Maths Wise

Books 3, 4, and 5Teaching Guide

Shamlu Dudeja

Contents

Table of contents ........................................................................................... iiiTopic / unit objectives / skills learnt .......................................................... ivUsing the guide ........................................................................................... xiv A. Introduction ......................................................................................1 B. Teaching Guide for Maths Wise 3, 4, and 5 ..................................2 1. Skills acquired by children .......................................................3 2. Maths laboratory ........................................................................5 C. Maths Wise Books 3, 4, and 5 .........................................................7 1. Numbers ......................................................................................7 2. Number operations ................................................................. 10 3. Factors and multiples ............................................................. 14 4. Fractions .................................................................................. 16 5. Decimal fractions .................................................................... 20 6. Money ....................................................................................... 23 7. Percentage ................................................................................ 24 8. Measurements ......................................................................... 28 a. Time ................................................................................... 28 b. Length ............................................................................... 32 c. Weight ............................................................................... 32 d. Capacity ............................................................................ 33 e. Temperature ...................................................................... 34 9. Graphs ...................................................................................... 34 10. Area and perimeter ................................................................ 37 11. Geometrical concepts ............................................................. 42 D. A suggested lesson plan ................................................................ 48 E. Answers to Book 3 ........................................................................ 50 F. Answers to Book 4 ........................................................................ 80 G. Answers to Book 5 ...................................................................... 107

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Topics / Objectives / Skills learnt

Book 3 Book Pages

Unit 1 Assess and review 1 Unit Objectives: To reinforce lessons learnt in Maths Wise

Book 2. Skills learnt: Reinforcement of some of the concepts taught

in the preceding year

Unit 2 Numbers 9 Unit Objectives: To recognize, read, and write Roman

numbers p10; to identify even/odd numbers within a given sequence p16; to work with place values up to 6-digits and numbers up to 100,000 p17; to work with place values and expanded form p21; to compare numbers and put them in a sequence p28

Skills learnt: To identify commonly used Roman numerals; to recognize even/odd numbers up to 99 in a given sequence; to understand place-value concept with 6-digit numbers; to use >, <, = symbols; to understand number orders and sequencing

Unit 3 Number operations 31 Unit Objectives: To learn horizontal and vertical addition

p32; to learn addition without and with carrying over p32; to learn subtraction without p40 and with borrowing p42; to develop multiplication tables from 2 to 5 and the 10s table p46; to learn tables from 6 to 9 p50; to learn to multiply 2-digit numbers with 1-digit numbers p55; to learn long division p59 and short division p62; to solve word problems related to the four operations p63

Skills learnt: To work with the four number operations; to work out larger sums; to use multiplication tables up to 10; to carry out vertical and horizontal additions; to do mental sums; to solve word problems related to real-life

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Unit 4 Fractions 65 Unit Objectives: To learn common fractions and match

them with related figures p67; to solve equivalent fractions p71; to work with proper and improper fractions p75; to compare different fractions p75; to add and subtract fractions that have the same denominator p77

Skills learnt: To work with different types of fractions and relate them to everyday objects and situations; to compare different fractions and learn to add and subtract fractions

Unit 5 Measurements 81 Unit Objectives: To learn concepts of measurements and

their units: length p82; addition/subtraction of length p87; weight p91; addition/subtraction of weight p95; capacity p98; addition/subtraction of capacity p99

Skills learnt: To use units of measurement for length, weight, and capacity; to solve real-life problems related to measurements

Unit 6 Time 101 Unit Objectives: To learn a.m. and p.m., and midnight to

midday to midnight sequence p102; to read and write time from analogue and digital clocks p103; to add and subtract hours p106; to recall calendar months p109; to read and write dates from a calendar p110

Skills learnt: To differentiate between a.m. and p.m. times; to calculate time, before or after a given hour, using simple sums and word problems; to be able to remember the calendar sequence; to read dates from a calendar and write dates correctly

Unit 7 Geometry 113 Unit Objectives: To learn concepts of points, line segments,

and rays p114; to draw triangles and quadrilaterals p115; to draw a circle and to recognize its components p118;

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to learn the concept of perimeter and to solve problems related to it p121

Skills learnt: To differentiate between a line and line segments; to recognize the qualities of a point; to draw triangles and quadrilaterals; to recognize circles in nature; to work with the components of a circle; to learn the methods of drawing a circle; to measure perimeter; to work with related word problems

Unit 8 Graphs 125 Unit Objectives: To learn to make pictographs Skills learnt: To use symbols; to arrange and interpret

data

Unit 9 Assess and Review 2 129 Unit Objectives: To help children assess and review the

lessons learnt in this book Skills learnt: To reinforce concepts by doing review

exercises

Book 4 Book Pages

Unit 1 Assess and Review 1 1 Unit Objectives: To reinforce lessons learnt in Maths Wise

Book 3 Skills learnt: Reinforcement of some of the concepts learnt

in the preceding year

Unit 2 Numbers and Arithmetic Operations 7 Unit Objectives: To revise 6-digit numbers and learn the

Pakistani place value of numbers p8; to learn place value up to 9-digits p9; to compare and order big numbers p11; to learn addition of big numbers p12; to learn properties of addition p13; to learn subtraction of big numbers p15; to learn to multiply big numbers and properties of multiplication p17; to learn to divide p19; to learn properties of division p20

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Skills learnt: Students will be able to differentiate between the International and the Pakistani place-value chart when writing bigger numbers; they will learn about millions and up to 9-digit numbers; they will learn to add, subtract, multiply, and divide big numbers and the properties of these number operations

Unit 3 Factors and Multiples 23 Unit Objectives: To learn divisibility tests and how they

apply to numbers p24; to tell the difference between prime and composite numbers p26; to learn factors p28 and multiples p29; to learn prime factorization by the Listing Method p31 and by the Tree Method and short division p32; to learn to calculate HCF by the Venn diagram and by prime factorization p33; to learn to calculate LCM by common multiples, prime factorization, and short division p37

Skills learnt: Learning divisibility rules helps when dividing by certain numbers; an understanding of prime numbers and composite numbers; important mathematical concepts of factors and multiples with a clear understanding of the methods to calculate these. Exercises that include both numerical and word problems will consolidate understanding

Unit 4 Fractions 41 Unit Objectives: To learn types of fractions: unit, mixed,

proper, and improper fractions p44, equivalent fractions p46, comparing like and unlike fractions p49; to add p52 and subtract fractions p56 and some rules associated with them; to learn multiplication of fractions p59 and division of fractions p62

Skills learnt: In Maths Wise Book 3, the students were introduced to the types of fractions; in this book, they learn some more types. They should be able to perform number operations involving fractions

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Unit 5 Decimal Fractions 65 Unit Objectives: To learn that decimal fractions are

another way of writing fractions p66; to learn to add and subtract decimal fractions p73; to learn to multiply decimal fractions p75; to learn to divide decimal fractions p77

Skills learnt: The students should be able to work with decimal fractions using all the number operations

Unit 6 Measurements: Length, Weight, and Capacity 79 Unit Objectives: To learn conversion of units of length p80,

units of weight p83, and units of capacity p85 Skills learnt: Students will learn to convert units of

measurement using real-life examples

Unit 7 Time 87 Unit Objectives: To learn conversion of units of time p88;

to learn addition and subtraction of hours p91 Skills learnt: By the end of this unit, students will be able

to convert various units of time including days to hours, hours to seconds, months to years and vice versa. They will know how to add and subtract time

Unit 8 Geometry 95 Unit Objectives: To introduce the components of the

geometry box and their uses p96; to learn to measure lines p98; to draw and measure lines in cms and mms p100; to measure a curved line p101; to differentiate between horizontal and vertical lines p102; to draw a vertical line on a horizontal line using a ruler p103; to differentiate between parallel and non-parallel lines, and intersecting lines p104; to learn to draw parallel lines using a set square and ruler p106; to draw angles using an angle flipper, learn angle components: arms, vertex, types of angles p110; to learn to use a protractor to measure angles p113; to use the protractor to draw angles p117; to recognize a circle and its components p121; to

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draw a circle using a compass and ruler p123; to measure the circumference p125; to learn some terms related to quadrilaterals and types of quadrilaterals p126; to construct squares and rectangles using a set square and ruler p128

Skills learnt: Students should be skilled in using the geometry box to draw geometrical shapes—angles, circles, squares, and rectangles. They should be able to identify the components of these geometrical shapes

Unit 9 Information Handling 131 Unit Objectives: To learn the parts of a graph: label, X- and

Y-axis, scale, and title p132; to be able to draw bar graphs p134 and line graphs p135, pictographs p136; to learn what a pie chart is p136; to understand the information given and derive data from it to make graphs using symbols

Skills learnt: Students will be able to use symbols and arrange and interpret data to make pictographs

Unit 10 Assess and Review 2 139 Unit Objectives: assess and review lessons learnt in the

book p140 Skills learnt: Students recall what they have learnt; concepts

are reinforced by doing review exercises

Book 5 Book Pages

Unit 1 Assess and Review 1 1 Unit Objectives: To reinforce lessons learnt in Maths Wise

Book 4 Skills learnt: Reinforcement of some of the concepts learnt

in the preceding year

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Unit 2 Numbers and Arithmetic Operations 11 Unit Objectives: To revise large numbers, and the Pakistani

place value of numbers p12; to learn place value up to 10-digits and introduce billion p13; comparing and ordering numbers up to 10-digits p15; to learn addition and subtraction of big numbers p16; to learn to multiply with 3-digit numbers p21; to learn to divide by 2- and 3-digit numbers p22 and by 10, 100, and 1000 p23; to learn to do BODMAS operations p25

Skills learnt: Students will be able to differentiate between the International Place value Chart and the Pakistani system of writing bigger numbers; they will learn about billions and up to 10-digit numbers; they will learn to add, subtract, multiply, and divide big numbers and the properties of these number operations; they will learn to perform BODMAS operations

Unit 3 HCF and LCM 27 Unit Objectives: Remembering factors and multiples p28;

to review prime and composite numbers HCF and LCM p29, and some divisibility tests p31; to learn to calculate HCF by prime factorization p34, by long-division p35; finding LCM of 4 numbers p36; to learn what square numbers are p38

Skills learnt: Students will recall some concepts learnt in the previous year and also learn some more rules of divisibility; they will learn about square numbers

Unit 4 Fractions 43 Unit Objectives: To recall various types of fractions p44;

to learn to add and subtract unlike fractions p45; to learn to multiply fractions and whole numbers p50; to learn to multiply fractions with fractions p51; to learn to divide fractions by a whole number p55; to learn to divide a whole number by a fraction p56; to learn to divide a fraction by a fraction p56; to apply BODMAS on fractions problems p59; to learn properties related to fractions p60

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Skills learnt: By the end of this unit, students will have learnt to perform arithmetic operations involving fractions. They will also learn to solve word problems related to fractions

Unit 5 Decimal Fractions and Percentages 61 Unit Objectives: To learn about like and unlike decimal

fractions p63; to learn to add decimal fractions p64, and subtract decimal fractions p65; to learn to multiply decimal fractions p67, and to divide decimal fractions p69; to learn to apply BODMAS on decimal fractions p72; to learn to convert decimal fractions to fractions p73, and fractions into decimal fractions p73; to learn to round off decimal fractions p74; to learn that percentage is a special kind of fraction p77; to learn to convert fractions and decimal fractions into percentage p80; to solve related word problems p81

Skills learnt: By the end of this unit, students will be able to perform arithmetic operations on decimal fractions and percentages. They will also be able to perform conversions related to these

Unit 6 Measurements: Distance, Time, and Temperature 83 Unit Objectives: To learn conversion of units of length p84;

to learn to use smaller or bigger units p85; to learn to do addition and subtraction by converting unlike units into like units p86; to learn conversion related to time p91; to learn to add time p92 and to subtract time p93; to introduce the topic of temperature p98; to learn about the Celsius scale p99; to learn about the Fahrenheit scale and conversion between the two scales p101

Skills learnt: By the end of this unit, students will be able to convert units of measurements and be able to solve real-life problems related to these

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Unit 7 Unitary Method; Ratio and Proportion 103 Unit Objectives: To learn the unitary method p104;

to learn about ratio p105, and direct and inverse proportion p106

Skills learnt: By the end of this unit, students will know the concepts of ratio and proportion and be able to solve problems related to these topics

Unit 8 Geometry 109 Unit Objectives: To learn to use a protractor to construct

a right angle p111, a straight angle, and a reflex angle p112; to learn about angle pairs—adjacent, complementary p113, and supplementary angles p114; to learn the definition of a triangle p116; to classify triangles according to their sides p116, and according to the measure of their angles p117; to learn to construct triangles p120; to learn that the measure of the angles of a quadrilateral add up to 360° p123; to learn about different kinds of quadrilaterals p124, and to construct squares and rectangles using a protractor or a set square and a ruler p125

Skills learnt: Students will be able to differentiate between complementary and supplementary angles. They will know about adjacent angles and they will learn to construct triangles and quadrilaterals

Unit 9 Perimeter and area 127 Unit Objectives: To learn about perimeters and to solve

related problems p128; to use a formula to calculate the perimeter of a rectangle and a square p130; to learn about area p132; to use a formula to calculate area p134; to learn the use of calculating area in real-life situations p136; to learn about perimeter and area of irregular shapes p138

Skills learnt: By the end of this unit, students will know the concepts of perimeter and area and will be able to solve related problems

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Unit 10 Information Handling 141 Unit Objectives: To learn about averages p142; to learn

about bar / column graphs p148; to learn to construct pie charts p154

Skills learnt: By the end of this unit, students will know about the concept of average and how to calculate it. Students will be able to read and interpret data presented in bar graphs and pie charts

Unit 11 Assess and review 2 155 Unit Objectives: Assess and review lessons learnt in the

book p156 Skills learnt: Students recall what they have learnt; concepts

are reinforced by doing review exercises

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Using the Guide

This Teaching Guide combines suggestions for lessons from Maths Wise Books 3, 4, and 5 for primary classes. Teachers from different schools can easily adapt the suggestions given in the Guide to their existing teaching methods.Teaching at this level changes according to the requirements of individual schools and the kinds of students in each class. ‘Be prepared with a lesson plan each day, but keep teaching open-ended’ is the best motto.Objectives for each topic and skills learnt are given at the beginning of each book in the Table of Contents. These are also reproduced in this Guide to help teachers plan lessons. They can also assess the level of skills each child has acquired after each topic is completed.In the Guide, each topic is explained in detail. Additional work, in the form of activities using the objects listed in the Maths laboratory, is also suggested. The answer key for each book is given at the end.A sample lesson plan is suggested which can form a base for similar lesson plans on other topics.

Shamlu Dudeja

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A. Introduction

Mathematics has always been the best food for the enquiring mind, of a growing child. In today’s world of changing lifestyles, where IT, electronic gadgetry, and finding logical solutions to problems in daily life have become the needs of the day, employers are increasingly looking for thinking minds. It has become imperative that mathematics plays a significant role in education, right from the very beginning.Teachers of pre-primary levels and classes 1 and 2 have already laid a foundation for open and active minds in children. Maths Wise 3, 4, and 5 continue to use similar informal teaching methods, in order to imbue in children, keener mathematical skills. The transition from a ‘child’ to a ‘pupil’ becomes easy and smooth.It is recommended that pupils (up to class 5) are not put through rigid examinations. The teacher should be able to assess the progress of pupils with the help of a regular, weekly record of their work.

IMPORTANTThe ideal pupil-to-teacher ratio is around 8 children to 1 teacher. This is rarely possible. In a situation where a teacher may have a large class, there are 2 strategies, which may help:1. Willing mothers may be invited to help during classes, as ‘Buddy

Teachers’ (instead of assistant teachers). Many mothers will be willing to help, as they enjoy this activity. Some may wish to remain with the class, even after their children have moved on. It will require a week’s orientation before a mother is able to come in as a ‘Buddy teacher.’

2. Divide students into small groups so that they can work cooperatively; they will not require constant teacher attention.

The class starts with a review of previous day’s lesson using a fun activity. It could be a short quiz or a round of mental maths. It is useful to revise tables every day. A game involving the use of hands to find answers makes tables interesting! Teachers of Maths Wise Introductory Books 1, 2, and 3 may also find this useful.

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B. Teaching Guide for Maths Wise 3, 4, and 5

Maths Wise 3, 4, and 5 are for 8, 9, and 10 year old children, and have been written fully in line with the requirements of the National Mathematics Curriculum, and children’s levels of understanding and capability.As they grow older, children must be encouraged to think independently, explore their surroundings more boldly, and ask questions. Maths Wise 3, 4, and 5 provide children to use opportunities to explore, relate numbers to daily life situations and letters of the alphabet, use arithmetic operations (+, –, × and ÷), look for patterns in numbers and number formations, and other objects in their environment, and find answers for themselves whenever possible.New vocabulary, new topics, and new concepts are introduced by means of pre-topic discussions (or story-telling) and practical activities. At every step, concepts are developed using examples that smoothly flow into a series of relevant exercises. Hands-on work, in addition to exercises in the books, further consolidates these concepts and encourages independent thinking.The books provide a range of activities including puzzles, crosswords, coded message, brainteasers, and fun pages to guarantee the retention of interest and involvement of every child. There is sufficient drill for the students and challenging sums at the end of each topic and sub topic to extend the students. Samples of maze paths and blank cross number grids are given, which a child may use to create a puzzle for class fellows to solve.One of the greater needs for a teacher, as children grow older, is to recognize differing abilities, and to address them separately in each class. The minds of some children need to be stretched and, their capabilities exercised to the full, often independently of the teacher. The less mathematically able children need greater direction and support to ensure that they do not feel left out. The activities and problems in these books are of varied levels of difficulty, to meet these requirements.The Teaching Guide for Maths Wise books 3, 4, and 5 contains lot of suggestions for activities which lead to lateral thinking within the confines of a school syllabus. The activities and challenges are exciting for children who have learnt to enjoy maths. It is still not too late to develop in most children a liking for the subject by encouraging them to think just a little

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outside the textbooks. This can be great fun both for the teacher and pupils.

