
57 58 30 57 58 30
53 15
54
27 28 15 27 28
53 54
39 40
59 60
59 60
55 56
29 30
55 56
1741 42
41 42
29 30
14 38 8 20
15391640
62
Math G4 TG.indb 62
Mathematics Grade 4 Teacher’s Guide
TerM 1
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2. The learners complete numbers in 3 × 3 squares taken from a
calendar for February 2012. They should realise that they must
work with consecutive natural numbers and , for example,
differences of 7. Let them check their solutions on a calendar
and explain the patterns they notice.
a) 5 6 7
b) 9 10 11
c) 12 13 14
12 13 14
19 20 21
16 17 18
19 20 21
23 24 25
26 27 28
3. Learners discover that they must add and subtract constant
numbers when counting forwards and backwards. The terms
are not multiples of the numbers they count in; they count in
intervals of these numbers.
a) 5; 16; 27; 38; 49; 60; 71; 82; 93
Add 11; count in intervals of 11.
b) 4; 11; 18; 25; 32; 39; 46; 53; 60; 67
Add 7; count in intervals of 7.
c) 1 000; 850; 700; 550; 400; 250; 100
Count 150 back.
d) 200; 185; 170; 155; 140; 125; 110; 95
Count 15 back.
e) 100; 92; 84; 76; 68; 60; 52; 46; 38; 30
Count 8 back.
unit 26 number groups and patterns
Mental Maths
Learner’s Book page 41
Ask the learners to multiply the pairs to see if they get 18. Let
learners give the factor pairs of the given numbers. List the
numbers 1 to 24 on the board. They have to work systematically,
starting from 1 and exploring which numbers form pairs to give
the products 12, 15, 16, 20 and 24. Ask them to describe any
patterns they notice. They should realise that 24 has more pairs
(factors) than the other numbers.
1 × 12 = 12 12 × 1 = 12
2 × 6 = 12 6 × 2 = 12
3 × 4 = 12 4 × 3 = 12
1 × 16 = 16
2 × 8 = 16
4 × 4 = 16
16 × 1 = 16
8 × 2 = 16
1 × 20 = 20 20 × 1 = 20
2 × 10 = 20 10 × 2 = 20
5 × 4 = 20 4 × 5 = 20
1 × 24 = 24
2 × 12 = 24
3 × 8 = 24
4 × 6 = 24
24 × 1 = 24
12 × 2 = 24
8 × 3 = 24
6 × 4 = 24
1 × 15 = 15 15 × 1 = 15
3 × 5 = 15 5 × 3 = 15
Mathematics Grade 4 Teacher’s Guide
Math G4 TG.indb 63
TerM 1
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2012/09/14 5:32 PM
Activity 26.1
Learner’s Book page 41
The learners continue to look for relationships in number
sequences and calculations. They study the numbers in the diagram
to find that the numbers in the row make a sum of 8 and in the
column a sum of 10.
Solutions
1. a)
c)
e)
7
5
b)
12
9
11 15
9
8
21
23
d)
13 11 24
16
4
12 15 27
7
8
31
35
25
9
34
2. Encourage learners to look for relationships in the numbers in
the 5, 4, 8 and 9 times tables. This will allow them to predict
and judge the reasonableness of solutions when they multiply
or divide by these numbers. Ask the learners to extend the
sequences to see whether the patterns are consistent.
a) 5 times table: 0; 5; 0; 5; 0; 5; ...
The units are always 0 or 5.
b) 4 times table: 0; 4; 8; 2; 6; 0; ...
There is a difference between the digits is 4; 4; 6; 4; 6; ...
c) 8 times table: 0; 8; 6; 4; 2; 0; 8; 6; 4; 2; 0
There is a repetition in the digits. The difference between the
digits is 8; 2; 2; 2; 2; 8; 2; 2; 2; 2.
d) the 9 times table: 0; 9; 8; 7; 6; 5; 4; 3; 2; 1; 0
The value of the digits decrease by 1 (excluding the first 0)
so that the digits form descending consecutive counting
numbers.
3. Learners should see that for the 5 times table the units are
0; 5; 0; 5 and after 100, the hundreds and tens are 10; 12; 15; 20.
Let them extend the sequence to see how the pattern develops
further.
4. a) 0 = zero groups of 25
25 = 1 group of 25
50 = 2 groups of 25, and so on
64
Math G4 TG.indb 64
Mathematics Grade 4 Teacher’s Guide
TERM 1
2012/09/14 5:32 PM
b) Add 0 when multiplying the numbers by 10, which will go
into the thousands. Let them record the sequence.
0; 25; 50; 75; 100; 125; 150; 175; 200
0; 250; 500; 750; 1 000; 1 250; 1 500; 1 750; 2 000
c) Let them find out how many groups of 250 there are in each
number.
0 = zero groups of 250
250 = 1 group of 250
500 = 2 groups of 250
750 = 3 groups of 250, and so on
250 + 250 = 2 × 250 = 500
250 + 250 + 250 = 3 × 250 = 750
250 + 250 + 250 + 250 = 4 × 250 = 1 000
250 + 250 + 250 + 250 + 250 = 5 × 250 = 1 250
250 + 250 + 250 + 250 + 250 + 250 = 6 × 250 = 1 500
250 + 250 + 250 + 250 + 250 + 250 + 250 = 7 × 250 = 1 750
250 + 250 + 250 + 250 + 250 + 250 + 250 + 250
= 8 × 250 = 2 000
h) 500 + 500 = 2 × 500 = 1 000
i) 500 + 500 + 500 = 3 × 500 = 1 500
j) 500 + 500 + 500 + 500 = 4 × 500 = 2 000
5. a)
b)
c)
d)
e)
f)
g)
Mental Maths Learner’s Book page 42
This activity assists in the development of skills needed in work
on number patterns.
1. Let the learners investigate the numbers. They should notice
that the top numbers are multiplied by themselves. The bottom
numbers are 1 more and 1 less than the top ones and the
answers are 1 fewer. Let them explore more examples. Suggest
that they include 1 × 1 to observe that 1 × 1 = 1 and 0 × 1 = 0,
so the pattern is consistent. Ask the learners why they think the
pattern works. Let them experiment with numbers bigger than
10. Give them calculators. Below are more examples:
1×1=1
2×2=4
4 × 4 = 16
5 × 5 = 25
6 × 6 = 36
7 × 7 = 49
2×0=0
3×1=3
5 × 3 = 15
6 × 4 = 24
7 × 5 = 35
8 × 6 = 48
Mathematics Grade 4 Teacher’s Guide
Math G4 TG.indb 65
TERM 1
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2012/09/14 5:32 PM
2. Learners explore numbers in the 9 times table and should
note that when read across (horizontally), the digits are
swapped. The sum of the digits always adds up to 9. We do
not normally write numbers as 09; we only write it this way
for the purpose of this investigation. We do insert a 0 when
for example, we write down dates with 1-digit numbers:
09/08/13, 9 August 2013.
9 × 5 = 45
9 × 6 = 54
9 × 4 = 36
9 × 7 = 63
9 × 3 = 27
9 × 8 = 72
9 × 2 = 18
9 × 9 = 81
9 × 1 = 09
9 × 10 = 90
Patterns
• The multipliers decrease by 1 in the columns.
• The sum of the two digits in the solutions is always 9.
• S
ubtracting the digits in the products always gives an
uneven number:
5 – 4 = 1; 6 – 3 = 3; 7 – 2 = 5; 8 – 1 = 7, except for 9 – 0.
Zero is not an odd number.
• M
ultiplying the two digits in the solutions results in a
pattern of even numbers:
4 × 5 = 20; 3 × 6 = 18; 2 × 7 = 14; 1 × 8 = 8; 9 × 0 = 0.
• T
he tens digits decrease by 1 (actually 10) and the units
increase by 1 in the first column. The opposite happens in
the second column.
Assessment task 4: numeric patterns
Investigate patterns while you complete number sequences, work
with calendar squares, create factors of numbers, find missing
numbers in a table and create your own patterns.
66
Math G4 TG.indb 66
Mathematics Grade 4 Teacher’s Guide
TERM 1
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Assessment task 4 Numeric patterns
1. Complete each number pattern.
a) n; n; n; 12; 16; 20; n; n; n; 36
b) 39; n; n; 30; n; n; n; 18; n; n
c) n; n; 150; n; n; 75; 50; n; n
d) 0; 250; n; n; 1 000; n; n; 1 750; n
e) 90; n; n; 63; 54; n; n; 27; 18; n; n(10)
2. Complete the calendar squares.
a)
b)
22
c)
18
29
7
26
d)
e)
21
19
(10)
28
3. Give the multiplication pairs for each number.
a) 8
b) 15(4)
4. Create two different number patterns using the
pattern below.
4; 8; n; n; n; n; n(2)
5. Study the numbers in the multiplication triangle. Find out
which numbers should be written in the empty squares.
1
2
3
4
6
9
4
12
5
6
1
10
12
2
15
16
25
3
24
4
30
5
6
(4)
Total [30]
Mathematics Grade 4 Teacher’s Guide
Math G4 TG.indb 67
TERM 1
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2012/09/14 5:32 PM
Assessment task 4 Numbers and place value
Solutions
1. The number sequences involve multiples of 4, counting back in
multiples of 3, 25 and 9 and counting on in multiples of 250.
a) 0; 4; 8; 12; 16; 20; 24; 28; 32; 36
b) 39; 36; 33; 30; 27; 24; 21; 18; 15; 12
c) 200; 175; 150; 125; 100; 75; 50; 25; 0
d) 0; 250; 500; 750; 1 000; 1 250; 1 500; 1 750; 2 000
e) 90; 81; 72; 63; 54; 45; 36; 27; 18; 9; 0
(10)
2. The difference between numbers in a column is 7.
a)
d)
22
23
29
30
11
12
18
19
b)
e)
18
19
25
26
20
21
27
28
c)
7
8
14
15
(10)
3. Learners should work systematically, starting with 1 × 8 = 8.
a) 1 × 8 = 8
2 × 4 = 8
b) 1 × 15 = 15
3 × 5 = 15
4. The learners might create various sequences. Let them explain
how they created the sequences after the assessment task.
Suggestions that involve constant and non-constant
differences:
4; 8; 16; 32; 64; ... (times 2)
4; 8; 12; 16; 20; ... (add 4)
4; 8; 16; 20; 28; ... (add 4; 8; 4; 8)
(2)
5.
1
2
3
4
5
6
1
4
6
8
10
12
2
9
12
15
18
3
16
20
24
4
25
30
5
36
6
(4)
68
Math G4 TG.indb 68
Mathematics Grade 4 Teacher’s Guide
Total [30]
TERM 1
2012/09/14 5:32 PM
Whole numbers: multiplication and division
Unit 27 Multiplication by grouping and
repeated addition
This week learners continue to work with multiplication and
division. Ask questions as you refer to the concepts and skills in the
unit. The pictures should help learners remember what they learnt
earlier. Tell them that they will be assessed on what they have learnt
at the end of the week.
Mental Maths Learner’s Book page 44
This session serves as revision of previous work. The knowledge
and skills involve repeated addition and subtraction (which they
should solve by multiplying and dividing at this stage). Learners
work with larger numbers and multiples of 10 and 100. By now,
they should have developed a rule for multiplying and dividing by
0. They work with multiples of 2, 3, 25 and realise how counting
links to tables, doubling and halving. The questions they answer in
this session are based on the concepts listed in the introduction.
Solutions
1. 10 × 8 = 80; 10 × 7 = 70; 10 × 9 = 90; 10 × 17 = 170;
10 × 25 = 250 dots
2. 60 ÷ 10 = 6; 100 ÷ 10 = 10; 300 ÷ 10 = 30; 450 ÷ 10 = 45;
500 ÷ 10 = 50 strips
3. The learners do repeated addition or multiplication. If
they want to use repeated addition, they will soon realise
that it is problematic to add 10 one hundred times. Let
learners explain 10 × 2. Some of them will probably say
it is 10 + 10. Ask them how they will solve 10 × 20 – will
they add 10 twenty times? The activity forces learners to use
multiplication instead of repeated addition. Help learners
generate a rule for multiplying by 10 if they have not
discovered the rule, yet. Ask them to look for a relationship
in the numbers in the first problem. They should notice that
a 0 is added to 2 to give 20.
10 × 2 = 20 stacks
10 × 20 = 200 stacks
10 × 35 = 350 stacks
10 × 55 = 550 stacks
10 × 100 = 1 000 stacks
4. There is a relationship between the numbers in the first
problem. Ask learners what they notice about 80 ÷ 10 = 8.
Mathematics Teacher’s Guide Grade 4
Math G4 TG.indb 69
TERM 1
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2012/09/14 5:32 PM
They should realise that the 0 has been removed. If some
learners suggest that the problem should be solved by
repeated subtraction (80 – 10 – 10 – 10 – 10 – 10 – 10 – 10 –
10 = 0) and reason that 8 children could each get 10 sweets,
ask them how they would then solve 200 ÷ 10. They should
realise that 200 ÷ 10 = 20 and note that the 0 was removed
from 200 to give 20. If some learners do not understand
the context of the first problem, use smaller numbers (such
as: If there are 20 sweets, how many children can each get
10 sweets? If there are 10 sweets, how many children can
each get 10 sweets?) Ask learners to look for patterns in the
solutions. For example, they should observe the doubling
from 20 to 40, and 40 to 80.
80 ÷ 10: 8 children can get 10 sweets
200 ÷ 10: 20 children can get 10 sweets
400 ÷ 10: 40 children can get 10 sweets
800 ÷ 10: 80 children can get 10 sweets
5. The learners must use division. If there are learners who
struggle, use smaller numbers, for example, ask how many
people have 20 fingers altogether. Learners could also do this
practically by using their own fingers. Allow the learners to
look for relationships in the calculations to generalise a rule
for division by 10.
50 ÷ 10: 5 people have 50 fingers
500 ÷ 10: 50 people have 500 fingers
900 ÷ 10: 90 people have 900 fingers
1 000 ÷ 10: 100 people have 1 000 fingers
6. a) 10 × 10 = 100
b) 40 ÷ 10 = 4
10 × 20 = 200
140 ÷ 10 = 14
10 × 34 = 340
400 ÷ 10 = 40
10 × 78 = 780
440 ÷ 10 = 44
10 × 150 = 1 500
7. 6 × 10 = 60 slices
6 × 100 = 600 slices
6 × 1 000 = 6 000 slices
8. Ask the learners to count in 2s up to 16. Write down the
numbers on the board as they count. Let them find out how
many 2s they counted. They may count on their fingers. Ask
them to solve the multiplication by 2 problems. Let them
determine the number of groups of 2 in the numbers given.
They halve and double numbers and explain what they
notice about the different concepts. They should realise that
counting, multiplication, grouping, halving and doubling all
have to do with 2. The learners use counting in 25s and 3s to
notice related concepts about 25 and 3. Encourage them to
look for relationships and patterns.
70
Math G4 TG.indb 70
Mathematics Teacher’s Guide Grade 4
TERM 1
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Count in 2s up to 16: 2; 4; 6; 8; 10; 12; 14; 16
a) eight 2s
b) 2 × 8 = 16
2×4=8
2×3=6
2 × 7 = 14
2 × 9 = 18
c) 12 = 6 groups of 2
d) Half of 28 = 14
20 = 10 groups of 2
Half of 26 = 13
30 = 15 groups of 2
Half of 22 = 11
50 = 25 groups of 2
Half of 32 = 16
100 = 50 groups of 2
Half of 36 = 18
e) Double 6 = 12
f) Answers will differ.
Double 4 = 8
Double 7 = 14
Double 9 = 18
Double 11 = 22
Double 14 = 28
Double 16 = 32
9. Counting in 25s from 25 to 200:
25; 50; 75; 100; 125; 150; 175; 200
a) eight 25s
b) 50 = two 25s
100 = four 25s
200 = eight 25s
300 = twelve 25s
400 = sixteen 25s
Find out if the learners notice the doubling in these
problems. Encourage them to realise that they have to
add the number of 25s in 100 and 200 to get the number
of 25s in 300.
c) 2 × 25 = 50
3 × 25 = 75
4 × 25 = 100
8 × 25 = 200
10. Counting in 3s from 3 to 30: 3; 6; 9; 12; 15; 18; 21; 24; 27; 30
a) ten 3s
b) 6 = two 3s
c) 13 = four 3s remainder 1
12 = four 3s
16 = five 3s remainder 1
15 = five 3s
18 = six 3s
21 = seven 3s
You could extend this activity by asking how many 2s there
are in different uneven numbers and how many 25s there are in
numbers that are not multiples of 25. Ask them why there are
remainders. They should understand that uneven numbers or
non-multiples of 2 divided by 2 would always leave remainders.
Mathematics Teacher’s Guide Grade 4
Math G4 TG.indb 71
TERM 1
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2012/09/14 5:32 PM
Activity 27.1
Learner’s Book page 45
The learners should understand the relationship between repeated
addition, equal groups and multiplication. They should realise that
multiplication is a short way for addition. Once they understand
the concept of multiplication, they will find it easier to memorise
and recall the basic multiplication facts. Demonstrating the concept
practically also helps to develop understanding. Once they have
grasped the multiplication concept, allow them to work on a more
abstract level. You could also address their understanding of
commutative property and inverse operations again using example
such as:
3 × 4 = 12 and 4 × 3 = 12
12 = 4 × 3 and 12 = 3 × 4
12 ÷ 4 = 3 and 12 ÷ 3 = 4
4 = 12 ÷ 3 and 3 = 12 ÷ 4
Ask the learners to explain what 3 × 6 means. Let them draw or use
cubes, counters or bottle tops.
Lead a class discussion about how the cubes are arranged. Let
them work with cubes, demonstrate the different groups and write
multiplication number sentences.
Solutions
1. a) 2 groups of 8 = 16
b) 3 groups of 4 = 12
d) 2 groups of 7 = 14
f) 3 groups of 8 = 24
h) 4 groups of 9 = 36
j) 7 groups of 4 = 28
c)
e)
g)
i)
6 groups of 2 = 12
4 groups of 6 = 24
5 groups of 2 = 10
6 groups of 3 = 18
2. Allow those learners who struggle to use cubes if they are not
ready to calculate repeatedly. Also allow those who can do
multiplication without repeated addition to do so.
a) 8 + 8 = 16
2 × 8 = 16
b) 4 + 4 + 4 = 12
3 × 4 = 12
c) 6 + 6 = 12
6 × 2 = 12
d) 7 + 7 = 14
2 × 7 = 14
e) 6 + 6 + 6 + 6 = 24
4 × 6 = 24
f) 8 + 8 + 8 = 24
3 × 8 = 24
g) 2 + 2 + 2 + 2 + 2 = 10
5 × 2 = 10
h) 9 + 9 + 9 + 9 = 36
4 × 9 = 36
i) 3 + 3 + 3 + 3 + 3 + 3 = 18
6 × 3 = 18
j) 4 + 4 + 4 + 4 + 4 + 4 + 4 = 28
7 × 4 = 28
72
Math G4 TG.indb 72
Mathematics Teacher’s Guide Grade 4
TERM 1
2012/09/14 5:32 PM
3. Ask the learners to use their own number of groups and to write
multiplication number sentences such as the following:
2 groups of 5: 2 × 5 = 10
6 groups of 6: 6 × 6 = 36
7 groups of 7: 7 × 7 = 49
8 groups of 8: 8 × 8 = 64
9 groups of 9: 9 × 9 = 81
9 groups of 4: 9 × 4 = 36
7 groups of 1: 7 × 1 = 7
0 groups of 0; 0 × 0 = 0
4. Learners may make drawings or use cubes or counters to help
them solve the word problems, if necessary.
a) 7 × 9 = 63, or 7 × 3 × 3 = 21 × 3 = 63 blocks
b) 8 × 6 = 48, or 8 × 3 × 2 = 24 × 2 = 48 learners
Unit 28 Know the multiplication tables
Mental Maths Learner’s Book page 46
Learners write their answers for question 1 on their Mental
maths grids. Encourage those who struggle to use doubling when
adding repeatedly. Make up an example before they answer the
questions, for example:
2+2+2+2=n
4×2=8
4+4=8
2×4=8
7 + 7 = 14
2 × 7 = 14
Solutions
1. a)
c)
e)
g)
i)
4 + 4 + 4 = 12
3 + 3 + 3 + 3 = 12
6 + 6 + 6 = 18
8 + 8 + 8 = 24
9 + 9 = 18
b)
d)
f)
h)
j)
3 × 4 = 12
4 × 3 = 12
3 × 6 = 18
3 × 8 = 24
2 × 9 = 18
2. As learners to make up a rule for multiplying by 1, and by 2.
a) 1 × 2 = 2
1×3=3
1×4=4
b) 2 × 2 = 4
2×3=6
2×4=8
c) 1 × 5 = 5
1×6=6
1×7=7
d) 2 × 5 = 10
2 × 6 = 12
2 × 7 = 14
e) 1 × 8 = 8
1×9=9
1 × 10 = 10
f) 2 × 8 = 16
2 × 9 = 18
2 × 10 = 20
3. Answers will differ.
4. Give the learners copies of the tables. They work on their
own. You could also ask them to complete the tables at home.
They should realise that doubling the 2, 3, 4 and 5 times
tables results in the 4, 6, 8 and 10 times tables.
5. They triple the 3 times table to get the 9 times table.
Mathematics Teacher’s Guide Grade 4
Math G4 TG.indb 73
TERM 1
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2012/09/14 5:32 PM
6. They should realise that there are no whole numbers you can
double or triple to get 7.
7. As soon as the learners have developed a conceptual
understanding of multiplication, you should expect them to
learn the tables by heart. Let them take their copies home and
learn the tables this way. Show them that some of the factors
in the tables are included in others so that they can eliminate
some. They should now know that any number multiplied by
zero is 0 and any number multiplied by 1 is 1.
If they know the 2, 3, 4, 5 and 10 times tables they only have
to learn the following:
7 × 7 = 49
8 × 7 = 56
9 × 7 = 63
10 × 7 = 70
8 × 9 = 72
9 × 9 = 81
Activity 28.1
Learner’s Book page 47
Ask the learners to look at drawings that show an understanding of
multiplication. Let them explain what they notice.
Solutions
1. The groups of objects in the examples are single objects – they
are not grouped to form a countable unit. This makes drawing
multiplication by 0 problematic. Learners will probably struggle
with this idea.
If 2 × 3 is demonstrated as n n n + n n n how would they
demonstrate 0 × 3 using repeated addition?
2. To show multiplication by 0, group the objects so that the group
is a countable unit as in the illustration. The learners should
now be able to see that 0 × 3 = n + n + n, this is three empty
groups.
3. a) 4 + 4 + 4 = 12
3 × 4 = 12
b) 3 + 3 + 3 = 9
3×3=9
c) 2 + 2 + 2 = 6
3×2=6
d) 1 + 1 + 1 = 3
3×1=3
e) 0 + 0 + 0 = 0
3×0=0
f) 4 + 4 + 4 + 4 = 4 × 4 = 16
g) 4 + 4 + 4 = 3 × 4 = 12
h) 4 + 4 = 2 × 4 = 8
i) 4 = 1 × 4 = 4
j) 0 = 0 × 4 = 0
4. a)
0
1
2
3
4
5
6
7
8
9 10
×8
b)
×9
74
Math G4 TG.indb 74
0
8
16
24
32
40
48
56
64
72
80
0
1
2
3
4
5
6
7
8
9
10
0
9
18
27
36
45
54
63
72
81
90
Mathematics Teacher’s Guide Grade 4
TERM 1
2012/09/14 5:32 PM
5. Ask learners how they find the input if the output value is
given. They should realise that they have to perform the inverse
operation.
5 5
35 35 7 7
42 42
b)
a)
3574
42 42
65 × 7 × 7 35
24 42
24
47 × 6 × 6 42
7
7
49
9
54
4
4
24 24 7
7
56 56
46
2475
24 24
64 × 4 × 4 24
56
67
c)
68
4249
49 49
76 × 7 × 7 42
499
d)
2456
32 32
86 × 4 × 4 24
24
49 × 6 × 6 24
54 54
54
56
40 40
57 × 8 × 8 56
40
48 48
65 × 8 × 8 40
8 should find that
326
6 the multiplier
48
8
32 breaking
48 (the
6. The learners
up
second number in a multiplication problem) might make it
easier to multiply. Multiplying by 8 can be easier if you multiply
by 4 and then by 2 (doubling – a concept know). This could
be even easier if you see 8 as 2 × 2 × 2 so that, for example,
6 × 8 = 6 × 2 × 2 × 2 = 12 × 2 = 24 × 2 = 48. This concept needs
practice.
Learners can use these strategies when they do mental
calculations.
a)
9
×6
54
9
×3
×2
54
8
×9
72
8
×3
×3
72
6
9
×8
×4
48
36
6
9
×4
×2
×2
×2
48
36
b) The answers in both columns are the same.
Unit 29 Round off and estimate in real life
Mental Maths Learner’s Book page 48
To make estimates, learners will have to round off the numbers
to the nearest 10. Tell the learners that we often make estimations
in real life. Ask them if they know what estimation means.
They should understand that estimations are not wild guesses.
We round off numbers when we make estimations. When we
make estimates, we normally say: I think ..., It’s about ..., or
Roughly ... Let the learners use their own strategies to calculate
the estimated amounts.
Mathematics Teacher’s Guide Grade 4
Math G4 TG.indb 75
TERM 1
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2012/09/14 5:32 PM
Solutions
1. The prices look cheaper, for example, R9 looks cheaper than
R10. We do not have 1-cent coins anymore so you do not get
1c change.
2. It saves space if R is not included and everybody knows that
the prices are given in rand.
3. Soap: R10
Body spray: R20
Shower gel: R40
Mouthwash: R70
Bio-oil: R80
Facial crème: R120
Shampoo: R50
4. a) herbal soap and facial crème: R10 + R120 = R130
b) shampoo and mouthwash: R50 + R70 = R120
c) shampoo and bio-oil: R50 + R80 = R130
d) body spray, shower gel and bio-oil:
R20 + R40 + R80 = R80 + R20 + R40 = R140
After the learners have solved this problem, show them how to
group numbers for calculations using the associative property
of numbers (for example, R80 + R20 = R100). It is easier to
work with multiples or powers of 10. Learners can regroup the
numbers in question 4(d).
Learners can use the commutative property for addition and
multiplication, but not for subtraction and division. The property
is used effectively, for example, when you have to justify why
0 × 1 = 0. You could claim that 1 × 0 = 0 so therefore 0 × 1 = 0.
The associative property involves grouping and the distributive
property is often used in multiplication and division problems
where we distribute or regroup the numbers so that we use
numbers more economically.
The use of the properties shows effective number sense
development and sophisticated thinking and reasoning.
Using the properties of numbers helps us manipulate numbers
so that they are easier to use. When using the distributive law or
property, we split numbers to make calculations easier. If you
do not know what 4 × 7 is, you could break up the numbers into
combinations or facts that you can manage, for example:
(2 × 7) + (2 × 7) = 14 + 14 = 28.
Activity 29.1
Learner’s Book page 49
1. Learners estimate where the numbers should be written on the
number lines. If they need help, ask them to find the midpoint on
each line first, then the quarter and three-quarter points.
a)
0
100
200
300
400
500
600
800
700
900
1 000
950
76
Math G4 TG.indb 76
Teacher’s
Guide Grade 4500TERM 1
125
250
0 Mathematics
750
1 000
2012/09/14 5:32 PM
100
0
0
200
100
300
200
400
400
300
600
500
600
500
800
700
900
800
700
1 000
900
1 000
b)
950
125
0
125
0
250
500
250
750
500
950 1 000
750
1 000
c)
0
1 000
0
1 000
37
38
37
38
255
256
255
256
2 000
4 000
6 000
2.2 000
a) 43 → 40 4 000
49 → 50
45 → 50
39
40
41
42
43
38 → 40
39
40
41
42
43
44 → 40
51 → 50
257
258
259
260
Activity 29.2
257
258
259
260
261
261
8 000
6 000b)
44
45
44
262
262
45
263
263
46
46
264
264
000
257 → 8260
266 → 270
269 → 270
47
48
265 → 270
47
48
264 → 260
265
265
266
266
9 000
9 000
49
49
267
268
50
10 000
10 000
51
50
51
269
270
Learner’s Book page 49
267
268
269
270
Ask the learners to explain their estimates by referring to the
position of the needle on the petrol gauge in each case. The learners
use knowledge of halving and fractions to make the estimates and
they count in 10s and 20s.
Solutions
1. Learners should reason that the gauge is about halfway between
0 and 45. The reading cannot be 10 ℓ, 15 ℓ or 20 ℓ because half
of 45 ℓ is about 23 ℓ, and so the reading is 25 ℓ (D).
2. a) If you divide the gauge into three equal parts the reading
should be about 20 ℓ. They count in tens to get the accurate
reading. The gauge is at about 25 ℓ because it is the closest
1
to 22 2 .
b) If you divide the gauge into five equal parts, the reading
should be about 10 ℓ. They count in tens to check their
estimates.
c) If you divide the gauge into four equal parts, the reading is
about 60 ℓ. They count in twenties to check their estimates.
Unit 30 Use grouping and sharing
Mental Maths Learner’s Book page 50
Encourage learners to judge their calculations and estimations to
check if they are reasonable.
Solutions
1. a)
c)
e)
g)
13 × 2 → 6
29 × 2 → 28
60 × 7 → 0
30 × 8 → 0
b)
d)
f)
h)
15 × 6 → 0
35 × 5 → 5
28 × 6 → 8
37 × 9 → 3
Mathematics Teacher’s Guide Grade 4
Math G4 TG.indb 77
TERM 1
77
2012/09/14 5:32 PM
2. After learners have completed the assessment activity, you
can ask them to solve the problems they said are correct.
Then ask them to solve the problems where the answers are
incorrect.
a) 24 × 7 = 168 Yes, 4 × 7 = 28
b) 18 × 6 = 84
No, 8 × 6 = 48 18 × 6 = 108
c) 15 × 7 = 80
No, 5 × 7 = 35 15 × 7 = 105
d) 11 × 9 = 91
No, 1 × 9 = 9
11 × 9 = 99
207 ÷ 7 = 29 remainder 4
e) 207 ÷ 7 = 27 No, 7 ÷ 7 = 1
f) 14 × 4 = 56
Yes, 4 × 4 = 16
3, 4. The learners should relate to these problems because they
work with money in real life. In calculating the number of
10c and 50c coins in the amounts, learners should discover
that they could use addition, multiplication, division and
doubling.
50c = 10c + 10c + 10c + 10c + 10c
R10 = 100 × 10c
50c = 5 × 10c
R10 = 50 × 20c
R1 = 10 × 10c
R10 = 20 × 50c
R1 = 5 × 20c
R20 = 200 × 10c
R1 = 2 × 50c
R20 = 100 × 20c
R2 = 20 × 10c
R20 = 40 × 50c
R2 = 10 × 20c
R2 = 4 × 50c
R5 = 50 × 10c
R5 = 25 × 20c
R5 = 10 × 50c
Activity 30.1
Learner’s Book page 50
Ask learners what division means and where and which things
they divide in real life. Learners use counters, cubes, and so on
to demonstrate their understanding of division and drawings to
show their understanding of equal grouping. Drawings can be used
to show intuitive understanding of concepts. Learners who have
developed a good understanding might use repeated subtraction
while others can share objects into equal groups. Ask learners to
check their solutions by reversing division by multiplying.
Solutions
1. a) Each child gets four biscuits.
Subtracting 4:
16 – 4 = 12 – 4
=8
–4
=4
–4
=0
shows that you take away 4 four times to give four children
four biscuits each and there will be no biscuits left over.
Learners can draw the 16 biscuits and separate them into
four groups by using circles to group four biscuits; they will
get four groups with four biscuits in each.
78
Math G4 TG.indb 78
Mathematics Teacher’s Guide Grade 4
TERM 1
2012/09/14 5:32 PM
b) Four children can each get five biscuits.
Learners could also do addition or counting on instead
of division (5 + 5 + 5 + 5 = 20) to work out that four
children can each get four biscuits. Learners who know the
multiplication tables might divide 20 by 5.
c) Some learners might draw six plates and do one-to-one
correspondence – put one biscuit at a time on each plate
until each plate has six biscuits.
Giving learners the opportunity to illustrate their own
understanding will also allow you to assess their levels of
development so that you can plan activities to address their
mathematical needs.
2. Ask the learners to explore the two strategies and tell you if
they would group the cubes differently. Let them explain their
thinking and reasoning for solving the division calculations. The
one strategy uses equal grouping and the other one uses repeated
subtraction. Learners can make drawings of the practical
representations and use written strategies so that they work from
the concrete to develop abstract thinking and reasoning.
a) 24 ÷ 6 = n
24 – 6 – 6 – 6 – 6 = 0
24 = 4 groups of 6
24 ÷ 6 = 4
b) 28 ÷ 7 = 4
c) 18 ÷ 6 = 3
d) 32 ÷ 4 = 8
e) 36 ÷ 9 = 4
Activity 30.2
Learner’s Book page 51
Learners can start by using practical representations to explore the
relationship between the numbers and then explore the relationship
between the numbers in the two calculations. They should, for
example, notice that 12 ÷ 4 = 3 and 12 ÷ 3 = 4.
Solutions
1. a) 8 ÷ 4 = 2
8÷2=4
b) 12 ÷ 4 = 3
12 ÷ 3 = 4
c) 15 ÷ 5 = 3
15 ÷ 3 = 5
d) 18 ÷ 3 = 6
18 ÷ 6 = 3
e) 20 ÷ 4 = 5
20 ÷ 5 = 4
2. The answers to the two problems in each set are inversely
related.
3. Emphasise the importance of knowing the multiplication tables.
If learners know 9 × 9 = 81, they will know that 81 ÷ 9 = 9 (the
inverse operation).
a) 14 ÷ 7 = 2
14 ÷ 2 = 7
b) 21 ÷ 7 = 3
21 ÷ 3 = 7
c) 24 ÷ 8 = 3
24 ÷ 3 = 8
d) 30 ÷ 6 = 5
30 ÷ 5 = 6
e) 36 ÷ 9 = 4
36 ÷ 3 = 12
Mathematics Teacher’s Guide Grade 4
Math G4 TG.indb 79
TERM 1
79
2012/09/14 5:32 PM
4. Learners can make their own representation of the apples
(circles or tally marks) to show the equal groupings and write
the division calculation for each problem. They should realise
that if there are 0 children, there is no need to divide or share.
You cannot do it, so there is no answer; it is impossible and
24 ÷ 0 = 0 is false; learners should write the answer as
24 ÷ 0 = impossible.
You can extend the activity by using 36 apples.
a) 24 ÷ 2 = 12
12 + 12 = 24
24 – 12 – 12 = 0
24 is two groups of 12
2 learners can get 12 apples
b) 24 ÷ 2 = 12 children
c) 24 ÷ 6 = 4 children
d) 24 ÷ 4 = 8 children
e) 24 ÷ 8 = 3 children
f) 24 ÷ 3 = 8 children
g) 24 ÷ 1 = 24 children
h) 24 ÷ 24 = 1 child
5. Learners discuss possible answers.
6. Allow learners who are still at the level of using concrete
materials to use such methods to work out the answers.
Learners who can solve problems using only numbers should
do additional exercises, or they can play games such as Division
Bingo (see the example). They can use multiplication to check
their answers.
a) 24 ÷ 8 = 3
24 ÷ 3 = 8
b) 24 ÷ 6 = 4
24 ÷ 4 = 6
c) 24 ÷ 1 = 24
24 ÷ 24 = 1
d) 24 ÷ 12 = 2
24 ÷ 2 = 12
Unit 31
Division facts and rules
Mental Maths Learner’s Book page 52
Learners write the solutions to question 1 on the Mental maths
grid. The numbers are switched in the multiplication calculations
so that the answers are the same (commutative property
of numbers). If necessary ask learners to write down other
calculations in this way.
Solutions
1. a)
c)
e)
g)
i)
80
Math G4 TG.indb 80
7 × 4 = 28
20 ÷ 5 = 4
9 × 6 = 54
35 ÷ 5 = 7
72 ÷ 8 = 9
Mathematics Teacher’s Guide Grade 4
b)
d)
f)
h)
j)
4 × 7 = 28
20 ÷ 4 = 5
6 × 9 = 54
35 ÷ 7 = 5
72 ÷ 9 = 8
TERM 1
2012/09/14 5:32 PM
2. Encourage learners to work systematically. Start with 3 and
multiply it by all the other numbers; then with 4, and so
on. They include calculations that show the commutative
property.
3 × 4 = 12
3 × 6 = 18
3 × 8 = 24
4 × 3 = 12
4 × 6 = 24
4 × 8 = 32
6 × 3 = 18
6 × 4 = 24
6 × 8 = 48
8 × 3 = 24
8 × 4 = 32
8 × 6 = 48
3. Halving is the same as dividing by 2; so, half of 16 is the
same as 16 ÷ 2.
a) Half of 16 = 8
16 ÷ 2 = 8
b) Half of 18 = 9
18 ÷ 2 = 9
c) Half of 24 = 12
24 ÷ 2 = 12
d) Half of 32 = 16
32 ÷ 2 = 16
e) Half of 50 = 25
50 ÷ 2 = 25
4. Multiplying by a number and dividing by the same number
results in the original number. Learners can write number
sentences to reinforce the concept (for example, 63 ÷ 9 = 7
and 7 × 9 = 63). Learners practise division by 6, 3, 8 and 4
by filling in the output numbers.
b)
a)
8
7
8
7
4
8
8
7
÷2
×2
2
÷3
2
×3
16
16
21
32
32
5
÷8
×8
4
c)
8
÷7
e)
8
×7
48
÷ ÷2 6
3
8
16
1621
832
7
4
74
32
5
1624
21
4
× ×2 6
36
3
8 ÷ ÷8
7
3263
5
÷ ÷8 9
7
563
× ×8 9
7
4 78
Activity 31.1
2124
4
8
÷ ÷7 9
d)
× ×8
36
69
7
7 ÷8
÷
f)
× ×7 9
87
4
4
××
23 ÷ 2
36
÷
23
×2 ÷÷
÷2
8
7
×2
36 ÷ 3
6
×
×3
6
÷6
16
2124
21
24
24
463
32
5
5
63
63
24
4
87
69 ÷ 7
×8
7
9
×
×7
9
÷9
78
63
÷9
×9
7
Learner’s Book page 53
Mathematics Teacher’s Guide Grade 4
TERM 1
×
7
Ask the learners to explain what they observe in the counter
arrangements. This work is important for developing understanding
of calculating area and perimeter.
In the first example, there are 4 columns with 2 counters each
and 8 counters with 4 columns of 2 counters each; so, 4 × 2 = 8
and 8 ÷ 4 = 2. Learners can write multiplication and division
calculations to explain the arrays to show relationships between
different arrangements of the same number of objects as well as the
relationship between multiplying and dividing the objects. Learners
work with inverse operations and the commutative property.
Math G4 TG.indb 81
×
81
2012/09/14 5:32 PM
Solutions
1. a)
b)
c)
d)
e)
f)
g)
h)
5 × 3 = 15
3 × 5 = 15
3 × 6 = 18
6 × 3 = 18
4 × 5 = 20
5 × 4 = 20
7 × 3 = 21
3 × 7 = 21
15 ÷ 5 = 3
15 ÷ 5 = 3
18 ÷ 3 = 6
18 ÷ 6 = 3
20 ÷ 4 = 5
20 ÷ 5 = 4
21 ÷ 7 = 3
21 ÷ 3 = 7
2. a)
b)
c)
d)
e)
f)
g)
h)
i)
j)
4 × 4 = 16
2 × 9 = 18
8 × 9 = 72
7 × 6 = 42
8 × 7 = 48
63 ÷ 7 = 9
45 ÷ 5 = 9
49 ÷ 7 = 7
81 ÷ 9 = 9
48 ÷ 8 = 6
16 ÷ 4 = 4
18 ÷ 2 = 9
72 ÷ 8 = 9
42 ÷ 6 = 7
48 ÷ 7 = 8
7 × 9 = 63
5 × 9 = 45
7 × 7 = 49
9 × 9 = 81
8 × 6 = 48
3. a)
b)
28
28
32
32
24
÷4
÷4
24
56
56
72
28
28
32
÷4
÷4
32
24
24
56
56
72
÷8
÷8
72
64
64
7c)
78
81
81
27
98
8
20
35
35
86
6
7
79d)
4. a)
54
15
15
20
6
7
79
÷5
÷5
96
6
36
6
7
79
÷5
÷5
96
6
÷6
6
1
12
2
18
3
24
4
30
5
36
6
42
7
48
8
54
9
60
10
÷3
0
0
3
1
6
2
9
3
12
4
15
5
18
6
21
7
24
8
27
9
30
10
÷8
0
0
8
1
16
2
24
3
32
4
40
5
48
6
56
7
64
8
72
9
80
10
÷4
0
0
4
1
8
2
12
3
16
4
20
5
24
6
28
7
32
8
36
9
40
10
c)
d)
Math G4 TG.indb 82
20
35
35
÷9
÷9
0
0
b)
82
98
8
9
93
36
9
93
÷9
÷9
27
54
54
15
15
20
81
81
27
27
54
6
7
79
÷8
÷8
72
64
64
7
78
86
Mathematics Teacher’s Guide Grade 4
TERM 1
2012/09/14 5:32 PM
Activity 31.2
Learner’s Book page 54
This problem requires application addition, subtraction, division and
multiplication.
1. Learners can work in groups. They should discover that the balls
(column 1) cost R4 each (16 ÷ 4 = 4). In column 2, the two balls
should cost a total of R8. So, 32 – 8 = 24, divided by 2 means
the dolls cost R12 each. Then learners can look at column 4:
42 – 12 = 30, and 30 ÷ 10 = 10. So, each book costs R10.
2. In column 3: 36 – 4 – 10 – 12 = 10 and the missing object must
be a book.
Learners use the table to check their solutions.
R4
R4
R4
R4
R16
R4
R4
R12
R12
R32
R12
R10
R10
R4
R36
R10
R10
R10
R12
R52
They can calculate the total sum of the objects. You can ask
questions such as:
• How much will ten balls cost?
• What would you pay in total for four dolls and six books?
Mathematics Teacher’s Guide Grade 4
Math G4 TG.indb 83
TERM 1
83
2012/09/14 5:32 PM
Assessment task 5 Multiplication and division
Complete this assessment task to find out what you have learnt
about multiplication and division.
1. How many of each amount in each total?
a) 50c coins in R2,00
b) 10c coins in R2,00
c) 5c coins in R2,00
d) 20c coins in R2,00
(4)
2. How many of each number in each total?
a) 2s in 28
b) 3s in 30
c) 4s in 16
d) 6s in 36
e) 7s in 42
f) 8s in 64
(6)
3. There are 18 sweets. How many children can each get the
following number of sweets?
a) 9 sweets
b) 3 sweets
c) 6 sweets
d) 1 sweet
e) 0 sweets
(5)
4. Write a multiplication calculation for each problem.
a) 3 + 3 + 3 + 3 + 3 =
b) 4 + 4 + 4 =
c) 8 + 8 + 8 + 8 =
d) 9 + 9 =
e) 10 + 10 + 10 + 10 + 10 + 10 =
f) 7 + 7 + 7 =
(6)
5. Work out the answers.
a) 4 × 9 = n
b) 28 ÷ 7 = n
(4)
n÷4=9
7 × n = 28
6. Complete a copy of the flow diagram.
Input
Output
2
4
6
8
×4
÷2
10
20
(6)
Total [31]
84
Math G4 TG.indb 84
Mathematics Teacher’s Guide Grade 4
TERM 1
2012/09/14 5:32 PM
Assessment task 5 Multiplication and division
Solutions
1. a)
b)
c)
d)
R1,00 = 2 × 50c
R1 = 10 × 10c
R1 = 20 × 5c
R1 = 5 × 20c
R2,00 = 4 × 50c coins
R2 = 20 × 10c coins
R2 = 40 × 5c coins
R2 = 10 × 20c coins
2. a)
b)
c)
d)
e)
f)
28 ÷ 2 = 14
30 ÷ 3 = 10
16 ÷ 4 = 4
36 ÷ 6 = 6
42 ÷ 7 = 6
64 ÷ 8 = 8
(6)
3. a)
b)
c)
d)
e)
18 ÷ 9 = 2 children
18 ÷ 3 = 6 children
18 ÷ 6 = 3 children
18 ÷ 1 = 18 children
18 ÷ 0 is impossible
(5)
4. a)
b)
c)
d)
e)
f)
3 + 3 + 3 + 3 + 3 = 5 × 3 = 15
4 + 4 + 4 = 3 × 4 = 12
8 + 8 + 8 + 8 = 4 × 8 = 32
9 + 9 = 2 × 9 = 18
10 + 10 + 10 + 10 + 10 + 10 = 6 × 10 = 60
7 + 7 + 7 = 3 × 7 = 21
(6)
5. a) 4 × 9 = 36
b) 28 ÷ 7 = 4
36 ÷ 4 = 9
7 × 4 = 28
6. Input
Output
2
4
4
8
6
8
10
×4
÷2
(4)
(4)
12
16
20
20
40
(6)
Total [31]
Mathematics Teacher’s Guide Grade 4
Math G4 TG.indb 85
TERM 1
85
2012/09/14 5:32 PM
Time
Unit 32 Revision of Grade 3 work
Mental Maths Learner’s Book page 55
Although learners are not expected to work with multiples of 12,
24 and 60, they should be able to add 12s, 24s and 60s mentally.
Write down the multiples on the board as learners count in
multiples. They should notice that multiples of 12 are double the
multiples of 6 and multiples of 24 are double the multiples of 12.
Multiples of 60 are multiples of 6 × 10.
Solutions
1. a)
b)
c)
d)
e)
0; 5; 10; 15; 20; 25; 30; 35; 40; 45; 50; 55; 60
0; 7; 14; 21; 28; 35; 42; 49; 56; 63; 70; 77; 84; 91; 98
0; 12; 24; 26; 48; 60; 72; 84; 96; 104; 120; 132; 144
0; 24; 48; 72; 96; 120
0; 60; 120; 180; 240; 300
2. 5 relates to the 5-minute intervals that we use to count
divisions on analogue clocks or watches.
7 relates to days in a week.
12 relates to months in a year.
24 relates to hours in a day.
60 relates to minutes in an hour and 60 seconds in a minute.
Activity 32.1
Learner’s Book page 55
Learners revise what they learnt about time in Grade 3 (including
units of time and leap years) and use their knowledge of whole
numbers and fractions to solve number problems related to time.
They read and represent time on clocks.
Solutions
1. a) There are 60 minutes in an hour and 60 seconds in a minute.
b) There are 24 hours in a day and 7 days in a week.
c) There are approximately 4 weeks in a month and 12 months
in a year.
d) There are 365 days in one year.
e) There are 366 days in a leap year and every fourth year is a
leap year.
f) The next leap year will be in 2016 (or 2020 if it is now after
2016).
g) the Olympic Games
2. a) 120 minutes
b) 36 hours
c) 31 days
d) 122 days
(30 + 31 + 30 + 31 = 62)
86
Math G4 TG.indb 86
Mathematics Teacher’s Guide Grade 4
TERM 1
2012/09/14 5:32 PM
e) (365 × 8) + (366 × 2) = 3 652 days
There could be three leap years in ten years if the first of the
ten years is a leap year.
f) (75 × 365) + (25 × 366) = 36 525
Learners can work out what the maximum number of leap
years can be in 100 years.
3. a) multiples of 5: 5; 10; 15; 20; 25; 30; 35; 40; 45; 50; 55; 60
b) 60 ÷ 5 = 12
c) 5 × 12 = (5 × 10) + (5 × 2)
= 50 + 10
= 60
1
d) 2 of 60 = 30
f) 43 of 60 = 60 ÷ 4 × 3
e) 14 of 60 = 60 ÷ 4 × 1
= 15 minutes
= 15 × 3
= 45 minutes
4. a) 3:30: half past three
b) 9:30 half past nine
c) 2:00: two o’clock
d) 6.00: six o’clock
e) 9.00 p.m.: nine o’clock in the evening
f) 0.00: midnight
5. Learners work in pairs and draw times on clock faces. Let them
check each other’s work.
Unit 33 24-hour time
Revise reading time on analogue watches and clocks including the
two hands (hour and minute). Let learners predict the reading on a
digital clock when they are shown an analogue clock, and the other
way around. Use watches and clocks that have two hands.
Demonstrate how to draw a timeline and use a ruler to make the
markings for the divisions. Teach learners how to use and read a
stopwatch. Divide the class into groups (depending on how many
stopwatches you have) and make sure learners understand what to
do.
In the assessment activity that follows, assess only using a 24-hour
clock as this is the focus of the exercise.
Mental Maths Learner’s Book page 56
Use an analogue clock that is big enough for learners to see well.
(If you do not have a clock, borrow one from the Foundation
Phase teachers.) You can also make a clock out of cardboard
with hands that can move. Assess which learners still struggle
to read time. Ask them to name the morning times (a.m.) and
show you on the clock. Let them record the times in numbers on
the board starting with the examples below. If you have enough
clocks, learners can work in pairs and check each other’s ability
to read time. Learners can also make cardboard clocks to use for
this exercise.
Mathematics Teacher’s Guide Grade 4
Math G4 TG.indb 87
TERM 1
87
2012/09/14 5:32 PM
Examples
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
quarter past three: 3:15 or 03:15
two o’clock: 2:00 or 02:00
ten minutes to ten: 9:50 or 09:50
half past seven: 6:30 or 06:30
quarter to eleven: 10:45
twenty past twelve: 12:20
five to six: 5:55 or 05:55
eight o’clock: 8:00 or 08:00
twenty-five past four: 4:25 or 04:25
half past one: 12:30
Activity 33.1
1. a)
b)
c)
d)
Learner’s Book page 57
3.20 a.m. and 3:20
3.20 p.m. and 15:20
8.55 a.m. and 8:55
8.55 p.m. and 20:55
2. a)
b)
c)
d)
16:15
9:29
12:00
17:55
3. Learners write times on clocks. They can check each other’s
work.
4. The learners can use counting on and addition to solve the
problems.
a) Start of match: 12:30
Duration of matches: 20 + 20 + 10 = 50 min.
End of matches: 12:30 + 30 + 20 = 13:20
b) School ends: 13:20
Duration of walk home: 20 min.
Time she gets home: (13:20 – 10) + 10 = 13:30 + 10
= 13:40
c) Start of practice: 18:00
Warming up and practice: 20 min. + 1 h = 1 h 20 min.
Parents to arrive: 18:00 + 1 h = 19:00
19:00 + 20 = 19:20
d) Departure: 22:50
Arrival: 00:40
Duration of flight:
22:50 to 23:50 = 1 h
23:50 to 00:00 = 10 min.
00:00 to 00:30 = 30 min.
00:30 to 00:40 = 10 min.
1 h + 10 min. + 30 min. + 10 min. = 1 h 50 min.
5. a) 07:30 to 08:10: 07:30 to 08:00 = 30 min.
08:00 to 08:10: 10 min.
Duration of one period: 30 min. + 10 min. = 40 min
b) Duration of two periods for English:
40 + 40 = 80 min., or 2 × 40 = 80 min.
88
Math G4 TG.indb 88
Mathematics Teacher’s Guide Grade 4
TERM 1
2012/09/14 5:32 PM
c) Duration of first Mathematics period: 08:50 to 09:20:
08:50 to 09:00 = 10 min.
09:00 to 09:20 = 20 min.
10 min. + 20 min. = 30 min.
d) Duration of break 1: 09:20 to 09:40:
09:20 to 09:30 = 10 min.
09:30 to 09:40 = 10 min.
10 + 10 = 20 min.
Duration of break 2: 11:40 to 12:00
11:40 to 11:45 = 5 min.
11:45 to 12:00 = 15 min.
5 min. + 15 min. = 20 min.
Duration of the two breaks: 2 × 20 = 40 min.
e) Time when school day ends: 12:40 + 40 min.
12:40 + 20 min. = 13:00
13:00 + 20 min. = 13:20
Unit 34 Read time in 5-minute intervals
Mental Maths Learner’s Book page 58
Some learners would probably use finger counting to work out
the answers. This is fine.
Solutions
1.
3.
5.
7.
9.
0 to 15: 3
0 to 60: 12
0 to 55: 11
10 to 55: 10
30 to 60: 6
Activity 34.1
2. 0 to 30: 6
4. 0 to 45: 9
6. 0 to 40: 8
8. 15 to 45: 6
10. 45 to 60: 3
Learner’s Book page 59
Learners use stopwatches for question 5. If you do not have enough
stopwatches let learners work in larger groups. Ideally, they should
work in pairs of four.
Solutions
1. a) ten past two
b) five to one
c) twenty-five past nine
d) twenty to seven
e) quarter past six
f) half past six
g) twenty past five
h) ten to ten
2. 6:30 a.m., 7:30, 13:25, 4:00 p.m., 18:00, 8:15 p.m.
3. Learners draw a timeline.
4. a) 12:05 and 00:05, or 12:05 a.m. and 12:05 p.m.
b) 6:35 and 18:35, or 6:35 a.m. and 6.35 p.m.
c) 8:55 and 20:55, or 8:55 a.m. or 8:55 p.m.
d) 2:20 and 14:20, 2:20 a.m. or 2.20 p.m.
5. Learners use stopwatches to time events.
Mathematics Teacher’s Guide Grade 4
Math G4 TG.indb 89
TERM 1
89
2012/09/14 5:32 PM
Revision and consolidation
Learners work on their own to show what they have learnt about
time in the last four units. They will convert between units of time
using knowledge of fractions and whole numbers, match digital
times to times given in words and change p.m. times to 24-hour
formats.
Solutions
1. a)
b)
1
2
3
4
min. = 12 of 60 s = 60 ÷ 2 × 1 = 30 s
min. = 43 of 60 = 60 ÷ 4 × 3 = 15 × 3 = 45 s
c) 112 min. = (1 × 60) + 30 = 90 s
d)
e)
1
2
1
4
day = 24 ÷ 2 = 12 h
day = 24 ÷ 4 = 6 h
f) 72 s = 72 ÷ 60 min. = 1 remainder 12 = 1 h 12 s
g) 2 min. 30 s = (2 × 60) + 30 = 120 + 30 = 150 s
h) 112 min. = (1 × 60) + 30 = 90 s
2. a)
b)
c)
d)
e)
10:30 – half past ten
23:45 – quarter to twelve
00:20 – twenty minutes past twelve
16:25 – twenty-five minutes past four
6:15 – quarter past six
3. a) 17:30
c) 20:15
d) 23:15
b) 18:20
d) 22:25
Unit 35 Read calendars
Learner’s Book page 61
90
Math G4 TG.indb 90
Do lots of practical work and oral calculations with calendars and
timetables before allowing the learners to complete the activities
in Units 35 and 36. They must be totally confident about how
to calculate the number of days between two dates, or a time
interval by the time they start writing. Use timetables from your
school, local bus and train timetables and any other locally used
timetables (such as church services, sports practice) to give learners
more opportunities to practise reading times and calculating time
intervals. Let them do this in pairs and groups, before they work on
their own.
If learners do not have diaries, copy a year calendar from a diary
and let them paste it into their books before they answer questions.
Make sure there is a very big wall calendar in your class.
Also put a birthday calendar for your register class on the wall and a
calendar that shows religious holidays of all denominations.
Add questions about date intervals that relate to local events (such
as how many weeks and days there are until the school fête or
Sports Day).
Mathematics Teacher’s Guide Grade 4
TERM 1
2012/09/14 5:32 PM
To help learners write down how old they are to the day, work
through a few examples with them:
2013 year 10 month 24 day
– 2004 year 09 month 16 day
9 years 01 month 08 days
Show learners how to carry from years to months to days. From a
year to a month, carry 12 and from a month to a day, carry 30 or 31
(depending on the month).
Mental Maths Learner’s Book page 61
The learners work in groups and answer questions 1 to 9 without
using calendars. Ask them how the information for February
would be different in a leap year.
Solutions
1.
2.
3.
4.
5.
6.
7–9.
10.
April, June, September, November
January, March, May, July, August, October, December
February
Leap year
52
four
Answers will differ.
Learners use calendars to check their answers to questions
1 to 9.
Activity 35.1
Learner’s Book page 61
The learners explore timetables and answer questions about bus
times and school day activities. Make sure that the class timetable
is structured neatly and positioned strategically. Learners use the
pie chart and the key to compare data and answer questions about
a school’s weekly extramural activities.
Month
Number of weeks and days in the month Round off to the
nearest week
January
February
March
April
May
June
July
August
September
October
November
December
4 weeks and 3 days
4 weeks (and 1 day in leap years)
4 weeks and 3 days
4 weeks and 2 days
4 weeks and 3 days
4 weeks and 2 days
4 weeks and 3 days
4 weeks and 3 days
4 weeks and 2 days
4 weeks and 3 days
4 weeks and 2 days
4 weeks and 3 days
4 weeks
4 weeks
4 weeks
4 weeks
4 weeks
4 weeks
4 weeks
4 weeks
4 weeks
4 weeks
4 weeks
4 weeks
Mathematics Teacher’s Guide Grade 4
Math G4 TG.indb 91
TERM 1
91
2012/09/14 5:32 PM
Activity 35.2
Learner’s Book page 62
Answers differ from year to year.
14 January
14 days
3 days + 21 days = 23 days
7 days + 28 days + 18 days = 53 days
19 January + 7 = 23 January
23 January + 7 = 30 January
30 January + 7 = 6 February
6 February + 7 = 13 February
13 February + 7 = 20 February
20 February + 4 = 24 February
24 February
So, there are 6 × 7 = 6 weeks and 4 days
7. Leap years are on every second even number. So, 2000 is a
leap year. The next one is 2004, then 2008 and then 2012. 2023
will not be a leap year – it is not an even year. In normal years,
February has 28 days, but in leap years it has 29 days. So, if
February has 29 days, that year is a leap year.
(Learners who are interested can find out if 1800 and 1900 were
leap years and if 2100 will be a leap year.)
8. 24 + 7 = 31 March
31 March + 7 = 7 April
School will start on 7 April.
9–14. Answers will differ from year to year.
1.
2.
3.
4.
5.
6.
Unit 36 Read timetables
Mental Maths Learner’s Book page 63
Learners calculate lapsed time in hours and minutes. They have
to work without clocks and calculate the time mentally. Ask them
to explain their counting strategies.
Solutions
1.
3.
5.
7.
9.
4 hours
30 minutes
40 minutes
35 minutes
32 minutes
2. 1 hour
4. 2 hours 30 minutes
6. 40 minutes
8. 18 minutes
10. 35 minutes
Activity 36.1
Learner’s Book page 64
1. a) 7 o’clock
b) quarter past 7
c) 1 hour
d) Beach Road
e) 3 hours and 15 minutes
2. Answers will vary from school to school.
3. a) Cricket takes the most time.
b) 60 + 55 = 115 minutes or 1 h 55 min.
c) 75 – 50 = 25 min.
d, e) Answers will differ.
92
Math G4 TG.indb 92
Mathematics Teacher’s Guide Grade 4
TERM 1
2012/09/14 5:32 PM
Unit 37 History of time
Learner’s Book page 64
Collect additional information about the history of time. Draw a
timeline to indicate the development of the measurement of time
described in this unit.
Ask learners to collect information about, for example, how
their parents and grandparents tell the approximate time, using
information in the environment around them. (For example, ‘When
I hear a train passing I know it is ten minutes past the hour.’ ‘When
the birds start singing I know it is about 5 a.m. in summer, or about
7 a.m. in winter.)
Discuss the photographs and pictures on pages 64 an 65 in the
Learner’s Book with the class, and explain how instruments to
measure time work. Bring a candle, pendulum and hourglass to
school if possible.
Take learners outside and make a shadow stick. Place stones around
the stick for every hour from 8 a.m. to 1 p.m. Compare the shadows
the next day. Take the learners outside and ask them to read the time
the shadow stick shows.
Activity 37.1
Learner’s Book page 66
1
3
H
O
U
R
5
6
7
B
S
I
U
8
W
R
N
A
D
J
T
D
T
E
2
M
G
4
L
U
H
I
E
C
A
L
E
N
D
A
R
9
S
E
Y
A
A
E
A
L
C
U
M
A
S
P
B
N
S
E
R
O
C
K
Y
L
N
The mystery word is calendars.
Revision
Learner’s Book page 66
Learners can complete this activity for homework.
Solutions
1. a)
b)
c)
d)
e)
f)
g)
02:55
14:55
Learners draw an analogue clock that shows 25 to two.
30 days + 31 days + 15 days = 76 days
60 min. + 60 min. +30 min. = 150 min.
52 weeks × 3 = 156 weeks
12 hours and 30 minutes
2. a) 40 min.
b) 20 min.
c) 12:40 + 40 min. = 13:20
Mathematics Teacher’s Guide Grade 4
Math G4 TG.indb 93
TERM 1
93
2012/09/14 5:32 PM
Assessment task 6 Time
1. Draw a clock and draw the hands on it to show quarter to
eleven.(2)
2. Write down the missing words to make the sentences true.
a) There are ... days in a year and every ... year is a
leap year.
b) The short hand on the clock is called the ... hand.
c) There are ... hours in two and a half days and ... months
in three years.
d) March has ... days and ... is the seventh month of
the year.
e) 156 weeks are ... years
(8)
3. Write the given times in words.
a) 02:15
b) 23:55(2)
4. Write the following times as 24-hour time.
a) 20 past 1 in the morning
b) 25 to 9 in the evening
(2)
5. Write down the number of hours and minutes.
a) from 08:40 to 14:25
b) from 23:55 to 00:27
(2)
6. Write a number sentence and then calculate the answer.
a) Anoushka took 3 hours and 55 minutes to complete
a cross country race that started at 06:00. At what time
did she finish the race?
b) A clock loses four seconds every three hours.
How many minutes does it lose in two days?
(4)
Total [20]
94
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Mathematics Teacher’s Guide Grade 4
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Assessment task 6 Numbers and place value
Solutions
1. Watch showing quarter to eleven.
(2)
2. a) 365; fourth
b) hour
c) 60 hours; 36 months
d) 31 days; July
e) three(8)
3. a) quarter past two
b) five to twelve
(2)
4. a) 01:20
b) 20:35(2)
5. a) 5 hours 45 minutes
b) 32 minutes
(2)
6. a) 06:00 + 3 h 55 min. = 09:55.
She finished the race at five to ten.
b) There are 24 hours in a day.
24 ÷ 3 = 8
4 s × 8 = 32 s
4 × 8 = 32 s/d (seconds per day)
32 × 2 = 64 s
The clock loses one minute and four seconds in
two days.
(2)
(4)
Total [20]
Collecting and organising data
In Grade 3, learners worked through the whole data cycle of
collecting, sorting, representing and analysing data. Now they will
revise the various elements in the data cycle and extend the nature
or content of the data to both personalised contexts, as well as social
awareness contexts. For example, the contexts for Grade 4 data
handling include recycling, an important social and environmental
awareness issue.
Unit 38 Use tally marks
Learner’s Book page 67
This unit helps revise using tally marks and tables when collecting
information in order to sort and organise data. Some learners
will remember how to use tally marks if they were taught this in
previous years, but others will need to be shown how to do this.
Tally marks and tables
In order to help learners understand the significance of breaking up
tally marks into groups of five, you can let them do an exercise in
which they have to count about 40 vertical strokes in one row. Then
let them count 40 written as tally marks in eight groups of five.
Mathematics Teacher’s Guide Grade 4
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Mental Maths Learner’s Book page 68
The learners should understand that tally marks involve
grouping in 5s or multiples of 5. Ask them to count forwards
and backwards in multiples of 5. They name the groups of 5 in
different numbers.
Solutions
1. a) 0; 5; 10; 15; 20; 25; 30; 35; 40; 45; 50
b) 100; 95; 90; 85; 80; 75; 70; 65; 60; 55; 50; 45; 40; 35;
30; 25; 20; 15; 10; 5; 0
c) 20
d) eight
e) 30: 6; 45: 9; 60: 12; 55: 11; 85: 17; 105: 21
f) 100: 20; 150: 30; 200: 40; 250: 50
2. a)
b)
c)
d)
e)
f)
g)
h)
Activity 38.1
Learner’s Book page 68
This activity helps the learners revise and practise using tally marks
and tally tables.
Solutions
1. three
3. Family D
2. six
4. Family A
Unit 39 Draw up a tally table
Learner’s Book page 68
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The example helps learners read tally marks and understand data in
a table.
Method for gathering and recording information:
• Take the learners outside or to an area where there is enough
space. Learners should each have a piece of paper and a pencil to
recor data they collect.
• Organise learners so that they form an inner and an outer circle
– there must be the same number of learners in each circle and a
pair of learners – one from each group – must face each other.
• Let each learner mark the favourite colour of the learner he or she
faces. Then ask the inner circle of learners to step to the right so
that they each face another learner. Each learner then records the
new partner’s colours. They continue to step to the right to face a
new partner and ask for the learner’s favourite colour until they
have collected data from all the learners in the other circle.
Mathematics Teacher’s Guide Grade 4
TERM 1
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• Now ask the learners in each circle to collect data from the other
learners in their circle. Each learner takes a turn to go around
the circle and collect the data from all the other learners. Once
everyone has their data, the learners can go back to their seats
and then organise the information.
Mental Maths Learner’s Book page 69
Learners interpret data in the tally table. You could also discuss
questions such as the following with them:
• How many learners are there altogether in the Grade 4 class?
• What is the difference between the number of learners who
like Jolly Juice and those who like Flayva?
Solutions
1.
Cooldrink
Tally marks
Number of learners
JollyJuice
Coolio
Fizz-Fun
Water Wonders
Flayva
Mix-Tricks
2. 5
5. Mix-Tricks
8
5
2
4
3
1
3. 1
6. 23
4. Jolly Juice
Activity 39.1
Learner’s Book page 69
1. This question helps the learners think about and practise
organising data using tally marks and tables. They must work
out how many rows the table will need and then use tally marks
to fill in the data.
If learners struggle, take them through the worked example
about favourite colours, which isolates the different steps they
should follow when organising collected information. Also
ask them specific guiding questions about the data they should
organise, for example: How many types of sport are there? In
which column will you write each sport?
These key questions should guide learners to think logically so
that they can solve the problem.
a)
Sports
Soccer
Tennis
Rugby
Volleyball
Tally marks
Netball
b) four
Numbers
5
3
3
2
4
c) three
d) soccer
Mathematics Teacher’s Guide Grade 4
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2. Answers will differ. It will probably be best for you to
coordinate the data-collection part of the activity so that it is
done in an orderly way.
The answers will depend on the data learners collected. Ensure
that learners understand how to organise the data and answer the
questions correctly.
Suggested informal assessment questions to ask yourself
•
•
•
•
How well are the learners able to collect the data?
Do they use tally marks correctly when collecting data?
Can they create a tally table that represents the data?
How well are learners able to answer questions about their
tables?
Unit 40 Show data in pictographs
Learners worked with pictographs in Grade 3. Briefly refresh
their memories about the differences between a tally table and a
pictograph. Use the example in the Learner’s Book and draw up a
tally table with the same data.
After learners have completed the activity, assess their work by
asking questions such as the following:
• How well are the learners able to draw a pictograph correctly to
show data?
• How well do they understand that each symbol or picture in the
pictograph represents one counted item or person?
Mental Maths Learner’s Book page 70
The learners contribute to making a class pictograph of the
class’s favourite fruit. After they have followed the steps in the
activity, you should have a large class pictograph, which the
learners should then use to answer the questions.
Suggested informal assessment questions to ask yourself
• How well do the learners follow what is happening as you are
building a class pictograph?
• Do they place their pictures in the correct row?
• Can they answer questions about the class pictograph easily
enough?
Ask the learners how they can count the number of learners in the
class in the most effective way.
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Activity 40.1
Learner’s Book page 71
1. a) five
b) strawberry, chocolate, vanilla, mint, banana
c) Three children like strawberry, four like chocolate, five like
vanilla, two like mint and two like banana.
If the learners need more practice, you could ask questions
such as the following:
• How many children liked strawberry ice cream?
• How many children liked chocolate ice cream?
• How many children liked vanilla ice cream?
• Which other two flavours do children like?
• Which flavour is most popular?
• Which flavour is least popular?
• If you were to buy two flavours of ice cream for this class,
which flavours would you buy? Why?
2. a) Learners compare pictographs.
b) sizes 1, 2, 3, 4 and 5
c) four children
d) three children
e) one child
f) size 5
g) size 1
h) sizes 2 and 3
Unit 41
Show data in bar graphs
Do not assume that all learners will automatically understand how
a bar graph works, no matter how simple it may look.
The example in the Learner’s Book shows step by step how to
draw a bar graph using data in a tally table. Make sure that the
learners understand the different parts of a bar graph and how to
draw each bar.
Mental Maths Learner’s Book page 72
Learners count in 5s or multiply by 5 to find each total. Let them
explain how they could change the tally marks in questions 8 to
10 to make more groups of 5. You could use this opportunity to
help learners who are still dependant on using repeated addition
to use multiplication.
Solutions
1. 5 + 5 + 2 = 12
2. 5 + 5 + 5 + 2 = 17
3. 5 + 5 + 5 + 4 = 19
4. 5 + 5 + 5 + 5 + 2 = 22
5. 5 + 5 + 5 + 5 + 4 = 24
6. 5 + 5 + 5 + 5 + 5 + 3 = 28
7. 5 + 5 + 5 + 5 + 5 + 4 = 29
8. 5 + 5 + 5 + 5 + 5 + + 5 + 5 + 2 = 38
9. 5 + 5 + 5 + 5 + 5 + 5 + 2 = 32
10. 5 + 5 + 5 + 5 + 5 = 25
Mathematics Teacher’s Guide Grade 4
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Project
Learner’s Book page 73
Making a bar graph with the class will help the learners to put into
practice the ideas shown in the example in the Learner’s Book.
1–2. When you collect the class data, learners can take turns to give
their answers to the class. Ask different learners to use tally
marks on the board to record the data.
3.
Ask other learners to work out the total for each category and
complete the tally table.
4.
Guide the process of drawing the bar graph. Ask the learners
to tell you which steps to follow next and involve them in
drawing the bar graph.
5.
Help the learners summarise the data in the bar graph in a short
paragraph, for example: The graph shows the class’s favourite
subjects. Most children like Maths best and they like Social
Sciences the least.
Activity 41.1
Learner’s Book page 74
Having completed a bar graph with the class, the learners should
now feel more confident when they draw a bar graph on their own.
Once learners have completed the activity, ask yourself questions
such as the following:
• How well are the learners able to differentiate which data should
be shown on each axis of the bar graph?
• How well have they plotted the bars?
• How well are they able to answer questions about the bar graph?
Solutions
1. Learners discuss their bar graphs.
2. a) blue
b) red
c) blue, pink, purple, yellow, green (or green then yellow), red
Unit 42 Explain data
Learner’s Book page 74
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Math G4 TG.indb 100
So far this term, the learners have had to analyse data in the form of
tables and graphs to show their understanding of how the tables and
graphs worked. In the activities in this unit they will practise reading
and analysing data in more detail.
Learners are often put off when presented with paragraphs of text,
especially in activities in Mathematics. These activities will help
them work through the text, find the relevant information and
present it in a different way.
If it will help learners understand the text and respond more fully to
it, translate the paragraphs into the home languages that the learners
speak and read best.
Mathematics Teacher’s Guide Grade 4
TERM 1
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Mental Maths Learner’s Book page 75
1. four
(The learners can underline each different kind of material
once – this will give them the number of things to list in the
rows in the table.)
2. plastic, cardboard, cans, glass
3.
Material
Tally
Number
Plastic
Cardboard
Cans
Glass
3
5
2
1
4. cardboard
5. glass
Activity 42.1
Learner’s Book page 75
1. four
2. plastic, newspapers, cardboard, cans
3. The table does not have to include tally marks because the totals
for each item are already given in the paragraph.
Material
Plastic
Newspapers
Cardboard
Cans
Number of items
10
8
7
12
4. cans
5. cardboard
Suggested informal assessment questions to ask yourself
• How well are the learners able to identify or extract relevant
data from sentences and paragraphs?
• How well are they able to represent the data in another form,
such as a table?
• How well are they able to answer questions about the data in
the new form?
Unit 43 Data from pictographs
Mental Maths Learner’s Book page 76
Remind the learners that they have worked with pictographs
before. Ask them to explain how data is recorded and displayed
on pictographs. Let them work together as a class to complete
the task.
Mathematics Teacher’s Guide Grade 4
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Solutions
1.
3.
5.
6.
plastic bottles
Sipho, Bernice, Claire and Jabu
Bernice
Sipho, Claire, Jabu, Bernice
Activity 43.1
2. four
4. Sipho
7. 47
Learner’s Book page 76
1. It shows the number of newspapers that different learners
collected.
2. five
3. Charl, Maryam, Lucy, Pieter, Mali
4. Lucy
5. Pieter
6. Lucy, Charl, Mali, Maryam, Pieter
7. 29
Unit 44 Data from pie charts
Learner’s Book page 77
Learners are not expected to draw pie charts yet, but they should
know how to read one. Tell them that pie charts show different
proportions of things that make up a whole. Pie charts with simple
fractions are used in this unit so that the learners can also practise
and reinforce their understanding of fractions.
Mental Maths Learner’s Book page 78
1. Learners at Sunflower Primary School
2. 52
3. 53
4. No, 52 is smaller than 53 so there are fewer girls than boys at
the school.
Activity 44.1
1. a)
b)
2. a)
c)
3. a)
e)
Learner’s Book page 78
Learners at Oak Memorial School
1
c) 24
d) Foundation Phase
4
2
Electricity and candles
b) 3
1
d) More people use candles.
4
1
c) 62
d) 63
six portions b) 6
Washing and cleaning
Suggested informal assessment questions to ask yourself
• How well do the learners understand that pie charts show
portions or fractions of a whole?
• How well do they understand the fractions into which pie
charts in the exercises are divided?
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Unit 45 Data from bar graphs
In the Mental maths question, the number intervals on the vertical
axis increase in 2s, so learners will have to practise identifying the
values that are not shown when necessary. You may want to point
out that the numbers along the number axis work in 2s and that a
point halfway between two numbers is equal to one.
