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Book Title: Imagine Mathematics Teacher Manual 4
ISBN: 978-81-984519-0-3
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 and 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. These teacher manuals are designed to be indispensable companions for educators, providing well-structured guidance to make teaching mathematics both effective and enjoyable. With a focus on interactive and hands-on learning, the manuals include a variety of activities, games, and quizzes tailored to enhance conceptual understanding. By integrating these engaging strategies into the classroom, teachers can foster critical thinking and problem-solving skills among students. Moreover, the resources emphasise creating an enriched and enjoyable learning environment, ensuring that students not only grasp mathematical concepts but also develop a genuine interest in the subject.
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 Mathematics and join us in nurturing the next generation of thinkers, innovators and problem-solvers. Embark on this exciting journey with us and let Imagine Mathematics be a valuable resource in your educational adventure.
Numbers up to 8 Digits 1
Imagine Mathematics Headings: Clear and concise lessons, aligned with the topics in the Imagine Mathematics book, designed for a seamless implementation.
Alignment
C-1.1:
C-4.3:
Numbers up to 8 Digits 1
2
Imagine Mathematics Headings
Place Value, Face Value and Expanded Form
Indian and International Number Systems
Comparing and Ordering Numbers
Numbers up to 8 Digits 1
Learning Outcomes: Clear, specific and measurable learning outcomes that show what students should know, understand, or do by the end of the lesson.
Learning Outcomes
Students will be able to:
Rounding–off Numbers
Learning Outcomes
Students will be able to: write the place value, face value, expanded form and number names for numbers up to write numbers up to 8 digits in the Indian and International number system. compare numbers up to 8 digits and arrange them in ascending and descending order. round off numbers up to 8 digits to the nearest 10, 100 and 1000.
Alignment to NCF
Numbers up to 8 Digits 1
C-1.1: Represents numbers using the place value structure of the Indian number system, numbers, and knows and can read the names of very large numbers
C-4.3: Selects appropriate methods and tools for computing with whole numbers, such as 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 the history of its transmission to the world, and its modern impact on our lives and in all technology
Let’s Recall
write the place value, face value, expanded form and number names for numbers up to 8 digits. write numbers up to 8 digits in the Indian and International number system. compare numbers up to 8 digits and arrange them in ascending and descending order. round off numbers up to 8 digits to the nearest 10, 100 and 1000.
Alignment to NCF
Place Value, Face Value and Expanded Form
Recap to check if students know how to write the place value, expanded form and number 6-digit numbers.
Indian and International Number Systems
Ask students to solve the questions given in the Let’s Warm-up section.
Comparing and Ordering Numbers
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
Alignment to NCF: Learning Outcomes as recommended by NCF 2023.
Vocabulary
Rounding–off Numbers
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
Learning Outcomes
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
Let’s Recall
expanded form: writing a number as the sum of the values of all its digits order: the way numbers are arranged estimating: guessing an answer that is close to the actual answer rounding off: approximating a number to a certain place value for easier calculation
Teaching Aids
Recap to check if students know how to write the place value, expanded form and number names for 6-digit numbers.
Students will be able to: write the place value, face value, expanded form and number names for numbers up to 8 digits. write numbers up to 8 digits in the Indian and International number system. compare numbers up to 8 digits and arrange them in ascending and descending order. round off numbers up to 8 digits to the nearest 10, 100 and 1000.
Ask students to solve the questions given in the Let’s Warm-up section.
Vocabulary
Chart papers with empty place value chart drawn; Buttons; Beads; Bowls; Digit
Alignment to NCF
4
5 number cards with a 7-digit or 8-digit number written on them; Two bowls with number 8-digit numbers in one bowl and rounded-off places in another bowl
expanded form: writing a number as the sum of the values of all its digits order: the way numbers are arranged estimating: guessing an answer that is close to the actual answer rounding off: approximating a number to a certain place value for easier calculation
Teaching Aids
Let’s Recall: Recap exercises to check the understanding of prerequisite concepts before starting a topic.
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-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
Let’s Recall
Chart papers with empty place value chart drawn; Buttons; Beads; Bowls; Digit cards; Bowls with 5 number cards with a 7-digit or 8-digit number written on them; Two bowls with number cards having 8-digit numbers in one bowl and rounded-off places in another bowl
Recap to check if students know how to write the place value, expanded form and number names for 6-digit numbers. Ask students to solve the questions given in the Let’s Warm-up section.
Vocabulary
expanded form: writing a number as the sum of the values of all its digits order: the way numbers are arranged estimating: guessing an answer that is close to the actual answer rounding off: approximating a number to a certain place value for easier calculation
Teaching Aids
Chart papers with empty place value chart drawn; Buttons; Beads; Bowls;
cards; Bowls
Learning Outcomes
Students will be able to: write the place value, face value, expanded form and number names for numbers up to 8 digits. write numbers up to 8 digits in the Indian and International number system. compare numbers up to 8 digits and arrange them in ascending and descending order. round off numbers up to 8 digits to the nearest 10, 100 and 1000.
Numbers up to 8 Digits 1
Numbers up to
Alignment to NCF
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-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
QR Code: Provides access to digital solutions and other interactive resources.
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
Learning Outcomes
Let’s Recall
Recap to check if students know how to write the place value, expanded form and number names for 6-digit numbers.
Ask students to solve the questions given in the Let’s Warm-up section.
Vocabulary
Students will be able to: write the place value, face value, expanded form and number names for numbers up to 8 digits. write numbers up to 8 digits in the Indian and International number system. compare numbers up to 8 digits and arrange them in ascending and descending order. round off numbers up to 8 digits to the nearest 10, 100 and 1000.
Alignment to NCF
expanded form: writing a number as the sum of the values of all its digits order: the way numbers are arranged estimating: guessing an answer that is close to the actual answer rounding off: approximating a number to a certain place value for easier calculation
Teaching Aids
Vocabulary: Helps to know the important terms that are introduced, defined or emphasised in the chapter.
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-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
Chart papers with empty place value chart drawn; Buttons; Beads; Bowls; Digit cards; Bowls with 5 number cards with a 7-digit or 8-digit number written on them; Two bowls with number cards having 8-digit numbers in one bowl and rounded-off places in another bowl
Let’s Recall
expanded form and number names for numbers up to 8 digits. Indian and International number system. arrange them in ascending and descending order. nearest 10, 100 and 1000.
place value structure of the Indian number system, compares whole names of very large numbers and tools for computing with whole numbers, such as mental pencil calculation, in accordance with the context zero in India and the Indian place value system for writing numerals, world, and its modern impact on our lives and in all technology
Teaching Aids: Aids and resources that the teachers can use to significantly improve the teaching and learning process for the students.
Chapter: Numbers up to 8 Digits
Chapter: Numbers up to 8 Digits
Recap to check if students know how to write the place value, expanded form and number names for 6-digit numbers.
Ask students to solve the questions given in the Let’s Warm-up section.
Vocabulary
expanded form: writing a number as the sum of the values of all its digits order: the way numbers are arranged estimating: guessing an answer that is close to the actual answer rounding off: approximating a number to a certain place value for easier calculation
Teaching Aids
Chart papers with empty place value chart drawn; Buttons; Beads; Bowls; Digit cards; Bowls with 5 number cards with a 7-digit or 8-digit number written on them; Two bowls with number cards having 8-digit numbers in one bowl and rounded-off places in another bowl
Place Value, Face Value and Expanded Form Imagine Maths Page 2 Learning Outcomes
write the place value, expanded form and number names for given in the Let’s Warm-up section.
the sum of the values of all its digits close to the actual answer to a certain place value for easier calculation
Place Value, Face Value and Expanded Form
Learning Outcomes
Teaching Aids
Imagine Maths Page 2
Students will be able to write the place value, face value, expanded form and number names for numbers up to 8 digits.
Students will be able to write the place value, face value, expanded form and number names for numbers up to 8 digits.
Chart papers with empty place value chart drawn; Buttons; Bowls; Digit cards
Activity
Teaching Aids
Chart papers with empty place value chart drawn; Buttons; Bowls; Digit cards
Activity
Instruct the students to work in small groups. Distribute the teaching aids among the groups. Keep number cards (0–9) in a bowl. Pick a card from the bowl and say the digit aloud along with a place, for example “3 in the thousands place.” Instruct the groups to place as many buttons as the digit in that place on the place value chart. Repeat this to form a 7-digit or 8-digit number. You can repeat digits in more than 1 place. Once the 7-digit or 8-digit number is formed, have each group record the expanded form and number names in their notebooks. Discuss the face value and place value of a few digits in the class. Ask the students to say the number names aloud.
Extension Idea
Activity: A concise and organised lesson plan that outlines the activities and extension ideas that are to be used to facilitate learning.
Instruct the students to work in small groups. Distribute the teaching aids among the groups. Keep number cards (0–9) in a bowl. Pick a card from the bowl and say the digit aloud along with a place, for example “3 in the thousands place.” Instruct the groups to place as many buttons as the digit in that place on the place value chart. Repeat this to form a 7-digit or 8-digit number. You can repeat digits in more than 1 place. Once the 7-digit or 8-digit number is formed, have each group record the expanded form and number names in their notebooks. Discuss the face value and place value of a few digits in the class. Ask the students to say the number names aloud.
Ask: In a number 1,67,48,950, if we interchange the digit in the ten thousands place with the digit in the crores place, then what is the difference in the place values of the digits in the ten thousands place?
Extension Idea
Say: On interchanging the digits, the new number is 4,67,18,950. Difference in the place values = 40,000 – 10,000 = 30,000.
Ask: In a number 1,67,48,950, if we interchange the digit in the ten thousands place with the digit in the crores place, then what is the difference in the place values of the digits in the ten thousands place?
Extension Idea: A quick mathematical-thinking question to enhance the critical thinking skill.
Indian and International Number Systems Imagine Maths Page 5
Say: On interchanging the digits, the new number is 4,67,18,950. Difference in the place values = 40,000 – 10,000 = 30,000.
chart drawn; Buttons; Beads; Bowls; Digit cards; Bowls with number written on them; Two bowls with number cards having rounded-off places in another bowl
Learning Outcomes
Students will be able to write numbers up to 8 digits in the Indian and International number system.
Indian and International Number Systems Imagine Maths Page 5
Teaching Aids
Learning Outcomes
Chart papers with empty place value chart drawn; Buttons; Beads
Students will be able to write numbers up to 8 digits in the Indian and International number system.
Activity
Teaching Aids
Chart papers with empty place value chart drawn; Buttons; Beads
Activity
Arrange the class in groups of 4, with each group split into teams of 2 students. Distribute the teaching aids among the groups. Guide the students to form 7-digit or 8-digit numbers on the chart using buttons and beads as in the previous lesson. Ask them to place buttons for the Indian number system on one chart and beads for the International number system on the other. The two teams should record the expanded form of the numbers using commas in their notebooks and number names. Ask the groups to find the similarities and differences between the two systems. Discuss the answers with the class.
Extension Idea
Arrange the class in groups of 4, with each group split into teams of 2 students. Distribute the teaching aids among the groups. Guide the students to form 7-digit or 8-digit numbers on the chart using buttons and beads as in the previous lesson. Ask them to place buttons for the Indian number system on one chart and beads for the International number system on the other. The two teams should record the expanded form of the numbers using commas in their notebooks and number names. Ask the groups to find the similarities and differences between the two systems. Discuss the answers with the class.
Ask: How many lakhs are there in 10 million?
Answers: Answers, provided at the end of each chapter, for the questions given in Do It Together and Think and Tell sections of the Imagine Mathematics book.
Answers
Say: To find out, we divide 10,000,000 by 100,000. So, there are 100 lakhs in ten million.
Extension Idea
Ask: How many lakhs are there in 10 million?
Say: To find out, we divide 10,000,000 by 100,000. So, there are 100 lakhs in ten million.
Period Plan
The teacher manuals corresponding to Imagine Mathematics books for Grades 1 to 8 align with the recently updated syllabus outlined by the National Curriculum Framework for School Education, 2023. These manuals have been carefully designed to support teachers in various ways. They provide recommendations for hands-on and interactive activities, games, and quizzes that aim to effectively teach diverse concepts, fostering an enriched learning experience for students. Additionally, these resources aim to reinforce critical thinking and problem-solving skills while ensuring that the learning process remains enjoyable.
In a typical school setting, there are approximately 180 school days encompassing teaching sessions, exams, tests, events, and more. Consequently, there is an average of around 120 teaching periods throughout the academic year.
The breakdown of topics and the suggested period plan for each chapter is detailed below.
Chapters No. of Periods
1. Numbers up to 6 Digits 11
2. Addition and Subtraction 14
3. Multiplication 12
Break-up of Topics
Place Value, Face Value and Expanded Form
5-digit Number Names
Place Values and Expanded Form in 6-digit Numbers; Face Value in 6-digit Numbers
6-digit Number Names
Comparing Numbers
Ordering Numbers
Forming Numbers
Rounding off Numbers
Revision
Simple Vertical Addition
Simple Horizontal Addition
Adding with Regrouping
Word Problems on Adding Numbers
Subtraction without Regrouping
Subtraction with Regrouping
Word Problems on Subtracting Numbers
Simplifying Numbers
Word Problems on Simplifying Numbers
Estimating the Sum
Estimating the Difference
Revision
Horizontal Method
Multiplying by Expanding the Bigger Number
Vertical Method
Properties of Multiplication; Multiplication by 10, 100 and 1000
Using the Column Method
Quick Multiplication
Multiplication by a 3-digit Number
Word Problems
Estimating the Product
Revision
4. Division 10
5. Multiples and Factors 8
Division by 1-digit Numbers
Properties of Division
Dividing by Tens
Division of Numbers up to 4 Digits
Dividing by Multiples of 100s and 1000s
Estimating the Quotient
Word Problems
Revision
Using Skip Counting, Using Multiplication
Finding Common Multiples
Finding Factors Using Factor Tiles; Finding Factors Using Multiplication
Finding Factors Using Division Finding Common Factors
Prime and Composite Numbers
Revision
Halves, Quarters and Thirds
Equivalent Fractions
Comparing and Ordering Like Fractions
6. Fractions 10
7. Lines and 2-D Shapes 6
8. Representing 3-D Shapes 5
9. Patterns and Symmetry 11
Comparing and Ordering Unlike Fractions with Same Numerators
Proper and Improper Fractions
Adding Like Fractions
Subtracting Like Fractions
Revision
Points, Rays and Lines
Line Segments
Open and Closed Figures, Simple vs Non-simple
Polygons
Circles and Their Parts; Constructing Circles
Revision
Views of Objects
Nets of 3-D Shapes
Maps
Revision
Repeating Patterns
Rotating Patterns
Growing and Reducing Patterns
Number Patterns
Tiling Patterns and Tessellations
Coding and Decoding Patterns
Symmetry
Reflection
Revision
Measuring Length
Converting Between Units of Length
Measuring Weight
Converting Between Units of Weight
Measuring Capacity
Converting Between Units of Capacity
Revision
Finding Perimeter Using a String
Finding Perimeter Using Squared Paper
Finding Perimeter of Polygons
Finding Area Using Squared Paper;
Area of Irregular Shapes
Revision
Reading Time on a Clock
Reading a 24-hour Clock
Changing 12-hour to 24-hour Clock Time;
Changing 24-hour Clock to 12-hour Clock Time
Elapsed Time
Time in Days, Weeks, Months and Years
Revision 13. Money
Express Money in Words
Conversion Between Rupees and Paise
Reading Bills
Making Bills
Expense List
Word Problems on Money
Revision
Organising Data
Creating Pictographs
Interpreting a Pictograph
Creating Bar Graphs
Interpreting Bar Graphs
Pie Charts
Revision
Numbers up to 6 Digits 1
Learning Outcomes
Students will be able to:
write the place value, face value and expanded form for 5-digit numbers and represent them using the correct periods.
write the numer names for 5-digit numbers. write the place value, face value and expanded form for 6-digit numbers and represent them using the correct periods.
write the number names for 6-digit numbers. compare numbers up to 6 digits.
arrange numbers up to 6 digits in ascending or descending order. form 6-digit numbers using the given digits with/without repeating digits. round off numbers up to 6 digits to the nearest 10, 100 or 1000 using a number line.
Alignment to NCF
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-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
Let’s Recall
Recap to check if students know how to find the place value of a digit in a number. Ask students to solve the questions given in the Let’s Warm-up section.
Vocabulary
round off: to write the closest 10, 100 or 1000 number instead of the actual number
Teaching Aids
Place value charts; Number cards for 5-digit numbers and their corresponding number name cards; Number cards (0–9) in yellow, blue and red; Comma cards; Number cards for 6-digit numbers and their corresponding number name cards; 6-rod abacus; Numbers cards for 2-digit, 3-digit and 4-digit numbers
Chapter: Numbers up to 6 Digits
Place Value, Face Value and Expanded Form
Learning Outcomes
Imagine Maths Page 2
Students will be able to write the place value, face value, expanded form and represent 5-digit numbers using the correct periods.
Teaching Aids
Place value charts
Activity
Help students recall how the numbers are to be placed in a place value chart. Divide the class into small groups and provide each group with a place value chart. Assign a 5-digit number to each group (for example 32487).