1. SKILLS ACQUIRED BY CHILDREN

The activities undertaken in Classes 1 and 2 will help children to achieve higher levels of comprehension and higher standards of work in Classes 3, 4, and 5.1. Concentration becomes automatic when children participate in

practical work using objects from daily life. This helps them to relate their school work to the world around them.

2. Memory is honed and new concepts are stored into quick-recall memory, through work such as tables and sequences. Mnemonics have been suggested to help memorize sequences of objects/activities. For example: BODMAS and work with 5– or 6–digit numbers which draw on recall of work done with 2– and 3–digit numbers.

3. Recognition increases as children are exposed to more ideas, such as number patterns, fractions, factors, and shapes (including animals and cartoons). Later, there are situations where they need to recall these.

4. Association occurs when children apply knowledge gained in earlier years to newer concepts. Memory and recognition are used to associate one object with another through a common characteristic. For example: a hexagon has 6 sides, a beehive has hexagonal cells.

5. The study of mathematics depends upon logic and it comes from concentration, memory, recognition, and association.

a. Bees use hexagonal cells and not circular ones to make a hive, because in hexagonal tessellate, there is no wastage of space.

b. Use of comparative language such as long, longer, longest, comes from logic.

As Mathematics gets more formal, it is mandatory that the interest of the children is kept alive by continuing with outdoor / indoor activities, colourful charts, making up a story to introduce a new topic and practical demonstrations, whenever possible.If the interest is kept alive, success will follow. Not only does learning become fun for children, the teachers will enjoy their teaching as well.Three painful ‘Ps’ which should not exist in a teacher’s vocabulary are:1. Partiality to one child kills initiative in 10. So, please no partiality to

any child.

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2. Pointing out mistakes in front of others is a definite no. It is best to look out for the best traits using positive language. Coming up from Class 2, children are still very sensitive as they settle in to a more formal style of schooling.

3. Punishment is ruled out. There are no children who are beyond gentle cajoling, a smile or a hug of a teacher. Punishment, like a ‘slap on the hand,’ only makes matters worse, and children tend to become stubborn. Milder punishment like standing outside the classroom may become necessary for the unruly student and can be very effective.

The positive ‘Ps’ which must exist in a teacher’s vocabulary are:1. Praise is positive: employ a ‘yes’ attitude as often as possible. Praising

good work and good behaviour will encourage other children to follow suit.

2. Patience: there is no virtue like patience, especially in a teacher. This means not losing one’s temper.

3. Parent-like attitude is very warming. Teachers should know when to respond to attention-seeking behaviour and when to ignore it; the bottom line is the underlying sense of security a child feels.

The height of tables and chairs must be correct for the students. Emphasis needs to be laid on correct posture, when children write. If attention is not paid to this now, it can lead to a bad posture permanently and back problems.A little exercise to relax those load-carrying shoulders helps muscles relax, and motor control improves.With straight backs, hands on hips, forward and backward bending is helpful.Then, the same posture, children put both hands straight ahead and start writing numbers 0 to 9 with their hands in air, first both hands going in the same direction and then the two hands going in opposite directions, one clockwise and the other anticlockwise. (Here is an excellent opportunity to introduce these new words into their vocabulary. Does the tap open in a clockwise or an anticlockwise direction? The screwdriver and the lock on the door are further examples.

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2. MATHS LAB

A maths lab for classes 3 to 5 must contain some of the items included in the earlier classes. Some extra items are suggested here:• some soft-drink bottle caps, strings of 10 bottle caps strung together

and a group of 10 strings knotted together to represent one hundred. Sets of such strings can be used for explaining numbers, addition, and subtraction.

• strings for measuring lengths of objects or a child’s height• weighing scales of 4 different types: a spring scale, an ordinary

balance, a regular scale with a vertical circular dial, and a ground level weighing scale on which children can weigh themselves. Children can be taken on a field trip to the station to observe weighing scales on which cars and other heavy objects are weighed.

• tape measures and rulers of different sizes• a trundle wheel• shells, small stones, beads in groups of 10s, 100s and 1000s, 10000s

wrapped securely in cloth bags• Several sets of 4 almost identical objects, one with a very slight

difference, to improve observation activities• colourful pictures or charts of shops displaying fruit and vegetables,

toys, and a rack of clothes, all with price tags• sudoku puzzles of differing levels• fabrics or strong paper, to make different objects• solid shapes in the form of wooden blocks, balls (spheres), egg-

shapes, dice shapes (cubes), box shapes (cuboids), cans (cylinders) and cones

• cubes, cuboids, cylinders, and cones made from thick card, which can be opened out and laid flat

• flat shapes cut out from thick card or wood, such as circles, squares, and triangles, so the students can feel the flat surface and count the corners and the edges. It will be useful to have flat shapes which are equal to the sides of the solids, so that children can explore the relationship between solids and their faces.

• rolls of cords and ribbon• plastic or steel tins, jars, bowls of different sizes for comparing

capacity. Bowls made of halves of dried coconut shells or bamboo segments split in halves may be used.

• pencils and crayons of different colours and lengths

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• charts corresponding to different concepts studied• solids made from play dough which have 2 (or more) lines of

symmetry, so that they can be cut into halves along 2 axes• squares of reflecting plastic surfaces (avoid keeping glass mirrors)• 3-piece jigsaw cards with a number and corresponding multiplication

and division sums; e.g. 8, 324 and 4 × 2, domino and flashcards

• a giant number square 1 to 100 on the wall and several sheets with blank squares for children to work on

• a horizontal wooden rod with several pegs, wooden numbers hang from these

• number tabs, up to 4–, 5–, 6–, and 7– digit figures• analogue and digital clocks• abacus and calculators• 12 pages to make up a calendar; sunshine, rain and cold weather to

be depicted by symbols on each day. Reinforces counting, association between weather and appropriate symbols, clothes which people wear and food that people eat during these seasons

• plastic baskets or trays to keep assorted objects• a fraction wall, with fractions such as 1/2s, 1/3s, 1/4s, and 1/5s• plastic cakes / pizzas / fruits / jars of water to demonstrate fraction

and percentage• gem clips, rubber bands• a stopwatch• a set of geometrical instruments• waste bins marked PLASTICS, GLASS, and PAPER• attractive charts and other child-friendly displays on walls for use as

learning aids• a soft board covered with chamois leather on which children can stick

numbers or pictures• to make learning enjoyable, a patch of garden in the school yard, with

different shrubs and pets such as rabbits, white mice, and tortoises, a fish aquarium and an aviary, would be useful. These also help create awareness of the environment.

Each Maths Wise book begins with a detailed review of the previous year’s work. It is important to check that each child has mastered concepts learnt in the previous year and is handling these independently, with confidence.An interesting way to do it may be to conduct a quiz following the pattern

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C. Maths Wise 3, 4, and 5

1. NUMBERS

Step-by-step, numbers up to hundreds of thousands are introduced. The concept has been introduced based on the students’ prior knowledge. The comparison of place values has been done pictorially to aid the visual learning.It must be emphasized here that if a student is working well with 3-digit numbers, going further to 5-, 6-, or 7-digit numbers would be easy. The language used, the methodology, and the techniques are the same for carry over and grouping or borrowing.The concept of 4-, 5-, and 6-digit numbers is best explained by using the terms ‘house of thousands’ and ‘house of tens’, with a comma between the houses.Less than (<), greater than (>)A crocodile’s mouth drawn on the board, always ready to grab the bigger number, can be used. Similarly, the left hand with the thumb held horizontally and the forefinger held straight up, makes an angle to show less than. Similarly, the right hand can be used to show greater than.Students are introduced to Pakistani and international numbering systems.

Activity

Team games are an excellent way to present problems involving large numbers in expanded form, ascending/descending orders, identify ‘before and after’ a given number and skip counting.One half of the class collects statistics with large numbers, such as populations of various countries, heights of mountains, and distances of stars from the Earth.For example, they find that the population of a country is 4,56,890 (456,890) and write the number on the board. The other half reads the numbers aloud:‘Four lakh fifty-six thousand, eight hundred and ninety,’ or‘Four hundred and fifty-six thousand, eight hundred and ninety.’

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Another topic on everyone’s lips is global warming. Statistics say 2 trillion tonnes of ice melted into the seas between 2003 and 2008. These are international issues and the numeration is international. This is a good chance to talk about conservation, to reduce the greenhouse effect in 10 years time; to slow down the global warming and its hazardous effects on the lives of the students, who will be adults in 10 years.

Activity

The board has a list of the numerical value of the letters of the alphabet.A = 1 J = 10 (1) S = 19 (1)B = 2 K = 11 (2) T = 20 (2)C = 3 L = 12 (3) U = 21 (3)D = 4 M = 13 (4) V = 22 (4)E = 5 N = 14 (5) W = 23 (5)F = 6 O = 15 (6) X = 24 (6)G = 7 P = 16 (7) Y = 25 (7)H = 8 Q = 17 (8) Z = 26 (8)I = 9 R = 18 (9)

Words formed have different values, according to the position of the numbers. For example, CAB = 312 or three hundred and twelve.Write the following words:1. 1-letter words with the highest and the lowest number values. (I is 9,

A is1)2. 2-letter words with the highest and the lowest number values

(IF … 96, AS …11) and values in between (AT = 12, SO = 16, BE = 25, PI = 79 and so on)

It is quite a challenge to work with 4–, 5–, and 6–digit numbers. Teachers need to identify some words before putting this activity to the class.

SOME NUMBER PUZZLES

Puzzle 1Which 1-digit number, when multiplied by itself, gives a number which is the reverse of the number, doubled.

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Answer9

9 × 9 = 81; 9 + 9 = 18

Puzzle 2a. If 9358 stands for SINK, what does 358 stand for?b. If 456789 stands for SAMPLE, what does 56789 stand for?c. If 4321 stands for LAMP, what does 1342 stand for?d. If 7531 stands for GOLF, what does 1357 stand for?

Answera. INKb. AMPLEc. PALMd. FLOG

Puzzle 3 (Tick the correct choice.)1 1 1 1

1 2 3 4

1 3 5 7

1 4 7 ?

Choicesa. 8b. 9c. 10

Sudoku puzzles

9 1 1611 4 87 13 5

7 17 2 15 Hint: a total of 35 in each box.

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2. NUMBER OPERATIONS

The four operations +, –, ×, and ÷ have been handled up to 3-digit numbers. Working with larger numbers will not be difficult as long as students have understood their work with 3-digit numbers.Discuss the phrases ‘sum of 2 or more numbers’ for addition, ‘difference between 2 numbers’, ‘subtract the smaller number from the bigger one’ for subtraction, ‘product of 2 or more numbers’ for multiplication and ‘quotient’ and ‘divide the bigger number by the smaller one’ for division.Once again, it is emphasized that addition and multiplication are associative but subtraction and division are not.

45 × 396 = 396 × 4534 + 598 + 213 = 598 + 213 + 34

109 – 98 is not the same as 98 – 10998 ÷ 7 is not the same as 7 ÷ 98

• This requires a lot of practice. Additional worksheets must be prepared to supplement the exercises given in the book. Once again, emphasis must be laid on setting out the sums in neat straight columns. Maths exercise copies with squares may be used for this purpose.

• Some children have difficulty in setting out the sums involving large numbers. They may set out digits in the wrong columns if not guided. Plenty of addition / subtraction sums are horizontally set out on the board. The students are required to set them out vertically in their exercise copies and find the answers. The teacher must ensure that the digits are placed in the correct columns.

Example

3897 – 26

Students may write this as: 3 8 9 7 – 2 6

This is incorrect.

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In the initial stages, it is worthwhile writing out problems like the one given below:

Find the difference between:

Th H T U T U 3 8 9 7 – 2 6

Th H T U 3 8 9 7– 2 6 This is the correct way to write the sum. 3 8 7 1

• Remember that children often forget to carry the 1 from Units to Tens or Tens to Hundreds columns.

Example

Th H T U 1 1 1

4 3 5 6+ 2 7 8 9 7 1 4 5

Now observe the subtraction using the same numbers above:

Th H T U 3 12 14 1

4 3 5 6– 2 7 8 9 1 5 6 7

Here, 9 cannot be taken away from 6, so 5 (Tens) is regrouped into 10 Units and 4 Tens (or 1 is borrowed from the 5 in the Tens column). This gives 16 Units, and 4 Tens. (16 – 9 = 7). Similar regrouping is done in the other columns, and subtraction is carried out easily.This can be displayed, practically, in bundles of 10 in every column and one group of 10 is opened when necessary as shown in the textbook.

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When a number is multiplied by 10, 100 or 1000, the same digits appear in the product but the place value of each number changes. The number has shifted, so to speak, 1 place to the left, when multiplied by 10 (1 zero) or 2 places to the left for 100 (2 zeroes).

Example

Th H T U TTh Th H T U 4 5 7 × 1 0 = 4 5 7 0 4 5 7 × 1 0 0 = 4 5 7 0 0

When multiplied by 10:7 in U, shifts 1 place to the left to T.5 in T, shifts 1 place to the left to H.4 in H, shifts 1 place to the left to Th.

The entire number, 457 when multiplied by 10, shifts 1 place left (10 has 1 zero), and 2 places to the left when multiplied by 100 (100 has 2 zeroes).This simple but important rule will help students master the tricks of mental multiplication by 10, 100, and 1000.Multiplicands of 2-digits are introduced as a 2-part multiplication, noting down the products at both stages, as shown in the book, in 2 different lines (with correct shift to the left). The sum of the 2 products gives the final product.

Activity

Students work in groups of 8.Students have circular cardboard discs of about 4 cm diameter. Alternatively, children can draw circles in their exercise copies.Each circle is divided into parts, like a clock, but with only 9 parts, with numbers written from 1 to 9 around the circumference.Students write down tables from 2s to 15s as follows:1 × 3 = 3 1 × 8 = 82 × 3 = 6 2 × 8 = 16 (7)3 × 3 = 9 3 × 8 = 24 (6)4 × 3 = 12 (1 +2 = 3) 4 × 8 = 32 (5)

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5 × 3 = 15 (6) 5 × 8 = 40 (4)6 × 3 = 18 (9) 6 × 8 = 48 (13)7 × 3 = 21 (3) 7 × 8 = 56 (11)8 × 3 = 24 (6) 8 × 8 = 64 (10)9 × 3 = 27 (9) 9 × 8 = 72 (9)10 × 3 = 30 (3) 10 × 8 = 80 (8)

Students join the numbers on separate discs, in the order of the numbers: for the 3s tables: 3, 6, 9, 3, 6, 9,… and for the 8s tables: 8, 7, 6, 5, 4, 13,…Each group of 8 has 16 separate discs with the patterns formed by different tables. It is interesting to look at tables for 1 and 10; 2, 7, and 11; 9 and 18; 3, 6, and 15, 4, 5 and 13; and so on. These activities help students to learn tables.For division by 10, 100, and 1000, the concept of shift remains the same. The number shifts to the right, according to the number of zeroes in the divisor.Short-division or long-division, with or without remainder is handled practically, through repeated subtractions.As in multiplication, so also in division, practical work with beads, marbles, and buttons is necessary. Division as repeated subtraction (also with a remainder) becomes clearer with practical examples.Long-division can be easily mastered if the child understands that he must record each stage of the operation carefully in the appropriate column. The process should be started with known multiplication facts without remainders, slowly moving on to more complicated divisions, with remainders.BODMAS is the accepted method to simplify a sequence of numbers with different operations.

Example

I must pay Club bill of Rs 2000 on Monday – 2000I have Rs 500 in my wallet + 500My 3 friends each give me Rs 1000 at the end of the week (3 × 1000) + 3000The money I have after paying my bill is Rs 1500

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Presenting problems with two operations would be better explained on the board. Pupils point out the part that is to be solved first. Moving on to 3rd and 4th operations (i.e. simplification). Discuss the order of operations, first. Brackets are introduced as helpers which simplify the job of solving problems.

3. FACTORS AND MULTIPLESThe concepts of factors and multiples start with multiplication tables. The beginning of factors can be shown as intersections of sticks or as a branch (of a tree) bearing new leaves.

ActivityStudents start with prime factors. They go back to the tables; they see that in 3 × 5 = 15, or 5 × 3 = 15, 5 and 3 are factors of 15.On the board, number 15 is shown with 3 and 5 as leaves; similarly 35 is written with 7 and 5 shown as leaves.It is important that students use phrases, such as, ‘1, 2, 3, and 6 are factors of 6’ and ‘6 is a multiple of 1, 2, 3, and 6’.Later, with larger numbers, such as 12, the first step could be 2 branches, showing 6 and 2, and the next step shows 6 dividing into 2 further branches of 3 and 2.Students work with different numbers in their exercise copies.

IMPORTANTEach number is a multiple of 1 and each number has 1 as a factor.Also, each number is a multiple of the number itself, and the number is a factor of itself.

ActivityA Venn diagram is introduced to groups of students.The teacher calls out names of children who have sports and art as extra curricular activities. There are bound to be some who take both. Draw two intersecting circles on the floor, to explain how to show students who take part in both activities.Later, a Venn diagram helps students visualize the idea of common multiples. First, the teacher reproduces this exercise on the board and then the students work on Venn diagrams of their own in their exercise copies.