In Activity 45.1, the intervals are in the thousands.
Mental Maths Learner’s Book page 79
1. five (Other refers to items that do not fall into one of the
other categories (such as clothing or old toys).)
2. paper
3. glass
4. paper: 14 boxes; cans: 9 boxes; plastic: 12 boxes; glass:
3 boxes; other: 4 boxes
5. 42 boxes
6. a) recycling
b) paper; plastic
Activity 45.1
1. five
4. 7 000
7. 24 500
Learner’s Book page 80
2. one week
5. 1 500
3. School C
6. C, B, E, A, D
Unit 46 Draw your own bar graph
Learners will work through the whole data cycle on their own to
create and explain their own bar graph.
Mental Maths Learner’s Book page 80
Learners count the number of learners in the class in groups of
2, 3, 4, 5 and 6. They should explain for which groups there are
remainders.
Activity 46.1
Learner’s Book page 81
1. Learners can decide on their own topics for the graph or they
can choose from the topics suggested in the Learner Book. You
could also give the class a list of possible topics that you think
are relevant, and let them choose from your list.
2. Help the learners collect data where necessary. If necessary, help
them set up the structures they may need to be able to collect
the information. For example, if they are collecting data about
the amount of recycling materials that different classes collect,
arrange with the other class teachers for your learners to go to
their classes and collect the data.
3. The learners should create their own tally tables.
Mathematics Teacher’s Guide Grade 4
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Activity 46.2
Learner’s Book page 81
1. The criteria for the bar graphs are given in the Learner’s Book.
Remind the learners to check that they have drawn the bar graph
correctly.
2. Help the learners to briefly describe the information shown by
their bar graphs. If they struggle to write the description, ask them
to explain orally to you or to the class what the graph is about.
Suggested informal assessment questions to ask yourself
• How well are learners able to collect data?
• How well are they able to organise data using tables?
• Do they know how to draw a bar graph to represent
information?
• How well are they able to answer questions about bar graphs?
Revision
Learner’s Book page 82
You could ask the learners to complete the revision task for
homework if time is limited. Learners compare the data to find the
most and least common type of vehicle. They answer questions
about data represented in a pictograph and draw a bar graph to
represent the data. They explore the data in a pie chart and answer
true and false questions.
Solutions
1. a)
Type of vehicle
Bakkie
Car
Bus
4×4
Taxi
Bicycle
Tally mark
Number of vehicles
5
10
2
4
3
2
b) Cars were most common and bicycles were least common.
2. a) pictograph
b) soccer
c) golf
d) six
e) 21
3. Learners compare graphs.
4. a) False, it shows the portion of used paper sent for recycling.
b) False, 53 of all the paper used is sent for recycling.
c) False, more paper is sent for recycling than thrown away.
d) True.
Remedial activities
• Give the learners practice in using tally marks when counting
items. Let them work in pairs to sort and count a variety of
objects placed randomly on a table or in a bag. Use everyday
objects that the learners can recognise, see and touch (such as
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pencils, pens, crayons and erasers). Let the learners repeat similar
sorting, counting and recording activities and then let them use
pictures of items and then their favourite items.
• If learners need more help with pictographs, they can do activities
that are similar to the tally mark activities above, but draw a
picture for each item instead of using tally marks.
• Show learners who struggle to draw bar graphs step by step how
to draw bar graphs from the data they recorded when practising
to use tally tables and pictographs. Show the same data on a tally
table, a pictograph and a bar graph. Help the learners see the
connection between the concrete sets of objects they sorted and
counted, and the bar graphs that show the same data.
• Help the learners understand pie charts better by giving them
paper or cardboard circles that are cut into different fractions.
Make sure that each portion of the circle has the fraction written
on it. Let the learners fit the portions of the circle together to form
the full circle.
Extension activities
• The learners can practise drawing graphs that convert one graph
type to another. For example, give them a pictograph and ask
them to draw the same data in the form of a bar graph or a table.
• Let the learners work through the whole data cycle for a question
of their own about the class or the school. Let them use a tally
table to collect the data, then draw a graph to reflect the data, then
write a sentence or two to summarise the data, and then present
the data to the class.
Self-assessment
How well can I do these things?
I can ...
Yes, easily Usually
Sometimes I need a lot
of help
read and draw tally marks
draw up a table of data
collect data from my family or
classmates
answer questions from tables
draw a pictograph
draw a bar graph
answer questions about a pictograph
answer questions about a bar graph
answer questions about a pie chart
answer questions about data given in a
sentence or paragraph
describe the data in a pictograph, bar
graph and pie chart
Mathematics Teacher’s Guide Grade 4
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Properties of 2-D shapes
Introduction
In this section, the learners revise their knowledge about 2-D
shapes. They further investigate the properties of shapes by
engaging in practical sorting, shape-making and shape-drawing
activities. We also introduce two new shapes: pentagons and
hexagons.
Working with concrete, cut-out shapes
Learners need concrete apparatus for shape and space work. At the
very least, have enough cardboard shapes cut out for all the learners
to use as they work with the different shapes and solids.
There are templates of the shapes the learners will need in the back
of this guide. You can copy these shapes, paste them onto cardboard
from food packaging and then cut them out. The learners can help
you do this. You need enough copies each shape for each learner to
have a set.
Learners need to start with concrete shapes. This will help them
to better understand the abstract concepts of each shape and its
properties. They can do more abstract work later.
Give the learners ample opportunities to play and investigate how
shapes can be organised into different groups, and how various
shapes can work together to make patterns. The activities in the
Learner’s Book aim to help learners move progressively from the
concrete to the abstract. Some learners will be able to make the leap
quite easily. Others will need more time. Be aware of this and give
these learners more time where they need it. If they struggle to draw
a shape from memory, let them see and feel the cut-out shape first,
and then copy it.
Learners will encounter many new terms. Encourage them to talk
about the shapes and parts of shapes in their home languages, and
relate the new English terminology to familiar home-language
descriptions, so that learners can easily integrate the new terms into
their conceptual vocabulary.
Unit 47 Different shapes
Learner’s Book page 83
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Learners classify different shapes according to the categories open
and closed shapes, and shapes with curved and/or straight sides.
This will help them recognise and describe polygons (closed shapes
with straight sides). Make sure they understand each classification
of 2-D shapes before moving on to the next one.
Mathematics Teacher’s Guide Grade 4
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Open and closed shapes
Do not assume learners will automatically know the difference
between a closed and an open shape. Some learners may struggle to
understand this concept unless they actually see and touch examples
of open and closed shapes. Let them draw many different examples
of each type of shape until they can confidently say what the
difference is and draw it.
Closed shapes with curved sides only
The learners should be able to differentiate between a curve and a
straight line, so they should not have too much trouble recognising
the curved and straight sides of closed shapes. Let them draw
examples to demonstrate that they understand the difference.
Closed shapes with curved and straight sides
The semicircle is perhaps the most common 2-D shape that learners
have come across so far that has both a curved and a straight side.
Give them opportunities to draw a semicircle, and to invent other
shapes with this combination of lines. They should label the curved
and straight sides on each shape.
Closed shapes with straight sides only
Once you have worked through the above classifications, you can
isolate the group of 2-D shapes with straight sides only – polygons.
Ask which shapes learners recognise from Grade 3 work, and what
they can tell you about the properties of triangles, squares and
rectangles, as a way to assess how well prepared they are to build on
the Grade 3 concepts.
Polygons
The learners are already familiar with triangles, squares and
rectangles. In Grade 4 they begin to look closely at polygons
with more sides, and they identify these 2-D shapes in pictures
and in their surroundings. Remind them that a quadrilateral is
any four-sided shape. So, squares and rectangles are examples of
quadrilaterals.
If they grasp the concepts pentagon and hexagon easily, you may
want to extend the work by talking about heptagons (seven sides),
octagons (eight sides) and decagons (ten sides), and let the learners
investigate how to draw these shapes. However they are only required
to work with shapes up to the level of the hexagon in Grade 4.
seven sides
heptagon
eight sides
octagon
nine sides
nonagon
Mathematics Teacher’s Guide Grade 4
Math G4 TG.indb 107
ten sides
decagon
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Activity 47.1
Learner’s Book page 84
1. triangle: 3 sides
quadrilateral: 4 sides
pentagon: 5 sides
hexagon: 6 sides
heptagon: 7 sides
octagon: 8 sides
2. penta-: 5; hexa-: 6
3. A triangle is a closed shape with three straight sides.
A quadrilateral is a closed shape with four straight sides.
A pentagon is a closed shape with five straight sides.
A hexagon is a closed shape with six straight sides.
4. It is a square. All four sides are the same length.
5. There are numerous examples in the picture; they include:
• triangles in bridge support structures
• squares and rectangles in the shapes of windows in buildings
• hexagons and pentagons on some soccer balls
• hexagons in the paving pattern.
6. Answers will differ.
Suggested informal assessment questions to ask yourself
• How well are the learners able to name different 2-D shapes?
• How well do they understand the relationship between the
name of the shape and its number of sides?
• How well can they differentiate between shapes based on the
number of sides?
• How well can the learners identify different 2-D shapes?
• How well can they recognise 2-D shapes in their surroundings
and from pictures?
Polygons around us
Now that learners can identify the mathematical shapes, they
should be able to recognise triangles, quadrilaterals, pentagons and
hexagons in pictures and in their environments. Identify examples
of these shapes in local buildings, vehicles, machinery, artworks and
other objects that learners see every day.
Unit 48 Triangles and quadrilaterals
Learner’s Book page 85
Learners investigate the properties of triangles and quadrilaterals
through practically manipulating an elastic band on a geoboard as
well as by drawing shapes on grid paper. Both types of activity will
help learners begin to develop an understanding of the properties of
2-D shapes.
Make geoboards
If you have not been able to buy geoboards, you can make them
using a square piece of wood and nails. Draw a grid of about five
vertical and five horizontal lines on the wood, at intervals of 1 cm.
Then hammer nails in on each intersection of the lines.
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Mental Maths Learner’s Book page 85
Check whether learners recognise the rectangle and square as
quadrilaterals – they are special quadrilaterals – and whether
they classify the square as a rectangle. They should understand
that a square is a special rectangle – it has four right angles
and the opposite sides are equal just as in a rectangle. Let them
count the number of sides in all the triangles, rectangles and
quadrilaterals. The exercise allows them to practise counting
in multiples of 3 and 4. Some learners might use multiplication
rather than counting in 3s and 4s.
1.
2.
3.
4.
5.
triangles: B, E, F, G, J
square: I
rectangles: H, I
quadrilaterals: A, C, D, H, I
a) triangles: 4 × 3 = 12 sides
b) rectangles: 2 × 4 = 8 sides
c) quadrilaterals: 5 × 4 = 20 sides
Activity 48.1
Learner’s Book page 85
1. Give the learners sufficient time to explore the shape-making on
the geoboard. It is through this experimentation and exploration
that they become familiar with properties such as angles,
vertices and sizes of 2-D shapes. Once they have done the
practical manipulation on a geoboard, they can copy the shape
onto dotted paper. This will help them to start moving from the
practical to the abstract.
2. a) A: quadrilateral; B: triangle
b) There is one option for C, but there are a number of options
for D – discuss these with the class.
c) There are numerous options for E and F; three options are
shown below for each shape.
d) The learners can experiment with their own shapes, as long
as the sides are straight and they have three or four sides.
Suggested informal assessment questions to ask yourself
• How easily are the learners able to distinguish between
triangles, squares, rectangles and other quadrilaterals?
• How well are they able to create the various shapes on a
geoboard?
• How well are they able to complete given shapes by drawing
them on dotted paper?
Mathematics Teacher’s Guide Grade 4
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2012/09/14 5:32 PM
Unit 49 Pentagons and hexagons
Remind the learners what the prefixes penta- and hexa- mean to
help them remember the number of sides to count for each polygon.
Mental Maths Learner’s Book page 86
Ask learners to name polygons they know and give the number
of sides of each one. Draw five squares on the board. Ask
learners how many sides they see altogether. Most learners
would probably use multiplication to find the total. Check if
some learners use repeated addition. Ask learners to explain
and compare their counting strategies. They should record their
solutions on their Mental maths grids and use multiplication and
addition to find the number of sides.
Solutions
3 × 3 = 9 sides
9 × 4 = 36 sides
6 × 4 = 24 sides
(4 × 3) + (7 × 4)
= 12 + 28
= 40 sides
9. (4 × 4) + (6 × 4)
= 16 + 24
= 40 sides
8 × 3 = 24 sides
7 × 5 = 35 sides
9 × 6 = 54 sides
(5 × 5) + (6 × 6)
= 25 + 36
= 61 sides
10. (10 × 3) + (10 × 5)
= 30 + 50
= 80 sides
1.
3.
5.
7.
2.
4.
6.
8.
Activity 49.1
Learner’s Book page 87
1. pentagons: B, F, G, H
hexagons: A, C, D, E
2. As with the previous activity, allow the learners sufficient time
to explore the shapes on a geoboard and by drawing the shapes
on dotted paper.
3. a) A: pentagon; B: hexagon
b) There is only one option for shape C as four of the five sides
are already given. There are a number of options for D –
discuss options with the class.
c) There are a number of options for each shape – discuss
options with the class.
d) The learners can experiment with their own shapes, as long
as the sides are straight and they have five or six sides.
Suggested informal assessment questions to ask yourself
• How easily are the learners able to identify and distinguish
between hexagons and pentagons?
• How well are learners able to create the various shapes on the
geoboard?
• How well are they able to complete given shapes by drawing
them on dotted paper?
110
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Unit 50 Put shapes together
Learner’s Book page 88
Now that the learners are able to identify and draw examples of
isolated 2-D shapes, they can begin to understand how two or more
shapes can be put together to create composite shapes. Squares and
triangles are the simplest shapes to use to demonstrate the idea, so
the examples shown use these shapes.
Activity 50.1
Learner’s Book page 89
The learners will need a set of cardboard cut-out shapes to work
with as they put the smaller shapes together to build bigger ones.
Allow sufficient time for this activity. It is essential that learners
are able to move shapes around into new positions so that they can
begin to develop an understanding of the concept of transformation,
which they will learn about later. It will help them develop clearer
mental or abstract images of 2-D shapes if they first work with
concrete objects.
Encourage the learners to copy the shapes shown in the examples
in the Learner’s Book before experimenting to create composite
shapes.
Solutions
1–3. Let the learners experiment with building any their own shapes
and then drawing the outline of each shape.
4. Learners compare their pictures.
Suggested informal assessment questions to ask yourself
• How well do the learners understand the idea of building
bigger shapes with smaller ones?
• How easily are they able to put shapes together to build bigger
shapes?
Revision
Mental Maths Learner’s Book page 89
The learners have to visualise the shapes as you describe them.
You could ask them to draw the shapes as they see them in their
minds. Learners who develop effective visualisation skills often
display effective mental calculation skills.
Solutions
1.
2.
3.
4.
5.
triangle
circle
rectangle
pentagon
hexagon
Mathematics Teacher’s Guide Grade 4
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2012/09/14 5:32 PM
Revision activity
Learner’s Book page 90
1. a) triangles: J, L
b) quadrilaterals: B, C, D
c) pentagons: F, M
d) hexagons: G, H, K
e) square: B
f) rectangle: C
g) circle: Q
h) open shapes: A, I, N
2. At this stage, learners do not have to make drawings on dotted
paper, but let them use dotted paper if they would rather do so.
They do not have to draw regular polygons – any shape with the
correct number of sides is acceptable.
3. The learners create composite shapes. If necessary, let them use
cardboard shapes before they make the drawings.
Remedial activities
Check where the problems lie for learners who are struggling.
Use the grid below to help you assess their work and identify any
problem areas.
How well can the learners do the following?
Yes,
Most
easily times
Sometimes Need a lot
of help
tell the difference between closed and open shapes
tell the difference between straight and curved sides
identify circles among other curved shapes
recognise triangles, rectangles and squares
recognise other quadrilaterals
recognise pentagons and hexagons
put shapes together to build bigger shapes
• Explain to learners who need help with the first two questions
above, the difference between open and closed shapes, and
straight and curved lines. Show them more drawn examples of
each type of shape and let them practise identifying the ones you
describe.
• Discuss each type of shape, define and then give learners more
opportunities to identify a shape. It may help learners to start by
working with cardboard cut-outs of the shapes rather than with
drawings.
• The learners will definitely need to work with cardboard shapes
and place them inside drawn outlines of bigger shapes. Start
with simple shapes at first and then gradually build up to more
complex composite shapes.
Extension activities
• Challenge learners who are able to work through the activities
easily to build smaller shapes and build more complex composite
shapes.
• Let learners create their own shape outlines and challenge a
partner to build the composite shape using smaller shapes.
112
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Additional class activity
1. Let each learner draw a big triangle, quadrilateral, pentagon and
hexagon on dotted paper (the shape should almost fill the page).
2. Ask them to colour the shapes in and cut them out.
3. Divide the class into two groups. Put all the shapes learners
made on two heaps – one for each group.
4. Have a competition to see which group can sort their shapes into
triangles, quadrilaterals, pentagons and hexagons the quickest.
Learners must take turns to pick a shape from the heap and put it
in the correct group.
Assessment task 7 Properties of 2-D shapes
Match each shape with its description in the second column.
1.
2.
square
rectangle
A
B
3.
circle
C
4.
quadrilateral
D
5.
pentagon
E
6.
hexagon
F
a flat shape with one curve
a shape with five straight
sides
a shape with six straight
sides
a quadrilateral with all four
sides the same length
a quadrilateral with
opposite sides the same
length
a shape with four straight
sides
Total [6]
Assignment
Find shapes around you that match the 2-D shapes you know. Write
down where you saw each shape and complete a table such as the
one below.
Shape
Drawing of shape
Where I saw the shape
Square
Rectangle
Circle
Quadrilateral (not a
square or rectangle)
Pentagon
Hexagon
Mathematics Teacher’s Guide Grade 4
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Assessment task 7 Properties of 2-D shapes
1. D
4. F
2. E
5. B
3. A
6. C
Solutions
Total [6]
Assignment
In this assignment, the learners will have to find examples of shapes
in their surroundings. Suggest that the shapes could be found in:
• nature
• their home environments or shops and other places they visit
• items used at religious or traditional occasions.
Mark allocation:
Give one mark for identification of the correct shape.
Give one mark for describing where they saw the shape.
Total marks: 12
Self-assessment
How well can I do these things?
I can ...
Yes, easily
Usually
Sometimes
I need a lot
of help
tell the difference
between closed
and open shapes
tell the difference
between straight
and curved sides
find circles
among other
curved shapes
say which shapes
are triangles,
rectangles and
squares
say which
shapes are other
quadrilaterals
other than
rectangles and
squares
say which shapes
are pentagons and
hexagons
put shapes
together to build
bigger shapes
114
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Whole numbers: multiplication and division
Unit 51
Equal sharing and multiples
Mental Maths Learner’s Book page 91
Learners record their solutions on the Mental maths grid. The
problems involve equal sharing and repeated addition.
Solutions
1. a) 12 sweets equally between 4 children:
12 ÷ 4 = 3 sweets each
b) 32 sweets equally between 8 children:
32 ÷ 8 = 4 sweets each
c) 28 sweets equally between 7 children:
28 ÷ 7 = 4 sweets each
d) 24 sweets equally between 6 children:
24 ÷ 6 = 4 sweets each
d) 36 sweets equally between 6 children:
36 ÷ 6 = 6 sweets each
2. a) 3 packets: 8 × 3 = 24 sweets
b) 5 packets: 8 × 5 = 40 sweets
c) 7 packets: 8 × 7 = 56 sweets
d) 4 packets: 8 × 4 = 32 sweets
e) 6 packets: 8 × 6 = 48 sweets
Activity 51.1
Learner’s Book page 91
1. The learners should be able to recall the multiples of all numbers
from 2 to 10. They should fill in the missing multiples in a copy
of the table and look for patterns.
a)
b)
c)
d)
e)
f)
g)
h)
i)
0
0
0
0
0
0
0
0
0
2
3
4
5
6
7
8
9
10
4
6
8
10
12
14
16
18
20
6
9
12
15
18
21
24
27
30
8
12
15
20
24
28
32
36
40
10
15
20
25
30
35
40
45
50
12
18
24
30
36
42
48
54
60
14
21
28
35
42
49
56
63
70
16
24
32
40
48
56
64
72
80
18
27
36
45
56
63
72
81
90
20
30
40
50
60
70
80
90
100
2. Patterns may differ.
3. Allow the learners to try to work out the answer before you
intervene. They have to look for multiples of 4 and 5, which, if
they add 3 and 4 will give the same number. They might try to
Mathematics Teacher’s Guide Grade 4
Math G4 TG.indb 115
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2012/09/14 5:32 PM
solve the problem by trial and improvement and start by listing
multiples of 4 and 5: 19 – (4 × 4) = 3 and 19 – (5 × 3) = 4.
0; 4; 8; 12; 16; 20; ...
0; 5; 10; 15; 20; ...
16 + 3 = 19
15 + 4 = 19
Unit 52 Multiplication and division strategies
Mental Maths Learner’s Book page 92
Learners demonstrate their ability to recall multiplication facts
instantly.
Solutions
1.
3.
5.
7.
9.
3 × 4 = 12 chickens
3 × 8 = 24 apples
4 × 9 = 36 strawberries
5 × 7 = 35 cats
5 × 5 = 25 beetles
Activity 52.1
2.
4.
6.
8.
10.
3 × 6 = 18 sweets
4 × 7 = 28 pencils
9 × 4 = 36 rabbits
7 × 3 = 21 snails
5 × 6 = 30 oranges
Learner’s Book page 93
The learners study the short cuts used to multiply smarter. Learners
may ask why they should use these methods if they can recall
the solutions instantly. Tell them that they will work with bigger
numbers in Term 2 and these strategies could then be useful. If you
have to multiply or divide by 8, you can multiply by 4 and then by
2 (this is doubling), which they should know. For multiplying or
dividing by 6, you can multiply by 3 and then by 2. This can be used
for, for example 280 ÷ 8. It is easy to divide by 4, and dividing by 2
is halving: 280 ÷ 4 and 70 ÷ 2.
Solutions
1. Answers will differ.
2. a) 7 × 8
7 × 2 × 2 × 2 = 14 × 2 × 2 or
7 × 4 = 28 × 2
= 28 × 2 28 × 2 = 56
= 56
b) 5 × 6
5 × 3 = 15
or 5 × 2 = 10 or
5 × 3 = 15
15 × 2 = 30 10 × 3 = 30 15 + 15 = 30 (15 × 2)
c) 9 × 5
d) 8 × 9
9 × 10 = 90 8 × 6 = 48
90 ÷ 2 = 458 × 3 = 24
24 + 48 = 72
e) 36 ÷ 6
f) 48 ÷ 6
36 ÷ 2 = 18 48 ÷ 2 = 24
18 ÷ 3 = 6 24 ÷ 3 = 8
116
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Mathematics Teacher’s Guide Grade 4
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3. The area model helps learners practise inverse operations and
the commutative property. Refer to the arrays the learners
worked with earlier.
a) 5 × 6 = 30
30 ÷ 6 = 5
6 × 5 = 30
30 ÷ 5 = 6
b) 6 × 7 = 42
42 ÷ 6 = 7
7 × 6 = 42
42 ÷ 7 = 6
c) 4 × 6 = 24
24 ÷ 6 = 4
6 × 4 = 24
24 ÷ 4 = 6
d) 6 × 3 = 18
18 ÷ 6 = 3
3 × 6 = 18
18 ÷ 3 = 6
e) 3 × 8 = 24
24 ÷ 3 = 8
8 × 3 = 24
24 ÷ 8 = 3
f) 4 × 5 = 20
20 ÷ 4 = 5
5 × 4 = 20
20 ÷ 5 = 4
4. The exercise helps learners develop effective multiplication
calculation strategies and an understanding of how to perform
the correct order of operations, using brackets and the
distributive property.
a) 4 + 6 + 4
b) 5 + 7 + 5
= (4 × 3) + 2
= (5 × 3) + 2
= 12 + 2
= 15 + 2
= 14
= 17
c) 8 + 8 + 10 + 8 + 8
= (8 × 5) + 2
= 40 + 2
= 42
Unit 53 Basic multiplication and division facts
Mental Maths Learner’s Book page 94
Learners recall basic multiplication and division facts in
contexts. Although they have to perform mental calculations,
allow those who struggle to write down calculations if this will
help them. You should, however, encourage learners to memorise
the division and multiplication tables. They record their answers
on their Mental maths grids.
Solutions
1.
3.
5.
7.
9.
32 ÷ 4 = 8 children
21 ÷ 3 = 7 sweets
6 × 5 = 30 crayons
48 ÷ 6 = 8 children
81 ÷ 9 = 9 children
2. 24 ÷ 4 = 6 children
4. 8 × 8 = 64 counters
6. 36 ÷ 6 = 6 learners
8. 7 × 7 = 49 counters
10. 63 ÷ 9 = 7 sweets
Mathematics Teacher’s Guide Grade 4
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2012/09/14 5:32 PM
Activity 53.1
Learner’s Book page 95
The learners solve the problems and look for patterns. These
activities help learners practise multiplication and division facts.
The learners practise division with the same dividend and divisor to
deduce that a number divided by itself always gives and answer of 1
(except 0 ÷ 0, which is undefined or not allowed).
Solutions
1. a) Division where the dividends are becoming smaller to help
learners understand that 0 × 4 = 0 and not 4.
6 ÷ 6 = 1
5 ÷ 5 = 1
4 ÷ 4 = 1
3 ÷ 3 = 1
2 ÷ 2 = 1
1 ÷ 1 = 1
b) 6 × 4 = 24
5 × 4 = 20
4 × 4 = 16
3 × 4 = 12
2 × 4 = 8
1 × 4 = 4
0 × 4 = 0
c) Multiply to find out which different numbers multiplied
together give the same product.
2 × 9 = 18
3 × 6 = 18
2 × 10 = 20
5 × 4 = 20
3 × 4 = 12
2 × 6 = 12
d) The answers to these calculations are called square numbers.
Draw a 4 × 4 or a 5 × 5 array to help learners understand
this concept. The number of dots or blocks in the rows and
columns are the same and they form a square. If learners ask
why 0 × 0 is not included, ask them how they would draw a
square with zero sides.
10 × 10 = 100
9 × 9 = 81
8 × 8 = 64
7 × 7 = 49
6 × 6 = 36
5 × 5 = 25
4 × 4 = 16
3 × 3 = 9
2 × 2 = 4
1 × 1 = 1
118
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Mathematics Teacher’s Guide Grade 4
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2. Some learners might just count the number of triangles, squares
and dots. Allow them to do this but emphasise the importance
of explaining and recording effective calculation strategies.
The exercise requires the application of counting, doubling,
multiplication, addition, the distributive and the commutative
and associative properties.
a) triangles: 3 + 3 + 2 = (2 × 3) + 2 = 6 + 2 = 8
squares: 3
b) triangles: 5 + 5 + 2 = (2 × 5) + 2 = 10 + 2 = 12
squares: 5
c) triangles: 7 + 7 + 2 = (2 × 7) + 2 = 14 + 2 = 16
squares: 7
Total number of triangles: (12 + 8) + 16
= 20 + 16
= 36 triangles
Total number of squares: 3 + 5 + 7
=7+3+5
= 10 + 5
= 15
3. Check whether there are learners who count the black dots in 1s.
Encourage them to use more effective counting strategies. Also
check which learners use brackets effectively.
black dots: 7 + 16 + 10 = 23 + 10 = 33
pink dots:
(2 × 5) + (5 × 5) + (3 × 5)
= 10 + 25 + 15
= 25 + 15 + 10
= 40 + 10
= 50
or, 10 × 5 = 50
Total number of dots: 50 + 33 = 83 pink and black dots
Unit 54 Multiplication and division flow diagrams
Mental Maths Learner’s Book page 95
Give the learners copies of the flow diagrams to complete and consolidate
multiplication and division facts.
Solutions
1. Input Input
2. Input
× 4 Output
÷ 8 Output
Input
÷ 7 Output
Input
× 4 Input
Output
Input
÷ 8 Input
Output
Input
÷ 7 Output
× 4 Output
÷ 8 Output
÷3.
7 Output
0
0
00
0 0 0
0
00
0 0 0
0
00
0
0
1
1
14
4 16 4 16
162
2 7 2
7
71
1
1
3
3
312
12 32 12 32
324
4 21 4 21
213
3
3
6
6
624
24 40 24 40
405
5 35 5 35
355
5
5
8
8
832
32 56 32 56
567
7 42 7 42
426
6
6
9
9
936
36 72 36 72
729
9 63 9 63
639
9
9
Mathematics Teacher’s Guide Grade 4
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Activity 54.1
Learner’s Book page 96
Learners complete copies of the flow diagrams. This will help them
practise division and multiplication facts.
1.
33
66
3.
64
64
49
49
49
5.
7.
33
66
64
64
49
49
888 88
9.
×
× 33
× 33 ×
×
× 55
× 55 ×
÷
÷ 88
÷ 88 ÷
÷ 77
÷ 777 ÷
÷
÷
× 33
× 333 ×
×
×
99
30
30
30
888
777
99
30
30
88
77
24
24 24
24
24
2. 2 22 × 4 ×× 44
2
×4
4. 00 00 ×× 22 ×× 22
6. 999 99 ××× 444 ×× 44
8. 444 44 ××× 555 ×× 55
÷
÷ 11
÷ 11 ÷
×
× 44
× 44 ×
÷
÷ 66
÷ 66 ÷
÷ 22
÷ 222 ÷
÷
÷
10.111 11 ××× 444 ×× 44 ÷÷÷ 222 ÷÷ 22
88
000
666
10
10
10
88
00
66
10
10
222 22
Unit 55 Number rules for multiplication and
division
The learners use number properties to multiply and divide more
easily. They use inverse operations, break up multipliers into factors
and the distributive property to regroup multiplicands. These
strategies will help learners develop more sophisticated ways to
multiply and divide.
Mental Maths Learner’s Book page 96
Ask the learners to record the solutions to the problems on their
Mental maths grids.
Solutions
8 × 4 = 32
2. (5 × 4) + (5 × 4) = 40
32 ÷ 8 = 4
5 × 8 = 40
3.
12 × 4 = 48
4. 63 ÷ 7 = 9
12 × 2 × 2 = 48 9 × 7 = 63
5.
9 × 6 = 54
9 × 3 × 2 = 54
6. Lead a class discussion about what learners notice about the
calculations. They should notice the relationship between
multiplication and division (inverses).
Multiplication is the ‘opposite’ of division, so, 8 × 4 = 32 and
32 ÷ 8 = 4. They should also realise that 5 × 8 = 40 and
(5 × 4) + (5 × 4) = 40; 8 is regrouped (distributed) to multiply
by 4 twice (learners do not have to know or use the term
distributive property, although that is what they are using
here). They should also notice that 12 × 4 = 12 × 2 × 2; 4 is
broken up into its smaller factors.
1.
120
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Mathematics Teacher’s Guide Grade 4
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Activity 55.1
Learner’s Book page 97
Let learners explore the strategies used in the examples. Revise
division with 2-digit numbers by 1-digit numbers with remainders.
Pose simple calculations with 1-digit numbers, for example:
7÷2=n
8÷3=n
12 ÷ 5 = n
14 ÷ 6 = n
8 ÷ 7 = n.
1. Learners should notice that the product in the multiplication and
the dividend in the division calculations is the same number.
2. Help learners understand that multiplying by 2 and 4, or 2 and 3
could be easier than multiplying by 8 or 6. Using factors of the
multipliers makes it easier to multiply. Learners can also break
up (distribute) numbers to solve the problems.
3. Discuss the examples of regrouping numbers with the class. Let
them use the distributive property to solve the problems.
Solutions
1. a) 50 ÷ 8
b) 38 ÷ 5
8 × 6 = 48 5 × 7 = 35
50 – 48 = 2 38 – 35 = 3
50 ÷ 8 = 6 remainder 2 38 ÷ 5 = 7 remainder 3
Check:Check:
(8 × 6) + 2 = 48 + 2 = 50 (5 × 7) + 3 = 35 + 3 = 38
c) 69 ÷ 7
d) 65 ÷ 6
7 × 9 = 63 6 × 10 = 60
69 – 63 = 6 65 – 60 = 5
69 ÷ 7 = 9 65 ÷ 6 = 10 remainder 5
Check: Check:
(7 × 9) + 6 = 63 + 6 = 69 (6 × 10) + 5 = 60 + 5 = 65
e) 32 ÷ 3
f) 103 ÷ 10
3 × 10 = 30 10 × 10 = 100 – 100 = 3
32 – 30 = 2 103 ÷ 10 = 10 remainder 3
32 ÷ 3 = 10 remainder 2 Check:
Check: (10 × 10) + 3 = 100 + 3
(3 × 10) + 2 = 32
= 103
g) 98 ÷ 10
h) 48 ÷ 5
10 × 9 = 90 5 × 9 = 45
98 – 90 = 8 48 – 45 = 3
98 ÷ 10 = 9 remainder 8 48 ÷ 5 = 9 remainder 3
Check: Check:
(10 × 9) + 8 = 90 + 8 = 98 (5 × 9) + 3 = 45 + 3 = 48
Mathematics Teacher’s Guide Grade 4
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i) 34 ÷ 4
j) 79 ÷ 9
4 × 8 = 32 9 × 8 = 72
4 – 32 = 2 79 – 72 = 7
34 ÷ 4 = 8 remainder 2 79 ÷ 9 = 9 remainder 7
Check: Check:
(4 × 8) + 2 = 32 + 2 = 34 (9 × 8) + 7 = 79
2. a) 36 × 6
= 36 × 2 × 3
= 72 × 3
= (70 × 3) + (2 × 3)
= 210 + 6
= 216
b) 47 × 8
= 47 × 2 × 4
= 94 × 4
= (90 × 4) + (4 × 4)
= 360 + 16
= 376
c) 43 × 9
= 43 × 3 × 3
= 129 × 3
= (120 × 3) + (9 × 3)
= 360 + 27
= 387
d) 55 × 12
= 55 × 2 × 6 (or × 3 × 4)
= 110 × 6
= 660
= 210 + 6
= 216
e) 32 × 15
= 32 × 3 × 5
= 94 × 5
= (90 × 5) + (4 × 5)
= 450 + 20
= 470
f) 22 × 18
= 22 × 3 × 6
= 66 × 6
= (60 × 6) + (6 × 6)
= 360 + 36
= 396
3. a) 43 × 8 = n
(40 × 8) + (3 × 8)
= 320 + 24
= 344
b) 69 × 7 = n
(60 × 7) + (9 × 7)
= 420 + 63
= 483
c) 87 × 9 = n
(80 × 9) + (7 × 9)
= 720 + 63
= 783
d) 58 × 6 = n
(50 × 6) + (8 × 6)
= 300 + 48
= 348
e) 535 ÷ 5 = n
(500 ÷ 5) + (35 ÷ 5)
= 100 + 7
= 107
f) 654 ÷ 6 = n
(600 ÷ 6) + (54 ÷ 6)
= 100 + 9
= 109
g) 472 ÷ 8 = n
(400 ÷ 8) + (72 ÷ 8)
= 50 + 9
= 59
h) 567 ÷ 7 = n
(560 ÷ 7) + (7 ÷ 7)
= 80 + 1
= 81
4. The learners use inverse operations to check solutions.
Encourage them to use the closest 3-digit multiples of the
divisors. They apply the distributive property. Remind learners
that they should get the multiplicands and dividends above as
answers.
122
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a) 344 ÷ 8 = n
(320 ÷ 8) + (24 ÷ 8)
= 40 + 3
= 43
b) 483 ÷ 7 = n
(420 ÷ 7) + (63 ÷ 7)
= 60 + 9
= 69
c) 783 ÷ 9 = n
(720 ÷ 9) + (63 ÷ 9)
= 80 + 7
= 87
d) 348 ÷ 6 = n
(300 ÷ 6) + (48 ÷ 6)
= 50 + 8
= 58
e) 107 × 5 = n
(100 × 5) + (7 × 5)
= 500 + 35
= 535
f) 109 × 6 = n
(100 × 6) + (9 × 6)
= 600 + 54
= 654
g) 59 × 8 = n
(50 × 8) + (9 × 8)
= 400 + 72
= 472
h) 81 × 7 = n
(80 × 7) + (1 × 7)
= 560 + 7
= 567
Unit 56 Ratio and rate
Mental Maths Learner’s Book page 97
1. Ask learners to record the solutions on their Mental maths
grids.
a) 6 × 1 × 2 = 12
b) 0 × 7 × 2 = 0
c) 5 × 1 × 4 = 20
d) 1 × 8 × 3 = 24
e) 2 × 9 × 1 = 18
f) 36 ÷ 6 × 0 = 0
g) 49 ÷ 7 × 1 = 7
h) 16 ÷ 4 × 0 = 0
i) 81 ÷ 9 × 0 = 0
j) 64 ÷ 8 × 1 = 8
2. Learners should notice that, no matter which number you
multiply or divide by, if you multiply the result by 0 the
answer is 0. They should have discovered this rule earlier
this term. They should also have generated the rule that a
number multiplied by 1 stays the same.
Activity 56.1
Learner’s Book page 98
The concept of ratio might be new to the learners. A ratio refers
to the relationship between two groups of objects or amounts or
quantities. Ratio shows how much bigger or more one group or
quantity is than another. There is one flower for three birds. There
are three girls for four boys. The ratio of birds to flowers is 3 to 1,
or you can write this as 3 : 1. You can also say the ratio of flowers to
birds is 1 to 3 or 1 : 3.
Learners can reason that there are 3 motorbikes for each car so that
the ratio is 3 : 1 which is equivalent to 6 : 2. Question 3, the ratio
of drawers to knobs can be seen as 1 : 2 which is equivalent to 3 : 6
and in question 5, the ratio of grey tiles to purple tiles is 15 : 30 or
Mathematics Teacher’s Guide Grade 4
Math G4 TG.indb 123
TERM 1
123
2012/09/14 5:32 PM
3 : 6. Learners can look for relationships in the equivalent ratios
and realise, for example, that 3 : 6 and 15 : 30 are the same as 1 : 2.
Knowledge of equivalent fractions is a basis for this understanding.
We often use ratios in real life. When people mix paint to get
specific colours, they mix certain amounts of the different colours
to get just the right balance. Use the following examples to enhance
learners’ understanding of ratio.
Below is an example of a label
on a stain remover.
Sinks
and
basins
Use 30 ml fluid to each 5 ℓ
of cold water and rinse well.
Stains
and
removal
Use 5 ml to each liter of cold
water, soak for 5 minutes to
15 minutes, rinse.
Below is an example
of a label on a skirt.
Donella
2 parts wool
8 parts cotton
Made in South Africa
Solutions
1. a) ratio of cars to motorbikes: 2 to 6 or 2 : 6.
b) ratio of motorbikes to cars: 6 to 2 or 6 : 2.
2. ratio of books to crayons: 4 to 3 or 4 : 3.
3. ratio of drawers to knobs: 3 to 6 or 3 : 6.
4. ratio of purple tiles to grey tiles: 30 to 15, or 30 : 15.
5. Learners will probably count all the beads and reason, for
example, in (a), that the ratio of green beads to yellow beads is
5 to 25.
a) ratio of yellow to green beads: 1 : 5; 2 : 10; 3 : 15; 4 : 20 or
5 : 25
b) ratio of pink beads to black beads: 5 : 8; 10 : 16; 15 : 24 or
20 : 32
6. a) ratio of brown beads to orange beads: 1 : 1; 2 : 2; 3 : 3, and
so on
b) For each orange bead there is a brown bead. There are 35
brown and 35 orange beads, so, the ratio is 35 : 35.
7. The label on a bottle of concentrated drink shows how much
water and how much cool drink you should use. Let the learners
look at the water and orange squash mixture the boys made.
The picture should help them understand that the boys use six
1-ℓ jugs of water for four bottles of squash. Ask them what the
ratio is of bottles to jugs, and jugs to bottles. They have to find
out which mixtures are the same as Joe and Jabu’s mixture.
a) ratio of bottles to jugs: 4 : 6 or 2 : 3
b) ratio of jugs to bottles: 6 : 4 or 3 : 2
8
124
Math G4 TG.indb 124
The mixture in D has the same strength: 2 : 3.
Mathematics Teacher’s Guide Grade 4
TERM 1
2012/09/14 5:32 PM
Activity 56.2
Learner’s Book page 100
The exercises develop an understanding of rate. We normally work
with rate when working with concepts such as speed or tempo, pay
or salary (for example, you want to find out how much someone
earns per hour or per day at a specific rate or payment).
1. Learners must first find out what Tasneem earns each Saturday
and then calculate the salary she gets if there are four Saturdays
in a month.
3. Instead of working with traditional calculations, the learners
could use doubling effectively to solve the problems. Learners
should use their knowledge of multiplication with multiples
of 10.
Solutions
1. 1 hour: R25,50; 2 hours: R51,00 (double R25,50);
4 hours: R102,00 (double R51,00)
Tasneem earns R102,00 on a Saturday.
First Saturday: R102,00
Second Saturday: R204,00
(double R102,00)
Third Saturday: R306,00
(R102 × 3)
Fourth Saturday: R408,00
(R102 × 4, or double R204,00)
If the month has four Saturdays, Tasneem earns R408,00 per
month.
2. a) Thabo delivers:
80 (Saturday) + 160 (Sunday)
= 40 + 40 + 160
= 240 newspapers
b) He earns for delivering 80 newspapers:
R40 × 4 = R160
He earns for delivering 160 newspapers:
R40 × 8 = R320
Total earnings = R160 + R320 = R480
c) Payment for delivering one newspaper:
R40 ÷ 20 = R2
3. Earnings for working 2 hours: R50,50
Earnings for working 4 hours: R101,00
Earnings for working 6 hours: R151,50
(double R50,50)
(R50,50 + R151,50)
Mathematics Teacher’s Guide Grade 4
Math G4 TG.indb 125
TERM 1
125
2012/09/14 5:32 PM
Assessment task 8 Multiplication and division
1. Write multiplication a number sentence for each problem
a) 8 × 8 × 8 × 8 × 8 = n
b) 9 + 9 + 9 = n
c) 4 + 4 + 4 + 4 + 4 + 4 = n
d) 6 + 6 + 6 + 6 + 6 = n
e) 0 + 0 + 0 + 0 = n(5)
2. How many oranges altogether in each number of bags?
a) 4 bags with 7 oranges in each
b) 7 bags with 4 oranges in each
c) 8 bags with 8 oranges in each
d) 6 bags with 7 oranges in each
e) 6 bags with 0 oranges in each
(5)
3. Work out the answers.
a) 7 × 8 = n
b) 6 × 3 = n
c) 4 × 8 = n
d) 5 × 9 = n
e) 1 × 6 = n(5)
4. Complete copies of the flow diagrams.
9 ×6
a)
b)
6
×8
c)
9
×3
×2
d)
6
×4
×2
(4)
5. Write a division number sentence for each problem.
a) 25 – 5 – 5 – 5 – 5 – 5 = 0
b) 24 – 8 – 8 – 8 = 0
(2)
6. Write a number sentence for each problem.
If there are 24 apples, how many children can each get the
following number of apples?
a) 8
b) 4
c) 3
d) 6(4)
7. Complete each problem.
a) 5 × 7 = n
7×5=n
b) 4 × 8 = n
8×4=n
c) 36 ÷ 9 = n
9×4=n
d) 42 ÷ 7 = n
7×6=n
e) 8 ÷ 1 = n
1 × 8 = n(5)
126
Math G4 TG.indb 126
Mathematics Teacher’s Guide Grade 4
TERM 1
2012/09/14 5:32 PM
8. Complete a copy of the table.
0
1
2
3
4
5
×6
÷3
(12)
9. How many packets with four tomatoes each can you fill if
you have 20 tomatoes?
(1)
10. There are six packets of tomatoes with eight tomatoes in
each packet. How many tomatoes are there altogether?
(1)
11. Write the opposite (inverse) operations.
a) Multiply by 100
b) Divide by 10
c) 6 × 7 = n
d) 36 ÷ 9 = n
e) 12 – 4 = n
f) 23 + 7 = n(6)
Total [50]
Mathematics Teacher’s Guide Grade 4
Math G4 TG.indb 127
TERM 1
127
2012/09/14 5:32 PM
Assessment task 8 Multiplication and division
Solutions
Tell the learners that they will now perform an assessment task to
demonstrate knowledge of the multiplication and division concepts
they have developed over the past lessons. They will display
understanding of repeated addition as a short way for multiplication,
repeated subtraction as a short cut for division, knowledge of
the multiplication and division tables, the commutative property,
inverse operations and properties of 1 and 0. Learners must show
all their calculations so that you can check their understanding. This
will help you plan activities to develop concepts that they still find
difficulty with.
Solutions
1. The learners write multiplication number sentences for
repeated addition.
a) 8 × 8 × 8 × 8 × 8
b) 9 + 9 + 9
= 5 × 8
=3×9
= 40
= 27
c) 4 + 4 + 4 + 4 + 4 + 4
= 6 × 4
= 24
d) 6 + 6 + 6 + 6 + 6
=5×6
= 30
e) 0 + 0 + 0 + 0
= 4 × 0
= 0
(5)
2. a)
b)
c)
d)
e)
4 bags with 7 oranges
7 bags with 4 oranges
8 bags with 8 oranges
6 bags with 7 oranges
6 bags with 0 oranges
3. a)
b)
c)
d)
e)
7 × 8 = 56
6 × 3 = 18
4 × 8 = 32
5 × 9 = 45
1 × 6 = 6
4 × 7 = 28
7 × 4 = 28
8 × 8 = 64
6 × 7 = 42
6 × 0 = 0
(5)
(5)
4. The learners complete the flow diagrams with single and
double function machines. They should notice that × 8 is
the same as × 4 × 2, for example.
9 ×6
a)
128
Math G4 TG.indb 128
b)
6
×8
c)
9
×3
×2
d)
6
×4
×2
Mathematics Teacher’s Guide Grade 4
(4)
TERM 1
2012/09/14 5:32 PM
5. a) 25 – 5 – 5 – 5 – 5 – 5 = 0
25 ÷ 5 = 5
b) 24 – 8 – 8 – 8 = 0
24 ÷ 8 = 3
(2)
6. a)
b)
c)
d)
8 apples
4 apples
3 apples
6 apples
24 ÷ 8 = 3 children
24 ÷ 4 = 6 children
24 ÷ 3 = 8 children
24 ÷ 6 = 4 children
(4)
7. a)
b)
c)
d)
e)
5 × 7 = 35
4 × 8 = 32
36 ÷ 9 = 4
42 ÷ 7 = 6
8÷1=8
7 × 5 = 35
8 × 4 = 32
9 × 4 = 36
7 × 6 = 42
1 × 8 = 8
(5)
8. Complete a copy of the table.
×6
÷3
0
0
0
1
6
2
2
12
4
3
18
6
4
24
8
(12)
9. 20 ÷ 4 = 5 packets of tomatoes
(1)
10. 6 × 8 = 48 tomatoes altogether
(1)
11. a)
b)
c)
d)
e)
f)
Multiply by 100
Divide by 10
6 × 7 = 42
36 ÷ 9 = 4
12 – 4 = 8
23 + 7 = 30
Divide by 100
Multiply by 10
42 ÷ 7 = 6
4 × 9 = 36
8 + 4 = 12
30 – 7 = 23
Mathematics Teacher’s Guide Grade 4
Math G4 TG.indb 129
5
30
10
(6)
Total [50]
TERM 1
129
2012/09/14 5:32 PM
Math G4 TG.indb 130
2012/09/14 5:32 PM
TERM
Whole numbers
Whole numbers: multiplication
Unit 1
Count and order
Unit 24 Basic multiplication facts
Unit 2
Compare and represent
numbers
Unit 25 Multiplication strategies
Unit 3
Place value
Unit 26 Problem-solving with
multiplication
Unit 4
Estimate and round off
Unit 27 Multiplication and estimation
Whole numbers: addition and
subtraction
Unit 5
Add and subtract multiples
of 10
Unit 6
Strategies for adding and
subtracting
Unit 7
More strategies for adding
and subtracting
Unit 8
Add and subtract with 3- and
4-digit numbers
Common fractions
Unit 9
Order and compare fractions
Unit 28 Patterns in multiplication
Unit 29 More multiplication methods
Properties of 3-D objects
Unit 30 Flat and curved surfaces
Unit 31 Shapes and faces of
3-D objects
Unit 32 Straight, flat faces: polyhedra
Investigation
Revision
Geometric patterns
Unit 33 Explore geometric patterns
Unit 10 Represent fractions
Unit 34 Identify and extend patterns
Unit 11 Equal sharing
Unit 35 Extend patterns
Unit 12 Calculations with fractions
Unit 36 Input and output numbers
(values)
Unit 13 Equivalent fractions
Unit 14 Count and calculate
fractions
2
Symmetry
Unit 37 What is symmetry?
Investigation
Length
Unit 15 Revision of Grade 3 work
Revision
Unit 16 Work with centimetres (cm)
and millimetres (mm)
Whole numbers: addition and
subtraction
Unit 17 Tricky measurements
Unit 38 Round off to add and
to subtract
Unit 18 Understand units of
measurement
Unit 19 Convert between kilometres,
metres and millimetres
Unit 20 Convert between centimetres
and metres
Unit 21 Convert between millimetres
and centimetres
Unit 22 Round off measurements
Unit 23 Problem-solving with distance
and length
Unit 39 Different ways to add
Unit 40 Different ways to subtract
Unit 41 Check addition and
subtraction calculations
Whole numbers: division
Unit 42 Solve story problems
Unit 43 Division with and without
remainders
Unit 44 Division with remainders
Revision and consolidation
Unit 45 Division with 3-digit numbers
and remainders
Revision
Unit 46 Problem-solving with division
131
Math G4 TG.indb 131
2012/09/14 5:32 PM
Whole numbers
In Term 2, the learners will:
• count forwards and backwards in 2s, 3s, 5s, 10s, 25s, 50s and
100s between 0 and at least 10 000
• round off numbers to the nearest 10, 100 and 1 000
• order, compare and represent at least 4-digit numbers
• recognise place value of whole numbers to at least 4-digit
numbers.
The CAPS schedule allocates only one hour to these concepts in
this term. However, you can use mental mathematics time for this
work because of the importance of developing these concepts. The
concept of place value is integrated into calculations involving the
four basic operations where learners break up numbers to develop
and illustrate understanding.
The concepts dealt with in the Mental maths activities should be
linked to the concepts that will be developed in the main lesson. You
should read the learners’ reflections regularly to find out what they
think they know and do not know.
Unit 1
Count and order
Mental Maths Learner’s Book page 102
1–4. The learners count the number of learners in the class in
2s. If the number of learners is an odd number such as 41,
ask learners if they can ordered the numbers in rows with
an equal number of learners in each group. Ask the learners
to arrange themselves in rows with an equal number of
learners in each row (for example, rows of 10, 5, 8, 20). If
there is an odd number of learners, they should check the
remainder for each arrangement. This can be a game where
learners try to not be a remainder.
When learners count fingers and eyes, you can ask them
how the totals will change if five learners join the class.
5, 6. Learners should find the quickest way to count the number
of objects in the pictures. Check their strategies and suggest
ones they do not use as in the solutions. Ask the class which
strategy they find the easiest to work with. In this way you
do not impose strategies and force learners to use prescribed
strategies, but rather allow them to compare strategies and
use the most effective ones or the ones they prefer. The
learners should realise how effective it is to use doubling to
solve certain problems.
132
Math G4 TG.indb 132
Mathematics Teacher’s Guide Grade 4
TERM 2
2012/09/14 5:32 PM
7.
They should also understand why it is important to be able
to calculate with 10, multiples of 10 and powers of 10.
Draw the number lines on the board and let learners
estimate and plot the numbers on the number lines. Once
they have estimated the distance between 0 and 50 (for
(a)), they should find it easy to plot the other numbers.
Remember, the distances do not have to be perfect; they are
estimates. Let them count in 1 000s to bridge 10 000. Ask
them to count from 9 500 in 50s and 100s to bridge 10 000
and from 9 800 in 25s to bridge 10 000.
If there are calculators, show the learners how to program
them to do counting. For example, to count in 100s: key in
.
The calculator automatically adds 100 each time you press
.
Solutions
1. Answers will differ.
2. 400 ÷ 10 = 40 learners
3. 90 ÷ 2 = 45 learners
4. Yes. They can make three rows of 9 because 3 × 9 = 27.
5. Using distributive property, doubling, multiplying by
multiples of 10 and using factors will make it easier to work
out the answers.
a) (10 × 5) × 2 + (5 × 5) = n. Learners are not expected to
write this number sentence. Allow them to use their own
strategies.
R5 × 10 = R50
R50 × 2 = R100
R100 + R25 = R125
b) 9 × 50 = 9 × 5 × 10
= R450
c) Using doubling makes calculations easier. Break up 6
into addition bonds.
(25 × 6) + 10 = n
25 × 3 = 75
or (25 × 4) + (25 × 2) + 10
75 + 75 = 150 (double 75)
= 100 + 50 + 10
150 + 10 = 160 pencils
= 160
or (20 × 6) + (5 × 6) + 10
= 120 + 30 + 10
= 160
d) 11 × 100 = 1 100 books
e) 13 × R10 = R130
Mathematics Teacher’s Guide Grade 4
Math G4 TG.indb 133
TerM 2
133
2012/09/14 5:32 PM
6. Learners explain their strategies.
7. a)
0
50
250
400 450 500
750
900 950 1 000
0
50
250
400 450 500
750
900 950 1 000
b)
9 500
0
1 000
5 000
6 000
8 000
0
1 000
5 000
6 000
8 000
10 500
11 500
9 500 10 500 11 500
10 000 11 000 12 000
10 000
Activity 1.1
11 000 12 000
Learner’s Book page 103
1. Ask learners to create equal spaces for the intervals. Assist
them if they struggle to bridge the powers of 10 (100; 1 000 and
10 000).
a)
280
280
280
280
280
285
285
285
285
285
290
290
290
290
290
295
295
295
295
295
300
300
300
300
300
305
305
305
305
305
310
310
310
310
310
315
315
315
315
315
320
320
320
320
320
325
325
325
325
325
280
280
830
280
830
830
830
830
285
285
840
285
840
840
840
840
290
290
850
290
850
850
850
850
295
295
860
295
860
860
860
860
300
300
870
300
870
870
870
870
305
305
880
305
880
880
880
880
310
310
890
310
890
890
890
890
315
315
900
315
900
900
900
900
320
320
910
320
910
910
910
910
325
325
920
325
920
920
920
920
840
840
800
840
800
800
800
800
850
850
825
850
825
825
825
825
860
860
850
860
850
850
850
850
870
870
875
870
875
875
875
875
880
880
900
880
900
900
900
900
890
890
925
890
925
925
925
925
900
900
950
900
950
950
950
950
910
910
975
910
975
975
975
975
920
11920
000
920
000
11 000
1 000
000
775
d)
775
600
775
600
600
600
600
800
800
700
800
700
700
700
700
825
825
800
825
800
800
800
800
850
850
900
850
900
900
900
900
950
950
11 400
950
400
11 400
400
1 400
975
975
11 500
975
500
11 500
500
1 500
1 000
111 000
600
1 000
600
11 600
600
1 600
600
e)
11600
000
600
000
11 000
1 000
000
700
22700
000
700
000
22 000
2 000
000
800
33800
000
800
000
33 000
3 000
000
1 400
119 400
000
9 400
000
99 000
9 000
000
1 500
11 500
10
000
500
10
000
10
000
10
10 000
000
1 600
11 600
11
000
600
11
000
11
000
11
11 000
000
b)
c)
830
830
775
830
775
775
775
775
875
900
925
875
900
925
11 000
11 300
87511 100
000
10090011 200
200 925
300
11 000
1
100
1
200
000 11 100
100 11 200
200 111 300
300
1 000
300
900
44900
000
900
000
44 000
4 000
000
1 000
115 000
000
5 000
000
55 000
5 000
000
1 000 2 000 3 000 4 000
9 100
99 200
99 300
100
2003 000
300
f) 2 000
1199 000
44 000
100
9
200
99 300
000
2
000
3
000
000
100
99 200
99 100
200
9 300
300
5 000
99 400
400
000
99 55400
000
400
9 400
6 000 7 000 8 000 9 000 10 000
9 500
99 600
99 700
99 800
500 7 000
6008 000
700
800
6699 000
99 000
10
500
9
600
9
700
9 800
000
7
000
8
000
000
10 000
000
500
99 600
99 700
800
99 500
600
700 99 800
11 000
10
000
10 000
000
11
10
11
000
10 000
000
10
000
9 100
10 100
100
9910
100
10
100
10100
100
10
100
g)
9 200
9 950
950
9999 200
950
200
950
99 950
9 300
9 900
900
9999 300
900
300
900
99 900
9 400
9 850
850
9999 400
850
400
850
99 850
9 500
9 600
600
9999 500
600
500
600
99 600
10 000
99 550
550
10
000
99 550
10
000
550
9 550
10 100
10
100
10 000
000
100
10
000
10
000
10 000
9 950
999975
950
975
950
999975
975
9 975
9 900
99 950
99 900
950
900
99 950
950
9 950
9 850
9 600
9 750
9 700 9 650
99 925
9
900
9
875
9
850
99 825
99 800
9
850
9
600
9
750
925
9 850 99 700
825 99 650
800
9 85099 900
9 60099 875
9 750
700
650
99 925
900
875
99 850
99 825
9 800
925
9
900
9
875
850
825
9 925 9 900 9 875 9 850 9 825 99 800
800
9 550
775
9999 550
775
99 550
775
9 775
775
10 000
10
10 000
000
9 975
99 975
975
9 950
99 950
950
9 925
99 925
925
9 775
99 775
775
h)
1 100
116 100
000
6 100
000
66 000
6 000
000
9 900
99 900
900
1 200
117 200
000
7 200
000
77 000
7 000
000
1 300
118 300
000
8 300
000
88 000
8 000
000
9 600
9 750
750
9999 600
750
600
750
99 750
9 875
99 875
875
9 850
99 850
850
9 700
9 700
700
9999 700
700
700
700
99 700
9 825
99 825
825
9 800
9 650
650
9999 800
650
800
650
99 650
9 800
99 800
800
2. a) 10 000; 9 995; 9 990; 9 985; 9 980; 9 975; 9 970; 9 965;
9 960
b) 9 986; 9 988; 9 990; 9 992; 9 994; 9 996; 9 998; 10 000;
10 002; 10 004
134
Math G4 TG.indb 134
Mathematics Teacher’s Guide Grade 4
TERM 2
2012/09/14 5:32 PM
c) 10 050; 10 040; 10 030; 10 020; 10 010; 10 000; 9 990;
9 980; 9 970; 9 960
d) 9 225; 9 325; 9 425; 9 525; 9 625; 9 725; 9 825; 9 925;
10 125; 10 225
e) 9 993; 9 996; 9 999; 10 002; 10 005; 10 008; 10 011; 10 014;
10 017; 10 020
3. The learners use the area model to count the total number of
dots by counting the number of dots in a row and the number of
rows and then multiplying the numbers. Check which learners
use repeated addition. Let them compare their strategies to find
the quickest way to count the dots. You could ask how many
dots there would be if the number of dots were doubled in each
array.
a) 10 × 8 = 80
b) 10 × 10 = 100
c) 12 × 10 = 120
d) 5 × 9 = 45
4. a) There are four ways to create arrays for 40 counters or
dots. If you consider the commutative property, there are
eight ways of doing it; for example, 1 × 40 and 40 × 1. The
number of dots is the same, but the arrangement is different
(one row with 40 dots and 40 rows with one dot).
1 × 40 = 40
2 × 20 = 40
4 × 10 = 40
5 × 8 = 40
b) The learners draw a chocolate slab with 12 blocks arranged
in four different ways. Let them find the factors that give
a product of 12. Ask them to record the factors and their
commutative pairs. The learners should know by now that
they have to work systematically. Start with 1:
1 × 12 = 12
2 × 6 = 12
3 × 4 = 12
12 × 1 = 12
6 × 2 = 12
4 × 3 = 12
c) Ask the learners to represent 24 in six different ways. Let
them use both pairs of factors that give a product of 24:
1 × 24 = 24
2 × 12 = 24
3 × 8 = 24
4 × 6 = 24
24 × 1 = 24
12 × 2 = 24
8 × 3 = 24
6 × 4 = 24
5. Explain how the Dienes blocks are built up with
1 cm × 1 cm × 1 cm (1 cubed centimetre (1 cm3)) cubes and
let learners find out how many small cubes there are in each
representation.
Units learners use.
• When they measure length, they work with centimetres (cm).
Mathematics Teacher’s Guide Grade 4
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• W
hen they measure area, they work with square centimetres
(cm2).
• When they work with volume, they work with cubic
centimetres (cm3).
In Grade 4, we use Dienes blocks to develop understanding of
place value and the representation of numbers.
1 000 + 100 + 10 + 1 = 1 111 blocks
Unit 2
Compare and represent numbers
Mental Maths Learner’s Book page 104
The learners now start working with numbers that are bigger
than 1 000. They use single digits to create numbers. Discuss the
difference between reading four digits as a year and as a number
of years. Learners record their solutions to question 4 on their
Mental maths grids.
Solutions
1. a) Biggest: 5 421
Smallest: 1 245
b) Biggest: 7 531
Smallest: 1 357
c) Biggest: 2 200
Smallest: 2 002
d) Biggest: 5 411
Smallest: 1 145
2. a) 5 421 – 1 000 = 4 421
b) 7 531 – 1 000 = 6 531
c) 2 200 – 1 000 = 1 200
d) 5 411 – 1 000 = 4 411
3. Ask the learners to listen carefully when they watch a movie
with American actors. The Americans read a number such
as 1 922 as nineteen twenty-two (they group the thousands
and hundreds and do not say ‘hundred’). Ask the learners to
read 1 922 as we do. We read years as dates, for example, for
2011 we say two thousand and eleven, but people also read it
as twenty eleven. The year 1922 is read as nineteen hundred
and twenty-two or nineteen twenty-two. Learners can find
out in which years their family members were born and they
read the years.
4. Answers will differ.
5. a) 3 003
b) 4 500
c) 9 999
d) 305
e) 1910
f) 6 450
g) 5 000
h) 8 008
i) 8 550
j) 2020
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Activity 2.1
Learner’s Book page 105
1. The learners would probably count the number of cubes in a
linear manner. You could show them this way of adding the
number of cubes as an alternative strategy. The vertical column
strategy could prepare them for column calculations that they
will formally engage with in Grade 5. Learners can read each
number aloud.
a) 1 000 + 1
1 000 + 1
1 000 + 1
4 000 + 3 = 4 003
b) 1 000 + 100 + 10 + 1
1 000 + 100 + 10
1 000 + 100 + 10
1 000 + 100 + 10
5 000 + 400 + 40 + 1 = 5 440
c) 1 000 + 100 + 70 = 1 170
d) 1 000 + 100 + 10 + 6
1 000 + 100 + 10
100 2 000 + 300 + 20 + 6 = 2 326
e) 1 000 + 100 + 10 + 1 = 1 111
f) 1 000 + 10
1 000 + 10
1 000 + 10
1 000 + 10
10
4 000 + 50 = 4 050
2. a)
b)
c)
d)
e)
f)
9 060: nine thousand and sixty
345: three hundred and forty-five
8 455: eight thousand four hundred and fifty-five
2 826: two thousand eight hundred and twenty-six
1 203: one thousand two hundred and three
7 006: seven thousand and six
3. Learners will have to work systematically to avoid confusion.
Learners can read the numbers they created to the class.
1 035, 1 305, 1 350, 1 503, 1 530
3 015, 3 051, 3 105, 3 150, 3 150
5 013, 5 031, 5 103, 5 301, 5 310
4. The learners investigate numbers that would fit the descriptions.
They will probably solve the problems by trial and improvement.
a) 36
b) 11
c) 25
5. 102, 120, 210
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Unit 3
Place value
Mental Maths Learner’s Book page 106
This task can be quite challenging because the places of the
digits in the clue cards are mixed up. Allow the learners to
work in pairs. Then tell them to write down the numbers as
indicated by the clue cards before they search for the numbers.
Give learners copies of the number grid. They have to write the
number of the question next to each number they identify and
also circle the number.
1
3
9
5
6
3
8
1
7
4
1
1
1
1
3
3
6
0
5
3
7
3
2
7
6
1
8
2
3
9
3
7
9
4
3
2
3
9
0
3
7
9
0
2
2
5
1
1
4
4
4
0
8
9
3
3
1
2
6
2
2
5
3
1
1
1
4
5
9
2
9
1
6
8
1
Activity 3.1
2
1
3
8
8
0
8
0
2
4
7
3
1
3
0
7
2
0
3
1
3
2
8
0
5
5
3
9
8
3
4
7
2
8
4
Learner’s Book page 107
1. Make copies of the place value scatter board and give each
group ten beans. Read the rules with the class and check that
learners know how to play the game.
10 000
100
1
10
10
1 000
1
100
2, 3. Remember that learners’ ability to identify the place value
of digits in numbers is not proof that they have conceptual
understanding of place value. Learners who have developed
a good sense of number and place value can identify, for
example, that there are 38 hundreds in 3 800, 380 hundreds in
38 000 and 380 tens in 3 800. You could help learners develop
this understanding by asking questions such as the following:
• How many boxes with 10 biscuits each can be filled from
198 biscuits?
138
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Mathematics Teacher’s Guide Grade 4
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• H
ow many bags of wood with 10 pieces each can be filled
from 233 pieces of wood?
• The bakery baked 3 456 cookies. How many tins with
100 cookies each can they fill?
2,3. a)
b)
c)
d)
e)
f)
g)
h)
i)
Unit 4
4 324: 20
6 304: 6 000
1 567: 500
7 822: 2
97: 90
123: 100
1 001: 1
3 450: 3 000
9 090: 90
4 000 + 300 + 20 + 4
6 000 + 300 + 4
1 000 + 500 + 60 + 7
7 000 + 800 + 20 + 2
90 + 7
100 + 20 + 3
1 000 + 1
3 000 + 400 + 50
9 000 + 90
Estimate and round off
Mental Maths Learner’s Book page 107
The learners worked with estimation in Term 1. Ask them why
it us useful to be able to round off numbers. We often estimate
physical quantities in real life. Examples include population
figures, measurements, people’s ages and the cost of items
we buy. Ask the learners where they use estimates in real life.
We often use the words “I think it’s about . . .” when we make
estimates. In working with numbers, we try to make good
(accurate) estimates. Developing the skill to make accurate
estimations could assist learners in calculating more effectively.
They can use estimation to predict the size of solutions,
check them and judge their reasonableness. Learners should
develop the knowledge and skill to round off numbers to make
estimations.
1. Lead a class discussion about using estimates in everyday
life.
2. The ship is about 50 m long.
3. a) 23 → 20; 35 → 40; 55 → 60; 96 → 100
b) 105 → 100; 223 → 200; 451 → 500; 555 → 600;
996 → 1 000
c) 1 005 → 1 000; 2 223 → 2 000; 4 451 → 4 000;
5 555 → 6 000; 9 996 → 10 000
Mathematics Teacher’s Guide Grade 4
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4. Rounding off to the nearest 10, 100 and 1 000
Number
Round off to the nearest:
10
100
1 000
15
0
0
15
23
20
0
0
35
40
0
0
51
50
100
0
55
60
100
0
96
100
100
0
105
110
100
0
223
220
200
0
451
450
500
0
555
560
600
1 000
996
1 000
1 000
1 000
1 005
1 010
1 000
1 000
2 223
2 220
2 200
2 000
4 451
4 450
4 500
4 000
5 555
5 560
5 600
6 000
9 996
10 000
10 000
10 000
5. 498 + 435 = 500 + 435 (compensation)
= 935 – 2
= 933
920 is the closest estimate.
6. a)
b)
c)
d)
e)
f)
g)
h)
i)
j)
100 + 40 = 140
170 – 120 = 50
500 + 460 = 960
3 600 – 2 100 = 1 500
2 400 + 3900 = 6 300
70 × 10 = 700
3 800 + 3 200 = 7 000
450 + 500 = 950
800 ÷ 20 = 40
200 × 40 = 8 000
97 + 43 ≈ 140
167 – 115 ≈ 50
503 + 455 ≈ 960
3 626 – 2 056 ≈ 1 500
2 423 + 3 879 ≈ 6 300
73 × 14 ≈ 700
3 799 + 3 199 ≈ 7 000
445 + 495 ≈ 950
836 ÷ 21 ≈ 40
239 × 43 ≈ 8 000
Activity 4.1
Learner’s Book page 109
Rounding off to the nearest 10 normally gives more accurate
estimates than rounding off to the nearest 100 or 1 000. Divide the
learners into three groups. Let each group use a different rounding
strategy. Let them calculate the accurate solutions to check whose
estimates are more accurate. Let them solve the word problems by
estimation. Learners can then round off numbers to the nearest 10,
100 and 1 000. Ask them to solve the real-life problems by rounding
off and estimation.
140
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1. Learners compare estimates with calculated answer
a) 97 + 43 = 97 + 3 + 40
= 140
(Equivalent to the estimate 140)
b) 167 – 115 = 160 – 115 + 7
= 145 + 7
= 152
(2 more than the estimate 150)
c) 503 + 455
= 958 (2 less than the estimate 960)
d) 3 626 – 2 056
= 1 570
(70 more than the estimate 1 500)
e) 2 423 + 3 879
= 6 302
(2 more than the estimate 6 300)
f) 73 × 14 = (70 × 10) + (70 × 4) + (3 × 10) + (3 × 4)
= 700 + 280 + 30 + 12
= 1 022 (322 more than the estimate 700;
70 × 10 is not a good estimate.)
2. a) 23 × 65
20 × 70 = 1 400
They pay about R1 400.
b) 23 × 99
20 × 100 = 2 000
The T-shirts cost about R2 000.
c) 555 × 65
600 × 70 = 42 000
The total cost of the tickets is about R42 000.
d) 23 × 15
20 × 20 = 400
The bus tickets cost about R400.
3.
Number
10
a)
b)
c)
d)
e)
Round off to the nearest:
100
1 000
1 325
1 330
1 300
1 000
646
650
600
1 000
7 578
7 580
7 600
8 000
3 299
3 300
3 300
3 000
10 768
10 770
10 800
11 000
4. 8 × R29,99: 8 × 30 = 240 The material costs about R240.
5. 39 × R10,95: 40 × 11 = 440 The petrol costs about R440.
6. 6 × R14,95: 6 × 15 = 90 The socks cost about R90.
Mathematics Teacher’s Guide Grade 4
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Assessment task 1 Counting, place value and estimation
1. How many objects are there? Show how you count the objects.
a)
b)
c)
d)
50
50
Fifty
rand
50
50
Fifty
rand
50
Fifty
rand
50
50
50
20
50
50
Fifty
50 Fifty
50 Fifty
Fifty
Fifty
Fifty
20
20 Twenty
Twenty
Twenty
rand
rand
rand
rand 50
rand 50
rand 50
50
50
50
rand
rand
rand
20
20
20
20
20
20
20
Twenty
20 Twenty
20 Twenty
Twenty
Twenty
Twenty
rand
rand
rand
rand 20
rand 20
rand 20
20
20
20
(4)
2. Give the number that is represented by each set of blocks.
a)
b)
c)
142
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Mathematics Teacher’s Guide Grade 4
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d)
(4)
3. Write down the missing numbers.
a) 7; 17; __; __; 47; __; __; 77; 87
b) __; 250; 275; __; 325; __; 375; __; 425
c) 445; 545; 645; __; __; 945; __; __; 1 245
d) 172; 1 172; 2 172; __; __; 5 172; 6 172; 7 172; __; __
(4)
4. Write the number name for each number.
a) 108
b) 625
c) 777
d) 3 056
(4)
5. Expand each number.
a) 428
c) 999
(4)
b) 709
d) 5 341
6. Which number is shown on each set of flard cards?
a) 1 000
300
60
6
1 000
300
60
6
17 000 400300 10 60 2 000
6
6
b) 17 000 400300 10 60 2 000
7
400
2 000
50
5 000 10
7
400
10 50
2 000
50
5
000
50
c) 50
5
000
50
950000 5 000
900 50
9
9 000
900
9
900
9
d) 99 000
(4)
000
900
9
7. What is the value of the underlined digit in each number?
a) 7 145
b) 2 678
c) 5 914
c) 8 395(4)
8. a) Round off to the nearest 10: 56 and 342
b) Round off to the nearest 100: 418 and 1 055
c) Round off to the nearest 1 000: 634 and 6 678
(3)
9. Round off the numbers and estimate the answers.
Do not work out the answers.
a) 63 + 75 = n
b) 1 244 – 567 = n
c) 38 × 42 = n
d) 110 × 13 = n(4)
Total [35]
Mathematics Teacher’s Guide Grade 4
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Assessment task 1 Counting, place value and estimation
Solutions
1. The learners should show their counting strategies. Encourage
them to use brackets to show the calculations they do first.
a) (3 × 6) + 3
b) (4 × 4) + 3
= 18 + 3
= 16 + 3
= 21 circles
= 19 squares
c) (5 × 5) + (4 × 2)
= R25 + R8
= 33 rectangles
d) (6 × 50) + (3 × 20)
= R300 + R60
= R360
(4)
2. a) 129
c) 560
b) 308
d) 2 444
(4)
3. a)
b)
c)
d)
7; 17; 27; 37; 47; 57; 67; 77; 87
225; 250; 275; 300; 325; 350; 375; 400; 425
445; 545; 645; 745; 845; 945; 1 045; 1 145; 1 245
172; 1 172; 2 172; 3 172; 4 172; 5 172; 6 172; 7 172; 8 172;
9 172(4)
4. a)
b)
c)
d)
108: one hundred and eight
625: six hundred and twenty-five
777: seven hundred and seventy-seven
3 056: three thousand and fifty-six
(4)
5. a)
b)
c)
d)
428 → 400 + 20 + 8
709 → 700 + 9
999 → 900 + 90 + 9
5 341 → 5 000 + 300 + 40 + 1
(4)
6. a) 1 366
c) 5 550
b) 2 417
d) 9 909
(4)
7. a) 7 145: 100
c) 5 914: 10
b) 2 678: 2 000
d) 8 395: 5
(4)
8. a) Round off to the nearest 10: 56 → 60 and 342 → 340
b) Round off to the nearest 100: 418 → 420 and 1 055 → 1 000
c) Round off to the nearest 1 000:
634 → 1 000 and 6 678 → 7 000
(3)
9. a) 63 + 75 → 60 + 80 = 140
b) 1 244 – 567: 1 200 – 600 = 600
or, 1 240 – 600 = 640
or, 1 240 – 570 = 1 240 – 500 – 70
= 740 – 70
= 670
c) 38 × 42 → 40 × 40 = 1 600
d) 110 × 13 → 110 × 10 = 1 110
144
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Mathematics Teacher’s Guide Grade 4
(4)
Total [35]
TERM 2
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Whole numbers: addition and subtraction
Unit 5
Add and subtract multiples of 10
Mental Maths Learner’s Book page 110
Learners add and subtract small numbers from multiples of 10.
