Imagine_Maths_CB_Grade4_SS_AY25 by Uolo

Master Mathematical Thinking

MATHEMATICS

Master Mathematical Thinking

Acknowledgements

Academic Authors: Muskan Panjwani, Animesh Mittal, Anjana AR, Anuj Gupta, Gitanjali Lal, Simran Singh

Creative Directors: Bhavna Tripathi, Mangal Singh Rana, Satish

Book Production: Sanjay Kumar Goel, Vishesh Agarwal

Project Lead: Neena Aul

VP, Learning: Abhishek Bhatnagar

All products and brand names used in this book are trademarks, registered trademarks or trade names of their respective owners.

First impression 2024

Second impression 2025

This book is sold subject to the condition that it shall not by way of trade or otherwise, be lent, resold, hired out, or otherwise circulated without the publisher’s prior written consent in any form of binding or cover other than that in which it is published and without a similar condition including this condition being imposed on the subsequent purchaser and without limiting the rights under copyright reserved above, no part of this publication may be reproduced, stored in or introduced into a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior written permission of both the copyright owner and the above-mentioned publisher of this book.

Book Title: Imagine Mathematics 4

ISBN: 978-81-979482-9-9

Published by Uolo EdTech Private Limited

Corporate Office Address: 85, Sector 44, Gurugram, Haryana 122003

CIN: U74999DL2017PTC322986

Illustrations and images: www.shutterstock.com, www.stock.adobe.com and www.freepik.com

All suggested use of the internet should be under adult supervision.

Preface

Mathematics is an essential tool for understanding the world around us. It is not just another subject, but an integral part of our lives. It shapes the very foundation of our understanding, personality and interaction with the world around us. Studies from across the globe have shown that proficiency in mathematics significantly influences career prospects and lifelong learning.

According to the NEP 2020, mathematics and mathematical thinking are crucial for empowering individuals in their everyday interactions and affairs. It focuses on competencies-based education, which essentially means actively and effectively applying mathematical concepts in real life. It also encourages innovative approaches for teaching maths, including regular use of puzzles, games and relatable real-world examples to make the subject engaging and enjoyable.

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 math exciting, relatable and meaningful for children.

Imagine Mathematics 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 NCF 2023 and other literature in learning educational pedagogies. Subsequent pages elaborate further on this approach and its actualisation in this book.

This book incorporates highly acclaimed, learner-friendly teaching strategies. Each chapter introduces concepts through real-life situations and storytelling, connecting to children’s experiences and transitioning smoothly from concrete to abstract. Clear explanations and simple steps are provided for problem-solving. Engaging facts, error alerts and fun activities are integrated throughout to keep learning interesting and holistic. Importantly, concepts are interconnected across topics and grades, providing a scaffolding that ensures comprehensive and meaningful learning.

This book supports learners at all levels, providing opportunities to build critical thinking skills through questions and activities aligned with Bloom’s Taxonomy. For those seeking a greater challenge, the book includes thought-provoking questions that push learners to apply, analyse and evaluate. Additionally, the problems are rooted in real-world contexts, making the learning experience both relatable and meaningful.

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 worksheets, among other things. Interactive exercises on the digital platform make learning experiential and help in concrete visualisation of abstract mathematical concepts. We invite educators, parents and students to embrace Imagine and join us in nurturing the next generation of thinkers, innovators and problem-solvers. Embark on this exciting journey with us and let Imagine be a valuable resource in your educational adventure.

Product Package at a Glance

Welcome to our comprehensive learning package designed to enhance educational experiences through three key components: print materials, digital resources and assessment tools. Our print materials provide in-depth and accessible information in a pedagogically suitable way, our digital resources offer interactive and engaging learning experiences, while our assessment tools ensure thorough understanding and progress tracking.

PRINT

Engaging Textbooks

Teacher Manuals

ASSESSMENT

Competency-based Model Assessments

Question Paper Generator

Student and Teacher Apps

Video Solutions

Interactive Tasks & Exercises

Byte Size Lesson Modules

Key NEP Recommendations

The National Education Policy (NEP) 2020, introduced by the Government of India, represents a transformative shift in the country’s education system. It aims to create a more holistic, dynamic and multidisciplinary approach to education. NEP 2020 focuses on fostering conceptual understanding, skills and values that align with the demands of the 21st century, while also preserving India’s rich cultural heritage. UOLO is fully committed to actualising the vision of NEP 2020 by meticulously adhering to its outlined recommendations.

1. Focus on conceptual understanding

2. 21st century skills, values and dispositions

3. Critical thinking and problem-solving

4. Application in real life

5. Holistic and integrated learning

6. Experiential learning

7. Enjoyable and engaging

8. Computational and mathematical thinking

9. Technology-based solutions

10. Factoids on India

Competency-based Education

NEP Pages 12, 17, and 22

Teaching and Learning Pedagogy

NEP Pages 3, 11, 12, 18, and 27

National Pride

NEP Pages 15, 16, and 43

11. Assessment of core concepts and application skills Assessments

NEP Pages 12, 18, and 22

Our Key Features: Aligning with the NEP

The GRR Approach

Pedagogical approach that empowers students to learn by the teacher progressively transitioning the responsibility to the students.

Competency-based Assessments

Test papers designed to evaluate understanding of core concepts and application of skills.

3 8 11

Contextual Learning

Introducing concepts through real-life situations and connecting them to students’ experiences.

Classroom Activity

A hands-on classroom activity to investigate and understand mathematical concepts in an engaging and concrete way.

Recall and Revisit

Introductory page with a quick recall of concepts done in previous grades.

Translating Words into Action!

Critical Thinking Questions

Intellectually stimulating questions designed to enhance problem-solving and analytical-thinking skills, promote deeper understanding and foster independent thinking.

Real-Life Math Problems

Scenario-based questions designed to help in applying theoretical knowledge to real-world scenarios, reasoning skills and improves reading comprehension & analytical abilities.

Common Misconceptions

Concise snippets of information designed to caution against potential errors and misconceptions

Think, Analyse and Answer

A quick, mathematical-thinking question

Fun Facts

Facts related to the concepts, or bite-sized information about the contribution of Indian scholars to mathematics

Gradual Release of Responsibility

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

Real Life Connect

Theoretical Explanation

I do

Examples

Understanding Large Numbers

Dhruv was reading a newspaper. He came across news about different states in India that participated in the COVID vaccination drive and the number of vaccinations given until August 2023. Given below is the data of four states.

Delhi Haryana Sikkim Goa 37409161 45546836 1360477 2874477

The temperature can change with different weather conditions. On a sunny day, the air feels warm, and the thermometer shows a higher temperature. On a cloudy or rainy day, the air feels cooler, and the temperature on the thermometer drops. Weather conditions like rain, wind and sunlight can all affect the temperature of the air.

All About 7-Digit and 8-Digit Numbers

Temperature Range (°C) Weather

While reading the news, Dhruv got confused and could not read the numbers given in the data. The numbers of vaccinations given were either 7-digit numbers or 8-digit numbers.

Below 0 Very

Temperature Range (°C)

Did You Know?

Sikkim Goa Delhi Haryana 1360477 2874477 37409161 45546836

Did You Know?

Drass in Jammu and Kashmir is the second coldest inhabited region on Earth.

Above 40 Very Hot

7-digit Numbers 8-digit Numbers

Example 8: Aliya’s body temperature was 2.3°C higher than normal. What was Aliya’s body temperature?

Normal body temperature = 37°C.

Place Value, Face Value and Expanded Form Reading and Writing 7-Digit and 8-Digit Numbers

Drass in Jammu and Kashmir is the second coldest inhabited region on Earth.

Example 8: Aliya’s body temperature was 2.3°C higher than normal. What was Aliya’s body temperature?

Aliya’s body temperature = 37°C + 2.3°C = 39.3°C

Let us help Dhruv understand 7-digit and 8-digit numbers!

Normal body temperature = 37°C.

So, Aliya’s body temperature was 39.3°C.

Aliya’s body temperature = 37°C + 2.3°C = 39.3°C

We know that the greatest 6-digit number is 999999. Now, if we add 1 to this number, we get 1000000. 999999 + 1 = 1000000

Example 9: In a city, the highest temperature was 37.5°C, and the lowest was 21.2°C. What is the difference between these two temperatures?

So, Aliya’s body temperature was 39.3°C.

To find the difference, subtract the lowest temperature from the highest temperature.

37.5°C – 21.2°C = 16.3°C

Example 9: In a city, the highest temperature was 37.5°C, and the lowest was 21.2°C. What is the difference between these two temperatures?

The temperature difference is 16.3°C.

1000000 is the smallest 7-digit number and is read as “Ten Lakhs”. We saw in the news article that the number of vaccinations administered in Sikkim was 1360477. Let us try to place this 7-digit number in the place value chart.

To find the difference, subtract the lowest temperature from the highest temperature.

Write the readings of the given thermometers.

37.5°C – 21.2°C = 16.3°C

The temperature difference is 16.3°C. Write the readings of the given thermometers.

Temperature = 45°C Temperature = Temperature = Temperature =

Converting Between Units of Temperature

Temperature = 45°C Temperature = Temperature = Temperature =

Celsius (°C) is like the universal language of temperature. Fahrenheit (°F) is another way to talk about temperature. We learnt that water freezes at 0°C and boils at 100°C. In Fahrenheit, water freezes at 32°F and boils at 212°F.

Converting Between Units of Temperature

GRR Steps Unit Component

Do It Yourself

You do

Chapter Checkup Challenge

Case Study

Do It Yourself 14B

1 Identify the shape for which the net is drawn.

Do as directed.

a Write the greatest 7-digit number that has the smallest odd digit at its hundreds, ten thousands and lakhs place.

b Write the smallest 8-digit number that has the digit 7 at all its odd positions, starting from the ones place.

2 Look at the net and identify the object it belongs to.

3 Draw the net of the given shapes.

Critical Thinking & Cross Curricular

The Kumbh Mela is a major pilgrimage where Hindu pilgrims take the holy bath in the Ganges. The number of people who visited the Kumbh Mela in 1980 were 20,356,817 and those who visited in 1989 were 29,304,871. In which year did less than 25 crore people visit to the Kumbh Mela?

a 1980 b 1989 c Both years d None of the year

4 Rishi and Megha made the net of a square-based pyramid. Who made the net correctly? Explain your answer with reasons.

Comparing, Ordering and Rounding-Off Large Numbers

a Rishi’s drawing

Chapter Checkup

b Megha’s drawing

During COVID, India offered support to 150 affected countries in the form of vaccines, medical equipment and medicines. Given below is the data of the number of vaccine doses supplied by India to four different countries.

Guess the best units of length (m or cm) and weight (kg or g) for the given objects. a b c

5 Draw the net of a hexagonal prism.

6 Show using nets how a rectangular prism is different from a rectangular pyramid.

Comparing and Ordering Numbers

Measure the objects.

Rahul: Which country did India supply the greatest number of vaccine doses to?

Challenge

Critical Thinking

Bran: We could compare the numbers to find the country to which India supplied the greatest number of vaccine doses.

Comparing Numbers

1 Sanya wants to solve a 7-digit secret code in a safe. Use the given clues to help Sanya solve the secret code.

a The digit in the hundreds and ones place is 6.

What if Rahul wanted to compare the number of vaccines sent to Nepal and Australia? Let us find out.

b The digit in the lakhs place is 4 less than the digit in the ones place.

c The digit in the ten lakhs and ten thousands place is the smallest odd number.

Since 94,99,000 has 7 digits and 3,09,13,200 has 8 digits, 3,09,13,200 > 94,99,000.

d The face value of the digit in the thousands place is 5.

Thus, Australia was donated more vaccine doses.

Remember!

e The digit in the tens place is the biggest 1-digit number. What is the secret code?

Now, what if we want to compare two numbers with the same number of digits? Let us consider 4,13,23,456 and 4,13,23,657.

A number with more number of digits is always greater.

200

James has some cotton candy which is 3 cm longer than the cotton candy shown below. How long is James’ cotton candy?

2 Write the greatest 8-digit odd number using only 5 digits. Do not repeat any digit more than twice.

Chapter 1 • Numbers up to 8 Digits

Case Study

How much longer is the red straw than the blue straw?

Cross Curricular

Population of Countries

The population of different countries is shown using a table. Read the data carefully and answer the questions.

Country

The population of different countries is shown using a table. Read the data carefully and answer the questions.

Convert the lengths. a 8 m into km b 4 hm 35 m into hm c 1232 m into mm d 897 m into dam

Convert the weights.

United Kingdom 67,736,802

a 5 kg into g b 4 g 64 cg into g c 5487 g into mg d 43 kg 7 dag into kg

United

Poland 41,026,067

A baby koala is called a joey. A young joey weighs about 0.38 kg. How much is that in g?

A candle weighs 125 g. How much is it in mg?

1 Which country has the least population?

a Italy b Germany c Poland d United Kingdom

2 Which country has the greatest population?

a Poland b United Kingdom c Italy d Germany

Pearson, P. D., & Gallagher, G. (1983). Contemporary Educational Psychology. Fisher, D., & Frey, N. (2021). Better learning through structured teaching: A framework for the gradual release of responsibility. Fisher, D., & Frey, N. (2014). Checking for understanding: Formative assessment techniques for your classroom.

a Poland b United Kingdom c Italy d Germany

3 Which country has approximately double the population than that of Poland?

4 Arrange the countries in ascending order as per their population.

5 If all the digits in the population of each country is rearranged to form the greatest number, then which country will have the greatest population?

The NEP Tags

The National Education Policy (NEP), 2020, outlines essential skills, values, dispositions and learning approaches necessary for students to thrive in the 21st century. This textbook identifies and incorporates these elements throughout its content, activities and exercises. Referred to as “NEP Tags,” they are defined as follows:

Art Integration

Bringing creativity and fun into learning by combining art with maths. Students construct and demonstrate understanding through an art form.

Collaboration

Working effectively with others. Includes clear communication, teamwork, active listening and valuing diverse approaches.

Cross Curricular

Integrating mathematical concepts with other subjects to see the real-life applications of maths.

Experiential Learning

Gaining knowledge and skills through direct, hands-on experiences rather than a traditional classroom setting.

Communication

Explaining mathematical ideas, processes and solutions to others, whether through verbal explanations, written work or visual representations.

Value Development

Promoting ethics, and human & constitutional values, like empathy, respect for others, cleanliness, courtesy, democratic spirit, spirit of service, respect for public property, responsibility and equality.

Creativity

Using imagination and original thinking to solve problems and explore mathematical concepts in innovative ways. Also includes finding unique solutions, and designing original problem-solving strategies.

Education Standards as per the NCF 2023

The National Curriculum Framework for School Education (NCF), released in 2023, is based on the vision of the National Education Policy (NEP), 2020, and enables its implementation. The NCF provides guidelines for designing school syllabi and textbooks in India. It aims to improve the quality of education by making it more relevant, engaging, inclusive and learner-centric. To achieve this, the NCF has articulated precise Learning Standards through well-defined Curricular Goals and Competencies. These statements serve to harmonise the syllabus, content, pedagogical practices and assessment culture, ensuring a cohesive and comprehensive educational experience.

Curricular Goals: Statements that give direction to curriculum development and implementation in order to achieve the Curricular Aims. They are also specific to a School Stage (e.g., the Foundational Stage) and a Curricular Area (e.g., Mathematics).

Competencies: Learning achievements that are observable and can be assessed systematically. These Competencies are derived from the Curricular Goals and are expected to be attained by the end of a Stage.

Curricular Goals

CG-1

Understands numbers (counting numbers and fractions), represents whole numbers using the Indian place value system, understands and carries out the four basic operations with whole numbers, and discovers and recognises patterns in number sequences

CG-2

Analyses the characteristics and properties of two- and three-dimensional geometric shapes, specifies locations and describes spatial relationships and recognises and creates shapes that have symmetry

Competencies

C-1.1 Represents numbers using the place value structure of the Indian number system, compares whole numbers, and knows and can read the names of very large numbers

C-1.2 Represents and compares commonly used fractions in daily life (such as ½, ¼) as parts of unit wholes, as locations on number lines and as divisions of whole numbers

C-1.3 Understands and visualises arithmetic operations and the relationships among them, knows addition and multiplication tables at least up to 10×10 (pahade) and applies the four basic operations on whole numbers to solve daily life problems

C-1.4 Recognises, describes, and extends simple number patterns such as odd numbers, even numbers, square numbers, cubes, powers of 2, powers of 10, and Virahanka–Fibonacci numbers.

C-2.1 Identifies, compares and analyses attributes of two- and three-dimensional shapes and develops vocabulary to describe their attributes/properties

C-2.2 Describes location and movement using both common language and mathematical vocabulary; understands the notion of map (najri naksha)

C-2.3 Recognises and creates symmetry (reflection, rotation) in familiar 2D and 3D shapes

C-2.4 Discovers, recognises, describes and extends patterns in 2D and 3D shapes

The above is an abridged version of the curricular goals and competencies relatioship in Maths for the Foundational Stage (NCF 2023, pages 340–341). The next section shows the coverage of all these competencies across the chapters.

Mapping with NCF 2023

CG-1

C-1.1

Represents numbers using the place value structure of the Indian number system, compares whole numbers, and knows and can read the names of very large numbers

C-1.2 Represents and compares commonly used fractions in daily life (such as ½, ¼) as parts of unit wholes, as locations on number lines and as divisions of whole numbers

C-1.3 Understands and visualises arithmetic operations and the relationships among them, knows addition and multiplication tables at least up to 10 × 10 (pahade) and applies the four basic operations on whole numbers to solve daily life problems

C-1.4 Recognises, describes and extends simple number patterns such as odd numbers, even numbers, square numbers, cubes, powers of 2, powers of 10, and Virahanka–

Fibonacci numbers.

C-2.1 Identifies, compares and analyses attributes of twoand three-dimensional shapes and develops vocabulary to describe their attributes/properties

C-2.2 Describes location and movement using both common language and mathematical vocabulary; understands the notion of map (najri naksha)

C-2.3 Recognises and creates symmetry (reflection, rotation) in familiar 2D and 3D shapes

C-2.4 Discovers, recognises, describes and extends patterns in 2D and 3D shapes

CG-2

Analyses the characteristics and properties of twoand threedimensional geometric shapes, specifies locations and describes spatial relationships, and recognises and creates shapes that have symmetry

Measures in non-standard and standard units and evaluates the need for standard units

CG-3

C-3.1

Understands measurable attributes of objects and the units, systems and processes of such measurement, including those related to distance, length, weight, area, volume and time using nonstandard and standard units

C-3.2 Uses an appropriate unit and tool for the attribute (like length, perimeter, time, weight, volume) being measured

C-3.3 Carries out simple unit conversions, such as from centimetres to metres, within a system of measurement

C-3.4 Understands the definition and formula for the area of a square or rectangle as length times breadth

C-3.5 Devises strategies for estimating the distance, length, time, perimeter (for regular and irregular shapes), area (for regular and irregular shapes), weight, and volume and verifies the same using standard units

C-3.6 Deduces that shapes having equal areas can have different perimeters and shapes having equal perimeters can have different areas

C-3.7 Evaluates the conservation of attributes like length and volume, and solves daily-life problems related to them

C-4.1 Solves puzzles and daily-life problems involving one or more operations on whole numbers (including word puzzles and puzzles from ‘recreational’ areas, such as the construction of magic squares)

C-4.2 Learns to systematically count and list all possible permutations or combination given a constraint, in simple situations (e.g., how to make a committee of two people from a group of five people)

C-4.3 Selects appropriate methods and tools for computing with whole numbers, such as mental computation, estimation or paper-pencil calculation, in accordance with the context

C-5.1 Understands the development of zero in India and the Indian place value system for writing numerals, the history of its transmission to the world, and its modern impact on our lives and in all technology

CG-4

Develops problem-solving skills with procedural fluency to solve mathematical puzzles as well as daily-life problems, and as a step towards developing computational thinking.

CG-5

Knows and appreciates the development in India of the decimal place value system that is used around the world today

1 Numbers up to 8 Digits

Let’s Recall

Numbers are used everywhere in our daily lives. These numbers are formed using the digits 0 to 9 and are written using commas after every period, starting from the one’s period.

Chapter at a Glance: Walkthrough of Key Elements

For example, let us say the pin code of your area is 201301. This is a 6-digit number. It can be written using commas by representing it in a place value chart.

1 Numbers up to 8 Digits

Number Lakhs Thousands Ones Lakhs (L) Ten Thousand (TTh) Thousands (Th) Hundreds (H) Tens (T) Ones (O) 2,01,301 2 0 1 3 0 1

periods place

Each of these digits has a place value and a face value. Let us write the face value, place value, expanded form and number name for 201301.

Let’s Recall

Numbers are used everywhere in our daily lives. These numbers are formed using the digits 0 to 9 and are written using commas after every period, starting from the one’s period.

Let’s Recall: Introductory page with a quick recall of concepts learnt in the previous grade

For example, let us say the pin code of your area is 201301. This is a 6-digit number. It can be written using commas by representing it in a place value chart.

Expanded form: 2,00,000 + 1000 + 300 + 1

Dhruv was reading a newspaper. He came across news about different states in participated in the COVID vaccination drive and the number of vaccinations given August 2023. Given below is the data of four states.

Number name: Two lakh one thousand three hundred one Letʹs Warm-up Fill in the blanks.

Each of these digits has a place value and a face value. Let us write the face value, place value, expanded form and number name for 201301.

Let’s Warm-up: Quick 4–5 questions to test the pre-knowledge

1 The place value of 8 in 8,60,765 is 2 The number 4,36,536 can be written in words as:

3 The place value of the digit in 4,15,124 and 4,67,890 is the same.

4 8,76,504 has 6 in the place.

Haryana Sikkim Goa

All About 7-Digit and 8-Digit Numbers

I scored out of 4.

While reading the news, Dhruv got confused and could not read the numbers given data. The numbers of vaccinations given were either 7-digit numbers or 8-digit

Letʹs Warm-up Fill in the blanks.

Connect: Concept introduction with a

Dhruv was reading a newspaper. He came across news participated in the COVID vaccination drive and the August 2023. Given below is the data of four states.

Temperature Range (°C) Weather

Place Value, Face Value and Expanded Form

Drass in Jammu and Kashmir is the second coldest inhabited region on Earth.

Reading and Writing 7-Digit and 8-Digit Numbers

Did You Know?

Example 8: Aliya’s body temperature was 2.3°C higher than normal. What was Aliya’s body temperature?

Normal body temperature = 37°C.

Below 0 Very Cold 0–10 Cold 11–20 Mild 21–30 Warm 31–40 Hot

Let us help Dhruv understand 7-digit and 8-digit numbers! We know that the greatest 6-digit number is 999999. Now, if we add 1 to this number, we get 1000000.

Drass in Jammu and Kashmir is the second coldest inhabited region on Earth.

Above 40 Very Hot

Aliya’s body temperature = 37°C + 2.3°C = 39.3°C

999999 + 1 = 1000000

Example 8: Aliya’s body temperature was 2.3°C higher than normal. What was Aliya’s body temperature?

So, Aliya’s body temperature was 39.3°C.

Normal body temperature = 37°C.

Examples: Solved problems showing the correct method and complete solution

Aliya’s body temperature = 37°C + 2.3°C = 39.3°C

So, Aliya’s body temperature was 39.3°C.

All About 7-Digit and 8-Digit Numbers

While reading the news, Dhruv got confused and could data. The numbers of vaccinations given were either

Example 9: In a city, the highest temperature was 37.5°C, and the lowest was 21.2°C. What is the difference between these two temperatures?

1000000 is the smallest 7-digit number and is read as “Ten Lakhs”. We saw in the news article that the number of vaccinations administered in Sikkim 1360477. Let us try to place this 7-digit number in the place value chart.

To find the difference, subtract the lowest temperature from the highest temperature. 37.5°C – 21.2°C = 16.3°C

Example 9: In a city, the highest temperature was 37.5°C, and the lowest was 21.2°C. What is the difference between these two temperatures?

The temperature difference is 16.3°C.

To find the difference, subtract the lowest temperature from the highest temperature.

Write the readings of the given thermometers.

Do It

37.5°C – 21.2°C = 16.3°C

Together: Guided practice for learners with partially solved questions

The temperature difference is 16.3°C. Write the readings of the given thermometers. Temperature = 45°C

Converting Between Units of Temperature

7-digit Numbers 8-digit

Place Value, Face Value and Expanded Form

Reading and Writing 7-Digit and 8-Digit Numbers

Let us help Dhruv understand 7-digit and 8-digit numbers!

We know that the greatest 6-digit number is 999999.

Converting Between Units of Temperature

Now, if we add 1 to this number, we get 1000000.

1000000 is the smallest 7-digit number and is read

We saw in the news article that the number of vaccinations 1360477. Let us try to place this 7-digit number in

What

Let

Fill in the blanks to convert 5 m 230 mm to m. mm = 1 m 1 mm = 1 m 5 m 230 mm = 5 m + 230 × 1 m = 5 m + m = m

Word Problems on Length

Did You Know: Interesting facts related to the topic

Do It Yourself: Exercise at the end of each topic with practice questions

After making a pair of trousers with 2 m of cloth, Sana thought of buying 1 m 55 cm of cloth for making a shirt. Let us see what length of cloth she has together.

Cloth bought by Sana for making a pair of trousers = 2 m

Cloth required by Sana for making a shirt = 1 m 55 cm = 1.55

So,

NEP Tags: To show alignment with NEP skills and values

Error Alert: Caution against misconceptions

1 Identify the shape for which the net is drawn.

2 Look at the net and identify the object it belongs to.

3 Draw the net of the given shapes.

4 Rishi and Megha made the net of a square-based pyramid. Who made the net correctly? Explain answer with reasons.

a Rishi’s drawing b Megha’s drawing

Chapter Checkup: Chapter-end practice exercises aligned to different levels of Blooms Taxonomy

5 Draw the net of a hexagonal prism.

6 Show using nets how a rectangular prism is different from a rectangular pyramid.

Picture-based Questions: Questions featuring visual stimuli to foster comprehension and interpretation

Challenge

Challenge: Critical thinking questions to enhance problem-solving and analytical-thinking skills

Case Study: Scenario-based questions designed to help apply theoretical knowledge to real-world situations

Real-life Questions: Questions that help make connections with real life or other subjects

Critical Thinking

1 Sanya wants to solve a 7-digit secret code in a safe. Use the given clues to help Sanya solve the secret code.

a The digit in the hundreds and ones place is 6.

b The digit in the lakhs place is 4 less than the digit in the ones place.

1 Sanya wants to solve a 7-digit secret code in a safe. Use the given clues to help Sanya solve the secret code.

c The digit in the ten lakhs and ten thousands place is the smallest odd number.

a The digit in the hundreds and ones place is 6.

d The face value of the digit in the thousands place is 5.

b The digit in the lakhs place is 4 less than the digit in the ones place.

e The digit in the tens place is the biggest 1-digit number.

What is the secret code?

c The digit in the ten lakhs and ten thousands place is the smallest odd number.

d The face value of the digit in the thousands place is 5.

1 Sanya wants to solve a 7-digit secret code in a safe. Use the given clues to help Sanya solve the secret code.

e The digit in the tens place is the biggest 1-digit number. What is the secret code?

2 Write the greatest 8-digit odd number using only 5 digits. Do not repeat any digit more than twice.

a The digit in the hundreds and ones place is 6.

b The digit in the lakhs place is 4 less than the digit in the ones place.

2 Write the greatest 8-digit odd number using only 5 digits. Do not repeat any digit more than twice.

Case Study

c The digit in the ten lakhs and ten thousands place is the smallest odd number.

Case Study

What is the secret code?

d The face value of the digit in the thousands place is 5.

e The digit in the tens place is the biggest 1-digit number.

Population of Countries

2 Write the greatest 8-digit odd number using only 5 digits. Do not repeat any digit more than twice.

Cross Curricular

Population of Countries

The population of different countries is shown using a table. Read the data carefully and answer the questions. Country Population

The population of different countries is shown using a table. Read the data carefully and answer the questions.

Country Population

Population of Countries

The population of different countries is shown using a table. Read the data carefully and answer the questions.

1 Which country has the least population?

a Italy b Germany c Poland d United Kingdom

2 Which country has the greatest population?

Which country has the greatest population?

a Poland b United Kingdom c Italy d Germany

3 Which country has approximately double the population than that of Poland?

Poland b United Kingdom c Italy d Germany

4 Arrange the countries in ascending order as per their population.

1 Which country has the least population?

3 Which country has approximately double the population than that of Poland?

5 If all the digits in the population of each country is rearranged to form the greatest

Cross Curricular

1 Numbers up to 6 Digits

Letʼs Recall

We know that numbers are basic units of mathematics and are used for counting, measuring and comparing quantities. Each digit in a number has a specific value, which we refer to as its place value.

Let us take the 2-digit number 39 as an example. The digit 3 on the left is at the tens place, while the digit 9 on the right is at the ones place.

Similarly, in the number 2548, the digit 5 represents 5 hundreds, or 500. However, in the number 56, the digit 5 represents 5 tens, or 50. Therefore, even if a digit is the same, its value always depends on where appears within the number. The position determines whether it represents tens, hundreds, or other units.

Number name: two thousand five hundred forty-eight

Letʼs

Warm-up

Write the correct place value of the coloured digits and the number name.

Numbers Beyond 9999

Nikhil: Hello daddy, I found this letter that has your name and the address of our home.

Father: Thank you, Nikhil.

Nikhil: What is this big number, 781005, written on the letter?

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

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

All About 5-digit Numbers

To help Nikhil understand 6-digit numbers, let us 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

10000 is read as “Ten Thousand”.

Let us learn more about 5-digit numbers!

Place Value, Face Value and Expanded Form

Remember!

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

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 and write it in a place value chart.

Did You Know?

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

To: Ajay Shukla, 12, Dispur, Guwahati - 781005

Example 1: 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 93421 in the place value chart.

Think and Tell

Are 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.

Example 2: Write the place value of each digit 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.

We can also write the number in expanded form in the following way:

Write the place value of each digit in the number 54319. What is the face value of the digit in the thousands place? Also write the number in the expanded form.

Let us find the place value using the chart for the number 54319.

Face value of the number at the thousands place = .

We can also write the number in expanded form as:

5-digit Number Names

When we have really big numbers, it is important to know where each digit belongs.

“Periods” help us do this.

In the Indian numbering system convention, Ones, Tens and Hundreds are grouped together in one period. Similarly, Thousands and Ten Thousands are grouped together in another period.

Let us understand this by using the Place Value Chart.

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

13,435

The number name of a 5-digit number can be read using the periods as given below:

13,435

thirteen thousand

four hundred thirty-five

Error Alert!

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

Thirteen Thousand Four Hundred Thirty-Five

Hence, the number name is “thirteen thousand four hundred thirty-five.”

Example 3: Represent the number 64819 using the correct period. Also write the number name.

The correct representation of the number is 64,819.

Number name: Sixty four thousand eight hundred nineteen.

Write the number 63109 using the place value table and use commas. Write its number name.

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

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

Mark the period in the correct position: 6 3 1 0 9

Number name is: Sixty-three

Which of the following numbers has 7 tens?

a 36789 b 47690 c 32478 d 67698

2 Which of the following numbers has the greatest value in the thousands place?

a 45687 b 65690 c 78483 d 96152

3 Write the place value and the face value of the underlined digit. Also, write the expanded form of the numbers.

a 56938 b 65899 c 25401 d 89376

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

a 17372 b 43890 c 74065 d 80379

5 Write the following in numerals using commas.

a Twelve thousand three hundred twenty-one

b Thirty-four thousand six hundred

c Seventy-eight thousand five d Fifty thousand ten

6 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

7 The distance between India and USA is 13568 km. Write the number using periods, and then write it in words.

8 During 2023, 28,965 people visited the Isle Royal National Park. Write the number in words and in expanded form.

1 Identify the number which has 4 in the tens place and 8 in the thousands place. The digit in the ones place is half the sum of the digits in the tens and thousands places, and the digit in the hundreds place is six less than the digit in the ones place. Challenge Critical Thinking

All About 6-digit Numbers

Now that we have learnt about 5-digit numbers, let us help Nikhil 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

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.

Place Value, Face Value and Expanded Form

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 Lakhs.

Let us write “781005” in the place value chart. A new lakhs column will be added.

Example 4: Write the place value of each digit and the expanded form of the number 801246. Also write the face value of the digit at the thousands place.

Let us find the place value using the chart for the number 801246.

Expanded Form = 8 × 100000 + 0 × 10000 + 1 × 1000 + 2 × 100 + 4 × 10 + 6 × 1 = 800000 + 1000 + 200 + 40 + 6

The face value of the digit at the thousands place = 1.

To: Ajay Shukla, 12, Dispur, Guwahati - 781005

Write the place value of each digit and expanded form of the number 172909. Also write the face value of the digit at the ten thousands place.

The face value of the digit at the ten thousands place = .

6-digit Number Names

Let us continue to learn about the “periods” convention in 6-digit numbers. We learnt that the Thousands period includes the Ten Thousands and Thousands places. The Ones period includes the Hundreds, Tens and Ones places.

In 6-digit numbers, the Lakhs place falls within the Lakhs Period.

Lakhs Ten Thousands (TTh)

We can therefore represent 781005 as:

7,81,005

The number name of a 6-digit number can be read using the periods as given below: 7,81,005

seven lakh eighty-one thousand five

Hence, the number name is “seven lakh eighty-one thousand five.”

Example 5: Represent the number 342381 with the correct periods. Also write the number name.

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

The correct representation of the number is 3,42,381.

The number name is three lakh forty-two thousand three hundred eighty-one.

Write the number 905721 using the correct periods. Also write the number name.

Let us first write the number in a place value chart.

Mark the period in the correct position: 9 0 5 7 2 1

The number name is

Do It Yourself 1B

Write the place value of each digit and the expanded form of the following numbers. Also, write the face value of the digits at the lakhs place.

a 584736

c 370943

2 Write True or False.

b 704391

d 985401

a The place value of the digit 5 in the number 205649 is five hundred.

b In the number 342658, the place value of the digit 3 is 30000 × 20.

c The difference of the place values of the digit 5 in the number 849553 is 450.

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

4 Write the following in numerals.

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

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

a 197637 b 365021

c 632845 d 824137

6 The approximate distance to the Moon is 3,84,400 kilometers. Write this number in its expanded form.

7 A pencil can draw a line that is almost 1,84,800 feet long. Find the place value for each of the digits and write the number in words.

