Imagine_Maths_CB_Grade4_Sample by Uolo

Imagine Mathematics seamlessly bridges the gap between abstract mathematics and realworld relevance, offering engaging narratives, examples and illustrations that inspire young minds to explore the beauty and power of mathematical thinking. Aligned with the NEP 2020, this book is tailored to make mathematics anxiety-free, encouraging learners to envision mathematical concepts rather than memorize them. The ultimate objective is to cultivate in learners a lifelong appreciation for this vital discipline.

Imagine Mathematics

About This Book

MATHEMATICS

Key Features • Let’s Recall: Helps to revisit students’ prior knowledge to facilitate learning the new chapter • Real Life Connect: Introduces a new concept by relating it to day-to-day life • Examples: Provides the complete solution in a step-by-step manner • Do It Together: Guides learners to solve a problem by giving clues and hints • Think and Tell: Probing questions to stimulate Higher Order Thinking Skills (HOTS) • Error Alert: A simple tip off to help avoid misconceptions and common mistakes • Remember: Key points for easy recollection • Did You Know? Interesting facts related to the application of concept • Math Lab: Fun cross-curricular activities • QR Codes: Digital integration through the app to promote self-learning and practice

About Uolo Uolo partners with K-12 schools to provide technology-based learning programs. We believe pedagogy and technology must come together to deliver scalable learning experiences that generate measurable outcomes. Uolo is trusted by over 10,000 schools across India, South East Asia, and the Middle East.

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NEP 2020 based

NCF compliant

CBSE aligned

26/09/23 7:16 PM

MATHEMATICS Master Mathematical Thinking

Grade 4

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Fo re wo rd

Mathematics is not just another subject. It is an integral part of our lives. It shapes the very foundation of our understanding, personality and interaction with the world around us. However, due to the subject’s abstract nature, the stress of achieving high academic scores and complex teaching methods, most children develop a fear of mathematics from an early age. This fear not only hinders their mathematical thinking, logical reasoning and general problem solving abilities, but also negatively impacts their performance in other academic subjects. This creates a learning gap which widens over the years. The NEP 2020 has distinctly recognised the value of mathematical thinking among young learners and the significance of fostering love for this subject by making its learning engaging and entertaining. Approaching maths with patience and relatable real-world examples can help nurture an inspiring relationship with the subject. It is in this spirit that Uolo has introduced the Imagine Mathematics product for elementary grades (1 to 8). This product’s key objective is to eliminate the fear of mathematics by making learning exciting, relatable and meaningful for children. This is achieved by making a clear connection between mathematical concepts and examples from daily life. This opens avenues for children to connect with and explore maths in pleasant, relatable, creative and fun ways. This product, as recommended by the NEP 2020 and the recent NCF draft, gives paramount importance to the development of computational and mathematical thinking, logical reasoning, problem solving and mathematical communication, with the help of carefully curated content and learning activities. Imagine Mathematics strongly positions itself on the curricular and pedagogical approach of the Gradual Release of Responsibility (GRR), which has been highly recommended by the NEP 2020, the latest NCF Draft and other international educational policies. In this approach, while learning any new mathematical concept, learners first receive sufficient modelling, and then are supported to solve problems in a guided manner before eventually taking complete control of the learning and application of the concept on their own. In addition, the book is technologically empowered and works in sync with a parallel digital world which contains immersive gamified experiences, video solutions and practice exercises among other things. Interactive exercises on the digital platform make learning experiential and help in concrete visualisation of abstract mathematical concepts. In Imagine Mathematics, we are striving to make high quality maths learning available for all children across the country. The product maximizes the opportunities for self-learning while minimising the need for paid external interventions, like after-school or private tutorial classes. The book adapts some of the most-acclaimed, learner-friendly pedagogical strategies. Each concept in every chapter is introduced with the help of real-life situations and integrated with children’s experiences, making learning flow seamlessly from abstract to concrete. Clear explanations and simple steps are provided to solve problems in each concept. Interesting facts, error alerts and enjoyable activities are smartly sprinkled throughout the content to break the monotony and make learning holistic. Most importantly, concepts are not presented in a disconnected fashion, but are interlinked and interwoven in a sophisticated manner across strands and grades to make learning scaffolded, comprehensive and meaningful. As we know, no single content book can resolve all learning challenges, and human intervention and support tools are required to ensure its success. Thus, Imagine Mathematics not only offers the content books, but also comes with teacher manuals that guide the pedagogical transactions that happen in the classroom; and a vast parallel digital world with lots of exciting materials for learning, practice and assessment. In a nutshell, Imagine Mathematics is a comprehensive and unique learning experience for children. On this note, we welcome you to the wonderful world of Imagine Mathematics. In the pages that follow, we will embark on a thrilling journey to discover wonderful secrets of mathematics—numbers, operations, geometry and measurements, data and probability, patterns and symmetry, algebra and so on and so forth. Wishing all the learners, teachers and parents lots of fun-filled learning as you embark upon this exciting journey with Uolo. ii

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1 6-digits Let's Recall We know that numbers are basic units of mathematics and are used for counting, measuring and comparing quantities. We have also previously learned that each digit in a number has a value, which we call the place value. Let us take a 2-digit number as an example! The digit on the left is at ten’s place while the digit on the right is at one’s place.

K ey El emen t s o f a C h apt e r— a Q u i c k G lanc e

Similarly, in the number 2548, 5 represents 5 hundreds, or 500. However, in the number 56, 5 represents 5 tens, or 50. Therefore, even if a digit is same, its value always depends on where it is in the number. Th

4 Tens

8 Ones

2 Thousands 5 Hundreds

tiples

Let's Warm-up

Introductory

Concept

Write the correct place value of the coloured numbers.

page with a 5 multiples introduction Find the first of 6. Check__________ by dividing if 92 and 96 are multiples of 6. 1 32

Do It Together

ng holidays is always fun!quick You first decide on a place to visit, warm-up 2 548 you want__________ 1 2 esearch what you can do there, and then finally make arrangements. 3 876 __________

4563 s going to Ooty. The train departs every second day.

7 14

This is too much! Sanju gets confused and asks his father to help.

__________

9958

6×1=6

6 × 2 = ____

6 × ____ = 24

6 × ____ = ____

Sanju’s father helps him find the number of words using multiplication.

Multiplication by 1-digit Number We know that the number obtained from multiplication is the product.

Add 12,344 and 1115.

A quick-thinking

21 question 22

6 × ____ = ____

Do It Together

What do you think do

TTh

even number?

Find the factors of 36 using the division method. What do we get?

Check the remainder

=31 ×

multiplicand

TTh

multiplier

36 ÷ 3

1= 2= 3= 4= 5=

3 6 9 12 15

____

On multiplying a number by 1, the product is always the number itself.

On multiplying a number by 0, the product is always zero.

product

Step 1

Step 2

Step 3

Multiply by ones.

Multiply by tens.

Multiply by hundreds.

Multiply 3 and 3 ones.

Multiply 3 and 2 tens.

123 × 3 = 369

123 × 3 = __ __9

123 × 3 = __69

Did You Know?

Rinne Tsujikubo of Japan broke The productthe of 123 and 3 is 369.world record for Guinness fastest mental arithmetic on Chapter 3 • Multiplication January 17, 2023 by correctly adding 15 sets of three-digit numbers in 1.62 seconds.

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The total number of fiction books = 1219 The total number of non-fiction books = 1567

Are the numbers factors of 36?

____

Let us multiply 123 and 3.

What do we need to find?

dividing 92 by weby get 2 remainder. On dividing 96 by 6, we get ____ remainder. 36 ÷ 1 36 the number 0 ultiples of a number areOn the products we6,get multiplying by 1,Yes2, Face Value Numbers Thinkinand Telltotal number =6-digit The of fiction books 36 ÷ 2 ____ 0 ____ nd so on. The total number of non-fiction books Do we need+of to go6. So, 92 _______________________________ of 6. So, 96 ___________________________

les of 3 can be found by using multiplication tables36as follows: ÷4 ____

= 62

We can find the product of two numbers by placing them horizontally next to each other.

What do we know?

We can multiply two numbers in any order. The product always remains the same.

Horizontal Method

Let us start finding the total number of books step by step.

n travel on the following dates - 2, 4, 6, 8, 10, 12, 14, 16 and so on. 6 92 Divide numbers by 36

to the concept

ing Multiples

In the school library, there are 1219 fiction 9/11/2023 4:24:58 PM books and 1567 non-fiction books. How many books are there in total?

Are the multiples of an even number always an

Do It Together

Fun fact, related

Think and Tellhave in common?

Story Sums

all the circled numbers

Multiplication Rules

The number to be multiplied is multiplicand and the number by which we multiply is multiplier.

Thinkofand Tell I scored out12,344 of 5. + 1115 = _____________. The first five multiples 6 are 6, ____, ____, 24, ____._________ So, 17

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2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 +2 + 2 + 2 + 2………

He has learnt 14 words already! Hurray! Sanju struggles to find how many words will he learn in whole January.

4 5 6 7Sunday 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 Saturday

2 + 2 + 2 + 2 + 2 + 2 + 2 = 14

example

0 1 Friday 2 3 Thursday

with a real-life

__________

y Tuesday Wednesday

Understanding Multiplication Real Life Connect

The total number of books in the library

We learnt beyond about the concept of face value in the previous section. as the 2 7 It8is defined 6 6=to1219 find + 1567 numericalmore value ofofthe factors 36? digit on a particular place in a number. Solve to find the answer.

out 36 ÷ 5 7 1 ____ For example, the Pincode on Ajay’s 3 × 6 = An 18important So,Pointing the total number of books in the library is 2786. ____ 0 Yes letter was 781005. The face value of It Yourself 5A 36 ÷ 6 keep in commonly made 3 × 7 = point 21 toDo Error Alert! The city NGO organised a two-day donation drive. the Lakhs placeOnisthe simply 7.of the drive, 1366 clothes were Remember! Remember! first day So, the factors of 36 are ___________________________________________________________________. 3 × 8 = mind 24 mistakes and When we add 1000 to a collected. On the second day of the drive, 1000 Never confuse Face Value with Place Value! For A number is a multiple of itself too. Similarly, we say that the face value of 4-digit number, only the clothes were collected. How many clothes were 3 × 9 = 1 27Colour the balloons that are multiples of 2. howinto avoid example, in 781005, the Value of the Lakhs digit Face in the thousands For example, multiples of 5 are 5, total? the digit on thecollected Ten Thousands place place changes. place is 7 and the Place Value is 7 × 100000 = 700000. Do15, It Yourself 3 × 10 = 30 10, 20, 25,5C 30 and so on! them 4

Round off each dividend to the nearest 10s and 100s and then divide. To the nearest 10s

Example 4

To the nearest 100s

a 1147 ÷ 2

b 4589 ÷ 3 c 6478 ÷ 6

d 8974 ÷ 7

is 8.

4 6 5 8 7 Show 10 in different arrangements. Then, list the factors of 10.

e 5555 ÷ 5

What do we know?

860 people have been invited to a banquet. The caterer is arranging tables. Each table can seat 10 Number of clothes collected during the 5first day of the drive = 1366 people. About how many tables are needed?

[Round off the dividend to the nearest hundred]. Number of clothes collected duringin thethe second day ofplace the drivein = 1000 What is the face value of the number lakhs 348673? Show 18 in different arrangements. Then, list the factors of 18.

2 n also check if a number is a multiple of a number using division. If the remainder Word Problems What do we need to find? Show 20 in different arrangements. Then, list the factors of 20. We know that face value is the numerical value of the digit in a particular place. Let us hen the bigger number is a multiple of the other3number. For example: Example 6

ves no remainder

15 – 15 00

3 11

1 238 children went to a school camp. If one tent can be shared by 4 children, number of clothes collected = 1366 + 1000 Find the factors of the following numbers using multiplication. write the placeTotal value chart for 348673:

5 d 3916 e 340 18 13 – 17 15 Find the factors of the following numbers using division. 5 b 11 1 c 12 d 1301 e 15 9 leavesa remainder a 14

b 21

c 36

19 L

g 48

h 50

20 30 TTh

approximately how many tents will be needed for the camp?

₹5734 is distributed between 2 groups of students. Each group has 11 students.

Estimate the amount of money that each student gets.

Points to Remember HOTS: end Is 18 a factor of 126? Explain your answer. 6 The face value ofChapter the number in the Lakhs place is 3. viding 15 by 5, we get 0 remainder. On dividing 16 by 5, we get remainder 1. Is of 6 a the factorgiven of 64? Explain your answer. 7 Find the first five multiples numbers. 2 • The number being divided is called the dividend. The number by which we divide is Applicative summary called the divisor. The result of the division is called the quotient. The number left over Do It is a multiple of 5. So, 16 8is not multiple 5.10 have exactly TWO factors? Which a numbers betweenof 1 and Thousands division is called the place remainder. in 800234? a 7 b 8 c 9 dTogether e 11value of the number in the after 10 What is the face and analytical • To check if our answer after division is correct, we can use: Dividend = (Quotient × UM24CB_G4.indb 30

Which number has the greatest number of factors between 5 and 15? What is the smallest number that has exactly three factors?

5 • Multiples and Factors

h 14

Points to Remember m

Word Problems

place value chart for 800234: j 16 15 Let us write the

n 20

•

The place value table is divided into groups called periods.

•

5-digit numbers have 2 periods - Thousands Period and Ones Period.

•

Face value is the numerical value of the digit in a particular place in a number.

________

Tina bought 16 eggs. She wants to arrange them into a tray. In how many Chapter 5 • Multiples and Factors • 16-digit numbers have 3 periods - Lakhs Period, Thousands Period and Ones Period. • • •

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________

When a number is divided by 1, the quotient is always the number itself.

•

When a number is divided by itself, then the quotient is 1.

•

When 0 is divided by any number, then the quotient is always 0.

•

When a number is divided by 10, the digit at the ones place forms the remainder and the remaining digits form the quotient.

•

When a number is divided by 100, the digit in the ones place and tens place forms the remainder and the remaining digits form the quotient.

Raman, a baker, baked biscuits. He wants to place arranged from thehas smallest to the72 largest are said to be in an ascending order. the same number 2Numbers

________

thousands place form the remainder and the remaining digits form the quotient.

of biscuits each packet. are possible? Numbers arranged in from the largest to theWhat smallestdifferent are said to bearrangements in a descending order.

Representing 6-digit Numbers

Rounding numbers is helpful when we need an estimate and when we want to convey 9/11/2023 4:26:02 PM numbers in an easier way.

Chapter 4 • Division

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Place value Scavenger Hunt:

to interactive

Materials Required: Newspapers, Magazines, or the Internet

For 6-digit numbers, the Lakhs place falls in the Lakhs Period.

Setting: Groups of 4

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digital resources 9/11/2023 4:26:15 PM

Divide the entire class into groups of 4.

classroom

Each group can be dedicated a particular category like City population, State population,

Lakhs Period

activity

Each group collects the data on their respective category from newspapers, magazines, or

Each groups’ data must include at least 5 numbers in their category.

Lakhs

Each group discusses and deduces the following for each number in their data:

Followers of celebrities, Number of speakers of a language, and Car and Bike prices. the internet.

Place value and face value of each digit.

Thousands Period Ten Thousands (TTh)

Ones Period

Thousands (Th)

Hundreds (H)

Tens (T)

Ones (O)

We can therefore represent our 6-digit number 781005 as:

Correct number representation. Correctly written number names.

Lakhs Period

Correctly order the numbers in ascending and descending order Round off the numbers to the nearest 10s, 100s and 1000s.

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We learnt that the Thousands period includes Ten Thousands and Thousands places. The QR Code: Access Ones period includes the Hundreds, Tens and Ones places.

Chapter 1 • Numbers up to 6-digits

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7,81,005

Thousands Period

Ones Period

iii 9

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The face value of the number in the thousands place is _________. • When a number is divided by 1000, the digits in the ones place, tens place and

ways can she arrange the eggs?

Math Lab

multidisciplinary and fun

________

Let us continue to learn about the4:26:03 “periods” convention for large numbers. 9/11/2023 PM

o TTh 25

Divisor) + Remainder.

•

  

k 17

g 13

  

f 12 questions

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G rad ual R e le ase of Re spon si bi li t y

The Gradual Release of Responsibility (GRR) is a highly effective pedagogical approach that empowers students to learn progressively by transitioning the responsibility from the teacher to the students. This method involves comprehensive scaffolding—including modelling, guided practice, and ultimately fostering independent application of concepts. GRR, endorsed and promoted by both the NEP 2020 and NCF, plays a pivotal role in equipping teachers to facilitate age-appropriate learning outcomes and enabling learners to thrive. The GRR methodology forms the foundation of the IMAGINE Mathematics product. Within each chapter, every unit follows a consistent framework: 1. I Do (entirely teacher-led)

2. We Do (guided practice for learners supported by the teacher) 3. You Do (independent practice for learners) GRR Steps

Unit Component

Snapshot

Numbers Beyond 9999 Ajay: Hello daddy, I found this letter, it is for you.

Real Life Connect

Father: Thank you. Ajay: The letter has your name and the address of our home. Father: Yes.

Real Life Connect

Ajay: But, what is this big number 781005?

From: Ajay Shukla, 12, Hathipol,

Guwahati - 781005

Father: It is a special code, also called the postal code. This code helps in finding the exact location in a city.

Theoretical explanation

Ajay: Okay daddy. But it has 6-digits, and I find it hard to read!

Facts about Multiples

• Every number is a multiple of 1 and the number itself.

ForAbout example, 55-digit × 1 = 5. Here, 5 is a multiple of 1 and 5. All Numbers!

• Every multiple is either greater than or equal to the number itself.

ToFor help Ajay understand 6-digit numbers, let's first learn about 5-digit numbers. example, the multiples of 8 are 8, 16, 24, 32, 40, … and so on. Here, each multiple is

to that or greater 8. thousand nine hundred ninety-nine is the greatest 4-digit Weequal know 9999than – nine • Every number has an unlimited number of multiples. number.

I do

For example, the multiples of 7 are 7, 14, 21, 28, 35, …, 70, 77, …, 7000, …, 70000, …,

Now, we add 1 to this, we unlimited. get 10000. and when so on. Here, multiples of 7 are Example 1

Find the 5 multiples of 4. 9999 + 1first = 10000

Remember!

We can find the multiples of 4 by using the number line showing jumps of 10000 4. is the smallest 5-digit

10000 is read as “Ten Thousand”. 1

Let us learn more about 5-digit numbers! 0

number. 99999 is the greatest 5-digit number.

9 10 11 12 13 14 15 16 17 18 19 20

Place Values and Expanded Form in 5-digit Numbers We can find the multiples of 4 by using multiplication tables as follows:

We4 know that a 4-digit number has 4 places on the place value chart - ones, tens, ×1= 4 hundreds and thousands. The place on the left to the Thousands place is called the Ten Did You Know? 4×2= 8 Leap years are always Thousands place. 4 × 3 = 12

Examples

multiples of 4. For example

× 4 take = 16a 5-digit number 13435. The place value chart Let4 us for this can be written the years 2016, number 2020, 2024, ... as:4 × 5 = 20 are all leap years. The five multiples TThfirstTh H T of 4Oare 4, 8, 12, 16 and 20. Example 2

Did You Know?

1 3 4 3 5 Find the first 5 multiples of 5. Check by dividing whether 95 is a multiple of 5. The number 4 is the 1 2 3 4 5 only number with the same number of Always remember that place value is the letters as its value in value of the digit in a number based on the English language. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 its0position in the given number.

The place 100 5= ×400. 5 ×value 1 = 5 of “4” 5 × in 2 =13435 10 5is × 34 =×15 4 = 20

5 × 5 = 25

The first five multiples of 5 are 5, 10, 15, 20 and 25. 5

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leaves remainder 0

95 – 95 00

19 On dividing 95 by 5, we get 0 remainder. So, 95 is a multiple of 5.

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Find the factors of 36 using the division method.

Do It Together

GRR Steps

Divide numbers by 36

Unit Component

What do we get?

Check the remainder

36 ÷ 1

36 ÷ 2

____

Yes

36 ÷ 4

____

Think and Tell Do we need to go

Find 20 and 30. Also, common factor of both 36 ÷ 3the common 12 factors of ____ ____ find the highest beyond 6 to find the numbers. more factors of 36?

Do It Together

Find the method. Step 1 factors7of 36 using1 the division 36 ÷ 5 ____

Do It Together

Factors of 20 36 ÷ 6 Divide numbers by 36

We do

Are the numbers

factors of 36? Snapshot

1 2 0 What do we get?

____

4 5 10 20 Yes Check the Are the numbers remainder factors of 36?

Step 2 So, the factors of 36 are ___________________________________________________________________.

Do It Together

36 ÷of130 Factors

Yes

Step363÷ 2

____

Common 36 ÷ 3Factors

____

Do It Yourself 5C

of 20 and 30 3610 ÷ in 4 different arrangements. ____ Show Then, list 0 the factors of 10. ____

Show Then, list 1 the factors of 18. ____ 3618 ÷ in 5 different arrangements. 7

Do we need to go beyond 6 to find more factors of 36?

The common factors of 20 and 30 are ____________________________________________________. Show 20 in different arrangements. Then, list the factors of 20.

3 4

Think and Tell

36 ÷ 6

____

Yes

Math Lab of the following Find the factorscommon numbers multiplication. The highest factor of 20 using and 30 is __________.

So,a the b 21 of 36c are d 39 e 40 f 42 g 48 h 50 i 77 14 factors 36 ___________________________________________________________________.

Board Game of Multiples

Find the factors of the following numbers using division.

Materials Required: Number grid asnumbers. shown below, dice, crayons Find the common factors of the following 1 18 Is a factor of 126? Explain your answer.

Do It Yourself 5D

Setting:bIn groupsc of 4 a 9

d 13

e 15

8, 10 of 64? Explainbyour 12, answer. 15 IsMethod: 6a a factor

c 13, 16

Do It Yourself 5C

d 14, 20

e 16, 18

10 different the factors of 10. 45 j 72, f numbers g h Then, i 54, 64 12 2681 Each player chooses their 20, 30 in 33, 10 44 35, 50list 1Show between 1 arrangements. and have colour. exactly TWO factors? 8 1 Which

number. 4 The common player chooses multiple that Also, find the lowest and the highest 3 Find factors ofathe pairsnumbers of of numbers. the factors of the following using multiplication. 4 3 Find the 18 20 22 48 Word common Problems number on Show the board and shades it with factors. it with a diagram. a 14 b 21 c 36 d 39 e 40 21 f 42 30 17g 4822 their colour.

Do It Yourself

a is6the b arrangements. c Then, d 3 40 8that has exactly 12 factors? number three Show 20smallest in different list the factors of 20. 50 e 436 10 3 What

Word Problems

a 16 and 24

a common factor of allyour the numbers. Isc 6 0aisfactor of 64? Explain answer. _______ d 15 and 25 havebetween a total of 1 3 and common factors. _______ Which numbers 10 have exactly TWO factors? e 6 is number a common factor 18, 30 and 66. _______ Which has theof greatest number of factors between 5 and 15?

f The lowest common factor of 20, 34, 39 and 42 is 1. _______ Chapter Checkup 10 What is the smallest number that has exactly three factors? 5

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Radhe says, “The number 14 has a greater number of factors than 45.” Is he correct? Verify your answer.

Word Find theProblems factors.

c 72 d 88 eggs. She wants to arrange them into a tray. iIn how many g 16 1201 Tina bought 156 200 h 180

ways can she arrange the eggs?

222

Raman, a baker, has baked 72 biscuits. He wants to place the same number 9/11/2023 4:26:19 PM

Which of these pairs of numbers have the common factor of 4? 92

a 5 and 20

b 20 and 100

c 12 and 36

d 60 and 200

c 15

d 23

Write the first 5 multiples. a 7

b 11

e 30

Write the smallest number which is a common multiple of the given number pairs.

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a 2 and 5

e 10 and 15

e 98 95

Find the common factors of the given pairs of numbers.

a 7 and 14 c 9arrangements of biscuits b in each packet. are possible? 24 and 30 What different and 12 d 20 and 25

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of biscuits in each packet. What different arrangements are possible?

aChapter 45 5 • Multiples andbFactors 66 2

13 have no common _______ aand factor ofwho 126? Explain factors. yourmost answer. The player colours the number of multiples on the board is the winner. 5Isb 1811

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f 818

92 8

Chapter Checkup

h 50 11 60

b 21 and 42 c 63 and 18 d 55 and 100 e 48 and 84 Tina bought 16 eggs. She wants to arrange them into a tray. In how many

Inthe case a player gets 1 on the dice, they 67 factors of the following numbers using division. 7 5 4 4Find ways can she arrange the eggs? Write if True or False. can choose any number on the board. b 11 c 72 dHe13 e is 15 9 12biscuits. 18 a baker, has baked wants to36 place samefnumber The biggest common factor of numbers 24 and 3.the _______ 2aa Raman, (Do you know why?)

You do

18 inhas different arrangements. Then, list the of number therolls greatest of factors between 5factors and 15?the One player the number diceare and sees Which of the following numbers factors ofthe 78 and 96? Circle correct 9 2 Which 5 18. 24 option. 15 Verify 42your answer. 25 35 2 2Show

b 3 and 7 f

10 and 25

c 5 and 8

g 11 and 22

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d 4 and 14

h 20 and 24

Find the following. a Multiples of 4 that are smaller than 30.

Pearson, P. D., & Gallagher, G. (1983). Contemporary Educational Psychology.

b Multiples of 6 that are smaller than 50.

Fisher, D., & Frey, N. (2021). Better learning through structured teaching: A framework for the gradual release of responsibility. c Multiples of 8 that are greater than 30 but smaller than 80.

Fisher, D., & Frey, N. (2014). Checking for understanding: Formative assessment techniques for your classroom.

Chapter 5 • Multiples and Factors

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C o nt e nt s

Numbers up to 6 Digits �� 1 • Numbers Beyond 9999 • Comparing and Rounding off Numbers

2 12

Addition and Subtraction �� 25 • Addition of Numbers Beyond 999 • Subtraction of Numbers Beyond 999 • Addition and Subtraction of Numbers Beyond 999 • Estimation

26 33

39 42

Multiplication �� 48

Division �� 62

• Understanding Multiplication • Estimation

• Division by 1-digit and 2-digit Numbers • Estimation

49 58

63 74

81 87

100 114

Lines and 2-D Shapes �� 122 • Understanding Basic Terms 123 • Understanding More Geometrical Figures 129 • Circles and Its Parts 135

143

Patterns and Symmetry �� 156

Length, Weight and Capacity �� 178

• Patterns Around Us • Symmetry and Reflections

• Length • Weight • Capacity

157 169

179 184 188

Perimeter and Area �� 195

Time �� 213

Money �� 232

Data Handling �� 249

Fractions �� 99 • Understanding Fractions • Operations on Fractions

• Representing 3-D Shapes as 2-D Shapes

Multiples and Factors �� 80 • Multiples • Factors

Representing 3-D Shapes �� 142

• Understanding Perimeter and Area

• Reading Time • Time and Events

• Counting Money • More on Money

• Organising Data • Pictographs • Bar Graphs • Pie Charts

196

214 221

233 236

250 254 263 270

Answers �� 278

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Numbers up to 6-digits

Let's Recall We know that numbers are basic units of mathematics and are used for counting, measuring and comparing quantities. We have also previously learned that each digit in a number has a value, which we call the place value. Let us take a 2-digit number as an example! The digit on the left is at tens place while the digit on the right is at ones place. Similarly, in the number 2548, 5 represents 5 hundreds, or 500. However, in the number 56, 5 represents 5 tens, or 50. Therefore, even if a digit is same, its value always depends on where it is in the number. Th

4 Tens

8 Ones

2 Thousands 5 Hundreds

Let's Warm-up

Write the correct place value of the coloured numbers. 1

__________

548

__________

876

__________

4563

__________

9958

__________

I scored _________ out of 5.

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Numbers Beyond 9999 Ajay: Hello daddy, I found this letter, it is for you.

Real Life Connect

Father: Thank you. Ajay: The letter has your name and the address of our home. Father: Yes. Ajay: But, what is this big number 781005?

From: Ajay Shukla, 12, Hathipol,

Guwahati - 781005

Father: It is a special code, also called the postal code. This code helps in finding the exact location in a city. Ajay: Okay daddy. But it has 6-digits, and I find it hard to read!

All About 5-digit Numbers! To help Ajay understand 6-digit numbers, let's first learn about 5-digit numbers. We know that 9999 – nine thousand nine hundred ninety-nine is the greatest 4-digit number. Now, when we add 1 to this, we get 10000. 9999 + 1 = 10000

Remember!

10000 is read as “Ten Thousand”. Let us learn more about 5-digit numbers!

10000 is the smallest 5-digit number. 99999 is the greatest 5-digit number.

Place Values and Expanded Form in 5-digit Numbers We know that a 4-digit number has 4 places on the place value chart - ones, tens, hundreds and thousands. The place on the left to the Thousands place is called the Ten Thousands place. Let us take a 5-digit number 13435. The place value chart for this number can be written as: TTh

Always remember that place value is the value of the digit in a number based on its position in the given number.

Did You Know? The number 4 is the only number with the same number of letters as its value in the English language.

The place value of “4” in 13435 is 4 × 100 = 400. 2

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Let us now try to write the number in the expanded form. We know that the expanded form of a number can be written by the help of place value. Notice the new Ten Thousands place! TTh (× 10000)

Th (× 1000)

H (× 100)

T (× 10)

O (× 1)

The expanded form is the sum of the place value of each of its digits. + 3 × 10

Hundreds

+ 5×1

Tens

  

Example 1

Thousands

4 × 100

  

Ten Thousands

  

+ 3 × 1000

  

1 × 10000

Ones

Write the place value of each place in the number “43276”. Also write the number in the expanded form. Let us find the place value using the chart for the number 43276. TTh (× 10000)

Th (× 1000)

H (× 100)

T (× 10)

O (× 1)

40000

3000

200

We can also write the number in an expanded form in the following way: 43276 = 4 × 10000 + 3 × 1000 + 2 × 100 + 7 × 10 + 6 × 1 Do It Together

Write the place value of each place in the number “54319”. Also write the number in the expanded form. Let us find the place value using the chart for the number 54319. TTh (× ________)

Th (× 1000)

H (× _______)

T (× 10)

O (× 1)

_________

4000

_______

We can also write the number in an expanded form as given below: 54319 = _______ × _______ + 4 × 1000 + 3 × _______ + 1 × 10 + 9 × 1

Chapter 1 • Numbers up to 6-digits

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Face Value in 5-digit Numbers The face value of a digit in a number is the numerical value of the digit itself. For example, in 13435, the face value of the digit 4 is simply 4. We can also say that the face value of the digit on the hundreds place in 13435 is 4. Similarly, the face value of the digit on the Ten Thousands place in 13435 is 1. Example 2

What is the face value of the number in the Ten Thousands place in 93421? We know that face value is the numerical value of the digit in a particular place. Let us write the place value chart for 93421: TTh

Think and Tell

Is the place value and face value of any digit in the ones place always the same?

The face value of the number in the Ten Thousands place is 9. Do It Together

What is the face value of the number in the Thousands place in 17034? Let us write the place value chart for 17034: TTh

_____

The face value of the number in the Thousands place is ____.

Representing 5-digit Numbers When we have really big numbers, it’s important to know where each digit belongs. We should be able to accurately read, write and understand large numbers, as well as understand the value of each digit within the number. “Periods” help us in doing that. As per the Indian convention, Ones, Tens and Hundreds are put together in one period. Similarly, Thousands and Ten Thousands are put together in one period. Let us understand this by the Place Value Chart. Thousands Period Ten Thousands (TTh)

Thousands (Th)

Ones Period Hundreds (H)

Tens (T)

Ones (O)

Thousands period consists of ten thousands and thousands. One’s period consists of hundreds, tens and ones places. 4

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We can therefore represent our 5-digit number 13435 as:

13,435

Represent the number “64819” using the correct period.

Example 3

Let us first represent the number in a place value chart. Thousands Period

Ones Period

TTh

We know that the Ones Period includes Ones, Tens, and Hundreds places. We therefore mark the period after the Hundreds place. The correct representation of the number is 64,819. Do It Together

Represent the number "13709" with the correct periods. Let us first represent the number in a place value chart. Thousands Period

Ones Period

TTh

______

Mark the period in the correct position:

1 3 7 0 9

5-digit Number Names Let us now learn how to read and write 5-digit numbers as a number name. Let us use the number 13435 to understand this better.

Step 1

Thousands Period

  

13,435   

Write the number using the correct period.

Ones Period

Step 2 We first read the numbers in the Thousands place together. So, for 13,435 the Thousands period will be read as “Thirteen Thousand”.

Step 3 The part of the number in the Ones Period can be read as we already know. So, for 13,435 it will be read as “Four Hundred Thirty-Five”. Chapter 1 • Numbers up to 6-digits

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Step 4

Error Alert!

Now read the full number. The number can be read as “Thirteen Thousand

Do not use ‘and’ while writing the number name. Thirteen Thousand and Four Hundred Thirty-Five  Thirteen Thousand Four Hundred Thirty-Five 

Four Hundred Thirty-Five”. Example 4

Name the number “26572” in words.

Let us first write the number in the place value chart. Thousands Period

Ones Period

Ten Thousands (TTh)

Thousands (Th)

Hundreds (H)

Tens (T)

Ones (O)

We can name the part of the number in the Thousands period first and then the rest of the number in the Ones period. The number 26,572 is written in words as: “Twenty-six thousand five hundred seventy-two” Do It Together

Write the number 20567 in words. Let us first write the number in the place value chart. Thousands Period

Ones Period

Ten Thousands (TTh)

Thousands (Th)

Hundreds (H)

Tens (T)

Ones (O)

______

Remember! The place value of 0 is always 0, it does not depend on the place it occupies.

The number in words is: __________________________________ five hundred __________________________________

Do It Yourself 1A 1

Rewrite the numbers using periods, and then write them in words. a 17372

b 43890

c 74065

d 80379

e 33510

Write the following in numerals. a Twelve thousand three hundred twenty-one b Thirty-four thousand six hundred c Seventy-eight thousand five d Fifty thousand ten

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Write the place value and the face value of the underlined digit. a 56938

b 65899

b 47,690

c 32,478

d 67,698

Which of the following numbers has the greatest value in the thousands place? a 45,687

d 89376

Which of the following numbers has 7 tens? a 36,789

c 25401

b 65,690

c 78,483

d 96,152

Write the following numbers in the short form using commas. a 40000 + 6000 + 300 + 20 + 2 b 50000 + 0 + 700 + 50 + 7 c 70000 + 3000 + 0 + 60 + 1 d 90000 + 6000 + 400 + 0 + 8

During the holiday season, 38437 people visited The National Park.

Identify the number which has 4 in tens place and 8 in thousands place. The digit in ones place is half

Write the number in words and in expanded form.

the sum of the digits in tens and thousands, and the digit in hundreds place is six less than the digit in ones place. Find the number.

All About 6-digit Numbers! Now, we have learnt about the 5-digit numbers, let us help Ajay with the 6-digit number - 781005.

We now know that 99999 – Ninety-nine thousand nine hundred ninety-nine is the greatest 5-digit number. When we add 1 to this, we get 100000. 99999 + 1 = 100000

From: Ajay Shukla, 12, Hathipol,

Guwahati - 781005

100000 is read as “One Lakh”. Let us learn more about 6-digit numbers!

Remember! 100000 is the smallest 6-digit number. 999999 is the greatest 6-digit number.