They can do these problems mentally without writing anything
down. They should have developed knowledge of the basic
addition and subtraction facts based on 10 and multiples of 10 by
now. Allow the learners to check each other’s solutions. Learners
work with up to 5-digit numbers. The activity involves counting
forwards and backwards in 2s, 20s, 25s, 40s, 50s and 200s.
Starting number
Do this:
10
+2
12
14
16
18
20
50
+ 20
70
90
110
130
150
Activity 5.1
500
– 25
475
450
425
400
375
70
+ 40
110
150
190
230
270
1 000
– 50
950
900
850
800
750
2 000
– 200
1 800
1 600
1 400
1 200
1 000
Learner’s Book page 110
Some learners might be able to solve these problems mentally. Let
them do the written calculations anyway so it will be easier for them
to notice the relationships. They start with units and then add and
subtract multiples of 10 and powers of 10. Allow learners to talk
about the patterns they notice.
Solutions
1. a) 6 + 7 = 13
60 + 70 = 130
600 + 700 = 1 300
6 000 + 7 000 = 13 000
b) 9 + 5 = 14
90 + 50 = 140
900 + 500 = 1 400
9 000 + 5 000 = 14 000
c) 13 – 8 = 5
130 – 80 = 50
1 300 – 800 = 500
13 000 – 8 000 = 5 000
d) 17 – 9 = 8
170 – 90 = 80
1 700 – 900 = 800
17 000 – 9 000 = 8 000
e) 12 – 7 = 5
120 – 70 = 50
1 200 – 700 = 500
12 000 – 7 000 = 5 000
Mathematics Teacher’s Guide Grade 4
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Unit 6
2. a) 10 – 7 = 3
100 – 7 = 93
1 000 – 7 = 9 993
b) 20 – 6 = 14
200 – 6 = 194
2 000 – 6 = 1 994
c) 30 + 13 = 43
300 + 113 = 413
3 000 + 1 113 = 4 113
d) 60 + 26 = 86
600 + 260 = 860
6 000 + 2 260 = 8 260
e) 40 – 12 = 28
400 – 120 = 280
4 000 – 1 120 = 1 880
f) 70 + 40 = 110
700 + 400 = 1 100
7 000 + 4 000 = 11 000
g) 50 – 15 = 35
500 – 150 = 350
5 000 – 1 500 = 3 500
h) 80 + 90 = 170
800 + 900 = 1700
8 000 + 9 000 = 17 000
Strategies for adding and subtracting
Mental Maths Learner’s Book page 111
1. Let the learners study the explanations and find out what
the original calculation is. This could be challenging, but
give them time to battle with the problems. Guide them if
necessary. In (a) the learner broke up 7 and first took away 6
from 56 and then subtracted 1. The problem is 57 – 7 = n.
In (b) the learner had to calculate 28 + 29 = n. He
built up both numbers to the nearest multiples of 10
and then subtracted the numbers he had added (he used
compensation).
In (c) the original problem is 37 + 26 = n. The learner broke
up both numbers and added the units and then the tens.
2. Ask the learners to explain how they will solve the problems.
Allow learners to use their own strategies. If learners do not
use the strategies given below, suggest these strategies as
alternatives. Let them compare the strategies.
a)
57 + 36 = n
57 + 3 + 33 = 60 + 33
(building up)
= 93
b) 68 – 9 = n
68 – 8 = 69 – 10
= 59
(add 1 to both numbers)
c)
92 – 27 = n
92 – 2 – 25 = 90 – 25
= 90 – 20 – 5
= 65
(breaking down)
d)
49 + 47 = n
49 + 1 + 46 = 50 + 46
= 96
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e)
36 + 48 = n
36 + 4 + 44 = 40 + 44
= 84
f)
74 – 46 = n
74 – 4 – 42 = 70 – 42
= 70 – 40 – 2
= 28
(compensation)
g)
27 + 59 = n
27 + 3 + 56 = 30 + 56
= 86
h)
55 – 27 = n
55 – 5 – 22 = 50 – 22
= 50 – 20 – 2
= 28
(compensation)
The learners should have realised that the most
effective strategy for subtraction is probably adding the
same number to both numbers. The strategy used for
question 2(h) (above) is easier than breaking down the
numbers as shown below.
55 – 27 = 58 – 30
(add 3 to both sides)
= 28
55 – 27 = 55 – 5 – 22
= 50 – 20 – 2
= 30 – 2
= 28
Activity 6.1
(breaking down)
Learner’s Book page 111
Ask the learners to use their own methods to solve the problems
with 3- and 4-digit numbers. Allow them to compare strategies. If
they do not use the strategies below, suggest them as alternatives.
Solutions
1. a) 467 + 518 = n
467 + 13 = 480
480 + 505 = 985
b) 523 – 294 = n
(decomposition)
500 + 20 + 3 400 + 110 + 13
– 200 + 90 + 4
– 200 + 90 + 4
= 200 + 20 + 4
523 – 294 = 224
c) 735 – 386 = n
(decomposition)
700 + 30 + 5 600 + 120 + 15
– 300 + 80 + 6
– 300 + 80 + 6
= 300 + 40 + 9
735 – 386 = 349
Mathematics Teacher’s Guide Grade 4
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d) 603 – 452 = n
600 – 400 = 200
200 – 50 = 150
150 – 2 + 3 = 148 + 3
603 – 452 = 151
e) 1 223 – 1 134 = n
1 200 – 1 100 = 100
100 – 30 – 4 = 66
66 + 23 = 89
1 223 – 1 134 = 89
f) 3 426 – 2 357 = n
3 400 + 2 300 = 1 100
1 043 + 7 + 19 = 1 069
3 426 – 2 357 = 1 069
g) 2 526 + 2 338 = n
2 500 + 2 300 = 4 800
4 800 + 26 + 4 = 4 830
4 830 + 34 = 4 864
2 526 + 2 338 = 4 864
h) 5 104 + 1 316 = n
5 104 + 16 = 5 120
5 120 + 1 300 = 6 420
5 104 + 1 316 = 6 420
(breaking down)
(compensation)
(compensation)
(building up)
(breaking down)
2. Learners explain how they worked out answers.
3. The learners only need to look at the units in the solutions to be
able to say that a calculation is incorrect. Check that they use
decomposition to justify their reasons when subtracting.
a) 7 + 9 = 16
The unit should be 6; not 5.
b) 17 – 8 = 9
The unit should be 9; not 1.
c) 3 + 8 = 11
The unit should be 1; not 5.
d) 15 – 9 = 6
The unit should be 6; not 4.
4. Learners explain their strategies.
a) 427 + 239 = 666
b) 517 – 238 = 279
c) 1 203 + 1 478 = 2 681
d) 2 435 – 1 469 = 966
Unit 7
More strategies for adding and
subtracting
Mental Maths Learner’s Book page 112
The learners write down the solutions in the first exercise on
their Mental Maths Grids. They should discover that the units
in the calculations add up to 10 each time (they are bonds of
10). They explain how they will solve the problems in the next
exercise. Check whether they recognise that the tens and units
are now addition and subtraction bonds of 10s and 100s.
148
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Solutions
1. a)
c)
e)
g)
i)
17 + 3 = 20
75 + 5 = 80
82 + 8 = 90
299 + 1 = 300
593 + 7 = 600
b)
d)
f)
h)
j)
36 + 4 = 40
61 + 9 = 70
124 + 6 = 130
1 995 + 5 = 2 000
1 996 + 4 = 2 000
2. a)
125 + 75 = n
100 + 25 + 75 = 100 + 100 = 200
b)
273 + 27 = n
273 + 7 + 20 = 280 + 20 = 300
c) 470 – 230 =240
d) 580 – 310 = 270
e)
1 890 – 445 = n
1 800 – 400 – 90 – 45 = 1 400 – 45 = 1 455
Activity 7.1
Learner’s Book page 112
Learners use their own methods to solve the 4-digit number
calculations. Allow them to discuss and compare their methods.
They then solve problems by adding the numbers needed to both
numbers to fill up thousands, and then subtract.
Solutions
1. a) 4 657 + 2 243 = 6 900
c) 6 448 + 3 352 = 9 800
e) 2 591 + 2 319 = 4 910
b) 3 280 – 1 212 = 2 068
d) 5 390 – 4 235 = 1 155
2. He did this to round up 1 800 to 2 000 so that it would be easier
to subtract.
3. a) (5 394 + 250) – (1 750 + 250)
= 5 644 – 2 000
= 3 644
b) (4 536 + 400) – (2 600 + 400)
= 4 936 – 3 000
= 1 936
c) (8 178 + 100) – (5 900 + 100)
= 8 278 – 6 000
= 2 278
d) (7 729 + 200) – (4 800 + 200)
= 7 929 – 5 000
= 2 929
e) (9 467 + 500) – (3 500 + 500)
= 9 967 – 4 000
= 5 967
Mathematics Teacher’s Guide Grade 4
Math G4 TG.indb 149
TERM 2
149
2012/09/14 5:32 PM
Unit 8
Add and subtract with 3- and
4-digit numbers
Mental Maths 1. a)
×2
b)
÷2
Learner’s Book page 112
25
35
45
65
75
125
250
750
50
70
90
130
150
250
500 1 500
24
240
380
440
560
670
880
950
12
120
190
220
280
335
440
475
2. Answers will differ.
Activity 8.1
Learner’s Book page 113
The learners study and explain the two methods. They apply the
near-doubles strategy for addition and rounding off to the nearest
10 for subtraction. Ask them to estimate the answers to the word
problems before they solve them. Let them compare their estimates
to the actual solutions.
Solutions
1. Learners’ explanations will differ.
2. a) 358 + 359 = n
350 + 350 = 700
8 + 9 = 17
700 + 17 = 717
b) 447 + 449 = n
440 + 440 = 880
7 + 9 = 16
880 + 16 = 896
c) 1 254 + 1 257 = n
1 250 + 1 250 = 2 500
4 + 7 = 11
2 500 + 11 = 2 511
d) 839 – 536 = n
830 – 530 = 300
9–6=3
300 + 3 = 303
e) 2 463 – 1 251 = n
2 400 – 1 200 = 1 200
60 – 50 = 10
3–1=2
1 200 + 10 + 2 = 1 212
150
Math G4 TG.indb 150
Mathematics Teacher’s Guide Grade 4
TERM 2
2012/09/14 5:32 PM
f)
2 546 – 2 554 = n
2 500 + 2 500 = 5 000
40 + 50 = 90
6 + 4 = 10
5 000 + 90 + 10 = 5 100
3. a)
b)
c)
d)
223 + 154 = 477
576 + 569 = 1 145
1 829 – 934 = 895
5 800 – 2 478 = 3 322
4. a) estimate: 1 400 + 560 = 1 960
1 355 + 555 → 1 350 + 500 = 1 850
1 850 + 50 = 1 900
1 900 + 10 = 1 910
They planted 1 910 seeds altogether.
b) estimate: 5 800 – 2 600 = 3 200
5 800 – 2 575 → 5 800 – 2 500 = 3 300
3 300 – 100 = 3 200
3 200 + 25 = 3 225
They still need to print 3 225 pages.
c) estimate: R6 600 – R4 000 = R2 600
R6 575 – R3 999 → 6 575 – 4 000 = 2 575
2 575 + 1 = 2 576
You pay R2 576 more for the laptop.
d) estimate: 3 660 – 1 250 = 2 410
3 657 – 1 250 → 3 650 – 1 250 = 2 400
2 400 + 7 = 2 407
The shop ordered 2 407 CDs in April.
Assessment task 2: addition and subtraction
The learners perform the assessment task at the end of week
2. They display knowledge of addition and subtraction based
on facts involving multiples and powers of 10. They will need
this skill for when they have to use the place values of numbers
in calculations. They are expected to use knowledge of place
value in breaking up numbers to illustrate understanding of
calculations where they use different strategies. Learners also
develop awareness of the importance of consistently looking
for relationships or patterns. In questions 1 and 3, the learners
solve number sentences by identifying the relationship between
the numbers. In question 2, they use their own methods to solve
the problems. In question 4, they copy and complete the flow
diagrams and in question 5, they apply knowledge of calculating
with multiples and powers of 10 in real-life situations.
Mathematics Teacher’s Guide Grade 4
Math G4 TG.indb 151
TERM 2
151
2012/09/14 5:32 PM
Assessment task 2 Addition and subtraction
1. Solve the number sentences.
a)
8+5=n
80 + 50 = n
800 + 500 = n
8 000 + 5 000 = n
b)
7+9=n
70 + 90 = n
700 + 900 = n
7 000 + 9 000 = n
c)
14 – 8 = n
140 – 80 = n
1 400 – 800 = n
14 000 – 8 000 = n
d)
15 – 7 = n
150 – 70 = n
1 500 – 700 = n
15 000 – 7 000 = n
2. Use your own methods to solve each problem.
a) 76 + 34 = n
b) 955 + 675 = n
c) 80 – 69 = n
d) 700 – 548 = n
3. Solve each number sentence.
a)
81 + 9 = n
810 + 90 = n
8 100 + 900 = n
b)
53 + 7 = n
530 + 70 = n
5 300 + 700 = n
c)
70 – 6 = n
700 – 60 = n
7 000 – 600 = n
d)
90 – 4 = n
900 – 40 = n
9 000 – 400 = n
(16)
(4)
(16)
4. Copy and complete each flow diagram.
a)
74
164
344
+6
484
130
530
152
+ 70
Mathematics Teacher’s
830Guide Grade 4
TERM 2
1 030
Math G4 TG.indb 152
2012/09/14 5:32 PM
164
164
344
344
+
+ 66
484
484
b)
130
130
530
530
830
830
+
+ 70
70
11 030
030
c)
30
30
300
300
700
700
–– 77
11 000
000
d)
50
50
500
500
11 500
500
–– 18
18
22 550
550
(16)
5. Read the problems carefully and use your own methods to solve
the problems.
Simphiwe and Lindiwe’s father sells firewood. They help their
father to earn pocket money.
a) Siphiwe and Lindiwe chopped 228 pieces of firewood during
one week. The next week they chopped 152 more pieces.
How many pieces of firewood did they chop altogether
during two weeks?
b) Siphiwe and Lindiwe want to chop 1 000 pieces of firewood
by the end of the month. They have already chopped 657
pieces of wood. How many more pieces do they have to
chop?
(8)
Total [60]
Mathematics Teacher’s Guide Grade 4
Math G4 TG.indb 153
TERM 2
153
2012/09/14 5:32 PM
Assessment task 2 Addition and subtraction
Solutions
1. The learners solve the number sentences. Keeping the equals
signs aligned will help learners recognise the patterns.
a)
8 + 5 = 13
80 + 50 = 130
800 + 500 = 1 300
8 000 + 5 000 = 13 000
b)
7 + 9 = 16
70 + 90 = 160
700 + 900 = 1 600
7 000 + 9 000 = 16 000
c)
14 – 8 = 6
140 – 80 = 60
1 400 – 800 = 600
14 000 – 8 000 = 6 000
d)
15 – 7 = 8
150 – 70 = 80
1 500 – 700 = 800
15 000 – 7 000 = 8 000
(16)
154
Math G4 TG.indb 154
2. The problems involve building up 100s and subtracting
from multiples of 10 and 100.
a) 76 + 34 = 110
b) 955 + 675 = 1 630
c) 80 – 69 = 11
d) 700 – 548 = 152
(4)
3. Learners solve the following calculations that require
understanding of addition and subtraction with multiples
and powers of 10.
a)
81 + 9 = 90
810 + 90 = 900
8 100 + 900 = 9 000
b)
53 + 7 = 60
530 + 70 = 600
5 300 + 700 = 6 000
c)
70 – 6 = 64
700 – 60 = 640
7 000 – 600 = 6 400
d)
90 – 4 = 86
900 – 40 = 860
9 000 – 400 = 8 600
(16)
Mathematics Teacher’s Guide Grade 4
TERM 2
2012/09/14 5:32 PM
4. a)
74
74
164
164
344
344
484
484
b)
130
130
530
530
830
830
11 030
030
c)
30
30
300
300
700
700
11 000
000
d)
50
50
500
500
+
+ 66
+
+ 70
70
–– 77
80
80
170
170
350
350
490
490
200
200
600
600
900
900
11 100
100
23
23
293
293
693
693
993
993
32
32
482
482
–– 18
18
11 500
11 482
500
482
22 550
22 532
550
532
(16)
5. Assist learners who have reading problems so that they
understand the context of the word problems.
a)
228 + 152 = n
228 + 2 + 150 = 230 + 150
= 380
They chopped 380 pieces of firewood.
b)
1 000 – 657 = n
1 000 – 600 – 50 – 7 = 400 – 50 – 7
= 350 – 7
= 343
They need to chop 343 more pieces of firewood.
(8)
Total [60]
Mathematics Teacher’s Guide Grade 4
Math G4 TG.indb 155
TERM 2
155
2012/09/14 5:32 PM
Common fractions
Learner’s Book page 114
In Grade 3, the learners worked with unitary fractions where the
numerator is 1 and non-unitary fractions where the numerator is
more than 1. These fractions included halves, quarters, eighths,
thirds, sixths and fifths. They recognised fractions in diagrammatic
form and wrote fractions as 1 half, 2 thirds, and so on. They
recognised that 3 thirds make 1 whole and that 2 quarters is
equivalent to 1 half.
This year, the learners will build on their previous knowledge. In
Term 2:
• the fraction parts they work with are extended to include
sevenths, ninths and tenths
• they compare and order fractions with different denominators
• they describe and compare fractions in diagrammatic form
• they recognise the equivalence of fractions
• they solve problems in contexts involving grouping and equal
sharing.
They will practise fraction concepts or be introduced to new
concepts in Mental maths. They will work with a range of models to
develop a good understanding of the fraction concept, for example,
paper folding, diagrammatic representations, number lines, fraction
chains, counters, and multi-fix cubes.
Unit 9
Order and compare fractions
Mental Maths Learner’s Book page 114
1. Hand out strips of paper of equal length and width. Ask
learners to fold the strips into equal parts to make halves,
thirds, quarters, fifths, sixths and eighths.
2. Let them draw dashed lines on the folds and write the
fraction symbols on the different equal parts. Help them
write the fraction words and symbols. Ask them if they can
describe what a fraction is. Also ask them where they use
fractions in real life.
3. Learners should count the fraction parts, for example, one
fifth and two fifths. Write the symbols and words on the
board.
4. Learners should recognise that 2 halves or 22 = 1 or
one whole.
Learners can draw the different fraction strips they made to
practise drawing equal parts in a whole.
156
Math G4 TG.indb 156
Mathematics Teacher’s Guide Grade 4
TERM 2
2012/09/14 5:33 PM
Solutions
1. Practical exercise
2. a) half
d) fifth
3. a) 2 halves
d) 5 fifths
4. a) 2 halves = 22 = 1
b)
e)
b)
e)
c) 4 quarters = 44 = 1
third
c) quarter
sixth
f) eighth
3 thirds
c) 4 quarters
6 sixths
f) 8 eighths
b) 3 thirds = 33 = 1
d) 5 fifths = 55 = 1
e) 6 sixths = 66 = 1
f) 8 eighths = 88 = 1
Activity 9.1
Learner’s Book page 115
Ask the learners to copy the fraction strips from the Mental maths
activity. Let learners use the strips to compare the sizes of the
fractions. Let the learners discuss the situation of the chocolate bars
and give their opinions. They should realise that the size of the units
is important, but not generalise.
Solutions
1. Practical exercise
2. a)
1
2
> 13
b)
1
5
< 13
c)
1
4
<
f)
1
3
> 61
g)
1
5
<
h)
1
4
> 61
1
2
1
2
d)
1
6
> 81
e)
1
8
<
i)
1
8
< 13
j)
1
2
> 15
1
4
3. Learners should realise that the bigger the numerator, the
smaller the fraction part.
1 1 1 1 1 1
8 ; 6 ; 5 ; 4; 3 ; 2
4. The learners should notice that 12 of 8 is 4 blocks and 13 of 12 is
also 4 blocks but 12 of 12 is 6 blocks so that 13 is smaller than 12 .
Show them that 84 >
4
8
Unit 10
≠
4
12
4
12
and 12 > 13 .
or 12 ≠ 13
Represent fractions
Mental Maths Learner’s Book page 116
1. The learners should look carefully at the circle that is divided
into thirds as learners often experience problems with this
representation. Let them practise dividing a circle into 3, 4,
6 and 8 equal parts. Dividing the circle into 5 equal parts is
a bit challenging with freehand drawings.
2. Draw blank number lines on the board and ask the learners
to estimate the positions of the different fractions. Let them
write down the fraction symbols where they indicate their
positions.
Mathematics Teacher’s Guide Grade 4
Math G4 TG.indb 157
TERM 2
157
2012/09/14 5:33 PM
Solutions
1. a)
1
2
b)
4
5
c)
2
3
d)
3
4
e)
5
6
f)
5
6
g)
1
2
h)
2
3
i)
3
4
j)
4
5
k)
1
2
l)
3
4
m) 78
n)
7
8
o)
5
6
p)
4
5
q)
2
3
r)
s)
3
4
t)
5
6
1
2
2. Fractions that are smaller than 12 are closer to 0 and those
greater than 12 are closer to 1.
a)
0
1
4
b)
0
c)
1
6
d)
0
3
6
1
3
1
5
e)
f)
1
1
4
1
2
0
0
1
1
2
1
8
0
1
3
5
2
8
1
4
4
8
3
8
1
2
6
8
5
8
7
8
1
4
b) 81 ;
1
4
c) 61 ; 13
d)
1
5
e) 81 ;
2
8
f)
1 3
4; 8
4. a)
1
2
b) None
c)
3
6
3 4
5; 5
e) 84 ; 86 ; 78
f)
1 5 3
2; 8 ; 4
Activity 10.1
1
1
3
4
3. a)
d)
1
4
5
Learner’s Book page 116
1. The learners have worked with fractions represented in single
units. They will now work with units that are subdivided into
smaller units. You could allow them to count the number of
squares in each half to check. Where diagonals are drawn,
learners will have to count the half squares to make wholes.
Give them copies of the grids or let them draw them (this might
take up too much time) and shade them in the different ways as
indicated.
158
Math G4 TG.indb 158
Mathematics Teacher’s Guide Grade 4
TERM 2
2012/09/14 5:33 PM
Solutions
1. a) Shading any 12 squares means that you have shaded half of
the 24 squares.
b) The learners explain their strategies.
c) 12 of 24 = 12
2. The exercise helps learners develop logical and creative thinking
and data-sorting skills. Give the learners square grid sheets of
paper. They shade the fractions of the wholes as indicated. Here
are some of the ways they could do the shading. You should
expect different solutions. Let them write down the calculation
for each diagram.
3. a)
1
4
of 16 = 4
b)
1
3
of 12 = 4
d)
1
6
of 18 = 3
e)
1
8
of 16 = 2
c)
1
5
of 15 = 3
4. Give learners copies of the diagrams. They continue to work
with fractions of wholes in which the whole is one single unit
and wholes consisting of quantities of smaller units. Possible
answers are shown below. Ask learners to write down a
calculation for each diagram.
a)
3
4
of 12 = 9
b)
2
3
of 1 =
c)
3
5
of 1 = 53
d)
5
8
of 1 = 85
e)
5
6
of 18 = 15
f)
2
5
of 10 = 4
2
3
Mathematics Teacher’s Guide Grade 4
Math G4 TG.indb 159
TerM 2
159
2012/09/14 5:33 PM
Unit 11
Equal sharing
Mental Maths Learner’s Book page 117
The learners share units equally. Make drawings on the board as
learners name the fraction parts. Let them make mental images
for each question before you draw the diagram for each one. Ask
questions such as, how many halves in one bar? Write down and
let them calculate the answers. They should realise that 1 bar for
1 child = 1, and 1 bar for 0 children is impossible because you
will not divide it if there are 0 children.
You should have an example of a sweet bar (such as a fizzer),
a chocolate bar and a chocolate slab to show the learners. They
work with one whole that consists of one unit to share in different
parts. The learners develop understanding of the concept of
fractions, addition, equivalent, mixed and improper fractions.
You could use one bar and cut it to share equally between three
learners. Make a representation on the board, for example:
1
3
+
1
3
+
1
3
=
3
3
Learners should realise that adding the different parts make a
whole again. They should understand 91 for example, as 1 of
9 equal parts. Revise the names of the fraction parts, i.e. the
numerator, denominator and the division line. You could also
relate the fraction concept to division; so, 13 is the same as
1 ÷ 3 = 13 .
Solutions
Ask the learners to identify and explain the patterns they notice
when sharing 1 bar shared equally between each number of
children.
1. 2 =
2.
1
2
3. 3 = 13
1
3
+ 13 + 13 = 1
4. 5 = 15
+ 15 + 15 + 15 + 15 = 1
Math G4 TG.indb 160
Mathematics Teacher’s Guide Grade 4
1÷2=
1
4
1
4
1
4
1÷4=
1
6
1
2
+ + 14 + 14 = 1
5. 6 =
1 ÷ 5 = 15 160
+ 12 = 1
4. 4 =
1 ÷ 3 = 13 1
5
1
2
1
4
1
6
1
6
+ + 61 + 61 + 61 + 61 = 1
1÷6=
1
6
TERM 2
2012/09/14 5:33 PM
6. 7 =
1
7
1
7
1
7
+ + 71 + 71 + 71 + 71 + 71 = 1
1÷7=
1
7
7. 8 = 81
1
8
+ 81 + 81 + 81 + 81 + 81 + 81 + 81 = 1
1 ÷ 8 = 81
8. 9 =
1
9
1
9
1
9
+ + 91 + 91 + 91 + 91 + 91 + 91 + 91 = 1
1÷9=
1
9
9. 1 = 1
1÷1=1
10. 0 is impossible. 1 ÷ 0 is not allowed.
Activity 11.1
Learner’s Book page 117
Let the learners explore Linda’s drawing. They should notice that
she gave each friend 1 whole bar and then divided the remainder
into 5 equal parts to get fifths. Each friend gets 1 15 or one and
one fifth. You can tell them that these fractions are called mixed
fractions because they consist of whole numbers and fractions.
A fraction such as 15 is called a proper fraction. It consists of a
numerator (naming the number of equal parts) and a denominator
(the number of equal parts the whole is divided into). Write the new
terminology cards and paste them on the board.
Ask the learners to make their own drawings to illustrate how
to share the sweet bars equally. They should not colour in the
drawings because this is not an art lesson, but focus on the
mathematics. Sharing 5 sweet bars between 2 children will give
2 12 each. Let them add 2 12 + 2 12 to get 5.
Some learners may also say that each child gets 52 if they divided
all the bars in half. If they do not do this, show them the strategy as
an alternative. You can then introduce them to improper fractions
and let them name the difference between the proper, mixed and
improper fractions. Allow the learners to compare their drawings
and solutions.
Show them an example with an illustration such as the one below.
Share 6 candy bars among 4 friends:
1
4
+ 14 + 14 +
1
4
1
4
+ 14 + 14 +
1
4
Each friend gets 1 + 14 + 14 = 1 24 .
Mathematics Teacher’s Guide Grade 4
Math G4 TG.indb 161
TERM 2
161
2012/09/14 5:33 PM
You can now show them the equivalence of 12 and 24 so that they
understand that 1 24 is the same as 1 12 .
1
4
+ 14 + 14 +
1
4
14243 14243
1
2
+
1
2
1
4
+ 14 + 14 +
1
4
14243 14243
1
2
+
1
2
Solutions
The learners make their own drawings. Allow them to record their
own intuitive fraction calculations and illustrations. You could share
the following strategies with them if they do not include them.
1. a) Share five sweet bars equally between 2 children.
2 + 2 = 4
12 + 12 = 1
Each child gets 2 12 sweet bars.
You could also show learners that all the bars can be divided
into halves to develop understanding of proper, improper
and mixed fractions:
•
1
2
is a proper fraction
•
2
5
is an improper fraction
• 2 12 is a mixed fraction.
1
2
1
2
1
2
1
2
1
2
= 52 = 2 12
1
2
1
2
1
2
1
2
1
2
= 52 = 2 12
b) Share five sweet bars equally between 3 children.
1 + 1 + 1 = 3
Divide the remaining 2 bars into thirds.
13 + 13 + 13 = 1
13 + 13 + 13 = 1
One child gets: 1 + 13 + 13 = 1 23
c) Share five sweet bars equally between 4 children.
1 + 1 + 1 + 1 = 4
Divide remaining 1 sweet bar into quarters.
14 + 14 + 14 + 14 = 1
Each friend gets: 1 14
162
Math G4 TG.indb 162
Mathematics Teacher’s Guide Grade 4
TERM 2
2012/09/14 5:33 PM
2. a) Share 6 candy bars equally among 3 friends.
1+ 1 + 1 + 1 + 1 + 1 = 6 (no remainders)
Each one gets 2 whole bars
b) Share 6 candy bars equally among 4 friends.
1 + 1 + 1 + 1 = 4
Divide remaining 2 bars into quarters.
14 + 14 + 14 + 14 = 1
14 + 14 + 14 + 14 = 1
Each friend gets 1 24 .
c) Share 6 candy bars equally among 5 friends.
1 + 1 + 1 + 1 + 1 = 5
Divide remaining 1 into fifths.
15 + 15 + 15 + 15 + 15 = 1
Each child gets 1 52 .
d) Share 6 candy bars equally among 6 friends.
1 + 1 + 1 + 1 + 1 + 1 (no remainders)
Each friend gets 1 bar.
3. a) Share 13 candy bars equally among 5 friends.
2 + 2 + 2 + 2 + 2 = 10
Divide remaining 3 into fifths.
1 = 15 + 15 + 15 + 15 + 15
1 = 15 + 15 + 15 + 15 + 15
1 = 15 + 15 + 15 + 15 + 15
Each friend gets 2 15 .
b) Share 13 candy bars equally among 6 friends.
2 + 2 + 2 + 2 + 2 + 2 = 12
Divide remaining 1 into sixths.
61 + 61 + 61 + 61 + 61 + 61 = 1
Each friend gets 2 61 .
c) Share 13 candy bars equally among 7 friends.
1 + 1 + 1 + 1 + 1 + 1 + 1 = 7
Divide remaining 6 into sevenths.
71 + 71 + 71 + 71 + 71 + 71 + 71 = 1
71 + 71 + 71 + 71 + 71 + 71 + 71 = 1
71 + 71 + 71 + 71 + 71 + 71 + 71 = 1
71 + 71 + 71 + 71 + 71 + 71 + 71 = 1
71 + 71 + 71 + 71 + 71 + 71 + 71 = 1
71 + 71 + 71 + 71 + 71 + 71 + 71 = 1
Each friend gets 1 76 .
Mathematics Teacher’s Guide Grade 4
Math G4 TG.indb 163
TERM 2
163
2012/09/14 5:33 PM
d) Share 13 candy bars equally among 8 friends.
1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 = 8
Divide remaining 5 into eighths.
81 + 81 + 81 + 81 + 81 + 81 + 81 + 81 = 1
81 + 81 + 81 + 81 + 81 + 81 + 81 + 81 = 1
81 + 81 + 81 + 81 + 81 + 81 + 81 + 81 = 1
81 + 81 + 81 + 81 + 81 + 81 + 81 + 81 = 1
81 + 81 + 81 + 81 + 81 + 81 + 81 + 81 = 1
Each one gets 1 85 .
e) Share 13 candy bars equally among 9 friends.
1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 = 9
Divide remaining 4 into ninths.
91 + 91 + 91 + 91 + 91 + 91 + 91 + 91 + 91 = 1
91 + 91 + 91 + 91 + 91 + 91 + 91 + 91 + 91 = 1
91 + 91 + 91 + 91 + 91 + 91 + 91 + 91 + 91 = 1
91 + 91 + 91 + 91 + 91 + 91 + 91 + 91 + 91 = 1
Each friend gets 1 94 .
f) Share 13 candy bars equally among 10 friends.
1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 = 10
Divide remaining 3 into tenths.
101 + 101 + 101 + 101 + 101 + 101 + 101 + 101 + 101 + 101 = 1
101 + 101 + 101 + 101 + 101 + 101 + 101 + 101 + 101 + 101 = 1
101 + 101 + 101 + 101 + 101 + 101 + 101 + 101 + 101 + 101 = 1
Each friend gets 1103 .
Unit 12
Calculations with fractions
Learners should be able to solve addition with unitary fractions
calculations after all the work they have done with fractions.
Mental Maths Learner’s Book page 118
Encourage learners to convert between improper and mixed
fractions.
Solutions
164
Math G4 TG.indb 164
1.
1
2
+ 12 = 22 or 1
2.
1
4
+ 14 + 14 + 14 = 44 or 1
3.
1
6
+ 61 + 61 + 61 + 61 + 61 = 66 or 1
4.
1
3
+ 13 + 13 + 13 = 43 or 1 13
Mathematics Teacher’s Guide Grade 4
TERM 2
2012/09/14 5:34 PM
5.
1
2
+ 12 + 12 = 23 or 1 12
6.
1
4
+ 14 + 14 + 14 + 14 + 14 = 64 or 1 24
7.
1
5
+ 15 + 15 + 15 + 15 + 15 + 15 = 75 or 1 52
8.
1
10
9.
1
7
+ 101 + 101 + 101 + 101 + 101 + 101 + 101 + 101 + 101 = 10
10 or 1
+ 71 + 71 + 71 + 71 =
5
7
10. 81 + 81 + 81 + 81 + 81 + 81 + 81 + 81 + 81 + 81 + 81 = 118 or 1 83
Activity 12.1
Learner’s Book page 118
Ask the learners to explore Nolwazi’s strategy to calculate fractions
of whole numbers using cubes. They should understand that the
and
can be used
representations
for calculations. Learners can draw the representations – this will
help them conceptualise the concept. Ask them if they notice any
patterns.
Solutions
1. a)
1
3
of 12 = 4
b)
2
3
of 12 = 8
2. a)
1
6
of 12 = 2
b)
5
6
of 12 = 10
3. a)
1
5
of 20 = 4
b)
2
5
of 20 = 8
d)
4
5
of 20 = 16
e)
5
5
of 20 = 20
4. a)
1
4
of 20 = 5
b)
2
4
of 20 = 10
d)
3
4
of 20 = 15
e)
4
4
of 20 = 20
5. a)
1
8
of 16 = 2
b)
2
8
of 16 = 4
c)
3
8
of 16 = 6
d)
4
8
of 16 = 8
6. a)
1
2
of 16 = 8
b)
2
4
of 16 = 8
c)
3
4
of 16 = 12
d)
4
4
of 16 = 16
7. a)
1
6
of 18 = 3
b)
2
6
of 18 = 6
c)
1
3
of 18 = 6
d)
2
3
of 18 = 12
8. a)
1
5
of 15 = 3
b)
2
5
of 15 = 6
c)
3
5
of 15 = 9
d)
4
5
of 15 = 12
c)
3
5
of 20 = 12
c)
1
2
of 20 = 10
9. Learners should notice that 12 of a number is half the number
and 44 of a number is the whole number because 44 = 1, and
the relationship between the fractions and the solutions, for
example:
1
3
of 12 = 4 and 23 of 12 = 8 and 4 × 2 = 8.
Mathematics Teacher’s Guide Grade 4
Math G4 TG.indb 165
TERM 2
165
2012/09/14 5:34 PM
Ask learners to create calculations to show how the numbers
work. They should note that:
1
3
2
3
of 12 = 4 → 3 × 4 = 12 and 1 × 4 = 4
of 12 = 8 → 3 × 4 = 12 and 2 × 4 = 8
8 is double 4 so 23 is double 13 .
Learners explore these relationships in the other groups of
fractions of whole numbers so that they are able to deduce for
example that, if 15 of 20 = 4, then 54 of 20 is 4 × 4 = 16. They
should also notice that 12 of 20 is the same as 24 of 20 which is
10 and 62 of 18 is the same as 13 of 18 which is 6. They should
connect these concepts to equivalent fractions that they will
engage with in the next few lessons.
Unit 13
Equivalent fractions
Learners work with number lines and a fraction wall to explore
equivalent fractions (fractions that have the same value). Ask
them to name such fractions. Let them use the fraction wall to
find equivalent fractions. Do not show them a rule to calculate
equivalent fractions at this stage. Make copies of the fraction circles
on card – use a different colour for each different fraction part.
Mental Maths Learner’s Book page 120
1. The learners count in fractions to complete the missing
fractions on the number lines. Ask them to explain what they
observe. They should notice the fractions that appear in the
same positions on the number lines.
They should notice for example, that 12 , 24 , 63 and 105 are
the same distance from 0 and 1 on the number lines; 62 , 13 and
3
9 are in the same positions, and so on. Explain that these are
equivalent fractions – they have the same value. They should
notice that there are no equivalent fractions for sevenths.
Solutions
1. a) 0
b) 0
166
Math G4 TG.indb 166
2
4
1
4
c) 0
d) 0
1
1
2
1
3
1
6
Mathematics Teacher’s Guide Grade 4
2
6
3
4
1
2
3
3
6
4
6
1
5
6
1
TERM 2
2012/09/14 5:34 PM
e) 0
f)
1
5
1
10
0
g) 0
h)
2
5
2
10
1
9
3
10
2
9
1
7
0
3
5
4
10
4
9
2
7
6
10
5
10
3
9
4
5
7
10
5
9
3
7
6
9
4
7
1
8
10
7
9
5
7
9
10
1
8
9
1
6
7
1
You can use the number lines to compare fractions. For
example, ask the learners which fraction is bigger, 72 or 62 ; 23 or
3
6 . Let them use the number lines to justify their solutions.
2. Below are examples of drawings that learners might create to
show their understanding.
a)
4 circles
1
4
b)
10 squares
1
10
c)
5 stars
1
5
of 5 = 1 star
d)
7 hearts
1
7
of 7 = 1 heart
Activity 13.1
Learner’s Book page 120
1. Give the learners copies of the fraction wall. They complete the
fraction wall and fill in the missing equivalent fractions. You
could also ask them to use the number lines in Mental maths to
complete the equivalent fractions.
a)
c)
e)
1
2
1
3
3
5
= 24 = 63 = 84 = 105
b)
= 62 =
d)
= 106
3
9
f)
3
6
4 = 8
1
2
9 = 10
4
2
10 = 5
Mathematics Teacher’s Guide Grade 4
Math G4 TG.indb 167
TERM 2
167
2012/09/14 5:34 PM
2. Ask learners to look at Aneesa’s strategy to make 1 12 . Ask them
if they can make 1 12 using other fraction parts. Let them illustrate
different ways to make two wholes using different pieces. They
will come up with various combinations. This activity assists in
eliminating common misconceptions about fractions such as that
1
2
+ 12 = 24 .
Let learners write down the calculations they create. Below are a
few examples.
1 whole
1
2
1 whole
1
+
1 whole
1
12
12
4
1
4
1
1
4
14
41
14
4
4
1 4
2
1
4
1
1
4
14
41
14
4 1
4
11
44
1
41
14
1
12
2
4
1
2
+
2
4
1 whole
+
1
8 421
418
1 8
1
8
81 1
8
8 11
1 88
8 1
1 8
8
=
11
44
1
421
144
4
1 whole
1 +
1 whole
= 1 12
11
44
1
41
14
+
1
4
1 24
=
1
8
1
8
1
8
11
88
1
8
6
4
1 1
8 8
1
1
8
81
1
1
8
81
1
8
1
8
1
8
1
8
8 8
8
8
4
8
1
8
1
8
1
8
1
8
11
88
1
8
1
1
8
81
8
1
8
1
8
+
= 24 + 24 + 84 =1+
= 44 + 84 = 1 12
= 1 12
1
8
1
8
1
8
11
88
1
8
4
8
1
2
= 1 12
Unit 14
Count and calculate fractions
Mental Maths Learner’s Book page 121
1. Learners can copy the fraction chains and fill in the missing numbers. Check
whether they fill in improper or mixed fractions. Ask them to write mixed
fractions for improper fractions and improper fractions for mixed fractions.
Let them count in the different fraction intervals.
2. Learners draw a fraction chain to show how to add sevenths.
Solutions
1. a)
a)
b)
168
Math G4 TG.indb 168
0
0
1
+2
1
2
412
1
+4
1
112
+2
3 12
1
2
+ 4
1
+ 2
4
1
4
+ 4
1
1
1
1
+ 2
+ 2
1
1
1
Mathematics
2 1Teacher’s
+ Guide2 Grade +4 TERM
1 32
4
4
c)
0
+
1
1
4
+
1
2
4
+
1
1
1
+ 2
2
+ 2
3
+ 2
2 12
3
4
+ 4
1
+ 4
112
+ 4
114
11
+
1
1
+ 4
1
1
+
1
1
1
+ 2
1
1
2012/09/14 5:34 PM
a)a)
00
1
+ +2 1
2
1
22
+ +2 1
2
+ +2 1
2
33
+ +2 1
2
1
2 122 1
+ +4 1
4
1
3
3
4
4
+ +4 1
4
1
11
+ +4 1
4
+ +4 1
4
1 431 3
+ +4 1
4
11211
+ +4 1
4
1
11411
+ +3 1
3
2
2
3
3
+ +3 1
3
1
11
+ +3 1
3
1
1 131 1
+ +3 1
3
+ +3 1
3
2 232 2
+ +3 1
3
2321
+ +3 1
3
1
22
+ +3 1
3
1 231 2
+ +5 1
5
1
1
5
5
+ +5 1
5
1
2
2
5
5
+ +5 1
5
1
3
3
5
5
+ +5 1
5
1
4
4
5
5
+ +5 1
5
15414
+ +5 1
5
15313
+ +5 1
5
15212
+ +5 1
5
11511
+ +5 1
5
11
+ +6 1
6
1
1
6
6
+ +6 1
6
1
1
3
3
+ +6 1
6
1
1
2
2
+ +6 1
6
1
2
2
3
3
+ +6 1
6
11211
+ +6 1
6
11311
+ +6 1
6
16111
+ +6 1
6
1
11
+ +6 1
6
1
5
5
6
6
+ +7 1
7
2
2
7
7
+ 7+ 1
7
3
3
7
7
1
+ 7+ 1
4
4
7
7
+ 7+ 1
1
7
5
5
7
7
1
+ 7+ 1
1 731 3
+ +7 1
7
1
1 721 2
1
+ 7+ 1
1 711 1
+ 7+ 1
7
1
11
1
+ 7+ 1
6
6
7
7
+ +2 1
2
1
1
2
2
41241
00
1
+ +2 1
2
11211
+ +2 1
2
44
+ +2 1
2
1
3 123 1
+ +4 1
4
1
1
4
4
+ +4 1
4
1
1
1
2
2
2 124 1
+ +4 1
4
22
+ +3 1
3
1
1
3
3
1
33
1
4
c)c)
d)d)
00
00
1
5
e)e)
00
1
2
2.
00
1
11
2
b)b)
1
+ +2 1
2
1
7
1
1
1
3
1
5
1
1
3
1
7
1
1
1
1
7
2
4
1
3
5
2
1
1
1
1
6
7
7
Activity 14.1
2
3
1
5
1
2
1
4
1
3
1
1
1
7
7
Learner’s Book page 122
1. The learners use the objects in the pictures to determine
fractions of whole numbers. You can show them how to
calculate the solutions when they are not working with pictures
or drawings.
a) 65 of 18
= (18 ÷ 6) × 5
= 3 × 5
= 15
b)
of 40
= (40 ÷ 5) × 3
=8×3
= 24
c) 23 of 12
= (12 ÷ 3) × 2
= 4 × 2
= 8
d)
of 16
= (16 ÷ 4) × 3
=4×3
= 12
e) 14 of 8
= (8 ÷ 4) × 1
= 2 × 1
= 2
f)
of 25
= (25 ÷ 5) × 4
=5×4
= 20
g) 43 of 8
= (8 ÷ 4) × 3
= 2 × 3
= 6
h)
of 18
= (18 ÷ 3) × 1
=6×1
=6
3
5
3
4
4
5
1
3
Mathematics Teacher’s Guide Grade 4
Math G4 TG.indb 169
TERM 2
169
2012/09/14 5:35 PM
i) 14 of R8
= (8 ÷ 4) × 1
= 2 × 1
= 2
of 25
= (25 ÷ 5) × 2
=5×2
= 10
j)
2
5
Assist the learners in understanding the contexts of the problems
in questions 2 to 4. They have done this type of equal sharing
problems before.
2. Some learners may draw 20 sweet bars and show the sharing.
You could show them the more abstract way after they have
used their own intuitive strategies. They should realise that
each friend will get 2 whole bars and share the 2 remainders
equally.
2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 = 18
1
9
+ 91 + 91 + 91 + 91 + 91 + 91 + 91 + 91 = 1
1
9
+ 91 + 91 + 91 + 91 + 91 + 91 + 91 + 91 = 1
Each friend gets 2 92 sweet bars.
3. Learners can make drawings if they would like to do so. Show
them the following strategy if they do not use it. Each friend
gets 2 whole bars and the remaining 3 bars are shared equally.
2 + 2 + 2 + 2 + 2 + 2 + 2 = 14
1
7
+ 71 + 71 + 71 + 71 + 71 + 71 = 1
1
7
+ 71 + 71 + 71 + 71 + 71 + 71 = 1
1
7
+ 71 + 71 + 71 + 71 + 71 + 71 = 1
Each friend gets 2 73 sweet bars
4. The ingredients must be increased 5 times to find out how much
of each ingredient Solly needs for 5 cakes. Learners can use
repeated addition and work with improper and mixed fractions.
1
4
cup margarine × 5 = n
= 14 + 14 + 14 + 14 +
1
4
= 1 cup + 14 cup or 54 cups
= 1 14 cups of margarine
1 egg × 5 = n
1 × 5 = 5 eggs
1 12 cups of flour
1 + 1 + 1 + 1 + 1 + 12 + 12 + 12 + 12 +
= 5 cups + 1 + 1 +
1
2
1
2
= 7 12 cups of flour
170
Math G4 TG.indb 170
Mathematics Teacher’s Guide Grade 4
TERM 2
2012/09/14 5:35 PM
1
4
teaspoon salt × 5 = n
1
4
+ 14 + 14 + 14 +
=
5
4
1
4
= 1 14 teaspoonfuls of salt
1
2
cup of sugar × 5 = n
1
2
+ 12 + 12 + 12 +
=1+1+
1
2
1
2
= 2 12 cups or 52 cups of sugar
3
4
cup of milk × 5 = n
3
4
+ 43 + 43 + 43 +
3
4
= 43 + 14 (or 1 cup) + 43 + 14 (or 1 cup) + 43 + 14 (or 1 cup)
+ 43 + 14 (or 1 cup) + 43 + 14 (or 1 cup) + 43 + 14 (or 1 cup)
=6+
3
4
= 6 43 cups of milk
2 12 teaspoons baking powder × 5 = n
2 12 + 2 12 + 2 12 + 2 12 + 2 12
= 2 + 2 + 2 + 2 + 2 + 12 + 12 + 12 + 12 +
= 10 + 1 + 1 +
1
2
1
2
= 12 12 teaspoonfuls of baking powder
5. Learners should realise that they cannot break off 13 , 43 and 85
because there would not be an equal number of cubes; 10 cannot
be divided by 3, 4 and 8 without remainders or 3, 4 and 8, which
are not factors of 10. Groupings with remainders do not give
equal parts. They can break off 12 s, 15 s, and 101 s.
Mathematics Teacher’s Guide Grade 4
Math G4 TG.indb 171
TERM 2
171
2012/09/14 5:35 PM
Assessment task 3 Common fractions
1. Fill in the missing words:
a) If you divide a whole into 7 equal parts, each part is called ...
b) If you divide a whole into 9 equal parts, each part is a
called ...
c) If you divide a whole into 5 parts, each part is called ...
d) If you divide a whole into 10 parts, each part is called ...
e) If you divide a whole into 8 parts, each part is called ...
(5)
2. Fill in two fractions.
a) A half is bigger than ... and ...
b) One third is bigger than ... and ...
c) Three quarters are bigger than ... and ...
d) One fifth is smaller than ... and ...
e) One tenth is smaller than ... and ...
(10)
3. Five friends share 22 fizzers equally. How many does
each friend get?
(3)
4. Six friends want to share two pizzas equally.
How can they do it?
(3)
5. Draw each shape and shade half of it.
a) rectangle
b) circle
c) square
d) trapezium
e) triangle (isosceles)
(5)
6. Divide each shape into four equal parts.
b)
a)
c)
d)
172
Math G4 TG.indb 172
Mathematics Teacher’s Guide Grade 4
(4)
Total [30]
TERM 2
2012/09/14 5:35 PM
Assessment task 3 Common fractions
Solutions
The learners perform the assessment task to display knowledge of
the concept of fractions, comparing and representing fractions, equal
sharing with remainders to be shared too and repeated addition with
fractions.
c) one fifth 1 15 2
1. a) one seventh 1 71 2 b) one ninth 1 91 2
d) one tenth 1101 2
e) one eighth 1 81 2(5)
2. The learners fill in two fractions to show understanding of
comparing sizes of fractions. Solutions may differ. Here are
some examples:
b) 15 and 91
c) 12 and 14
a) 13 and 14
d)
1
3
and
1
2
e)
1
5
and 14 (10)
3. Learners should realise that 5 × 4 = 20, so each of the five
friends gets 4 whole bars and 2 bars need to be shared equally
between them. Learners could make a drawing or use numbers:
1
1
1
1
1
5 + 5 + 5 + 5 + 5 =1
1
1
1
1
1
5 + 5 + 5 + 5 + 5 = 1.
(3)
Each friend gets 4 52 of the fizzers. 4. To divide the pizzas, learners could draw two circles and divide
each one into six equal parts or they could work more abstractly:
1
1
1
1
1
1
6 + 6 + 6 + 6 + 6 + 6 =1
1
1
1
1
1
1
6 + 6 + 6 + 6 + 6 + 6 = 1.
Each friend gets 62 of the pizzas. 5. Solutions may differ. Below are examples.
b)
a)
d)
(3)
c)
e)
(5)
6. Answers may differ. Below are examples.
b)
a)
c)
d)
(4)
Total [30]
Mathematics Teacher’s Guide Grade 4
Math G4 TG.indb 173
TERM 2
173
2012/09/14 5:35 PM
Measurement: length
For the work on measuring length, make sure that you have tape
measures, rulers, metre sticks and trundle wheels in class – at least
one of each. You could make your own metre stick by tying a rope
or thick string into lengths of 1 m up to 5 m in length. This will
make it possible for the learners to measure a distance longer than a
metre without a trundle wheel.
Allow all the learners in the class to take part in the practical aspects
of this lesson. They should get a feel for the different distances.
Demonstrate to the learners how to read the builder’s tape.
Make sure the measurements are accurate on your height chart. If
you do not have one available, ask a clinic or pharmacy to donate
an old chart to the school. If you make your own chart, do not
start at 0; start at 75 cm so that the chart does not have to be fixed
to the wall from floor level. Measure 75 cm from the floor up the
wall, and fix the chart to the wall at this point. It can be fixed semipermanently, as you will measure the learners at the end of the year
again to measure by how much they have grown.
Unit 15
Revision of Grade 3 work
Mental Maths Learner’s Book page 123
Ask the learners to count in multiples of 10, 100 and 1 000. They
often work with powers of 10 (1, 10, 100, 1 000, 10 000 and so
on) in measurement. Ask them to round off the numbers to the
nearest 10 and 100. We also often round off numbers to make
estimations in measurement.
Solutions
1. a–f) Learners count in multiples of 10, 100 and 1 000.
2. a) 55: nearest 10: 60; nearest 100: 100
b) 78: nearest 10: 80; nearest 100: 100
c) 143: nearest 10: 140; nearest 100: 100
d) 92: nearest 10: 90; nearest 100: 100
e) 427: nearest 10: 430; nearest 100: 400
Activity 15.1
1.
Learner’s Book page 123
ruler – shoe; tape measure – bent elbow;
metre stick – large table; trundle wheel – rugby field
2–6. Answers will differ.
7.
174
Math G4 TG.indb 174
Practical work.
Mathematics Teacher’s Guide Grade 4
TERM 2
2012/09/14 5:35 PM
Unit 16
Learner’s Book page 126
Work with centimetres (cm) and
millimetres (mm)
Again, make sure that you demonstrate how to use a builder’s tape.
Learners may look at a centimetre mark somewhere along the tape,
where the metre value to the left is 1 m or 2 m, not 0. They must
know that the total length they have measured to that point includes
all the metres already laid out from the start of the tape, as well as
the centimetre and millimetre markings at the point where they are
reading off the end measurement.
Give learners plenty of practice with measuring small objects that
are laid so that they start further along the ruler or tape measure than
the 0 mark. It may take a while for learners to understand that they
must subtract the ‘unused’ length of tape or ruler before the start
of the object from the value they read off at the end of the object’s
length.
In Grade 4 learners do not work with decimal numbers. They can
show measured lengths using centimetres and millimetres (for
example, 3 cm + 4 mm), or express them using common fractions
(for example, 12 cm + 3 mm = 12 14 cm). Give learners plenty of
practice in expressing measurements in all these ways. When they
have learnt to do conversions from one measuring unit to another
(see next unit), they can also express measurements in this way.
Mental Maths Learner’s Book page 126
Learners work in groups.
1. Estimates will differ.
2. a) 4 cm or 40 mm
b) 1,5 cm or 15 mm
c) 6,5 cm or 65 mm
d) 3,5 cm or 35 mm
e) 12 cm or 120 mm
3. Learners compare and discuss estimates.
Activity 16.1
Learner’s Book page 127
1. 10
2. a) A: 16 mm
C: 34 mm
b) 9 mm
B: 25 mm
D: 68 mm
c) 9 mm
3. a) pencil: 4 cm 3 mm or 43 mm
b) hairbrush: 5 cm 7 mm or 57 mm
4. a) egg: 42 mm
c) pocket knife: 87 mm
b) rose: 109 mm
5. Answers will differ.
6. a) egg: 4 cm 2 mm
c) pocket knife: 8 cm 7 mm
b) rose: 10 cm 9 mm
Mathematics Teacher’s Guide Grade 4
Math G4 TG.indb 175
TERM 2
175
2012/09/14 5:35 PM
Unit 17
Tricky measurements
Learners practise measuring the lengths of objects from the 10 mm
mark instead of the 0 mm mark on a ruler. Tell them that we
sometimes have to use rulers or tape measures that are broken or
have faded numbers and calibrations. Learners must explain their
strategy when measuring from numbers other than 0.
Mental Maths Learner’s Book page 128
Learners convert between centimetres (cm) and millimetres (mm)
and they add lengths in millimetres (mm) and centimetres (cm).
Solutions
1. a)
b)
c)
d)
3 cm = 30 mm
5 cm = 50 mm
10 mm = 1 cm
40 mm = 4 cm
2. a)
b)
c)
d)
e)
f)
(4 cm – 2 cm) + 7 cm = 9 cm
(15 mm – 8 mm) + 8 mm = 15 mm
25 mm + 9 mm = 36 mm
9 cm + 12 cm = 21 cm
15 cm – 7 cm = 8 cm
20 cm – 13 cm = 7 cm
Activity 17.1
Learner’s Book page 128
1. a) crocodile – length (end of tail to tip of lower jaw):
15 cm 4 mm
b) donkey – height (from left front hoof to top of shoulder): 4 cm
donkey – length (from forehead to tip of tail):
7 cm 3 mm and 5 mm
c) shoe – length: 4 cm 8 mm
d) snake – length: 5 cm 6 mm and 8 mm
2. Answers depend on the objects that learners measure.
Unit 18
Understand units of measurement
Mental Maths Learner’s Book page 129
Let the learners explore the measurement facts and solve the
problems. They multiply and divide by 10, 100 and 1 000. If
you give learners scraps of paper and ask them to write down
their answers and hold up their pieces of paper when they are
ready, they won’t shout the answers out and everyone will have
the opportunity to work out the answers. You can also see which
learners depend on others to give answers.
176
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Mathematics Teacher’s Guide Grade 4
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Solutions
1. a)
c)
e)
g)
i)
k)
80 mm = 8 cm
4 cm = 40 mm
500 cm = 5 m
7 m = 700 cm
6 km = 6 000 m
5 000 m = 5 km
b)
d)
f)
h)
j)
20 mm = 2 cm
12 cm = 120 mm
900 cm = 9 m
2 000 m = 2 km
8 km = 8 000 m
2. Explanations will differ.
Activity 18.1
Learner’s Book page 130
1. a) 1 cm = 10 mm
c) 1 12 cm = 15 mm
e) 30 mm = 3 cm
b) 2 cm = 20 mm
d) 5 mm = 12 cm
f) 35 mm = 3 12 cm
2. a) 2 m = 200 cm
c) 1 12 m = 150 cm
e) 250 cm = 2 12 m
b) 10 m = 1 000 cm
d) 50 cm = 12 m
f) 300 cm = 3 m
3. a)
c)
e)
g)
1 km = 1 000 m
10 km = 10 000 m
2 000 m = 2 km
2 500 m = 2 12 km
b)
d)
f)
h)
2 km = 2 000 m
1 12 km = 1 500 m
8 000 m = 8 km
500 m = 12 km
4. a)
c)
e)
g)
i)
1 234 m < 1 324
1 m > 99 cm
1 m 30 cm = 130 cm
2 km 360 m < 2 036 km
1 12 km > 1 250 m
b)
d)
f)
h)
j)
624 > 342
2 m = 200 cm
1 km > 1 000 mm
5 cm = 50 mm
500 m = 12 km
5. 60 cm; 1 m; 2 000 mm; 203 cm; 900 cm
6. 1 m 45 cm; 132 cm; 1 m 16 cm; 109 cm
7. a) 7,5 km
c) straight to airstrip
e) 1,5 km
Unit 19
b) 5,5 km
d) via Mahlala hide
f) 27 km
Convert between kilometres, metres
and millimetres
Mental Maths Learner’s Book page 133
Lead a class discussion about converting between units of length.
Learners have worked with these conversions recently (with
whole numbers). Ask them to tell you how to convert lengths
in millimetres (mm) to centimetres (cm), centimetres (cm) to
millimetres (mm), and metres (m) to kilometres (km).
Mathematics Teacher’s Guide Grade 4
Math G4 TG.indb 177
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Go through the facts for conversions with them. Let them use the
strategies to do the conversion. Remind them that they worked
with fractions of whole numbers in previous units. Ask them to
find the fractions of metres. They also convert kilometres (km) to
metres (m), and metres (m) to kilometres (km).
Solutions
1. a) 7 km = 700 m
c) 17 km = 17 000 m
2. a)
c)
e)
g)
b) 9 km = 900 m
d) 23 km = 23 000 m
5 000 m = 5 km
11 000 m = 11 km
7 000 m = 7 km
35 000 m = 35 km
b)
d)
f)
h)
8 000 m = 8 km
20 000 m = 20 km
10 000 m = 10 km
23 000 m = 23 km
3. a) 3 m = 3 000 mm
c) 9 m = 9 000 mm
b) 8 m = 8 000 mm
d) 10 m = 10 000 mm
4. a) 4 000 mm = 4 m
c) 7 000 mm = 7 m
b) 6 000 mm = 6 m
d) 12 000 mm = 12 m
5. a)
c)
6. a)
c)
1
5 of 10 m = 2 m
1
10 of 1 km = 100
1
10
7
10
b)
m
d)
of 1 m = 100 mm
b)
of 1 m = 700 mm
d)
7. a) 3 km 212 m = 3 212 m
c) 15 km 9 m = 15 009 m
3
5
1
2
of 40 m = 24 m
of 1 km = 500 m
4
5 of 1 m = 800 mm
5
10 of 1 m = 500 mm
b) 5 km 50 m = 5 050 m
d) 20 km 5 m = 20 005 m
Activity 19.1
Learner’s Book page 134
1. a) 4 m = 4 000 mm
b) 3 m 47 mm = 3 047 mm
c) 2 12 m = 2 500 mm
d) 1 m 6 mm = 1 006 mm
e) 1 km 855 m = 1 866 000 mm
f) 6 m = 6 000 mm
g) 1 000 mm = 1 m
h) 5 000 mm = 5 m
i) 500 mm = m
j) 250 mm = 14 m
k) 750 mm = m
l) 13 000 mm = 13 m
1
2
3
4
2. a) 3 km 245 m = 3 245 m
178
Math G4 TG.indb 178
b) 12 km 426 m = 12 426 m
c) 6 km 200 m = 6 200 m
d) 5 km 3m = 5 003 m
e) 27 km 19 m = 27 019 m
f) 8 km 8 m = 8 008 m
g) 1 000 m = 1 km
h) 1 500 m = 1 12 km
i) 750 m = 43 km
j) 3 000 m = 3 km
k) 6 500 m = 6 12 km
l) 10 000 m = 10 km
Mathematics Teacher’s Guide Grade 4
TERM 2
2012/09/14 5:35 PM
Unit 20 Convert between centimetres
and metres
Mental Maths Learner’s Book page 134
The learners discuss the conversions between metres (m) and
centimetres (cm) and do the conversions accordingly. They
multiply and divide by 100 and write fractions of metres (m) in
centimetres (cm). Remind the learners that the work they do in
number is important and applied in various topics in mathematics.
That is why they have to know the basic operations and facts
about multiplication and division by 10, 100 and 1 000.
Solutions
1. a) 7 m = 700 cm
c) 17 m = 1 700 cm
b) 9 m = 900 cm
d) 23 m = 2 300 cm
2. a) 700 cm = 7 m
c) 1 500 cm = 15 m
b) 1 000 cm = 10 m
d) 3 000 cm = 30 m
3. a)
c)
1
5 m = 20 cm
1
10 m = 10 cm
4. a) 2 m 15 cm = 215 cm
c) 11 m 5 cm = 1 105 cm
b)
d)
4
5 m = 80 cm
9
10 m = 90 cm
b) 9 m 10 cm = 910 cm
d) 8 m 8 cm = 808 cm
Activity 20.1
Learner’s Book page 135
1. a) 1 m 30 cm = 130 cm
c)
3 12
m = 3 500 cm
b) 5 m 15 cm = 515 cm
d) 10 12 m = 1 050 cm
e) 100 cm = 1 m
f) 320 cm = 3 m 20 cm
g) 170 cm = 1 m 7 cm
h) 506 cm = 5 m 6 cm
2. a) 1 km – 700 m = 1 000 m – 700 m = 300 m
b) 500 m + 500 m = 1 km
c) 800 m + 700 m = 1 500 m = 1 12 km
d) 2 km – 800 m = 2 000 m – 800 m = 1 200 m
e) 1 400 m + 1 600 m = 3 000 m = 3 km
f) 1 m – 80 cm = 100 cm – 80 cm = 20 cm
g) 50 cm + 50 cm = 100 cm = 1 m
h) 2 m – 75 cm = 200 cm – 75 cm = 1 m 25 cm
i) 1 12 m + 150 cm = 150 cm + 150 cm = 300 cm = 3 m
j) 170 cm + 230 cm = 400 cm = 4 m
Mathematics Teacher’s Guide Grade 4
Math G4 TG.indb 179
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Unit 21
Convert between millimetres and
centimetres
Mental Maths Learner’s Book page 135
The learners convert between centimetres (cm) and millimetres
(mm). Work through the strategies with them. They multiply and
divide by 10 and write fractions in centimetres and millimetres.
Solutions
1. a)
c)
2. a)
c)
40 mm = 4 cm
150 mm = 15 cm
5 cm = 50 mm
29 cm = 290 mm
b)
d)
b)
d)
100 mm = 10 cm
300 mm = 30 cm
12 cm = 120 mm
150 cm = 1 500 mm
3. a)
1
10
8
10
cm = 1 mm
b)
cm = 8 mm
d)
1
5
4
5
c)
cm = 2 mm
cm = 8 mm
Activity 21.1
Learner’s Book page 136
1. a)
c)
e)
g)
i)
10 cm = 100 mm
100 cm = 1 000 mm
1101 cm = 11 mm
27 mm = 2 cm 7 mm
543 mm = 54 cm 3 mm
b)
d)
f)
h)
j)
47 cm = 470 mm
1 12 cm = 15 mm
12 mm = 1 cm 2 mm
350 mm = 35 cm
15 mm = 1 12 cm
2. a)
c)
e)
g)
i)
30 mm = 3 cm
350 cm = 3 12 m
5 43 m = 575 cm
302 m = 30 m 2 cm
3 12 km = 3 500 m
b)
d)
f)
h)
j)
128 cm = 1 280 mm
23 km = 23 000 m
2 500 m = 2 12 km
2 km = 2 000 m
16 14 m = 1 625 cm
Unit 22 Round off measurements
Mental Maths Learner’s Book page 136
The learners round off numbers to the nearest 10, 100 and 1 000.
They should realise that rounding off to the nearest centimetre
(cm), meter (m) and kilometre (km) is the same as rounding off
whole numbers.
Solutions
1. a)
c)
2. a)
c)
3. a)
180
Math G4 TG.indb 180
17 → 20
35 → 40
103 → 100
256 → 300
1 005 → 1 000
Mathematics Teacher’s Guide Grade 4
b)
d)
b)
d)
b)
12 → 10
121 → 120
134 → 100
328 → 300
2 588 → 3 000
TERM 2
2012/09/14 5:35 PM
Activity 22.1
Learner’s Book page 137
1. a) 24 mm ≈ 2 cm
c) 223 mm ≈ 22 cm
e) 107 mm ≈ 11 cm
b) 5 mm ≈ 1 cm
d) 15 mm ≈ 2 cm
f) 12 cm ≈ 1 cm
2. a) 35 mm ≈ 0 m
c) 1 675 cm ≈ 17 m
e) 199 cm ≈ 2 m
b) 149 cm ≈ 1 m
d) 213 cm ≈ 2 m
f) 7 cm ≈ 0 m
3. a) 2 000 m ≈ 2 km
c) 3 999 m ≈ 4 km
e) 3 499 m ≈ 3 km
b) 2 438 m ≈ 2 km
d) 15 m ≈ 0 km
f) 6 12 km ≈ 7 km
Unit 23 Problem-solving with distance
and length
Mental Maths Learner’s Book page 138
Learners do addition and subtraction calculations with
measurement units. Let them discuss and explain the examples.
Ask a few learners to calculate the solutions to the problems on
the board and explain their thinking.
Solutions
1.
2.
3.
4.
5.
6.
7.
8.
356 m + 568 m = 924 m
5 l, 324 m – 2 km 596 m = 2 km 728 m
5 m – 3 m 40 cm = 1 m 60 cm
16 km + 5 km 799 m = 21 km 799 m
37 m × 6 = 222 m
90 km × 5 = 450 km
550 km ÷ 5 = 110 km
8 m ÷ 4 = 200 cm
Activity 23.1
Learner’s Book page 138
1. (6 × 5) + (5 × 2) + 110 × 3 12 2 m = 30 + 10 + 35
= 75 m
The total length of the poles is 75 m.
2. 1 km 200 m = 300 m
Each girl runs 300 m.
3. 432 – 67 = 432 – 32 – 35
= 400 – 35
= 365 cm
He jumped 365 cm.
4. (30 × 2) + 1 12 of 302 = 60 + 15
= 75 km
Janico’s farm is 75 km from town.
Mathematics Teacher’s Guide Grade 4
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5. It takes 1 hour (60 min.) to drive 120 km.
It takes 12 hour (30 min.) to drive 60 km.
It takes 14 hour (15 min.) to drive 30 km.
It takes 15 minutes to get to the farm.
6. The car uses 1 ℓ to go 10 km (or 60 ÷ 10 = 6 ℓ).
The car uses 2 ℓ to go 20 km.
The car uses 3 ℓ to go 30 km.
The car uses 6 ℓ to go 60 km.
The car will use 6 ℓ of petrol.
7. 1 ℓ will cost R10 (or 6 × R10 = R60)
2 ℓ will cost R20.
4 ℓ will cost R40.
6 ℓ will cost R60.
The trip will cost R60.
8. Janico’s farm is 75 km from town.
A trip to and from the farm is 150 km.
To drive 10 km the car will need 1 ℓ of petrol. This will
cost R10.
To drive 100 km the car will need 10 ℓ of petrol. This will
cost R100.
To drive 50 km the car will need 5 ℓ of petrol. This will
cost R50.
To drive 150 km the car will need 15 ℓ of petrol. This will
cost R150.
One trip to the farm will cost R75. If you use 1 ℓ of petrol (R10)
for every 10 km, it will cost you R1 for the petrol to drive 1 km.
Revision and consolidation
Learner’s Book page 139
The learners will perform two revision tasks to display knowledge
of concepts they have developed in the last few units. In the first
task they illustrate their ability to measure and estimate lengths
accurately, convert between units and do simple addition and
subtraction calculations involving length.
Solutions
1. a) 49 mm
b) 8 cm
2. a)
b)
c)
d)
4 mm + 3 mm = 7 mm
245 cm – 198 cm = 47 cm
100 cm or 1 000 mm
2 m 9 mm = 2 009 mm
3. a) A 1 m
b) B 4 mm
182
Math G4 TG.indb 182
Mathematics Teacher’s Guide Grade 4
TERM 2
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4. a)
b)
c)
d)
e)
f)
5 km 70 m = 5 070 m
3 cm = 30 mm
5 000 cm = 50 m
7 cm = 70 mm
20 cm + 33 mm + 2 m = 2 233 mm
2 354 mm = 235 cm 4 mm
Revision
Learner’s Book page 139
Learners show how accurately they can measure, convert between
units, compare lengths and solve problems in measurement contexts.
Solutions
1. a) 4 cm 7 mm
b) 26 mm
2. a) 1 cm = 10 mm
b) 5 cm = 50 mm
c) 3 12 cm = 35 mm
d) 2 m = 2 000 mm
e) 4 km = 4 000 m
f) 30 mm = 3 cm
g) 15 mm = 1 12 cm
h) 110 mm = 11 cm
i) 150 cm = 1 12 m
j) 500 m = 12 km
3. a) 100 cm = 1 000 mm
b) 2 12 m > 2 250 mm
c) 2 km 360 m > 2 360 m
d) 300 cm = 3 m
4. a) 800 m
b) 3 12 times
c) 480 cm
Mathematics Teacher’s Guide Grade 4
Math G4 TG.indb 183
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Assessment task 4 Length
1. Measure the following lines.
a)
b)
c)
(3)
2. Convert the following lengths as indicated.
a) 15 km = n m
b) 15 mm = n cm
c) 3 000 cm = n m
(3)
3. Complete each calculation.
a) 2 km – 1 346 m = n m
b) 334 mm × 9 = n mm
c) 3 245 km + 658 km = n km
d) 108 cm ÷ 6 = n cm
(4)
4. Study each problem. Write an open number sentence
and then complete each calculation.
a) Susan has completed half of the race and she must
still run 1 250 m. How long is the race? Write your
answer in kilometres and metres.
(2)
b) On the first day of a race the cyclists have to ride
2 450 m, on the second day they must to ride
1 km 750 m and on the last day they must ride 955 m.
What is the total length of the race?
(5)
c) Lindiwe bought a square table cloth with sides
75 cm long. She wants to sew a ribbon around
the edge of the table cloth. How long will the
ribbon be? (3)
Total [20]
184
Math G4 TG.indb 184
Mathematics Teacher’s Guide Grade 4
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Assessment task 4 Length
1. a) 8 cm 1 mm
b) 5 cm 2 mm
c) 10 cm 3 mm
(3)
2. a) 15 km = 15 000 m
b) 15 mm = 1 12 cm
c) 3 000 cm = 30 m
(3)
3. a)
b)
c)
d)
2 km – 1 346 m = 346 m
334 mm × 9 = 3 006 mm
3 245 km + 658 km = 3 903 km
108 cm ÷ 6 = 18 cm
(4)
4. a) 1 250 m + 1 250 m = n
1 250 m + 1 250 m = 2 500 m
= 2 km 500 m
(2)
b) 2 450 m + 1 km 750 m + 955 m = n
2 450 m + 1 km 750 m + 955 m
= 2 450 + 1 750 m + 955 m
= 5 155 m
= 5 km 155 m
(5)
c) 4 × 75 cm = n
(2 × 75) + (2 × 75)
= 150 + 150
= 300 cm
Mathematics Teacher’s Guide Grade 4
Math G4 TG.indb 185
Solutions
(2)
Total [20]
TERM 2
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Whole numbers: multiplication
Learners will work with multiplication this week. They multiplied
with 1-digit numbers in Term 1. Now they will learn to multiply
with 2-digit numbers. They will also multiply by 10, 100 and 1 000
and multiples of 10.
Unit 24 Basic multiplication facts
Mental Maths Learner’s Book page 141
Learners study the two learners’ reasoning. Guide them to
understand that 40 × 50 is (4 × 10) + (5 × 10), which is
(4 × 5) × 100 because 10 × 10 = 100. This understanding is
enforced with the help of flow diagrams.
Solutions
1. Answers will differ.
2. a) 5 × 6 = 30
b) 3 × 8 = 24
c) 7 × 5 = 35
d) 5 × 9 = 45
e) 4 × 7 = 28
f) 8 × 8 = 64
50 × 60 = 3 000
30 × 80 = 2 400
70 × 50 = 3 500
50 × 90 = 4 500
40 × 70 = 2 800
80 × 80 = 6 400
Activity 24.1
Learner’s Book page 141
Ask the learners to copy the flow diagrams and complete the
numbers. They copy the tables and complete them. Here they start
multiplying non-multiples of 10 by multiples of 10.