1 The digits in the Tens and the Thousands place of a number are 3 and 9, respectively. The digit in the Ones and the Ten Thousands places are 3 and 4 more than the digit in the Tens place. If the digit in the Hundreds place is three less than the digit in the Ten Thousands place and the digit in the lakhs place is the greatest one-digit number, find the number.

Comparing and Rounding-off Numbers

Jai and Tina are doing a research project on the prices of different cars and bikes of different brands in India. Let us take a look at the data table provided by Jai and Tina:

Vehicle Image

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

Comparing and Ordering Numbers

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

Comparing Numbers

Let us compare the prices of the two bikes which is 78,957 and 74,801. We know that we compare two numbers by comparing the digits at the same place.

78,957

Same digit

Different digits (8 > 4)

Hence, 78,957 > 74,801.

74,801

Remember!

The number with more digits is always greater.

Example 6: Compare 32,751 and 1,52,631.

As 32,751 is a 5–digit number and 1,52,631 is a 6–digit number, 32,751 < 1,52,631.

Example 7: Compare 1,47,213 and 1,43,507.

Both the numbers are 6–digit numbers and the digits at the lakhs place and ten thousands place are the same.

Comparing the digits at the thousands place: 7 > 3

Hence, 1,47,213 > 1,43,507.

Compare 7,53,278 and 7,22,271.

48517 > 153142 Error Alert!

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

48517 < 153142

Both the numbers are 6–digit numbers and the digits at the place are the same.

Moving to the next place = place >

Hence > .

Ordering Numbers

Now, what if Jai and Tina want to sort the cars in order of their prices?

In this case, we will have to order the prices of the cars in ascending or descending order. Let us look at the table again.

The prices of cars of brands C, D and E are 9,63,890, 7,47,871 and 8,29,860. As all the prices are 6–digit numbers, so we can compare the prices by comparing the digits at the lakhs place.

As 7 < 8 < 9, the prices can be arranged in ascending and descending order as:

Ascending order: 7,47,871 < 8,29,860 < 9,63,890

Descending order: 9,63,890 > 8,29,860 > 7,47,871

Example 8: Arrange the following numbers in ascending and descending order.

72,510, 51,068, 94,321, 1,86,344

1,86,344 is the largest number as it is a 6–digit number and the rest are 5–digit numbers. On comparing the digits at the ten thousands place in the rest of the numbers, we get 5 < 7 < 9

Ascending order: 51,068 < 72,510 < 94,321 < 1,86,344

Descending order: 1,86,344 > 94,321 > 72,510 > 51,068

Arrange the following numbers 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: .

Forming Numbers

Tina and Jai started playing a game with the prices of the vehicles. They picked the digits in the price of the Brand B vehicle which is:

7 4 8 0 1 and formed different 5-digit numbers, as shown below.

48,017 78,401 17,480

48,071 78,104 17,408

48,107 78,140 17,804

Think and Tell

Example 9: Which are the largest and the smallest numbers that Tina and Jai could have formed using the digits 7, 4, 8, 0, and 1?

To form the largest number using the given digits, arrange the digits in decreasing order that is from greatest to smallest.

8 7 4 1 0

So the largest number that we can form using these digits is 87,410. We need the smallest digit in the leftmost places to form smaller numbers. 0 is the smallest digit among the given digits but we cannot have a 0 in the leftmost place

because then it becomes a 4-digit number. So 1 will be the leftmost digit, and the digits will be arranged in ascending order as:

1 0 4 7 8

So the smallest number that we can form using these digits is 10,478.

Example 10: Write the smallest and largest 6–digit number using the digits 5, 7, 3, 9, 1 by repeating exactly 1 digit.

To form the largest 6–digit number, we will repeat the largest digit (9), and arrange the rest of the digits in ascending order as: 9,97,531.

To form the smallest 6–digit number, we will repeat the smallest digit (1), and arrange the rest of the digits in descending order as: 1,13,759.

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

1 5-digit number with no repetition of digits.

Largest number: 9 __ __ __ __ 1

Smallest number: __ __ __ __ __ 9

2 6-digit number with exactly 1 repeating digit.

Largest number: 9 __ __ __ __ __ 1

Smallest number: 1 __ __ __ __ __ 9

Think and Tell

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

Did You Know?

A number that reads the same when read forward and backwards is called a palindrome. 11, 121, 333, 4554, 78987 and 876678 are all examples of palindromes. Can you think of any?

Do It Yourself 1C

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 f 4,35,071 and 4,35,261

2 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

3 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

4 Write the smallest and the greatest 6-digit number by repeating exactly 1 digit.

a 2, 1, 7, 4, 9

c 6, 9, 1, 2, 7

b 3, 8, 5, 0, 1

d 8, 1, 0, 9, 7

5 Exercise keeps us fit and healthy by burning unwanted calories. Supriya and her brother exercise on a regular basis. Supriya burnt 15,248 calories, while her brother burnt 18,396 calories this week. Who burnt more calories this week?

6 Anna wants to buy some books for her library. Her father has given her `11,200. The books cost `11,700. Does she have enough money to buy the books?

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

8 Create a word problem on comparing two 6-digit numbers.

1 Form the smallest 6-digit number using the digits 2, 0 and 5. Each digit has to be used at least once and a digit can be repeated any number of times.

Rounding-off Numbers

Do you remember, the price of the car of Brand C was INR 9,63,890?

Let us consider the scenario where we want to convey the price of Brand C’s car, which is 9,63,890. However, saying this exact number aloud can be difficult.

Instead, we can round it off to approximately 9,60,000. This gives us an approximate price of the car. This is called rounding off a number.

When we round off, we use terms like “about” and “approximately” to convey that the number is close to the exact number.

Rounding off to the Nearest 10

How can we round off 83 to the nearest 10?

83 is between 80 and 90, but closer to 80.

Therefore, 83 will be rounded off to 80.

Let us check for 78,957. We apply the same concept here.

78,957 is between 78,950 and 78,960, but it is closer to 78,960.

So, 78,957 can be rounded off to 78,960.

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.

Remember!

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

Therefore, 271 will be rounded off to 300.

Similarly, 78,957 is between 78,900 and 79,000, but it is closer to 79,000. So, 78,957 can be rounded off to 79,000.

Rounding off to the Nearest 1000

Now, let us learn how to round numbers off to the nearest 1000.

How shall we round 7842 to the nearest 1000?

7842 is between 7000 and 8000. It is closer to 8000. So, 7842 can be rounded off to 8000.

Similarly, 78,957 is between 78,000 and 79,000, but it is closer to 79,000. So, 88,957 can be rounded off to 79,000.

Example 11: 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.

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

Rounding off to the 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.

Round-off 90,135 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 1D

Round off the following numbers to the nearest 10.

2 Round off the following numbers to the nearest 100.

3 Round off the following numbers to the nearest 1000.

4 Ramesh, a garden designer, has been tasked with creating a new garden similar to the one at a monument. The monument currently has 23,912 plants. How many plant saplings should Ramesh order approximately, knowing that some plants may not grow well? (Hint: Round off to the nearest 1000.) How do you take care of the plants around you?

5 The Earthʼs circumference is approximately 40,075 kilometres. What is the Earthʼs circumference when rounded to the nearest 1000?

Critical Thinking

1 A 6-digit number when rounded off to the nearest 100, gives 2,56,300. The number when rounded off to the nearest 10, gives 2,56,290. The sum of all the digits of the number is 30. What is the number?

Points to Remember

• The place value table is divided into groups called periods.

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

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

• 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 descending order.

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

Math Lab

Setting: In groups of 4

Experiential Learning & Collaboration

Place Value Scavenger Hunt

Materials Required: Newspapers, Magazines or the Internet

Divide the entire class into groups of 4.

Each group can be given a particular category like City population, State population, Followers of celebrities, Number of speakers of a language, and Car and Bike prices.

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

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

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

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.

Chapter Checkup

1 Write the place value of each digit and the expanded form of the following numbers.

a 48361 b 87109 c 458320 d 692042

2 Spot the error and fix it.

a 685486 = 6 × 100000 + 85 × 10000 + 4 × 100 + 8 × 10 + 6 × 1

b 213548 = 200000 + 1000 + 30000 + 50 + 400 + 8

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

a 38237 b 456321 c 970540 d 806399

4 Write the following as numerals.

a Forty-eight thousand three hundred twenty-one b One lakh thirty-four thousand six hundred c Seventy-eight thousand six hundred ten d Nine lakh ten thousand forty-five

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

a 80000 + 2000 + 300 + 20 + 2 b 300000 + 50000 + 0 + 700 + 50 + 7

c 200000 + 70000 + 3000 + 0 + 60 + 1 d 700000 + 90000 + 6000 + 400 + 0 + 8

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

a 64,614 and 51,700 b 85,592 and 81,320

c 48,184 and 48,157

d 2,18,720 and 3,14,265

e 7,84,126 and 7,84,510 f 4,35,893 and 4,35,893

7 Arrange the following numbers in ascending and descending order.

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

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

c 64,393; 64,520; 64,905; 64,012 d 8,26,750; 3,58,801; 3,95,701; 93,854

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

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

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

a 3429 b 6126 c 39,887 d 53,475

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

10 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

a Revanth wanted to buy a saree that cost the least. Arrange the sarees in ascending order of their costs.

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

11 The table shows the top four online languages:

a Which language is used the most?

b Which language is used the least?

c Write the order of the language from the least to the one used the most. (Hint: Arrange in ascending order.)

12 Create a word problem on ordering four 6-digit numbers.

Challenge

1 Suhani has six digits: 9, 0, 2, 4, 7, 1. She uses each digit once to make the smallest number with 2 in the hundreds place. What would be the place value and face value of the digit in the thousands place?

2 Write the largest 6-digit number using a minimum of 3 digits that ends with the digit 5 and reads the same, forward and backwards. (For example, 782287).

Case Study

Cross Curricular

Global Population Project

Rahul and Megha are twins and study in the same class. They and their classmates got a project where they had to find the population of countries around the world. The teacher assigned each student some countries and asked them to collect the population data of the assigned countries. The population data collected by Rahul and Megha can be given below. Read the data collected and answer the questions.

Rahul’s Data

Megha’s Data

Bhutan 7,87,424

Macao 7,04,149

Malta 5,35,064

Maldives 5,21,021

Iceland 3,75,318

Suriname 6,23,236

Fiji 9,36,375

Guyana 8,13,834

1 What is the difference in the place value of 4 in the population of Macao? a 3996 b 3960 c 396 d 360

2 Which is the most populated country in Rahul’s data?

3 Which is the least populated country in Megha’s data?

4 Combine the data of Rahul and Megha and arrange the countries in ascending order as per their population.

5 What is the population of Fiji rounded off to the nearest 1000?

2 Addition and Subtraction

Letʼs Recall

We perform addition and subtraction in our everyday lives!

Suppose, there are 125 students in Grade 1 and 120 student in Grade 2. We know that, to find the total number of students, we will have to add. Therefore, the total number of students in the two grades will be 125 + 120 = 245, i.e. addition.

Addition helps us to put items together or find the total of two or more numbers.

Now, let us say 15 students were absent from the 2 grades. Then the number of students that were present was 245 – 15 = 230. Hence, subtraction helps us when we want to take something away from a particular group.

Letʼs Warm-up

1 Match the following.

a 692 – 30

b 355 + 345

c 556 – 120

d 666 – 66

e 722 + 18

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

Adding and Subtracting Numbers Beyond 999

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

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

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

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

`1462 4562

01/05/2025 `1325

We already know how to add two numbers. Let us add the given numbers.

Simple Vertical Addition

We arrange the given numbers vertically in the correct places, then add the ones, tens, hundreds and finally thousands.

Did You Know?

The sum of 1462 and 1325 is 2787.

Bhaskara II, a great Indian mathematician, wrote books that included simple and clear explanations of basic arithmetic operations, helping to make addition and subtraction easier to understand for students and scholars. Addend Addend Sum

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, we also write the ten thousandths place.

So, 83,471 + 12,304 = 95,775.

Add 65,234 and 2345.

TTh Th H T O 6 5 2 3 4 + 2 3 4 5 7 7

So, 65,234 + 2345 = .

Simple Horizontal Addition

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

Let us find the sum of two numbers by placing them horizontally.

Let us add 6712 and 1235.

So, 6712 + 1235 = 7947.

Example 2: Find the sum of 5810 and 4142.

So, 5810 + 4142 = 9952.

Add 12,344 and 1115.

So, 12,344 + 1115 = .

Adding with Regrouping

We have already learnt about “regrouping”—a case in which the sum of the 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 1

Add the ones:

• 1 ones + 9 ones = 10 ones

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

• Carry over 1 to the tens place.

Step 2

Add the tens:

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

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

• Carry over 1 to the hundreds place.

Step 3

Add the hundreds:

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

Step 4

Add the thousands:

1 thousands + 8 thousands = 9 thousands

So, 1371 + 8459 = 9830.

Example 3: Find the sum of 13,431 and 56,718.

So, 13,431 + 56,718 = 70,149.

Add 84,467 and 2893.

So, 84,467 + 2893 = .

Word Problems on Adding Numbers

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?

The total number of fiction books = 1219

The total number of non-fiction books = 1567

What do we need to find?

Total number of books in the library = Fiction books + Non-fiction books = 1219 + 1567

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?

Number of clothes collected on the first day = 1366

Number of clothes collected on the second day = 1000

What do we need to find?

Total number of clothes collected = 1366 + 1000

Solve to find the answer.

The total number of clothes that were collected is 2366.

A home baker made 13,456 chocolates in January and 24,257 chocolates in February. How many chocolates did she make in these two months?

What do we know?

Chocolates produced in January = 13,456

Chocolates produced in February = 24,257

What do we need to find?

Solve to find the answer.

So, the home baker made __________ chocolates in the two months.

Find the sum of the following numbers horizontally.

Find the sum of the given numbers.

+ 1234

+ 34,789

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

Each shape represents a number as given.

Prashant is a volunteer at the national animal rescue shelter. He and his team rescued 1000 animals last year. This year, the team rescued 1145 more animals than the previous year. How many animals were rescued in all?

A car company produced 45,821 cars, in 2021. It produced 1208 more cars in 2022 than in 2021. How many cars did it produce in 2022?

Every year tulips are imported from the Netherlands to display at Shanti Path for the Delhi Tulip festival. In one year around 80,000 tulips bloomed at first. If 8574 tulips bloomed later, how many tulips were displayed in total?

Subtracting 4-digit and 5-digit Numbers

Do you remember Diya visiting her grandmother in Dehradun during the summer holidays?

Diya bought a gift for her grandmother for ₹1247 from her pocket money. How much money was left with Diya if she had ₹2468 initially?

Subtraction without Regrouping

Let us subtract 1247 from 2468.

We arrange the given numbers vertically in the correct places, then perform subtraction for each digit starting from the ones place.

Minuend

Subtrahend Difference

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.

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, we also subtract the digits in the ten thousands place.

TTh Th H T O

9 1 8 9 7 – 4 1 2 9 0 5 0 6 0 7

So, 91,897 – 41,290 = 50,607.

Solve: 75,234 – 3121.

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.

So, 75,234 – 3121 = .

Subtraction with Regrouping

What if we had to subtract 1978 from 2586?

Step 1

Subtract the ones:

• We cannot take away 8 ones from 6 ones.

• We will borrow one ten from the tens place.

• So, 8 tens becomes 7 tens, and 6 ones becomes 16 ones.

• Now, subtract 8 ones from 16 ones.

Step 2

Subtract

Step 3

Subtract the hundreds:

• We cannot subtract 9 hundreds from 5 hundreds.

• We borrow 1 thousand from the thousands place.

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

• Now, subtract 9 hundreds from 15 hundreds.

Step 4

Subtract the thousands:

• 1 thousand – 1 thousand = 0 thousands

So, 2586 – 1978 = 608.

Example 6: Find the difference of 87,821 and 45,586.

So, 87,821 – 45,586 = 42,235.

So, 78,131 – 9993 = . Do It Together

Find the difference between 78,131 and 9993.

Remember!

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

Word Problems on Subtracting Numbers

Rani had ₹17,845 in her bank account. She withdraws ₹3230 for shopping. How much money is left in her account?

Let us find the amount left in her account step by step. 1. What do we know?

Amount Rani had in her bank account = ₹17,845

Amount Rani withdrew for shopping = ₹3230

Chapter 2 • Addition and Subtraction

2. 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

3. Solve to find the answer.

= ₹17,845 – ₹3230

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

4. Check your answer.

`14,615 + `3,230 = `17,845

`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 left.

Check your answer.

2890 + 350 = 3240 notebooks

So the answer is correct.

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 = 8455

Stamps with her sister = What do we need to find?

More stamps with Sunaina than her sister.

Solve to find the answer.

8455 – 6712

So, Sunaina has more stamps than her sister.

Check your answer.

6712 + = 8455

So, the answer is correct.

Solve to find the answer.

Subtract the given numbers.

Each shape represents a number as given.

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?

The Amazon is the largest tropical rainforest in the world. It has around 16,000 different tree species. Out of these around 1600 tree species store carbon in them. How many of them do not store carbon?

Create a word problem on subtracting a 4-digit number from another.

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 X?

Addition and Subtraction Together

Simplifying Numbers

Let us solve a problem that has both addition and subtraction.

Simplify: 3500 – 1890 + 1255

We first subtract 1890 from 3500 and then add 1255 to the difference.

So, 3500 – 1890 = 1610

1610 + 1255 = 2865

So, 3500 – 1890 + 1255 = 2865.

Example 8: Simplify.

4321 – 788 – 621

We can simplify this expression by subtracting in the given order.

4321 – 788 = 3533

3533 – 621 = 2912

So, 4321 – 788 – 621 = 2912.

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 = .

Word Problems on Simplifying Numbers

Let us solve some word problems on simplifying numbers.

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 = 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 – Number of bricks already used – Number of bricks delivered = 8976 – 3412 – 2587

Solve to find the answer.

8976 – 3412 = 5564

5564 – 2587 = 2977

So, 2977 bricks are still required for the project.

Example 9: A grocery store has 7654 cartons of milk and 5231 cartons of juice. The store received a new shipment of 2376 milk cartons and sold 3487 cartons of juice. How many cartons of milk and juice are there in total at the store now?

Number of cartons of milk at the grocery store = 7654

Number of cartons of juice at the grocery store = 5231

Number of cartons of milk received in the new shipment = 2376

So, the total number of cartons of milk at the grocery store = 7654 + 2376 = 10,030

Number of cartons of juice sold = 3487

So, the number of cartons of juice left at the grocery store = 5231 – 3487 = 1744

Total number of cartons of milk and juice left at the store = New total number of cartons of milk + New total number of cartons of juice = 10,030 + 1744 = 11,774

A stadium in Ahmedabad has a capacity of 54,000 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 = 54,000

Number of men watching the match = , Number of women = What do we need to find?

Solve to find the answer.

Calculate the following.

Each shape represents a number as given.

Find the

Sarah has `8752 in her bank account. She withdrew `3256 to buy a gift for her grandparents. Then, she deposited `9823 in her account. How much money does Sarah have in her bank account now?

Hum, Croatia is the least populated town in the world with a population of 30 people. The population of Werdenberg, Switzerland is 41,284. The population of Norton City, Virginia is 3627. How much more is the poulation of Werdenberg than the total population of Norton City and Hum?

A farmer harvested 6543 kilograms of wheat, and 4298 kilograms of rice. If he sold 3785 kilograms of wheat and 1932 kilograms of rice, how much grain is left with the farmer?

Challenge

Statement: 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.

Which of the conclusions is true for the given statement?

Conclusion I: A total donation of 11,246 books was made to the library.

Conclusion II: The library finally had fewer than 18,000 books after the annual painting.

Options:

1 Only conclusion I is true.

2 Only conclusion II is true.

3 None of the conclusions is true.

4 Both conclusions are true.

Estimation

The Taj Mahal has the most number of visitors during the weekends.

Manager: How many tickets were sold on Saturday and Sunday?

Ticket Seller: On Saturday, around 44,799 tickets were sold. On Sunday, around 53,878 tickets were sold.

Manager: This means there were about 99,000 visitors during the weekend.

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 adding them to find the estimated sum. Let us learn about this further!

Round off the addends. Add the rounded off numbers.

Rounded off to the nearest thousand

Number of visitors on Saturday = 44,799 45,000

Rounded off to the nearest thousand

Number of visitors on Saturday = 53,878 54,000

45,000 + 54,000 = 99,000

The sum of 45,000 and 54,000 is 99,000. That is how the manager estimated the attendance so quickly!

Example 10: Estimate the sum of 28,894 and 13,894 by rounding off to the nearest ten thousand.

Rounded off to the nearest ten thousand

28,894 30,000

Rounded off to the nearest ten thousand

13,894 10,000

Let us find the sum: 30,000 + 10,000 = 40,000

So, the estimated sum of 28,894 and 13,894 by rounding off to the nearest ten thousand is 40,000.

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.

13,567

28,082

Rounded off to the nearest thousand

The estimated sum = + = .

So, the estimated sum of 13,567 and 28,082 is .

Estimating the Difference

Now, what if the manager wanted to estimate the difference in the attendance on Saturday and Sunday?

He would have to estimate the difference of 53,878 and 44,799.

Round off the numbers. Find the difference.

Rounded off to the nearest thousand

Number of visitors on Saturday = 44,799 45,000

Rounded off to the nearest thousand

Number of visitors on Saturday = 53,878 54,000

54,000 – 45,000 = 9,000

The estimated difference of 44,799 and 53,878 is 9000.

Example 11: Round off to the nearest 100 and estimate the difference: 69,894 – 51,124. Compare with the actual answer. 69,894

Rounded off to the nearest hundred

Let us find the difference: 69,900 – 51,100 = 18,800

So, the estimated difference of 69,894 and 51,124 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.

Example 12: 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.

Rounded off to the nearest thousand

The estimated difference to the nearest thousand = 2000 − 1000 = 1000

So, about 1000 more cupcakes were produced in the evening than in the morning.

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

Rounded off to the nearest thousand

78,111 78,000 21,991

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

Do It Yourself 2D

Round off the numbers to the nearest 100. Find the estimated sum or difference.

+ 2456

Round off the numbers to the nearest 1000. Find the estimated sum or difference.

2 islands in Canada have areas as given below. What is an approximate total area of both the islands, rounded off to the nearest 1000? Axel Heiberg Island: 43,178 sq. km; Melville Island: 42,149 sq. km

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. Write one benefit of physical exercise such as walking.

Challenge

Two friends, Zain and Riyan were estimating the sum of numbers by rounding off to the nearest ten thousand.

Zain says the estimated sum of 72,374 and 16,773 is more than the estimated sum of 67,124 and 28,974, while Riyan says that the estimated sum of 67,124 and 28,974 is more. Who is right, Zain or Riyan?

Points to Remember

• While adding, if the sum of the digits in a certain place is 10 or more, we can carry over to the next place on the left.

• 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 to the right of the desired place.

Math Lab

Money Money!

Collaboration & Experiential Learning

Objective: Practice addition and subtraction in a real-world context.

Materials: Fake money, printed price tags, small items or pictures of items.

Activity:

• Set up a mini shop with items labelled with prices up to 4 digits.

• Give each student a set amount of fake money.

• Students ʺbuyʺ items by adding prices together and subtracting the total from their given money amounts.

• They record their transactions and check their calculations.

Chapter Checkup

Solve.

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

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

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

Compute: 81,654 – 53,217 – 2345.

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

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

The construction company ordered 8327 bricks for one project, and 9912 bricks for another project. Estimate the total number of bricks ordered for both projects, by rounding off to the nearest hundred.

The length of the river Ganga is 2520 kilometres, while the length of the Yamuna is 1376 kilometres. Approximately, what is the total length of these rivers combined? About how much longer is the Ganga than the Yamuna? (Find out by rounding off to the nearest 100.)

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

a 2357 and 1235 b 5457 and 1108 c 3347 and 3567

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

a 6790 and 5667 b 7890 and 3889 c 8103 and 4899

Create one word problem involving both addition and subtraction. Think of the situations from your daily life.

On rounding off 34,873 to the nearest ten thousand, and 35,289 to the nearest thousand, what will be the sum of the estimated sum and the estimated difference?

An online store sold 17,645 pounds of chocolates, and 24,891 pounds of candies last month. While doing the accounting, it was estimated that the total weight of chocolates and candies sold, to the nearest thousand, was more than 40,000 pounds. Is that a correct estimate?

Case Study

Value Development

Forest Conservation

In a small village near a dense forest, the villagers noticed that many trees were being cut down. The local school decided to help by organising a project to plant new trees. They also wanted to keep track of how many trees were saved and how many were planted.

The villagers observed that every week, 1,250 trees were cut down. The school decided to plant 2,750 new trees every week. The students started recording the number of trees cut down and planted over 4 weeks.

Answer the questions.

1 How many trees were planted in total over 2 weeks?

a 4000

c 5500

b 3000

d 2750

2 What is the difference of the number of trees planted and the number of trees cut down over 4 weeks?

a 1500 trees

c 5000 trees

b 6000 trees

d 7500 trees

3 True or False: After 4 weeks, the forest has fewer trees than before the project started.

4 If the village decides to increase the number of trees they plant by 500 each, and the rate of trees being cut down remains the same; calculate the total number of trees in the forest after 4 weeks. How does this change affect the overall forest compared to the original scenario?

3 Multiplication

Let’s Recall

Multiplication can also be understood as the repeated addition of the same number 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 pencils + 6 pencils + 6 pencils = 18 pencils. (Packet 1) (Packet 2) (Packet 3)

This is the same as saying 3 packets × 6 pencils per packet = 18 pencils.

In the above example, the packets can be called “groups” containing the same number of units.

3 groups of 6 = 18 or 3 × 6 = 18.

Let’s Warm-up

Write True or False.

1 8 + 8 + 8 + 8 + 8 = 8 × 4

2 23 × 0 = 0

3 30 is the product of 10 and 5.

4 7 multiplied by 11 is 77.

5 There are 9 petals in each flower. We have 9 such flowers. Therefore, we have 72 petals in all.

I scored out of 5.

Understanding Multiplication

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, and now Sanju is trying to recall how many words he has learnt. He starts adding quickly.

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

He has already learnt 14 words! Hurray!

Sanju struggles to find how many words will he learn throughout 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 for help.

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

Multiplication by a 1-digit Number

We know that the number obtained from multiplication is the product.

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

Horizontal Method

31 2 × = 62 multiplicand multiplier product

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

Step 1

Multiply by ones.

Multiply 3 and 3 ones.

123 × 3 = __ 9

The product of 123 and 3 is 369.

Step 2

Multiply by tens

Multiply 3 and 2 tens.

123 × 3 = 6 9

Example 1: Find the product of 2123 and 3.

2123 × 3 = __ __ __ 9 2123 × 3 = __ __ 6 9

The product of 2123 and 3 is 6369.

Step 3

Multiply by hundreds.

123 × 3 = 3 6 9

the product of 4132 and 2. 4132 × 2 =

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. This is also called the box method.

Multiply 170 by 5.

Step 1

Expand the bigger number.

170 = 100 + 70 + 0

Step 3

Multiply the smaller number.

Multiply 5 by 100, 70 and 0.

100 70 0

5 5 × 100 = 500 5 × 70 = 350 5 × 0 = 0

The product of 170 and 5 is 850.

Example 2: Multiply 4287 by 2.

Expand the bigger number.

4287 = 4000 + 200 + 80 + 7

2 2 × 4000 = 8000 2 ×

Add all the products.

8000 + 400 + 160 + 14 = 8574

The product of 4287 and 2 is 8574.

Step 2

Write the numbers.

100 70 0 5

Step 4

Add all the products.

500 + 350 + 0 = 850

Multiply the given numbers by expanding the bigger number.

639 × 9

The expanded form of 639 = 600 + + 600

9 9 × 600 = 5400

The product is 5400 + + =

Remember!

On multiplication by 10, 20, 30… 90, there is always 0 in the ones place.

Did You Know?

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

The product is + + + =

Vertical Method

We can multiply using the vertical method of multiplication. Here, we place the multiplier vertically below the multiplicand, as per the place value of the digits.

Let us multiply 365 by 2.

Step 1

Multiply by ones.

2 × 5 ones = 10 ones

10 ones = 1 tens + 0 ones

Carry over 1 tens.

Step 2

Multiply by tens.

2 × 6 tens = 12 tens

Add carried over 1 ten.

12 tens + 1 tens = 13 tens

13 tens = 1 hundreds + 3 tens

Carry over 1 hundreds.

Step 3

Multiply by hundreds.

2 × 3 hundreds = 6 hundreds

Add carried over 1 hundreds.

6 hundreds + 1 hundreds = 7 hundreds

So, 365 × 2 = 730

Example 3: Find the product of 1134 and 8.

Multiply 8 by the ones, tens, hundreds and thousands.

The product of 1134 and 8 is 9072.

Multiply the given numbers mentally.

a 233 × 2 b 622 × 4 c 2001 × 7 d 4011 × 9

2 Find the product by expanding the bigger number.

a 313 × 3 b 802 × 9 c 1002 × 2 d 2908 × 4

3 Multiply using the vertical method. a 193 × 3 b 563 × 4

1225 × 3

4687 × 8

4 Look at the value of each shape. Draw shapes to show 3210 × 2. = 10 = 100 = 1000

5 Creativity Create your own question on multiplying a 3-digit number by a 1-digit number.

Challenge

1 Richard thought of a number. Find the number using the hints.

a The number is the sum of my age and my father’s age.

b My father’s age is 3 times my brother’s age.

c My brother’s age is 4 more than 3 times 3.

d My age is 3 years less than my brother’s age.

Critical Thinking

Multiplication by a 2-digit Number

Sanju learnt 2 words each for 7 days. His sister learnt 7 words each day for 2 days. Who learnt more words? Let us find the answer.

Properties of Multiplication

Multiplication by 1

The product of any number multiplied by 1 will always be the number itself.

For Example: 6 × 1 = 6

Multiplication by 0

The product of any number multiplied by 0 will always be 0.

For Example: 8 × 0 = 0

Order Property Grouping Property

Two numbers can be multiplied in any order. The product will always be the same.

For Example: 2 × 7 = 14 and 7 × 2 = 14

Two or more numbers can be grouped in any way. The product will be the same.

For Example: (2 × 7) × 5 = 14 × 5 = 70 and 2 × (7 × 5) = 2 × 35 =70

Distributive Property of Multiplication over Addition

The product of a sum of two or more numbers is equal to the sum of the products of two numbers.

For Example: 2 × (4 + 3) = (2 × 4) + (2 × 3) = 8 + 6 = 14 and 2 × (4 + 3) = 2 × 7 = 14

Example 4: Find the product using the correct property.

a 0 × 6 = 6 b 1 × 4 = 4

c 2 × 4 × 3 = 2 × (4 × 3) = 24

Fill in the blanks.

3 × (2 + 4) = (3 × 2) + (3 × 4) = 6 + 12 = 18

Multiplying by 10, 100 and 1000

To multiply a number by 10, put one zero to the right of the number. 3 × 10 = 30

To multiply a number by 100, put two zeroes to the right of the number.

To multiply a number by 1000, put three zeroes to the right of the number.

Example 5: Fill in the blanks. 1 5 × 10 = 50

To multiply by 10, add 1 zero to the right of 5.

To multiply by 100, add 2 zeroes to the right of 6.

To multiply by 1000, add 3 zeroes to the right of 8.

Using the Column Method

Let us multiply 245 by 25.

Step 1

Multiply 245 by 5 ones.

245 × 5 = 1225

Step 2

Multiply 245 by 2 tens or 20.

245 × 20 = 4900

The product of 245 and 25 is 6125.

Example 6: Multiply 179 by 18.

Step 1: Multiply by ones

Step 2: Multiply by tens

Step 3: Add the products

The product of 179 and 18 is 3222.

Complete the multiplication.

1 4 8

3 9 1 0 3 3 2 +

Step 1: Multiply by ones

Step 2: Multiply by tens

Step 3: Add the products

The product of 1148 and 39 is .

Quick Multiplication

Let us see some tricks for easy multiplication.

Multiplying 2-digit numbers by 11.

Let us now see how to multiply two 2-digit numbers.

Step 3

Add the products. 1225 + 4900 = 6125

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

Split 26 into tens and ones

Example 7: Use the trick to multiply 52 by 11.

5 × 1 = 5 and 2 × 1 = 2

5 + 2 = 7

So the product of 52 × 11 is 572.

Example 8: Multiply 24 × 13 without splitting.

2 × 1 = 2 and 4 × 3 = 12

4 × 1 = 4 and 2 × 3 = 6, 6 + 4 = 10

Think and Tell

If 46 × 11 = 506, how will you use the trick to find 39 × 11?

2 10 12: Add 1 and 2 in the hundreds place, add 0 and 1 in the tens place

So, 312 is the product of 24 × 13.

Find the product of 62 and 18 by splitting the numbers.

1 18 = _________ + 8

2 62 × 18 = 62 × _________ + 62 × = 620 + _________ =

Do It Yourself 3B

Write True or False.

Use the column method to multiply.

3 4

Complete the given multiplication.

Find the product using an quick multiplication tricks.

The weight of a baby elephant is around 91 kg. If there are 11 baby elephants, then find their weight using an appropriate trick.

Think and Tell

Can the product of a 4-digit number and a 2-digit number be a 7-digit number?

Hint: Check for the greatest numbers!

Era distributes sandwiches and juice bottles to 11 orphanages every year on her daughter's birthday. If she distributes 45 sandwiches and 25 juice bottles to each orphanage, how many of each does she distribute? Have you ever visited an orphanage?

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

Multiplication by a 3-digit Number

Let us learn to multiply numbers by 3-digit numbers.

Find the product of 144 and 273.

The product of 144 and 273 is 39,312. Now, what if you want to multiply 345 by 400?

Split the number into multiples of 4 × 100.

We can multiply 345 only by 4 and add zeroes to the right of the product.

Product of 345 × 4 = 1380 1380 × 100 = 1,38,000

So, the product of 345 × 100 is 1,38,000.

Example 9: Multiply 432 by 317.

The product of 432 and 317 is 1,36,944.

Step 1: Multiply 432 by 7 ones

Step 2: Multiply 432 by 1 tens or 10

Step 3: Multiply 432 by 3 hundreds or 300

Step 4: Add the products

Multiply 752 by 417.