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Activity

Working with sticks, 5 horizontally and 3 vertically, intersecting at 15 points, also shows that factors of 15 are 3 and 5 or for a number like 12, formations of 3 and 4, 2 and 6 are possible.This presents many interesting possibilities.

1 × 12 = 12

For prime numbers, the formations will be like this.

5 = 1 × 5 = 5 × 1 3 = 1 × 3 = 3 × 1 2 = 1 × 2 = 2 × 1

3 × 5 = 15 2 × 6 = 12 3 × 4 = 12

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Once the concept of factors is clear, students will have little trouble in distinguishing prime and composite numbers.Team games will help the class learn the divisibility test. The class can be divided into 4 teams. Each team stands for divisibility of a number, such as 2, 3, 5, 7 or 11. The teacher reads out random numbers; the relevant team, which stands for a factor, responds with the correct answer.

4. FRACTIONS

Students have worked with fractions in earlier books. A fraction wall chart is a useful tool to explain the various aspects of fractions, especially to review concepts such as:1 whole =2 × 2

1 (2 halves) = 22

3 × 31 (3 one-thirds) = 3

3

4 × 41 (4 quarters) = 4

4

5 × 51 (5 one-fifths) = 5

5

Also, that the following fractions are equal.

1 × 21 = 2

1

2 × 41 = 4

2

3 × 61 = 6

3

In addition to the fraction wall, students cut strips of coloured paper into halves, thirds, quarters, and fifths.Emphasis is laid on the correct usage of fraction terms such as one-fifth instead of one over five and 99 hundredths instead of 99 over 100.Later, for a fraction such as 2000

3 , it is acceptable to say 3 upon 2000. Choice of language is also important for introduction of numerator and denominator.

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a. To introduce improper fractions

Shehla made strawberry jam and filled it in a jar. The jar was full with some jam remaining. She took another jar and it was half full. How many jars of jam does Shehla have?

1 + 21 = three halves = 2

3 = 1 21

23 is an ‘improper fraction’ because it is more than a whole.

In the same manner, a strip, which is equal to 1 whole and a quarter on the fraction wall, is obviously larger than one whole. This will be written as:

1 + 41 = five quarters = 4

5 = 1 41

Activity

Students write the following fractions on the board and separate improper fractions from proper fractions also showing if any of these can be converted to mixed fractions.

32 ,

45 ,

61 ,

67 ,

109 ,

1011 ,

89

b. Number operations (+, – , ×, ÷ ) involving fractions

Once students are comfortable with the fraction wall, fraction equations like the following are easy to understand.

21 + 2

1 = 1, 1 – 21 = 2

1 and 21 of 2

1 = 41 = 2

1 × 21

Many children encounter a problem in multiplication and division while working out simplifications of fractions. A great deal of confusion is removed by paying close attention to the language in which these concepts are introduced. When students begin to multiply fractions, it is essential that they understand exactly what is happening. Rules are to be looked at after they have understood the concepts. The fact is that ‘of ’ and ‘×’ translate to the same operation, but they have different significance in BODMAS.Multiplication of fractions is also repeated addition. Students will see that:

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72 + 7

2 + 72 = 3 × 7

2 = 13 × 7

2 = 3 × 12 × 7 = 7

6

(Multiply numerators and denominators)

When a fraction is multiplied by another fraction, a sequence diagram is ideally explained on the board. Each stage can then be explained separately.

Example

21 of 4

3 = 83

1. 21 of 4

3

In a diagram, it is easy for a child to see that 21 of 4

3 = 83

83 = 2

1 × 43 = 2 4

1 3#

#

Multiply the numerators and denominators separately.

2. 21 × 5

4 = 104 = 5

2

Here, it is possible to explain the method of reducing the number to its lowest form before multiplication is carried out.

21 × 5

4 (can be divided by 1 or 22 without changing the value)

21 × 5

4 (div by 2)

19

= 11 × 5

2

= 52 (as shown in the diagram)

It is essential to reinforce the division concept using very simple steps.

Look at this 41 × 1 or 1 × 4

1

This is translated into how much a quarter of a whole is or what 1 times a quarter is. The answer is simple, if students go back to the earlier fraction strips and multiplication by 1.How many sixths (1/6ths) make 2 wholes? How many twelfths (1/12ths) make 9 wholes? By the time the students have completed the first few exercises, it will be clear that they have been using multiplication to solve division problems and that division is the inverse of multiplication.This is the time to introduce the idea of reciprocal fractions. Students will now proceed quickly through the various stages of division of fractions, applying the language of division.Once this is understood, multiplication of mixed numbers and reducing a set of fractions to the lowest terms is easy.

Example

5 71 × 2 8

5 ÷ 1 207

= 736 × 8

21 ÷ 2027

div by 4 div by 7

= 736 × 8

21 × 2720

div by 7 div by 4

20

div by 9 div by 2

= 19 × 2

3 × 2720

div by 2 div by 9

div by 3

= 1 × 3 × 310

div by 3

= 1 × 1 × 10= 10

IMPORTANT

It must be explained that ‘of ’ from BODMAS (Brackets, Of, Division, Multiplication, Addition, Subtraction) is the same as ×. In a simplification problem, ‘of ’ must be worked out before ‘division’ whereas × is worked out after ‘division’. Sometimes, it will not affect the answer, but at other times, it does.

Examples

2 54 ÷1 3

2 × 75

= 514 × 5

3 × 75

= 56 = 1 5

1

2 54 ÷ 1 3

2 of 2 54

= 514 ÷ 3

14

= 514 × 14

3 = 53

5. DECIMAL FRACTIONS

The most significant aspects of decimals fractions are the facts that:a. numbers with decimal points are, in fact, fractions.b. decimal notation is an extension of the ‘tens’ concept of the place

value of whole numbers.

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The decimal point simply makes it clear where the whole number ends and the decimal fraction begins, each digit having one-tenth the value, as it goes to the right.To use decimal fractions confidently in everyday life, with money and measurements, pupils need to be comfortable with the decimal notation, its equivalence with fractions and the place value of each number to the right of the decimal.

Activity

Students work with different objects such as a loaf of bread, a cucumber, 100-square grids, and cubes made from play dough.Different groups cut these objects into 10 equal parts: 10 slices from each loaf and each cucumber, 10 strips from each paper grid, and 10 slabs from each cube.Students are familiar with the fact that each of these equal parts is equal to 10

1 or a tenth of the whole.The teacher explains that there is another method of writing one-tenth of a whole, or 10

1 . One-tenth is written as 0.1; the decimal point ‘.’ is used to separate wholes and tenths. (In 0.1, 0 is for wholes and .1 stands for tenths. The prefix deci stands for a tenth; deca stands for 10 times.)This is how decimal points are used to indicate fractions:

101 = 0.1

102 = 0.2

103 = 0.3

1010 = 1.0

(It is important to explain that 01 = 1 or 005 = 5. So also, 0.10 or 0.100 is equal to 0.1)These can be shown on a square sheet of paper with 10 strips, or the decimal bar.

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Further, each of the tenths is cut into 10 equal parts. A ‘tenth’ strip from the square grid is cut into 10 squares, and the slab of play dough is cut into 10 equal rods.

Activity

This is a good time to review the number of faces a cube has.Groups of 4 students each have a ball of play dough. Each group has a plastic knife.The teacher asks, ‘What is the minimum number of cuts you need to make to get a perfect cube out of this ball?’Students make cuts and eventually come to the conclusion that since a cube has 6 faces, it would need 6 cuts at right angles to make a perfect cube of any size from the sphere.

Activity

Through practical work, students understand that there are 10 strips or 100 small squares in the 100-square grid.Students count and colour the strips and squares on the grids as they work with decimal fractions.

They see that one small square (coloured) is equal to 1001 of the whole.

They write this as:

1001 = 0.01

1003 = 0.03

1009 = 0.09 and so on

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On the board, this is written as:Tens Units. Tenths Hundredths00.10(‘1 Tenth’ or ‘10 Hundredths’; read as ‘zero point one’)

00.30(‘3 Tenths’ or ‘30 Hundredths; read as ‘zero point three’)

00.05(‘5 Hundredths’; read as ‘zero point zero five’)

0.1 = 101 0.3 = 10

3

6. MONEY

All children learn to use money early in life, so this is an easy topic to study.A little introduction about how man used beads and shells and even cocoa beans as currencies was given in classes 1 and 2 and students worked with simple addition and subtraction of whole rupees.Currencies of various countries are discussed and it is seen that all countries use decimal systems, up to hundredths (or 2 places of decimal). ‘Euro’ is introduced as the common currency of most states of the European Union. The Dollar on the other hand, is used by America, Singapore, and Australia, but in every country the value of a dollar is different. Children learn about currencies of countries such as India and Indonesia, where a ‘rupee’ is called a ‘rupaiyah’.

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A collection of dollars, pounds, yens, cents, pennies etc. is useful for general knowledge. Comparative values, at a very simple level, can be discussed. A talk about ‘Dollar stores’ in the US and other countries is interesting, where everything is available for $1. Questions such as ‘If a toy car costs $1, how much will it cost in Pakistani rupees?’ generate active interest.Finer points, such as ‘Re’ is written for 1 rupee and ‘Rs’ is written for more than 1 rupee, need to be explained.Once the decimal fractions up to ‘hundredths’ place are understood, everything falls into place, when they pay 25 p for a marble, and write it as Re 0.25.

Activity

Money is the ideal instrument for teaching 10ths and 100ths. Card money with 25 p coins, 50 p coins, and Re 1 coins explains the concept very clearly. Students set up a little shop with:1 lollipop ....................... 75 p1 shampoo sachet ......... Rs 2.501 pencil ........................... Rs 2and so on. They make out bills and practise addition and subtraction.

Activity

Assuming you have enough coins of all denominations, 25p and 50p, in how many ways will you be able to fill your money box with change amounting to Re 1.Answer: four 25 p coins, two 50 p coins. Similarly, give other amounts that can be divided into the smaller currency unit.Each student can write the different combinations and then a comprehensive list is created in the form of a chart.

7. PERCENTAGE

Again, students have some idea of percentages, because of marks obtained in class, and 50% fares for children and 30% off at sales in clothing stores before the season changes.Students have learnt concepts of fractions and decimal fractions, and now is the time to establish the link between fractions, decimal fractions, and percentages.

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Students need to understand the fact that percentages are another way of writing decimal numbers with 2 places of decimals, and fractions with denominator 100. It is important to relate 1 whole to 100%, which again is not difficult, because they have experienced 100% marks when all sums have been correct. ‘A complete correct exam paper’ gets you full marks or 100%.Students use their knowledge of ‘cancellation’ or ‘reducing to lowest form’ here.

Activity

Students work with various real-life problems.1. 4 sandwiches in a tiffin box is 100%. 2 are eaten. Therefore 50% of

the sandwiches have been eaten and 50% of the sandwiches are left.

2. 1 whole piece of ribbon is 100%. When 20% or 51 is cut off it leaves

80% or 54 .

3. The cost of a fabric is Rs120 per metre, with 30% off. 30% or 103

of Rs 120 is Rs 36. Therefore, the cost of the fabric will be Rs 120 – Rs 36 = Rs 84 per metre.

4. The population of Pakistan is 172,800,048. 60% of the people turned up for voting while 40% or 10

4 did not vote. 4/10 of 172,800,048 is 69,120,019.

Work with the percentage of girls and boys in the class, 40% girls and 60% boys. The whole class is 100%.

Fractions such as 21 and 5

1 can be written as fractions with 10 or 100 as denominator.

21 = 10

5 = 10050 = 50% = 0.5 = 0.50

51 = 10

2 = 10020 = 20% = 0.2 = 0.20

This needs a great deal of practice so that pupils are able to convert one form into another with ease and speed.This practise should precede the introduction of the word ‘percentage’ and the special symbol %.

26

Percentages are so much a part of our daily life that plenty of activities are easy to arrange (a mock sale in the class room: % of sale, % profit, % loss, % articles not sold). A visit to a shop with a reduction sale on is also interesting.

Activity

Children have 10 × 10 square grids. They are required to colour certain percentages of the whole in different manner:

1. 20%

2. 60%

More activities with 50%, 72%, and 0% of the square grid can be patterned similarly.

27

Once the decimal notation is fully understood, students will have no difficulty handling more places of decimal such as 1000ths, at a later stage.The concept of metric units may also be introduced, simultaneously. There is plenty of opportunity for practical work with decimal fractions and metric measurements.If students can understand what happens when they multiply or divide with decimal fractions or handle calculations confidently, multiplication and division by 10, 100, 1000 and so on must also be understood easily. Some oral exercises or team games designed to test this ability are useful:1. What is 492 × 100?2. Which is more: 25% of 200 or 50% of 1003. 30% off in a sale for a crystal vase is Rs 90, what was the full price?4. 10

7 of a number is 700. What is 35% of the number?5. Which is greater 40% of 400 or 100% of 160?After this, application of the same basic rules to decimal numbers should be easy; written work must be reinforced with oral/mental exercises. Multiplication with decimal numbers should proceed carefully according to the stages suggested.A 1-decimal place number with a multiplier with tenths only is the same as multiplying two whole numbers and then dividing the product by 100.

For example: 0.3 × 107 = 10

3 × 107 = 100

21 = 0.21The next step is to multiply a decimal number with tenths and hundredths.

For example: 0.5 × 107 × 100

3 = 1005 × 10

7 × 1003 = 1000

105 = 0.105If we turn each decimal number into a common fraction, it is easy to see that a decimal with tenths multiplied by a decimal with tenths and hundredths will produce a decimal product with tenths, hundredths, and thousandths.Dividing by a decimal may pose a few problems; supplementary worksheets are useful.Beginning with problems such as these helps consolidate concepts about multiplication and division of fractions, be they decimals, percentage or simple fractions.

28

2 ÷ 50 = 2 × 501 = 50

2 = 251

5 ÷ .0 41 = 5 × .

10 4 = 2

7.5 ÷ .2 51 = 7.5 × 2.5 = 75 × 25 ÷ 100 = 100

1875 = 18.75

8. MEASUREMENT

Concepts of length, weight, capacity, and time are introduced with 3 decimal places at this stage. The chapters have been designed in such a way that there is plenty of scope to relate to real-life objects. A lot of charts or pictures, fieldwork, and group work is required while doing measurements be it length, weight or capacity.The entire metric table depends upon 3 places of decimals: a prefix of ‘kilo’ means 1000 times. Kilometre, kilogram, kilolitre follow this pattern.

a. TIME

Time is the only form of measurement, which is not calculated in decimal fractions. The number of days it takes for the Earth to go around the Sun is 365 and 4

1 days and it cannot be converted into tenths and hundredths.A day has been further divided into 60 minutes (and not 100 minutes) and a minute into 60 seconds (and not 100 seconds).Pupils should already know how to make a clock face by dividing it into 60 parts, as was shown in Maths Wise Book 1. Counting in 5s has also been done, right from the beginning, on a number line.The 24-hour clock can now be introduced, with the clock face divided into 12 and numbers written in 2 circles, 1 to 12, and 13 to 24.

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The concept of time is mainly related to real life and word problems can easily be written to further develop students’ understanding.Students hear about ‘time management’ such as the school timetable. Based on that, they prepare a timetable for their daily activities once they arrive home from school.When these timetables are compared, it becomes obvious that students spend various amounts of time watching TV, doing homework, playing sports or interacting with parents and siblings. A balance is essential.It is necessary for students to understand the concepts of one hour, one minute, and one second. A stopwatch helps. Questions listed below, help in assessment:1. For how many hours are we in school? Or How many hours do we

spend doing homework?2. What can you do in one minute? (For example, how many words can

one speak in one minute, how many words can one write, how many bites of a sandwich can one eat in one minute, and so on.)

3. How many times can you clap your hands in one second?4. Ahmed has read a paragraph from Alice in Wonderland; how many

minutes did he take?5. How long does it take to sing a particular song, or our national

anthem?The following conversion table is important to understanding time:60 seconds = 1 minute60 minutes = 1 hour24 hours = 1 day30 days = 1 month12 months = 1 year100 years = 1 century10 centuries = 1 millennium

Activity

Students make charts with names or pictures of activities, which might take, say, 30 seconds, 60 seconds, 5 minutes, 15 minutes or 1 hour.The class is divided into 3 or 4 teams. They play games to reinforce concepts and skills of telling time.

30

ActivityFind the names of the days hidden in the puzzle:Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday SA M O N D A Y T AT L K W I K F U TH E N R N E UU F D D S RR R N A D DS I E A AD D S U N D A YA Y A D S E U T Y Y A Y

A similar activity can be designed with the names of the months.

Activity

Worksheets with clock faces are fun. Clock faces have numbers only and no hands. Students write the time alongside, or draw hands of the clock to match the time given.

Activity

Match the times in column 1 with those in column 2.1. 14:50 a. 2:30 a.m.2. 2:30 b. 6:50 p.m.3. 12:00 c. 7:55 a.m.4. 7:55 d. 12 o’clock5. 6:00 e. 6 a.m.6. 22:10 f. 2:50 p.m.7. 18:50 g. 10:10 pm8. 5:15 h. five fifteen

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Activity

In 3 hours, the time will be:

12:30 + 3:00 = 15:30 or 3:30 pm

3:15 + 3:00 =

4:45 + 3:00 =

6: 25 + 3:00 =

Some sample exercises related to time are given here which can be given for further practice.The time now is 7:10 a.m. 4 hours ago the time was .It is 11:00 p.m. now. In 5 hours the time will be .