Chapter 1 • Numbers up to 6-digits

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Place Values and Expanded Form in 6-digit Numbers We know about the places in 5-digit numbers. In 6-digit numbers a place is added on the left. This new place is called the Lakhs place. Let us place “781005” in the place value chart. A new Lakhs column will be added. Lakh

TTh

Let us now try to write the number in the expanded form. We know that the expanded form of a number can be written by the help of place value. Notice the new Lakhs place! L (× 100000)

TTh (× 10000

Th (× 1000)

H (× 100)

T (× 10)

O (× 1)

The expanded form is the sum of the place value of each of its digits, as we already know for 4-digit numbers. Let us write the number 71,3435 in its expanded form. Hundreds

Tens

  

Example 5

Thousands

4 × 100 + 3 × 10 + 5 × 1

  

Lakhs Ten Thousands

  

+ 3 × 1000

  

7 × 100000 + 1 × 10000

Ones

Write the place value of each place in the number “801246”. Also write the number in the expanded form. Let us find the place value using the chart for the number 801246. L (× 100000)

TTh (× 10000)

Th (× 1000)

H (× 100)

T (× 10)

O (× 1)

800000

1000

200

We can also write the number in an expanded form in the following way: 801246 = 8 × 100000 + 0 × 10000 + 1 × 1000 + 2 × 100 + 4 × 10 + 6 × 1 Do It Together

Write the place value of each place in the number “172909”. Also write the number in the expanded form. Let us find the place value using the chart for the number 172909. L (× _________)

TTh (× _________)

Th (× 1000)

H (× _________)

T (× 10)

O (× 1)

_________

2000

_________

We can also write the number in an expanded form in the following way: 172909 = _____ × _____ + _____ × _____ + 2 × 1000 + _____ × _____ + 0 × 10 + 9 × 1

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Face Value in 6-digit Numbers We learnt about the concept of face value in the previous section. It is defined as the numerical value of the digit on a particular place in a number. For example, the Pincode on Ajay’s letter was 781005. The face value of the Lakhs place is simply 7.

Error Alert! Never confuse Face Value with Place Value! For example, in 781005, the Face Value of the Lakhs place is 7 and the Place Value is 7 × 100000 = 700000.

Similarly, we say that the face value of the digit on the Ten Thousands place is 8. Example 6

What is the face value of the number in the lakhs place in 348673? We know that face value is the numerical value of the digit in a particular place. Let us write the place value chart for 348673: L

TTh

The face value of the number in the Lakhs place is 3. Do It Together

What is the face value of the number in the Thousands place in 800234? Let us write the place value chart for 800234: L

TTh

________

The face value of the number in the thousands place is _________.

Representing 6-digit Numbers Let us continue to learn about the “periods” convention for large numbers. We learnt that the Thousands period includes Ten Thousands and Thousands places. The Ones period includes the Hundreds, Tens and Ones places. For 6-digit numbers, the Lakhs place falls in the Lakhs Period. Lakhs Period Lakhs

Thousands Period Ten Thousands (TTh)

Ones Period

Thousands (Th)

Hundreds (H)

Tens (T)

Ones (O)

We can therefore represent our 6-digit number 781005 as:

Chapter 1 • Numbers up to 6-digits

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  

Lakhs Period

7,81,005

Thousands Period

Ones Period

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Example 7

Represent the number “342381” with the correct periods. Let us first represent the number in a place value chart. Lakhs Period

Thousands Period

Ones Period

TTh

We now know about the Ones, Thousands and Lakhs period. The correct representation of the number is 3,42,381. Do It Together

Write the number “905721” using the correct periods. Let us first write the number in a place value chart. Lakh Period

Thousands Period

Ones Period

TTh

________

Mark the period in the correct position:

9 0 5 7 2 1

6-digit Number Names Let us now learn how to read and write a 6-digit number as a number name. Let us use the number 723942 to understand this better. Step 2

Step 1 Write the number using the correct periods.

7,23,942

We read the numbers in the Lakhs period first. So for 7,23,942 the Lakhs period will be read as “Seven Lakh”.

Step 3

Step 4

The rest of the number, which is a 5-digit number

The Ones Period is read last.

can also be read as we have learnt in the previous section. First the Thousands Period.

In this case it will be “Nine Hundred Forty-Two”.

In this case it will be “Twenty-Three Thousand”.

Step 5 Now read the complete number. The number can be read as “Seven Lakh Twenty-Three Thousand Nine Hundred Forty-Two”.

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Example 8

Write the number “502102” in words. Let us first write the number in the place value chart. Lakhs Period

Thousands Period

Ones Period

Lakhs (L)

Ten Thousands (TTh)

Thousands (Th)

Hundreds (H)

Tens (T)

Ones (O)

We first read the Lakhs Period, then the Thousands Period, and then Ones Period at last. The number name is “Five lakh two thousand one hundred two”. Do It Together

Write the number “723421” in words. Let us first write the number in the place value chart. Lakhs Period

Thousands Period

Ones Period

Lakhs (L)

Ten Thousands (TTh)

Thousands (Th)

Hundreds (H)

Tens (T)

Ones (O)

______

The number 7,23,421 written in words is: Seven lakh ________________________________ four hundred ________________________________

Do It Yourself 1B 1

Rewrite the numbers using periods and write them in words.

Write the following in numerals.

a 197637

b 365021

c 632845

d 824137 a

a Four lakh eighteen thousand three hundred

b Six lakh twenty thousand

c Eight lakh five thousand two hundred sixty-four

d Seven lakh twenty thousand fifty

Write if true or false.

a The place value of the digit 5 in the number 2,05,649 is five hundred.

b In the number 3,42,658, the place value of the digit 3 is 30,000 × 20.

c The place value of the digit 5 in the number 5,49,853 is 4,90,214 more than 9,786.

Write the place value and the expanded form of the following numbers.

Write the following numbers in the short form using commas.

a 5,84,736

b 7,04,391

a 400000 + 10000 + 8000 + 200 + 20 + 2 c 700000 + 40000 + 9000 + 0 + 20 + 1

Chapter 1 • Numbers up to 6-digits

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c 3,70,943

d 9,85,401

b 500000 + 40000 + 0 + 100 + 40 + 7

d 900000 + 80000 + 2000 + 900 + 0 + 2

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A pencil can draw a line that is almost 184800 feet long. Find the place value for

The digits in the Tens and the Thousands place of a number are 3 and 9,

each of the digits and write the number in words.

respectively. The digit in the Ones and the Ten Thousands places are 3 and 4 more than the digit in Tens

place. If the digit in the Hundred place is three less than the digit in ten thousands place, find the number.

Comparing and Rounding off Numbers Real Life Connect

John and Tina are doing a research project on the prices of different cars and bikes in India. They have put together the following data table. BP

Jeep

Car 1

Breva

Vehicle Type

Bike

Car

Price (INR)

88,957

74,801

9,63,890

7,47,871

8,29,860

Vehicle Image

Tina: This is such an interesting exercise that we have done. John: We now have so much information about these cars and bikes.

Comparing and Ordering Numbers What if Tina and John want to find the most expensive bike in their research? Since there are only two bikes, they will have to compare the prices.

Comparing Numbers Here, we are comparing the prices of the two bikes. So, is 88,957 greater than 74,801? We already know how to compare 4-digit numbers. Let us learn how to compare 5-digit and 6-digit numbers. Step 1 Check the number of digits in both the numbers. The number with the greater number of digits is greater.

Remember! The number with more number of digits is always greater.

Both numbers have the same number of digits. So, we can’t decide yet. 12

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Step 2 We already know that if both the number of digits are

Remember!

the same, we check the first digit from the left. The one

If the number of digits in two numbers are the same, we check the digits in the leftmost place. 1,82,848 AND 1,81,812

with the bigger digit, is the bigger number. In this case, is it true? 88,957 and 74,801 8>7

So, BP is the expensive bike because 88,957 > 74,801. Example 9

Compare 32,751 and 1,52,631. Step 1 Check the number of digits in both the numbers. One is a 5-digit number and the other is a 6-digit number. The 6-digit number is always greater than a 5-digit number. So, 1,52,631 > 32,751.

Example 10

Compare 47,213 and 43,507. Check the number of digits in both the numbers. Step 1

Step 2

Both the numbers are 5-digit numbers.

Check the first digit from the left.

So, cannot decide yet.

4 7, 2 1 3

4 3, 5 0 7

Both the first digits are the same. Still cannot decide!

Step 3 Check the next digit from the left. 4 7, 2 1 3

4 3, 5 0 7

7 is greater than 3. So, 47,213 > 43,507. Do It Together

Compare 7,53,278 and 7,43,271. Step 1 The digits in the two numbers are the same. Both are 6-digit numbers. So we cannot decide yet.

Error Alert!

Never compare the leftmost digits if the number of digits in the number are not the same. Similarly, never move to the next place if the digits in the leftmost places are not the same.

Step 2 We check the digits in the __________ place. The digits in the ________ place are the same. So we cannot decide yet.

Step 3 We now move to the next place, which is the _________ place. Here, ____ > ____ Therefore, 7,53,278 ______ 7,43,271.

Chapter 1 • Numbers up to 6-digits

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Ordering Numbers Now, what if John and Tina want to sort the cars in order of their prices? In this case, we will have to order the price of the cars. Let us look at the table again. BP

Jeep

Car 1

Breva

Vehicle Type

Bike

Car

Price (INR)

88,957

74,801

9,63,890

7,47,871

8,29,860

Vehicle Image

To order the prices of the three cars—Jeep, Car 1 and Breva, we will have to compare by taking two at a time. Step 1 Let us compare prices of Jeep and Car 1. Is 9,63,890 less than or greater than 7,47,871? Both prices have the same number of digits. So we will compare the leftmost digits. Since the digit 9 in 9,63,890 is greater than digit 7 in 7,47,871; 9,63,890 > 7,47,871. Thus, the price of Jeep is more than that of Car 1. Price of Jeep > Price of Car 1

Step 2 Let us compare prices of Jeep and Breva. Is 9,63,890 less than or greater than 8,29,860? Both prices have the same number of digits. So we will compare the leftmost digits. Since 9 > 8, 9,63,890 > 8,29,860. Jeep > Breva

Step 3 Let us compare prices of Car 1 and Breva. Is 7,47,871 less than or greater than 8,29,860? Both prices have the same number of digits. So we will compare the leftmost digits. Therefore, 7,47,871 < 8,29,860. Price of Car 1 < Price of Breva The above steps tell us that: Price of Jeep > Price of Breva > Price of Car 1 This type of order where the values go from higher to lower from left to right is called descending order. Or Price of Car 1 < Price of Breva < Price of Jeep This type of order where the values go from lower to higher from left to right is called ascending order. 14

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Example 11

Arrange the following numbers in ascending and descending order. 72,510, 51,068, 94,321, 1,86,344 We first need to check the sizes of the numbers. Step 1 Check whether the numbers have the same digits. So, we keep placing the higher numbers on the right. 1,86,344 is the largest number because it has the highest number of digits.

Step 2 For the other three numbers with the same number of digits, we compare the leftmost digits. 72510

51068

94321

Step 3 We conclude that numbers can be arranged as: 51,068 < 72,510 < 94,321 < 1,86,344 This is the ascending order.

Step 4 For descending order, we could also place the numbers in the following way: 1,86,344 > 94,321 > 72,510 > 51,068 This is the descending order. Do It Together

Arrange the following number in ascending and descending order. 3,68,109, 75,045, 1,76,902, 60,438 Two numbers here are 6-digit numbers and two numbers are 5-digit numbers.

3,68,109 ____ 1,76,902 and 75,045 ____ 60,438 So, the numbers can be sequenced as ____________________________________________________. The ascending order is: ___________________________________________________________________. The descending order is: _________________________________________________________________.

Do It Yourself 1C 1

Compare the following numbers using the symbols >, <, = . a 24,614 and 41,700

b 50,092 and 51,320

c 72,184 and 72,157

d 3,15,720 and 4,13,265

e 8,74,126 and 8,24,510

4,35,071 and 4,35,261

g 3,54,680 and 3,54,680 Chapter 1 • Numbers up to 6-digits

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Arrange the following numbers in ascending order. a 40,765, 14,390, 79,430, 37,935

b 66,773, 27,880, 59,573, 32,860

c 8,64,853, 4,67,943, 4,88,392, 8,33,067

d 7,48,546, 7,59,404, 7,20,157, 7,06,583

The given table shows the deepest points of the oceans of the world in feet (ft.). Which is the deepest ocean? Arrange the depths in descending order. Atlantic Ocean

Arctic Ocean

Indian Ocean

Pacific Ocean

Southern Ocean

27,840

18,264

23,810

36,161

23,740

Supriya ate 15,248 calories this week and her brother ate 18,396 calories. Who ate more calories this

Students voted on their favourite ice-cream flavours. The results are shown in the table below.

week?

Flavour

Number of Students

Vanilla

450

Chocolate

230

Strawberry

684

Mango

420

Order the flavour from the most favourite to the least favourite flavour.

Word Problem 1

Anna wants to buy some books for her library. Her father has given her `11,200. The

books cost `11,700. Does she have sufficient money to buy the books?

Forming Numbers Tina: Do you want to play a game, John? John: Sure! Which game? Tina: From our data, my favourite bike is the Hero Splendor. It is so attractive! John: Yes, and its price is ₹74,801. Tina: For our game, let’s try to form numbers using digits of this price. It will be fun! John: Ok, we have to use all digits exactly once - so that there is no repetition of digits! And we will form 5-digit numbers.

74801

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Let us jumble these digits and list some of the numbers that we can form!

Example 12

48,017

78,401

17,480

Think and Tell

48,071

78,104

17,408

Form at least 5 new numbers

48,107

78,140

17,804

that are not listed here.

Which is the largest number that Tina and John could have formed using the digits 7, 4, 8, 0, and 1 We need the highest digits on the leftmost places to form bigger numbers. Then let us try to arrange the digits in the order of big to small going from left to right.

87410 So the largest number that we can form using these digits is 87,410. Example 13

Which is the smallest 5-digit number that Tina and John could have formed using the digits 7, 4, 8, 0, and 1 We need the lowest digits on the leftmost places to form smaller numbers. Then let us try to arrange the digits in the order of big to small going from left to right. We cannot have a 0 in the leftmost place because then it becomes a 4-digit number and Tina and John want to form a 5-digit number.

10478 So the smallest number that we can form using these digits is 10,478. Now, what if Tina and John wanted to form a 6-digit number using the same digit such that only one digit could repeat?

Remember! We cannot use “0” in the Ten Thousands place, because then the number will become a 4-digit number.

Let us try to form numbers!

74801 Repeating the digit 7

Repeating the digit 4

Repeating the digit 0

7,74,810

4,74,801

7,48,001

7,48,710

4,47,810

7,04,801

7,84,701

4,87,104

7,00,148

Think and Tell Form at least 5 more new numbers that are not listed here.

Which is the largest 6-digit number that Tina and John could have formed such that only one digit repeats? We need the highest digits on the leftmost places to form bigger numbers. So, 8 should repeat.

887410

So, the largest number that we can form in this way is 8,87,410. Chapter 1 • Numbers up to 6-digits

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Which is the smallest number that Tina and John could have formed using the digits?

Think and Tell

We need the lowest digit in the leftmost place to form the smallest number. As we know, we cannot have a 0 in the leftmost place. But being the lowest, 0 should repeat!

100478

What are the largest and the smallest 6-digit numbers possible if any digit could repeat any number of times?

So, the smallest number that we can form using these digits is 1,00,478. Do It Together

Form the largest and the smallest numbers using the digits 9, 4, 1, 5, 6 in the following cases: 1

5-digit numbers with no repetition of digits.

6-digit numbers with exactly 1 repeating digit.

Part 1: Forming 5-digit numbers first, with no repeating digits. We need the highest digit in the leftmost place to form bigger numbers.

9 __ __ __ 1 The highest 5-digit number that can be formed is ______________. We need the smallest digit in the leftmost place to form smaller numbers.

1 __ __ __ 9 The lowest 5-digit number that can be formed is ______________. Part 2: Let us form 6-digit numbers first, with 1 repeating digit. We need the highest digit in the leftmost place to form bigger numbers. So ___________ should repeat!

9 __ __ __ __ 1 The highest 6-digit number that can be formed is ______________. We need the smallest digit in the leftmost place to form smaller numbers. So, ________ should repeat!

1 __ __ __ __ 9 The lowest 6-digit number that can be formed is ______________.

Did You Know? A number that reads the same when read forwards and backwards is called a Palindrome. 11, 121, 333, 4554, 78987, 876678 are all examples of Palindromes. Can you think of any?

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Do It Yourself 1D 1

Form the smallest and the greatest numbers using the following digits without repetition. a 4, 2, 7, 6, 5

b 6, 1, 3, 7, 8

c 5, 0, 2, 1, 7, 4

d 8, 6, 2, 5, 9

e 2, 3, 9, 8, 0, 1

Write the smallest and the greatest 6-digit number by repeating exactly 1 digit. a 2, 1, 7, 4, 9

b 3, 8, 5, 0, 1

c 6, 9, 1, 2, 7

d 8, 1, 0, 9, 7

e 4, 9, 1, 2, 0

List four 6-digit numbers that start and end with the number 5 and read the same, forward and

Form the smallest and the greatest 5-digit numbers using the digits 6, 2, 0, 8 such that exactly one

Raghav arranged the digits 8, 6, 3, 5, 7, and 1 in ascending order, using each digit only once, to form

backward.

digit is allowed to repeat. Also, find the difference of the numbers formed.

a number. At the same time, Ishaan also arranged the same digits in descending order to form a number. Whose number is the largest?

Rounding off Numbers

Tina: John, we now have information about cars and bikes. We should start thinking about how will we present in class. John: Yes. When we present, telling the prices in big number names would be difficult. Tina: Y es! “Eighty-eight thousand nine hundred fifty-seven” (88,957) is a very long phrase! John: And imagine, such long prices for each vehicle. Rounding off the numbers could greatly help Tina and John in their presentation. Rounded off numbers can be communicated better and understood easily. We already know how to round off numbers, let us recap!

Rounding to the Nearest 10 How can we round off 83 to the nearest 10? 83 is between 80 and 90, but closer to 80.

Remember! 80 81 82 83 84 85 86 87 88 89 89 Therefore, 83 will be rounded off to 80. Now, how will Tina and John round off 88,957 to the nearest 10? Chapter 1 • Numbers up to 6-digits

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If the number is exactly in between, it is rounded off to the higher ten.

9/23/2023 3:31:02 PM

We apply the same concept. 88,957 is between 88,950 and 88,960, but it is closer to 88,960. So, 88,957 can be rounded off to 88,960. So, upon rounding off to the nearest 10, it can be read as “Eighty-eight thousand nine hundred sixty”.

Rounding off to the Nearest 100 How can we round off 271 to the nearest 100? 271 is between 200 and 300, but closer to 300.

271

200 210 220 230 240 250 260 270 280 290 300 Therefore, 271 will be rounded off to 300. Now, how will Tina and John round off 88,957 to the nearest 100? 88,957 is between 88,900 and 89,000, but it is closer to 89,000. So, 88,957 can be rounded off to 89,000. So, upon rounding off to the nearest 100, it can be read as “Eighty-nine thousand”.

Rounding off to the Nearest 1000 Now, let us learn how to round numbers off to the nearest 1000. How shall we round 7,842 to the nearest 1000? 7,842 is between 7,000 and 8,000. It is closer to 8000. So, 7,842 can be rounded off to 8000.

7000

7100

7200

7300

7400

7842 7500

7600

7700

7800

7900

8000

Now, how will Tina and John round off 88,957 to the nearest 1000? 88,957 is between 88,000 and 89,000, but it is closer to 89,000. So, 88,957 can be rounded off to 89,000. So, upon rounding off to the nearest 100, it can be read as “Eighty-nine thousand”. Example 14

Round off 63,241 to the nearest 100 and nearest 1000. Rounding off to the nearest 100. 63,241 is between 63,200 and 63,300, but is closer to 63,200.

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So, 63,241 can be rounded off to 63,200. 63241 63200

63250

63300

Rounding off to nearest 1000. 63,241 is between 63,000 and 64,000, but is closer to 63,000. So, 63,241 can be rounded off to 63,000. 63241 63,000 Do It Together

63,500

64,000

Round off 90135 to the nearest 100 and 1000. Rounding off to the nearest 100. 90,135 is between 90,100 and ________, but is closer to ________. So, 90,135 can be rounded off to ________. Rounding off to the nearest 1000. 90,135 is between ________ and 91,000, but is closer to ________. So, 90,135 can be rounded off to ________.

Do It Yourself 1E 1

Round off the following numbers to the nearest 10. a 134

d 1468

e 47121

b 1653

c 7610

d 2447

e 23492

d 87301

e 90123

Round off the following numbers to the nearest 1000. a 1653

c 161

Round off the following numbers to the nearest 100. a 174

b 569

b 6573

c 34784

Ramesh, the landscaper, is being contacted to replicate the garden in a monument. According to the records of the monument, there are 23,912 plants in the whole complex. How many approximate

number of plant saplings should Ramesh order, assuming some plants always go bad? (Hint: you can round off to the nearest 1000s.) 5

Ria and Sia went to the park. They spent 92 minutes there. About how long were they at the park?

Chapter 1 • Numbers up to 6-digits

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Points to Remember •

The place value table is divided into groups called periods.

•

5-digit numbers have 2 periods - Thousands Period and Ones Period.

•

6-digit numbers have 3 periods - Lakhs Period, Thousands Period and Ones Period.

•

Face value is the numerical value of the digit in a particular place in a number.

•

Numbers arranged from the smallest to the largest are said to be in an ascending order.

•

Numbers arranged from the largest to the smallest are said to be in a descending order.

• Rounding numbers is helpful when we need an estimate and when we want to convey numbers in an easier way.

Math Lab Place Value Scavenger Hunt: Materials Required: Newspapers, Magazines, or the Internet Setting: Groups of 4 1

Divide the entire class into groups of 4.

Each group can be dedicated a particular category like City population, State population,

Each group collects the data on their respective category from newspapers, magazines, or

Each groups’ data must include at least 5 numbers in their category.

Each group discusses and deduces the following for each number in their data:

Followers of celebrities, Number of speakers of a language, and Car and Bike prices. the internet.

Place value and face value of each digit. Correct number representation. Correctly written number names. Correctly order the numbers in ascending and descending order Round off the numbers to the nearest 10s, 100s and 1000s.

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Chapter Checkup 1

Rewrite the numbers using periods and write them in words. a

456321

970540

Forty-eight thousand three hundred twenty-one

One lakh thirty-four thousand six hundred

Seventy-eight thousand six hundred ten

Nine lakh ten thousand forty-five

806399

Write the place value and the expanded form of the following numbers. 48361

87109

458320

692042

937676

Write the place value and the face value of the underlined digit. a

Write the following in numerals.

38237

569385

165899

254010

Spot the error and fix it. a

6,85,486 = 6 × 1,00,000 + 85 × 10,000 + 4 × 100 + 8 × 10 + 6 × 1

2,13,548 = 2,00,000 + 1,000 + 30,000 + 50 + 400 + 8

Write the following numbers in the short form using commas. a

80000 + 2000 + 300 + 20 + 2

300000 + 50000 + 0 + 700 + 50 + 7

200000 + 70000 + 3000 + 0 + 60 + 1

700000 + 90000 + 6000 + 400 + 0 + 8

Compare the following numbers using the symbols >, <, = . a

64,614 and 51,700

85,592 and 81,320

48,184 and 48,157

2,18,720 and 3,14,265

7,84,126 and 7,84,510

4,35,893 and 4,35,893

Arrange the following numbers in ascending and descending order. a

46,773; 37,880; 69,573; 42,860

25,409; 28,540; 23,752; 24,431

64,393; 64,520; 64,905; 64,012

8,26,750; 3,58,801; 3,95,701; 93,854

7,13,725; 7,58,645; 7,89,371; 7,26,890

5,87,206; 5,88,205; 5,80,723; 5,81,945

Round off the numbers to the nearest 10, 100, and 1000. a

342

6126

39,887

53,475

10 Rohan says, “On rounding off 4,85,345 to the nearest 1000, we get 4,85,300” Is he correct? Why?

Chapter 1 • Numbers up to 6-digits

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Word Problems 1

The cost of sarees at a store are listed below. Answer the questions. Saree 1

Saree 2

Saree 3

Saree 4

`25,907

`97,463

`54,768

`25,879

Revanth wanted to buy a saree that costs the least. Arrange the sarees in

What is the approximate cost of each saree? Round off to the nearest 1000.

ascending order of their cost.

The table shows the top four online languages: Language

Online Users

Chinese

1,56,810

English

3,67,185

Japanese

85,674

Spanish

78,342

Which language is used the most?

Which language is used the least?

Write the order of the language from the least to the one used the most.

(Hint: Arrange in ascending order.)

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Addition and Subtraction

Let's Recall We perform addition and subtraction in our everyday life! Suppose, there are 25 girls and 20 boys in your class. We know that, to find the total number of students in your class, you will have to add the number of girls and the number of boys. Therefore, the total number of students in your class will be 25 + 20 = 45, i.e. addition.

Addition helps us to put items together or total two or more numbers. Now, let us say there were 5 students absent on Monday, then the number of students that were present on Monday was 45 – 5 = 40. Hence, subtraction helps us when we want to take away from a particular group.

Let's Warm-up 1

Fill in the blanks with ‘+’ and ‘–‘. a

455 _____ 5 = 450

987 _____ 13 = 1000

Match the following. a

692 – 30

436

355 + 345

662

556 – 120

700

I scored _________ out of 5.

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Addition of Numbers Beyond 999 Real Life Connect

Diya and her mom are going to visit her grandmother in Dehradun during the summer holidays. 3588

Diya: Mom! I am so excited to see grandmother. I miss her so much! Have we booked our tickets?

xoxoxoxoxoxoxoxoxoxoxox xoxoxoxoxoxoxoxoxoxoxox xoxoxoxoxoxox

Mom: Yes, Diya! They are on the table. You can see them, but please don’t lose them.

15/04/2025

`1462

4562

Diya: Mom, the ticket from Delhi to Dehradun costs ₹1462 and the return ticket costs ₹1325.

xoxoxoxoxoxoxoxoxoxoxox xoxoxoxoxoxoxoxoxoxoxox xoxoxoxoxoxox

01/05/2025

`1325

Adding 4-digit and 5-digit Numbers If Diya wants to find the total cost of travel, how can she do that? She can do that by adding the two numbers! We already know how two numbers are added. Let us see the steps together.

Simple Vertical Addition Math Connect

Step 1

Step 2

Let us first arrange the numbers

Add the ones: 2 ones + 5 ones = 7 ones

vertically in correct place.

Step 4

Step 3 Add the tens: 6 tens + 2 tens = 8 tens

Add the hundreds:

4 hundreds + 3 hundreds = 7 hundreds

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Step 5 Add the thousands: 1 thousands + 1 thousands = 2 thousands

Addend

Sum

Remember! The sum of two numbers can never be smaller than the numbers.

Properties of Addition The order in which the numbers or addends are added does not change the sum. Changing the grouping of numbers when adding more than 2 numbers does not change the sum. Adding zero to any number does not change the value of that number. The sum of a number and 0 is the number itself.

The sum of 1462 and 1325 is 2787. So, the cost of the entire journey is `2787! Now, what changes when we try to add 5-digit numbers? The process of addition remains the same. Just a step for the Ten Thousands place is added. Let us see this with an example: Example 1

Find the sum of 83,471 and 12,304. 83,471 and 12,304 are 5-digit numbers. 5-digit numbers also have the ten-thousands place. Therefore, while adding 5-digit numbers:

Think and Tell Are there any ten-thousands in a 4-digit number?

• Add the ones • Add the tens • Add the hundreds • Add the thousands • Add the ten-thousands

TTh

So, 83,471 + 12304 = 95775. Chapter 2 • Addition and Subtraction

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Do It Together

Add 65,234 and 2,345.

Error Alert!

TTh

DO NOT arrange the digits of the 4-digit number incorrectly when adding it with a 5-digit number.

So, 65,234 + 2345 = ______________.

Vertical Addition with Regrouping We also know about “regrouping” - a case in which the sum of numbers in a place is more than 10. We regroup and carry over 10 to the next place. Let us add 1371 and 8459. Step 2

Step 1 Add the ones:

Add the tens:

• 1 one + 9 ones = 10 ones

• 1 ten (carried over) + 7 tens + 5 tens = 13 tens

• 10 ones will get regrouped as 1 ten + 0 ones

• 13 tens will get regrouped as 1 hundred + 3 tens

• Carry over 1 to the tens place.

• Carry over 1 to the hundreds place.

1 8

H 3 4

T 1

7 5

O 1

Th Carry

9 0

1 8

Step 3

Step 4

Add the hundreds:

Add the thousands:

• 1 hundred (carried over) + 3 hundreds + 4 hundreds = 8 hundreds Th

1 8

4 8

5 3

9 0

• 1 thousand + 8 thousands = 9 thousands

O 1

So, 1,371 + 8,459 = 9,830. 28

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Example 2

Find the sum of 13,431 and 56,718. Step 1: Add the ones Step 2: Add the tens

TTh

Step 3: Add the hundreds Step 4: Add the thousands

Step 5: Add the ten-thousands Step 6: Regroup and carry over wherever required

So, 13,431 + 56,718 = 70,149. Do It Together

Add 84,467 and 2893. TTh

So, 84,467 + 2893 = ______________.

Adding Numbers Horizontally Let us find the sum of two numbers by placing them horizontally. Let us add 6712 and 1235. Step 1: Add the ones. 2 ones + 5 ones = 7 ones Step 2: Add the tens. 1 ten + 3 tens = 4 tens Step 3: Add the hundreds. 7 hundreds + 2 hundreds = 9 hundreds Step 4: Add the thousands. 6 thousands + 1 thousand = 7 thousands Th

So, 6712 + 1235 = 7947.

Example 3

Find the sum of 5810 and 4142. Th

So, 5810 + 4142 = 9952. Chapter 2 • Addition and Subtraction

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Do It Together

Add 12,344 and 1115. TTh

TTh

So, 12,344 + 1115 = _____________.

Story Sums In the school library, there are 1219 fiction books and 1567 non-fiction books. How many books are there in total? Let us start finding the total number of books step by step. What do we know?

Did You Know? Rinne Tsujikubo of Japan broke the Guinness world record for fastest mental arithmetic on January 17, 2023 by correctly adding 15 sets of three-digit numbers in 1.62 seconds.

The total number of fiction books = 1219 The total number of non-fiction books = 1567 What do we need to find? The total number of books in the library = The total number of fiction books + The total number of non-fiction books

= 1219 + 1567

9 6

Solve to find the answer. So, the total number of books in the library is 2786. Example 4

The city NGO organised a two-day donation drive. On the first day of the drive, 1366 clothes were collected. On the second day of the drive, 1000 clothes were collected. How many clothes were collected in total? What do we know?

Remember! When we add 1000 to a 4-digit number, only the digit in the thousands place changes.

Number of clothes collected during the first day of the drive = 1366 Number of clothes collected during the second day of the drive = 1000 What do we need to find? Total number of clothes collected = 1366 + 1000 30

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Solve to find the answer. The total number of clothes that were collected during the two-day donation drive is 2366.

6 6

A chocolate factory produced 3456 chocolates in the morning and 4257 chocolates in the evening. How many chocolates did they produce in a day?

Do It Together

What do we know? What do we need to find? Solve to find the answer. So, the factory produced __________ chocolates during the day. Th

H 4

O 6

Do It Yourself 2A Fill in the sum.

+ 2

Find the sum of the following numbers horizontally: a 1239 + 3740

b 23,423 + 1231

c 1232 + 12,361

d 51,773 + 40,126

Find the sum of the given numbers. a 5684 + 1234

b 10,000 + 34,789

c 18,720 + 12,003

d 36,734 + 4999

e 2467 + 16,398

Chapter 2 • Addition and Subtraction

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2678 + 12,468 31

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A number exceeds 56,122 by 3411. What is that number?

Each shape represents a number as given. = 17,803

Find: a

= 34,618 b

= 28,671

= 11,190 6

Jim and Steve found the sum of 23,166 and 6189. Who added the numbers correctly and why? Jim 2

+ 2 7

Steve

6 6

Complete the addition pyramid. (Hint: Add two numbers to produce the third number. For example: 1032 + 1340 = 2372)

2372 1032

2790 1340

1008

1782

Word Problems 1

1000 animals were rescued by the national animal rescue shelter last year. This year, they

A car company produced 45,821 cars in 2021. It produced 1208 more cars in 2022 than

There were 81,232 wild buffaloes in a national park. During the year, 7342 calves were

rescued 1145 more animals than the last year. How many animals were rescued this year? in 2021. How many cars did it produce in 2022?

born. What was the population of buffaloes in the national park by the end of the year?

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Subtraction of Numbers Beyond 999 Real Life Connect

Raghu is a farmer and grows seasonal fruits. Every year, he has an option to sell his produce in the fruit market or sell it to a fruit juice company producer. This year Raghu’s farm produced 2468 kilograms of mangoes during the summer season. Raghu sold 1247 kilograms to a supermarket.

Subtracting 4-digit and 5-digit Numbers We saw that Raghu has sold his produce in the supermarket. Does he have any stock left? To find the part of his stock left, we will have to subtract the amount of mangoes sold from the total mangoes produced.

Simple Vertical Subtraction Let us subtract 1247 from 2468. Step 1

Step 2

To subtract 4-digit numbers, first arrange the

Subtract the ones: 8 ones – 7 ones = 1 one

numbers vertically in correct place.

–

Step 3

Step 4

Subtract the tens: 6 tens – 4 tens = 2 tens

Subtract the hundreds:

–

6 2

8 1

Chapter 2 • Addition and Subtraction

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4 hundreds – 2 hundreds = 2 hundreds

–

8 1

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Step 5 Subtract the thousands: 2 thousands – 1 thousand = 1 thousand

–

2 1

4 2

6 2

Minuend Subtrahend

Difference

Remember! • Minuend – Subtrahend = Difference • Subtrahend + Difference = Minuend

So, 2468 – 1247 = 1221. Let us check the answer using addition! We found that: 2468 – 1247 = 1221 Let us find 1221 + 1247.

So, 1221 + 1247 = 2468.

1 8

We can say that the sum of 1221 and 1247 will give 2468. Example 5

Find the difference of 91,897 and 41,290. 91,897 and 41,290 are 5-digit numbers. 5-digit numbers also have the ten-thousands place. Therefore, while subtracting 5-digit numbers: • Write the larger number as the minuend • Write the smaller number as the subtrahend • Subtract the ones

TTh

• Subtract the tens

–

• Subtract the hundreds • Subtract the thousands • Subtract the ten-thousands Do It Together

So, 91,897 – 41,290 = 50,607.

Find the difference of 75,234 and 3121.

–

TTh

So, 75,234 – 3121 = _________________.

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Vertical Subtraction with Regrouping What if we had to subtract 1978 from 2586? Step 1 Write the subtraction statement vertically.

–

Step 2 Subtract the ones: • We cannot take away 8 ones from 6 ones!

Minuend

Subtrahend Difference

We will regroup the ones.

• We will borrow from the tens place. • So, 8 tens becomes 7 tens and 6 ones becomes 16 ones.

• Now, subtract 16 ones – 8 ones. Th

Step 3 Subtract the tens:

–

• 7 tens – 7 tens = 0 tens Th

–

8 8

Step 4 Subtract the hundreds: • We cannot subtract 9 hundreds from 5 hundreds! • We will regroup the hundreds and will borrow 1 thousand from the thousands place.