Step 1: Multiply by 7 ones

Step 2: Multiply by 1 tens

Step 3: Multiply by 4 hundreds

Step 4: Add the products

The product of 752 and 417 is __________________ .

Find the product of

Find the product.

348 × 300 =

850 x 707 =

Fill in the missing digit such that both the products are equal. 860 × 150 = 375 × 34

The daily water consumption per person of a family of 4 is about 200 litres. How many litres of water do they consume in a year? What do you do to save water?

1 The product of two numbers is 600, and their sum is 50. What are the numbers?

Word Problems

Suresh has to print the posters for the upcoming debate competition at school. There are 8 packets of printing paper available, and each packet has 145 sheets. 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 sheets in each packet = 145

What do we find? Total number of posters printed =

Solve to find the answer.

145 × 8 = 1160

1160 posters were printed.

Example 10: A local bus covered 145 km in a day. How much distance it covered in the year 2023?

Distance the bus covers in a day = 145 km.

Number of days in the year 2023 = 365 (there are 365 days in a year)

Distance covered in 2023 = 145 × 365

The bus covered 52,925 km in the year 2023.

Example 11: Kabir has ₹9000 for shopping. He buys 4 T-shirts for ₹460 each and 3 pairs of jeans for ₹987 each. How much money does he have left?

Amount with Kabir for shopping = ₹9000

Cost of 4 T-shirts = 460 × 4 = ₹1840

Cost of 3 pairs of jeans = 987 × 3 = 2961

Total amount spent by Kabir = 1840 + 2961 = ₹4801

Amount left with Kabir = ₹9000 − ₹4801 = ₹4199

So, ₹4199 is left with Kabir.

A free meal service provides food for 6315 people every day. For how many people will it provide food in 92 days?

The number of people for whom food is provided in a 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

The entry fee per person at the club is ₹423. A group of 9 tourists visited 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? Do you exercise everyday?

All 28 members of the reader’s club are going on a holiday. They have a budget of ₹1,00,000 for the tickets. If one plane ticket costs ₹3879, will the total cost be within their budget?

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

a What is the total number of people who watched Horror Story in all the shows, if all the seats were occupied?

Fun with Mary

b What is the total number of people who watched Fun with Mary in all the shows, if 250 seats were unused in each show.

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

How much money does Karan have now?

A school purchased 350 pens in February 2017 and 265 pens in November 2017. If the cost of each pen is ₹9, how much money did the school spend on pens in total?

Challenge

Critical Thinking

1 In a certain warehouse, there are crates that each contain boxes of small items. The crates are stacked in rows, columns, and layers.

There are 213 rows of crates.

Each row contains 187 crates.

Each crate contains 126 boxes of items.

If 1 10 of the boxes are defective and need to be replaced, how many boxes are not defective?

Estimation

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 11 times in a month.

Sameer: Wow! How far is Amritsar from Delhi?

Uncle: It is about 448 km and I travel 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 multiple to get the estimated product.

Let us find the estimated product of 448 and 11 to the nearest ten.

Step 1

Round off both the numbers.

448 rounded up to 450. 11 rounded down to 10.

Step 2

Multiply the rounded off numbers. 450 × 10 = 4500

The estimated product of 448 and 11 is 4500.

Remember!

An estimation is used to find the approximate or near around products. It makes the calculations quicker and easier.

Example 12: Find the estimated product of 627 and 456 by rounding off both the numbers to the nearest hundred.

627 is rounded down to 600. 456 is rounded up to 500.

600 × 500 = 3,00,000

The estimated product is 3,00,000.

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

148 is rounded (up/down) to 150. 879 is rounded (up/down) to .

150 × =

The estimated product of 148 and 879 is .

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

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 = f 655 × 845 =

Find the estimated product by rounding off the given numbers to the nearest hundred.

a 169 × 74

b 518 × 96 c 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.

a 109 × 54

b 444 × 777 c 976 × 862

The marathon is one of the sporting events in the Olympics. If each of the 110 athletes had to cover a distance of 43 km, estimate (to the nearest 10) the total distance covered by all. Compare the actual and estimated answers.

Workers at the stadium put 1 water bottle at each seat. There are about 17 stands in the stadium, and each stand has 238 seats. Approximately how many water bottles will they put out? Round off the numbers to the nearest 10. Do you throw the used bottles in the bin in public places?

Challenge

Critical Thinking

1 Rina wants to buy 39 notebooks for her class each costing ₹83. If she goes to the market carrying the estimated amount, will it be enough for her to buy the notebooks? Why or why not? Estimate to the nearest 10.

Points to Remember

• The number to be multiplied is the multiplicand. The number by which we multiply is the multiplier. The number obtained from multiplication is the product.

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

• 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 exact numbers. We round off the numbers before multiplying.

Math Lab

Setting: In groups of 3

Chit Game for Multiplication!

Materials Required: Connecting cards showing multiplication problems and their answers, pencil, paper, chart paper and glue.

Method:

Work in groups to solve the multiplication problems mentally. Paste the multiplication card and its connecting answer card next to each other on chart paper. The

that finishes first wins.

Chapter Checkup

1 Find the product by using properties or tricks.

a 42 × 100 b 54 × 11 c 63 × 21 d 172 × 300

2 Multiply using the horizontal method. Check the answer by using the vertical method.

a 410 × 7 b 844 × 2 c 8023 × 3 d 9101 × 8

3 Find the product by expanding the bigger number.

a 564 × 4 b 492 × 6 c 7397 × 9 d 593 × 7

4 Write the numbers in columns and multiply.

a 141 × 84 b 389 × 40 c 378 × 65 d 7041 × 33

e 9672 × 96 f 4356 × 75 g 638 × 500 h 204 × 630

5 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)

6 The Qutub Minar in Delhi, India, has 379 steps. What is the total number of steps climbed by a worker who goes up and then comes down the stairs?

7 Swati runs 750 m every day for 15 days for a fun competition. How far will she run throughout the competition?

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

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

10 Formula One (F1) is a global motor-sport competition. Each driver has to cover a minimum distance of 305 km in the race. Each driver has to make 55 laps of 14 km each. How much distance will each driver cover?

11 447 books can be placed on a shelf in the National Library. How many books can be placed in the library with 345 shelves? Do you borrow books from the library? How can you take care of the library books?

12 There are 24 schools in a town. Each school gets 3 pieces of equipment for the science lab. The cost of each piece of equipment is ₹394. How much does all the equipment cost?

13 Around 467 people visit the exhibition at the National Gallery every day. Estimate how many people visit the exhibition in a year by rounding off to the nearest 100.

2 Write a multiplication word problem with one 3-digit number and one 2-digit number. Critical Thinking & Creativity

1 The vowels are written as even numbers from 0 to 8, and the other letters are written as odd numbers. What is the product of DOG and BE as English letters?

Cross Curricular & Value Development

A Day with an Archaeologist!

An archaeologist is a specialist who analyses artifacts, monuments, and many other things to get insights about our history.

Rakesh is an archaeologist. There are 45 members, including him, on the team.

1 The taxi fare from their office to the railway station is ₹82. If the team required 5 cabs, they spent on cabs.

2 Write True or False.

a The cost of 1 train ticket is ₹450. The cost of 45 train tickets is ₹20,050.

b The cost of a train ticket is ₹450. The cost of 45 train tickets is ₹20,250.

3 The team was supposed to inspect a monument made of stone. Each stone weighs 52 kg. 812 stones were used to construct the monument. What is the total weight of the stones used?

4 The team takes 42 days to study a mural inside the monument. If there are 267 murals that they have to study, how many days will the team need to do it?

5 Rakesh and his team travel 1780 km in a month. Estimate the distance they will travel in 19 months. Estimate the distance to the nearest 100 and the months to the nearest 10.

6 Did you like visiting historical places? What should be kept in mind while visiting these places?

4 Division

Letʼs 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?

We will start by placing 1 garment in each slot and continue until we have no garments left.

This equal grouping of items is called division.

We grouped 8 items of clothing into 4 slots. We finally got 2 items in each slot.

This can be written as: 8 divided by 4 is equal to 2.

Letʼs Warm-up

Answer the following.

Division by 1-digit and 2-digit Numbers

Ramu, the milkman, supplies milk to different shopkeepers and customers.

Ramu has a large drum of milk with a capacity of 210 litres. He pours the milk into smaller containers, each with a capacity of 5 litres. This way, he can efficiently distribute the milk to his customers equally.

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

Division of a 3-digit Number by a 1-digit Number

We need to divide 210 L by 5 L to find out how many small containers Ramu can fill from the drum.

Therefore, 210 L ÷ 5 L

While doing division, always go from left to right.

1 Write the dividend and the divisor in the division house.

2 Compare the digit at the hundreds 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.

3 As 2 < 5, so now consider the number at hundred and tens place that is 21 and then use multiplication to find the nearest quotient.

4 Subtract

Thus, Ramu can fill 42 containers.

6 Repeat steps 3 and 4 until you get 0 as the remainder or a number less than the divisor.

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.

Quotient × Divisor = 42 × 5 = 210.

So, we get Dividend = Quotient × Divisor.

So, our answer is correct.

Divide 864 by 6. Verify the quotient.

Think and Tell

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

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 Capacity of each container = 3575 5

We divide a 4-digit number by a 1-digit number the same way as we divide a 3-digit number by a 1-digit number.

3575 ÷ 5 = 715.

So, 715 containers can be filled.

Example 1: Divide 1488 by 8. Verify the quotient.

0 Thus, 864 ÷ 6 = . Verify the quotient. = ( × Divisor). 864 = ( × ). Divide 2133 by 9. Do It Together 2

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

Properties of Division

0 division by a Number

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

0 ÷ 4 = 0

0 ÷ 41 = 0

0 ÷ 128 = 0

Division Facts

Division by 1

When a number is divided by 1, the quotient is always the number itself. 6 ÷ 1 = 6 18 ÷ 1 = 18

÷ 1 = 194

Division by Itself

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

÷ 5 = 1

÷ 14 = 1

÷ 184 = 1

Division and multiplication are reverse operations. Every multiplication fact has 2 division facts.

Multiplication Fact Division Fact

÷ 1 = 5

5 × 1 = 5

4 × 6 = 24

÷ 5 = 1

÷ 4 = 6

÷ 6 = 4

Never try dividing any number by 0. Division by 0 is not defined. 3 ÷ 0 = 0 3 ÷ 0 = Not defined Error Alert!

Write 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 rule states: Dividend = Quotient × Divisor + Remainder

Fill in the boxes with the missing numbers.

Find the quotient.

Divide the numbers. Verify the answer.

a 280 ÷ 5 b 672 ÷ 8 c 1656 ÷ 9

The Mahabharata is one of the epics of ancient India. ‟The Complete Mahabharata in English” is a translated book by Kisari Mohan Ganguly. The book has 18 chapters and 4900 pages. If the pages are divided equally for each chapter, how many pages are left for the foreword, acknowledgement and bibliography?

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? Do you know that bananas are rich in vitamin B6 and potassium? Do you like eating bananas?

What must be added to 4587 so that it can be divided by 5 with no remainder? 1 Challenge Critical Thinking

Division by 2-digit Numbers

Ramu, the milkman, gets an order to deliver milk to a sweet shop. Let us see how he plans his deliveries.

Dividing by Tens

The sweet shop has ordered 500 litres of milk. How many 10 L containers does Ramu need?

Number of containers required = Total milk Capacity of each container = 500 ÷ 10 = 500 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 digits make 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 and so on.

What if we want to divide 500 by 20?

Quick Way

When both the dividend and the divisor have 0 in the ones place, we can apply a trick to divide quickly.

25 20 500 – 40 100 – 100 0

Step 1

Cancel the zeroes.

500 20 = 50 2

Example 2: 546 ÷ 10 = ?

Step 2

Now mentally divide the remaining number by 2. If you are unable to divide mentally, use the long division method.

So, 50 ÷ 2 = 25

We get the same answer.

The digit in the ones place makes up the remainder, and the rest of the digits make up the quotient.

546 ÷ 10 gives a quotient of 54 and a remainder of 6.

Example 3: 840 ÷ 40 = ?

÷ 40 = 84 ÷ 4

Divide mentally. 1 9300 by 30 2 1200 by 40 9300 30 = 9300 30 =

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? How many such containers can be filled?

Step 1

Divide 45 by 21.

The result is 2 (21 × 2 = 42)

The remainder is 3 (45 – 42 = 3).

Write 2 directly above 5 and bring down the next digit, 3 along the remainder 3, making it 33. 2 21 4536 – 42 33

Step 2

Divide 33 by 21.

The result is 1 (21 × 1 = 21)

The remainder is 12 (33 – 21 = 12).

Write 1 directly above 3 and bring down the next digit, 3 along the remainder 12, making it 126.

Step 3

Divide 126 by 21.

The result is 6 (21 × 6 = 126)

The remainder is 0 (126 – 126 = 0).

Write 6 directly above 6, and the remainder 0 at the bottom. 216 21 4536 – 42 33 – 21

Divisor

Quotient Dividend

Remainder

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

Verify the quotient.

The Dividend should be equal to (Quotient × Divisor) + Remainder.

Dividend = 4536

Quotient = 216 Divisor = 21

= 0

(Quotient × Divisor) + Remainder = (216 × 21) + 0 = 4536

So, we get Dividend = (Quotient × Divisor) + Remainder.

Our answer is correct!

Example 4: Divide 8210 by 18.

Think and Tell

Will the division of a 4-digit number by a 2-digit number always be a 2-digit number?

1548 by 18.

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 litre capacity each? How many drums does he require?

Number of drums required = Total capacity of tanker Capacity of each drum = 5000 100 = 50.

When a number is divided by 100, the digits in the tens and ones places make up the remainder and the rest of the digits make up the quotient.

Dividing by 100 is easy. Look at the divisions given below:

546 ÷ 100

Quotient = 5

Remainder = 46

1456 ÷ 100

Quotient = 14

Remainder = 56

9842 ÷ 100

Quotient = 98

Remainder = 42

Similarly, 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.

300 ÷ 30

Cancel out the zeroes.

Divide

300 30 = 30 3 = 10

Example 5: Divide 8210 by 100.

4500 ÷ 300

Cancel out the zeroes.

Divide

4500

300 = 45 3 = 15

The digits in the tens and ones place will make up the remainder and the rest of the digits will make up the quotient.

Thus, 8210 ÷ 100 gives Quotient = 82 and Remainder = 10

1 Divide 4980 by 200.

4980

200 =

= 498 ÷ ______ =

Quotient = , Remainder =

Estimating the Quotient

8000 ÷ 4000

Cancel out the zeroes.

Divide

8000

4000 = 8 4 = 2

Example 6: Divide 9583 by 3000.

The quotient is 3 (3000 × 3 = 9000) and the remainder is 583 (9583 – 9000 = 583).

Thus, 9583 ÷ 3000 gives Quotient = 3 and Remainder = 583.

2 Divide 6541 by 2000.

Quotient is 3 (2000 × 3 = )

Remainder is (6541 − = .)

So, 6541 ÷ 2000 gives Q = , R =

Find the estimated quotient when 9000 is divided by 19 rounded to the nearest 10.

Step 1

Round off the numbers.

There is no need to round off 9000.

19 rounded off to the nearest 10 = 20

Step 2

Divide the numbers and estimate the quotient.

9000 ÷ 20 = 450

Estimated quotient for 9000 ÷ 20 is 450.

Actual quotient for 9000 ÷ 19 is 473 with remainder = 13.

Hence, the estimated answer and the actual answer are close to each other.

Think and Tell Why did we not round 9000 off?

473

19 9000 – 76 140 – 133 70 – 57 13

Do It Together

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 7: Estimate the quotient for 9635 ÷ 41.

9635 rounded off to the nearest hundreds is 9600.

41 rounded off to the nearest tens is 40.

9600 ÷ 40

So, 9600 ÷ 40 = 240.

Estimate the quotient

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

1 1210 by 300 2 5897 by 2000 1210 can be estimated to 1200. 5897 can be estimated to . 1200 300 = 2000 =

Do It Yourself 4B

Find the quotient and the remainder without using long division.

a 486 ÷ 10 b 986 ÷ 100 c 9765 ÷ 10

d 3479 ÷ 100 e 7894 ÷ 1000 f 5555 ÷ 1000

Find the quotient and the remainder.

a 1547 ÷ 30 b 7946 ÷ 300 c 8764 ÷ 2000

Fill in the missing numbers. a 7

Divide the numbers and verify the answer.

a 443 ÷ 12

b 4897 ÷ 24

c 9876 ÷ 49

Round off the bigger number to the nearest 100, and the smaller number to the nearest 10, and find the estimated quotient.

a 478 ÷ 97

b 879 ÷ 48

c 2736 ÷ 31

Bandipur forest is one of the largest habitat of elephants. The park organises safaris for tourists. 855 people have come for the safari ride and 19 people can be accommodated on each ride. About how many rides are needed? [Round off the dividend to the nearest hundred and the divisor to the nearest 10].

How many hours are there in 1200 minutes?

Colour the boxes with possible answers that you can get on dividing a 4-digit number by 100.

Manu has saved ₹5460. He has 6 notes of ₹10, 9 notes of ₹100 and the rest are ₹500 notes. How many ₹500 notes does he have? Do you save the money given to you?

Word Problems

Ramu has to feed his cows to make sure they are healthy and produce milk. He purchases 1032 kg 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.

1 Circle the numbers.

Ramu has to feed hay to the cows. He purchases 1032 kg 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?

2 Underline the question.

4 Evaluate:

C: Circle the numbers.

U: Underline the question.

B: Box the keywords.

E: Evaluate/draw.

S: Solve and check.

3 Box the keywords.

Hay given to each cow = Total hay purchased Total number of cows = 1032 43

Creativity

5 Solve and Check:

Thus, Ramu gives 24 kg of hay to each cow.

Check the answer:

Dividend = (Quotient × Divisor) + Remainder

Quotient = 24, Divisor = 43, Remainder = 0

Dividend = (24 × 43) + 0

Dividend = 1032

Thus, the answer is correct.

Example 8: 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.

Example 9: Create a word problem on dividing a 3-digit number by a 1-digit number.

There can be many word problems that can be written. One example can be as given below:

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?

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.

1 Circle the numbers.

2 Underline the question.

3 Box the keywords.

5 Solve and Check:

4 Evaluate. Thus, Rohit makes such groups.

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

Ravi collects stamps and pastes them in a notebook. He has a total of 1240 stamps. He pastes 31 stamps on each page. How many pages were used altogether?

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

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

A shopkeeper buys 45 packets of candy. Each packet has 45 candies. If he repacks the candies in smaller packets containing 15 candies each, then how many packets will he get?

The municipal corporation ran a drive to plant trees and provide habitats for various species of birds, insects and other wildlife. The total number of trees planted was 4560. If the drive went on for 5 days, how many trees were planted on one day?

Create a word problem on dividing a 4-digit number by a 1-digit number.

Challenge

Critical Thinking

A apartment has 24 floors with 13 rooms on each floor. If there are 12 housekeepers, how many rooms will each housekeeper clean?

Points to Remember

• The number left after division is called the remainder.

• To check if our answer after division is correct, we can use the formula: 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.

Dice Game

Setting: In groups of 3

Materials Required: 3 dice

Method:

Roll the three dice.

Note the numbesr on the dice.

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

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 who gets the sum of the remainder equal to or more than 30, wins.

Chapter Checkup

Find the quotient.

a 47 ÷ 10 b 489 ÷ 10 c 145 ÷ 145

d 4000 ÷ 100 e 4789 ÷ 1000 f 8500 ÷ 1000

Divide.

a 459 by 3 b 958 by 10 c 855 by 19

d 7848 by 4 e 7894 by 100 f 9984 by 48

Find the quotient and the remainder.

a 987 ÷ 8 b 945 ÷ 23 c 2129 ÷ 9

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

a 1459 ÷ 4 b 779 ÷ 13 c 8797 ÷ 16

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

a 489 ÷ 9 b 1548 ÷ 52 c 6987 ÷ 49

A small solar panel generates 108 kilowatt-hours (kWh) of energy in a week. If each household needs 12 kWh per week, how many households can the solar panel provide energy for?

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

A class collected ₹1540 to distribute equally among 14 children of an orphanage. How much money will each child 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?

1 A farmer packs 1200 kg of tomatoes equally in two different types of boxes. The weight of one tomato is 100 g. The first type of box has 15 tomatoes each, and the second type of box has 25 tomatoes each. How many boxes in total did he pack?

2 Solve.

a What should be added to 341 so that, on dividing by 17, we get no remainder?

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

Case Study

Let’s Farm!

Agriculture is one of the major industries in India. India has set up a few agricultural universities to help farmers yield more crops. An agricultural university experiments on different varieties of seeds, plants to improve the quality and quantity of the agricultural products.

1 Nisha is doing research on different varieties of oranges. The total number of oranges she produced is 2100. There are 10 varieties in all. How many oranges of each variety did she collect?

2 The college asks 375 students to create innovative projects to increase the yield of fruits while using less water. The students are divided into 25 groups. Write whether the given statements are true or false.

a The college will get 25 innovative projects from the groups.

b The college will get 15 innovative projects from the groups.

3 The university decides to sell the excess oranges to the public. Each bag can hold a dozen oranges. There are 1656 oranges. bags are required to pack the oranges.

4 A farmer asks Nisha to give him banana saplings to yield more bananas with less water. He wants to plant 936 saplings in 18 rows. Estimate the number of saplings that each row will have by rounding off the dividend to the nearest 100 and divisor to the nearest 10.

Cross Curricular

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 each box Total 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

Boxes Chocolates in each box Total Chocolates

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

Let’s Warm-up

I scored out of 5.

Multiples

Sahil and his family are planning a trip to Ooty. Sahil’s father asked him to look for the trains to Ooty so that they can decide the dates and book the tickets.

Finding Multiples and Common Multiples

Using Skip Counting

Sahil checked online and marked the dates on a calendar for trains departing for Ooty, as shown below.

Think and Tell

What do you think 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.

The dates that have been marked show skip counting by 2.

The numbers encircled can be called the multiples of 2.

Example 1: Find the first 5 multiples of 4 by skip counting on the number line.

We can find the multiples of 4 by using the number line showing jumps of 4 as:

Write the first six multiples of 7.

7, , , 28, ,

Using Multiplication

We learnt that when we skip count by the same number, we get multiples of that number. The multiples of a number are the products we get on 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 3 × 6 = 18

3 × 2 = 6 3 × 7 = 21

3 × 3 = 9 3 × 8 = 24

3 × 4 = 12 3 × 9 = 27

3 × 5 = 15 3 × 10 = 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:

leaves no remainder leaves remainder 1

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

Facts About Multiples

On dividing 16 by 5, we get a remainder of 1. So, 16 is not a multiple of 5. 5 15 3 – 15 00 5 16 3 – 15 01

• 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, the number of multiples of 7 is unlimited.

Example 2: Find the first five multiples of 5 using multiplication. Check by dividing whether 95 is a multiple of 5.

5 × 1 = 5 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.

leaves remainder 0

5 19 95 95 00 –

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

Example 3: Find the first eight odd multiples of 9.

Multiples of 9 = 9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99, 108, 117, 126, 135 ... .

First 8 odd multiples of 9 are: 9, 27, 45, 63, 81, 99, 117 and 135.

Think and Tell

Are the multiples of an even number always even numbers?

Find the first five multiples of 6 using multiplication.

6 × 1 = 6 6 × 2 = 6 × = 6 × = 24 6 × =

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

Finding Common Multiples

Do 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 possible common days.

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

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 us see some common multiples using a number line.

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:

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

Example 4: Find the first 5 common multiples of 3 and 6.

of 3 3 6

of 6 6

The first 5 common multiples of 3 and 6 are 6, 12, 18, 24 and 30.

Example 5: Is 54 a common multiple of 4 and 6? Explain your answer. Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56. As 54 is not a multiple of 4, it cannot be a common multiple of 4 and 6.

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:

Multiples of 2 2 6 8 14

Multiples of 5 5 25

Common multiples of 2 and 5 = , We can find the canteen and party hall together on 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 the highest common multiple.

Do It Yourself 5A

Colour the balloons that are multiples of 4 red, multiples of 7 green and multiples of 9 blue.

2 Find the first five multiples of the given numbers.

a 8 b 10 c 11 d 12 e 13 f 14 g 15 h 16 i 20 j 25

3 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 f 5th multiple of 25

4 Find the first six even multiples of 12.

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

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

a 5, 75 b 7, 68 c 8, 64 d 11, 88

7 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 3 and 5 e 6 and 9 f 10 and 15

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

9 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?

10 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?

2 Fill in the given 3 × 3 grid where each cell must contain a unique number from 1 to 9 such that the sum of each row, column and diagonal must be a multiple of 5.

Critical Thinking 5

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

Factors

Priya and Anshu are siblings. Their father bought them a tiling game. The game has tiles of different colours along with a grid. Each player needs to select an equal number of tiles of different colours and arrange them to create rectangular shapes of different lengths and widths.

Finding Factors and Common Factors

Finding Factors Using Square Tiles

Priya and Anshu started playing the game. They both took 12 tiles of each colour and arranged them on the grid as shown below.

Let us note down the arrangement of tiles in terms of multiplication sentences. We will consider the length as the number of columns occupied and the breadth as the number of rows occupied.

Tile: 1 × 12

Tile: 12 × 1

Tile: 3 × 4

Tile: 4 × 3

Tile: 6 × 2

Tile: 2 × 6

Factors can be defined as the are numbers that you can multiply together to get another number. So, the factors of 12 are 1, 2, 3, 4, 6 and 12.

Let us play the tilling game with Anshu and Priya. Colour the tiles to show different rectangular arrangements of 8 tiles.

Tile: 1 × 8

Tile: ×

Finding Factors Using Multiplication

When it comes to finding factors, drawing tiles can be challenging. Instead, we can use multiplication to identify the factors of a 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.

• 1 and the number itself are always the factors of the given numbers.

Example 6: Find the factors of 18 using multiplication.

Multiply by 1 1 × 18 = 18

Multiply by 2 2 × 9 = 18

Multiply by 3 3 × 6 = 18

Multiply by 4 4 × ? = Not possible

Multiply by 5 5 × ? = Not possible Stop!

We have already found 6 to be a factor of 18.

So, the factors of 18 are 1, 2, 3, 6, 9 and 18.

Find the factors of 20 using multiplication.

Multiply by 1 × =

1 20 20

Multiply by 3 Not possible

Multiply by 5 × =

Should we multiply further?

Multiply by 2 × =

Multiply by 4 × =

Multiply by 6

So, the factors of 20 are .

Finding Factors Using Division

To determine the factors of a number, we can also look for numbers that divide the given number exactly, leaving no remainder behind.

Let us find out which numbers divide 30 completely. 2 and 15 are factors of 30. 3 and 10 are

4 and 7 are NOT factors of 30.

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.

Facts About Factors

Every number is a factor of itself.

Factors of 5 1, 5

1 is a factor of every number.

Example 7: Is 7 a factor of 42?

Remember!

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

Every number has a limited number of factors.

5 has 2 factors, and 8 has 4 factors.

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.

To check whether a number is a factor or not, we divide.

Divide 42 by 7.

On dividing 42 by 7, the remainder is 0.

So, 7 is a factor of 42.

Find the factors of 36 using the division method.

Think and Tell

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

So, the factors of 36 are .

Finding Common Factors

As the name suggests, common factors are the numbers that are common among the factors of two or more numbers.

Let us find the common factors of 4 and 8. Factors of 4

of 8

Common factors of 4 and 8 = 1, 2 and 4.

Example 8: Find the common factors of 15 and 18.

Factors of 15

Factors of 18

Common factors of 15 and 18 are 1 and 3.

Find the common factors of 20 and 30.

Factors of 20

Factors of 30

Remember!

1 is a factor of all the numbers. So, it is also a common factor of any two numbers.

The common factors of 20 and 30 are .

Prime and Composite Numbers

We have learnt about factors and how to find the factors of numbers using various methods. We also noticed that each number does not have a fixed number of factors. These numbers can be classified as prime or composite numbers based on the number of factors, as shown below.

Prime Numbers: 2 factors – 1 and the number itself.

Composite Numbers: More than 2 factors

Find the factors of 14, 19, 45 and 59. Classify as prime or composite.

Factors of 14 = 1, 2, , ; Composite Number

Factors of 19 = ,

Factors of 45 = ,

Factors of 59 = , _________________ 1

Remember!

The number 1 is neither prime nor composite

Did You Know?

Prime and composite numbers were discovered, by a Greek mathematician, astronomer, and poet, Eratosthenes.

Do It Yourself 5B

1 Show 24 in different arrangements using circles. Then, list the factors of 24.

2 Find the factors of the following numbers using multiplication.

3 Find the factors of the following numbers using division.

4 Is 18 a factor of 126? Explain your answer.

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

6 Find the common factors of the following numbers. a 14, 20 b 16, 18 c 35, 50 d 54, 64

7 Which of the following numbers are common factors of 78 and 96? Circle the correct option. Verify your answer.

8 Write True or False.

a 11 and 13 have no common factors.

b 0 is a common factor of all the numbers.

c 15 and 25 have a total of 3 common factors.

d Both 41 and 49 are prime numbers.

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

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

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

12 Create a word problem to find factors of 2 numbers.

Challenge

Critical Thinking & Value Development

1 Write 34 as a sum of: a 4 prime numbers b 4 composite numbers

2 Bhanwar Lal is a farmer. He always distributes the first few saplings to the children in his village. 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

Factors are numbers that divide another number without leaving a remainder. We generally use division to find factors.

The number of factors of a number are countable or limited.

We generally use multiplication to find multiples. Multiples of a number are found by multiplying a given number by other numbers.

We can have any number of multiples of a number. Multiples are unlimited.

Factors are either equal to or smaller than the given number. Multiples are either equal to or greater than the given number.

Common factors of two or more numbers are the numbers that completely divide all the given numbers without leaving a remainder.

A number that is a multiple of two or more numbers is a common multiple of those numbers.

Math Lab

Setting: In groups of 4

Experiential Learning & Collaboration

Board Game of Multiples

Method:

1 Each player chooses their own colour.

Materials Required: Number grid as shown below, dice, crayons

2 One player rolls the dice and notes the number.

3 The player chooses a multiple of that number on the board and shades it with their chosen colour.

4 In case a player gets 1 on the dice, they can choose any number on the board. (Do you know why?)

5 The player who colours the greatest number of multiples on the board is the winner.

Chapter Checkup

1 Write the first 5 multiples of the given numbers.

a 7 b 17

e 21 f 23 g 30 h 32

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

a 2 and 5 b 3 and 8 c 5 and 8 d 4 and 14

e 10 and 15 f 10 and 25 g 11 and 22 h 20 and 24

3 Find the factors.

a 50 b 66

e 98 f 120 g 156 h 180

4 Find the following.

a Multiples of 4 that are smaller than 30.

b Multiples of 8 that are greater than 30, but smaller than 80.

c Multiples of 7 between 7 and 85 that are divisible by 2.

5 • Multiples and Factors

5 Write the common factors.

a 4 and 8

c 9 and 15

e 25 and 60

g 36 and 81

6 Write Yes or No for the following statements.

a 1 and 6 are factors of 7.

c 1 is the smallest and only factor of 31.

e 6 and 9 are factors of 54.

b 6 and 10

d 12 and 15

f 28 and 42

h 41 and 87

b 61 is a prime number.

d 2 and 4 are factors of 8.

f Both 91 and 93 are prime numbers.

7 The bells at Church 1 ring after 60 minutes, while at Church 2, they ring after 45 minutes. At what time will the bells at the 2 churches ring together next if both start ringing simultaneously?

[Hint: Find the common multiples.]

8 Megha wants to greet her class teacher with a bouquet made of lilies and roses on her birthday. She went to a florist who sells roses in groups of 5 and lilies in groups of 4. What is the least number of each kind of flower Megha should buy so that she has an equal number of roses and lilies in the bouquet?

[Hint: To know the number of flowers to be purchased, we should look for multiples.]

9 Create a word problem to find common multiples of 2 numbers.

Challenge

1 Identify the number using the hints.

a The number is between 1 and 80.

b It is a multiple of 2 and 3 and a factor of 90.

c The sum of the digits of the number is 9

Critical Thinking

2 Naina is organising her toys. She 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. What is the maximum number of toy groups that Naina can create?

[Hint: Look for the biggest common factor!]

Case Study

Cross Curricular & Value Development

Nikita joined an afforestation NGO that focuses on planting trees to restore forests (reforestation). The NGO also plants trees in designated areas that were previously not forested, which involves selecting appropriate tree species, preparing the land and planting saplings. Read the questions and answer them.

1 The volunteers planted oak saplings on every 3rd day of July and pine saplings on every 4th day of July. On which days of July did they plant both the saplings? Hint: (Find common factors of 3 and 4).

a 6th July and 8th July b 12th July and 16th July

c 12th July and 24th July d 15th July and 16th July

2 The NGO has 20 volunteers and wants to create teams with an equal number of volunteers in each team. Which of the following options do not represent possible team sizes?

a 4 teams with 5 volunteers in each team.

b 2 teams with 10 volunteers in each team.

c 3 teams with 7 volunteers in each team.

d 5 teams with 4 volunteers in each team.

3 The volunteers were instructed to plant a total of 36 trees in a specific area in rows and columns with an equal number of trees in each row and column, what are the possible configurations for the rows and columns?

4 Besides planting trees, in what other ways can we care for the Earth and protect our environment?

6 Fractions

Letʼs Recall

This is a piece of paper. Let this piece of paper represent one whole.

When we cut the paper into 2 equal parts, each equal part of the whole represents a fraction.

Each part is called a half and is written as 1 2 .

Now, if we cut another piece of paper of the same size into 4 equal parts, each one out of the four parts is called one-fourth and is written as 1 4 .

All these parts of a whole are called fractions.

Notice that each fraction has a number on top. This signifies how many parts we are talking about. This number at the top is called the numerator.

The number at the bottom tells how many equal parts a whole is divided into. This is called the denominator.

1 2 Numerator Denominator

Letʼs

Warm-up

Look at the picture and fill in the blanks.

1 There are fish in the aquarium.

2 The fraction of yellow fish is of all the fish.

3 The fraction of pink fish is of all the fish.

4 The fraction of red fish is of all the fish.

5 The fraction of purple fish is of all the fish.

Understanding Fractions

Shalu, Karan, Raghav and Pooja went on a picnic. All of them brought snacks with them.