Solutions
1. a) 8
6
9
7
5
4
2. a)
×2
× 10
11
22
220
18
36
360
×3
× 10
1
3
30
2
6
60
b)
186
Math G4 TG.indb 186
8
800
6
600
9
900
× 10 × 10
× 10 × 10
7
700
5
500
4
400
Mathematics Teacher’s Guide Grade 4
b)
800
8
600
6
900
9
700
7
500
5
400
4
23
46
460
3
9
90
34
68
680
4
12
120
8
6
9
7
5
4
× 100× 100
45
90
900
5
15
150
800
600
900
700
500
400
800
600
900
700
500
400
56
73
84
112 146 168
1 120 1 460 1 680
6
18
180
7
21
210
8
24
240
9
27
270
TERM 2
2012/09/14 5:35 PM
c)
10
40
400
×4
× 10
20
30
40
50
60
70
80
90
80 120 160 200 240 280 320 360
800 1 200 1 600 2 000 2 400 2 800 3 200 3 600
Unit 25 Multiplication strategies
Draw a Th, H, T, U table on the board. The objective is to assist
learners to develop an understanding of the behaviour of the
numbers when they are multiplied by 10, 100 and 1 000. Let them
explain how they calculate the number of dots in the rectangle.
This skill will help them understand the concepts of perimeter and
area in measurement. They multiply the number of dots in a row
by the number in a column. Check which learners still use repeated
addition. Let learners share strategies and convince each other that
certain strategies are quicker to use than others.
Mental Maths 1. a)
b)
c)
d)
e)
23 × 10 = 230
36 × 10 = 360
47 × 10 = 470
53 × 10 = 530
64 × 10 = 640
Learner’s Book page 142
23 × 100 = 2 300
36 × 100 = 3 600
47 × 100 = 4 700
53 × 100 = 5 300
64 × 100 = 6 400
2. a) 150 dots
c) 560 dots
23 × 1 000 = 23 000
36 × 1 000 = 36 000
47 × 1 000 = 47 000
53 × 1 000 = 35 000
64 × 1 000 = 64 000
b) 300 dots
d) 240 dots
Activity 25.1
Learner’s Book page 143
Learners study the double and single flow diagrams to understand
that bigger numbers can be broken up into factors to multiply
smarter and quicker. Next, they study the strategies for multiplying
a 1-digit number by a 2-digit number involving non-multiples of 10.
Let them compare their methods and select one to use for solving
the problems. You could let them work in pairs.
Solutions
1. a) 3 × 5 = 15
15 × 10 = 150
c) 75× 8 =×56
4
×3
56 × 10 = 560
8
× 44
× 32
×
2. 5 × 45× 3 =× 60
× 48
× 232
854 = 60
×
×
5 × 12
× 84
× 23
a) 485
×
×
5
64
5
b) 5 × 6 = 30
30 × 10 = 300
d) 85× 3 =× 12
24 5
24 × 10 = 240
58
8
××12
64
5
54
64
58
××12
8
54
64
60
64
4
855
××12
12
8
×
60
64
4
b)
542
× 485
×
× 326
×
60
4
60
52
30
× 12
60
4
60
c)
258
× 545
×
× 63
×
120
60
582
15
× 12
30
120
60
d)
28
×5
× 63
60
120
28
× 30
15
60
120
8
×5
×3
120
8
× 15
120
Mathematics Teacher’s Guide Grade 4
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3. a) The methods show repeated addition, doubling, breaking
up numbers and the distributive property to multiply by 15.
They use all three strategies to solve 7 × 15. Allow them to
discuss the different strategies and to decide which is the
most effective one.
b) Calculations with the three methods are shown below.
A 7 × 15 = 15 + 15 + 15 + 15 + 15 + 15 + 15
= (10 + 10 + 10 + 10 + 10 + 10 + 10 + 10)
+ (5 + 5 + 5 + 5 + 5 + 5 + 5)
= 70 + 35
= 105
B 7 × 15 = n
3 × 15 = 45
3 × 15 = 45
1 × 15 = 15
45 + 45 + 15 = 90 + 15
= 105
C 7 × 15 = n
7 × 10 = 70
7 × 5 = 35
70 + 35 = 105
4. The learners should realise that strategy A could be timeconsuming. Doubling and breaking up numbers could work
more effectively.
a) 3 × 18 = (3 × 10) + (3 × 8) (strategy C)
= 30 + 24
= 54
b) 4 × 21 = (2 × 21) + (2 × 21)
= 42 + 42
= 84
c) 5 × 32 = (5 × 30) + (5 × 2)
= 150 + 10
= 160
d) 4 × 16 = (2 × 16) + (2 × 16)
= 32 + 32
= 64
e) 7 × 35 = (2 × 35) + (2 × 35) + (2 × 35) + (1 × 35)
= 70 + 70 + 70 + 35
= 210 + 35
= 245
188
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Mathematics Teacher’s Guide Grade 4
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Unit 26 Problem-solving with multiplication
Mental Maths Learner’s Book page 144
This session involves problem-solving and investigation.
Read the instructions to the class and make sure learners know
what to do.
Solutions
1. Ask learners to write the operations on the board as you say
them: n × 10 → n + 7 = n double → n × 2 → + n. Then
they can list the answers and their original numbers. They
should discover that there is a difference of 14 between the
2-digit number they started with and the final answer.
2. Learners use multiplication to find the different number of
objects stated. They could use the distributive property to
make the multiplication easier. They should manipulate by
associating numbers that are easier to get answers for (thus
using the associative property).
Learners should only write the number sentences during
the Mental maths session; they can solve the problems for
homework.
a) Number of beds:
12 × 7 × 6 = n
12 × 42 = (12 × 40) + (12 × 2)
= 480 + 24
= 504
b) Number of chairs:
16 × 7 × 6 = n
16 × 42 = (16 × 40) + (16 × 2)
= 640 + 32
= 672
c) Number of tables:
3 × 6 × 7 = n
3 × 42 = (3 × 40) + (3 × 2)
= 120 + 6
= 126
d) Number of wards: 7 × 6 = 42
Activity 26.1
Learner’s Book page 144
Learners can work in pairs. Ask them to write the number sentences
first before they calculate the answers. They can calculate the total
mass of the potatoes and solve the problems about the equipment in
the school. They use their own strategies. Lead a discussion during
which learners can compare their strategies and solutions.
Mathematics Teacher’s Guide Grade 4
Math G4 TG.indb 189
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Solutions
1.
pink: R15 × 5 = n
(10 × 5) + (5 × 5) = 50 + 25
= R75
yellow: R25 × 4 = R100
beige: R30 × 3 = R90
2.
16 × 15 = n
(15 × 10) + (15 × 6) = 150 + 90
= 240 kg
3. a) Number of cupboards:
3 × 6 × 4 = n
3 × 4 × 6 = 12 × 6
= 72
b) Number of writing boards:
2 × 6 × 4 = n
2 × 4 × 6 = 8 × 6
= 48
c) Number of tables:
20 × 6 × 4 = n
= (10 × 24) × 2
= 240 × 2
= 480
d) Number of chairs:
24 × 6 × 4 = n
(20 × 24) + (4 × 24) = 480 + (4 × 20) + (4 × 4)
= 480 + 80 + 16
= 560 + 16
= 576
Unit 27 Multiplication and estimation
Mental Maths Learner’s Book page 145
This activity helps learners check and judge the reasonableness
of solutions. They have to look at the unit digits and know which
unit each digit will result in when multiplied. They do not have
to calculate the solutions. They could judge, for example, that
7 and 1 in 37 × 11 will give a 7 as a unit. They can solve the
problems for homework. Learners can use inverse operations to
check their answer.
Solutions
1. a) 37 × 11 = 407 (407 is smaller than 1 000 and
ends in 7.)
b) 42 × 24 = 1 008(1 008 is between 1 000 and 1 010 and
ends in 8.)
c) 24 × 11 = 264
(264 is smaller than 500 and ends in 4.)
190
Math G4 TG.indb 190
Mathematics Teacher’s Guide Grade 4
TERM 2
2012/09/14 5:35 PM
d) 39 × 37 = 1 443(1 443 is smaller than 1 500 and
ends in 3)
e) 24 × 39 = 936(936 is between 900 and 1 000 and
ends in 6.)
2.
Last unit of multiplier
Last unit of multiplicand
×
0
1
2
3
4
5
6
7
8
9
0
0
0
0
0
0
0
0
0
0
0
1
0
1
2
3
4
5
6
7
8
9
2
0
2
4
6
8
0
2
4
6
8
3
0
3
6
9
2
5
8
1
4
7
4
0
4
8
2
6
0
4
8
2
6
5
0
5
0
5
0
5
0
5
0
5
6
0
6
2
8
4
0
6
2
8
4
7
0
7
4
1
8
5
2
9
6
3
8
0
8
6
4
2
0
8
6
4
2
9
0
9
8
7
6
5
4
3
2
1
3. a) 16 × 34 = 544
b) 21 × 27 = 567; 22 × 27 = 594; 23 × 27 = 621
24 × 27 = 648; 25 × 27 = 675; 26 × 27 = 702
27 × 27 = 729; 28 × 27 = 756; 29 × 27 = 783
c) Answers will differ.
Activity 27.1
Learner’s Book page 146
The learners study the strategies for estimating solutions.
Learners can calculate the actual solutions and compare them
with their estimates. They could subtract the products to see what
the difference is – how far or close the estimation was from the
real answer.
Solutions
1. a)
b)
c)
d)
e)
24 × 40 = 960
39 × 10 = 390
42 × 40 = 1 680
37 × 40 = 1 480
42 × 10 = 420
2. a) 24 × (40 – 1)
= (24 × 40) – (24 × 1)
= 960 – 24
= 936
b) 37 × (10 + 1)
= (37 × 10) + (37 × 1)
= 370 + 37
= 407
c) (37 × 30) + (37 × 7)
= 1 110 + 259
= 1 369
d) (37 × 40) + (37 × 2)
= 1 480 + 74
= 1 554
Mathematics Teacher’s Guide Grade 4
Math G4 TG.indb 191
TERM 2
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2012/09/14 5:35 PM
Unit 28 Patterns in multiplication
Mental Maths Learner’s Book page 146
Learners worked with the multiplication by 11 and 99 when they
explored number patterns in Term 1. They explore the patterns
in the solutions to realise that, for example, in 12 × 11, the 1 and
2 become the units and hundreds and you add 1 and 2 and insert
3 as tens so that the answer is 132. They look for patterns when
multiplying 99. For example, 99 × 5, is 454 – the hundreds and
units digits are the answer to 5 × 9 = 45, and 5 is the tens digit.
Solutions
1. a)
d)
2. a)
d)
191
495
297
396
b)
e)
b)
e)
264
693
594
792
c)
f)
c)
f)
Activity 28.1
374
902
891
990
Learner’s Book page 147
The learners work with rectangular shapes to estimate the area. Let
them use their own strategies to calculate the answers.
Solutions
1. a)
b)
c)
d)
20 × 40 = 800 square metres
60 × 70 = 4 200 square metres
50 × 90 = 4 500 square metres
40 × 70 = 2 800 square metres
2. a) 900
d) 3 500
b) 2 400
e) 2 400
c) 7 200
3. (1) a) 924 square metres
c) 3 956 square metres
b) 3 685 square metres
d) 2 698 square metres
(2) a) 924
c) 7 719
e) 2 356
b) 2 464
d) 3 551
Unit 29 More multiplication methods
Mental Maths Learner’s Book page 148
The learners record the solutions to the problems on their Mental
maths girds. Check how well they are able to multiply multiples
of 10.
Solutions
1. 6
5. 48
9. 4 000
192
Math G4 TG.indb 192
2. 60
6. R120
10. 3 600
Mathematics Teacher’s Guide Grade 4
3. 600
7. 300
4. R45
8. 240
TERM 2
2012/09/14 5:35 PM
Activity 29.1
Learner’s Book page 148
Learners break up both numbers, multiply their values and then add
the products. They complete copies of the tables and work out the
sum for each problem.
Solutions
1. a) 600 + 120 + 60 + 12
= 792
20
2
30
600
60
40
6
120
12
8
c) 1 400 + 120 + 560 + 48
= 2 128
70
20
8
2. a)
b)
c)
d)
e)
f)
b) 42 400 + 480 + 120 + 24
= 3 024
48
3
2 400 120
480
24
d) 2 000 + 450 + 280 + 63
= 2 793
6
1 400 120
560
60
40
50
7
9
2 000 450
280
63
1 073
5 074
5 032
4 512
5 607
3 705
3. Learners use calculators to check their solutions.
Assessment task 5: multiplication
The learners complete this assessment task at the end of Week 6,
Term 2. They demonstrate knowledge of multiplication of 1- and
2-digit numbers. The learners multiply by 10, 100 and 1 000
using flow diagrams. They multiply numbers based on objects
arranged in arrays, write number sentences before solving
problems and use repeated addition and the distributive property
to display understanding of multiplication. They solve contextual
and non-contextual multiplication problems and use short cuts to
multiply by 11 and 99.
Mathematics Teacher’s Guide Grade 4
Math G4 TG.indb 193
TERM 2
193
2012/09/14 5:35 PM
Assessment task 5 Multiplication
1. Copy and complete the flow diagrams.
a) Input
Output
0
0
1
1
2
2
3
Input
0
1
2
3
× 10
4
8
9
Input
4
4
8
8
× 10
× 10
Output
0
1
1
2
2
3
3
× 10
Output
9
9
(7)
Input
Input
Output
Output
Output b) Input
Input
Output
0
0
0
0
1
1
1
1
2
2
2
2
3
3
× 10
× 10
× 10
3
× 10
× 10
3
× 100
4
4
4
4
8
8
8
8
9
9
9
9
(7)
c) Input
Output
Output
0
Input
× 100
3
× 100
4
4
8
8
9
(7)
2. Solve each number sentence.
a) 43 × 10 = n
43 × 100 = n
43 × 1 000 = n
b) 39 × 10 = n
39 × 100 = n
39 × 1 000 = n(6)
9
194
Math G4 TG.indb 194
Mathematics Teacher’s Guide Grade 4
TERM 2
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3. Copy and complete each number sentence.
a) 4 × 15 = n + n + n + n = n + n = n
b) 4 × 15 = (2 × n) + (2 × n) = n + n = n
c) 4 × 15 = (4 × n) + (n × 5) = n + n = n(17)
4. Solve each problem.
a) 6 × 23 = n
b) 4 × 12 = n
c) 3 × 14 = n
d) 5 × 24 = n(4)
5. Use your own methods to solve each problem.
a) 23 × 26 = n
b) 44 × 36 = n(2)
6. Solve these problems without doing calculations.
a) 45 × 11 = n
19 × 11 = n
b) 99 × 6 = n
99 × 5 = n(4)
7. Write a number sentence for each problem and then solve it.
a) A chocolate slab has 24 blocks. How many blocks are
there in eight chocolate slabs?
b) Fifteen friends went to the movies. A movie ticket
costs R28. How much did the friends pay altogether?
c) A ladybird has eight spots. How many spots are there
on 14 ladybirds altogether?
(6)
Total [60]
Mathematics Teacher’s Guide Grade 4
Math G4 TG.indb 195
TERM 2
195
2012/09/14 5:35 PM
Assessment task 5 Multiplication
1. a)
Solutions
Input
Output
Input
0
0
0
0
1
10
1
100
2
20
2
200
30
3
4
40
4
400
8
80
8
800
Input
Input
Input
0
Output
Input
Output
0
0
1
100
2
200
3
× 10
Output
× 10
300
× 10
(7)
9
9
90
900
Input
Output
0
0
1
2
3
× 10
4
8
9
Output
0
0
1
100
2
200
× 100
300
4
400
8
800
9
Output
Output
0
0
0
0
0
0
10
1
10
100
1
10
100
1
100
2
2
20
200
2
20
200
200
32
3
30
300
× 10
× 10
× 10
3
30
300
× 10
× 10
300
× 100
43
4
40
400
4
40
400
4
400
8
8
80
800
8
80
800
8
800
9
9
90
900
(7)
9
90
900
9
900
Input
3
Input
Output
b)
c)
3
× 100
300
4
400
8
800
(7)
9
900
900
2. a) 43 × 10 = 430
43 × 100 = 4 300
43 × 1 000 = 43 000
b) 39 × 10 = 390
39 × 100 = 3 900
39 × 1 000 = 39 000
3. a) 4 × 15 = 15 + 15 + 15 + 15 = 30 + 30 = 60
b) 4 × 15 = (2 × 15) + (2 × 15) = 30 + 30 = 60
c) 4 × 15 = (4 × 10) + (4 × 5) = 40 + 20 = 60
(6)
(17)
4. a) 6 × 23 = 138
b) 4 × 12 = 48
c) 3 × 14 = 42
d) 5 × 24 = 120
(4)
5. a) 23 × 26 = 598
b) 44 × 36 = 1 584
(2)
6. a) 45 × 11 = 495
19 × 11 = 209
b) 99 × 6 = 594
99 × 5 = 495
(4)
7. a) 24 × 8 = n
b) R28 × 15 = n
= 192 blocks
= R420
c) 14 × 8 = n
= 112
196
Math G4 TG.indb 196
Mathematics Teacher’s Guide Grade 4
(6)
Total [60]
TERM 2
2012/09/14 5:35 PM
Properties of 3-D objects
Learners learn to recognise and name rectangular prisms, spheres,
cylinders, cones and square-based pyramids. They distinguish
between properties of 3-D objects by investigating flat and curved
surfaces as well as the face shapes of polyhedra. They also build
3-D objects by working with cardboard polygons.
It is essential for learners to work with concrete, physical
3-D objects so that they can develop a better understanding of
3-D properties when they see them represented on paper.
Differences between 2-D and 3-D
Some people have difficulty in understanding the difference
between 2-D and 3-D. At this stage, learners should first familiarise
themselves with examples of 3-D objects. An informal way to
explain the difference is to say that 2-D shapes are flat on a page
– they have no height – while 3-D objects are things you can walk
around and see from different positions.
A more formal explanation could be expressed like this: Everything
in the real world that can be measured is 3-D (it has three
dimensions). It has height, width and length. Even a sheet of paper,
no matter how thin it is, has some height, and so it is 3-D. We can
measure its height, width and length.
The only things that are 2-D are drawings. There is no such thing
as a rectangle or triangle that we can pick up and feel. These shapes
exist only as 2-D drawings. As soon as we cut out the shape of a
triangle or square from cardboard, they are no longer 2-D shapes.
They become 3-D objects because we can measure their length,
width and height. (For the sake of convenience we still refer to these
cut-out shapes as triangles, squares, and so on. Technically, they are
very flat prisms.)
Unit 30 Flat and curved surfaces
Learner’s Book page 149
Explain that the word surface refers to the outside area of an object.
Start by describing the flat surface of a desk, table, board and
wall. Run your hand over the surfaces and let the learners do the
same. Make sure you point out all the surfaces, for example the top
surface, side and bottom surface of a table and desk. Then choose
other objects that have smaller surfaces, such as a school bag, a
book and an eraser. Then choose objects that are not necessarily flat,
such as a ball, the pot of a pot plant or a vase.
Mathematics Teacher’s Guide Grade 4
Math G4 TG.indb 197
TERM 2
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2012/09/14 5:35 PM
Curved surfaces
Explain the difference between curved and flat surfaces. Take balls
of different sizes to class so that learners can touch them.
Curved and flat surfaces
Learners should have learnt about cones and cylinders in previous
grades. However, reinforce the concept of curved and flat surfaces.
Also take examples of conical and cylindrical objects, such as icecream cones and party hats for cones, and food cans or flasks for
cylinders to class.
Some learners may be confused by the fact that circles have curves,
yet in the examples says that circles are flat surfaces. You can use
two fabric cut-outs of circles and place one circle on a flat surface,
such the board or a table. Even though the edges of the circle are
round, the surface of the circle is flat. Then place the second cut-out
circle on a round surface, such as a ball. Now the surface is curved;
it is not flat. This will help the learners to distinguish between a flat
and a curved surface.
Flat surfaces only
This year learners work with rectangular prisms and pyramids
– learners need to learn to identify these objects. Provide home
language support so that learners can describe the meaning of each
name in words that they understand well. They will investigate the
properties of these 3-D objects later in this section.
Mental Maths Learner’s Book page 150
Learners play Feely bag. The game will help them describe the
attributes of 3-D objects. Use a bag or a box so that learners cannot
see the objects in the container. Put different spheres, cones, cylinders,
prisms and pyramids in the bag. If you do not have a set of these
3-D objects, borrow some from the Foundation Phase teachers or use
real-life containers learners collected.
Show the class an example. Put your hand into the bag or box without
looking into it. Select one object and describe it to the learners. For
example: I feel an object with a curved surface and a flat surface. There
is a sharp point on top and the bottom is a circle. The learners try to
identify the object. If they do not recognise it, take the object out of the
bag and repeat your description. The learners would probably use more
informal terminology to describe the objects. Allow them to do this
but also emphasise the formal terminology and write new terms on the
board.
198
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Mathematics Teacher’s Guide Grade 4
TERM 2
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Activity 30.1
Learner’s Book page 150
Ask learners to bring a variety of objects with curved and flat
surfaces to class. Also take examples to class. Examples of everyday
objects include:
• curved surfaces only: balls, marbles, globe of the earth
• flat surfaces: books, files, boxes
• curved and flat surfaces: pens, glue tubes, cups, glasses.
Solutions
1–3. Practical work
Suggested informal assessment questions to ask yourself
• How well do learners understand the word surface?
• How well can the learners distinguish between flat and curved
surfaces?
Unit 31
Learner’s Book page 150
Shapes and faces of 3-D objects
In this unit, learners are introduced to the different parts of a
3-D object, namely face, edge and corner. Although they do not
work with corners or angles of 3-D objects as yet, they need to
know what a face or edge of an object is because the surface of an
object is divided into different faces by edges.
Shapes of real objects
Here, the learners relate the mathematical shapes of 3-D objects
they have learnt about to the shapes of real objects.
Mental Maths Learner’s Book page 150
Ask learners to describe the candles in the illustration. Remind
learners that there are boxes that are shaped like pyramids and
that pyramids also have flat faces and that they cannot roll!
Encourage learners to give accurate descriptions. For example,
prisms are objects that have the identical flat faces on the top and
the bottom. If you turn a prism upside down, it will stand on a
surface and not fall over. Learners should identify the triangularbased prism and the square- or rectangular-based prisms.
1–5. The learners can work in pairs to spot the various shapes in
the picture.
Activity 31.1
Learner’s Book page 151
1. Learners can work on their own or in pairs to find suitable
objects.
2. Each time the learners show the class their objects, let the class
decide whether they agree or disagree that an object is the shape
that the learner says it is.
Mathematics Teacher’s Guide Grade 4
Math G4 TG.indb 199
TERM 2
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2012/09/14 5:35 PM
Faces of 3-D objects
The learners distinguish the parts of 3-D objects. They should know
what the face, edge and corner of a 3-D object is.
Activity 31.2
1
5
face
corner
Learner’s Book page 151
2
6
edge
face
3
7
corner
edge
4
8
edge
face
Suggested informal assessment question to ask yourself
• How well are the learners able to identify the faces of a
3-D object?
Unit 32 Straight, flat faces: polyhedra
Learner’s Book page 151
Until now, the learners have worked with 3-D objects that had both
flat and curved surfaces. Now the focus is on polyhedra, 3-D objects
with straight, flat faces; or 3-D objects whose faces are made of
polygons.
Have models of different polyhedra available in the classroom for
the learners to handle and examine.
Faces of polyhedra
The ideal would be to have a rectangular prism with different
coloured faces to match those in the Learner’s Book in class. You
could paint a shoe box. Using a real box will make it easier for the
learners to make sense of the picture in the example.
Mental Maths Learner’s Book page 152
1. a) sphere
c) square-based pyramid
2. six
3. five
b) rectangular prism
d) cylinder
Activity 32.1
Learner’s Book page 152
1. a) face 5
c) face 6
2.
Shape
A cube
B rectangular prism
C square-based
pyramid
200
Math G4 TG.indb 200
Mathematics Teacher’s Guide Grade 4
b) face 4
Number of faces
6
6
5
Shape of faces
All six are squares.
All six are rectangles.
One square; four
rectangles
TERM 2
2012/09/14 5:35 PM
Suggested informal assessment questions to ask yourself
• How well are the learners able to identify the number of faces
of a polyhedron?
• How well can learners identify the shapes of the faces of
different polyhedra are?
• How well can learners distinguish a cube from other
rectangular prisms?
Building models of polyhedra
Building models of polyhedra helps the learners to familiarise
themselves with the different shapes and number of faces that make
up the polyhedra. In this way, they further explore the properties of
3-D shapes.
Activity 32.2
Learner’s Book page 153
The faces of the polyhedra are shown in the Learner’s Book. The
number next to each shape shows how many of that shape needed to
build a particular polyhedron.
Note:
• The shapes are drawn on dotted paper so that the learners can
easily copy them onto dotted paper. The shapes in the Learner’s
Book may be a little too small for the learners to handle
comfortably, so learners can enlarge the shapes. They can just
double the number of dots when they draw the shapes. If it is too
difficult for the learners to do this, you can give them enlarged
copies of the shapes.
• Learners can then stick the paper onto cardboard, cut out the
shapes and stick them together to make a 3-D shape.
• It is easier for the learners to use adhesive tape rather than glue to
build each shape.
Suggested informal assessment questions to ask yourself
• How easy is it for the learners able to make models of
polyhedra?
• How well do they understand the process they are following
when they build the models?
Naming prisms and pyramids
Make sure you have a prism and a square-based pyramid in class
when you discuss this section with the learners.
The focus in Grade 4 is only on square-based pyramids. If you
think your class is able to understand extensions of this concept,
you can introduce triangular-based and pentagonal-based pyramids
to illustrate the fact that the shape of the base of a pyramid can be
different polygons.
Mathematics Teacher’s Guide Grade 4
Math G4 TG.indb 201
TERM 2
201
2012/09/14 5:35 PM
Suggested informal assessment question to ask yourself
• How well can the learners recognise 3-D objects and match
them to the shapes of everyday items?
Investigation
Learner’s Book page 154
The learners will have to cut out and measure shapes with the same
and different lengths, and then come to a conclusion.
Answer to investigation question:
• The height of the triangles that make up the sides of the
pyramid all have to be the same so that they can meet at one point
at the top.
• The length of the sides of the triangles that will meet at the base
of the pyramid must be the same as the length of the base’s sides.
Revision
Learner’s Book page 154
1. A: cone – flat and curved surfaces
B: sphere – curved surface only
C: rectangular prism – flat surfaces only
D: square-based pyramid – flat surfaces only
E: cylinder – flat and curved surfaces
2. a)
b)
c)
d)
e)
C: prism; D: pyramid
C: rectangular prism; D: square-based pyramid
C: 6 faces; D: 5 faces
C: 6 rectangles; D: 1 square and 4 triangles
C: 8; D: 5
3. a) square-based pyramid
b) 4 rectangles and 2 squares, or 6 rectangles
Remedial activities
Learners who struggle with work in this unit will usually have
a poor mental picture of 3-D objects. It is thus essential to have
models of the 3-D objects in class for the learners to pick up, feel
and turn around when they work on this section.
• Let the learners work through similar activities to those in
the Learner’s Book, but give them more time to handle the
3-D models as they build up their concepts and work through
the activities.
• Let learners who struggle, work together and build many models
of the same 3-D objects. Initially, provide the cut-out shapes that
they must use. Then later, let them copy and select the shapes
they need to build their own models.
202
Math G4 TG.indb 202
Mathematics Teacher’s Guide Grade 4
TERM 2
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Extension activity
Challenge the learners to build a model of a rectangular prism using
only squares (a cube) and then to build one using only two squares
and rectangles. Let them work out the measurements.
Project
The learners could work in groups of four to complete the project.
Make a toy mobile of at least four 3-D objects.
1. Use cardboard to build 3-D objects.
2. Decorate your 3-D models with colours and patterns you like.
3. Tie two sticks with string or wool to make a cross shape.
4. Thread a piece of wool or string through each model. Knot the
one side of the wool to keep it tied to the model.
5. Tie the other side of the wool onto your crossed sticks. Make the
threads different lengths.
6. Hang the mobiles from the classroom ceiling. Or tie a piece of
string across the classroom and hang the mobiles from the line.
You can use the following checklist to assess the learner’s projects
and give them a mark out of 40.
Criteria
4
Mark allocation
3
2
0–1
Choose the correct faces to build
model 1
Choose the correct faces to build
model 2
Choose the correct faces to build
model 3
Choose the correct faces to build
model 4
Size the faces correctly for model 1
Size the faces correctly for model 2
Size the faces correctly for model 3
Size the faces correctly for model 4
Finish and decorate the models
attractively
Make a stable hanger with two sticks
Attach the models to the hanger
suitably and securely
Total mark out of 40
Mathematics Teacher’s Guide Grade 4
Math G4 TG.indb 203
TERM 2
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Geometric patterns
Grade 4 learners work with geometric patterns to look for
relationships and patterns. They describe relationships or rules
observed in their own words.
The patterns involve:
• physical or diagrammatic representations
• sequences not limited to a constant difference or ratio
• patterns of learners’ own creation.
Learners repeat 2-D and 3-D patterns and shapes that grow or
decrease in different ways. They copy and extend patterns and
describe the rules in their own words. The learners represent
patterns they noticed in input and output flow diagrams, tables and
number sentences.
Remind the learners of the number patterns they worked with in
Term 1. Ask them to describe the relationships in a few number
patterns, for example, 6; 16; 26; 36 . . . or 0; 9; 18; 27 . . . and
let them extend the patterns. Learners will work with geometric
patterns that involve 2-D shapes and 3-D objects this week. They
will work in groups and pairs to discuss and describe relationships
(connections or links) between shapes, objects and numbers in
patterns.
Unit 33 Explore geometric patterns
In Term 1 you worked with number patterns. In this unit you will
work with geometric patterns. In Grade 3 you created patterns using
objects and drawings. You described and copied geometric patterns
formed in nature, real life and culture. This term you will build on
the work done in Grade 3.
Mental Maths Learner’s Book page 155
Drumbeats have rhythms that we can translate into patterns.
Discuss this with the learners African people have used drums
for communication for many centuries, but nowadays drums are
mostly used to create musical rhythms. Let the learners identify
other places where there are patterns in real life. Examples
include paving, bathroom tiles and fabrics. Learners can make up
patterns using home-made drums, spoons and other instruments.
A pattern is a repetition that displays harmony amongst its
elements. Allow learners to clap their hands and stamp their feet
to create patterns, and use numbers to keep time.
204
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The learners have to observe and understand the pattern units
(groups of objects) that are repeated in the number and picture
patterns, and not focus on a single shape or object in a pattern.
Solutions
1. Learners practise drum patterns.
2. Learners discuss the patterns.
a)
b)
OOn
n
nO
n
n
nn
OOn
nnO
nO
OOn
n
nO
n
n
n
OOn
OOn
n
nO
nn
n
n
OOn
nO
OOn
n
nO
n
n
OOn
nn
nO
OOnnnOnnnOOnnnO
OOnnnOnnnOOnnnO
c)
d)
e)
f)
Activity 33.1
Learner’s Book page 156
Write down the names of new shapes such as parallelogram,
rhombus, hexagon and pentagon. Ask learners if there is a difference
between the rhombus and the square. They should realise that the
shapes are the same; the rhombus shown here is a rotated square.
Let them describe Beauty’s patterns. You can extend this lesson to
space and shape and let learners use transformation terminology
such as slide (translate), flip (reflect) and turn (rotate) to describe
how a pattern was created. Make copies of the 2-D shapes and
let the learners cut them out. They create and describe their own
patterns. Use a space on the wall to display their work. Encourage
learners to describe how the shapes are positioned using formal
names if possible. Let them study the bead and seed jewellery.
They copy and extend the patterns. Tell them something about each
pattern and show them on the map where Swaziland and Kenya
are situated. Give learners string or cotton, and beads of different
colours or 3-D objects. They can also use buttons or macaroni and
create different necklaces or bracelets and ask other learners to
describe, draw and extend their patterns.
Mathematics Teacher’s Guide Grade 4
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Unit 34 Identify and extend patterns
Mental Maths Learner’s Book page 157
Let the learners study the shapes in the pattern. Make copies
and let them cut out the strips below. They have to decide
how the strips fit together to form the pattern above. During
this experience you should encourage them to use the formal
terminology. Write down new words such as octagon.
Order of patterns: F, C, E, D, B, G, A, H
Activity 34.1
Learner’s Book page 157
Learners should describe the relationships they notice. They
could extend the number patterns to include the next five numbers.
They describe the even numbers, odd numbers and multiples of 3
and 4. You can ask learners to create cube patterns and ask their
partners to describe the relationships. For question 2, you could ask
them to describe how the patterns grow. In the next exercise, they
study the growing patterns created with squares and triangles. They
should observe that each square has a triangle on top and below and
an extra triangle on the side. In pattern 1, the relationship between
the square and triangle is (1 × 2) + 2, which gives 4 triangles. In
pattern 2, the number of triangles is (2 × 2) + 2 and in pattern 3 it
is (3 × 2) + 2. Learners should copy the patterns and draw the next
three patterns. Do not give them the rule, rather let them explore
and discover it. You could ask them what operation they have used
to perform on the squares to work out the number of triangles. Ask
them to copy and complete the flow diagram and the table using the
rule or relationship they have discovered.
Solutions
1. a)
b)
c)
d)
2; 4; 6; 8; 10
1; 3; 5; 7; 9
3; 6; 9; 12
4; 8; 12; 16
even numbers or multiples of 2
odd numbers
multiples of 3
multiples of 4
2. a) pattern 4: four squares and 10 triangles
pattern 5: five squares and 12 triangles
pattern 6: six squares and 14 triangles
b) Rule: × 2 + 2
206
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TERM 2
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c)
Squares
Triangles
1
2
3
4
5
10
21
4
6
8
10
12
22
44
Rule
×2+2
d) Squares
Triangles
1
4
2
6
3
8
4
10
5
12
6
14
7
16
12
26
20
42
Unit 35 Extend patterns
Mental Maths Learner’s Book page 158
The learners study the pattern with black and green beads.
Without drawing pattern 4, they have to work out how many
beads in total there will be in a pattern. This will help them find
a rule they can use to calculate the number of green beads if they
know how many black beads there are in a pattern. They do not
draw the extended patterns, but create mental images to use to
work out how many beads there will be in a pattern. To calculate
the number of green beads using the number of black beads, the
pattern is (1 × 2) + 1 = 3; (2 × 2) + 1 = 5; (3 × 3) + 1 = 10; and so
on. To calculate the number of black beads from a given number
of green beads: (31 – 1) ÷ 2 = 30 ÷ 2 = 15. Remind learners to
use inverse operations to check calculations. Allow learners to
discuss their observations and solutions.
Solutions
13
9; 11; 13 (odd numbers)
7; 8; 9; 10 (one black bead is added to every pattern)
1; 2; 3; 4; 5; 6; 7; 8; 9; 10 (natural numbers)
3; 5; 7; 9; 11; 13; 15; 17; 19; 21 (odd numbers)
black: 20; 21; 22
green: 41; 43; 45
7. Rule: × 2 + 1
8. (31 – 1) ÷ 2 = 30 ÷ 2 = 15
9. (101 – 1) ÷ 2 = 100 ÷ 2 = 50
10. 200 × 2 + 1 = 401
1.
2.
3.
4.
5.
6.
Mathematics Teacher’s Guide Grade 4
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Activity 35.1
Learner’s Book page 159
Learners have been introduced to square numbers before. Ask
them if they can see why these numbers are called square numbers
(they create perfect squares). Learners should realise that the
numbers are created by multiplying the pattern number by itself
(they square the numbers so that 1 = 1 × 1; 4 = 2 × 2; 9 = 3 × 3, and
so on). Next, they study the patterns created by dots and rhombi
(rhombuses) where they should realise that a rhombus (often called
a diamond) is not a square. A square is always a rhombus, but a
rhombus is not always a square. In the pattern for the number of
dots (4; 7; 10; 13; ..., 3 is added to each pattern to find the next one).
The rhombus pattern involves the numbers 1; 2; 3; 4; 5; ... (natural
numbers). Ask them to complete the flow diagram and describe the
numbers in the pattern sequences.
Solutions
1. a) pattern 4 has 16 dots
pattern 5 has 25 dots
pattern 6 has 36 dots
b) square numbers: the product of a number multiplied by itself
c) 1; 4; 9; 16; 25; 36; 49; 64; 81; 100
d) Pattern
Number
number
1
2
3
4
7
9
10
of dots
Rule
number
×
number
1
4
9
16
49
81
100
2. a) Pattern 4 will have 4 rhombi, with a dot at each corner;
pattern 5 will have 5 rhombi, and so on.
b) 4; 7; 10; 13; 16; 19; 22; 25; 28; 31 (add 3)
c) 1; 2; 3; 4; 5; 6; 7; 8; 9; 10
d) The numbers in (c) are counting numbers.
e) Number
Number
of rhombi
1
2
3
4
7
9
10
208
Math G4 TG.indb 208
of dots
Rule
×3+1
Mathematics Teacher’s Guide Grade 4
4
7
10
13
22
28
31
TERM 2
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Unit 36 Input and output numbers (values)
Ask the learners to explain what input and output values are and
how the operator influences these values. The operator could be any
operation or combination of operations. The operator is given for
question 1, but not for the other questions. Do not give the rules to
the learners. They should discover rules for themselves.
Mental Maths 44
64 6 4
86 8 6
10810 8
12
101210
121612
16
1.
÷÷22
÷ 2÷ 2
40
164016
150
40
15040
150
150
180
180
180180
3.
11
31 3 1
53 5 3
75 7 5
87 8 7
98 9 8
9 9
++1010
+ 10
+ 10
Learner’s Book page 160
22
32 3 2
43 4 3
54 5 4
65 6 5
86 8 6
2.
20820 8
75
207520
759075
90
90 90
1111
13
111311
15
131513
151715
17
171817
18
19
181918
19 19
1111
111511
15
152115
21
213021
30
39
303930
397139
71
71
107
10771
107
1001
107
1001
1001
1001
4.
11
21 2 1
32 3 2
43 4 3
54 5 4
65 6 5
6 6
Activity 36.1
––22
– 2– 2
99
13913 9
131913
19
192819
28
37
283728
376937
69
69
107
10769
107
1001
107
1 001
1 001
1001
××33++66
× 3×+36+ 6
99
12912 9
15
121512
151815
18
182118
21
212421
24
24 24
Learner’s Book page 161
The learners should realise after exploring and discovering that the
rule to determine the number of red beads is × 2 + 2. They complete
the table to enforce this understanding. If there are 20 black
beads, there will be (20 × 2) + 2 = 42 red beads. They use the flow
diagrams and fill in input and output values.
Solutions
1. a)
Number of black beads
Number of red beads
1
4
2
6
3 4 5 6 7 8 9 10
8 10 12 14 16 18 20 22
b) × 2 + 2
c) 42 red beads
2. Black beads
Red beads
2
×2
+2
6
4
×2
+2
10
6
×2
+2
14
Mathematics Teacher’s Guide Grade 4
Math G4 TG.indb 209
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2012/09/14 5:36 PM
Black beads
Red beads
8
×2
+2
18
10
×2
+2
22
20
×2
+2
43
25
×2
+2
52
33
×2
+2
68
50
×2
+2
102
209
×2
+2
420
3. The learners use their own input values and the rules to create
output values.
Assessment task 6: patterns
The learners perform an assessment task involving geometric
patterns. They study the squares pattern and extend it. To
calculate the number of white squares, calculate (1 × 2) + 1.
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Assessment task 6 Patterns
1. Look at these patterns.
Pattern 1
Pattern 2
Pattern 3
a) Draw pattern 4.
b) How many white squares will there be in pattern 5?
c) Write a number sentence to show how to calculate the
number of white squares.
d) Complete a copy of the table.
1 2 3 4 5 6 7 8 9 10
Grey squares
White squares
(17)
2. Give the next three numbers in each number pattern.
a) 1; 4; 7; 10; 13; 16; …; …; …
b) 2; 7; 12; 17; 22; 27; …; …; …
c) 3; 13; 23; 33; 43; 53; …; …; …
d) 0; 8; 16; 24; 32; 40; …; …; …
(12)
3. Complete a copy of each flow diagram.
a)
×4
12
b)
÷6
5
c) (3)
+ 93
100
4. Complete a copy of the table.
1
2
3
4
5
6
10
20
Input
Output
(8)
Total [40]
Mathematics Teacher’s Guide Grade 4
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Assessment task 6 Patterns
Solutions
1. a) Two white and one grey square are added to make each new
pattern.
Pattern 1
Pattern 2
Pattern 3
Pattern 4
b) The number of grey squares is equal to the pattern number.
Pattern 5 will have double the number of grey squares
plus one.
c) Number of grey squares × 2 + 1 = number of white squares
White squares in pattern 4: 5 × 2 + 1 = 11
d) The numbers of white squares form the sequence of odd
numbers.
1 2 3 4 5 6 7 8 9 10
Grey squares
White squares 3 5 7 9 11 13 15 17 19 21
(17)
2. a) 1; 4 ;7; 10; 13; 16l 19; 22; 25
(intervals of 3)
b) 2; 7; 12; 17; 22; 27; 32; 37; 42
(intervals of 5)
c) 3; 13; 23; 33; 43; 53; 63; 73; 83
(intervals of 10)
d) 0; 8; 16; 24; 32; 40; 48; 56; 64
(multiples of 8) (12)
3. a)
3
×4
12
12 ÷ 4 = 3
b)
30
÷6
5
5 × 6 = 30
c)
7
+ 93
100
100 – 93 = 7
(3)
4. The learners should discover that they have to multiply by 2 and
add 2 to get the output values. The rule is × 2 + 2.
1
2
3
4
5
6
10
20
Input
4
6
8
10
12
14
22
42
Output
(8)
Total [40]
212
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Symmetry
Introduction
The learners learnt bout symmetry in Grade 3. This section revises
and consolidates the concept of line symmetry. Learners identify
lines of symmetry in pictures and shapes where these lines are not
necessarily vertical.
Unit 37 What is symmetry?
Learner’s Book page 162
Revise what symmetrical shapes or pictures are. Make sure learners
know what a line of symmetry is. If they are not sure, draw a simple
symmetrical diagram or pattern on a sheet of paper, such as two
dots. Fold the sheet in half, then open it again. Show the learners
and point to the fold. Remind them that this is the line of symmetry
that divides the picture into two mirror images.
Do the same exercise to show shapes that do not have symmetry.
This time, place the dots asymmetrically on the sheet of paper so
that they do not fit over one another when you fold the paper.
Mental Maths Learner’s Book page 162
The learners work in pairs to recognise the shapes that are
symmetrical. They identify the lines of symmetry and explain
their reasoning. Encourage them to use terms such as horizontal,
vertical and diagonal lines.
Solutions
1. B, C, D, E, F, G, H
2. The sketches show lines of symmetry.
A
B
C
D
E
F
G
H
3. Learners explain patterns.
Mathematics Teacher’s Guide Grade 4
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Activity 37.1
Learner’s Book page 163
1. See sketches in Mental maths.
2. Most learners should manage well enough on their own when
creating their symmetrical figures. If they need help, suggest that
they start with simple outlines and then progress to more and
more interesting symmetrical shapes.
Suggested informal assessment questions to ask yourself
• How easily are the learners able to identify symmetrical shapes
and pictures?
• Are learners able to create their own symmetrical shapes and
figures?
Lines of symmetry
Some learners may have realised that some shapes have more than
one line of symmetry, but clarify this point for all the learners.
Activity 37.2
Learner’s Book page 164
1. The triangle and square as they appear on page 163 with their
lines of symmetry
2. Learners do an investigation.
3. a) All except for shape E are symmetrical.
b) Learners compare and discuss the lines of symmetry they
have drawn.
c) B has one line of symmetry.
A, C, D and H each have two lines of symmetry.
F and G each have three lines of symmetry.
Investigation
Learner’s Book page 164
Learners discuss which letters are symmetrical. Some learners may
also realise that the letter O has many lines of symmetry (this can
also depend on the font and whether it is shaped as a circle).
Drawing symmetrical shapes
Graph paper or dotted paper can help the learners to keep track
of the lengths and direction of the lines they need to draw when
creating symmetrical shapes.
Activity 37.3
Learner’s Book page 164
1, 2. Learners copy and complete the patterns.
3. There are numerous ways in which the shapes can be
completed so that they are not symmetrical. Discuss examples
with the class.
214
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Suggested informal assessment questions to ask yourself
• How well are the learners able to identify more than one line of
symmetry in a shape or picture?
• How easily are they able to complete shapes that are
symmetrical and shapes that are not symmetrical?
Revision
Learner’s Book page 165
1. A: yes; the line is a line of symmetry.
B: no; there is only a chimney on one side of the house.
C: no; the two parts are not mirror images.
D: Only the horizontal line is a line of symmetry as it divides the
picture into two mirror images. The vertical line does not create
two mirror images.
E: The vertical line is a line of symmetry as it divides the left
and right halves into mirror images. The horizontal line is not a
line of symmetry as it does not create mirror images of the top
and bottom halves.
F: Both lines are lines of symmetry.
2. Learners to draw their own shapes and assess their partners.
Remedial activities
If learners find it difficult to determine whether pictures are
symmetrical just by looking at them, let them copy the picture onto
a sheet of paper, and then fold the paper to see whether there is a
line of symmetry.
Let the learners make symmetrical shapes with their bodies. They
can work in pairs and help each other.
Extension activities
Let the learners do mirror writing, where they write their names
upside down and then use a mirror to read the writing right side up.
Challenge the learners to create shapes that have more than one line
of symmetry.
Mathematics Teacher’s Guide Grade 4
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Whole numbers: addition and subtraction
Unit 38 Round off to add and to subtract
Mental Maths Learner’s Book page 166
Ask the learners what they remember about rounding off. Let
them look at the rules in the speech bubbles and ask if they
agree.
Solutions
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
Number
4 324
1 296
5 655
2 075
9 212
6 879
1 558
7 006
3 004
2 947
10
4 320
1 300
5 660
2 080
9 210
6 880
1 560
7 010
3 000
2 950
100
4 300
1 300
5 700
2 100
9 200
6 900
1 600
7 000
3 000
3 000
Activity 38.1
1 000
4 000
1 000
6 000
2 000
9 000
7 000
2 000
7 000
3 000
3 000
Learner’s Book page 166
1. The learners should have realised that rounding off to the nearest
10 give estimates closest to the accurate solutions. Check which
strategies they use. Estimations will differ.
2. a) 1 019 + 1 914 = 2 933
c) 7 478 – 4 489 = 2 989
e) 4 675 + 2 386 = 7 061
b) 2 224 + 2 318 = 4 542
d) 9 010 – 5 675 = 3 335
Unit 39 Different ways to add
Mental Maths Learner’s Book page 167
Let the learners work together as a class to discuss the three
strategies for the addition problem. They will probably prefer
one of the shorter strategies. Ask learners if they have different
methods to solve the problems. Let them explain their methods
to the class.
216
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The strategies involve repeated addition, building up multiples of
10 and breaking up numbers. Remind the learners that working
with multiples of 10 helps them calculate easier and smarter. It
is easier to add and subtract numbers to and from multiples of
10. They could also use the commutative property by swapping
numbers to calculate easier.
Solutions
1. Responses will differ.
2. a) 58 + 37 = 58 + 2 + 35
b) 36 + 37 = 36 + 4 + 33
= 60 + 35
= 40 + 33
= 95
= 73
c) 49 + 38 = 49 + 1 + 37
d) 24 + 69 = 69 + 1 + 23
= 50 + 37
= 70 + 23
= 87
= 93
e) 75 + 18 = 75 + 5 + 13
f) 29 + 56 = 29 + 1 + 55
= 80 + 13
= 30 + 55
= 93
= 85
g) 43 + 28 = 28 + 2 + 41
h) 37 + 44 = 37 + 3 + 41
= 30 + 41
= 40 + 41
= 71
= 81
i) 54 + 39 = 39 + 1 + 53
j) 39 + 52 = 39 + 1 + 51
= 40 + 53
= 40 + 51
= 93
= 91
Activity 39.1
Learner’s Book page 167
The first two problems have been solved using the second method.
Encourage the learners to try to keep the = signs below each other
so that the numbers are aligned according to their place values.
This will help learners enhance the place value concept and when
calculations entail carrying. It could also assist them in developing
understanding of the column calculation strategy.
Solutions
1.
789 + 1 356 = 3 145
2. 3 472 + 3 589 = 7 061
1 000 + 1 000 = 2 000 3 000 + 3 000 = 6 000
700 + 300 = 1 000 400 + 500 = 900
80 + 50 = 130
70 + 80 = 150
9+6=
15
2+9=
11
3 145
7 061
3. 5 627 + 3 484 = 9 111
4. 4 584 + 3 767 = 8 351
5. 6 375 + 3 869 = 10 244
6. 4999 + 3 999 = 8 998
7. 2 555 + 2 555 = 5 110
8. 4 069 + 3 087 = 7 156
9. 3 490 + 2 609 = 6 099
10. 7 277 + 2 488 = 9 765
Mathematics Teacher’s Guide Grade 4
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Unit 40 Different ways to subtract
Mental Maths Learner’s Book page 168
Learners should realise that the first strategy is time-consuming,
but if there are learners who are comfortable with this strategy,
allow them to use it. Write the problems on the writing board
and record learners’ strategies to compare if they use different
strategies.
Solutions
1. Responses will differ.
2. a) 76 – 59 = 76 – 50 – 9
b) 64 – 27 = n
= 26 – 9 60 – 20 = 40
= 17 4 – 7 = –3
40 – 3 = 37
c) 87 – 38 = n
d) 93 – 69 = 93 – 60 – 9
87 – 30 = 57
= 33 – 9
57 – 8 = 49
= 24
e) 55 – 39 = n
f) 46 – 27 = 46 – 20 – 7
50 – 30 = 20
= 26 – 7
5 – 9 = – 4
= 19
20 – 4 = 16
g) 63 – 38 = n
h) 78 – 59 = 78 – 50 – 9
60 – 30 = 30
= 28 – 9
3 – 8 = –5
= 19
30 – 5 = 25
i) 65 – 48 = n
j) 82 – 69 = 82 – 60 – 9
60 – 40 = 20
= 22 – 9
5 – 8 = –3
= 13
20 – 3 = 17
Activity 40.1
Learner’s Book page 169
The learners study and discuss the strategies for the same
subtraction problem. Let them decide which method they find
more effective. Ask them to use one of the strategies to solve the
subtraction problems, but they can also use their own methods.
The first four problems have been solved using methods 2 and 3.
These strategies involve breaking up numbers in the place values
and using the closest multiples of 100 and 1 000 for the second
number. Remind the learners to align the = signs and the numbers
according to their place values. Method 3 will assist with developing
an understanding of column addition. Some learners might be able
to solve the problems mentally, especially where decomposition is
not required.
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Solutions
1. a) 2 768 – 1 436 = n
2 000 – 1 000 = 1 000
700 – 400 = 300
60 – 30 =
30
8–6=
2
1 332
b) 6 527 – 4 216 = n
6 527 – 4 200 = 2 327
2 327 – 16 = 2 311
c) 4 523 – 2 446 = n
4 000 – 2 000 = 2 000
500 – 400 = 100
= 2 100
23 – 46 = –23
2 100 – 23 = 2 100 – 20 – 3
= 2 080 – 3
= 2 077
d) 8 684 – 4 573 = n
8 684 – 4 500 = 4 184
4 184 – 73 = 4 184 – 70 – 3
= 4 114 – 3
= 4 111
e) 9 734 – 8 645 = 1 089
g) 3 678 – 1 567 = 2 111
i) 6 450 – 4 500 = 1 950
f) 5 250 – 2 500 = 2 750
h) 7 800 – 5 900 = 1 900
j) 4 230 – 2 330 = 1 900
2. Assist learners who have reading problems to understand the
context of the word problems. Let the learners write number
sentences before solving the problems. They use their own
strategies and estimate the solution before calculating. Below
are strategies they might use.
a) 155 + 125 + 135 = n
b) R3 575 + R685 = n
100 + 100 + 100 = 300
3 000 + 0 = 3 000
50 + 20 + 30 = 100
500 + 600 = 1 100
5 + 5 + 5 = 15
70 + 80 = 150
415 roses were delivered
5+5=
10
Andile earns R 4 260
c) 2 011 – 1 957 = n
1 957 + 3 = 1 960
1 960 + 40 = 2 000
2 000 + 11 = 2 011
Her age was 3 + 40 + 11 = 54
Mathematics Teacher’s Guide Grade 4
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Unit 41
Check addition and subtraction
calculations
Mental Maths Learner’s Book page 169
Let the learners study Sipho’s strategy to check addition by
subtraction. They should recognise that he added 3 to both
numbers in the subtraction calculation. Ask them to explain how
they will solve the addition problems. Record the strategies on
the board. Let them use the strategy to check the solutions of the
addition problems.
Solutions
1. Learners discuss a strategy for adding and subtracting.
2. The first five problems have been solved using the suggested
strategy, i.e. adding the same number to both numbers to do
subtraction to check addition calculations.
a) 34 + 57 = 91
Check: 91 – 57 = n
94 – 60 = 34
b) 56 + 29 = 85
Check: 85 – 29 = n
86 – 30 = 56
c) 28 + 49 = 77
Check: 77 – 49 = n
78 – 50 = 28
d) 45 + 38 = 83
Check: 83 – 38 = n
85 – 40 = 45
e) 37 + 49 = 86
Check: 86 – 49 = n
87 – 50 = 37
f) 49 + 23 = 72
g) 58 + 34 = 92
h) 37 + 48 = 85
i) 26 + 57 = 83
j) 39 + 53 = 92
Activity 41.1
Learner’s Book page 170
The learners solve addition problems with bigger numbers.
They use the inverse operation to check the solutions. Ask them
to look at the magic square. They should understand that the
numbers in the rows, columns and diagonals add up to the same
sum. Explain what a diagonal is if they do not know. Let them
check the sum of the numbers in the rows, columns, diagonals
and the four corners of the Indian magic square. Ask them to write
down the numbers as indicated to create two magic squares. Let
them draw two 3 × 3 squares. They should arrange the numbers
so that there is a magic number for each square. Let them struggle
with the problems but if they get stuck, give them a clue. Each line
of three numbers must add up to one third of the total of all the
numbers used in a square. In the first square they created, the set of
nine numbers must total 54, so that the magic number is 18. In the
second square the magic number is 19.
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Solutions
345 + 478 = 823
557 + 359 = 916
1 450 + 1 460 = 2 910
4 956 + 4 276 = 9 232
2 329 + 1 737 = 4 066
3 470 + 2 630 = 6 100
1. a)
b)
c)
d)
e)
f)
2. a) Remind the learners to use effective mental calculation
strategies to find the sum of the numbers in the rows,
columns and diagonals. They should use the commutative
and associative properties, for example,
1 + 14 + 15 + 4 = 15 + 15 + 4 = 34. The magic number is 34.
34 34 34 34
1
14 15
4
34
12
7
6
9
34
8
11 10
5
34
13
2
16
34
34
3
34
b) 1 + 4 + 13 + 16 = 16 + 4 + 13 + 1
= 20 + 14
= 34
c) Below is one solution. The magic number is 18. There are
several solutions. The learners should notice that the number
in the centre is half of the sum of the numbers on its sides,
for example, 11 + 1 = 12; 9 + 3 = 12; 4 + 8 = 12, and so on.
The number in the centre is one third of the magic number.
4
11
3
5
6
7
9
1
8
d) They create a magic square in the same way as in (c).
Mathematics Teacher’s Guide Grade 4
Math G4 TG.indb 221
TERM 2
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Whole numbers: division
This week learners will learn to divide 3-digit numbers by 1-digit
numbers using the knowledge they developed about division in
Term 1. Remind learners about the importance of knowing the
multiplication tables. If they know the multiplication facts, doing
division is easy because of the connection between multiplication
and division.
Unit 42 Solve story problems
Mental Maths Learner’s Book page 171
Ask the learners to explain what division means. Make a table
on the board with the calculations in one column and the reason
for each calculation in the second column. Write down their
understanding of each division calculation. Ask the learners to
record the solutions to (b) on their Mental maths grid. Let them
give the inverse operations for each problem. Write these on the
board as they name them.
Solutions
1. a)
d)
g)
j)
30 ÷ 6 = 5
24 ÷ 8 = 3
64 ÷ 8 = 8
27 ÷ 3 = 9
b) 40 ÷ 8 = 5
e) 42 ÷ 6 = 7
h) 32 ÷ 4 = 8
c) 35 ÷ 7 = 5
f) 81 ÷ 9 = 9
i) 20 ÷ 4 = 5
2. a)
d)
g)
j)
18 ÷ 3 = 6
36 ÷ 9 = 4
56 ÷ 8 = 7
49 ÷ 7 = 7
b) 15 ÷ 5 = 3
e) 32 ÷ 8 = 4
h) 63 ÷ 9 = 7
c) 16 ÷ 4 = 4
f) 42 ÷ 7 =6
i) 72 ÷ 8 = 9
3. a)
d)
g)
j)
6 × 3 = 18
9 × 4 = 36
7 × 8 = 56
7 × 7 = 49
b) 3 × 5 = 15
e) 8 × 4 = 32
h) 7 × 9 = 63
c) 4 × 4 = 16
f) 7 × 6 = 42
i) 9 × 8 = 72
Activity 42.1
Learner’s Book page 171
Learners have to understand that the numbers get 10 times bigger
and move one place to the left as they are multiplied by bigger
powers of 10. They explore the numbers and place value of the
digits showing division by powers of 10. They should realise that
the numbers become smaller and the digits move one place to the
right as they divide by 10. This is important basic knowledge for the
development of understanding of decimal numbers that they will
deal with in Grade 6. The learners solve problems involving division
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of multiples of 10 by 10. After making sense of these problems they
should deduce that you remove the 0 from the multiple. Ask them to
investigate the strategy to divide multiples of 10 by multiples of 10.
They should make sense of the fact that you first divide by 10 and
then by the tens digits. Later they could provide a rule and reason
that you remove a 0 from both numbers and divide by the tens unit.
Solutions
1. a, b) 1 × 3 = 3
10 × 3 = 30
100 × 3 = 300
1 000 × 3 = 3 000
10 000 × 3 = 30 000
c) There is the same number of 0s on each side of the equal
sign.
2. a, b) 23 000 ÷ 10 = 2 300
2 300 ÷ 10 = 230
230 ÷ 10 = 23
c) The number is ten times smaller (has one less 0).
3. a) 350 ÷ 10 = 35
c) 930 ÷ 10 = 93
e) 1 870 ÷ 10 = 187
b) 860 ÷ 10 = 86
d) 4 600 ÷ 10 = 460
f) 7 000 ÷ 10 = 700
4. First divide by 10.
5. a)
c)
e)
g)
i)
(450 ÷ 10) ÷ 9 = 5
(480 ÷ 10) ÷ 8 = 6
(350 ÷ 10) ÷ 5 = 7
(810 ÷ 10) ÷ 9 = 9
(560 ÷ 10) ÷ 8 = 7
b)
d)
f)
h)
j)
(630 ÷ 10) ÷ 7 = 9
(420 ÷ 10) ÷ 6 = 7
(240 ÷ 10) ÷ 4 = 6
(490 ÷ 10) ÷ 7 = 7
(300 ÷ 10) ÷ 3 = 10
Unit 43 Division with and without remainders
Mental Maths Learner’s Book page 172
The learners explore the strategies given for dividing 3-digit
multiples of 10 by 1-digit numbers. They first divide by 10 or
divide by the divisor and multiply by 10. Later they should
conclude that you remove the 0 and divide the hundreds and tens
units. Let them use both strategies and decide which one they
prefer to use. Some learners might not know the tables by heart.
Refer them to the multiplication tables but also encourage them
to memorise the tables.
Solutions
1. a)
b)
c)
d)
200 ÷ 5 = (200 ÷ 10) × 2 = 20 × 2 = 40
320 ÷ 8 = (320 ÷ 10) ÷ 8 = 32 ÷ 8 × 10 =40
360 ÷ 6 = (360 ÷ 10) ÷ 6 = 36 ÷ 6 × 10 = 60
250 ÷ 5 = (250 ÷ 10) ÷ 5 = 25 ÷ 5 × 10 = 50
Mathematics Teacher’s Guide Grade 4
Math G4 TG.indb 223
TERM 2
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2012/09/14 5:36 PM
e) 120 ÷ 6 = (120 ÷ 10) ÷ 6 = 12 ÷ 6 × 10 = 20
f) 810 ÷ 9 = (810 ÷ 10) ÷ 9 = 81 ÷ 9 × 10 = 90
g) 480 ÷ 6 = (480 ÷ 10) ÷ 6 = 48 ÷ 6 × 10 = 80
h) 480 ÷ 6 = (480 ÷ 10) ÷ 6 = 48 ÷ 6 × 10 = 60
i) 480 ÷ 6 = (480 ÷ 10) ÷ 6 = 48 ÷ 6 × 10 = 60
j) 480 ÷ 6 = (480 ÷ 10) ÷ 6 = 48 ÷ 6 × 10 = 40
2. Discuss learners’ use of Nadia’s method with the class.
3. a) 180 ÷ 9 = (18 ÷ 9) × 10 = 20
b) 180 ÷ 9 = (18 ÷ 9) × 10 = 90
c) 180 ÷ 9 = (18 ÷ 9) × 10 = 90
d) 180 ÷ 9 = (18 ÷ 9) × 10 = 50
e) 180 ÷ 9 = (18 ÷ 9) × 10 = 90
f) 180 ÷ 9 = (18 ÷ 9) × 10 = 80
g) 180 ÷ 9 = (18 ÷ 9) × 10 = 80
h) 180 ÷ 9 = (18 ÷ 9) × 10 = 90
i) 180 ÷ 9 = (18 ÷ 9) × 10 = 90
j) 180 ÷ 9 = (18 ÷ 9) × 10 = 50
4. Answers will differ.
Activity 43.1
Learner’s Book page 173
The learners solve problems involving 3-digit multiples divided
by 1-digit numbers. Ask them to explore the cube arrangements in
the pictures. You could let them build the arrangements. Tell them
that they will now work with remainders. They work with small
2-digit numbers to develop the concept of remainders. They create
cube constructions to solve the problems. Tell them to write number
sentences for the problems. They might need the multiplication
facts to solve the 2-digit division problems. Help them understand
that, for example, for 27 ÷ 5 they should know that 25 is the closest
multiple of 5 and 5 goes into 27 five times with a remainder of 2.
Let them check the solutions by reversing the operations and adding
the remainders.
Solutions
1. a) 120 ÷ 2 = 60
b) 180 ÷ 6 = 30
c) 200 ÷ 5 = 40
d) 240 ÷ 6 = 40
e) 240 ÷ 8 = 30
f) 320 ÷ 4 = 80
g) 300 ÷ 6 = 50
h) 270 ÷ 3 = 90
i) 280 ÷ 4 = 70
j) 360 ÷ 9 = 40
2. a) 16 ÷ 3 = 5 remainder 1
b) 14 ÷ 4 = 3 remainder 2
c) 19 ÷ 6 = 3 remainder 1
d) 23 ÷ 5 = 4 remainder 3
e) 29 ÷ 3 = 9 9 remainder 2
3. a) 30 ÷ 7 = 4 remainder 2
b) 26 ÷ 6 =4 remainder 2
c) 27 ÷ 5 = 5 remainder 2
d) 38 ÷ 7 = 5 remainder 3
e) 44 ÷ 6 = 7 remainder 2
f) 46 ÷ 9 = 5 remainder 1
g) 35 ÷ 8 = 4 remainder 3
h) 30 ÷ 9 = 3 remainder 3
i) 65 ÷ 7 = 9 remainder 2
j) 76 ÷ 8 = 9 remainder 4
4. Learners check their solutions.
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Unit 44 Division with remainders
Mental Maths Learner’s Book page 174
The learners get more practice in working with remainders in
2-digit numbers. Allow them to use the multiplication tables,
counters or cubes if they are still dependent on these. Let them
check their solutions by multiplication.
Solutions
1. a)
c)
e)
g)
i)
12 ÷ 5 = 2 remainder 2
27 ÷ 6 = 4 remainder 3
36 ÷ 7 = 5 remainder 1
50 ÷ 7 = 7 remainder 1
85 ÷ 9 = 9 remainder 4
b)
d)
f)
h)
j)
17 ÷ 4 = 4 remainder 1
33 ÷ 6 = 5 remainder 3
47 ÷ 6 = 7 remainder 5
67 ÷ 8 = 8 remainder 3
76 ÷ 8 = 9 remainder 4
2. Learners use multiplication to check solutions.
Example: 8 × 9 + 4 = 72 + 4 = 76
Activity 44.1
Learner’s Book page 175
Ask learners to study the strategies to divide with dividends that will
leave remainders. They should understand that they have to find the
closest multiples of the divisors. Make sure that they understand the
strategy. Also help them understand how to check by multiplying
and adding to get the original dividend as the answer. They use the
given strategy to solve the problems.
Solutions
1. The learners solve division with 3-digit numbers without
remainders. Some of the learners would probably be able to
solve the problems mentally by identifying the multiples of the
divisors. The strategy below can be used for 3-digit number
division without remainders. The processes and checks are
shown for the first two problems.
a) 255 ÷ 5 = n
50 × 5 = 250
255 – 250 = 5
1 × 5 = 5
5–5=0
255 ÷ 5 = 51
Check: 51 × 5 = 255
b) 287 ÷ 7 = n
40 × 7 = 280
287 – 280 = 7
1 × 7 = 7
7–7=0
287 ÷ 7 = 41
Check: 41 × 7 = 287
c)
e)
g)
i)
248 ÷ 8 = 31
284 ÷ 4 = 71
369 ÷ 9 = 41
426 ÷ 6 = 71
d)
f)
h)
j)
306 ÷ 6 = 51
273 ÷ 3 = 91
357 ÷ 7 = 51
183 ÷ 3 = 61
Mathematics Teacher’s Guide Grade 4
Math G4 TG.indb 225
TERM 2
225
2012/09/14 5:36 PM
2. Ask learners to check each solution by multiplication. The
strategies are shown for the first two problems below. Learners
should realise that they have made a mistake when the
remainder is more than the divisor.
a) 157 ÷ 5 = n
30 × 5 = 150
157 – 150 = 7
1 × 5 = 5
7–5=2
157 ÷ 5 = 31 remainder 2
Check: 31 × 5 + 2 = 155 + 2
= 157
b) 204 ÷ 8 = n
20 × 8 = 160
204 – 160 = 44
5 × 8 = 40 44 – 40 = 4
204 ÷ 8 = 25 remainder 4
Check: 25 × 8 + 4 = 200 + 4
= 204
c)
d)
e)
f)
g)
h)
i)
j)
307 ÷ 5 = 61 remainder 2
238 ÷ 5 = 47 remainder 2
344 ÷ 6 = 57 remainder 2
246 ÷ 9 = 25 remainder 1
335 ÷ 8 = 41 remainder 7
330 ÷ 9 = 36 remainder 6
265 ÷ 7 = 37 remainder 6
179 ÷ 8 = 22 remainder 3
Unit 45 Division with 3-digit numbers and
remainders
Mental Maths Learner’s Book page 175
Learners play a division game.
Activity 45.1
Learner’s Book page 175
1. The learners solve problems in which they have to determine
the missing quotients, dividends and divisors. In some 3-digit
number division problems, learners have to perform inverse
operations to calculate the solutions.
2. Learners use inverse operations to check solutions to 3-digit
number division problems. They use the distributive property to
multiply effectively.
Solutions
1. a) 250 ÷ 10 = 25
b) 810 ÷ n = 9
810 ÷ 90 = 9
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Mathematics Teacher’s Guide Grade 4
810 ÷ 9 = 90
TERM 2
2012/09/14 5:36 PM
c) n ÷ 80 = 4
d) 488 ÷ 8 = 61
e) 364 ÷ 9 = 40 remainder 4
f) 244 ÷ n = 61
61 × 4 = 244
244 ÷ 4 = 61
g) n ÷ 40 = 50
200 ÷ 40 = 50
h) 217 ÷ 7 = 31
i) 13 ÷ n = 2 remainder 3
13 ÷ 5 = 2 remainder 3
j) 246 ÷ 4 = 61 remainder 2
40 × 50 = 200
2 × 5 + 3 = 13
2. a) 489 ÷ 4 = 122 remainder 1
122 × 4 + 1 = 488 + 1
= 489
b) 305 ÷ 7 = 43 remainder 4
43 × 7 + 4 = (40 × 7) + (3 × 7) + 4
= 280 + 21 + 4
= 305
c) 448 ÷ 6 = 74 remainder 4
74 × 6 + 4 = (70 × 6) + (4 × 6) + 4
= 420 + 24 + 4
= 448
d) 289 ÷ 5 = 57 remainder 4
57 × 5 + 4 = (50 × 5) + (7 × 5) + 4
= 250 + 35 + 4
= 289
e) 669 ÷ 7 = 95 remainder 4
95 × 7 + 4 = (90 × 7) + (5 × 7) + 4
= 630 + 35 + 4
= 669
Unit 46 Problem-solving with division
Mental Maths Learner’s Book page 176
1. The learners should identify that they apply inverse
operations in every second diagram.
Learners determine the number of equal groups and the
remainders in numbers.
a)
7
×8
+2
58
b)
58
–2
÷7
8
c)
25
×5
+4
129
Mathematics Teacher’s Guide Grade 4
Math G4 TG.indb 227
TERM 2
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2012/09/14 5:36 PM
d)
129
–4
÷5
25
e)
12
× 20
+7
247
f)
247
–7
÷ 20
12
2. a)
b)
c)
d)
e)
33 has 6 groups of 5 remainder 3
649 has 81 groups of 8 remainder 1
810 has 9 groups of 90 remainder 0
460 has 23 groups of 20 remainder 0
490 has 70 groups of 7 remainder 0
Activity 46.1
Learner’s Book page 176
The learners solve problems in context. Allow them to use their own
strategies. Here are some strategies that they might use. In questions
2 and 3, they might use trial and improvement to determine the
number of equal groups if there is no divisor. They should realise
that the divisor cannot be an even number. Some learners might
identify 155 as a multiple of 5. In question 3 the number of bags of
potatoes can be any number. The problem does not state that there
should be no remainders. They should however be realistic because
bags are normally not be filled with 2, 3, 4 or 5 potatoes.
Solutions
1. 268 ÷ 8 = n
30 × 8 = 240
268 – 240 = 28
3 × 8 = 24 28 – 24 = 4
268 ÷ 8 = 33 remainder 4
The greengrocer can fill 33 bags and 4 tomatoes will be
left over.
2. 155 ÷ n = n
155 ÷ 3 = 51 remainder 2
(not possible)
155 ÷ 5 = 31
The grocer puts 31 oranges in one bag so that he has 5 bags. He
can also put 5 oranges in a bag so that he has 31 bags.
3. 283 ÷ n = n
40 × 7 = 280
30 × 8 = 240
5 × 8 = 40
30 × 9 = 270
1×9=9
28 × 10 = 280
228
Math G4 TG.indb 228
283 – 280 = 3 H
e can use 40 bags filled
with 7 potatoes each.
283 – 240 = 43
43 – 40 = 3 He can use 35 bags filled
with 8 potatoes each.
283 – 270 = 13
13 – 9 = 4 He can use 31 bags filled
with 9 potatoes each.
283 – 280 = 3 He can use 28 bags filled
with 10 potatoes each, and
so on.
Mathematics Teacher’s Guide Grade 4
TERM 2
2012/09/14 5:36 PM
4. 127 ÷ 8 = n
10 × 8 = 80 127 – 80 = 47
5 × 8 = 40 47 – 40 = 7
127 ÷ 8 = 15 remainder 7
The grocer needs 15 boxes.
5. 235 ÷ 6 = n
30 × 6 = 180
235 – 180 = 55
9 × 6 = 54 55 – 54 = 1
235 ÷ 6 = 39 remainder 1
The greengrocer can fill 39 bags and 1 squash remains.
Assessment task 7: whole numbers (division)
The learners perform this assessment task at the end of Term 2,
week 9. They demonstrate knowledge of division with 2- and
3-digit numbers and display knowledge of division by 10 and
multiples of 10. The learners show understanding of division with
remainders and use the quotients of problems to do multiplication
to check the reasonableness of solutions. They further solve
division problems in context.
Mathematics Teacher’s Guide Grade 4
Math G4 TG.indb 229
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Assessment task 7 Whole numbers (division)
1. Complete each number sentence.
a) 21 ÷ 7 = n
b) 45 ÷ 9 = n
c) 49 ÷ 7 = n
d) 32 ÷ 4 = n
e) 64 ÷ 8 = n(5)
Input
2. Copy
and complete each flowOutput
diagram.
Input
Output
a)
20
20
200
200
250
250
22 000
000
÷
÷ 10
10
22 550
550
22 500
500
b)
Input
Input
Output
Output
420
420
340
340
570
570
610
610
÷
÷ 10
10
860
860
11 010
010
(12)
3. Use short cuts to solve the problems.
a) 420 ÷ 20 = n
b) 320 ÷ 80 = n
c) 360 ÷ 60 = n
d) 280 ÷ 70 = n
e) 270 ÷ 30 = n(5)
4. Solve the problems.
a) 21 ÷ 6 = n
b) 26 ÷ 3 = n
c) 16 ÷ 3 = n(3)
5. Solve these problems.
a) 18 ÷ 9 = n
b) 20 ÷ 9 = n
c) 28 ÷ 7 = n
d) 30 ÷ 7 = n
e) 40 ÷ 8 = n
f) 45 ÷ 8 = n(6)
230
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Mathematics Teacher’s Guide Grade 4
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2012/09/14 5:36 PM
6. Use your own method to solve these problems.
a) 408 ÷ 4 = n
b) 360 ÷ 9 = n
c) 525 ÷ 5 = n
d) 427 ÷ 7 = n
e) 168 ÷ 8 = n(5)
7. Use multiplication to show that these number sentences
are true.
a) 205 ÷ 6 = 34 remainder 1
b) 357 ÷ 5 = 71 remainder 2
c) 169 ÷ 4 = 42 remainder 1
d) 247 ÷ 7 = 30 remainder 7
(4)
8. Solve these problems. Write a number sentence for
each problem.
a) How many boxes with 6 eggs each can be filled
from 246 eggs?