Instruct the groups to write the digits of the number in their proper place value. Ask students to write the face values of the digits in the number. Also ask students to place the commas in the place value chart in the correct period. Next, ask them to write the expanded form of the number. For example: 32487 = 30000 + 2000 + 400 + 80 + 7.
Give students more numbers to practise and ask them to write the place value, face value and expanded form.
Extension Idea
Ask: Suppose the digits in the tens and thousands places in the number 15,483 are interchanged. Will the place value and face value of the digits change?
Say: After interchanging the digits, the new number is 18,453. The place value of the digits would change but not the face value.
5-digit Number Names
Learning Outcomes
Students will be able to write the number names for 5-digit numbers.
Teaching Aids
Number cards for 5-digit numbers and their corresponding number name cards
Activity
Imagine Maths Page 4
Demonstrate how to write the number name of a 5-digit number. Create a set of number cards with the corresponding names for 5-digit numbers.
Ask students to work in groups. Distribute 10 number cards and their corresponding number name cards to each group.
Instruct the groups to shuffle the cards and arrange them facing down on a big table. Ask 1 student from the group to flip over 2 cards at a time. The number card should match the card with its corresponding number name. If the number matches its name, the player keeps the cards and plays another turn. If not, then the next student gets the chance. Continue this till all the students in a group get a turn. The student who gets the maximum number of matches wins.
Place Value, Face Value and Expanded Form
Learning Outcomes
Students will be able to write the place value, face value, expanded form and represent 6-digit numbers using the correct periods.
Teaching Aids
Number cards (0–9) in yellow, blue and red; Comma cards
Activity
Divide the class in groups. Distribute the number cards and comma cards to each group. Give a 6-digit number to each group. For example: 723560.
Assign a different coloured card to the digits in each period. For example, use yellow cards for the ones period, blue for the thousands period and red for the lakhs period.
Ask students to use the cards to show the number assigned to each group.
Ask students to write the place value and face value of the number in their notebooks.
Now, ask students to use the comma cards and place them in the correct period.
Instruct students to write the number using commas and its expanded form, in their notebooks.
Ask: What is the expanded form of a number which has six thousands less than 84,236?
Say: Six thousands less than 84,236 = 84,236 – 6000 = 78,236. So, the expanded form of 78,236 = 70,000 + 8000 + 200 + 30 + 6.
6-digit Number Names Imagine Maths
Learning
Outcomes
Students will be able to write the number names for 6-digit numbers.
Teaching Aids
Number cards for 6-digit numbers and their corresponding number name cards
Activity
Demonstrate how to write the number name of a 6-digit number. Create a set of number cards with numbers and their corresponding names for 6-digit numbers.
Ask students to work in groups. Distribute 10 number cards and their corresponding number name cards to each group.
Instruct the groups to shuffle the cards and scatter all the cards on a big table facing down. Ask 1 student from the group to flip over 2 cards at a time. The number card should match the card with its corresponding number name. If the number matches its name, the player keeps the cards and plays another turn. If not, then the next student gets the chance. Continue this till all the students in a group get a turn.
The student who gets the maximum number of matches wins.
Comparing Numbers
Learning Outcomes
Students will be able to compare numbers up to 6 digits.
Teaching Aids
6-rod abacus
Activity
Ask students to form 3 groups. Provide each group with two 6-rod abacuses. Instruct students to represent a 6-digit number on one abacus (say, 4,14,358) and another 6-digit number on the other abacus (say, 4,56,125).
Ask them to first compare the beads in the lakhs place of each abacus, then say if the number of beads is the same and finally compare the beads in the ten thousands place. Explain that if the number of beads is different, then the number with more beads in the same place is greater than the other number. Ask students to write the comparison of the numbers in their notebooks using the appropriate signs. Give students more 6-digit numbers to compare to practise.
Extension Idea
Ask: If the hundreds and the ten thousands digits in the number 8,15,367 are interchanged, which will be greater: the original number or the new number?
Say: The original number is 8,15,367 and the new number is 8,35,167. 8,35,167 > 8,15,367. Thus, the new number is greater than the original number.
Ordering Numbers
Learning Outcomes
Students will be able to arrange numbers up to 6 digits in ascending or descending order.
Teaching Aids
Number cards for 6-digit numbers (Same as used in lesson above)
Activity
Take the students to the playground and make a number line on the ground (make sure it covers a broad range of 6-digit numbers).
Shuffle and arrange the number cards with 6-digit numbers, face down, in the playground.
Ask students to form groups. Instruct 1 student from each group to pick 1 number card and place it at the correct position on the number line. Then, the next student from each group should pick a card and place it on the number line. Continue this activity until all the students in the group get a chance.
Back in the classroom, give some numbers to the students and ask them to arrange them in ascending and descending order.
Forming Numbers
Learning Outcomes
Students will be able to form 6-digit numbers using the given digits with/without repeating digits.
Teaching Aids
Number cards (0–9) in yellow, blue and red
Activity
Ask students to form groups. Distribute number cards (0–9) to each group.
Instruct students to form the largest 6-digit number using the number cards when no digit is repeated. Hint that they should use the 6 largest digits to make the required number.
Ask students to form the smallest 6-digit number when no digit is repeated and then form the smallest 6-digit number when 1 digit can be repeated once.
Ask students to note down the numbers so formed in their notebooks.
Ask questions like: What is the smallest 6-digit number that can be formed using any of the digits from 0 to 9 when no digit is repeated?
Extension Idea
Ask: Use the digits, 7, 9, 0 and 2 to make the smallest 6-digit number. The digits 0 and 2 can be repeated once. Say: The given digits are 7, 9, 0, 2 and the digits 0 and 2 can be repeated once. So, the smallest number will be 2,00,279.
Rounding off Numbers
Learning Outcomes
Students will be able to round off numbers up to 6 digits to the nearest 10, 100 or 1000 using a number line.
Teaching Aids
Number cards for 2-digit, 3-digit and 4-digit numbers
Activity
Take the students to the playground and draw large number lines on the ground for numbers 10 to 100, 100 to 1000 and 1000 to 10,000. Divide the class into groups. Make number cards with different numbers.
Call 1 student from each group and give a number card to each of them. Ask them to go and stand near the number on the number line. Instruct the remaining students to identify the nearest 10, 100 or 1000 and round off the numbers. If the number rounds up, the students standing on the number line can jump forward and if the number rounds down, they can jump backwards. Repeat this activity for all the students in the group by assigning different number cards to each student.
Extension Idea
Ask: A number when rounded to the nearest 1000 gives 156,000. How many such numbers are possible? Say: Numbers that can be rounded to the nearest 1000 as 156,000 lie between 155,500 and 156,499.
Answers
1. Place Value, Face Value and Expanded Form
Think and Tell
Yes, the place value and face value of the ones digit in any number is always the same.
Do It Together
(× 1) 5 4 3 1 9
50000 4000 300 10 9
Face value of the number in the thousands place = 4.
Mark the period in the correct position: 6 3 , 1 0 9
Number name is: Sixty-three thousand one hundred nine
3. Place Value, Face Value and Expanded Form
Do It Together
Form
= 100000 + 70000 + 2000 + 900 + 9
The face value of the digit in the ten thousands place = 7
4.
6-digit Number Names
Do It Together Lakhs
Mark the period in the correct position: 9 , 0 5 , 7 2 1
The number name is nine lakh five thousand seven hundred twenty-one
5. Comparing Numbers
Do It Together
Both the numbers are 6–digit numbers and the digits in the lakhs place are the same.
Moving to the next place = ten-thousands place 5 > 2
Hence 7,53,278 > 7,22,271.
6.
Ordering Numbers
Do It Together
3,68,109 > 1,76,902 and 75,045 > 60,438
So, the numbers can be sequenced as 60,438; 75,045; 1,76,902; 3,68,109.
The ascending order is: 60,438 < 75,045 < 1,76,902 < 3,68,109.
The descending order is 3,68,109 > 1,76,902 > 75,045 > 60,438.
7. Forming Numbers
Think and Tell
14,807; 14,870; 14,780; 87,140; 87,410
Think and Tell
Largest number: 9,99,999
Smallest number: 1,00,000
Do It Together
1. 5-digit number with no repetition of digits
Largest number: 9 6 5 4 1
Smallest number: 1 4 5 6 9
2. 6-digit number with exactly 1 repeating digit
Largest number: 9 9 6 5 4 1
Smallest number: 1 1 4 5 6 9
8. Rounding-off Numbers
Do It Together
Rounding off to the nearest 100. 90,135 is between 90,100 and 91,200, but is closer to 90,100 So, 90,135 can be rounded off to 90,100.
Rounding off to the nearest 1000 90,135 is between 90,000 and 91,000, but is closer to 90,000 So, 90,135 can be rounded off to 90,000.
Addition and Subtraction 2
Learning Outcomes
Students will be able to:
add two or more 4-digit or 5-digit numbers vertically (without regrouping). add two or more 4-digit or 5-digit numbers vertically (with regrouping). add two or more 4-digit or 5-digit numbers horizontally (without regrouping). solve word problems on adding 4-digit or 5-digit numbers. subtract a 4-digit or 5-digit number from a 5-digit number (without regrouping). subtract a 4-digit or 5-digit number from a 5-digit number (with regrouping). solve word problems on subtracting 5-digit numbers. solve expressions that have both addition and subtraction of 5-digit numbers. solve story problems that have both addition and subtraction of 5-digit numbers. estimate the sum of two 5-digit numbers. estimate the difference of two 5-digit numbers.
Alignment to NCF
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-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
Let’s Recall
Recap to check if students know addition and subtraction of numbers up to 3 digits. Ask students to solve the questions given in the Let’s Warm-up section.
Vocabulary
estimate: approximate value of a number or a sum
Place value blocks of 1000s, 100s, 10s and 1s; Place value charts; Counters; Word problem sheets; 5-rod abacus; Sets of 3 puzzle interlocking cards with expressions on 2 cards and the answer on 1 card; Number lines showing 10 divisions drawn on a sheet of paper
Chapter: Addition and Subtraction
Simple Vertical Addition
Learning Outcomes
Imagine Maths Page 20
Students will be able to add two or more 4-digit or 5-digit numbers vertically (without regrouping).
Teaching Aids
Place value blocks of 1000s, 100s, 10s and 1s
Activity
Ask the students to work in groups. Distribute the place value blocks to each group. Instruct them to add the numbers 1317 and 3181 using the blocks.
Then, ask them to write the 2 numbers, one below the other, by writing the digits as per their place value. Instruct them to first add the digits in the ones place, then the tens, the hundreds and finally the thousands place. Ask them to write the sum below the addend digits.
Instruct the students to check the vertical addition result with the result they got using the place value blocks. Repeat the activity with the numbers 58,124 and 1525.
Teacher Tip: If resources are unavailable, cut-outs showing the different place value blocks can be distributed.
Simple Horizontal Addition
Learning Outcomes
Imagine Maths Page 21
Students will be able to add two or more 4-digit or 5-digit numbers horizontally (without regrouping).
Teaching Aids
Place value charts; Counters
Activity
Ask the students to work in groups. Distribute 3 place value charts to each group.
Instruct them to show the numbers 53,423 and 1404 on separate place value charts by placing the same number of counters as the face value of the digits in each number.
Then, ask them to count the total number of counters in the ones column of both place value charts and write the digit for the total in the ones place of the third place value chart to show the addition. Instruct them to repeat these steps for the counters in the other places of both numbers. Explain that the result they got is the sum of the 2 numbers.
Repeat the activity with the numbers 45,214 and 23,675. Once the students are done, ask them to replace the counters with digits and then add.
Ask questions like: Will you get the same answer if the digits on both grids are written vertically?
Adding with Regrouping
Learning Outcomes
Imagine Maths Page 22
Students will be able to add two or more 4-digit or 5-digit numbers vertically (with regrouping).
Teaching Aids
Place value blocks of 1000s, 100s, 10s and 1s
Activity
Ask the students to work in groups. Distribute place value blocks to each group. Instruct them to add the numbers 1825 and 2386 using the blocks. Tell them that if the sum of the number of blocks in any place exceeds 9, they should regroup to the next higher place. Explain that ten 1s make one 10, ten 10s make one 100 and ten 100s make one 1000.
Then, instruct the students to write the numbers one below the other with the digits as per their place value. Ask them to first add the ones digits, then the tens, the hundreds and finally the thousands. Tell them that if the sum of the digits in any place exceeds 9, they should regroup to the next higher place.
Instruct the students to check the vertical addition result with the result they got using the place value blocks. Repeat the activity with the numbers 48,635 and 32,478.
Extension Idea
Ask: When do you get a 6-digit number on adding two 5-digit numbers?
Say: When the sum of the digits in the ten thousands place of the 2 numbers is more than 9, we get a 6-digit number.
Word Problems on Adding Numbers
Learning Outcomes
Students will be able to solve word problems on adding 4-digit or 5-digit numbers.
Teaching
Aids
Word problem sheets
Activity
Imagine Maths Page 23
Distribute the word problem sheets to the students. Instruct them to read the problem: A construction company used 14,567 bricks to build one part of a building and 8345 bricks for another part. How many bricks did they use in total? Discuss what they understand about the question. Ask them to write down what they know, what they need to find and then solve to find the answer using the column method. Once the students are done, discuss the answer with the whole class.
Ask questions like: How many bricks would the construction company have used if they used 256 more bricks in another part?
Extension Idea
Ask: Create your own word problem on adding 5234, 1845 and 1056.
Say: There can be multiple word problems on adding three 4-digit numbers. One such problem could be: A school purchased 5234 notebooks, 1845 pens and 1056 markers for the classrooms. How many school supplies did they buy in total?
Subtraction without Regrouping
Learning Outcomes
Students will be able to subtract a 4-digit or 5-digit number from a 5-digit number (without regrouping).
Teaching Aids
5-rod abacus
Activity
Ask the students to work in groups. Distribute an abacus to each group.
Instruct them to subtract 4230 from 75,650 using the abacus by first showing the bigger number on it and then removing the same number of beads as the smaller number.
Then, ask them to write the numbers, one below the other, by writing the digits as per their place value. Instruct them to first subtract the digits in the ones place, then the tens, hundreds, thousands and finally the ten thousands place. Ask them to write each difference below the 2 digits they subtract with.
Instruct the students to check the vertical subtraction result with the result they got using the abacus. Repeat the activity to subtract 12,356 from 34,466.
Teacher Tip: If abacuses are unavailable, students can draw an abacus in their notebooks and show the beads by drawing circles.
Ask questions like: Can the difference of 2 numbers be greater than the minuend? Why or why not?
Subtraction with Regrouping
Learning Outcomes
Students will be able to subtract a 4-digit or 5-digit number from a 5-digit number (with regrouping).
Teaching Aids
5-rod abacus
Activity
Ask the students to work in groups. Distribute an abacus to each group.
Instruct the groups to subtract 5257 from 98,512 by first showing the bigger number on the abacus and then taking away the same number of beads as the smaller number. Tell them that if the number of beads to be taken away in any place is more than the number of beads available to subtract from, then they should remove 1 bead from the next higher place and add 10 beads to the lower place.
Then, instruct the students to write the numbers one below the other with the digits lined up as per their place value and then subtract. Tell them that if the digit in the minuend is less than the digit in the subtrahend, they should regroup from the next higher place.
Instruct the students to check the vertical subtraction result with the result they got using the abacus. Repeat the activity with the numbers 81,135 and 32,127.
Extension Idea
Ask: Can you think of any 2 numbers which give the difference as 13,254 on subtraction?
Say: We can find the numbers by first taking any number. If the number taken is greater than 13,254, subtract 13,254 from it to get the second number. If the number taken is less than 13,254, add it to 13,254 to get the second number. Let us take the number 24,165. 24,165 – 13,254 = 10,911. Hence 24,165 and 10,911 are two such numbers which give 13,254 on subtraction.
Word Problems on Subtracting Numbers
Learning Outcomes
Students will be able to solve word problems on subtracting 5-digit numbers.
Teaching Aids
Word problem sheets; 5-rod abacus
Activity
Distribute the word problem sheets and 5-rod abacuses.
Instruct the students to read the problem: The distance between two cities is 3456 miles. A car has travelled 2198 miles so far. How many more miles does it need to travel to reach the destination? Discuss what they understand about the question. Ask them to write down what they know and what they need to find in their notebooks and then solve to find the result using the abacus.
Once the students are done, discuss the answer with the whole class.
Have them solve 1 more problem with 5-digit numbers.
Extension Idea
Ask: Create your own word problem where you need to subtract a 4-digit number from a 5-digit number.
Say: There can be multiple word problems on subtracting a 4-digit number from a 5-digit number. One such problem could be: Emily had ₹15,678 in her bank account. She withdrew ₹4849 to pay her house rent. How much money did she have left in her account?
Simplifying Numbers
Learning Outcomes
Students will be able to solve expressions that have both addition and subtraction of 5-digit numbers.
Teaching Aids
Sets of 3 puzzle interlocking cards with expressions on 2 cards and the answer on 1 card
Activity
Begin by taking an expression, like 25,635 + 31,458 – 23,147, that involves both addition and subtraction. Explain that we need to first add the 2 numbers and then subtract the third number from the sum.
Ask the students to work in groups. Distribute the sets of puzzle interlocking cards to each group.