Shalu: I have brought 2 apples, 4 oranges and 3 packets of biscuits.

Karan: I will eat half of the number of apples.

Raghav: I want to have a quarter of the number of oranges.

Pooja: I will eat one-third of the number of packets of biscuits that we have.

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) into equal parts gives us fractions.

Let us revise this again.

Fraction of a Whole

When a whole is divided into two equal parts, each part is called a half.

It is denoted as 1 2 .

Two halves make a whole. 1 whole 1 2 or half

When a whole is divided into four equal parts, each part is called a quarter or one-fourth. It is denoted as 1 4 .

Four quarters make a whole. 1 4 or one-fourth

Example 1: State True or False.

One-third is smaller than 1 2. True

One-quarter is bigger than 1 3. False

Half is represented as 1 4. False

When a whole is divided into three equal parts, each part is called one-third.

It is denoted as 1 3 .

Three one-thirds make a whole.

1 3 or one-third

Did You Know?

Only 3 100 of Earthʼs water is freshwater, available for drinking, agriculture, and other human needs.

State True or False.

One-fourth can also be written as 1 4. True

Half can also be written as 1 3.

One-third is more than 1 4.

Fraction of a Collection

We can also find the fraction of a collection.

1 2 of 12 butterflies

Butterflies in one group = 6

1 3 of 12 butterflies

Butterflies in one group = 4

Let us now try to find 2 3 of 12 butterflies.

1 Identify the numerator and the denominator.

2 Denominator = 3, so divide 12 butterflies equally into 3 groups.

3 Numerator = 2, count the number of butterflies in 2 groups = 8.

4 So, 2 3 of 12 = 8.

Example 2: What fraction of the given flowers are. a white? b pink?

Total number of flowers = 7

Number of white flowers = 3

Fraction of white flowers = 3 7

Number of pink flowers = 4

Fraction of pink flowers = 4 7

Example 3: What is 3 5 of 35 cupcakes?

Number of cupcakes = 35

Divide 35 cupcakes equally into the,same number of groups as the denominator, which is 5 groups. Each group has 7 cupcakes. Count the number of cupcakes in 3 groups, which is 21.

Hence, 3 5 of 35 = 21 cupcakes.

1 4 of 12 butterflies

Butterflies in one group = 3

2 3 of 12 butterflies

Butterflies in 2 groups = 8

An ice-cream seller has 36 ice creams. He sold 5 6 of the ice creams. How many ice creams did the ice-cream seller sell?

Total number of ice creams = 36

Fraction of ice creams sold =

There are a total of groups. There are ice-creams in each group.

Number of ice creams in 5 groups =

Thus, number of ice creams sold = 5 6 of 36 = .

Write the fraction represented by the shaded part of each image. a b c d

Circle and find the number of butterflies in each collection.

a 1 3 of the collection = butterflies.

b 4 9 of the collection = butterflies.

c 7 9 of the collection = butterflies.

Find.

a 1 4 of 32 b 1 8 of 24 c 1 5 of 40 d 1 6 of 36 e 2 3 of 33 f 5 6 of 36 g 3 7 of 35 h 5 8 of 32

Draw 12 balls and circle 2 3 of the total balls.

Manya made an apple pie. She divides the apple pie into 10 equal slices and eats 5 of them. What fraction of the apple pie was eaten by Manya?

There are about 50 snow leopards in Uttarakhand, India. If there are half the number of snow leopards in Sikkim as in Uttarakhand, then how many snow leopards are there in Sikkim?

There are 48 pencils in a box. If 3 4 of the pencils are blue and the rest are black, how many pencils are blue?

There are 75 students in a class. 2 5 of the students did not go on the class picnic. How many students went on the picnic?

Anna runs a flower shop. She has 48 roses left in her shop. 1 4 of the roses wilted. 3 4 of 48 roses were sold. How many roses are left?

Challenge

Critical Thinking & Value Development

Prakhar decided to distribute rice bags at an old-age home and blind school on his birthday. He distributed 3 5 of the rice bags at the old-age home. Then he gave 5 8 of the remaining bags to the blind school. If Prakhar had 40 rice bags initially, then how many rice bags were left with him? 1

Equivalent Fractions

Let us show 3 fractions on fraction circles and shade the same amount. 1 out of 2 parts

2 out of 4 parts 4 out of 8 parts

Fractions that represent the same or equal value but have different numerators and denominators are called equivalent fractions. Here 1 2, 2 4 and 4 8 are equivalent fractions.

Finding Equivalent Fractions

To find equivalent fractions, we multiply the numerator and denominator by the same number.

Equivalent Fractions of 5 6

Simplest Form

A fraction is said to be in its simplest form when the denominator and the numerator have no common factors other than 1.

Let us find the simplest form of the fraction 9 45 . Remember!

A common factor is a factor shared by two or more numbers. For example: 7 is a common factor of 14 and 21.

Divide the numerator and the denominator by one of the common factors until it cannot be divided further. We can divide the numerator and the denominator by either 3 or 9.

Divide by common factor 3.

9 ÷ 3

45 ÷ 3 = 3 15

Think and Tell

Why do we not divide the numerator and the denominator by common factor 1?

3 15 can be further divided by 3 as 3 ÷ 3 15 ÷ 3 = 1 5 9 ÷ 9

In both cases, the simplest form of 9 45 = 1 5 .

Divide by common factor 9.

45 ÷ 9 = 1 5 1 5 cannot be divided further.

Example 4: Look at the figures and write them in the form of equivalent fractions.

3 out of 4 parts are shaded 6 out of 8 parts are shaded

Example 5: Find the simplest form of 15 30 .

Factors of 15 = 1, 3, 5 and 15

Factors of 30 = 1, 2, 3, 5, 6, 10, 15 and 30

Common factors of 15 and 30 = 1, 3, 5 and 15

Divide the numerator and the denominator by 15.

15 ÷ 15

30 ÷ 15 = 1 2

1 2 cannot be divided further. Hence, the simplest form of 15 30 = 1 2 .

Find the equivalent fraction of 5 25. Also, find its simplest form.

Equivalent fraction of 5 25 = 5 × 2 25 × =

The simplest form of 5 25 can be given as:

Factors of 5 = 1, Factors of 25 = , 5,

Common factors of 5 and 25 = , 5 ÷ 25 ÷ = 1 . OR 3 4 6 8

Do It Yourself 6B

Shade the second figure to make it equivalent to the first figure.

Write four equivalent fractions for each of the given fractions. a 3 4 b 2 7 c 1 5 d 1 4

Write the fractions in their simplest form. a 10 50 b 30 90 c 2 26 d 5 65 e 15 45 f 6 42

Fill in the missing numerator or denominator in each of these equivalent fractions.

Circle the fractions that are in their simplest form.

The literacy rate in Kerala is the highest in India, which is 47 50 when given as a fraction. Write an equivalent fraction for the literacy rate in Kerala out of 100.

Pandas are fascinating creatures with many unique characteristics. They spend most of their day sleeping. A panda sleeps for 7 12 of a day, eats for 1 3 of a day and does other activities for 1 12 of a day.

Represent the number of hours dedicated to each activity by shading a grid accordingly.

Create a word problem to write 2 equivalent fractions of a fraction.

Look at the pattern. Identify the fraction that is not part of the pattern.

Like and Unlike Fractions

Like Fractions

Comparing and Ordering 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.

Think and Tell Can equivalent fractions be like fractions?

Example 6: Which is greater 6 17 or 12 17 ?

Numerator of 6 17 = 6

Numerator of 12 17 = 12

As 12 > 6; the fraction 12 17 > 6 17 .

Example 7: Arrange the given set of fractions in descending order: 8 12 , 9 12 , 7 12 and 4 12 .

Arrange the numerators in descending order: 9 > 8 > 7 > 4.

Descending order: 9 12 > 8 12 > 7 12 > 4 12 .

Arrange the given set of fractions in ascending order: 4 9 , 2 9 , 7 9 and 8 9 .

Arrange the numerators in ascending order: 2 < < < 8.

Ascending order: 2 9 < < < .

Comparing and Ordering Unlike Fractions with the Same Numerator

We know that the denominators of unlike fractions are different. To compare two or more fractions with the same numerator but different denominators:

• We compare their denominators.

• The greater the denominator, the smaller the fraction.

Example 8: Which is smaller 5 9 or 5 12 ?

Denominator of 5 9 = 9

Denominator of 5 12 = 12

As 12 > 9; the fraction 5 12 < 5 9 .

Example 9: Arrange the fractions in ascending order: 2 5 , 2 3 , 2 7 and 2 9 .

Since the numerators are the same, we arrange the denominators in descending order: 9 > 7 > 5 > 3.

Ascending order: 2 9 < 2 7 < 2 5 < 2 3 .

Arrange the fractions in descending order: 7 12 , 7 5 , 7 10 , 7 3 and 7 15 .

Since the numerators are the same, we arrange the denominators in ascending order:

Descending order:

Testing for Equivalence by Cross-multiplying

Let us now learn how to compare two fractions with different numerators and denominators.

Step 1

Write the fractions next to each other. For example, 3 7 and 2 9

Step 2

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.

Step 4

Compare the products. The greater fraction will have the greater product below it.

Here, 27 > 14.

Therefore, 3 7 > 2 9 .

Example 10: Compare the fractions 2 5 and 4 10 .

Compare the fractions 4 9 and 7 10 . Step 1 4 9

Name the

as like or unlike.

Compare the fractions and put the < , > or = sign in the box.

Circle the smallest fraction in the group.

Arrange the fractions in ascending order.

The water content in apples, bananas, oranges and watermelons

, respectively. Arrange the fruits according to their water content in descending order.

Create a word problem on ordering like fractions.

1 Rajiv ate 5 8 of an apple pie, while Arti ate a few slices out of some other apple pie, which had 9 equal slices. What could be the maximum number of slices that Arti ate such that she ate a smaller amount of apple pie than Rajiv?

Proper and Improper Fractions

Numerator is smaller than the denominator.

Numerator is bigger than the denominator.

Converting Improper Fractions to Mixed Numbers

Let us convert 9 4 into a mixed number.

Step 1

Divide the numerator by the denominator and identify the quotient, remainder and divisor.

Divisor 4 2 Quotient

Remainder 9 8 1

Step 2

Mixed number = Quotient

Remainder Divisor

Hence, 9 4 = 2 1 4

Example 11: Convert 13 4 to a mixed number.

Divisor 4 3 Quotient

Remainder 13 12 1

Mixed number = Quotient Remainder Divisor

Therefore, 13 4 = 3 1 4 .

Do It Together

Convert 56 12 to a mixed number.

As, Mixed number = Quotient Remainder Divisor Therefore, 56 12 =

Converting Mixed Numbers to Improper Fractions

Let us convert 2 3 4 into an improper fraction.

Example 12: Convert 5 2 3 to an improper fraction.

Writing a Whole Number as a Fraction

When the numerator is equal to the denominator, the fraction represents 1 whole.

When the denominator divides the numerator completely with no remainder, the fraction is a whole number.

Example 13: What is the value of the fraction?

The fraction can be written as 9 3 = 3.

Sort

Convert the improper fractions into mixed numbers.

Convert the mixed numbers into an improper fractions.

Simplify the fractions.

Operations on Fractions

Vishal helps his parents with household chores. Today, he is helping his mother to prepare dinner.

Mother: Vishal, could you please bring some sugar from that box?

Vishal: How much sugar should I bring?

Mother: I need 2 4 cups of sugar for the cake batter and 1 4 cups for the milkshake.

Vishal brings the required amount of sugar and gives it to his mother.

Adding and Subtracting Like Fractions

Vishal brought the required amount of sugar by adding what was needed for both recipes.

Adding Like Fractions

Let us find out how much sugar Vishal brought.

Adding Two Like Numbers: 1 4 2 4 3 4 = Same denominator

Add the numerators

Adding Mixed Numbers

Let us add 1 1 6 and 2 1 6 .

Convert to an improper fraction. Add the numerators. Reduce to its simplest form.

Example 13: Add. 1 11 16 and 3 16 11 16 + 3 16 = 11 + 3 16 = 14 16 = 7 8

Convert to a mixed number.

Did You Know?

Brahmagupta, an Indian mathematician, wrote down the rules for fractions around 1500 years ago.

Rohan bought 2 1 4 kg of bananas and 1 1 4 kg of apples. What is the total weight of the fruit bought by Rohan?

Weight of bananas = 2 1 4 kg = 9 4 kg

Weight of apples = 1 1 4 kg = kg

Total weight = 9 4 + =

Converting kg to a mixed number, we get .

Subtracting Like Fractions

Let us subtract 3 6 from 5 6 .

Subtract the numerators

Same denominator

Subtracting Mixed Numbers

Let us subtract 2 1 4 from 3 3 4 .

Convert to an improper fraction.

Subtract the numerators.

Reduce to its simplest form.

Convert to a mixed number. 2

Example 14: Subtract.

Riya bought 2 1 5 m of ribbon. She used 2 5 m of it. How much ribbon does she have left? Length of the ribbon bought = 2 1 5 m = Length of the ribbon used = 2 5 m. Length of the ribbon left = –

Do It Yourself 6E

Ria has 1 5 8 m of cloth. She used 7 8 m to cover a chair. What length of cloth does she have left?

Manya had 1 2 8 packets of cookies. She ate 7 8 packets of cookies. What fraction of cookies is she left with?

A jogging track is 2 2 8 km long. A cycling track is 3 4 8 km longer than the jogging track. How long is the cycling track?

Running strengthens muscles and increases bone density. Sunil ran 1 1 6 km on Saturday. On Sunday, he ran 2 3 6 km. How much farther did Sunil run on Sunday than on Saturday?

a word problem on the addition of

Challenge

Critical Thinking

1 Jeeshan, Raghav, Animesh and Priya had pizzas with them. The total combined pizza with all of them was 1 7 12 . Jeeshan had 3 12 of a pizza. Raghav had twice as much as Jeeshan’s fraction. Animesh and Priya had equal fractions of pizza. Determine the fraction of pizza each person had.

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.

• 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.

• To add and subtract two like fractions, we add or subtract the numerators and keep the denominator the same.

Setting: In groups of 4

Exploring Fractions!

Materials Required: Fraction cards or pieces of paper with proper, improper or mixed fractions written on them, pen and timer

Method:

1 Distribute the fraction cards or pieces of paper among the groups.

2 Ask the groups to sort the fraction cards into 3 categories: proper fractions, improper fractions and mixed numbers.

3 Track the time each group takes to sort the fractions.

4 The group with the greatest number of correct answers in the least time wins!

Shade or draw the fractions.

Find the fraction of a collection of objects.

a 1 6 of 18 flowers b 1 3 of 27 cakes c 1 4 of 36 boxes d 1 5 of 50 balloons

Write four equivalent fractions for the given fractions. a 5 6 b 7 8

Complete the equivalent fractions.

Reduce each fraction to its simplest form.

Compare the fractions and put the < , > or = sign in the box.

Arrange the fractions in ascending and descending order.

Convert the mixed numbers to improper fractions and the improper fractions to mixed numbers.

On Sunita's birthday, her mother was baking a cake for her. She cut the cake into 20 equal pieces.

Sunita distributed 3 5 of the pieces among some poor children. How many pieces did Sunita distribute?

Anuj, a shopkeeper, got a contract to deliver rice at a party venue. He had 28 kg of rice. He delivers 5 7 of the rice. How much rice does Anuj have left?

Most frogs can jump from 10 to 20 times their body length. A frog took two jumps. The first jump was 2 9 m long, and the second jump was 3 9 m long. How far did the frog jump in total?

Sudha has 2 5 of 50 rupees, and Ravi has 1 2 of 50 rupees. Who has more money?

A farmer has 56 cows. 3 7 of them are grazing in the field, and the rest are in the barn. How many cows are in the barn?

A vessel contains 2 1 4 litres of milk. John drinks 3 4 litres of milk. How much milk is left in the vessel?

Mohit is travelling from Mumbai to Pune by road. He drove 7 1 5 km on Monday, and 5 3 5 km on Tuesday. How far did he travel on both days?

Challenge

1 Is the fraction 12 16 is equivalent to the fraction shown in the figure? Shade another equivalent fraction for the given figure.

2 Look at the number puzzle. The sum of the fractions in each row, column and across the diagonals is 15 17 What is the sum of all the missing values?

Cross Curricular & Value Development

Land Distribution Across the Continents

The total area of land on Earth is unevenly distributed among its seven continents. Each continent has its share of human population, and resources like coal, oil, crops, etc. In order to use these resources well and distribute them evenly amongst the world population, we need to learn to manage them well.

Read the table on the land area of the continents and answer the questions.

1 Which continent has the largest land area? Use equivalent fractions to compare.

a Asia

c Africa

2 Which continent has the least land area?

a Europe

c Australia

b Europe

d Australia

b North America

d Antarctica

3 Arrange the continents in the ascending order of their land areas.

4 What is the difference between the land areas of North America and South America?

5 How can continents with larger land areas (e.g., Asia and Africa) utilise their resources better to support both their populations and the global community?

7 Lines and 2-D Shapes

Let’s Recall

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, these shapes are present everywhere!

We already learnt about sleeping, standing and curved lines, triangles, circles, squares and rectangles.

Let’s recall what we learnt about flat shapes.

Let’s Warm-up

Understanding Basic Terms

Ritu is playing a game of joining the numbered dots.

Ritu: Look Mom! I have joined the dots and made the picture of a hut.

Mother: Wow! This looks quite nice!

Points, Rays and Lines

Notice the figure that Ritu has just drawn. She joined the numbered dots in order. The lines formed an interesting shape, that of a hut.

Let us now learn about the different elements of geometry.

Points

A point shows the exact position of an object. It is represented by a dot (.).

A point by itself cannot be measured, as it doesn’t have any length, breadth, or height. However, it can be used to describe a location or position of an object.

Points are named using capital English letters, such as A, B, etc. The map below shows the position of the hotel, the house, and the museum using three different points A, B and C.

Rays

What if Ritu draws a straight path from point A to point B, as shown below, and keeps extending it in one direction?

When a straight path starts at a point and extends endlessly in the other direction, it is called a ray.

A ray has a starting point but 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. Light emitted by the Sun is an example of rays.

Ray AB is different from ray BA. In one, A is the starting point while in the other, B is the starting point. Error Alert!

Line

What if Ritu keeps on drawing the straight line beyond each of the two points, so that it extends endlessly on both sides?

When a straight path extends in both directions and has no end points, it is called a line.

It is represented with the help of small letters like l, m, n, or points (like AB) that fall on it and is represented as AB

Let us see some examples of types of pairs of lines.

Parallel Lines

Intersecting Lines

When two lines intersect at an angle of 90°, they are called perpendicular lines.

Example 1: State whether the following statements are true or false.

a The light from a torch is an example of a ray. True

b A line extends endlessly in both directions. True

c A point has only length. False

Example 2: Mark the correct geometric elements in the figure.

O, A, B, C, D and E are points.

AB and CD are lines.

OE OA, OB, OC and OD are a ray.

Example 3: Circle the correct representation of the rays: Think and Tell

How many lines 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 .

The line intersecting FC is .

Do It Yourself 7A

State whether the following statements are true or false.

a A ray has no end points.

b A line has 2 end points.

c Only one line can pass through a point.

Name the given figures as point, ray or line.

Name the points, rays and lines in the

The light emitted by stars is an example of rays and cross roads are an example of intersecting lines. What type of lines are the zebra crossings?

Identify sets of lines that appear to be parallel or intersecting.

Line Segments

Remember how Ritu joined the dots on the paper to form a hut?

When she joined two points, she drew something called a Line Segment.

So, the straight path between any two points that has a definite length is called a Line Segment.

A line segment is the shortest distance between two points.

The shown line segment can be represented as PQ or simply PQ.

Think and Tell Can a ray and a line segment be parts of the same line?

In the world around us, we can find many examples of line segments: 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).

Did You Know?

Bhaskara II was a renowned mathematician and astronomer. He wrote a book called Lilavati where he talked about adding, subtracting, shapes, and how to measure them.

Let us learn to use the ruler and measure the line segment AB as shown:

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. We can see that the point B is on the mark 7 on the ruler. Thus, the length of the line segment is 7 cm.

Remember!

NEVER put the zero mark of the ruler at any other point of the segment except the starting point.

Example 4: Measure the lengths of the line segments using a ruler. 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.

Look at the objects placed along the edge of a ruler. Read their lengths carefully (in cm) on the scale and fill in the blanks.

Did You Know?

A ‘smoot’ is a funny unit of measurement named after a person, Oliver R Smoot, in 1958, laid 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!

The length of the pencil is 6 cm.

The length of the comb is cm.

Drawing Line Segments

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.

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

Think and Tell

Can a line be measured using a ruler whose zero mark is missing?

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 ruler for accurate results.

Use a ruler to draw a line segment of length 10 cm.

Measure the length of the highlighted edges of the objects using a ruler.

Edge of the book = cm

Edge of the deck of cards = ___________ cm

The figure consists of line segments. Determine the length of each segment.

Look at the two pencils.

Tina’s pencil:

Sheena’s pencil:

What will be the total length of the two pencils? Measure the length of your pencil and draw the same length of pencil in your notebook. Use

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 the line segments.

Understanding More Geometrical Figures

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!

Classifying Figures and Shapes

There are different types of figures and shapes that the girls can draw. Let us see some.

Open vs Closed Figures

On 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”.

Open Figures

There is a gap in the boundary of the figure.

Closed Figures

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.

Example 5: Name 3 letters from the English alphabet which are open figures?

Letters C, U, and S from the English alphabet are open figures.

Look at the figures shown below. Classify these as open or closed figures.

Simple vs Non Simple

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:

Example 6: Recognise the simple closed figures.

The shapes 1, 3, and 4 are simple closed figures as they do not cross themselves at any point.

Look at the figures shown below. Classify these as simple or non-simple figures.

Polygons

Now, let us learn about special type of simple closed figures called a polygon.

Simple closed figures that are made up of only line segments are called polygons.

Did You Know?

A polygon with 1 million sides is known as a Megagon.

The line segments that form the polygon are called its sides. The point 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.

3 sides and 3 vertices

4 vertices

6 sides and 6 vertices

7 sides and 7 vertices

8 sides and 8 vertices

9 sides and 9 vertices

10 sides and 10 vertices

Example 7: Recognise whether the shapes are polygons or non-polygons.

Simple Closed Figure? Yes

Polygon?

Reason

Simple and closed, all sides are line segments.

Simple and closed, but part of the shape is a curve.

Join the dots with line segments and name the polygon formed. Do It Together

Not a simple closed figure.

Sort the following shapes as open or closed figures.

a b c d

Sort the following images as simple figures or non-simple figures.

a b c d

Categorise each of the following figures as simple closed, non-simple closed, simple open or non-simple open.

a b c d e f

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.

Write 3 letters from the English alphabet that are closed figures?

Draw any 2 simple figures and 2 non-simple figures.

Identify the polygons. For the ones that are not polygons, substitute it by drawing a possible polygon. a b c d

polygons.

Circles

Have you ever tried to draw a circle in a field, in the 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.

Circles and Their Parts

A circle is a perfectly round shape, with no corners or edges. Circles are shapes that we see in clocks, wheels, and even in nature.

Parts of a Circle

Let us now identify the different parts of a circle.

Centre

A point inside the circle, called the centre (O), is at equal distance from any point on the circle.

Circumference

The length of the boundary of a circle is called its circumference.

Semicircle

A diameter cuts the circle into two equal parts. Each half is called a semicircle

Radius

The distance between the centre of a circle and any point on it is called a radius. OA, OB, OC are the radii

Diameter

The line segment (AB) passing through the centre with both its ends lying on the boundary of the circle is called the diameter.

A diameter is twice the radius of a circle.

Diameter = 2 × Radius, or Radius = 1 2 × Diameter

When two circles have the same centre but different radii, they are called concentric circles. In the figure, Circle 1 and Circle 2 are concentric circles.

Remember!

All the radii (plural of radius) of a circle are of the same length.

Think and Tell

How many diameters can a circle have?

Example 8: Identify and write the name of the radius, centre and diameter of the circle.

Centre of the circle: O is the centre

Radii: OA, OB, OC, OD, OE, OF

Diameters: AB and EF

Example 9: Find the diameter of a circle where the radius is 6 cm.

We know that, Diameter = 2 × Radius or Radius = Diameter ÷ 2

So, in this case,

Diameter = 6 cm × 2 = 12 cm

Think and Tell

Example 10: A pot has a radius of 7 cm. If the lid has a diameter of 2 cm more than the diameter of the pot, what is the radius of the lid?

Radius of the pot = 7 cm. So, diameter of the pot = 7 × 2 = 14 cm.

Diameter of the lid = 14 + 2 = 16 cm.

So, the radius of the lid = 16 ÷ 2 = 8 cm.

Identify the centre and every radius and diameter of the circle.

Constructing Circles

We use an instrument called the compass to construct a circle. A compass has 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.

Insert a pencil into the pencil holder of the compass. Place the pointed end of the compass on the paper as a fixed end.

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!

Example 11: Construct a circle of radius 4 cm.

To draw a circle with compass, NEVER displace the tip of the compass from its position and ALWAYS tighten the screw of the compass else it can result in an incomplete or imperfect circle. Error Alert!

Construct a circle of diameter 6 cm.

Fill in the blanks.

a Every point on the boundary of circle is at the same distance from the .

b All the radii of a circle are in length.

c A circle can have a/an number of diameters.

d The length of the boundary of a circle is called its .

e A circle has only centre.

Draw a circle with centre O and label its centre, radius and diameter.

Choose the correct answer.

a A circle with a diameter of 10 cm is drawn. What will be its radius?

b A diameter divides a circle into equal parts.

c What is the relation between the radius (R) and diameter (D) of a circle?

Construct circles of the given radii.

The diameter of the planet Mars is 6780 km. What is its radius?

Kanchi was playing with a circular frisbee. She wanted 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.

A is the centre of Circle 1. B is the centre of Circle 2. What is the relation between the radii of the 2 circles?

Points to Remember

• A point shows an exact location and is represented by a dot.

• A ray starts at a point and extends in one direction.

• A line extends endlessly in both directions and has no fixed length.

• A line segment has 2 end points and has a fixed length.

• Shapes with two different endpoints are open shapes. Shapes with the same endpoint are closed shapes.

• Polygons are closed shapes made up of 3 or more line segments.

• A circle is a perfectly round shape, with no corners or edges.

Math Lab

Setting: Individual

Circle Designs

Experiential Learning & Creativity

Materials Required: A compass, sharpened pencils of different colours

Method:

Step 1: Draw a circle of any radius, say, 3 cm at the centre of a sheet of 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.

Chapter Checkup

Tick () the correct answer.

a AB represents a i ray ii line segment iii line iv point

b A dot made with a pen is an example of a i circle ii polygon iii point iv ray

c How many line segments are there in the figure?

i 10 ii 11 iii 13 iv 12

d An octagon has line segments.

i 10 ii 9 iii 8 iv 7

e Which of the following is an example of a ray?

i an arrow released from a bow ii a javelin iii light from a torch iv railway tracks

Identify the letters and numbers as closed, open, simple, or non-simple figures.

B C D U 7 0 8 S

Write the names and number of sides of the following polygons.

Contruct circles of the given measurements.

a diameter = 4 cm b radius = 4 cm

c radius = 5 cm

= 6 cm

Mary is standing at point A. She wants to get to point B by choosing the shortest route. Which route should she take?

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?

Surbhi 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

Find the distance between point A and D in the figure. Given that AB = 12 cm and OP = 4 cm.

A horse is tied to a post 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 the final shape of the area that has no grass left?

Read the assertion and reason and choose the correct option.

Assertion: Ansh is running around a circular field. The distance from the centre of the field to its boundary is 16 m. The diameter of the field is 32 m.

Reason: The diameter is half the radius.

Options:

a Both A and R are true and R is the correct explanation of A.

b Both A and R are true but R is not the correct explanation of A.

c A is true but R is false.

d A is false but R is true.

Case Study

Value Development & Art Integration

The Track and Field Relay Race

In a school's annual sports meet, an event of the Track and Field Relay Race is organised. This relay race involves teams of four runners each, passing a baton to the next runner in line. The track for this race is a set of concentric circles, with each team starting at different positions along the circles.

Answer the following questions:

1 How would you describe the shape of the track used in the Track and Field Relay Race?

2 In the Track and Field Relay Race, why do teams start at different positions along concentric circles?

a To make the race more challenging b To ensure fairness and equal distance c To confuse the runners d To reduce the number of laps required

3 Draw 2 concentric circles of radius 3 cm and 4 cm in your notebooks.

4 Why are sports important in our lives?

8 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 us recall the shapes that we have learnt 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.

Solid shapes are called 3-D shapes. All objects around us are 3-D shapes. We can measure the length, width, thickness or depth of a 3-D shape.

Let’s Warm-up

Label each of these shapes as a 2-D or a 3-D shape.

This dice is in the shape of a cube. This duster is in the shape of a cuboid.

This water bottle is in the shape of a cylinder.

This party hat is in the shape of a cone.

This football is in the shape of a sphere.

Representing 3-D Shapes as 2-D Shapes

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:I will draw the picture while standing on the terrace. They completed their drawings and showed them to each other. Real Life Connect

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.

Views of Objects

All the objects can be seen from 3 different views.

1 Top view—Looking at the object from the top.

2 Side view—Looking at the object from the side.

Think and Tell

Do all objects look different when seen from different views?

3 Front view—Looking at the object from the front.

Riya drew the picture while sitting at the window. Side view Pooja drew the picture while standing on the terrace. 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?

The different views of the dice can be given as:

Top view

Front view

Side view

Hence, while looking from the side, Suhani will see 1 dot.

Chapter 8 • Representing 3-D Shapes

Front view

Riya’s drawing

Pooja’s drawing

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:

Therefore, the front view of an ice cream cone looks like a triangle.

Observe each image and draw its respective view.

Do It Yourself 8A

) the correct view that is seen in these pictures.

1 Look at the figure and answer.

a How many candles will be seen in the top view?

b How many candles will be seen in the front view?

c How many candles will be seen in the left-side view?

Given below are the 3 views of some objects. Tick () the side view of the given objects.

Draw and colour a 3-D shape of your choice and show its front, side and top view.

The picture shows the National War Memorial at India Gate in New Delhi. It is a war memorial to 74,187 soldiers of the Indian Army who died in the First World War. Draw the front view, top view and side view of the memorial.

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. Such shapes, which are used to make a 3-D shape, are called nets.

Let us look at the net of a sweet box.

The net of the sweet box looked like this.

Below are some more nets of a cuboid.

Similar to a cuboid, we have the nets of a cube. Unfolding a cube box along its edges gives the net shown below.

Always observe the size of the opposite faces of the cuboid. They can never be different. Error Alert!

Given below are some more nets of a cube.

Example 3: Observe the net of a paper cube and answer the given questions.

Which number will never be next to the number 3? — 6

Which number will be on the opposite side of 1? — 5

Which number will you see if you turn right from 5? — 4

Example 4 : Draw the net of the given cuboid.

On unfolding the box, we will get a net, as shown below.

The net of a cuboid is shown. Draw and colour the remaining faces of the cube, considering that the colour of the opposite faces is the same.

Match the cubes/cuboids with their nets.

Draw the net of the following shapes. 4 Art Integration

5 Cross Curricular

Toblerone is a famous Swiss chocolate brand. Theodor Tobler came up with unique triangular chocolate and packaging. Draw the net of the picture shown.

Challenge

Critical Thinking

1 Opposite faces on a dice add up to 7. Fill in the net of the cube with dots to make a dice.

Maps

Mom called Riya and asked her to buy some medicines from the chemist.

Riya: Mom, I don’t know the route from the sweet shop to the chemist.

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 chemist from the sweet shop.

Did You Know?

The first realistic world map is credited to Ptolemy, who created it in the 2nd century. Ptolemy plotted places from Britain to Asia to North Africa.

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 chemist.

Step 2

Look at the possible routes to reach the destination. Here, the possible route to the chemist can be shown as:

Did You Know?

In ancient times, people used star maps to go from one place to another. A star map is a map of the night sky!

Step 3

Think about the directions in terms of left, right, straight and front. To reach the chemist, Riya will take a left turn, then move straight. She will then take the first right turn and move straight. She will again take the first right turn and will move straight. The chemist will be in front of her.

Think and Tell

Will Riya follow the same directions if she is standing outside the supermarket?

Example 5: Look at the map shown and answer the questions.

1 Where is the playground when you enter through Gate B?

The playground will be to our right when we enter from Gate B.

2 What is in front of you when you are sitting on the chair?

The game zone is in front of the chair.

Mohan can reach Neerja’s house using 4 different routes. Two routes are marked for you. Mark the other two routes on the map.

How many times does Meera turn left if she walks to school along the path shown?

Mohan’s House

Neerja’s House

Mohan’s House

Neerja’s House

a Suhani’s house is on the second road.

b If Kavita steps out of her house on to 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 on to the second road.

Look at the map shown and fill in the blanks. 3

a If Rani is on Mall Road facing Rose Street, the police station will be to her

b Sam’s house is on road.

c The restaurant is in front of house.

d is at the centre of all the roads.

e Sam’s house is in front of park.

Suraj is standing outside the pharmacy shown by the red mark. He wants to reach the bank. Write the possible route to reach the bank.

The first fire truck service was started in Bombay in the year 1803 by the Police Force. The given map shows the way to a burning house the fire truck can take. Which road will the fire truck NOT cross to reach the burning house?

Draw a map showing the way from your classroom to the library.

1 Mohan wants to reach the bus stop from his house. Follow the directions below and draw a map.

a Walk straight from the house and turn right.

b Walk straight for 100 m, you will reach a supermarket.

c Cross the road and walk left.

d Turn right and walk for 100 m. You will reach the bus stop.

Points to Remember

• All objects can be seen from 3 different views. Top view, side view and front view.

• Two-dimensional shapes that are folded to make three-dimensional shapes are called nets.

• A cuboid can have 54 different nets.

• A cube can have 11 different nets.

• A map is a drawing of an area made on a flat surface like a sheet of paper.