• 2 thousands becomes 1 thousand and 5 hundreds becomes 15 hundreds.

• Now, subtract 15 hundreds – 9 hundreds.

Step 5 Subtract the thousands:

• 1 thousands – 1 thousand = 0 thousand

–

So, 2586 – 1978 = 608.

Chapter 2 • Addition and Subtraction

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–

Remember! If we don’t have enough in a certain place to subtract, we can borrow from the next place on the left.

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Example 6

Find the difference of 87,821 and 45,586. Step 1: Subtract the ones

TTh

Step 2: Subtract the tens Step 3: Subtract the hundreds Step 4: Subtract the thousands

–

Step 5: Subtract the ten-thousands

So, 87,821 – 45,586 = 42,235. Do It Together

Carry

Find the difference of 78,131 and 9993. TTh

–

O Carry

Error Alert! DO NOT FORGET to borrow from the next higher place on the left.

So, 78,131 – 9993 = ________________.

Story Sums ani had ₹17,845 in her bank account. She withdrew ₹3230 for shopping. How much R money is left in her account? Let us find the amount left in her account step by step. What do we know?

Amount Rani had in her bank account = ₹17,845 Amount Rani withdrew for shopping = ₹3230 What do we have to find?

Amount left in Rani’s bank account = Amount Rani had in her bank account – Amount Rani withdrew for shopping Solve to find the answer. = ₹17,845 – ₹3230 TTh

– 1

Rani is left with ₹14,615 in her account. 36

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Check your answer. To check the answer, add the amount left with Rani and the amount she spent on shopping `14,615 + `3,230 = `17,845

TTh 1

H 6

`17,845 is the amount that Rani had in her bank account. So, the answer is correct. Example 7

The school stationery store had 3240 notebooks. During the academic year, students bought 2890 notebooks. How many notebooks are remaining in the store? What do we know? Number of notebooks in the stationery store = 3240 Number of notebooks sold = 2890 What do we need to find? Number of notebooks left = 3240 – 2890 Solve to find the answer.

–

So, the school stationery store has 350 notebooks remaining. Check your answer. Do It Together

Sunaina and her sister are collecting stamps. Sunaina collected 8455 stamps and her sister collected 6712 stamps. How many more stamps does Sunaina have than her sister? What do we know? Stamps with Sunaina = _______________ Stamps with her sister = _______________ What do we need to find? Total number of stamps with Sunaina and her sister Solve to find the answer. 8455 – 6712 So, the total stamps with Sunaina and her sister are ______________. Check your answer. Chapter 2 • Addition and Subtraction

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Do It Yourself 2B 1

Solve to find the answer. a

Subtract the given numbers. a 5312 – 1101

b 34,789 – 10,000

c 28,324 – 19,812

d 38,004 – 19,999

e 29,234 – 2898

How much does 52,032 exceed 8972 by?

Each shape represents a number as given.

90,002 – 19,991

= 17,803 Find: = 34,618

–

= 28,671 b

–

= 11,190 5

What should be added to 13,456 to get 57,801?

What should be subtracted from 17,890 to get 1829?

Which number is 2335 less than 12,345?

Find the missing addend in 13,467 + ____________ = 76,512.

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Word Problems 1

Rahul thinks of a number X. Rahul’s friend, Amit thinks of

another number Y which is 1234 more than Rahul’s number. If Y is 61,020, then what is the number thought by Rahul?

A bulb manufacturing company produced 1672 more bulbs than the previous year. If this year the company produced 78,770 bulbs, how many bulbs were produced in the previous year?

Addition and Subtraction of Numbers Beyond 999 Real Life Connect

Mr Sameer owns a book publishing house. He displayed 3500 books in a two-day book fair. On the first day of the fair, 1890 books were sold. On the second day of the fair, 1255 books were sold. I still have some books left unsold that I need to take back to the shop. Let us find out how many!

Addition and Subtraction Together Simplify Let us first find the total number of books sold in the two days. It is: 1890 + 1255 = 3145

We need to subtract this sum from 3500 to find the books left. 3500 – 3145 = 355 So, 355 books will have to be carried back to the book store. Chapter 2 • Addition and Subtraction

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–

3 0

0 5

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Example 8

Simplify. 4321 – 788 – 621 We can simplify this expression by performing the subtraction operations in the given order. 4321 – 788 = 3533 3533 – 621 = 2912 So, 4321 – 788 – 621 = 2912.

Do It Together

Simplify to find the answer. 46,798 + 1457 – 21,020 Let us find the sum of 46,798 + 1457 46,798 + 1457 = _____________ Let us now subtract 21,020 from the sum. _____________ – 21,020 = _____________ So, 46,798 + 1457 – 21,020 = _____________.

Story Sums A construction project requires 8976 bricks. 3412 bricks were already used. 2587 more bricks were delivered to the project site. How many bricks are still needed? Let us find the number of bricks still needed by performing the operations in order. What do we know? Number of bricks required for the construction project = 8976 Number of bricks already used for the project = 3412 Number of bricks delivered to the project site = 2587 What do we need to find? Number of bricks still needed = Number of bricks required for the construction project – Number of bricks already used for the project – Number of bricks delivered to the project site = 8976 – 3412 – 2587 Solve to find the answer. = 2977 So, 2977 bricks are still required for the project. 40

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Example 9

The city library has 7654 fiction books and 5231 nonfiction books. 2376 new fiction books were added and 3487 non-fiction books were borrowed by people. How many books are there in total at present in the library?

Fiction and Non fiction

Number of fiction books in the library = 7654 Number of non-fiction books in the library = 5231 Number of new fiction books added to the library = 2376 So, the new total number of fiction books = 7654 + 2376 = 10,030 Number of non-fiction books borrowed by people = 3487 So, the number of non-fiction books left in the library = 5231 – 3487 = 1744 The number books in the library = The new total number of fiction books + The number of non-fiction books left in the library = 10,030 + 1744 = 11,774 Do It Together

A stadium has a capacity of 45,000 people. 25,765 men and 11,567 women were watching a match in the stadium. How many seats were left empty? What do we know? Total capacity of the stadium = ____________ Number of men watching the match = ____________ Number of women watching the match = ____________ What do we need to find? Solve to find the answer. Check your answer.

Do It Yourself 2C 1

Calculate the following. a 1299 + 8772 – 1001

b 1661 + 571 – 1006

c 15,679 – 1654 + 20,865

d 9283 – 7724 + 882

e 17,711 – 77,241 – 28,978

Chapter 2 • Addition and Subtraction

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67,002 – 16,621 – 70,918

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Each shape represents a number as given. = 17,803

= 28,671

= 34,618

= 11,190

Find the value: +

–

A farmer harvested 6543 kilograms of wheat and 4298 kilograms of rice. If he sold 3785 kilograms of

Sarah has `8752 in her bank account. She withdrew `3256 to buy a gift for her grandparents. Then,

Asifa had 12,345 stickers in her collection. She gave 3789 stickers to her friend. Then, she received a

wheat and 1932 kilograms of rice, how much is left with the farmer?

she deposited `9823 in her account. How much money does Sarah have in her bank account now?

gift of 6543 stickers from her cousin. How many stickers does Asifa have in her collection now?

Word Problem 1

A public library has 8236 books. A donation of 1534 books was made to the library. Then, a group of citizens made a generous donation of 9712 books to the library.

However, during the annual library painting, 672 books got damaged and had to be discarded. How many books does the library finally have?

Estimation Real Life Connect

The Taj Mahal has the most number of visitors during the weekends. Manager: How many tickets were sold on Saturday? Ticket Seller: On Saturday, around 44,799 tickets were sold. Manager: Okay! How many tickets were sold on Sunday?

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Ticket Seller: On Sunday, around 53,878 tickets were sold. Manager: This means there were about 99,000 visitors during the weekends. Ticket Seller: Yes, that’s correct.

Estimating the Sum How did the manager find the sum so quickly! He did so by rounding off the addends and then added them to find the estimated sum. Let us learn this further! Step 1 Rounding off the addends. Number of visitors on Saturday = 44,799

Rounded off to the nearest thousand

45,000

Number of visitors on Saturday = 53,878

Rounded off to the nearest thousand

54,000

Step 2 Add the rounded off numbers. 45,000 + 54,000 = 99,000 The estimated sum of 45,000 and 54,000 is 99,000. That is how he estimated the attendance so quickly!

Example 10

Estimate the sum of 33,134 and 13,894 by rounding off to the nearest ten-thousand. 33,134

Rounded off to the nearest thousand

30,000

13,894

Rounded off to the nearest thousand

10,000

Let us find the sum: 30,000 + 10,000 = 40,000 So, the estimated sum of 33,134 and 13,894 by rounding off to the nearest ten-thousand is 40,000. Do It Together

Think and Tell To round off a number to the nearest ten-thousand, we look at the digits at which place?

Estimate the sum of 13,567 and 28,082 by rounding off to the nearest thousand. Also find the actual sum. 13,567

Rounded off to the nearest thousand

28,082

___________ ______________

The estimated sum = ___________ + ___________ = ___________.

Chapter 2 • Addition and Subtraction

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Estimating the Difference Now, what if the Manager wanted to estimate the difference between the attendance on Saturday and Sunday? He would have to estimate the difference between 53,878 and 44,799. Let us follow the same process, but at the end find the difference. Step 1 Rounding off the numbers Number of visitors on Saturday = 44,799

Rounded off to the nearest thousand

45,000

Number of visitors on Saturday = 53,878

Rounded off to the nearest thousand

54,000

Step 2 Find the difference between numbers. 54,000 – 45,000 = 9,000 The estimated difference between 45,000 and 54,000 is 9,000. Example 11

Estimate the sum of 69,894 – 51,124 by rounding off to the nearest hundred. 69,894

Rounded off to the nearest thousand

51,124

69,900 51,100

Let us find the difference: 69,900 – 51,100 = 18,800 So, the estimated difference of 69,894 and 51,124 by rounding off to the nearest hundred is 18,800. Let us also find the actual difference. 69,894 – 51,124 = 18,770 So, the actual difference of 69,894 and 51,124 is 18,770. Do It Together

Estimate the difference of 78,111 – 21,991 by rounding off to the nearest thousand. Also, find the actual difference. 78,111 21,991

Rounded off to the nearest thousand

78,000

______________

The estimated difference = 78,000 – ______________ = ______________ The actual difference of 78,111 – 21,991 = ______________. 44

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Do It Yourself 2D 1

Find the estimated sum of 1245 and 2456 by rounding off to the nearest 100.

Find the estimated difference of 9013 and 3578 by rounding off to the nearest 100.

Find the estimated sum of 23,456 and 56,771 by rounding off to the nearest thousand.

Find the estimated difference of 97,761 and 87,112 by rounding off to the nearest thousand.

Which is more - the estimated sum of 72,374 and 16,773 or the estimated sum of 67,124 and 28,974? [Round off to the nearest ten thousand.]

Word Problems 1

A cupcake factory produced around 1346 cupcakes in the morning and 2313

cupcakes in the evening. About how many more cupcakes were produced in the

evening than in the morning? Estimate the difference by rounding off each to the nearest thousand. 2

Sarah walked 2347 steps in a day. About how many more steps should she walk to

complete 10,000 steps? Find the estimated number of steps by rounding off to the nearest thousand.

Points to Remember • Adding zero to any number does not change the value of that number. • •

The grouping of the numbers does not change the sum.

The order in which the numbers or addends are added does not change the sum.

• While subtracting, if we don’t have enough in a certain place, we can borrow from the next place on the left.

• We can verify the answer of a subtraction statement by adding the difference to the subtrahend. • When rounding off numbers to a given place, we look at the digits on the place right to the desired place.

Chapter 2 • Addition and Subtraction

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Math Lab Just About Right! Instructions: Divide the class into groups of 4. Materials Required: Number chits from 0 to 9. Write each number from 0 to 9 on number cards. Step 1: Each group picks any 2 random cards. Step 2: The group then decides a particular “Ten Thousand” number for the group. For

example, if the group picked “6” and “7”, the

group can choose either 67,000 or 76,000 as the group’s number.

Step 3: The group then finds 3 pairs of

numbers whose sum is almost equal to the group number. (round off to the nearest thousand) Step 4: The group then finds 3 pairs of numbers whose difference is almost equal to the group number. (round off to the nearest thousand)

Step 5: The group that completes these tasks first, wins!

Chapter Checkup 1

Solve. a

+ c

+ e

–

Find the sum of the largest 5-digit number and the smallest 4-digit number.

Find the sum of 8542 + 3721.

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Find the difference: 13,567 – 12,357.

What is the difference between the sum and difference of the number 45,998 and 1321?

Find the sum of 45,223, 12,678, and 16,941.

Compute: 81,654 – 53,217 – 2345.

Find the estimated sum of 5678 + 1665 when rounded off to the nearest thousand. Also, find the actual

Find the estimated difference sum of 1835 – 1346 when rounded off to the nearest thousand. Also,

sum.

find the actual difference.

10 The estimated sum of two numbers A and B when rounded off to the nearest hundred is 6600. Which of the following set of numbers could be A and B? a

2357 and 1235

5457 and 1108

3347 and 3567

11 The estimated difference of two numbers C and D when rounded off to the nearest thousand is 4000. Which of the following set of numbers could be C and D? a

6790 and 5667

7890 and 3889

8103 and 4899

Word Problems 1

The length of the river Ganga is 2520 Kilometres, while the length of Yamuna is

1376 Kilometers. Approximately what is the total length of these rivers combined? About how much longer is Ganga than Yamuna? (find by rounding off to the nearest 100)

The construction company ordered 8327 bricks for one project and 9912 bricks for

A grocery store sold 17,645 pounds of bananas and 24,891 pounds of apples last

another project. Estimate the total number of bricks ordered for both projects. month. Estimate the total weight of bananas and apples sold.

Chapter 2 • Addition and Subtraction

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Multiplication

Let's Recall Multiplication can also be understood as repeated addition of the same units or quantities.

For example, if there are 3 packets of 6 pencils each, we can use repeated addition to find the total number of pencils, as follows: 6 (packet 1) + 6 (packet 2) + 6 (packet 3) = 18 pencils. This is the same as saying 3 packets × 6 pencils per packet = 18 pencils. In the above example, the packets can be called as “groups” containing the same number of units.

Let's Warm-up Write True or False. 1

8 + 8 + 8 + 8 + 8 = 8 × 4

_______________

23 × 0 = 0

_______________

30 is the product of 10 and 5.

_______________

7 multiplied by 11 is 77.

_______________

here are 9 petals in each flower. We have 9 such flowers. T We therefore have 72 petals in all.

_______________

I scored _________ out of 5.

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Understanding Multiplication Real Life Connect

Sanju and his father play a newspaper game where he learns 2 new words each day. They have been playing this game for a week. Sanju is trying to find all the words he has learnt. He starts adding quickly. 2 + 2 + 2 + 2 + 2 + 2 + 2 = 14 He has learnt 14 words already! Hurray! Sanju struggles to find how many words will he learn in whole January. 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 +2 + 2 + 2 + 2……… This is too much! Sanju gets confused and asks his father to help. Sanju’s father helps him find the number of words using multiplication.

Multiplication by 1-digit Number We know that the number obtained from multiplication is the product. The number to be multiplied is multiplicand and the number by which we multiply is multiplier. 31 × multiplicand

multiplier

Multiplication Rules We can multiply two numbers in any order. The product always remains the same. On multiplying a number by 1, the product is always the number itself. On multiplying a number by 0, the product is always zero.

= 62 product

Horizontal Method We can find the product of two numbers by placing them horizontally next to each other. Let us multiply 123 and 3. Step 1

Step 2

Step 3

Multiply by ones.

Multiply by tens.

Multiply by hundreds.

Multiply 3 and 3 ones.

Multiply 3 and 2 tens.

123 × 3 = 369

123 × 3 = __ __9

123 × 3 = __69

The product of 123 and 3 is 369. Chapter 3 • Multiplication

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Example 1

Find the product of 2123 and 3. Step 1

Step 2

Multiply by ones.

Multiply by tens.

2123 × 3 = __ __ __ 9

2123 × 3 = __ __ 69

Step 3

Step 4

Multiply by hundreds.

Multiply by thousands.

2123 × 3 = __ 369

2123 × 3 = 6369

Remember!

The product of 2123 and 3 is 6369. Do It Together

On multiplication by 10, 20, 30… 90, there is always 0 in

Find the product of 4132 and 2.

the ones place.

Step 1

Step 2

Multiply by ones.

Multiply by tens.

4132 × 2 = __ __ __ 4

Step 3

Step 4

Multiply by hundreds.

Multiply by thousands.

4132 × 2 = __ __ __ 4

The product of 4132 and 2 is _________.

Multiplying by Expanding the Bigger Number We can also multiply large numbers by 1-digit numbers by expanding the bigger number. Multiply 170 and 5. Step 1

Step 2

Expand the bigger number.

Write the numbers.

170 = 100 + 70 + 0

100

Step 3

Step 4

Multiply the smaller number.

Add all the products.

Multiply 5 by 100, 70 and 0.

500 + 350 + 00 = 850

100

5 × 100 = 500

5 × 70 = 350

5×0=0

The product of 170 and 5 is 850. 50

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Example 2

Multiply 4287 and 2.

Remember!

Step 1

This method helps in multiplying numbers mentally.

Expand the bigger number. 4287 = 4000 + 200 + 80 + 7

Step 2

4000

200

4000

200

2 × 4000 = 8000

2 × 200 = 400

2 × 80 = 160

2 × 7 = 14

Place the numbers.

Step 3 Multiply the smaller number.

Multiply 2 by 4000, 200, 80 and 7.

Step 4

Did You Know?

Add all the products.

In 1980, Shakuntala Devi from India correctly multiplied two 13-digit numbers in 28 seconds.

8000 + 400 + 160 + 14 = 8574

The product of 4287 and 2 is 8574.

Do It Together

Multiply the given numbers by expanding the bigger number. 1

639 × 9

The expanded form of 639 = 600 + _______ + _______ 600 9

____

9 × 600 = 5400

The product is 5400 + _______ + _______ = __________ 2

3281 × 3 3000

200

The product is _______ + _______ + _______ = __________

Chapter 3 • Multiplication

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Vertical Method We can multiply using the vertical method of multiplication. Here, we place the multiplier vertically below the multiplicand. Let us multiply 365 and 2. Step 1

Step 2

Step 3

Multiply by ones.

Multiply by tens.

Multiply by hundreds.

2 × 5 ones = 10 ones

2 × 6 tens = 12 tens Add carried over 1 ten. 12 tens + 1 ten = 13 tens 13 tens = 1 hundred + 3 tens Carry over 1 hundred.

2 × 3 hundreds = 6 hundreds

10 ones = 1 ten + 0 one Carry over 1 ten.

Add carried over 1 hundred. 6 hundreds + 1 hundred = 7 hundreds

2 0

So, 365 × 2 = 730 Example 3

Find the product of 1134 and 8.

Step 1

Step 2

Multiply by ones.

Multiply by tens.

8 2

Step 3

Step 4

Multiply by hundreds.

Multiply by thousands.

Th 1

H 2

T 3

The product of 1134 and 8 is 9072.

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Do It Together

Complete the multiplication. 1

1 4

4 3

414 × 3 = __ __ __ 2

1 0

4306 × 2 = __ __ 12

8 4

1268 × 4 = __ __ __ 2

Do It Yourself 3A Horizontally multiply the given numbers.

a 233 × 2

b 501 × 6

c 622 × 4

d 3322 × 3

e 2001 × 7

4011 × 9

d 1002 × 2

e 2908 × 4

1249 × 8

Find the product by expanding the bigger number.

a 313 × 3

b 802 × 9

c 529 × 6

Complete the given multiplication.

3 3

3 4

5 3

7 8

Multiply the numbers using any method. Check if your answer is correct using another method.

a 947 × 8

b 734 × 9

c 4059 × 2

d 3274 × 3

e 7251 × 4

1149 × 7

Multiplication by a 2-digit Number Let us multiply 245 by 25. Step 3

Step 1

Step 2

Multiply by ones.

Multiply by tens.

Add the products.

Multiply 245 by 5 ones.

Multiply 245 by 2 tens or 20.

1225 + 4900 = 6125

245 × 5 = 1225

245 × 20 = 4900

1 4

2 0

2 0 2

5 0 5

The product of 245 and 25 is 6125.

Chapter 3 • Multiplication

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Example 4

Multiply 179 and 18. 1

1 1

4 7

7 1

9 8

3 9

Error Alert! In vertical multiplication, make sure that the digits are aligned as per their place value.

Step 1: Multiply by ones Step 2: Multiply by tens Step 3: Add the products

2 0

184 ×24 000 7360 36800 44160

The product of 179 and 18 is 3222.

Do It Together

Complete the multiplication. ×

4 3

8 9

184 24 736 3680 4416

Step 1: Multiply by ones Step 2: Multiply by tens Step 3: Add the products

The product of 1148 and 39 is __________.

Do It Yourself 3B 1

Multiply the given numbers. a 47 × 13 f

777 × 96

c 102 × 40

d 113 × 27

g 3290 × 36

h 2954 × 38

e 589 × 64

6492 × 59

8926 × 88

Complete the given multiplication. a × +

b 73 × 52

___

3 0

___

5 ___ 7 8

___

___ 7

8 ___ 8

× +

___

7 1

___

7 6

___

6 6

___ ___

___

4 6

9 ___

Given below is an incomplete multiplication. Ravi has to use the digits 4, 6, and 8 only once to complete it. What is highest product he can have? Help him find it! ×

___

0 ___

2 ___

Thinkand andTell Tell Think

Can the product of a 4-digit number and a 2-digit number be a 7-digit number? Hint: Check for the greatest numbers!

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Multiplication by a 3-digit Number Let us learn to multiply numbers by 3-digit numbers. Find the product of 144 and 273. Step 1

Step 2

Multiply by ones. 1 2

4 7 3

Multiply by tens. 4 3

Step 3

× 1 2

0 8

4 0 8

4 7 3 8 0

4 0

4 7 3 8

4 3 2 0

Step 4

Multiply by hundreds. 1 2

1 2

Add the products. 4 3

2 0 0

1 2

4 7

4 3

4 0

3 8 1

2 0 2

The product of 144 and 273 is 39312. Example 5

Multiply 432 and 317. ×

1 1

2 3

3 4 9 6

4 3

3 1

2 7

0 3 9

2 2 4

4 0

Step 1: Multiply 432 by 7 ones Step 2: Multiply 432 by 1 ten or 10 Step 3: Multiply 432 by 3 hundreds or 300 Step 4: Add the products

The product of 432 and 317 is 136944. Do It Together

Multiply 752 and 417. ×

7 4

5 1

2 7

Step 1: Multiply by 7 ones Step 2: Multiply by 1 ten Step 3: Multiply by 4 hundreds Step 4: Add the products

The product of 752 and 417 is __________________. Chapter 3 • Multiplication

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Do It Yourself 3C 1

Find the product.

a 348 × 300 = ____________

b 207 × 400 = ____________

c 460 × 239 = ____________

d 850 x 707 = ____________

e 302 × 119 = ____________

g 261 x 589 = ____________

h 665 x 648 = ____________

i 254 × 983 = ____________

Fill in the missing digit such that both the products are equal. 8

6 5

0 0

Complete the given multiplication.

Multiply 452 with 718. Show all the steps

775 × 405 = ____________

3 3

7 4

___

239 ×139 1

Word Problems Suresh has to print the posters for the upcoming debate competition at school. There are 8 packets of printing papers available and each packet has 145 papers. What is the total number of posters printed? Let us start finding the total number of posters step by step. What do we know?

Packets of printing paper = 8 Printing papers in each packet = 145

image

What do we find?

Total number of posters printed = Printing papers in each packet × Packets of printing paper = 145 × 8

Solve to find the answer. ×

5 8 0

1160 posters were printed. 56

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Example 6

The cost of a science book is ₹128. Ankit’s school orders 689 copies for all the students of grade 7. What is the cost of all the books? 6 Cost of 1 book = ₹128 1 × Number of books = 689

Total cost of the books = 128 × 689

The total cost of the books is ₹88,192. Do It Together

1 6 8

3 8

7 1

8 2

9 8

1 0 9

2 0 2

A free meal service provides food for 6315 people every day. For how many people will it p rovide food in 92 days? Number of people for whom food is provided in a 6 3 1 5 day = 6315 × Number of days = ___________

Total number of people for whom food is provided = 6315 × _________

The free meal service provides food for ____________ people.

Do It Yourself 3D 1

The entry fee at the Club is ₹423. A group of 9 tourists visit the club. How much will they pay?

Jupiter has 4333 days in a year. How many days are there in 5 Jupiter years?

Riti burns 427 calories by jogging every day. How many calories will she burn in January?

All the 28 members of the reader’s club are going on a vacation. They have a budget of ₹1,00,000 for

A local bus covers 145 km in a day. How much distance will it cover in the year 2025?

An auditorium has 755 seats. The number of shows run is given in the table below. Read the table and

the tickets. If one plane ticket costs ₹3879, will the total cost be in their budget?

answer the questions. Show

Numbers of Shows

Horror Story

Fun with Mary

a What is the total number of people who watched

the Horror Story in all the shows, if all the seats were occupied?

b What is the total number of people who watched the Fun

with Mary in all the shows, if 250 seats were unused?

Karan has a bundle of ₹200 notes. There are 86 notes. He gives 24 notes out of these to Ramesh. How

Kabir has ₹9000 for shopping. He buys 4 t-shirts for ₹460 each and 3 pairs of jeans for ₹987 each. How

A school purchased 350 pens in February 2017 and 265 pens in November 2017. If the cost of each

much money does Karan have now? much money does he have left?

pen is ₹9, how much money did the school spend on pens in total?

Chapter 3 • Multiplication

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Estimation Real Life Connect

Sameer’s uncle is a train driver. Sameer: Kamal uncle! Where do you go when you are driving the train? Uncle: I drive the train between Delhi and Amritsar. I do that 14 times in a month!! Sameer: Wow! How far are Delhi and Amritsar apart? Uncle: It is about 448 km. Sameer: Oh! You travel so much! Uncle: Yes! It is roughly about 4500 km! Sameer: Really!!! How did you calculate that so fast?

Estimating the Product Kamal uncle estimated the numbers and multiplied them quickly! We round off the multiplicand and the multiplier to get the estimated product. Let us find the estimated product of 448 and 14 to the nearest ten. Step 1

Remember! Estimation is used to find the approximate or near around products. It makes the calculations quicker and easier.

Round off both the numbers. 448 rounded up to 450.

14 rounded down to 10.

Step 2

Think and Tell

448 rounded off to the nearest ten is 450 and to the nearest hundred

Multiply rounded off numbers. 450 × 10 = 4500

is 400. What if we rounded off 487?

The estimated product of 448 and 14 is 4500. Example 7

Find the estimated product of 627 and 456 by rounding off both the numbers to the nearest hundred. Step 1

Step 2

Round off both the numbers.

Multiply rounded off numbers.

627

600 × 500 = 300000

456

600 500

The estimated product is 300000. 58

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Do It Together

Find the estimated product of 148 and 879 by rounding off both the numbers to the nearest 10. Step 1

Error Alert!

Round off both the numbers. 148

150

879

While rounding off to the nearest ten, never round down when the digit in the ones place is 5 or more.

_______

Step 2

Multiply rounded off numbers.

145

150 × _______ = __________

140

145

150

The estimated product of 148 and 879 is ___________.

Do It Yourself 3E 1

Find the estimated product by rounding off the given numbers to the nearest ten. a 235 × 13 = _____________

b 582 × 84 = _____________

c 809 × 96 = _____________

d 409 × 962 = _____________

e 849 × 167 = _____________

Find the estimated product by rounding off the given numbers to the nearest hundred. a 169 × 74

655 × 845 = _____________

b 518 × 96

c 222 × 668

d 874 × 228

Find the estimated product of the numbers rounded off to the nearest ten. Also, find the estimated product rounded off to the nearest hundred. Check which is closer to the actual product. a 109 × 54

b 182 × 95

c 444 × 777

d 976 × 862

Round off the numbers 616 and 717 to the nearest hundreds. Multiply them to get the estimated

Workers at the stadium put 1 water bottle at each seat. There are

product. Compare the estimated product with the actual product.

about 17 stands in the stadium and each stand has 238 seats. Approximately how many water bottles will they put out?

Chapter 3 • Multiplication

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Points to Remember •

The number to be multiplied is the multiplicand.

•

The number obtained from multiplication is the product.

•

364

The number by which we multiply is the multiplier.

• On multiplying a number by 1, the product is always the number itself.

multiplicand

= 5460

multiplier

product

• On multiplying a number by 0, the product is always 0.

• On multiplying two numbers in any order, the product always remains the same.

• Estimation is used when we want to quickly multiply without using the exact numbers. We round off the numbers before multiplying.

Math Lab Chit Game for Multiplication! Setting: In groups of 3. Materials Required: Number chits from 0 to 9. Method: 1

Make chits for numbers 0 to 9.

Player 1 picks 3 different chits and forms a 3-digit number.

Player 2 picks 3 chits from the remaining ones and forms a 3-digit number.

Everyone must pick chits randomly! Hint: Try forming the biggest possible numbers!

Player 3 finds the product of these numbers correctly.

The team with the highest product wins. If the winning teams have the same products, they can repeat the game to find the final winner!

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Chapter Checkup 1

Multiply using the horizontal method. Check the answer by using the vertical method. a 410 × 7

d 9101 × 8

b 492 × 6

c 7397 × 9

d 593 × 7

a 141 × 84

b 389 × 40

c 378 × 65

d 7041 × 33

4356 × 75

g 638 × 500

h 204 × 630

Find the product.

f 4

c 8023 × 3

Find the product by expanding the bigger number. a 564 × 4

b 844 × 2

911 × 117

e 9672 × 96 j

835 × 469

Estimate the product as mentioned. a 893 × 84 (to the nearest ten)

b 768 × 111 (to the nearest ten)

c 143 × 78 (to the nearest hundred)

d 862 × 376 (to the nearest hundred)

Qutub Minar in Delhi, India has 379 steps. What is the total number of steps climbed by a worker who

Ruby walks 5567 steps in a day. How many steps will she walk in a week?

S wati runs 750 m every day for 15 days for a fun competition. How far will she run throughout the

447 books can be placed on a shelf in the National Library. How many books can be placed in the

Ratan deposits ₹4555 in his bank account every month. How much money will he deposit in 12 months?

goes up and then comes down the stairs?

competition?

library with 345 shelves?

10 A small town produces 314 kg of waste every day. How much waste do they produce from March to May?

Word Problems 1

T here are 24 schools in a town. Each school gets 3 pieces of equipment for the science

Around 467 people visit the exhibition at the National Gallery every day. Estimate how

lab. The cost of each piece of equipment is ₹394. How much does all the equipment cost? many people visit the exhibition in a year.

Chapter 3 • Multiplication

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Division

Let's Recall We should always keep our belongings organised! Suppose we have 8 folded items of clothing as shown. Our almirah has 4 slots in which we need to keep these clothes. How can we arrange these clothes in these slots so that each slot has an equal number of clothes? Let us start with 1 garment in each slot. We now have 4 items of clothing left. We can again distribute these clothes in the slot! Now no items of clothing are left and all clothes have been equally distributed in the slots! This equal grouping of items is called Division. We grouped 8 items of clothing into 4 slots. And we finally got 2 items in each slot. This can be written as: 8 divided by 4 is equal to 2.

↓ Dividend

↓ Divisor

↓ Quotient

Let's Warm-up

Answer the following. 9÷3

= __________

10 ÷ 5

= __________

12 ÷ 6

= __________

18 ÷ 9

= __________

I scored ___________ out of 4.

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Division by 1-digit and 2-digit Numbers Real Life Connect

Ramu, the milkman, supplies milk to different shopkeepers and customers. Ramu has a big drum of milk with a capacity of 210 litres. He also pours the milk into smaller containers of 5 litres capacity, so that he can supply the milk to customers.

Division by 1-digit Numbers Let us see how many small containers Ramu can fill from the drum. Number of containers = Capacity of the drum ÷ Capacity of each container

Remember!

= 210 L ÷ 5 L

The Dividend is the number which is divided. The Divisor is the number by which the dividend is divided. The result of the division is called the quotient.

Let us solve the problem using long division. While doing division, always go from left to right. 5 210

5 210

5 210 1

rite the dividend W and the divisor in the division house.

ompare the digit at the hundreds C place of the dividend with the divisor. Here, 2 < 5. There are not enough hundreds. So, write 0 in the hundreds place in the quotient or do not write anything there.

se multiplication to find the U nearest quotient.

5 210

– 20

042

Divisor

5 210

ubtract to find S the leftover tens.

Bring down the ones.

Dividend

– 20

– 10

Quotient

Remainder

epeat steps 3 and 4 until you R get 0 as the remainder.

Thus, Ramu can fill 42 containers from the drum. Chapter 4 • Division

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Let us check to see if the quotient is correct. When there is no remainder, the Dividend should be equal to Quotient × Divisor. Here, the Dividend = 210.

Remember! The Remainder is what is left over after performing the division.

Quotient × Divisor = 42 × 5 = 210. So, we get Dividend = Quotient × Divisor.

Think and Tell

So, our answer was correct.

Will the division of a 2-digit number by a 1-digit number always be a 1-digit number?

Division Facts 0 division by a Number

Division by 1

Division by Itself

When 0 is divided by any number then the quotient is always 0.

When a number is divided by 1, the quotient is always the number itself.

When a number is divided by itself then the quotient is 1.

0÷4=0 0 ÷ 41 = 0 0 ÷ 128 = 0

6÷1=6 18 ÷ 1 = 18 194 ÷ 1 = 194

5÷5=1 14 ÷ 14 = 1 184 ÷ 184 = 1

Division and multiplication are reverse operations. Every multiplication fact has 2 division facts. 5÷1=5 5×1=5 5÷5=1

Error Alert! Never try dividing by 0. Division by 0 is not defined. For example, 0 ÷ 3 is 0, but 3 ÷ 0 is not defined.

24 ÷ 4 = 6 4 × 6 = 24 24 ÷ 6 = 4 50 ÷ 10 = 5 10 × 5 = 50 50 ÷ 5 = 10

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Division of a 4-digit Number by a 1-digit Number How many containers of 5 litres can be filled from a tanker with a capacity of 3575 litres? Number of containers =

Capacity of the tanker 3575 = Capacity of each container 5

5 3575

5 3575 1

rite the dividend W and the divisor in the division house.

– 35

ompare the digit at the thousands C place of the dividend with the divisor. Here, 3 < 5. There are not enough thousands. So, write 0 in the thousands place in the quotient or do not write anything there.

se multiplication to find the U nearest quotient.

5 3575

0715

5 3575

– 35

5 3575

– 35

– 5

– 25

ubtract to find S the leftover tens.

Bring down the ones.

epeat steps 3 and 4 until you R get 0 as the remainder.

Thus, 715 containers can be filled from the tanker. Example 1

Divide 864 by 6. Verify the quotient. 144 6 864 – 6 26 – 24 24 – 24 0

Thus, 864 ÷ 6 = 144. Verify the quotient.

Example 2

Divide 1488 by 8. 186 8 1488 – 8 68 – 64 48 – 48 0

Thus, 1488 ÷ 8 gives the quotient = 186 and remainder = 0.

Dividend = (Quotient × Divisor). 864 = (144 × 6). Chapter 4 • Division

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Do It Together

Divide 2133 by 9.

2 9 2133 – 18 3 –

9 × 2 = _________

–

9 × __ = _________

Thus, 2133 ÷ 9 = _________.