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It is 3:45 p.m. In 2 hours 30 minutes, the time will be .3 hours 30 minutes ago, the time was 2:10 p.m. What is the time now?

b. LENGTH

Activity

To reinforce the concept of length, some activities like the one given below would be useful.Fill in the blanks choosing from the list given below:metre, taller, taller, shorter, inches, kilometres1. The electricity tower is 12 high.2. The lamp post is than a man.3. The Muslim Commercial Bank tower is than the Habib

Bank Plaza.4. I am 5 feet 2 . My brother is 6 feet, I am than

my brother.5. The motorway from Lahore to Islamabad is 765 long.

Or ask the students questions like:1. Is the doorway longer or shorter than the edge of a carpet?2. Which object length in this room is just a little longer/shorter than

a metre?

c. WEIGHT

Weight concepts are introduced like concepts for length, with relevant vocabulary:1. Which of these two (a large bath sponge and a small stone) do you

think is heavier? (The students hold the two objects and assess.)2. Are you heavier than Sadiq or lighter?3. Does your school bag weigh more or less than 4 kg?4. In this weighing scale (pan balance), which side will go down: the one with 6 apples or the one with 6 bananas?6. On a spring balance, a packet of peas shows 1 kg. Will 2 dictionaries

weigh more or less?”7. If there is an elephant on one side of a see-saw and 2 children on the

other, which side will go down?

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Wherever possible, demonstrate with real objects. Some exercises like the last one, require students to use logic.

Activity

Students work in groups and make 4 charts:

Less than 1 kg More than 2 kg About 5 kg More than 10 kg

Less than 1 kg may be a pin, a feather, a pencil or a half-full bottle of water. More than 10 kg may be a car, an aircraft or the Statue of Liberty. The students should identify a variety of objects that are heavier than 10 kg.

d. CAPACITY

Once again, during a dialogue or an experiment about ‘litres’, it is necessary to use vocabulary related to capacity and litres. These are:full, half-full, empty, nearly full, nearly empty, liquids, container, litre, half-a-litre,

Example

1. How many litres of water or milk or juice can a container hold? If a container holds 1 litre of water, it will also hold 1 litre of juice or 1 litre of milk.

2. If a half-full jug contains 1 litre of juice, how much would a full jug contain?

Activity

Students see an array of containers such as a watering can, a bucket, a bottle, a water jug, a can, and a tetra-pack. They also observe the height of 1 litre of water in a flat dish, in a broad container like a water jug, or in a tall container, like a slim vase.Then, they compare 2 or more containers.1. How much water does a cup hold?2. Does a teapot hold more water than a water jug?3. How many litres of water does a bucket hold?

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Students make charts of objects that may hold:less than 1 litreabout 5 litresmore than 10 litresmore than 1000 litresWater reservoirs, lakes, and ponds can hold several hundreds of thousands of litres.

TEACHER’S NOTE

Social issues such as pollution (students can find the amounts of poisonous gas in a certain capacity of air in different parts of the city or village, amount of waste in the river water and sewage water), conservation of water (wastage of water in daily house-hold chores, dripping taps, and methods of minimizing it, the need to conserve water), cutting trees, after-effects of war, burning petrol, pollution caused by 2-wheelers and open fires, global warming, population explosion, and the extremely harmful effects of plastic waste can be discussed using metric measurements.

e. TEMPERATURE

This topic is dealt with in Maths Wise Book 5. Its an important concept as children hear about it daily in weather reports or if they fall sick, they use the thermometer and the Centigrade or Fahrenheit scales.

Activity

The students can be given a home exercise to note down the temperatures of different cities of Pakistan on a particular day from the weather report on TV.Some more examples of conversion between the two scales can be given.

9. GRAPHS

This section deals with the pictorial representation of data and its analysis. The section also has lots of activities which could be used as group projects. It would not only encourage and hone practical thinking amongst pupils but also develop positive group dynamics and leadership qualities.

35

Graphs are pictorial representations of a daily-life situation.When counting the animals on a farm, a graph can be drawn to illustrate:

dog rabbit duck cow horse hen cat

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f an

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s

Activity

Students work in groups and make graphs on chart paper using cut-out pictures of animals.

36

Initially, 1 animal is represented by 1 animal picture. The same chart can be made with a key where 1 animal would mean 5 animals, in which case it will represent:10 dogs, 30 rabbits, 35 ducks, 5 cows, 10 horses, 25 hens, 15 cats

dog rabbit duck cow horse hen cat

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mb

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f an

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Key: 1 animal picture = 5 animals

In later classes, actual picture graphs become bar graphs, and then line graphs, with an X-axis and a Y-axis.Some students have difficulty in remembering which is X-axis and which is Y-axis, so you can always refer that X axis is a ‘X’ (a cross).

Activity

The number of students in various classes in the school can be represented as a block graph, either vertically or horizontally.Other examples of pictorial graphs, after visiting and collecting data from the relevant areas are: colours of cars in the parking lot, types of trees in the park, (organize a tree planting week as a conservation project), snacks available in the canteen; amounts of different fruits available at the fruit seller’s stall (here, it will be necessary to say, ‘1 orange = 30 oranges, 1 banana = 30 bananas, and so on)

37

Students work in groups and prepare bar or column graphs of the listed items and compare the two.

10. AREA AND PERIMETER

The concepts of area and perimeter need to be clearly understood.The students paint the surface of any shape. For example, the surface of a cardboard. Students give examples of other surfaces, which have area:The wall, the floor, the windowpane, the surface of the teacher’s desk, the face of a block of wood, the inside of a cup, the outside of a cup, the shell of a tortoise, these are all surfaces. A surface occupies space and has an area.The area of the top of teacher’s desk is less than the area of a classroom, the area of a classroom is less than the area of a school building. The area of a school building is less than the area of the zoo. The area of the zoo is less than the area of Karachi, and the area of Karachi is less than the area of Sind. The area of Sindh is less than the area of Pakistan and so on.If grass needs to be laid in the school garden, one needs to know the area of the garden; if a fence is to be put around the garden, one needs to know the perimeter of the garden. The length of the wire required needs to be found.The perimeter of the school ground, a garden, a classroom, a desk, a library or a hotel lobby can be found by using a measuring tape. Area can also be found by using squared sheets of paper.

Activity

A group of students finds the perimeter of a verandah by using a measuring tape. Other groups find the lengths of each side of the classroom. The four lengths are added and the two groups compare the results.The essential point for students to understand is that the perimeter is the measurement of a line, whether straight or curved, enclosing a space.

38

Activity

On a geo-board dot paper, the teacher works with dots on the board and explains that the distance between 2 dots is one unit. Students draw lines between points (vertically or horizontally) and count the number of units in each line. They record this below each line.

Length of AB = 3 units

Length of AC = 2 units

Students count the ‘units’ in each side of a square or a rectangle and record the lengths of the straight lines.Students make squares and rectangles on a geo-board paper and count the number of units along the perimeter of each shape. They record the sizes below each shape.The next activity can be worked on the board with the help of two students. The teacher calls out a number ‘16 units’ and says, ‘Draw a rectangle with perimeter of 16 units.’ Asim draws a rectangle with a perimeter of 16 units.Various rectangles are possible. Students participate in a discussion on how the size of the sides is arrived at. Halve the perimeter ( 2

16 = 8) and find different combinations where the sum is 8; 7 + 1, 6 + 2, 5 + 3, 4 + 4. Asim draws one or two of these. Students with geo-board papers draw these rectangles and write the length of each side:7 units, 1 unit6 units, 2 units5 units, 3 units4 units, 4 units

1 unit 1 unit 1 unitA B

A

C

1 unit

1 unit

39

Students learn to tabulate these:

Perimeter Half of Length of Width of perimeter rectangle rectangle 16 units 8 units 7 units 1 unit 20 units 10 units 6 units 4 units

This tabulation continues, with different numbers until students are fully confident about the relationship of the lengths of sides to the perimeter of a rectangle.

CHALLENGE

The rectangles and squares given above have perimeters which are even numbers. Students try and make rectangles with perimeters which are odd numbers.

CHALLENGE

On a geo-board paper, make 2 units = 1 UNIT. Draw rectangles and squares with perimeters which are odd numbers.

Activity

To introduce area, the teacher’s desk is covered with square sheets of paper. If 15 sheets of paper cover the desktop, then the area of the table top is equal to the area of 15 sheets of paper. If the same desktop is covered by 5 sheets of glazed paper, then the area of the 5 sheets of glazed paper is equal to the area of the 15 sheets of squared paper.

Activity

Each student holds a loop of string with the 2 forefingers and the 2 thumbs of his hands. He moves his fingers in the loop (stretched all the time). He finds that he can enclose different sizes of area with any fixed perimeter.If he brings the thumb and the forefinger of each hand close together, he will ‘squeeze’ all the area out but the perimeter will remain the same.

Activity

Students work on squared paper. They colour squares or number them to find the area of the sheet.

40

1 11 21 31 41 51 61 71 81 912 12 22 32 42 52 62 72 82 923 13 23 33 43 53 63 73 83 934 14 24 34 44 54 64 74 84 945 15 25 35 45 55 65 75 85 956 16 26 36 46 56 66 76 86 967 17 27 37 47 57 67 77 87 978 18 28 38 48 58 68 78 88 989 19 29 39 49 59 69 79 89 99

10 20 30 40 50 60 70 80 90 100

The area of this sheet of paper is equal to 100-square units.Students estimate areas of different spaces vis-à-vis the squares. An area can be coloured and the tip of a finger, or the palm of a hand can be moved across an area.

Activity

On a square sheet of paper (or a geo-board paper), students have drawn rectangles and squares with given perimeters. Now, they draw rectangles and squares of given areas:

This is a 3 × 5 rectangle This is a 4 × 6 rectangleArea = 15 square units Area = 24 square units

41

Students draw different rectangles on their sheets of square paper, and colour them. It is useful if they put a dot in each square as they count the area.They learn to tabulate areas, as they did for perimeters. A rectangle with 24 square units can be any of these:24 units, 1 unit12 units, 2 units 8 units, 3 units 6 units, 4 units

Students learn to tabulate these:

Area of Length of Width of rectangle rectangle rectangle

12 square units 12 square units 1 square unit 12 square units 6 square units 2 square units 12 square units 4 square units 3 square units 20 square units 20 square units 1 square unit

Activity

(on square sheets of paper)

Squares in a row: Squares in a row: Number of rows: Number of rows:

× = × = Area = sq. units Area = sq unitsPerimeter = units Perimeter = units

42

The students then draw rectangles of various sizes on blank square paper sheets, and just write the following:

× = Area = square unitsPerimeter = units

Activity

Students are given sheets of paper with different shapes drawn on them, such as different quadrilaterals, pentagons, and hexagons. The name of each shape, and the measurement of each side is written next to it.

For example:

Draw different pentagons with the following measurements: 4 cm, 5 cm, 6 cm, 3 cm, and 7 cm (measurements written next to each side.)Students find the perimeter and write the answers below each shape.

Activity

11. GEOMETRICAL CONCEPTS

Open and closed shapes

The following work reinforces concepts of shapes and prepares the students to use the geometry set.The teacher draws 12 to 20 dots arranged in a circle on the board. The students copy this in their exercise books.

43

The teacher connects dots drawn in a circle on the board to make open shapes. On their individual papers, students join any 2 dots with a ruler and a pencil. They continue to join dots to create open shapes.Finally, the 2 end-points of the open shape are joined together to make a closed shape.Students count the number of sides each closed shape has and write it below.

A wooden board with 16 or 20 nails driven in to form a circle is a very interesting tool. A cord wound around various nails produces an amazing variety of figures such as:equilateral triangles, isosceles triangles, right-angled triangles, obtuse-angled triangles, quadrilaterals such as squares, trapeziums, rectangles, pentagon hexagons, pentagonal stars, and even, a 20-sided figure.

Activity

Angles

Asma comes to the front of the class and stands with her 2 arms stretched facing the class. The teacher asks her to turn round clockwise on the same spot, till she comes back to the original position.‘How many times has Asma gone around?’, the teacher asks.The students answer, ‘One.’Asma turns again and the students say ‘Two.’Asma makes more anticlockwise turns.

44

In this way, clockwise and anticlockwise concepts are reinforced. When one moves around in the direction of the hands of a clock, it is moving in a ‘clockwise’ direction. The opposite direction is known as ‘anticlockwise’ direction.Students find out whether the fans in the classroom move in a clockwise or an anticlockwise direction: What about the following objects which can move in both directions?• a door knob turning to open or close a door?• a key locking or unlocking a lock?• A screwdriver driving a screw or unscrewing in it?

CHALLENGE

Students look for more objects which move in clockwise and anticlockwise directions. Taps, numbers on locks of suitcases, blades of a helicopter are some examples.

Activity

Students have worked with the movements of the 2 hands of a clock, to show times. From the position of 12 o’clock, as the minute hand moves clockwise, an angle is formed. The minute hand goes through a full rotation (in a clockwise direction) before it comes back to the number 12.Asim stands with his arms stretched, facing the class and turns clockwise through a right angle (using the tracing of a paper right angle on the floor). Asim goes through a 2nd right angle, turning clockwise again. He turns a third and fourth right angle, before he comes back to the original position, facing the class.On a clock, it is 12 hours in one round of the hour hand or one hour by the minute hand. Asim turns through 4 right angles or 360°, when he completes one rotation. The teacher works with a wooden protractor before children work with their own.It can be useful to draw a protractor on the floor, where students can see angles marked from 0° through 90°, 180°, and 270° and back to 360°.Students learn to use the protractor from their geometry box to draw right angles. They draw 4 right angles in such a way as to reinforce the concept that 4 right angles makes 360°.

45

CHALLENGE

N

E W

S

Junaid draws this diagram, in bold letters, on a sheet of an old newspaper. He puts the paper on the floor, with N pointing towards north in the classroom.To go from N, through W, S, and E, back to N, he needs to move in a clockwise direction.What happens to this movement when the sheet of paper is ‘stuck’ to the ceiling (or held upside down above Junaid’s head)! (The teacher needs to work this out before putting it to the students.)

Activity

Before the students draw the angles as given in Maths Wise Book 5, the teacher needs to work on the board with a large ruler and a protractor.She draws a line segment AB on the board, centres the protractor on A, marks 10°, as point C. She joins C to A, and writes ∠CAB = 10°.She goes through 10°, 20°, through 90° to 180°, back to 360°, reinforcing words such as an acute angle, a right angle, an obtuse angle, a straight angle, a reflex angle and a full rotation. This is done in both clockwise and anticlockwise directions, first on the board and then in the student’s in their exercise books.They use the right-angled corner in a set square to identify acute angles and obtuse angles, in addition to construction of right angles.Students need to be fully confident with the various aspects of angles; the working with polygons becomes very easy if they do.It is essential that the students can estimate the sizes of other angles with some accuracy, and identify right angle and parallel lines. This is extremely useful in everyday life when sewing hanging picture frames or curtain rods, carpentry or making painting.

46

Activity

A large wall chart shows different angles, without sizes, with a few pairs of complementary angles and supplementary angles. Students make verbal assessments and tabulate the first two columns. They then measure these angles and the rest of the columns are filled in.

Real Angle Assessment Measure Supplement Complement

1. ∠PQR 35° 38° 2. ∠LMN 50° 52° 3. ∠ABC 150° 142°

Plenty of work is necessary with the various instruments in the geometry set in order to develop students’ confidence.

CHALLENGE

QuestionI am a quadrilateral. I have 2 sets of opposite sides equal. I have 2 pairs of opposite angles equal. What am I? Is there another like me?

Answer a parallelogram and a rectangle

REMEMBER

A square is a quadrilateral, but all quadrilaterals are not squares.A rectangle is a parallelogram, but all parallelograms are not rectangles.A diamond (or a rhombus) is a kite, but all kites are not diamonds.How many more statements can the students make like the ones given above?

47

CHALLENGE

Individual worksheets 1 cm grid, with these figures for students:

Imagine that each polygon, starting with a pentagon, is rolled along the line. Estimate the distance each polygon will roll after one full turn. Mark this point on the line running along the shape.Discuss your findings with your teacher.Which shape has the smallest perimeter, the largest perimeter?Which shape has the smallest area, the largest area?

48

D. A Suggested Lesson Plan

Periods required: 8Skills acquired: Listening, responding to questions, application of

the subject.Objects from the biscuits, bread sticks, coloured square papers,Maths Table: a fraction booklet, and other related objects.

Content Method ActivitiesIntroduceFractions

a) Demonstration

b) Interaction

c) Brainstorm

d) Practical demonstration

a) The teacher begins by asking the children, “What is a fraction?” (Many students may know what a fraction is but not what it looks like and where it is used.)

b) The teacher takes a biscuit or a bread stick and breaks it in half. She asks a child to eat one half.

c) The teacher asks the children what she did and hears their responses. (Broke it in half or in pieces.)

d) The teacher now goes on to explain that things can be broken into equal parts and those parts are fractions of the whole item.

e) The teacher can now give each child one biscuit or bread stick and have them break it in half and then quarters.

The teacher asks the children questions like:

i) How many pieces did you have when you broke the biscuit into halves?

ii) How many pieces did you have when you broke the biscuit into quarters?

f) The teacher hands out a square, coloured sheet and asks the children in how many different ways can they fold it in half?

49

Content Method Activities

e) Explanation

f) Application

g)

The same activity can be done for

41 or 3

1

g) The teacher explains that the coloured square was folded into two congruent or equal parts. Each part is half or one half.

h) A fraction booklet has been made for each child. Some of the concepts covered in the booklet are listed below:

i) Identifying fractions – 21 , 4

1 , 31

of a given shape. ii) To circle or colour half of a

collection of 20 balls or quarter of a collection of 8 oranges.

iii) Introduce the two terms of f r ac t i on — nu me r ator and denominator—with the help of examples.

iv) What fraction of the given figure is shaded / unshaded?

v) Fraction facts they need to know.