Explain that 2 expressions have the same answer and that students have to join those 2 expressions with their answer. Ask them to use their notebooks to find the result. The group that solves all the puzzles first, wins.
Ask questions like: Does the result change if we perform the subtraction operation first?
Word Problems on Simplifying Numbers
Learning Outcomes
Students will be able to solve story problems that have both addition and subtraction of 5-digit numbers.
Teaching Aids
Word problem sheets
Activity
Distribute the word problem sheets to the students. Instruct them to read the problem: Suhani saved ₹5078 in January, spent ₹2845 in February and then saved ₹3256 in March. What is her total savings at the end of March? Discuss what they understand about the question. Ask them to write down what they know and what they need to find in their notebooks. Discuss what operation needs to be carried out first. Ask them to perform the operation using vertical addition or subtraction. Once they have solved the problem, discuss the answer with the whole class.
Extension Idea
Ask: Create your own word problem for 12,562 + 14,258 – 19,365.
Say: Multiple word problems can be written for this expression. One such problem could be: A clothing company produced 12,562 shirts in a month and 14,258 shirts in another month. If the company sold 19,365 shirts in the 2 months, how many shirts was it left with?
Learning Outcomes
Students will be able to estimate the sum of two 5-digit numbers.
Teaching Aids
Number lines showing 10 divisions drawn on a sheet of paper
Activity
Recall how to round off 2-digit or 3-digit numbers to the nearest 10 and 100. Divide the class into small groups. Distribute the number line sheets to each group.
Instruct the students to find the estimated sum of 24,145 and 65,712 by first marking the 2 numbers on the number line, then finding the nearest thousand for each number, marking the rounded off number and finally, adding the rounded off numbers.
Then, ask the students to write the question and its answer in their notebooks and discuss the digit that needs to be checked while rounding off to the nearest 1000.
Ask questions like: Is the sum after rounding off to the thousands and to the ten thousands the same? Give the students 1 more question to solve using both methods.
Learning Outcomes
Students will be able to estimate the difference of two 5-digit numbers.
Teaching
Aids
Number lines showing 10 divisions drawn on a sheet of paper Activity
Ask the students to work in groups. Distribute the number line sheets to each group.
Instruct the students to find the estimated difference of 24,362 and 15,236 by first marking the 2 numbers on the number line, then finding the nearest thousand for each number, marking the rounded off number and finally, subtracting the rounded off numbers.
Then, ask the students to find the answer in their notebooks and compare the answers that they got using both methods.
Repeat the process with more 5-digit numbers.
Answers
1. Simple Vertical Addition Do It Together
So, 65,234 + 2345 = 67,579
2. Simple Horizontal Addition
Do It Together
3. Adding with Regrouping Do It Together
4. Word Problems on Adding Numbers Do It Together
What do we know?
Chocolates produced in January = 13,456
Chocolates produced in February = 24,257
What do we need to find?
Number of chocolates made in two months. Solve to find the answer.
So, the home baker made 37,713 chocolates in two months.
To round off a number to the nearest 10,000, check the digit in the thousands place to decide whether to round up or round down. If the thousands are 5000 or more, round up. If they are 4999 or less, round down.
Do It Together
Rounded off to the nearest thousand
Round off to the nearest thousand
The estimated sum = 14,000 + 28,000 = 42,000
So, the estimated sum of 13,567 and 28,082 is = 13,567 + 28,082 = 42,000
11. Estimating the Difference
Do It Together 78,111 21,991 Rounded off to the nearest thousand
Round off to the nearest thousand
The estimated difference = 78,000 – 22,000 = 56,000
The actual difference of 78,111 – 21,991 = 56,120.
Multiplication 3
Learning Outcomes
Students will be able to:
multiply a 3-digit or 4-digit number by a 1-digit number by writing them horizontally. multiply a 3-digit or 4-digit number by a 1-digit number by expanding the bigger number. multiply a 3-digit or 4-digit number by a 1-digit number by writing them vertically. apply the properties of multiplication while multiplying 2 numbers. multiply a 3-digit or 4-digit number by a 2-digit number by writing them vertically. multiply 2-digit numbers using quick multiplication tricks. multiply a 3-digit or 4-digit number by a 3-digit number by writing them vertically. solve word problems on multiplying numbers up to 4 digits by numbers up to 3 digits. estimate the product of 2 numbers.
Alignment to NCF
C-1.3: Understands and visualises arithmetic operations and the relationships among them, knows addition and multiplication tables at least up to 10 x 10 (pahade) and applies the four basic operations on whole numbers to solve daily life problems
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
Let’s Recall
Recap to check if students know how to use repeated addition to multiply. Ask students to solve the questions given in the Let’s Warm-up section.
Vocabulary
multiplicand: number to be multiplied multiplier: number by which we multiply
Teaching Aids
product: number obtained from multiplication
Price list of 3 items; Number cards with 3-digit, 4-digit and 1-digit numbers; Bill template showing 4 grocery items with the cost per kg and the quantity bought; 1 example card for each property; Property name cards; Multiplication cards; Bingo cards; Puzzle cards; Answer cards; Chart paper; Glue stick; Word problem cards
Chapter: Multiplication
Horizontal Method
Learning Outcomes
Imagine Maths Page 40
Students will be able to multiply a 3-digit or 4-digit number by a 1-digit number by writing them horizontally.
Teaching Aids
Price list of 3 items
Activity
Divide the class into groups of 4. Provide each group with a price list of 3 items.
Ask 1 student in each group to pose as a shop owner and the remaining students as customers. Instruct 1 student in the group to find the total price of rice, the second student to find the total price of dal and the third student to find the total price of wheat. Tell them that they need to find the product using horizontal multiplication. The shop owner should check the calculations. Ask the students to write the answers in their notebooks.
Extension Idea
Ask: What will be the total cost of all the items if, along with the listed items, 2 L of oil is also purchased at ₹110 per litre?
Say: Cost of 3 kg of rice = ₹399, cost of 2 kg of dal = ₹282, cost of 3 kg of pulses = ₹669, cost of 2 L of oil = ₹220. So, the total cost = ₹399 + ₹282 + ₹669 + ₹220 = ₹1570.
Multiplying by Expanding the Bigger Number
Learning Outcomes
Imagine Maths Page 41
Students will be able to multiply a 3-digit or 4-digit number by a 1-digit number by expanding the bigger number.
Teaching Aids
Number cards with 3-digit, 4-digit and 1-digit numbers
Activity
Demonstrate the multiplication of a 3- or 4-digit number by a 1-digit by expanding the bigger number. Ask the students to work in groups. Distribute the number cards to each group (cards with bigger numbers like 543, 4122, 3548 and smaller numbers like 5, 6, 7, 8, 9).
Ask the students to expand the bigger number and find the product using the boxes, as shown in the example for 326 × 7.
Discuss the answers and ask the students to write them in their notebooks.
326 × 7
300 × 7 20 6
Learning Outcomes
Students will be able to multiply a 3-digit or 4-digit number by a 1-digit number by writing them vertically.
Teaching Aids
Bill template showing 4 grocery items with the cost per kg and the quantity bought
Activity
Demonstrate how to multiply a 3- or 4-digit number by a 1-digit number vertically. Ask the students to work in groups of 4. Distribute the bill templates (example below) to each group.
Instruct the students to complete the bill for 4 items and also find the total bill amount. Instruct each student from the group to calculate the total cost of 1 item and then add the amounts to get the total bill amount. Invite some groups to come up and present their bills and discuss the answers.
Extension Idea
Ask: Which purchase will cost more: 5 kg of walnuts at ₹449 per kg or 3 kg of hazelnuts at 634 per kg?
Say: 5 kg of walnuts at ₹449 per kg will cost more.
Properties of Multiplication;
Multiplication by 10, 100 and 1000
Learning Outcomes
Students will be able to apply the properties of multiplication while multiplying 2 numbers.
Teaching Aids
1 example card for each property; property name cards
Activity
Discuss the properties of multiplication in class. Divide the class into groups. Distribute the property cards, number cards and symbol cards to each group.
Ask the students to match the property name cards with the example card for the same property. Ask the students to write 1 example of their own for each property in their notebooks. Ask questions such as: Are the properties for multiplication and addition same or different?
Multiplication by 1 4 × 1 = 4
Extension Idea
Ask: What is the product of 98 and 1000?
Say: To multiply by 1000, just add 3 zeroes to the right of 98. So, the product of 98 and 1000 is 98,000.
Learning Outcomes
Students will be able to multiply a 3-digit or 4-digit number by a 2-digit number by writing them vertically.
Teaching Aids
Multiplication cards; Bingo cards
Activity
Demonstrate how to multiply a 3- or 4-digit number by a 2-digit number vertically. Make multiplication cards (for example: 329 × 43, 566 × 25, 728 × 12) and Bingo cards (showing answers to the corresponding multiplication cards).
Divide the class into groups and distribute the Bingo cards, with numbers in each cell, to each group.
Put the multiplication cards inside a box and shuffle these. Pick a card and read out the multiplication problem. Instruct the students to solve it and check whether the answer is on their Bingo card. If so, they should cross out the number. The group that crosses out a row vertically or horizontally first wins. Ask the students to write the answers in their notebook.
Extension Idea
Instruct: Arrange the following items in increasing order of their costs: 25 handkerchiefs at ₹266 per unit, 18 packets of toffees at ₹399 per packet, 12 knives at ₹165 per piece.
Say: 12 knives at 165 per piece < 25 handkerchiefs at 266 per unit < 18 packets of toffees at 399 per packet.
Quick Multiplication
Learning Outcomes
Students will be able to multiply 2-digit numbers using quick multiplication tricks.
Teaching Aids
Puzzle cards; Answer cards
Activity
Imagine Maths Page 46
Discuss the different methods that can be used to multiply 2 numbers and how to multiply a number by 11 easily.
Divide the class into groups. Distribute the puzzle card and the answer card to each group. Instruct the students to use only the quick multiplication method. Give them 5 minutes for each puzzle.
Ask each group to match the puzzle with the answer card. The group to match all the cards quickly will win a point.
Ask question such as: How do we get the middle number in a 3-digit product using quick multiplication? Can we use exact numbers here as used in the activity?
Multiplication by a 3-digit Number Imagine Maths Page 47
Learning Outcomes
Students will be able to multiply a 3-digit or 4-digit number by a 3-digit number by writing them vertically.
Demonstrate how to multiply a 3- or 4-digit number by a 3-digit number vertically. Make multiplication cards (for example: 311 × 1743, 534 × 825, 456 × 912) and answer cards (showing the answers to the corresponding multiplication cards).
Ask the students to work in groups. Distribute the multiplication cards, answer cards and chart paper to each group.
Instruct the groups to pick a multiplication card, solve the multiplication problem, match it with the correct answer card and paste them on the chart paper. Ask them to do this for all the cards. Ask the students to also write the answers in their notebooks.
Word Problems Imagine Maths Page 48
Learning Outcomes
Students will be able to solve word problems on multiplying numbers up to 4 digits by numbers up to 3 digits.
Teaching Aids
Word problem cards
Activity
Ask the students to work in groups. Distribute the word problem cards to each group. Ask them to read the problem.
Ask the students to write down what we know and what we need to find. Discuss how to solve the problem. Instruct them to solve the problem and discuss the answer.
Extension Idea
Instruct: Create your own word problem where the number of rows and flowers to be planted in each row are known.
Say: There can be multiple such problems. One such problem could be:
Seema has a garden. She has planted roses in 234 rows, and each row contains 2587 roses. How many roses are there in total in Seema’s garden?
Maria has a flower garden. She has planted sunflowers in 387 rows, and each row contains 4592 sunflowers. How many sunflowers are there in total in Maria’s garden?
What do we know?
What do we need to find?
Solve to find the answer.
Estimating the Product
Learning Outcomes
Students will be able to estimate the product of 2 numbers.
Teaching Aids
Puzzle cards
Activity
Begin by recalling the term estimation. Discuss some real-life scenarios where we use estimation.
Ask the students to work in groups. Distribute the puzzle cards to each group. (Puzzle cards can be made and printed beforehand. Ensure that the puzzle shows only 1 correct option.)
Tell the students that the number in the green part shows the estimated product of one of the problems in the yellow parts.
Instruct the groups to estimate the answers for all the problems and match with the answer in the green part.
Ask the students to write the answers in their notebooks.
1. Horizontal Method
Answers
Vertical Method Do It Together
2. Multiplying by Expanding the Bigger Number
Properties of Multiplication
5. Multiplying by 10, 100 and 1000
Do It Together
6. Using the Column Method
Do It Together 1 1 4 8 × 3 9 1 0 3 3 2 + 3 4 4 4 4 4 7 7 2
The product of 1148 and 39 is 44,772
7. Quick Multiplication
Think and Tell
3 × 1 = 3 and 9 × 1 = 9
3 + 9 = 12
Add the carry over 1 to 3, 3 + 1 = 4
So, the product of 39 and 11 is 429.
Do It Together
1. 18 = 10 + 8
2. 62 × 18 = 62 × 10 + 62 × 8 = 620 + 496 = 1116
Think and Tell
Greatest 4-digit number = 9999 and greatest 2-digit number = 99. Product of 9999 and 99 = 9,89,901. Thus, the product of a 4-digit number and a 2-digit number cannot be a 7-digit number.
8. Multiplication by a 3-digit Number
Do It Together
The product of 752 and 417 is 3,13,584.
9. Word Problems
Do It Together
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 ×
5,80,980
The free meal service provides food for people.
10. Estimating the Product
Do It Together
148 is rounded up to 150. 879 is rounded up to 880. 150 × 880 = 1,32,000
The estimated product of 148 and 879 is 1,32,000.
Division 4
Imagine Mathematics Headings CB Page
Division by 1-digit Numbers 56
Properties of Division 58
Dividing by Tens 59
Division of Numbers up to 4 Digits 60
Learning Outcomes
Students will be able to:
Imagine Mathematics Headings CB Page
Dividing by Multiples of 100s and 1000s 61
Estimating the Quotient 62 Word Problems 64
divide a 3-digit or 4-digit number by a 1-digit number using the long division method. apply the properties of division to divide 2 numbers quickly. divide numbers up to 4 digits by 10s.
divide a 4-digit number by a 2-digit number using the long division method. divide numbers up to 4 digits by 100s and 1000s.
estimate the quotient of 3-digit or 4-digit numbers divided by 2-digit numbers. solve word problems on dividing numbers up to 4 digits by numbers up to 2 digits.
Alignment to NCF
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-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
Let’s Recall
Recap to check if students know how to divide a 1-digit or 2-digit number by a 1-digit number. Ask students to solve the questions given in the Let’s Warm-up section.
Vocabulary
estimation: the process of guessing an answer close to the actual answer dividend: the number that is to be divided divisor: a number that divides another number
Teaching Aids
Place value blocks of 100s, 10s and 1s; Property cards; Number cards; Symbol cards; Place value charts; Counters; Chart paper; Pieces of a jigsaw puzzle with division problems and estimated quotients written on them; Word problem written on a sheet of paper with space given for each element of the CUBES strategy
Chapter: Division
Division by 1-digit Numbers
Learning Outcomes
Students will be able to divide a 3-digit or 4-digit number by a 1-digit number using the long division method.
Teaching Aids
Place value blocks of 100s, 10s and 1s
Activity
Begin by recalling division as the process of equally sharing items among groups. Use the problem 448 ÷ 4 to illustrate the concept. Ask the students to work in groups. Distribute place value blocks to each group. Instruct them to use the place value blocks to represent the dividend, 448. Ask them to divide each place value into 4 groups ensuring that each group has the same number of each type of blocks. Ask the students to count the number of 100s, 10s and 1s blocks to find the quotient. Show the same division using the long division method. Instruct the students to solve 452 ÷ 4 using the place value blocks and the long division method.
Extension Idea
Ask: Is there a way to check if the quotient obtained for the division problem 452 ÷ 4 = 113 is correct? Say: Yes. When there is no remainder, the dividend should be equal to Quotient × Divisor. Here, Dividend = 452. Quotient × Divisor = 113 × 4 = 452. We get Dividend = Quotient × Divisor. So, our answer is correct.
Properties of Division Imagine Maths Page 58
Learning Outcomes
Students will be able to apply the properties of division to divide 2 numbers quickly.
Teaching Aids
Property cards; Number cards; Symbol cards
Activity
Discuss the different properties of division in the class. Divide the class into groups. Distribute the property cards, symbol cards and number cards. Ask the students to start with the property 0 division by a number. Ask them to place the property card on the table and ask the group to make an example for the property using the number cards and symbol cards.
0 division by a number
0 ÷ 16 = 0
Ask the students to repeat the process for all the other properties. Ask question such as: Can a number be divided by 0? Why or why not?
Learning Outcomes
Students will be able to divide numbers up to 4 digits by 10s.
Teaching Aids
Place value charts; Counters
Activity
Ask the students to work in groups. Distribute the place value charts and counters. Ask the students to represent the number 140 by placing counters on the place value chart.
Discuss how dividing by 10 means moving each counter one place to the right on the chart. Let the students physically move each counter one place to the right on the chart while discussing the division process. As they move the counters, emphasise that the 1 counter representing 100 moves from the 100s column to the 10s column, and the 4 counters representing 40 move from the 10s column to the 1s column, showing that 140 ÷ 10 = 14.