9 × 3 = _________

Do It Yourself 4A 1

Find the quotient. a 440 ÷ 1

b 432 ÷ 2

c 963 ÷ 3

d 999 ÷ 9

e 392 ÷ 8

2792 ÷ 8

g 6170 ÷ 5

h 5382 ÷ 6

c 1656 ÷ 9

d 280 ÷ 5

Divide the numbers. Verify the answer. a 1776 ÷ 4

b 672 ÷ 8

Fill in the boxes with the missing numbers. a

7 9 2 6

– 7 – –

8 3 9 1 2

– 3

– 6

1 2

–

1 72 0

Write if true or false. a Dividing any number by zero gives the same number as the quotient. b Dividing any number by the number itself gives 1 as the answer. c If zero is divided by a number, the answer is always zero. d The division algorithm states: Dividend = Quotient x Divisor + Remainder

What must be added to 4587 so that it can be divided by 5 with no remainder?

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Word Problems 1

A book is divided into 6 chapters. The total number of pages in the book is 1410. If

there are an equal number of pages in each chapter, then how many pages are there in each chapter?

Raghu, a fruit seller, sells bananas. He packs bananas in small boxes. He packs 145

dozen bananas in boxes. Each box has 6 bananas. How many boxes of bananas does he pack?

Division by 2-digit Numbers Ramu, the milkman, gets an order to deliver milk to a sweet shop. The order is big, and he has to deliver the milk in bigger containers! Let’s see how he plans his deliveries.

Dividing by Tens The sweet shop requires milk containers of 10 litres each. The full order is for 500 litres. How many containers does Ramu need? Number of containers required =

Total milk 500 = Capacity of each container 10

= 50 containers

When a number is divided by 10, the digit in the ones place comes up as the remainder and the rest of the digit makes up the quotient. Dividing by 10 is easy. Look at the following divisions: 54 ÷ 10, Quotient = 5 and Remainder = 4. 543 ÷ 10, Quotient = 54 and Remainder = 3. 5432 ÷ 10, Quotient = 543 and Remainder = 2. Let us now divide by 20, 30, 40, … What if we want to divide 500 by 20? Quick Way When both the Dividend and the Divisor have a 0 in the ones place, we can apply a trick to divide quickly.

Chapter 4 • Division

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10 Litres Milk

25 20 500 – 40 100 – 100 0

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Step 1

Step 2

Cancel the zeroes.

Now mentally divide the remaining number by 2. If

500 50 = 2 20

method.

you are unable to divide mentally, use the long division

2 50

–

– 10

50 ÷ 2 = 25

We get the same answer.

Example 3

546 ÷ 10 = ? The digit in the ones place makes up the remainder and the rest of the digits make up the quotient. 546 ÷ 10 gives Quotient = 54 and Remainder = 6.

Example 4

840 ÷ 40 = ? Step 1

Step 2

Cancel the zeroes in the

Divide the remaining digits mentally or using the

numbers.

84 ÷ 4 = 21

ones place of both the 840 ÷ 40 = 84 ÷ 4

Do It Together

long division method. Thus, 840 ÷ 40 = 21

04 –4

9300 by 30 9300 = 30

4 84 – 8

Divide mentally. 1

1200 by 40

1200 = = 40

Division of Numbers Up to 4-digits Ramu is filling milk containers from a tanker of 4536 litres capacity. What if Ramu has to fill containers that each have a capacity of 21 litres? 21 litres. How many such containers can be filled? Step 1 Divide 45 by 21. The result is 2 (21 × 2 = 42) and the remainder is 3 (45 – 42 = 3). Put 2 directly above 5 and bring down the next digit 3 along the remainder 4, making it 33.

21 4536 – 42

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Step 2 Divide 33 by 21.

21 4536

The result is 1 (21 × 1 = 21) and the remainder is 12 (33 – 21 = 12). Put 1

– 42

directly above 3 and bring down the next digit 6 along with the remainder

12, making it 126.

– 21

126

Step 3

216

Divide 126 by 21.

21 4536

Divisor

The result is 6 (21 × 6 = 126) and the remainder is 0

Quotient

Dividend

– 42

(126 – 126 = 0). Put 6 directly above 6 and write the

remainder 0 at the bottom.

– 21

126

–126

Thus, Ramu has to fill 216 containers each with a capacity of 21 litres.

Remainder

Verify the quotient. The Dividend should be equal to (Quotient × Divisor) + Remainder. Dividend = 4536 Quotient = 216 Divisor = 21 Remainder = 0 (Quotient × Divisor) + Remainder = (216 × 21) + 0

× + + =

= 4536 So, we get Dividend = (Quotient × Divisor) + Remainder. Our answer is correct! Example 5

Verification: The Dividend should be equal to (Quotient × Divisor) + Remainder. Dividend = 8210 = (456 × 18) + 2 = 8208 + 2 = 8210 = Dividend So, our answer is correct! Chapter 4 • Division

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1 2 1 2 3

2 3 5

4 4

6 1 6 6

Think and Tell

Will the division of a 4-digit

Divide 8210 by 18 and verify the answer. Thus, 8210 ÷ 18 gives Quotient = 456 and remainder = 2.

456 18 8210 – 72 101 –90 110 – 108 2

number by a 2-digit number always be a 2-digit number?

× + + =

4 3 4 8

6 5 2

5 1 4 6 0

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Do It Together

Divide 1548 by 18.

18 1548 – 144 0 0 –

Dividing by Multiples of 100s and 1000s What if Ramu has a tanker with a capacity of 5000 litres and he wants to pour milk into drums of 100 litres capacity each? How many drums does he require? Total capacity of tanker 5000 Number of drums required = = = 50. Capacity of each drum 100 When a number is divided by 100, the digit in the tens and ones place make up the remainder and the rest of the digits make up the quotient. Dividing by 100 is easy. Look at the divisions on the side: Dividing by Multiples of 100

546 ÷ 100

1456 ÷ 100

9842 ÷ 100

Quotient = 5

Quotient = 14

Quotient = 98

Remainder = 46

Remainder = 56

Remainder = 42

When a multiple of 10, 100, or 1000 is divided by a multiple of 10, 100 or 1000, we cancel out the zeroes and then solve.

Example 6

300 ÷ 30

4500 ÷ 300

8000 ÷ 4000

Cancel out the zeroes.

Divide

300 30 = = 10 30 3

4500 45 = = 15 3 300

8000 8 = =2 4000 4

Divide 8210 by 100.

Example 7

Divide 9583 by 3000.

Step 1

Cancel out the common zeroes between the

Divide 9583 by 3000.

dividend and divisor to get 8210 ÷ 100 = 821 ÷ 10

Step 2 When a number is divided by 10, the digit in the

tens place makes up the remainder and the rest

The result is 3 (3000 × 3 = 9000) and the

remainder is 583 (9583 – 9000 = 583). Put 3 directly above 3 and the remainder is 483. Thus, 9583 ÷ 3000 gives Quotient = 3 and Remainder = 583.

of the digits make up the quotient.

821 ÷ 10 gives Quotient = 82 and Remainder = 1. 70

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Do It Together

Divide mentally. 1200 by 300

4565 by 2000

1200 = 300

4565 = 2000

Do It Yourself 4B 1

Find the quotient and the remainder. a 440 ÷ 10

b 1547 ÷ 30

e 4530 ÷ 15

d 4879 ÷ 98

g 7946 ÷ 300

h 8764 ÷ 2000

c 4897 ÷ 24

d 9876 ÷ 49

Divide the numbers and verify the answer. a 443 ÷ 12

952 ÷ 100

c 487 ÷ 21

b 149 ÷ 11

Fill in the missing numbers in the boxes. a

21 796

19 2496

166

– 6 –

– 19 –

– 19

Find the quotient and the remainder without using long division. a 486 ÷ 10

b 9765 ÷ 10

c 986 ÷ 100

d 3479 ÷ 100

e 7894 ÷ 1000

5555 ÷ 1000

How many hours are there in 1200 minutes?

Word Problems 1

There are 1025 students in a school containing 25 sections. If there is an equal

Manu has saved ₹5460. He has 6 notes of ₹10, 9 notes of ₹100 and rest of ₹500 notes.

number of students in each section, find the number of them in each section. How many ₹500 notes does he have?

Chapter 4 • Division

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Word Problems Remember Ramu who was delivering milk to different shopkeepers and customers? He has to feed hay to the cows. He purchases 1032 pounds of hay for his 43 cows. If each cow eats an equal amount of hay then how much hay does he give to each cow? Let us apply the CUBES method to solve the problem.

Circle the numbers. C: Circle the numbers.

Ramu has to feed hay to the cows. He purchases 1032 pounds of hay for his 43 cows. If he gives each cow an equal amount of hay, then how much hay does he give to each cow?

U: Underline the question. B: Box the keywords. E: Evaluate/draw.

S: Solve and check. 2

Underline the question.

Evaluate:

Hay given to each cow = 5

Box the keywords.

Total hay purchased 1032 = Total Number of cows 43

Solve and Check:

Thus, Ramu gives 24 pounds of hay to each cow. Check the answer: Dividend = (Quotient × Divisor) + Remainder

24 43 1032 – 86 172 – 172 0

Quotient = 24 Divisor = 43 Remainder = 0 Dividend = (24 × 43) + 0 Dividend = 1032 Thus, the answer is correct.

× + + =

9 0

2 4 7 6 3

4 3 2 2

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Example 8

Rahul has 123 pencils. He packs the pencils in boxes. If he packs 3 pencils in each box, then how many boxes does he pack? Rahul has 123 pencils. He packs the pencils in boxes. He packs 3 pencils in each box. How many boxes does he pack? 1

Circle the numbers – 123 and 3.

Underline the question – How many boxes does he pack?

Box the keywords – each.

Evaluate:

Total number of boxes = = 5

Solve and Check:

Total number of pencils Number of pencils in each box

123 3

Check the answer:

41 3 123 – 12 03 –3 0

Dividend = (Quotient × Divisor) + Remainder Quotient = 41 Divisor = 3

Hence, Rahul packs 41 boxes of pencils.

Remainder = 0

× =

4 1

1 3 3

Dividend = (41 × 3) + 0 Dividend = 123 Thus, the answer is correct.

Example 9

A packet of crayons contains 24 crayons. If 5489 crayons are to be packed, how many packets are required to pack all the crayons? Total number of crayons = 5489. Number of crayons in each packet = 24. Total number of packets required =

Total number of crayons Number of crayons in each packet

5489 24 So, there are 228 packets of crayons required but 17 crayons are left over. =

These 17 crayons also need to be packed in one packet. So, total packets required = 228 + 1 = 229. Hence, the total number of packets required to pack 5489 crayons is 229. Chapter 4 • Division

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Rohit has 525 marbles. He wants to make groups of 25 each. How many such groups can he make? Use the CUBES method to solve.

Do It Together

Circle the numbers.

Underline the question.

Box the keywords.

Evaluate.

Solve and Check:

Thus, Rohit makes ____ such groups.

Do It Yourself 4C 1

A bottle factory produces 644 bottles in 46 days. How many bottles will the factory produce in one day?

Ravi collects stamp and pastes them in a notebook. He has a total of 1240 stamps. He pastes 31

A person gets ₹1500 in the month of April. How much money did he get each day if he gets an equal

In the library, there are 8255 books. The books are kept on shelves. If 13 books are kept on each shelf,

A shopkeeper buys 45 packets of candies. Each packet has 45 candies. If he repacks the candies in

stamps on each page. How many pages were used altogether? amount of money each day?

then how many shelves are there?

smaller packets containing 15 candies each, then how many packets will he get?

Estimation Real Life Connect

Sam and Preeto’s school has organised a sports event. The winners will be awarded with cash prizes. The teacher announces that there will be 19 events and the school will distribute a total amount of ₹9000 among the winners. Sam: How much money will each person get? Preeto: Each person will get roughly ₹450. Sam: Really! How did you calculate it so fast?

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Estimating the Quotient Preeto estimated the numbers and divided them quickly to find the approximate answer. Round off the dividend and the divisor to estimate the answer. Find the estimated quotient when 9000 is divided by 19 rounded to the nearest 10.

Think and Tell

Step 1

9000 off?

Why did we not round

Round the numbers off.

473

There is no need to round 9000 off.

19 9000

19 rounded to the nearest tens = 20.

– 76

Step 2

–

Divide the numbers and estimate the quotient. 9000 ÷ 20

140 133

– 57

Thus, the estimated quotient for 9000 ÷ 20 is 450. Check the actual answer and its closeness to the estimated answer.

9000 ÷ 19 Thus, 9000 ÷ 19 gives quotient = 473 and remainder = 13. Hence, the estimated answer and the actual answer are close to each other.

Error Alert!

Remember! An estimate is a smart guess about something. Estimation is used to find the approximate quotients. It makes the calculations quicker and easier.

Example 10

While rounding off to the nearest hundred, never round down when the digit in the tens place is 5 or more. 355 → 300

355 → 400 Rounded off to the nearest 100

Estimate the quotient for 184 ÷ 4. Step 1 Round off the numbers. Compare the answer with the actual answer. 184 rounded off to the nearest tens is 180.

Chapter 4 • Division

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Step 2

Step 3

Divide the numbers and estimate the quotient.

Find the actual quotient.

180 ÷ 4

184 ÷ 4

4 180

4 184

– 16 –

– 16

–

So, 180 ÷ 4 = 45.

So, 184 ÷ 4 = 46.

The estimated quotient is close to the actual quotient. Example 11

Estimate the quotient for 9635 ÷ 41. Step 2

Round off the numbers.

Divide the numbers and estimate

9635 rounded off to the nearest hundreds is 9600. 41 rounded off to the nearest tens is 40.

Do It Together

240

Step 1

the quotient. 9600 ÷ 40

40 9600 – 80 –

So, 9600 ÷ 40 = 240.

160 160

–0

A geometry box can hold only 11 pens. Estimate the number of boxes required to contain 572 pens.

Do It Yourself 4D 1

Estimate the quotient for the given problems by rounding off to the nearest tens. By how much did the exact answer vary? a 149 ÷ 9

c 434 ÷ 31

d 955 ÷ 39

Round off the numbers to the nearest hundred and find the estimated quotient. Compare the answer with the actual answer. a 478 ÷ 97

b 897 ÷ 18

b 2799 ÷ 31

c 879 ÷ 48

d 8744 ÷ 58

Divide 6012 ÷ 6. a Estimated quotient = ____________

b Actual quotient = ____________

c Compare the estimated quotient with the actual quotient. Which is greater? ____________

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Round off each dividend to the nearest 10s and 100s and then divide. To the nearest 10s

To the nearest 100s

a 1147 ÷ 2 b 4589 ÷ 3 c 6478 ÷ 6 d 8974 ÷ 7 e 5555 ÷ 5

860 people have been invited to a banquet. The caterer is arranging tables. Each table can seat 10 people. About how many tables are needed?

[Round off the dividend to the nearest hundred].

Word Problems 1

238 children went to a school camp. If one tent can be shared by 4 children,

₹5734 is distributed between 2 groups of students. Each group has 11 students.

approximately how many tents will be needed for the camp? Estimate the amount of money that each student gets.

Points to Remember • The number being divided is called the dividend. The number by which we divide is called the divisor. The result of the division is called the quotient. The number left over after division is called the remainder. • To check if our answer after division is correct, we can use: Dividend = (Quotient × Divisor) + Remainder. •

When a number is divided by 1, the quotient is always the number itself.

•

When a number is divided by itself, then the quotient is 1.

•

When 0 is divided by any number, then the quotient is always 0.

• When a number is divided by 10, the digit at the ones place forms the remainder and the remaining digits form the quotient. • When a number is divided by 100, the digit in the ones place and tens place forms the remainder and the remaining digits form the quotient. • When a number is divided by 1000, the digits in the ones place, tens place and thousands place form the remainder and the remaining digits form the quotient. Chapter 4 • Division

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Math Lab Dice Game Setting: In groups of 3. Materials Required: 3 dice. Method: 1

All three dice are rolled.

The numbers on the dice are noted.

The largest number formed using all three digits that appear on the dice becomes

The sum of all three digits that appear on the dice becomes the divisor.

Divide the numbers and note down the remainders obtained.

The student to get the sum of remainder equal or more than 30, wins.

the dividend.

Chapter Checkup 1

Find the quotient. a d

489 ÷ 10

145 ÷ 100

4789 ÷ 1000

8500 ÷ 1000

459 by 3

7848 by 4

958 by 10

7894 by 100

855 by 19

9984 by 48

945 ÷ 23

Find the quotient and the remainder. a

4000 ÷ 100

Divide. a

47 ÷ 10

987 ÷ 8

2129 ÷ 9

2460 ÷ 13

Fill in the missing numbers in the spaces. a

7 1548

6 5487

– 14 –

1 –

– 54 –

–2

14 3487 – 28 – –

127 12

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Solve. a b c d

What should be subtracted from 1745 so that on dividing by 9, we get no remainder? What should be added to 341 so that on dividing by 17, we get no remainder?

What should be subtracted from 7134 so that on dividing by 26, we get no remainder?

Estimate the quotient by rounding off the dividend to the nearest tens. a

What should be added to 154 so that on dividing by 5, we get no remainder?

448 ÷ 9

1459 ÷ 4

779 ÷ 13

4577 ÷ 20

8797 ÷ 16

Estimate the quotient by rounding off the dividend to the nearest hundreds and divisor to the nearest tens. a

112 ÷ 11

489 ÷ 9

1548 ÷ 52

6987 ÷ 49

₹1540 is to be distributed equally among 14 friends. How much money will each friend get?

A shopkeeper gets 45 boxes of 19 chocolates each and 53 boxes of 27 chocolates each. He packs all the chocolates in smaller boxes each having 18 chocolates. How many boxes does he pack?

10 A florist has 1260 flowers. He makes bunches of flowers. For half of the flowers, he makes bunches of 14 flowers each and for the other half he makes bunches of 15 flowers each. How many bunches did he make in total?

Word Problems 1

A bus can hold 108 passengers. If there are 12 rows of seats on the bus, how

Mark baked 195 cookies and divided them equally into 13 packs. How many

here are 1025 students in a school containing 25 sections. If there is an equal T number of students in each section, find the number in each section.

many seats are there in each row?

cookies did Mark put in each pack?

32 oranges are packed in boxes with each having 43 oranges. Estimate the 4 number of orange boxes packed.

How many hours are there in 7200 seconds?

farmer packs 1200 kg of tomatoes equally in two types of boxes. The first type A of box has 15 tomatoes each and the second type of box has 25 tomatoes each. How many boxes does he pack?

Chapter 4 • Division

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35 Multiples and Factors Let's Recall Multiplication is the same as repeated addition. Using multiplication, we can find the total number of items that are there in groups of equal items. For example, if we have 5 boxes of chocolates, each containing 4 chocolates, the total number of chocolates is 5 × 4 = 20.

5 × 4 = 20

Boxes

Chocolates in Total each box Chocolates

Similarly, division is the same as repeated subtraction. Using division, we can find the number of items in each group from the total number of items and number of groups. Similarly, we can find the number of groups from the total items and items in each group. For example, if 20 chocolates are to be put in 5 boxes, each box would get 20 � 5 = 4.

20 ÷ 5 = 4

Total Chocolates

Boxes

Chocolates in each box

So, we can see that multiplication and division are actually closely related to one another. In this chapter, let’s learn about new concepts that are linked to multiplication and division.

Let's Warm-up

Match the following. a

50 × 5

9×9

30 + 3

363 ÷ 11

250

250 ÷ 50

930

310 × 3

90 – 9 I scored _________ out of 5.

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Multiples Real Life Connect

Planning holidays is always fun! You first decide on a place you want to visit, then research what you can do there, and then finally make arrangements. Sahil is going to Ooty. The train departs every second day. Monday Tuesday Wednesday

Thursday

Friday

Saturday

Sunday

Think and Tell What do you think do all the circled numbers have in common?

He can travel on the following dates - 2, 4, 6, 8, 10, 12, 14, 16 and so on.

Finding Multiples

The multiples of a number are the products we get by multiplying the number by 1, 2, 3,... and so on. Multiples of 3 can be found by using multiplication tables as follows: 3×1= 3×2= 3×3= 3×4= 3×5=

3 6 9 12 15

3×6= 3×7= 3×8= 3×9= 3 × 10 =

18 21 24 27 30

Remember! A number is a multiple of itself too. For example, multiples of 5 are 5, 10, 15, 20, 25, 30 and so on!

You can also check if a number is a multiple of a number using division. If the remainder is 0, then the bigger number is a multiple of the other number. For example: 5 leaves no remainder

15 – 15 00

On dividing 15 by 5, we get 0 remainder. So, 15 is a multiple of 5.

Chapter 5 • Multiples and Factors

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5 leaves remainder 1

16 – 15 01

On dividing 16 by 5, we get remainder 1. So, 16 is not a multiple of 5.

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Facts about Multiples

• Every number is a multiple of 1 and the number itself. For example, 5 × 1 = 5. Here, 5 is a multiple of 1 and 5.

• Every multiple is either greater than or equal to the number itself.

For example, the multiples of 8 are 8, 16, 24, 32, 40, … and so on. Here, each multiple is equal to or greater than 8.

• Every number has an unlimited number of multiples.

For example, the multiples of 7 are 7, 14, 21, 28, 35, …, 70, 77, …, 7000, …, 70000, …, and so on. Here, multiples of 7 are unlimited.

Example 1

Find the first 5 multiples of 4.

We can find the multiples of 4 by using the number line showing jumps of 4. 1

9 10 11 12 13 14 15 16 17 18 19 20

We can find the multiples of 4 by using multiplication tables as follows: 4×1=

4×2=

Did You Know?

Leap years are always

4 × 3 = 12

multiples of 4. For example

4 × 4 = 16

the years 2016, 2020, 2024, ...

4 × 5 = 20

are all leap years.

The first five multiples of 4 are 4, 8, 12, 16 and 20. Example 2

Find the first 5 multiples of 5. Check by dividing whether 95 is a multiple of 5. 1

5×1=5

9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

5 × 2 = 10

5 × 3 = 15

5 × 4 = 20

5 × 5 = 25

The first five multiples of 5 are 5, 10, 15, 20 and 25. 5 leaves remainder 0

95 – 95 00

19 On dividing 95 by 5, we get 0 remainder. So, 95 is a multiple of 5.

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Do It Together

Find the first 5 multiples of 6. Check by dividing if 92 and 96 are multiples of 6. 1

6×1=6

9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

6 × 2 = ____

6 × ____ = ____

6 × ____ = 24

6 × ____ = ____

The first five multiples of 6 are 6, ____, ____, 24, ____.

Think and Tell

Are the multiples of an even number always an even number?

On dividing 92 by 6, we get 2 remainder.

On dividing 96 by 6, we get ____ remainder.

So, 92 _______________________________ of 6.

So, 96 ___________________________ of 6.

Do It Yourself 5A 1

Colour the balloons that are multiples of 2.

Find the first five multiples of the given numbers. a 7

b 8

c 9

d 10

g 13

h 14

m 19

n 20

k 17

Chapter 5 • Multiples and Factors

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e 11 j

o 25

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Solve to find.

a 6th multiple of 10

b 9th multiple of 13

c 11th multiple of 9

d 5th multiple of 12

e 4th multiple of 15

Find the first 8 odd multiples of 9.

Find the first 6 even multiples of 12.

Check if the second number is a multiple of the first number.

Write multiples of 8 and 11 that are greater than 75 but less than 150.

a 5, 75

b 7, 68

c 8, 64

d 11, 88

e 12, 96

5th multiple of 25

17, 52

Word Problem 1

Juhi is collecting flowers and leaves stickers for her album. She bought

a pack of 50 such stickers. To her surprise, every 5th sticker in the pack

was a special sticker with glitter. Can you find out which numbers in the pack have glittery stickers?

Common Multiples You remember how Sahil’s train to Ooty departed every second day? He has not booked his ticket yet. His cousin Ashima from a nearby city also wants to travel to Ooty! The train from her city leaves every third day. Since they both want to leave and reach Ooty on the same day, they decided to mark their calendars to check for the possible common days of travel, as follows: Monday Tuesday Wednesday

Thursday

Friday

Saturday

Sunday

Sahil marked in blue and Ashima marked in red. They figured out that they can travel together on 6th, 12th, 18th, 24th and 30th of the month. 84

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In the above example, we looked for common multiples of 2 and 3 to find the common days of travel. The numbers that are circled in red and blue together are common multiples of 2 and 3. A number that is a multiple of two or more numbers is a common multiple of those numbers. Let’s take another example! Multiples of 3 are 3, 6, 9, 12, 15, 18, 21, 24, 30 and so on. Multiples of 4 are 4, 8, 12, 16, 20, 24, 28, 32 and so on. This can be shown on a number line as: 3

9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Here, common multiples of 3 and 4 are 12 and 24.

Example 3

Find the first 5 common multiples of 3 and 6. Step 1: Find multiples of the first number. Multiples of 3

3×1

3×2

3×3

3×4

3×5

3×6

3×7

3×8

3×9

3 × 10

Step 2: Find multiples of the second number. Multiples of 6

6×1

6×2

6×3

6×4

6×5

6×6

6×7

6×8

6×9

6 × 10

Step 3: Identify the multiples that are common for both the numbers. Multiples of 3 and 6

Step 4: Write the first five common multiples of the two numbers. The first five common multiples of 3 and 6 are 6, 12, 18, 24 and 30.

Chapter 5 • Multiples and Factors

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Do It Together

Every second floor of a building has a canteen. In the same building, every fifth floor has a party hall. The building has a total of 20 floors. On which floors, can residents find both the canteen and the party hall? In order to find floors with both the canteen and the party hall, we need to find common multiples of 2 and 5. We can solve the problem as follows: Step 1: Find multiples of 2 to find floors with the canteen. Multiples of 2 2×1

2×2

2×3

2×4

2×5

2×6

2×7

2×8

2×9

2 × 10

Step 2: Find multiples of 5 to find floors with the party hall. Multiples of 5 5×1

5×2

5×3

5×4

Think and Tell

Why did we stop at 20?

Step 3: Find the common floors by finding the common multiples. Multiples of 2 and 5

Step 4: Tell the answer in complete sentence. We can find the canteen and party hall together on the floors ______________________.

Remember! We know that a number can have unlimited multiples. It is not possible to find its highest multiple. So, two numbers cannot have a highest common multiple.

Do It Yourself 5B 1

Find the first 2 common multiples of the following pairs of numbers. a 2 and 3

b 3 and 7

c 2 and 9

d 2 and 6

e 3 and 5

6 and 9

g 5 and 10

h 10 and 15

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Find the first three common multiples of the following pairs of numbers. a 4 and 5

b 2 and 6

c 4 and 6

d 10 and 20

Is 27 a common multiple of 8 and 9? Explain your answer.

Is 54 a common multiple of 4 and 6? Explain your answer.

I am a multiple of 11 but less than 160. I am an even multiple of 7 too. What number am I?

Word Problem 1

Raj goes to meet his grandparents every 4th day. He also goes to visit a dog shelter every 6th day. On which dates in this month will he go to both the places? Monday Tuesday Wednesday

Thursday

Friday

Saturday

Sunday

Factors Real Life Connect

Sheela aunty wants to distribute food to children. She bought 8 apples. She is thinking how many plates of food she can make to distribute these apples. She tries the following arrangements.

1 plate with all the 8 apples 1x8=8

Chapter 5 • Multiples and Factors

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2 plates with 4 apples in each 2x4=8

4 plates with 2 apples in each 4x2=8

8 plates with 1 apple in each 8x1=8

Error Alert!

Think and Tell If Sheela aunty wants to distribute apples to the largest number of

Sheela aunty cannot make plates with 3 apples in them. That’s because 8 apples cannot be distributed equally in 3 plates.

children, which arrangement should she go with?

Finding Factors In the above example, we can say that 8 is a multiple of 1, 2, 4 and 8. We can also say that 1, 2, 4 and 8 are all factors of 8.

Remember!

When two numbers are multiplied, the result is called their product. The numbers that are multiplied are called the factors of the product.

1 ×

8 =

factors

8 product

2 ×

4 =

factors

8 product

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Facts about Factors Every number is a factor of itself.

Every number has a limited number of factors. 5 has 2 factors and 8 has 4 factors.

Factors of 5 1, 5

1 is a factor of every number.

Factors of 8 1, 2, 4, 8 Every factor is either less than or equal to the given number. Factors of both 5 and 8 are less than the numbers.

Finding Factors Using Multiplication In order to find the factors of a number, we look for numbers whose product is the given number. For example, the number 8 is the product of the following numbers: 1 × 8 or 8 × 1

2 × 4 or 4 × 2

Therefore, 1, 2, 4 and 8 are factors of the number 8. When finding factors by multiplication, always: • Start with multiplying by 1.

• Stop when any factor starts repeating.

Think and Tell Using multiplication for finding factors is a more time-consuming method. Can you tell why? Can you think of a quicker method to find factors?

Example 4

List factors of 12. Show different arrangements possible for 12.

1 × 12 = 12

2 × 6 = 12 Chapter 5 • Multiples and Factors

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3 × 4 = 12 89

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4 × 3 = 12

12 × 1 = 12

6 × 2 = 12

We can see that 12 can be shown as 1 × 12, 2 × 6, 3 × 4, 4 × 3, 6 × 2 and 1 × 12. Therefore, 1, 2, 3, 4, 6, and 12 are factors of 12. Example 5

Find the factors of 18 using multiplication. Multiply by 1

1 × 18 = 18

Multiply by 3

3 × 6 = 18

Multiply by 2 Multiply by 4 Multiply by 5 Stop!

2 × 9 = 18 4 × ? = Not possible 5 × ? = Not possible

We have already found 6 to be a factor of 18. So, the factors of 18 are 1, 2, 3, 6, 9, and 18. Do It Together

Find the factors of 20 using multiplication. Multiply by 1

1 × _____ 20 = _____ 20 _____

Multiply by 2

_____ × _____ = _____

Multiply by 3

Not possible

Multiply by 4

_____ × _____ = _____

Multiply by 5

_____ × _____ = _____

Multiply by 6

Should we multiply further? _______________________________________________________________ So, the factors of 20 are ___________________________________________________________________. 90

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Finding Factors Using Division In order to find the factors of a number, we can also look for numbers that divide that number exactly, leaving no remainder behind! Let’s find out which numbers divide 30 completely. 2

30 – 2 10 – 10 00

Remainder is 0.

30 – 28 02

5 Remainder is 2.

4 and 7 are NOT factors of 30.

Remainder is 0.

3 and 10 are factors of 30.

2 and 15 are factors of 30. 4

30 – 3 00 – 00 00

30 – 30 00

6 Remainder is 0.

5 and 6 are factors of 30.

Therefore, 1, 2, 3, 5, 6, 10, 15 and 30 are factors of the number 30.

Error Alert! NEVER include 0 as a factor of any number because we cannot divide any number by 0.

Example 6

Remember!

The factors of a number will always divide it completely, leaving 0 as the remainder!

Is 7 a factor of 42? To check whether a number is a factor or not, we divide the two numbers. Divide 42 by 7. On dividing 42 by 7, remainder is 0. So, 7 is a factor of 42.

Chapter 5 • Multiples and Factors

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7 Remainder is 0.

42 – 42 00

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Do It Together

Find the factors of 36 using the division method. Divide numbers by 36

What do we get?

Check the remainder

Are the numbers factors of 36?

36 ÷ 1

Yes

36 ÷ 2

____

36 ÷ 3

____

36 ÷ 4

____

36 ÷ 5

____

36 ÷ 6

____

Yes

Think and Tell Do we need to go beyond 6 to find more factors of 36?

So, the factors of 36 are ___________________________________________________________________.

Do It Yourself 5C 1

Show 10 in different arrangements. Then, list the factors of 10.

Show 18 in different arrangements. Then, list the factors of 18.

Show 20 in different arrangements. Then, list the factors of 20.

Find the factors of the following numbers using multiplication. a 14

b 21

c 36

d 39

e 40

g 48

h 50

Find the factors of the following numbers using division.

a 9

b 11

c 12

d 13

e 15

Is 18 a factor of 126? Explain your answer.

Is 6 a factor of 64? Explain your answer.

Which numbers between 1 and 10 have exactly TWO factors?

Which number has the greatest number of factors between 5 and 15?

What is the smallest number that has exactly three factors?

Word Problems 1

ina bought 16 eggs. She wants to arrange them into a tray. In how many T ways can she arrange the eggs?

aman, a baker, has baked 72 biscuits. He wants to place the same number R of biscuits in each packet. What different arrangements are possible?

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Common Factors Do you remember Sheela aunty who had bought 8 apples to distribute among children? Now she has bought 4 mangoes. She has decided to distribute both apples and mangoes to children. But she has a condition—she wants to distribute the fruits in an identical manner. That means, each plate should have the same number of apples as the others, and each plate should have an equal number of mangoes as the others. All the children should get identical fruit plates. Let’s see what she does. If she distributes the fruit only to 1 child, that child will get 1 plate with 8 apples and 4 mangoes. 1 exactly divides both 8 and 4. Remainder is 0.

If she distributes the fruit to 2 children, then both of them will get plates with 4 apples and 2 mangoes. 2 exactly divides both 8 and 4. Remainder is 0.

If she distributes the fruit to 3 children, then neither the apples nor the mangoes can be divided in an identical manner among them. 3 does not divide 8 exactly. The remainder is 2. 3 also does not divide 4 exactly. The remainder is 1.

If she distributes the fruit to 4 children, then all four of them will get plates with 2 apples and 1 mango. 4 exactly divides 8 and 4. Remainder is 0.

Sheela aunty wanted to distribute the fruit to the greatest number children. So, she ended up making 4 plates with 2 apples and 1 mango in each. All the children were happy!

Think and Tell Why did Sheela aunty stop after making 4 plates?

Chapter 5 • Multiples and Factors

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Common factors of two or more numbers are the numbers that divide the numbers completely, leaving no remainder. For example, as we saw in the previous example, the common factors of 4 and 8 are 1, 2 and 4. These numbers divide both 4 and 8 completely. Follow the given steps to find the common factors of two numbers. Step 1: Find factors of the first number. Factors of 4

Step 2: Find factors of the second number. Factors of 8

Think and Tell Which is the lowest common factor

Step 3: Identify the common factors for both the

of 4 and 8? Which is the highest

numbers.

Factors of 4 and 8

common factor of 4 and 8? 1

Step 4: Write the common factors of the two numbers. The common factors of 4 and 8 are 1, 2 and 4. Example 7

Find the common factors of 15 and 18. Step 1 Factors of 15

Remember!

Step 2 Factors of 18

Step 3 Common Factors of 15 and 18

1 is a factor of ALL the numbers. So, it is also a common factor of any two numbers.

The common factors of 15 and 18 are 1 and 3.

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Do It Together

Find the common factors of 20 and 30. Also, find the highest common factor of both the numbers. Step 1 1

Factors of 20

Step 2 Factors of 30

Step 3 Common Factors of 20 and 30

The common factors of 20 and 30 are ____________________________________________________. The highest common factor of 20 and 30 is __________.

Do It Yourself 5D 1

Find the common factors of the following numbers. a 8, 10

b 12, 15

c 13, 16

d 14, 20

20, 30

g 33, 44

h 35, 50

f 2

72, 81

b 8

c 12

d 3

e 4

ind the common factors of the pairs of numbers. Also, find the lowest and the highest F common factors. Show it with a diagram. a 16 and 24

Which of the following numbers are factors of 78 and 96? Circle the correct option. Verify your answer. a 6

54, 64

e 16, 18

b 21 and 42

c 63 and 18

d 55 and 100

e 48 and 84

Write if True or False. a The biggest common factor of numbers 24 and 36 is 3. _______ b 11 and 13 have no common factors. _______ c 0 is a common factor of all the numbers. _______ d 15 and 25 have a total of 3 common factors. _______ e 6 is a common factor of 18, 30 and 66. _______ f

The lowest common factor of 20, 34, 39 and 42 is 1. _______

adhe says, “The number 14 has a greater number of factors than 45.” Is he correct? Verify your R answer.