E.g. 21 dozen = ?

or 21 a score = ?

or 41 of a day = ? hours

Each page of the booklet needs to be done after a quick brainstorming session.Revise the lesson.

50

E. Answers to Book 3

Unit 1: Assess and Review 1

Exercise 1

1. ones 2. tens 3. tens 4. hundreds5. ones 6. ones 7. ones 8. tens9. hundreds 10. ones

Exercise 2

1. 36 2. 235 3. 519 4. 405. 176 6. 904 7. 21 8. 1009. 998 10. 20 11. 508 12. 97613. 710

Exercise 3

1. 264, 265, 266, 267, 268, 269, 2702. 599, 699, 799, 8993. 37, 47, 57, 67, 77, 87, 97, 1074. 62, 825. 152, 162,172, 182,192

Exercise 4

1. 07, 23, 61, 75, 82, 94 and 94, 82, 75, 61, 23, 072. 128, 287, 348, 475, 711 and 711, 475, 348, 287, 1283. 504, 524, 554, 564, 594 and 594, 564, 554, 524, 5044. 600, 601, 603, 606, 609 and 609, 606, 603, 601, 6005. 227, 337, 777, 887, 997 and 997, 887, 777, 337, 227

Exercise 5

1. 247, two hundred and forty-seven2. 617, six hundred and seventeen3. 689, six hundred and eighty-nine4. 495, four hundred and ninety-five

51

5. 944, nine hundred and forty-four6. 160, one hundred and sixty7. 926, nine hundred and twenty-six8. 116, one hundred and sixteen9. 600, six hundred10. 1400, one thousand four hundred

Exercise 6

1. 12, twelve2. 84, eighty-four3. 393, three hundred and ninety-three4. 432, four hundred and thirty-two5. 386, three hundred and eight-six6. 666, six hundred and sixty-six7. 32, thirty-two8. 106, one hundred and six9. 47, forty-seven10. 610, six hundred and ten

Exercise 7

1. 8 2. 21 3. 30 4. 605. 18 6. 20 7. 130 8. 609. 102 10. 120

Exercise 8

1. 2 cars each 2. 5 sweets each 3. 4 pencils each4. 5 teddies each 5. 3 coins each

Exercise 9

circle, square, triangle, rectangle

Exercise 10

cube, sphere, cuboid, cylinder, pyramid,

52

Exercise 11

Exercise 12

Exercise 13

1. 6.1 cm 2. 3.4 cm 3. 5.1 cm 4. 8 cm5. 5.2 cm

Exercise 14

1. 5 minutes past 2 4. half past 92. 20 minutes past 3 5. 12 o’clock3. 7 o’clock 6. quarter past 11

Exercise 15

1. 841 books 2. Rs 24 3. 182 days4. Rs 18 5. answers will vary 6. 56 kg7. 8 m 8. 3 hrs, evening 9. 410. 12 cans of juice, 30 sandwiches

Puzzle

There can be many combinations:2 + 8, 5 +5, 6 + 4, 3 + 7, 7 + 3, 20 ÷ 2 etc.Similarly, combinations can be made for other numbers.

53

Unit 2: Numbers

Exercise 1

1. Children draw three flowers.2. 2 marks3. 5 marks

Exercise 2

2. III 3. V 4. IV 5. VIII

Exercise 3

1, 7, 4, 9 5, 2, 6, 10

Exercise 4

Roman Words

X + I = XI eleven

X + II = XII twelve

X + III = XIII thirteen

X + IV = XIV fourteen

X + V = XV fifteen

X + VI = XVI sixteen

X + VII = XVII seventeen

X + VIII = XVIII eighteen

X + IX = XIX nineteen

X + X = XX twenty

Exercise 5

1. XVI, 16 2. XX, 20

54

Exercise 6

1. VII seven2. IX nine3. C 1004. XX twenty5. XI eleven6. IV four

Activity

• LID, MILD, DILL, MILL, CIVIC, CIVIL, etc.• Cross out LONG, VI will be left.• XI (It becomes eleven.)

Exercise 7

Students colour the grid as instructed.

Exercise 8

1. 56√, 57 , 58√, 59 , 60√2. 87 , 88√, 89 , 90√, 913. 201 , 202√, 203 , 204√, 205√4. 444√, 445 , 446√, 447 , 448√5. 1234√, 1235 , 1236√, 1237 , 1238√

Exercise 9

HTh TTh Th H T O number names

1. 3, 1 7 4 three thousand, one hundred and seventy-four2. 2, 0 5 8 two thousand and fifty-eight3. 5 6, 3 6 7 fifty six thousand, three hundred and sixty-seven4. 2 4 3, 0 9 8 two hundred and forty-three thousand and ninety-eight5. 8 7 0, 4 9 6 eight hundred and seventy thousand, four hundred and ninety-six

55

Exercise 10 2. 2 ones 3 tens 7 hundred 8 thousand 0 ten thousand 6 hundred

thousand 600,000 + 8000 + 700 + 30 + 2

3. 8 one 4 ten 9 hundred 5 thousand 5,000 + 900 + 40 + 8

4. 2 one 8 ten 9 hundred 8 thousand 0 ten thousand 6 hundred thousand

600,000 + 8000 + 900 + 80 + 2

5. 8 one 7 ten 2 hundred 6 thousand 7 ten thousand 70,000 + 6000 + 200 + 70 + 8

6. 7 one 3 ten 4 hundred 8 thousand 6 ten thousand 9 hundred thousand

900,000 + 60,000 + 8000 + 400 + 30 + 7

Exercise 11

HTh TTh Th H T O number names

2. 7 6 4 3 seven thousand six hundred and forty-three3. 9 1 5 9 0 nine one thousand five hundred and ninety4. 1 8 1 2 0 7 one hundred eight-one thousand, two hundred and seven

Exercise 12

1. H 2. TTh 3. Th, T 4. HTh5. O 6. T

Exercise 13

Answers will vary.

56

Exercise 14

2. 3108; 3208; 3308; 3408; 3508; 36083. 13,009; 13,019; 13,029; 13,039; 13,049; 13,0594. 210,345; 220,345; 230,345; 240,345; 250,345; 260,3455. 980,819; 980,820; 980,821; 980,822; 980,823; 980,8246. 35,909; 36,909; 37,909; 38,909; 39,909; 40,909

Exercise 15

4678 5678 6678 7678 8678 9678 21 121 221 321 421 52118,101 19,101 20,101 21,101 22,101 23,101 24,101 709,543 710,543 711,543 712,543 713,543 714,543 79,677 179,677 279,677 379,667 479,667 134,257 134,357 134,457 134,557 134,657 134,757 134,857

Activity

Across1. 53,067 4. 123 7. 897,653 9. 64,91010. 5796 11. 6363 Down2. 3527 3. 718,329 5. 300,000 6. 980,1548. 66,666

Exercise 16

1. < 2. > 3. > 4. =5. > 6. =

Exercise 17

Only the second pair is correct.

57

Exercise 18

1. ascending; 367, 921; 368,9212. descending; 77,249; 77,2393. descending; 214,291; 214,2814. ascending; 526,344; 627,3445. ascending; 220,024; 230,025; 240,026

Exercise 19

1. 76,431 largest; 13,467 smallest2. 98,620 largest; 20689 smallest3. 984,210 largest; 102,489 smallest4. 764,321; 123,467 smallest

Exercise 20

1. 1000 2. 999,999 3. 99,999 4. 100,0005. 10,000

Ascending order: 1000; 10,000; 99,999; 100,000; 999,999

Unit 3: Number operations

Exercise 1

1. 9976 2. 5844 3. 8867 4. 65895. 4599 6. 10,983

Exercise 2

1. 9889 2. 3688 3. 8678 4. 3444

Exercise 3

1. 6230 2. 4968

58

Exercise 4

1. 7467 2. 9459 3. 9319 4. 94445. 4489 6. 2213

Exercise 5

1. 8139 2. 6919 3. 7493 4. 1494

Exercise 6

1. 6914 2. 6578 3. 3550 4. 94705. 7931

Exercise 7

1. 60 2. 70 3. 80 4. 605. 60 6. 65 7. 30 8. 399. 58 10. 95

Exercise 8

1. 62 2. 96 3. 76 4. 605. 83 6. 29 7. 92 8. 849. 40 10. 61

Activity

Hoopla 5 and 25; 10 and 20; 10, 10, 10;

Wheel of fortune 100, 100, 100, 100, 100, 100 100, 150, 250 200, 300 150, 350 100, 400

Darts 50, 150 50, 50, 100 100,100

59

Exercise 9

1. 4221 2. 2242 3. 8322 4. 42235. 4083 6. 6743 7. 5203 8. 22009. 3000

Exercise 10

1. 7051 bees 2. 1111 pages 3. 2413 men

Exercise 11

1. 2156 2. 2519 3. 4175 4. 75605. 3427 6. 5782

Exercise 12

1. 2744 2. 2950 3. 2881 4. 42035. 8775 6. 1652

Exercise 13

1. 882 cards 2. 1498 people 3. Rs 70244. 7095 bangles 5. 2486 bottles

Exercise 14

1. 40 2. 40 3. 40 4. 525. 25 6. 10 7. 14 8. 389. 7 10. 16 11. 38 12. 2013. 32 14. 50

Exercise 15

1. 35 2. 56 3. 41 4. 485. 36

60

Activity

Spider with 8 legs is the correct choice, since in the first row each animal has legs in the multiples of 2. (snail, 0 legs; kiwi, 2 legs; squirrel, 4 legs; beetle, 6 legs)

Exercise 16

1. 16 2. 54 3. 7 4. 65. 11 6. 9 7. 9 8. 569. 7 10. 8 11. 6 12. 8813. 64 14. 9

Exercise 17

1. 170 2. 402 3. 116 4. 7126. 360 6. 245 7. 396 8. 328

Exercise 18

1. Rs 369 2. 160 legs 3. 72 students4. 144 dozen 5. 100 crayons 6. 343 days

Exercise 19

1. 70 2. 830 3. 990 4. 3405. 48 6. 93 7. 99 8. 889. 216 10. 108 11. 648 12. 18913. 32 14. 0

Activity

5 2

10

2 5

4 8

32

8 4

5 9

45

9 5

61

6 4

24

4 6

7 9

63

9 7

10 10

100

10 10

Exercise 20

2. 10 ÷ 5 = 2 3. 12 ÷ 6 = 2 4. 14 ÷ 2 = 7 5 × 2 = 10 6 × 2 = 12 2 × 7 = 14

Exercise 21

1. 5 2. 2 3. 4 4. 45. 6 6. 4

Exercise 22

2. 4, 6 3. 6, 5 4. 4, 12 5. 6, 8

Exercise 23

1. 18 2. 151 3. 28 R1 4. 492 R15. 12 R4 6. 52 R4 7. 8 R3 8. 64 R2

Exercise 24

1. 24 2. 16 R2 3. 60 R4 4. 50 R45. 133 R1 6. 46 R7

Exercise 25

1. 6 seeds 2. 7 buttons 3. 16 kg 4. Rs 525. 152 students

62

Exercise 26

1. 21 2. 23 3. 11 4. 315. 81 6. 21 7. 20 8. 219. 103 10. 500 11. 247 R1 12. 40

Activity

The secret message is:

DIVISION IS FUN

Unit 4: Fractions

Exercise 1

Exercise 2

1.

2. 62

62

62+ + = 1 whole

Exercise 3

2. 3.

63

4. 5.

6.

Exercise 4

1. 21 of 8 = 4 2. 2

1 of 12 = 6 3. 21 of 6 = 3

4. 21 of 14 = 7 5. 4

1 of 20 = 5 6. 41 of 8 = 2

7. 41 of 8 = 2 8. 4

1 of 12 = 3 9. 52 of 10 = 4

10. 52 of 20 = 10 11. 5

2 of 15 = 6 12. 52 of 30 = 12

Exercise 5

1. 42 2. 8

4 3. 63 4. 2

1

5. 52 6. 10

5

Exercise 6

1. 93

31= 2. 15

1032= 3. 12

261= 4. 16

461=

64

Exercise 7

2. 6,

3. 4,

4. 6,

Exercise 8

65

Exercise 9

All fractions are equivalent except 2 and 5.

Exercise 10

Exercise 11

1. 42 2. 6

4 3. 186 4. 24

20

5. 22

Exercise 12

Proper: , , , ,83

662

675

561

687 Improper: , , , ,3

45

779

247

156

34

Exercise 13

1. 73

75< 2. 9

295< 3. 11

10117> 4. 5

351>

5. 152

156<

66

Exercise 14

1. , ,41

43

44 2. 0, ,10

4105 3. , ,5

153

55 4. , ,7

174

78

5. , ,111

114

115

Exercise 15

1. 22 2. 5

3 3. 85 4. 7

5

5. 86 6. 10

5 7. 34 8. 11

10

9. 1211 10. 9

10

Exercise 16

1. 41 2. 5

1 3. 72 4. 8

6

5. 61 6. 11

2 7. 51 8. 12

5

9. 137 10. 9

3

Exercise 17

1. + 2. – 3. – 4. +5. +

Exercise 18

1. 41 eaten, 4

3 left 2. 61 eaten, 6

5 left

3. 83 eaten, 8

5 left 4. 31 eaten, 3

2 left

Exercise 19

1. 43 of the pencils remain.

2. Amir ate 4 pieces and 4 pieces are left.3. 25 apples were unripe; 75 apples were ready to eat.

67

Game page 80

Unit 5: Measurements

Exercise 1

1. metre 2. kilometre 3. centimetre4. centimetre 5. centimetre 6. centimetre

68

Exercise 2

2. 80 cm 3. 9 cm 4. 8 cm5. kilometre 6. kilometre

Exercise 3

1st section1. 4.5 cm 2. 3.5 cm 3. 1.5 cm 4. 2.6 cm

2nd section1. The example shows a ruler measuring the line as 6 cm, which is not

drawn to scale. Allow the children to use their rulers to draw actual measurements as given for the next three exercises.

Exercise 4

1. 18 cm 2. 9 cm 3. 9 cm 4. 17 cm5. 34 cm

Exercise 5

1. 7 m 2. 38 m 3. 17 m 4. 9 m5. 13 m

hill, tower, lamp post, house, tree, ladderladder, tree, house, lamp post, tower, hill

Exercise 6

home superstore ice-cream parlour school

Exercise 7

2. 355 m 3. 54 km 4. 118 cm 5. 145 m

Exercise 8

2. 217 m 3. 41 km 4. 23 cm 5. 108 m

69

Exercise 9

16 cm

Exercise 10

12 cm

Exercise 11

Children draw a line 5 cm long.

Exercise 12

2. 7 cm, 16 cm 3. 9 cm, 21cm 4. 79 cm, 171 cm5. 27 cm, 64 cm

Exercise 13

2. car 3. truck 4. dog5. can of juice 6. fox

Exercise 14

1. 4 kg 2. 450 g 3. 250 g 4. 3 21 kg

5. 25 g 6. 60 kg

Exercise 15

2. mg 3. g 4. kg 5. kg6. g

Exercise 16

less than 1 kg: teddy, pencil, scissors, photo frame, jar of sweets, CDsmore than 1 kg: laptop, television, fish bowl, books

70

Exercise 17

The weights that the shopkeeper can use are:2. 250 g, 10 g 3. 30 g, 20 g, 20 g, 10 g4. 10 kg, 3 kg 5. 300 g, 25 g6. 500 g, 250 g, 10 g

Exercise 18

2. 1102 g 3. 68 kg 4. 499 g 5. 178 g

Exercise 19

2. 130 g 3. 120 g 4. 4 kg 5. 39 g6. 12 kg

Exercise 20

2. 16 kg 3. 462 g 4. 39 kg 5. 67 g

Exercise 21

Total = 201 kg; overweight by 101 kg

Exercise 22

1. jug 2. thermos 3. larger pack4. bigger bowl 5. bottle 6. can of juice

Exercise 23

1. ml 2. litre 3. ml 4. litre5. litre 6. ml

Exercise 24

1. 2 l 2. 250 ml 3. 25 l 4. 51 l5. 100 ml 6. 13 l

71

Exercise 25

2. 443 ml 3. 1330 ml 4. 15212 l 5. 60 l7. 519 ml 8. 156 ml 9. 52 l 10. 27 l

Exercise 26

1. 27 l 2. 28 l 3. 26 ml, 5 ml 4. 1650 ml.

Unit 6: Time

Exercise 1

1. 2 a.m., 4 a.m., 10 a.m., 12 noon, 3 p.m., 4 p.m., 5 p.m., 7 p.m.2. a. a.m. b. p.m. c. a.m. d. p.m. e. a.m.3. a. a.m. b. a.m. c. p.m. d. p.m. e. p.m.

Exercise 2

2. 5 minutes to 2 3. 21 past 9

4. 15 minutes to 4 5. 15 minutes past 76. 10 minutes to 8

Exercise 3

2. eleven thirty 3. two twenty-eight4. twelve three 5. six one6. four fifty two

Exercise 4

1.

72

2.

Exercise 5

1.

2.

3.

4.

5.

7:10

11:50

1:00

2:30

3:10

73

Exercise 6

1.

2.

3.

4.

5.

Exercise 7

2. 154 hours 3. 697 hours 4. 456 hours5. 1746 hours 7. 43 hours 8. 26 hours9. 26 hours 10. 105 hours

Exercise 8

1. 6 hours 2. 2 hours 3. 12 noon4. 3 hours 5. Bilal, Emad, Amir, 8 minutes, 3 min

2:25

3:45

11:05

8:40

6:20

74

Exercise 9

1. January, June, July 2. 30 3. April4. 366 5. Friday 6. 307. Friday, Monday, Saturday, Sunday, Thursday, Tuesday, Wednesday8. 156

Exercise 10 and Exercise 11

The answers to these exercises would be best given using the current year’s calendar. The teacher can adapt Exercise 11 to the current year’s month of December.