Ask questions like: What is 8200 ÷ 20? Collect student responses.
Explain how students can cancel the zeroes in the ones place of both numbers and then divide the remaining digits mentally or using the long division method.
Division of Numbers up to 4 Digits
Learning Outcomes
Maths Page 60
Students will be able to divide a 4-digit number by a 2-digit number using the long division method.
Teaching Aids
Chart paper
Activity
Demonstrate how to draw an area model on the chart paper to represent a division problem such as 1224 ÷ 12. Ask the students to break 1224 into 2 numbers that are divisible by 12 and that are easy to work with.
For example: 1200 + 24 = 1224. Then, let them find the side lengths of each of the smaller rectangles. Finally, explain that the missing side of the first rectangle is 100, because 12 × 100 = 1200; that the missing side of the second rectangle is 2, because 12 × 2 = 24; and that on adding the side lengths, we get the quotient.
Ask the students to work in groups. Give them a division problem, such as 5025 ÷ 25, and let the groups construct their own area models. Instruct them to also divide 5025 by 25 using the long division method and then compare the answers that they got using the 2 methods.
Ask students to verify if the quotient and the remainder so obtained are correct using the equation, Dividend = (Quotient × Divisor) + Remainder.
Learning Outcomes
Students will be able to divide numbers up to 4 digits by 100s and 1000s.
Teaching Aids
Place value chart; Counters
Activity
Ask the students to work in groups. Distribute the place value chart and counters. Ask the students to represent the number 2140 on the place value chart using counters. Instruct them to divide 2140 by 100.
Ask the students to move the counters to the right as many times as the number of zeroes in the divisor. Explain how on moving the counters to the right by 2 places, there will be 2 tens and 1 ones which gives 21 as the quotient. The 4 tens will move out of the place value chart to give 4 tens or 40 as the remainder.
Give the students 1 more division problem and ask them to solve it using the place value chart.
Extension Idea
Ask: What is 4884 divided by 400?
Say: The quotient is 12 (400 × 12 = 4800) and the remainder is 84 (4884 – 4800 = 84). Thus, 4884 ÷ 400 gives Quotient = 12 and Remainder = 84. Estimating
Learning Outcomes
Students will be able to estimate the quotient of 3-digit or 4-digit numbers divided by 2-digit numbers.
Teaching Aids
Pieces of a jigsaw puzzle with division problems and estimated quotients written on them
Activity
Begin by recalling the term estimation in division. Discuss some real-life scenarios where we use estimation. Show the students how to estimate the quotient of 257 ÷ 14 using a number line drawn on the board.
Ask the students to work in groups. Distribute the jigsaw puzzle pieces (at least 5 sets) to each group. Instruct the groups to first round off the dividend and divisor in the division problems on the puzzle pieces, estimate their quotient to the highest place and join the problem pieces with the correct answer pieces.
Extension Idea
Ask: Which 2 problems will give the same estimated quotient as 3956 ÷ 43? A. 4367 ÷ 39 B. 7544 ÷ 82 C. 8368 ÷ 86 D. 8464 ÷ 92
Say: Options A and B will give the same quotient as 3956 ÷ 43.
Learning Outcomes
Students will be able to solve word problems on dividing numbers up to 4 digits by numbers up to 2 digits.
Teaching Aids
Word problem written on a sheet of paper with space given for each element of the CUBES strategy
Activity
Distribute the sheet with the word problem to the students.
Instruct them to circle the numbers, underline the question and box the key words. Discuss what they need to find. Ask them to evaluate the problem, solve it using the vertical method and write the answer. Ask questions like: How will you find out if your answer is correct?
Extension Idea
Circle the Numbers
There are 7200 pencils in a stationery store. If they are bundled into packs of 45, how many complete packs will there be?
Underline the Question Box the Key Words
Evaluate/Draw
Solve and Check!
Ask: Create your own word problem where you need to divide 1372 by 14. Say: There can be several word problems on the division of 1372 by 14. One such problem could be: Mary has 1372 apples. She wants to distribute them equally in 14 baskets. How many apples will each basket contain?
Answers
1. Division of a 3-digit Number by a 1-digit Number
Think and Tell
No, the division of a 2-digit number by a 1-digit number can give a 2-digit number. For example, 85 ÷ 5 = 17.
Students will be able to: find the multiples of a number using the multiplication tables. find the common multiples of 2 or more numbers. find the factors of 2 numbers using square tiles and multiplication. find the factors of 2 numbers using division. find the common factors of 2 numbers and then their highest common factor. identify prime and composite numbers.
Alignment to NCF
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-1.2: Discovers, identifies, and explores patterns in numbers and describes rules for their formation (e.g., multiples of 7, powers of 3, prime numbers), and explains relations between different patterns
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)
Let’s Recall
Recap to check if students know that multiplication and division are closely related. Ask students to solve the questions given in the Let’s Warm-up section.
Vocabulary
multiple: product that we get when one number is multiplied by another number factor: number that divides another number completely common factor: number that can evenly divide a set of 2 or more numbers
Teaching Aids
Number grid 1–100; Crayons; Counters; Number cards with numbers 7, 8, 11 and 15
Chapter: Multiples and Factors
Using Skip Counting; Using Multiplication
Learning Outcomes
Students will be able to find the multiples of a number using the multiplication tables.
Teaching Aids
Number grid 1–100; Crayons
Activity
Ask the students to work in pairs. Distribute the number grids and crayons to each pair.
Instruct the students to take jumps of 4 and shade the numbers on the grid using a particular colour. Once they have shaded the multiples of 4, have a discussion on what they noticed. Bring out the fact that the numbers shaded appear in the multiplication table of 4 and are called multiples of 4. Ask them to write the multiples of 4 in their notebooks.
Then, instruct the students to find the first five multiples of 11 on the same number grid using another colour. Ask them to write the first five multiples of 11 in their notebooks.
Ask questions like: What are the first five multiples of 11? Is 57 a multiple of 11? Why?
Extension Idea
Ask: How many multiples does a number have? Is 1 a multiple of every number?
Say: Every number has an infinite number of multiples. No, 1 is not a multiple of every number.
Finding Common Multiplies
Learning Outcomes
Students will be able to find the common multiples of 2 or more numbers.
Teaching Aids
Number grid 1–100; Crayons
Activity
Ask the students to work in pairs. Distribute the number grids and crayons to each pair.
Instruct the students to imagine that the number grid shows the numbered floors of a building where they run the canteens and the party halls. To show that every second floor has a canteen, they should write ‘C’ on every second number on the grid. To show that every fifth floor has a party hall, they should write ‘P’ on every fifth number on the grid. Then, in their notebooks, they should write down the numbers on which they wrote C and P. Finally, ask them to circle the numbers that had both C and P written on them.
Ask questions like: What numbers had both C and P written on them?
Explain that the numbers that are circled are called the common multiples of 2 and 5.
Ask the students to find the multiples and common multiples of 6 and 8 using the grid by shading the numbers.
Finding Factors Using Square Tiles;
Finding Factors Using Multiplication
Learning Outcomes
Students will be able to find the factors of 2 numbers using square tiles and multiplication.
Teaching Aids
Counters
Activity
Discuss the different ways in which 4 counters can be arranged in equal rows and equal columns.
Ask the students to work in groups. Distribute 20 counters to each group.
Instruct them to arrange the 20 counters in different ways so that they are placed in equal rows and equal columns. Each student in a group should show 1 arrangement. Then, in their notebooks, they should write the multiplication sentence for each arrangement.
Ask questions like: What arrangements did you get for 20 counters and what were their multiplication sentences?
Bring out the fact that the numbers that form the multiplication sentences are called the factors of that number.
Extension Idea
Ask: Is there any number that has the same factor as the number itself?
Say: All numbers are their own factors. For example, the number 6 has factors 1, 2, 3 and 6. Here 6 is a factor of itself.
Finding Factors Using Division
Learning Outcomes
Students will be able to find the factors of 2 numbers using division.
Teaching Aids
Counters
Activity
Maths Page 76
Discuss with the students how, if we divide 15 apples into 3 groups and get 5 apples in each group, then 3 and 5 are factors of 15 because, on dividing 15 by 3 or 5, we get no remainder.
Ask the students to work in groups. Distribute 24 counters to each group.
Instruct the groups to arrange an equal number of counters in 2 rows, in 3 rows and then in 4, 6, 8 and 12 rows. Then, in their notebooks, they should write the division sentences and the factors, and list the numbers that leave no remainder.
Bring out the fact that if they are able to arrange the counters in the specified number of rows, that number would be a factor of the given number of counters.
Extension Idea
Ask: What is the smallest number that has exactly 3 factors?
Say: The smallest number that has exactly 3 factors is 4. The factors of 4 are 1, 2 and 4. 4 24 6 – 24 00 4 rows and 6 columns
Learning Outcomes
Students will be able to find the common factors of 2 numbers and then their highest common factor.
Teaching Aids
Counters
Activity
Ask the students to work in pairs. Distribute the counters to each pair.
Instruct the students to find the factors of 12 and 15 using different arrangements of counters. Then, in their notebooks, they should list the factors, colour the common factors of both numbers and circle the common factor that is the highest.
Factors of 12 = 1, 2, 3, 4, 6, 12
Factors of 15 = 1, 3, 5, 15
Highest Common Factor = 3
Next, instruct students to find the factors of 20 and 30, write their common factors and find their highest common factor.
Extension Idea
Ask: When finding the common factors of 2 numbers, can one of the numbers be the common factor of itself and the other number?
Say: Yes, a number can be a common factor of itself and another number. For example, for the numbers 5 and 15, the number 5 is a common factor of 5 and 15.
Prime and Composite Numbers
Learning Outcomes
Students will be able to identify prime and composite numbers.
Teaching Aids
Number cards with numbers 7, 8, 11 and 15; Counters
Activity
Discuss the method to find the factors of numbers.
Ask the students to work in groups of 4. Distribute the resources.
Instruct each student in the group to pick one number card and the same number of counters as the number. They will then arrange the counters in different ways so that they are placed in equal rows and equal columns. Then, in their notebooks, they should write the factors of the number.
Finally, in their groups they will discuss what they notice about the number of factors in each number.
Ask question such as: How many factors do each number have?
Explain what prime and composite numbers are and how to identify them.
Answers
1. Using Skip Counting
Think and Tell
All the circled numbers are part of the multiplication table of 2.
Do It Together
7, 14, 21, 28, 35, 42
2. Using Multiplication
Think and Tell
Yes, the multiples of even number are always even.
The first five multiples of 6 are 6, 12, 18, 24, 30
3. Finding Common Multiples
Do It Together
Multiples of 2 2 4 6 8 10 12 14 16 18 20
Multiples of 5 5 10 15 20 25 30 35 40 45 50
Common multiples of 2 and 5 = 10, 20
We can find the canteen and party hall together on floors 10 and 20
4. Finding Factors Using Square Tiles
Do It Together
5. Finding Factors Using Multiplication
Do It Together
Multiply by 1 × =
Multiply by 2 × =
Multiply by 3 Not possible
Multiply by 4 × =
Multiply by 5 × =
Multiply by 6 not possible
Should we multiply further?
So, the factors of 20 are
6. Finding Factors Using Division
Think and Tell
No, we do not need to go beyond 6 to find more factors of 36.
Do It Together
1, 2, 4, 5, 10 and 20 4 1, 2, 3, 4, 6, 9, 12, 18 and 36
So, the factors of 36 are .
7. Finding Common Factors
Do It Together
Factors of 20
Factors of 30
The common factors of 20 and 30 are . 1, 2, 5 and 10
8. Prime and Composite Numbers
Factors of 14 = 1, 2, 7, 14; Composite Number
Factors of 19 = 1, 19; Prime Number
Factors of 45 = 1, 3, 5, 9, 15, 45; Composite Number
Factors of 59 = 1, 59; Prime Number
Fractions 6
Learning Outcomes
Students will be able to: identify halves, quarters and thirds of a whole or a collection. find equivalent fractions and write a fraction in its simplest form. identify like and unlike fractions and compare them. compare and order unlike fractions. convert mixed numbers to improper fractions and vice versa. add 2 like fractions. subtract 2 like fractions.
Alignment to NCF
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
Let’s Recall
Recap to check if students know the concepts of a whole, halves and quarters. Ask students to solve the questions given in the Let’s Warm-up section.
Vocabulary
numerator: number of parts of a whole chosen denominator: total number of equal parts that a whole is divided into equivalent fraction: a fraction obtained by multiplying or dividing the numerator and denominator by the same non-zero number
Teaching Aids
Buttons; Circle cut-outs divided into 4 equal parts and 3 equal parts; Crayons; Glue stick; Pairs of circle cut-outs divided into 4 and 8 parts; Rectangular strips; Rectangular strips of equal length divided into 5, 7, 9 and 11 parts; Sheets with unshaded fraction circles divided into 6 parts; Fraction strips
Chapter: Fractions
Halves, Quarters and Thirds
Learning Outcomes
Students will be able to identify halves, quarters and thirds of a whole or a collection.
Teaching Aids
Buttons; Circle cut-outs divided into 4 equal parts and 3 equal parts; Crayons; Glue stick
Activity
Imagine Maths Page 85
Ask students to work in groups. Distribute buttons (say, 24 buttons) and circle cut-outs divided into 4 equal parts to each group.
Ask students to find 1 4 of the total number of buttons. Instruct them to divide the total number of buttons into 4 equal groups and ask the value of one group.
Instruct students to shade 1 4 of the circle cut-out and stick 1 4 of the buttons on the shaded part.
Instruct students to write the fraction inside the circle. Repeat the activity to find 2 3 of the buttons.
Extension Idea
Ask: If a quarter of the buttons are further divided into halves, then how many buttons will there be in each equal part?
Say: The total number of buttons is 24. 1 4 of 24 is 6 and half of 6 is 3. So, there will be 3 buttons in each equal part.
Equivalent Fractions
Learning Outcomes
Imagine Maths Page 88
Students will be able to find equivalent fractions and write a fraction in its simplest form.
Teaching Aids
Pairs of circle cut-outs divided into 4 and 8 parts
Activity
Divide the class into groups of 4. Distribute the circle cut-outs to each group. Instruct students to shade 1 4 , 2 4 , 2 8 and 4 8 of the circle cut-outs and write the shaded fraction inside the circle. Ask them to compare the 1 4 shaded cut-out with the 2 8 shaded cut-out and the 2 4 shaded cut-out with the 4 8 shaded cut-out. Bring out the fact that the compared fractions are equivalent fractions as they represent the same value. Help them find more equivalent fractions for 2 8 and 4 8 by multiplying or dividing the numerator and denominator by the same whole number in their notebooks. Ask students whether they can divide the numerator or denominator of the fraction 1 4 further by any number. Discuss the simplest form of the fractions.
Comparing and Ordering Like Fractions
Learning Outcomes
Students will be able to identify like and unlike fractions and compare them.
Teaching Aids
Rectangular strips; Crayons
Activity
Imagine Maths Page 91
Begin by discussing like and unlike fractions. Bring out the fact that like fractions have the same denominator and unlike fractions have different denominators.
Ask students to work in groups. Distribute rectangular strips of equal lengths to each group. Instruct students to represent 4 7 and 5 7 on the rectangular strips by drawing and shading the fractions.
Instruct them to place the rectangular strips one below the other and compare the shaded parts. Ask them to say which fraction is bigger by looking at the strips.
Explain that for like fractions, greater the numerator, greater the fraction. Instruct students to write the answer in their notebooks. Give them some more problems to solve.
Comparing and Ordering Unlike Fractions
Imagine Maths Page 91 with Same Numerators
Learning Outcomes
Students will be able to compare and order unlike fractions.
Teaching Aids
Rectangular strips of equal length divided into 5, 7, 9 and 11 parts; Crayons
Activity
Ask students to work in groups. Distribute rectangular strips of equal length divided into 5, 7, 9 and 11 parts to each group.
Instruct students to shade 2 parts in each rectangular strip and write the fractions for the shaded parts of these strips in their notebooks.
Then, ask them to arrange all the rectangular strips, one below the other, and look at the shaded parts. Ask students to compare and order the fractions and write the answers in the notebooks. Explain that for unlike fractions with the same numerators, smaller the denominator, greater the fraction.
Extension Idea
Ask: Riya ate 4 8 of a pizza and Swati ate 3 6 from the same size of pizza. Who ate more pizza?
Say: Simplest form of 4 8 = 1 2 and simplest form of 3 6 = 1 2 . Hence, both of them ate the same amount of pizza.
Proper and Improper Fractions
Learning Outcomes
Students will be able to convert mixed numbers to improper fractions and vice versa.
Teaching Aids
Sheets with unshaded fraction circles divided into 6 parts
Activity
Ask students to work in groups.
Distribute sheets with the circles divided into 6 equal parts to each group.
Imagine Maths Page 94
Ask students to show 4 6 by shading the first circle and 10 6 by shading the second and third circles. Ask them to write the fractions below the circles. Discuss proper and improper fractions based on the shaded circles.
Help students write the mixed fraction as a combination of a whole number and a fraction. Discuss the division and multiplication methods to convert an improper fraction to a mixed fraction and vice versa.
Repeat the activity for some other fractions. Ask students to make the conversions and write the answers in their notebooks.
Extension Idea
Ask: How many 2 5 s are there in 1 3 5 ?