Math Lab

Setting: In groups of 5

Method:

Experiential Learning & Collaboration

Exploring Nets of Cubes and Cuboids

Materials Required: Paper containing the drawings of cubes and cuboids along with measures of their lengths; cardstock paper, rulers, scissors, markers, tape or glue

1 Each group will make gift boxes of different sizes.

2 Each group will first draw the nets of the gift box.

3 Cut the net carefully and fold it into a box.

4 Secure the edges of the box using glue or tape.

5 Display the box and discuss the differences between the boxes created.

Look at the map shown here and fill in the blanks.

a The is between John’s house and Rohit’s house.

b The is nearest to the school.

c The grocery store is nearest to house.

d If Rohit is on First Road with the mall to his left, he needs to take a turn to reach the Pizza house.

Observe the map and answer the questions.

a There are 3 houses on the map. Whose house is the farthest from the school?

b Whose house is not opposite to Anand Garden?

c How many roads meet at the Central Chowk?

Cleopatraʼs Needle in London commemorates the victory of Lord Nelson at the Battle of Niles. Draw the front view of the structure.

Draw the map from your house to the school. Mention 3 places or shops on the way.

Arrange A, B, C and D so the car reaches the house. Draw to show your answer.

Cross Curricular

Rita's house

Mohit's house

Mina's house Bus stop Railway station School

Central Chowk

Anand garden Market Bus stop

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.

Case Study

Cross Curricular

Police Check Post

Let’s Visit Sikkim!

Sikkim is a famous tourist place in India. Below is a rough map of some places in Sikkim.

Assembly House

Chogyal Palden Thedyp Memorial

Ropeway Point

1 Draw the direction from Assembly House to reach Nathula Pass.

2 Write True or False.

a Nathula is in the left direction from Tsomgo Lake.

b Chogyal Palden Thedyp Memorial Park is between the ropeway point and the assembly house.

3 A flower pot in Chogyal Palden Thedyp Memorial Park is in the shape of a cube. Draw the net of the pot.

4 David is standing at the police check post facing the assembly house. On what side does the ropeway point to David?

Nathula Pass Tsomgo Lake

9 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.

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. This is a Shibori design which is a traditional Japanese dyeing technique.

Seema: The black, white, and brown lines create a repeating pattern across the entire cloth. It is a very interesting pattern.

Mother: Yes, a very interesting pattern!

Extending and Creating 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 the arrangement of shapes or objects in a regular and repeating manner. These patterns can be based on various factors such as size, shape, colour or direction.

Repeating Patterns

In the figure below, the green square and the red square are arranged in an order and are repeating.

What kind of pattern do you see in the print on the cloth? Think and Tell

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 repeating over and over again in an order, are called repeating patterns.

The craftsman in the image is making a pattern with a wooden block on cloth. The wooden block is dipped in different colours to form different patterns. This art is called block printing

Example 1: Complete the following pattern:

Complete the following repeating patterns.

Rotating Patterns

A rotating pattern occurs when shapes or objects turn around a centre point and repeat.

Terms that make up a pattern can look different or the same after you rotate them.

Observe the rotating pattern. Extend the pattern for two more units.

Growing and Reducing Patterns

Notice how the pattern in the figure grows: it starts with 2 squares, then becomes 4 squares, then 6 squares and finally 8 squares. These are called growing patterns

Rule: The number of squares is increasing by 2 each time.

Opposite to growing patterns, in reducing patterns, the units reduce through the steps.

Rule: The number of circles is reduced by 4 each time.

Example 2: Complete the following pattern for one more step. Also, identify the type of pattern.

Notice that the stars are increasing with each steps. Therefore, this is a growing pattern.

Did You Know?

Euclid, a Greek mathematician, studied shapes and discovered patterns in their sides and angles. This helped us classify shapes and understand their properties better.

Example 3: Complete the following pattern and recognise the type of pattern.

The number of triangles decreases as the steps increase. This forms a reducing pattern: starting with 5 triangles, then 4, then 3, losing one triangle at each step.

Extend the following patterns for one more step. Also, identify the type of pattern.

Extend the pattern by drawing 2 more shapes.

Draw the missing shapes in the following growing pattern. 2 Art Integration

Art Integration

Observe how the figures are rotating. Extend the pattern by drawing the next two units.

Draw the picture that comes next in

Use these shapes to create a growing pattern of your own.

1 The given figure is rotated as shown by the arrows. Draw the figure that will replace the question mark. Challenge Critical Thinking & Art Integration

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 Y89 X89 W89 22 44 88

Rule: Multiply the previous number by 2. Rule: Letters of the alphabet are in reverse order, and the number remains the same.

We form magic squares and magic triangles using numbers 1–9 along with the rule.

Let us look at some triangle and square number patterns.

Numbers on each side of the square add up to 15.

Example 4: Observe the tower and complete it. Here the numbers are arranged in tower form. We add 2 numbers below to get the number in the box above them.

Numbers on each side of the triangle add up to 9.

Example 5: Use numbers 1–9 to fill in the stars so that each line adds up to 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 the sum adding up to 107.

The rule of the pattern states that the sum of 2 numbers is 47.

First from the left and First from the right

Second from the left and Second from the right

Third from the left and Third from the right

= 107

Thus, the 6 numbers are , , , , , .

Complete the pattern. a 10, 30, 50, 70, , b 13, 26, 39, 52, , c 84, 74, 64, 54, , d 1, 2, 4, 7, 11, , Complete the number tower. Colour the block having the largest number.

1 Fill in the squares, so that each line adds up to 26.

Tiling Patterns and Tessellations

Have you ever looked around at the walls and floors? Do you observe any pattern in them?

Yes, if you look closely, the brick structure that we see when walls are built is actually a pattern.

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 structure is also an example of tessellation! Hexagonal shapes are arranged together to form the honeycomb.

Such patterns are created by fitting together identical shapes (tiles) without any gaps or overlaps to cover a flat surface, such as a wall, and are called tessellations. The following are some tessellating and non-tessellating patterns.

A tessellating shape

Tangrams

A non-tessellating shape

Look at the collection of shapes in the square. Interestingly, the individual shapes do not necessarily repeat, like in a pattern. However, 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.

Circle the figures which show tessellation. Do It Together

Critical Thinking & Creativity Count and name all the shapes in the following tangram.

1 Can you use this block to create a tessellating pattern? If this block can’t tessellate on its own, modify it (cut, add pieces) to make it a tessellating block. Then, design a pattern using your new block!

Coding 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.

Now, what if we want to say GET WELL SOON? How can we say it using the above code?

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 coding.

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.

This code means GOOD MORNING.

This process of finding meaning from a secret code is called the process of decoding.

Decode the message: 7 15 7 18 5 5 14

The message for the code 7 15 7 18 5 5 14 is .

Creativity

Do It Yourself 9D

Refer to the letter-digit code table. Decode the following sentences. Create one sentence on your own using the code.

a Go to (19-3-8-15-15-12)

b Find the (17-21-9-26)

c Click quiz now (1-20-20-5-13-16-20)

d and answer the question (18-5-1-4)

Using the code on page 147, write the codes for the following messages the students created for Environment Week.

a KEEP IT UP b SAVE WATER c PLANT TREES

d FANTASTIC WORK e REDUCE REUSE RECYCLE

Challenge

Critical Thinking

1 In a certain coding system, CAMEL is coded as 25106, and LION is coded as 6793. How will you code NAME in this coding system?

Symmetry and Reflections

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 a 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 about 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.

Now, let us try to create a symmetrical figure using a sheet of paper.

Step 1

Line of symmetry

Step 2

Cut along the dotted lines

Line of symmetry

Step 3

Fold along the line of symmetry

Symmetry exists all around us. The following shapes are also symmetrical:

In the figures given below, the line of symmetry does not divide the shapes into two identical halves. Therefore, these figures are not symmetrical and so they are called asymmetrical.

Tick () the symmetrical figures and cross out () the asymmetrical figures.

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

Vertical symmetry means that if you divide an object or shape down the middle, the left side will look exactly like the right side. This dividing line is called the vertical line of symmetry.

Horizontal symmetry

Horizontal symmetry means the top and the bottom of an object look the same when divided in the middle. The dividing line is called the horizontal line of symmetry.

Reflection

Every morning, we look 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 they are placed in front of a mirror.

The line between the object and the reflection is called the mirror line or the line of reflection

The mirror line or line of reflection can be of two types—vertical or horizontal. You can see the example below.

Reflection can be seen in nature as well. Look at the beautiful reflection of the mountains.

Draw the reflection of this shape along the mirror line.

Do It Yourself 9E

Are the given figures symmetrical along the lines marked? Write Yes or No.

Draw the line of symmetry for each of the following figures, wherever possible.

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.

Draw the reflection of each of the shapes.

1 Use the line of symmetry to complete the missing part of the pattern.

2 Draw all the possible lines of symmetry for this polygon. Explain why each of them is a line of symmetry.

Points to Remember

• A pattern is a sequence or arrangement of figures, things, numbers or letters in a fixed repetitive way.

• 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.

• The line between an object and its image is called the mirror line, or the line of reflection.

Setting: Individual

Symmetrical Design

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 to clean

Method:

Take the white paper. Fold it in half along the horizontal line of symmetry.

Now, unfold the paper and spill a few drops of ink on one half.

Press the two halves together.

The resulting figure will be a symmetrical figure.

Try with another sheet of paper. Fold the paper along the vertical line of symmetry.

Try out different combinations of colours to make it colourful.

Follow the instructions as given.

Does the dotted line on each shape represent a line of symmetry? Write Yes or No.

Draw a line of symmetry on each shape.

Draw the other half of each symmetrical shape.

Complete the number pattern.

a 110, 130, 150, 170, _, , b 111, 122, 133, , , c 890, 780, 670, 560, , , d 140, 131, 122, 113, , ,

Understand the code and complete the pattern.

Write the following words using the code in the above table:

Draw the shapes following the vertical line of symmetry. The first one is done for you.

7 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?

8 Create your own code using any rule. Ask your friends to decode the code.

Challenge

Imagine planting a garden where each row's flowers follow a sequence: the first and the second rows have 1 flower each, the third row has 2 flowers, the fourth row has 3 flowers, the fifth row has 5 flowers and so on.

If you continue this pattern, how many flowers will be there in the 6th row?

What will be the correct letter code for the shape or pattern given at the end of the line? 2

Case Study

Imagine, Mr Jones, your neighbour is blind. He uses Braille to study. Braille is a special code used by people who are blind or visually impaired to read and write. Braille uses raised dots in a rectangular cell to represent each letter, number and symbol.

Answer the following questions:

1 How many raised dots are there in Braille to represent the letter “d”? a One b Two c Three d Four

2 How many dots are used to form each Braille character?

3 True or False: The Braille system is used only for writing numbers.

4 Suppose a friend visited Mr Jones and left their name as a coded message using Braille dots on a piece of paper. Each dot feels like a small bump. Hereʼs the message: What is the name of the friend?

5 You notice Mr Jones struggling to cross a busy road. How can you help Mr Jones safely cross the street?

10 Length, Weight and Capacity

Letʼs 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).

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).

Letʼs Warm-up

1 Match the following.

Length

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.

Vibhu reads the signboard which says, “Free delivery up to 3 km. Order now!”.

Vibhu asks his Mother, “Mom, how far is 3 kilometres?”

Let us help Vibhu to find it out!

Measuring Length

“Fresh bread and buns. I must hurry back home.”

We often need to know how far places are, or how long or tall things are. This is referred to as lengths or distances. 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.

Converting Between Units of Length

Metres (m) and kilometres (km) are units for measuring length.

Kilometres are a larger unit than metres and are used to measure longer lengths, like the length of a road or the distance between 2 cities.

To measure shorter lengths, we use millimetres (mm) and centimetres (cm), like the length of a pencil or a paper strip.

Remember!

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 the length in centimetres (cm) and metres (m).

Units of length: Millimetre (mm) < Centimetre (cm) < Metre (m) < Kilometre (km)

Did You Know?

The Ganga river is about 2525 km long. It is the longest river in India.

To get the correct and simple measurement, sometimes we need to change the units of length from one to another. Let us learn the relationship between different units of length.

1 cm = 10 mm 1 m = 100 cm 1 km = 1000 m

Changing to larger units

÷1000 km m cm mm ÷100 ÷10

Changing to smaller units ×1000 km m cm mm ×100 ×10

To change to a larger unit, we divide. 10 mm = 1 cm. So, 1 mm = 1 10 cm

To change to smaller units, we multiply. 1 km = 1000 m. So, 3 km = 3 × 1000 = 3000 m. 1 m = 100 cm. So, 2 m = 2 × 100 = 200 cm. 1 cm = 10 mm. So, 4 cm = 4 × 10 = 40 mm.

From the above table, we know that 1 km = 1000 m

Hence upto 3 km free delivery is = 3 × 1000 = 3000 m.

Example 1: Suggest the right tool and find the length of the line.

This is a short line on paper. A ruler or a measuring tape can be used to measure the length. Let us use a ruler to measure the length of the line.

Place the ruler against the line. Make sure it starts at 0.

Note the length of the line in millimetres and centimetres. This line is 8 cm 2 mm long.

Example 2: Convert.

1 255 cm to m and cm.

255 cm = 200 cm + 55 cm

Remember, 100 cm = 1 m

So, 200 cm + 55 cm = 2 m 55 cm

Convert.

1 6 km to m.

2 1092 m to km and m.

1092 m = 1000 m + 92 m

Remember, 1000 m = 1 km

So, 1000 m + 92 m = 1 km 92 m

1 km = m

So, 6 km = × m = m

1000 6

2 3 m to cm. 1 m = cm

So, 3 m = × cm = cm

Solving Problems on Length

Let us solve some problems related to length in our daily lives.

Example 3: 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?

250 m

The distance from the home to the dairy = 250 m

The distance from the dairy to the home = 250 m

So, the total distance Priya has walked = 250 m + 250 m = 500 m

Example 4: The Red Fort is about 1500 metres away from the Jama Masjid. How far is it in kilometres and metres?

1500 m = 1000 m + 500 m = 1 km + 500 m

So, the Red Fort is 1 km and 500 m away from the Jama Masjid.

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?

The gardener needs to measure a length of cm.

We know that, cm = 1 m

So, 600 cm = m

The garden needs to use a metre long stick times to mark the lines.

What would you use to measure the length of these objects—a ruler or a measuring tape?

a A grain of rice

c Length of a dining table

b The height of a door

d Width of a book

Look at the pictures and write how long these objects are in centimetres (cm) and millimetres (mm).

Express the lengths given in larger unit to smaller unit and from smaller unit to larger unit.

a 9 m b 1200 cm

Express in kilometres and metres.

1500 cm

a 1400 m b 1600 m c 2200 m

e 1475 m f 1925 m

2125 m

13 m

1336 m

4250 m

Ramanʼs uncle drives 2 km to the office every day. How much distance (in metres) does he cover when driving to the office and back?

A leopard is about 2 m 10 cm in length, and a tiger is about 3 m 60 cm in length. Write their lengths in centimetres.

A tailor cuts 4 pieces of cloth, each measuring 30 cm, to make a pair of trousers.

a How much cloth has he cut in total?

b What is the total length of the cut pieces of cloth in metres?

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. Write about one benefit of jogging.

[Hint: Convert the distance into metres and then add.]

Challenge Critical Thinking

1 Imagine a race between a 5-centimetre-long caterpillar and a 10-centimetre-long earthworm. The race trackʼs length equals the total length of both creatures. How much longer is the total length otf the track compared to the caterpillarʼs length? Also, if the caterpillar walks at a speed of 2 millimetres per second, how long will it take to complete the race track?

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 a rock! “It is so heavy,” he cried. “How heavy is it?” he wondered.

Measuring Weight

Chintu could use a weighing scale to measure the weight of the rock. We measure weight in many other situations.

A seller measures the weight of fruit and vegetables on a weighing scale.

A doctor measures our body weight on a weight machine before a check-up.

Converting Between Units of Weight

On a weighing scale, different stones or bars are used. Look at the picture of kilogram (kg) bars. It is a measuring unit for weight.

Did You Know?

Galileo was an Italian scientist who studied how objects move and how gravity works. His experiments helped us understand how to measure weight, which tells us how heavy something is.

Kilogram (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.

We usually use grams as a unit to measure smaller weights, such as spices. We use kilograms to measure our body weight.

Bars of different weights

Changing to Larger Units

1 g = 1 1000 kg

5000 g = 5000 1000 kg = 5 kg 1 mg = 1 1000 g 6000 mg = 6000 1000 g = 6 g

Changing to Smaller Units × 1000 × 1000

g = 5 × 1000 mg = 5000 mg

Example 5: Read the weight of an object on a weighing machine.

Remember!

Units of Weight: Milligram (mg) < Gram (g) < Kilogram (kg)

Make sure the needle of the scale is at 0 when you start.

Place the object (a watermelon) on the machine.

Now, note where the needle is on the scale.

The weight of this watermelon is 2 kg.

Example 6: Change the units of weight.

1 Change 1050 mg to g and mg.

1050 mg = 1000 mg + 50 mg

Remember, 1000 mg = 1 g

So, 1000 mg + 50 mg = 1 g 50 mg

Convert.

1 4 kg to g.

1 kg = g

So, 4 kg = × _______ g = g

2 Change 2090 g to kg and g.

2090 g = 2000 g + 90 g

Remember, 1000 g = 1 kg

So, = 2000 g + 090 g = 2 kg 90 g

2 6 g to mg. 1 g = mg

So, 6 g = × mg = mg

Solving Problems on Weight

Let us solve some problems on weight in our daily lives.

Example 7: Nishant 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

Weight of 7 oranges = 7 × 100 g = 700 g

Example 8: Mihir's uncle bought 8 packs of pulses, each weighing 500 g. What is the weight of the pulses he has bought in kilograms (kg)?

Weight of 1 pack of pulses = 500 g

Weight of 8 packs of pulses = 8 × 500 = 4000 g

We know that 1000 g = 1 kg

So, 4000 g = 4000 ÷ 1000 = 4 kg

Raj is buying fruit from a shop. He is checking the weight of each fruit on a weighing machine.

1 Write the weight of each fruit.

2 What is the total weight of the fruit that Raj bought today?

Total weight of fruit = 1 kg + kg + 1 2 kg = kg

So, Raj bought a total of kg fruit today.

3 The price of 1 kg of pears is ₹220. How much would Raj pay for pears if he bought 2 kg of pears?

1 kg of pears = ₹220

The price of 2 kg of pears = 2 × ₹220 = ₹

Look at the picture and write the weight of the objects in kilograms (kg) and grams (g).

a b c

Change to grams (g). a 3000 mg b 7000 mg c 10,000 mg d 2467 mg

Change to kilograms (kg). a 5000 g b 4500 g c 6557 g

Change to milligrams (mg).

5 g

Change to grams (g).

81 2 g c 4 g 102 mg

9782 g

15 g 770 mg

a 17 kg b 10 kg 500 g c 5 kg 10 g d 15 kg 25 g

By the time a puppy is about 3 months old, it weighs about 4 kg. What is its weight in grams?

A vegetable seller sold 145 kg 500 g of onions today. If the price of 1 kg of onions is ₹30, what is the total earning of the vegetable seller today on the sale of onions?

Sudhir uses 500 g of flour to make 65 pancakes. How many pancakes can he make in 5 kg of flour?

Sunita weighs 9 kg 500 g. Sunitaʼs father weighs 9 times Sunitaʼs weight. What is her fatherʼs weight?

Create a word problem on converting between units of weight.

There are 6 boxes of marbles with each weighing 600 g. All the marbles in those 6 boxes have been divided equally among 36 children. What would be the weight of marbles that each child gets?

Rima is visiting a supermarket. She wants to buy some juice for a party. There are different bottles with different amounts of juice. She wants to buy the bottles with the greatest amount of juice. Let us help her!

Measuring Capacity

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.

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. Thus, they have different capacities.

Converting Between 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 with a litre capacity. Medicinal droppers, cups, and syringes measure liquid syrups and medicines in mL.

Let us learn the relationship between different units of measuring capacity.

1 L = 1000 mL

Let us see how to convert between units of weight.

Example 9: Measure 1 litre of water using a measuring jug.

Remember!

Capacity is the amount of liquid a container (bottle, glass, etc.) can hold.

Slowly pour the water into the jug.

Stop when it reaches the mark of 1 litre.

Let the water settle and read again.

This jug contains 1 litre (L) of water.

Example 10: Convert.

1 3050 mL to L.

Remember, 1000 mL = 1 L

So, 3050 mL = 3000 mL + 50 mL = 3 L 50 mL

2 8 L to mL.

Remember, 1 L = 1000 mL

So, 8 L = 8 L × 1000 mL = 8000 mL

Rima opened a bottle of 2 L orange juice. Then, she 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 Rehaan 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 amount of the juice remaining in the bottle.

Solving Problems on Capacity

Let us apply our learnings in solving some problems on capacity in our daily lives.

Example 11: This evening, Anju’s mother is making mango shake for 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.

So Anju’s mother needs to prepare 1250 mL of shake.

Example 12: Simran is buying water bottles at a market. Each bottle has 250 mL of water. She needs 2 L of water. How many bottles does she need to buy?

We know that 1 L = 1000 mL

2 L = 2000 mL

If 250 mL = 1 bottle

Then, 2000 mL = 2000 ÷ 250 = 8 bottles

So, to get 2 litres of water, Simran needs to buy 8 bottles.

Rehaan poured different amounts of coloured water in different jugs.

1 Write down the amount of coloured water Rehaan poured into each of these jugs.

2 How much coloured water does he have in total?

400 mL + mL + mL + mL = mL

3 To get 1000 mL of coloured water in one jug, Rehaan, would need to empty jug and completely.

Look at the pictures and write the amount of liquid in these measuring jugs.

Express the capacity in litres (L) and millilitres (mL).

A tea cup holds 100 mL of tea. How much tea is required to fill 6 such cups?

Coco is helping her mother by filling bottles to store in the fridge. She fills 3 water bottles each having a capacity of 2 litres. How much water did she use to fill all three?

Neha bought 4 small packs of 250 mL each and 2 bottles of 1 L each of shampoo. How much shampoo has she bought in total?

JJ and her sister are selling ‘Fresh Lemonade’ at their school fair to send money for a charity. 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?

Challenge

1 A vendor selling oil has only a 3 L and a 5 L measuring jar. How will he measure 4 L of oil for a customer?

Points to Remember

• Millimetre (mm), centimetre (cm), metre (m), and kilometre (km) are units of measuring length.

Math Lab

Setting: In pairs

• Milligram (mg), gram (g) and kilogram (kg) are units of measuring weight.

• Millilitre (mL) and litre (L) are units of measuring capacity.

Experiential Learning & Collaboration

The Guess Game!

Materials Required: Containers such as bottles, jugs, cups, pitchers; cards with capacities written on them, measuring tools such as measuring cups.

Method:

1 Look at each container and guess which capacity group it belongs to. Does it hold about 250 mL, 500 mL, or 1 litre?

2 Use the measuring tools to fill each container with water until it is full. This will help you see how much each container can really hold.

3 After measuring, compare what you guessed with the actual capacity of each container.

4 Discuss with your partner how the different shapes of the containers made it harder or easier to guess their capacities correctly.

Convert as directed:

a into metres (m) and centimetres (cm)

b into kilometres (km) and metres (m) i 1205 m

Express the following weight in kilograms (kg) and grams (g).

Convert the given weights into grams (g).

a 2kg 500 g b 4 kg 600 g c 5 kg 750 g d 12 kg 500 g

Express the capacity in litres (L) and millilitres (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 smaller jars does Tara use?

The price of 1 kilogram of sugar is ₹60. Find the price of sugar for the weights given.

Imagine you went on a trip to the moon. Here on earth, you weigh 60 kg. The weight of a person on earth is 6 times the weight on the moon. What would be your weight on the moon? Would you feel lighter, heavier or the same?

You have a seesaw and two buckets. In one bucket, you place 5 rocks, each weighing 100 grams. In the other bucket, you add sand until the seesaw balances perfectly. If you estimate that there are 20 grams of sand in each handful, how many handfuls of sand did you need to add to balance the weight of the rocks?

Meira, a healthcare provider in a village, has 4 litres of hand sanitizer in her clinic. There is a request for four 500 mL bottles, five 200 mL bottles, and twelve 100 mL bottles of sanitizer for the village. Does she have enough?

Read the statements and answer the questions.

Assertion (A): A 1-kilogram bag of rice weighs more than a 500-gram bag of sugar.

Reason (R): 1 kilogram is equal to 1000 grams.

a Both A and R are true, and R is the correct explanation of A.

b Both A and R are true, but R is not the correct explanation of A.

c A is true, but R is false.

d A is false, but R is true.

Case Study

Ancient India's Length Units

Ancient India had a rich history of measurement systems. The people of ancient India used various units to measure length, which was essential for various purposes, such as construction, trade, and daily life.

Here are some key units of measurement used in ancient India: Angula (Finger breadth): A basic unit of length.

Hasta (Hand): Approximately equal to 24 angulas.

Answer the following questions:

Dhanus or Danda (Rod): Approximately equal to 96 angulas.

1 One hasta is equivalent to how many angulas?

2 How many angulas are in 7 hastas?

a 168 b 144 c 128 d 196

3 If the height of a sand castle is 2 hastas, how many angulas is it?

4 If the width of a sand castle is 3 hastas, it is equal to 72 angulas. (True or False)

5 How many hastas is equal to 1 Dhanus or Danda?

Cross Curricular

11 Perimeter and Area

Letʼs Recall

We have learnt in the previous chapter that the length is the distance or measurement between two points.

We can measure lengths 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, the 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, such as the distance from your bedroom to the kitchen.

We use kilometres to measure long distances, such as the distance between two cities.

We have also learnt how to convert one unit of length to another.

Letʼs Warm-up

Convert the following.

Understanding Perimeter and Area

It is Rita’s parents’ 15th wedding anniversary! She wants to give them a family photo, but the frame has a plain brown edge. Rita doesn’t like it and decides to paste a colourful ribbon over it. How will she know how much ribbon is needed?

Perimeter

Rita needs to cut a piece of ribbon that is the same length as the edge of the 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. It can be measured in millimetres (mm), centimetres (cm) and metres (m).

Some real-life examples where we need to find the perimeter include:

• Making or decorating the edges of a photo frame, a gift box or a scarf.

• Putting fencing around a house, building, park, farm or field.

Decorated

Finding Perimeter Using a String

Let us see how we can measure the length of the boundary of an irregular shape.

Step 1

Place the thread along the edge of the shape, as shown in the picture.

Step 2

Mark the end point on the thread where it meets the starting point. You may mark or cut the thread at this point.

Step 3

Use a ruler to measure the length of the piece of thread. The length of the thread is the perimeter of the shape. Find the perimeter of the given shapes. Which shape has a greater perimeter?

Shape A

Shape B

Shape has a greater perimeter than shape .

Finding Perimeter Using Squared Paper

We learnt to find the perimeter of irregular shapes. What if we need to find the perimeter of shapes with straight edges? We can do so by using squared paper.

Below are two polygons on squared paper. To find the perimeter, we count the number of squares around the polygons.

Figure A

Perimeter = 20 units

A polygon is a closed figure that is made up of only line segments.

Figure B

Perimeter = 20 units

Both the figures have different shapes but the same perimeter of 20 units.

Example 1: Find the perimeter of the following shapes.

Find the perimeter of these shapes.

Perimeter = units

Finding Perimeter of Polygons

Perimeter = units

To find the perimeter of polygons such as a triangle, square or rectangle, we can simply add the lengths of all the sides.

For example: The perimeter of figure A and figure B can be given as:

What if the perimeter of a figure is given and we need to find one of the missing sides? To do so, we add the lengths of the sides given and then subtract them from the perimeter. Let us 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

This means,

4 cm + 5 cm + 8 cm + the length of the missing side = 23 cm

17 cm + the length of the missing side = 23 cm

So, the length of the missing side = 23 cm − 17 cm = 6 cm

Example 3: Sunita took 2 rounds of the given rectangular park. How much distance did she cover in total?

Distance covered in one round = Perimeter of the park = 150 m + 100 m + 150 m + 100 m = 500 m

Distance covered in two rounds = 2 × 500 m = 1000 m

1 Find the perimeter.

Use a piece of thread and a ruler to measure the perimeters of the given figures. Which figure has the longer perimeter? Figure X Figure Y

Find the perimeter of the following figures, where 1 unit = 1 cm.

Look at the figures. The length of the side of each shape is given. Find their perimeters.

The perimeter of each figure is given. Find the length of the missing sides in each figure.

Find the perimeter of each of the following figures. 5

Seema's mother runs a tailoring shop. Seema helps her mother by sewing borders for different covers. They received an order to customise 3 pillows covers of length as shown in the figure. What length of sewing thread should Seema buy from the market?

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. In what other way can the tiles be joined? Show by drawing the tiles.

Area

Draw 2 shapes with perimeters of 22 cm and 24 cm. 1

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 wall that needs to be painted.

• Area of a floor to check if it can be covered with carpets or 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).

Finding Area Using Squared Paper

We learnt to find the perimeter of a polygon using squared paper. We can also find the area of a polygon on a square grid.

Let us look at the figure below and find out its area, where the side of 1 unit = 1 cm.

The area of 1 square = 1 unit × 1 unit = 1 square unit.

We can find the area by counting the number of squares covered by the shape .

Did You Know?

Honeybees use hexagons to build their hives so that their storage space (area) is maximised, and the perimeter is minimised.

Number of units covered by the shape = 9.

1 unit = 1 cm. So, the area of the shape = 9 square cm.

What is the area of the given figure?

Number of units covered by the shape = Area of the shape = 1 2

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. Half

More than half squares Less than half squares

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.

2 All half squares are counted as 1 2 .

3 All more than half squares are counted as 1.

4 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

Shape B

Think and Tell

Can you trace some other shape on the grid, and find its area?

Example 4: Find the area of the shapes, where the side of each square = 1 cm.

Find the area of the shapes, where the side of each square = 1 unit.

Find the area of these figures. 1 side of the square = 1 unit.

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.

c Which school has the bigger playing field? 3 Cross

To teach the importance of a balanced diet, Schools A and B planted a vegetable patch in their schools. They also constructed a playing field for various outdoor sports, as playing is a natural way to reduce stress and anxiety. Look at the area of the fields and answer the questions.

Here, the side of each square = 1 unit.

a Find the area of the playing field in School A.

b Find the area of the vegetable patch in School B.

Curricular

Vegetable patch

Playing field School A

Playing field

Vegetable patch School B

Rohit bought a chocolate and ate part of it, as shown. Each square of the bar = 1 unit.

a Find the area of the chocolate bar he ate.

b Find the area of the complete chocolate bar he had before eating.

Create two figures on a square grid having the same area but different perimeters.

Challenge

Critical Thinking & Creativity

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? 1

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.

• To find the perimeter of a polygon, we can either count the number of units around a polygon on a grid, or add the lengths of all the sides of the polygon.

• The area is the total space covered by a closed figure. Area is measured in square units, such as sq. cm or sq. units.

• 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.

Math Lab

Setting: In groups of 2

Experiential Learning & Collaboration

Measure Your Hand!

Materials Required: Square-grid paper, pencils, erasers, colours, a roll of thread, a pair of scissors

Take a sheet of square-grid paper.

Method: Both partners will follow these steps:

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

Find the perimeter of each shape in units. Which shape has a smaller boundary?

Fill in the blanks.

Find the perimeter of each of the given figures.

The perimeter of figure A is cm.

The perimeter of figure B is cm.

The perimeter of figure C is cm.

Figure _____ and figure have the same perimeter.

Find the length of the missing side.

Each of the figures have 1 unit squares. Find the area and perimeter of each figure and answer the questions.

a Figure and figure have the same area but different perimeters.

b Figure and figure have the same perimeter but different areas.

c Figure and figure have the same area and perimeter.

Find the area of the given shapes.

Draw two different shapes, each with an area of 8 sq. units.

Add squares to the shape given on the right to make it into a square. What is the area of the square?

Manya drew an owl on square grid paper for her art and craft activity. What is its area if the side of each square is 1 unit?

Kanti wants to fence the boundary of his field of length 120 m and breadth 150 m He already has a fence that can cover a boundary of 150 m. How much more fencing does he need to buy?

Regular walking keeps the heart healthy and strong. Every morning, Suhani takes two rounds of a rectangular park of length 200 m and breadth 170 m, while Rishabh takes 3 rounds of a square park of length 150m. Who covers more distance and how much?

Challenge

Critical Thinking & Art Integration

How many rectangles can you draw with a perimeter of 20 units? Draw and write the side lengths.

Divide the 5 × 5 cell into equal areas using different coloured pencils such that each area has one dot inside it.

Case Study

Value Development & Creativity

Mr Sharma’s Dream House

Mr Sharma has purchased a new flat. The architect gave the layout of the flat on a square grid, so that the interiors can also be finalised. Look at the layout and answer the questions, considering the length of 1 square = 1 metre.

1 Mr Sharma wants to carpet his living room. How many square metres of carpet does he need to order?

a 24 square metres b 44 square metres

c 28 square metres d 40 square metres

2 Mr Sharma plans to install an electrical wire around the roof of the kitchen for lighting. How many metres of electrical wire will he require?

a 28 metres b 22 metres

c 30 metres d 24 metres

3 Mr Sharma wants his parentsʼ room to be the biggest in the house to ensure their comfort. Which bedroom should be given to his parents?

4 Which portions of the house have the same perimeter and area?

5 Create the layout of a flat having the same area as the above flat.

12 Time

Letʼs 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 film 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 to regulate traffic flow.

A second is a smaller unit of time. One minute is equal to 60 seconds.

A handshake typically lasts just a couple of seconds. The notification sound on the mobile phone is usually 1 or 2 seconds long.

Letʼs Warm-up

Fill in the blanks.

1 6 hours is the same as minutes.

2 360 seconds is equal to minutes.

3 There are seconds in 2 hours.

4 40 minutes is 1 hour. (less than/greater than/equal to)

5 2 hours is 130 minutes. (less than/greater than/equal to) I scored out of 5.

Time on a Clock

Noor was very tired yesterday. When he went to bed, it was 8 o’ clock on 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.

12-hour Clock

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 us find out how the wall clock shows the same time twice each day.

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.

Reading Time on a Clock

We already know how to read time 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 number to the next, 5 minutes have passed.

The time on the clock is 7:22.

Hours—Since the number just before the hour hand is 7, 7 hours have passed.

a.m. and p.m.

When the minute hand moves from one small marking to the next, 1 minute has passed.

Minutes—Skip count by 5 as 5, 10, 15, 20. Then count forward 21, 22.

So, 22 minutes have passed.

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.