(3)
b) The farmer picked 480 avocado pears. How many
bags can she fill with 10 avocado pears each?
(2)
Total [45]
Mathematics Teacher’s Guide Grade 4
Math G4 TG.indb 231
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Assessment task 7 Whole numbers (division)
1. a)
b)
c)
d)
e)
2. a)
21 ÷ 7 = 3
45 ÷ 9 = 5
49 ÷ 7 = 7
32 ÷ 4 = 8
64 ÷ 8 = 8
b)
(5)
20
20
200
200
250
250
22 000
000
22
20
20
25
25
200
200
÷
÷ 10
10
22 550
550
22 500
500
255
255
250
250
420
420
340
340
42
42
34
34
570
570
610
610
Solutions
57
57
61
61
÷
÷ 10
10
860
86
860
86
11 010
101
010
101
(12)
3. The learners use shortcuts to divide 3-digit numbers by
multiples of 10. They apply knowledge of basic division facts.
a) 420 ÷ 20 = 21
b) 320 ÷ 80 = 4
c) 360 ÷ 60 = 6
d) 280 ÷ 70 = 4
e) 270 ÷ 30 = 9
(5)
4. a) 21 ÷ 6 = 3 remainder 3
b) 26 ÷ 3 = 8 remainder 2
c) 16 ÷ 3 = 5 remainder 1
(3)
18 ÷ 9 = 2
20 ÷ 9 = 2 remainder 2
28 ÷ 7 = 4
30 ÷ 7 = 4 remainder 2
40 ÷ 8 = 5
45 ÷ 8 = 5 remainder 5
(6)
5. a)
b)
c)
d)
d)
e)
232
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Mathematics Teacher’s Guide Grade 4
TERM 2
2012/09/14 5:36 PM
6. The learners use their own strategies to solve problems
involving division with 3-digit numbers without remainders.
a) 408 ÷ 4 = 102
b) 360 ÷ 9 = 40
c) 525 ÷ 5 = 505
d) 427 ÷ 7 = 61
e) 168 ÷ 8 = 21
(5)
7. The learners use inverse operations to check the solutions
to problems (dividing 3-digit numbers with remainders).
They use the distributive property for multiplication.
a) 205 ÷ 6 = 34 remainder 1
34 × 6 + 1 = (30 × 6) + (4 × 6) + 1
= 180 + 24 + 1
= 205
b) 357 ÷ 5 = 71 remainder 2
71 × 5 + 2 = (70 × 5) + (1 × 5) + 2
= 350 + 7
= 357
c) 169 ÷ 4 = 42 remainder 1
42 × 4 + 1 = (40 × 4) + (2 × 4) + 1
= 160 + 9
= 169
d) 247 ÷ 7 = 35 remainder 2
35 × 7 + 7 = (30 × 7) + (5 × 7) + 2
= 210 + 37
= 247
(4)
8. They solve division problems in context. They should know
that they have to write number sentences before calculating.
They use their own strategies.
a) 246 ÷ 6 = n
240 ÷ 6 = 40
6 ÷ 6 = 1
41 egg boxes can be filled.
(3)
b) 480 ÷ 10 = 48
48 bags can be filled.
Mathematics Teacher’s Guide Grade 4
Math G4 TG.indb 233
(2)
Total [45]
TERM 2
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TERM
Capacity and volume
Unit 22 Investigate polygons
Unit 1
What do you remember?
Unit 2
Measure capacity and volume
Unit 3
Understand volume and capacity
Unit 4
Estimate and round off
Collect, organise and present data
Unit 5
Calculations with litres
and millilitres
Unit 24 Use tally marks
Unit 6
Calculate capacity with fractions
Revision
Common fractions
Investigation
Unit 23 Patterns and pictures
with 2-D shapes
Unit 25 Show data on graphs
Unit 26 Explain data
Unit 27 More graphs
Project
Unit 7
Recognise fraction parts
Unit 8
Fractions of whole numbers
Numeric patterns
Unit 9
Equivalent fractions
Unit 28 Patterns in number grids
Unit 10 Equal sharing and problemsolving
Unit 11 Count, order and calculate with
fractions
Whole numbers: adding
and subtracting
Unit 12 Rules for operations
Unit 13 Count, compare, represent
numbers and place values
Unit 14 Addition and subtraction facts
Investigation
Unit 29 Finding rules
Unit 30 Rules for number patterns
Whole numbers: addition
and subtraction
Unit 31 Quick calculations
Unit 32 Count, order and compare
numbers and place value
Unit 33 Problem-solve with
whole numbers
Whole numbers: multiplication
Unit 15 Problem-solving
Unit 34 Multiplication strategies
Unit 16 D
ouble, halve and round off for
estimations and calculations
Unit 35 Basic multiplication facts
Unit 17 Different strategies for
calculations
View objects
Unit 18 Side views and top views
Investigation
Unit 19 Side views and plan views
3
Unit 36 Round off and solve problems
Number sentences
Unit 37 Write number sentences
Unit 38 Balance and inspect
number sentences
Unit 39 Equations and problem-solving
Properties of 2-D shapes
Transformations
Unit 20 Sort 2-D shapes
Unit 40 Make new shapes
Unit 21 Investigate circles
Unit 41 Tangrams
Revision
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Capacity and volume
Unit 1
Learner’s Book page 178
What do you remember?
Introduction
Tell learners the day before they start with this unit to bring
containers of different shapes and sizes to school. Do many
experiments with different containers to measure volumes.
Examples include finding out how many teaspoons are there in a
dessert spoon and how many cups can be filled from one-litre bottle.
Make sure you have lots of different shapes and sizes of containers
in the classroom for learners to work with (examples include short,
fat bottles and tall thin bottles).
Show learners a 5-ml syringe and fill it with water. Ask the class if
the liquid in the syringe would fill a teaspoon, dessert spoon and so
on. Let them find out by doing the experiment if their estimations
are correct.
Use food dye to colour the water that you use in these experiments
so that it is easier to see the level of the liquid. For example, pour
blue and yellow dye together – the liquid will be green.
Check that learners are able to estimate a number of litres that is
sensible – they should not just guess, but must be able to reason
about why they estimated as they did (for example, the water will
fill the bottle about halfway, so it must be about 500 ml).
Mental Maths Learner’s Book page 178
1. Volume and capacity both used to measure the size of 3-D
objects or liquids – both are measured in millilitres, litres,
kilolitres, cubic centimetres and cubic metres:
• Capacity generally refers to the amount that a container
will hold when has been filled.
• Volume typically refers to the amount of space a liquid
takes up in a container.
So, for example, the capacity of a one-litre bottle that is half
full of milk is 1 ℓ and the volume of the milk in the bottle is
500 ml (or half a litre).
2. Examples include cool drink, milk and medicine.
3. If containers are filled completely (to capacity), it will be
difficult not to spill the liquids when the container is opened.
4. Learners count in 250s and 500s and they look for the
relationship between these numbers and counting in 25s and
50s. You could ask questions such as: How many 250s are
there in 750?, How many 500s are there in 2 000?
236
Math G4 TG.indb 236
Mathematics Teacher’s Guide Grade 4
TERM 3
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a) 250; 500; 750; 1 000; 1 250; 1 500; 1 750; 2 000
b) six 250s
c) 500; 1 000; 1 500; 2 000; 2 500; 3 000; 3 500; 4 000;
4 500; 5 000
d) ten 500s
Activity 1.1
Learner’s Book page 178
1. a) teaspoon: 5 ml
c) cup: 250 ml
e) dessert spoon: 20 ml
b) tin (cool drink): 375 ml
d) litre bottle: 1 ℓ
2. Help learners practise rounding off. Remind them to round off
up or down depending on whether the volume is above or below
the halfway level in the container. They should also recognise
that if there is less than half a litre (500 ml) in a container, it
does not make sense to round off to 0 ml – this would mean that
there is no liquid in the container. We use smaller measuring
units to measure quantities that are less than a litre.
In Grade 4 learners do not work with decimals. But they will see
decimal fractions such as 0,5 or 1,5 on some packaging. Explain
that 0,5 is another way of writing the fraction 12 , and 0,5 is the
same as 12 or 500 ml.
Unit 2
Measure capacity and volume
Mental Maths Learner’s Book page 179
By Grade 4, most learners should understand it when you say
that one container holds more than another container. The
concept of capacity or volume for solid objects might not be as
easy to understand. The apparent volumes of solid objects might
be misleading; a method to compare such volumes could also be
difficult.
Solutions
1–5. Practical work.
Activity 2.1
Learner’s Book page 179
1. Make sure learners understand the difference between volume
and capacity.
Container
Capacity of
container
Volume of water in container
Green bottle
Cooldrink bottle
Milk bottle
Bottle with blue cap
10 ℓ
2ℓ
1ℓ
5ℓ
The bottle looks full (10 ℓ).
1ℓ
1ℓ
about 4 ℓ
Mathematics Teacher’s Guide Grade 4
Math G4 TG.indb 237
TERM 3
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2012/09/14 5:36 PM
2. milk bottle, cooldrink bottle, bottle with blue cap, green bottle
3. Smallest volume of water: the milk bottle and the colddrink
bottle – both have about 1 ℓ of liquid in them.
4. the green bottle
Unit 3
Understand volume and capacity
Mental Maths Learner’s Book page 180
The learners would probably know that 1 litre (ℓ) = 1 000 millilitre
(ml). Ask learners to name products that are packaged with
capacities of less than and more than a litre.
Solutions
1. a) 112 ℓ
b) 1 ℓ
c) 2 ℓ
d)
3
4
ℓ
2. a) 4 cups
c) 250 ml
b) 1 ℓ = 1 000 ml
4 × 250 ml = 1 000 ml
d) 125 ml
3. a) 4 bottles
b) 50 × 10 = 500 ml
500 × 4 = 2 000 ml
2 000 ml = 2 ℓ
4. a) 2 ℓ = 2 000 ml
c) 7 ℓ = 7 000 ml
b) 4 ℓ = 4 000 ml
d) 10 ℓ = 10 000 ml
Activity 3.1
Learner’s Book page 180
Learners can complete some of the questions for homework.
Solutions
1. a)
c)
e)
g)
i)
1 ℓ = 1 000 ml
5 ℓ = 5 000 ml
314 ℓ = 3 250 ml
4 000 ml = 4 ℓ
1 500 ml = 1,5 ℓ
b)
d)
f)
h)
j)
3 ℓ = 3 000 ml
212 ℓ = 2 500 ml
2 000 ml = 2 ℓ
10 000 ml = 10 ℓ
6 750 ml = 643 ℓ
2. a) 250 ml, 300 ml, 750 ml, 1 ℓ, 1,5 ℓ, 243 ℓ
b) 250 ml
300 ml
750 ml
1 ℓ = 1 000 ml
112 ℓ = 1 500 ml
243 ℓ = 2 750 ml
c) (250 + 300 + 750 + 1 000 + 1 500 + 2 750) ml = 6 550 ml
d) 6 ℓ and 550 ml
3. a) 500 ml + 500 ml = 1 000 ml = 1 ℓ
b) 1 ℓ + 250 ml = 1 000 ml + 250 ml = 1 250 ml
238
Math G4 TG.indb 238
Mathematics Teacher’s Guide Grade 4
TERM 3
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c) 500 ml + 500 ml = 1 ℓ; so, 500 ml = 12 ℓ
d) 250 ml + 250 ml + 250 ml + 250 ml = 1 000 ml;
so, 250 ml = 14 ℓ
e) 250 ml + 250 ml + 250 ml + 250 ml + 250 ml + 250 ml
+ 250 ml + 250 ml = 2 000 ml
2 000 ml ÷ 250 ml = 8
f) 2 ℓ – 250 ml = 1 750 ml
4. a) 112 ℓ or 1 500 ml
c) 2,5 ℓ or 2 500 ml
e) 750 ml
b) 1 750 ml or 143 ℓ
d) 200 ml
5. a) cooldrink: 12 × 300 = 3 600 ml
b) 12 × R 6,50 = R78,00
6. a) 2 000 ml + 625 ml = 2 625 ml or 2,625 ℓ
b) 750 ml – 392 ml = 358 ml
c) 2 000 ml – 625 ml – 392 ml = 983 ml
7. a)
b)
c)
d)
Unit 4
500 ml ÷ 50 ml = 10 glasses
10 × 200 ml = 2 000 ml
15 × 200 ml = 3 000 ml or 3 ℓ
15 × 50 ml = 750 ml
Estimate and round off
Mental Maths Learner’s Book page 182
The learners compare and estimate litre and millilitre readings on
containers. Explain the concepts rounding up and rounding down
to them in the context of volume and capacity. They should
understand that approximately means more or less, about or
almost.
Solutions
1. a) 4 ℓ
b) 214 ℓ
c) closer to 2 ℓ
2. a) 500 ml
b) 200 ml
c) The level of the liquid is exactly between 200 ml and
300 ml.
3. a)
c)
e)
g)
1 ℓ 250 ml ≈ 1 ℓ
4 ℓ 499 ml ≈ 4 ℓ
365 ml ≈ 0 ℓ
1ℓ 999 ml ≈ 2 ℓ
b)
d)
f)
h)
2 ℓ 500 ml ≈ 3 ℓ
600 ml ≈ 1 ℓ
2 ℓ 16 ml ≈ 2 ℓ
16 ℓ 450 ml ≈ 16 ℓ
Activity 4.1
1. a) 7 ℓ
e) 1 ℓ
Learner’s Book page 183
b) 4 ℓ
f) 2 ℓ
c) 1 ℓ
g) 5 ℓ
Mathematics Teacher’s Guide Grade 4
Math G4 TG.indb 239
d) 2 ℓ
h) 2 ℓ
TERM 3
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2012/09/14 5:36 PM
Remind learners to divide by 1 000 to convert millilitres to
litres.
a) 7 000 ml ÷ 1 000 = 7 ℓ
b) 3 654 ml ÷ 1 000 = 3 ℓ 654 ml ≈ 4 ℓ
c) 999 ml ÷ 1 000 = 0 ℓ 999 ml ≈ 1 ℓ
d) 2 020 ml ÷ 1 000 = 2 ℓ 20 ml ≈ 2 ℓ
e) 1 007 ml ÷ 1 000 = 1 ℓ 7 ml ≈ 1 ℓ
f) 1 907 ml ÷ 1 000 = 1 ℓ 907 ml ≈ 2 ℓ
g) 4 500 ml ÷ 1 000 = 4 ℓ 500 ml ≈ 5 ℓ
h) 1 616 ml ÷ 1 000 = 1 ℓ 616 ml ≈ 2 ℓ
2. Some learners might be able to perform the calculations
mentally but you should insist that they explain their thinking
processes. They can use strategies such as building up and
breaking down numbers and number properties such as the
commutative and associative properties to calculate smarter.
They break up 8 into smaller factors to solve the division
problem. Remind them to round off the solutions to the nearest
500 ml.
a) 2 ℓ 500 ml + (400 ml + 600 ml)
= 2 ℓ 500 ml + 1 000 ml
= 3 ℓ 500 ml
3 ℓ 500 ml ≈ 3 500 ml
b) 1 ℓ 850 ml + 150 ml + 400 ml + 150 ml
= 2 ℓ + 550 ml
= 2 550 ml
2 550 ml ≈ 2 500 ml
c) 2 ℓ – 750 ml = 2 000 – 750
= 1 250 ml
1 250 ml ≈ 1 000 ml
d)
5 ℓ ÷ 8 = 5 000 ÷ 4 ÷ 2
1 000 ÷ 4 = 250
250 × 5 = (200 × 5) + (50 × 5)
= 1 000 + 250
= 1 250 ÷ 2
= 625 ml
625 ml ≈ 500 ml
3. a) 112 = 12 + 12 + 12
Each one gets 12 ℓ of the fruit juice.
b) 1 whole = 13 + 13 + 13
Capacity of the bottle:
350 + 350 + 350 = 900 + 150
= 1 050 ml
c) If 100 ml = 52 , then 15 = 50 ml
3
= 50 × 3
5
= 150 ml
The capacity of the bottle is: 150 ml + 100 ml = 250 ml
240
Math G4 TG.indb 240
Mathematics Teacher’s Guide Grade 4
TERM 3
2012/09/14 5:36 PM
d) 2 : 5 is 2 parts juice nectar and 5 parts water.
If you mix 2 ℓ of juice nectar with 5 ℓ of water you will have
7 ℓ of juice.
4 : 10 is 4 parts juice nectar and 10 parts water:
2 × 2 = 4 and 2 × 5 = 10.
If you mix 4 ℓ of juice nectar with 10 ℓ of water, you will
have 14 ℓ of juice.
Unit 5
Calculations with litres and millilitres
Mental Maths Learner’s Book page 184
The learners will work with volume calculations that involve
the four basic operations. They solve contextual problems
(some problems may seem non-contextual, but as they involve
millilitres, so they are problems in the context of measurement).
Solutions
1. a)
b)
c)
d)
5 ℓ 529 ml or 5,529 ℓ or 5 529 ml
2 ℓ 544 ml or 2,544 ℓ or 2 544 ml
1 ℓ 345 ml or 1,345 ℓ or 1 345 ml
3 359 ml or 3 ℓ 357 ml or 3,357 ℓ
2. a) 8 672 ml
c) 1 750 ml
b) 2 151 ml
d) 3 000 ml
3. a) 250 ml + 620 ml + 330 ml + 1 000 ml = 2 200 ml
3 000 ml – 2 200 ml = 800 ml of lemonade must be
added.
b) 2 000 ml – 1 280 ml = 1 720 ml
= 1 ℓ 720 ml
Activity 5.1
Learner’s Book page 185
Revise multiplication tables for at least 10 minutes before starting
this activity. Revise multiplication of 2-digit numbers with 1-digit
numbers, and 2-digit numbers with 2-digit numbers. Do at least four
calculations on the board before learners do the exercise below.
Solutions
1. a) 25 ml
c) 2 ℓ 230 ml
b) 25 ℓ
d) 1 ℓ 750 ml
2. a) 112 ℓ = 1,5 ℓ = 1 500 ml
1 500 × 7 = 10 500 ml = 10 ℓ 500 ml
b)
52
× 12
104
+ 520
624 ℓ
Mathematics Teacher’s Guide Grade 4
Math G4 TG.indb 241
TERM 3
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2012/09/14 5:36 PM
Activity 5.2
Learner’s Book page 185
1. a) 196 ÷ 7 = 28 ℓ
c) 996 ml ÷ 6 = 166 ml
b) 1 872 ml ÷ 6 = 312 ml
d) 12 144 ml ÷ 4 =
3 036 ml
= 3 ℓ 36 ml
2. a) 2 592 km ÷ 8 = 324 ℓ
He will have to buy 324 litres of petrol.
b) 324 ℓ × R10 = R3 240
3. 36 ℓ 240 ml ÷ 12
= 3 ℓ 20 ml
Each lamb drinks 3 ℓ 20 ml per day.
Unit 6
Calculate capacity with fractions
Mental Maths Learner’s Book page 186
Learners use their knowledge of fractions to add and subtract
capacities. For some problems, they have to convert different
units to the same unit.
Solutions
1. 643 ℓ
5.
414
ℓ
2. 754 ℓ
6.
112
ℓ
3. 582 ℓ
4. 2 ℓ
7. 6 000 ml
8. 4 ℓ 250 ml
Activity 6.1
Learner’s Book page 186
This activity can be done for homework.
Solutions
1. 212 ℓ + 114 ℓ = 343 ℓ
2. 2 500 ml
3. a) 250 ml + 750 ml = 1 000 ml or 1 ℓ
b) 500 ml
4. 1 250 ml
5. a) 2 ℓ 500 ml
b) 25 ℓ
Revision
c) 25 000 ml
Learner’s Book page 187
1. a) 500 ml
b) 330 ml
c) 10 ml
2. a) 2 000 ml ÷ 250 ml = 8 cups
b) 5 000 ml ÷ 500 ml = 10 bottles
c) 250 ml ÷ 5 ml = 50 teaspoons
d) 212 ℓ = 2 500 ml
3. a) 3 ℓ = 3 000 ml
b) 1 50 ml = 112 ℓ
c) 2 ℓ 15 ml = 2 015 ml
4. a) 500 ml + 250 ml + 2 ℓ = 2 750 ml
b) 5 ℓ - 800 ml + 3 ℓ 27 ml = 7 ℓ 227 ml
c) 7 ℓ 227 ml ≈ 7 ℓ
d) 16 ℓ × 9 = 144 ℓ
5. a) 192 ÷ 8 = 24 bottles
b) 24 × 500 ml = 12 000 ml
= 12 ℓ
242
Math G4 TG.indb 242
Mathematics Teacher’s Guide Grade 4
TERM 3
2012/09/14 5:36 PM
Assessment task 1 Measurement
1. A water jug with a capacity of 250 ml is filled to 175 ml.
a) What is the capacity of the beaker?
b) What is the volume of the water?
c) How much more water can be poured into the jug?
d) How many beakers will you have to fill if you want to
use 2 ℓ of water? (4)
2. Convert the following measurements as shown.
a) 1 ℓ = n ml
b) 512 ℓ = n ml
c) 4 000 ml = n ℓ
d) 3 250 ml = n ℓ
e) 43 ℓ = n ml
f) 10 500 ml = n ℓ n ml
3. What fraction is 500 ml of one litre? Mathematics Teacher’s Guide Grade 4
Math G4 TG.indb 243
(6)
(2)
Total [12]
TERM 3
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2012/09/14 5:36 PM
Assessment task 1 Measurement
Solutions
1. a) 250 ml
b) 175 ml
c) 75 ml
d) 8 × 250 = 2 000 ml
(4)
2. a) 1 ℓ = 1 000 ml
b) 512 ℓ = 5 500 ml
c) 4 000 ml = 4 ℓ
d) 3 250 ml = 314 ℓ
e)
3
4
ℓ = 750 ml
f) 10 500 ml = 10 ℓ 500 ml (6)
3. 12 (2)
Total [12]
244
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Mathematics Teacher’s Guide Grade 4
TERM 3
2012/09/14 5:36 PM
Common fractions
Unit 7
Recognise fraction parts
Learners revised the fraction concepts they learnt in Grade 3 this
week. The fraction names and symbols learners know have been
extended to include sevenths, ninths and tenths. In this section,
learners will extend their knowledge and develop procedural and
conceptual understanding of common fractions.
Mental Maths Learner’s Book page 188
Learners play Fraction dominoes.
2
7
4
5
4
6
Fraction dominoes is a game for two, three, four or more players.
It is played like ordinary dominoes. Learners shuffle the cards.
If there are four players, each player takes seven cards. The
player with the card that has 72 as a fraction symbol starts. The
next player can add the diagrammatic representation for 72 on
the left or the fraction symbol for the picture of 54 . A player who
does not have one of these cards knocks and skips a round. The
winner is the player who plays all his or her cards first. The
rest of the players finish the game. The game allows learners to
identify fraction symbols and their diagrammatic representations.
Learners count the fraction parts while they play. Encourage the
learners to name the fractions, for example, two-sevenths and not
two over seven.
Activity 7.1
Learner’s Book page 188
1. Introduce learners to the parts of a fraction – the numerator
(number of shaded parts) and denominator (number of equal
parts). Ask the learners to identify the number of shaded parts
in the diagrams and to name the fractions. Encourage them
to use the formal language and write down, for example, two
equal parts of three equal parts = two thirds or 23 .
Guide the learners so that they understand that 15 + 15 + 15 + 15 + 15
= 55 or 1 whole.
2. Learners should see that there are wholes and fraction parts in
some of the diagrams. Introduce the learners to the different
types of fraction: 23 is a proper fraction, 53 is an improper fraction
and 123 is a mixed fraction.
Solutions
1. a) four-fifths or 54
c) four-sixths or 64
b) one-third or 13
d) three-ninths or 93
Mathematics Teacher’s Guide Grade 4
Math G4 TG.indb 245
TERM 3
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2012/09/14 5:36 PM
e) five-eights or 85
f) three-quarters or 43
g) three-fifths or 53
or 105
b)
e) 214 or 94
f)
2. a)
Unit 8
1
2
7
10
17
10
c) 123 or 53
or 1107
g)
d)
7
9
4
7
Fractions of whole numbers
Mental Maths Learner’s Book page 189
1, 2. Learners use the different fruit to find different fractions
of each kind of fruit. You can make an enlargement of the
pictures or draw them on the board. The learners draw
circles around the objects to show the fraction parts. Write
down the number sentences as they illustrate the fraction
parts, for example, 12 of 6 = 3.
3. Ask the learners to write the solutions on their Mental maths
grids. The focus is on identifying equivalent fractions. They
should observe, for example, that 15 of the 10 strawberries
is the same of 102 of 10 strawberries. Lead a class discussion
about these fractions later. Ask the learners to identify all
the fractions that give the same number of strawberries.
They should notice the equivalent fractions.
Solutions
1. a)
b)
c)
2. a)
b)
c)
d)
e)
f)
g)
h)
3. a)
b)
c)
d)
e)
246
Math G4 TG.indb 246
1
of 6 apples is 3 apples
2
1
of 6 apples is 2 apples
3
1
of 6 apples is 1 apple
6
1
of 8 oranges is 4 oranges
2
1
of 8 oranges is 2 oranges
4
3
of 8 oranges is 6 oranges
4
1
of 8 oranges is 1 orange
8
2
of 8 oranges is 2 oranges
8
4
of 8 oranges is 4 oranges
8
5
of 8 oranges is 5 oranges
8
6
of 8 oranges is 6 oranges
8
1
of 10 strawberries is 5
2
5
of 10 strawberries is 5 strawberries
10
1
of 10 strawberries is the same as 102 strawberries
5
2
of 10 strawberries is 2 strawberries
10
2
of 10 strawberries is 4 strawberries
5
Mathematics Teacher’s Guide Grade 4
TERM 3
2012/09/14 5:36 PM
f)
g)
h)
i)
j)
4
of 10 strawberries is 4 strawberries
10
3
of 10 strawberries is 6 strawberries
5
6
of 10 strawberries is 6 strawberries
10
4
of 10 strawberries is 8 strawberries
5
8
of 10 strawberries is 8 strawberries
10
Activity 8.1
Learner’s Book page 190
1. Ask the learners to draw six apples (like the ones in Mental
maths) and shade the fraction parts indicated. Learners could
also look for patterns. They should notice that 13 = 2 apples and
2
= 4 apples ( 13 of the apples) is double 13 of the apples. They
3
should also notice that 23 of the apples = 4 and 64 of the apples
= 4. Ask learners to explain why this is so. Show them the
calculations that are related to their drawings.
b) 64 of 6 = 6 ÷ 6 × 4
a) 23 of 6 = 6 ÷ 3 × 2
= 4
=4
d) 62 of 6 = 6 ÷ 6 × 2
c) 13 of 6 = 6 ÷ 3 × 1
= 2
=2
f) 63 of 6 = 6 ÷ 6 × 3
e) 12 of 6 = 6 ÷ 2 × 1
= 3
=3
2. Learners should notice equivalent fractions for halves, quarters
and eighths. Ask them to write down calculations related to the
drawings.
a) 12 of 8 = 8 ÷ 2 × 1 b) 24 of 8 = 8 ÷ 4 × 2 c) 84 of 8 = 8 ÷ 8 × 4
= 4
= 4
=4
e) 82 of 8 = 8 ÷ 8 × 2
d) 14 of 8 = 8 ÷ 4 × 1
= 2
=2
g) 86 of 8 = 8 ÷ 8 × 6
f) 43 of 8 = 8 ÷ 4 × 3
= 6
=6
3. The learners explain their observations – either in writing or
verbally.
4. Learners look at the drawings and they represent different
fractions that are related to either the whole or a fraction part of
the objects in the pictures. For example, in a) there are 6 carrots
in the whole bunch, so they have to draw 3 carrots to represent
1
of the carrots. The activity requires more intensive thinking
2
and reasoning than previous exercises. Let them work in their
groups.
Learners must not make fancy drawings; they should focus on
the mathematics. Let them struggle with the problems before
you give them any help. Share the calculations with later.
They use their knowledge of repeated addition, multiplication,
division and doubling.
Mathematics Teacher’s Guide Grade 4
Math G4 TG.indb 247
TERM 3
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2012/09/14 5:36 PM
a) If the whole bunch of carrots is 6, they draw 3 carrots.
1 whole = 6
12 of 6 = 6 ÷ 2 × 1
= 3 carrots
2
b) If 4 of all the oranges is 4, they draw 8 oranges.
24 or 12 = 4 oranges
4 + 4 = 8
or
4 × 2 = 8 oranges
c) If 14 of the sandwiches left on a tray is 2, they draw
8 sandwiches.
14 of 8 = 2 sandwiches
14 + 14 + 14 + 14 = 44
2 + 2 + 2 + 2 = 8
or
4 × 2 = 8 sandwiches
d) If all the sweets are 9, they draw 3 sweets.
or
9÷3=3
13 of 9 = 9 ÷ 3 × 1
= 3 sweets
e) If all the tomatoes are 12, they draw 10 tomatoes.
5
or
of 12 = 12 ÷ 6 × 5
61 of 12 = 2
6
62 of 12 = 4
= 10
64 of 12 = 8
65 of 12 = 2 + 8 = 10 tomatoes
Unit 9
Equivalent fractions
Mental Maths Learner’s Book page 191
In this lesson you could formally introduce equivalent fractions.
Learners study the fractions in the examples and they should
notice the fractions that are of equal size – equivalent fractions.
You can also record the learners’ explanations, write them on
strips of paper and put them on a notice board. You could refer
learners to the fraction wall they worked with earlier. You could
also make a simple drawing to enhance the concept of equivalent
fractions, for example:
2 of 6 equal parts = 62
1 of 3 equal parts = 62 = 13
Ask the learners to make similar drawings to represent 14 and 82 ,
for example.
Learners can look for more equivalent fractions in the examples.
248
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Mathematics Teacher’s Guide Grade 4
TERM 3
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1. The fractions on the number line are examples of fractions in
measurement form.
2. The learners study the fraction parts in the cube
arrangements. They have to count the number of cubes
to find the denominator in each diagram and then decide
how many of the green cubes are equivalent to the number
of colour cubes. You should help them understand that 1
cube is not equal to 2 cubes. This could create a possible
misconception. They have to understand that both groups of
cubes are divided into three equal parts so that 62 is the same
as 13 . They therefore consider the equal parts that represent
the whole each time.
Solutions
1. a)
1
6
= 122 ; 62 = 124 ; 64 = 128 ; 65 = 10
12
b)
1
7
= 142 ; 72 = 144 ; 73 = 146 ; 74 = 148 ; 75 = 10
; 6 = 12
14 7
14
c)
1
9
= 182 ; 92 = 184 ; 93 = 186 ; 94 = 188 ; 95 = 10
; 6 = 12
, and so on
18 9
18
2. a)
1
3
= 62 ; 23 = 64 ; 33 = 66 = 1
b)
1
5
= 102 ; 52 = 104 ; 53 = 106 ; 54 = 108 ; 55 = 10
=1
10
c)
3
9
= 13 ; 69 = 23 ; 99 = 33 = 1
d)
2
12
= 61 ; 124 = 62 ; 126 = 63 ; 128 = 64 ; 10
= 65 ; 12
= 66 = 1
12
12
Activity 9.1
Learner’s Book page 192
1. Learners can use the cube arrangements they worked with in
Mental maths to solve the following equivalent fractions. They
could also use number lines, the fraction wall and fraction
circles for practical experience.
a)
6
8
= 43
b)
1
2
d)
4
6
= 23
e)
8
10
g)
1
3
= 93
h)
j)
4
5
= 106
2. a)
1
of
2
= 24
c)
1
3
= 93
= 54
f)
8
12
= 64 = 23
6
6
= 33
i)
12
12
= 66
k)
6
8
= 23
l)
5
6
= 10
12
24 = 12
b)
1
3
of 24 = 8
c)
2
3
of 24 = 16
d)
1
4
of 24 = 6
e)
2
4
of 24 = 12
f)
3
4
of 24 = 18
g)
1
6
of 24 = 4
h)
3
6
of 24 = 12
i)
2
6
of 24 = 8
j)
1
8
of 24 = 3
k)
2
8
of 24 = 6
l)
4
8
of 24 = 12
m)
1
12
of 24 = 2
n)
4
12
of 24 = 8
o)
6
12
of 24 = 12
3. Learners identify and write down the equivalent fractions they
notice.
Mathematics Teacher’s Guide Grade 4
Math G4 TG.indb 249
TERM 3
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Unit 10
Equal sharing and problem-solving
Mental Maths Learner’s Book page 192
1. R100 ÷ 4 = R25
2.
3. R50 ÷ 2 = R25
4.
5. R50 ÷ 5 = R10
6.
7. R100 ÷ 5 = R20
8.
9. R200 ÷ 4 = R50
10. of R200 = R50
1
4
1
2
1
5
1
5
1
4
Activity 10.1
of R100 = R25
of R50 = R25
of R50 = R10
of R100 = R20
Learner’s Book page 193
The learners solve calculations with remainders that also have
to be shared equally. Encourage them to make simple drawings
as they did earlier this year. They could also solve the problem
by looking at each slice as: 33 so 11 × 3 = 333. Dividing it by 3 is
11
, which is equal to 11 ÷ 3 = 3 remainder 2 = 323 .
3
2, 3. Ask the learners to draw a rectangle or use square grid paper
to find 65 of 18. There are 6 columns so they shade 5 of the six
columns so that 65 of 18 = 15. They use the diagram to shade
and find more fraction parts of 18. They should realise that 12 of
18 = 63 of 18 = 9.
4. Ask the learners to choose whether they prefer 15 of R100 or 14
of 100. They could argue that 14 is larger than 15 . Ask them to
use the semi- and concrete fraction materials they used before
or make a drawing to support and justify their reasoning.
Show them the relationship between equal sharing and division
(100 ÷ 5 = 20 and 100 ÷ 4 = 25). They should understand
that the more parts the whole is divided into, the smaller the
fraction becomes.
5. The learners compare the same wholes divided into different
equal parts. Strip B is divided into fifths and Strip C into tenths.
They use the diagram to reason that 23 > 53 and give a reason
for their argument. They should observe that 13 > 103 and give a
reason.
6. They order the fractions in the list from smallest to largest. Ask
them to explain how they do it and which fractions they found
the most difficult to order.
1.
Solutions
1. a) 11 ÷ 3 = 3 remainder 2
3 + 3 + 3 = 9
13 + 13 + 13 = 1
13 + 13 + 13 = 1
11 ÷ 3 = 323 slices
Each child gets 323 slices of bread.
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b) Each piece is 81 of the roll of cheese.
c) 15 ÷ 4 = 343 slices
d) 17 ÷ 5 = 352 bars
e) 11 ÷ 8 = 1114 slices
2. Divide 18 in 6 equal groups.
Each group will consists of 3 parts.
Count the five parts: 5 × 3 = 15
1
of 18 = 15
3
3. Learners make copies of the diagram and shade fractions of 18.
Let them check their drawings by doing the calculations. Below
are some examples of how the learners might shade the fraction
parts.
b)
a)
d)
c)
1
4
1
4
1
5
1
5
1
4
1
4
1
5
1
5
4. The learners should know by now that, the smaller the
denominator in a unitary fraction, the bigger the fraction. You
could however ask them to make a1 drawing to justify this
4
understanding, for example:
1
5
1
4
They
1 could support their reasoning with calculations, for
5
example:
100 ÷ 5 × 1 = 20 and 100 ÷ 4 × 1 = 25
R25 is more than R20.
5. a) fifths
b) tenths
c)
d)
6.
2
> 53
3
3
< 101
10
1 2 3 2 3 5 7
; ; ; ; ; ;
4 5 7 3 4 6 8
Mathematics Teacher’s Guide Grade 4
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Unit 11
Count, order and calculate with
fractions
Mental Maths 1.
2.
3.
4.
5.
6.
1
4
0
3
4
1
3
0
0
2
5
1
7
1
9
2
7
2
9
1
3
5
3
7
3
9
4
9
114
1
2
3
1
5
0
0
2
4
Learner’s Book page 194
5
9
5
7
6
9
1 23
1
115
5
5
6
5
6
7
7
9
134
1 13
4
5
4
7
124
2 24
213
2
2 34
2 23
3
7
5
8
5
127
137
10
5
9
5
147
157
11
5
167
1
10
2
10
3
10
4
10
5
10
6
10
177
1 119 129 139 149 159 169 179 189 2
8
9
5
110
1
0
3
135
117
7
7
2 14
2
7
10
8
10
9
10
10
10
11
10
12
10
13
10
14
10
15
10
Lead a discussion in which the learners explore the relationship
between improper and mixed fractions. Let them use informal
strategies to convert between the two types of fraction. Ask them how
to convert a mixed fraction such as 114 into an improper fraction 154 2.
They might be able to reason that 1 × 4 + 1 = 54 or 5 ÷ 4 = 114 . Do
not show them the algorithm for the conversion now; they should
construct their own meaning of the relationship.
Activity 11.1
Learner’s Book page 194
Learners find groups of equivalent fractions. Ask them to
explain how and why they grouped certain fractions. You could
ask them to make drawings to support their reasoning or use a
fraction wall.
2. Learners will soon realise that they can never list all the
fractions between 0 and 1. Every whole number can be divided
into fractions. They should realise that the larger the number,
the smaller are the equal parts, for example if you divide 100
into 100 equal parts each part would be smaller than the equal
parts you get when you divide 10 into 10 equal parts.
3, 4. Learners to identify which fractions of a whole are shaded.
5. Learners study the diagrams showing addition and subtraction
of fractions with the same denominators. The first fraction is
shaded in the first diagram. You fill up this diagram with parts
of the second fraction and continue shading the remaining parts
in the second diagram. To find the answer, count the shaded
parts in both diagrams. To subtract, shade each fraction in the
calculation in the two diagrams.
1.
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Solutions
1.
1
2
2
3
1
3
3
4
2
5
1
5
= 84 = 105 = 147
= 69 = 128
= 155
= 129
= 104 = 156
= 102
2. Learners will probably give familiar fractions. Make sure they
understand that a whole could be divided into any number of
1
and 1 0001 000 .
parts and that you can have fractions such as 501 , 300
3.
7
s
18
4.
3
5
of the windows are shaded.
of the fence is painted.
5. a)
5
9
= 195
b)
+ 99 = 149
4
7
+ 76 = 107
= 173
c)
6
7
– 74 = 72
d)
7
8
+ 85 = 128
e)
7
10
f)
9
10
= 184
11
18
+ 10
= 10
= 1108
– 105 = 104
Assessment task 2: common fractions
The learners perform the assessment task to display knowledge of
fraction representation, calculating fractions of whole numbers,
equivalent fractions, counting fractions (entailing improper and
mixed fractions) and equal sharing in contextual problems.
Mathematics Teacher’s Guide Grade 4
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Assessment task 2 Common fractions
1. Which fraction of each diagram is shaded?
b)
a)
c)
d)
e)
f)
(6)
2. Shade the fraction part on a copy of each diagram.
a)
2
5
of a pentagon
b)
7
8
of an octagon
3
7
of the circles
d)
5
9
of the stars
c)
e)
3
10
of the rectangle
(5)
3. How many squares is each fraction of the eight squares?
254
Math G4 TG.indb 254
a)
1
2
of 8 the squares
b)
1
8
of 8 the squares
c)
1
4
of 8 the squares
d)
3
4
of 8 the squares
e)
7
8
of 8 the squares
Mathematics Teacher’s Guide Grade 4
(5)
TERM 3
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4. Use a fraction wall, number lines or fraction circles if you need
help finding equivalent fractions.
1
2
1
3
1
4
1
5
1
6
1
7
1
8
a)
1
2
=
=
=
b)
1
4
=
=
c)
2
4
=
=
=
d)
3
4
=
e)
1
3
=
f)
2
3
=
g)
2
7
=
(7)
5. Which fractions are missing on the number lines?
a)
1
2
0
2 12
1
3
b)
3
4
1 14
1 24
c)
0
1
5
2
5
125
135
(16)
6. Solve these problems. You can make drawings to help you.
a) Five friends share 6 chocolate bars equally. How
much chocolate does each one get?
(3)
b) Seven friends share 10 fizzers equally. How many
fizzers does each one get?
(5)
c) What would you prefer to receive? Explain.
(3)
12 of R100 or 105 of R100?
Total [50]
Mathematics Teacher’s Guide Grade 4
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Assessment task 2 Common fractions
1. a)
1
2
b)
8
12
c)
2
3
d)
5
6
Solutions
e) 183 (6)
2. The learners shade the fraction parts in the shapes.
3. a)
1
2
of 8 = 4
b)
1
8
of 8 = 1
c)
1
4
of 8 = 2
d)
3
4
of 8 = 6
e)
7
8
of 8 = 7
(5)
(5)
4. Other solutions can also be correct.
a)
1
2
= 24 = 84 = 63 = 126
b)
1
4
= 82 = 123
c)
2
4
= 84 = 105 = 126
d)
3
4
= 86 or 129
e)
1
3
= 93 or 124 or 62
f)
2
3
= 64 or 69 or 128
g)
2
7
= 144 (7)
5. The learners identify the missing fractions on the number lines.
a)
1
2
0
1 12
1
2 12
2
3
b)
1
4
0
2
4
3
4
1
1 14
1 24
1 34
2 14
2
c)
1
5
0
2
5
3
5
4
5
5
5
115
125
135
1 45
1 55
2 15
(16)
6. a) 1 + 1 + 1 + 1 + 1 = 5
15 + 15 + 15 + 15 + 15 = 1
Each learner gets 115 of the chocolate bars.
(3)
b) 1 + 1 + 1 + 1 + 1 + 1 + 1 = 7
71 + 71 + 71 + 71 + 71 + 71 + 71 = 1
71 + 71 + 71 + 71 + 71 + 71 + 71 = 1
71 + 71 + 71 + 71 + 71 + 71 + 71 = 1
Each friend gets 173 of the fizzers.
c)
1
2
of R100 = R50
105 is the same as 12 so 105 of R100 = R50
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Math G4 TG.indb 256
(5)
Mathematics Teacher’s Guide Grade 4
(3)
Total [50]
TERM 3
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Whole numbers: adding and subtracting
The learners will only work with counting, ordering, comparing and
representing whole numbers for a few days this term. However, they
have to do Mental maths every morning. The activities provided
here are sufficient for the entire week.
Unit 12
Rules for operations
Mental Maths Learner’s Book page 196
1. Ask the learners to identify the pattern of each string of
beads using the number sentences below. Let them point
out the beads that show each number sentence. You can
draw each string of beads on the board. Ask learners to give
number sentences for the other strings of beads. They can
focus on the relationships between the numbers, for example
where numbers are swapped or turned around (commutative
property) and the relationship between addition and
subtraction (inverse operations). This exercise might take
more than ten minutes. The learners could do the rest of the
activities in the Mental maths sessions during this week. Also
ask the learners to do some of the activities for homework.
2. Learners use the commutative and associative properties,
inverse operations, order of operations and brackets to
create number sentences using the bead arrangements. They
should realise that we do addition and subtraction from left
to right, in order of appearance. They should also realise the
difference between:
20 – 5 – 5 – 2 = 8
20 – 5 + 5 + 2 = 22
20 – (5 + 5 + 2) = 8
Solutions
1. Class discussion
2. Possible answers include the following.
a) 3 + 8 = 11
b) 1 + 9 = 10
8 + 3 = 11
9 + 1 = 10
11 – 3 = 8
10 – 1 = 9
11 – 8 = 3
10 – 9 = 1
c) 5 + 5 + 2 + 3 = 15
3 + 2 + 5 + 5 = 15
15 – 5 – 5 – 2 = 3
15 – 3 – 2 – 5 = 5
Mathematics Teacher’s Guide Grade 4
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d) 5 + 4 + 1 + 5 = 15
5 + 1 + 4 + 5 = 15
15 – 6 = 9
15 – 9 = 6
15 – 5 – 1 = 9
15 – 1 – 5 = 9
15 – (5 + 5 + 2) = 3
15 – 5 – 5 – 2 = 3
Activity 12.1
e) 5 + 5 + 2 + 3 + 5 = 20
5 + 3 + 2 + 5 + 5 = 20
20 – 3 – 5 = 12
20 – 5 – 3 = 12
20 – (5 + 5 + 2) = 8
20 – (10 + 2) = 8
Learner’s Book page 196
1. Learners create their own number sentences using the strings of
beads. Examples:
5 + 5 + 5 + 5 = 20
10 + 10 = 20
5 + 15 = 20
20 = (2 × 5) + (2 × 5)
4 × 5 = 20
2 × 10 = 20
15 + 5 = 20
20 = (3 × 5) + (1 × 5)
20 – 5 – 5 – 5 – 5 = 0
20 – 5 = 15
20 ÷ 5 = 4
20 – 15 = 5
2. Ask learners to explain their number sentences. They
should use repeated addition and subtraction and connect
it to multiplication and division, use the commutative and
distributive properties and inverse operations. They can also
use doubling, halving and repeated addition. The list of number
sentences cannot be exhausted. Examples:
25 – 4 = 21
double 12 = 24
half of 60 = 30
42 ÷ 2 = 21
12 × 2 = 24
60 ÷ 2 = 30
half of 42 = 21
48 ÷ 2 = 24
double 15 = 30
7 + 7 + 7 = 21
half of 48 = 24
10 + 10 + 10 = 30
3 × 7 = 21
8 + 8 + 8 = 24
50 – 20 = 30
15 + 6 = 21
30 – 6 = 24
5 + 5 + 10 = 30
3. Learners write open number sentences to show how to work out
the number of hidden dots if they use the number of dots and
the given totals. Discuss the example with the class. Learners
should use the placeholder to show the unknown in different
positions. They use effective counting strategies to determine
the number of dots in the arrays and the commutative property
and inverse operations to construct number sentences and solve
the problems.
They first have to realise that they must subtract the number of
dots from the total below (12 – 8 = 4). They use this calculation
to create other number sentences (12 – 4 = 8; 4 + 8 = 12 and
8 + 4 = 12). This gives them further practice in using the
commutative property and inverse operations.
4. Give the learners copies of the calculation diagrams or they can
draw copies. They practise addition and subtraction bonds of
34. You could also give learners blank copies and use different
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numbers in the centre so that they can practise addition and
subtraction bonds for 2- or 3-digit numbers. Learners can also
use the diagrams to practise doubling and halving – write double
or halve in the centre circle and write numbers in the squares for
them to double or halve.
5. Learners could use doubling and multiplication for repeated
addition. Strategies they could use are shown in the possible
solutions below. (Learners can convert millilitres to litres after
they have worked out the solutions.)
The mathematics lessons for this week involve capacity/
volume. This exercise should assist the learners in developing
or enhancing their knowledge of capacity. If they have the
basic number concepts they will be able to make sense of the
measurement concepts because they do not have to struggle with
the number knowledge required. They practise counting in 5s,
25s, 250s, 500s, 350s and 1 000s. This allows them to see the
connection or relationships between number and measurement.
Solutions
1, 2. Examples are given above.
3. a) 16 + 8 = 24; 24 – 16 = 8
8 + 16 = 24; 24 – 8 = 16
b) 25 + 15 = 40; 40 – 15 = 25
15 + 25 = 40; 40 – 25 = 15
c) 16 + 14 = 30; 30 – 14 = 16
14 + 16 = 30; 40 – 25 = 15
d) 18 + 18 = 36; 36 – 18 = 18
e) 15 + 25 = 40; 40 – 25 = 15
25 + 15 = 40; 40 – 15 = 25
f) 20 + 15 = 35; 35 – 15 = 20
15 + 20 = 35; 35 – 20 = 15
4. a)
23
28
29
27
25
– 11
–7
–9
–6
34
– 13
–5
–8
– 16
26
21
18
b)
45
40
39
41
+ 11
+7
+9
+6
34
+ 13
+5
+8
+ 16
42
47
50
5. a) 250 + 250 + 250 + 250
= 500 + 500
= 1 000 ml
b) 500 + 500 + 500
= 1 000 + 500
= 1 500 ml
c) 5 + 5 + 5 + 5 + 5 + 5
= 6 × 5
= 30 ml
d) 25 + 25 + 25 + 25 + 25 + 25
= 100 + 50
= 150 ml
Mathematics Teacher’s Guide Grade 4
Math G4 TG.indb 259
43
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e) 350 + 350 + 350 + 350 + 350 + 350
= 700 + 700 + 700
= (3 × 7) × 100
= 21 × 100
= 2 100 ml
f) 1 000 + 1 000 + 1 000 + 1 000 + 1 000 + 1 000
= 1 000 × 6
= 6 000 ml
Unit 13
Count, compare, represent numbers
and place values
Mental Maths Learner’s Book page 199
1. Assist the learners in understanding that they have to explore
different numbers that would fit the description.
2. Ask as many questions as you can to model the type of
questions the learners have to ask when it is their turn to find
a number. When you have found the number, ask the learners
to find a number you think of. They could also work in pairs
or groups to find someone’s number. Extend the exercise to
thinking of a number between 500 and 1 000 and follow the
same procedures.
3. Ask a few learners to count the number of dots on the strips
without counting in 1s. Focus on learners who may struggle
with some concepts to establish whether they can count the
groups of dots.
4. Ask the learners if they know the story of Jack and the
Beanstalk. If they do not know it, you could tell it to them
in a language lesson or ask the language teacher to tell it
to them. Also encourage them to get the book from the
library if possible. Jack did not know numbers as we do
today. Let them read Jack’s numbers aloud. They have to
work out which word Jack used for the number twenty by
linking his numbers to our natural or counting numbers. This
exercise gives learners the opportunity to make sense of the
development of the number system. You could ask them to
create an imaginary number system of their own.
Solutions
1. 140
104
401
410
230
203
302
320
2. Answers will differ.
3. Some learners may use repeated addition while others use
multiplication. Lead a discussion about counting strategies
so learners can convince each other that some strategies
(multiplication) are more effective than others (repeated
addition). We always try to use short cuts in mathematics.
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a) 10 + 10 + 10 + 10 + 8 = 48
4 × 10 + 8 = 48
b) 10 + 10 + 10 + 10 + 10 + 3 = 53
5 × 10 + 3 = 53
c) 10 + 10 + 10 + 10 + 10 + 10 + 10 + 5 = 75
7 × 10 + 5 = 75
4.
fot → 5
feefot → 1 + 5 = 6
fiefot → 2 + 5 = 7
foefot → 3 + 5 = 8
fumfot → 4 + 5 = 9
fotfot → 5 + 5 = 10
13 → 3 + 10 → foefotfot
14 → 4 + 10 → fumfotfot
15 → 5 + 10 → fotfotfot
16 → 6 + 10 → feefotfot
17 → 7 + 10 → fiefotfot
18 → 8 + 10 → foefotfot
19 → 9 + 10 → fumfotfot
20 → 10 + 10 → fotfotfotfot (5 + 5 + 5 + 5)
Activity 13.1
Learner’s Book page 200
1. Learners can work in groups or you can work through the
activity with the whole class. They can make cards with the
numbers 1 to 4 and then make as many 4-digit numbers as
possible. Let them do this on their own first, but you might
want to encourage them to work systematically. You could
write the numbers 1 to 4 on the board and help learners work
systematically to create the numbers and order the digits. Let
them list these numbers in columns.
• They start with 1 234 and make combinations that start with
1 (1 243; 1 234; 1 324; 1 342; 1 423 and 1 432).
• They then make numbers that start with 2 (2 134; 2 143;
2 314; 2 341; 2 413 and 2 431, and so on).
The activity develops the concepts of ordering, place value and
data handling (organising digits systematically). Ask the learners
to find out how many even and odd numbers there are and what
the sum of the digits is. They will find that it is the same for
all the 4-digit numbers they make with the digits 1 to 4. Ask
each learner to select ten numbers from the list and write these
numbers in words.
2. Learners can identify and name the different intervals and the
multiples they count in.
4. The activity develops problem solving skills, knowledge of
representing numbers, division and multiplication.
5. Learners will have to remember the place value of the numbers.
Mathematics Teacher’s Guide Grade 4
Math G4 TG.indb 261
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Solutions
1. a) 1 234; 1 243; 1 324; 1 342; 1 423; 1 432; 2 134; 2 143;
2 314; 2 341; 2 413; 2 431;
3 124; 3 142; 3 214; 3 241; 3 412; 3 421; 4 123; 4 132;
4 213; 4 231; 4 312; 4 321
b) 12
c) 12
d) 10
e) Ask the learners to select the numbers from the list and write
these numbers in words.
2. a) 3; 8; 13; 18; 23; 28; 33
(Count in intervals of 5.)
b) 1; 2; 4; 8; 16; 32; 64; 128
(Multiply by 2. The numbers are powers of 2.)
c) 81; 78; 75; 72; 69; 66; 63; 60
(Count back in multiples of 3.)
d) 10 000; 9 750; 9 500; 9 250; 9 000; 8 750; 8 500
(Count back in multiples of 250.)
e) 989; 990; 991; 992; 993; 994; 995; 996; 997; 998; 999;
1 000
(Count on in 1s.)
f) 6 004; 6 003; 6 002; 6 001; 6 000; 5 999; 5 998; 5 997;
5 996; 5 995
(Count back in 1s.)
g) 8 040; 8 030; 8 020; 8 010; 8 000; 7 990; 7980; 7 970;
7 960; 7 950
(Count back in 10s.)
h) 7 500; 8 000; 8 500; 9 000; 9 500; 10 000; 10 500; 11 000;
11 500; 12 000
(Count on in 50s.)
3. a)
b)
c)
d)
e)
f)
137 → 30
300 → 300
13 → 3
3 024 → 3 000
31 546 → 30 000
4 536 → 30
4. The learners multiply multiples of 10. Allow them to use their
own strategies to calculate the number of different coins and
notes in R200. You could share the strategy below with them
to illustrate a systematic approach to problem-solving. Ask
them to describe the patterns they observe in the calculations
and solutions. They should notice the process of halving in the
solutions.
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Mathematics Teacher’s Guide Grade 4
TERM 3
2012/09/14 5:37 PM
a)
b)
c)
d)
e)
f)
g)
h)
i)
j)
k)
l)
5c
10c
20c
50c
R1
R2
R5
R10
R20
R50
R100
R200
20 in R1
10 in R1
5 in R1
2 in R1
200 in R1
50 in R100
20 in R100
10 in R100
5 in R100
2 in R100
1 in R100
1 in R200
20 × 200
10 × 200
5 × 200
2 × 200
1 × 200
2 × 50
2 × 20
2 × 10
2×5
2×2
1×2
1×1
4 000 in R200
2 000 in R200
1 000 in R200
400 in R200
200 in R200
100 in R200
40 in R200
20 in R200
10 in R200
4 in R200
2 in R200
1 in R200
5. The learners add 1 100 to each number to observe how the
values of the digits increase and the place values change. They
get practise in working with 5-digit numbers. Some of your
learners might solve the problems mentally while others might
apply expanded notation.
a) 300 + 1 100
b) 900 + 1 100
= 1 000 + 300 + 100
= 1 000 + 900 + 100
= 1 400
= 2 100
c) 2 900 + 1 100
d) 9 800 + 1 100
= 3 000 + 900 + 100
= 10 000 + 800 + 100
= 4 000
= 10 900
e) 9 900 + 1 100
f) 7 900 + 1 100
= 10 000 + 900 + 100
= 8 000 + 900 + 100
= 11 000
= 9 000
g) 11 900 + 1 100
h) 19 900 + 1 100
= 12 000 + 900 + 100
= 20 000 + 900 + 100
= 13 000
= 21 000
Assessment task 3: Counting and place value
1. Learners solve problems that involve repeated addition and
doubling. Assess how well they use grouping to calculate the
numbers used in volume and capacity.
2. Learners complete the numbers on the number lines. These
involve counting in even numbers, intervals of 10 and 3,
counting on and back in 1s, 25s, 50s and 100s to bridge
multiples of 100, 1 000 and 10 000.
3. Learners identify the values of digits up to 5-digit numbers.
4. Learners identify the numbers represented by dots in the place
value tables and numbers represented by the Dienes blocks.
Mathematics Teacher’s Guide Grade 4
Math G4 TG.indb 263
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2012/09/14 5:37 PM
Assessment task 3 Counting and place value
Work on your own.
1. Work out the answers.
a) 250 + 250 + 250 + 250 + 250 = n
b) 500 + 500 + 500 = n
c) 1 000 + 1 000 + 1 000 + 1 000 + 1 000 = n
d) 25 + 25 + 25 + 25 + 25 = n
e) 350 + 350 + 350 + 350 = n(5)
2. Fill in the missing numbers on the number lines.
a)
b)
c)
d)
e)
f)
g)
h)
■
2
4
■
■
14
1
4
8
12
■
■
7
■
■
■
■
■
■
■
960
■
■
990
1 000
1 010
1 005
1 004
■
■
■
■
3 050
3 025
■
■
■
■
9 850
9 900
■
■
■
■
10 300
10 200
■
■
■
■
(19)
3. Give the value of the underlined digits.
a) 405
b) 7 777
c) 22 893
d) 9 085
e) 10 060(5)
4. Write down the number represented in each place value table.
264
Math G4 TG.indb 264
a)
H
l
l
l
l
T
U
l
l
b)
Th
l
l
l
H
l
T
l
l
l
Mathematics Teacher’s Guide Grade 4
U
TERM 3
2012/09/14 5:37 PM
c)
Tth
l
l
Th
H
T
l
l
l
l
U
l
l
l
l
l
d)
Tth
l
l
l
l
l
l
Th
l
l
l
H
l
T
l
l
l
l
U
e)
Tth
l
l
l
l
l
Th
H
l
l
l
T
l
l
l
l
U
f)
Tth
l
l
l
l
l
l
Th
H
l
l
l
T
U
l
l
l
(6)
5. Write down the numbers represented by the Dienes Blocks.
b)
a)
c)
d)
e)
(5)
Total [40]
Mathematics Teacher’s Guide Grade 4
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Assessment task 3 Counting and place value
Solutions
1. a) 250 + 250 + 250 + 250 + 250
= 500 + 500 + 250
= 1 250
b) 500 + 500 + 500 = 1 500
c) 1 000 + 1 000 + 1 000 + 1 000 + 1 000 = 5 000
d) 25 + 25 + 25 + 25 + 25
= 50 + 50 + 25
= 125
e) 350 + 350 + 350 + 350
= 700 + 700
= 1 400
2. a)
b)
c)
d)
e)
f)
g)
h)
2
4
4
1
6
24
14
4
8
10
7
14
12
34
(5)
16
44
13
54
16
19
960
970
980
990
1 000
1 010
1 005
1 004
1 003
1 002
1 001
999
3 050
3 025
3 000
2 975
2 950
2 925
9 850
9 900
9 950
10 000
10 050
10 100
10 300
10 200
10 100
10 000
9 900
9 800
(19)
3. a) 405 → 400 and 5
c) 22 893 → 20 000 and 800
e) 10 060 → 10 000 and 60
266
Math G4 TG.indb 266
b) 7 777 → 7 000 and 70
d) 9 085 → 9 000 and 5
4. a)
b)
c)
d)
e)
f)
400 + 2 = 402
3 000 + 100 + 30 = 3 130
2 000 + 40 + 5 = 2 045
60 000 + 3 000 + 100 + 40 = 63 140
50 000 + 300 + 40 = 50 340
60 000 + 300 + 3 = 50 303
5. a)
b)
c)
d)
e)
200 + 10 = 210
1 000 + 20 + 2 = 1 022
1 000 + 200 + 10 + 3 = 1 213
2 000 + 300 + 40 + 4 = 2 344
3 000 + 200 + 6 = 3 206
Mathematics Teacher’s Guide Grade 4
(5)
(6)
(5)
Total [40]
TERM 3
2012/09/14 5:37 PM
unit 14
Addition and subtraction facts
Learners will work with addition and subtraction this week. They
will work with up to 4-digit numbers in calculations, but also
practise basic addition and subtraction facts in fun activities.
Learners write an assessment task at the end of the week.
Mental Maths
Learner’s Book page 201
Game board
Answer sheet
6
Bingo!
5+5
0+6
6+6
15 + 7
7+9
10
12 22 16
8+8
6+7
17 + 9
9+9
5+8
16 13 28 18 13
5+6
8+9
5+9
15 + 5
6+8
11 17 14 20 14
7+7
16 6
18 + 8
5+7
19 + 9
14 22 26 12 28
15 + 6 18 + 9 16 + 7
7+0
0+8
21 27 23
7
8
Give each learner a copy of the answer sheet for Addition Bingo.
Learners use the calculation board for the game. You use the
board to select and pose the questions. Ask the questions in
random order (use the order given in the answers below). The
learners should complete as many questions as possible before
they have crossed out numbers in an entire row, column or
diagonal. They shout Bingo! when they have crossed out five
numbers in a row, column or diagonal. The game allows them to
practise the basic addition facts they need for calculations with
larger numbers.
Addition Bingo
1. 5 + 5 = n
4. 18 + 8 = n
7. 5 + 7 = n
10. 7 + 9 = n
13. 15 + 5 = n
16. 6 + 7 = n
19. 0 + 6 = n
22. 19 + 9 = n
25. 18 + 9 = n
Activity 14.1
2.
5.
8.
11.
14.
17.
20.
23.
17 + 9 = n
0+8=n
9+9=n
5+8=n
7+0=n
16 + 6 = n
15 + 7 = n
15 + 6 = n
3.
6.
9.
12.
15.
18.
21.
24.
5+6=n
6+6=n
7+7=n
8+9=n
8+8=n
6+8=n
5+9=n
16 + 7 = n
Learner’s Book page 201
1. Remind the learners of the importance of looking for patterns
and relationships in numbers – this is what mathematicians do!
Learners should notice, for example, that 78 is 10 more than
68 and therefore the answer to 594 + 68 should be 10 more
than 672 (the solution to 594 + 78). Learners should not work
out the answers, but solve them by using the relationships or
connections.
Mathematics Teacher’s Guide Grade 4
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2. Ask the learners who has been to a fair or fun park. Let them
describe what one does at a fair. Learners practise adding three
numbers at a time.
3. Remind the learners that working with 10, 100 and 1 000 is
usually easy if they know the basic calculation facts. For this
question, they should realise, for example, that knowing that
4 + 8 = 12 allows them to solve the problem 4 000 + 8 000.
There are 12 ways to get 50:
10 + 14 + 26 = 50
10 + 17 + 23 = 50
10 + 19 + 21 = 50
11 + 18 + 21 = 50
12 + 15 + 23 = 50
12 + 17 + 21 = 50
13 + 14 + 23 = 50
13 + 18 + 19 = 50
11 + 13 + 26 = 50
14 + 15 + 21 = 50
14 + 17 + 19 = 50
15 + 17 + 18 = 50
Solutions
1. a) 594 + 78 = 672
594 + 68 = 662
594 + 88 = 682
594 + 28 = 622
b) 399 + 57 = 456
399 + 47 = 446
399 + 27 = 426
399 + 77 = 476
c) 745 – 67 = 678
745 – 77 = 668
745 – 57 = 688
745 – 87 = 658
d) 553 – 59 = 494
553 – 69 = 484
553 – 79 = 474
553 – 99 = 454
2. 10 + 14 + 26 = 50
10 + 17 + 23 = 50
10 + 19 + 21 = 50
11 + 13 + 26 = 50
11 + 18 + 21 = 50
12 + 17 + 21 = 50
13 + 14 + 23 = 50
13 + 18 + 19 = 50
14 + 17 + 19 = 50
15 + 17 + 18 = 50
(10 less)
(10 more)
(50 less)
(10 less)
(30 less)
(20 more)
(10 less)
(10 more)
(20 less)
(10 less)
(20 less)
(40 less)
12 + 15 + 23 = 50
14 + 15 + 21 = 50
3. You can write the solutions on the board as they are given
below. This will help learners realise that they are working with
place value.
a)
4 + 8 = 12
b)
5 + 7 = 12
40 + 80 = 120
50 + 70 = 120
400 + 800 = 1 200 500 + 700 = 1 200
4 000 + 8 000 = 12 000 5 000 + 7 000 = 12 000
268
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c)
8 + 9 = 17
d)
12 – 7 = 5
80 + 90 = 170
120 – 70 = 50
800 + 900 = 1 700 1 200 – 700 = 500
8 000 + 9 000 = 17 000 12 000 – 7 000 = 5 000
e)
16 – 9 = 7
f)
14 – 8 = 6
160 – 90 = 70
140 – 80 = 60
1 600 – 900 = 700 1 400 – 800 = 600
16 000 – 9 000 = 7 000 14 000 – 8 000 = 6 000
Investigation
Learner’s Book page 201
2 + 5 + 32 + 193 + 18 + 10 + 100 = 360
Unit 15
Problem-solving
Mental Maths Learner’s Book page 202
Subtraction Bingo is played in the same way as Addition Bingo.
Learners cross out the answers on the answer sheets as you pose the
subtraction questions and shout out Bingo! when they have crossed
out five numbers in a row, column or diagonal. Ask the questions
in random order. Use the order as in the number sentences with the
solutions below.
Subtraction Bingo
1. 10 – 5 = n
4. 10 – 8 = n
7. 13 – 5 = n
10. 16 – 0 = n
13. 13 – 4 = n
16. 15 – 6 = n
19. 15 – 8 = n
22. 10 – 9 = n
25. 20 – 7 = n
Activity 15.1
2. 10 – 0 = n
5. 13 – 9 = n
8. 12 – 9 = n
11. 12 – 5 = n
14. 20 – 8 = n
17. 20 – 9 = n
20. 18 – 9 = n
23. 15 – 9 = n
3. 10 – 6 = n
6. 17 – 9 = n
9. 16 – 6 = n
12. 13 – 8 = n
15. 16 – 7 = n
18. 17 – 8 = n
21. 17 – 7 = n
24. 15 – 7 = n
Learner’s Book page 202
1. The learners solve problems with small numbers. This exercise
is not about the numbers, but rather about understanding the
structure of the problem (about knowing how to manipulate the
numbers). Learners can use cubes, bottle-tops, counters, number
lines, and so on.
2. Learners write open number sentences to show their
understanding of the structure of the problems. Allow them
to use their own strategies and share strategies and solutions.
Below are strategies learners might use or that you could share
with the learners.
3. Learners make copies of the number triangles. The sum of two
numbers in the bottom row is written above the two numbers.
To work out what the two numbers below a number are, learners
need to do logical reasoning.
Mathematics Teacher’s Guide Grade 4
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Solutions
1. Some learners might think that the problems are very easy. They
will use what they learn in this exercise when working on the
next exercise where they work with 3- and 4-digit numbers.
a) 9 – 6 = n
b) 9 – 3 = n
9 – 6 = 3 cubes left 9 – 3 = 6 cubes more
c)
9–6=n
9 – 6 = 3 cubes more
(9 + 9) – (6 + 7) = n
18 – 13 = 5 cubes more
d) 5 + n = 12
e) 9 + 8 = n
12 – 5 = 7 cubes added 9 + 8 = 17 cubes altogether
2. a) The numbers are broken up into place values and
decomposed to subtract the larger digits. Help learners who
struggle with decomposition by using smaller numbers such
as 43 – 39 and 324 – 157.
2 453 – 1 565 = n
2 453 → 2 000 + 400 + 50 + 3
– 1 565 → 1 000 + 500 + 60 + 5
1 300 + 140 + 13 (decompose 2 000,
– 500 + 60 + 5 400 and 50)
800 + 80 + 8
= 888
Farmer Brown has 888 chickens left.
b) Breaking up numbers and compensation are applied.
2 675 – 889 = n
2 000 – 800 = 1 200
600 – 90 = 510
75 + 1 = 76
1 200 + 510 + 76 = 1 786
Farmer Nelson has 1 786 more cows than Farmer Louw.
c) The learners should perform multi-operations, such as
addition and subtraction in one problem. They could break
down the numbers in place value parts to add. They could
also use counting on and compensation to subtract.
(756 + 467) – (678 + 489) = n
700 + 400 + 50 + 60 + 6 + 7 = 1 100 + 110 + 13
= 1 200 + 23
= 1 223
Farmer Anele’s workers picked 1 223 oranges
600 + 400 + 70 + 80 + 8 + 9 = 1 000 + 150 + 17
= 1 167
Farmer Andile’s workers picked 1 167 oranges
270
Math G4 TG.indb 270
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1 223 – 1 167 = n
1 000 – 1 000 = 0
167 + 3 = 170
170 + 30 = 200
30 + 3 + 23 = 30 + 26
= 56
Farmer Anele’s workers picked 56 more apples.
d) They could use the inverse operation, work with the closest
multiples of 100 and use compensation to subtract.
1 545 + n = 2 320
2 320 – 1 545 → 2 300 – 1 300 = 1 000
1 000 – 200 = 800
800 – 40 – 5 = 760 – 5
= 755
755 + 20 = 775
The farm workers picked 775 apples on Tuesday.
e) The learners could use breaking up numbers to perform
addition.
1 205 + 1 415 = n
1 000 + 1 000 = 2 000
205 + 415 = 620
2 000 + 620 = 2 620
The chickens laid 2 620 eggs altogether.
3. a)
96
40 56
16 24 32
10 14 18
4
6
8 10
c)
107
59 48
29 30 18
11 18 12 6
4 7 11 1 5
b)
96 96
107 107
154 154
40 40
56 56
59 59
48 48
75 75
79 79
24 32
41 38
34 34
16 16
24 32
18 18
29 29
30 30
41 38
6
14
11
6
15
19
18
12
10
18
6 10 14 18
11 18 12 6
15 19 22 22
16 1
7 11
2 17
5
2 17
5 11
8 104 74 11
1 51 5 13 13
2 24 46 68 10
154
75 79
34 41 38
15 19 22 16
13 2 17 5 11
Mathematics Teacher’s Guide Grade 4
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Unit 16
Double, halve and round off for
estimations and calculations
Mental Maths Learner’s Book page 203
1, 2. Learners solve number puzzles that involve doubling and
halving. They have to use inverse operations and/or trial
an improvement to solve the puzzles. Give them scraps
of paper to write number sentences and solve them to
prevent shouting out and give other learners time to think
for themselves. Let learners work in pairs. Learners show
the solutions after you have read a puzzle. Discuss the
strategies in the solutions below with them and show them
the substitution in the number sentences.
3. Ask different learners to demonstrate how they would
solve 67 + 18. They compare and justify their strategies
and negotiate which is the most effective. Let them
explore the calculation in the example. The learner in the
picture rounded off both numbers and then subtracted
(compensation). Ask the learners to compare their strategies
with this one.
4. Ask the learners to solve 36 + 49. Check whether they
use rounding off and compensation as in the example.
They explore John and Tom’s estimation strategies for the
problem. They should notice that John rounded off to the
nearest 5 and 10. Tom rounded off to the nearest 10 only.
Ask the learners to compare the estimates with the accurate
solution and explain which estimation strategy is more
effective.
Solutions
1,2 a) 2 × n – 3 = 15
b)n ÷ 2 + 7 = 27
15 + 3 = 18 27 – 7 = 20
18 ÷ 2 = 9 20 × 2 = 40
2 × 9 – 3 = 15 40 ÷ 2 + 7 = 27
c) n × 2 + 50 = 550
d)n ÷ 2 – 3 = 42
550 – 50 = 500 42 + 3 = 45
500 ÷ 2 = 250 45 × 2 = 90
250 × 2 + 50 = 550 90 ÷ 2 – 3 = 42
e) n × 2 + 4 = 70
70 – 4 = 66
66 ÷ 2 = 33
33 × 2 + 4 = 70
3. Learners explain their strategies.
4.
272
Math G4 TG.indb 272
36 + 49 = 40 + 50
= 90
90 – 4 – 1 = 85
Mathematics Teacher’s Guide Grade 4
TERM 3
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5. John’s strategy is more effective than Tom’s. He rounded
36 off to the nearest 5 so that the estimate is the accurate
solution. He subtracted 1 and added 1 (+ 1 – 1 are additive
inverses) this does not influence the solution.
Activity 16.1
Learner’s Book page 204
The learners estimate the solutions to addition and subtraction
calculations with 3- and 4-digit numbers. They have to decide on
the best strategy – rounding off to nearest 10, 100 or 1 000. Some
learners may prefer to round off to the nearest 5. They use their own
strategies to calculate the accurate solutions and compare them with
their estimates. You could show them the estimates below if they
have not used them.
Solutions
1. a) 345 + 475 = 820
(round off to the nearest 5)
b) 570 + 425 = 1 000(round off one number to the
nearest 10 and the other to the
nearest 5)
c) 910 + 600 = 1 510(round off to the nearest 10 and
100)
d) 1 070 + 1 490 = 2 560 (round off to the nearest 10)
e) 2 420 + 1 590 = 4 010 (round off to the nearest 10)
f) 5 900 + 3 175 = 9 075(round off to the nearest 10 or 100
and 5)
g) 720 – 460 = 260
(round off to the nearest 10)
h) 1 800 – 700 = 1 100
(round off to the nearest 100)
i) 4 340 – 2 290 = 2 050 (round off to the nearest 10)
j) 8 720 – 5 860 = 2 860 (round off to the nearest 10)
2. Accurate solutions are given below.
a) 343 + 476 = 819
b) 578 + 425 = 1 003
c) 912 + 594 = 1 506
d) 1 067 + 1 485 = 2 552
e) 2 423 + 1 589 = 4 012
f) 5 899 + 3 174 = 9 073
g) 721 – 456 = 265
h) 1 796 – 689 = 1 107
i) 4 343 – 2 287 = 2 056
j) 8 720 – 5 856 = 2 864
The differences between estimates and accurate solutions
depend on learners’ work.
Mathematics Teacher’s Guide Grade 4
Math G4 TG.indb 273
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Unit 17
Different strategies for calculations
Mental Maths Learner’s Book page 204
1. Copy cards for the addition and subtraction game, I have ...
There are 40 cards. If there are fewer than 40 learners,
give some learners each two cards. If there are more than
40 learners, let some learners work in pairs. Take a card and
start the chain. Read the statement, for example, ‘I have 8.