Say: 1 3 5 = 8 5 . So, there are four 2 5 s in 1 3 5 .
Adding Like Fractions
Learning Outcomes
Students will be able to add two like fractions.
Teaching Aids
Fraction strips; Crayons
Activity
Imagine Maths Page 97
Ask students to work in groups. Distribute fraction strips to each group. Instruct them to divide the fraction strip into 6 equal parts. Ask them to find 1 6 + 4 6 by shading 1 6 of the fraction strip with one colour and 4 6 more in the same strip with another colour.
Ask students what fraction of the total strip is shaded. Instruct them to write the answer in their notebooks. Give students a word problem. Raju has 1 8 m of cloth and he buys 5 8 m more. What is the total length of cloth with Raju? Instruct students to use the fraction strips and write the answers in their notebooks.
Extension Idea
Ask: What is 2
+ 5
?
Subtracting Like Fractions
Learning Outcomes
Students will be able to subtract 2 like fractions.
Teaching Aids
Fraction strips; Crayons
Activity
Imagine Maths Page 98
Ask students to work in groups. Distribute fraction strips to each group. Instruct students to divide the fraction strip into 8 equal parts and then to find 5 8 –2 8
Tell them to shade 5 8 on the fraction strip, and then cross out 2 8 of the shaded part.
Then, ask students what fraction of the total strip is shaded. Instruct them to write the answers in their notebooks. Give students a word problem. Sid had 5 6 litres of paint. He used 1 6 litres of paint. How much paint was he left with?
Instruct students to use the fraction strips to find the amount of paint left. Ask them to write the answers in their notebooks.
Extension Idea
Ask:
Answers
1. Fraction of a Whole
Do It Together
Half can also be written as 1 3 . False
One-third is more than 1 4 . True
2. Fraction of a Collection
Do It Together
Total number of ice creams = 36
Fraction of ice creams sold = 5 6
There are a total of 6 groups. There are 6 ice creams in each group.
Number of ice creams in 5 groups = 30
Thus, number of ice creams sold = 5 6 of 36 = 30.
3. Finding Equivalent Fractions; Simplest Form
Think and Tell
We do not divide the numerator and denominator by the common factor 1 because it does not have any effect after division as any number divided by 1 is the number itself.
No, equivalent fractions cannot be like fractions because like fractions have the same denominator and equivalent fractions have different denominators (but the same value).
Do It Together
Arrange the numerators in ascending order: 2 < 4 < 7 < 8
Ascending order:
2 9 < 4 9 < 7 9 < 8 9
5. Comparing and Ordering Unlike Fractions with the Same Numerator
Do It Together
Since the numerators are the same, we arrange the denominators in ascending order:
3 < 5 < 10 < 12 < 15
Descending order:
7 3 > 7 5 > 7 10 > 7 12 > 7 15
6. Testing for Equivalence by Cross-multiplying
Do It Together
7. Converting Improper Fractions to Mixed Numbers
Do It Together
As, Mixed number = Quotient Remainder Divisor
Therefore, 56 12 = 4 8 12 = 4 2 3
8. Converting Mixed Numbers to Improper Fractions
Do It Together 4 3 6 = (6 � 4) + 3 6 = 27 6 Add Multiply
9. Writing a Whole Numbers as a Fraction
Do It Together 6
[ 6 ] = 1 [ 20 ] 5 = 4
10. Adding Like Fractions
Do It Together
Weight of bananas = 2 1 4 kg = 9 4 kg
Weight of apples = 1 1 4 kg = 5 4 kg
Total weight = 9 4 + 5 4 = 14 4 = 7 2 kg
Converting 7 2 kg to a mixed number, we get 31 2 kg.
11.
Subtracting Like Fractions
Do It Together
Length of ribbon bought = 2 1 5 m = 11 5 m
Length of ribbon used = 2 5 m
Length of ribbon left = 11 5 –2 5 = 9 5 m
Converting 9 5 m to a mixed number, we get 1 4 5 m.
Lines and 2-D Shapes
Learning Outcomes
Students will be able to: identify and draw a point, a ray and a line. measure and draw line segments in centimetres. identify types of curves as open, closed, simple and non-simple. classify the different types of polygons based on their sides and angles. identify the different parts of a circle and construct circles with given radii.
Alignment to NCF
C-2.1: Identifies, compares, and analyses attributes of two- and three-dimensional shapes and develops vocabulary to describe their attributes/properties
Let’s Recall
Recap to check if students know how to identify 2-D shapes and find them in 3-D shapes. Ask students to solve the questions given in the Let’s Warm-up section.
Vocabulary
radius: distance from the centre to the edge of a circle diameter: a straight line that joins two points on the edge of a circle and passes through the centre circumference: the line that forms the outside edge of a circle
Teaching Aids
2 sets of cards each representing points, lines, line segments and rays; Straws of lengths 5 cm and 6 cm; Ruler; Sheets with numbered points; 4 sets of cards with details about the shape, its name, the number of sides and the number of angles, respectively; Circle cut-outs; Drawing sheets; Thread; Glue stick; Compass; Pencil
Chapter: Lines and 2-D Shapes
Points, Rays and Lines
Learning Outcomes
Students will be able to identify and draw a point, a ray and a line.
Teaching Aids
2 sets of cards each representing points, lines, line segments and rays
Activity
Make stacks of 8 cards with 2 each representing points, lines, line segments and rays. Ask students to work in groups of 4. Shuffle and distribute a stack of cards to each group.
Instruct group members to discuss what they notice about each card. Ask them to sort the cards based on the feature they noticed. Help them list the features in their notebooks.
Then, ask students to draw points, line segments, lines and rays in their notebooks and label them. Discuss where we may see these in real life.
Line Segments
Learning Outcomes
Students will be able to measure and draw line segments in centimetres.
Teaching Aids
Straws of lengths 5 cm and 6 cm; Ruler
Activity
Distribute straws of lengths 5 cm and 6 cm to students.
Instruct students to measure the lengths of the straws using a ruler and write the measurements in their notebooks. Once they are done, ask them to draw line segments of the same lengths as the straws in their notebooks and mark the initial and end points of each line segment with letters such as A and B.
Ask them to measure some more objects around them such as pencils, erasers etc. and draw line segments showing those lengths.
Extension Idea
Ask: Suppose you measured the pencil’s length with one end at 3 cm on the ruler and the other end at 11 cm. What is the actual length of the pencil?
Say: The actual length of the pencil is 11 cm – 3 cm = 8 cm.
Open vs Closed Figures;
Simple vs Non-simple
Learning Outcomes
Students will be able to identify types of curves as open, closed, simple and non-simple.
Teaching Aids
Sheets with numbered points
Activity
Lead students to the playground. Instruct them to form a circle by joining hands. Once the circle is established, say that the circle is complete and closed. Ask some students to step out while the rest remain in place, creating an incomplete circle. Say that the circle is now open.
Explain that if the starting point coincides with the end point, the curve is considered closed; conversely, if they differ, the curve is open.
Lead students back to the classroom and ask them to work in pairs. Distribute the sheets with numbered points to each pair.
Explain that you will call out numbers from 1 to 12, one by one, and students have to join the points without lifting their pencils. Call out the numbers in such a way so as to form one simple and one non-simple curve. Direct students to identify the type of figure created through this observation and to record their findings in their notebooks.
Extension Idea
Ask: Draw a non-simple open curve.
Say: This is an example of a non-simple open curve.
Learning Outcomes
Students will be able to classify the different types of polygons based on their sides and angles.
Teaching Aids
4 sets of cards with details about the shape, its name, the number of sides and the number of angles, respectively
Activity
Create 4 sets of cards for each shape, including one with a picture of the shape, a second with the name, a third with the number of sides and corners and a fourth with one unique feature. Divide the class into 4 groups. Distribute all the name cards to one group, picture cards to the second group, side/corner cards to the third group and unique feature cards to the fourth group. Instruct students to move around and find the other 3 cards that match their shape. The first group of 4 students to correctly form a set wins. Repeat the activity by shuffling the cards among students. Instruct students to write the details of the respective shapes in their notebooks.
Learning Outcomes
Students will be able to identify the different parts of a circle and construct circles with given radii.
Teaching Aids
Circle cut-outs; Drawing sheets; Thread; Glue stick; Compass; Pencil
Activity
Ask students to work in pairs. Distribute a circle cut-out, thread, glue stick, drawing sheet, compass and pencil to each pair.
Instruct students to fold the circle cut-out in half and then unfold it. Ask them to rotate the cut-out and fold it in half once more and then unfold it. Explain that the point where the creases intersect is identified as the centre of the circle, and that the crease itself represents the diameter. Explain that the radius of a circle is half of the diameter. Ask students to paste the thread at a point at one end of the diameter, encircling the cut-out until it returns to the starting point. Explain that the looping thread is referred to as the circumference. Ask students to mark the discussed parts on the cut-out.
After the pairs are done with their labelling, demonstrate how to use a compass to draw a circle. Discuss why it is important to hold the needle of the compass steady at one point. Instruct students to set the compass width to 3 cm using a ruler and draw a circle on the sheet. Ask them to label all the parts of a circle, discussed so far, on the circle they have drawn.
Extension Idea
Ask: The distance between the centre and the circumference of a circle is 5 cm. What is the diameter of the circle?
Say: The distance between the centre and the circumference of a circle is the radius. If the radius is 5 cm, then the diameter is 2 × 5 cm = 10 cm.
Answers
1. Points, Rays and Lines
Think and Tell
One line can pass through 2 given points.
Do It Together
The points are: A, B, C, D, E and F
The figure has rays: DE , AB and CF .
The lines in the figure are: Ɩ and m
Line Ɩ is the same as FC and line m is the same as AD.
The line intersecting FC is m.
2. Measuring Line Segments
Think and Tell
Yes, a ray and a line segment can be parts of the same line.
Do It Together
The length of the comb is 7 cm.
3. Drawing Line Segments
Think and Tell
Yes, a line segment can be measured using the ruler whose zero mark is missing. We just need to keep in mind the point where it starts from.
Do It Together
4. Open vs Closed Figures
Do It Together
5. Simple vs Non Simple
Do It Together
6. Polygons
Do It Together
7. Parts of a Circle
Think and Tell
A circle can have infinite diameters.
Think and Tell
Yes, the radius of a circle does impact its circumference. The greater the length of the radius, the greater the circumference. Do It Together
8. Constructing Circles
Do It Together Radius = 6 cm
Representing 3-D Shapes
Learning Outcomes
Students will be able to: draw different views of objects. identify the nets of a cube and a cuboid and draw them. read a map and answer questions on it.
Alignment to NCF
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)
Let’s Recall
Recap to check if students know about basic 2-D and 3-D shapes. Ask students to solve the questions given in the Let’s Warm-up section.
Vocabulary
nets: 2-D representations of solid shapes map: a picture of a place printed on a flat surface
Teaching Aids
Glass; Book; Water bottle; Pencil box; Cake box shaped like a cube; Cake box shaped like a cuboid; Thick chart paper; Scissors; Glue stick; Word cards labelled with names of places like Hotel, Park, Hospital, Market, etc.; Chart paper
Chapter: Representing 3-D Shapes
Views of Objects
Learning Outcomes
Students will be able to draw different views of objects.
Teaching Aids
Glass; Book; Water bottle; Pencil box
Activity
Imagine Maths Page 123
Show students a glass and turn it around to demonstrate how it looks from different sides. Ask questions on how the glass looks from the top, side and front. Draw its 3 views on the board. Divide the class into groups of 3. Distribute 3 objects to each group (any other object available in class can also be used). Instruct the groups to place the objects, one by one, on the table and draw their different views. Each student in the group should draw the side view of one object, the top view of the second object and the front view of the third object. Allow them to turn the objects around to see the view that they are drawing.
Ask students to draw the different views in their notebooks.
Extension Idea
Ask: What is one object that has the same front, side and top views?
Say: Any object that is shaped like a sphere or a cube has the same front, side and top views. For example, a sugar cube.
Nets of 3-D Shapes
Learning
Outcomes
Students will be able to identify the nets of a cube and a cuboid and draw them.
Teaching Aids
Maths Page 126
Cake box shaped like a cube; Cake box shaped like a cuboid; Thick chart paper; Scissors; Glue stick
Activity
Show students the cube-shaped cake box and open it out. Ask students what they see. Then, close it up to form the box again.
Discuss how a 3-D shape can be represented in 2-D form and what nets are.
Ask students to work in groups of 4. Distribute the chart paper and cuboidal boxes to students.
Ask them to look at the box, visualise how the box would look when opened out and draw its net on a thick chart paper. Each student should draw a net and then compare the nets with the other members of their group. They should then cut and fold the net to see the box that they get. Finally, they should open out the cuboidal box to see if the net that they have drawn has the same 2-D shapes that are there in the box.
Ask questions like, “Is it possible to draw different nets for the same solid shape?”
Extension Idea
Ask: If you join 2 cubes end to end, what will be the net of the new shape formed?
Say: When two cubes are joined end to end, a cuboid is formed. So, the net will be that of a cuboid.
Learning Outcomes
Students will be able to read a map and answer questions on it.
Teaching Aids
Word cards labelled with names of places like Hotel, Park, Hospital, Market, etc.; Chart paper; Glue stick
Activity
Ask students to work in groups. Distribute word cards labelled with different places to each group. For example, distribute cards labelled Hotel, Park, Hospital, Post Office, School, Market, etc.
Hotel Park Hospital Post Office School Market
Instruct students to create a map by pasting the word cards on chart paper. Give them instructions on where to place the word cards such that all the groups get the same map, like the given sample.
Ask questions like, “How will you reach the school from the park? Which direction is the hotel from the post office?” Instruct students to write the answers in their notebooks. Ask them to create 2 questions, per group based on the map, exchange the questions with other groups and answer them. Discuss the importance of maps.
Extension Idea
Ask: What would be the actual distance between the hotel and the hospital if the distance between them is 5 cm on the map? 1 cm on paper = 3 km.
Say: 1 cm = 3 km. So, the actual distance is 5 × 3 = 15 km.
Answers
1. Views of Objects
Think and Tell
No, not all objects look different when seen from different views. Some objects shaped like a cube or a sphere look the same from all the views.
Do It Together
2. Nets of 3-D Shapes
Do It Together
3. Maps
Think and Tell
No, Riya will not follow the same route if she is standing outside the supermarket. Do It Together
Mohan's House
Neerja's House
School Park Hotel Hospital
Post Office Market
Patterns and Symmetry 9
Learning Outcomes
Students will be able to: identify the rule in a repeating pattern and extend the pattern. identify the rule in a rotating pattern and extend the pattern. identify the rule in a growing or reducing pattern and extend the pattern. identify the rule in a number pattern and extend the pattern. identify shapes that tile and extend a tiling pattern. decode puzzles or secret messages and solve them. identify and draw lines of symmetry in shapes and figures. draw the reflection of a shape or figure along the mirror line.
Alignment to NCF
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-2.3: 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
Let’s Recall
Recap to check if students know how to extend simple patterns. Ask students to solve the questions given in the Let’s Warm-up section.
Vocabulary
number pattern: sequence of numbers in a fixed repetitive way tessellations: patterns made by repeating and fitting together shapes without any gaps or overlaps symmetry: an arrangement where a line can be drawn dividing a figure into 2 identical parts
Teaching Aids
2 buntings showing a pattern and a non-pattern made with Christmas decorations; Christmas decorations (baubles, stars, bells, etc.); Pieces of string; Shape cards (in different directions); Cut vegetables such as okra, onion, mushroom, broccoli, etc.; Paint in different colours; Sheets of paper; Sudoku grids; Cardboard cut-outs of different shapes; Blank slips of paper; Cards depicting everyday objects including shapes and figures; Grid paper; Small mirrors
Chapter: Patterns and Symmetry
Repeating Patterns
Learning Outcomes
Students will be able to identify a repeating pattern and extend the pattern.
Teaching Aids
2 buntings showing a pattern and a non-pattern made with Christmas decorations; Christmas decorations (baubles, stars, bells, etc.); Pieces of string
Activity
Show the 2 buntings—one forming a pattern and the other forming a non-pattern. Ask students to say what they notice about the 2 buntings.
Discuss what patterns and non-patterns are and how the pattern shows repeating decorations. Ask the students to work in groups. Distribute the Christmas decorations and pieces of string to each group.
Instruct the students to create their own festive buntings using the decorations and string, focusing on creating repeating patterns.
Discuss the variety of patterns created by the groups and celebrate the creativity within class. Teacher Tip: If decorations are not available, give the students cardboard cut-outs with holes punched.
Rotating Patterns
Learning Outcomes
Students will be able to identify the rule in a rotating pattern and extend the pattern.
Teaching Aids
Shape cards (in different directions)
Activity
Discuss what is a rotating pattern and how to find its rule. Divide the class into groups and distribute the shape cards to each group.
Ask the students to form different rotating patterns with the given shapes. Ask them to note the rule in their notebooks. Instruct the students to draw 1 rotating pattern in their notebook and write the rule for the same.
Ask question such as: How is this pattern different from the repeating patterns?
Growing and Reducing Patterns
Learning Outcomes
Students will be able to identify the rule in a growing or reducing pattern and extend the pattern.