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.

Did You Know?

The word “meridian” is from the Latin language. In Latin, ‘meridies’ means ‘midday’ or ‘noon’.

Example 1: Read the time on the given clocks.

Do It Together

Example 2: 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.

Look at Manu’s daily routine. Draw hands of the clock to show the time. Fill in the blanks.

Wakes up at 6:10 Eats breakfast at Comes back from school at Goes cycling at Goes to bed by 9:07 p.m.

Every morning, Manu wakes up at 6:10 . 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 . Every evening, Manu goes cycling at . He goes to bed by .

Do It Yourself 12A

Write the time in a.m. and p.m. One is done for you

a Evening 5 o’ clock - 5:00 p.m.

b At 10:00 in the morning -

c At 04:30 in the afternoon - d At 10:00 in the night -

Read and write the time on the given clocks.

Akhil goes for his football practice at 11:30 in the morning. How will you write the time using a.m. or p.m.?

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. -

Draw the hands of the clock for the given times.

A Gurudwara is a place of worship for the Sikhs. Isha goes to volunteer at a Gurudwara at 10:30 a.m. and comes back 2 hours later. Write the time when Isha comes back as a.m. or p.m.

Critical Thinking

Use the clues to find the correct time from the 8 options.

I am between 8:30 in the morning and 2:30 in the afternoon. My number of minutes is odd. I am closer to 5:30 p.m. than to 5:30 a.m.

What time am I? a 8:27 a.m. b 2:50 p.m. c 1:33 p.m. d 8:41 p.m. e 10:03 a.m. f 7:35 p.m. g 11:40 a.m. h 9:54 p.m.

24-hour Clock

Divya is on 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.

Divya: “Dad, what kind of clock is this? It is not like our wall clock or watch! It is showing 16:00.”

Dad: “This is a digital clock, Divya. It shows time in 24 hours.”

“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. Let us learn about this different format of reading time.

Reading a 24-hour Clock

A digital clock is a 24-hour clock. It shows time 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 23 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 noon or 12 p.m.

On a digital clock, we read time in ‘hours’ units. For example, the time on this clock will be read as 14:05 hours.

Error Alert!

Time after 23:59 is read as 00:00 and not 24:00.

24:00 00:00

Remember!

Time in the 24-hour format can be read as 13:00 hours or 1300 hours. Both ways are correct and mean the same.

Example 3: A digital clock shows the time as given. How do you read this in the 24-hour format? Clock 10:15 2:45 13:17 24–hour format 10:15 hours 02:45 hours 13:17 hours

The prayer break at a boarding school is at 18:00. Is that in the afternoon or evening?

Time on the clock = 18:00

A 24-hour clock uses the numbers to to show hours after 12 noon. So, 18:00 is the time in the .

Changing 12-hour Clock to 24-hour Clock Time

Railway and airline timetables use 24-hour formats or digital clocks. But our regular analogue clocks use the 12-hour format. Therefore, it is important to learn how to convert time in the 12-hour format into the 24-hour format.

Changing time in a.m. to 24-hour clock:

Let us change 9:30 a.m. into 24-hour format.

Step 1

Keep the same hour value. For 9:30 a.m., the hour value is 9 or 09.

Step 2

Write the minutes as they are. For 9:30 a.m., the minute are 30.

So, 9:30 in 24-hour time will be written as 09:30.

Step 3

Now, replace a.m. with hours. Thus, time in 24-hour format is 09:30 hours.

Changing time in p.m. to the 24-hour clock:

Now, let us learn to change 10:30 p.m. into the 24-hour format.

Remember!

In a 24-hour format, 12 midnight is written as 00:00, and 12 noon is written as 12:00 hours.

Step 1

Add 12 to the hour value.

Here, hour value = 10. So, the value = 10 + 12 = 22.

Step 2

Write the minutes as they are. Herethe minutes are 30. So, in the 24-hour format 10:30 = 22:30.

Example 4: Change 3:30 a.m. to 24-hour time.

Keep the same hour value : 3

Write the minutes as they are : 3:30

Replace a.m. with hours : 03:30 hours

Example 5: Change 6:45 p.m. to 24-hour time.

Add 12 to the hour value : 6 + 12 = 18

Write the minutes as they are : 18:45

Replace p.m. with hours : 18:45 hours

Step 3

Replace p.m. with hours.

Thus, the time in the 24-hour format = 22:30 hours.

Example 6: Liftoff for a school’s drama “Mission to Mars” is scheduled for 9:15 p.m.

Express this time in the 24-hour format so the spaceship’s clock can be set correctly.

Add 12 to the hour value: 9 + 12 = 21

Write the minutes as they are: 21:15

Replace p.m. with hours: 21:45 hours

Thus, the time in the 24-hour format is 21:45 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?

To know if Nira is on time, we need to change 5:50 p.m. to 24-hour time.

Add 12 to the hour value: 5 + 12 =

Write the minutes as they are:

Replace p.m. with hours: hours

Hence, .

Changing 24-hour Clock to 12-hour Clock Time

Let us learn how to change the 24-hour clock time to 12-hour clock time.

Step 1

Look at the first two digits of the time.

• It is a.m., if the number is less than 12.

• It is p.m., if the number is more than 12.

Example 7: Change 11:30 hours into 12-hour time.

Step 2

For a.m. time, keep the hour value as the first two digits. For p.m. time, subtract 12 from the first two digits.

The first two digits of the time are 11. Since it is less than 12, it is a.m. time.

For a.m. time, keep the same hour value as the first two digits, which is 11.

Write down the minutes as they are. So 11:30 hours by a 12-hour clock is 11:30 a.m.

Step 3

Write down the minutes as they are.

Example 8: Change 13:00 hours into 12-hour time.

The first two digits of the time are 13. Since it is more than 12, it is p.m. time.

For p.m. time, subtract 12 from the first two digits: 13 – 12 = 1.

Write down the minutes as they are:

So 13:00 hours by a 12-hour clock is 1:00 p.m.

Example 9: Astronauts on the International Space Station (ISS) follow a 24-hour clock. If dinner is served at 19:00 hours, what time is that in the 12-hour format?

24-hour clock time = 19:00 hours

Time in hours = 19 – 12 = 7.

Time in minutes = Remains the same = 00

So, 19:00 hours by a 12-hour clock is 7:00 p.m.

Divya saw the time on the railway station clock as 18:30 hours. What is the time on a 12-hour clock?

Time on 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 = .

Change the time into 24-hour time.

a 03:28 p.m. b 11:56 p.m. c 12 midnight d 11:59 p.m.

Change the time into 12-hour time.

a 22:40 hours b 18:25 hours c 23:24 hours d 13:03 hours

The International Space Station completes one orbit around the Earth every 90 minutes. If it starts its orbit at 4:30 a.m. in the 12-hour clock format, what time would it finish in the 24-hour clock format?

The Rajdhani Express is a high-speed train service in India. It departs from the New Delhi Railway Station at 16:55 hours and arrives next day at the Mumbai Central Station at 8:35 hours. Write its schedule in the 12-hour clock format.

The flight from New Delhi to Goa departs at 14:45 hours. The boarding pass will be given 2 hours before departure. At what time will the boarding passes be given by a 12-hour clock?

Challenge

Critical Thinking

Samaira is video chatting with a friend in London who uses a 24-hour clock. Her call starts at 20:00 hours in India. If London is 5 hours 30 minutes behind, what time will it be for her friend? Since London is behind, is it already night time for her friend too, or is it still day?

Elapsed Time

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.

“How much time did the bus take to reach school today?” Siya wonders.

08:00 a.m. 08:40 a.m.

The start time is 8:00 a.m. and the end time is 8:40 a.m. The time that the bus takes to reach the school is 40 minutes.

Siya attends a music class from 4:30 p.m. to 6:30 p.m. every Sunday. The duration of her class can be found by calculating the time between these two points. 1 hour 1 hour

4:30 p.m.

Start time

5:30 p.m.

6:30 p.m.

End time

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.

Hours to Minutes

1 hour = 60 minutes

2 hours = 2 × 60 = 120 minutes

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.

Example 10: 1 Change 4 hours to minutes.

1 hour = 60 minutes

So, 4 hours = 4 × 60 minutes = 240 minutes.

to Hours 1 minute = 1 60 hour 120 minutes = 120 × 1 60 = 2 hours

Error Alert!

Add or subtract hours and minutes separately. 2 hours + 2 minutes = 4 hours OR = 4 minutes 2 hours + 2 minutes = 2 hours and 2 minutes

2 Change 186 minutes to hours and minutes.

60 minutes = 1 hour

So, 186 minutes = 186 ÷ 60 = 3 hours 6 minutes.

Example 11: 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. How many minutes will be left for writing the answers?

Total time for the test = 2 hours

2 hours in minutes = 2 × 60 = 120 minutes

Time for reading the questions = 20 minutes

Time left for writing the answers = 120 – 20 = 100 minutes

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 =

10:30 p.m.

Duration in minutes = hours × 60 = minutes

Do It Yourself 12C

Change minutes to hours and minutes.

a 340 minutes b 450 minutes

c 560 minutes d 675 minutes

Change hours to minutes. Then, compare and write the appropriate symbol ( >, = or < ) in the blanks.

a 7 hours 420 minutes b 3 hours 115 minutes

c 6 hours 360 minutes d 10 hours 360 minutes

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

When India gained independence on August 15, 1947, the oath-taking ceremony of the first Prime Minister, Jawaharlal Nehru, began at 9:40 a.m. If the ceremony lasted for 2 hours 15 minutes, at what time did it finish?

A doctor starts his patient visits at 10:15 a.m. and returns to his cabin by 1 p.m. How much time did he spend on the patient visits?

Reading helps you learn new things and improves your imagination. Nihit starts reading a book at 16:30 hours. He reads for 45 minutes. What time does he stop reading? Do you read books at home?

your

Challenge

Neha starts studying at 4:00 p.m. It takes her 30 minutes to do the maths homework, 15 minutes to write the English notes and 70 minutes to revise all other subjects for the upcoming test. If her friend Pooja comes to play when Neha finishes studying, will Pooja arrive before or after 6:00 p.m.?

Time on a Calendar

Every student in the class is thrilled to hear the news of an upcoming one-day trip.

“We will be going on the 15th of this month. It is two weeks from now!” the teacher informs them.

“15th of this month? 2 weeks from today? How many days later, exactly?” Divya is confused.

Time in Days, Weeks, Months and Years

Let us look at the calendar to understand the different units like days, weeks, months and years to see how they are related.

Days in Week

7 days make a week. In the calendar above, the week starts on a Sunday and ends on a Saturday.

Have a look at the calendar for the month of May. The first full week of this month starts from Sunday, May 1 and ends on Saturday, May 7. This makes a week.

The same days of the week repeat every 7 days. So, if April 5 is a Tuesday, then the next Tuesday will be 5 + 7 = 12 April.

In the same way, if 18 July is a Monday, then the previous Monday 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!

The same day repeat after every 7 days. For example, if today is Monday, it will be Monday again after 7 days.

Days in a Month

We know that there are 12 months in a year. These months have 28, 29, 30 or 31 days.

Did You Know?

You can count the number of days in a month using your fist! The top of the knuckles 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.

Days in a Year

There are 365 days in a 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 a 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. So, the year 2022 ends on 31 December 2022, and the new year begins on 1 January, 2023.

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 form using the format Date Month Year. For example, 18 July 2023 can be written as 18.07.2023.

Example 12: 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 + 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 13: 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

Days in February = 12 days

So, total days = 13 + 12 = 25 days.

Rahul is going to 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.

The date of return = January, .

Do It Yourself 12D

Fill in the blanks.

There are days in a leap year.

b If 03.03.23 is a Friday, then the next Sunday will be on ___________ .

c ___________ is a month with 28 or 29 days.

d 2 years = months

Write these dates in short form — Date.Month.Year.

a 19 November 1996 b 15 August 1947 c 29 July 2023 d 28 February 2004

Find the number of days between the given dates.

a 30 June and 23 July b 5 September and 2 November

c 12.05.2020 and 10 June 2020 d 07.06.2023 and 23.07.2023

Human Rights Day is celebrated every year on December 10 to honor the United Nations General Assemblyʼs adoption of the Universal Declaration of Human Rights. If 10 December is a Friday, on which day will the New Year begin?

Jay takes 2 weeks of leave from school to attend a wedding. The leave begins on 5 March. When will he return to school?

Riya’s birthday is on the third Monday after December 25. When is her birthday if December 25 is on a Monday?

Create a question on finding duration on your own.

Challenge

Ria and Manas went to buy chocolates.

They saw that the chocolate was made on December 12, 2023, and the label said, “Best before 18 months.”

Ria says the chocolate expires on June 12, 2024. Manas says it will expire on June 12, 2025. Who is right, Ria or Manas? Why?

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 at midnight, and p.m. for the next 12 hours of the day, starting at noon.

• Digital clocks use a 24-hour time format. These clocks are commonly used in railway, airline and military timetables.

• 1 hour = 60 minutes. We change hours to minutes by multiplying by 60. We change minutes to hours by dividing by 60.

Math Lab

Setting: In groups

My Monthly Planner

Materials Required: Chart paper, ruler, pencil, coloured pencils or crayons

Method:

• Form 12 groups. Each group is to prepare a creative calendar for one month of the year.

• Draw a table on the paper with 7 columns. Add days of the week at the top, starting with Monday.

• Discuss in your groups and mark birthdays of the students of your class and other important events on this calendar.

• Paste the calendars on the class notice board.

Read the given clocks and write the time.

a b c d

Fill in the blanks with the correct time in a.m. or p.m.

a This morning, Emily woke up at 7 .

b She took 45 minutes to get ready, then it was a.m.

c She had her lunch at 12:30 in the cafeteria with her friends.

Write the correct time using either a.m. or p.m.

a 2 hours after 4:30 in the morning - b 3 hours after 8:45 in the evening -

c 1 hour after 10:00 at night - d 4 hours after 1:20 in the afternoon -

Change the time into 24-hour clock time.

a 06:30 a.m. b 07:55 a.m. c 01:03 p.m. d 09:15 p.m.

Change the time into 12-hour clock times.

a 14:20 hours b 15:45 hours c 21:12 hours d 04:30 hours

Find the duration between the times.

a 08:00 a.m. to 02:45 p.m.

b 07:00 p.m. to 11:30 p.m.

c 02:00 p.m. to 06:45 p.m. d 09:30 a.m. to 05:15 p.m.

Change the following time between hours and minutes.

a 3 hours and 30 minutes to minutes. b 550 minutes to hours and minutes.

c 1 hour and 15 minutes to minutes. d 90 minutes to hours and minutes.

Siya began to colour at 17:30 hours. If she finished colouring after 100 minutes, at what time did she finish?

Meditation helps in feeling calm and focused. Sravan meditates for 30 minutes every day. How many hours does he spend on meditation in 4 days?

The largest multiplex in India is the 16-screen multiplex Mayajaal in Chennai. A film starts at 15:30 hours and ends at 18:15 hours. If the interval is of 20 minutes, what is the duration of the film?

A bus departs from Cochin at 08:45 p.m. and reaches Bangalore at 04:45 a.m. What is the total duration of this journey in minutes?

Today is 10 January. Anil’s birthday is in 45 days. On which date will his birthday be?

Six children took part in a puzzle solving challenge. They all started at 12:30 p.m. and finished as shown on their clocks. Read the time and answer the following questions.

Getu: 1:10 p.m.

Shyam: 1:00 p.m.

Ritu: 1:20 p.m.

Ravi: 1:35 p.m.

a Who took 30 minutes to solve the puzzle?

b Who took more than 1 hour to solve the puzzle?

c How long did Getu take to solve the puzzle?

Jia: 12:45 p.m.

Ahmed: 1:30 p.m.

d Who took the least time in the competition?

Jacob lives on Earth, and his friend Luna lives on a planet called Zaria. On Earth, a month has 30 days. On Zaria, a month has 42 days. Suppose a school year in Zaria lasts 9 Zarian months. If a school year on Earth is 11 months long, which planet’s school year is longer?

How much longer?

Study

The Great Penguin Race!

Emperor penguins are incredible swimmers! A recent study tracked emperor penguin chicks as they waddled from the colony to the ocean for their first swim. The average penguin chick took 52 days and 12 hours to complete this journey.

Scientists want to understand the penguin chicksʼ travel patterns better. They recorded the starting time for a group of chicks and want to predict when they might reach the ocean.

Group Start Time: November 10, 2024, at 2:30 p.m.

1 About how many weeks will it take the penguin chicks to reach the ocean?

a 9 weeks

c 10 weeks

b 7 weeks

d Impossible to determine

2 True or False: The penguin chicks will reach the ocean on December 1, 2024.

3 To find the estimated arrival date at the ocean, we need to add the number of travel days to the .

4 The penguins stopped to bask two weeks after the journey. They stopped some time around November

Cross Curricular

13 Money

Let’s Recall

Money helps us buy things that we need in our daily lives. 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 of India. Similarly, the USA has its dollar, the UK has its pound sterling, and Japan has its yen.

The INR has many currency notes.

Dollar Pound Yen

What notes can we use if we want to buy a water bottle that is priced at ₹80?

We can use 8 notes of ₹10.

We can use 4 notes of ₹20.

We can use 1 note of ₹50, 1 note of ₹20 and 1 note of ₹10.

We can also use 1 note of ₹50 and 3 notes of ₹10.

Let’s Warm-up

Fill in the blanks.

1 We can exchange ₹10 notes for a ₹50 note.

2 We can exchange ₹20 notes for a ₹100 note.

3 For a book that costs ₹120 we can give note of ₹100 and notes of ₹10.

I scored out of 3.

Rupee

Counting Money

Mother took Rohan to the nearby ATM to withdraw some money.

Rohan: Why have we come here, Mom?

Mother: 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?

Mother: 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 learn how to read the amount.

The number on the left of the dot shows rupees.

Express Money in Words

The dot separates the rupees and paise. ₹648.65

The number on the right of the dot shows paise.

When reading the amount in words, we read the left part in rupees, and the right part in paise. So, ₹648.65 can be expressed in words as “six hundred forty-eight rupees and sixty-five paise”.

Example 1: How will you write these amounts in words?

Did You Know?

₹255.84 = Two hundred fifty-five rupees and eighty-four paise.

Rupees Paise

₹157.06 = One hundred fifty-seven rupees and six paise. ₹255.84 Rupees Paise

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

Example 2: Write three hundred fifty-eight rupees and nine paise in figures.

Three hundred fifty-eight rupees = ₹358

Nine paise = 09

Three hundred fifty-eight rupees and nine paise = ₹358.09

Write True or False.

1 ₹354.20 = Three hundred fifty-four rupees and twenty paise. True

2 ₹368.02 = Three hundred sixty-eight rupees and twenty paise.

3 ₹517.26 = Five hundred seventeen rupees and twenty-six paise.

4 ₹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

Remove the dot and ₹ sign.

₹1229.54 = 122954

Step 2

Write paise with the number. Therefore, ₹1229.54 = 122954 paise.

We can also convert paise into rupees. Let us convert 151236 paise into rupees.

Step 1

Remove the word ‘paise’ and put a dot after counting 2 numbers from the right of the number.

Example 3: Convert ₹4236.25 to paise.

Step 1

Remove the dot and ₹ sign.

₹4236.25 = 423625.

Example 4: Convert 745623 paise to rupees.

Step 1

Remove the word ‘paise’ and put a dot after counting 2 numbers from the right of the given number. 745623 paise can be written as 7456.23.

Step 2

Put the sign of ₹ before the number.

Therefore, 151236 paise = ₹1512.36.

Step 2

Write paise with the number. Therefore, ₹4236.25 = 423625 paise.

Step 2

Put the sign of ₹ before the number. Therefore, 745623 paise = ₹7456.23.

Fill in the blanks. In Rupees In Paise

1 ₹635.23

2 ₹ 85205 paise

3 ₹4126.24 412624 paise

4 ₹5386.15

5 ₹ 825632 paise

Do It Yourself 13A

One rupee is represented by a rectangle and one paisa is represented by a circle. Represent ₹3.05 using rectangles and circles.

Express the amount in words.

a ₹154.56

d ₹469.05

Write the amount in numerals.

a Five hundred forty-two rupees and eighty-three paise.

b Six hundred fifty-two rupees and thirty-nine paise.

c Eight hundred sixty-three rupees and seventy-seven paise.

d Nine hundred seventy-four rupees and three paise.

Convert the amount into paise.

a ₹578.24

d ₹945.37

Convert the amount into rupees.

a 63512 paise

d 97456 paise

b 74624 paise

e 112564 paise

₹1247.69

c 84761 paise

f 135489 paise

The Euro (€) is the official currency of 20 out of the 27 member states of the European Union. If 1 Euro = ₹90.78 (as of 12 July 2024) then how many paise are there in one Euro?

Raj’s grandfather showed him a black slate and told him that this slate cost 32 paise in his time.

= 1 paise = 1 anna

If = , how many circles will you draw to show the cost of the slate in anna?

More on Money

Sam and his family went to the restaurant for lunch. Everyone chose their favourite dish and placed the order. At the end of the lunch, Sam’s father asked the waiter for the bill. Sam’s father read the bill and made the payment.

Bills

Let us now learn how to read bills. Here is the bill that Sam’s father received. Real

A bill shows the details of how much we need to pay for items or services. The waiter in the above case gave Sam’s father the bill for the food that the family ate together.

Reading Bills

Restaurant Name

Anna’s Restaurant

No.

1. Idli

Item: Shows the items bought. For example, idly, dosa, etc. were bought.

5. Mango juice

6. Water bottle

S. No. —Serial Number. It tells about the number of items ordered. Here, 6 items were ordered.

Rate: This shows the cost of each item. For example, the cost of 1 plate of idli is ₹30.

Bill No. A156

Bill Date: Jun 12, 2024

Bill Number and Bill Date on which the items were bought.

Cost: This column shows the total cost of the quantity ordered. For example 2 plates of idly cost ₹60.00.

Total: It shows the total amount to be paid. So, ₹801 was to be paid.

We can find the cost of multiple items when the cost of 1 item is given. Similarly, we can find the cost of 1 item when the cost of multiple items is given. This is called the unit cost.

Cost of 1 apple = ₹5

Find the cost of multiple items.

Cost of 7 apples = ₹5 × 7 = ₹35

Cost of 12 pears = ₹84

Find the cost of 1 item or unit cost.

Cost of 1 pear = ₹84 ÷ 12 = ₹7

Remember!

0 after the dot means 0 paise ₹256.00 = ₹256 only.

Example 5: Read the bill given above on page above and answer the questions.

1 What will be the cost of 4 water bottles?

Cost of 1 water bottles = ₹20

Cost of 4 water bottles = ₹20 × 4 = ₹80

2 If Sam’s family ordered 6 more glasses of mango juice, what would be the new bill?

Cost of 6 glasses of mango juice = 6 × ₹50 = ₹300

Total bill = ₹801 + ₹300 = ₹1101

Sudha bought some items at the stationery shop. Look at the bill and answer the questions.

Adarsh Stationery

1 What is the cost of 1 notebook?

Cost of notebooks = ₹180.00

Cost of 1 notebook = ₹180 = ₹

2 What is the total amount that Sudha needs to pay?

Total amount to be paid = ₹80.00 + + ₹180.00 + ___________ + = ₹ .

3 How much will Sudha get if she gives a ₹500 note to the shopkeeper?

Amount of change that Sudha will get = ₹500 – _____________ = ₹ .

Making Bills

We have learnt to read a bill. Let us now learn how to make a bill. Vivek went to the market to buy some fruit and vegetables. He purchased 2 kg of apples at ₹120.00 per kg, 1 kg of bananas at ₹40.00 per kg, 3 kg of potatoes at ₹25.00 per kg, 2 kg of tomatoes at ₹50.00 per kg, 1 kg of onions at ₹29.50 per kg and 1 2 kg of garlic at ₹200.00 per kg. Prepare a bill and find out how much he paid.

Step 1

Make a table as shown. Write the S. No., item names, quantity of each item and rate of each item.

Step 2

Find the amount of each item.

Amount = Quantity × Rate per item.

Therefore, Vivek paid ₹584.50 for all the fruit and vegetables.

Example 6: It is Jane’s birthday! She calls an ice cream parlour and orders some ice creams for the guests:12 vanilla cones at ₹32.00 per cone, 8 chocolate cones at ₹38.00 per cone, 9 strawberry cones at ₹35.00 per cone, and 10 choco bars at ₹40.00 per bar. Prepare a bill for Jane. Bill No. 981/IZ

Aby’s Ice-cream Parlour

Mahi went to a bakery and made the following purchase. Prepare a bill for Mahi.

5 cheese sandwiches for ₹45.00 each.

10 samosas for ₹15.00 each.

12 vegetable rolls for ₹30.00 each. 1 2 kg wafers for ₹130.00 per kg. 1 2 kg cookies for ₹240.00 per kg.

Do It Yourself 13B

Suhaas purchased some items for the new-year celebration. Read the bill and answer the questions. 1

Bill No. 1242/A

Jen’s Party Shop

S. No. Item

Bill Date: 01/11/2024

1. Balloons 8 packets 40.00

2. Party blowers 10 ?

3. Cupcakes 25 30.00

4. Gift bags 15

5. Juice 20 bottles

a What is the cost of 1 party blower?

b What is the cost of 30 gift bags?

c What is the cost of 2 packets of balloons?

d How much did Suhaas spend on the celebration?

The pencil was invented by Nicholas Jacques Contre in 1795. Renuka orders different types of pencils. She orders 2 packs of HB pencils for ₹72.00 each, 3 packs of H pencils for ₹69.00 each, 2 packs of B pencils for ₹90.00 each and 1 pack of dustless chalk for ₹180.00 each. Prepare a bill for Renuka.

Ravi has ₹1000. He purchases 1 2 kg of chillies for ₹40.00 per kg, 1 kg of potatoes for ₹31.50 per kg, 2 kg of oranges for ₹50.00 per kg and 1 4 kg of cherries for ₹120.00 per kg. Prepare a bill for Ravi and find the amount he has left after the purchase. Do you help your parents in buying vegetables and fruits?

Sam has ₹750.00 with him. He goes to a toy shop and buys 1 teddy bear for ₹125, 2 building blocks for ₹175 per block, 3 toy cars for ₹45.00 per car and 1 jigsaw puzzle for ₹215.50.

Prepare a bill for Sam. Will he be able to purchase all the items? If not, how much money does he require to purchase all the items?

Neha gets medicine delivered twice a month for her grandmother. The rate of Vitamin D increased by ₹15 and the price of eye drops decreased by ₹27 in the second delivery. The bill for the first delivery is shown.

a Make the latest bill for the second delivery.

b Neha paid a total of ₹400 for the 2 deliveries. Did she pay the correct amount? If not, what is the difference. Explain your answer.

Expense List

Sam’s father notes down his daily expenses in a diary. An expense is money spent on different items. Shown below is the expense list made by Sam’s father for one day! Remember!

Expense lists can also be used to find the savings. Savings = Total earning – Total expenditure.

An expense list can be made on a weekly, monthly or yearly basis as well!

Example 7: Shown below is Shikha’s weekly expense list. Find the total amount she spent in a week.

Rohan notes down his monthly expenses in a diary. If he had ₹10,000 at the start of the month, find out how much he saved.

Total expenditure = Amount saved = The total amount spent by Shikha = ₹680.25 + ₹1250.00 + ₹742.00 + ₹175.50 = ₹2847.75.

Do It Yourself 13C

Read the expense list given below and write True or False.

Create an expense list for the expenses

a The total expenditure is more than ₹10,000.

b The expenditure on education is more than the expenditure on food.

c The expenditure on transportation is less than the expenditure on clothing.

d The total expenditure is less than ₹12,000.

Kapil earns ₹18,000 per month. Given below is his monthly expenditure. How much money did he save at the end of the month?

Allahabad Museum and Rani Durgavati Museum are famous art museums in India. Below are the biweekly expenses of the two museums. Draw the expense list for the museums. Whose expense is bigger?

Allahabad Museum’s Expense List

Utilities—₹5500, Housekeeping—₹7000, Maintenance—₹4000, Wages—₹9000

Rani Durgavati Museum’s Expense List

Utilities—₹7500, Housekeeping—₹6500, Maintenance—₹5000, Wages—₹9500

Kunal prepared his monthly expense list as shown. He saved ₹2765 at the end of the month. How much money did he have at the start of the month?

Collect information on the monthly expenses at your home and create an expense list.

A small food truck serves idli and dosa for breakfast. The expenses for the ingredients used to make idli and dosa to serve 20 people are given. If they had to serve 40 more people, how much extra expenses would they have to pay?

4 kg urad dal for ₹190.00 per kg, 1 kg rice flakes for ₹32.00 per kg, 5 kg boiled rice for ₹97.00 per kg, 1 kg fenugreek for ₹74.50 per kg, 2 kg toor dal for ₹145.00 per kg

Word Problems on Money

We can solve word problems on money by using 1 or more operations.

Sam and his family spent ₹680 on lunch. On the way back home, they purchased vegetables for ₹512 and groceries for ₹436. Sam’s father had ₹2000 with him. How much money did he have left?

As we need to find the money Sam's father has left, we first need to find out how much money he spent.

Amount spent on lunch = ₹680

Amount spent on vegetables = ₹512

Amount spent on groceries = ₹436

Total money spent = ₹680 + ₹512 + ₹436 = ₹1628

Amount Sam's father has left = Total amount – Amount spent = ₹2000 – ₹1628 = ₹372

Example 8: Each student of a class contributed ₹115 for flood victims. If there are 37 students in the class, what is the total amount of money collected?

Money contributed by each student = ₹115

Total number of students in the class = 37

Total amount of money collected = ₹115 × 37 = ₹4255

Example 9: Suhani bought 3 dresses for ₹999. Megha bought 2 dresses of the same type and gave ₹1000 to the shopkeeper. How much change will Megha get back from the shopkeeper?

Suhani bought 3 dresses for ₹999.

Cost of 3 dresses = ₹999

Cost of 1 dress = ₹999 ÷ 3 = ₹333

Megha bought 2 dresses of the same type.

Cost of 2 dresses = 2 × cost of 1 dress = 2 × ₹333 = ₹666

Megha gave ₹1000 to the shopkeeper, therefore change received = ₹1000 – ₹666 = ₹334.

Mayra’s monthly salary is ₹20,000. She spends ₹5000 on rent, ₹1800 on paying her electricity bill, ₹7000 on food and ₹2000 on miscellaneous expenses each month. The remainder is her savings. How much will Mayra save in a year?

To find the savings in a year, we will first find the savings in a month.

Amount spent on rent = ₹5000

Amount spent on electricity bill = ₹1800

Amount

on food = ₹

on miscellaneous = ₹

Total amount spent in a month = ₹

Savings per month = Money earned – Money spent = ₹

Savings per year = ₹

Did You Know?

Prasanta Chandra Mahalanobis, an Indian expert in data analysis, is called the father of statistics. He was awarded the Padma Vibushan in 1968.

Do It Yourself 13D

A man pays a rent of ₹99 a day. How much will he pay in the month of January?

Madhavi wanted to give birthday treat to her friend. She bought 12 vanilla pastries and 15 plum pastries. A vanilla pastry costs ₹25, and a plum pastry costs ₹35. How much did she pay?

Mysore Zoo is one of the oldest zoos. A ticket for an adult costs ₹60, and a ticket for a child costs ₹30. What would be the cost of tickets for 5 adults and 3 children?

Aryan paid ₹186 for two chocolate bars and 1 ice cream. If the cost of 1 ice cream is ₹60, find the cost of 1 chocolate bar.

Mahi gets ₹385 pocket money per week. How much pocket money does she get per day?

Aarav buys four toys that cost ₹225 each. How much change will he get back if he gives the shopkeeper ₹1000?

Rashmi’s school decided to take the students out for a trip. The school collected a total of ₹5000 from 25 students for the trip. If each student received ₹20 back after the trip, how much money was spent on the trip?

Create a word problem based on the addition of costs.

Challenge

1 Critical Thinking

Sara went to the market with some money in her purse. She bought 2 bracelets for ₹142 each and 2 pairs of earrings for ₹55 each. She got back ₹166 as change. How much money was Sara carrying?

Points to Remember

• While reading the amount in words, we read the left part in rupees and the right part in paise.

• To convert rupees into paise, remove the dot and ₹ sign and write paise.

• To convert paise into rupees, put a dot after counting 2 numbers from the right and put a ‘₹’ sign with the given number.

• A bill shows the details of how much we need to pay for the items or services.

• The expense list is the list of items or services bought and the money spent on them.

• Savings = Total earning – Total expenditure

Math Lab

Setting: Groups of 4

Materials Required:

Experiential Learning & Collaboration

Exploring Expenses

Pen and paper, Price tags for various items and their costs in rupees and paise, Play money

Method:

Set up a market with various items with their price tags in class. Each group gets a fixed amount of money for the purchase.

The groups need to purchase the items with the amount they have in hand.

Each group prepares an expense list for their purchase.

The group that purchases the greatest number of items with the given amount wins!

Chapter Checkup

Three hundred fifty-six rupees and seventy-two paise

Five hundred eighty-two rupees and thirty paise

Seven hundred ninety rupees and fifty-two paise

Convert the given amount to paise. a ₹236.45 b ₹345.78 c ₹598.14 d ₹894.69 e ₹1054.54 f ₹1568.17 g

Fill in the blanks.

a 3651 paise = ₹ b 4865 paise = ₹ c 5631 paise = ₹ d 7856 paise = ₹ e 12567 paise = ₹ f 36574 paise = ₹

Sunita purchased the following items from the stationery shop. Read the bill and answer the questions.

Rima’s Stationery

a What is the cost of 1 packet of pencils? b What is the cost of 10 erasers? c What is the cost of 2 notebooks? d What is the total bill amount?

e If Sunita had ₹800 with her, what amount does she have left?

Rohan wanted to learn how to ride a bicycle. His father rents bicycle for ₹55 a day. How much money will he pay for 2 weeks?

India is the largest manufacturer of cotton clothes. Mahi purchased a famous Bengali Tant saree for ₹895 and a Khadi for ₹1263. She still has an amount of ₹1526 left. How much money did she have initially?

Kunal had ₹5000 with him. He gave ₹1550 to Suhani and divided the rest of the amount equally among his 3 cousins. How much did each cousin get?