Chapter 5 • Multiples and Factors

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Word Problem Bhanwar Lal is a farmer. He has 36 apple tree saplings and 48 orange tree saplings. He wants to give these saplings to the children in his village. But, he wants to distribute them in a way that each child gets the same number of apple tree saplings, as well as the same number of orange tree saplings. What is the largest number of children he can distribute the saplings to, in an identical manner, without any leftovers?

Points to Remember Factors

Multiples

We generally use division to find factors. Factors of a number are perfect divisors of that number.

We generally use multiplication to find multiples. Multiples of a number are found when that number is multiplied by other numbers.

8÷2=4

2×4=8

2 is a factor of 8.

8 is a multiple of 2.

The number of factors of a number are countable or limited.

We can have any number of multiples of a number. Multiples are unlimited.

Factors of 8 are 1, 2, 4 and 8 only.

Multiples of 8 are 8, 16, 24, 32, 40 and so on.

Factors of a number are either equal to or smaller than the given number.

Multiples of a number are either equal to or greater than the given number.

1, 2, 4 and 8 are factors of 8. The first three are smaller than 8.

2, 4, 6 and 8... are multiples of 2. Other than 2, all are greater than 2.

Common factors of two or more numbers are the numbers that divide the numbers completely, leaving no remainder.

A number that is a multiple of two or more numbers is a common multiple of those numbers.

5 is a common factor of 10 and 20.

20 is a common multiple of 5 and 10.

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Math Lab Setting: In groups of 4

Board Game of Multiples

Materials Required: Number grid as shown below, dice, crayons Method: 1

Each player chooses their colour.

One player rolls the dice and sees the number.

The player chooses a multiple of that number on the board and shades it with their colour.

In case a player gets 1 on the dice, they can choose any number on the board. (Do you know why?)

The player who colours the most number of multiples on the board is the winner.

Chapter Checkup 1

Find the factors. a 45 f

120

200

e 98 j

222

b 24 and 30

c 9 and 12

d 20 and 25

b 20 and 100

c 12 and 36

d 60 and 200

b 11

c 15

d 23

e 30

Write the smallest number which is a common multiple of the given number pairs. a 2 and 5

e 10 and 15

h 180

d 88

Write the first 5 multiples. a 7

c 72

Which of these pairs of numbers have the common factor of 4? a 5 and 20

g 156

Find the common factors of the given pairs of numbers. a 7 and 14

b 66

b 3 and 7 f

10 and 25

c 5 and 8

g 11 and 22

d 4 and 14

h 20 and 24

Find the following. a Multiples of 4 that are smaller than 30. b Multiples of 6 that are smaller than 50. c Multiples of 8 that are greater than 30 but smaller than 80.

Chapter 5 • Multiples and Factors

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d Multiples of 7 between 7 and 85 that are divisible by 2. e Multiples of 10 that are smaller than 200 but divisible by 3. f 7

Multiples of 12 that are bigger than 50 but smaller than 200.

Write the common factors. a 4 and 8

b 6 and 10

c 9 and 15

d 12 and 15

e 25 and 60

28 and 42

g 36 and 81

h 41 and 87

Write Yes or No for the following statements. a 1 and 6 are factors of 7. _____

b 1 is the smallest factor of 21. _____

c 1 is the smallest and only factor of 31. ______

d 2 and 4 are factors of 8. ______

e 6 and 9 are factors of 54. ______

g 306 is a multiple of 9. ______

h 16 is the biggest common factor of 32 and ______.

246 is a multiple of 3. ______

Find the multiples of all the odd numbers lying between 11 to 20 (including 11 and 20).

10 The bells at Church 1 ring after 60 minutes while at Church 2 after 45 minutes. At what time will the

bells at the 2 churches ring together next? [Hint: Find the common multiples.]

Word Problems 1

aina is organising a collection of toys. She N has 18 cars and 24 teddy bears. She wants to arrange them into groups with equal numbers of cars and teddy bears in each group. How many such toy groups can Naina create if she wants to put highest number of toys in each of them? Hint: Look for the biggest common factor!

florist is creating flower arrangements for a special A event. They have two types of flowers: roses and lilies. Roses look best when arranged in groups of 5, while lilies are best arranged in groups of 4. What is the smallest number of flowers the florist needs to create arrangements with both roses and lilies? Hint: To know the number of flowers in the arrangement, we should look for multiples.

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Fractions

Let's Recall This is a piece of paper. Let this piece of paper represent one whole.

One Whole

When we cut the paper into 2 equal parts, each equal part of a whole represents a fraction. 1 Each part is called half and is written as . 2

1 2

Now, if we cut another piece of paper of the same size into 4 equal parts, 1 each one out of the four parts is called one-fourth and is written as . 4 All these parts of a whole are called fractions.

1 4 1 4

1 4

otice that each fraction has a number on top. This signifies how many parts we are N talking about. This number on the top is called the numerator. 1 Numerator The number at the bottom is how many equal parts a whole is divided into. This is called the denominator.

Denominator

Let's Warm-up

Look at the picture and fill in the following blanks. 1

There are __________ fishes in the aquarium.

The fraction of yellow fish is __________ of all the fish.

The fraction of pink fish is __________ of all the fish.

The fraction of red fish is __________ of all the fish.

The fraction of purple fish is __________ of all the fish.

I scored _________ out of 5.

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Understanding Fractions Real Life Connect

Shalu, Karan, Raghav and Pooja went on a picnic. All of them brought snacks with them. Shalu: I have brought 2 apples and 4 oranges. Karan: I have 2 sandwiches. Raghav: I have 2 packets of biscuits and 4 packets of chips with me. Pooja: I brought 2 pancakes and a yogurt cup. The friends divided the snacks among themselves and enjoyed their day!

Halves, Quarters and Thirds We already learnt that dividing a whole (a shape or collection of objects) in equal parts gives us fractions. Let us revise it again.

Half • When a whole is divided into two equal parts, each part is called a half. 1 • It is written as . 2 • Two halves make a whole. 1 or half 2

1 whole 1 or half 2

Quarter

• When a whole is divided into four equal parts, each part is called a quarter or one-fourth. 1 or one-fourth 1 or one-fourth 1 • It is written as . 4 4 4 1 or one-fourth 1 or one-fourth • Four quarters make a whole. 4 4

One-Third

• When a whole is divided into three equal parts, each part is called one-third. 1 • It is written as . 3

• Three one-thirds make a whole.

1 or one-third 3 1 or one-third 3

1 or one-third 3

100

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Fraction of a Collection We know that we can also find the fraction of a collection. Remember! To find the fraction of a collection of objects, we identify Collection = Group of objects the numerator and the denominator of the given fraction. We make the same number of equal groups as the denominator and then put aside the same number of groups of objects as the numerator. We count the number of objects to get the fraction of the collection. Let us recall. 1 • T o find of a collection, we divide the objects equally into 2 groups. The number of 2 1 objects in each group is . 2 1 • T o find of a collection, we divide the objects equally into 3 groups. The number of 3 1 objects in each group is . 3 1 • T o find of a collection, we divide the objects equally into 4 groups. The number of 4 1 objects in each group is . 4 Numerator Number of parts chosen We also know that Fractions = = Denominator Total number of parts Let us learn other fractions of a number or collection of objects.

We already know how to find the fraction of a collection or a number. 1 For example, let us try to find of 24 objects. 6 Divide 24 into the same number of equal groups as the denominator 6. Then count the 1 objects in one group. There are 4 objects in each group. So, of 24 = 4. 6 But what if both the numerator and denominator are greater than 1? 2 Let us try to find of 27 butterflies. 3 We follow these steps: Step 1 Identify the numerator and the denominator. Divide 27 butterflies equally into the same number of groups as the denominator, 3.

Number = 27; Numerator = 2, Denominator = 3

Step 2 Since the numerator is 2, count the number of butterflies in 2 groups, which is 18. So,

3 of Earth’s water 100 is freshwater, available for drinking, agriculture, and other human needs. Only

2 of 27 = 18. 3

Chapter 6 • Fractions

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Did You Know?

101

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Example 1

What fraction of the given flowers are: a

white

pink

Total number of flowers = 7 Number of white flowers = 3 3 Fraction of white flowers = 7 Number of pink flowers = 4 Fraction of pink flowers =

Example 2

What is Step 1

4 7

3 of 35 cupcakes. 5

Identify the numerator and the denominator. Number of cupcakes = 35 Numerator = 3, Denominator = 5 Divide 35 cupcakes equally into the same number of groups as the denominator, which is 5 groups.

Step 2 Since the numerator is 3, count the number of cupcakes in 3 groups, which is 21. So, 3 of 35 = 21. 5 3 Hence, of 35 = 21 cupcakes. 5 Do It Together

5 of the ice creams. How many ice 6 creams did the ice-cream seller sell? Draw circles to show the ice creams in equal groups. An ice-cream seller has 36 ice creams. He sold

Total number of ice creams = 36 Fraction of ice creams sold = __________ Number of ice creams sold =

5 of 36. 6

Step 1

Step 2

Total number of ice creams = 36;

Hence, the ice-cream seller sold _________ ice creams.

Denominator = _________

102

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Do It Yourself 6A 1

Write the fraction represented by the shaded part of each image. a

Write the fractions for the numerators and denominators. a Numerator = 11, Denominator = 18

b Numerator = 10, Denominator = 30

c Numerator = 15, Denominator = 25

d Numerator = 8, Denominator = 16

Find and write the fraction for the collection of butterflies. 1 of the collection = __________ butterflies. a 3 4 of the collection = __________ butterflies. b 9 7 of the collection = __________ butterflies. c 9

Find.

1 of 32 4 2 of 33 e 3 4 of 75 i 5

1 of 56 8 5 of 90 f 6 2 of 49 j 7

1 of 65 5 3 g of 84 7 4 of 81 k 9 c

Anna has 48 roses left in her flower shop. roses are left?

1 of 72 6 5 of 96 h 8 3 of 88 l 8 d

1 3 of the roses wilted. of 48 roses were sold. How many 4 4

Word Problems 1

Manya is colouring a paper strip. She divides the paper strip into 10 equal parts and

Ranita has a bunch of 48 flowers, out of which

shades 5 parts out of it. What is the fraction represented by the shaded part of her strip? 1 are tulips. How many tulips are there? 6 3 Jane has a basket of 30 apples. If of the apples are green and the rest are red, how 5 many green apples are there? There are 50 pencils in a box. If

There are 75 balloons at a party. If

many pencils are blue?

how many balloons are white?

Chapter 6 • Fractions

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3 of the pencils are blue and the rest are black, how 4

2 of the balloons are red and the rest are white, 5

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Equivalent Fractions Let us take a circle and divide it into 2 equal parts. Colour divide it into 4 equal parts as shown below. 1 2

1 of the part, and then further 2

2 4

1 out of 2 parts 2 out of 4 parts 1 2 Here, and are fractions with different denominators and numerators but they 2 4 represent the same portion of shaded parts in the whole. Such kinds of fractions are called equivalent fractions. Equivalent fractions are fractions that have different numerators and denominators but represent the same value. 1 2 Here, and are equivalent fractions. 2 4 How to find equivalent fractions? To find equivalent fractions, we multiply the numerator and denominator by the same number. 1 In the above case, the numerator and denominator of the fraction is multiplied with 2 as: 2 1 2 1×2 2 × = = 2 2 2×2 4 We can also find multiple equivalent fractions of a fraction. Let us find 3 equivalent 5 fractions of . 6 5 To find 3 equivalent fractions of , multiply the numerator and denominator with any 6 three numbers. 5 Here we are multiplying the numerator and denominator of with numbers 2, 3 and 4. 6 First equivalent fraction of 5 2 5 × 2 10 × = = 6 2 6 × 2 12

5 6

Second equivalent fraction of 5 3 5 × 3 15 × = = 6 3 6 × 3 18

5 6

Third equivalent fraction of 5 4 5 × 4 20 × = = 6 4 6 × 4 24

5 6

Simplest Form of a Fraction A fraction is said to be in its simplest form when the denominator and numerator have no common factors other than 1. Let us find and learn the steps to find the 9 simplest form of a fraction . 45

Remember! A common factor is a factor shared by multiple numbers. Example: 7 is a common factor of 14 and 21.

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Step 1 List the common factors of the numerator and denominator of the fraction. Factors of 9 = 1, 3, 9

Think and Tell

Factors of 45 = 1, 3, 5, 9, 15, 45

Why we have not divided the numerator

Common factors of 9 and 45 = 1, 3, 9

and denominator by common factor 1?

Step 2

Divide the numerator and denominator with one of the common factors until it cannot be divided further. We can divide the numerator and denominator by either 3 or 9. Division by common factor 3

Division by common factor 9

9÷3 3 = 45 ÷ 3 15

9÷9 1 = 45 ÷ 9 5

3 3÷3 1 can be further divided by 3 as = 15 15 ÷ 3 5

1 cannot be divided further. 5

9 1 = . 45 5 Look at the figures and write them in the form of equivalent fractions. In both the cases; the simplest form of

Example 3

= 3 (3 out of 4 parts are shaded) 4 6 The fraction for the second figure can be given as (6 out of 8 parts are shaded) 8 3 6 The figures can be written in the form of equivalent fractions as = . 4 8 15 Find the simplest form of . 30 Step 1 The fraction for the first figure can be given as

Example 4

Factors of 15 = 1, 3, 5, 15 Factors of 30 = 1, 3, 5, 10, 15, 30 Common factors of 15 and 30 = 1, 3, 5, 15

Step 2

Divide the numerator and denominator with the one of the common factors. Let us divide them by 15. 15 ÷ 15 1 = 30 ÷ 15 2

1 15 1 cannot be divided further. Hence, the simplest form of = . 2 30 2

Do It Together

Find the equivalent fraction of Equivalent fraction of

Chapter 6 • Fractions

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5 . Also find its simplest form. 25

5 5 × 2 = 25 25 × __

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5 can be given as: 25 Factors of 5 = 1, ________ The simplest form of

Factors of 25 = ________, 5, ________ Common factors of 5 and 25 = ________, ________ Dividing the numerator and denominator with the common factor ________ we get, 5 ÷ __ 1 = . 25 ÷ __ __

Do It Yourself 6B 1

Shade an equal part and write an equivalent fraction. a

______________ = ______________ c

______________ = ______________ d

______________ = ______________

______________ = ______________ 2

Write four equivalent fractions for each of the given fractions. a

3 = _____, _____, _____, _____ 4

2 = _____, _____, _____, _____ 7

1 = _____, _____, _____, _____ 5

1 = _____, _____, _____, _____ 4

Write the simplest form of the given fractions. 30 15 12 a c b 90 45 24 2 5 6 g e f 26 65 42

10 50 8 h 26 d

Fill in the missing numerator or denominator in each of these equivalent fractions. 1 3 3 × = 2 3 [ ] 15 5 3 ÷ = e 25 [ ] [ ]

1 [ ] 3 × = 5 3 [ ] 2 [ ] 8 × = f 7 4 [ ]

[ ] 2 1 ÷ = 12 [ ] [6] 5 6 [ ] g × = 8 [ ] 48 c

Circle the fractions that are in their simplest form. 20 3 12 9 2 a c e b d 35 8 30 21 41

5 24

1 7 [ ] × = 3 [ ] [21] 27 [ ] 3 ÷ = h 45 9 [ ] d

8 14

7 42

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Like and Unlike Fractions Observe the given set of figures and the fractions they represent.

Think and Tell

Can equivalent fractions be

1 4

2 4

3 4

like fractions?

4 4

We can see that the denominators of the given fractions are the same. Fractions with the same denominator are called like fractions. Observe another set of figures and the fractions they represent.

1 2

1 3

1 4

1 6

We can see that the denominators of the given fractions are different. Fractions with different denominators are called unlike fractions.

Ordering and Comparing Like Fractions We know that the denominators of like fractions are the same. To compare two or more like fractions: • we compare their numerators.

• the greater the numerator, the bigger the fraction. For the above set of like fractions, on comparing the numerators, the fractions can be arranged in ascending form as:

< 1 4

Chapter 6 • Fractions

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< 2 4

< 3 4

4 4

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Example 5

6 12 or ? 17 17 As the denominators of the given fractions is the same, the given fractions are like fractions. 6 12 Numerator of = 6; Numerator of = 12 17 17 We know for the like fractions, the greater the numerator, the bigger the fraction. Which is greater

As 12 > 6; the fraction Example 6

12 6 > . 17 17

8 9 7 4 , , , and . 12 12 12 12 As the denominators of the given set of fractions are the same, the given set of fractions are like fractions. Compare and arrange the given set of fractions in descending order:

On comparing and arranging the numerators in descending order, we get: 9 > 8 > 7 > 4. The given fractions can be arranged in descending order as: Do It Together

9 8 7 4 > > > . 12 12 12 12

4 2 7 8 Compare and arrange the given set of fractions in ascending order: , , , and . 9 9 9 9 As the denominators of the given set of fractions are the same, the given set of fractions are like fractions. On comparing and arranging the numerators in ascending order, we get: 2 <___ <___ < 8. 2 The given fractions can be arranged in ascending order as: <___ <___ <___. 9

Ordering and Comparing Unlike Fractions with Same Numerators We know that the denominators of unlike fractions are different. To compare two or more fractions with the same numerator but different denominator: • we compare their denominators.

• the greater the denominator, the smaller the fraction. For the given set of unlike fractions, on comparing the denominators, the fractions can be arranged in ascending order as:

< 1 6

< 1 4

< 1 3

1 2

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Example 7

Which is smaller

5 5 or ? 9 12

As the denominators of the given fractions are different, the given fractions are unlike fractions. 5 5 Denominator of = 9; Denominator of = 12 9 12 We know for unlike fractions with the same numerator, the greater the denominator, the smaller the fraction. 5 5 As 12 > 9; the fraction < . 12 9 Example 8

2 2 2 2 Compare and arrange the fractions in ascending order: , , , and . 5 3 7 9 As the denominators of fractions are different, the fractions are unlike fractions.

Since the numerators are the same, when we compare and arrange the denominators in descending order, we get: 9 > 7 > 5 > 3. 2 2 2 2 The fractions can be arranged in ascending order as: < < < . 9 7 5 3 Do It Together

Compare and arrange the fractions in descending order:

7 7 7 7 7 , , , , and . 12 5 10 3 15

As the denominators of the fractions are different, the fractions are unlike fractions. Since the numerators are the same, when we compare and arrange the denominators in ascending order, we get: ______________________ The fractions can be arranged in descending order as: ______________________.

Testing for Equivalence by Cross-multiplying We have learnt the comparison of fractions when either the denominators or the numerators are the same. How would you compare two fractions with different numerators and denominators? Let us learn how to do it. To compare two fractions with different numerator and denominator: Step 1 Write the fractions next to each other.

Step 2

5 7

3 9

Multiply the numerator of the first fraction with the denominator of the second fraction and write the product below the first fraction.

Step 3 Multiply the denominator of the first fraction with the numerator of the second fraction and write the product below the second fraction.

Chapter 6 • Fractions

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5 7

3 9

5 7

3 9

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Step 4 Compare the products. The greater fraction will have the greater product below it. Here, 45 > 21. 5 3 Therefore, > . 7 9 Example 9

Compare the fractions Step 1 2 5

Do It Together

4 10

Compare the fractions Step 1 4 9

2 4 and . 5 10

Step 2

Step 3

2 5 20

4 10

4 7 and . 9 10

______

Step 3

[ ] 9

4 9

______

As, 20 = 20

4 10 20

Step 2 [ ] 10

Step 4

[ ] 10

______

Therefore,

2 4 . = 5 10

Step 4 As, ______ < ______ Therefore, ______ < _____.

Do It Yourself 6C 1

Name the fractions as like or unlike. a c

_______________

13 14 1 12 , , , – _______________ 15 15 15 15

2 6 10 9 , , , – 11 11 11 11 4 4 4 4 , , , – 8 14 8 12

Compare the fractions and put < or > or = sign in the box. 1 3 4 2 2 a c b 3 9 8 8 6 e

1 4 7 5 , , , – 9 9 9 9

8 10

5 15

5 7

5 9

Circle the smallest fraction in the group.

_______________ _______________

1 7

4 9

7 9

9 16

3 6 1 5 2 7 , , , , , 9 9 9 9 9 9

2 2 2 2 2 2 , , , , , 7 5 3 4 8 10

2 6 3 5 7 4 , , , , , 7 7 7 7 7 7

5 5 5 5 5 5 , , , , , 9 8 7 11 6 10

8 6 9 5 11 7 , , , , , 13 13 13 13 13 13

7 7 7 7 7 7 , , , , , 9 12 11 15 8 13

Circle the largest fraction in the group. a

2 4 3 5 1 7 , , , , , 8 8 8 8 8 8

4 4 4 4 4 4 , , , , , 6 5 11 9 8 10

5 8 3 9 7 10 , , , , , 11 11 11 11 11 11

3 3 3 3 3 3 , , , , , 9 15 7 8 6 5

8 6 4 5 2 7 , , , , , 9 9 9 9 9 9

5 5 5 5 5 5 , , , , , 9 10 11 7 8 13

3 7 7 15

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Arrange the fractions in ascending order. a

8 4 3 6 1 7 , , , , , 9 9 9 9 9 9

5 5 5 5 5 5 , , , , , 6 7 11 9 13 10

4 8 2 9 3 7 , , , , , 10 10 10 10 10 10

2 2 2 2 2 2 , , , , , 4 10 7 9 6 3

11 9 4 15 12 7 , , , , , 16 16 16 16 16 16

7 7 7 7 7 7 , , , , , 8 10 11 12 8 9

Arrange the fractions in descending order. a

9 4 3 10 1 7 , , , , , 12 12 12 12 12 12

7 7 7 7 7 7 , , , , , 12 15 11 9 8 10

15 8 13 9 7 10 , , , , , 17 17 17 17 17 17

3 3 3 3 3 3 , , , , , 9 10 7 4 6 5

1 6 4 5 2 7 , , , , , 8 8 8 8 8 8

5 5 5 5 5 5 , , , , , 15 6 11 7 8 10

Proper and Improper Fractions Fractions which are less than 1 whole are called proper fractions. In a proper fraction, the numerator is smaller than the denominator.

1 3

6 9

3 4

1 6 3 The fractions , and are proper fractions. 3 9 4

1 is a unit fraction. 3 Fractions which are equal to or more than 1 whole are called improper fractions. Proper fractions with numerator 1 are called unit fractions.

In an improper fraction, the numerator is greater than the denominator. Whole

4 4

Fraction

and

3 4

7 4

7 is an improper fraction. 4 Improper fractions can also be written as mixed numbers as shown below. The fraction

A mixed number or mixed fraction is a combination of a whole number and a proper fraction. Whole

Chapter 6 • Fractions

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Fraction

and 3 1 4

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Changing Improper Fractions to Mixed Numbers 9 into a mixed number. 4

Let us convert Step 1

Divide the numerator by the denominator and identify the quotient, remainder and divisor. Divisor

Quotient

Remainder

4 9 8

Step 2

Mixed number = Quotient Hence, Example 10

9 1 =2 . 4 4

Convert Step 1 Divisor

Convert Divisor

Remainder Divisor

The fractional part of the mixed number can never be an improper fraction.

13 to a mixed number. 4 3

4 13 12

Step 2

Quotient

Mixed number = Quotient Therefore,

Remainder

Do It Together

Error Alert!

13 1 =3 . 4 4

Remainder Divisor

56 to a mixed number. 12 _____

_____ 56 48 _____

Quotient

As, Mixed number = Quotient Therefore,

Remainder

56 = ________. 12

Remainder Divisor

Changing Mixed Numbers to Improper Fractions Let us convert 2 Step 1

3 into improper fraction. 4

Multiply the whole number part and the denominator of the mixed number. Whole number part × Denominator = 2 × 4 = 8

Step 2

Add the numerator to the product obtained in step 1. Product + Numerator = 8 + 3 = 11

Step 3

Write the sum as the numerator and retain the denominator of the mixed number. Product + Numerator 11 = Denominator 4

Hence, 2

3 11 = . 4 4

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Example 11

Convert 5 Step 1

Do It Together

2 to an improper fraction. 3

Step 2

Step 3

Whole number part × Denominator

Product + Numerator

Improper fraction

= 5 × 3 = 15

= 15 + 2 = 17

Convert 4

Product + Numerator 17 = Denominator 3

3 to an improper fraction. 6

Step 1

Step 2

Step 3

Whole number part × Denominator

Product + Numerator

Improper fraction

= _______ × 6 = _______

= _______ + 3 = _______

Product + Numerator [ ] = Denominator 6

Whole Number as a Fraction • Whole numbers can be written as fractions with denominator 1. 5 7 10 For example, 5 = , 7 = , 10 = . 1 1 1 • When the numerator is equal to the denominator, the fraction represents 1 whole.

3 =1 3

7 =1 7

9 =1 9

• When the numerator divides the denominator completely without leaving a remainder, the fraction represents a whole number. Example 12

What is the value of the fraction?

12 =3 4

The fraction can be written as

Do It Together

9 = 3. 3

Express the figures in the form of fractions. 1

6 = ______ [ ] Chapter 6 • Fractions

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Do It Yourself 6D 1

Sort the given fractions into proper, improper and mixed. 2 15 2 a c 3 b 3 8 6 e 2

86 10

g 2

52 7

8 10

5 7

Simplify the fractions. 6 24 a b 3 8 e

3 7

1 9

65 9

Convert the mixed numbers into improper fractions. 1 5 2 a 5 c 7 b 2 3 8 6 e 3

Convert the improper fractions into mixed numbers. 8 26 25 a c b 3 8 6 e

6 10

20 10

Susan had 1

35 7

16 9

13 6

49 9

91 6

d 4

1 9 2 6

g 5

5 9

h 9

c 3

30 6

36 9

18 6

45 9

3 of the apple pie. Write the amount of apple pie Susan has as an improper fraction. 4

Operations on Fractions Real Life Connect

Vishal helps his mother with household chores. Today, he is helping his mother to prepare the dinner. Mother: Vishal, could you please bring some sugar from that box? Vishal: How much sugar should I bring? 2 Mother: I need cup of sugar for the cake batter 4 1 and cup for the milkshake. 4 Vishal brings the required amount of sugar and gives it to his mother.

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Adding and Subtracting Like Fractions We saw that Vishal brought the required amount of sugar for his mother. Do you know how? He simply added the amount of sugar needed for both the recipes and brought the required amount.

Adding Fractions

Error Alert!

Let us find how much sugar Vishal brought.

Do not add denominators.

Adding Two Like Numbers: Step 1 Add the numerators and keep the denominator the same. Total cups of sugar =

Step 2

2 1 2+1 3 + = = cups. 4 4 4 4

3 2 5 + = 9 9 18

3 2 5 + = 9 9 9

Reduce the fraction to its simplest form. 3 In the above case, the fraction is already in its simplest form. 4

Adding Mixed Numbers: 1 1 Let us add 1 and 2 . 3 3 Step 1

Step 2

Convert the mixed fraction to an improper fraction.

Add the numerators and keep the denominators

1 4 1 7 = , and 2 = 3 3 3 3

the same.

4 7 4 + 7 11 + = = 3 3 3 3

Step 3

Reduce the fraction to its simplest form and convert the improper fraction to mixed numbers. 11 11 2 The fraction is in its simplest form. Convert to a mixed fraction = 3 . 3 3 3 Example 13

Find the sum of: As

11 3 and . 16 16

11 3 and are like fractions, add the numerators and keep the denominator the same. 16 16

11 3 11 + 3 14 + = = 16 16 16 16 Reducing

14 to its simplest form gives: 16

14 14 ÷ 2 7 = = . 16 16 ÷ 2 8 Chapter 6 • Fractions

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Example 14

2 1 Add: 1 and . 6 6 2 As 1 is a mixed fraction, changing it to an improper fraction gives: 6 2 8 = 6 6

8 1 and are like fractions. Adding the numerators and keeping the denominator the 6 6 same gives: Now

8 1 9 + = 6 6 6 Reducing the above fraction to its simplest form gives: 9 9÷3 3 1 = = =1 . 6 6÷3 2 2

Do It Together

1 1 kg of bananas and 1 kg of apples. What is the total weight of the 4 4 fruit bought by Rohan? 1 9 Weight of bananas = 2 kg = kg (Converting mixed number to improper fraction) 4 4 1 Weight of apples = 1 kg = _______ kg 4 1 Total weight = + _______ = _______ 2 Converting _______ kg to an improper fraction, we get _______. Rohan bought 2

Subtracting Fractions Subtracting Like Fractions 3 5 Let us subtract from . 6 6 Step 1 Subtract the numerators and keep the denominator same. 5 3 5–3 2 – = = 6 6 6 6

Step 2 Reduce the fraction to its simplest form. 2 2÷2 1 = = 6 6÷2 3

5 3 1 – = . 6 6 3 Subtracting Mixed Numbers Hence,

Let us subtract 2 Step 1

1 3 from 3 . 4 4

Convert the mixed number to an improper fraction. 2

1 9 3 15 = , and 3 = . 4 4 4 4

Step 2 Subtract the numerators and keep the denominators the same.

15 9 15 – 9 6 – = = . 4 4 4 4

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Step 3 Reduce the fraction to its simplest form and convert the improper fraction to a mixed number. 6 6÷2 3 1 = = =1 . 4 4÷2 2 2

Subtract

Example 15

As 2 2

5 2 from 2 . 12 12

2 is a mixed number, 12

2 26 = 12 12

26 5 26 – 5 21 – = = 12 12 12 12 21 21 ÷ 3 7 3 = = =1 12 12 ÷ 3 4 4

Do It Together

1 2 m of ribbon. She used m of it. How much ribbon does she have left? 5 5 1 Length of ribbon bought = 2 m = ______ (Converting mixed number to improper fraction). 5 2 Length of ribbon used = m. 5 2 Length of ribbon left = ______ – = ______ m. 5 Converting ______ to an improper fraction, we get ______. Riya bought 2

Do It Yourself 6E 1

Add. a

2 5 + 3 3

e 2

5 2 – 9 9

c 3

2 5 + 6 6

d 1

3 7 + 5 5

g 2

1 5 +3 9 9

13 7 – 8 8

1 5 + 9 9

11 1 +1 6 6

7 2 – 10 10

Solve. a 2

1 2 – 3 3

b 3

e 3

1 4 –2 9 9

Chapter 6 • Fractions

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11 5 + 8 8

Subtract. a

6 2 +1 8 8

5 8 –1 9 9

c 3

2 4 – 5 5

d 1

3 4 – 5 5

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4 5

3 2 of his garden with tomatoes and of his garden with cucumbers. How much of his 6 6 garden is planted in total? 3 1 Suhani spent of an hour reading a book, and then she spent an additional of an hour drawing. How 5 5 much time did she spend on both activities? John planted

Word Problems 1 2 3 4 5 6

5 7 Ria has 1 m of cloth. She used m to cover a chair. What length of cloth does she has left? 8 8 1 2 Varun read of the pages of his book on Monday and of the pages on Tuesday. 4 4 What fraction of the book did he read in all? 1 3 Sunil ran 1 km on Saturday. On Sunday, he ran 2 km. How much farther did Sunil 6 6 run on Sunday than on Saturday? 2 7 Manya had 1 packet of cookies. She ate of the cookies. What fraction of cookies is 8 8 she left with? 2 4 A jogging track is 2 km long. A cycling track is 3 km longer than the jogging track. 8 8 How long is the cycling track? 4 1 A construction worker used of a bag of cement on one project and of the same 5 5 bag on another project. How much cement is left in the bag?

Points to Remember •

Fractions are equal parts of a whole or collection.

• Equivalent fractions are the fractions that have different numerators and denominators but represent the same value.

• To find equivalent fractions of a fraction, multiply the numerator and denominator by the same number. • A fraction is said to be in its simplest form when the denominator and numerator have no common factors other than 1. • •

Like fractions have the same denominator. Unlike fractions have different denominators. A fraction in which the numerator is smaller than the denominator is a proper fraction.

• A fraction in which the numerator is equal to or greater than the denominator is an improper fraction. • An improper fraction can be represented as mixed numbers. Mixed numbers have two parts: The whole number part and the fractional part.

• To add and subtract two like fractions, we add or subtract the numerators and keep the denominator the same.

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Math Lab Exploring Fractions! Setting: In groups of 4. Materials Required: Fraction cards or pieces of paper

with fractions written on them (include a mix of proper

fractions, improper fractions, and mixed numbers), Paper and pen, Timer.

2 6

3 7

1 2

1 4

Method: Distribute the fraction cards or pieces of paper among the groups. Ask the groups to sort the fraction cards into 3 categories: proper fractions, improper fractions, and mixed numbers.

Track the time each group takes to sort the fractions. The group with greatest number of correct answers in the least time wins!

Chapter Checkup 1

Shade or draw the fractions. a

1 of 18 flowers 6

7 10

3 8

4 7

1 of 27 cakes 3

1 of 36 boxes 4

1 of 50 balloons 5

Write four equivalent fractions for the following. a

5 = _______, _______, _______, _______, 6

7 = _______, _______, _______, _______, 8

3 = _______, _______, _______, _______, 9

2 = _______, _______, _______, _______, 7

Complete the equivalent fractions. a

1 [ ] = 2 8

4 [ ] = 3 27

7 21 = 9 [ ]

8 16 = 12 [ ]

2 8 = 5 [ ]

5 [ ] = 6 24

6 30 = 11 [ ]

5 [ ] = 13 39

Chapter 6 • Fractions

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Find the fraction of a collection of objects. a

5 9

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Reduce each fraction to its simplest form. a

50 90

25 80

6 24

5 35

24 56

16 76

15 70

27 36

5 6

1 7

2 9

5 7

6 9

2 3

14 16

11 15

Compare the fractions and put < or > or = sign in the box. a

13 17

7 17

8 10

9 10

2 7

3 7

8 11

8 15

Arrange the fractions in ascending and descending order. a

8 4 3 6 1 7 , , , , , 11 11 11 11 11 11

7 7 7 7 7 7 , , , , , 8 12 11 9 13 10

4 8 2 9 3 15 , , , , , 17 17 17 17 17 17

4 4 4 4 4 4 , , , , , 12 10 7 9 6 8

11 9 4 15 12 7 , , , , , 19 19 19 19 19 19

3 3 3 3 3 3 , , , , , 8 10 11 12 8 9

Convert the improper fractions into mixed fractions. a

16 3

26 5

32 6

53 4

92 5

65 7

75 8

88 6

Convert the mixed numbers into improper fractions. a

1 4

3 7

1 6

3 5

5 10

5 6

5 8

2 5

2 1 + 5 5

11 5 + 7 7

2 5 + 4 4

1 5 + 8 8

3 7 + 8 8

1 4 +3 9 9

15 1 +1 2 2

13 5 – 9 9

1 3 – 4 4

9 5 – 11 11

5 2 –2 6 6

18 2 –2 5 5

10 Add.

3 2 +1 6 6

11 Subtract. a

5 4 – 8 8

2 3 – 4 4

4 6 –3 7 7

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Word Problems 1 2

3 4 5 6 7 8 9

3 other cut a cake into 20 equal pieces. Sunita ate of the pieces. How many M 5 pieces did Sunita eat? 2 3 A frog took two jumps. The first jump was m long and the second jump was 9 9 m long. How far did the frog jump altogether?? 5 A shopkeeper has 28 kg of rice. He sells kg of the rice. How much rice does the 7 shopkeeper have left? 4 Kavya cuts m of wire from a 10 m long piece of wire. How much wire does 5 Kavya have left? 1 3 A vessel contains 2 litre of milk. John drinks litres of milk. How much milk is 4 4 left in the vessel? 1 3 Mohit drove 7 km on Monday and 5 km on Tuesday. How far did he travel on 5 5 both days? 5 There are 36 students in a class. On Friday students were absent. How many 12 students were absent on Friday? 3 A farmer has 56 cows. of them are grazing in the field, and the rest are in the 7 barn. How many cows are in the barn? aria bought a pizza and divided it into 8 equal slices. She ate 3 slices, and her M brother ate 2 slices. What fraction of the pizza did they eat altogether?