Activity page 111

Similarly for this activity, use dates for the year in which the book is being taught.

Exercise 12

1. Naveen 2. Danish 3. June 4. 3 5. Naveen

Unit 7: Geometry

Page 115

points: A, B, O, M, N, P, R, Sline segments: MN, RSray: PA, OXstraight line: AB

Activity page 117

1. 2.

75

Exercise 1

1. rectangle 2. pyramid 3. circle 4. kite5. oval 6. cube

Exercise 2

kite 3, square 2, rectangle 1, diamond 1, arrowhead 2, triangle 5

Exercise 3

Help the students draw the picture.

Exercise 4

Students colour as indicated.

Exercise 5

Exercise 6

2. 5 + 3 + 5 + 3 = 16 cm3. 5 + 2 + 3 + 2 + 2 + 4 = 18 cm4. 3 + 2 + 3 + 4 + 6 + 2 = 19 cm

Exercise 7

44 m

76

Exercise 8

80 m

Exercise 9

24 cm

Exercise 10

1. 170 m 2. 70 m 3. 240 m 4. m

Exercise 11 For exercises 2 to 4, students draw different figures and calculate the perimeters accordingly.

Activity page 124

13.3 cm approximately

Unit 8: Graphs

Exercise 1

1. favourite flavours of ice cream2. an ice cream cone3. 10 scoops4. fruity5. orange6. six7. 115

Exercise 2

1. favourite pet 2. a circle dived into quarters 3. 564. dog 5. 14 6. spider7. rat and tortoise

77

Exercise 3

Fairy Tales

Horror Stories

Funny Stories

Science Fiction

Comic Books

Number of Children

= 2 Children

Favorite Types of Books____________________

Unit 9: Assess and Review 2

Exercise 1

HTh TTh Th H T O Expanded Form Number Names

2 7 0 6 9 2 200,000 + 70,000 + 600 + 90 + 2 two hundred and seventy thousand, six hundred and ninety two

3 0 2 0 1 30,000 + 200 + 10 + 1 thirty thousand, two hundred and one

3 0 9 0 1 2 300,000 + 9000 + 10 + 2 three hundred and nine thousand and twelve

4 0 1 9 3 40,000 + 100 + 90 + 3 forty thousand one hundred and three

2 0 4 0 0 7 200,000 + 4000 + 7 two hundred and four thousand and seven

5 9 8 0 5000 + 900 + 8 + 0 five thousand, nine hundred and eighty

6 6 4 7 9 60,000 + 6000 + 400 + 70 + 9 sixty-six thousand, four hundred and seventy-nine

9 0 0 0 9 8 900,000 + 90 + 8 nine hundred thousand and ninety-eight

7 4 5 8 3 9 700,000 + 40,000 + 5000 + 800 + 30 + 9 seven hundred and forty-five thousand, eight hundred and thirty-nine

Favourite Types of Books

78

Exercise 2

1. 654,426; 654,522; 654,562; 655,426; 655,466; 655,526 655,526; 655,466; 655,426; 654,562; 654,522; 654,426

2. 30,039; 30,309; 33,009; 90,303; 93,300; 309,903 309,903; 93,300; 90,303; 33,009; 30,009; 30,309; 30,039

3. 77,770; 707,070; 707,777; 770,770; 770,777; 777,707; 777,707; 770,777; 770,770; 707,777; 707,070; 77,770

4. 23; 222; 232; 3323; 23,332; 223,323 223, 323; 23,332; 3323; 332, 222, 23

5. 10; 100; 999; 1000; 99,999; 999,999 999,999; 99,999; 1000; 999; 100; 10

Exercise 3

1. 19,104 2. 1858 3. 383 4. 1135. 76,574 6. 403 7. 180 R = 4 8. 72,9949. 7555 10. 96 11. 25437 12. 3852413. 686 14. 56 R = 6

Exercise 4

1. 96 ,

128 ,

1510 ,

1812 ,

2114 2.

148 ,

2112 ,

2816 ,

3520 ,

4224

3. 2422 ,

3633 ,

4844 ,

6055 ,

12066 4.

2612 ,

3918 ,

5224 ,

6530 ,

7836

5. 4024 ,

6036 ,

8048 ,

10060 ,

12072

Exercise 5

1. 56 (I) 2. 13

4 (P) 3. 1912 (P) 4. 9

7 (P)

5. 1413 (P) 6. 56

11 (P)

79

Exercise 6

1. 43 ,

65 ,

76 ,

31 2.

99 ,

1212

Exercise 7

1. length 2. straight line 3. point 4. two5. triangle 6. sides, corners 7. square 8. opposite9. curved 10. diameter 11. four

Exercise 8

Check the students’ work.

Exercise 9

Check the students’ work.

Exercise 10

1. 1289 km 2. Rs 43,508 3. 945 km, 4 p.m.4. 72 m, 216 m 5. Rs 90 6. 730 kg7. 115 cartons in each truck, 1035 bottles8. 5

2 9. 50 ml 10. 2 June

Exercise 11

1. footfall in the cafeteria2. days: horizontal axis; number of students: vertical axis3. Friday4. Saturday5. 756. 6757. 125

For questions 8 and 9, help the children to complete the graphs.

Exercise 12

kite, square, rectangle, circle, triangle, rectangle, arrowhead

80

F. Answers to Book 4

Unit 1: Assess and Review 1

HTh TTh Th H T O Expanded Form Number Names

3 7 0 8 9 2 300000 + 70000 + 800 + 90 + 2 three hundred and seventy thousand eight hundred and ninety-five

3 0 2 1 1 30,000 + 200 + 10 + 1 thirty thousand two hundred and eleven

3 0 0 0 4 6 300000 + 40 + 6 –

800000 + 9000 + 10 + 2 eight hundred and nine thousand and twelve

4 0 1 2 3 40,000 + 100 + 20 + 3 –

2 4 0 0 7 – twenty four thousand and seven

5 9 8 6 5000 + 900 + 80 + 6 –

60,000 + 6000 + 400 + 70 + 9 six hundred and six thousand, four hundred and seventy-nine

9 0 0 0 9 – ninety thousand and nine

7 4 5 8 3 9 – seven hundred forty-five thousand eight hundred and thirty-nine

Exercise 2

1. 650,426; 650,522; 654,502; 655406; 655,5062. 300,390; 303,090; 309,030; 309,903; 330,090 3. 77,777; 77,707; 70,777; 70,770; 77704. 3320; 22,332; 23,232, 23,332; 230,2225. greatest 6D, greatest 5D, smallest 4D, greatest 3D, smallest 3D

Exercise 3

1. 19,104 2. 1,611 3. 4,620 4. 111 R = 15. 30,096 6. 8,454 7. 780, R = 4 8. 18,8449. 78,040 10. 810 R = 2 11. 25,699 12. 18,71613. 11,067 14. 167 R = 7

81

Exercise 4

1. 64 2. 14

8 3. 2422 4. 39

2

5. 4024

Exercise 5

1. 55 (unit) 2. 13

9 (P) 3. 192 (P) 4. 9

9 (unit)

5. 143 (P) 6. 56

47 (P)

Exercise 6

1. ,76

77 and , ,7

273

74 2. ,2

233

Exercise 7

1. 51 2. 60 3. 200 4. 2175. 126

Exercise 8

1. length 2. straight line 3. point 4. two5. triangle 6. sides/vertices 7. square/rhombus8. opposite 9. curved 10. diameter 11. four

Exercise 9

Check that the students mark the correct components of the circle.

Exercise 10

1. 1289 km2. Rs 44,506, Rs 44,2563. 1,035Km, 9 p.m.4. 72 m, 360 m5. Rs 90, equal amounts were contributed by the boys and girls6. 1,600 kg

82

7. 1,150 crates each, 20,200 cans8. 15

652=

9. 50 ml, 16 pens, 2 ml left10. 9th June, Monday

Unit 2: Numbers

Exercise 1

1. 645,762, 645,0002. 500,000 + 60,000 + 1000 + 90 + 73. four hundred and fifty-six thousand, eight hundred and seven4. thousands5. 682,511

Activity

999,999 place value: Hundred thousand

Exercise 2

1. 123,453,298 2. 892,046,710 3. 40,097,012 4. 6,337,027

Exercise 3

1. two million nine hundred and sixty-six thousand eight hundred and fifty

2. three hundred and fifty million, nine hundred and seventy-six thousand, two hundred and twenty-five

3. thirty-four million, eight hundred and seventy-three thousand, three hundred

4. five million, eight thousand four hundred and fifty

Exercise 4

2. 5,67,88,004 (five crore, sixty-seven lac, eighty-eight thousand and four)3. 6,75,43,098 (six crore, seventy-five lac, forty-three thousand and

ninety eight)4. 4,67,63,005 (four crore, sixty-seven lac, sixty-three thousand and

five)

83

Exercise 5

1. < 2. > 3. > 4. <

Exercise 6

1. 8,014,300; 18,320,200; 81,630,4502. 1,573,694; 2,516,019; 4,532,4813. 9,208,751; 9,240,715; 9,248,5174. 4,035,812; 4,053,612; 4,530,216

Activity

The given activities require students to search for the figures on the internet or in the school library.

Exercise 7

1. 499,699 2. 533,577 3. 900,8114. 761,911 5. 899,962 6. 998,9597. 1,040,167 8. 189,085 9. 708,12210. 1,108,008 11. 1,092,140 12. 1,174,05313. 925,454 14. 3,646,121 15. 974,498

Exercise 8

1. Rs 152,000 2. 1,108,030 3. 1,031,3474. 1,572,204 5. 1,071,500, 6. 264,4647. 322,200 8. 1,585,750 9. 513,32810. 40,100

Exercise 9

1. 521,162 2. 696,994 3. 35,9994. 397,985 5. 236,182 6. 407,9497. 383,495 8. 153,434 9. 111,11310. 135,177 11. 912,960 12. 530,299

84

Exercise 10

1. 676,400 2. 853,200 3. 747,500

Exercise 11

1. 64,319 2. 727,109

Exercise 12

1. 127,765 2. 470,464 3. 120,7784. 236,745 5. 175,049 6. 42,5997. 88,690 8. 264,319 9. 48,83410. 4,069,500

Exercise 13

1. 27,184,815 2. 33,316,650 3. 47,574,1354. 22,756,000 5. 23,909,904

Exercise 14

1. 15,237,600 2. Rs 6500 3. 9,424,250 m or 9424 km4. 276,480 5. Rs 28,050 6. 127,0207. Rs 9,088,625 8. 80,000 9. Rs 202,08010. 4,657,500

Exercise 15

1. 533 R = 9 2. 155 R = 41 3. 153, R = 6 4. 100 R = 9

Exercise 16

2. Rs 494 3. 82 km 4. 655. 1030 6. 73, 16 left 7. 200, 158. Rs 205 9. 302

Number Puzzle: a dozen

85

Unit 3: Factors and multiples

Exercise 1

2. NYN 3. NYN 4. YYY 5. YYY

Exercise 2

1. 2,5,10 2. odd 3. remainder 4. 35. 10

Exercise 3

Solved table from the book

YNY, NYY, NNN, NYY, NNN, YYY, YNN, YNY

Exercise 4

1. 2, 3, 5, 72. 14 (40, 42, 44, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58 , 60)3. 83, 894. 97 – 2 = 955. 1016. 27. 288. 23039. 13210. No, it is divisible by 3

Exercise 5

1. a. 3, 5, 1, 15 b. 1, 20, 2, 10, 4, 5 c. 1, 21, 3, 7 d. 1, 35, 5, 7 e. 1, 18, 2, 9, 3, 6

86

2. a. 1, 5 b. 1, 12, 3, 4, 2, 6 c. 1, 16, 2, 8, 4 d. 1, 27, 3, 9 e. 1, 30, 2, 15, 3, 5, 6 f. 1, 32, 2, 16, 4, 8 g. 1, 36, 2, 18, 9, 4, 6 h. 1, 40, 2, 10, 8, 5, 4 i. 1, 48, 2, 24, 6, 8, 3, 4, 12 j. 1, 3, 7, 9, 21, 63

3. yes 4. no 5. yes

Exercise 6

1. 3, 6, 9, 12, 15 2. 12, 18, 24, 30, 36, 42, 48 3. 196

Exercise 7

1. multiples of 4 2. multiples of 113. multiples of 8 4. multiples of 9

Exercise 8

1. a. 1, 2 , 3 , 6, 9, 18 b. 1, 2 , 11 , 22 c. 1, 2 , 3 , 4, 6, 9, 12, 18, 36 d. 1, 2 , 3 , 4, 6, 8, 12, 16, 24, 48 e. 1, 2 , 5 , 10, 25, 50

2. a. 1, 11 b. 2, 3 c. 2 d. 7, 5 e. 5, 2

3. a. 3, 5 b. 3, 2 c. 2, 3 d. 7 e. 2, 3, 5

Exercise 9

1. a. 3 b. 1, co-prime numberc. 2 d. 2 e. 5

2. a. 8 b. 2 c. 2 d. 1, co-prime number e. 2

87

Exercise 10

1 a. 20 l b. 9 times and 10 times2. 8 3. 9 cm 4. 6 cm

Exercise 11

1. a. 20 b. 18 c. 36 d. 40 e. 55

2. a. 84 b. 60 c. 48 d. 48 e. 72

3. a. 30 b. 100 c. 72 d. 16 e. 300

Exercise 12

1. 9:00 a.m. 2. 60 3. 180 4. 90 sec

Activity

Product of Numbers HCF LCM HCF × LCM

54 3 18 3 × 18 = 54 50 5 10 5 × 10 = 50 24 2 12 2 × 12 = 24 36 3 12 3 × 12 = 36 108 3 36 3 × 36 = 108 168 2 84 2 × 84 = 168 80 2 40 2 × 40 = 80 28 1 28 1 × 28 = 28 48 2 24 2 × 24 = 48

88

Unit 4: Fractions

Exercise 1

1. 41 2. 16

5 3. 83 4. 8

3

5. 43

Exercise 2

1. Colour 6 boxes 2. Colour 8 boxes3. Colour 7 boxes 4. Colour 3 boxes5. Colour 1 box

Exercise 3

1. unit fractions: 55

, 44 proper fractions:

42 ,

43 ,

75

improper fractions: 45 ,

37 ,

23 mixed fractions:

312 , 5

127 ,

221

2. a. 222 b. 3

31 c. 241 d. 5

21

e. 514 f. 3

42 g. 433 h. 5

32

i. 332 j. 5

23

3. a. 311 b. 4

21 c. 316 d. 2

5

e. 419 f. 3

4 g. 417 h. 2

13

i. 49 j. 5

21

89

Exercise 4

1. 93 2. 16

12 3. 9027 4. 1

1

5. 21 6. 5

3 7. 54 8. 15

10

Exercise 5 1. 4

1 2. 65 3. 4

3 4. 64

5. 53 6. 2

1 7. 32 8. 9

8

9. 32 10. 2

1 11. 83

Exercise 6

1. 1, 6, 30, 3, 2, 5 2. 24, 4, 9, 24, 18, 603. 30, 10, 12, 60, 6, 20 4. 7, 36, 56, 9, 21, 545. 16, 6, 24, 18, 6, 3

Activity

Sidra: 64 Nadir: 3

2

Exercise 7

1. , , , ,97

4863

64

32

10090

109

52

2510

86

43= = = = =

2. Hammad

3. a. , ,41

52

107 b. , ,7

2148

43

4. a. , ,65

32

95 b. , ,5

4107

21

90

Exercise 8

1. 22 1= 2. 5

3 3. 85 4. 7

5

5. 86

43= 6. 10

521= 7. 3

4 8. 1110

9. 1211 10. 9

10

Exercise 9

1. 65 2. 24

11 3. 185 4. 12

11

5. 112

1 6. 2419 7. 12

15 8. 1

9. 2111

8 10. 1615

Exercise 10

1. 53 2. 56

23 3. 75 ,

72 4. 10

6

5. 3029

Exercise 11

1. 41 2. 5

1 3. 72 4. 8

643=

5. 61 6. 11

2 7. 51 8. 12

5

9. 139 10. 9

331=

Exercise 12

1. 116 2. 10

7 3. 2411 4. 60

17

91

5. 202

101= 6. 18

5 7. 185 8. 20

3

9. 61 10. 10

1

Exercise 13

1. 1 2. 101 3. 60

23 4. 75

6. 61

Exercise 14

1. 61 2. 24

7 3. 121 4. 4

1

5. 125

Challenge:

1.

84

83

87

81

82

83

83

81

84

2.

21

31

65

61

61

31

62

61

63

Exercise 15

1. ( )P354 2. ( )I12

35 3. ( )P211 4. ( )P24

1

5. ( )P61 6. ( )P77

13 7. ( )P101 8. ( )P40

3

9. ( )P31 10. ( )P4

3 11. ( )P356 12. ( )P21

2

13. ( )P283 14. ( )P63

10 15. ( )P83

92

Exercise 16

1. 40, 60 2. 9 3. Rs 61 4. 8

1

5. 514

Activity

;175

1712

Exercise 17

1. 23 2. 1 3. 1

3 4. 15

5. 514

Exercise 18

1. 3635 2. 4

1 3. 211 4. 7

6

5. 65 6. 4

33 7. 1312 8. 6

1

9. 4

Exercise 19

1. 98 2. 28 3

2 3. 1 76 4. 7

5

5. 32

Activity

26 branches

93

Unit 5: Decimal Fractions

Exercise 1

1. 103 , 0.3 2.