Naina is trying to save money for a dress. She saves ₹75 per week. If the dress costs ₹900, how many weeks will it take to save enough to buy the dress? Do you also save your pocket money to buy your favourite things?

A family has monthly earnings of ₹22,000. Given below is their monthly expense list. Read the list and answer the questions.

a How much does the family spend on medicine?

b How much does the family spend on groceries and milk?

c What is the total expenditure of the family?

d What is their monthly saving?

Shown below are the expense lists of Rohan and Mohit. If Rohan earns ₹22,000 whereas Mohit earns ₹23,000 a month, who saves more at the end of the year and by how much?

’s Expense list

Rohit went to the market to purchase groceries. After his purchase he received the following bill. Is the bill correct? If not, make the correct bill.

The ink of a printer in a shop adds black spots to a few places in a bill. Look at the bill given below.

a Find the numbers in the blackened spots.

b Change the quantity and draw the bill to bring the total bill amount to ₹2235.50.

Raju bought some junk from the junk collector. He paid ₹943 and six 50 paise coins.

Five friends went on a trip and paid ₹98 each. Rahul joined them on the trip and paid some money. The total money paid by the 5 friends and Rahul was ₹600. How much money did Rahul pay? 1 2

Statement 1: Raju paid ₹946 in total to the junk collector.

Statement 2: Raju paid for the junk using 6 notes of ₹100, 6 notes of ₹50, 1 note of ₹20, 4 coins of ₹5 and three ₹1 coins.

a Only Statement 1 is true

b Only Statement 2 is true.

c Both statements 1 and 2 are true. d Both statements 1 and 2 are false.

Case Study

Cross Curricular & Value Development

The science teacher conducts an experiment to demonstrate a closed circuit to the students. She uses a small wooden board that costs ₹385.00, a switch costing ₹40.00, 3 small cut wires each costing ₹15.00, a battery that costs ₹196.50, a bulb costing 14200 paise.

1 The cost of the bulb in rupees is .

2 The cost of 2 batteries is a ₹293.00 b ₹199.00 c ₹393.50 d ₹393.00

3 Make a bill for the given items.

4 There are 30 students in a class. The class is divided in groups of 5. How much money is required to buy all the items for the experiment if each group gets one set of equipment?

5 How do you save electricity at home?

14 Data Handling

Letʼs Recall

Don’t we all love going to a toy store? It is so nice to see so many different toys!

The toy stores often have a large selection of toys to choose from.

Have a look at this picture:

Let us count the toys!

2 2 1 4

We can also write:

There are 9 toys in total.

The highest number of toys is teddy bears.

The least number of toys is the football.

There are an equal number of dolls and cars.

From the above data, we can conclude that the toy store has the highest quantity of teddy bears, the lowest quantity of footballs, and equal quantities of dolls and cars.

Letʼs Warm-up

1 There are lions.

2 There is cat.

3 There are giraffes. This is 1 less than the .

4 There are elephants. This is 1 more than the .

5 There are crocodiles. This is more than the .

I scored out of 5.

Organising Data

Rahul has a habit of saving money by collecting coins in his money bank.

Today he opened his money bank to count the coins he had saved.

He started piling the coins, one upon the other, in the following order:

₹1, ₹2, ₹5, ₹10, ₹1, ₹10, ₹2, ₹5, ₹1, ₹10, ₹1, ₹1, ₹1, ₹2, ₹10, ₹5, ₹5, ₹2, ₹1, ₹1, ₹2, ₹10, ₹1, ₹1, ₹2, ₹5, ₹2.

If we have to count each type of coin from this list, it will be hard. Is there an easier way? Let us learn that.

Tables and Tally Marks

We can organise the types of coins that Rahul had in a tally marks table. Let's do it step by step.

Step 1

Write each unique type of coin in a table.

Step 2

Each time you take a coin put a mark next to its type, as shown in the table.

The table below shows the first four entries: ₹1, ₹2, ₹5, ₹10. Coins Tally Marks

₹1

These marks are called tally marks.

Step 4

Finally, put the totals in a new column. Coins Tally Marks Total Coins

₹1 |||| |||| 10

₹2 |||| || 7

₹5

Step 3

Put in the tally marks one by one for each coin.

Note that when the number of tally marks exceeds 4, we use |||| and not |||||.

So, bunches of |||| form 5s. This makes it easier to count and find totals in tables.

Similarly, any kinds of data can also be organised with the help of pictures, graphs, tables, etc. This way of organising data is called data handling.

Data handling is the process by which data is arranged in a systematic way.

Example 1: Rahul visited a zoo on Sunday with his parents. He used tally marks to count and record the number of different types of animals he saw there. Look at the table and answer the following questions.

1 Which animal did he see the greatest number of in the zoo?

Rahul saw the greatest number of elephants.

2 How many tigers and deer were there in the zoo?

In the zoo, there were 3 tigers and 4 deer.

3 What is the difference in the number of leopards and zebras that he saw?

Number of leopards = 12

Number of zebras = 6

The difference in the number of leopards and zebras = 12 – 6 = 6.

At a school, class 4 students voted for their favourite sport. Use the tally chart to answer the following questions.

How many students voted for cricket?

How many students voted for basketball?

7 students voted for football. Represent this in tally marks:

How many students in total voted for their favourite sport?

Do It Yourself 14A

Tick () the tally marks count which shows the number 25. 1

a b

2 Cross Curricular Health Drink Number of People Green tea

A health drink has vitamins, minerals, and other good ingredients that help you stay energised. The tally chart shows the data collected on the types of health drinks people take. Which is the most popular health drink?

a Lemon water b Green tea

c Coconut water d Beetroot juice

Lemon water

Coconut water

The image shown below shows different kitchen appliances. Count the number of each kind of appliance and draw the tally chart.

Nitin has some vegetables at home. Create a tally chart and answer the following questions.

a How many carrots are there?

c How many potatoes are there?

b How many pumpkins and capsicums are there?

d What is the total number of vegetables at home?

The data table shows the number of households that segregate different types of waste. Read the data and answer the following questions.

a Create tally marks to represent the number of households segregating each type of waste.

b How many more households segregate wet waste compared to e-waste?

c If 3 more households start segregating sanitary waste, how many will there be in total?

d Which type of waste has the least number of households segregating it?

e Do you or your family do waste segregation at home? If so, how do you do it? If not, why not?

Read the statements and choose the correct option.

Assertion (A): Consider the marks scored by 10 students in a class test. 9, 9, 8, 10, 10, 8, 7, 9, 6, 8. The tally marks for 8 and 9 marks together can be given as | | | | | |.

Reasoning (R): Each group of tally marks represents five items.

a Both A and R are true, and R is the correct explanation of A.

b Both A and R are true, but R is not the correct explanation of A.

c A is true, but R is false.

d A is false, but R is true.

Pictographs

Drawing and Reading Pictographs

It was Rahul's birthday. He wanted to treat his 10 friends to a pizza party. He told them about the three types pizza toppings that they could choose for their pizza: cheese, vegetable, and paneer. He then conducted a survey to see how many of his friends liked each topping. After that, he wanted to represent this data as pictures. How do you think he would do that? Let us find out.

Creating Pictographs

Rahul created a pictograph based on the above data. Study the pictograph below.

Toppings

Cheese

Paneer

Vegetable

= 1 friend

Number of Friends

Rahul thought of conducting the same kind of survey in his class that has 32 students. The type of pizza liked by each student was recorded as follows.

Cheese, Paneer, Chicken, Mushroom, Vegetable, Paneer, Vegetable, Cheese, Paneer, Chicken, Vegetable, Mushroom, Chicken, Paneer, Cheese, Chicken, Vegetable, Paneer, Cheese, Chicken, Paneer, Vegetable, Cheese, Mushroom, Vegetable, Paneer, Chicken, Chicken, Paneer, Cheese, Mushroom, Chicken.

Let us draw a pictograph for the data.

The key for this pictograph could be:

= 2 students

We use a key to denote the value of the symbol.

The key helps us to represent large values of data easily on a pictograph.

The symbols that are drawn in a pictograph should be of the same size. Error Alert!

Toppings Number of Students Who Like Pizza

Cheese

Paneer

Vegetable

Mushroom

Chicken = 2 students

Did You Know?

Pictographs are often used as road signs as people who speak different languages understand them better.

Example 2: The number of students in a class using different soap brands is shown in this table. Represent the data in the form of a pictograph.

Key = 1 student

Number of students using different soap brands:

Brand

Brand A

Brand B

Brand C

Brand D

Brand E

Number of Students

The number of wall clocks manufactured by a factory in a week is shown below. Represent the data using a pictograph.

One represents 50 wall clocks.

Monday → 300 clocks = 300 ÷ 50 = 6

Tuesday → 350 clocks = 350 ÷ 50 = 7

Wednesday → 250 clocks =

Thursday → 400 clocks =

Friday → 300 clocks =

Saturday → 200 clocks =

Interpreting Pictographs

Number of Clocks

Monday

Tuesday

Wednesday

Thursday

Friday

Saturday

Do you remember that Rahul had to order pizzas on his birthday for 10 friends? He wanted to identify the different types of pizzas and how many pizzas of which type he had to order.

How do you think he would have done that?

= 1 friend

Number of cheese pizzas = 3

Number of paneer pizzas = 3

Number of vegetable pizzas = 4

Example 3: The following picture shows how many cars are washed at the washing centre of a service station in four days of a week. Study the pictograph and answer the given questions.

Day

Number of Cars Washed

Key: = 5 cars

1 On which day are the most cars washed? How many? The most cars are washed on Tuesday. Number of cars washed on Tuesday = 8 × 5 = 40.

2 On which day are the least number of cars washed? How many?

The least number of cars are washed on Monday. Number of cars washed on Monday = 4 × 5 = 20.

The following pictograph represents the number of candy boxes sold by a shop during the Diwali week. Study the data and answer the given questions.

1 How many more boxes were sold on Monday than on Tuesday?

Number of boxes sold on Monday = 5 × 5 = 25

Number of boxes sold on Tuesday = 1 × 5 =

Difference in the number of candy boxes sold on Monday and on Tuesday =

Thus, more candy boxes were sold on Monday than on Tuesday.

2 How many candy boxes were sold in the entire week?

Total candy boxes =

So, the number of candy boxes sold in the entire week =

Do It Yourself 14B

Write if each statement is True or False.

a A pictograph is a way to represent data using images and symbols.

b A pictograph makes the data representation visually interesting and easy to understand.

c Pictographs cannot be used to compare two quantities.

Look at the pictograph and answer the questions.

How many students scored a higher grade than the number of students who scored a C grade?

Here is a pictograph showing the rainfall levels in different cities (named A, B, C, D, E and F) in a year. Look at the pictograph and answer the following questions.

a How much rainfall was recorded in city A and city D?

b Compare the rainfall levels of city B and city E.

Key: = 25 cm rainfall City

The total numbers of tigers in the sanctuaries in four states in India are as follows:

Bihar: 30 Andhra Pradesh: 70

Prepare a pictograph for the animals using the symbol which represents 10 animals. 4

Chhattisgarh: 50 Rajasthan: 60

The pictographs show the number of apples and oranges consumed by students of three government schools in a city. Read the pictographs and answer the questions. Schools Fruit Consumed

Key: = 35 students = 45 students

a In which school is the total number of students that consumed oranges more than those who consumed apples?

b What is the difference in the number of students who consumed apples at schools 1 and 3?

c How many children consumed each kind of fruit at school 2?

Write a new question for the pictographs given in Q5.

Challenge

Nandini collects data on the favourite fruits in her class. She lists the data in a table. Look at the data.

a Create a pictograph for the data using the key of 1 picture = 1 fruit.

b Use the same data to create another pictograph if the key is 1 circle = 2 fruits.

c If the key shows 1 circle = 4 fruits, how many circles will you draw for grapes?

Bar Graphs

Rahul went to watch a cricket match. He noticed that different runs are made in different overs. He wants to compare the runs scored in each over. He wondered if there was any way by which he could compare the runs quickly and effectively.

The chart below shows the runs made in the first 6 overs.

Drawing and Reading Bar Graphs

Rahul realised that data representation by pictograph is not only time-consuming but at times difficult as well.

He wanted to find an easier and better visual form of representation of the data. What could be an easier way to do this? Bar graphs!

Creating Bar Graphs

We know that bar graphs are one of the simplest ways to represent data by using numbers and rectangular bars.

Steps to make the bar graph for the runs scored:

1 Draw horizontal and vertical axes.

2 On the horizontal axis, put the overs.

3 On the vertical axis, put the runs in intervals of 1.

4 Draw the rectangular bars.

Remember!

The title explains what the graph is about (Example: runs per over). The scale is the number that shows the units used (Example: 1 division = 1 run).

Labels show what kind of data is shown (Example: runs, overs).

Example 4: The table below shows information about the marks (out of 50) obtained by five students in a recent test.

Name of the Students Ria Sam Sonia Anita Tania

To create a bar graph for this data, we need to:

• firstly, draw horizontal and vertical lines;

• then, on the horizontal axis, we put the names of the students and on the vertical axis, we put the marks obtained by the students;

• and finally, we take 1 unit length to represent 5 marks.

The table shows the number of students in a class.

Complete the bar graph to represent the following information.

Label of the horizontal axis:

Label of the vertical axis:

Scale – 1 division = 10 children.

Interpreting Bar Graphs

Remember, Rahul wanted to identify the most and least runs scored in an over!

From the graph,

Most runs in an over = the tallest bar = 9 runs

Over in which the most runs are scored = 6th over

Least runs in an over = the shortest bar = 4

Over in which the least runs are scored = 1st over

Sam Sonia Anita Tania

of the Students

Example 5: The following bar graph shows the monthly expenditure of a family on vegetables (in ₹). Read the graph and answer the questions below.

Monthly Expenditure of a Family

1 Which month has the least expenditure?

From the graph, the height of the bar is the lowest in the month of September.

Thus, in September, the family’s expenditure is the least.

2 What is the expenditure in the month of March?

The expenditure in the month of March = ₹1800.

The bar graph shows cars parked in a parking lot on different days of the week. Read the bar graph and answer the following questions.

1 How many cars were parked in the lot on Wednesday?

2 On which day were the most cars parked?

3 Find the difference in the number of cars parked in a lot on Tuesday and Thursday.

Number of cars parked on Tuesday =

Number of cars parked on Thursday =

in the number of cars

in a lot on Tuesday and Thursday =

Do It Yourself 14C

Fuel is a substance that is burned to produce energy. We use this energy to make things work, like cars, buses and airplanes. Read the bar graph and answer the questions.

a Which fuel is used in the greatest number of houses?

b How many houses are using coal as fuel?

The table represents the sale of refrigerators in the first six months of the year.

Draw a bar graph for the given information.

The number of people in various age groups in a village is given in the following table. Draw a bar graph to represent the information (1 unit = 1000).

A person records his family’s monthly expenditure (in ₹) on various things as shown below.

Draw a bar graph to represent the given information.

Children of Grade 1 to Grade 5 participate in the school’s Annual Day Function. The bar graph shows the number of children from each grade who participated. Study the graph and answer the questions.

a From which grade did the lowest number of children participate?

b From which grade did only 60 children participate?

c How many fewer children participated from Grade 3 than Grade 1?

d How many more children participated from Grade 5 than Grade 2?

Create two questions based on the bar graph created in Q4.

Challenge

Sanjay checked his bookshelf and found the books as shown in the bar graph.

a If Sanjay donates one-half of his books, how many books will he have left on his bookshelf?

b Why do you think organising one's books is important?

Pie Charts

The teacher assigns a recycling project to the class that promotes a clean environment.

As part of a recycling project, Aarti and her two friends collect plastic holders.

The three friends collect 40 holders altogether. Aarti wonders how many holders each of them collected.

The number of holders collected is shown using a pie chart or circle graph.

Aarti has collected 1 4 of the total holders while Mohan has collected 1 2 of the total holders.

Number of holders collected by Aarti = 1 4 of 40 = 40 ÷ 4 = 10

Number of holders collected by Rahul = 1 4 of 40 = 40 ÷ 4 = 10

Number of holders collected by Mohan = 1 2 of 40 = 40 ÷ 2 = 20

A pie chart is a pictorial representation of data in the form of a circular chart or pie where the slices of the pie show the fraction of each data category out of the total.

Example 6: The teacher does a survey among students to find their favourite ice-cream flavours. She makes a pie chart with the gathered information. Study the pie chart and answer the questions.

1 Which ice cream is the most popular among the students? − Vanilla

2 What fraction of the students chose chocolate? − 1 4

3 Which flavour(s) is/are preferred by fewer students compared to chocolate? − Blueberry and strawberry

Example 7: Look at the pie chart, Jay and his classmates were asked about their favourite outdoor games. If 72 students were surveyed, how many students voted for running?

Fraction of students who voted for running = 1 3 .

Now, find 1 3 of 72.

1 3 of 72 = 72 3 = 24.

Thus, 24 students voted for running.

The pie chart represents the sale of different sizes of T-shirts in a month. Study the pie chart and answer the questions.

1 Which size is sold the most?

In the circle chart, the size covers the largest part of the circle.

So, the size is sold the most.

Rahul Aarti Mohan

10 + 10 + 20 = 40 holders

BlueberryStrawberry Vanilla Chocolate

2 Which sizes are sold in equal numbers?

In the circle chart, the small and sizes cover equal parts of the circle.

So, small (S) and sized T-shirts are sold in equal numbers.

3 If 600 people buy T-shirts, find the number of people who buy large-sized T-shirts.

Fraction of large-sized T-shirts sold in the month = 1 2 .

So, the number of large-sized T-shirts sold = 1 2 of 600 = .

Do It Yourself 14D

1 Which list is arranged from the smallest to the biggest category of donors?

A charitable company kept track of the amount of funding (money) it received from three categories of donors over a year. The pie chart shows this data.

a Category B, Category C, Category A

b Category B, Category A, Category C

c Category A, Category C, Category B

d Category A, Category B, Category C

Look at the circle chart of the seasons liked by students of class 4 and write True or False for each statement.

a A greater number of children in class 4 like summer than autumn.

b The most preferred season is summer.

c Spring is liked by more students than autumn.

d Winter is liked by more students than spring.

The circle chart shows the games school children like to play. Observe the pie chart and select the correct options.

b The fraction of the children who do not like to play hockey is: i One-half ii One-third iii One-fourth iv Three-fourth 3 Hockey Lawn Tennis

a The fraction of the children who like to play hockey is: i One-half ii One-third iii One-fourth iv Three-fourths

c If there are 60 children, write the number of children who:

i Like to play lawn tennis ii Do not like to play lawn tennis

The pie chart shows the land area occupied by different continents of the world. Look at the pie chart and answer the questions that follow.

a Which continent is the second-largest in terms of area?

b What fraction of the total area is covered by Europe?

c What fraction of the total area is covered by Australia and South America?

d Which continent has an area less than that of South America but more than that of Europe?

Create a question based on the pie chart in Q4.

Challenge

Critical Thinking

c What fraction of the people said that cycling was their favourite activity? 1

Ajay surveyed 100 people on their favourite outdoor activities on Sports Day. Their choices are shown on the pie chart.

a If 10 people chose walking as their favourite activity and 15 people chose skating, how many chose cycling?

b If 10 people changed their vote from walking to skating, what fraction of the people like skating?

Points to Remember

• Data organisation means arranging numbers in order to understand them better.

• Tally marks help to organise data in tables.

• Pictographs use pictures to show numbers.

• Bar graphs show numbers with bars of different heights.

• Bar graphs need a scale, title, axes, and labels.

• Circle or pie charts also show and compare numbers.

Setting: In groups of 4

A Graphical Gathering!

Materials Required: Paper, squared paper, pencil, pen, ruler, origami sheets

Method:

Count the number of people in the family of any five of your friends or classmates and record the data in a table.

Now, take a sheet of squared paper and draw horizontal and vertical lines on it.

Mark all your friends' names on the horizontal axis and the number of family members on the vertical axis.

Cut strips from the origami sheets of appropriate heights and paste them on the squared paper to represent the number of members in each friend’s family.

Chapter Checkup

How many paintings did the painter sell in March?

a 20

b 40

c 10

The tally chart shows the distribution of elephants across zoos in different states in India.

a How many elephants are there in the zoo of Assam?

b Which two states have an equal number of elephants?

The given pictograph shows the number of pumpkins harvested by three friends.

a Who harvested 200 pumpkins?

b How many pumpkins did Hari harvest?

In a school, a survey was conducted about the favourite activities of the students in Grades 3 and 4 as shown below.

How many more students in Grade 4 prefer dancing than the students in Grade 3?

The following tally chart shows the number of bicycles sold during a period of five weeks. Study the tally chart and answer the questions that follow.

Week 1 |||| |||| |||| |||| ||

Week 2 |||| |||| |||| |||| |||| ||||

Week 3 |||| |||| |||| |

Week 4 |||| |||| ||||

Week 5 |||| |||| |||| |||| ||||

This bar graph is prepared by the teacher to find out how many children are interested in going to the Zoo, the National Museum, the Rail Museum, or Adventure Island. Look at the bar graph and answer the questions.

a How many children would like to go to the National Museum?

b How many children would like to go to Adventure Island?

a How many bicycles were sold in the first week?

b How many bicycles were sold in the first and fourth week?

c How many bicycles were sold in the 5 weeks altogether?

c How many more children would like to go to the Zoo than the Rail Museum?

d How many students were surveyed in total?

Maya asked 40 friends to vote for their favourite board game. Complete the circle graph showing the fraction of students who voted for each game.

Create a question based on the pie chart in Q7.

500 students were asked how they travel to school every day. The collected data is shown in the circle graph.

Find the fractions of the students who do not travel to school by car.

Aditya has a bakery. He made the bar graph showing the number of cakes sold over four days last week.

On day 5, he sold twice as many cakes as he sold on day 1. On day 6, he sold half as many cakes as he sold on day 3. Find out how many cakes Aditya sold in total over these 6 days.

Case Study

The Plastic Problem

Plastic pollution is a big problem for our environment. Every year, millions of plastic bottles end up in landfills and oceans, causing harm to wildlife and our planet. Read the bar graph showing the number of plastic bottles that were used at different events last year and answer the questions.

1 How many plastic bottles were used in total at the School Fair and Sports Day combined?

2 Which event used the most plastic bottles? a School Fair b Sports Day c Town Festival d Community Picnic

3 True or False: The Community Picnic used fewer plastic bottles than the Sports Day.

4 The event that used 350 plastic bottles was the .

5 The total number of plastic bottles used at all four events was .

6 Why should we avoid using single-use plastics? Give one alternative to using plastic bags.

Answers

Chapter 1

Letʼs Warm-up 1. 2; thirty-two

2. 8; five hundred forty-eight

3. 800; eight hundred seventy-six

4. 60; four thousand five hundred sixty-three

5. 9000; nine thousand nine hundred fifty-eight

Do It Yourself 1A

1. c 2. c

3. a. 900 and 9 Expanded form = 50000 + 6000 + 900 + 30 + 8

b. 60,000 and 6 Expanded form = 60000 + 5000 + 800 + 90 + 9

c. 5000 and 5 Expanded form = 20000 + 5000 + 400 + 0 + 1

d. 6 and 6 Expanded form = 80000 + 9000 + 300 + 70 + 6

4. a. 17,372; Seventeen thousand three hundred seventy-two

b. 43,890; Forty-three thousand eight hundred ninety

c. 74,065; Seventy-four thousand sixty-five

d. 80,379; Eighty thousand three hundred seventy-nine

5. a. 12,321 b. 34,600 c. 78,005 d. 50,010

6. a. 46,322 b. 50,757 c. 73,061 d. 96,408

7. 13,568; thirteen thousand five hundred sixty-eight.

8. Twenty-eight thousand nine hundred sixty-five; 20000 + 8000 + 900 + 60 + 5

Challenge 1. 8046

Do It Yourself 1B

1. a. Place value of digit 5 is 500000, 8 is 80000, 4 is 4000, 7 is 700, 3 is 30, 6 is 6; Expanded form: 5,00,000 + 80,000 + 4000 + 700 + 30 + 6; face value of digit at lakhs place = 5 b. Place value of digit 7 is 700000, 4 is 4000, 3 is 300, 9 is 90, 1 is 1; Expanded form: 7,00,000 + 4000 + 300 + 90 + 1; face value of digit at lakhs place = 7 c. Place value of digit 3 is 3,00,000, 7 is 70,000, 9 is 900, 4 is 40, 3 is 3; Expanded form: 3,00,000 + 70,000 + 900 + 40 + 3; face value of digit at lakhs place = 3 d. Place value of digit 9 is 900000, 8 is 80000, 5 is 5000, 4 is 400, 1 is 1; Expanded form: 9,00,000 + 80,000 + 5,000 + 400 + 1; face value of digit at lakhs place = 9 2. a. False b. False c. True

3. a. 4,18,222 b. 5,40,147 c. 7,49,021 d. 9,82,902

4. a. 4,18,300 b. 6,20,000 c. 8,05,264 d. 7,20,050

5. a. 1,97,637; One lakh ninety-seven thousand six hundred thirty-seven b. 3,65,021; Three lakh sixty-five thousand twenty-one c. 6,32,845; Six lakh thirty-two thousand eight hundred forty-five d. 8,24,137; Eight lakh twenty-four thousand one hundred thirty-seven 6. 3,84,400 = 3,00,000 + 80,000 + 4000 + 400 + 0 + 0 7. The place value of 1 is 100000, 8 is 80000, 4 is 4000, 8 is 800, 0 is 0, 0 is 0, One lakh eighty-four thousand eight hundred.

Challenge 1. 9,79,436

Do It Yourself 1C

1. a. < b. < c. > d. < e. > f. <

2. a. 14,390 < 37,935 < 40,765 < 79,430 b. 27,880 < 32,860 < 59,573 < 66,773 c. 4,67,943 < 4,88,392 < 8,33,067 < 8,64,853 d. 7,06,583 < 7,20,157 < 7,48,546 < 7,59,404

3. a. 24,567 and 76,542 b. 13,678 and 87,631

c. 1,02,457 and 7,54,210 d. 25,689 and 98,652

4. a. 1,12,479 and 9,97,421 b. 1,00,358 and 8,85,310

c. 1,12,679 and 9,97,621 d. 1,00,789 and 9,98,710

5. Supriyaʼs brother 6. No 7. Pacific Ocean; 36,161 > 27,840 > 23,810 > 23,740 > 18,264

8. Answer may vary. Sample answer. A book company released two popular books. Book A sold 5,23,782 copies, and Book B sold 527,914 copies. Which book sold more copies?

Challenge 1. 2,00,005

Do It Yourself 1D

1. a. 130 b. 570 c. 160 d. 1470 e. 47,120 2. a. 200 b. 1700 c. 7600 d. 2400 e. 23,500 3. a. 2000 b. 7000 c. 35,000 d. 87,000 e. 90,000 4. 24,000 saplings

5. 40,000 km

Challenge 1. 2,56,287

Chapter Checkup

1. a. Place value of digit 4 is 40000, 8 is 8000, 3 is 300, 6 is 60 and 1 is 1. Expanded form of 48,361 = 40000 + 8000 + 300 + 60 + 1 b. Place value of digit 8 is 80000, 7 is 7000, 1 is 100, 0 is 0 and 9 is 9. Expanded form of 87,109 = 80000 + 7000 + 100 + 9 c. Place value of digit 4 is 400000, 5 is 50000, 8 is 8000, 3 is 300, 2 is 20 and 0 is 0. Expanded form = 4,58,320 = 400000 + 50000 + 8000 + 300 + 20 + 0 d. Place value of digit 6 is 600000, 9 is 90000, 2 is 2000, 0 is 0, 4 is 40 and 2 is 2. Expanded form = 6,92,042 = 600000 + 90000 + 2000 + 40 + 2

2. a. 6 × 100000 + 8 × 10000 + 5 × 1000 + 4 × 100 + 8 × 10 + 6 × 1 b. 200000 + 10000 + 3000 + 500 + 40 + 8

3. a. 38,237; Thirty-eight thousand two hundred thirty-seven. b. 4,56,321; Four lakh fifty-six thousand three hundred twentyone. c. 9,70,540; Nine lakh seventy thousand five hundred forty. d. 8,06,399 Eight lakh six thousand three hundred ninety-nine. 4. a. 48,321 b. 1,34,600 c. 78,610 d. 9,10,045 5. a. 82,322 b. 3,50,757 c. 2,73,061 d. 7,96,408 6. a. > b. > c. > d. < e. < f. = 7. a. 37,880 < 42,860 < 46,773 < 69,573; 69,573 > 46,773 > 42,860 > 37,880 b. 23,752 < 24,431 < 25,409 < 28,540; 28,540 > 25,409 > 24,431 > 23,752 c. 64,012 < 64,393 < 64,520 < 64,905; 64,905 > 64,520 > 64,393 > 64,012 d. 93,854 < 3,58,801 < 3,95,701 < 8,26,750; 8,26,750 > 3,95,701 > 3,58,801 > 93,854 e. 7,13,725 < 7,26,890 < 7,58,645 < 7,89,371; 7,89,371 > 7,58,645 > 7,26,890 > 7,13,725 f. 5,80,723 < 5,81,945 < 5,87,206 < 5,88,205; 5,88,205 > 5,87,206 > 5,81,945 > 5,80,723

8. a. 3430, 3400, 3000 b. 6130, 6100, 6000 c. 39,890, 39,900, 40,000 d. 53,480, 53,500, 53,000 9. No. On rounding off 4,85,345 to the nearest 1000, we get 4,85,000. 10. a. Saree 4 < Saree 1 < Saree 3 < Saree 2 b. Saree 1 = ₹ 26,000 Saree 2 = ₹ 97,000 Saree 3 = ₹ 55,000 Saree 4 = ₹ 26,000

11. a. English b. Spanish c. Spanish < Japanese < Chinese < English 12. Answers may vary. Sample answer

Four cities are competing to host the next international sports event. The population of each city is as follows: City A: 534,892; City B: 679,213; City C: 425,678; City D: 796,054

To determine which city is the largest, can you arrange the populations from smallest to largest?

Challenge 1. Place value = 4000; Face value 4 2. 598895

Case Study

1. b 2. Bhutan 3. Iceland 4. Iceland < Maldives < Malta < Suriname < Macao < Bhutan < Guyana < Fiji 5. 9,36,000

Chapter 2

Letʼs Warm-up 1. a. 662 b. 700 c. 436 d. 600 e. 740

2. a. – b. + c. + d. − e. −

Do It Yourself 2A

1. a. 9341 b. 69,553 c. 89,979 2. a. 4979 b. 24,654

c. 13,593 3. a. 6918 b. 18,865 c. 15,146 d. 44,789

e. 41,733 f. 30,723 4. 59,533

5. a. 52,421 b. 57,664 6. 3145 animals

7. 47,029 cars 8. 88,574

Challenge 1. 1032 2372 2348 2790 5138 4720 9858 1340 1008 1782

Do It Yourself 2B

1. a. 76,370 b. 42,112 c. 50,822 d. 12,446 2. a. 4211 b. 24,789 c. 8512 d. 18,005 e. 26,336 f. 70,011

3. a. 16,815 b. 17,481 4. 44,345 5. 16,061

6. 10,010 7. 14,400 species

8. Answers may vary. Sample answer:

Mother bought 2589 balloons for my birthday party. 1294 were blue and the rest were red. How many balloons were red?

Challenge 1. 59,786

Do It Yourself 2C

1. a. 9070 b. 1226 c. 34,890 d. 2441 e. 30,552 f. 27,295

2. 23,750 3. ₹15,319 4. 37,627 people 5. 5124 kg

Challenge 1. Only conclusion I is true.

Do It Yourself 2D

1. a. 3700 b. 6200 c. 5400 d. 2100

2. a. 6000 b. 80,000 c. 11,000 d. 14,000

3. 85,000 sq. km

4. 8000 steps

Challenge 1. Riyan

Chapter Checkup

1. a. 84,640 b. 12,226 c. 92,983 d. 77,497

e. 60,836 f. 37,137 2. 8000, 7343 3. 1000, 489

4. 74,842 5. 26,092 6. 1,00,999 7. 2642

8. 18,200 bricks 9. 3900 km, 1100 km 10. b 11. b

12. Answers may vary. Sample answer: I have 123 marbles, I gave 23 to my friends and bought 30 more from a shop. How many marbles do I have now?

Challenge 1. 70,000 2. Yes

Case Study

1. c. 5500 2. b. 6000 trees 3. False

4. If the village plants 3250 trees per week, after 4 weeks, the forest will have 8000 more trees than it started with, which is 2000 more trees than the original scenario where only 2750 trees were planted each week. This change results in a greater number of trees in the forest.

Chapter 3

Letʼs Warm-up 1. False 2. True 3. False 4. True

5. False

Do It Yourself 3A

1. a. 466 b. 2488 c. 14,007 d. 36,099

2. a. 939 b. 7218 c. 2004 d. 11,632

3. a. 579 b. 2252 c. 3675 d. 37,496

4. 6420 × 2 = 6420

5. Answers may vary. Sample answer: Rohit bought 345 pencils each costing 8 rupees. How much did Rohit pay for the pencils?

Challenge 1. 49

Do It Yourself 3B

1. a. False b. True c. False d. True 2. a. 3796 b. 4080 c. 3051 d. 37,696 e. 74,592 f. 1,12,252 g. 1,18,440 h. 7,85,488 3. a. 4,2,0; 1,4,5 b. 1,9,0; 5,9,4,0,9

4. a. 891 b. 1134 c. 2480 5. 1001 kg

6. 495 sandwiches, 275 juice bottles.

Challenge 1. 51,328

Do It Yourself 3C

1. a. 37,800 b. 1,18, 800 c. 3,19,200 d. 5,63,200

2. a. 1,04,400 b. 35,938 c. 1,09,940 d. 6,00,950

e. 3,13,875 f. 2,49,682

3. 860 × 150 = 1,29,000 and 375 × 344 = 1,29,000. So the missing digit is 4. 4. 2,92,000 L

Challenge 1. Numbers are 20 and 30

Do It Yourself 3D

1. ₹3807 2. 21,665 days 3. 13,237 calories

4. No, the total cost of the tickets won’t fit in their budget.

5. a. 24,160 people b. 22,725 people

6. ₹12,400 7. ₹5535

Challenge 1. 4,516,835 boxes

Do It Yourself 3E

1. a. 2400 b. 46,400 c. 81,000 d. 3,93,600 e. 1,44,500 f. 5,61,000 2. a. 20,000 b. 50,000 c. 1,80,000

3. a. 5500; 10,000 b. 3,43,200; 3,20,000 c. 8,42,800

4. Estimated product = 4400

Actual product = 4730 Actual product is greater than the estimated product 5. 4800 bottles approx.