10 Sudha has

Chapter 6 • Fractions

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2 1 of 50 rupees, Ravi has of 50 rupees. Who has more money? 5 2

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Lines and 2-D Shapes

Let's Recall When we open our eyes to the world around us, we can find fascinating examples of shapes and patterns around us in our everyday lives. From the wheels of a bicycle to the slices of pizza we enjoy, the shapes are present everywhere! We have already learnt about sleeping, standing and curved lines, triangles, circles, squares and rectangles. Let’s recall what we have learnt previously about flat shapes.

This is a TRIANGLE

This is a SQUARE

This is a RECTANGLE

This is a CIRCLE

Let's Warm-up Match the following objects with the shape that they resemble. Objects Shapes a

Circle

Triangle

Square

Rectangle I scored ___________ out of 4.

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Understanding Basic Terms Real Life Connect

Ritu is playing a game of joining the numbered dots.

Ritu: Look Mom! I have joined the dots and got a picture of a hut.

Mother: Wow! This looks quite nice! 5

Points, Rays and Lines

Notice the figure that Ritu has just drawn. She joined the numbered dots or points in order. The lines formed an interesting shape of a hut. Now, when we learn geometry, “points” and “lines” have specific meanings. Let us learn about them.

Points

A point shows the exact position of an object. It is represented by a dot (.). Points are named using capital English letters, such as A, B, etc. The figure shows Point A.

A point by itself cannot be measured, as it doesn’t have any length, breadth, or height. But, it can be used to describe a location or position of an object. The map below shows the position of the hotel, the house, and the museum using three different points.

Rays What if Ritu draws a straight path from point A to point B, as shown, and extends it in one direction? When a straight path starts at a point and extends endlessly in the other direction, it is called a ray. Chapter 7 • Lines and 2-D Shapes

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A ray has a starting point. But it has no endpoint. Therefore, it has no fixed length either. In the ray shown, A is the starting point. We can represent the above ray as AB. Real-life examples of rays are very limited because they are unending at one end. Light emitted by the Sun is an example of rays.

Error Alert! Ray AB is different from ray BA. In one, A is the starting point while in the other, B is the starting point.

Line What if Ritu keeps on drawing the straight path beyond two points, so that both the ends are unending, as shown in the image of the hut? When a straight path extends in both directions and has no end points, it is called a line. Just like a ray, the length of a line cannot be measured as it is unending. But as opposed to a ray, it is unending in both directions.

A line therefore has no starting point and no end point. It is represented with the help of points that fall on it. The line shown can be represented as XY. A line can also be represented by a small letter, say l, m, or n.

Real-life examples of lines are almost impossible to find because they have no ends. Example 1

Mark the correct geometric elements in the figure. O, A, B, C, and D are points.

A C O

AB and CD are lines. OA, OB, OC, and OD are rays.

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Circle the correct representation of the rays:

Example 2

B Q

Do It Together

or or

Think and Tell

How many line(s) can pass through two given points?

Look at the figure and fill in the blanks.

The points are: A, _____, _____, _____, _____, and _____.

The figure has rays: DE, _____, and _____. The lines in the figure are: l and _____. Line l is the same as FC and line m is the same as _____.

C D A

Do It Yourself 7A 1

State whether the following statements are true or false. a A ray has no end points. ______________ b A line has no end points. ______________ c Only one line can pass through a point. ______________ d The light from a torch is an example of a ray. ______________ e A line extends endlessly in both directions. ______________ f

A point has only length. ______________

Name the points, rays, lines, and line segments in the figures. Q

c Q

P __________________ 3

__________________

Name the following in the figure.

a Name all the points.

b Name all the rays.

c Name all the lines.

Name any 2 points, 2 rays, and 2 lines in the figure. H E

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K F

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Line Segments Remember how Ritu joined the dots on the paper to form a hut? When she joined two points, she drew something called the “Line Segment”. So, the straight path between any two points is called a Line Segment. A line segment is a piece of the line that has a definite length. Always remember, the line segment is the shortest distance between two points. The shown line segment can be represented as PQ or simply PQ. In the world around us, we can find many examples of line segments:

Think and Tell

Can a ray and a line segment be parts of the same line?

A straight tight rope

A tubelight

Let us learn more about measuring and drawing line segments.

Measuring Line Segments We measure the length of a line segment using a ruler or scale. The large number markings that you see on the scale are centimetres (cm) as shown below. 5 cm 4 cm 3 cm 2 cm 1 cm

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Let us learn to use the ruler and measure the segment AB as shown: A

Step 1

Place the edge of the ruler along the line segment AB so that the zero

mark of the ruler is at A. Hold the ruler firmly along the line for accurate measurement.

Step 2 Read the mark on the ruler at point B. It can be seen that the end point B reaches the mark 7 on the ruler.

Thus, the length of the line segment is 7 cm.

Error Alert! NEVER put the zero mark of the ruler at any other point of the segment except the starting point which is zero.

Measure the lengths of the line segments using a ruler.

Example 3

The starting point of the segment is at the 0 mark of the ruler. The end point B comes to the “8” marking. Therefore, the length of PQ = 8 cm. P

Do It Together

Did You Know? A ‘smoot’ is a funny unit of measurement named after a person, Oliver R Smoot in 1958, who lay down repeatedly on a bridge to measure its length. The bridge was approximately 364.4 smoots long! Needless to say, hardly anyone in the world uses this unit to measure length!

ook at the objects placed along the edge of a ruler. Read their lengths carefully (in cm) L on the scale and fill in the blanks.

Think and Tell The length of the pencil is 6 cm. The length of the comb is ___________ cm.

Can a line be measured using a ruler whose zero mark is missing?

Drawing a Line Segment Let us now learn to draw line segments using a ruler. Let us assume that we want to draw a line segment that is 5 cm long. Chapter 7 • Lines and 2-D Shapes

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Step 1 Place the ruler firmly on the paper and mark a point

with a sharpened pencil against the zero mark of the ruler. Name the point as, say, A.

Step 2 Starting from the point A, move the pencil along the edge of the ruler and draw a

line segment of the required length, i.e. 5 cm. Name the other point as B. The drawn line segment AB is 5 cm long.

Always remember to position your eye directly above the measurement markings on the rule for accurate results. Position of the eye Correct

Incorrect Pencil

Incorrect Ruler

Do It Yourself 7B 1

Measure the length of the highlighted edges of the objects. a

Edge of the book = ___________ cm 2

Use a ruler to draw line segments of the lengths given. a 6 cm

Edge of the deck of cards = ___________ cm

b 9 cm

c 10 cm

d 14 cm

The figure has _________ line segments. Measure their lengths.

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Look at the two pencils. Tina’s pencil: Sheena’s pencil: What will be the total length of the two pencils?

Anu draws a line segment of length 6 cm. Jiya draws a line segment which is 4 cm more than Anu’s. What is the length of Jiya’s line segment? Draw both of the line segments.

Word Problems 1

Look at the image and find the correct length of the plank.

Hint: The distance between the starting point of the plank is 1 cm.

Understanding More Geometrical Figures Real Life Connect

Shaarvi and Kavya love to scribble and draw! Shaarvi: What shall we draw today, Kavya? Kavya: We will draw free-hand shapes today! Shaarvi: Wow! I want to try this!

Open and Closed Figures In their first try, the girls drew the following shapes: Do you notice anything in these shapes that the girls have drawn? Shapes A and B are “open”. This means that the starting point of the shape and the end point are not the same. Shapes C and D on the other hand are “closed”. They are continuous and while they were being drawn, the pen/pencil would not have been lifted before completing. Chapter 7 • Lines and 2-D Shapes

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Open Figures

Do It Together

Closed Figures

There is a gap in the boundary of the figure.

The figure is continuous and there is no gap in the boundary.

The figure has a different starting point and ending point.

These figures have no end points.

Look at the figures shown below. Classify these as open or closed figures. 1

Closed figure

Open figure

_____________

Do It Yourself 7C 1

Sort the following shapes as open or closed figures. a

Which 3 letters from the English alphabet are open figures?

Which 3 letters from the English alphabet are closed figures?

Which of the following is NOT a characteristic of an open shape?

a It has different start and end points. b There is a gap in the boundary of the figure. c The figure is continuous.

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Simple and Non-simple Figures What if Shaarvi and Kavya drew the following shapes? How are they different from each other? Let us learn!

What do you notice in these shapes? In shape B, the boundaries do not cross over at any point, while in shapes A, C and D, the boundaries of the shapes cross over one another. The figures that do not cross at any point are called simple figures. In this case, only shape B is a simple figure. Shapes A, C and D are non-simple figures. Let us look at more examples of simple and non-simple figures: Simple Figures

Do It Together

Non-simple Figures

Look at the figures shown below. Classify these as simple or non-simple figures. 1

_____________ Chapter 7 • Lines and 2-D Shapes

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_____________

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Do It Yourself 7D 1

Sort the following images as simple figures or non-simple figures. a

Simple figures -

_________________________________________________

Non-simple figures - _________________________________________________ 2

Draw any 2 simple figures and 2 non-simple figures.

Which of the following is the feature of a non-simple curve. a The boundaries do not cross themselves. b Simple figures can be closed or open figures. c The starting point and the end point of the figures are always the same.

Akhil was arranging sticks to make different shapes for his holiday homework. He has 6 sticks. Draw 3 simple and 3 non-simple figures that he can create using the sticks. Hint: He may or may not use all the stick at once.

Can a figure be non-simple and open? If yes, draw the figure to show it.

Classifying Figures We have learnt that figures can be open or closed. We also learnt that they can be simple or non-simple. Let us look at different possible types of figures: Simple and Closed

Simple and Open

Non-simple and Closed

Non-simple and Open

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Polygons Now, let us learn about a special type of simple closed figures - Polygons. Simple closed figures that are made of only line segments are called polygons. Some common polygons are a triangle, a square, and a rectangle.

Did You Know?

The line segments that form the polygon are called its sides. The points where two sides meet is called a vertex. A polygon is named on the basis of the number of sides it has. The table shows the different types of polygons.

Example 4

A polygon with 1 million sides is known as a Megagon.

Triangle 3 sides and 3 vertices

Quadrilateral 4 sides and 4 vertices

Pentagon 5 sides and 5 vertices

Hexagon 6 sides and 6 vertices

Heptagon 7 sides and 7 vertices

Octagon 8 sides and 8 vertices

Nonagon 9 sides and 9 vertices

Decagon 10 sides and 10 vertices

Recognise the simple closed figures. 1

The shapes 1, 3, and 4 are simple closed figures as they do not cross themselves at any point. Example 5

Recognise whether the shapes are polygons or non-polygons.

Simple Closed Figure?

Yes

Polygon?

Yes

Reason

Simple and closed, Simple and closed, Simple and closed, all sides are line all sides are line but part of the segments. segments. shape is a curve.

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Not a simple closed figure.

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Do It Together

Join the dots with line segments and write the name of the polygon formed. 1

Triangle

Do It Yourself 7E 1

Write if true or false. a All simple closed shapes are polygons. ______________ b A shape that crosses itself is not a simple closed shape. ______________ c A polygon can be formed with two lines. ______________ d A hexagon has 7 sides. ______________

Categorize each of the following figures as simple closed, non-simple closed, simple open or nonsimple open. a

d Hexagon

e Heptagon

Draw the following polygons. a Triangle

Which of these are simple figures? a

b Rectangle

c Pentagon

Identify the polygons. For the ones that are not polygons, substitute with a possible polygon. a

Word Problem 1

Shanaya is making a bracelet using beads. She wants to create a bracelet in the

shape of a closed figure. If she can only use 8 beads to make the bracelet, what is the maximum number of sides the closed figure can have?

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Circles and Its Parts Real Life Connect

Have you ever tried to draw a circle in a field, on sand, or on a beach? As shown in the image, you can sit at a place and try to draw with a stick in your hand by rotating your body. The shape that you can get is almost a circle!

Constructing Circles What you drew on the ground was ‘almost’ a circle. How can we construct a perfect circle? We use an instrument called the compass. As shown in the figure, a compass consists of two movable arms joined together where one arm has a pointed end, and the other arm holds a pencil. Now let us learn how to draw circles. Step 1 Insert a pencil into the pencil holder of the compass. Make sure to tighten the nut so that the pencil remains fixed.

Step 2 Place the pointed end of the compass on the paper as a fixed end.

Step 3 Rotate the pencil about the pointed end which is now fixed on the paper. The shape that is drawn on the paper is a circle!

Error Alert! EVER displace the tip of the compass from its position and N ALWAYS tighten the screw of the compass else it can result in an incomplete or imperfect circle.

Remember! The point where the needle of a compass is placed becomes the centre of the circle.

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Parts of a Circle The Centre While constructing a circle, we placed the pointed end of the compass on the paper. The circle was drawn around this point. This is the centre of the circle.

The centre of a circle is a point inside the circle so that the distance between any point on the circle and the centre is equal. In the circle, point ‘C’ is called the centre of the circle.

The Radius The radius of the circle decides how big the circle will be. The distance between the centre of a circle and any point on it is called the radius of the circle.

If we want to construct a circle of a specific radius, we open the compass to a measured distance before drawing the circle. Say, we want to draw a circle of 3 cm radius. O

We first put the pointed end of the compass at the 0-mark of the ruler. Extend the arm holding the pencil up to 3 cm (radius of the circle) on the ruler as shown. And then we draw the circle as described previously.

The Diameter The line passing through the centre with both its ends lying on the circle is known as the diameter of the circle. Here, AB is the diameter of the circle.

A diameter cuts the circle into two equal parts. Each half is called a semicircle. A diameter is twice the radius of a circle. Diameter = 2 × Radius, or Radius =

1 2

Think and Tell How many diameters can a circle have?

× Diameter

Remember! All the radii (plural of radius) of a circle are of the same length.

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The Circumference Circumference

The length of the boundary of a circle is called its circumference. Identify and write the name of the radius, centre and diameter of the circle.

Example 6

Centre of the circle: O is the centre Radii: OA, OB, OC, OD, OF, OE

Diameters: AB and EF Find the diameter of a circle of which radius is 6 cm.

Example 7

So, in this case,

O B

We know that, Diameter = 2 × Radius or Radius = Diameter ÷ 2

Think and Tell

Diameter = 6 cm x 2 = 12 cm.

Does the radius of a circle impact its

Construct a circle of radius 4 cm.

Example 8

circumference?

Step 1 Fix a pencil in the hole of the compass.

Step 2 Place the pointed end of the compass on the paper as a fixed end.

Step 3 Take a ruler and place its 0 mark at the pointy hand of the compass. Using the ruler, open the other hand to a measure of 4 cm.

Step 4 Keeping the pointy hand fixed, rotate the hand with pencil, starting and finishing at the same point.

Step 5

4 cm

Label the centre of the circle at the point where the tip of the needle was resting as O. Join O to any point on the circle. Label the point as P. OP is the radius. Do It Together

Identify the centre and every radius and diameter of the circle. Centre

Radii

Diameter

____

OA, ____, ____ and ____

____

O D

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Do It Yourself 7F 1

Fill in the blanks. a Every point on a circle is at the same distance from the _______________. b All the radii of a circle are _______________ in length. c A circle can have _______________ number of diameters. d The length of the boundary of a circle is called its _______________. e A circle has only _______________ centre.

Label the parts of the given circle.

Choose the correct answer. a A circle with a diameter of 10 cm is drawn. What will be its radius?

i. 20 cm

ii. 4 cm

iii. 5 cm

iv. 6 cm

b A diameter divides a circle into _______________ equal parts.

i. 3

ii. 2

iii. 4

iv. 8

c What is the relation between the radius and diameter of a circle?

ii. Radius =

iii. Diameter = Radius 2

iv. Radius = Diameter 2

Draw the circles of the given radii. a 2 cm

2 Diameter

i. Radius = 2 × Diameter

b 5 cm

c 6 cm

d 8 cm

Draw a circle so that: a its radius is 7 cm.

b its diameter is 8 cm.

Ansh is running on a circular field. The distance from the centre of the ground to its boundary is 16 m.

Draw a circle of which diameter is 6 cm and mark its centre as C. Draw a radius CD.

Find the diameter of the field.

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Points to Remember •

A point shows an exact location.

•

A point is represented by a dot (.) using a capital letter.

•

A point has no dimensions, i.e., length, breadth, or height.

•

A line extends endlessly in both the directions. The length of a line cannot be measured.

•

A line segment is a part of a line. It is the shortest distance between two points.

• A ray has one starting point, also called an end point and extends in one direction. It has no fixed length. • A shape or a figure that begins and ends at two different points is called an open shape or open figure. A shape of figure that begins and ends at the same point is called a closed figure. •

A simple closed figure does not cross itself at any point.

•

Simple closed figures made of line segments only are called polygons.

• The distance from the centre of a circle to its circumference is called the radius of the circle. •

The diameter of a circle is twice the radius.

• The geometrical instrument used for drawing a circle of a given radius is called a compass.

Math Lab Circle Designs Setting: Individual Materials Required: A compass, sharpened pencils of different colours Method: Creating circle designs can be a fun and creative process. Let’s create the following design using a compass. Step 1: Draw a circle of any radius, say, 3 cm in the centre of a paper. Step 2: Using the same length of radius, draw three more circles so that they pass through the centre of the first circle.

Step 3: Colour the design using your favourite colours. Try making more designs using a compass.

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Chapter Checkup 1

Tick () the correct answer. a

Which of the following represents a line?

AB represents a

How many line segments are there in the figure?

A line segment has ________ end point (s).

Which of the following is an example of a ray?

i. ray i. 10 i. 1

ii.

iii.

iv.

ii. line segment

iii. line

iv. point

ii. 11 ii. 2

i. an arrow released from a bow iii. light from a torch

A dot made with a pen is an example of

The letter C is an example of a/an

i. circle

ii. polygon

iii. 13

iv. 12

iii. 3

iv. 4

ii. a javelin

iv. railway tracks iii. point

i. closed figure ii. simple closed figure iii. open figure

iv. ray iv. non-simple closed figure

An octagon has _________ line segments.

The line joining the centre of a circle to any point on its boundary is called the __________ of the circle.

i. 10

ii. 9

i. centre

iv. 7

ii. diameter

iii. circumference j

iii. 8

iv. radius

The diameter of a circle whose radius is 5 cm is ___________. i. 12 cm

ii. 9 cm

iii. 10 cm

iv. 15 cm

What will be the diameter of the circle that is drawn with the following compass?

Identify the letters and numbers as closed, open, simple, or non-simple figures.

Which of the following is a polygon?

What is the name of the polygon that has two more sides than the given shape?

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Write the names and number of sides of the following polygons. Polygon Name Number of sides

Draw the following circles: a c

7 8 9

diameter = 4 cm radius = 5 cm

b d

radius = 2 cm

diameter = 6 cm

Mary is standing at point A. She wants to get to point B by choosing the shortest

E D

route. Which route should she take?

Find the distance between points A and D in the figure. Given that AB = 12 cm and OP = 4 cm.

4 cm

12 cm O

A jar has a diameter of 16 cm. A man wants to buy a lid for the jar so that its radius is 2 cm more than the jar. What should be the radius of the lid?

10 Kanchi was playing with a circular frisbee. She thought to make a frisbee at home with a radius of 2

cm less than that of the original frisbee. If the diameter of the original frisbee is 18 cm, then construct a circle of the size of the frisbee that Kanchi made.

Word Problems 1 S urbhi wants to buy some stationery. Find the shortest distance she has to travel, if the shop is located at the other end of the park as shown in the image.

Surbhi's house

Stationery shop

2 A horse is tied in a field full of grass. The length of the rope is 10 metres. If the horse starts grazing everywhere that it can reach, what will be final shape of the area that has no grass left?

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Representing 3-D Shapes

Let's Recall We have learnt about different types of shapes. We know that the lines that form a shape are called sides. The point where the sides meet is called a corner. Let’s recall the shapes that we’ve learnt about previously. Flat shapes are called 2-D shapes. We can measure only their length and width. Flat shapes do not have any thickness or depth. Let us look at some flat shapes.

SQUARE

CIRCLE

TRIANGLE

RECTANGLE

SEMICIRCLE

Solid shapes are called 3-D shapes. All objects around us are 3-D shapes. Of these we can measure the length, width and thickness or depth.

This die is in the This duster is in the shape of a cube. shape of a cuboid.

CUBE

CUBOID

This water bottle is in the shape of a cylinder.

This party hat is in This football is in the the shape of a cone. shape of a sphere.

CONE

CYLINDER

SPHERE

Let's Warm-up Label each of these shapes as a 2-D or a 3-D shape. a

I scored _________ out of 5.

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Representing 3-D Shapes as 2-D Shapes Real Life Connect

Riya and her sister Pooja are fond of drawing pictures. Both of them decided to draw a picture of their car. Riya: I will draw it while sitting at the window.

Pooja’s drawing

Pooja: I will draw the picture while standing on the roof. They completed their drawings and showed them to each other. Both of them wondered how they had drawn different pictures of the same car! Let us see who has drawn the correct picture of the car.

Riya’s drawing

Views of Objects All the objects can be seen from 3 different views. 1

Top view – Looking at the object from the top.

Side view – Looking at the object from side.

Front view – Looking at the object from the front.

Think and Tell Do all objects look different when seen from different views?

Riya drew the picture while sitting at the window.

Side view

Pooja drew the picture while standing on the roof.

Top view

Therefore, both Riya and Pooja drew the picture of the car correctly. Example 1

Suhani is rolling a dice. What number will she see on the dice if she is looking at it from the side view? The different views of the dice can be given as: Top view side view Front view Chapter 8 • Representing 3-D Shapes

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Example 2

What is the shape of an ice cream cone when seen from the front? The front view of the ice cream cone can be given as:

Front view

Therefore, the front view of an ice cream cone looks like a triangle. Do It Together

Observe each image and draw its respective view. Front view

Top view

side view side view

Do It Yourself 8A 1

Tick () the correct view that is seen in these pictures.

Top / Front / Side

Observe each image and identify the view.

View

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Match the objects with their top views.

Given below are the 3 views of some objects. Tick () the side view of the given objects.

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Draw the front view of the given objects. b

5 Kg

Nets of 3-D Shapes Riya and Pooja went to the market to buy some sweets. The shopkeeper took a paper cutout and folded it into a box. Riya wondered how a cutout turned into a box. Pooja explained that the cutout was the net of the box. A three-dimensional shape can be made by folding two-dimensional (2-D) shapes. Those 2-D solids which are used to make a 3-D shape are called nets. Let us see the net of a cuboid box. 1

The net of the sweet box looked like this. Below shown are a some more nets of a cuboid. 1

Similar to a cuboid, we have nets of a cube. Unfolding a cube box along its edges gives the net shown below. 1

Error Alert! Always observe the size of the opposite faces of the cuboid. They can never be different.

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Given below are some more nets of a cube. 1

Example 3

Observe the net of a dice and answer the given questions. Which number will never be next to the number 3? - 6

Which number will be at the opposite side of 1? - 5 Which number will you see if you turn right from 5? - 4

Example 4

1 3 5 6

Draw the net of the given cuboid. On unfolding the box, we will get a net as shown below.

Do It Together

The net of a cuboid is shown below. Draw and colour the remaining faces of the cube considering that the colour of the opposite faces is the same.

Do It Yourself 8B 1

Tick the nets of a cube. a

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Which of the following are NOT the nets of a cuboid? a

Match the cubes/cuboids with their nets. a

Draw the net of the following figures. a

Opposite faces on a dice add up to 7. Fill in the net of the cube with dots to make a dice.

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Maps Mom called Riya and asked her to buy some medicines from the hospital. Riya: Mom, I don’t know the route from the sweet shop to the hospital. Mom: Don’t worry Riya, I will tell you the route. Mom tells her the route but Riya gets completely confused! Sometimes, it is difficult to reach a place when someone tells us the route. Another way to find our way around is by using maps. A map is a drawing of an area made on a flat surface, like a sheet of paper. Let us look at the map and help Riya reach the hospital from the sweet shop.

Hospital Hotel

Museum

Town hall

Shopping centre

Post office Theatre

Supermarket

School

Sweet shop

Bus station

Riya is standing here

Step 1 Mark the place you are standing at and the destination you want to reach. In this case, Riya is standing at the sweet shop and wants to reach the hospital.

Hospital Hotel

Museum

Town hall

Shopping centre

Supermarket

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Post office

Sweet shop

Did You Know? Theatre School

Bus station

In ancient times, people used star maps to go from one place to another. A star map is a map of the night sky!

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Step 2

Step 3

Look at the possible routes to reach the destination.

Think about the directions in terms of left / right /

Here the possible route to the hospital can be shown as:

To reach the hospital Riya will take a left turn,

then move straight. She will then take the first

Hospital Hotel

straight and front.

Museum

right turn and move straight. She will again take the first right turn and will move straight. The

Post office Theatre

Supermarket

hospital will be in front of her.

School

Think and Tell Town hall

Example 5

Shopping centre

Sweet shop

Will Riya follow the same directions if she

Bus station

Look at the map shown and answer the questions.

is standing outside the supermarket?

Gate A

Chair Water fountain

Playground

Game Zone

Ice cream corner

Gate B Office

Game Zone

here is the playground when you W enter through Gate B?

Let us mark Gate B and the playground.

Let us mark the chair and look what is in front of it.

Clearly, the playground will be at our right when we enter from Gate B.

The game zone is in front of the chair.

Gate A

Chair Water fountain

playground

Game Zone

hat is in front of you when you are W sitting on the chair?

Ice cream corner

Gate A

Chair Water fountain

playground

Game Zone

Ice cream corner

Gate B Walking Path

Game Zone

Gate B Walking Path

Game Zone

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Do It Together

Mohan can reach Neerja’s house using 4 different routes. Two routes are marked for you. Mark the other two routes on the map. Mohan's House

Mohan's House

Neerja's House

Do It Yourself 8C 1

How many times does Meera turn left if she walks

Write if true or false.

School

to school along the path shown?

Factory

Temple

Bank

Meera

Market

Hospital

Second Road

School

First Road Lisa's

Park

Restaurant

Post Office

House a Lisa’s house is on the second road. b If Lisa steps out of her house on the first road, the bank will be to the left of her house. c The post office is the nearest place to the factory.

d The restaurant is in front of the park.

e To reach the factory, one has to go to the second road.

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Look at the map shown below and fill in the blanks. a If Rani is on mall road facing Rose Street,

Our Community

Brooke Park

the Police station will be near her

Rose Street

Police Station

_________________.

Janpath

c The restaurant is in front of _________________ house. d _________________ is in the centre of all the roads.

Sam's House

Supermarket

MG Road

Link Road

Mall Road

b Sam’s house is on _________________ road.

Rani's House

Restaurant

e Sam’s house is in front of _________________ park.

Bob is standing outside the pharmacy shown by the red mark. He wants to reach the bank. Write the possible routes to reach the bank. RESTAURANT

ELECTRONICS

BARBER SHOP

PHARMACY

DAIRY STORE

PIZZA

Bank

CAKE SHOP

FRUIT STORE

Bank

BANK

Bank

Bank Bank

Bank Which road will the truck NOT cross to reach house? Bankthe burning

Main Street

Maple Street Bank

Green Road

Farm Road

Green Park Crow Road Oak Lane New Street

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Points to Remember •

All the objects can be seen from 3 different views. Top view, side view, and front view.

•

A cuboid can have 54 different nets.

•

A map is a drawing of an area made on a flat surface like a sheet of paper.

• Two-dimensional shapes that are folded to make three-dimensional shapes are called nets. •

A cube can have 11 different nets.

Math Lab Exploring Nets of Cubes and Cuboids

Setting: In groups of 5 Materials Required:

• Paper containing the drawings of cube and cuboid along with measure of their lengths. • Cardstock paper

• Rulers

• Scissors

• Markers

• Tape or glue

Method:

Distribute the drawings, cardstock paper, rulers, scissors, and markers to each group. The groups draw the nets of cubes and cuboids of given length on the cardstock paper with the help of a ruler.

After drawing the nets, the teams carefully cut them out. The teams can use tape or glue to secure the edges. The team who makes the submission first wins!

Chapter Checkup 1

Draw the top view of the objects. a

Tick () the correct view to look at these things. a

Top / Front / Side

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Top / Front / Side

Circle the objects that look the same when looked at from side or front views.

D 4

Which side would be opposite to the purple side when the net is folded to

Draw the net of the following figures.

make a box?

See the net on the side. Colour the net in such a way that the opposite sides of the

Arrange A, B, C and D, so the car reaches the house.

cube have the same colour.

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Look at the map shown below and fill in the blanks. a

The ____________ is between John’s house and Rohit’s

The grocery store is nearest to ____________ house.

If Rohit is on First Road with the mall to his left, he

needs to take a ____________ turn to reach the Pizza

First Road

____________ is next to the school.

Mall

Fourth Road School Park Third Road Ice cream shop

John's House

Restaurant

house. 9

Kavya's House

Stationery Shop

house. b

Grocery Store

Pizza house

Rohit's House

Second Road

Hospital

Nita's House

Observe the map and answer the questions. Mohit's house Anand garden

Rita's house

Mina's house

School

Bus stop

Central Chowk

Bus stop

Market

Railway station

There are 3 houses on the map. Whose house is the farthest from the school?

Whose house is not opposite to Anand Garden?

How many roads meet at the Central Chowk?

10 Shown are three views of a cube and 6 faces of the same cube.

Design this cube using the net below. There can be more than 1 correct answer.

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Patterns and Symmetry

Let's Recall The world around us is full of patterns! A pattern is formed when something is repeated in a particular sequence or order. The repeating units can be symbols, numbers, shapes or objects. The following are examples of simple patterns.

The following is an example of number patterns.

1, 3, 5, 7, 9, 11, …….. Let's Warm-up

Extend the following patterns. 1 2 3

____________, ____________ 1, 11, 111, 1111, ______, _______

____________, ____________ A, BB, CCC, DDDD, _______, ________ I scored ___________ out of 4.

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Patterns Around Us Seema and her Mother are buying garments together. Seema: Mom! Look at that scarf! The design on it is unique! Mother: Yes Seema, I like it too. This is a Shibori design. Seema: Shibori! Look at the black, white and brown lines. They keep repeating on the entire cloth. Mother: Yes, a very interesting pattern!

Revisiting Patterns Seema and her Mother are discussing the pattern on the cloth above. But what does a “pattern” mean in geometry? In geometry, a pattern refers to a repeating arrangement of shapes, objects, or elements in a systematic and predictable manner. These arrangements can be based on various properties, such as size, shape, colour, or orientation.

Think and Tell

What kind of a pattern do you see in the print on the cloth?

Repeating Patterns In the figure below, the green square and the red square are arranged in an order and are repeating.

Did You Know? Pine cones have a spiral pattern.

In this case the repeating unit is Patterns like these in which a certain unit keeps on repeating over and over again in an order are called repeating patterns. The unit that repeats to form the pattern is called the repeating unit. The craftsman in the image is making a pattern with a wooden block on a cloth. The wooden block is dipped in different colors to form different patterns. This art is called block printing.

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Some more examples of repeating patterns are:

Example 1

Example 2

Do It Together

Complete the following pattern:

Step 1

Step 2

Identify the smallest unit that is repeating.

Draw/write the smallest unit over and over.

Complete the following pattern:

Step 1

Step 2

Identify the repeating unit.

Draw/write the repeating unit over and over.

Complete the following repeating pattern. _____ _____ _____ _____ _____ _____ _____ _____

_____ _____ _____ _____ _____ _____ _____ _____

Growing and Reducing Patterns Notice how in the adjoining figure, the pattern grows. From 2 squares first, we get 4 squares and then 6 squares and then 8 squares. These patterns are called growing patterns.

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A growing pattern is a type of pattern in which units increase through the steps.

Converse to growing patterns, in reducing patterns, the units reduce through the steps. In the adjoining figure, we start with 12 circles, which then become 8 and then finally become 4. This is an example of a reducing pattern.

Complete the following pattern for one more step. Also identify the type of pattern.

Example 3

Step 1 Identify the smallest unit that is repeating.

Step 2 Notice that the stars are increasing with steps. Therefore this is a growing pattern.

Step 3 Draw the pattern with the repeating and the growing pattern.

Example 4

Complete the following pattern and recognise the type of pattern. Step 1 Identify the smallest unit

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Step 2

Step 3

Notice that the number of triangles are reducing

The number of triangles are reducing from 5 to 4

pattern.

pattern accordingly.

as the steps increase. Therefore this is a reducing

5 Do It Together

to 3, i.e. reducing one each step. We extend the

Extend the following patterns. Also, identify the type of pattern. 1 2

Do It Yourself 9A 1

Extend the pattern by drawing 2 more shapes. Write whether it is a growing or reducing pattern. a

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Complete the missing shapes in the following growing pattern. a

Draw the picture that comes next in the pattern. a

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Number Patterns Patterns can be seen in numbers. They can be formed using skip counting, addition, subtraction, multiplication or division. We can show patterns using numbers and letters of the alphabet. 11

Z89

Rule: Multiply the previous number by 2.

We form magic patterns and magic triangles using numbers 1 – 9 along with the rule.

5 4

Example 5

W89

Error Alert! Always observe that no number comes twice in each line.

Numbers on each side of the triangle add up to 9.

X89

Rule: Letters of the alphabet are in reverse order and the number remains the same.

Let us look at some triangle and square number patterns.

Y89

Numbers on each side of the square add up to 15.

Observe the tower below and complete it. 24 3

Here the numbers are arranged in the tower form. We add 2 numbers below to get the number in the box above them.

12 60 24

24 + 36 = 60 36

15 + 21 = 36 12

6 + 9 = 15 162

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Example 6

Use numbers 1 – 9 to fill the stars so that each line adds up to 15. 3 + 5 + 7 = 15

Do It Together

4 + 5 + 6 = 15

Look at the pattern formed with numbers 21 to 26. Read and understand the rule. Write 6 more numbers that form a similar pattern with sum adding up to 107. 21

The rule of the pattern states that the sum of 2 numbers add up to 47. First from the left and First from the right

Second from left and Second from the right

Third from the left and Third from the right

Do It Yourself 9B 1

Complete the pattern. a 13, 26, 39, 52, _____, _____

b 10, 30, 50, 70, _____, _____

c 1, 2, 4, 7,11, _____, _____

d 84, 74, 64, 54, _____, _____

e 10, 15, 25, 40, _____, _____

Use numbers from 1 – 9 to fill in the magic triangle so that each line adds up to 12.

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Complete the number tower.

Fill in the squares, so that each line adds up to 26.

Complete the table. 1

2 4 5

+ + +

3 5 6

= =

Can you see any pattern in the sum? What is the pattern?