Who has 4 more?’ The learner with the card who has the
answer to the question reads his or her statement next (I have
12. Who has half of this?), the chain continues. Tell learners
to concentrate and listen carefully, and speak loudly and
clearly. Discourage the shouting out of answers. The chain
ends when your original statement is the answer to the last
question (I have 8). Learners can play the game as often as
there is time. The game develops listening skills, mental
calculation skills and concepts such as more than, less
than, halving, doubling, plus and minus. It also encourages
learners to work together. All learners have to do the
calculations because they might have to answer next.
Activity 17.1
Learner’s Book page 204
1. The problems involve simple vertical column addition. Tell the
learners that they will use this calculation method in Grade 5
with large numbers. This activity introduces this method and
also looks for patterns and relationships. It develops or enhances
awareness or identification of unit digits in numbers with a sum
of 10 or 0 as the unit. They can use this knowledge to check
and justify solutions. Explain how to carry 10s. If they make
mistakes, use breaking up to help them understand (for example:
10 + 6; 20 + 4; 30 + 10 = 40).
3. Learners study Ann’s solutions and look at the units to explain
that the answers are wrong because adding the units should
result in 0 as a unit. They should discover that Ann subtracted
instead of adding the units.
4. Ask the learners to use Thami’s strategy to solve addition
problems with 4-digit numbers that involve carrying. Let them
use subtraction to check the solutions.
5. Learners use use Faizel’s method to subtract 4-digit numbers.
They use addition to check the solutions. Let them use methods
they prefer to do the inverse operation.
Solutions
1. a) 40 (all answers)
b) 21 (all answers)
c) 70 (all answers)
2. Learners discuss what they notice about the calculations.
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3. The units are incorrect in Ann’s solutions. She subtracted the
units instead of adding them. The units in the solutions should
all be 0 because they add up to 10 in each calculation.
4. a) 2 567
2 000
+ 500 + 60 + 7
3 984 + 3 000
+ 900 + 80 + 4
5 000 + 1 400 + 140 + 11
= (5 000 + 1 000) + (400 + 100) + (40 + 10) + 1
= 6 551
b) 4 738
4 000
+ 700 + 30 + 8
3 595 + 3 000
+ 500 + 90 + 5
7 000 + 1 200 + 120 + 13
= (7 000 + 1 000) + (200 +100) +(20 + 10) + 3
= 8 333
c) 5 149
5 000
+ 100 + 40 + 9
2 963 + 2 000
+ 900 + 60 + 3
7 000 + 1 000 + 100 + 12
= (7 000 +1 000) + 100 + 10 + 2
= 8 112
d) 8 376
8 000
+ 300 + 70 + 6
1 776 + 1 000
+ 700 + 70 + 6
9 000 + 1 000 + 140 + 12
= (9 000 + 1 000) + 100 + (40 +10) + 2
= 10 000 + 100 + 50 + 2
= 10 152
e) 7 927
7 000
+ 900 + 20 + 7
1 284 + 1 000
+ 200 + 80 + 4
8 000 + 1 100 + 100 + 11
= (8 000 + 1 000) + (100 + 100) + 10 + 1
= 9 000 + 200 + 10 + 1
= 9 211
5. Faizel first built up numbers by adding 2 to both numbers. He
then counted on to solve the problem. Faizel used addition to
solve subtraction problems. Ask the learners to use Faizel’s
strategy to solve the 4-digit subtraction problems. Let them
check the solutions by doing the inverse operations using
strategies of their choice.
2 465 – 1 678 = n
2 467 – 1 680 → 1 680 + 20 = 1 700
1 700 + 300 = 2 000
2 000 + 467 = 2 467
467 + 20 + 300 = 787
2 465 – 1 678 = 787
Mathematics Teacher’s Guide Grade 4
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a) 2 593 – 1 678 = n
2 595 – 1 680 → 1 680 + 20 = 1 700
1 700 + 300 = 2 000
2 000 + 595 = 2 593
595 + 5 + 300 + 15 = 915
2 593 – 1 678 = 915
b) 5 426 – 2 457 = n
5 429 – 2 460 → 2 460 + 40 = 2 500
2 500 + 2 500 = 5 000
5 000 + 429 = 5 429
2 500 + 429 + 40 = 2 969
5 426 + 2 457 = 2 969
c) 8 217 – 6 329 = n
8 218 – 6 330 → 6 330 + 70 = 6 400
6 400 + 600 = 7 000
7 000 + 1 000 + 218 = 8 218
1 000 + 600 + 218 + 70 = 1 888
8 217 – 6 329 = 1 888
6. Allow learners to use their own strategies and find short cuts
or the easiest ways to do this. If they do not notice that they
can calculate the sum using a systematic method and a pattern,
show them this strategy. Add numbers in pairs starting with the
first number and the last number, then the second number and
the second-last numbers, and so on. This is called the Gauss
method, named after the famous mathematician who invented it.
a) 1; 2; 3; 4; 5; 6; 7
1 + 7 = 8; 2 + 6 = 8; 3 + 5 = 8; 4
3 × 8 + 4 = 28
b) 2; 4; 6; 8; 10; 12; 14
2 + 14 = 16; 4 + 12 = 16; 6 + 10 = 16; 8
3 × 16 = (3 × 10) + (3 × 6) + 8 = 30 + 18 + 8 = 56
c) 1; 3; 5; 7; 9; 11; 13
1 + 13 = 14; 3 + 11 = 14; 5 + 9 = 14; 7
3 × 14 = (3 × 10) + (3 × 4) = 30 + 12 + 7 = 49
276
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View objects
Learner’s Book page 206
Introduction
Learners start visualising single objects and then a bigger area or
a collection of objects from different points of view. First they
practise recognising top views of single items and then work with
plan views of areas such as classrooms and schools with which
learners are familiar. Include other examples that will be familiar
to the learners in your class. Examples can include a sports field, a
church, mosque or temple they see often.
Unit 18
Side views and top views
Let the learners look at a variety of items from the top. Create
distance so that they can view the items from good top positions,
such as standing on chairs, tables, ladders or from other outdoor
positions such as balconies.
Mental Maths Learner’s Book page 206
This is an activity for groups of players. The learners practise
reading with understanding. They develop problem-solving,
logical and critical reasoning skills. The activity promotes
communication and group dynamics and learners develop
effective co-operative learning skills. They also develop spatial
sense and the perspective skills they need for working with space
and shape. They interpret and describe the views of 3-D objects.
The activity helps the learners prepare for the concepts that are
developed in the main lesson.
Rules:
• Each group receives a pack of clue cards and a set of six
cubes: two yellow, two blue, one red and one green cube.
• Learners work together to determine the position of the cubes
in the tower.
• They have to read the clue cards carefully to find out exactly
where each cube goes.
Tell the learners that they have to construct (build) a tower with
the cubes. Learners should realise that they have to read all the
cards and work systematically. They should read with insight and
develop knowledge about position. They should also discover
that above does not imply that a cube must be immediately
above another. The learners cannot perform this task without a
set of cubes. Below is the order in which learners should read
and interpret the clues, from left to right. Do not give them this
order – they need to discover that the clues should be used in a
certain order.
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There are six
cubes in a tower
that is six cubes
high. There is a
yellow cube on
top of the tower.
In a set of cubes,
there are two
yellow, two blue,
one green and
one red cube.
Each blue cubes
shares a face with
the green cube.
No two cubes of
the same colour
touch each other.
One of the yellow
cubes is above
the green cube.
The other yellow
cube is below it.
The red cube is
above the green
cube.
Solutions
yellow
red
blue
green
blue
yellow
Investigation
Learner’s Book page 207
Learners’ own work
Activity 18.1
Learner’s Book page 207
The learners should be able to draw fairly accurate shapes from both
the side and the top view. Remind them that they do not need to
include all the detail when they draw top views. They merely need
to show an accurate shape – the way something looks from above.
Solutions
A.
B.
C.
D.
E.
4
3
2
5
1
Suggested informal assessment questions to ask yourself
• Are the learners able to differentiate between a side view and a
top view?
• How easily can they match side views to top views of single
objects?
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Unit 19
Side views and plan views
Now that learners have had practice in recognising single objects
from different views, they can recognise a group of objects, first in a
classroom setting and then in a school setting.
Most learners will be able to move from viewing a single object
from the top to viewing a bigger area from the top. If they struggle
with this, increase the complexity gradually by having first two
objects, then three objects, and so on, arranged on a surface and
letting learners identify them from a top view.
Activity 19.1
Learner’s Book page 208
1. Let the learners point to the corresponding places and people in
the two views.
2. He or she was probably close to the door – let learners go and
stand at the classroom door to see what the person who made the
drawing would see from that position.
3. a) D
b) Y
4. front
5. L
6. Their seats are close to the windows.
7. The learners will need help with this activity. They could tackle
it as a class first before they attempt it on their own. Ask them
questions such as those below to help them, and fill in the
answers on a rectangle drawn on the board. (Erase the drawing
before learners work on their own on the activity.)
• Where is the board?
• Where is door and where are the windows?
• Where is the teacher’s table?
• Where are the desks?
• How much space is there between the desks?
• How far away from the board or door are the desks?
Activity 19.2
1.
2.
3.
4.
Learner’s Book page 209
Let the learners point to the places on the plan view.
the toilets
It is closer to the classrooms.
The learners engage further with reading the plan view of the
school by working with a partner. Let them take turns to write
questions and write answers.
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Suggested informal assessment questions to ask yourself
• How well are the learners able to recognise places on the plan
views?
• How well are they able to locate corresponding places or
people in the side views and the plan views?
• How good is the learners’ understanding of distance between
places shown on the plan views?
Remedial activities
• If learners struggle to read plan views, first do a few activities
in which you place a group of objects on the floor and let the
learners look at the whole group of objects from the top. Let them
practise identifying the objects and describing where each object
is in relation to other objects.
• Let the learners draw top views of different groups of objects by
looking at the groups of objects from the top.
• Draw simple plan views on the board, starting with two buildings
or areas of a school. Then let the learners slowly familiarise
themselves with the plan view. Add one building at a time, until
the learners are comfortable with looking at a plan view that
shows at least five different buildings or areas.
Extension activities
• Let the learners make a model of their school or classroom.
They need only show the main items in the classroom or main
buildings and areas in the school.
• Challenge the learners to draw a simple plan view of their school.
If the school is big, choose a section of the school for them to
draw.
• Let learners use the plan and explain in words how to get from
one place to another at the school.
Project
Draw a plan view of your school.
1. Look carefully to see which buildings you should show and
where these buildings are in relation to each other.
2. Take rough measurements to help you.
3. Notice the shapes of the buildings and what they would look
like from the top.
4. Draw a plan view of your school.
5. Fill in the names of the different buildings and areas.
6. Write four questions about your map and ask a partner to answer
them.
7. Answer a partner’s questions and check each other’s answers.
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Assessment Project
Let the learners work in pairs or on their own.
Use the following rating scale to assess each learner’s skills.
Criteria
4
3
2
0–1
The learner draws a plan view as
opposed to a side view.
The learner places relevant
buildings and other areas on the
plan.
The learner places the buildings
and areas correctly in relation to
one another.
The learner uses relevant shapes
to represent the buildings and
other areas.
The learner fills in the names of
the different buildings and areas,
or provides a key.
The learner writes four questions
about the map.
The learner writes the answers to
the questions.
The learner answers a partner’s
questions correctly.
Totals
Add up the totals of the rating
scale to give the learners a mark
out of 30.
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Properties of 2-D shapes
Unit 20 Sort 2-D shapes
Learner’s Book page 210
The next three units revise and consolidate work that the learners
did on 2-D shapes in Term 1. They differentiate between different
shapes and investigate more aspects that are related to the properties
of 2-D shapes.
As in Term 1, the learners need many different opportunities to work
with 2-D shapes (including recognising shapes from cardboard cutouts; mathematical drawings; pictures; their surroundings; and they
work with shapes on cardboard cut-outs and geoboards; they draw
shapes on dotted paper or on blank paper, and trace around shapes).
Revise the differences between closed and open shapes, and shapes
with straight and/or curved sides.
Mental Maths Learner’s Book page 210
If learners struggle to name the groups, help them by suggesting
one kind of grouping. They should then be able to use the
suggestion to develop other sorting principles along the same
lines.
Solutions
1. a) Answers may differ.
Two obvious groups are open and closed shapes.
• open shapes: C, E, H, K
• closed shapes: A, B, D, F, G, I, J, L, M.
b, c) Answers will differ.
2. a) Answers may differ.
Three groups could be:
• shapes with curves only: D, E, K, L.
• shapes with straight lines only: B, C, G, F, H, J
• shapes with curves and straight sides: A, I, M.
b) Answers will differ.
3. Answers may differ.
a) Five groups would most likely be as follows:
• open shapes with curves only: E, K
• open shapes with straight sides only: Z, H
• closed shapes with curves only: D, L
• closed shapes with straight sides only: B, G, F, J
• closed shapes with both curves and straight sides:
A, I, M.
b) Answers will differ.
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Activity 20.1
Learner’s Book page 210
1. Give the learners enough time to experiment with manipulating
the string or wool to make their chosen shapes. Try not to
skip over this activity as it will help the learners engage with
properties such as open and closed, and straight and curved in a
practical way. It also helps them to begin developing concepts of
shape size and perimeters or side lengths.
2. A: quadrilaterals
B: triangles
C: pentagons
D: hexagons
Suggested informal assessment questions to ask yourself
• How well are the learners able to identify open and closed
shapes?
• How well can they identify shapes with curved and/or straight
sides?
• How easily can they name the 2-D shapes?
Unit 21
Investigate circles
Learners have been exposed to circles from Grade R and so they
will be able to recognise a circle quite easily. In this unit, learners
practise recognising circles in objects in everyday life. They also
start to identify circles that are used in composite circular shapes.
Activity 21.1
Learner’s Book page 211
1. A: The centre of the daisy is circular. There is one circle.
B: The iris of the eye is circular, and so is the pupil inside it.
There are two circles.
C: The two wheels of the bicycle are circular, but so are the
front and rear chain rings on the bicycle. Learners may
identify four circles.
D: The outer rim of the CD creates one circle, then there are two
smaller circles towards the inside of the CD. There are three
circles altogether.
2. a) A: There are two circles next to each other. The broken lines
show where they overlap.
B: There are three circles that overlap. One circle is on
the left, another on the right, and a third circle is at the
bottom.
C: There are two circles that overlap. One circle is on the
left and a smaller one is on the right.
D: There are two circles that overlap. One is at the bottom
and a smaller one is at the top. In this shape, part of both
circles are not shown.
b) Shapes C and D consist of circles of different sizes.
c) Learners use circles to create composite shapes and patterns.
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Unit 22 Investigate polygons
In this unit, the learners investigate polygons as they use geoboards,
draw polygons on grid paper, and do an investigation about the
corners of a quadrilateral.
Remind the learners about the polygons they learnt about previously.
Mental Maths Learner’s Book page 212
1. Learners should realise that the number of sides (given
in ascending order) form consecutive counting or natural
numbers. Ask learners to find out what shapes with 9 and 10
sides are called (nonagons and decagons).
2. The learners will have learnt about prefixes in the language
class. Let learners explain what they think each prefix means.
They should relate the prefixes to the names of the polygons.
You can relate the prefixes to words used in everyday life
(such as tricycle, triathlon (a sport event consisting of three
activities), pentagram (a star with five points), Pentateuch (the
first five books in the Bible) and octopus).
3. Learners should realise that some shapes share sides. They
should use the associative property (grouping) to calculate the
number of sides in (a) easily. In (b) they could use repeated
addition or multiplication and brackets.
Solutions
1.
Shape
Number of sides
triangle
quadrilateral
pentagon
hexagon
heptagon
octagon
3
4
5
6
7
8
2. a) hepta-: seven
c) hexa-: six
e) octo-: eight
b) tri-: three
d) quad-: four
f) penta-: five
3. a) 6 + 5 + 5 + 4 + 3
= (6 + 4) + (5 + 5) + 3
= 10 + 10 + 3
= 23 sides
5
b) 3 + (5 × 2)
5
= 3 + 10
3
6
= 13 sides
4
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6
3
5
4
2
5
3
2
3
2
2
2
2
2
2
2
2
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Activity 22.1
1–3. A
B
C
D
E
Learner’s Book page 212
hexagon
rectangle or quadrilateral
pentagon
triangle
quadrilateral
Activity 22.2
Learner’s Book page 213
Since they drew polygons in Term 1, learners have learnt about
symmetry – the activity incorporates this concept as well.
1. Practical exercise
2. Square has four lines of symmetry.
A rectangle two lines of symmetry. A fold along the diagonal of
a rectangle is not a line of symmetry.
Suggested informal assessment questions to ask yourself
• How well can the learners describe 2-D shapes in terms of the
number of sides they have?
• How well can the learners draw 2-D shapes on grid or dotted
paper?
Investigation
Learner’s Book page 213
A quadrilateral is a closed shape with four straight sides.
The learners do not need to know that the sum of the angles of any
quadrilateral equals 360°. However, the investigation is included
to provide enrichment work for learners and help start thinking
about the properties of 2-D shapes. The learners will find that the
four corners of any quadrilateral will always fit together around
one point.
Unit 23 Patterns and pictures with 2-D shapes
The learners work with concrete shapes in this unit.
Mental Maths Learner’s Book page 214
Tell the learners that they have to visualise and name the
2-D shapes according to the descriptions. You could let them
draw the shapes to check their solutions. They also work out how
many triangles are in one shape and how many squares there are
in another shape. First let them give the solutions. If they did
not realise it, you should tell them that there are many shapes
embedded in the two shapes in question 7. They can work with
square numbers to find the number of squares in the shape in
question 7(b).
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Solutions
1. rectangle
2. triangles
3. A square can be divided into 4, 9, 16, 25 number of squares.
4. Yes, it could have more than one line of symmetry.
5. five
6. four
7. a) five triangles (four small ones and one big one)
b) nine 1 × 1 squares
four 2 × 2 squares
one 3 × 3 square
9 + 4 + 1
= 9 + 1 + 4
= 14 squares
Activity 23.1
Learner’s Book page 214
Make sure that the triangles are equilateral and have sides the same
length as the sides of the squares. This will help them to build
shapes more easily at this stage, when they mix the two shapes
together to form composite shapes.
Solutions
1. a) Learners use squares to build shapes.
b) A and 2; B and 4; C and 1; D and 3
c) 2 and 3
d) A has four sides; it is a quadrilateral (or rectangle).
B has eight sides; it is an octagon (but learners do not need
to be able to name octagons yet).
C has eight sides; it is an octagon (but learners do not need
to be able to name octagons yet).
D has 12 sides.
e) Learners to create their own composite symmetrical shapes.
2. a) Learners to build the shapes on their own or working with a
partner.
b) A and 4; B and 3; C and 1; D and 2
c) A, B and D are symmetrical.
d) A has three sides; it is a triangle.
B has four sides; it is a quadrilateral.
C has six sides; it is a hexagon.
D has six sides; it is a hexagon.
e) Learners to create three composite symmetrical shapes.
3. Let the learners experiment with creating hexagons and
pentagons.
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Suggested informal assessment questions to ask yourself
• How easily can the learners build bigger composite shapes
with smaller shapes?
• How well are learners able to make composite shapes that are
symmetrical?
Remedial activities
• If learners struggle with the work in this section, give them
problems similar to the ones they did in Term 1. The activities
in the Learner’s Book have been specifically designed to help
the learners work through concepts progressively. So follow the
format and sequence of the activities in this section, but adapt
the questions slightly so that the learners get more practice in
working with these concepts.
• Remember to let the learners work with concrete objects as
this will help them understand certain concepts easier. Drawing
shapes on blank or dotted paper will also be useful.
Extension activities
• As an extension of the investigation in this section, encourage the
learners to investigate whether the corners of a triangle also fit
around one point. (Of course, they will not fit around a point, but
they will form a straight line.)
• Encourage the learners to build more composite shapes that are
also symmetrical.
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Collect, organise and present data
Learner’s Book page 215
In Term 1, the learners collected, organised, represented and
analysed data. In this term, they continue to do so, but the level at
which they work increases slightly in complexity. Now learners
begin to think more critically about the data because, in most of the
activities, they are given two sets of data that they need to collect,
organise, represent or compare.
Unit 24 Use tally marks
Remind the learners about how to use tally marks and tables to
collect and organise data.
Work through the example in the Learner’s Book with the learners.
Point out that there are two sets of data in the list: one set for boys’
T-shirts and one set for girls’ T-shirts.
Mental Maths 1.
Learner’s Book page 215
Girls’ favourite subjects
Social Sciences
Languages
Mathematics
Natural Sciences
Tally marks
Number
3
4
5
4
Boys’ favourite subjects
Social Sciences
Languages
Mathematics
Natural Sciences
Tally marks
Number
2
4
4
5
2. a) Mathematics
b) Natural Sciences
Activities 24.1 and 24.2
Learner’s Book page 216
The learners should be able to collect data and organise data into
tables. They should also be able to answer questions about the data.
In Activity 24.2 help them choose movies that most of the children
in the class will have seen. Repeat these activities with different data
questions if the learners need more practice. Choose questions about
topics that interest the learners, for example, local sports teams and
weekend activities.
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Suggested informal assessment questions to ask yourself
• How well are the learners able to collect data?
• How accurately are they able to sort through and organise the
data?
• Do they present clear tables of data that make sense?
Unit 25 Show data on graphs
In this unit, the learners represent data in the form of pictographs
and bar graphs.
Mental Maths Learner’s Book page 216
This activity gives the opportunity to check learners’
understanding of the different kinds of graph they have learnt
about and worked with – pictographs, bar graphs and pie charts.
Solutions
1.
2.
3.
4.
5.
B
B
B
C
A
bars
rectangle
circle
pictures
pie chart
Pictographs
The learners should not have trouble understanding how to create
a pictograph. The only trouble some learners may experience is
deciding on the categories of data to present in each row.
Activity 25.1
Learner’s Book page 217
Although the questions in this activity do not give step-by-step
guidance for creating pictographs, learners should be able to work
out that they must start by using the data to draw up tables and then
they can present the data in the form of a pictograph. If learners
get stuck or struggle to solve the problem revise the steps to follow
when making a pictograph.
Solutions
1. Learners compare pictographs.
2. pies
3. cakes
4. chips
5. The learners may first want to create a table in which they add
together the snacks of the two classes.
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Snack
pies
samoosas
chips
cakes
Sifiso’s class
7
5
8
6
Priya’s class
9
6
6
4
Both classes
16
11
14
10
The most popular snack in both classes is pies.
Bar graphs
Remind the learners that we can use a bar graph to compare the
amounts of different items in a set of data. We can get a general
overview or picture of a set of data by comparing the heights of
the bars.
Activity 25.2
Learner’s Book page 217
1, 2. Learners complete the bar graphs.
3. Area B has fewer services, because fewer schools have running
water and electricity than in area A.
Unit 26 Explain data
Learner’s Book page 218
Learners get further practice in analysing data in the form of words,
pictographs, bar graphs and pie charts. The main context of the data
is again awareness of social and environmental issues (recycling).
Ensure that the learners understand that data presented in lists of
words or in sentences and paragraphs are more easy to understand
when presented in the form of tables and graphs first.
Mental Maths Learner’s Book page 218
1. a) Items Hannelie collected:
Type of material
Tally marks
Paper and cardboard
Plastic
Glass
Cans
Number of items
11
5
1
10
b) paper and cardboard
c) glass
2. a) Items Daniel collected:
Type of material
Tally marks
Paper and cardboard
Plastic
Glass
Cans
b) plastic
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Number of items
8
13
3
6
c) glass
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3. They collected similar items and amounts – they both collected
the fewest number of glass items, However, Hannelie collected
more paper and cardboard, and cans than Daniel and he collected
more plastic items than Hannelie.
Suggested informal assessment questions to ask yourself
• How easily are the learners able to change data from words
into the form of a table or graph?
• What areas do they struggle with?
• Do they need more practice in extracting data from words or
do they need practice in accurately representing the data in
another format?
Activity 26.1
Learner’s Book page 219
Learners clean up litter, record the litter they found, and do a
presentation.
Activity 26.2
Learner’s Book page 219
1. a) five
b) hot chocolate: five; tea: three; juice: ten; cool drink: 5;
water: 3
c) juice
2. a) hot chocolate
b) No.
c) In graph B, more children like tea; and fewer children like
juice and cool drink than in graph A.
d) Perhaps it was summer when the data were collected
for graph A and winter when the data were collected for
graph B. This may account for more children preferring
warmer drinks such as hot chocolate and tea in winter and
juice and cool drink in summer. Encourage the learners to
suggest many possible reasons and discuss whether they
agree with what other learners suggest.
Unit 27 More graphs
Mental Maths 1. a)
b)
c)
d)
2. a)
d)
g)
Learner’s Book page 221
four
ten
No, more bicycles than taxis drove past the school.
cars, bicycles, taxis, bakkies (or bakkies, taxis), buses
ten
b) three
c) eight
cars
e) trucks
f) bicycles
It could be Ibrahim’s school because there seems to be a
lot of trucks and bakkies that drove past his school.
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Activity 27.1
Learner’s Book page 221
1. a) They walk to school.
b) 43
c) The same number of children use both forms of transport.
2. a) by taxi
b) 83
c) The same number use both forms of transport.
d) Taxi, car, train, bus, walk and bicycle
3. There is no definite answer to this question. It is probable that
school A is in the small town, but there is no way to be sure of
this from the given data.
4. School A: The pie chart shows that most of the learners walk to
school. This makes up six-eighths of the learners. The rest go to
school by car, bus and taxi.
School B: The pie chart shows that most of the learners travel
to school by taxi and then next by car. The fewest number of
learners walk or ride bicycles to school.
Suggested informal assessment questions to ask yourself
• How well can the learners answer questions from pictographs,
bar graphs and pie charts? Can they find a particular fact (for
example, a fraction, or a number of items represented by
symbols on the pictograph)?
• Do learners need more practice in reading any of the types of
graph?
• Do learners make errors when reading the graphs or are their
fundamental problems with their understanding of the concept?
Do they understand that information is shown in a symbolic
form on a graph, and that the symbols can be explained using
words and numbers?
Project
Learner’s Book page 222
The learners should be able to work on their own to create a
pictograph.
Help learners who struggle by showing them the next step they need
to take. Set aside class time for each stage of the project:
• collecting data (give the learners a homework task if they need to
find information from people at home)
• organise the data in the form of a table
• draw the pictograph
• write a summary of the information in the pictograph.
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Suggested informal assessment questions to ask yourself
• How well are the learners able to work on their own to collect,
organise and present data? Which parts of the process do they
find easiest, and most difficult? (Revise the activity elements
that some learners struggle with, such as deciding how many
pictograph symbols to draw.)
• How well are learners able to summarise data in words?
Remedial activities
If the learners find it difficult to read or analyse graphs, you may
want to give them similar graphs, but with topics that are easier
for them to relate to. For example, use more topics that relate to
personal data, such as number of family members, number of girls
and boys in class, favourite colours, sports, animals and food. Do
not worry too much if you seem to repeat topics. This will help
learners who struggle. When the learners understand how to read
the different kinds of graphs, they can work with topics such as the
suggested environmental and socio-economic topics.
Extension activities
Learners are meant to start considering data sources in Grade 5,
but if the learners are confident at working through the whole data
cycle at Grade 4 level, you may want to let them start thinking about
questions such as:
• Where did the data come from?
• When were the data collected?
• What factors can make a difference in the value of the data
collected? (For example, did people understand the question?
Did they answer the questioner honestly? How was the data
recorded?)
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Numeric patterns
Learner’s Book page 223
Remind the learners about their work with number patterns in
Term 1. This week they will learn more about number patterns
as they study sequences in number grids, bead work patterns and
a tiling pattern. They will work with flow diagrams and number
sentences to explore and create number sequences. They will also
perform an assessment task at the end of the week.
Unit 28 Patterns in number grids
Mental Maths Learner’s Book page 223
1. The learners work with two 100-grids. One has space for
numbers 1 to 100 and the other for numbers 0 to 99. The
learners should have seen these grids before. They have to
look for patterns and count to find the numbers that go in the
green and yellow squares. To find the second number in the
second row, for example, they should understand that 11 goes
below 1. The number next to it is 12. In the second grid, to
find the number below 9, they must add 10 to 9; the number
is thus 19. Ask them to explain how they found the numbers
that go in the coloured blocks.
1
10
12
0
10
9
19
28
24
27
35
50
43
48
57
55
64
61
71
77
83
89
96
83
100
90
99
2. Learners describe the patterns they see in the numbers in the
rows of counting numbers (0 is included). You can encourage
learners to use the term consecutive counting numbers. In
row 2, they could describe the numbers as consecutive natural
numbers because 0 is not included in this row. For example,
ask the to name the consecutive counting or natural numbers
from 70 to 79; the number that is 3 less than 81; and the
consecutive even numbers between 39 and 51. Praise learners
for identifying even simple patterns. The numbers in the
diagonals from the top right starting with 9 to the bottom left
are multiples of 9. Ask learners to explore the digits in the
numbers. The units decrease by 1 and the tens increase by 10.
294
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TERM 3
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You could help learners notice that the sum of the digits in
these numbers is always 9. The difference between these
digits always gives an uneven number, and so on. The
diagonal from the top left starting with 0 to the bottom right
gives multiples of 11. The digits in the numbers are the same
(11; 22; 33; ...). If you add the digits in consecutive multiples
of 11, you will get a new sequence of even numbers (0; 2; 4;
6; ...) Revise multiplication by 11 (32 × 11 = 352 – split the
two digits and insert the sum of the digits between the other
digits).
3. The multiples of 6 are spread across the grid and not in
straight line as with the multiples of 9.
0
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19
20 21 22 23 24 25 26 27 28 29
30 31 32 33 34 35 36 37 38 39
40 41 42 43 44 45 46 47 48 49
50 51 52 53 54 55 56 57 58 59
60 61 62 63 64 65 66 67 68 69
70 71 72 73 74 75 76 77 78 79
80 81 82 83 84 85 86 87 88 89
90 91 92 93 94 95 96 97 98 99
4. The multiples of 8 behave like the multiples of 9 – the units
decrease and the tens increase. If you subtract the digits in
the consecutive numbers, you will get a sequence of even
numbers. If you get the product of the digits you will get:
0; 7; 12; 15; 16; 15; 12; 7; 0. The digits are reversed after
44 (17 and 71; 26 and 62; 35 and 53 and 08 and 80). Ask
the learners to explore whether the same situation will occur
if they shade numbers with digits that have a sum of 7, 10
and 12. They should realise that exploring numbers in the
100-grid allows them to identify a wide variety of number
patterns.
6. Draw the square with the four numbers on the board.
Learners use addition, subtraction, multiplication and
division to explore relationships between the numbers (for
example, 4 + 10 = 14 and 5 + 10 = 15; 4 × 5 = 20 and 14 ×
15 = 210 (10 more); 4 × 15 = 60 and 5 × 14 = 70 (10 more);
15 ÷ 3 = 5 and 14 ÷ 2 = 7).
Mathematics Teacher’s Guide Grade 4
Math G4 TG.indb 295
TERM 3
295
2012/09/14 5:37 PM
7. The learners explore the numbers in the 100-grid with
numbers 1 to 100. They should notice that the numbers in
the rows and columns are multiples of the numbers 1 to 10.
Count in multiples of each number in the first column to find
the rest of the numbers in that column. Ask questions such as
which multiple of 5 is between 40 and 50, name a multiple
of 6 that is smaller than 60 but bigger than 50. Learners can
use the four basic operations to explore the patterns that are
formed by the digits in multiples. If you, for example, add
the digits in the multiples of 3, you get a new sequence:
3; 6; 9; 3; 6; 9; 3; 6; 9; 3. The following pattern is also
interesting:
3; 6; 9; 12; 15; 18; 21; 24; 30.
3+6=9
3 + 9 = 12
6 + 9 = 15
3 + 12 = 15
9 + 12 = 21
6 + 15 = 21, and so on
8. The learners should notice that the numbers in the two
shaded rows and columns form fractions that are equivalent
to 12 (so, 12 = 24 = 147 = 168 and so on). The numbers in the first
and the third row or column form fractions equivalent to 13
and so on). The numbers in the first
(so, 13 = 62 = 93 = 155 = 10
30
and the fourth row or column form fractions equivalent to 14
(so, 14 = 82 = 246 = 369 and so on). You could use this grid in the
units that teach common fractions to help learners explore
equivalent fractions.
9. Remind learners that patterns are not only created by adding
or subtracting numbers. They should also check whether
multiplication or division was used to create the term.
a) 10; 20; 40; ...
Rule: add 10
b) 10 000; 1 000; 100; 10; 1 Rule: divide by 10
c) 50; 40; 30; 20; ...
Rule: subtract 10
d) 20 000; 2 000; 200; 20; 2 Rule: divide by 10
e) 0; 1; 3; 6; 10; 15; ... Rule: add 1, then 2, then 3
and so on
f) 1; 4; 7; 10; 13; ...
Rule: add 3
g) 1; 2; 4; 8; 32; ...
Rule: multiply by 2
h) 1; 2; 4; 7; 11; 16; ... Rule: add 1, then 2, then 3
and so on
Activity 28.I
Learner’s Book page 224
In some of the patterns the difference or ratio is no consistent, but
the learners should not have problems interpreting the relationships.
They explore the colours of the beads to investigate and complete
the flow diagrams.
296
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TERM 3
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Solutions
1. The number of black beads is odd numbers and the orange beads
are arranged in even numbers. They write a number sequence
for each bead string.
a) 1; 2; 3; 4; 5; ...
b) 2; 4; 2; 4; 2; ...
c) 1; 3; 1; 3; ...
d) 2; 6; 2; 6; 2; ...
e) 5; 3; 5; 3; ...
2. Learners should imagine that the bead patterns continue. In (a),
for example the last beads are 5 black beads. If the pattern
continues there will be 6 orange beads. They should also
visualise in (b), for example that if the pattern is extended and
there are 8 green beads, there will be 16 red beads. Ask the
learners to describe the input and output numbers in the flow
diagrams.
a) Black
beads
Orange
b) Green
beadsbeads
Red
beads
5
6
2
4
3
4
2
4
1
2
2
4
Rule for orange beads: + 1
c) Blue
beads
Rule for red beads: × 2
Yellow
d) Brown
beadsbeads
Blue
beads
1
3
2
6
1
3
2
6
1
3
2
6
Rule for yellow beads: × 3
e) Purple
beads
Rule for blue beads: × 3
Pink
beads
5
3
5
3
5
3
Rule: Subtract consecutive even numbers
(5 – 2 = 3; 10 – 4 = 6 and 15 – 6 = 9)
Mathematics Teacher’s Guide Grade 4
Math G4 TG.indb 297
TERM 3
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2012/09/14 5:37 PM
Unit 29 Finding rules
Mental Maths Learner’s Book page 226
1. Learners should notice that 3 is subtracted from a term to
find the next term. The next five terms in the sequence: 18;
15; 12; 9; 6.
2. Let them investigate the numbers in the flow diagram and
find the next two terms in the sequence: 53; 109
3. They explore the sequence 7; 15; 23 and find the rule is add
8. The next three terms: 31; 39; 47
4. The learners explore the function machines to find the rule
for creating the sequence 1; 3; ... They should find out which
operations they have to perform to 1 to get 3. They can do
this by eliminating possibilities. Rule A gives the output
values 1; 3; ... which are the first two terms of the sequence.
Input
Rule A
Output
1
×2–1
1
2
3
3
5
4
7
Input
Rule B
Output
1
×2+1
3
2
5
3
7
4
9
Input
Rule C
Output
1
×4–3
1
2
5
3
9
4
13
5. Learners create two number sequences and flow diagrams to
show the rules for creating the terms in their sequences.
Activity 29.1
Learner’s Book page 226
1. The learners might look, for example, in (c) for the rule + 4
because 4 is added to create the next terms in the sequence
1; 5; 9; 13; 17; 21. This is correct. This should lead to a
discussion and you should tell them that there can be different
descriptions for the same sequence. The additional rule for the
sequence in (b): + 7 – 3.
3. Learners look at the number cards that Alex created and the flow
diagrams he constructed. They use the rules to create five flow
diagrams.
298
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TERM 3
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Solutions
1. a) Rule: + 5
c) Rule: + 7 → – 3
e) Rule: × 2 → – 1
2. a)
b)
c)
d)
e)
b) Rule: – 6
d) Rule: × 2 → + 3
2; 7; 12; 17; 22; 27; 32; 37; 42
64; 58; 52; 46; 40; 34; 28; 22; 16
1; 5; 9; 13; 17; 21; 25; 29; 33
5; 13; 29; 61; 125; 253; 509; 1 021; 2 045
2; 3; 5; 9; 17; 33; 65; 129; 257
3. a)
1
×2
2
+3
2
–1
2
b)
1
–1
9
×2
1
+3
2
c)
1
–1
9
+3
1
×2
2
d)
1
+3
1
×2
2
–1
2
e)
1
+3
1
–1
1
×2
2
Unit 30 Rules for number patterns
Mental Maths Learner’s Book page 226
Ask the learners to explore the relationships between the
numbers in the tiling pattern. (Tile patterns like this one are
also called tessellations. In a tessellation shapes are arranged so
that there are no gaps between them and they do not overlap.)
Learners should explore the numbers in the rows and columns
and also in the diagonals. Tell them to imagine that the pattern
continues and there are numbers that they cannot see.
Solutions
1. Count in intervals of 3 in the rows:
• first row: 1; 4; 7; ...;
• second row: 2; 5; 8; ...
• third row (multiples of 3): 3; 6; 9; ...
Odd numbers in intervals of 6 are on yellow tiles in rows:
1; 7; 13; 19; ...
Odd numbers in intervals of 4 are on diagonal tiles:
1; 5; 9; ...
Reading from bottom left, consecutive odd numbers are on
yellow diagonal tiles: 9; 11; 13; ...
Even numbers in intervals of 6 are on green tiles in the rows:
4; 10; 16; ...
Reading from the left on the middle row, consecutive even
numbers are on green tiles in diagonals: 2; 4; 6; 8; ...
Mathematics Teacher’s Guide Grade 4
Math G4 TG.indb 299
TERM 3
299
2012/09/14 5:37 PM
Reading from the left on the middle row, even numbers with
a difference of 4 are on alternative green tiles on diagonals:
2; 6; 10; 14; ...
Numbers in the columns are consecutive natural numbers.
2. multiples of 3: 24; 27; 30; 60; 90; 120
3. even numbers: 22; 28; 38; 52; 70; 90
4. multiples of 6: 24; 30; 42; 48; 60; 300
Activity 30.1
Learner’s Book page 227
1. Learners study the number pattern.
2. Assist them in understanding that the sequence is arranged in a
vertical order with the rule for creating each term next to each
term. They should understand that a sequence is a range of terms
in a horizontal structure. A series is a range of number sentences
with number sequences arranged vertically.
Solutions
1. a) 7; 9; 11; 13; 15; 17
b) consecutive odd numbers starting with 7
c) 7 = (1 × 2) + 5
9 = (2 × 2) + 5
11 = (3 × 2) + 5
13 = (4 × 2) + 5
3. a) 5; 9; 13; 17; 21
b) 5 = (1 × 4) + 1
9 = (2 × 4) + 1
13 = (3 × 4) + 1
17 = (4 × 4) + 1
21 = (5 × 4) + 1
25 = (6 × 4) + 1
29 = (7 × 4) + 1
33 = (8 × 4) + 1
37 = (9 × 4) + 1
41 = (10 × 4) + 1
4. 4 = (1 × 3) + 1
7 = (2 × 3) + 1
10 = (3 × 3) + 1
13 = (4 × 3) + 1
16 = (5 × 3) + 1
19 = (6 × 3) + 1
22 = (7 × 3) + 1
25 = (8 × 3) + 1
28 = (9 × 3) + 1
31 = (10 × 3) + 1
34 = (11 × 3) + 1
37 = (12 × 3) + 1
300
Math G4 TG.indb 300
Mathematics Teacher’s Guide Grade 4
TERM 3
2012/09/14 5:37 PM
Assessment task 4 Number patterns
Learners complete this assessment task after Unit 31.
1. These are the tens digits in the 8 times and 9 times tables.
Fill in the missing unit digits.
1...; 2...; 3...; 4...; 5...; 6...; 7...; 8...
1...; 2...; 3...; 4...; 5...; 6...; 7...; 8... (7)
2. What is the rule for creating the numbers in each sequence?
a) 3; 9; 15; 21; 27; ...
b) 94; 74; 54; 34; 14; ...
c) 1; 5; 9; 13; 17; ...
d) 110; 99; 88; 77; ...
e) 7; 17; 27; 37; 47; ...
(5)
3. Complete each flow diagram.
a)
a)
a)
a)
a)
b)
a)
b)
b)
b)
b)
c)
b)
c)
c)
c)
c)
d)
c)
d)
d)
d)
d)
e)
d)
e)
e)
e)
e)
f)
e)
f)
f)
f)
f)
f)
f)
222
22
2
666
66
7677
77
7
× 66
×
×
66
×
×
6
×
×
666
×
×
6
×
× 666
×
×
6
×
6
×
× 666
×
×
66
×
×
6
×
×
666
×
×
6
×
× 666
×
×
× 66
×
× 66
––– 333
–– 33
–– 33
–– 33
–– 33
––– 333
–– 33
–– 33
–– 33
–– 33
–– 33
–– 33
–– 33
–– 33
–3
21
21
21
21
21
27
21
27
27
27
27
27
57
57
57
57
57
57 (6)
4. a) Write down the number pattern or sequence for these
numbers up to the 10th term.
5 = (1 × 6) – 1
11 = (2 × 6) – 1
17 = (3 × 6) – 1
23 = (4 × 6) – 1
... = ... – 1
b) What is the twelfth term in the sequence?
Mathematics Teacher’s Guide Grade 4
Math G4 TG.indb 301
(6)
(2)
Total [26]
TERM 3
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2012/09/14 5:38 PM
Assessment task 4 Number patterns
1. 16; 24; 32; 40 or 48; 56; 64; 72; 80
18; 27; 36; 45; 54; 63; 72; 81
2. a)
b)
c)
d)
e)
3. a)
a)
a)
a)
a)
b)
a)
b)
b)
b)
b)
c)
b)
c)
c)
c)
c)
d)
c)
d)
d)
d)
d)
e)
d)
e)
e)
e)
e)
f)
e)
f)
f)
f)
f)
f)
f)
Rule: + 6
Rule: – 20
Rule: + 4
Rule: – 11
Rule: + 10
2
222
232
33
3
4433
44
6644
66
7766
77
797
99
99
9
× 66
×
×
66
×
×
66
×
×
66
×
×
66
×
×
66
×
×
66
×
×
66
×
×
66
×
×
66
×
×
66
×
×
66
×
×
66
×
×
66
×
×
6
×
× 66
3; 9; 15; 21; 27; ...
94; 74; 54; 34; 14; ...
1; 5; 9; 13; 17; ...
110; 99; 88; 77; ...
7; 17; 27; 37; 47; ...
–3
––– 333
–– 33
–– 33
–3
––– 333
–– 33
––– 333
–– 33
––– 333
–– 33
–– 33
–– 33
–– 33
–3
Math G4 TG.indb 302
(5)
(6)
b) twelfth term: 12 × 6 – 1 = 71
Mathematics Teacher’s Guide Grade 4
(7)
9
999
99
21
21
21
21
21
27
21
27
27
27
27
33
27
33
33
33
33
39
33
39
39
39
39
57
39
57
57
57
57
57 (6)
4. a) 5 = (1 × 6) – 1
11 = (2 × 6) – 1
17 = (3 × 6) – 1
23 = (4 × 6) – 1
29 = (5 × 6) – 1
35 = (6 × 6) – 1
41 = (7 × 6) – 1
47 = (8 × 6) – 1
52 = (9 × 6) – 1
58 = (10 × 6) – 1 302
Solutions
(2)
Total [26]
TERM 3
2012/09/14 5:38 PM
Whole numbers: addition and subtraction
Unit 31
Quick calculations
Mental Maths Learner’s Book page 228
1. Make copies of the game cards for the addition and
subtraction game, I have ... The learners played the game
earlier this term and so they should know the rules. The game
allows the learners to practise basic addition and subtraction
facts.
2. Ask the learners to explore the shortcut strategies the learners
in the pictures use to add and subtract 9, 99 and 999. These
learners use compensation by building up to 10, 100 and
1 000.
If time permits, revise the short cut for multiplying by 11 and
show learners the short cuts for multiplying by 25 and 125.
Examples:
28 × 25 = 28 × 100 ÷ 4
= 2 800 ÷ 4
= 700
32 × 125 = 32 × 1 000 ÷ 8
= 32 000 ÷ 8
= 4 000
Solutions
1. Learners play I have ...
2. a) 67 + 9
= 67 + 10 – 1
= 76
c) 148 – 9
= 148 – 10 + 1
= 139
e) 137 – 99
= 137 – 100 + 1
= 38
g) 88 + 999
= 88 + 1 000 – 1
= 1 087
i) 2 000 – 999
= 2 000 – 1 000 + 1
= 1 001
b) 76 – 9
= 76 – 10 + 1
= 67
d) 89 + 99
= 89 + 100 – 1
= 188
f) 1 004 – 99
= 1 004 – 100 + 1
= 905
h) 1 250 + 999
= 1 250 + 1 000 – 1
= 2 249
j) 8 000 – 999
= 8 000 – 1 000 + 1
= 7 001
Mathematics Teacher’s Guide Grade 4
Math G4 TG.indb 303
TERM 3
303
2012/09/14 5:38 PM
Activity 31.1
Learner’s Book page 229
The learners practise short cuts to add and subtract 9, 99 and 999.
Explore the strategies with them. Refer to the strategies given with
the Mental maths solutions.
Solutions
1. 57 + 9 = 66
234 + 9 = 233
2. 74 – 9 = 63
157 – 9 = 148
3. 78 + 99 = 177
576 + 99 = 675
4. 140 – 99 = 41
986 – 99 = 887
5. 85 + 999 = 1 086
874 + 999 = 1 873
6. 1 540 – 999 = 541
7 605 – 999 = 6 606
88 + 9 = 97
3 582 + 9 = 3 591
96 – 9 = 85
1 743 – 9 = 1 734
123 + 99 = 222
1 986 + 99 = 2 085
444 – 99 = 345
2 421 – 99 = 2 322
357 + 999 = 1 356
5 642 + 999 = 6 641
2 127 – 999 = 1 128
9 304 – 999 = 8 305
Activity 31.2
Learner’s Book page 229
1. The learners should realise that the numbers in each calculation
have been reversed (such numbers are called palindromes).
They explore the solutions to find that in (a) the answers are
always 99. Ask learners to explain why they think this is so.
They should observe that both numbers in the calculations
are multiples of 9. In (b) the numbers are multiples of 8 and
solutions result in multiples of 11. In (c) the pattern is even
more interesting. Some solutions result in for example 444,
666, and so on, while others result in 484, 646, and so on. Ask
the learners to create more 3-digit numbers to find out how this
works.
2. You could also encourage them to experiment with 4-digit
numbers.
3. Let learners explore whether subtracting numbers with reversed
digits works in the same way. For subtraction learners have
to use a big number and reverse it, for example: 62 – 26 = 36;
84 – 48 = 36, and so on. This type of activity helps to create
excitement and experiencing mathematics as fun. The learners’
interest in mathematics will be enhanced and they will develop
an appreciation for numbers.
Solutions
1. a) 99 (All answers are 99.)
b) 77; 66; 121; 88; 99; 132; 55; 110 (All answers are multiples
of 11.)
c) 444; 484; 585; 666; 888; 545; 646
(Notice that the first and the last digits are the same.)
2. Class discussion
304
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Mathematics Teacher’s Guide Grade 4
TERM 3
2012/09/14 5:38 PM
3.
3 814
+ 4 183
7 997
The first and last digits are the same if the original number’s first
and last digit is smaller than 5.
4.
521
72
543
– 125
– 27
– 345
396
55
198
The pattern is not as strong as the addition problems.
Unit 32 Count, order and compare numbers
and place value
Mental Maths Learner’s Book page 229
1. Give the learners copies of the magic square. They worked
with magic squares in Term 2. The numbers in the rows,
columns and diagonals should add up to 33. Learners should
realise that they have to start with the row with two numbers
(add 10 and 19 and subtract the sum from 33) to find the
third number in that row.
2. You could let the learners do this activity when they work
with time. They explore the arrangements of numbers in a
calendar. The difference between the numbers in the rows
is 1, in the columns the difference is 7 and in the diagonals
the difference is 6. Let learners calculate the sums of the
numbers in the rows, columns and diagonals and look for
relationships.
3. When they have completed the activity, encourage learners
to create their own calendar squares and order the numbers
as they would appear on a calendar. Give them a calendar to
explore in which the month’s numbers are arranged as in the
squares they created.
Solutions
1.
18 3 12
5
11 17
10 19 4
30 33 36
2. This is not a magic square.
The middle row, middle column
and the two diagonals have a sum
of 33. All the sums are multiples
of 3. 3
4
5
12
10
11
12
33
17
18
19
54
33
Mathematics Teacher’s Guide Grade 4
Math G4 TG.indb 305
33
TERM 3
305
2012/09/14 5:38 PM
b)
c) 15
14
12 14
13
14 15
16 13
12 14
13 15
14 15
15 16
16 13
13 15
16
14
12 13
16 16
3. a)
19 21
22
21
23 20
19 21
21 22
22 20
20 22
21 22
21 23
20
21 23
20 21
19 20
21 22
26 27
28 30
29
27 28
28 30
29 27
30 28
27 19
28 19
26 28
27 28
28 29
26 27
19
28
4. The learners create their own calendar squares and explain
their processes. You could also ask them to create magic
squares. They should realise that the number in the centre
square should be one third of the magic number. In question
1, for example the magic number is 33 and the number in the
centre is 11; 11 is 13 of 33.
Activity 32.1
Learner’s Book page 230
This activity allows the learners to practise place value, addition and
subtraction. They can make their own cards and place value tables,
or give them copies of the cards. Learners need two sets of cards
marked 0 to 9.
Activity 32.2
Learner’s Book page 230
Learners follow the instructions in the Learner’s Book and play the
game 15.
Unit 33 Problem-solve with whole numbers
Activity 33.1
Learner’s Book page 231
The learners solve word problems involving 4-digit amounts of
money. They have to select the correct number sentences that show
how they would solve the problems.
Solutions
1.
2.
3.
4.
5.
B
C
E
A
D
R5 600 – R3 375 = R2 225
R2 760 – R1 880 = R880
R2 345 + R1 585 = R3 930
R6 355 – R3 285 = R3 070
R1 056 + R985 + R1 955 = R3 996
Assessment task 5: addition and subtraction
In this assessment task, learners use shortcuts to add and subtract
9, 99 and 999 by applying compensation. They solve 3- and
4-digit addition and subtraction calculations using their own
strategies. They solve word problems. They solve addition
and subtraction problems involving multiples of 10, 100 and
1 000. The learners solve basic calculations by filling in missing
numbers in the diagram. They should use effective mental
calculation strategies.
306
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TERM 3
2012/09/14 5:38 PM
Assessment task 5 Addition and subtraction
1. Use shortcuts to calculate the answers.
a) 78 + 9 = n
b) 127 + 99 = n
c) 346 + 999 = n
d) 65 – 9 = n
e) 248 – 99 = n
f) 1 874 – 999 = n(6)
2. Work out the answers.
a) 456 + 458 = n
b) 1 380 + 1 465 = n
c) 4 750 + 4 250 = n
d) 987 – 532 = n
e) 2 290 – 1 310 = n
f) 5 423 – 3 674 = n(6)
3. Lerato and Lolly downloaded songs from the Internet.
a) Lerato downloaded 1 025 songs. Lolly downloaded
845 songs. How many more songs did Lerato download
thanLolly?
b) Then Lolly downloaded another 275 songs. How many
songs did she download altogether?
c) Then Lerato downloaded more songs. She now has
1 476 songs. How many more songs did she download? (6)
4. Calculate the answers.
a) 6 + 9 = n
60 + 90 = n
600 + 900 = n
6 000 + 9 000 = n
b) 8 – 3 = n
80 – 30 = n
800 – 300 = n
8 000 – 3 000 = n
c) 14 – 5 = n
140 – 50 = n
1 400 – 500 = n
14 000 – 5 000 = n(12)
5. Complete the diagrams.
b)
a)
–5
– –105
– 710
–7
+5
++85
++48
+4
– 11
– 11 –53
53
9
–9
+6
+ 6 + 67
67
10
+ 10
–8
––68 – –156
– 15
+9
++39
+7
++73
(16)
Total [46]
Mathematics Teacher’s Guide Grade 4
Math G4 TG.indb 307
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2012/09/14 5:38 PM
Assessment task 5 Addition and subtraction
Solutions
If the learners do not use the strategies suggested below, you should
discuss them with the learners.
1. a) 78 + 9
= 78 + 10 – 1
(compensation)
= 87
b) 127 + 99
= 127 + 100 – 1
= 226
c) 346 + 999
= 346 + 1 000 – 1
= 1 345
d) 65 – 9
= 65 – 10 + 1
= 56
e) 248 – 99
= 248 – 100 + 1
= 149
f) 1 874 – 999
= 1 874 – 1 000 + 1
= 875
(6)
2. a) 456 + 458
= 450 + 450 + 6 + 8
(using near doubles)
= 900 + 14
= 914
b) 1 380 + 1 465
= (1 300 + 1 400) + (80 + 20) + 45
(breaking down)
= 2 700 + 100 + 45
= 2 845
c) 4 750 + 4 250
= 8 000 + (750 + 250)
(associative property)
= 9 000
d) 987 – 532
= (980 – 530) + (7 – 2)
(breaking down)
= 450 + 5
= 455
e) 2 290 – 1 310
= 2 300 – 1 320
(add 10 to both numbers)
= 1 000 – 20
= 980
f)
5 423 5 000 + 400 + 20 + 3
– 3 674 3 000 + 600 + 70 + 4
4 000 + 1 300 + 110 + 13 (decomposition)
– 3 000 + 600 + 70 + 4
1 000 + 700 + 40 + 9
= 1 749
(6)
308
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TERM 3
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3. a) 1 025 – 845 = 180
b) 845 + 275 = 1 120
c) 1 476 – 1 025 = 451
(6)
4. a)
6 + 9 = 15
60 + 90 = 150
600 + 900 = 1 500
6 000 + 9 000 = 15 000
b)
8–3=5
80 – 30 = 50
800 – 300 = 500
8 000 – 3 000 = 5 000
c)
14 – 5 = 9
140 – 50 = 90
1 400 – 500 = 900
14 000 – 5 000 = 9 000
5. a)
48
48
43
43
46
(12)
b)
46 72
– 5 ––510 – 10
–7 –7
42
–4211 – 11
53
53
9
– 9 – 44
45
47
47
38
75
75
71
71
+ 5 ++58 ++84 + 4
44 73
– 8 ––86 ––615 – 15
45
72
73
6
+ 6 +67
67
+ 10 + 10
77
77
+ 9 ++93 ++37 + 7
38 76
76
70
70
74
74
(16)
Total [46]
Mathematics Teacher’s Guide Grade 4
Math G4 TG.indb 309
TERM 3
309
2012/09/14 5:38 PM
Whole numbers: multiplication
Learners use and build on the knowledge they acquired in Term 2.
Unit 34 Multiplication strategies
Mental Maths Learner’s Book page 232
1. Ask learners to explain how to use short cuts to multiply by
11, 99, 25 and 125. They should remember that, for example,
14 × 11 = 154. Separate the digits in 14 and write the sum
of the two digits between them: for example, for 3 × 99,
multiply 3 by 9 and write 9 between them: 3 × 99 = 297.
2. Learners play Multiplication Bingo in the same way as
Addition and Subtraction Bingo.
Activity 34.1
Learner’s Book page 233
Learners practise strategies to multiply easier and smarter. They use
the dot arrays to understand, for example, that 6 × 8 = (5 × 8) + 8;
9 × 6 = (10 × 6) – 9 and 3 × 24 = (3 × 20) + (3 × 4) use the
distributive property of numbers. The learners solve multiplication
with 1- and 2-digit numbers by exploring shorter ways to do so.
Encourage them to make arrays to show their understanding.
Solutions
1. a) 7 × 5 = (2 × 5) + (5 × 5)
b) 9 × 6 = (3 × 6) + (6 × 6)
= 10 + 25
= 18 + 36
= 35
= 36 + 4 + 14
= 54
c) 7 × 4 = (2 × 7) + (2 × 7)
d) 3 × 13 = (3 × 10) + (3 × 3)
= 14 + 14 = 30 + 9
= 28
= 39
e) 4 × 23 = (4 × 20) + (4 × 3) f) 6 × 25 = (4 × 25) + (2 × 25)
= 80 + 12
= 100 + 50
= 92
= 150
g) 9 × 7 = (2 × 7) + (7 × 7)
h) 4 × 8 = (2 × 8) + (2 × 8)
= 14 + 49
= 16 + 16
= 49 + 1 + 13
= 32
= 63
2. a) 9 × 16 = (10 × 16) – 9
b) 9 × 27 = (10 × 27) – 9
= 160 – 9
= 270 – 9
= 151
= 261
310
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Mathematics Teacher’s Guide Grade 4
TERM 3
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c) 9 × 38 = (10 × 38) – 9
d) 9 × 42 = (10 × 42) – 9
= 380 – 9 = 420 – 9
= 371
= 411
e) 5 × 34 = (10 × 34) ÷ 2
f) 5 × 42 = (10 × 42) ÷ 2
= 340 ÷ 2
= 420 ÷ 2
= 170
= 210
g) 5 × 68 = (10 × 68) ÷ 2
h) 5 × 84 = (10 × 84) ÷ 2
= 680 ÷ 2
= 840 ÷ 2
= 340
= 420
i) 20 × 13 = (10 × 13) × 2
j) 20 × 15 = (10 × 15) × 2
= 130 × 2
= 150 × 2
= 260
= 300
k) 20 × 24 = (10 × 24) × 2
l) 20 × 43 = (10 × 43) × 2
= 240 × 2
= 430 × 2
= 480
= 860
Activity 34.2
Learner’s Book page 233
Explore the short cuts with the learners. By now they should have
realised that it is very easy to multiply by 10 or powers of 10 (100
and 1 000, for example). Let them use the strategies to solve the
problems.
Solutions
1. Learners discuss strategies.
2. You could add a step in the process for multiplying by 19 for
learners who struggle with subtraction. They subtract 20 instead
of 19 and then add 1 (so, they use compensation as in the
example below).
a) 19 × 14
= (20 × 14) – 19
= (14 × 2) × 10 – 19
= 280 – 20 + 1
= 261
b) 19 × 23 = 437
c) 19 × 24 = 456
d) 19 × 32 = 608
e) 19 × 44 = 836
f) 50 × 22
= (22 × 100) ÷ 2
= 2 200 ÷ 2
= 1 100
g) 50 × 24 = 1 200
h) 50 × 42 = 2 100
i) 50 × 44 = 2 200
j) 50 × 62 = 3 100
k) 25 × 16
= (16 × 100) ÷ 4
= 1 600 ÷ 4
= 400
l) 25 × 48 = 1 200
m) 25 × 28 = 700
Mathematics Teacher’s Guide Grade 4
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2012/09/14 5:38 PM
n) 25 × 36 = 900
p) 125 × 16
= (16 × 1 000) ÷ 8
= 16 000 ÷ 8
= 2 000
q) 125 × 40 = 5 000
o) 25 × 32 = 800
r) 125 × 48 = 6 000
Unit 35 Basic multiplication facts
Mental Maths Learner’s Book page 234
This activity will enhance learners’ fascination with numbers.
Tell them about the black hole described by astronomers.
They will work with number black holes to find out that they
mysteriously end the calculation procedure with the number
they started with. In Grade 5, they will find out exactly how the
number black hole works.
Draw the table on the board. Learners follow the instructions
for 7 and use their own numbers to find out that the answer is
always 4.
Activity 35.1
Learner’s Book page 234
The learners use their knowledge of basic multiplication facts to
solve the problems with multiples of 10 and multiples of powers
of 10.
1. 21
210
2 100
21 000
2. 32
320
3 200
32 000
3. 30
300
3 000
30 000
4. 48
480
4 800
48 000
5. 42
420
4 200
42 000
6. 56
560
5 600
56 000
7. 72
720
7 200
72 000
8. 63
630
6 300
63 000
9. 40
400
4 000
40 000
10. 54
540
5 400
54 000
Unit 36 Round off and solve problems
Mental Maths Learner’s Book page 235
1. Let the learners play I have ... to practice basic calculations.
2. Learners record their solutions on their Mental maths grids.
They multiply by multiples of 10 and determine unknowns
in different positions in number sentences. They could use
inverse operations but also solve the problems by inspection.
312
Math G4 TG.indb 312
Mathematics Teacher’s Guide Grade 4
TERM 3
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For example, some might ask how many 20s there are in
120, what you have to multiply by 20 to get 120, or reason
that that there are five 20s in 100; so 120 = six 20s. Let them
explain their calculation strategies.
a) 20 × 6 = 120
120 ÷ 20 = 6
b) 3 × 50 = 150
150 ÷ 50 = 3
c) 10 × 52 = 520
520 ÷ 10 = 52
d) 5 × 99 = 495
5 × 9 = 45
e) 15 × 20 = 300
(15 × 2) × 10 = 300
f) 50 × 40 = 2 000
(5 × 4) × 100 = 2 000
g) 20 × 21 = 420
(21 × 2) × 10 = 420 2 × 10 = 20
h) 30 × 40 = 1 200
(3 × 4) × 100 = 1 200
i) 25 × 8 = 200
25 × 4 = 100 double 100 is 200
j) 33 × 11 = 3633 + 3 = 6 between 33 in units and
hundreds
Activity 36.1
Learner’s Book page 235
1. The learners round off both numbers to estimate the solutions.
2. Learners compare the estimates with the accurate solutions.
Solutions
1. a)
b)
c)
d)
e)
f)
g)
h)
i)
j)
2.
33 × 21 ≈ 30 × 20 = 600
38 × 19 ≈ 40 × 20 = 800
16 × 32 ≈ 20 × 30 = 600
54 × 45 ≈ 50 × 50 = 2 500
76 × 28 ≈ 80 × 30 = 2 400
21 × 18 ≈ 20 × 20 = 400
48 × 34 ≈ 50 × 30 = 1 500
25 × 35 ≈ 30 × 40 = 1 200
43 × 39 ≈ 40 × 40 = 1 600
23 × 27 ≈ 20 × 30 = 600
Accurate solutions
a) 33 × 21 = 693
b) 38 × 19 = 722
c) 16 × 32 = 512
d) 54 × 45 = 2 430
e) 76 × 28 = 2 128
f) 21 × 18 = 378
g) 48 × 34 = 1 632
h) 25 × 35 = 875
i) 43 × 39 = 1 677
j) 23 × 27 = 621
Estimates
600
800
600
2 500
2 400
400
1 500
1 200
1 600
600
Differences
93
78
88
70
272
22
132
325
77
21
You could now ask learners to make a generalisation about how
effective estimates are. They could conclude that estimates with
differences less than 100 are good, but those with a difference
Mathematics Teacher’s Guide Grade 4
Math G4 TG.indb 313
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of less than 50 are even more effective. They should notice that
the biggest difference is 325 for the estimation 30 × 40. Ask
the learners why they think this estimate is not effective. They
should realise that there is a big difference between 25 and 30
and 35 and 40.
Activity 36.2
Learner’s Book page 235
1. CDs: 4 × R59,95 ≈ 4 × 60 = R240
2. Gift wrap: 8 × R11,75 ≈ 8 × 12
= (8 × 10) + (8 × 2)
= R96
3. Colour card: 20 × R4,55 ≈ 20 × 5 = R100
4. Paper: 15 × R39 ≈ 15 × 40
= (10 × 40) + (5 × 40)
= 400 + 200
= R600
5. Felt-tip pens: 25 × R24 ≈ 25 × 25
= (25 × 20) + (25 × 5)
= 500 + 125
= R625
6. 12 × R3 = R36
25 × R36 ≈ 25 × 40
= (25 × 40) + (25 × 10)
= 1 000 + 250
= R1 250
R1 250 – R625 = R325
The school saved about R325.
7. 8 boxes
16 boxes
24 boxes
R40
R80
R40 + R80 = R120
8. Colouring pencils: 16 × 19 ≈ 20 × 20 = 400 pencils
9. Colouring pencils: 10 × R15 = R150
100 × R15 = R1 500
10. 50 × R15 = (15 × 100) ÷ 2
= 1 500 ÷ 2
= R750
314
Math G4 TG.indb 314
Mathematics Teacher’s Guide Grade 4
TERM 3
2012/09/14 5:38 PM
Number sentences
Remind the learners about the work they did with number sentences
in Term 1. They have also worked with number sentences when they
have solved problems. Ask learners for examples of problems where
they have used number sentences.
Unit 37 Write number sentences
Mental Maths Learner’s Book page 236
The learners have played I think of a number before. They
should realise that they have to use inverse operations to find the
original number. Ask them to write a number sentence for each
problem to help them solve it. It would be useful to start with the
unknown first and use a place holder for it, for example in (1):
n × 6 = 54 so 54 ÷ 6 = 9 is the number. You could ask them to
make up similar problems. To check solutions, they substitute the
solutions in the number sentences.
Solutions
1. n × 6 = 54
2. n × 2 – 5 = 65
54 ÷ 6 = 9 65 + 5 ÷ 2 = 35
9 × 6 = 54 35 × 2 – 5 = 65
3.
n ÷ 2 × 3 = 75
4. n × 8 = 40
75 ÷ 3 × 2 = 50 40 ÷ 8 = 5
50 ÷ 2 × 3 = 75 5 × 8 = 40
5.
n – 4 = 999
6.
n × 6 = 120
999 + 4 = 1 003 120 ÷ 6 = 20
1 003 – 4 = 999 20 × 6 = 120
Activity 37.1
Learner’s Book page 236
1. Learners use their own methods to solve the problems. If some
of them still use repeated addition, let them compare methods
and also use multiplication. Encourage them to use brackets to
show which calculation is done first. They could, for example in
question 1, write the sentence as:
8 + 8 + 8 + 8 + 8 + 8 + 3 = (6 × 8) + 3 = 48 + 3 = 51.
2. Let learners explore the arrays and make sense of the number
sentences connected to each one. They should understand the
arrays that represent subtraction and division. Ask them to write
number sentences for the arrays in the exercise. Encourage
them to first write the number sentence with a place holder for
Mathematics Teacher’s Guide Grade 4
Math G4 TG.indb 315
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2012/09/14 5:38 PM
the unknown before they solve it. They get practice in using
brackets, and should understand that the calculations in brackets
are performed first, or in the order in which they appear when
there are two sets of brackets.
Solutions
1. a) (6 × 8) + 3 = 24 + 3
b) (7 × 4) + 2 = 28 + 2
= 27 dots
= 30 diamonds
c) (4 × 9) + 5 = 36 + 5
d) (3 × 0) + 4 = 0 + 4
= 41 butterflies
= 4 cats
e) (5 × 7) + 3 = 35 + 3
= 38 flowers
2. a) 5 × 5 = n
b)
5 × 5 = 25 c) (6 × 3) + (3 × 4) = n
d)
18 + 12 = 30
e) 4 × 7 = n
f)
4 × 7 = 28
g) 5 × 0 = n
h)
5 × 0 = 0
3. a) 6 × 9 = 54
c) 72 ÷ 8 = 9
e) 8 × 7 + 6 = 62
(4 × 9) ÷ 4 = n
36 ÷ 4 = 9
4×6=n
4 × 6 = 24
(4 × 8) – (1 × 8) = n
32 – 8 = 24
(3 × 6) + (4 × 5) = n
18 + 20 = 38
b) 100 – 39 = 61
d) 120 – 70 + 36 = 86
Unit 38 Balance and inspect number sentences
Mental Maths Learner’s Book page 238
1. Write the number sentences on the board. Learners might
use various combinations with different numbers to get the
answers, for example in (a), 20 + 10 + 2 = 32. You should,
however, explain that they have to use the same number for
the same place holder (15 + 15 + 2 = 32). In this way they
get practice in creating doubles. This concept is important for
the development of algebraic thinking (which learners will
do in later grades).
Solutions
1. a)
c)
e)
g)
i)
15 + 15 + 2 = 32
14 + 14 + 14 = 42
125 ÷ 5 ÷ 5 = 25
25 + 25 + 25 = 75
250 ÷ 5 ÷ 5 = 10
b)
d)
f)
h)
j)
20 + 20 + 20 + 5 = 65
6 × 6 = 36
50 + 50 + 50 + 50 = 200
150 – 25 – 25 = 100
64 = 4 × 4 × 4
2. Answers will differ.
316
Math G4 TG.indb 316
Mathematics Teacher’s Guide Grade 4
TERM 3
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Activity 38.1
Learner’s Book page 239
FIn this lesson the learners solve number sentences through
inspection and substitution by using balancing scales as models.
They study the balancing scale and should understand that the mass
of the objects on one side and the masses on the other side of the
scale balance (they have the same mass). They should understand
that the total mass of the four boxes is 88 kg. They could solve this
through trial and improvement by reasoning that the boxes could
have different masses that give a sum of 88 kg, for example
40 + 30 + 10 + 8 = 88. The size of the boxes is, however, the same.
Some learners could argue that it depends on the mass of the content
of the boxes. You should encourage debates like these. The strategy
suggests that the boxes have the same mass. Help the learners
understand that n represents one box so that n + n + n + n
represents four boxes – this means that their mass is the same. Help
learners understand the procedure to find the mass of one box. By
now they should know that division is the inverse of multiplication.
Ask them to use the suggested strategy to solve the problems.
Solutions
1.
n + n + n = 39
13 + 13 + 13 = 39
3 × 13 = 39
n = 13 kg
39 ÷ 3 = 13
2.
n + n + n + n + n = 105
21 + 21 + 21 + 21 + 21 = 105
5 × 21 = 105
n = 21 kg
105 ÷ 5 = 21
3.
n + n + n + n + n + n = 126
21 + 21 + 21 + 21 + 21 + 21 = 126
6 × 21 = 126
n = 21 kg
126 ÷ 6 = 21
4.
n + n + n + n = 128
32 + 32 + 32 + 32 = 128
4 × 32 = 128
n = 32 kg
128 ÷ 4 = 32
Mathematics Teacher’s Guide Grade 4
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Unit 39 Equations and problem-solving
Mental Maths Learner’s Book page 239
The learners have to solve problems by substitution. They have to
find out for example, what they should multiply by 10 and 4 to get the
same values on both sides of the equal sign. They do this by trial and
improvement and find out that there could be more than one solution
in some cases.
Solutions
1. n + 18 = 4 × 5
2 + 18 = 4 × 5
20 = 20
2. n + 12 = 3 × n
12 = 12
0 + 12 = 3 × 4
3 + 12 = 3 × 5
6 + 12 = 3 × 6
9 + 12 = 3 × 7, and so on
3. 7 × n + 6 = 3 × n + 5
4.5 × n – 1 = 3 × n + 3
7 × 2 + 6 = 3 × 5 + 5 5 × 2 – 1 = 3 × 2 + 3
20 = 20
9=9
5. 36 ÷ n + 4 = 10 ÷ n + 8
36 ÷ 4 + 4 = 10 ÷ 2 + 8
13 = 13
6. n + 15 = 5 × n
15 = 15
0 + 15 = 5 × 3
5 + 15 = 5 × 4
10 + 15 = 5 × 5, and so on
7. n × n = 4 × 9
8.
n ÷ 3 = 16 ÷ 2
6 × 6 = 4 × 9 24 ÷ 3 = 16 ÷ 2
36 = 36
8=8
9. 20 – n = 8 + n
10 = 10
10. 23 + n = 15 + n
24 = 24
318
Math G4 TG.indb 318
Mathematics Teacher’s Guide Grade 4
20 – 10 = 8 + 2
20 – 11 = 8 + 1
20 – 0 = 8 + 12
20 – 12 = 8 + 0, and so on
23 + 1 = 15 + 9
23 + 2 = 15 + 10
23 + 0 = 15 + 8
23 + 3 = 15 + 11, and so on
TERM 3
2012/09/14 5:38 PM
Activity 39.1
Learner’s Book page 240
1. Make sure that learners understand the contexts. Work
systematically through the example with them to help them
understand the procedure. All the problems involve unknown
values at the start. Let learners use the strategy shown in the
Learner’s Book to solve the problems. This type of problem is
often solved by trial and improvement. You should help learners
understand that they should use inverse operations to find the
solutions. They will discover that there can be more than one
solution. Below are some of the solutions.
2. The input value is the same and the output value is the same.
Learners solved this type of number sentence in Mental
maths ealier this term. Learners should understand that
(n × 5) + 6 = (n × 6) + 1 and solve the number sentences by
trial and error. Encourage them to use inverse operations to
check their solutions, for example: (5 × 5) + 6 = 31 and
(31 – 6) ÷ 5 = 5; 5 × 6 + 1 = 31 and (31 – 1) ÷ 6 = 5.
3. Each flow diagram has two rules. The input values are given
and learners have to use substitution (replace the place holders
with the correct numbers). Ask them to write number sentences
to solve the problems. They start with the rule in which the
numbers are completed. There are various numbers that could
replace the place holders in (b). Encourage the learners to find as
many numbers as possible. Below are examples.
Solutions
1. a) n + 33 = 3 × n
33 ÷ 3 = 11
There were 11 cows in the field.
Later there were 33 + 11 = 44 cows in the field.
b) n + 30 = 6 × n
6 + 30 = 6 × 6
36 = 36
n=6
There were 6 cars at the shopping centre.
36 is 6 times more than 6.
Later there were 36 + 6 = 42 cars at the shopping centre.
c) n + 45 = 5 × n
45 ÷ 5 = n
n=9
There were 9 elephants at the water hole.
Later there were 9 + 45 = 54 elephants at the water hole.
d) n + 250 = 5 × n
250 ÷ 5 = n
n = 50
There were 50 passengers on the ship.
Later there were 50 + 250 = 300 passengers on the ship.