Teaching Aids
Cut vegetables such as okra, onion, mushroom, broccoli, etc.; Paint in different colours; Sheets of paper
Activity
Discuss growing and reducing patterns as patterns where units increase and decrease, respectively, at each step.
Ask the students to work in groups. Distribute the cut vegetable to each group.
Instruct them to dip the vegetables in paint and imprint the shapes onto a sheet of paper, forming beautiful growing or reducing patterns. Encourage them to try different layouts such as vertical and horizontal. Initiate a class discussion and ask students to point out the unit that is added or reduced at each step.
Ask questions like: How do you know the pattern is growing? How do you know it is reducing?
Extension Idea
Ask: Can a pattern have both growing and reducing units? If yes, show such a pattern.
Say: Yes, a pattern can have both growing and reducing units. One such pattern could be:
Number Patterns
Learning Outcomes
Students will be able to identify the rule in a number pattern and extend the pattern.
Teaching Aids
Sudoku grids
Activity
Explain the rules of Sudoku of placing numbers 1 to 9 in each row, column and 3×3 sub-grid without repetition. Guide the students through the initial steps of solving a Sudoku puzzle, demonstrating how to identify and fill in numbers based on the rules.
Organise the students into groups. Distribute Sudoku grids to each group. Allow them to solve the puzzles independently, encouraging them to record their answers in the provided blank spaces. Offer help as needed.
Facilitate a brief discussion on the number patterns they observed during the activity.
Extension Idea
Ask: Create an increasing pattern starting from 4, where each number is double the previous number.
Say: The pattern would be 4, 8, 16, 32, ...
Tiling Patterns and Tessellations
Learning Outcomes
Students will be able to identify shapes that tile and extend a tiling pattern.
Teaching Aids
Cardboard cut-outs of different shapes
Activity
Begin with a discussion on tiling patterns and tessellations. Explain that tiling patterns with no gaps and overlaps become tessellations. Show them some examples of tiling patterns.
Ask the students to work in groups.
Distribute shape cut-outs to each group, such as hexagons, squares, random shapes, jigsaw puzzle pieces, etc.
Ask the students to arrange these cut-outs to create tilling patterns. Help them note whether their patterns are tessellations or non-tessellations and write their findings in their notebooks.
Discuss about tangrams in the class. Ask the students to make a cat or a fish using the shape cut-outs.
Coding and Decoding Patterns
Learning Outcomes
Students will be able to decode puzzles or secret messages and solve them.
Teaching Aids
Blank slips of paper
Activity
Begin by providing guidance on the key to encode, which involves assigning numerical values to each letter, following the pattern A = 1, B = 2 and so on, up to Z.
Ask the students to work in pairs. Distribute 2 blank slips of paper to each pair.
Instruct 1 student in each pair to write an affirmation like “BE CALM” or “BE HAPPY” on their slip. Instruct the other student in the pair to write an encoded number for a different affirmation on their slip. Ask them to exchange their slips, and encode or decode the message, depending on the slip they received.
Ask the students to record their answers in their notebooks. Engage them in a class discussion to share insights and decode the affirmations among themselves.
Extension Idea
Ask: If you were to share a secret message during a mission, what method or code would you choose to encode the message “MISSION ACCOMPLISHED”?
Say: You can use a code known as Caesar cipher. This involves shifting each letter of the message by a fixed number of positions in the alphabet. If we apply a Caesar cipher with a shift of 3 to the message “MISSION ACCOMPLISHED,” the encoded message would be: “PLVVLRQ DEERPSOLVKHG”
Learning Outcomes
Students will be able to identify and draw lines of symmetry in shapes and figures.
Teaching Aids
Cards depicting everyday objects including shapes and figures
Activity
Ask the students to work in groups. Provide them with a collection of cards featuring everyday shapes and objects such as leaves, traffic signs, cups, butterflies, flags, squares, rectangles, etc.
Instruct students to observe and identify whether each shape or figure has symmetry. Explain the concept of figures having more than 1 line of symmetry. Guide them to fold the cards to discover symmetry in the figures. Instruct them to draw lines of symmetry using a pencil.
Ask questions like: Which figure had symmetry? How did you find out?
Extension Idea
Ask: Can you draw a rangoli pattern with 2 lines of symmetry? If yes, draw 1 such pattern. Say: Yes, we can draw rangoli patterns with 2 lines of symmetry. There can be many such patterns. One could be:
Reflection
Learning Outcomes
Students will be able to draw the reflection of a shape or figure along the mirror line.
Teaching Aids
Grid paper; Small mirrors
Activity
Start by drawing a simple image on the board and showing its reflection using a mirror.
Ask the students to work in pairs. Give each student a sheet of grid paper with a line drawn down the centre and a small mirror.
Ask both students in the pair to draw a shape on one side of the line drawn. Once they are done, ask them to exchange drawings with their partners and let them complete the figure by drawing its mirror image on the other side of the line drawn.
Ask them to place the mirror along the line to check if their figure matches the reflection of the original shape.
1. Repeating Patterns
Think and Tell
The pattern on the print of the cloth is a repeating pattern.
Do It Together
2. Rotating Patterns Do It Together
3. Growing and Reducing Patterns Do It Together
5. Tiling Patterns and Tessellations Do It Together
. Number Patterns
Do It Together
91 + 16 = 107 92 + 15 = 107
93 + 14 = 107
Thus, the 6 numbers are 91, 16, 92, 15, 93, 14.
6. Coding and Decoding Patterns Do It Together
7. Symmetry Do It Together
8. Reflection
Length, Weight and Capacity
Learning Outcomes
Students will be able to: estimate and measure the length of an object. convert between metres and centimetres. estimate and measure the weight of an object. convert between grams and kilograms. estimate and measure the capacity of a container. convert between litres and millilitres.
Alignment to NCF
C-3.1: Measures in non-standard and standard units and evaluates the need for 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-3.5: 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-3.7: Evaluates the conservation of attributes like length and volume, and solves daily-life problems related to them
Let’s Recall
Recap to check if students know the units and tools to measure length, weight and capacity. Ask students to solve the questions given in the Let’s Warm-up section.
Vocabulary
estimate: to round off the answer that is close to the actual answer length: horizontal distance from one end to the other weight: measurement of the heaviness of an object capacity: the amount that can be held in a particular space
Teaching Aids
Ruler; Measuring tape; 1 m strips of paper; 10 cm strips of paper; Glue stick; Grains of rice; Small digital weighing scale; Weights of 100 g, 200 g, 250 g and 500 g; Measuring cups for 1 L and 100 mL; Water bottles of capacity 200 mL, 700 mL and 1 L with no labels; Bottle of water
Chapter: Length, Weight and Capacity
Measuring Length
Learning
Outcomes
Students will be able to estimate and measure the length of an object.
Teaching Aids
Ruler; Measuring tape
Activity
Start by showing the students lengths of 1 mm, 1 cm and 1 m using measuring tools such as rulers and measuring tapes. Give them some examples of long distances to introduce kilometre as a unit.
Ask the students to work in groups. Distribute a ruler and measuring tape to each group.
Instruct the students to look at various objects around the classroom. Ask them to estimate the unit that they can use to measure their lengths. Ask them to guess the lengths of their book, pencil and desk and write the object name and its estimated length in their notebooks. Then, instruct them to use the ruler to measure the lengths of the objects. Ask them to write the actual measures next to the estimated measures and see how close these are to their estimates.
Extension Idea
Ask: What would be the length of a line formed by placing 5 pencils, end to end, of length 12 cm each?
Say: Length of 1 pencil = 12 cm; length of 5 such pencils = 5 × 12 = 60 cm. So, the line will be 60 cm long.
Converting Between Units of Length
Learning
Outcomes
Students will be able to convert between metres and centimetres.
Teaching Aids
1 m strips of paper; 10 cm strips of paper; Ruler; Measuring tape; Glue stick
Activity
Ask the students to work in groups of 10. Distribute one 1 m strip of paper, ten 10 cm strips of paper, a ruler and a measuring tape to each group. Ask the groups to first measure the longer strip of paper using the measuring tape and then the 10 cm strips using a ruler. Discuss what measures they got in metres. Then, ask them to use the glue stick to paste the 10 cm strips across the length of the 1 m strip without leaving any gaps in between.
Ask questions like: What is the relation between metres and centimetres?
Give some conversion problems for students to solve in their notebooks. Discuss the answers.
Extension Idea
Ask: Find the relation between millimetres and centimetres using your ruler. How many mm is 4 cm?
Say: There are 10 mm lines between two cm lines, so 10 mm = 1 cm. 4 cm = 4 × 10 = 40 mm.
Learning Outcomes
Students will be able to estimate and measure the weight of an object.
Teaching Aids
Grains of rice; Small digital weighing scale
Activity
Set up a weighing station in the classroom with a digital weighing scale. Start by showing the students the weight of some objects like a few grains of rice, a pencil and a thick book on the scale. Allow students to come up and read the measures on the weighing scale.
Ask the students to work in groups.
Instruct them to look at various objects around the classroom such as a piece of paper, an eraser, a lunch box, a water bottle, etc. Ask them to estimate the unit that they can use to measure their weight, like mg, g or kg. Ask them to guess the weight of their Imagine Mathematics book, school bag and pencils and write the object name and its estimated weight in their notebooks.
Then, instruct the students to come up and use the weighing scale to measure the weight of the objects. Ask them to write the actual measures next to the estimated measures and see how close these were to their estimates.
Ask questions like: What is lighter: 1 kg of cotton or 1 kg of iron nails?
Converting Between Units of Weight
Learning Outcomes
Students will be able to convert between grams and kilograms.
Teaching Aids
Small digital weighing scale; Weights of 100 g, 200 g, 250 g and 500 g
Activity
Set up a weighing station in the classroom with a digital weighing scale.
Ask the students to work in groups.
Instruct the students of each group to come up to the weighing station and place the gram weights on the scale to make up an exact weight of 1 kg. Once a group gets 1 kg on the scale, ask them to write the measures of the weights used in their notebooks. For example, if a group uses two 500 g weights, they will write 500 g, 500 g = 1 kg. Once all the groups do the activity, ask them all to add the weights.
Discuss their answers and ask them to find a relation between grams and kilograms. Also, emphasise that there can be multiple combinations of grams that can be used to make 1 kg but the total weight in grams will always add up to 1000 g to make 1 kg.
Ask them to write the relation in their notebooks. Give the students some measures in kg like 4 kg and 7 kg. Ask them to convert the weights to grams in their notebooks. Discuss the answers.
Extension Idea
Ask: What is heavier: A bag weighing 5 kg 650 g or a bag weighing 5700 g?
Say: 5 kg 650 g = 5000 g + 650 g = 5650 g. 5700 > 5650; hence, the bag weighing 5700 g is heavier.
Learning Outcomes
Students will be able to estimate and measure the capacity of a container.
Teaching Aids
Measuring cups for 1 L and 100 mL; Water bottles of capacity 200 mL, 700 mL and 1 L with no labels; Bottle of water
Activity
Show the students a measure of 10 mL of water using the millilitre cup and then 500 mL of water and 1 L of water using a litre cup.
Ask the students to work in groups. Distribute the measuring cups, water bottles and empty bottles of different capacities to each group.
Instruct the students to first estimate the capacity of each empty bottle and write down their estimates in their notebooks. Next, they should measure the capacity of each bottle by filling it with water and then pouring the water into the measuring cups. Tell them to read the measure written on the measuring cup at the level up to where it is filled. Ask them to write the actual capacity next to the estimated one and see how close these were to their estimates.
Ask questions like: Which of these has a capacity of more than 500 litres: a bathtub, a bucket of water or a large swimming pool?
Converting Between Units of Capacity
Learning Outcomes
Students will be able to convert between litres and millilitres.
Teaching Aids
Measuring cups for 1 L and 100 mL; Bottle of water
Activity
Ask the students to work in groups. Distribute the measuring cups and a bottle of water to each group. Instruct the students to use the 100 mL cup to pour water into the 1 L cup as many times as it takes to fill the 1 L cup and count the number of times the 100 mL cup was used.
Discuss how many 100 mL cups were used to fill the 1 L cup. Ask the students to multiply the capacity of 1 small cup by the total number of cups used to find the relation between litres and millilitres in their notebooks.
Ask questions like: You have a water bottle that holds 500 mL and your friend has a bottle that holds 1 2 L. Whose bottle holds more water?
Extension Idea
Ask: What is more: 4 L 540 mL or 5440 mL?
Say: 4 L 540 mL = 4000 mL + 540 mL = 4540 mL. 5440 > 4540; hence; 5440 mL is more.
Answers
1. Converting Between Units of Length
1. 6 km to m.
1 km = 1000 m
So, 6 km = 6 × 1000 m = 6000 m
2. 3 m to cm.
1 m = 100 cm
So, 3 m = 3 × 100 cm = 300 cm
2. Solving Problems on Length
Do It Together
The gardener needs to measure a length of 600 cm.
We know that, 100 cm = 1 m
So, 600 cm = 6 m
The gardener needs to use a metre-long stick 6 times to mark the lines.
5. Converting Between Units of Capacity
Do It Together
Total juice in the bottle = 2 L
We know that 1 L = 1000 mL.
So, a bottle of 2 L capacity contains 2 × 1000 = 2000 mL.
So, if Rima poured 200 mL into a glass, the remaining juice in the bottle is:
2000 mL – 200 mL = 1800 mL
Remember, 1000 mL = 1 L.
So, 1800 mL = 1 L 800 mL. This will be the juice remaining in the bottle.
6. Solving Problems on Capacity
Do It Together
1.
3. Converting Between Units of Weight
1. 4 kg to g.
1 kg = 1000 g
So, 4 kg = 4 × 1000 g = 4000 g
2. 6 g to mg.
1 g = 100 mg
So 6 g = 6 × 100 mg = 600 mg
4. Solving Problems on weight
Do It Together
1.
2. Total weight of fruit = 1 kg + 3 kg + 2 1 2 kg
So, Raj bought a total of 6 1 2 kg fruit today.
3. 1 kg of pears = ₹220
The price of 2 kg of pears = 2 × ₹220 = ₹440
2. 400 mL + 500 mL + 700 mL + 300 mL = 1900 mL
3. To get 1000 mL of coloured water in one jug, Rehaan would need to empty jug c and d completely.
Perimeter and Area 11
Learning Outcomes
Students will be able to:
find the perimeter of different shapes using a piece of string. find the perimeter of different shapes on squared paper. find the length of the missing side of a polygon, when the perimeter and length of the other sides are given. find the area of irregular shapes on squared paper.
Alignment to NCF
C-3.1: Measures in non-standard and standard units and evaluates the need for 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-4.1: Discovers, understands, and uses formulae to determine the area of a square, triangle, parallelogram, and trapezium and develops strategies to find the areas of composite 2D shapes
Let’s Recall
Recap to check if students know how to convert length from one unit to another. Ask students to solve the questions given in the Let’s Warm-up section.
Vocabulary
perimeter: total distance covered along the boundary of a closed figure area: total space covered by a closed figure
Teaching Aids
Shape cutouts; String; Ruler; Triangle and rectangle cutouts; Squared paper with squares of 1 cm; Cutout of a figure drawn with lengths written inside and one missing length; Coloured pencils; Leaves from the playground
Chapter: Perimeter and Area
Finding Perimeter Using a String
Learning Outcomes
Students will be able to find the perimeter of different shapes using a piece of string.
Teaching Aids
Shape cutouts; String; Ruler
Activity
Imagine Maths Page 173
Instruct the students to pick up their Imagine Mathematics book. Ask them to move their fingers around the edges of the front of the book. Explain that this is the boundary of the front cover and that the distance around the boundary is called its perimeter. Discuss how a string can be used to find the perimeter of irregular shapes.
Divide the class into group of 3. Give each group a string and 3 shape cut outs.
Ask each group to take shape 1 cut out and place the string at one end of the shape, ask them to measure the length until the string reaches the starting point again.
Shape 1
Shape 2
Ask the students to note down the perimeter in their notebooks.
Shape 3
Ask the group to repeat the steps for shape 2 and shape 3. Discuss the answers that each group has got.
Ask question such as, “Which shape had a larger perimeter?”
Extension Idea
Ask: The length of a string used to measure shape A is 12 cm. If half the length of string is used to measure shape B, what is the perimeter of shape B?
Say: Length of sting used to measure perimeter of shape A is 12 cm. Half of 12 is 6. So, the perimeter of shape B is 6 cm.
Finding Perimeter Using Squared Paper
Learning Outcomes
Students will be able to find the perimeter of different shapes on squared paper.
Teaching Aids
String; Ruler; Triangle and rectangle cutouts; Squared paper with squares of 1 cm
Activity
Discuss with the students how to find the perimeter of shapes drawn on squared paper. Instruct the students to form groups of 3. Distribute the squared paper.
Imagine Maths Page 174
Ask the students to draw the outline of a figure on squared paper and pass that figure to the group next to them. Each group will then find the perimeter of the shape on the squared paper. Discuss the answers with the students. Then, distribute the shape cutouts. Instruct the students to measure the perimeter of the shapes using a piece of string and a ruler. Ask them to write the measures in their notebooks.
Extension Idea
Ask: How will you draw a figure with a perimeter of 20 units on squared paper? Draw figures to show.
Say: Different figures can be drawn with a perimeter of 20 units. E.g. we can draw a rectangle by shading 7 squares along the length and 3 squares along the width.