Challenge 1. No, the estimated amount will not be enough because the estimated amount is less than the actual amount. To buy something, you cannot give less money than the cost of the item.

Chapter Checkup

1. a. 4200 b. 594 c. 1323 d. 51,600

2. a. 2870 b. 1688 c. 24,069 d. 72,808 3. a. 2256 b. 2952 c. 66,573 d. 4151 4. a. 11,844 b. 15,560

c. 24,570 d. 2,32,353 e. 9,28,512 f. 3,26,700

g. 3,19,000 h. 1,28,520 5. a. 71,200 b. 84,700 c. 10,000

d. 3,60,000 6. 758 steps 7. 11,250 m 8. ₹54,660

9. 28,888 kg 10. 770 km 11. 1,54,215 books

12. ₹28,368 13. About 2,00,000 people

Challenge 1. OUEU

2. Answer may vary. Sample answer. Rishi has to pack 429 gift boxes with 10 ceramic cups in each. How many ceramic cups does Rishi need in total?

Case Study

1. ₹410 2. a.false b.true 3. 42,224 kg

4. 11,214 days 5. 36,000 km 6. Answers may vary.

Chapter 4

Letʼs Warm-up 1. 3 2. 2 3. 2 4. 2

Do It Yourself 4A

1. a. False b. True c. True d. True 2. a. 3, 2, 2, 2, 1, 6, 1, 4

b. 8, 9, 2, 7, 4 ,7, 2 3. a. 49 b. 216 c. 321 d. 349 e. 1234

f. 897 4. a. 56 b. 84 c. 184

5. Number of pages left = 4 pages 6. 290 boxes

Challenge 1. 3

Do It Yourself 4B

1. a. 48, 6 b. 9, 86 c. 976, 5 d. 34, 79 e. 7, 894 f. 5, 555

2. a. 51, 17 b. 26, 146 c. 4, 764 3. a. 3, 3, 1, 4, 7, 9

b. 3, 1, 5, 5, 7, 2, 6, 7 4. a. Q = 36; R = 11 b. Q = 204; R = 1

c. Q = 201; R = 27 5. a. 5 b. 18 c. 90

6. 45 7. 20 hours

8. 200 40

Challenge 1. 9 notes

Do It Yourself 4C

1. 14 bottles 2. 40 pages 3. ₹50 4. 635 shelves

5. 135 packets

6. 912 trees

7. Answer may vary. Sample answer. Rani had made 3860 cloth dolls. She sends her dolls to 5 different stores. How many dolls will each store receive?

Challenge 1. Each housekeeper will clean 26 rooms.

Chapter Checkup

1. a. 4 b. 48 c. 1 d. 40 e. 4 f. 8 2. a. Q = 153; R = 0

b. Q = 95; R = 8 c. Q = 45; R = 0 d. Q = 1962; R = 0 e. Q = 78; R = 94 f. Q = 208; R = 0 3. a. 123, 3 b. 41, 2 c. 236, 5

4. a. 365 b. 60 c. 550 5. a. 50 b. 30 c. 140

6. 9 households 7. 41 students 8. ₹110 9. 127 boxes

Challenge 1. 640 boxes 2. a. 16 b. 10 Case Study

1. d. 210 2. a. False b. True 3. 138 4. 45 saplings

Chapter

5

Let’s Warm-up 1. 250 2. 90 – 9 3. 30 + 3 4. 5 5. 930

Do It Yourself 5A

1. Colur 4, 8, 12, 16, 20 in red. 7, 14 in green and 9, 18 in blue.

2. a. 8, 16, 24, 32, 40 b. 10, 20, 30, 40, 50 c. 11, 22, 33, 44, 55 d. 12, 24, 36, 48, 60 e. 13, 26, 39, 52, 65 f. 14, 28, 42, 56, 70 g. 15, 30, 45, 60, 75 h. 16, 32, 48, 64, 80 i. 20, 40, 60, 80, 100 j. 25, 50, 75, 100, 125 3. a. 60 b. 117 c. 99 d. 60 e. 60 f. 125 4. 12, 24, 36, 48, 60, 72

5. The multiples of 8 are: 80, 88, 96, 104, 112, 120, 128, 136, 144

The multiples of 11 are: 77, 88, 99, 110, 121, 132, 143

6. a. Yes b. No c. Yes d. Yes

7. a. 6 and 12 b. 21 and 42 c. 18 and 36 d. 15 and 30 e. 18 and 36 f. 30 and 60

8. No, 27 is a multiple of 9 but not 8.

9. 5, 10, 15, 20, 25, 30, 35, 40, 45, 50

10. 12th and 24th

Challenge 1. 154

2. Answer may vary. Sample answer Do It Yourself 5B

1. 1 row of 24: 1 × 24 = 24; 24 rows of 1: 24 × 1 = 24; 2 rows of 12: 2 × 12 = 24; 12 rows of 2: 12 × 2 = 24; 3 rows of 8: 3 × 8 = 24; 8 rows of 3: 8 × 3 = 24; 4 rows of 6: 4 × 6 = 24; 6 rows of 4: 6 × 4 = 24; Factors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24.

2. a. 1, 2, 7, 14 b. 1, 3, 7, 21 c. 1, 2, 3, 4, 6, 9, 12, 18, 36

d. 1, 3, 13, 39 e. 1, 2, 4, 5, 8, 10, 20, 40 f. 1, 2, 3, 6, 7, 14, 21, 42 g. 1, 2, 3, 4, 6, 8, 12, 16, 24, 48 h. 1, 2, 5, 10, 25, 50

3. a. 1, 3, 9 b. 1, 2, 3, 4, 6, 12 c. 1, 3, 5, 15 d. 1, 2, 3, 5, 6, 10, 15, 30 4. Yes, as 126 leaves no remainder when divided by 18. 5. 12 6. a. 1, 2 b. 1, 2 c. 1, 5 d. 1, 2 7. 6, 3

8. a. False b. False c. False d. False

9. No, 45 has 6 factors and 14 has 4 factors.

10. Five ways

11. Number of Packets

Biscuits in each packet

12. Answers may vary. Sample answer. A teacher has 18 pencils and 24 erasers. She wants to distribute them to students in equal groups, with each group getting the same number of pencils and erasers. What are the factors of 18 and 24 that could represent the group sizes?

Challenge 1. a. 3 + 7 + 11 + 13 b. 4 + 8 + 10 + 12

2. 12 children

Chapter Checkup

1. a. 7, 14, 21, 28, 35 b. 17, 34, 51, 68, 85 c. 18, 36, 54, 72, 90 d. 19, 38, 57, 76, 95 e. 21, 42, 63, 84, 105 f. 23, 46, 69 , 92, 115 g. 30, 60, 90, 120, 150 h. 32, 64, 96, 128, 160

2. a. 10 b. 24 c. 40 d. 28 e. 30 f. 50 g. 22 h. 120

3. a. 1, 2, 5, 10, 25, 50 b. 1, 2, 3, 6, 11, 22, 33, 66

c. 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72 d. 1, 2, 4, 8, 11, 22, 44, 88 e. 1, 2, 7, 14, 49, 98 f. 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120 g. 1, 2, 3, 4, 6, 12, 13, 26, 39, 52, 78, 156 h. 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, 90, 180

4. a. 4, 8, 12, 16, 20, 24, 28 b. 32, 40, 48, 56, 64, 72

c. 14, 28, 42, 56, 70, 84

5. a. 1, 2, 4 b. 1, 2 c. 1, 3 d. 1, 3

e. 1, 5 f. 1, 2, 7, 14 g. 1, 3, 9 h. 1

6. a. No b. Yes c. No d. Yes e. Yes f. No

7. 180 minutes. 8. 20 flowers 1 6 8

9. Answers may vary. Sample answer.

A gardener waters one set of plants every 4 days and another set every 6 days. If both sets are watered today, in how many days will the gardener need to water both sets of plants again on the same day?

Challenge 1. 18 2. 6 groups

Case Study

1. c 2. c

3. 1 × 36, 36 × 1, 2 × 18, 18 × 2, 3 × 12, 12 × 3, 4 × 9, 9 × 4, 6 × 6

4. Answer may vary.

Chapter 6

Letʼs Warm-up 1. 10 2. 2 10 3. 1 10 4. 4 10 5. 3 10

Do It Yourself 6A

1. a. 3 4 b. 3 5 c. 3 8 d. 1 3 2. a. 3 b. 4 c. 7 3. a. 8

d. 6 e. 22 f. 30 g. 15 h. 20 4. 5. 5 10

6. 25 snow leopards 7. 36 pencils

8. 45 students 9. 0 roses

Challenge 1. 6 rice bags

Do It Yourself 6B

1. a. b. c. d.

2. Answer may vary. Sample answer. a. 6 8 , 9 12 , 12 16 , 15 20

b. 4 14 , 6 21 , 8 28 , 10 35 c. 2 10 , 3 15 , 4 20 , 5 25 d. 2 8 , 3 12 , 4 16 , 5 20

3. a. 1 5 b. 1 3 c. 1 13 d. 1 13 e. 1 3 f. 1 7 g. 4 13 h. 1 2 4. a. 6

b. 3, 15 c. 2, 2 d. 7, 7 e. 5, 5 f. 4, 28 g. 6, 30 h. 9, 5

5. 3 8 , 2 41 , 5 24 6. 94 100

7. Sleep Eat Other

8. Answers may vary. Sample answer.

Amir is pouring juice into glasses. He uses 3 5 of a liter of juice for the first batch. He needs to use an equivalent amount of juice for the next two batches but wants to express it in different fractions. Can you help Amir find two equivalent fractions for 3 5 of a liter of juice?

Challenge 1. 26 44

Do It Yourself 6C

1. a. Like fractions b. Like fractions c. Like fractions d. Unlike fractions

2. a. 4 8 > 2 8

g. 4 9 > 3 7 h. 9 16 > 7

5. Watermelons > Oranges > Apples > Bananas 6. Answers may vary. Sample answer.

Three athletes are practicing their swimming. Emma swam 9 15 of the pool length, Sam swam 11 15 , and Mia swam 7 15. Help the athletes by ordering the fractions from least to greatest to see who swam the shortest and longest distances.

Challenge 1. 5 slices

Do It Yourself 6D

1. a. Proper fraction b. Improper Fraction c. Mixed fraction d. Improper fraction e. Mixed fraction f. Proper fraction

2. a. 22 3 b. 31 4 c. 41 6 d. 54 9 e. 83 5 f. 73 7 3. a. 16 3 b. 21 8 c. 44 6 d. 37 9 e. 38 10 f. 47 7 4. a. 2 b. 3 c. 5

6. Answers may vary. Sample answer.

A painter is mixing colors for a mural. He has 4 1 2 cans of blue paint but needs to know how many half-cans of paint he has in total. Write 4 1 2 as an improper fraction?

Challenge 1. 15 12

Do It Yourself 6E

1. a. 2 3 b. 3 4 c. 5 6 d. 2 3 e. 23 5 f. 4

3. 3 4 m 4. 3 8 5. 5 3

7. Answer may vary. Sample answer. Lily baked a batch of cookies and decided to divide them among her friends. She gave 3 8 of the cookies to her friend Sarah, and 5 12 of the cookies to her friend Ben. How much of the batch did Lily give away in total?

Challenge 1. 3 12 , 6 12 , 5 12 , 5 12 respectively

Chapter Ceckup

Challenge 1. Yes 2. 110 17

Case Study

1. Option a 2. Option c

3. Australia, Europe, Antarctica, South America, North America, Africa and Asia.

4. 1 20 5. Answers may vary.

Chapter 7

Let’s Warm-up 1. Square 2. Rectangle 3. Triangle

4. Circle

Do It Yourself 7A

1. a. False b. False c. False 2. a. Ray PQ b. Line CD c. Point Q

3. Points: A, B, C, D, O, P, Q Rays: OA , OB , OC , OQ , OP Lines: BC , PQ

4. Parallel lines

5. Line a and Line b are parallel. Line x and Line y are perpendicular.

6. Answers may vary. Sample answers:

Challenge 1.

There are 4 pairs of parallel lines in this structure.

Do It Yourself 7B

1. a. 4

3. Measures may vary.

4. Not drawn to scale. Sample figures:

a. A B 6 cm b. A B 9 cm c. A B 10 cm

d. A B 14 cm

5. Not drawn to scale, sample figures: 10 cm A B 6 cm A B 10 cm

6. Answer may vary. Sample answer.

Using a ruler, draw a line segment that is 12 centimeters long. Label one endpoint as P and the other as Q

Challenge 1. Answers may vary. Sample answer. Ratna should connect the three points as shown.

Do It Yourself 7C

1. a. Open figure b. Open figure c. Closed figure

d. Closed figure 2. Simple figures - b, d Non-simple figures - a, c

3. Non-simple closed - a, c, f Simple open - b, d, e

4. a. False b. True c. False d. False

5. Answer may vary. Sample answer. B, D, O

6. Figures may vary. Sample figures.

Simple figures Non-simple figures

7. a. Figure a is a polygon. b.

c. d.

8. a. b. c. d.

Challenge 1. Octagon - 1, Pentagon - 2, Quadrilateral/Rectangle - 1, Triangle -1.

Do It Yourself 7D

1. a. centre b. equal c. infinite d. circumference e. one 2.

Centre

Diameter

3. a. iii b. ii c. iv

Radius

4. Not drawn to scale. Sample figures.

a. 2 cm O b. 5cm O c. 6 cm O A d. 3 cm O

5. 3390 km

6. Not drawn to scale. Sample figure. 7 cm 7 cm A

Challenge 1. The radii of both circles have the same length.

Chapter Checkup

1. a. i b. iii c. iv d. iii e. iii

2. B – closed and non-simple, C – open and simple, D – closed and simple, U – open and simple, 7 – open and simple, 0 – closed and simple, 8 – closed and non-simple, S – open and simple

3. a. Rectangle, 4. b. Triangle, 3. c. Pentagon, 5. d. Decagon, 10.

4. Not drawn to scale. Sample figures.

a. 2 cm O b. 4 cm O c. 5cm O d. 3 cm O

5. AB 6. 10 cm 7. 16 m

Challenge 1. 4 cm 2. Circle 3. c. A is true but R is false.

Case Study

1. b. Circle

2. b. To ensure fairness and equal distance

3. Students will construct concentric circles.

4. Answer may vary.

Chapter 8

Let’s Warm-up 1. 3-D shape 2. 2-D shape

3. 2-D shape 4. 2-D shape 5. 3-D shape

Do It Yourself 8A

4. Answers may vary. Sample answer: To reach the bank, Suraj will take the right turn and move straight, then he will take the first right turn and move straight. He will then take a left turn and move straight and stop in front of the bank.

5. Farm road. 6. Answer will vary.

Challenge 1. Bus Stop Super market House 100 m 100 m

Chapter Checkup

1. a. Top b. Front c. Side d. Side e. Top f. Front 2. a. b. c.

4. Side C

5. Answers may vary. Sample answers:

1. a. Top; b. Front; c. side

2. a. Top; b. Side; c. Front

3. a. b. c. d.

4. a. b. c.

Challenge 1. a. 14 b. 5 c. 8

Do It Yourself 8B

Challenge 1.

Do It Yourself 8C

5. Front View Side View Top View 6. Answer may vary. Sample answer. Top

1. a, c and d 2. b, c and f 3. a. b. c. d.

4. Answers may vary. Sample answers: a. b. c. 5.

7. a. Mall b. Park c. Kavya’s d. right 8. a. Rita’s house b. Mina’s house c. 4 roads

9. 10. Answer may vary, sample answer Fancy Store Footwear shop Bakery House School

Challenge 1.

2. Answers may vary. Sample answer:

1. 2 times. 2. a. True b. False c. False d. True e. True

3. a. right b. MG c. Rani’s d. Supermarket e. Brooke

Nathula Tsomgo Lake

Police Check Post

Assembly House

2. a. True b. False

Chogyal Palden

Thedyp Memorial Park

Ropeway Point

3. 4. a. left b. South

Chapter 9

Let’s Warm-up 1. , 2. 11111, 111111

3. , 4. EEEEE, FFFFFF

Do It Yourself 9A

1. a.

c. d.

4. a.

5. Answers may vary.

Challenge 1.

Do It Yourself 9B

1. a.

Challenge 1. Answers may vary. Sample answer.

Do It Yourself 9C

1. a. Yes b. No c. Yes d. No

2. a. b.

c. d.

3. 1 parallelogram, 5 triangles, 1 square

4. Answers may vary. Sample answer:

Challenge 1. No

Do It Yourself 9D

1. a. SCHOOL b. QUIZ c. ATTEMPT d. READ

2. a. 11-5-5-16 9-20 21-16 b. 19-1-22-5 23-1-20-5-18

c. 16-12-1-14-20 20-18-5-5-19

d. 6-1-14-20-1-19-20-9-3 23-15-18-11

e. 18-5-4-21-3-5 18-5-21-19-5 18-5-3-25-3-12-5

Challenge 1. 3510

Do It Yourself 9E

1. a. Yes b. No c. No d. Yes

2. a. b. c. d.

Answer

3. Vertical - A, M, T, U, V, W, Y Horizontal - B, C, D, E, K Both - H, I, O, X

4. a. b. c. d.

Challenge 1.

2. This is an 11-sided polygon with 11 lines of symmetry. Each of these lines divides the polygon into symmetrical halves.

Chapter 10

Letʼs Warm-up 1. a. more than 1 metre

b. less than 1 metre 2. Weighing balance 3. Jug

Do It Yourself 10A

1. a. Ruler b. Measuring tape c. Measuring tape d. Ruler

2. a. 4 cm or 40 mm b. 3 cm or 30 mm c. 10 cm or 100 mm

d. 7 cm or 70 mm 3. a. 900 cm b. 12 m c. 15 m d. 1300 cm

4. a. 1 km 400 m b. 1 km 600 m c. 2 km 200 m d. 1 km 336 m

e. 1 km 475 m f. 1 km 925 m g. 2 km 125 m h. 4 km 250 m

5. 4000 m 6. Leopard = 210 cm, Tiger = 360 cm

7. a. 120 cm b. 1 m 20 cm 8. 6000 metres

Challenge 1. 10 centimetres; 75 seconds

Do It Yourself 10B

1. a. 2 kg 500 g b. 500 g c. 1 kg 500 g 2. a. 3 g b. 7 g

c. 10 g d. 2 g 467 mg 3. a. 5 kg b. 4 kg 500 g c. 6 kg 557 g

d. 9 kg 782 g 4. a. 5000 mg b. 8500 mg c. 4102 mg

d. 15,770 mg 5. a. 17,000 g b. 10,500 g c. 5010 g

d. 15,025 g 6. 4000 g 7. ₹4365 8. 650 pancakes

Chapter Checkup

1. a. b.

c. d. – + ÷ × + ÷ × –

2. a. Yes b. Yes c. No d. No

e. f. g. h.

i. j. k. l.

3. a. 190, 210, 230 b. 144, 155, 166 c. 450, 340, 230 d. 104, 95, 86

4. 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51

a. 3-9-37-39 45-17-37-15-9-37

b. 31-23-1-27-39 1 39-35-9-9

c. 37-1-43-9 31-1-31-9-35 d. 35-9-5-49-5-23-9

5. a. b. c.

6. a. b. c.

7. Infinite 8. Answer will vary.

Challenge 1. 8 flowers 2. LZ

Case Study

1. c. Three 2. 6 dots 3. False 4. MAYA

5. Answer may vary.

9. 85 kg 500 g

10. Answers may vary. Sample answer: The weight of Rama's bag is 10 kg 500 g. What is the weight in grams?

Challenge 1. 100 g

Do It Yourself 10C

1. a. 250 mL b. 1 L 2. a. 3 1 2 L b. 5 L c. 10 1 2 L d. 17 L

3. a. 7 L 650 mL b. 8 L 235 mL c. 9 L 250 mL d. 11 L 300 mL

4. 600 mL 5. 6 L 6. 3 L 7. a. 2 L b. ₹200.

Challenge The oil vendor can measure 4L of oil with 3L and 5L jars:

1. Fill 3L jar, pour into 5L jar (leaving 2L space) leaving it empty.

2. Refill 3L jar, pour into 5L jar until full (1L left in 3L).

3. Discard oil from 5L jar.

4. Pour 1L from 3L jar into 5L jar.

5. Refill 3L jar, pour into 5L jar until empty (fills to 4L)

Chapter Checkup

1. a. (i) 2 m 5 cm (ii) 5 m 7 cm (iii) 7 m 64 cm (iv) 8 m 94 cm

b. (i) 1 km 205 m (ii) 5 km 763 m (iii) 6 km 49 m (iv) 7 km 777 m 2. a. 5 kg 65 g b. 4 kg 600 g c. 7 kg 450 g

d. 10 kg 500 g 3. a. 2500 g b. 4600 g c. 5750 g d. 12,500 g

4. a. 7 L 200 mL b. 8 L 660 mL c. 16 L 250 mL d. 17 L 600 mL

5. 475 mL 6. 12 jars 7. a. ₹120 b. ₹300 c. ₹750

d. ₹1530 8. 10 kg, Lighter 9. 25 handfulls

Challenge 1. No 2. Option a

Case Study

1. 24 angulas 2. Option a 3. 48 angulas

4. True 5. c. 4 hastas

Chapter 11

Let’s Warm-up 1. 250 2. 300 3. 630 4. 5000 5. 6040

Do It Yourself 11A

1. Figure Y 2. a. 16 cm b. 18 cm

c. 24 cm d. 36 cm e. 34 cm f. 18 cm 3. a. 16 cm

b. 20 cm c. 50 m d. 23 cm e. 135 m f. 36 cm

4. a. 8 cm b. 32 m c. 5 cm

5. a. 30 cm b. 70 mm 6. 450 cm 7. a. 600 cm b. 350 cm; Arrangement may vary.

Challenge 1. Answer may vary. Sample answer:

Do It Yourself 11B

1. a. 33 sq. units b. 43 sq. units c. 40 sq. units

2. a. 48 sq. units b. 27.5 sq. units c. 26 sq. units

d. 30 sq. units e. 25 sq. units f. 6 sq. units

3. a. 21 sq. units b. 6 sq. units c. School A

4. a. 7 sq. units b. 20 sq. units

5. Answers may vary. Sample answer:

Challenge 1. a. 6

b. Answers may vary. Sample answer:

The area of all figures will be the same but the perimeter will be different.

Chapter Checkup

1. a. 16 units b. 18 units; Shape a has smaller boundary.

2. 10; 14; 14; B, C 3. a. 12 cm b. 34 m c. 24 m

d. 30 cm e. 25 cm f. 39 m 4. a. 8 cm b. 3 cm

5. a. Figure Q and S b. Figure R and S c. Figure P and T

6. a. 56 sq. units b. 46 sq. units c. 22 sq. units d. 17 sq. units

e. 10 sq. units f. 15.5 sq. units

7. 8. 9 sq. units

9. 27 sq. units 10. 390 m

11. Rishabh covers more distance by 320 m.

Challenge 1. 5 rectangles Rectangle 1: Length = 1, Width = 9; Rectangle 2: Length = 2, Width = 8; Rectangle 3: Length = 3, Width = 7; Rectangle 4: Length = 4, Width = 6; Rectangle 5: Length = 5, Width = 5

Case Study

1. d 2. b 3. Bedroom 1

4. Garden, Bedroom 2 and Kitchen 5. Answer may vary.

Chapter 12

Letʼs Warm-up 1. 360 2. 6 3. 7200 4. less than 5. less than

Do It Yourself 12A

1. a. 5:00 p.m. b. 10 a.m. c. 4:30 p.m. d. 10:00 p.m.

2. a. 8:20 b. 11:47 c. 1:28 d. 4:44 3. 11:30 a.m.

4. a. 11:30 a.m. b. 02:15 a.m. c. 11:59 p.m. d. 6:44 p.m.

5. a. b.

c. d. 6. 12:30 p.m.

Challenge 1. c. 1:33 p.m.

Do It Yourself 12B

1. a. 15:28 hours b. 23:56 hours c. 00:00 hours

d. 23:59 hours 2. a. 10:40 p.m. b. 6:25 p.m. c. 11:24 p.m.

d. 1:03 p.m. 3. 06:00 hours 4. 4:55 p.m./8:35 a.m.

5. 12:45 p.m.

Challenge 1. 2:30 p.m.; Yes, it is still daytime for Samairaʼs friend

Do It Yourself 12C

1. a. 5 hours 40 minutes b. 7 hours 30 minutes

c. 9 hours 20 minutes d. 11 hours 15 minutes

2. a. = b. > c. = d. > 3. a. 12 hours 30 minutes

b. 5 hours 49 minutes c. 6 hours 20 minutes

d. 13 hours 15 minutes 4. 11:55 a.m. 5. 2 hours 45 minutes

6. 17:15 hours

7. Answers may vary. Sample answer: There are 2 shows that run in a theatre. The timings are 10:15 a.m. and 3:30 p.m. Write the timings in the 24-hour format.

Challenge 1. before 6:00 p.m.

Do It Yourself 12D

1. a. 366 b. 12.03.23 c. February d. 24

2. a. 19.11.96 b. 15.08.47 c. 29.07.23 d. 28.02.04

3. a. 23 days b. 58 days c. 29 days d. 46 days 4. Saturday

5. 19 March 6. 15 January

7. Answers may vary. Sample answer: Anaʼs mother left to go shopping at 4:15 p.m. and was back at 6:15 p.m. How much time was Anaʼs mother out of the house?

Challenge 1. Manas since 18 months from December 12, 2023 is June 12, 2024.

Chapter Checkup

1. a. 6:42 b. 7:29 c. 10:21 d. 5:14 2. a. a.m. b. 7:45

c. p.m. 3. a. 6:30 a.m. b. 11:45 p.m. c. 11:00 p.m.

d. 5:20 p.m. 4. a. 06:30 hours b. 07:55 hours

c. 13:03 hours d. 21:15 hours 5. a. 2:20 p.m. b. 3:45 p.m.

c. 9:12 p.m. d. 4:30 a.m. 6. a. 6 hours 45 minutes. b. 4 hours

30 minutes c. 4 hours 45 minutes. d. 7 hours 45 minutes. 7. a. 210 minutes b. 9 hours 10 minutes c. 75 minutes. d. 1 hour

30 minutes 8. 19:10 hours 9. 2 hours

10. 2 hours 25 minutes 11. 480 minutes 12. 24 February

Challenge 1. a. Shyam b. Ravi c. 40 mins d. Jia

2. Zaria; 48 days

Case Study

1. b. 7 weeks 2. False 3. start date 4. 24th

Chapter 13

Let’s Warm-up 1. 5 2. 5 3. 1; 2

Do It Yourself 13A

1. ₹3.05

2. a. One hundred fifty-four rupees and fifty-six paise

b. Two hundred seventeen rupees and eighty-five paise.

c. Three hundred ninety six rupees and forty-eight paise

d. Four hundred sixty-nine rupees and five paise.

e. Six hundred seventy-nine rupees and twenty-one paise.

f. Seven hundred forty-eight rupees and forty-nine paise.

3. a. ₹ 542.83 b. ₹652.39 c. ₹ 863.77 d. ₹974.03

4. a. 57,824 paise b. 64,712 paise c. 84,625 paise d. 94,537 paise e. 1,01,548 paise f. 1,24,769 paise

5. a. ₹635.12 b. ₹746.24 c. ₹847.61 d. ₹974.56 e. ₹1125.64 f. ₹1354.89 6. 9078 paise

Challenge 1. 8 circles

Do It Yourself 13B

1. a. ₹15 b. ₹3000 c. ₹80 d. ₹3320

2. Bill No. 1356/B

Hindustan Pencils Pvt Ltd

S. No.

Date: 02/07/2024

b. Total amount in the first delivery = ₹257

Total amount in the second delivery = ₹245

Total amount = ₹257 + ₹245 = ₹502

No, the total bill amount is ₹502 but she had paid ₹400. Neha has to pay ₹102 more for the bill.

Do It Yourself 13C

1. a. True b. False c. True d. True

2. S. No. Detail Amount(₹)

3. ₹5025

4. Allahabad Museum's Expense List

S. No. Detail Amount(₹) 1. Utilities 5500.00 2. Housekeeping 7000.00 3. Maintenance 4000.00 4. Wages 9000.00

Rani Durgavati Museum's Expense List

S. No. Detail Amount(₹) 1. Utilities 7500.00 2. Housekeeping 6500.00

3. Maintenance 5000.00

4. Wages 9500.00

The total expenditure of Rani Durgavati Museum is bigger.

5. ₹12,995 6. Answers may vary

Challenge 1. Extra expenses to be paid for 40 people = ₹1641.50 + ₹1641.50 = ₹3283.00

Do It Yourself

13D

1. ₹3069 2. ₹825 3. ₹390 4. ₹63 5. ₹55

6. ₹100 7 ₹4500

8. Answers may vary. Sample answer. The cost of a plastic chair is ₹459.00. The cost of small plastic table is ₹972.00. What is the total spent on chair a and table together?

Challenge 1. Sara was carrying ₹560 in her purse.

Chapter Checkup

1. Two hundred thirty-five rupees and forty-five paise; 356.72; 582.30; Six hundred forty-eight rupees and forty-seven paise; 790.52; Nine hundred seventy-eight rupees and sixty-five paise. 2. a. 23,645 paise b. 34,578 paise c. 59,814 paise d. 89,469 paise e. 1,05,454 paise f. 1,56,817 paise g. 1,86,458 paise h. 2,04,565 paise 3. a. ₹36.51 b. ₹48.65 c. ₹56.31 d. ₹78.56 e. ₹125.67 f. ₹365.74 4. a. ₹40 b. ₹100 c. ₹100 d. ₹730 e. ₹70

5. ₹770 6. ₹3684 7. ₹1150 8. 12 weeks 9. a. ₹1500 b. ₹8500 c. ₹18,400 d. ₹3600

10. Rohan, ₹500

11. Food Store

S. No.

12. ₹13,500

13. a. 1; ₹283.00; ₹75.00; ₹3543.25

b. Answer may vary. Sample answer.

Date: 28/11/2023

14. Answer may vary. Sample answer. Rishi had spent ₹585 on a toy car and Ria had spent ₹751 on a barbie doll. How much more did Ria spend than Rishi?

Challenge 1. Option a 2. ₹110 Case Study

1. ₹142.00 2. Option d

3. S.No. Item Cost per Item No. of Item Amount

1. Small wooden board ₹385.00 1 ₹385.00 2. Switch ₹40.00 1 ₹40.00

3. Cut wires ₹15.00 3 ₹45.00 4. Battery ₹196.50 1 ₹196.50

5. Bulb ₹142.00 1 ₹142.00 Total Cost ₹7808.50

4. ₹4851.00 will be required to buy experiment materials for all the groups.

5. Answers may vary Chapter 14

Letʼs Warm-up 1. 5, most 2. 1, least 3. 4, lions

4. 3, crocodiles 5. 2, cat Do It Yourself 14A 1. b 2. b

4. a. 8 b. 19 c. 6 d. 33

a. Type of Waste Number of Households Amount(₹) Wet Waste 8 |||| ||| Dry Waste 5 |||| Sanitary Waste 3 ||| E-waste 2 ||

b. 6 c. 6 d. E-waste e. Answer may vary

6. Answer may vary Sample answer. How many more tea pots are there than mixer grinders?

Challenge 1. Option d Do It Yourself 14B

1. a. True b. True c. False 2. Option c 3. a. 175 cm, 50 cm b. B > E

4. Key: 1 = 10 Tigers

States Number of Tigers

Bihar

Andhra Pradesh

Chhattisgarh

Rajasthan

5. a. School 2 and school 3 b. 245 students c. 160 students

6. Answer may vary. Sample answer. How many students consumed oranges in school 3?

Challenge

1. a. Solution/Answer: 1. Key: 1 = 1 fruit

Fruit Number of Fruits

Apples

Bananas

Cherries

Grapes

b. Key: 1 = 2 fruits

Fruit Number of Fruits

Apples

Bananas

Cherries

Grapes

c. two and one-fourth circles

Do It Yourself 14C

1. a. LPG b. 10

5. a. Grade 3 b. Grade 4 c. 10 students d. 30 students

6. Answer may vary. Sample answer.

a. How much did the family spent on education?

b. What is the total expenditure of the family?

Challenge 1. a. 13 books b. Answer may vary.

Do It Yourself 14D

1. Option b 2. a. False b. False c. False d. True

3. a. Three-fourths b. One-fourth c. i. 15 children ii. 45 children

4. a. Africa b. 7 100 c. 17 100 d. Antarctica

5. Answer may vary. Sample answer.

Which continent is the smallest in terms of area?

Challenge a. 25 people b. 1 4 c. 1 4

Chapter Checkup

1. Option a 2. a. 9 b. Andra Pradesh and Maharashtra

3. a. Madhav b. 275 pumpkins

4. 5 students 5. a. 22 bicycles b. 37 bicycles

c. 107 bicycles 6. a. 10 children b. 15 children c. 15 children d. 50 children

7. Favourite Board Game

Number of People

Scale:1

8. Answer may vary. Sample answer. How many friends vote for ludo?

Challenge 1. 3 4 2. 51 cakes Case Study

1. b. 750 2. c. Town Festival 3. True

4. Community Picnic 5. 1700 6. Answer may vary.

About the Book

Imagine Mathematics seamlessly bridges the gap between abstract mathematics and real-world 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.

Key Features

• Let’s Recall: Introductory page with a quick recall of concepts learnt in previous grades

• Real Life Connect: Introduction to a new concept related to real-life situations

• Examples: Solved problems showing the correct method and complete solution

• Do It Together: Guided practice for learners with clues and hints to help solve problems

• 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

• Challenge: Critical thinking questions to enhance problem-solving and analytical thinking skills

• Case Study: Scenario-based questions to apply theory to real-life situations

• QR Codes: Digital integration through the app to promote self-learning and practice

About Uolo

Uolo partners with K-12 schools to provide technology-enabled learning programs. We believe that pedagogy and technology must come together to deliver scalable learning experiences that generate measurable outcomes. Uolo is trusted by over 15,000+ schools across India, Southeast Asia and the Middle East.

ISBN 978-81-979482-9-9

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