105 , 0.5 3. .10

8 0 8= 4. 106 , 0.60

Exercise 2

1. 0.1 2. 0.4 3. 1.3 4. 5.9

Exercise 3

1. 102 , 0.2 cm 2.

106 , 0.6 cm 3.

107 , 0.7 cm 4.

103 , 0.3 cm

Exercise 4

0.3, 0.5, 0.8, 1.2, 1.5

Exercise 5

1. 0.34 2. 0.52 3. 0.79 4. 0.91

Exercise 6

1. 18 boxes shaded 0.82 unshaded2. 51 boxes shaded, 0.49 unshaded3. 7 boxes shaded, 0.93 unshaded4. 150 boxes shaded, 200

50 unshaded

5. 217 boxes shaded, 30083 unshaded

Exercise 7

2. 0.45 3. 1.33 4. 0.07 5. 2.456. 0.62

94

Exercise 8

1. 10015

203= 2. 100

5201= 3. 100

15023= 4. 100

2501=

5. 100375 15

4=

Exercise 9

1.

H T U . t7 4 3 . 6

2.

Th H T U . t h th1 0 0 2 . 1 5 8

3.

H T U . t h th5 0 3 . 0 4 5

4.

U . t h th0 . 8 9 2

5.

U . t h th0 . 9 0 1

Exercise 10

1. < 2. < 3. < 4. <5. <

95

Exercise 11

2. 1.4, 1.5, 1.6, 1.7 3. 8.81, 8.82, 8.83, 8.84, 8.85, 8.864. 3.010, 3.011, 3.012, 3.013, 3.0145. 7.17, 7.18, 7.196. 100.09, 100.10, 100.11

Activity

Different solutions are possible.

Exercise 12

1. 0.845 2. 11.547 3. 57.845 4. 167.02255. 67.988 6. 654.899

Exercise 13

1. 2.78 2. 919.92 3. 16.778 4. 44.225. 51.155 6. 76.001

Exercise 14

1. 2.81 2. 4.269 3. 79.638 4. 12. 842

Exercise 15

1. 472.993 2. 21.8 3. 26.544 4. Rs 2.885 Rs 113.57 6. 1.803 kg 7. 43.658 8. 3.278 km

Challenge

0.56 0.3 0.59

0.76 0.84 1.60

1.32 1.14 2.46

4.8 2.9 7.7

6.3 7.4 13.7

11.1 10.3 21.4

96

Exercise 16

1. 92.4 2. 1642.69 3. 3.174 4. 1.385. 0.294

Exercise 17

1. 72.8 2. 120.01 3. 2.016 4. 547.9365. 10.125

Exercise 18

1. 348.76 2. 193.7 3. 0.98 4. 3455. 0.1

Exercise 19

1. 18.72 m 2. Rs 3170 3. 1088.36 MB 4. Rs 26.04

Exercise 20

1. 3.467 2. 0.90678 3. 0.48623 4. 0.00345. 0.001009

Exercise 21

1. 0.4568 g 2. 1.245 m 3. 0.0235 mg4. 23.5 (Each friend gets 23 chocolates. Kabir keeps 5 chocolates for

himself.)5. 0.25 kg

Unit 6: Measurements

Exercise 1

1 cm 2. m 3. km 4. m5. m

97

Exercise 2

1. 1.55 m 2. 2.5 m 3. 13.04 m 4. 1209 m5. 3569 m 6. 0.034 km 7. 5.679 km 8. 0.283 km9. 8.006 km 10. 29.109 km 11. 3.4 cm 12. 5.12 cm13. 6.1 cm 14. 2372 cm 15. 45601 cm 16. 230 mm17. 7 mm 18. 361 mm 19. 128 mm 20. 429 mm

Exercise 3

1. 1.68 m 2. 29.12 m 3. 5.16 m 4. 247.69 m

Exercise 4

home → city X → city Z → city D

Exercise 5

1. 680000 m, 2770 m, 51500 m 2. 2.77 km3. 680 km 4. 734.27 km5. same

Exercise 6

100.84 m

Exercise 7

1 m 62 cm

Exercise 8

1. a. 1.342 kg b. 34.742 kg c. 0.889 kg d. 2.067 kg e. 67.005 kg2. a. 5000 g b. 8341 g c. 4091 g d. 56725 g e. 3005 g3. a. 9.779 kg b. 1.152 kg c. 12.832 kg d. 0.1 kg e. 8.024 kg

98

Exercise 9

1. 39.11 kg 2. 74.018 kg 3. 6.3 kg 4. 0.41 g5. 42.482 kg 6. 1 kg 500 g + 750 g + 250 g

Exercise 10

1. a. 4900 ml b. 3834 ml c. 7035 ml d. 46400 ml e. 7482 ml2. a. 0.49 l b. 2.222 l c. 0.098 l d. 75.806 l e. 126.004 l3. a. 25.562 b. 2.456 l c. 8.376 l d. 43.594 l e. 16.186 l

Exercise 11

1. 2.89 l 2. 446.5 l 3. 571.5 ml

Unit 7: Time

Exercise 1

1. 10:55 2. 4:25 3. 6:00 4. 2:155. 7:05 6. 12:40

Exercise 2

1. a. 180 sec b. 265 sec c. 1920 sec d. 402sec2. a. 5 min b. 240 min c. 9 min 2 sec d. 225 min3. a. 9 hr b. 72 hr c. 119 hr d. 1 hr4. a. 3 days b. 120 days c. 32 days d. 75 days5. a. 2 months b. 3 months 18 days c. 41 months d. 84 months6. a. 3 years b. 7 years c. 1 year 2 months d. 3 years 9 months

Exercise 3

Students’ answers may vary; help them write the correct a.m. or p.m. times.

99

Exercise 4

1. 19 min 53 sec 2. 3 hr 22 min 3. 10 days 20 hrs4. 8 min 8 sec 5. 52 hr 27 min 6. 2 days 11 hr7. 8 week 3 days 8. 4 years 3 months

Exercise 5

1. 6:50 a.m. 2. 3:00 p.m. 3. 10:36 a.m. 4. 7:45 p.m.5. 2:21 a.m.

Exercise 6

1. 11:20 a.m. in hall 2 2. 3 hr 40 min3. 1 hr 15 min 4. 2 hrs 50 min 5. 7:40 p.m.

Exercise 7

1. 7:35 a.m. 2. 10 hr 52 min 3. 8:12 a.m.4. 14 min 21 sec

Unit 8: Geometry

Exercise 1

1. 5.5 cm 2. 5.1 cm 3. 2.6 cm 4. 10.2 cm5 7.7 cm 6. 9.0 cm 7. 12.1 cm 8. 7.4 cm9. 11.8 cm 10. 11 cm

Exercise 2

Help the students with this exercise and check their work.

Exercise 3

1. 9.7 cm 2. 6.3 cm 3. 7 cm 4. 13 cm5. 9.9 (all approximate measures)

Exercises 4, 5, and 6

Help the students with these exercises and check their work.

100

Exercise 7

Parallel: 1, 4, 5, 8, 9 intersecting: 2, 3, 6, 7

Exercise 8 and 9

Help the students with these exercises and check their work.

Exercise 10

1. right angle 2. obtuse 3. acute 4. straight5. acute 6. complete 7. obtuse 8. reflex9. obtuse

Exercise 11

1. 55° 2. 15° 3. 80° 4. 102°5. 146° 6. 45°

Exercise 12

1. 45° 2. 122° 3. 140° 4. 50°5. 25° 6. 152°

Exercise 13

1. a. acute b. obtuse c. obtuse d. right e. acute f. obtuse g. obtuse h. straight i. acute j. obtuse2. a. 80°/acute b. 110°/obtuse c. 90°/right d. 145°/obtuse e. 10°/acute f. 170°/obtuse3. a. 30°, 60° (acute, acute) b. 75°, 150° (acute, obtuse) c. 90°, 180° (right, straight)4. a. 20°, 80°, 100° (obtuse) b. 45°, 85°, 130° (obtuse) c. 110°, 10°, 120° (obtuse) d. 145°, 15°, 160° (obtuse)

101

Activity1. a. b.

c. d.

Help the students measure the angles the clock hands make.2. 3 o’clock and 9 o’clock3. 6 o’clock

Activity

13

12

11

10

9

8

7

6

5

4

3

2

1

01 2 3 4 5 6 7 8 9 10 11 12 13

N

A

ostrich

oasis

camel

cactus

palm tree

102

Exercise 14

1. AE, EC, EB EW, EZ, EY, EX2. DG CD, XY, AB3. 1.8 cm 2 cm

Exercise 15

1. radius 2. circumference3. diameter 4. circumference

Exercise 16

1. Check the circles that students draw.2. a. 7.9 cm b. 15.3 mm c. 5.2 cm3. a. 11.6 cm b. 7.4 cm c. 9.8 cm

Exercise 17

Check the students’ work.

Unit 9: Information Handling

Exercise 1

1. There are 5 names, therefore 5 bars2. Fluffy3. Bobo 4. 5 + 8 + 6 + 2 + 1 = 225. 16. 17. Roger

Exercise 2

1. a. cricket b. tennis2. a. 12 b. 14 c. 63. hockey4. 2

103

5. Favourite Sports of the Maths Class y-axis: number of students x-axis: favourite sports

Exercise 3

1. a. 14 b. 19 c. 142. a. science b. language3. mathematics & social studies4. 68 5. 3

Exercise 4

1. Jan, 20; Feb, 55; Mar, 30; Apr, 45; May, 10; Jun, 25 July, 30; Aug, 75; Sept, 60; Oct, 302. a. August b. May3. July, 104. 3805. 306. 157. 270

Unit 10: Assess and review 2

Exercise 1

1. 40 million2. ten thousand3. 484. 25. 97

Exercise 2

1. 891 2. 3003 3. 2004 4. 999,999

104

Exercise 3

1. 63024 + 34871 = 978952. 87967 – 40300 = 47667

Exercise 4

1. > 2. > 3. <

Exercise 5

1. 8 2. 0

Exercise 6

1. 397.6 2. 0.037875 3. 5.94 4 4.3245. 1.445 6. 4.249

Exercise 7

1. Forty five point and zero nine six2. Three hundred forty-four million, eight hundred and ninety-three,

nine hundred and eighty3. Zero point zero zero four4. Three hundred million, four thousand and ten

Exercise 8

1. 1 2. 23 3. 2

9 4. 634

5. 72 6. 3 3

1

Exercise 9

1. 6.4 2. 600.203 3. 2004.07 4. 45020.009

Exercise 10

1. 1000345 2. 23100

65 3. 10025 4. 6 2

1

105

Exercise 11

19,000

Exercise 12

1. 763,566 2. 514,383

Exercise 13

79

Exercise 14

60

Exercise 15

19 jugs

Exercise 16

975,430

Exercise 17

1. 17,089 ml 2. 800 m 3. 60 months 4. 7.006 km5. 48 6. 7 7. 704 cm 8. 25,050 g9. 0.75 10. 100

15

Exercise 18

∠AOD, ∠AOB, ∠BOC, ∠COD, ∠AOC, ∠BOD

Exercise 19

1. 67° 2. 145° 3. 35° 4. 108°

106

Exercise 20

Check students’ answers.

Exercise 21

1. 9:35 p.m. 2. 12:00 p.m. 3. 12:10 a.m. 4. 4:30 p.m.

Exercise 22

1. 3.059 km 2. 3.721 km

Exercise 23

24 litres

Exercise 24

Check the students’ work.

Exercise 25

1. 2 2. 2 3. Tuesday & Saturday4. cloudy 5. True

107

G. Answers to Book 5

Unit 1: Assess and Review 1

Exercise 1

1. 104,260,000 2. 123,069,000 3. 260,5994. 103,944,000 5. 8 million

Exercise 2

1. 404,040 2. 200,048,503 3. 6,337,0274. 45,097,012

Exercise 3

1. 429,576 inches 2. 1,073,940 3. 10.91

Exercise 4

90,001

Exercise 5

0.001, 0.02, 0.112, 0.121

Exercise 6

1. 16:25 2. 00:10 3. 12:30 4. 21:18 Exercise 7

1. 1278 2. 12 3. 1419

Exercise 8

16,002 railway compartments

108

Exercise 9

8548 m

Exercise 10

468,505,600 sq.km

Exercise 11

1. 11:00 p.m. 2. 12:54 a.m. 3. 12:45 p.m.

Exercise 12

1324

Exercise 13

18

Exercise 14

535150

Exercise 15

Rs 423.20

Exercise 16

1. < 2. > 3. = 4. <5. > 6. > 7. <

Exercise 17

10.22 carats

109

Exercise 18

Sohail by 0.75 m

Exercise 19

orange

Exercise 20

630 kernels

Exercise 21

Siddiq by 101 km

Exercise 22

1. 3 & 98 2. 9 & 11

2 3. 10 & 63 4. 11 & 8

9

Exercise 23

1. 314 2. 10

67 3. 3100 4. 9

100

Exercise 24

Anwaar plays more.

Exercise 25

Check the angles the students draw.

Exercise 26

1. 12 2. 81 3. 1 4. 12

110

Exercise 27

obtuse angle, acute angle, right angle, acute angle

Exercise 28

1. 168 2. 48 3. 360 4. 1092

Exercise 29

96

Exercise 30

42, 84

Exercise 31

30

Exercise 32

19, 50

Exercise 33

26 kg 700 g

Exercise 34

1. 43 2. 7

6 3. 2715 4. 4

1

Exercise 35

1 and 4 are groups of like fractions.

Exercise 36

Tazeen got the biggest share and Maham the smallest.

111

Exercise 37

1. 24 2. 75 3. 0 4. 8

3

5. 121 6. 8

5

Exercise 38

13.95 kg

Exercise 39

1. 5700 minutes 2. 310 minutes 3. 4 minutes

Exercise 40

1. 5 hrs 59 mins 2. 7 hrs 30 mins 3. 8 hrs 59 mins

Exercise 41

40 mins 40 secs

Exercise 42

4320 flowers

Exercise 43

Check the lines the students draw.

Exercise 44

670, 532

Exercise 45

Check the lines the students draw.

112

Exercise 46

1. sqaure 2. parellogram 3. rectangle4. trapezium

Exercise 47

1. circumference 2. diameter 3. centre 4. equal

Unit 2: Numbers and arithmetic operations

Exercise 1

1. Only the number: 4,085,0002. 400, 000,000 + 50,000,000 + 8,000,000 + 500,000 + 60,000 + 1000 +

90 + 73. Two hundred and sixty-one million, four hundred and fifty-six

thousand and eight hundred and seven4. 2 is in 10 million’s place5. 345,682,510

Exercise 2

1. 123,453,298 2. 3,892,046,710 3. 2, 004,097,0124. 6,045,337,027 5. 25,040,015

Exercise 3

1. Five billion, seven hundred and sixty-two million, nine hundred and sixty-six thousand, eight hundred and fifty

2. Three hundred and fifty million, nine hundred and seventy-six thousand, two hundred and twenty-five

3. One billion, thirty-four million, eight hundred and seventy-three thousand, three hundred

4. Five billion, one hundred and twenty-three million, eight thousand, four hundred and fifty

5. Nine billion, eight million, forty thousand and five

113

Exercise 4

1. 1,256,788,004 2. 4,467,543,098 3. 106,763,0054. 46,583,930,400 5. 302,639,264

Exercise 5

1. > 2. < 3. > 4. >5. >

Exercise 6

1. 48,014,300; 418,320,200; 481,630,4502. 431,573,694; 542,516,019; 1,114,532,4813. 3,229,208,751; 6,479,248,517; 9,456,240,7154. 4,234,530,216; 4,256,053,612; 4,345,035,812

Exercise 7

1. 16,065,679 2. 13,093,421 3. 93,246,9564. 647,219,011 5. 51,386,961 6. 796,423,7167. 2,470,201,844 8. 149,470,717 9. 1,851,425,727

Exercise 8

1. 660,757,611 2. 71,936,787 3. 410,629,2094. 486,155,852 5. 49,089,134 6. 14,719,7087. 71,741,609 8. 51,469,060 9. 783,111,257

Exercise 9

1. 978,896,452 2. 89,439,794 3. 93,818,7024. 11,126,401 5. 77,535,805 6. 40,000,0017. 37,640 8. 8,645,336

Exercise 10

1. Karachi 2. 16,791,379 3. 8,917,5214. 18,201,147

114

Exercise 11

1. 140,584,815 2. 324,766,650 3. 529,664,1354. 350,756,000 5. 236,071,504

Exercise 12

1. 69,377 R78 2. 27,092 R 243 3. 6186 R 855 4. 9735. 1572

Exercise 13

1. Rs 212,925,000 2. Rs 275,053,450 3. 10014. 604,800 secs 5. Rs 1,338,525,000 6. 1500 mins7. 5669 min 8. 1,027,601

Exercise 14

1. 1119 2. 2530 3. 484 4. 05. 0 6. 226

Activity

3 4 12 9 3 37 5 6 2 8 16

21 20 2 18 24 177 4 10 40 8 48

22 48 26 2 16 1015 12 36 3 8 246 45 5 9 2 149 15 3 12 4 6

Unit 3: HCF and LCM

Exercise 1

1. 2 2. remainder 3. 18 4. 75. 4 6. multiples 7. 0 8. 999. 2 10. 195

115

Exercise 2

2. 24 3. 30 4. 84

Exercise 3

2. 7 3. 11 4. 6

Exercise 4

2. 6 3. 7 4. 8.