Tiling Patterns and Tessellations Have you ever looked around at the walls and floors? Do you observe any pattern in them?

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Yes, if you look closely, the brick structure that we see when walls are built is actually a pattern. The above pattern can be broken down in layers, as shown. Layer 1 Layer 2 Layer 3 Layer 4 Layers 3 and 4 are basically a repeat of layers 1 and 2. Also notice that once all these layers are attached to one another, there is no gap in the final pattern. Also, there are no overlaps of bricks anywhere.

Did You Know? A honeycomb is also an example of tessellation! Hexagonal shapes arranged together to form the honeycomb.

Such patterns that are created by fitting together identical shapes (tiles) without any gaps or overlaps to cover a flat surface, such as a wall, are called Tessellations. The following are some tessellating and non-tessellating patterns.

A Tesselating shape

Do It Together

A non-tesselating shape

A Tesselating shape

Which of these figures show tessellation?

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Tangrams

Look at the collection of shapes below. Interestingly, the individual shapes do not necessarily repeat, like in a pattern. But they fit together in a larger square. These “puzzles”, where individual pieces are put together to create various shapes and figures, are called Tangrams. Let us look at more shapes made using tangrams.

Do It Yourself 9C 1 1

Write 'Yes' if the pattern is a tessellating pattern.

Say yes to the pattern, if it is tessellating.

b a

Complete the following tessellations. Make the pattern grow by drawing the given shape. a

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Count and name all the shapes in the following Tangrams.

Encoding and Decoding Patterns

We learnt about patterns using shapes and objects. Wouldn’t it be fun to share messages as secret messages using numbers and letters of the alphabet? Let us learn how we can do this. Let us start by giving each letter of the alphabet a number. By using this understanding, we can write different code messages. A

Now what if we want to say GET WELL SOON? How can we say it using the above code? G

We can say: 7 5 20

23 5 12 12

19 15 15 14

This way of turning information or messages into special secret codes or hidden patterns is called Encoding. It’s like creating a secret language that makes communication fun and exciting! Chapter 9 • Patterns and Symmetry

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Now what if our friend – who knows the above code – sends us the following message? 7 15 15 4

13 15 18 14 9 14 7

How will we understand this? We refer to the same table above. 7

This code means GOOD MORNING. This process of finding meaning from a secret code is called the process of Decoding. Do It Together

Decode the message: 7 15

7 18 5 5 14

Step 1 Observe the code and understand the pattern.

Step 2 Identify the letters as per the code. 7

The message for the code 7 15

7 18 5 5 14 is ____________

Do It Yourself 9D

Using the above code, decode the following sentences. a

Go to ____________ (13-15-15-4-12-5)

Find the ____________ (17-21-9-26)

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Click ____________ quiz now (1-20-20-5-13-16-20)

____________ and answer the question (18-5-1-4)

Click ____________ (14-5-24-20)

Keep doing and click next when you ____________ (6-9-14-9-19-8)

Now you can see your grade and ____________ (1-14-21-23-5-18-19)

Click on your class name to return to class ____________ (16-1-7-5)

Using the above code, write the codes for the following messages. a

KEEP IT UP

SAVE WATER

FANTASTIC WORK

REDUCE REUSE RECYCLE

PLANT TREES

Symmetry and Reflections Real Life Connect

Riya: Look at that Butterfly, Raj! The wings have some lovely patterns. Raj: Yes. Butterflies are very special. I like them, their wings on both sides have the same size and similar pattern!

Symmetry Raj and Riya observe that the wings of a Butterfly have similar patterns. The two wings look very similar! The Butterfly is symmetrical. Let us learn more about Symmetry. Symmetrical and Non-symmetrical figures When a shape can be folded so that one half of it fits exactly on the other half along the fold line, the shape is said to be symmetrical. The fold line is called the line of symmetry. Let us take another example. In the adjoining figure, the fold line or the line of symmetry divides the ladybug into two equal halves. The ladybug is also symmetrical.

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Now let us try to create a symmetrical figure using a sheet of paper. Step 3 Line of symmetry Fold along the line of symmetry

Step 2 Cut along the dotted lines

Step 1 Line of symmetry

Symmetry exists all around us. The following shapes are also symmetrical:

We can say that some shapes can be split in half by a line of symmetry in the middle as shown in the figures. These figures are called symmetrical figures.

In the figures given below, the line of symmetry does not divide the shapes into two identical halves. Therefore, these figures are not symmetrical but asymmetrical.

Do It Together

Observe the following figures and find which of the following figures are symmetrical or asymmetrical.

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Types of Symmetry We saw previously that the line of symmetry divides the butterfly into two identical halves. Are there different types of lines of symmetry? Let us see! Vertical symmetry means that the left half of the object will be the same in size and shape as the right half when divided vertically. The line that divides an object or a shape into equal halves from left to right is called the vertical line of symmetry.

Horizontal symmetry means that the top half of the object will be the same in size and shape as the bottom half when divided horizontally. The line that divides an object or shape into equal halves from top to bottom is called the horizontal line of symmetry.

Do It Yourself 9E 1

Are the given figures symmetrical along the lines marked? Write Yes or No. a

Yes e

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Draw the line of symmetry for each of the following figures, wherever possible. a

Write the letters of the English alphabet and find out which of them are symmetrical. Also identify the type of line of symmetry they have.

Reflection

We have learnt about symmetry and the line of symmetry. Let us now learn about reflection. Every morning, we look ourselves in the mirror. What do we see? We see a reflection of ourselves in the mirror. We can also see our reflection on any shiny surface like glass or even water. Look at some more objects when placed in front of a mirror.

Object

Image

Object

Mirror

The line between the object and the reflection is called the mirror line or line of reflection.

Image

Mirror Line Shape

Reflection

The mirror line or line of reflection can be of two types: A mirror line can be vertical. In this case, the mirror line is called a standing line. You can see the example below. 172

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A mirror line can be horizontal. In this case the mirror line is called a sleeping line.

Do It Yourself 9F 1

Draw the reflection of each of the shapes. a

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Points to Remember • A pattern is a sequence or arrangement of figures, things, numbers or letters in a fixed repetitive way. •

A pattern that is completed by adding elements of their groups is a growing pattern.

•

A pattern that is completed by removing elements of their groups is a reducing pattern.

• Tessellation or tiling is when we cover a surface with a pattern of flat shapes so that there are no overlaps or gaps in between. • A tangram is a puzzle made up of seven shapes that can be arranged to form many different designs. • When a shape can be folded so that one half of it fits exactly over the other half along the fold line, the shape is said to be symmetrical. • A figure has a line of symmetry, if a line can be drawn dividing the figure into two identical parts. This is called the line of symmetry. •

The line between an object and its image is called the mirror line, or the line of reflection.

Math Lab Objective: To create an inkblot pattern

Materials required: White paper, Liquid ink or washable markers in various colours,

Plastic or disposable tablecloth, Small containers to hold the ink, Water and paper towels for clean-up 1

Take the white paper. Fold it in half along the

Now, unfold the paper and spill a few drops of ink

Press the two halves together.

The resulting figure will be a symmetric figure.

Try with another sheet of paper. Fold the paper

Try out different combinations of colours to make it colourful.

horizontal line of symmetry on one half.

along the vertical line of symmetry.

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Chapter Checkup 1

Complete the following pattern. a

+ ÷ − ×

÷ × + –

× – ÷ +

Understand the code and complete the pattern.

11 13 14

Write the following words using the code in the above table: a

BEST WISHES

PLANT A TREE

SAVE PAPER

RECYCLE

Complete the number pattern. a

110, 130, 150, 170, _____, _____, _____

234, 242, 251, 260, _____, _____, _____

140, 131, 122, 113, _____, _____, _____

890, 780, 670, 560, _____, _____, _____

111, 122, 133,_____, _____, _____

2304, 576, _____, 36, _____

Follow the instructions as given. Tell whether the dotted line on each shape represents a line of symmetry. Write yes or no. a

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Draw a line symmetry on each shape. e

Draw the second half of each symmetrical shape. i

Draw the shapes following the vertical line of symmetry. The first one is done for you. a

Reflect each of these shapes in the dotted lines. The first one has been done for you as an example. a

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Word Problem 1

Meenakshi uses her bangle to make a perfect circle. She wonders

how many lines of symmetry there are in a circle. Can you help her find out?

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Length, Weight and Capacity

Let's Recall

Have you ever noticed that your desk is longer than the pencil that you are using? That is because the length of your desk is more than the length of your pencil. Length is the distance or measurement between two points and can be measured in centimetres (cm), metres (m), and kilometres (km).

200 cm (2m)

Now, try holding your pencil in one hand and your notebook in the other. Does one of them feel heavier? This is because the notebook has more weight than the pencil. Weight is used to determine how heavy an object is and can be measured in grams (g) and kilograms (kg). Similarly, your water bottle can hold more water than a glass of water at home because the capacity of your water bottle is more than that of the glass. Capacity is the quantity of liquid a container can hold and can be measured in millilitres (mL) and litres (L).

Let's Warm-up 1

Match the following. LESS THAN 1 METER MORE THAN 1 METER

Tick () the one you would use to measure weight.

Tick () the one you would use to measure capacity.

I scored ___________ out of 4.

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Length Real Life Connect

Sharma Bakery is famous in the neighbourhood. Vibhu’s mom often sends him to get bread for the house. The shop is just a few metres away from their home. “Fresh bread and buns. I must hurry back home.”

Sharma Bakery

DELIVERY

Free delivery up to 3 km. Order Now!

Vibhu read the signboard. Vibhu asks his Mother, “Mom, how far is 3 kilometres? How far would it be?”

Measuring Length We often need to know how far places are, or how long or tall things are. This is referred to as measuring length. This is exactly what Vibhu is asking his mother about. For example, when Vibhu is walking from home to the neighbourhood bakery and back, he is walking a distance of a few metres. Similarly, when a delivery boy travels to another locality, he is travelling a distance of some kilometres.

Common Units of Length Metres (m) and kilometres (km) are units for measuring length. A metre is often used to measure how long or wide a piece of cloth is. We can also tell the length, width, and height of a room in metres. Kilometres are a larger unit than metres and are used to measure longer lengths. For example, how long a road is, how far two cities are, or how tall a mountain is. To measure shorter lengths, we use millimetres (mm) and centimetres (cm). These units are used to measure the lengths of small things, like a pencil or a paper strip.

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Remember! Units of length: Millimetre (mm) < Centimetre (cm) < Metre (m) < Kilometre (km)

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We can use a ruler or a measuring tape to measure different lengths. A ruler measures the length in millimetres (mm) and centimetres (cm), while a measuring tape measures in centimetres (cm) and metres (m).

Thus, a ruler is usually used to measure small objects such as a book, pen, or pencil. But for measuring longer lengths, such as a piece of cloth, furniture, and floors and walls of a room, we prefer to use a measuring tape. Example 1

Suggest the right tool and find the length of the line.

Did You Know?

The Ganga River is about 2525 km long. It is the longest river in India.

Think and Tell

Step 1 This is a short line on the paper. A ruler or a measuring tape can be used to measure the length. Let us use a ruler.

Would you use a scale or measuring tape to measure a road? Why?

Step 2 Place the ruler against the line. Make sure it starts at 0.

Step 3 Note the length of the line in millimetres and centimetres. This line is 8 cm long. Example 2

Priya is going to the dairy to get milk. If the dairy is 250 m away from her home, how far has she walked from home to the dairy and back home?

250 m The distance from house to dairy = 250 m The distance from dairy to house = 250 m So, the total distance Priya has walked = 250 m + 250 m = 500 m 180

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Do It Together

For a water pipeline setting, a plumber needs to cut 5 pieces of water pipe, each 7 m long. 1

Tick () the tool that he should use to measure the length of the pipe. a thread

a ruler

a measuring tape

What is the total length of the water pipes that the plumber needs to fit together? The length of 1 pipe = __________ m. The length of 7 pipes = 7 × __________ m = __________ m. So, the total length of the pipe that the plumber needs to fit is __________ m.

Interchanging the Units To get the correct and simple measurement, sometimes we need to change the units of length from one to another. Let’s learn the relationship between different units of length. Example 3

Change the units of length. To change to a larger unit, we divide. 1 10 mm = 1 cm. So, 1 mm = cm 10

Changing to larger units

÷1000 km

÷100 m

÷10 cm

×100 m

1 km 1000

1 km = 1000 m

×10 cm

1000 m = 1 km. So, 1 m =

1 m 100

To change to smaller units, we multiply.

Changing to smaller units

×1000

100 cm = 1 m. So, 1 cm =

1 m = 100 cm

1 cm = 10 mm

Change 255 cm to m.

Change 1092 m to km.

255 cm = 200 cm + 55 cm

1092 m = 1000 m + 92 m

Remember, 100 cm = 1 m

Remember, 1000 m = 1 km

So, 200 cm + 55 cm = 2 m 55 cm

So, 1000 m + 92 m = 1 km 92 m

Chapter 10 • Length, Weight and Capacity

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Example 4

Change 6 km to m.

Change 3 m to cm.

Remember, 1 km = 1000 m

Remember, 1 m = 100 cm

So, 6 km = 6 × 1000 m = 6000 m

So, 3 m = 3 × 100 cm = 300 cm

Rohan’s school is 2400 m away from his home. How far is it in kilometres and metres? 2400 m = 2000 m + 400 m

Remember! 1000 m = 1 km

= 2 km + 400 m So, Rohan’s school is 2 km and 400 m away from his home. Do It Together

In a park, two trees are 600 cm apart from each other. The gardener needs to mark a line between them. He has a metre-long wooden stick. How many times does he need to use the stick to measure the length between the trees?

600 cm

The gardener needs to measure a length of _____ cm. We know that, _____ cm = 1 m So, 600 cm = _____ The garden needs to use a metre long stick _____ time to mark the lines. 1m

600 cm 182

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Do It Yourself 10A 1

What would you use to measure the length of these objects – a ruler or a measuring tape? a A grain of rice

b The height of a door

c Length of a desk

d Width of a book

Look at the pictures and write how long these objects are in centimetres (cm) and millimetres (mm). a

Measure how long these lines are in millimetres (mm) and centimetres (cm). Use a ruler. a

Draw line segments of the given lengths using a ruler. a 4 cm

e 14 m 67 cm

925 cm

b 9m

e 40 km 175 m

d 11 m 34 cm

c 1200 cm

d 136 cm

15 m 22 cm

b 900 cm

g 1125 cm

h 1250 cm

b 9 km

c 13 km

d 16 km 165 m

c 2200 m

d 1336 m

e 475 cm

1520 cm

95 km 54m

Write/Express in kilometres and metres. a 1400 m f

1925 m

b 1600 m g 2125 m

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c 12 m

Change to metres. a 5 km

d 17 cm 5 mm

Write/Express in metres and centimetres. a 400 cm

c 11 cm 2 mm

Change to centimetres. a 2m

b 12 cm

h 4250 m

e 1475 m

7520 m

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Word Problems 1

Noor cut 5 ribbons of different colours to paste in a line on a wall. Each ribbon is 24 cm 5 mm long. What is the total length of the ribbon she cut?

A tailor needs to cut 4 pieces of 30 cm long cloth to make a coat.

a How much cloth has he cut in total? b What is the total length of the cut pieces of cloth in metres?

Raman's uncle drove the car for 4 km on a road. How far has he

For a project, Rajat measured how wide and tall a table is. It is 2

driven in metres?

m 30 cm wide and 1 m 12 cm tall. How wide and tall is the table in centimetres?

From Jiya’s home, the park is 1 km 500 m away. Every morning, her father jogs to

the park and back. How far has he jogged in 2 days? Write your answer in metres. Hint: Convert the distance into metres and then add.

Weight Real Life Connect

We often measure how heavy or light things are. This is referred to as weight. Chintu wants to know how strong he is. In the garden, he picked up a feather first. “It is so light,” he said cheerfully. But then he picked up piece of a rock! “It is so heavy,” he cried. “How heavy is it?” he wondered.

Measuring Weight Chintu could use a weighing scale or machine to measure the weight of the rock. We measure weight in many other situations.

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A seller measures the weight of fruit and vegetables on a weighing scale.

A doctor measures our body weight on a machine before a check-up.

Common Units of Weight On a weighing scale, different weighting stones or bars are used. Notice ‘kg’ written on these bars. It is a measuring unit for weight. Kilograms (kg), gram (g), and milligram (mg) are the units of measuring weight. A milligram (mg) is a small unit of weight measurement. A gram (g) is larger than a milligram (mg), and a kilogram (kg) is an even larger unit.

Bars of different weights

We usually use grams as a unit to measure the weight of fruit and vegetables. We use kilograms to measure our body weight. Example 5

Read the weight of an object on a weighing machine. Step 1

Make sure the needle of the scale is at 0 when you start.

1 2

Step 2 Place the object (a watermelon) on the machine.

Step 3 Now, note where the needle is on the scale.

The weight of this watermelon is 2 kg. Example 6

Remember! Units of Weight: Milligram (mg) < Gram (g) < Kilogram (kg)

Noor has 7 oranges of about the same size. The weight of 2 oranges is 200 g. What is the total weight of all the oranges? Weight of 2 oranges = 200 g Weight of 1 orange = 200 ÷ 2 = 100 g 100 2 200g – 2 0 –00 00

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Weight of 7 oranges = 7 × 100 g = 700 g kg ×

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Do It Together

Raj is buying fruit from a shop. He is checking the weight of each item of fruit on a weighing machine. 1

Write the weight of each fruit.

1 kg

_____ kg

_____

1 kg 2

What is the total weight of the fruit that Raj bought today? 1 Total weight of fruit = 1 kg + ______ kg + ______ kg = ______ kg 2 So, Raj bought a total of ______ kg fruit today.

he price of 1 kg of pears is ₹220. How much would Raj pay for pears if he bought 2 T kg of pears? 1 kg of pears = ₹220 The price of 2 kg of pears = 2 × ₹220 = ₹__________

Interchanging the Units Let’s learn the relationship between different units of weight and how to change from one unit to another. Changing to Larger Units

1 kg

÷ 1000

Changing to Smaller Units

1 mg ÷ 1000

1000 milligram = 1 gram

1000 gram = 1 kilogram

(1000 mg = 1 g)

(1000 g = 1 kg)

× 1000 1 kg

× 1000 1g

1 mg

1 1 kilogram 1 milligram = gram 1000 1000 1 1     kg g 1 g = 1 mg = 1000  1000   

1 gram =

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Example 7

Example 8

Change the units of weight. Change 1050 mg to g.

Change 2090 g to kg.

1050 mg = 1000 mg + 50 mg

2090 g = 2000 g + 90 g

Remember, 1000 mg = 1 g

Remember, 1000 g = 1 kg

So, 1000 mg + 50 mg = 1 g 50 mg

So, = 2000 g + 090 g = 2 kg 90 g

Change 4 kg to g.

Change 6 g to mg.

Remember, 1 kg = 1000 g

Remember, 1 g = 1000 mg

So, 4 kg = 4 × 1000 g = 4000 g

So, 6 g = 6 × 1000 mg = 6000 mg

Mihir uncle bought 8 packs of pulses, each weighing 500 g. What is the weight of the pulses he has bought in kilograms (kg)? kg

1 pack of pulses = 500 g 8 packs of pulses = 8 × 500 = 4000 g

We know that 1000 g = 1 kg So, 4000 g = 4000 ÷ 1000 = 4 kg Do It Together

500 8

4000

Meena weighs a few apples of the same size on a weighing machine. 4 apples weigh 1 kg 40 g. What would be the weight of one apple in gram? Total weight of 4 apples = 1 kg 40 g Remember, 1 kg = __________ g Total weight of 4 apples in g = __________ g + 40 g

2.400

= __________ g Now, the weight of 1 apple in gram would be: __________ g ÷ 4 = __________ g each apple

Do It Yourself 10B 1

Look at the picture and write the weight of the objects in kilograms (kg) and grams (g). a

Chapter 10 • Length, Weight and Capacity

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Change to grams (g). a 3000 mg

b 6000 g

Change to milligrams (mg). a 5g

c 10000 mg

d 1100 mg

e 2467 mg

5967 mg

c 10000 g

d 4500 g

e 6557 g

9782 g

1 c 8 g

d 4 g 102 mg

e 15 g 770 mg

9 g 802 mg

c 10 kg 500 g

d 5 kg 10 g

e 15 kg 25 g

16 kg 820 g

Change to kilograms (kg). a 5000 g

b 7000 mg

b 16 g

Change to grams (g). a 5 kg

b 17 kg

Word Problems 1

Jaya’s weight is 38 kg 300 g. What would the weight be in grams?

Rama aunty paid ₹150 to buy 1 kg of tomatoes. How much will she pay

The cost of a 500 g pack of rice is ₹65. What would Uncle Sudhir have to

A vegetable seller sold 145 kg 500 g of onions today. If the price of 1 kg

if she buys 2 and a half kg of tomatoes? pay if he needed 5 kg of rice?

of onions is ₹30, what is the total earning of the vegetable seller today on the sale of onions?

A farmer is packing wheat in sacks. He stores 9 kg 500 g of wheat in 1 sack. How much grain would he use to pack 9 such sacks?

Capacity Real Life Connect

Rima is visiting a supermarket. She wants to buy some juice for a party. There are different bottles of different amounts of juice. She wants to buy the bottles with the greatest amount of juice. Let’s help her!

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500 Millilitres

250 Millilitres

We know how to measure the weight of solids, such as fruit and vegetables, but we do not measure liquids such as water, milk and juice in the same way.

1 Litre

Measuring Capacity

Rima needs to check the ‘capacity’ of each bottle written on its labels to know how much juice it holds. Bottles, packets, glasses, and other utensils are made to hold different amounts of liquid in them. Thus, they have different capacities.

Common Units of Capacity

Different measuring cups or jars are used to measure different capacities. Millilitre (mL), litre (L), and kilolitre (kL) are the units of measuring capacity.

A millilitre (mL) is a small unit of capacity. Litre is bigger than mL, and kilolitre (kL) is even greater. We usually find water bottles, cans, and jars in litre capacity. Medicinal droppers, cups, and syringes measure liquid syrups and medicines in mL. Example 9

Measure 1 litre of water using a measuring jug.

Step 1 Slowly pour the water into the jar.

Step 2 Stop when it reaches the mark of 1 litre.

Step 3 Let the water settle down and read again.

Remember! Capacity is the amount of liquid a container (bottle, glass, etc.) can hold.

This jug contains 1 litre (L) of water. Example 10

This evening, Anju’s mother is serving milkshake to Anju and 4 of her friends. Each glass can hold 250 mL of mango shake. How much shake does she need to prepare to serve Anju and her friends? 1 glass = 250 mL For 5 people, we need 5 glasses of shake. 5 glasses would contain = 5 × 250 mL = 1250 mL.

250 5

1250

So Anju’s mother needs to prepare 1250 mL of shake.

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Do It Together

Rehaan poured different amount of orange-coloured water in different jugs. 1

Write down the amount of coloured water Rehaan poured into each of these jugs. a

_____mL 2

_____mL

How much coloured water does he have in total? 400 mL + ______ mL + ______ mL +______ mL = ______ mL

o get 1000 mL of coloured water in one jug, Rehaan, would need to empty jug ______ T and ______ completely.

Interchanging the Units Let’s learn the relationship between different units of measuring capacity and how to change from one unit to another. Change Millilitres to Litres

Change Litres to Millilitres

× 1000

÷ 1000

Litres

Millilitres

1 Litre = 1000 millilitres (1 L = 1000 mL)

Millilitres

Litres

1 Litre 1000 1   L 1 mL = 1000  

1 millilitre =

Let’s learn how to change L into mL and vise-versa. Change 3050 mL to L.

Change 8 L to mL.

Remember, 1000 mL = 1 L

Remember, 1 L = 1000 mL

So, 3050 mL = 3000 mL + 50 mL

So, 8 L = 8 L × 1000 mL

= 3 L 50 mL

= 8000 mL

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Example 11

Rima is buying juice in a market. Each bottle has 250 mL juice. She needs 2 L of juice. How many bottles does she need to buy? We know that 1 L = 1000 mL

8 250 2000 – 2000 0000

2 L = 2000 mL If 250 mL = 1 bottle Then, 2000 mL = 2000 ÷ 250 = 8 bottles

So, to get 2 litres of juice Rima needs to buy 8 bottles. Do It Together

Rehaan opened a bottle of 2 L orange juice. Then, he filled a 200 mL glass with it. How much juice is left in the bottle now, in litres and millilitres? Total juice in the bottle = 2 L We know that 1 L = _______ mL. So, a bottle of 2 L capacity contains 2 × 1000 = _______ mL. So, if Rima poured 200 mL into a glass, the remaining juice in the bottle is: _______ mL – 200 mL = 1800 mL Remember, 1000 mL = 1 L. So, 1800 mL = _______ L _______ mL. This will be the juice remaining in the bottle.

Do It Yourself 10C 1

Look at the pictures and write the amount of liquid in these measuring jugs. a

Litre

__________mL 2

__________L

Change from millilitres (mL) to litres (L). a 3500 mL

b 5000 mL

c 10500 mL

Express the capacity in litres (L) and millilitres (mL). a 1700 mL d 8235 mL

b 5286 mL e 9250 mL

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mL mL mL mL mL mL mL mL mL mL mL

c 7650 mL f

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Neha bought 4 small packs of 250 mL and 2 bottles of 1 L each of juices. How much

A cup holds 100 mL of tea. How much tea is needed in the pot to fill 6 cups of tea?

juice has she bought in total?

Word Problems 1

Coco is filling 3 water bottles to keep them in the fridge. Each bottle can hold 2 L of

Rima is serving a glass of milkshake to each of her friends. Each glass can hold

water. How much water will she use to fill all three?

200 mL, and the jug of milkshake has 2 L in it. How many glasses can be filled with one full jug of milkshake?

JJ and her sister are selling ‘Fresh Lemonade’ at their school fair. They are selling it for ₹25 per glass. Each

glass can hold 250 mL of lemonade. So far, they have sold 8 glasses.

a How much lemonade have they sold so far in litres (L)? b How much have they earned so far by selling lemonade?

Points to Remember Length

Weight

Capacity

Millimetre (mm), centimetre (cm), metre (m), and kilometre (km) are units of measuring length.

Milligram (mg), gram (g) and kilogram (kg) are units of measuring weight.

Millilitre (mL) and litre (L) are the units of measuring capacity.

1 mm < 1 cm < 1 m < 1 km

1 mg < 1 g < 1 kg

1 mL < 1 L

1 km = 1000 m

1 kg = 1000 g

1 L = 1000 mL

1 m = 100 cm

1 g = 1000 mg

1 cm = 10 mm

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Math Lab Setting: Pair Materials Required: An empty water or soft-drink bottle; 4-5 utensils of regular use— cup, bowl, glass, mug, etc. Method 1

Read the label on the empty bottle to find out its capacity.

Fill it with water up to the brim.

One by one, pour water from the bottle into each utensil.

With each pour, check how much water has been reduced in the bottle.

Then, discuss with your partner and guess the capacity of the utensil.

Chapter Checkup 1

Draw the lines of the lengths. Use a ruler or a measuring tape. a

205 cm

507 cm

764 cm

1205 m

5763 m

6049 m

5065 g

4600 g

7450 g

10500 g

2kg 500 g

4 kg 600 g

5 kg 750 g

12 kg 500 g

12500 mL

Change from millilitres (mL) to litres (L). a

25 cm 5 mm

Express the following weight in grams (g). a

Express the following weight in kilograms (kg). a

15 cm 8 mm

Express the following lengths in kilometres (km) and metres (m). a

Express the following lengths in metres (m) and centimetres (cm). a

12 cm 6 mm

2000 mL

700 mL

Express the capacity in litres (L) and millilitres (mL). a

7200 mL

8660 mL

16250 mL

From 1 L of milk, mother gave 350 mL to me and 175 mL to my brother. How much milk was left?

2 jars of cooking oil have a capacity of 3 L each. Tara pours the oil out of these 2 jars into smaller jars each with a capacity of 500 mL. How many jars does Tara use?

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Word Problems 1

The price of 1 kilogram of sugar is ₹ 60. Find the price of sugar of the weights given. a

2 kg

5 kg

12 kg and 500 g

25 kg and 500 g

ima opened a bottle of 1 L orange juice. Then she filled a 200 mL glass from it. R How much juice is left in the bottle now?

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Perimeter and Area

11 Let's Recall

We have learnt in the previous chapter that length is the distance or measurement between two points. We can measure length in millimetres (mm), centimetres (cm), metres (m) and kilometres (km). Let us say we have to measure the tip of a pencil. Which unit of length will we use? We use millimetres (mm) to measure very short lengths. However, to measure the length of a whole pencil, we will use centimetres (cm). We use centimetres (cm) to measure short lengths or short heights for example: Height of a chair. Now let us say we have to measure the length of a blackboard. We will have to use metres (m). We use metres (m) to measure long lengths, long heights or short distances. For example: the distance from your bedroom to the kitchen. We use kilometres to measure long distances. For example, the distance between two cities. We have also learnt how to convert one unit of length to another. 1 km = 1000 m

1 m = 100 cm

1 cm = 10 mm

Let's Warm-up

Convert the following. 1

25 cm = __________ mm

3 m = __________ cm

6 m 30 cm = __________ cm

5 km = __________ m

6 km 40 m = __________ m

I scored _________ out of 5.

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Understanding Perimeter and Area Real Life Connect

It is Rita’s parents’ 15th wedding anniversary! She wants to give them a family photo, but the frame has plain brown edge. Rita doesn’t like it and decides to paste a colourful ribbon over it. For this, she needs to cut a piece of ribbon from a roll. How will she know how much ribbon to cut?

Perimeter Rita needs to cut a piece of ribbon that is the same length as the edge of the frame. So, she needs to measure the total length of the brown frame. The total length of the frame around the photograph is the perimeter of this photo frame. The perimeter is the total distance covered along the edges of a closed figure or shape. The perimeter can be measured in millimetres (mm), centimetres (cm), and metres (m). We need to know the perimeter in the following situations: • To make and decorate the edges of a photo frame, a gift box, a table, or a blackboard. • To put fencing around a house, buildings, park, farm, or field. • To add decorative borders around a piece of cloth, such as a scarf or handkerchief.

Decorated boundary

Fencing

Border of a scarf

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We can use a thread, a ruler, or a measuring tape to find the perimeter of shape, such as a circular disc. Let’s learn to measure the perimeter of a disc. Step 1 Place the thread along the edge of the circular disc, as shown in the picture.

Step 2 Mark the end point on the thread where it meets the starting point and circles the edge of the whole disc. You may mark or cut the thread at this point.

Step 3 Use a ruler or measuring tape to measure the length of the piece of

Think and Tell

thread. The length of the thread is the perimeter of the circular disc.

Can you find the perimeter of shapes such as a square, rectangle, and triangle using only a ruler?

Perimeter of Polygons To find the perimeter of simple shapes such as a triangle, square, or rectangle, we can simply add the lengths of all the sides. To find the perimeter of polygons on a dot grid, we need to count the number of units around a polygon on the grid. Let's measure the perimeter of the following polygons:

Remember! A polygon is a closed figure that is made up of only line segments.

1 unit

5 cm

5 cm Figure A

Figure B

Figure C

In figure A, we simply add the lengths of all the sides of the shape or polygon.

5 cm

Perimeter of this figure = 5 cm + 5 cm + 5 cm = 15 cm. Chapter 11 • Perimeter and Area

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1 unit

In figures B and C, we count the number of square units around the polygons. 1 2 3 4

5 6 7 8 9 10

20 19 18 17 16 15

87 6 910 11

19 18 17

12 1 unit

16 15 14 13

14 13 12 11 Figure B

Figure C

Perimeter = 20 units

These shapes are different but have the same perimeter. Example 1

Find the perimeter of the following shapes. cm

1 unit

5 cm

1 unit

Shape B

2 16

1718

Shape C 3

1415

8 10

12 11

7 10

4 cm

Shape A 1

4 cm

5 cm

The length of the edges of Shape A is 16 units.

The length of the edge of Shape B is 18 units.

The total length of the sides = 5cm + 4 cm + 2 cm + 2 cm + 4 cm = 17 cm.

So, its perimeter is 16 units.

So, its perimeter is 18 units.

So, its perimeter is 17 cm.

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Do It Together

Find the perimeter of these shapes. 6m

1 unit

A Perimeter = ______ units

Perimeter = ______ units

Perimeter = ______ m

Finding the Missing Side If the perimeter is given, we can find the length of the missing side. First, we add the lengths of the sides given. Then we subtract it from the perimeter. Let’s understand this with examples. Example 2

Find the missing length of the side if the perimeter of this shape is 23 cm. Given, the perimeter = 23 cm.

4 cm 5 cm

4 cm + 5 cm + 8 cm + the length of the missing side = 23 cm 17 cm + the length of the missing side = 23 cm

8 cm

Example 3

This means,

So, the length of the missing side = 23 cm – 17 cm = 6 cm

The perimeter of this figure is 52 cm. Find the length of the missing side in the figure. 5 cm

Here, the perimeter is 52 cm. This means,

13 cm 17 cm

Or, 40 cm + the length of the missing side = 52 cm

5 cm

So, the length of the missing side = 52 cm – 40 cm ?

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5 cm + 13 cm + 5 cm + 17 cm + the length of the missing side = 52 cm

= 12 cm

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Do It Together

Find the length of the missing side in the given figure. Here, the perimeter is 34 cm.

2 cm

5 cm

2 cm

Perimeter = 34 cm 5 cm

2 cm

3 cm

This means, 3 cm + ____ cm + ____ cm + ____ cm + 2 cm + ____ cm + ____ cm + the length of the missing side = 34 cm This is same as ____ cm + the length of the missing side = 34 cm So, the length of the missing side = 34 cm – ____ cm = ____ cm

Do It Yourself 11A 1

Use a piece of thread and a ruler to measure the perimeters of the given figures. Which figure has the longer perimeter?

Figure X 2

Figure Y

Find the perimeter of the following figures, where 1 unit = 1 cm. a

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4 cm

3 cm

4 cm

3 cm

14 m

5 cm

6 cm

20 m

15 m

30 m

6 cm

11 cm

20 m 17 m

5 cm

Perimeter = 46 cm 7 cm

17 m

1 cm

12 m

3 cm

2 cm

6 cm

11 cm 6 cm

Find the perimeter of each of the following figures. 4 cm 4 cm

15 mm 7 cm

3 cm

7 mm 8 mm

12 mm

? ?

Draw 2 shapes with perimeters of 22 cm and 24 cm.

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8 cm

Perimeter = 128 m

3 cm

4 cm

figure.

Perimeter = 32 cm

8 cm

The perimeter of each figure is given. Find the length of the missing lengths of the sides in each

30 m

8 cm

5 cm

4m 4m

Look at the figures. The length of the sides of each shape is given. Find their perimeters.

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Word Problems Look at the picture. How much distance would you cover if you walked around this park once?

Park

150 m

100 m

100 m 45 cm

Look at the sides of the pillow cover. What length of sewing 3 pillow covers of this size?