Finding Perimeter of Polygons
Learning Outcomes
Imagine Maths Page 175
Students will be able to find the length of the missing side of a polygon, when the perimeter and length of the other sides are given.
Teaching Aids
Cutout of a figure drawn with lengths written inside and one missing length
Activity
Discuss how to find the length of a missing side when the lengths of the remaining sides and the perimeter are given.
Distribute the cutouts of a figure drawn with lengths written inside and one missing length. Instruct the students to find the length of the missing side in the cutout and write the answer in their notebooks.
Ask questions like, “What is the missing length? How did you find it?”
Finding Area Using Squared Paper; Area of Irregular Shapes
Learning
Outcomes
Imagine Maths Page 179
Students will be able to find the area of irregular shapes on squared paper with squares of 1 cm.
Teaching Aids
Squared paper; Coloured pencils; Leaves from the playground
Activity
Distribute sheets of squared paper to the students. Instruct the students to form groups of 3. Ask them to go out to the playground and pick 2 leaves. In the class, ask them to place the leaves on the squared paper, draw their outlines by tracing them, and shade the figures. Then, they will find the area of each figure drawn by taking the full and more than full shaded squares as 1, two halves as 1 and ignoring the less than half-shaded squares. Finally, they will write the area in their notebooks.
Extension Idea
Ask: The given figure shows a grass patch around a swimming pool. Which of these have more area?
Say: The area of the swimming pool is 6 square units and the area of the grass patch is 10 square units. So, the grass patch has more area.
Answers
1. Finding Perimeter Using a String
Shape A has a greater perimeter than shape B.
2. Finding Perimeter Using Squared Paper
A: Perimeter = 12 units
B: Perimeter = 16 units
3. Finding Perimeter of Polygons
1. Perimeter = 5 cm + 13 cm + 5 cm + 17 cm + 12 cm = 52 cm
2. Length of missing side + 3 cm + 5 cm + 2 cm + 2 cm + 2 cm + 5 cm + 3 cm = 34 cm
4. Finding Area Using Squared Paper
5. Area of Irregular Shapes
Think and Tell Yes, we can trace any small shape on the grid and find its area by counting squares inside the shape. Do It Together
1.
Number of units covered by the shape = 18
Area of the shape = 18 square cm
Learning Outcomes
Students will be able to: read time to the nearest minute and write it in 2 ways. read time on a 24-hour clock and write it as 00:00. convert time from a 12-hour clock to a 24-hour clock and vice versa. find elapsed time in hours and minutes and convert between hours and minutes. find elapsed time in days, weeks, months and years.
Alignment to NCF
C-8.10: Performs simple measurements of time in minutes, hours, days, weeks, and months
Let’s Recall
Recap to check if students know that different activities can be measured in hours, minutes and seconds. Ask students to solve the questions given in the Let’s Warm-up section.
Vocabulary
analogue clock: a clock that shows the time by the hour, minute and second hands duration: the time from the start to the end of an activity elapsed time: time that has passed from the start of an event till the end of an event
Teaching Aids
Large clock with movable hands; Paper plates; Markers; Big and small straws; Glue sticks; Analogue clock in 24-hour format with movable hands; Analogue clock in 12-hour format with movable hands; Puzzle interlocking cards; Clock with minute markings; 2 sets of cards with the time written in minutes on one set and the corresponding time in hours on the other set; Classroom calendar; School’s event calendar
Chapter: Time
Reading Time on a Clock
Learning Outcomes
Students will be able to read time to the nearest minute and write it in 2 ways.
Teaching Aids
Large clock with movable hands; Paper plates; Markers; Big and small straws; Glue sticks
Activity
Show students the clock with the time set to 12 o’clock. Discuss how to read the time while moving the hour and minute hands. Show a few times, such as 5:35, on the clock.
Ask questions like: How do you know if the given time is 5:35 in the morning or evening? Discuss how to read time with a.m. and p.m. and their meanings.
Ask students to work in groups of 3. Distribute paper plates, markers, big and small straws and glue sticks to each group.
Instruct the groups to mark the numbers as in a clock on the plate using markers. Ask them to show 5 o’ clock in the morning, 12:25 in the afternoon and 10:10 at night using the small and big straws as the hour and minute hands. Ask them to write the times in a.m. and p.m. in their notebooks.
Extension Idea
Ask: Tanya starts from home at 11:45 a.m. and reaches the market after 2 hours. Will she reach the market in a.m. time or p.m. time? What time will she reach?
Say: Two hours after 11:45 is 1:45, which is past noon. So, it will be p.m. time.
Reading 24-hour Clock
Learning Outcomes
Students will be able to read time on a 24-hour clock and write it as 00:00.
Teaching Aids
Analogue clock in 24-hour format with movable hands
Activity
Show students the 24-hour clock.
Ask questions like: What are the 2 formats of time that you see on this clock?
Imagine Maths Page 191
Discuss the differences and similarities in the time formats. Explain that this clock shows the time in the 12-hour as well as the 24-hour format. Write on the board: 1:00 p.m. = 13:00 hours.
Ask students to work in groups. Distribute the 12-hour clocks created by them in the previous lesson.
Instruct them to convert the 12-hour clock to a 24-hour clock. Ask them to tell time in the 24-hour format for different times in the 12-hour format, such as 6 o’ clock in the evening, 10 p.m. at night, 1 o’ clock after midnight, by looking at the clocks created. Instruct students to note down each 12-hour time and its respective 24-hour format in their notebooks.
Changing 12-hour to 24-hour Clock Time;
Changing 24-hour Clock to 12-hour Clock Time
Learning Outcomes
Imagine Maths Page 192
Students will be able to convert time from a 12-hour clock to a 24-hour clock and vice versa.
Teaching Aids
Analogue clock in 12-hour format with movable hands; Puzzle interlocking cards
Activity
Show students the 12-hour clock. Ask them if they can tell the time in the 24-hour format. Explain on the board how to change 11:30 a.m. and 11:30 p.m. into the 24-hour format and vice versa.
Ask students to work in groups of 3. Distribute the puzzle interlocking cards to each group.
Ask students to look at the time in the 24-hour format and look for the same time in the 12-hour format and word format. Ask them to join the 3 interlocking cards to complete one puzzle. The group that completes all the puzzles first wins!
Elapsed
Time
Learning Outcomes
Students will be able to convert time between hours and minutes.
Teaching Aids
Half past two 14:30 hours Quarter to 7 18:45 hours
Imagine Maths Page 195
Clock with minute markings; 2 sets of cards with the time written in minutes on one set and the corresponding time in hours on the other set
Activity
Show the time on a 12-hour clock. Explain that 1 minute passes when the minute hand moves from one marking to another. The minute hand moves 1 full cycle, while the hour hand moves 1 number to the next, showing that an hour has passed.
Ask students to count the number of minute markings on the clock.
Discuss how to change hours into minutes and vice versa. Take the students out to an open ground.
Ask students to work in 2 groups. Distribute 2 sets of cards with the time written in minutes on one set to group 1 and the corresponding time in hours on the other set to group 2. Ask students to look at the time card with them and look for their partner having the same time card in another unit. Ask them to stand with their partners once they have found them.
Repeat the activity by shuffling and exchanging the cards among the groups.
Extension Idea
Ask: Rohan’s school starts at 8:00 a.m. and ends at 2:30 p.m. How many minutes does he spend in school? Say: There are 6 hours 30 minutes between 8:00 a.m. and 2:30 p.m. So, Rohan spends 6 × 60 minutes + 30 minutes = 390 minutes in school.
Time in Days, Weeks, Months and Years
Learning Outcomes
Students will be able to find elapsed time in days, weeks, months and years.
Teaching Aids
Classroom calendar; School’s event calendar
Activity
Begin by showing students a classroom calendar.
Ask questions like: How many weeks are there in a month? How many months are there in a year? How many days are there in a year? Instruct students to write the relations between days, weeks, months and years in their notebooks. Ask students to work in groups. Distribute the school’s event calendar to each group.
Instruct them to look at the calendar and find the duration of the events. Ask questions like: For how many days will the Science Fair last? For how many weeks will the Art & Craft Show last? How many months are there between the Science Fair and the Music and Dance Show?
School Events
Science Fair
Starting Monday, June 25th to July 2nd
Maths Projects Exhibition
Starting Tuesday, August 20th to August 25th
Art & Craft Show
Starting Thursday, Nov. 26th to Dec. 8th
Music and Dance Show
Starting Monday, Jan. 29th to Feb. 4th
Instruct students to find the elapsed time in each case and write it in their notebooks.
Extension Idea
Ask: If it was a Wednesday on January 3, 2024, what was the date, month and year a week before?
Say: Move back on the calendar by 7 days. It was December 27, 2023 a week before.
Answers
1. Reading Time on a Clock
Do It Together
Wakes up at 6:10 a.m
Eats breakfast at 8:20 a.m. Comes back from school at 1:15 p.m Rides his bicycle at 4:15 p.m Goes to bed at 9:07 p.m.
Every morning, Manu wakes up at 6:10 a.m. He gets ready for school and eats breakfast by 8:20 a.m. Then, he walks to school. Manu’s school starts at 9 a.m. and ends at 1:15 p.m. Every evening, Manu rides his bicycle at 4:15 p.m. He goes to bed by 9:07 p.m.
2. Reading a 24-hour Clock
Do It Together
A 24-hour clock uses the numbers 13 to 23 to show the hours after 12 noon.
So, 18:00 is the time when it is 6 p.m.
3. Changing 12-hour to 24-hour Clock Time
Do It Together
Add 12 to the hour value: 5 + 12 = 17
Write the minutes as they are: 50
Replace p.m. with hours: 17:50 hours
Hence, Nira is on time
4. Changing 24-hour Clock to 12-hour Clock Time
Do It Together
Time on the railway station clock = 18:30 hours. It is more than 12. So, it is p.m. time.
Now, we subtract 12 from the first 2 digits: 18 − 12 = 6
Write down the minutes as they are. Thus, the time by a 12-hour clock is = 6:30 p.m
Total minutes in a week = 168 × 60 = 10,080 minutes
Do It Together
Total camping days = 15 days
Total days in December = 31 days. So, remaining days in December = 31 – 24 = 7 days.
In the next month of January, remaining days for camping = 15 – 7 = 8 more days.
The date of return = 8 January 2024
Learning Outcomes
Students will be able to: write money amounts in words and figures. convert between rupees and paise. read a bill and answer questions based on it. make a bill when the item name, unit cost of each item and number of items bought are given. make an expense list for the expenses incurred over a given period of time. solve word problems on adding, subtracting, multiplying and dividing money amounts.
Alignment to NCF
C-8.11: Performs simple transactions using money up to INR 100
Let’s Recall
Recap to check if students know about currency notes and coins, and how to count them to get an amount. Ask students to solve the questions given in the Let’s Warm-up section.
Vocabulary
money: the coins or paper notes of a country used to buy things bill: details of how much we need to pay for items or services bought expense: money spent on buying different things or services
Teaching Aids
5 envelopes with play money of 5 different amounts; Cards with rupee and paise amounts written; A stationery bill for 3 items; Printouts of a blank bill template; 5 items such as marbles, pencils, flowers, bottles and candies, with price tags; Slips with expenses written; Word problem sheets
Chapter: Money
Express Money in Words
Learning
Outcomes
Students will be able to write money amounts in words and figures.
Teaching Aids
5 envelopes with play money of 5 different amounts
Activity
Show students play money notes and coins adding up to ₹20.50. Discuss how to write the amount in 2 different ways—in figures as ₹20.50 and in words as twenty rupees and fifty paise.
₹20.50
Twenty rupees and fifty paise
Ask questions like: What do the numbers on the left and right of the dot show?
Ask students to work in groups. Distribute play money envelopes to each group. Instruct students to count the money in each envelope, one by one, and write the amount in words and figures in their notebooks. Ensure that each student in a group counts the money in one envelope.
Conversion Between Rupees and Paise
Learning
Outcomes
Students will be able to convert between rupees and paise.
Teaching Aids
Cards with rupee and paise amounts written
Activity
Imagine Maths Page 206
Ask: I want to buy a toffee for 100 paise. How much will it cost in rupees? Explain that 100 paise make ₹1 and 200 paise make ₹2.
Ask students to work in groups. Distribute the rupee cards and paise cards to each group.
Instruct students to pick a rupee card and write the amount in their notebooks. Then, they should convert the rupee amount to the paise amount and look for a card with a paise amount that matches the rupee amount. They must match all the rupee amount cards to their corresponding paise amount cards showing the same amount.
The group that matches all the cards first wins. Instruct students to write all the rupee and paise amounts in their notebooks.
Reading
Learning Outcomes
Students will be able to read a bill and answer questions based on it.
Teaching Aids
A stationery bill for 3 items
Activity
Show students the stationery bill. Discuss the different elements of a bill. Ask students to work in groups. Distribute a bill to each group. Ask them to read the details of the bill.
Ask questions like: When was this bill created? How many items were bought from the store? What was the cost of each brush? Which item cost the least? Instruct students to discuss these questions in groups and find the correct detail on the bill to answer each question. Also, ask them to find the total bill amount.
Teacher Tip: Students can be asked to bring a bill of purchase made by their parents to class and then read the bill in class to answer some general questions.
Extension Idea
Ask: If the total cost of 5 pencil cases is ₹500, what is the rate of 1 pencil case?
Say: The rate of 1 pencil case = ₹500 ÷ 5 = ₹100.
Making Bills
Learning Outcomes
Students will be able to make a bill when the item name, unit cost of each item and number of items bought are given.
Teaching Aids
Printouts of a blank bill template; 5 items such as marbles, pencils, flowers, bottles and candies, with price tags
Activity
Display a banner for ABC Stationers. Place the items on the table. Pretend to be the shopowner. Explain that the students will be the shoppers. Distribute the bill templates.
Ask students to work in groups of 5. Each group will send 1 student up to the table to pick 3 items that they need to buy.
Instruct each student to make a bill for the 3 items selected for their group. Ask them to discuss if they all got the same bill amount. Discuss the bill amounts at the end of the activity.
Extension Idea
Ask: If the cost of 2 kg of potatoes is ₹60, what is the cost of half a kg of potatoes?
Say: The cost of 2 kg of potatoes is ₹60, which means the cost of 1 kg is ₹30. So, the cost of half a kg of potatoes is ₹15.
Learning Outcomes
Students will be able to make an expense list for the expenses incurred over a given period of time.
Teaching Aids
Slips with expenses written
Activity
Instruct students to read about how to make expense lists on Page 212 of the Imagine Mathematics book, and draw a template in their notebooks.
Ask students to work in groups. Distribute the expense slips to each group. Instruct each group to imagine that they are going on a trip and have ₹2000. Ask them to make their own expense list using at least 3 expense slips they want to include in their list, making sure that the total amount does not exceed ₹2000.
Ask questions like: How much did you save after all the expenses?
Extension Idea
Ask: You have spent ₹1725 on a holiday. What would be your total expense if you go on 3 such trips, spending the same amount on each trip?
Say: Expense on 1 trip = ₹1725. So, the expense of 3 such trips would be 3 × ₹1725 = ₹5175.
Word Problems on Money Imagine Maths Page 214
Learning Outcomes
Students will be able to solve word problems on adding, subtracting, multiplying and dividing money amounts.
Teaching Aids
Word problem sheets
Activity
Distribute sheets with the given word problem written on each: I bought a bottle for ₹125, a lunch box for ₹100 and a book for ₹225. How much did I spend in all?
Instruct students to discuss in their groups: What is given? What is asked? How do we solve?
Ask students to solve the problem on the sheet. Repeat the activity with a new word problem.
I bought a bottle for ₹125, a lunch box for ₹100 and a book for ₹225. How much did I spend in all?
What is given?
What do we need to find?
How do we find?
1. Express Money in Words
Do It Together
1. True 2. False 3. True 4. False
2. Conversion Between Rupees and Paise Do It Together In Rupees In Paise
₹635.23 63523 paise
₹852.05 85205 paise
₹4126.24 412624 paise
₹5386.15 538615 paise
₹8256.32 825632 paise
3. Reading Bills
Do It Together
₹480.00
4. Making Bills
Do It Together
Bakery
1. Cost of 6 notebooks = ₹180.00 Cost of 1 notebook = ₹180 ÷ 6 = ₹30
2. Total amount to be paid = ₹80.00 + ₹50.00 + ₹180.00 + ₹70.00 + ₹100.00 = ₹480.00
3. Amount of change that Sudha will get = ₹500 – ₹480.00 = ₹20.00
Students will be able to: collect and organise data in a tally marks table. draw a pictograph to show data. read and interpret a pictograph and answer questions based on it. draw a bar graph to show data. read and interpret a bar graph and answer questions based on it. draw and interpret a pie chart.
Alignment to NCF
C-5.2: Selects, creates and uses appropriate graphical representations (e.g., pictographs, bar graphs, histograms, line graphs, and pie charts) of data to make interpretations
Let’s Recall
Recap to check if students know how to read and interpret a data table. Ask students to solve the questions given in the Let’s Warm-up section.