Exercise 5

1. by 3 6543; 20,058; 67,800; 12,609; 456,9842. by 4 29,612; 48,232; 67,800; 456,9843. by 6 20,058; 67,800; 456,9844. by 9 6543; 12,609; 456,9845. by 11 1870; 15,686; 70,202; 29,612; 456,984

Exercise 6

1. 3 2. 8 3. 34. 9 5. 7 6. 35

Exercise 7

1. 900 2. 180 3. 96 4. 1325. 84

Exercise 8

1. LCM: 36; HCF 3 product of 12 and 9: 108; product of LCM and HCF: 108

2. not possible3. not possible4. not possible

Exercise 9

1. 3 2. 60 3. 20

116

Exercise 10

1. 11:30 a.m.2. 1853. 364. 20 children, 4 books, 5 toys5 a. 20 ml b. 9, 10 times6. after 30 secs, 2nd time after 60 secs7. 168. after 6 days

Activity

1. D I V I 4. S I B L E I V

2. F 3. P X 6. O N E O R 7. T N U I E 8. Z9. R E M A I N D E R

E R10. F A C T O R

11. T W O

Unit 4: Fractions

Exercise 1

proper 155 , 5

2 , 31 , 9

1

improper ,716

1010

117

mixed 7125 , 3 2

1 , 7131 , 9 9

6

unit 1010

Exercise 2

1. 7 21 2. 3 7

2 3. 12 92 4. 316

3

Exercise 3

1. 313 2. 9

100 3. 977 4. 2

45

Exercise 4

1. 20 2. 63 3. 104 4. 5

Exercise 5

1. 54 2. 12

11 3. 211 4. 5

2

Exercise 6

1. 21 , 2

3 , 213 2. 4

1 , 1 41 , 4

23

Exercise 7

1. < 2. > 3. =

Exercise 8

1. 117 2. 28

2141= 3. 0 4. 1

Exercise 9

2. 97 3. 4

3 4. 3114

118

Exercise 10

1. 43 2. 24

41 3. 1 92 4. 1 4

1

5. 53 6. 5 35

17 7. 10121 8. 2 6

1

9. 5 2419 10. 3 34

7 11. 14 4229 12. 20

7

13. 1 601 14. 2 20

3 15. 157

Exercise 11

1. a. 65 b. 6

1 2. 2 43

3. Rs 51 21 4. 8 cm 5. Arman, 8

5 more

6. m6 32 7. cm5 2

1 8. 4 81 m

Activity

3 43

2 41 1 1

21 1

4 1 21

13 43

8 5 43

4 43 3 4

1 2 12

Exercise 12

1. 158 2. 14

3 3. 61 4. 3 4

1

5. 65 6. 22

3 7. 84 203 8. 4 3

2

Exercise 13

1. 16 hours 2. 25 seconds 3. 3 days 4. 65. 80 paisa 6. 75 cm 7. 500 ml 8. 6 days

119

Exercise 14

1. 20 m 2. 51 3. 21

2 4. 245

5. 3 eggs

Exercise 15

1. 54 2. 15

8 3. 154 4. 2 7

4

5. 2625

Exercise 16

1. 4 21 2. 22 2

1 3. 6 pieces 4. 25 pieces

Challenge

15 boards

Activity

THEY ALREADY HAVE BILLS.

Exercise 17

1. 1 207 2. 9

5 3. 0 4. 1 41

5. 21 6. 4

Unit 5: Decimal Fractions and Percentages

Exercise 1

1. 102 = 0.2 2. 100

75 = 0.75

Exercise 2

1. 1.9 2. 0.45 3. 57.3 4. 10.9

120

Exercise 3

2. 0.55 m 3. 0.575 m 4. 0.05 m

Exercise 4

2. 43 3. 25

2 4. 3 21 5. 10 4

3

Exercise 5

1. 2831.47 2. 303.2 3. 1004.308 4. 4.05

Exercise 6 1. > 2. > 3. = 4. <

Exercise 7

1. 8.854 2. 461.18 3. 67.425 4. 111.9685. 104.895

Exercise 8

1. 12.74 2. 19.09 3. 3.378 4. 54.0015. 361.125

Exercise 9

1. 11.8 2. 6.02 3. 10.293

Exercise 10

1. 12.4 2. 17.269 3. 5.0784 4. 05. 1.125

Exercise 11

1. 100 2. 1000 3. 10 4. 1005. 100

121

Exercise 12

1. 68.25 m 2. Rs 217.00

Exercise 13

1. 450 cm 2. 15,250 g 3. 510 cm 4. 1500 ml5. 55 mm

Exercise 14

1. 3.467 2. 0.90678 3. 0.48623 4. 0.00345. 0.001009 6. 3.24 7. 12.5 8. 46.59. 110 10. 1

Exercise 15

2. 100 3. 1000 4. 100 5. 0

Exercise 16

2. 0.0175 3. 0.0175 4. 17.5 5. 0.175

Exercise 17

1. 4.5 2. 0.90 3. 34.52 4. 6.5005. 9.500 6. 2.360 7. 20.50

Exercise 18

1. 198.583 2. 3.3 3. 0 4. 0.465. 2.25

Exercise 19

1. a. 3.8 b. 60.0 c. 196.92 a. 8.97 b. 37.90 c. 500.003. b. 4.98 c. 116.67 d. 73.89 d. 49.02

Exercise 20

1. Rs 15.10 2. 18 m 3. 300.2 kg 4. 10.23 cm5. 75.3 km 6. 45.3 7. Hafiz by 0.3 sec 8. Rs 75 9. 6 cups 10. 176.85 kg

122

Challenge 1

7 ÷ 0.7 × 7 ÷ 0.7 = 100

Challenge 2

9.95, 11.08, 12.21 (+1.13)3.4, 2.9, 3.3 (+ 0.4, – 0.5)0.47, 0.60, 0.57 (+ 0.13, – 0.03)0.8, 8, 80 (× 10)

Exercise 21

2. 32% 3. 45% 4. 80% 5. 60%6. 150%

Exercise 22

Per cent Fraction Decimal Fraction

35% 10035 0.35

50% 10050 0.50

45% 10045 0.45

75% 10075 0.75

5% 1005 0.05

1% 1001 0.01

129%100129 1.29

245% 100245 2.45

525% 100525 5.25

Exercise 23

1. , %, %10065 65 35 2. 100

43 , 43%, 57%

123

Exercise 24

2. 21 3. 20

1 4. 41 5. 4

6. 23

Exercise 25

2. 0.07 3. 0.02 4. 0.15 5. 0.906. 1.25

Exercise 26

2. 87% 3. 1% 4. 56% 5. 23%6. 225%

Exercise 27

Only Exercise 27.4 is incorrect, the correct answer is 0.015

Exercise 28

1. 6% 2. 175% 3. 36% 4. 75%5. 40% 6. 6% 7. 180% 8. 412.5%

Exercise 29

1. 60% 2. 170% 3. 50% 4. 254

Exercise 30

1. 30% 2. maths 90% 3. 21 , 5

1 , 54 , 25

3

4. a. 20% b. 80%5. a. 60% b. 312 c. 2086. a. 53% b. 47%

Activity

cricket (40%), football (20%), basketball (30%), badminton (10%)1. cricket 2. badminton 3. 160

124

Unit 6: Measurement: Distance, Time, and Temperature

Exercise 1

3. 0.8932 4. 9300 5. 0.326 6. 60847. 8.11 8. 34500 9. 4.050 10. 953011. 47.8 12. 45,000

Exercise 2

2. 1265 3. 1 km 567 m 4. 3 cm 9 mm 5. 30206. 12

Exercise 3

1. 15 m79 cm 2. 20 m 10 cm 3. 56 km 475 m4. 5 cm 2 mm 5. 56 m 5 cm

Exercise 4

1. 528.92 km 2. 42.05 m 3. 4.25 km4. 2.05 m 5. 1.015 km

Exercise 5

1. 141 km 34 m 2. 111 m 82 cm 3. 22 cm 2 mm4. 10 km 910 m 5. 9 km 91 m 6. 5 m 77 cm

Exercise 6

1. 784.241 km 2. 1.875 km 3. 6400.1 km4. 37,552 km 5. 2.5 km 6. 251.78 km

Exercise 7

1. 240 secs 2. 180 mins 3. 96 hrs 4. 250 mins5. 460 secs

125

Exercise 8

1. 5 mins 52 secs 2. 7 hrs 30 mins 3. 10 days 20 hrs4. 4 hrs 50 mins 5. 8 years 11 months.

Exercise 9

1. 2 hrs 5 mins 2. 11 mins 22 secs3. 5 weeks 1 day 4. 4 years 3 months

Exercise 10

1. 09.00 a.m. 2. 1 hour 3. 3 hours 15 minutes4. 1 hour 45 mins 5. 0900 hrs, 1000 hrs, 1600 hrs

Exercise 11

7 hrs 50 min

Exercise 12

1 min 46 secs

Exercise 13

2. 69 hrs 2 mins 3. 41 hrs 20 mins 4. 59 mins 47 secs5. 49 mins 12 secs 6. 81 mins 24 secs

Exercise 14

2. 3 hrs 34 mins 3. 10 hrs 34 mins 4. 5 mins 19 secs5. 4 mins 16 secs 6. 23 mins 48 secs

Exercise 15

1. 49 days 2. 40 weeks 3. 432 weeks 4. 8 weeks5. 96 months 6. 144 months 7. 1095 days 8. 5 months9. 120 days 10. 12 weeks

126

Exercise 16

1. 7:45 a.m. 2. 22:05 hours 3. 8 hrs. 45 min4. 4 hrs. 49 min 5. 4 hrs 35 min

Challenge

7 June

31/12/06, 11:52 p.m.

Exercise 17

1. 35°C 2. 39°C 3. 4°C 4. 45°C

Exercise 18

2.5°C

Exercise 19

70°C

Exercise 20

1. 12°C 2. 7°C 3. birds

Exercise 21

1. 50°C 2. 20°C 3. 2°C 4. 81.5°F5. 59°F 6. 140°F

Exercise 22

1. mild 2. 0°C 3. 15 degrees

Exercise 23

1. 27°C 2. 23°C

127

Unit 7: Unitary Method; Ratio and Proportion

Exercise 1

1. 1950 kg 2. 200 km 3. 25 kg 4. 16 days5. 500 men 6. 960 kg 7. 3 days 8. 150 pages9. 16 days 10. 16 nights

Exercise 2

1. proportion 2. ratio 3. directly 4. decrease

Unit 8: Geometry

Exercise 1

arms; vertex; acute, 90°; obtuse, straight; right; 360°

Exercise 2

1. <DCB = 26°, acute 2. <EFG = 125° obtuse3. <MNO = 55°, acute 4. <PQR 98°, obtuse

Exercise 3

Check the students’ work.

Exercise 4

2. any corner of the classroom3. any corner of your book4. any corner of the teacher’s desk

Exercise 5

300°, 340°, 250°, 310°

Exercise 6

Check the students’ work.

128

Exercise 7

<POR and <ROS ; <ROS and <SOQ<DOA and <AOC ; <AOC and <COB; <COB and <BOD; <BOD and <DOA

Exercise 8

1. complementary 2. supplementary3. supplementary 4. complementary5. complementary

Exercise 9

1. 57° 2. 124° 3. 102° 4. 68°5. 174° 6. 177° 7. 14° 8. 25°9. 72° 10. 12°

Exercise 10

3. 90° 4. 60° (30 × 2 = 60)

Exercise 11

equilateral, scalene, scaleneisosceles, scalene, equilateral

Exercise 12

right-angled triangle, right-angled triangle, acute-angled triangleobtuse-angled triangle, obtuse-angled triangle, acute-angled triangle

Exercise 13

Triangle Angle 1 Angle 2 Angle 3 Sum

ABC 90° 30° 60° 180°

PQR 90° 45° 45° 180°

XYZ 100° 50° 30° 180°

129

Exercise 14

The triangles PQR and DEC cannot be drawn as the sum of two of the sides in each triangle is less than the third side.

Exercise 15

Check the students’ work; triangle XYZ cannot be drawn with the given measurements.

Exercise 16

Check the students’ work.

Exercise 17

1. square 2. rhombus 3. parellelogram4. rectangle 5. trapezium

Exercise 18

Check the students’ work.

Exercise 19

Check the students’ work.

Activity

1. rhombus 2. parallelogram 3. trapezium4. rhombus 5. kite 6. an irregular quadrilateral

Unit 9: Perimeter and Area

Exercise 1

2. m 3. km 4. cm

Exercise 2

1. 22 cm 2. 24 cm 3. 24 cm 4. 19 cm

130

Exercise 3

1. 16 m2. a. 50 m b. Rs 37503. 1 km 4. 10 m 5. 16 m

Exercise 4

1. 8 cm² 2. 16 cm² 3. 12 cm²

Exercise 5

1. a. 4 cm² b. 70 cm² c. 15 cm²2. L = 8 cm, B = 2 cm, P = 20 cm, A= 16 cm²3. 64 m²4.

Length (cm) Breadth (cm) Perimeter (cm)2 (l + b)

Area (cm2)l × b

14 10 48° 140°

20 15 70° 300°

12.5 8.5 42 106.25

15 6 42 90

25 20 90 500

Exercise 6

1. Rs 6000 2. Rs 192,000 3. Rs 1.8 million

Exercise 7

1. P = 90 cm, A = 506.25 m²2. P = 112 m, A = 768 m²3. P = 150 cm, A = 1250 cm²4. a. B = 10 m b. A = 300 m² c. Rs 4500 d. P = 80 m e. Rs 20005. 210 m6. Sonia runs 1200 m (150 + 150 + 150 + 150) × 2 Nina runs 1080 m (100 + 80 + 100 + 80) × 3 Sonia runs 120 m

131

7. 12 m8. a. hexagons and equilateral triangles as their area is the easiest to

calculate b. because they leave gaps in between (the shape does not

tessellate)

Challenge

The first challenge shows two diagrams. Help the students calculate taking the hint.The second challenge is solved as shown below:

A B

A

C B

C

Unit 10: Information handling

Exercise 1

2. 45 3. 55 4. 3 5. 758 m6. 5

Exercise 2

1. Rs 2802. a. Younis: 113.6 Yousuf: 92.8, Imran: 74.8, Kamran: 83.6, Syed: 78.4,

Iqbal: 69.6 b. 69.6, 74.8, 78.4, 83.6, 92.8, 113.6 c. three players3. 66.8 grams4. 40 km5. 340 kg6. b. 2 + 4 = 6÷2 = 3 (odd); 24 + 26 + 50 ÷ 2 = 25 (odd) a. 3 + 5 = 12 ÷2 = 6 (even); 15 = 17 = 32 ÷ 2 = 16 (even)7. town A: 83°C, town B: 8.8°C—town A is colder.

132

Exercise 3

Laila

Kiran

Junaid

0 1 2 3 4 5 6

Exercise 4

The graph will look like the one given below. It can be either a horizontal bar graph or a vertical bar graph as shown.

1

0

2

3

4

5

6

7

8

9

10

11

12

13

14

Nu

mb

er o

f Sh

op

s /

Sto

res

Mall Shops

RibbonShop

HairpinShop

CheeseStore

PencilStore

HairbandShop

ShoelaceShop

The Southern City Mall___________________The Southern City Mall

133

Exercise 5

1. four bars2. labels3. the bar for the blue whale since it has the maximum length4. the bar for the humpback whale5. Check the graphs the students draw.

Exercise 6

1

0

2

3

4

5

6

7

8

9

Red Blue Pink Green

Colours

Nu

mb

er o

f Stu

den

ts

Yellow

Favourite Colours of Class 5______________________

Exercise 7

1. The graph shows information about the sports played by students of Class 5.

2. There are 5 bars, since 5 sports are listed.3. cricket

134

4. Hockey and volley ball5. 50 students6. 5:2

Exercise 8

The answers will vary.

Exercise 9

1. Rs 675 2. Rs 675

Unit 11: Assess and Review 2

Exercise 1

i. 20,480,503 2. 162,337,027 3. 45, 097,012

Exercise 2

1. 7080 kg 2. 5 3. 8 m 4. 605. 100, 111, 102 6. 36 7. 5-2/5 8. 500 secs9. Check the angles drawn by the students10. 10,050 m11. a. Rs 5.07 b. Rs 14. 58 c. Rs 0. 54 d. 5.67 m e. 333.450 km f. 56.560 kg g. 2.5 l h. 0.150 g12. 4 children13. 1.45 m14. Yes15. a. 23 days b. 72 days c. 9 days16. cabbage17. Skardu18. 10.87 m19. 15 hours20. 1. 57° 2. 86° 3. 130° 4. 240° 5. 320° 6. 160°21. a. windows b. 180° c. door22. 678, 411, 69323. a. 7 b. 9 c. 13

135

24. a. 96 b. 3420 c. 390 d. 36025. Rs 160,00026. Rs 720027. 857,750 bulbs, 17,155 cartons28. a. 28% b. 60% c. 100%29. a., d., f., and g. are complimentary b. and h. are supplementary c., e., i., and j. are neither 30. 64 l31. 130°32. a. 0.42 b. 0.77 c. 1.83 d. 0.8833. 1. 12.6 2. 29.1 3. 2.5 4. 3.4 5. 5.2 6. 89.434. a. 16.12 b. 100.43916 c. 36.694 d. 15.007 e. 6.145 f. 4 g. 48 h. 31.65

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Asian Circus__________

36. a. 30° b. 72° c. 180° d. 15°37. a. right angle b. obtuse angle c. acute angle d. reflex angle38. 1. 23.99 2. 3.71 3. 4.45 4. 44.04 5. 999.47 6. 7.44

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