Mathematics Teacher’s Guide Grade 4
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e) n + 440 = 4 × n
440 ÷ 4 = n
n = 110
There were 110 people at the concert.
Later there were 440 + 110 = 550 people at the concert.
2. a) (n × 8) + 4 = (n × 3) + 9 b) (n × 4) + 3 = (n × 5) – 3
(4 × 8) + 4 = (9 × 3) + 9(6 × 4) + 3 = (6 × 5) – 3
32 + 4 = 27 + 9
24 + 3 = 30 – 3
36 = 36
27 = 27
c) (n × 5) – 4 = (n × 4) + 5 d) (n ÷ 3) + 6 = (n ÷ 6) + 10
(9 × 5) – 4 = (9 × 4) + 5 (24 ÷ 3) + 6 = (24 ÷ 6) + 10
45 – 4 = 36 + 5
8 + 6 = 4 + 10
41 = 41 14 = 14
3. a) (◆ × n) + 10 = (◆ × 4) + 16
(2 × 7) + 10 = (2 × 4) + 16
14 + 10 = 8 + 16
14 = 14
b) (n × 3) + 6 = (n × ◆) + n
(3 × 3) + 6 = (3 × 4) + 3
9 + 6 = 12 + 3
15 = 15
(◆ × n) + 10 = (◆ × 4) + 16
(3 × 6) + 10 = (3 × 4) + 16
18 + 10 = 12 + 16
28 = 28
(n × 3) + 6 = (n × ◆) + n
(4 × 3) + 6 = (4 × 2) + 10
12 + 6 = 8 + 10
18 = 18
Transformations
Learner’s Book page 242
In earlier work in the term, the learners started putting smaller
shapes together to make bigger composite shapes. Now they have
more opportunities to build composite shapes. The activities include
working with tangram puzzles, which are excellent to help learners
improve their understanding of shape, space and transformations.
Unit 40 Make new shapes
Give the learners cut-out shapes that they can use to experiment.
Mental Maths 1.
2.
3.
4.
5.
320
Math G4 TG.indb 320
Learner’s Book page 242
two triangles or two squares or two smaller rectangles
two triangles or two rectangles
a square and a triangle
a rectangle or a square or a diamond
four
Mathematics Teacher’s Guide Grade 4
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Activity 40.1
Learner’s Book page 242
Before learners do this activity, let them build simpler shapes with
smaller triangles and squares.
When learners tackle this activity, they will also probably need cut
outs of the shapes if they cannot visualise how the shapes will fit
into the drawings. Learners who have more experience with similar
activities will find it easier to mentally fit the shapes together.
If the learners do not want to work with the concrete shapes, but
want to solve the problems mentally, let them work on scrap paper,
copy the drawings and shapes and then draw shapes and eliminate
shapes as they go along.
Solutions
1, 2. Learners discuss the solutions to the composite shapes.
3. Shapes A and C are symmetrical.
4. The learners can build any shape they wish as long as they use
all the smaller shapes. Encourage them to be creative.
Suggested informal assessment questions to ask yourself
• How well are the learners able to put together smaller shapes to
make bigger composite shapes?
• How well can they identify which composite shapes are
symmetrical and which are not?
Unit 41
Learner’s Book page 243
Tangrams
Tangrams are great fun and working with them can be very
challenging. They are excellent tools to help learners become
familiar with shape manipulation so that they can begin to visualise
moving shapes in space.
Mental Maths Learner’s Book page 243
Ask the learners to study the embedded shapes in the Chinese
tangram. They should imagine that the shapes are cut out. They
visualise and rotate shapes in their minds as they use smaller
shapes to make bigger shapes. There might be more than one
solution to the questions.
Solutions
1.
2.
3.
4.
5.
triangles 3 and 5
triangles 3 and 5
triangles 3, 5 and 7
square 4 and triangles 3 and 5
triangles 3 and 5
Mathematics Teacher’s Guide Grade 4
Math G4 TG.indb 321
TERM 3
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2012/09/14 5:38 PM
Activity 41.1
Learner’s Book page 243
The learners experiment with the tangram pieces. They have
manipulated the shapes mentally in Mental maths and now they
use the physical shapes to reconstruct the tangram square. They
construct the figure of the cat and use their imagination to create
figures of objects.
Activity 41.2
Learner’s Book page 244
Once they are familiar with the ways in which the shapes can be
manipulated, let learners continue with this activity, which is a more
focused activity in which they solve specific problems.
Solutions
1. Learners use combinations of shapes as indicated to construct
other shapes. They should realise, for example, that shapes 3 and
3
5 could be used to create a square or rhombus, a parallelogram
5
3
5
or a bigger triangle.
b) 3 3
a)
33 553 3
35 5 5
3 3 5 5
3
55
55
c)
4 4
5 5
5 5
3 3
55
5
3 533 3
3
5 5
4
4 4
44
53 3
4 4 43
45 5
3 3
5
55
5 5
3
55
3 3
77
7
2. Examples are shown below.
55
3 3 5
7
4
d)
4 47 7
4
3
5
3
44
3
4
3
7
33
3
4
3 34
3
3
3
5
3
5
55
5 4
4
3
5
3
5
5
3
5
4
4 to create different composite
3. They5use
all seven4 tangram
5 pieces
5 33
3
4
4 shapes.
4
3
Suggested informal assessment questions to ask yourself
• How well are the learners able to put together tangram pieces
to make other tangram pieces?
• How well can they build the given tangram figures using the
tangram pieces?
Revision
Learner’s Book page 244
1. a) two
c) six
b) two
d) four
2, 3. Learners build shapes.
322
Math G4 TG.indb 322
Mathematics Teacher’s Guide Grade 4
TERM 3
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Remedial activities
• Give the learners the outlines of shapes, together with shape
cut-outs. They must place the smaller shapes correctly inside
the bigger outline. Start with very simple shapes and build
progressively to more complex shapes.
• Let the learners use only two or three tangram pieces at a time to
build other shapes or pictures. Then let the learners gradually use
more and more pieces at a time.
Extension activities
• Have a class competition using tangram pieces. Let the learners
work in two or three teams and challenge them to put the tangram
pieces together to make various shapes. A learner from each team
can go to front of the classroom and build each shape as fast as
possible. The learner who finishes first scores a point for their
team.
• Ask the learners to make a class poster of tangram figures. They
can use black paper for the tangram figures and glue the pieces in
place on white cardboard. The contrast will give a striking finish
to the poster.
Assignment
Measure the heights of some friends and show the results in a table.
Group the heights like this:
less than 120 cm
120–125 cm
126–130 cm
131–135 cm
more than 135 cm.
• Draw a table to show how many learners belong in each height
group. Draw one row of the table for each height group.
• Write the tally marks and the total number of all the learners in
each height group.
Project
Work on your own to draw a pictograph and write about it.
1. Choose a question to ask your family, friends or classmates.
Here are some ideas:
• What is your favourite sport – soccer, cricket or swimming?
• What do you like to eat for breakfast – eggs, porridge, fruit or
bread?
• How do you get to school – car, taxi, walk, bus or train?
2. Collect your information. Use a table for tally marks.
3. Draw a pictograph to show your information.
4. Write down what your pictograph shows.
Mathematics Teacher’s Guide Grade 4
Math G4 TG.indb 323
TERM 3
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Math G4 TG.indb 324
2012/09/14 5:38 PM
TERM
Working with whole numbers
Unit 1
Revise rules for working with
numbers
Unit 2 Represent numbers and place
value
Unit 3 Problem-solving
Unit 4 Inverse operations
Unit 5 More calculations
Unit 6 Use estimating and
problem-solving
Whole numbers: division
Unit 19 Basic division facts
Unit 20 Divide by 10 and 100
Unit 21 Strategies for division
Perimeter, area and volume
Unit 22 Perimeter
Unit 23 Area
Unit 24 Volume
Revision and consolidation
Mass
Position and movement
Unit 7 Revision
Unit 25 Work with grids
Unit 8 Estimate
Unit 26 Grids on maps
Unit 9 More addition and subtraction
More transformations
Unit 10 More multiplication and
Unit 27 Tessellations
division
Unit 28 Describe patterns
Unit 11 Problem-solving
Geometric patterns
Properties of 3-D objects
Unit 29 Geometric patterns
Unit 12 Recognise and compare
Unit 30 Growing patterns
3-D objects
Unit 13 Faces and models of
3-D objects
Unit 14 Statements about 3-D objects
Common fractions
Unit 15 Order and compare fractions
4
Whole numbers: addition
and subtraction
Unit 31 Use place value to add
and subtract
Unit 32 Use 10-strips to add
and subtract
Unit 16 Calculate with fractions
Data handling
Unit 17 Fractions of whole numbers
Unit 33 Probability
Unit 18 Problem-solving with
Unit 34 Experiments and actual
fractions
outcomes
Revision
Revision
325
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2012/09/14 5:38 PM
Working with whole numbers
Unit 1
Revise rules for working with numbers
The learners revise knowledge of number properties they developed
in the first three terms.
Mental Maths Learner’s Book page 246
Learners use their knowledge of the commutative, associative
and the distributive properties as well as knowledge of the
properties of 0 and 1 to solve problems. Ask learners to explain
their solutions to the class. Learners do not have to use the terms
commutative, associative and the distributive properties in their
explanations.
Solutions
9 + 25 = 36
25 + 9 = 36
(commutative property)
17 + 6 + 13 + 14 = 50
(17 + 13) + (14 + 6) = 50 (associative property)
9 × 4 = (3 × 4) + (3 × 4) (distributive property)
6 × 5 = (2 × 5) + (4 × 5)
28 – 0 = 28(subtract a number from
0 = the number)
h) 35 – 6 + 6 = 35
(0 as the additive inverse)
i) 67 × 1 = 67
(1 as an multiplicative inverse)
j) 100 ÷ 1 = 100
(1 as a multiplicative inverse)
2. Learners explain their solutions.
1. a)
b)
c)
d)
e)
f)
g)
Activity 1.1
Learner’s Book page 246
1. Learners solve the number sentences or equations to justify
their answers. They work with the additive inverses of 0 and
1 and the multiplicative inverse of 1. Remember that 0 is not
a multiplicative inverse although, for example 7 × 0 = 0. Zero
(0) is the identity element for multiplication and addition (an
identity element does not have an influence on a number when
the commutative law is used). Encourage the learners to write
the equal signs below each other in solutions.
Solutions
1. a) 0 + 19 = 19 + 0
True
19 = 19
b) 23 – 0 = 0 – 23
False
23 ≠ 0 – 23
(Learners would probably say that you cannot subtract a big
number from a small number.)
326
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Mathematics Teacher’s Guide Grade 4
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c) 12 × 1 = 1 × 12
12 = 12
d) 5 ÷ 1 = 1 ÷ 5
5≠1÷5
e) 9 + 3 – 3 = 12
9 + 0 ≠ 12
f) 18 – 6 + 6 = 18 – 0
18 – 0 = 18 – 0
18 = 18
g) 456 × 0 = 0 × 456
0=0
h) 39 + 1 = 39
40 ≠ 39
i) 10 × 9 × 2 × 0 = 180
180 × 0 ≠ 180
0 ≠ 180
j) 7 × 2 × 1 = 1 × 2 × 7
14 × 1= 2 × 7
14 = 14
True
False
False
True
True
False
False
True
2. Learners use the commutative, associative and distributive
laws and brackets to show the order in which they will perform
calculations.
a) 26 + 7 + 4 + 13 + 9
b) 4 × 7 × 2 × 1
= (26 + 4) + (13 + 7) + 9
= (4 × 2 × 1) × 7
= 30 + 20 + 9
=8×7
= 59
= 56
c) 108 ÷ 6 ÷ 2
d) 378 ÷ 9
= 108 ÷ 2 ÷ 6
= (360 ÷ 9) + (18 ÷ 9)
= 54 ÷ 6
= 40 + 2
= 9
= 42
e) 95 × 5
f) 315 – 75 – 15
= (90 × 5) + (5 × 5)
= 315 – 15 – 75
= 450 + 25
= 300 – 75
= 475
= 225
g) 225 + 78 – 25
h) 67 – 10 = 77 – 20
= 225 – 25 + 78
57 = 57
= 200 + 78
= 278
i) 5 × 20 = 10 × 10
j) 9 × 5 = 15 × 3
100 = 100 45 = 45
Activity 1.2
Learner’s Book page 247
The learners use knowledge of doubling, halving, addition, division,
the distributive property, building up and breaking down numbers to
solve the problems.
Mathematics Teacher’s Guide Grade 4
Math G4 TG.indb 327
TERM 4
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2012/09/14 5:38 PM
1, 2. Help learners understand the structure of the cross. Learners
should find that 9 is half of the sum of the numbers on the
opposite sides of the red square.
3. Learners must make sure that they skip numbers immediately
above, left, right and below the centre number. Tell learners
that some numbers appear more than once on the big square.
Solutions
1. double 9 = 18
2. 3 + 15 = 18
15 + 3 = 18
3. a) Expect different solutions for 24.
double 24 = 2 × 24 or 24 + 24 = 48
8 + 40 = 48
18 + 30 = 48
48 ÷ 2 = 24
24 is half of 48.
b) Expect different solutions for 32.
double 32 = 2 × 32 or 32 + 32 = 64
16 + 48 = 48 + 2 + 14 = 50 + 14 = 64
24 + 40 = 64
64 ÷ 2 = 32
32 is half of 64.
c) double 56 = 2 × 56
or 56 + 56 = (2 × 50) + (2 × 6) = 100 + 12 = 112
40 + 72 = 40 + 60 + 12= 112
42 + 70 = 30 + 70 + 12 = 112
112 ÷ 2 = 56
56 is half of 112.
d) double 64 = 64 + 64
= (60 + 60) + (4 + 4)
= 128
48 + 80 = 80 + 40 + 8
= 128
48 + 80 = 128
128 ÷ 2 = 64
64 is half of 128.
4. Learners’ give written descriptions of their observations.
Unit 2
Represent numbers and place value
Mental Maths Learner’s Book page 248
Tell learners that they will work with consecutive numbers. Ask
them to explain what they understand about consecutive numbers
and to look at the examples of consecutive counting, even and
odd numbers. Explain the difference between counting numbers
(which include 0) and natural numbers again.
328
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Mathematics Teacher’s Guide Grade 4
TERM 4
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Whole numbers consist of natural and counting numbers.
Without 0, the numbers from 1 can be called natural or counting
numbers. Let learners give more examples of consecutive
numbers.
Learners have to look for three consecutive numbers that will
give the sums on the cards. Encourage them to use effective
strategies to find the sum of three numbers on the grid. For
example, for 25 + 26 + 27 they could add three 25s and add
the difference of 3: 25 + 25 + 25 + 1 + 2 = 78. Learners use the
closest multiples of 5 or 10 that make the calculations easier.
Solutions
1. 27 = 8 + 9 + 10
39 = 12 + 13 + 14
78 = 25 + 26 + 27
66 = 21 + 22 + 23
129 = 42 + 43 + 44
216 = 71 + 72 + 73
97 = 28 + 29 + 30
156 = 51 + 52 + 53
57 = 18 + 19 + 20
147 = 48 + 49 + 50
2. Divide the number by 3: that answer is the middle number of
the three consecutive numbers.
Activity 2.1
Learner’s Book page 248
1. You could make the number cards and give each pair of learners
a set to sort. Ask them to read the numbers aloud. You could also
ask them to write the numbers in words and in place value parts.
2. Ask learners to order the numbers from largest to smallest. They
can also write the numbers in words.
3. Learners write the numbers represented by the place value parts
in the expanded notations.
4. Learners write down the numbers represented on the place value
boards in expanded notation.
5. Learners can draw place value boards to represent the numbers.
They should notice that the numbers become bigger when they
multiply by powers of 10. For example, they should be able to
explain that 340 is 10 times more than 34 or ten 34s.
Solutions
1. Consecutive even numbers: 9 990; 9 992; 9 994; 9 996; 9 998;
10 000; 10 002; 10 004
Consecutive odd numbers: 9 991; 9 993; 9 995; 9 997; 9 999; 10
001; 10 003; 10 005
2. a) 1 + 1 + 1 + 1 + 1 = 5
5×1=5
Mathematics Teacher’s Guide Grade 4
Math G4 TG.indb 329
TERM 4
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2012/09/14 5:38 PM
b) 10 + 10 + 10 + 10 + 10 = 50
5 × 10 = 50
c) 100 + 100 + 100 + 100 + 100 + 5
(5 × 100) + 5 = 505
d) (5 × 1 000) + (5 × 10) + (5 × 1) = 5 000 + 50 + 5
= 5 055
3. a) 600 + 60 + 6 = 666
b) 40 + 3 + 20 000 + 500 = 20 543
c) 8 + 400 + 10 000 + 3 000 = 13 408
d) 1 000 + 50 000 = 51 000
4. a) 10 000 + 1 000 + 600 + 70 + 8 = 11 678
b) 7 000 + 60 = 7 060
c) 10 000 + 400 + 20 + 4 = 10 424
d) 20 000 + 1 000 + 300 + 3 = 21 303
5.
a)
b)
c)
d)
Tth
3
2
Th
4
5
H
0
1
T
0
2
U
0
0
6
0
1
1
0
0
0
0
Number in words
thirty-four thousand
five thousand one hundred and
twenty
six thousand one hundred
twenty thousand one hundred
Activity 2.2
Learner’s Book page 249
The learners will solve these problems mainly by trial and
improvement. Let them battle with the problems before you provide
them with number lines or 100-squares to help them solve the
problems.
1. Extend the activity by asking learners to find four consecutive
even numbers that will give a certain sum.
2. a) The answer is the 2-digit odd number, 15. Revise the terms
sum, difference, product and quotient if you have not done
so before.
d) There are at least six possible solutions. Ask the learners to
explore all options. Odd 3-digit numbers with a difference
of 4 between the digits: 105; 501; 703; 307; 905; 509; and
so on. Ask the learners to think logically. You could give
learners a hint if they get stuck. Tell them to work out which
two odd numbers less than 10 have a difference of 4 (5 and
1; 7 and 3 and 9 and 5). They then have to include 0 and
use the digits in different positions in the numbers. Some
learners might reason that 611 and 161 are possible solutions
because 6 – 1 – 1 = 4. These are acceptable solutions.
Solutions
1. 5 + 7 + 9 + 11
2. a) 15
c) 48
330
Math G4 TG.indb 330
Mathematics Teacher’s Guide Grade 4
b) 25
d) 951; 159; and so on
TERM 4
2012/09/14 5:38 PM
Unit 3
Problem-solving
Mental Maths Learner’s Book page 250
Learners use the addition and subtraction cards, I have ... to
practise mental calculations. They have played the game before
so they should be familiar with the rules.
Activity 3.1
Learner’s Book page 250
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
Remind learners that they have
to do calculations in brackets
first. They practise concepts
such as doubling, halving,
less than, more than, and so
on, and write the expressions
in words, in numbers and
use effective strategies and
number properties to solve
the expressions. If they solve
the statements and shade the
solutions correctly, they will
see the hidden figure.
Solutions
1. double 39
2. (eight 10s plus 20) minus 12
39 + 39 = 30 + 30 + 9 + 9 8 + 20 – 12 = 20 – 12 + 8
= 60 + 10 + 8
= 16
= 78
3. six 10s
4. twelve 5s plus two 2s
6 × 10 = 60 (12 × 5) + (2 × 2) = 60 + 4
= 64
5. 5 less than 100
100 – 5 = 95
6. (12 divided by 3) plus seven 10s
(12 ÷ 3) + (7 × 10) = 4 + 70
= 74
7. double 38
8. nine 8s plus 5
2 × 38 = (2 × 30) + (2 × 8) (9 × 8) + 5 = 72 + 5
= 60 + 16
= 77
= 76
9. 5 less than 90
10. (4 times 20) + (18 ÷ 3)
90 – 5 = 85 (4 × 20) + 6 = 80 + 6
= 86
Mathematics Teacher’s Guide Grade 4
Math G4 TG.indb 331
TERM 4
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11. 17 more than 70
12. half of 196
70 + 17 = 87 196 ÷ 2 = (180 ÷ 2) + (16 ÷ 2)
= 90 + 8
= 98
13. 100 minus 25
14. nine 7s plus 6
100 – 25 = 75 (9 × 7) + 6 = 63 + 6
= 69
15. double 35 plus 3
(2 × 35) + 3 = 70 + 3
= 73
Activity 3.2
Learner’s Book page 250
Ask the learners to work with their groups to solve the non-routine
problem. The learners should understand that they cannot solve
these kinds of problems by just performing the basic operations.
They have to apply logical and creative thinking skills and perform
investigations. You could suggest that the learners use the strategy
below after discussing options to solve the problem. They should
realise that there are four numbers on one page. Ask them to carry
on with the pattern series below to find the number of pages in
the book. They count backwards from 21 and forwards from 52
using pairs of numbers and should observe the patterns of even and
uneven numbers. The activity integrates with the content areas data
handling – sorting and recording data systematically and patterns,
functions and algebra.
Solution
22 and 51
21 and 52
20 and 53
19 and 54
18 and 55
17 and 56
16 and 57
15 and 58
14 and 59
13 and 60
12 and 61
11 and 62
10 and 63
9 and 64
8 and 65
7 and 66
6 and 67
5 and 68
4 and 69
3 and 70
2 and 71
1 and 72
The pages are numbered from 1 to 72, so there are 72 pages in the
book.
Unit 4
Inverse operations
Mental Maths Learner’s Book page 251
The learners solve vertical column addition and subtraction
problems without carrying mentally (they are expected to do
this in Grade 5). Learners should recognise the solutions without
doing serious calculations.
332
Math G4 TG.indb 332
Mathematics Teacher’s Guide Grade 4
TERM 4
2012/09/14 5:38 PM
Ask them to explain how they solve the problems.
Ask the learners to apply inverse operations to check the
solutions for question 2. They display knowledge of place value.
Solutions
1. a) 777
b) 416
c) 243
d) 7 583
e) 5 222
f) 6 897
g) 522
h) 3 333
i) 371
j) 9 100
2. Explanations will differ. (Learners should realise that they
added or subtracted in each column.)
Activity 4.1
Learner’s Book page 251
1. Ask the learners to explore the ancient Egyptian subtraction
calculation strategy of using addition to get the solution. The
problems do not involve carrying. Make sure they understand
the reasoning and use it to solve the problems.
2. The learners look at Aviwe and Alex’s reasoning. They should
realise that their procedures take place in their heads. You do not
add 1s but rather 10s and 100s.
3. Let them use the learners’ reasoning (knowledge of place value)
to solve the problems.
4. Learners work in groups to solve the non-routine problem. They
will start solving it by applying a trial and improvement strategy.
They should realise that the three girls ate different amounts of
sweets. There are four possible combinations.
Solutions
1. a) 1 + 4 = 5
785
b)
3 + 5 = 8
– 231
2 + 5 = 7
554
4+5=9
0+7=7
2+4=6
2+3=5
5 479
– 2 204
3 475
c) 2 + 2 = 4
894
d)
0 + 9 = 9
– 202
2 + 6 = 8
692
e) 0 + 3 = 3
0 + 4 = 4
9 743
2 + 5 = 7 – 7 200
7 + 2 = 9
2 543
2+6=8
3+4=7
1+5=6
5+1=6
6 678
– 5 132
1 546
2. Learners explain whether they agree with the boys’ reasoning.
3. a) 1 + 4 = 5
845
20 + 20 = 40
– 321
300 + 500 = 800
524
b) 0 + 9 = 9
769
0 + 60 = 60
– 400
400 + 300 = 700
369
Mathematics Teacher’s Guide Grade 4
Math G4 TG.indb 333
TERM 4
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2012/09/14 5:38 PM
c) 3 + 3 = 6
10 + 40 = 50
1 856
400 + 400 = 800
– 413
0 + 1 000 = 1 000
1 443
d) 2 + 5 = 7
0 + 80 = 80
5 687
200 + 400 = 600
– 3 202
3 000 + 2 000 = 5 000
2 485
e) 7 + 0 = 7
70 + 0 = 70
9 777
0 + 700 = 700
– 6 077
6 000 + 3 000 = 9 000
3 700
4. Learners should work systematically to find combinations of
three different even numbers with a sum of 20. The activity
integrates with data handling
2 + 4 + 14 = 20
2 + 6 + 12 = 20
2 + 8 + 10 = 20
4 + 6 + 10 = 20
Unit 5
More calculations
Mental Maths Learner’s Book page 252
1. The learners do mental calculations and use small numbers to
find out how much less or more the big numbers are. These
problems need consistent practice because some learners
cannot solve them mentally. Encourage them to use counting
on or back or building up 100s and 1 000s (for example, add
1 to 999 and add 4 to 1 000 to get 1 004 for 5 + 999).
2. Learners solve basic addition and subtraction calculations
instantly, recalling the number facts and looking for
relationships between the numbers. For example, 10 + 7 is 1
more than 9 + 7.
Solutions
1. a)
b)
c)
d)
e)
f)
g)
h)
i)
j)
334
Math G4 TG.indb 334
5 more than 999
3 less than 1 000
4 less than 10 003
8 less than 2 400
12 more than 3 550
9 more than 98
8 less than 231
6 more than 3 298
7 less than 2 001
2 more than 9 999
Mathematics Teacher’s Guide Grade 4
999 + 1 = 4 = 1 004
1 000 – 3 = 997
10 003 – 3 – 1 = 9 999
2 400 – 8 = 2 392
3 550 + 10 + 2 = 3 562
98 + 2 + 7 = 107
231 – 1 – 7 = 223
3 298 + 2 + 4 = 3 304
2 001 – 1 – 6 = 1 994
9 999 + 1 + 1 = 10 001
TERM 4
2012/09/14 5:38 PM
2. a)
b)
c)
d)
e)
f)
g)
h)
i)
j)
10 + 7 = 17
8 + 7 = 15
9 + 6 = 15
14 – 5 = 9
15 – 6 = 9
10 + 6 = 16
9 + 7 = 16
8 + 7 = 15
30 – 11 = 19
30 – 13 = 17
Activity 5.1
9 + 7 = 16
9 + 8 = 17
15 – 5 = 10
14 – 6 = 8
15 – 7 = 8
9 + 6 = 15
8 + 6 = 14
30 – 10 = 20
30 – 12 = 18
30 – 14 = 16
Learner’s Book page 252
Ask the learners to use breaking up numbers into place value parts
to calculate the addition with carrying problems. Let them check
their solutions by using the inverse operation.
Solutions
1. Learners should realise that all the solutions are multiples of
1 000.
a) (300 + 700) + (40 + 50) + (6 + 4)
= 1 000 + 90 + 10
= 1 000 + 100
b) (1 000 + 0) + (200 + 700) + (30 + 60) + (3 + 7)
= 1 000 + 900 + 90 + 10
= 1 900 + 100
= 2 000
c) (2 000 + 0) + (100 + 800) + (20 + 70) + (1 + 9)
= 2 000 + 900 + 90 + 10
= 2 900 + 100
= 3 000
d) (4 000 + 2 000) + (600 + 300) + (40 + 50) + (5 + 5)
= 6 000 + 900 + 90 + 10
= 6 900 + 100
= 7 000
e) (3 000 + 1 000) + (900 + 0) + (50 + 40) + (8 + 2)
= 4 000 + 900 + 90 + 10
= 4 900 + 100
= 5 000
f) (4 000 + 2 000) + (200 + 700) + (60 + 30) + (9 + 1)
= 6 000 + 900 + 90 + 10
= 6 900 + 100
= 7 000
g) (2 000 + 1 000) + (500 + 400) + (30 + 60) + (2 + 8)
= 3 000 + 900 + 90 + 10
= 3 900 + 100
= 4 000
Mathematics Teacher’s Guide Grade 4
Math G4 TG.indb 335
TERM 4
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2012/09/14 5:38 PM
h) (6 000 + 2 000) + (400 + 500) + (70 + 20) + (7 + 3)
= 8 000 + 900 + 90 + 10
= 8 900 + 100
= 9 000
i) (5 000 + 3 000) + (200 + 700) + (50 + 40) + (4 + 6)
= 8 000 + 900 + 90 + 10
= 8 900 + 100
= 9 000
j) (4 000 + 4 000) + (600 + 300) + (60 + 30) + (5 + 5)
= 8 000 + 900 + 90 + 10
= 8 900 + 100
= 9 000
2. Learners use subtraction to check their solutions for question 1.
Activity 5.2
Learner’s Book page 253
1. The learners solve a multi-operation problem. Tell them to break
down the numbers into place value parts.
2. Learners investigate bonds of 4-digit numbers with a sum of
3 000. They can use trial and improvement. Allow them to use
their own strategies and tell them later that they could subtract
any 4-digit number smaller than 3 000 from 3 000 to get the
bonds. So, they would be using inverse operations. Check which
learners use multiples of 10, 100 and 1 000 – these learners
think at a higher level because it is the easiest way to solve the
problem. For example:
3 000 – 1 000 = 2 000 and 2 000 + 1 000 = 3 000
3 000 – 1 500 = 500 and 500 + 1 500 = 3 000
3 000 – 1 750 = 1 250 and 1 250 + 1 750 = 3 000
3. Learners use inverse operations to check solutions.
Solutions
1. 1 500 – 355 – 645 = 500
or, 1 500 – (300 + 645) = 1 500 – 1 000 = 500
500 more tickets were sold.
2. Solutions will differ.
3. Learners use inverse operations to check solutions.
Unit 6
Use estimating and problem-solving
Mental Maths Learner’s Book page 253
Ask the learners to instantly recall as many basic addition and
subtraction number facts as they can in one minute. Tell them
when to start and when to stop while you keep time. You could
have them perform this exercise frequently over a period of time
to see whether they improve their knowledge of basic number
facts. Learners can make copies of the table.
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Solutions
Addition
1. 5 + 8 = 13
2. 7 + 6 = 13
3. 6 + 9 = 15
4. 4 + 8 = 12
5. 5 + 7 = 12
6. 14 + 7 = 21
7. 15 + 8 = 23
8. 18 + 12 = 30
9. 19 + 8 = 27
10. 17 + 7 = 24
One-minute calculations
Subtraction
11. 15 – 8 = 7
12. 14 – 6 = 8
13. 12 – 5 = 7
14. 16 – 9 = 7
15. 17 – 8 = 9
16. 21 – 7 = 14
17. 25 – 8 = 17
18. 23 – 7 = 16
19. 24 – 9 = 15
20. 26 – 7 = 19
Activity 6.1
Learner’s Book page 253
1. Remind the learners that they should round off effectively so
that their estimates are close to the accurate solutions. They
should decide whether rounding off to the nearest 10, 100 or
1 000 and even to the nearest 5 would be more effective, but
that could involve more complicated calculations. They should
realise that rounding off to the nearest 10 is more effective than
rounding off to the nearest 100 and 1 000.
2. Ask learners to use methods that they prefer to calculate the
accurate solutions. Let them compare the estimates to the
accurate solutions by finding the differences. Allow them to use
calculators to do this. (See the table below.) Below are strategies
that you could share with learners when discussing answers if
they have not used these strategies.
3. Learners use inverse operations to check solutions using
strategies that they are comfortable using.
Solutions
1. Example estimations where numbers are rounded off are given
in the table. In the table on the next page (below the calculations
of the answers), the estimates are compared with the accurate
solutions.
Addition
a) 2 345 + 2 453 = n
2 345 + 2 450 = 4 795
b) 1 764 + 1 764 = n
1 760 + 1 760 = 3 520
c) 3 836 + 2 475 = n
3 840 + 2 480 = 6 320
d) 3 078 + 2 906 = n
3 080 + 2 910 = 5 990
e) 1 857 + 1 267 = n
1 860 + 1 270 = 3 130
Subtraction
f) 2 454 – 1 134 = n
2 455 – 1 130 = 1 325
g) 3 150 – 1 075 = n
3 150 – 1 080 = 2 070
h) 4 236 – 2 677 = n
4 240 – 2 680 = 1 560
i) 6 003 – 4 008 = n
6 000 – 4 010 = 1 990
j) 5 500 – 3 432 = n
5 500 – 3 430 = 2 070
Mathematics Teacher’s Guide Grade 4
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2. a) 2 345 + 2 453 = n
2 300 + 2 400 = 4 700
45 + 53 = + 98
4 798
(breaking down to nearest 100)
b) 1 764 + 1 764 = n
1 700 + 1 700 = 3 400
60 + 60 =
120
4+4=+
8
3 528
(doubling)
c) 3 836 + 2 475 = n
3 800 + 2 400 = 6 200
36 + 4 + 71 = + 111
6 311
(breaking down)
(building up)
d) 3 078 + 2 906 = n
3 000 + 2 900 = 5 900
78 + 6 = + 84
5 984
(breaking down)
e) 1 857 + 1 267 = n
1 800 + 1 200 = 3 000
57 + 3 + 60 + 4 = + 124
3 124
f) 2 454 – 1 134 = n
2 450 – 1 130 = 1 320
4–4=–
0
1 320
g) 3 150 – 1 075 = n
3 000 – 1 000 = 2 000
150 – 75 = – 75
2 075
(breaking down)
(building up)
(breaking down)
(breaking down)
(halving)
h)
4 236 – 2 677 = n
4 259 – 2 700 → 4 000 – 2 700 = 1 300 (add 23 to
256 – 0 = 259 both sides)
1 559
i) 6 003 – 4 008 = n
6 008 – 4 008 = 2 000
(compensation)
2 000 – 5 = 1 995
j) 5 500 – 3 432 = n
5 500 – 3 400 = 2 100
(breaking down)
2 100 – 32 = 2 068
Compare estimates with accurate solutions.
a)
b)
c)
d)
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Accurate solution
4 798
3 528
6 311
5 984
Mathematics Teacher’s Guide Grade 4
Estimate
4 795
3 520
6 320
5 990
Difference
3
8
9
6
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e)
f)
g)
h)
i)
3 124
1 320
2 075
1 559
2 068
3 130
1 325
2 070
1 560
2 070
6
5
5
1
2
The learners could conclude that estimates with a difference
less than 10 are very effective.
3. The learners use inverse operations and strategies they prefer to
check the solutions for question 2.
Activity 6.2
Learner’s Book page 253
The learners work in their groups to solve the non-routine problem.
They should have realised by now that there are no fixed procedures
to use to solve this type of problem. They can start by trial and
improvement. They should assume a mass for one of the animals
and reason, for example that the rabbit weighs 5 kg. The rabbit and
cat weigh 23 kg together, so the cat should weigh 18 kg. The rabbit
and dog weigh 26 kg altogether, so the dog weighs 21 kg. The cat
and dog weigh 31 kg altogether. Below is an alternative solution.
Solutions
dog + cat = 31 kg
rabbit + cat = 23 kg
rabbit + dog = 26 kg
If the rabbit weighs 9 kg, the cat weighs 23 – 9 = 14 kg.
The dog weighs 26 – 9 = 17 kg.
Assessment task 1: whole number addition and subtraction
For this assessment task, learners write down numbers
represented by Dienes blocks and flard cards, practise doubling
and halving with 4-digit numbers, solve problems without
carrying and decomposing using vertical column calculations,
and use their own methods to solve problems with carrying and
decomposing.
Mathematics Teacher’s Guide Grade 4
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Assessment task 1 Whole number addition and subtraction
1. Write down the numbers that are represented by the Dienes
blocks.
a)
b)
c)
d)
(4)
2. Write the numbers that represented by the blocks in
question 1 in words.
(4)
3. Which numbers are represented by the flard cards?
a)
b)
000
33 000
3 000
40
40
40
c) 10
10 000
000
10 000
d)
77
7
50
50
50
66
6
10
10 000
000
10 000
66
6
99
9
300
300
300
77 000
000
7 000
70
70
70
300
300
300
55 000
000
5 000
(4)
4. Work out the answers.
a) double 3 425
b) half of 6 486
c) double 2 875
d) half of 8 648
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(4)
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5. Work out the answers. Use vertical column calculations.
a)
4 364
+ 2 325
b)
6 574
– 2 320
c)
5 429
– 3 216
d)
3 215
+ 6 434(4)
6. Use any method you prefer to work out the answers.
a) 7 254 + 1 867 = n
b) 8 445 – 5 678 = n(4)
Total [24]
Mathematics Teacher’s Guide Grade 4
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Assessment task 1 Whole number addition and subtraction
Solutions
1. a)
b)
c)
d)
4 056
2 034
5 256
1 601
(4)
2. a)
b)
c)
d)
four thousand and fifty-six
two thousand and thirty-four
five thousand two hundred and fifty-six
one thousand six hundred and one
(4)
3. a)
b)
c)
d)
3 356
10 046
15 309
7 707
(4)
4. a)
b)
c)
d)
double 3 425 → 3 425 × 2 = 6 850
half of 6 486 → 6 486 ÷ 2 = 3 243
double 2 875 → 2 875 × 2 = 5 750
half of 8 648 → 8 648 ÷ 2 = 4 324
(4)
5. a)
4 364
+ 2 325
6 689
b)
6 574
– 2 320
4 254
c)
5 429
– 3 216
2 213
d)
3 215
+ 6 434
9 649
(4)
6. a) 7 254 + 1 867 = 9 121
b) 8 445 – 5 678 = 2 767
(4)
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Total [24]
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Mass
Unit 7
Revision
Mental Maths Learner’s Book page 254
The learners order and compare whole numbers and fractions,
they calculate fractions of whole numbers in preparation for the
number work they will do when working with mass. Ask them to
explain their solutions.
Solutions
1. a) 750; 1 500; 2 000; 2 500; 2 750; 3 500
b)
c)
2. a)
b)
c)
d)
e)
f)
1 1 1 1 3 3
10 ; 8 ; 3 ; 2 ; 5 ; 4
1 12 ; 1 43 ; 1 54 ; 1 78 ; 2 81 ;
2 14
1
10 of 100 = 100 ÷ 10 × 1 = 10
1
5 of 20 = 20 ÷ 5 × 1 = 4
1
20 of 100 = 100 ÷ 20 × 1 = 5
3
5 of 500 = 500 ÷ 5 × 3 = 300
3
20 of 60 = 60 ÷ 20 × 3 = 9
7
10 of 1 000 = 1 000 ÷ 10 × 7 =
700
3. Learners explain their solutions.
Activity 7.1
Learner’s Book page 254
1. Ask the learners what they remember and understand about
mass and the units in which mass is measured. Let them name
some products that are measured in mass. They will estimate
and order mass and apply knowledge of fractions.
Learners collect their own items, estimate each item’s mass,
weigh the item and work out the difference.
2. 2,5 g; 2 12 kg; 2 14 kg; 2 000 g
3. Lerato could weigh herself, get on the scale holding the dog,
then subtract the first mass from the second one. The difference
is the dog’s mass.
4. a) 200 g
b) 1,6 kg
c) Learners explain their methods.
Mathematics Teacher’s Guide Grade 4
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Unit 8
Estimate
When we estimate the mass of an object, we usually round up or
down to the nearest kilogram.
Mental Maths Learner’s Book page 255
The learners estimate mass by rounding numbers up or down to
the nearest 1 kg, 10 g, 100 g and 1 000 g.
Solutions
1. a) 1 kg 350 g ≈ 1 kg
c) 499 g ≈ 0 kg
b) 2 kg 999 g ≈ 3 kg
d) 16 kg 872 g ≈ 17 kg
2. a) 160 g ≈ 160 g
c) 499 g ≈ 500 g
b) 1 234 g ≈ 1 230 g
d) 12 g ≈ 10 g
3. a) 160 g ≈ 200 g
c) 399 g ≈ 500 g
b) 1 234 g ≈ 1 200 g
d) 2 450 g ≈ 2 500 g
4. a) 1 600 g ≈ 2 000 g
c) 499 g ≈ 0 g
b) 1 234 g ≈ 1 000 g
d) 4 827 g ≈ 5 000 g
5. Explanations will differ.
Activity 8.1
Learner’s Book page 255
1. a) 240 g + 678 g = 240 + 640 + 38 (0 kg + 1 kg = 1 kg)
= 880 + 20 + 18
= 918 g
918 g ≈ 1 kg
b) 3 999 g – 2 541 g = 4 000 – 2 500 – 40 – 2 (4 kg – 3 kg = 1 kg)
= 1 500 – 40 – 2
= 1 458 g
1 458 g ≈ 1 kg
c) 1 499 g + 2 459 g = 1 500 + 2 460 – 2
= 3 038 g
3 038 g ≈ 3 kg
(1 kg + 2 kg = 3 kg)
d) 15 426 g – 989 g → 15 000 – 900 = 14 100
14 100 – 89 = 14 011
14 011 + 426 = 14 037 g
14 037 g ≈ 14 kg
The two methods give the same estimates.
(15 kg – 1 kg
= 14 kg)
2. 275 g × 5 = (200 × 5) + (70 × 5) + (5 × 5)
= 1 000 + 350 + 25
= 1 375 g
1 375 g ≈ 1 kg
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Unit 9
More addition and subtraction
In this unit, learners will do calculations that involve the four basic
operations with mass.
Mental Maths 1
a)
b)
c)
d)
Learner’s Book page 255
245 kg + 73 kg = 318 kg
1 kg 674 g + 3 kg 251 g = 4 kg 925 g
7 kg 320 g + 799g = 8 kg 119 g
8 563 g + 354 g = 8 kg 917 g
2. a) 3 846 g + 48 g + 890 g + 2145 g = 6 929 g = 6 kg 929 g
b) 4 304 g – 76 g = 4 228 g = 4 kg 228 g
3. Learners explained how they worked out the answers.
Activity 9.1
Learner’s Book page 256
1. Show the learners how to use inverse operations to work out the
answers. Examples are give below.
• 500 g + 450 g = 950 g
950 g – 550 g = 400 g
• 355 g + 25 g = 380 g
380 g – 285 g = 95 g
• 725 g – 250 g = 475 g
475 g – 35 g – 195 g = 245 g
It is important with calculations like these that the learners work
through the whole calculation after they have written down the
answer to make sure the number sentence is correct.
a) 500 g + 450 g = 550 g + 400 g
b) 285 g + 95 g = 25 g + 355 g
c) 195 g + 245 g + 35 g = 725 g – 250 g
2. Help learners use decomposition effectively. Below are
strategies you can share with learners when marking the answers
if they have not used them.
a) 973 kg – 389 kg
(900 + 70 + 3) – (300 + 80 + 9)
= (800 + 160 + 13) – (300 + 80 + 9)
= 500 + 80 + 4
= 584 kg
b) 7 kg 305 g – 2 kg 39 g
7 305 g 7 000 + 300
– 2 039 g 2 000 7 000 + 200
– 2 000 5 000 + 200
5 266 g = 5 kg 266 g
+ 0
+ 30
+ 90
+ 30
+ 60
+ 5
+ 9
+ 15
+ 9
+ 6
Mathematics Teacher’s Guide Grade 4
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c) 7 kg – 1 kg 367 g = 7 000 g – 1 367 g
1 367 + 3 = 1 370
1 370 + 30 = 1 400
1 400 + 600 = 2 000
2 000 + 5 000 = 7 000
5 000 + 600 + 30 + 3 = 5 633 g
5 633 g = 5 kg 633 g
d) 3 267 g – 99 g
3 267 – 100 + 1 = 3 167 + 1
= 3 168 g
3 168 g = 3 kg 168 g
3. Revise the meaning of the following words before learners do
the problems in this question:
• the sum of = add
• the difference = subtract
• the product = multiply
• the quotient = divide.
a) 4 kg 300 g + 16 kg 700 g
= (4 + 16) kg + (300 + 700) g
= 20 kg + 1 000 g
= 20 kg + 1 kg
= 21 kg
b) 47 kg 800 g – 43 kg 750 g
= (47 – 43) kg + (800 – 750) g
= 4 kg 50 g
Maleeha is 4 kg 50 g lighter than Patrick.
c) 12 kg of chalk is needed to mark 1 lane
1 kg of chalk is needed to mark 2 lanes
2 kg of chalk is needed to mark 4 lanes
3 kg of chalk is needed to mark 6 lanes
7 kg – 6 kg = 1 kg
He will use 3 kg of chalk to mark 6 lanes and there will be
1 kg of chalk left.
Unit 10
More multiplication and division
Mental Maths Learner’s Book page 256
The learners solve problems involving multiplication and
division with whole numbers and multiples of 10. Encourage
them to use effective calculation strategies and number
properties. They break up multipliers and divisors into smaller
factors and use the distributive property. They should know the
rule for multiplying by 10 and 100.
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Solutions
1. a) 34 kg × 10 = 340 kg
b) 310 g × 20
c) 1 kg 200 g × 4
= 310 × 10 × 2
= (4 × 1) + (4 × 200)
= 3 100 × 2
= 4 kg 800 g
= 6 200 g
d) 42 g ÷ 7 = 6 g
e) 75 g ÷ 3
f) 750 g ÷ 50
= (60 ÷ 3) + (15 ÷ 3)
= (750 ÷ 5) ÷ 10
= 20 + 5
= 150 ÷ 10
= 25 g
= 15 g
g) 45 kg × 100 = 4 500 kg
h) 200 g × 30
i) 2 kg 346 g × 0 = 0 g
= (2 × 3) × 10 × 10
= 6 × 100
= 600 g
j) 250 g ÷ 5
= (25 ÷ 5) × 10
= 5 × 10
= 50 g
2. Learners explain how they worked out the answers.
Activity 10.1
Learner’s Book page 257
The learners solve problems in and out of context. Ask them to
convert the solutions to grams and kilograms. They use what they
learnt in working with numbers to solve these problems (particularly
multiplication and division by multiples and powers of 10). Below
are strategies that you could share with learners when you mark the
answers if they have not used them.
Solutions
1. Learners multiply and divide 3- and 4-digit numbers and use
different operations.
a) 375 g × 10 = 3 750 g
b) 250 g × 4
250 × 2 = 500
250 × 4 = 1 000 g
c) (500 g × 2) + (375 × 2)
1 000 + (300 × 2) + (75 × 2)
= 1 000 + 600 + 150
= 1 750 g
1 750 ÷ 1 000 = 1 remainder 750
1 750 g = 1 kg 750 g
Tell the learners that in future they should perform the
last two steps (conversions between grams and kilograms)
mentally.
Mathematics Teacher’s Guide Grade 4
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d) (250 g × 5) + (3 × 500 g)
= (200 × 5) + (50 × 5) + (3 × 5 × 100)
= 1 000 + 250 + 1 500
= 2 750 g
2 750 g = 2 kg 750 g
e) (2 × 375 g) ÷ 30
= (300 × 2) + (75 × 2) ÷ 30
= (600 + 150) ÷ 30
= (600 ÷ 3 ÷ 10) + (150 ÷ 3 ÷ 10)
= (200 ÷ 10) + (50 ÷ 10)
= 20 + 5
= 25 g
Each bag will weigh 25 g.
f) (500 × 3) ÷ 20
= 1 500 ÷ 20
= (1 500 ÷ 2) ÷ 10
= 750 ÷ 10
= 75 g
Each bag will weigh 75 g
g) 250 g + 375 g + 500 g
= 500 + 375 + 25 + 225
= 500 + 400 + 225
= 900 + 100 + 125
= 1 125 g
1 125 g = 1 kg 125 g
h) Both weigh the same. You could also ask the learners what
they think weighs more, 100 kg of feathers or a 100 kg of
stones.
2. Remind the learners that they should break up the numbers into
place value parts to calculate easier. They apply the distributive
and associative properties to calculate smartly.
a) 37 kg × 3
b) 320 g × 7
= (30 × 3) kg + (7 × 3) kg
= (300 × 7) g + (20 × 7) g
= 90 kg + 21 kg
= 2 100 g + 240 g
= 90 kg + 10 kg + 11 kg
= 2 340 g
= 111 kg
2 340 g = 2 kg 340 g
c) 45 kg × 36
= (40 × 30) kg + (40 × 6) kg + (5 × 30) kg + (5 × 6) kg
= 1 200 kg + 240 kg + 150 kg + 30 kg
= (1 200 + 200 + 100) kg + (40 + 50 + 10) kg + 20 kg
= 1 500 kg + 100 kg + 20 kg
= 1 620 kg
1 620 × 1 000 = 162 000
1 620 kg = 162 000 g
Learners should do conversions (last two lines above)
mentally in future.
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In the next four problems, the learners divide 4-digit numbers.
Check whether their strategies are effective. They should work
with the closest multiples of the divisors.
d) 3 762 g ÷ 6
= (3 000 ÷ 6) g + (720 ÷ 6) g + (42 ÷ 6) g
= 500 g + 120 g + 7 g
= 627 g
e) 4 812 g ÷ 12
f) 6 000 g ÷ 20
= (4 800 + 12) g ÷ 12
= (6 000 ÷ 10) g ÷ 2
= 400 g + 1 g
= 600 g ÷ 2
= 401 g
= 300 g
g) 5 kg ÷ 25 g
= 5 000 g ÷ 25 g
= (50 ÷ 25) g × 100
= 2 g × 100
= 200 g
Unit 11
Problem-solving
Mental Maths Learner’s Book page 257
30 ÷ 6 = 5
300 ÷60 = 5
3 000 ÷ 600 = 5
24 ÷ 8 = 3
240 ÷ 8 = 30
2 400 ÷ 8 = 300
25 × 9 = 225
50 × 34 = 1 700
125 × 16 = 2 000
250 × 2 = 500
250 × 4 = 1 000
250 × 8 = 2 000
2 000 ÷ 500 = 4
9 000 ÷ 300 = 30 8 000 ÷ 200 = 4
R9,99 × 6
9. R5,50 × 8
= R10,00 × 6 – R0,01 × 6
= R5 × 8 + R0,50 × 8
= R60,00 – R0,06
= R40 + R4
= R59,94
= R44
10. R8,50 × 9
= R8,50 × 10 – R8,50
= R85,00 – R8,50
= R76,50
1.
2.
3.
4.
5.
6.
7.
8.
Activity 11.1
Learner’s Book page 258
The learners solve problems in the context of money and
measurement. They have practiced skills and knowledge they
will apply in Mental maths. Below are some strategies that you
could share with the learners. Remind them about how to write
open number sentences to show how they will solve the problems
involving rate and the application of the four basic operations. You
could let them work in groups and make sure that learners with
reading problems understand the structure of the problems.
Mathematics Teacher’s Guide Grade 4
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Solutions
1. a) 42 × 1 = n
1 standard brick weighs 4 kg.
42 × 4 = 168 kg
Zolani must push 168 kg to the site.
b) (2 000 ÷ 200) × 5 = n
= (2 000 ÷ 100 ÷ 2) × 5
= 20 ÷ 2 × 5
= 10 × 5
= R50
Or:
200 g of cheese cost R5.
400 g of cheese cost R10.
800 g of cheese costs R20.
1 600 g of cheese costs R40.
1 600 g + 400 g
= 2 000 g
= 2 kg
R40 + R10 = R50
2 kg of cheese cost R50.
c) 2 340 g × 8 = n
(2 000 g × 8) + (300 g × 8) + (40 g × 8)
= 16 000 g + 2 400 g + 320 g
= 18 720 g
18 720 g = 18 kg 720 g
8 buckets of sand will weigh 18 kg 720 g.
d) 43 kg × 2 = n
double 43 = 86 kg
86 kg = 86 000 g
e) (8 × 375) = (n ÷ 250)
375 × 8 = (300 × 8) + (70 × 8) + (5 × 8)
= 2 400 + (560 + 40)
= 3 000 g
3 000 g = 3 kg
8 tins of coffee weighs 3 kg.
3 000 ÷ 250
1 000 = 250 × 4
3 000 = 4 × 3
= 12
Check:
250 × 12
= (250 × 10) + (250 × 2)
= 2 500 + 500
= 3 000 g
12 boxes of tea will have the same mass as 8 tins of coffee.
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f) 4 kg ÷ 200 g
= 4 000 g ÷ 200
= (4 000 ÷ 2) g ÷ 100
= 2 000 g ÷ 100
= 20 g
Twenty 200 g slabs of cheese have the same mass as 4 kg.
2. a) 2 kg + 1 kg + 1 12 kg + 2 kg = n
2 + 2 + 1 + 1 12 = 6 12 kg
Mr James carries 6 12 kg.
b) 2 kg × R7,99 = n
2 kg potatoes = R7,99
= (R8 × 2) – 2c
= R16 – 2c
= R15,98
1 kg mince = R42,99
1 12 sugar = (1 × R7,50) + 1 12 of R7,502
= R7,50 + 1 12 of R72 + 1 12 of 50c2
= R7,50 + R3,50 + 25c
= R11,25
2 ℓ milk = R14,45/ℓ
= (2 × R14) + (2 × 45c)
= R28,90
Total for groceries
= R15,98 + R42,99 + R11,25 + R28,90
= R16 + R43 + R11 + R29 + 25 c – (2c + 1c + 10c)
= R29 + R11 + R43 + R7 + R9 + 12c
= R40 + R50 + R9,12
= R99,12
Mr James pays R99,12 for the groceries.
c) R50 + (2 × R20) + R5 + (2 × R2) + 10c + 2c
Mathematics Teacher’s Guide Grade 4
Math G4 TG.indb 351
TERM 4
351
2012/09/14 5:38 PM
Properties of 3-D objects
Unit 12
Recognise and compare 3-D objects
Activity 12.1
Learner’s Book page 259
1. a)
c)
e)
g)
i)
rectangular prism
rectangular prism
square-based pyramid
cone
cylinder
2. A
B
C
D
E
F
G
H
I
J
rectangular prism: matches
cone: ice cream cone
square-based pyramid: chocolate box
cylinder: paper towel
sphere: marbles
square-based pyramid: cheese grater
sphere: sponge ball
rectangular prism: cereal box
cone: hat
cylinder: tuna tin
b)
d)
f)
h)
j)
square-based pyramid
cone
cylinder
sphere
sphere
Activity 12.2
1.
Learner’s Book page 260
Features
Number of surfaces
Number of curved surfaces
Number of flat faces
Shapes of flat face(s)
Prism
6
0
6
rectangles
Cylinder
2
1
2
circles
2. The shapes all have curved surfaces.
3. They all have straight edges and flat surfaces.
4. Below are some of the categories that learners might use to
group the objects.
Curved surfaces: cone, sphere and cylinder
Flat surfaces: rectangular prism and square-based pyramid
Flat and curved surfaces: cone and cylinder
Surfaces meeting at a point: cone, prism and pyramid
Unit 13
Faces and models of 3-D objects
Remind the learners what a polyhedron is. Show them examples
of models of polyhedra. Let them notice the flat shapes that make
up the sides of the models. Also let them mention the names of the
shapes of the faces.
352
Math G4 TG.indb 352
Mathematics Teacher’s Guide Grade 4
TERM 4
2012/09/14 5:38 PM
Mental Maths 1.
5.
9.
D
F
B
2. E
6. A
10. G
Activity 13.1
Learner’s Book page 261
3. C
7. F
11. H
4. H
8. B
Learner’s Book page 261
By now, the learners should have sufficient knowledge and skill to
identify the faces they need to make a polyhedron, draw and cut out
the shapes of the faces and then glue them together to make suitable
models.
Activity 13.2
Learner’s Book page 262
This activity gives the learners another perspective from which to
make 3-D models. It helps them broaden their understanding of the
structure of polyhedra, as the focus shifts to building the polyhedron
by creating its edges.
Display all the models and give the learners turns to talk about the
problems they experience when building the models.
If learners struggle with this work:
• Let them work in pairs so that they can exchange ideas and
support each other.
• Encourage them to try their ideas even if they do not work.
Through making errors in building their models, they will
learn important information about the properties of the models.
For example, if the straws are not the correct lengths, the
polyhedron’s shape will be distorted.
Suggested informal assessment questions to ask yourself
• How well are the learners able to identify and match the
2-D shapes with the faces of polyhedra?
• How well can learners choose suitable faces to build a
3-D model?
• How accurately do learners work when creating a 3-D model?
Unit 14
Statements about 3-D objects
Mental Maths Learner’s Book page 262
The learners visualise and name objects according to their
attributes. They answer true and false questions about attributes
of 2-D shapes and count and calculate faces, edges and vertices
of multiple 3-D objects. They should use doubling, repeated
addition and multiplication. Learners are not allowed to use
pictures or objects to answer the questions. Allow them to use
pictures and objects to check their solutions.
Mathematics Teacher’s Guide Grade 4
Math G4 TG.indb 353
TERM 4
353
2012/09/14 5:38 PM
Solutions
1.
2.
3.
4.
5.
6.
7.
8.
cone or cylinder
rectangular prism, cube or square-based pyramid
triangular prism or pyramid
a) false
b) true
c) false
d) false
6 + 6 = 12, or double 6 is 12 faces
4 + 4 = 8, or double 4 is 8 edges
8 + 8 + 8 = 3 × 8 = 24 vertices
3 + 3 + 3 + 3 = 4 × 3 = 12 faces
Activity 14.1
Learner’s Book page 263
Yes.
b) Yes.
No, it has six flat faces.
d) Yes.
No, it has one curved surface and two flat surfaces.
Yes.
g) Yes.
No, a pyramid has only flat faces while a cone has a curved
surface.
i) Yes.
j) No, its base is a square.
They both have only flat faces that meet at the corners. The
rectangular prism only has rectangular faces while the pyramid
has some triangular faces and one rectangular (a square) face.
Learners choose objects to describe.
Learners find their examples of 3-D shapes in the environment.
rectangular prism: four rectangles and two squares
square-based prism: one square and four triangles
1. a)
c)
e)
f)
h)
2.
3.
4.
5.
Remedial activities
• Help the learners memorise the names of the different shapes if
they struggle to remember them. Use word cards and play word–
picture matching games to help them. Talk about the shapes in
the learners’ home languages so that they can use familiar words
and sentence structures to describe the properties of each shape.
• Use models and other real objects to help learners differentiate
between curved and flat surfaces.
• Let learners take models of polyhedra apart to see which shapes
are used to make them. Then let them rebuild each object again.
Extension activities
• Let the learners use cylinders to build models. Tell them that an
object (a cylinder) is made up of two circles and a rectangle that
is rolled up. Challenge learners to build a cylinder and explain
how they did it.
• Challenge learners to build a hexagonal-based pyramid. If they
understand how a square-based pyramid is named, they should
be able to work out which shapes and how many shapes of each
shape are used to make a hexagonal-based pyramid.
354
Math G4 TG.indb 354
Mathematics Teacher’s Guide Grade 4
TERM 4
2012/09/14 5:38 PM
Common fractions
Unit 15
Learner’s Book page 264
Order and compare fractions
Ask learners what they remember about fractions from the work
they did last term. They will learn more about ordering, comparing
and representing fractions and work with equivalent fractions
again. They will find fractions of whole numbers, work with equal
sharing and add fractions. They will use different models and reallife examples. They will complete an assessment task at the end of
the week.
Mental Maths Learner’s Book page 264
1–3. Learners use fraction strips to learn more about fractions
with different denominators that are equivalent. Let them
explore the fractions in the diagram. Ask them to name the
fraction parts that are shaded and not shaded. You could ask
them to give the sum of the shaded and unshaded parts of
the strips. Ask questions such as:
• How many sixths makes one whole?
• How many sevenths do you need to make two wholes?
• How many tenths do you need to make three halves?
4. Let learners use the fraction strips to determine whether the
statements that involve comparing fractions are true or false.
Find out which learners are able to do this without using
the fraction strips. Extend the activity by asking learners to
write the fraction names in ascending or descending order.
Ask them how the order will change if you add another two
shaded parts to each fraction part.
Solutions
1. A 1 whole
E
I
1
5
1
9
B
F
J
1
2
1
6
1
10
C
G
1
3
1
7
D
H
1
4
1
8
2. Each shaded part shows one of the equal parts into which the
whole strip has been divided.
3. The white parts make up the remainder of the whole.
4. a) True.
b) False.
c) False.
d) True.
e) False.
f) False.
g) False.
h) False.
i) True.
j) True.
k) True.
l) False.
m) False.
n) True.
o) True.
Mathematics Teacher’s Guide Grade 4
Math G4 TG.indb 355
TERM 4
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2012/09/14 5:38 PM
Activity 15.1
Learner’s Book page 264
1. Learners should shade the parts roughly – they can do it neatly
for homework.
4. Although the learners are not expected to subtract fractions
formally in Grade 4, you could use the strips to develop
understanding of subtraction if learners have a good
understanding of addition of fractions with the same
denominators.
Solutions
1. Learners compare the fractions they shaded.
2. Discuss all fractions that are equivalent to 23 .
3. Learners can take turns to write fractions that are more than
on the board.
Unit 16
4. a)
1
4
+
1
4
+
1
4
=
3
4
c)
1
9
+
1
9
+
1
9
+
1
9
e)
1
3
+
1
3
+
1
3
=1
g) 1 +
6
7
= 1 76
+
1
9
=
5
9
+
b)
1
5
+
d)
1
8
+ 81 + 81 + 81 + 81 + 81 + 81 = 78
1
5
1
5
+
f) 1 +
1
2
= 1 12
h) 2 +
4
6
= 2 64
1
5
=
1
2
4
5
Calculate with fractions
Mental Maths Learner’s Book page 265
1. Draw a table like the one below on the board. Ask the
learners to name the fractions from the list to write in each
column.
Less than
More than
1
2
1
2
2. As an example, ask learners what the difference is between
the whole numbers 2 and 5 and the fraction 52 . Learners
can make drawings to illustrate the difference. Below is an
example.
2
5
2
5
Solutions
1. Less than 12 : 13 ; 73 ; 82 ; 52 ; 14 ; 83 ; 94 ; 62 ; 81 ;
More than 12 : 54 ; 79 ; 43 ;
2.
356
Math G4 TG.indb 356
3
10
6 5 7 2 6 7 4 8
10 ; 6 ; 10 ; 3 ; 7 ; 8 ; 7 ; 9
4 5 2 6 7 8
5; 6; 3; 7; 8; 9
Mathematics Teacher’s Guide Grade 4
TERM 4
2012/09/14 5:38 PM
3. Examples:
2
8
3
8
1
8
+
+
+
2
8
1
8
3
8
=
=
=
1
4
4
8
4
8
=
=
=
1
1
2
4 + 4 = 4 =
2
1
3
6 + 6 = 6 =
3
2
5
10 + 10 = 10
1
2
1
2
1
2
1
2
1
2
=
1
2
4. Examples:
4
1
5
5 + 5 = 5 =
3
1
4
4 + 4 = 4 =
6
4
10
10 + 10 = 10
1
7
9
3
7
2
5
1
=1
+
+
+
2
9
4
7
3
5
9
9
7
7
5
5
=
=
=
=1
1
3
2
8
5
6
=1
=1
Activity 16.1
+
+
+
2
3
6
8
1
6
=
=
=
3
3
8
8
6
6
=1
=1
=1
Learner’s Book page 265
1. Let the learners look at the example. They add the fractions
by counting on (indicated by single jumps) to the first fraction
indicated by a long jump.
Notice what the learners do when they have to bridge wholes.
They could give either improper or mixed fractions as solutions.
Guide them to understand the relationship between these types
of fraction.
2. Learners represent the addition calculations on number lines and
solve the problems. Check who uses counting on and who does
straightforward addition. Encourage them to use both improper
and mixed fractions in their solutions.
3. Learners have to estimate the fraction parts that have been
eaten or drunk to decide how many parts were in the whole. In
question (c), for example, they should realise that there were
nine chocolates in each box.
Solutions
1. a)
b)
d)
2. a)
c)
2
1
1
1
6 + 6 + 6 + 6
4
3
7
10 + 10 = 10
12
7
5
8 – 8 = 8
1
6
3
4
+
+
4
6
6
4
=
=
5
6
9
4
= 65
or
2
6
+
or 2 14
i)
5
9 +
11
9 –
3
9
5
9
+
e)
b)
3
5
f)
–
11
4
7 = 17
2
6
4
3 = 3 – 3
+
1
5
=
4
5
+
4
7
7
8
h)
8
9
= 69
=
2
8
+
b) Remainder of the cupcakes:
3
8
5
6
c) Remainder of the chocolates:
6
7
5
8
–
=
=
10
7
2
8
2
3
or 1 73
=
+
5
8
2
6
7
6
= 1 65
11
2
9 = 19
cool drink: + = 33 = 1
cake: 43 + 43 = 64 = 1 24
8
9
1
3
+
=
3
9
2
3
=
Mathematics Teacher’s Guide Grade 4
Math G4 TG.indb 357
=
j) 1 54 – 52 = 1 52
3. a) Remainder of the pizza:
e) Remainder of the
–
6
7
2
3
d) 1 12 + 1 74 = 3
=
d) Remainder of the
5
6
5
7
6
3
c)
e) 1 43 + 1 43 = 2 64 or 3 12
g)
=
3
6
TERM 4
357
2012/09/14 5:39 PM
Unit 17
Fractions of whole numbers
Mental Maths Learner’s Book page 267
The learners practise finding fractions of whole numbers. They
identify the number of counters in the whole, the number of
equal parts it has been divided into and the number of counters
shaded to determine the fraction represented by the shaded
counters.
Learners should count the number of shaded dots in one segment
of the circles and multiply by 3 or 4 to get the whole number for
which they have to calculate a fraction.
You could use card board circles divided into 3 or 4 equal parts
to assist learners who still struggle to find fractions of whole
numbers. Let them use counters to represent the equal parts
and find the fractions of multiples of 3 and 4. In this way they
experience and practise the concept practically.
Solutions
1.
1
3
of 15 = 5
2.
2
3
of 18 = 12
3.
1
3
of 21 = 7
4.
1
3
of 24 = 8
5.
1
3
of 30 = 10
6.
3
4
of 12 = 9
7.
1
4
of 20 = 5
8.
2
4
or
9.
3
4
of 24 = 18
10.
1
4
of 32 = 8
1
2
of 8 = 4
Activity 17.1
Learner’s Book page 267
1. They have to overlook the shaded parts in the diagrams in
Mental maths when they refer to the diagrams for this question.
2. Do not tell the learners that the diagrams in the Mental maths
questions could help them solve these problems. Observe how
they solve the problems and which learners realise they could
use the diagrams. When you notice that they do not realise
the connection and that they struggle, encourage them to use
the diagrams. They should realise that 14 of R16 = R4 and
1
5 of R15 = R3, and then explain to which amount they prefer.
Last term, the learners learnt an informal strategy to calculate
fractions of whole numbers. Check which learners use this
strategy and which learners use the fraction circles in the Mental
maths for this unit.
3. Learners have to decide into how many equal sections to divide
a circle and how many counters to put into each section. They
should then shade the number of counters as indicated in the
problems.
Solutions
1 13 of 92 + 1 13 of 92 = 3 + 3 = 6
b) 1 13 of 182 + 1 23 of 182 = 6 + 12= 18
1. a)
358
Math G4 TG.indb 358
Mathematics Teacher’s Guide Grade 4
TERM 4
2012/09/14 5:39 PM
1 13
d) 1 13
e) 1 13
f) 1 43
c)
of 242 + 1 13 of 242 = 8 + 8 = 16
of 202 + 1 13 of 202 = 5 + 5 = 10
of 242 + 1 24 of 242 = 6 + 12 = 18
of 322 + 1 43 of 322 = 24 + 24 = 48
1
3
1
3
b) 13
23
c) 43
23
d) 13
14
e) 13
24
of R16 = R4
3. a)
1
4
of 40 = 10
b)
2
3
of 27 = 18
c)
1
5
of 15 = 3
2
5
of 20 = 8
e)
1
6
of 18 = 3
f)
5
6
of 12 = 10
2. a)
of R15 = R5
This gives more money.
of R24 = R8
of R18 = R12
This gives more money.
of R32 = R24
This gives more money.
of R30 = R20
of R12 = R4
of R16 = R4
The amounts are the same.
of R30 = R10
of R20 = R10
The amounts are the same.
d)
Unit 18
Problem-solving with fractions
Mental Maths Learner’s Book page 268
The learners play Fraction Snap. Make copies of the cards,
enough for the number of pairs of learners in your class.
Learners shuffle the cards and place them face-down on their
desks. Each player gets a turn to draw a card. If a player draws
a card that makes a sum of 2 with a card that was drawn earlier,
the player puts the pair of cards aside. They carry on drawing
cards until all the cards in the pack have been used. If they have
cards left that they have not matched, they shuffle these cards
again and continue playing. The player who has the most pairs of
cards is the winner.
Mathematics Teacher’s Guide Grade 4
Math G4 TG.indb 359
TERM 4
359
2012/09/14 5:39 PM
Activity 18.1
Learner’s Book page 268
1. If possible give the learners multi-link cubes – learners can work
in pairs or in small groups. Learners break the rod in half and
explain what they notice (for example, there are two equal parts;
1
2 of 18 = 9; 9 + 9 = 18 and double 9 = 18). They put the rod
together and break the rod into thirds and sixths. They should
explore which other equal pieces they could break the rod into
(for example, 1s to get 18 one-eighteenths and in 9 equal pieces
to show that 91 of 18 = 2).
2. Learners can work in groups. Learners have not learnt how to
multiply fractions yet and so they should use repeated addition.
In (a), for example, they add fractions to find out how many
lemons are needed to make seven lemon tarts. They should not
have trouble adding mixed fractions repeatedly. Let the learners
explain their thinking and the strategies they use.
Solutions
1. a) 9 + 9 = 18
12 of 18 = 9
1
2
+
1
2
=1
b) 6 + 6 + 6 = 18
13 of 18 = 6
1
3
+
1
3
+
1
3
=1
c) 3 + 3 + 3 + 3 + 3 + 3 = 18
61 of 18 = 3
1
3
+
1
3
+
1
3
+
d) 18 × 1 = 18
181 of 18 = 1
9 × 2 = 18
91 of 18 = 2
18
18
9
9
1
3
+
1
3
+
1
3
=1
=1
=1
2. Below are different strategies that the learners might use at
a more advanced level. Check which learners use doubling.
These learners are thinking and reasoning at a higher level.
Share the strategies below with the whole class if learners have
not used them.
a) 12 + 12 + 12 + 12 + 12 + 12 + 12 = 72 = 3 12 lemons
or 1 12 +
1
2
2 + 1 12 + 12 2 + 1 12 + 12 2 + 12 = 3 12 lemons
or 1 tart needs
360
Math G4 TG.indb 360
1
2
lemon; 1 half =
1
2
lemon
2 tarts need
1
2
+
1
2
lemons; 2 halves = 1 lemon
4 tarts need
2
2
+
2
2
lemons; 4 halves = 2 lemons
6 tarts need
3
2
+
3
2
lemons; 6 halves = 3 lemons
7 tarts need
6
2
+
1
2
lemons; 7 halves = 3 12 lemons
Mathematics Teacher’s Guide Grade 4
TERM 4
2012/09/14 5:39 PM
b) 1 child eats 1 14 apples
2 children eat 1 14 + 1 14 = 2 12 apples
4 children eat 2 12 + 2 12 = 5 apples
5 children eat 5 + 1 14 = 6 14 apples
Or, (5 × 1) + 1 14 + 14 + 14 + 14 + 14 2 = 5 + 54
= 5 + 1 + 14
= 6 14 apples altogether
c) 3 + 3 + 3 + 3 = 12 oranges
14 + 14 + 14 + 14 = 1 orange
14 + 14 + 14 + 14 = 1 orange
14 + 14 + 14 + 14 = 1 orange
3 43 + 3 43 + 3 43 + 3 43 = 15 oranges
Or, 15 ÷ 4 = 3 remainder 3
= 3 43
Each child gets 3 43 of the oranges.
d) 23 + 23 = 43 = 1 and 13 m is used to wrap 2 gifts
23 + 23 = 43 = 1 and 13 m is used to wrap 2 gifts
23 m is needed to wrap 1 gift
1 and 13 + 11 and 13 + 23 2 = 3 and 13 m is needed to wrap 5 gifts
e) 14 + 14 + 14 + 14 = 1 ℓ is for 4 guests
2 ℓ is for 8 guests
4 ℓ is for 16 guests
6 ℓ is for 16 + 8 = 24 guests
24 guests can each get 14 ℓ of cool drink if there are 6 ℓ of
cold drink.
Activity 18.2
Learner’s Book page 268
You can use this activity as an assessment task to check learners’
understanding of fractions.
Solutions
1. Learners look at the number of panes in the two windows and
determine which fraction of the window panes is not visible
behind the curtains. Allow them to make drawings if they
struggle with the concepts. Also encourage learners to use
concepts such as doubling.
a) The open curtains in picture A shows the whole window. In
picture B 13 of the panes is visible.
b)
1
3
23
= 4 panes
double
= 8 panes
double 4 = 8
1
3
=
2
3
2. The learners have to find out which fraction of an iceberg is
under water if only 71 of it is visible above the water 1 77 – 71 =
or 71 + 76 = 77 2.
Mathematics Teacher’s Guide Grade 4
Math G4 TG.indb 361
TERM 4
6
7
361
2012/09/14 5:39 PM
3. a) There are 16 stakes in the fence. Eight of them are painted.
So, 12 of the fence is painted and 12 of it has not been painted.
b) There are 20 stakes in the fence and 8 stakes have been
painted. So, 52 of the fence is painted and 53 of the fence has
not been painted.
4. a)
b)
c)
1
2
3
4
1
4
ℓ = 250 ml + 250 ml = 500 ml = 2 glasses
ℓ = 250 ml + 250 ml + 250 ml = 750 ml = 3 glasses
ℓ = 250 ml = 1 glass
The learners should know that 14 of a litre fills one glass
(250 ml).
5. a)
3
8
5
8
of the circles have been shaded.
of 24 = (24 ÷ 8) × 5
b)
=3×5
= 15
Revision
Learner’s Book page 270
1. Learners work in pairs or in groups to create the cube
construction with 20 cubes (5 layers of 4 cubes each).
2. Learners work on their own. They make drawings to illustrate
their thinking and reasoning. The learners should realise
that they first divide the whole number and then share the
remainder(s) equally. Allow learners who are still depended on
drawings to make drawings. Share the strategies below with
learners if they have not used them.