30 cm

thread would Seema’s mother require to sew straight borders on

30 cm

45 cm

Dhanush joined 6 square tiles in two different patterns, as shown in the figures below. Find the perimeter of each pattern if the side of each tile is 25 cm. a

Area Rita’s beautiful photo frame is ready! Now she needs to print a picture that fits perfectly in the frame. The space within the boundaries of the frame on which the photo needs to fit is the area of the photo frame. So, the area is the total space covered by a closed figure. We need to know the area in the following situations: • Area of a book or a notebook to add a cover on it. • Area of a wall that needs to be painted. • Area of a floor to check if it can be covered with carpets of certain sizes. • Area of floor while covering it with tiles of certain sizes. • Area of land while dividing it into parts for different uses. Area is measured in square units, that is square centimetres (sq. cm) or square metres (sq. m). 202

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The area of a polygon on a grid is measured by the number of squares needed to cover the shape. Let’s look at the figures and find out their areas, where the side of 1 unit = 1 cm. The area of 1 square = 1 unit × 1 unit = 1 square unit

1 cm

Step 1

Step 2

First, count the number of units that make

Area = 1 unit × 1 unit = 1 square unit.

up the shape.

1 2 3 4

1 2 3 4 5

6 7 8 9 10 11

12 13 14 15 16 17 18 19 20 21 22

5 6 7

8 9

1 cm

Number of units = 22

Number of units = 9

Area = 22 sq. units

Area = 9 sq. units

1 unit = 1 cm

So, area= 22 sq. cm.

So, area= 9 sq. cm.

Did You Know? Remember! Each side of the square is 1 unit.

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Honeybees use hexagons to build their hives so that their storage space (area) is maximised and the perimeter is minimised. Aren’t they skilled mathematicians?

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Area of Irregular Shapes Like other shapes, irregular shapes are those that are not made entirely of complete squares. Some squares are partly covered. For example, look at the shapes A and B given below. More than half squares

Half squares

Less than half squares

1 cm Shape A

Shape B

In shape A, there are half squares and complete squares, whereas in shape B, there are half squares, less than half squares, and more than half squares. In such cases, we can only find the approximate area using the rules: 1

Complete squares are counted as 1.

All half squares are counted as 1 . 2 3 All more than half squares are counted as 1. 2

All less than half squares are ignored.

The total count of all kinds of squares gives the approximate area of irregular shapes. Now, let us find the areas (approx.) of shapes A and B using these rules. Shape A Squares

Number of unit squares

Area (approx.)

Full squares ( )

Half squares ( )

More than half squares ( )

Less than half squares ( )

Total area (approx.)

27 sq. cm

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Shape B Squares

Number of unit squares

Area (approx.)

Full squares ( )

Half squares ( )

More than half squares ( )

Less than half squares ( )

Total area (approx.)

14 sq. cm

Think and Tell Can you trace some other shape on the grid, such as a leaf, and find its area?

Example 4

Find the area of the shapes, where side of each square = 1 unit. 1

1 unit

Both the shapes are irregular with some squares partly covered. Let’s list these squares in the tables. Squares

Number of unit squares

Area (approx.)

Full squares ( )

Half squares ( )

More than half squares ( )

Less than half squares ( )

Total area (approx.)

36 sq. cm

Chapter 11 • Perimeter and Area

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Squares

Do It Together

Number of unit squares

Area (approx.)

Full squares ( )

Half squares ( )

More than half squares ( )

Less than half squares ( )

Total area (approx.)

18 sq. cm

Find the area of the shapes, where the side of each square = 1 unit. 1

1 unit

Squares

1 unit

Number of unit squares

Area (approx.)

Full squares ( )

Squares

Number of Area unit squares (approx.)

Full squares ( )

Half squares ( )

More than half squares ( )

Less than half squares ( )

Total area (approx.)

Do It Yourself 11B 1

Find the area of these figures. 1 side of the square = 1 unit. a

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The figures below have some fully and some partly covered squares. Find the area of these figures, where the side of each square = 1 unit. a

Find the area of the figures. The side of 1 square = 1 cm. a

a How many rectangles can you draw with an area of 12 sq. units? b Draw and find their perimeters as well. Is the perimeter the same as the area?

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Look at the grounds of two different schools - A and B. Vegetable patch

Playing field

Playing field School A

Vegetable patch School B

Each of these schools has a vegetable patch and a playing field. a Find the area of the playing field in School A. b Find the area of the playing field in School B. c Which school has the bigger playing field?

Word Problem 1

Rohit bought a chocolate and ate part of it, as shown below.

a Find the area of the chocolate bar he ate. b Find the area of the complete chocolate bar he

had before eating.

Points to Remember • The perimeter is the total distance covered along the boundary of a closed figure. The unit of the perimeter is cm, m, mm, or units. • The area is the total space covered by a closed figure. Area is measured in square units, such as sq. cm or sq. units. • To find the perimeter of a polygon, we can also either count the number of units around a polygon on a grid, or add the lengths of all the sides of the polygon.

Perimeter

Area

• The area of any plane figure is the number of squares needed to cover the shape, where the side of 1 square = 1 unit or 1 cm.

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Math Lab Measure Your Hand! Setting: In groups of 2. Materials Required: Square-grid paper, pencils, erasers, colours, a roll of thread, a pair of scissors. Method: Both partners will follow the following steps: 1

Take a sheet of square-grid paper.

Trace your right or left hand on the grid paper.

Find the perimeter of your hands using thread and a ruler.

Then, find the area by counting squares.

Now, discuss the following with your partner: • Do both hands have the same area? • Do both hands have the same perimeter?

Chapter Checkup 1

Find the perimeter of each shape in units. Which shape has a smaller boundary? a

Fill in the blanks. 1 cm

1 cm

The perimeter of figure A is _____ cm. B

The perimeter of figure B is _____ cm. The perimeter of figure C is _____ cm. Figure _____ and figure _____ have

the same perimeter. Chapter 11 • Perimeter and Area

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Find the perimeter of each of the given figures. a

3 cm

2 cm

4 cm d

6 cm

1 cm e

8 cm 3 cm

8 cm

7 cm

4m 6m

8 cm

12 m

5 cm

2 cm

6m 3m

10 m

16 m

10 cm

Each shape has a missing side marked with a letter of the English alphabet. Find the missing length, given the perimeter. a

Perimeter = 39 cm

3 cm

Perimeter = 29 cm

5 cm E

2 cm

4 cm

7 cm

2 cm

4 cm

E 4 cm

9 cm

7 cm

10 cm 5

The figures given below are made up of 1 cm squares.

Which two shapes have the same area?

Which two shapes have the same perimeter?

Each of the figures have 1-unit squares. Answer the following questions.

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Find the area and perimeter of each of the figures by completing the table. Figure

Area (sq. units)

Perimeter (units)

P Q R S T b

Fill in the blanks. (i) Figure _____ and figure _____ have the same area but different perimeters. (ii) Figure _____ and figure _____ have the same perimeter but different areas. (iii) Figure _____ and figure _____ have the same area and perimeter.

Find the area of the given shapes. Find the perimeter of the curved shapes only. a

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How many rectangles can you draw with a perimeter of 20 units? Draw and write the side lengths.

Draw two different shapes, each with an area of 8 sq. units.

10 Add squares to the shape given on the right to make it into a square. What is the area of

the square?

Word Problems 1

Manya drew an owl on square grid paper for her art and craft activity. What is its area

Sneha made a robot for her art and craft activity. Find its area in sq. units and its perimeter in units.

if the side of each square is 1 unit?

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Time

Let's Recall Our day is full of activities! We measure the duration of different activities or events through time. How long is a day? A day is 24 hours long. What are some other events that are about an hour long? A T20 cricket match is typically 3 hours long. A movie is about 2 hours long. A minute is a smaller unit of time. One hour is equal to 60 minutes.

We brush our teeth in the morning. How much time does it take? It takes about 2 minutes. Similarly, traffic signals change every few minutes. An even smaller unit of time is a second. One minute is equal to 60 seconds. A handshake lasts a couple of seconds. A notification sound on the mobile phone lasts 1 or 2 seconds.

Let's Warm-up Fill in the blanks. 1

6 hours is the same as __________ minutes.

360 seconds is equal to __________ minutes.

There are __________ seconds in 2 hours.

40 minutes is __________ (less than, greater than or equal to) 1 hour.

2 hours 15 minutes is __________ (less than, greater than or equal to) 130 minutes. I scored ___________ out of 5.

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Reading Time Real Life Connect

Noor was very tired yesterday. When he went to bed, it was 8 o’clock by his bedroom clock. When he woke up in the morning, the clock was still showing 8 o’clock. “Oh! Has my clock stopped? Why is it showing 8 o’clock again like last night?” thought Noor.

a.m. and p.m. Yesterday, the clock in Noor’s room was showing 8 o’clock at night. Now, it is showing 8 o’clock in the morning. Let’s find out how the wall clock shows the same time twice each day.

08:00 morning

08:00 night

A full day has 24 hours in total while a clock shows only 12 hours. So, the hour hand on a clock completes two rounds. As a result, each time is shown twice a day. In the same way, a day can be divided into two equal halves to read the time. The new day by the clock starts at 12 o’clock at midnight. a.m. : We use a.m. (Ante-Meridian) for the first 12 hours. It includes time starting from midnight 12 o’clock to noon 12 o’clock.

Did You Know?

p.m. : We use p.m. (Post Meridian) for the next 12 hours of the day. It includes time starting from noon 12 o’clock to midnight 12 o’clock.

In Latin, ‘meridies’ means ‘midday’ or ‘noon’.

The word “meridian” is from the Latin language.

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p.m.

a.m.

We know how to read time to 5 minutes on an analogue clock. There are analogue clocks and digital clocks. Let us read time to the minute on an analogue clock. When the minute hand moves from one small marking to another 1 minute has passed.

5 10

When the minute hand moves from one number to another 5 minutes have passed.

15 20

Minutes – Skip count by 5 as 5, 10, 15, 20. Then count forward 21, 22.

Hours – Since the number just before the hour hand is 7, 7 hours have passed.

So, 22 minutes have passed.

The time on the clock is 7:22 Example 1

Write 6 o’clock in the morning and evening as a.m and p.m. 6 o’clock in the morning is 6 a.m. 6 o’clock in the evening is 6 p.m.

Example 2

Read the time on the given clocks. a

9:13 Chapter 12 • Time

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5:06

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Do It Together

Look at Manu’s daily routine. Draw hands of the clock to show the time. Fill in the missing blanks.

Wakes up at 6:10 _______.

Eats breakfast at _______

Comes back from school at _______.

Rides his bicycle at _______.

Goes to bed at 9:07 p.m.

Every morning, Manu wakes up at 6:10 a.m. He gets ready for school and eats breakfast by _________. Then, he walks to school. Manu’s school starts at 9 a.m. and ends at 1:15 _________. Every evening, Manu rides his bicycle _________. He goes to bed by _________.

Think and Tell

Can you think of any other method, other than a.m. and p.m., to show time?

Do It Yourself 12A 1

Write the time in a.m. and p.m. One is done for you. a At 4:30 in the morning

c At 04:30 in the afternoon e Evening 5 o’clock

04:30 a.m.

b At 10:00 in the morning

- ______________

d At 10:00 in the night

- ______________

______________

Write the exact time using a.m or p.m. a 2 hours after 4:30 in the morning

______________

b 4 hours after 10:00 in the morning

______________

c 2 hours before 01:30 in the night

______________

d 4 hours before 10:00 in the night

______________

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Write the time one hour before the time given. a 12:30 p.m

- ______________

b 03:15 a.m

- ______________

c 12:59 a.m

- ______________

d 07:44 p.m

- ______________

Read the time on the given clocks. a

c 10:24

d 12:14

Draw the hands of the clock for the given times. a 3:36

b 4:12

Word Problems 1

Akhil goes for his football practice at 11:30 in the afternoon. How will you write the

Isha goes to see a movie at 10:30 a.m. and comes back 2 hours later. Write the time

time using a.m. or p.m.?

when Isha comes back as a.m. and p.m.

24-hour Clock Real Life Connect

Divya is at a railway platform with her father. She knows that their train leaves at 5 p.m. “Are we on time?” Divya wants to know. She looks at the clock and feels confused. “Dad, what kind of clock is this? It is not like our wall clock or watch! It is showing 16:00.” Dad said, “This is a digital clock, Divya. It gives time in 24 hours.”

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“But how do we read time on this clock? How do we know if the time is a.m. or p.m.?” Divya is curious. And so are we! Let’s learn about this different format of reading time. Usually, we use clocks that display time with hour and minute hands. They use a 12-hour format. They are called analogue clocks. The clock Divya saw uses a 24-hour format of time. They are called digital clocks. A digital clock shows time in 4 digits, in the format of 00:00. The first two digits show the hours and the next two digits show the minutes. A 24-hour time clock uses numbers from 1 to 24 to show 24 hours of a day. 1 to 12 represent the first 12 hours of the day and 13 to 23 show hours after 12 o’clock noon or 12 p.m. On a digital clock, we read time in ‘hours’ unit. For example, the time in this clock will be read as 14:05 hours.

Error Alert! Do not use a.m. or p.m. when reading time in a 24-hour clock. It must be read in ‘hours’.

Remember! Time in 24-hour format can be read as 13:00 hours OR 1300 hours. Both ways are correct and mean the same.

Changing 12-hour to 24-hour Clock Time Railway and airline timetables use 24-hour time or digital clocks. But our regular analogue clocks use 12-hour time. Thus, we need to learn to change 12-hour time with a.m. and p.m. into 24-hour time. Changing from a.m. time: Let’s change 9:30 a.m. into 24 hour format. Step 1 Keep the same hour value. For 9:30 a.m., hour value = 9 or 09.

Step 2 Write the minutes as they are. For 9:30 a.m., minute are 30. Remember, we write 24-hour time as Hours:Minutes. So, 9:30 in 24-hour time will be written as 09:30.

Remember! In a 24-hour format, 12 midnight is written as 00:00 and 12 noon is written as 12:00 hours.

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Step 3 Now, replace a.m. with hours. Thus, time in 24-hour format is 09:30 hours.

Changing from p.m. time: Now, let’s learn to change 10:30 p.m. into 24-hour format. Step 1

Step 2

Step 3

Add 12 to the hour value.

Write the minutes as they are.

Replace p.m. with hours.

Here, hour value = 10. So, the value = 10 + 12 = 22.

Here, minute is 30. So, in 24hour format 10:30 = 22:30.

Change 3:30 a.m. to 24-hour time.

Example 3

Example 4

= 22:30 hours.

Change 6:45 p.m. to 24-hour time.

Follow the steps for changing from a.m. time—

Follow the steps for changing from p.m. time—

Step 1

Keep the same hour value

Add 12 to the hour value

Step 2 Write the minutes as they are

Replace a.m. with hours

: 6 + 12 = 18

Step 2 : 3:30

Write the minutes as they are

Step 3

Do It Together

Thus, the time in 24-hour format

: 18:45

Step 3 : 03:30 hours

Replace p.m. with hours

: 18:45 hours

Nira is at the airport. Her flight departs at 5:50 p.m. The clock at the airport reads 14:45. Is she on time by this time format? To know if Nira is on time, we need to change 5:50 p.m. to 24-hour time. 1 2 3

Add 12 to the hour value: 5 + 12 = __________ Write the minutes as they are: __________

Replace p.m with hours: __________ hours

Changing 24-hour Clock to 12-hour Clock Time Remember, at the railway station and airports we use the 24-hour clock. But we usually use the 12-hour clock. To be sure if we are on time, we need to change the time into the 12-hour format. Follow the steps given below to change the 24-hour into 12-hour clock time. Step 1

Step 2

Step 3

Look at the first two digits of the time.

For a.m. time, keep the hour

Write down the minutes

•

It is a.m. time if the number is less

value as the first two digits.

•

It is p.m time if the number is more

the first two digits.

than 12.

For p.m time, subtract 12 from

than 12.

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as they are.

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Example 5

Change 1130 hours into 12-hour time.

Example 6

Step 1

First two digits of the time are 11. So, it is a.m.

First two digits of the time are 13. It is more

Step 2

For a.m time, keep the same hour value as the

For p.m. time, subtract 12 from the first two

Step 3

Write down the minutes as they are. So the

Write down the minutes as they are: So the

time.

than 12, so it is p.m. time.

first two digits, which is 11.

digits: 13 – 12 = 1.

1130 hours by a 12-hour clock is 11:30 a.m. Do It Together

Change 1300 hours into 12-hour time.

1300 hours by a 12-hour clock is 1:00 p.m.

Divya saw the time on the railway station clock as 18:00 hours. What is the time in 12hour clock? Time in the railway station clock = _____ hours. It is _____ (less/more) than 12. So, it is _____ (a.m./p.m.) time. Now, we subtract 12 from the _____ (first/last) 2 digits: _____ - 12 = _____. Write down the minutes as they are. Thus, the time by a 12-hour clock is = _____.

Do It Yourself 12B 1

Change the time into 24-hour time. a 03:28 p.m.

c 12 Midnight

d 11:59 p.m.

c 23:24 hours

d 13:03 hours

Change the time into 12-hour time. a 22:40 hours

b 11:56 p.m.

b 18:25 hours

Match the following. 09:00 AM

1215

05:45 PM

2020

02:30 AM

2350

12:15 PM

1745

08:20 PM

0900

11:50 PM

0230

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A train starts from the station at 2315 hours and reaches its destination at 1730 hours. Write its

Rama’s train is arriving at the station at 20:00 hours. Her wrist watch is showing 7:45 p.m. Is she on

schedule in the 12-hour clock.

time by her wrist watch time? How? [Hint: Change the time on the wrist watch into 24-hour time.]

Word Problems 1

The Flight from New Delhi to Goa departs at 1445. The Boarding pass will be given 2 hours

Niru has to board the train at 16:45 hours. She reaches the station at 3:15 p.m. How

before departure. At what time will the boarding passes be given by a 12-hour clock? much time is remaining for the train to arrive at the station?

Time and Events Real Life Connect

Siya’s school is from Monday to Friday. On these days, Siya gets on the school bus at 8 a.m. The bus drops her at the school gate at 8: 40 a.m. It takes her 40 minutes to reach school each day. Today, Siya’s school bus is stuck in a traffic jam. Siya reaches school at 9:20 a.m. “How much time did the bus take to reach school today?” Siya wonders.

Time in Hours and Minutes

40 minutes Start time 08:00 a.m.

End time 08:40 a.m.

Siya needs to mark the start and end time to know the time taken by the bus to reach school. The time in between start and end time of Siya reaching the school is the duration. Let’s take another situation to learn about duration or time taken to complete an activity.

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Every Sunday, Siya goes to a music class. The class starts at 4:30 p.m and ends at 6:30 p.m. We can find the duration Siya spends in her music class as shown. 1 hour 4:30 p.m. Start time

1 hour

5:30 p.m.

Think and Tell

How many hours and minutes

6:30 p.m. End time

do you spend at school today?

Duration or time taken for music class = 2 Hours We can get duration in hours or minutes. We can also change time from minutes to hours and from hours to minutes. Change Time from Hours to Minutes. We know, 1 hour = 60 minutes. We can change hours to minutes by multiplying the hour by 60. Let’s see how to change 2 hours to minutes. 1 hour = 60 minutes So, 2 hours = 2 × 60 minutes = 120 minutes Converting Minutes to Hours. To convert minutes to hours, we divide the minutes by 60. Let’s see how to change 120 minutes to hours. 60 minutes = 1 hour 120 minutes in hours = 120 ÷ 60 = 2 hours

Error Alert! Add or subtract hours and minutes separately.

2 hours + 2 minutes = 4 hours OR = 4 minutes

Example 7

Remember! 1 hour = 60 minutes. Division is the opposite of multiplication. To change hours to minutes, we multiply the hour by 60. To change minutes to hours, we divide the minutes by 60.

2 hours + 2 minutes = 2 hours and 2 minutes

Change 4 hours to minutes. 1 hour = 60 minutes

Change 180 minutes to hours. 60 minute = 1 hour

So, 4 hours = 4 × 60 minutes = 240 minutes. So, 180 minutes in hours = 180 ÷ 60 = 3 hours. 222

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The teacher is telling the students about an upcoming test: “The test will be 2 hours long. The first 20 minutes will be for reading the questions.”

Example 8

Niru asked: How many minutes will be left for writing the answers?

Total time for exam = 2 hours 2 hours in minutes = 2 × 60 = 120 minutes Time for reading the questions = 20 minutes Time left for writing answers = 120 – 20 = 100 minutes Do It Together

Abhilash boarded the bus from Bangalore at 10:30 p.m. and reached Pune at 06:30 p.m the next day. What was the duration of his journey in minutes? Start time

___________

End time

___________

Duration in hours

___________

Duration in minutes =

___________ hours × 60 = ___________ minutes

Do It Yourself 12C 1

Fill in the blanks. a There are ______________ minutes in an hour. b There are ______________ hours in a day.

Change hours to minutes. Then, write the appropriate symbol ( >, = , < ) in the blank. a 7 hours ______________ 420 minutes

b 3 hours ______________ 115 minutes

c 6 hours ______________ 360 minutes

d 10 hours ______________ 360 minutes

Change minutes to hours. a 340 minutes

b 450 minutes

c 560 minutes

d 675 minutes

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Find the duration between the times. a 12:00 noon to 12:30 midnight

b 05:06 p.m to 10:55 p.m

c 14:25 hours to 20:45 hours

d 10:15 hours to 23:30 hours

The movie starts at 7:30 p.m. It is 2 hours 15 minutes long. At what time will the movie end?

Word Problems 1

A doctor starts his patient visits at 10:15 a.m. and returns to his cabin by 1 p.m. How

Neha takes 30 minutes to do her maths homework, 15 minutes to write the English

Nihit starts reading a book at 16:30. He reads for 45 minutes. What time does he

Rohan started to do his homework at 20:30 hours. He completed all his work after

A train left Mumbai at 2315 hours and reached Trivandrum at 1300 hours. How long

much time did he spend on the patient visits?

notes and 70 minutes to revise all other subjects. How much time does she study? stop reading?

50 minutes. At what time did he finish his homework? was the journey?

Time in Days, Weeks, Months and Years Real Life Connect

Every student in the class is thrilled on hearing the news of an upcoming one-day trip. “We will be going on the 25th of next month. It is two weeks from now!” the teacher informs them. “25th of next month? 2 weeks from today? How many days later, exactly?” Divya is confused. Let’s solve Divya’s confusion. We need to find out how many days there are in a week and in a month.

Days in Week A week starts with Monday and ends on Sunday. 7 days combined make a week. Have a look at this calendar for the month of April. The first full week of this month starts from Monday, 4 April and ends on Sunday, 10 April. This makes a week. The same days of the week repeat every 7 days. So, if 12 April is a Tuesday, then the next Tuesday will be 12 + 7 = 19 April. 224

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In the same way, if 18 July is a Saturday, then the previous Saturday was on 18 – 7 = 11 July.

Think and Tell

1 day has 24 hours. How many hours are there in a week? Can you tell the total minutes in a week too?

Remember! Same days repeat after every 7 days and all are multiples of 7.

Days in a Month We know that there are 12 months in a year. These months have 28, 29, 30 or 31 days. JANUARY

FEBRUARY

MARCH

APRIL

31 days

28 or 29 days

31 days

30 days

MAY

JUNE

JULY

AUGUST

31 days

30 days

31 days

SEPTEMBER

OCTOBER

NOVEMBER

DECEMBER

30 days

31 days

30 days

31 days

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Days in a Year As shown in the calendar above, 4 months have 30 days, 7 months have 31 days. February can have either 28 or 29 days. Adding all the days of all the twelve months, we get a full year. 4 × 30 days = 120 days

7 × 31 days = 217 days

1 × 28 days = 28 days

Adding all the days of the year: 120 + 217 + 28 = 365 days If it is leap year, February will have 29 days. Then, the year will have 366 days instead of 365 days.

Remember! A leap year is a year that has one extra day added to it. It occurs every fourth year. Number of days in a Leap Year = 365 days + 1 day = 366 days.

Each year starts with January and ends with December. Then, the twelve months repeat and the next year begins. For example, after 31 December 2023, the next year starts. So, the next day would be 1 January, 2024. A calendar is the record of all the days and months of a year. We read them as dates. We can write the date in short using the format Date Month Year. For example, 18 July 2023 can be written as 18.07.23. A calendar helps us to remember important events such as festivals, birthdays or anniversaries and plan for them. Example 9

Gokul’s birthday is two weeks after Independence Day which is on 15 August and happens to fall on Friday this year. When is Gokul’s birthday? Independence Day is on 15 August. 1 week = 7 days. So 2 weeks = 14 days Gokul’s Birthday= 15 August + 2 weeks OR 15 August + 14 days = 29 August Since 15 August is a Friday, 29 August will also be a Friday. Thus, Gokul’s Birthday is on Friday, 29 August.

Example 10

How many days are there between January 18 and February 12? Total days in January = 31 days Days remaining in January = 31 – 18 = 13 days

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Days in February = 12 days So, total days = 13 + 12 = 25 days. Do It Together

Rahul is going for a Winter Camp for 15 days on 24 December, 2023. When will he return? Total camping days = ______ days Total days in December = 31 days. So, remaining days in December = 31 – 24 = ______ days. In the next month of January, remaining days for camping = 15 – ______ = ______ more days.

November December

October

August September

June July

May

January February March April

The date of return = ______ January

Did You Know? You can count the number of days in a month using your fist! The top of the knuckles show shows months with 31 days and the hollows between the knuckles show months with 30 days and 28 or 29 days of February. We start with the top knuckle of our little finger.

Do It Yourself 12D 1

Fill in the blanks. a There are ___________ days in a leap year. b If 03.03.03 is a Friday, then the next Sunday will be on ___________ . c ___________ is a month with 28 or 29 days. d 2 years = ___________ months e 50 months = ___________ years and ___________ months.

Write these dates in short form — Date.Month.Year. a 19 November 1996

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b 15 August 1947

c 29 July 2023

d 28 February 2004

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Find the number of days between the given dates. a 30 June and 23 July

b 5 September and 2 November

c 12.05.20 and 10 June 2020

d 07.06.23 and 23.07.23

Ram spent 8 days of his vacation in Kashmir, 9 Days in Delhi and 4 days in Dehradun. How many

Joby takes 2 weeks of leave from school to attend a wedding. The leave begins on 5 March. When will

weeks of vacation did he have?

he return to school?

Word Problems 1

If 10 December is a Friday, on which day will the New Year begin?

Riya’s birthday is on the third Monday after December 25. When is her birthday if

The manufacturing date of a chocolate is written as 12.12.2023. The expiry date is

December 25 is on a Monday?

mentioned as “Best before 18 months.” When will the chocolate expire?

Points to Remember •

An analogue clock uses a 12-hour format and completes 2 rounds for the 24 hours in a day.

•

We use a.m. for the first 12 hours, starting from midnight, and p.m. for the next 12 hours of the day starting from noon.

•

Digital clocks use a 24-hour time format. These clocks are commonly used in railway, airline and military timetables.

•

We can convert p.m. times to 24-hour format by adding 12 to the hour value.

•

We can convert 24-hour time to p.m. time by subtracting 12 from the hour value.

•

1 Hour = 60 minutes. We change hours to minutes by multiplying by 60. We change minutes to hours by dividing by 60.

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Math Lab My Monthly Planner Setting: In pairs Materials Required: Paper, ruler, pencil. Method: •

Prepare a creative calendar for the month of February. MONTH SUNDAY

YEAR MONDAY

TUESDAY

WEDNESDAY

THURSDAY

FRIDAY

SATURDAY

•

Draw a table on the paper with 7 columns. Add days of the week at the top, starting with

•

January 25 is a Wednesday. Discuss with your partner how to find the first day of February.

•

Write all the dates for the month of February starting from the first day. Repeat the week

•

Discuss with your partner and mark any special or important dates on this calendar.

Monday.

for every 7th day.

You may prepare a similar planner for any other month!

Chapter Checkup 1

Fill in the blanks with the correct time in a.m. or p.m. a

This morning, Emily woke up at 7 ________.

She took 45 minutes to get ready, then it was ________ a.m.

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She had her lunch at 12:30 ________ in the cafeteria with her friends.

After spending 7 hours at school, Emily reached home at ________.

Read the given clocks. Write the time. a

Write the correct time using either a.m. or p.m. a

2 hours after 4:30 in the morning - __________

3 hours after 8:45 in the evening - __________

1 hour after 10:00 at night - __________

4 hours after 1:20 in the afternoon - __________

1 hour before 11:40 in the morning - __________

2 hours before 9:15 in the evening - __________

3 hours before 12:00 at noon - __________

1 hour after 7:55 at night - __________

2 hours before 2:30 in the afternoon - __________

3 hours after 6:10 in the morning - __________

Change the time into 24-hour clock time. a

06:30 a.m.

07:55 a.m.

09:25 a.m.

12:00 a.m.

01:03 p.m.

09:15 p.m.

10:24 p.m.

11:59 p.m.

Change the time into 12-hour clock times. a

14:20 hours

15:45 hours

21:12 hours

04:30 hours

10:10 hours

14:55 hours

20:18 hours

18:12 hours

23:06 hours

16:50 hours

Change the following time between hours and minutes. a

2 hours to minutes.

120 minutes to hours.

3 hours and 30 minutes to minutes.

450 minutes to hours and minutes.

1 hour and 15 minutes to minutes.

90 minutes to hours and minutes.

4 hours and 20 minutes to minutes.

300 minutes to hours.

5 hours and 45 minutes to minutes.

390 minutes to hours and minutes.

Find the duration between the times. a

08:00 a.m. to 02:45 p.m.

09:30 a.m. to 05:15 p.m.

07:00 p.m. to 11:30 p.m.

02:00 p.m. to 06:45 p.m.

10:15 a.m. to 01:30 p.m.

03:20 p.m. to 06:10 p.m.

11:00 a.m. to 02:45 p.m.

05:30 p.m. to 09:15 p.m.

08:45 a.m. to 11:30 a.m.

12:15 p.m. to 03:30 p.m.

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Siya began to colour at 17:30 hours. If she finished colouring after 100 minutes, at what time did she

Sravan meditates for 30 minutes every day. How many hours does he spend on meditation in 4 days?

finish?

10 Neetu does yoga for 45 minutes each day. How many hours does she do yoga every week?

Word Problems 1

The movie starts at 1530 hours and ends at 1815. If the interval is 20 minutes,

A bus departs from Cochin at 08:45 p.m. and reaches Bangalore at 04:45 a.m.

7 December is a Sunday. On which day will the New Year begin?

Today is 10 January. Anil’s birthday is in 45 days. On which date will his birthday

Chapter 12 • Time

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what is the duration of the movie?

What is the total duration of this journey in minutes?

be?

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Money

Let's Recall Money helps us buy things that we need in daily life. It is a medium of exchange through which the world works. The shopkeeper takes money from us and gives us what we need. All money is not the same! Different countries have different currencies. The Indian National Rupee or INR is the currency in India. Similarly, the USA has its Dollar, the UK has its Pound Sterling and Japan has its Yen.

Rupee

Dollar

Pound

Yen

The INR has many currency notes.

What notes can we use if we want to buy a water bottle that is priced at INR 80? We can use 8 notes of INR 10.

We can use 4 notes of INR 20. We can use 1 note of INR 50, 1 note of INR 20 and 1 note of INR 10. We can also use 1 note of INR 50 and 3 notes of INR 10.

Let's Warm-up Fill in the blanks. 1

We can exchange _______ ₹10 notes for a ₹50 note.

We can exchange _______ ₹20 notes for a ₹100 note.

For a book that costs ₹120 we can give _______ note of ₹100 and _______ notes of ₹10. I scored ___________ out of 3.

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Counting Money Real Life Connect

Mom took Rohan to the nearby ATM to withdraw some money. Rohan: Why have we come here, Mom? Mom: We have come to withdraw some money from the ATM. After the withdrawal, a slip came out of the machine. Rohan: What is written on this slip? Mom: The slip shows the money left in our bank account. Rohan saw that the amount left in the bank account was ₹648.65.

Reviewing Rupees and Paise Rohan was wondering how to read the amount. Let us see how to read the amount. ₹648.65 The dot separates the rupees

The number on the left of

and paise.

the dot shows rupees.

The number on the right of the dot shows paise.

Express Money in Words When reading the amount in words, we read the left part in rupees and the right part in paise. ₹648.65 can be expressed in words as “Six hundred forty-eight rupees and sixtyfive paise”.

Did You Know? Symbols below the date on the coin indicates where it was minted. Mint

Mint Mark

Identification

Mumbai

Diamond

Kolkata

No Mark

Hyderabad

Star

Noida

Dot

The star mark indicates that this coin was minted in Hyderabad.

Chapter 13 • Money

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How will you write ₹255.84 in words?

Example 1

Rupees

₹255.84

Write three hundred fifty-eight rupees and nine paise in figures.

Example 2

Three hundred fifty-eight rupees = ₹358.

Paise

Nine paise = 09.

₹255.84 = Two hundred fifty-five rupees and eighty-four paise. Do It Together

Three hundred fifty-eight rupees and nine paise = ₹358.09.

Write if true or false. 1

₹354.20 = Three hundred fifty-four rupees and twenty paise.

True

₹368.02 = Three hundred sixty-eight rupees and twenty paise.

___________

₹517.26 = Five hundred seventeen rupees and twenty-six paise.

___________

₹632.70 = Six hundred thirty rupees and seven paise.

False

Conversion between Rupees and Paise Money can be converted from rupees to paise and vice versa. Let us convert ₹1229.54 into paise. Step 1

Step 2

Remove the dot and ₹ sign.

Write paise with the number.

₹1229.54 = 122954

Therefore, ₹1229.54 = 122954 paise.

We can also convert paise into rupees. Let us convert 151236 paise into rupees. Step 1

Step 2

Remove the word ‘paise’ and put a dot after

Put the sign of ₹ before the number.

counting 2 numbers from the right of the

151236

number.

1512.36

`1512.36

Therefore, 151236 paise = `1512.36.

We have learnt about Indian currency and its conversion. But different countries have different currencies. The Indian Rupee for example is not accepted outside India. Let us see the currencies of different countries. Example 3

Country

Currency

United States (US)

US Dollar

United Kingdom

Sterling Pound

Germany Japan

China

Euro Yen

Yuan

Symbol $ € £ ¥ 元

Convert ₹4236.25 to paise. Step 1

Step 2

Remove the dot and ₹ sign.

Write paise with the number.

₹4236.25 = 423625.

Therefore, ₹4236.25 = 423625 paise.

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Convert 745623 paise to rupees.

Example 4

Step 1

Step 2

Remove the word ‘paise’ and put a dot after

Put the sign of ₹ before the number.

counting 2 numbers from the right of the given number.

Therefore, 745623 paise = ₹7456.23.

745623 paise can be written as 7456.23. Do It Together

Fill in the blanks. In Rupees

In Paise

₹635.23

____________

₹____________

85205 paise

₹4126.24

412624 paise

₹5386.15

______________

₹____________

825632 paise

Do It Yourself 13A 1

Express the money in words. a ₹154.56

b ₹217.85

c ₹396.48

d ₹469.05

e ₹679.21

₹748.49

e ₹1015.48

₹1247.69

Write the amount in numerals. a Three hundred eighty-nine rupees and sixty-three paise. b Five hundred forty-two rupees and eighty-three paise. c Six hundred fifty-two rupees and thirty-nine paise. d Seven hundred thirty-three rupees and forty-two paise. e Eight hundred sixty-three rupees and seventy-seven paise. f

Nine hundred seventy-four rupees and three paise.

Convert the amount in paise. a ₹578.24

c ₹846.25

d ₹945.37

Convert the amount in rupees. a 63512 paise

b 74624 paise

c 84761 paise

d 97456 paise

e 112564 paise

Chapter 13 • Money

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b ₹647.12

135489 paise

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Match the countries with the name of their currencies. India

Yen (¥)

United States

Rupees (₹)

China

Pound (£)

Japan

Euro (€)

United Kingdom

Dollar ($)

Germany

Yuan (元)