Solutions
1. a) 52 of 20 = (20 ÷ 5) × 2
b) 14 of 20 = (20 ÷ 4) × 1
= 8 yellow cubes
= 5 green cubes
d) 101 of 20 = (20 ÷ 10) × 1
c) 15 of 20 = (20 ÷ 5) × 1
= 4 blue cubes
= 2 orange cubes
e) 201 of 20 = (20 ÷ 20) × 1
= 1 brown cube
8 + 5 + 4 + 2 + 1 = 20 cubes
This is how they could shade the cubes.
yellow yellow yellow yellow
yellow yellow yellow yellow
362
Math G4 TG.indb 362
green green
blue orange
green green
blue orange
green
blue
blue
brown
Mathematics Teacher’s Guide Grade 4
TERM 4
2012/09/14 5:39 PM
2. Learners will need to use more abstract ways to calculate with
fractions in this question. Until now they used drawings and
repeated addition, but they should start working more effectively
with fractions. They will work with division with remainders
and should understand that the divisor is also the denominator in
the fraction.
a) 14 ÷ 6 = 2 remainder 2
= 2 62
Each child gets 2 and 62 of the pizzas.
You could let them check their solutions by letting them add
the equal parts.
2 62 + 2 62 + 2 62 + 2 62 + 2 62 + 2 62
= 12 126
= 12 + (12 ÷ 6)
= 12 + 2
= 14 pizzas
b) In 1 day, they use
3
4
In 2 days, they use
ℓ.
3
4
+
3
4
=
6
4
= 1 12 ℓ.
In 4 days, they use 1 12 + 1 12 = 3 ℓ.
In 7 days, they use 3 + 1 12 +
3
4
=4+
2
4
+
3
4
= 4 54 = 5 14 ℓ
They use 5 14 ℓ of milk in one week.
c) For 1 frame, he needs 13 m
For 3 frames, he needs 33 = 1 m
For 6 frames, he needs 63 = 2 m
For 9 frames, he needs 93 = 3 m
He can make 9 photo frames with 3 m of a wooden rod.
d) From 1 sausage, she gets 12 + 12 = 2 halves
From 30 sausages, she gets 30 × 2 = 60 halves.
She gets 60 half sausages if she has 30 sausages.
e) 19 ÷ 5 = 3 remainder 4
= 3 54
Each child gets 3 54 of the chocolate bars.
Mathematics Teacher’s Guide Grade 4
Math G4 TG.indb 363
TERM 4
363
2012/09/14 5:39 PM
Whole numbers: division
Unit 19
Learner’s Book page 271
Basic division facts
Although the learners last worked with division formally in Term 2,
they used division when working with multiplication and number
sentences last term. This term, learners will practise basic division
facts, solve division problems with remainders, divide by 10 and
100 and solve division with 3-digit numbers.
Mental maths Learner’s Book page 271
Make copies of the Reach for the moon game. The learners play
in pairs. They throw a dice to decide who goes first. The one
with the higher score starts at 1 and the other at 2. They also use
the two calculations that give 1 and 2 as solutions. Each learner
uses a counter to make moves. They complete the calculations.
The answers determine the number of places they can move.
The smiley faces show how many moves to go forward and
landing-on-the-moon faces send a player spaces back. The player
who reaches Home first is the winner. The game helps learners
practise basic calculations.
You should allow learners to play games that will help them with
mathematics. They give opportunities to develop mathematical
communication, teamwork, honesty, tolerance, and so on. Allow
learners who finish work early to play games.
Activity 19.1
Learner’s Book page 271
1. Tell learners when to start and when to stop (at the end of one
minute). Learners can repeat this often to see how they progress.
2. For this question, learners revise and practise division with
remainders.
3. Ask learners to write a number sentence for each problem. They
do this in preparation for division with 3-digit numbers.
Solutions
1.
a)
b)
c)
d)
e)
f)
g)
h)
i)
j)
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Math G4 TG.indb 364
24 ÷ 4 = 6
32 ÷ 4 = 8
18 ÷ 3 = 6
21 ÷ 7 = 3
35 ÷ 5 = 7
40 ÷ 8 = 5
56 ÷ 7 = 8
63 ÷ 9 = 7
36 ÷ 4 = 9
42 ÷ 6 = 7
One-minute division calculations
k) 72 ÷ 8 = 9
l) 48 ÷ 6 = 8
m) 25 ÷ 5 = 5
n) 64 ÷ 8 = 8
o) 0 ÷ 4 = 0
p) 45 ÷ 9 = 5
q) 54 ÷ 6 = 9
r) 27 ÷ 3 = 9
s) 35 ÷ 7 = 5
t) 81 ÷ 9 = 9
Mathematics Teacher’s Guide Grade 4
TERM 4
2012/09/14 5:39 PM
2. a)
b)
c)
d)
e)
27 ÷ 5 = 5 remainder 2: × 5 + 2 = 27
23 ÷ 3 = 7 remainder 2: 3 × 7 + 2 = 23
47 ÷ 7 = 6 remainder 5: 7 × 6 + 5 = 47
109 ÷ 10 = 10 remainder 9: 10 × 10 + 9 = 109
79 ÷ 8 = 9 remainder 7: 8 × 9 + 7 = 79
3. Drawings to show the following.
a) 37 ÷ 6 = 6 remainder 1
b) 47 ÷ 7 = 6 remainder 5
c) 67 ÷ 8 = 8 remainder 3
d) 85 ÷ 9 = 9 remainder 4
e) 50 ÷ 6 = 8 remainder 2
Unit 20 Divide by 10 and 100
Mental maths Learner’s Book page 272
The learners play Division Bingo. Make copies of the answer
sheets. Use the game board to pose the questions randomly to
prevent learners from finishing too soon.
Activity 20.1
Learner’s Book page 272
1–3. The learners have to use their understanding of place value
when dividing by 10 and 100. Ask them what happens when
you multiply by 10 and 100, and when you divide by 10 and
100. Let them discuss the reasoning of the learners in the
examples.
Learners solve the problems where the dividends are not
multiples of the divisors and solutions result in remainders.
Relate these exercises to the examples with the chocolate
boxes. Learners should understand that there are not enough
remainders to fill the boxes. Ask learners how many chocolates
are needed to fill boxes with one hundred chocolates for each
remainder. They can use multiplication to check their solutions.
Solutions
1. a) 480 ÷ 10 = 48 boxes
2. a)
c)
e)
g)
i)
k)
m)
o)
60 ÷ 10 = 6
610 ÷ 10 = 61
50 ÷ 10 = 5
570 ÷ 10 = 57
70 ÷ 10 = 7
740 ÷ 10 = 74
100 ÷ 10 = 10
200 ÷ 10 = 20
b) 485 ÷ 10 = 48 remainder 5
They can fill 48 boxes.
b)
d)
f)
h)
j)
l)
n)
p)
61 ÷ 10 = 6 remainder 1
613 ÷ 10 = 61 remainder 3
57 ÷ 10 = 5 remainder 7
577 ÷ 10 = 57 remainder 7
74 ÷ 10 = 7 remainder 4
749 ÷ 10 = 74 remainder 9
106 ÷ 10 = 10 remainder 6
208 ÷ 10 = 20 remainder 8
3. a) 4 000 ÷ 100 = 40 boxes
b) 4 089 ÷ 100 = 40 remainder 89
40 boxes can be filled.
Mathematics Teacher’s Guide Grade 4
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2012/09/14 5:39 PM
4. a)
b)
c)
d)
e)
f)
g)
h)
i)
j)
k)
l)
m)
n)
o)
300 ÷ 100 = 3
367 ÷ 100 = 36 remainder 7
3 000 ÷ 100 = 30
3 600 ÷ 100 = 36
3 634 ÷ 100 = 36 remainder 34
500 ÷ 100 = 5
518 ÷ 100 = 5 remainder 18
900 ÷ 100 = 9
946 ÷ 100 = 9 remainder 46
5 000 ÷ 100 = 50
5 500 ÷ 100 = 55
5 599 ÷ 100 = 55 remainder 99
9 000 ÷ 100 = 9
9 800 ÷ 100 = 98
9 876 ÷ 100 = 98 remainder 76
5. Learners check their answers.
Activity 20.2
Learner’s Book page 273
Learners work in groups for this investigation. Let them struggle
with the problem before you give them any hints. They could
choose any multiple of 5 for the first and second numbers and add
3 to the first and 4 to the second numbers (for example, 15 + 3 = 18
and 25 + 4 = 29; 18 + 29 = 47; 47 ÷ 5 = 9 remainder 2). Ask learners
to work systematically to find other numbers to fit the descriptions.
They start with 5 + 3 = 8 and 10 + 4 = 14; 8 + 14 = 22; 22 ÷ 5 = 4
remainder 2, and so on.
28 ÷ 5 = 5 remainder 3
29 ÷ 5 = 5 remainder 4
43 ÷ 5 = 8 remainder 3
44 ÷ 5 = 8 remainder 4
28 + 29 = 57
43 + 44 = 87
57 ÷ 5 = 11 remainder 2
87 ÷ 5 = 17 remainder 2
The remainder is always 2, because 3 + 4 = 7 and 5 goes into 7 once
with a remainder of 2.
366
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Unit 21
Strategies for division
Mental maths Learner’s Book page 273
Ask the learners to record their solutions on their Mental maths
grids.
1. 87 ÷ 10 = 8 remainder 7
2. 26 ÷ 5 = 5 remainder 1
3. 350 ÷ 10 = 35
4. 37 ÷ 9 = 4 remainder 1
5. 46 ÷ 10 = 4 remainder 6
6. 50 ÷ 8 = 6 remainder 2
7. 77 ÷ 10 = 7 remainder 7
8. 30 ÷ 7 = 4 remainder 2
9. 455 ÷ 10 = 45 remainder 5
10. 20 ÷ 3 = 6 remainder 2
Activity 21.1
Learner’s Book page 274
1. Ask the learners to explore the problem and the strategy that
involves breaking up of numbers into their place value parts.
They solve the problems using this strategy.
2. In preparation for division with dividends that are not multiples
of the divisors, explore and explain why 49 and 66 are or are not
multiples of 7 and 8.
3. Learners should look at the units to decide why a number is
a multiple of another number. 375 cannot be a multiple of 8
because no multiple of 8 ends in 5.
4. Learners look at the hundreds and tens in the numbers in
question 3 to find the closest multiples. They need this
knowledge to use the strategy shown in the example. Introduce
learners to the terms dividend, divisor and quotient if they do
not know them yet.
Solutions
1. The strategy learners should use is shown for the first two
solutions.
a) 135 ÷ 5 = n
b) 448 ÷ 8 = n
100 ÷ 5 = 20400 ÷ 8 = 50
30 ÷ 5 = 6 40 ÷ 8 = 5
5 ÷ 5 = 1 8 ÷ 8 = 1
2756
c) 366÷ 6 = 61
d) 108 ÷ 4 = 27
e) 369 ÷ 3 = 123
f) 185 ÷ 5 = 37
g) 848 ÷ 8 = 106
h) 606 ÷ 6 = 101
i) 777 ÷ 7 = 111
j) 428 ÷ 4 = 107
k) 749 ÷ 7 = 107
l) 927 ÷ 9 = 103
Mathematics Teacher’s Guide Grade 4
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m) 456 ÷ 8 = 57
o) 545 ÷ 5 = 109
n) 324 ÷ 6 = 54
2. John is right, because 7 × 7 = 49.
Peter is wrong, because 66 ÷ 8 = 8 remainder 2.
3. a)
b)
c)
d)
375 ÷ 8 = 46 remainder 7; 375 is not a multiple of 8
447 ÷ 6 = 74 remainder 3; 447 is not a multiple of 6
536 ÷ 5 = 107 remainder 1; 536 is not a multiple if 5
432 ÷ 9 = 48; 432 is a multiple of 9
4. a) 40
c) 55
b) 42
d) 45
5. a) 50 × 5 = 250
483 – 250 = 233
40 × 5 = 200
233 – 200 = 33
6 × 5 = 30 33 – 30 = 3
483 ÷ 5 = 96 remainder 3
b) 40 × 6 = 240
275 – 240 = 35
5 × 6 = 30 35 – 30 = 5
275 ÷ 6 = 45 remainder 5
c) 40 × 4 = 160
197 – 160 = 37
9 × 4 = 36 37 – 36 = 1
197 ÷ 4 = 49 remainder 1
d) 50 × 8 = 400
431 – 400 = 31
3 × 8 = 24 31 – 24 = 7
431 ÷ 8 = 53 remainder 7
e) 60 × 6 = 360
598 – 360 = 238
39 × 6 = 234238 – 234 = 4
598 ÷ 6 = 99 remainder 4
Assessment task 2: division
The learners work on their own to show what they have learnt
about division during the last lessons and what they remember
from previous lessons. They solve 2-digit and 3-digit by 1-digit
division problems with and without remainders, division with 10
and 100 as divisors and dividends that are not multiples of 10 and
100, do inverse operations and solve word problems involving
grouping.
368
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Assessment task 2 Division
1. Work out the answers.
a) 19 ÷ 4 = n
b) 29 ÷ 3 = n
c) 39 ÷ 6 = n
d) 50 ÷ 7 = n
e) 69 ÷ 8 = n(5)
2. Write a multiplication number sentence for each problem
in question 1.
(5)
3. Make drawings to show your understanding of each
calculation.
a) 15 ÷ 4 = n
b) 23 ÷ 5 = n
c) 15 ÷ 3 = n
d) 19 ÷ 6 = n
e) 27 ÷ 8 = n(5)
4. Work out the answers.
a) 57 ÷ 10 = n
b) 112 ÷ 10 = n
c) 634 ÷ 100 = n
d) 845 ÷ 10 = n
e) 1 324 ÷ 100 = n(5)
5. Solve the problems.
a) How many bags with 10 tomatoes each can you fill if
you have 143 tomatoes?
b) How many bags with 100 sweets each can you fill if
you have 728 sweets?
c) How many bags with 9 butternuts each can you fill if
you have 459 butternuts?
d) A mini-bus takes 15 learners to school. How many
trips will the mini-bus make if it takes 150 learners to
the public library?
6. a) How many groups of 8 are there in 72?
b) How many groups of 7 are there in 749?
Mathematics Teacher’s Guide Grade 4
Math G4 TG.indb 369
(8)
(2)
Total [30]
TERM 4
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2012/09/14 5:39 PM
Assessment task 2 Division
1. a)
b)
c)
d)
e)
19 ÷ 4 = 4 remainder 3
29 ÷ 3 = 9 remainder 2
39 ÷ 6 = 6 remainder 3
50 ÷ 7 = 7 remainder 1
69 ÷ 8 = 8 remainder 5
(5)
2. a)
b)
c)
d)
e)
4 × 4 + 3 = 19
3 × 9 + 2 = 29
6 × 6 + 3 = 39
7 × 7 + 1 = 50
8 × 8 + 5 = 69
(5)
3. Learners make drawings to show understanding of the
division calculations. They can do it as shown in
Activity 19.1.
a) 15 ÷ 4 = 3 remainder 3
b) 23 ÷ 5 = 4 remainder 3
c) 15 ÷ 3 = 5 remainder 0
d) 19 ÷ 6 = 3 remainder 1
e) 27 ÷ 8 = 3 remainder 3
Math G4 TG.indb 370
(5)
4. a)
b)
c)
d)
e)
57 ÷ 10 = 5 remainder 7
112 ÷ 10 = 11 remainder 2
634 ÷ 100 = 64 remainder 4
845 ÷ 10 = 84 remainder 5
1 324 ÷ 100 = 132 remainder 4
(5)
5. a)
b)
c)
d)
143 ÷ 10 = 14 bags with 3 tomatoes left.
728 ÷ 100 = 7 bags with 28 sweets left.
459 ÷ 9 = 51 bags zero butternuts left.
150 ÷ 15 = 10 trips
(8)
6. a) 72 ÷ 8 = 9 groups
b) 749 ÷ 7 = 107 groups
370
Solutions
Mathematics Teacher’s Guide Grade 4
(2)
Total [30]
TERM 4
2012/09/14 5:39 PM
Perimeter, area and volume
Unit 22 Perimeter
Learner’s Book page 275
It is extremely important for learners to distinguish between area
and perimeter. Perimeter is the measurement of the outer edge of a
shape (for example, the wall of the room). They must understand
that perimeter is a line, not a surface. Use Activity 22.1 to remind
learners about their experience of real perimeters, and then to work
with the abstract concept of a perimeter by calculating the length
around a geometric shape.
Mental maths Learner’s Book page 275
The learners calculate the perimeter of shapes mentally. They
should use repeated addition, multiplication and doubling. They
use the commutative, associative and distributive properties
to calculate smarter. Ask them to explain their strategies and
encourage them to use different strategies.
Solutions
1. Perimeter = 100 m + 100 m + 100 m + 100 m = 400 m
2–3. a) Perimeter = 30 mm + 30 mm + 30 mm = 90 mm
(double 30) + 30 = 60 + 30 = 90 mm
3 × 30 = 90 mm
b) Perimeter = 20 mm + 20 mm + 40 mm + 40 mm = 120 mm
(double 20) + (double 40) = 40 + 80 = 120 mm
(2 × 20) + (2 × 40) = 40 + 80 = 120 mm
c) Perimeter = 35 mm + 35 mm + 35 mm + 35 mm = 140 mm
(double 35) + (double 35) = 70 + 70 = 140 mm
(2 × 35) + (2 × 35) = 70 + 70 = 140 mm
4 × 35 = (4 × 30) + (4 × 5) = 120 + 20 = 140 mm
4. a,b) Perimeter of A = 4 + 3 + 4 + 3 = 14 mm
or, 1 cm 4 mm or (2 × 3) + (2 × 4) = 6 + 8 = 14 mm
Perimeter of B = 3 + 1 + 1 + 3 + 5 + 4 = 17 mm
= 1 cm 7 mm
Perimeter of C = 4 + 4 + 4 + 4 = 16 mm (or 4 × 4)
= 1 cm 6 mm
5. Perimeter of A = (2 × 3) + (2 × 4) = 6 + 8 = 14 mm
Perimeter of C = 4 × 4 = 16 mm
6. Perimeter of B = (2 × 3) + (2 × 1) + 5 + 4 = 6 +2 + 5 + 4
= 17 mm
Mathematics Teacher’s Guide Grade 4
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Activity 22.1
Learner’s Book page 276
The learners engage in a practical activity to help them understand
perimeter better. Take them outside and ask them to measure the
distance around the quad, netball, rugby or soccer field in paces.
They could do this in teams of four so that each team measures
one side of each area. They compare the number of paces and
apply problem-solving skills if their distances differ. Let the whole
class work together to calculate the total perimeter of the surface
mentally.
Learners measure the perimeter or circumferences of circular shapes
practically with string or wool. They measure in millimetres and
centimetres. Explain how to measure the perimeter of a circle. Have
the following ready in class for the next assignment: tape measure
and lots of wool or string, glue and scissors. Allow all learners to
measure shape A and then compare their measurements. Help the
learners who do not have steady hands.
Activity 22.2
Learner’s Book page 276
Explain that shapes cannot always be drawn at their real size. In
other words, on some shapes measurements will be written that do
not represent the actual measurements of the object. Use examples
to explain this concept. For example, a small drawing of a person is
not life size, but you can write the real height of the person on the
drawing.
Learners will use their knowledge of fractions and whole numbers
to calculate the perimeter of regular and irregular shapes.
Solutions
1. Practical activity
2. Perimeter of A:
(2 12 cm + 2 12 cm) × 2
=5×2
= 10 cm
10 cm × 10 = 100 mm
Perimeter of B:
(4 × 12 cm) + (2 × 2 cm) + (3 × 1 cm)
=2+4+3
= 9 cm
9 cm × 10 = 90 mm
Perimeter of C:
(2 14 cm + 2 14 cm + 2 14 cm)
= (2 × 3) + ( 14 +
1
4
+ 14 )
= 6 43 cm
372
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Mathematics Teacher’s Guide Grade 4
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6 cm × 10 = 60 mm
3
4 cm × 10 = 30 ÷ 4
= 7 24
= 7 12 cm
60 + 7 12 = 67 12 mm
Perimeter of D:
(3 × 2 cm) + 1 13 cm +
=6+
2
3
1
3
cm2 + 5 cm
+5
= 11 23 cm
11 cm × 10 = 110 mm
2
3
cm × 10 = 20 ÷ 3
= 6 mm remainder 2
(discard the remainder)
11 23 cm ≈ 110 mm + 6 mm = 116 mm
Perimeter in millimetres and centimetres:
A a) 100 mm
b) 10 cm
B a) 90 mm
b) 9 cm
C
D
a)
67 12 mm
b)
6 43 cm
a)
116 23 mm
b)
11 23 cm
3. a) Perimeter = (3 12 m + 3 12 m) + 4 m
=7m+4m
= 11 m
b) Perimeter = (3 km × 2) + (5 km 350 m × 2)
= 6 km + (2 × 5) + (2 × 350)
= 6 km + 10 km + 700 m
= 16 km 700 m
4. Learners will need trundle wheels, metre sticks, rulers, builder’s
tape measure (metal). Before measuring each item, they must
make sure they have selected the correct units.
5. a) P = (2 × 60) + (2 × 25) = 120 + 50 m = 170 m of wire
b) P = 3 × 170 = 510 m, so 510 m will be needed for three
strands of wire.
6. a) Perimeter = 320 m + 165 m + 90 m + 94 m + 131 m = 800 m
b) 3 × (187 + 90 +165 + 320) m
= 3 × 762 m
= 2 286 m or 2 km 286 m
Mathematics Teacher’s Guide Grade 4
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c) 5 × (131 + 94 + 187) m
= 5 × 412
= 2 060 m or 2 km 60 m
d) Total distance once = (90 + 165 + 320 + 187 + 131 + 94) m
= 987 m
Total distance twice = 987 m + 987 m
= 1 974 m
= 1 km 974 m
7. Allow the learners to work this out in any way they like as long
as they can show their calculation.
JKNL = 800 m
She runs at:
200 m/minute
400 m/2 minutes
600 m/3 minutes
800 m/4 minutes or, 800 ÷ 200 = 4 minutes
Unit 23 Area
Mental maths Learner’s Book page 278
Learners work with tiles in tessellations in preparation for
concepts they will learn about when working with area. Make
sure that they understand what tessellation means. Encourage
them to use effective mental calculation strategies. They could
use repeated addition, multiplication and properties of numbers
to count and calculate the number of tiles and find out which
pattern has the most and fewest tiles.
Solutions
A: 3 + 3 + 3 = 3 × 3 = 9 tiles
B: 3 + 3 + 2 + 2 = (2 × 3) + (2 × 2) = 10 tiles
C: 3 + 3 + 2 = 2 × 3 + 2 = 8 tiles
D: 3 + 3 + 1 = 2 × 3 + 1 = 7 tiles
1. Pattern B has the most tiles.
2. Pattern D has the fewest tiles.
3. Learners explain how they counted the tiles
Activity 23.1
Learner’s Book page 278
1. Give the learners copies of shapes C and D. They tessellate the
shapes to create patterns. Ask them to colour the patterns in for
homework.
2. Learners work with the tile shape.
374
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TERM 4
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Activity 23.2
1. a)
b)
c)
d)
Learner’s Book page 279
4 rows of 4 squares = 16 square units
2 rows of 5 + 1 = 10 + 1 = 11 square units
6 squares + 3 half squares = 7 12 square units
8 squares + 5 half squares = 10 12 square units
2. When you count blocks the following rules apply. You count the
whole blocks first. Then you count the blocks that are half or
more within the shape. Add them to the whole blocks and throw
the rest away. Hand out a page of grid paper to all the learners
that they will have to use throughout the week when area and
perimeter is done.
Show learners how to cut the paper so that there is no waste of
paper.
3. a) kitchen: 8 square metres; bedroom: 34 square metres;
lounge: 50 square metres; passage: 6 square metres
b) 8 + 34 + 6 + 50 = 108 square metres
c) 108 square metres × 9 tiles = 972 tiles
d) R50 × 108 = R5 400
4. a) There are 24 blocks.
b) There are 6 blocks in the length and 4 blocks in the width.
c) 6 × 4 = 24 blocks
Unit 24 Volume
Learner’s Book page 280
It is important to start this section on volume with a practical
approach. Have different coloured cubes or Dienes blocks in class,
and containers into which the cubes can be packed. Start explaining
by packing one row with blocks, then packing two rows on top
of each other, and then packing blocks so that some blocks are
behind other blocks. Let learners use the blocks to fill containers of
different capacities, and compare the number of blocks needed to fill
each container.
Remind learners of the difference between volume and capacity.
Learners explored this with liquid volume and now they will
investigate solid shapes that have volume, and 3-D containers that
have capacity.
Mental maths Learner’s Book page 280
1. three layers
2. four cubes
3. 12 cubes
Mathematics Teacher’s Guide Grade 4
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Activity 24.1
Learner’s Book page 281
1. A a) three cubes
B a) four cubes
C a) 12 cubes
2. C has the largest capacity.
3. A has the smallest capacity.
b) three cubes
b) four cubes
b) 12 cubes
Revision and consolidation
Learner’s Book page 281
1. 16 squares
2. a) 14 cm
b) 200 ml
3. Perimeter = 2 12 cm + 2 12 cm + 4 12 cm + 4 12 cm = 14 cm
4. 16 cubes
Project
Work in a group and tile your classroom floor with paper tiles.
1. Mark out a section of the classroom floor that you will tile. Your
teacher will help you.
2. Measure the floor area.
3. Design a tessellation pattern to tile the floor. Draw your design
on paper.
4. Make a design using A4 paper. Let each piece of paper have one
tile or part of a tile.
5. Colour or paint the paper and cut out your paper tiles. Make
enough for your floor area.
6. Lay your tiles on the floor using adhesive putty to hold the tiles
in place.
How well are you able to do the following?
I can ...
Yes, easily
Most times Some­times I need help
explain what a
tessellation is
say if a pattern is
tessellating or nontessellating
copy tessellating
patterns
describe patterns
in nature,
from everyday
surroundings and
from our cultural
heritage.
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Rubric to assess the project
The learners can decide on their own shape or shapes that they will
use to tile the given floor space.
You can use the following rubric to assess learners’ skills.
4
3
2
1
Mark
Very poor
Not very
measurement
accurate
measurement of the floor
space
of the floor
space
Very poor
Fairly accurate Poor
Accurate
measurements measurement measurement measurement
and cut-outs
and cut-outs
and cut-outs
and cut-outs
of the tiles
of the tiles
of the chosen of the tiles
tiles
Design of the Design of the Design of the Design of the
tiling on paper tiling on paper tiling on paper tiling on paper
to show ideas to show ideas to show ideas to show ideas
not done very done very
done fairly
done well
poorly
well
well
Tessellation
Tessellation
Tessellation
Tessellation
done poorly
done fairly
done well and done fairly
and it does not
well, but it
pattern covers well and
the whole area covers whole does not cover cover whole
area
whole area
area
Overall
Overall
Overall
Overall
look of the
look of the
look of the
look of the
tessellations
tessellations
tessellations
tessellations
very
not very
very attractive fairly
unattractive
attractive
attractive
Total (out of 20)
Accurate
measurement
of the floor
space
Fairly accurate
measurement
of the floor
space
Mathematics Teacher’s Guide Grade 4
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Position and movement
Learner’s Book page 282
Introduction
In Term 3, the learners practised looking at plan views of items and
places. Now they will learn how grids help us to locate or describe
the positions of places more easily. They will practise reading a grid
and then using a grid on a map.
Unit 25 Work with grids
In Grade 4, learners work with simple alpha-numeric grids, where
rows are labelled A to E and columns are labelled 1 to 5.
Mental maths Learner’s Book page 282
This activity helps learners understand the need for using grids to
help us locate items.
1. The learners will probably use words such as the following
to describe where the different items are: to the left of, to
the right of, below, above, further below, and next to. As a
further challenge, ask the learners to write down the position
of each item. Let learners use their home languages to
describe the positions, so that they understand the position
concepts well, and then help them use the relevant English
words as translations of the home language words.
2. Learners may point out that it is quite difficult to explain the
exact position of the shapes without a grid.
3. Learners use coordinates to describe the position of objects.
Understanding grids
Learners worked with rows and columns in units in the section on
numbers. Revise references on a grid. If necessary ask learners to
draw small grids with a given number of rows and/or columns, to
familiarise them with the concepts.
Activity 25.1
Learner’s Book page 283
The learners should be able to understand the purpose or usefulness
of using an alpha-numeric grid in locating places or items.
Solutions
1.
2.
3.
4.
378
Math G4 TG.indb 378
a) A1
b) C4
c) A3
a) cat
b) hen
c) nothing
a) C3; no
b) B3; no
The chicken is three blocks from the fish and the mouse is about
one and a half blocks from the fish.
Mathematics Teacher’s Guide Grade 4
TERM 4
2012/09/14 5:39 PM
Suggested informal assessment questions to ask yourself
• How well are the learners able to name the blocks in an alphanumeric grid?
• How easily are learners able to find items in blocks when the
name of the block is given?
Unit 26 Grids on maps
Learner’s Book page 284
The map in this unit is not a direct top view, as the focus in this unit
is on understanding how grids work. Showing the map as an oblique
view rather than a top view of an area will help the learners to
identify the places more easily and focus on the concept of grids.
Activity 26.1
Learner’s Book page 284
1. a) sports field
b) park
c) house
2. a) D7
b) A1 and A2
c) C1 and C2, and E1 and E2
d) A5
e) E5 and E6
3. sports fields, houses, park, clinic, library, church
Remedial activities
• Let the learners practise labelling empty grids. Let them label
the rows and columns, then let them label each block in the
grid. Start with a 3 × 3 grid, then move onto a 4 × 4 grid, and
eventually a 6 × 6 grid.
• Let the learners play a grid game outside. Draw a big 5 × 5 grid
on the ground. Label each row and column. Shout out the name
of a block in the grid and then the learners must run to it as fast
as they can. Once the learners get the hang of this, play the next
game. Let the first five learners to reach the block be safe while
the rest sit out the round. As you say a new block reference, only
four learners are safe, then only three, two and eventually one.
Play the game a few times, and let the learners play it in smaller
groups.
Extension activities
• Show the learners street maps of your local area from a street
atlas and let them find different places in different blocks of
the grid. Let them look for places they know from their own
experience.
• Let the learners use their pictures of a classroom or school that
they drew and draw and label their own grids over the pictures.
Then let them ask one another to find places on their maps using
the grid references.
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More transformations
Learner’s Book page 285
Introduction
Transformations were introduced last term where the focus was
on building composite shapes with smaller shapes. Now the focus
is on identifying and building patterns with shapes, focusing on
tessellation.
Make sure that you have shapes of different sizes in class. But
initially, have large cut-out shapes available and let the learners
use these big shapes when they investigate or manipulate them.
Let them use the floor as a surface to work on because they may run
out of desk space. For example, when doing tessellation patterns,
learners will have a lot of fun making beautiful patterns with their
cardboard shapes on the floor. These fun activities build the learners’
concepts of shape and space in important ways. You will find that
most learners will easily be able to transfer the same concepts to
a smaller area, such as the page of their notebook, once they have
grasped them.
Unit 27 Tessellations
Remind the learners what a tessellating pattern is.
Mental maths Learner’s Book page 286
1. A, B and C: squares E, F and G: triangles
2. A, E, G
3. C and E
4. B and D
Activity 27.1
Learner’s Book page 286
1. A: rectangles; B: diamonds; C: hexagons; D: rectangles and
squares; E: octagons and squares; F: hexagons and triangles
2. Learners complete this question in pairs.
3. Patterns with line symmetry: B, C, E and F
4. Learners complete this question on their own.
Activity 27.2
Learner’s Book page 287
The learners will realise that tiles will have to be cut in half in order
to completely fill the given space.
Suggested informal assessment questions to ask yourself
• How well can the learners explain what a tessellation is?
• How well can they identify tessellation patterns?
• How well can they tile an area in a tessellation pattern?
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Unit 28 Describe patterns
This unit focuses on recognising and describing patterns in nature,
patterns from learners’ surroundings and from our cultural heritage.
Mental maths Learner’s Book page 276
The learners can choose to describe different patterns they see.
Examples:
• Bridge: straight lines in the bridge railings with triangular
patterns.
• Basketry: overlapping lines at an angle to each other.
• Honeycomb: a tessellating pattern of hexagons; the shapes are
symmetrical.
• Geometric tortoise: hexagon or pentagon shapes on the shell;
patterns are not strictly symmetrical, but some of the shapes
look symmetrical.
• Beadwork: this varies depending on the way the beads are
strung and repeated.
• Paving: tessellating rectangles, which are rectangular prisms.
Activity 28.1
Learner’s Book page 288
Learners complete this activity on their own. Make sure they are
able to identify lines of symmetry in the patterns.
Suggested informal assessment questions to ask yourself
• How well are the learners able to identify patterns they see in
nature, in their daily surroundings and in examples of their
cultural heritage?
• How well are they able to describe the patterns?
• How well are they able to copy relevant shapes and patterns?
Remedial activities
• Let the learners do tessellations with large shapes on the floor.
Let them start with single simple shapes, then use the same shape
in two different colours, then let them work with more complex
shapes.
• Let the learners practise finding and describing various patterns
on objects around them such as on clothing fabric, containers and
brick walls. Let learners look in books to find suitable pictures
and to describe the pictures by referring to the types of line
they see, whether the shapes are open or closed, whether they
recognise any polygons, whether there are gaps or overlapping
sections in the patterns, and whether the shapes tessellate.
• Let the learners copy simple patterns that they find and help them
to describe these patterns.
Mathematics Teacher’s Guide Grade 4
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Extension activities
• Let the learners create tessellations with more complex shapes
on the floor or on a wall. Provide large cardboard cut-outs for
learners to use.
• Let the learners copy interesting patterns and work in a group to
put the patterns together on a poster. Ask them to present their
patterns to the rest of the class by describing what the patterns are
and how they were made.
Geometric patterns
Unit 29 Geometric patterns
Learner’s Book page 289
Remind learners that they worked with geometric patterns in
Term 2 and in space and shape units that included transformations.
Ask them what they remember about the concepts involved in this
topic. Let them give examples of patterns in nature, culture and
everyday life.
Mental maths Learner’s Book page 289
1. Ask the learners to study the pictures of a cushion, plant,
paving and the boy’s shirt. Let them describe the patterns
they observe by looking for regularity and repetition of
colours or shapes. Refresh their minds about the definition of
a pattern – regular forms that are displayed in the repetition
of shapes, objects or colours. Learners should, for example,
notice the repeating rhombi (diamonds) in the cushion,
the repeating rectangles and triangles on the shirt and the
arrangement of the leaves from small to big in the plant.
They should notice the transformation in the brick paving
that forms a tessellation (translation (slide) and rotation
(turn) and the translated rectangles and triangles on the shirt).
Allow the learners to use their own informal terminology
in their descriptions, but also use or introduce the formal
terminology and display the words on the board.
2. a) Learners should discover that the order of the shapes
alternates. In the second picture the triangle is second,
the circle third and the square first. In the third picture
the circle is first, the square second and the triangle
third and so on. The learners could also use the terms
beginning, middle, centre, last, end, start and so on.
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It is important that they use mathematical language to
describe the patterns. Encourage the learners to name the
shapes as they should follow and write the sequence on
the board: triangle, circle, square, square, triangle, circle,
circle, square, triangle, and so on.
b) The learners should look at the arrangement further on
in the pattern to realise that two chickens and two ducks
form the repeating units. If the learners argue that they
see only one chicken at the beginning, tell them that
these are repeating patterns and they should imagine
that they continue to the left and right. Only sections
of the patterns are shown. Ask the learners to write the
sequence on the board: chicken, duck, duck, chicken,
chicken, duck, duck and so on.
c) The learners would notice the unit of three cars and
think that the one motorbike is repeated. By inserting
two cars at the beginning, they will realise that the unit
of motorbikes include two of them so that the repeating
units are two motorbikes and three cars. They could
use numbers, words or circles to indicate the order for
example: 2; 3; 2; 3 or 3 cars; 2 motorbikes or drawings
such as n n n 7 7 n n n.
d) The order of the objects in the repeating units changes
so that the last object becomes the first one, etc. The
cylinder follows after the cone, then the cube, etc. Ask
the learners to name the repeating objects while you
record the words on the board: cone, cylinder, cube,
pyramid, pyramid, cone, cylinder, cube and so on.
3. Encourage the learners to use shapes or objects to create
repeating patterns. Give them concrete objects such as
counters, bottle tops, 3-D objects such as cubes and pyramids
or cut-out pictures to experiment and then make drawings of
the objects in the sequence. Let them first create the whole
pattern and then decide which elements they want to remove
so that a partner or another group investigate and extend their
patterns. Display learners’ work in the classroom.
Activity 29.1
Learner’s Book page 289
Learners will now work with patterns that repeat but also grow in
numbers (growing patterns).
1. Learners investigate the brown and green tile arrangements
– they are not allowed to count the brown tiles, but should
use the number of green tiles indicated and do calculations to
calculate the number of brown tiles by looking for a relationship
between the two colours of tiles. In (a), for example, they should
reason that 6 + 6 + 2 = 14 tiles altogether or double 6 + 2 = 14
or (2 × 6) + 2 = 12 + 2 = 14. Encourage learners to identify and
Mathematics Teacher’s Guide Grade 4
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express the rule, multiply the number of green tiles by 1 plus 2
to get the number of brown tiles (× 1 + 2 or simply number of
green tiles + 2 = number of brown tiles).
Solutions
1. The number of green tiles + 2 = number of brown tiles.
a) 6 + 2 = 8
b) 4 + 2 = 6
c) 8 + 2 = 10
d) 12 + 2 = 14
e) 7 + 2 = 9
f) 22 + 2 = 24
2. Learners use the rule or relationship to find the number of brown
tiles for each number of green tiles in the table.
Green tiles
Brown tiles
4
6
5
7
6
8
7
9
8
10
10
12
12
14
22
24
25
27
50
52
3. The learners complete the flow diagrams to find the number of
brown tiles for the given number of green tiles.
green
a)
2
brown
×1
+2
×1
+2
×1
+2
×1
+2
×1
+2
green
b)
9
brown
green
c)
d)
e)
f)
11
21
brown
green
99
13
brown
green
23
11
brown
green
19
4
25
brown
×1
+2
101
4. The learners should realise that they have to use the inverse
operations to calculate the number of green tiles for the given
number of brown tiles. The flow diagrams are reversed. Check
whether some learners argue that you could only subtract 2 and
not have to divide by 1.
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Math G4 TG.indb 384
a)
green
28
brown
b)
green
50
c)
green
72
÷1
–2
d)
green
104
÷1
–2
e)
green
100
÷1
–2
f)
green
260
÷1
–2
÷1
–2
30
brown
÷1
–2
52
brown
74
brown
106
brown
102
brown
Mathematics Teacher’s Guide Grade 4
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Unit 30 Growing patterns
Mental maths Learner’s Book page 291
1. Learners investigate and explain how to calculate the number
of pink tiles if the number of black tiles is given. Multiply
the number of black tiles by 2 and add 3 to get the number
of pink tiles. The rule is × 2 + 3. Let learners calculate the
number of pink tiles for each figure (for example, in the first
diagram it is (5 × 2) + 3 = 10 + 3 = 13 pink tiles).
2. Learners to use the rule they have discovered in question
1 to complete the flow diagrams. To get the number of
black tiles if the number of pink tiles are given, they use
inverse operations. Let the learners explore the numbers in
the solutions to describe the number sequences and extend
the patterns and fill in the missing number in the number
sequences that involve uneven and natural or counting
numbers.
Solutions
1. a)
b)
c)
d)
e)
2.
5 black tiles; so, (2 × 5) + 3 = 13 pink tiles
2 black tiles; so, (2 × 2) + 3 = 7 pink tiles
7 black tiles; so, (2 × 7) + 3 = 17 pink tiles
3 black tiles; so, (2 × 3) + 3 = 9 pink tiles
6 black tiles; so, (2 × 6) + 3 = 15 pink tiles
black tiles pink tiles
2
7
3
9
4
11
5
13
6
15
7
17
8
19
(Rule: black × 2 + 3)
Activity 30.1
black tiles pink tiles
9
21
10
23
11
25
10
31
20
43
23
49
53
53
(Rule: pink – 3 ÷ 2)
Learner’s Book page 291
1. Learners look at the relationship between the brown and yellow
tiles in each pattern. Learners count the number of tiles in the
rows and columns and multiply the numbers – they should
realise that they are working with square numbers. Remind
them that we call them square numbers because they are used
to create perfect squares (the number of shapes or objects in the
rows and columns is the same as in the sides of a square).
Check if there are still learners who count in 1s. Let the learners
compare strategies to find the most effective way to find the
number of squares.
Mathematics Teacher’s Guide Grade 4
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They can use these rules to calculated the number of squares:
• Pattern 1: 3 × 3 = 9
• Pattern 2: 5 × 5 = 25
• Pattern 3: 7 × 7 = 49
The number of rows and columns increases by 2.
The number of tiles on a side in pattern 4 is an odd number
(2 + 7 = 9) and so there are 9 × 9 = 81 tiles.
The learners investigate the number of brown tiles in each
pattern. They should find the following sequence, 5; 9; 13 and
realise that the number of brown tiles increases by 4 for each
pattern.
Let them investigate the number of yellow tiles:
• Pattern 1: 4 × 1 = 4 yellow tiles
• Pattern 2: (4 × 1) + (4 × 3) = 16 yellow tiles
• Pattern 3: (4 × 1) + (4 × 3) + (4 × 5) = 4 + 12 + 20 = 36
yellow tiles.
The number sequence for the number of yellow tiles is 4; 16; 36.
which involves the even square numbers (2 × 2 = 4; 4 × 4 = 16
and 6 × 6 = 36). The next number in the sequence is, therefore,
8 × 8 = 64 and the next number would be 10 × 10 = 100. The
sequence is 4; 16; 36; 64.
2. Learners use the rules they discovered for calculating the total
number of tiles (multiply the number of rows by itself), the
number of brown tiles (add 4) and the number of yellow tiles
(4 + 4 × the number of squares on one side of the square in each
pattern).
Pattern
number
1
2
3
4
5
6
Number of Pattern
tiles
number
9
25
49
81
121
169
Number of Pattern
brown tiles number
1
2
3
4
5
6
5
9
13
17
21
25
1
2
3
4
5
6
Number
of yellow
tiles
4
16
36
64
100
144
3. The learners use the rules and create new tiling patterns. Let
them display their patterns and explain how they created them.
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Whole numbers: addition and subtraction
Unit 31
Use place value to add and subtract
Mental maths Learner’s Book page 293
Give the learners copies of Addition Bingo answer sheets and let
them play the game again.
Activity 31.1
Learner’s Book page 293
Learners should note the way to carry values and explain how they
think this strategy works. The subtraction calculations involve
decomposing.
Solutions
1. a) 369 + 481 = 850
b) 3 544 + 2 878 = 6 422
2. Learners pack out flard cards to show the addition calculations
with carrying. Ask them to do the written calculations in their
workbooks. They break up numbers in place value parts. Show
them one of the calculations below.
a) 574 + 398 = n
500 + 70 + 4
300 + 90 + 8
800 + 60 + 2
(4 + 8 = 12; 70 + 90 = 160)
100 + 10 (carry 10 and 100)
900 + 70 + 2
= 972
b) 2 856 + 3 764 = n
2 000 + 800 + 50 + 6
3 000 + 700 + 60 + 4
5 000 + 500 + 10 + 0
(6 + 4 = 10; 50 + 60 = 110;
800 + 700 = 1 500)
1 000 + 100 + 10 (carry 10; 100 and 1 000)
6 000 + 600 + 20 + 0
= 6 620
c) 4 979 + 4 251 = n
4 000 + 900 + 70 + 9
4 000 + 200 + 50 + 1
8 000 + 100 + 20 + 0
(9 + 1 = 10; 70 + 50 = 120;
900 + 200 = 1 100)
1 000 + 100 + 10 (carry 10; 100 and 1 000)
9 000 + 200 + 30 + 0
= 9 230
Mathematics Teacher’s Guide Grade 4
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d) 5 186 + 2 948 = n
5 000 + 100 + 80 + 6
2 000 + 900 + 40 + 8
7 000 + 0 + 20 + 4
1 000 + 100 + 10 8 000 + 100 + 30 + 4
= 8 134
e) 6 234 + 2 876 = n
6 000 + 200 + 30 + 4
2 000 + 800 + 70 + 6
8 000 + 0 + 0 + 0
1 000 + 100 + 10 9 000 + 100 + 10 + 0
= 9 110
(8 + 4 = 12; 80 + 40 = 120;
100 + 900 = 1 000)
(carry 10; 100 and 1 000)
(4 + 6 = 10; 30 + 70 = 100;
200 + 800 = 1 000)
(carry 10; 100 and 1 000)
3. Answers will differ.
4. Learners pack out flard cards to show how to use decomposition
when subtracting. They break up numbers in place value parts.
Show them one of the strategies below.
a) 673 – 486 = n
600 + 70 + 3
400 + 80 + 6
500 + 160 + 13
(Take away 100 from 600
400 + 80 + 6
and 10 from 70.)
100 + 80 + 7
= 187
b) 3 142 – 1 363 = n
3 000 + 100 + 40 + 2
1 000 + 300 + 60 + 3
2 000 + 1 000 + 130 + 12 (Take away 1 000 from 3 000;
1 000 + 300 + 60 + 3
100 from 100 and 10 from 40.)
1 000 + 700 + 70 + 9
= 1 779
c) 2 536 – 1 787 = n
2 000 + 500 + 30 + 6
1 000 + 700 + 80 + 7
1 000 + 1 400 + 120 + 16 (Take away 1 000 from 2 000;
1 000 + 700 + 80 + 7 100 from 500; 10 from 30.)
700 + 40 + 9
= 749
d) 3 005 – 2 659 = n
3 000 + 0 + 0 + 5
2 000 + 600 + 50 + 9
2 000 + 900 + 90 + 15
(Take away 1 000 from 3 000;
2 000 + 600 + 50 + 9
100 from 1 000; 10 from 100.)
300 + 40 + 6
= 346
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e) 5 300 – 3 843 = n
5 000 + 300 + 0 + 0
3 000 + 800 + 40 + 3
4 000 + 1 200 + 90 + 10
3 000 + 800 + 40 + 3
1 000 + 400 + 50 + 7
= 1 457
(Take away 1 000 from 3 000;
100 from 300 and 10 from 100.)
Unit 32 Use 10-strips to add and subtract
Mental maths Learner’s Book page 294
1. Both strategies involve breaking down and building up
multiples of 10 and compensation, while the subtraction
strategy also uses inverse operations, and the associative
and distributive properties. The strategy helps learners
make sense of addition with carrying and subtraction with
decomposition. Ask the learners to discuss and explain the
strategies. Learners can shade the strips.
2. Tell learners that the blue and red strips represent addition
and the yellow and green strips represent subtraction.
3. Learners use the strategies and number properties that are
used in the illustrations.
Solutions
1. a) 46 + 39 = n
40 + 40 + 5 = 85
b) 57 – 28 = n
(40 + 17) – (20 + 8)
= (40 – 20) + (17 – 8)
= 29
2. a) 38 + 29 = n
30 + 30 + 7 = 67
b) 72 + 49 = n
70 + 50 + 1 = 121
c) 59 + 66 = n
50 + 70 + 5 = 125
d) 64 – 38 = n
(50 + 14) – (30 + 8)
= (50 – 30) + (14 – 8)
= 20 + 6
= 26
e) 53 – 37 = n
(40 + 13) – (30 + 7)
= (40 – 30) + (13 – 7)
= 10 + 6
= 16
f) 80 – 39 = n
(70 + 10) – (30 + 9)
= (70 – 30) + (10 – 9)
= 40 + 1
= 41
3. a) 45 + 38
= 40 + 40 + 3
= 38
b) 72 – 47
= (60 + 12) – (40 + 7)
= (60 – 40) + (12 – 7)
= 25
Mathematics Teacher’s Guide Grade 4
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c) 58 + 67
= 60 + 60 + 5
= 125
d) 84 + 39
= 80 + 40 + 3
= 123
e) 83 – 46
= (70 + 13) – (40 + 6)
= (70 – 40) + (13 – 6)
= 37
f) 96 – 68
= (80 + 16) – (60 + 8)
= (80 – 60) + (16 – 8)
= 28
g) 67 + 38
= 70 + 30 + 5
= 105
h) 88 + 29
= 90 + 20 + 7
= 117
Activity 32.1
Learner’s Book page 295
Make copies of the strips using the template. Ask the learners
to shade the strips in two colours of their choice. Let them use
the strips to represent and solve the 2-digit number addition and
subtraction involving carrying and decomposing.
Solutions
1. 49 + 87
= 50 + 80 + 6
= 136
2. 99 + 76
= 100 + 70 + 5
= 175
3. 98 – 59
= (80 + 18) – (50 + 9)
= (80 – 50) + (18 – 9)
= 39
4. 72 – 29
= (60 + 12) – (20 + 9)
= (60 – 20) + (12 – 9)
= 43
5. 86 + 57
= 90 + 50 + 3
= 143
6. 57 + 48
= 60 + 40 + 5
= 105
7. 76 + 65
= 80 + 60 + 1
= 141
8. 70 – 47
= (60 + 10) – (40 + 7)
= (60 – 40) + (10 – 7)
= 23
9. 60 – 29
= (50 + 10) – (20 + 9)
= (50 – 20) + (10 – 9)
= 31
10. 69 + 69
= 70 + 60 + 8
= 138
Activity 32.2
Learner’s Book page 295
This is an investigation. Let the learners work in their groups
to solve the non-routine problem. If there is no time to do these
investigations during the lesson, ask the learners to do them for
homework. You should have a class discussion first so that they
understand the context and structure. Doing the problems for
homework could promote family involvement as parents and
siblings assist learners in solving the problems.
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Solutions
There are 32 eyes altogether; so, there are 32 ÷ 2 = 16 animals.
There are 13 tails altogether; so, there are 13 mermaids and dolphins
altogether.
Octopuses do not have tails; so, there must be 16 – 13 = 3
octopuses.
There are 38 arms altogether and the octopuses have 8 arms each;
so, there are 3 octopuses (3 × 8 = 24).
The remaining arms (38 – 24 = 14) must be for the mermaids; so,
there are7 mermaids (14 ÷ 2 = 7) (dolphins do not have arms).
There are 16 – 7 – 3 = 6 dolphins.
There are 3 octopuses, 7 mermaids and 6 dolphins.
Data handling
Unit 33 Probability
Learner’s Book page 296
The concept of probability is quite complex for Grade 4 learners.
In order to understand it, learners need an understanding of the
possible outcomes of an event and then what percentage any
possible outcome is of the total number of possible outcomes. So,
learners in Grade 4 do not work with probability as such, but they
are introduced to it by doing trials where they list outcomes. The
main work in Grade 4 deals with possible and actual outcomes, as
an introduction to the complex concept of probability.
If necessary, provide home language support for the terms likely,
possible and certain outcomes so that the learners can understand
what the terms mean. Let them talk about situations in their own
lives where an outcome is likely, possible or certain, for example:
• It is certain that I will not go to school on Saturday.
• It is likely that I will see my teacher at school tomorrow.
• It is possible that it will be warm tomorrow.
Make sure that the learners know the difference between tails and
heads on a coin – on South African coins, heads is the side with the
South African coat of arms and South Africa, and tails is the side
with the value of the coin and a picture of an indigenous plant or a
wild animal.
For practical experience, you will need items such as coins, dice,
balls, cubes and counters in a bag or box and spinners.
Mathematics Teacher’s Guide Grade 4
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Mental maths
Learner’s Book page 296
Give each pair of learners a dice. Ask them what the chances are
that they will throw a six. They should realise that they have a
chance of one in six. Learners write their names in copies they
make of the table and take turns to throw the dice. They have
to count the number of throws they make before they throw a
six. Let them make tally marks to count their throws. Ask them
to study the tally marks to find out if there is a pattern in the
number of throws.
Activity 33.1
Learner’s Book page 297
1. There are only two possible ways that a drawing pin could land:
• It could land right side up.
• It could land upside down.
2. There are only two possible outcomes:
• a black ball
• a white ball.
3. The spinner could land on three numbers – 1, 2 or 3.
Suggested informal assessment questions to ask yourself
• How well can the learners explain what possible outcomes are?
• How well are learners able to list possible outcomes of the
events given?
Unit 32 Experiments and actual outcomes
Learner’s Book page 297
By now, the learners should have a good idea about various possible
outcomes of events. In this unit, they will use a coin and a dice
to list possible outcomes and then do experiments to list actual
outcomes.
The examples take the learners through the steps that they will need
to follow in the activities. Make sure they understand the steps.
Before learners tackle the experiments, do one or two experiments
with the class and record the outcomes on the board. Then discuss
the outcomes with the class.
Mental maths
Learner’s Book page 297
The learners apply the concept of multiplication to practise
possible and actual outcomes. They can work in pairs or in
groups if you do not have enough dice. (Learners could use the
nets of cubes and make their own dice.) Learners take turns to
roll two dice.
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They multiply the two numbers and write number sentences to
show the multiplicand, multiplier and product for each throw.
They determine after how many throws they can write the
number sentences 6 × 6 = 36 and how often they can write this
number sentence. They should realise that they have a 1 out of
21 chance of throwing 6 × 6.
The possible outcomes they could get are listed below:
1×1=1
2×2=4
3×3=9
1×2=2
2×3=6
3 × 4 = 12
1×3=3
2×4=8
3 × 5 = 12
1×4=4
2 × 5 = 10
3 × 6 = 18
1×5=5
2 × 6 = 12
1×6=6
4 × 4 = 16
5 × 5 = 25
6 × 6 = 36
4 × 5 = 20
5 × 6 = 30
4 × 6 = 24
Activity 34.1
Learner’s Book page 297
When the learners compare their results or repeat the experiment,
and compare their own sets of results, they start thinking about
probability. This will lead many learners to start thinking about
which possible outcomes seem to be actual outcomes more often
than other outcomes, and then they may try to predict what the
actual outcomes will be.
Activity 34.2
Learner’s Book page 298
Learners predict the outcomes and then conduct an experiment. This
question guides them to start thinking about or predicting probable
results. Their answers will be based on what the most frequent
actual outcomes were in the experiment in Mental maths.
Activity 34.3
Learner’s Book page 298
1–2. As with the coin-tossing experiments in Activity 34.1, rolling
the dice experiments help learners start to think about the
concept of probability. This time, the possible and actual
outcomes are on a bigger scale, so the learners have to keep
careful records of their results.
3. Learners may multiply each actual outcomes by 2 or 4 to help
them predict their outcomes for rolling the dice two or four
times more than in the first experiment. Their results will,
however, not be exactly the same as their predictions, and
they may even find that their results look nothing like their
predictions.
Mathematics Teacher’s Guide Grade 4
Math G4 TG.indb 393
TERM 4
393
2012/09/14 5:40 PM
Suggested informal assessment questions to ask yourself
• How well can the learners list possible outcomes of their
experiments?
• How well can learners accurately record the results or actual
outcomes of their experiments?
• How well can learners summarise or analyse their results?
• How good are learners’ predictions for further experiments
based on their actual results of a previous experiment, or are
they merely doing guesswork?
Revision
Learner’s Book page 299
1. a) 7
b) 9
c) 16
2. a) The table should take the following form.
Possible
outcomes
Tally marks
Number of actual
outcomes
Heads
Tails
b) The learners should state which outcome was the most
frequent – did the coin land more often on heads or on tails?
3. a)
b)
c)
d)
e)
twice
three times
twice
five
12 times
4. a)
Possible
outcomes
Tally marks
One
Two
Three
Four
Five
Six
Number of actual
outcomes
2
3
3
2
1
1
b) two and three
c) five and six
d) Megan doubled the outcomes of her first experiment. She
reasoned that if she doubled the number of times she rolled
the dice, then she would double each outcome she got the
previous time. Learners may argue that the outcomes of
the second 12 throws would not be the same as the first
12 outcomes – so the results cannot simply be doubled.
This is a good argument. In practice, we cannot predict the
outcomes of future throws based on past throws – each time
Megan throws the dice, all six outcomes are possible.
394
Math G4 TG.indb 394
Mathematics Teacher’s Guide Grade 4
TERM 4
2012/09/14 5:40 PM
Remedial activities
• Ensure that the learners know how to use tally marks. They will
need this skill to record the actual outcomes of their experiments.
• Let the learners work in pairs to do a number of coin-tossing
experiments only. This limits the number of possible outcomes
and, therefore, simplifies the activities. The learners can take
turns to toss the coin and together decide on where and how to
record the outcome in the table.
• Let the learners explain their own and others’ tables of outcomes,
stating what was done and what the outcomes were. For example,
the table shows that a coin was tossed 10 times; it landed on
tails 6 times and on heads 4 times. Once the learners are fairly
confident tossing coins and reading data from the tables, they can
do experiments where they roll dice.
Extension activities
Let the learners experiment by rolling a dice 20 to 30 times and
recording the outcomes. Then let them indicate the number of times
each outcome occurred. They can then compare the frequency
of outcomes with a partner’s results, and then with the results
of another three or four learners. Ask learners to summarise and
describe the results of their experiments.
Mathematics Teacher’s Guide Grade 4
Math G4 TG.indb 395
TERM 4
395
2012/09/14 5:40 PM
Math G4 TG.indb 396
2012/09/14 5:40 PM
5. Resources
Mental maths grid
Number lines
Flow charts
398
399
400, 401
Number chains, calculation diagrams, strips
402
Fraction walls 403
Fraction circles
Flard cards
404
405, 406
Dienes blocks 407
Place value boards, answer grid and multiplication grid 408
Number grid: 200 grid
409
Number grids: 99 grid, 100 grid, 109 grid 410
Shapes (1)
411
Shapes (2)
412
Square, triangle, pentagon and hexagon
Hexagons
413
414
Build shapes (Learner’s Book pages 242 and 243) 415
Square grid
416
Dotted grid
417
Triangle on dotted grid
418
Squares and rectangles on dotted grid
Symmetry: complete shapes
419
420
3-D objects (1) 421
3-D objects (2)
422
Six nets for a cube
Net for a cube
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424
Net for a rectangular prism
I have . . . (1)
426
I have . . . (2)
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425
Fraction snap 428
Fraction dominoes
429
Reach for the moon game board 430
Place value scatter board
431
Bingo games and answer sheets
432
Tangram 433
397
Math G4 TG.indb 397
2012/09/14 5:40 PM
Mental maths grid
Mental calculations
Name:
Task 1
Task 2
Task 3
Task 4
Task 5
Task 6
Task 7
Number
Shade the blocks below to show your progress.
10
9
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2
1
Reflection
What do I do well?
.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
What can I do better next time?
.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
398
Math G4 TG.indb 398
Mathematics Grade 4 Teacher’s Guide RESOURCES
This page may be photocopied.
2012/09/14 5:40 PM
Number lines
• • • • • • • • • • • • • • • • • • • •
• • • • • • • • • • • • • • • • • • • •
• • • • • • • • • • • • • • • • • • • •
• • • • • • • • • • • • • • • • • • • •
• • • • • • • • • • • • • • • • • • • •
This page may be photocopied.
Math G4 TG.indb 399
Mathematics Grade 4 Teacher’s Guide RESOURCES
399
2012/09/14 5:40 PM
Flow charts
Input
400
Math G4 TG.indb 400
Mathematics Grade 4 Teacher’s Guide RESOURCES
Output
This page may be photocopied.
2012/09/14 5:40 PM
Flow charts
144444424444443
144444424444443
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This page may be photocopied.
Math G4 TG.indb 401
Mathematics Grade 4 Teacher’s Guide RESOURCES
...
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401
2012/09/14 5:40 PM
Number chains, calculation diagrams, strips
Addition and subtraction strips
402
Math G4 TG.indb 402
Mathematics Grade 4 Teacher’s Guide RESOURCES
This page may be photocopied.
2012/09/14 5:40 PM
Fraction walls
This page may be photocopied.
Math G4 TG.indb 403
Mathematics Grade 4 Teacher’s Guide RESOURCES
403
2012/09/14 5:40 PM
Fraction circles
404
Math G4 TG.indb 404
Mathematics Grade 4 Teacher’s Guide RESOURCES
This page may be photocopied.
2012/09/14 5:40 PM
Flard cards
10000
1000
20000
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This page may be photocopied.
Math G4 TG.indb 405
Mathematics Grade 4 Teacher’s Guide RESOURCES
405
2012/09/14 5:40 PM
Flard cards
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406
Math G4 TG.indb 406
Mathematics Grade 4 Teacher’s Guide RESOURCES
This page may be photocopied.
2012/09/14 5:40 PM
Dienes blocks
This page may be photocopied.
Math G4 TG.indb 407
Mathematics Grade 4 Teacher’s Guide RESOURCES
407
2012/09/14 5:40 PM
Place value boards, answer grid and
multiplication grid
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×
Last unit of multiplicand
0
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H9
408
Math G4 TG.indb 408
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9
U
Mathematics Grade 4 Teacher’s Guide RESOURCES
Tth Th H
T
U
This page may be photocopied.
2012/09/14 5:40 PM
Number grid
200 grid
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This page may be photocopied.
Math G4 TG.indb 409
Mathematics Grade 4 Teacher’s Guide RESOURCES
409
2012/09/14 5:40 PM
Number grids
100 grid
99 grid
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410
Math G4 TG.indb 410
Mathematics Grade 4 Teacher’s Guide RESOURCES
This page may be photocopied.
2012/09/14 5:40 PM
Shapes (1)
This page may be photocopied.
Math G4 TG.indb 411
Mathematics Grade 4 Teacher’s Guide RESOURCES
411
2012/09/14 5:40 PM
Shapes (2)
412
Math G4 TG.indb 412
Mathematics Grade 4 Teacher’s Guide RESOURCES
This page may be photocopied.
2012/09/14 5:40 PM
Square, triangle, pentagon and hexagon
Square
Triangle
Pentagon
Hexagon
This page may be photocopied.
Math G4 TG.indb 413
Mathematics Grade 4 Teacher’s Guide RESOURCES
413
2012/09/14 5:40 PM
Hexagons
414
Math G4 TG.indb 414
Mathematics Grade 4 Teacher’s Guide RESOURCES
This page may be photocopied.
2012/09/14 5:40 PM
Build shapes (Learner’s Book pages 242 and 243)
This page may be photocopied.
Math G4 TG.indb 415
Mathematics Grade 4 Teacher’s Guide RESOURCES
415
2012/09/14 5:40 PM
Square grid
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Math G4 TG.indb 416
Mathematics Grade 4 Teacher’s Guide RESOURCES
This page may be photocopied.
2012/09/14 5:40 PM
Dotted grid
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Mathematics Grade 4 Teacher’s Guide RESOURCES
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417
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Triangle on dotted grid
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Math G4 TG.indb• 418
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418
Mathematics Grade 4 Teacher’s Guide RESOURCES
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2012/09/14 5:40 PM
Squares and rectangles on dotted grid
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Math G4 TG.indb 419
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419
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2012/09/14 5:40 PM
Symmetry: complete shapes
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420
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Math G4 TG.indb 420
A
E
Mathematics Grade 4 Teacher’s Guide RESOURCES
B
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D
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2012/09/14 5:40 PM
3-D objects (1)
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Math G4 TG.indb 421
Mathematics Grade 4 Teacher’s Guide RESOURCES
421
2012/09/14 5:40 PM
3-D objects (2)
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•Grade• 4 Teacher’s
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• Guide RESOURCES
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Mathematics
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• This•page may
• be photocopied.
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• •
422
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Math G4 TG.indb 422
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2012/09/14 5:40 PM
Six nets for a cube
A
B
C
D
E
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Math G4 TG.indb 423
Mathematics Grade 4 Teacher’s Guide RESOURCES
423
2012/09/14 5:40 PM
Net for a cube
424
Math G4 TG.indb 424
Mathematics Grade 4 Teacher’s Guide RESOURCES
This page may be photocopied.
2012/09/14 5:40 PM
Net for a rectangular prism
This page may be photocopied.
Math G4 TG.indb 425
Mathematics Grade 4 Teacher’s Guide RESOURCES
425
2012/09/14 5:40 PM
I have 20.Who
has double this plus 1?
I have 41.
Who as 2 fewer?
I have 39.
Who has 5 fewer?
I have 34.
Who has 1 more?
I have 35.
Who has this minus 2?
I have 33.
Who has twice as much?
I have 66. Who has
this minus 60, plus 5?
I have 11.
Who has 6 more?
I have 17.
Who has 3 fewer?
I have 31. Who has this
minus 1, divided by 3?
I have 8.
Who has 4 more?
I have 12.
Who has half of this?
426
Math G4 TG.indb 426
I have 6. Who as 1 fewer,
divided by 5?
I have 1. Who has
this multiplied by 9?
I have 9. Who has this
plus 3, divided by 4?
I have 3. Who has
double this plus 4?
I have 10.
Who has a dozen more?
Mathematics Grade 4 Teacher’s Guide RESOURCES
I have 22. Who has
this divided by 11?
I have 2. Who has this
minus 2, multiplied by 6?
I have 9. Who has
87 more divided by 2?
I have 18.
Who has 2 more?
I have 46. Who has
1 fewer divided by 5?
I have 37.
Who has 9 more?
I have 48.
Who has this minus 9?
I have 5. Who has this
plus 1, multiplied by 3?
I have 27.
Who has double this?
I have 30. Who has
this divided by 6?
I have 21.
Who has 10 more?
I have 14. Who has half
of this multiplied by 3?
I have 10.
Who has 17 more?
I have 54.
Who has this plus 6?
I have 40.
Who has 10 fewer?
I have 19. Who has this
plus 1 multiplied by 2?
I have 0.
Who has 19 more?
Distribute the pack of cards in the class. If there
are more cards than learners, give some learners
each two cards. If there are too few cards, some
learners can play in pairs and share a card. The
I have 24.
Who has double this?
I have 36. Who has half
of this plus 6?
I have 26.
Who has 10 more?
I have 13.
Who has double this?
I have 45.
Who has this plus 13?
I have 60.
Who has this minus 15?
I have . . . (1)
first player reads his or her card. The learner who
has the card with the answer, reads his or her
card. The game continues until the chain ends
with the first player answering the last question.
This page may be photocopied.
2012/09/14 5:40 PM
This page may be photocopied.
Math G4 TG.indb 427
I have 16.
Who has 9 fewer?
I have 15. Who has
double this minus 12?
I have 50.
Who has twice as much?
I have 23.
Who has 9 fewer?
I have 9.
Who has 4 more?
I have 7.
Who has this plus 8?
I have 42.
Who has 20 more?
I have 35.
Who has this minus 12?
I have 19.
Who has this minus 10?
I have 22.
Who has double this?
I have 27.
Who has 5 fewer?
I have 24.
Who has this minus 8?
I have 17.
Who has this plus 10?
I have 20. Who has
this, plus 3, minus 6?
I have 48.
Who has half of this?
I have 10.
Who has double this?
I have 6.Who has
1 fewer, plus 5?
I have 44.
Who has this, plus 4?
I have 12.
Who has half of this?
I have 8.
Who has 4 more?
I have 13.
Who has double this?
I have 14.
Who has double this?
I have 62.
Who has this minus 12?
I have 70.
Who has double this?
I have 59.
Who has 11 more?
I have 69.
Who has this minus 10?
I have 29.
Who has 40 more?
I have 46. Who has
half of this plus 6?
I have 53.
Who has 7 fewer?
I have 45.
Who has 8 more?
I have 26. Who has
half of this minus 5?
I have 28.
Who has 9 fewer?
I have 100. Who has
this, minus 60, plus 5?
I have 75.
Who has 40 fewer?
I have 90.
Who has this minus 15?
I have 140.
Who has this minus 50?
I have 49.
Who has 7 fewer?
I have 25. Who has
double this plus 1?
I have 51.
Who has 2 fewer?
I have 18.
Who has 7 more?
I have . . . (2)
Mathematics Grade 4 Teacher’s Guide RESOURCES
427
2012/09/14 5:40 PM
Fraction snap
1
2
1
2
3
Math G4 TG.indb 428
1
3
2
2
3
3
4
4
5
5
1
1
428
1 1
1
2
3
6
3
6
1
7
6
7
1
4
3
4
1
1 1
1
1
1
Mathematics Grade 4 Teacher’s Guide RESOURCES
2
4
2
6
2
4
4
6
3
5
2
5
3
8
5
8
7
10
3
10
This page may be photocopied.
2012/09/14 5:40 PM
Fraction dominoes: Enlarge and copy onto stiff card
This game is for two, three, four or more players.
Play it like dominoes that you play with 28 cards.
Each players gets the same number of cards
(seven each if there are four players). The player
who has two-sevenths starts playing. The next
player has to match the fraction symbol four-
fifths to the diagram next to two-sevenths. If the
next player does not have the matching card, he
or she knocks and loses a round. The first player
who has played all his or her cards, wins. The rest
of the players continue playing until they have
played all their cards.
1
2
4
5
4
6
3
4
6
9
1
2
1
4
2
5
1
10
2
3
4
8
1
6
2
8
6
8
2
6
5
10
1
9
2
4
3
7
4
7
1
3
3
6
1
7
3
9
5
6
2
10
1
5
1
8
This page may be photocopied.
Math G4 TG.indb 429
Mathematics Grade 4 Teacher’s Guide RESOURCES
429
2012/09/14 5:40 PM
Reach for the moon game board
Reach for
the moon
Start
Throw a dice
or choose
a calculation
and make
your move.
3 × 4 + 12
3×4÷6
27
5 × 9 – 40 forward
Home
9÷9
10 × 4 ÷ 20
26
100 ÷ 5 ÷ 5
7÷7
34
25
Go back
ten
spaces
22
35
Go two
spaces
forward
23
Go seven
spaces
24
Go back
six
spaces
33
0×1+1
28
112 + 1 12
21
16
43 – 39
15
Go back
three
spaces
1×1×2
14
Go six
spaces
forward
8÷8×1
13
Go back
seven
spaces
9×4÷6
Go two
spaces
forward
7 + 8 – 12
9
Go five
spaces
forward
4
10
21 ÷ 7
3
11
21 – 19
2
12
1
32
18 ÷ 6
Go back
three
spaces
31
Go two
spaces
29
15 + 16 – 25
30
14 – 11 + 0 forward
Go back
two
spaces
19
6
4×6÷8
13 + 14 – 21
7
18
8×3÷6
20
36 ÷ 9 + 2
27 ÷ 9
4×4÷4
17
8
5
2012/09/14 5:40 PM
Math G4 TG.indb 430
This page may be photocopied.
Mathematics Grade 4 Teacher’s Guide RESOURCES
430
Place value scatter board
1
10
1 000
10 000
1
100
100
10
This page may be photocopied.
Math G4 TG.indb 431
Mathematics Grade 4 Teacher’s Guide RESOURCES
431
2012/09/14 5:40 PM
Bingo games and answer sheets
Addition Bingo
Addition Bingo answer sheet
5+5
0+6
6+6
15 + 7
7+9
10
6
12
22
16
8+8
6+7
17 + 9
9+9
5+8
16
13
28
18
13
5+6
8+9
5+9
15 + 5
6+8
11
17
14
20
14
7+7
16 6
18 + 8
5+7
19 + 9
14
22
26
12
28
15 + 6
18 + 9
16 + 7
7+0
0+8
21
27
23
7
8
Subtraction Bingo
Subtraction Bingo answer sheet
10 – 5
16 – 0
13 – 5
15 – 6
17 – 7
5
16
8
9
10
15 – 8
17 – 9
20 – 9
12 – 5
10 – 9
7
8
11
7
1
16 – 7
13 – 4
10 – 6
15 – 9
13 – 9
9
9
4
6
4
10 – 8
15 – 7
20 – 8
17 – 8
16 – 6
2
8
12
9
10
20 – 7
18 – 9
12 – 9
13 – 8
10 – 0
13
9
3
5
10
Multiplication Bingo
Multiplication Bingo answer sheet
10 × 6
7×8
3×9
4×8
5×7
60
56
27
32
35
7×7
9×6
8×5
10 × 4
3×8
49
54
40
40
24
4×5
8×9
6×7
3×5
9×9
20
72
42
15
81
3×6
5×4
0×8
6×6
9×7
18
20
0
36
63
4×0
1×6
6×4
8×8
9×5
0
6
24
64
45
Division Bingo
Division Bingo answer sheet
5÷5
50 ÷ 10
24 ÷ 6
44 ÷ 11
1
5
4
3
4
48 ÷ 8
56 ÷ 7
36 ÷ 6 121 ÷ 11 28 ÷ 9
6
8
6
11
4
44 ÷ 4
18 ÷ 6
12 ÷ 12 66 ÷ 11
72 ÷ 8
11
3
1
6
9
35 ÷ 7
64 ÷ 8
90 ÷ 10
42 ÷ 7
45 ÷ 5
5
8
9
6
9
14 ÷ 2
12 ÷ 4
84 ÷ 12
63 ÷ 7
108 ÷ 9
7
3
7
9
12
432
Math G4 TG.indb 432
27 ÷ 9
Mathematics Grade 4 Teacher’s Guide RESOURCES
This page may be photocopied.
2012/09/14 5:40 PM
Tangram
1
3
2
4
5
6
This page may be photocopied.
Math G4 TG.indb 433
7
Mathematics Grade 4 Teacher’s Guide RESOURCES
433
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Math G4 TG.indb 434
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6. Documents
Add your own documents and notes, for example the CAPS document for Intermediate Phase
Mathematics, here.
Mathematics Grade 4 Teacher’s Guide
Math G4 TG.indb 435
435
2012/09/14 5:40 PM
Grade
4
Mathematics
Study & Master Mathematics has been specially developed by an
experienced author team to support the Curriculum and Assessment
Policy Statement (CAPS). This new and easy-to-use course not only
helps learners to master essential content and skills in the subject,
but gives them the best possible foundation on which to build their
Mathematics knowledge.
The comprehensive Learner’s Book provides:
• activities that develop learners’ skills and understanding
in each of the topics specified by the Mathematics
curriculum
• stimulating Mental Maths activities for all relevant topics
• examples based on learners’ own experiences.
The innovative Teacher’s Guide includes:
• a detailed daily teaching plan to support classroom
management
• teaching tips to guide teaching of the topics in the
learner material
• worked out answers for all activities in the Learner’s Book
• photocopiable record sheets and templates.
www.cup.co.za
SM_Maths_G4_TG_CAPS_Eng.indd 2
2012/09/12 11:07 AM