Vocabulary
data: a collection of facts such as numbers, words, measurements or observations data handling: a process of collection, organisation and representation of data in various forms tally mark: a vertical mark used to keep count pictograph: a table that shows the given data using pictures or symbols bar graph: a graph that shows information in the form of bars of different lengths pie chart: a graph in which a circle is divided into sectors and each represents a part of the whole
Teaching Aids
Sticky notes; Chart paper with a table drawn for collecting data; Square cut-outs in 4 different colours (brown, pink, green, orange); Chart paper; Chart paper with a pictograph to show app notifications of 4 people; Squared paper; Colour pencils; Sheets of paper with a bar graph drawn; Strips of paper; Pre-cut sectors of circles representing pie chart segments; Extra sections that do not fit into the pie chart; Glue
Chapter: Data Handling
Organising Data
Learning Outcomes
Students will be able to collect and organise data in a tally marks table.
Teaching Aids
Sticky notes; Chart paper with a table drawn for collecting data
Activity
Begin the activity by discussing the concept of tally marks and their use in data collection. Instruct students to work in groups.
Distribute sticky notes to students in each group.
Ask them to write down their favourite place out of Hill station, Sea Beach, Historical places and Grandparent’s home to visit. Read out each place written on the sticky notes, one by one, and instruct each group to work together to draw tally marks on the chart paper according to the data collected.
Once all the sticky notes have been read and tallied, review the completed tally chart as a group. Discuss the results, emphasising the place with the highest and lowest tallies.
Ask questions like: Which place got the maximum votes?
Creating Pictographs
Learning Outcomes
Students will be able to draw a pictograph to show data.
Teaching Aids
Square cut-outs in 4 different colours (brown, pink, green, orange); Chart paper
Activity
Start by discussing what a pictograph is and how it represents data using pictures or symbols.
Distribute square cut-outs of different colours representing ice creams to each student. Explain that each colour represents a different flavour: brown for chocolate, pink for strawberry, green for mint and orange for mango. Ask students about their favorite ice cream flavour. Instruct them to count the number of each flavour and record their counts in a pictograph. Ask students to draw the symbols for each flavour, according to the counts and to use a scale of 2 to draw the symbols.
Ask questions like: How many symbols did you draw to represent the number of students who liked chocolate ice cream? Strawberry ice cream? Mint ice cream?
Extension Idea
Ask: If 40 students like chocolate ice cream, and the scale changes to 1 symbol = 10 ice creams, how many symbols will you draw to show the data?
Say: You can draw 40 ÷ 10 = 4 symbols to show the number of students who like chocolate ice cream.
Pictograph
Learning Outcomes
Students will be able to read and interpret a pictograph and answer questions based on it.
Teaching Aids
Chart paper with a pictograph to show app notifications of 4 people
Activity
Instruct students to work in groups. Distribute 1 chart paper (with a pictograph drawn on it) to each group.
Instruct each group to examine the pictograph and discuss among themselves how it represents the app notifications for Gary, Phillip, Vivian and Shelly. Encourage them to count the symbols and make sense of the data.
Ask questions like: Which person had 24 notifications this week? How many notifications were received by Shelly and Gary? Which person had the most notifications this week?
Extension Idea
App Notifications in 1 Week
= 4 notifications
Ask: Create a question based on the pictograph drawn on the chart paper. Say: You can create multiple questions on the given pictograph. One such question could be: How many more notifications did Shelly receive than Vivian?
Creating
Bar Graphs Imagine Maths Page 229
Learning Outcomes
Students will be able to draw a bar graph to show data.
Teaching Aids
Squared paper; Colour pencils
Activity
Start by discussing how bar graphs are useful for displaying information. Introduce the various components of a bar graph. Draw a table on the board showing the miles run by Sarvesh.
Miles Run 15 20 25 30 22 18
Divide students into groups, ensuring each group has squared paper and colour pencils. Instruct each group to create a bar graph on the squared paper based on the provided data. Explain that each square on the grid represents 1 mile, and students should colour in the appropriate number of squares to represent the miles Sarvesh ran for each month. Emphasise the importance of labelling the graph accurately, including the months and a scale for the number of miles.
Gary
Phillip
Vivian
Shelly
Key:
Learning Outcomes
Students will be able to read and interpret a bar graph and answer questions based on it.
Teaching Aids
Sheets of paper with a bar graph drawn; Strips of paper
Activity
Review key components of a bar graph such as axes, bars and labels. Distribute 1 sheet of paper (with a bar graph drawn) to each group.
Ask students 2 questions based on the provided bar graph, such as: How many students chose Maths as their favourite subject? Which subject has the fewest number of students who chose it as their favourite?
Provide each group with 2 strips of paper. Instruct each group to create 2 questions related to the bar graph.
Have each group pass their 2 question strips to another group. Instruct each group to answer the questions they received from another group.
Pie Charts
Learning Outcomes
Students will be able to draw and interpret a pie chart.
Teaching Aids
Maths Page 233
Pre-cut sectors of circles representing pie chart segments; Extra sections that do not fit into the pie chart; Chart Paper; Glue
Activity
Start by discussing the purpose of pie charts in visually representing data and the importance of accurately constructing and interpreting them.
Instruct students to work in groups. Provide each group with pre-cut sectors of circles that represent pie chart segments. Include some extra sections that do not fit into the pie chart. Instruct students to arrange the sectors to construct a pie chart to show the favourite book genre of students where 1 4 like storybooks, 1 2 like science fiction, and the remaining like non-fiction.
Have students glue the sectors onto a chart paper to create their pie charts. Encourage creativity in colourcoding and labelling each section.
Ask each group to come up with 2 questions related to their pie chart, such as: What genre is the most popular among the surveyed individuals? What fraction of respondents prefer non-fiction books?
Groups swap questions with another group to answer. Instruct each group to answer the questions they received.
Extension Idea
Ask: If the total number of students surveyed was 80, how many students liked storybooks?
Say: Number of students who liked storybooks = 1 4 × 80 = 20 students.
Answers
1.
Organising Data
Do It Together
How many students voted for cricket? 17
How many students voted for basketball? 10
7 students voted for football. Represent this in tally marks:
How many students in total voted for their favourite sport? 43
2. Creating Pictographs
Do It Together
One represents 50 wall clocks.
Monday → 300 clocks = 300 ÷ 50 = 6
Tuesday → 350 clocks = 350 ÷ 50 = 7
4. Creating Bar Graphs Do It Together
Label at horizontal axis: Class
Label at vertical axis: Number of students
Wednesday → 250 clocks = Thursday → 400 clocks = Friday → 300 clocks = Saturday → 200 clocks = Day Number of Clocks Monday
250 ÷ 50 = 5
400 ÷ 50 = 8
300 ÷ 50 = 6
200 ÷ 50 = 4
3. Interpreting Pictographs
Do It Together
1. Number of boxes sold on Tuesday = 1 × 5 = 5
Difference in the number of candy boxes sold on Monday and on Tuesday = 25 – 5 = 20
Thus, 20 more candy boxes were sold on Monday than on Tuesday.
2. Total candy boxes = 23 × 5
So, the number of candy boxes sold in the entire week = 115
5. Interpreting Bar Graphs
Do It Together
2. On which day were the most cars parked? Wednesday
3. Number of cars parked on Tuesday = 4
Number of cars parked on Thursday = 6
Difference between the number of cars parked in a lot on Tuesday and Thursday = 6 – 4 = 2
6. Pie Charts
Do It Together
1. In a circle chart, the large (L) size covers the largest part of the circle.
So, the large size is sold the most.
2. In a circle chart, small and medium sizes cover equal parts of the circle.
So, small (S) and medium (M) sized T-shirts are sold in equal numbers.
3. So, the number of large-sized T-shirts = 1 2 of 600 = 300.
SolutionsAnswers
Chapter 1
Let’s Warm–up
1. 32 → 2 × 1 = 2
Number name = thirty-two
2. 548 → 8 × 1 = 8
Number name = five hundred forty-eight
3. 876 → 8 × 100 = 800
Number name = eight hundred seventy-six
4. 4563 → 6 × 10 = 60
Number name = four thousand five hundred sixty-three
5. 9958 → 9 × 1000 = 9000
Number name = nine thousand nine hundred fifty-eight
Do It Yourself 1A
1. a. 36789 has 8 tens.
b. 47690 has 9 tens.
c. 32478 has 7 tens.
d. 67698 has 9 tens.
Only 32478 has 7 tens. So, option c is correct.
2. a. Value in thousands place in 45,687 = 5
b. Value in thousands place in 65,690 = 5
c. Value in thousands place in 78,483 = 8
d. Value in thousands place in 96,152 = 6
Since 78,483 has the greatest value in the thousands place, option c is correct.
3. a. 56938
Place value of 9 = 9 × 100 = 900
Face value of 9 = 9
Expanded form = 50000 + 6000 + 900 + 30 + 8
b. 65899
Place value of 6 = 6 × 10000 = 60,000
Face value of 6 = 6
Expanded form = 60000 + 5000 + 800 + 90 + 9
c. 25401
Place value of 5 = 5 × 1000 = 5,000
Face value of 5 = 5
Expanded form = 20000 + 5000 + 400 + 0 + 1
d. 89376
Place value of 6 = 6 × 1 = 6
Face value of 6 = 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. Twelve thousand three hundred twenty-one = 12,321
b. Thirty-four thousand six hundred = 34,600
c. Seventy-eight thousand five = 78,005
d. Fifty thousand ten = 50,010
6. a. 40000 + 6000 + 300 + 20 + 2 = 46,322
b. 50000 + 0 + 700 + 50 + 7 = 50,757
c. 70000 + 3000 + 0 + 60 + 1 = 73,061
d. 90000 + 6000 + 400 + 0 + 8 = 96,408
7. Distance between India and USA = 13568 km 13568 can be written as 13,568 using periods.
13,568 = thirteen thousand five hundred sixty-eight
8. Number of people who visited the Isle Royal National Park = 28,965
In words: Twenty-eight thousand nine hundred sixty-five
Expanded form: 20000 + 8000 + 900 + 60 + 5
Challenge
1. The digit in tens place = 4
The digit in thousands place = 8
The digit at one’s place = (4 + 8) ÷ 2 = 12 ÷ 2 = 6
The digit at hundreds place = 6 – 6 = 0
So, the required number = 8046
Do It Yourself 1B
1. a. Number: 5,84,736
The place value of digit 5 is 500000, 8 is 80000, 4 is 4000, 7 is 700, 3 is 30 and 6 is 6.
4. a. Four lakh eighteen thousand three hundred = 4,18,300
b. Six lakh twenty thousand = 6,20,000
c. Eight lakh five thousand two hundred sixty-four = 8,05,264
d. Seven lakh twenty thousand fifty = 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. Approximate distance to the Moon = 3,84,400 kilometres
3,84,400 = 3,00,000 + 80,000 + 4000 + 400 + 0 + 0
7. Length of line that can be drawn by a pencil = 1,84,800 feet long
The place value of 1 = 100000, 8 = 80000, 4 = 4000, 8 = 800, 0 = 0, 0 = 0
In words: One lakh eighty-four thousand eight hundred.
Challenge
1. The digits in the tens and the thousands place is 3 and 9.
The digit in the ones place = 3 + 3 = 6
The digit in the ten thousand place = 4 + 3 = 7
The digit in the hundred place = 7 – 3 = 4
The digit in the lakhs place = 9
The number is 9,79,436
Do It Yourself 1C
1. a. 41,700 has more ten-thousands, so, 24,614 < 41,700.
b. 50,092 has the same number of ten-thousands as 51,320 but fewer thousands. So, 50,092 < 51,320.
c. 72,184 and 72,157 have the same number of ten thousands, thousands, hundreds. But 72,184 has more tens. So, 72,184 > 72,157.
d. 3,15,720 has less number of lakhs than 4,13,265. So, 3,15,720 < 4,13,265.
e. 8,74,126 and 8,24,510 have the same number of lakhs. But 8,74,126 has more ten thousands. So, 8,74,126 > 8,24,510.
f. 4,35,071 and 4,35,261 have the same number of lakhs, ten thousands and thousands.
But 4,35,071 has less number of hundreds. So, 4,35,071 < 4,35,261.
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. 4, 2, 7, 6, 5
Smallest number = 24,567
Greatest number = 76,542
c. 5, 0, 2, 1, 7, 4
Smallest number = 1,02,457
Greatest number = 7,54,210
4. a. 2, 1, 7, 4, 9
Smallest number = 1,12,479
Greatest number = 9,97,421
c. 6, 9, 1, 2, 7
Smallest number = 1,12,679
Greatest number = 9,97,621
5. Calories Supriya ate = 15,248
b. 6, 1, 3, 7, 8
Smallest number = 13,678
Greatest number = 87,631
d. 8, 6, 2, 5, 9
Smallest number = 25,689
Greatest number = 98,652
b. 3, 8, 5, 0, 1
Smallest number = 1,00,358
Greatest number = 8,85,310
d. 8, 1, 0, 9, 7
Smallest number = 1,00,789
Greatest number = 9,98,710
Calories Supriya’s brother ate = 18,396
Since, 15,248 < 18,396.
So, Supriya’s brother consumed more calories this week.
6. Amount given by father = 11,200
Amount required by Anna to buy books = 11,700
Since, 11,200 < 11,700
So, the amount Anna has is not sufficient to buy the books.
7. The depths arranged in descending order are 36,161 > 27,840 > 23,810 > 23,740 > 18,264
So, the Pacific Ocean is the deepest ocean.
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. To form the smallest 6-digit number using the digits 2, 0, 5, we will place the digits in ascending order. However, 0 is the smallest digit and will form a 5-digit number. Therefore, we will place the second largest number 2, repeat the smallest number = 0, and place the largest number at the end = 5. The answer will be 2,00,005.
Do It Yourself 1D
1. a. 134 is between 130 and 140, but is closer to 130. So, 134 rounded off to the nearest 10 is 130.
b. 569 is between 560 and 570, but is closer to 570. So, 569 rounded off to the nearest 10 is 570.
c. 161 is between 160 and 170, but is closer to 160. So, 161 rounded off to the nearest 10 is 160.
d. 1468 is between 1460 and 1470, but is closer to 1470. So, 1468 rounded off to the nearest 10 is 1470.
e. 47,121 is between 47,120 and 47,130, but is closer to 47,120.
So, 47,121 rounded off to the nearest 10 is 47,120.
2. a. 174 is between 100 and 200, but is closer to 200. So, 174 rounded off to the nearest 100 is 200.
b. 1653 is between 1600 and 1700, but is closer to 1700. So, 1653 rounded off to the nearest 100 is 1700.
c. 7610 is between 7600 and 7700, but is closer to 7600. So, 7610 rounded off to the nearest 100 is 7600.
d. 2447 is between 2400 and 2500, but is closer to 2400.So, 2447 rounded off to the nearest 100 is 2400.
e. 23,492 is between 23,400 and 23,500, but is closer to 23,500. So, 23,492 rounded off to the nearest 100 is 23,500.
3. a. 1653 is between 1000 and 2000, but is closer to 2000. So, 1653 rounded off to the nearest 1000 is 2000.
b. 6573 is between 6000 and 7000, but is closer to 7000. So, 6573 rounded off to the nearest 1000 is 7000.
c. 34,784 is between 34,000 and 35,000, but is closer to 35,000. So, 34,784 rounded off to the nearest 1000 is 35,000.
d. 87,301 is between 87,000 and 88,000, but is closer to 87,000. So, 87,301 rounded off to the nearest 1000 is 87,000.
e. 90,123 is between 90,000 and 91,000 but is closer to 90,000. So, 90,123 rounded off to the nearest 1000 is 90,000.
4. Number of plants at the monument = 23,912
23,912 can be rounded off to the nearest thousand as 24,000. Ramesh should order around 24,000 saplings.
5. Circumference of Earth = 40,075 kilometres 40,075 rounded off to the nearest 1000 is 40,000 km.
Challenge
1. As the number, when rounded off to the nearest 100, gives 2,56,300, the number is between 2,56,250 and 2,56,349. As the number, when rounded off to the nearest 10 gives 2,56,290. The number is between 2,56,285 and 2,56,294. A number within the given range that has a sum of all the digits as 30 is 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.
9. 4,85,345 rounded off to the nearest 1000 is 4,85,000. But according to Rohan, we get 4,85,300. That is why Rohan is not correct.
10. a. ₹25,879 < ₹25,907 < ₹54,768 < ₹97,463
Saree 4 < Saree 1 < Saree 3 < Saree 2
Saree 4 is the cheapest and Revathi can buy that saree
b. Approximate cost (Rounded off to the nearest 1000):
Saree 1 = ₹26,000
Saree 2 = ₹97,000
Saree 3 = ₹55,000
Saree 4 = ₹26,000
11. 78,342 < 85,674 < 1,56,810 < 3,67,185
a. English is the most used language.
b. Spanish is the least used language.
c. Spanish < Japanese < Chinese < English
12. Answer 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 population from smallest to largest?
Challenge
1. The smallest 6-digit number formed by using the digits 9, 0, 2, 4, 7 and 1 is 102479.
As 2 is placed in the hundreds place, the smallest 6-digit number that could be formed is: 104279.
Hence, the digit at the thousands place would be 4. The place value would be: 4 × 1000 = 4000, and the face value would be 4.
2. To make the largest 6-digit number that ends with digit 5 and reads the same forward and backward, the number will begin with the digit 5 only.
5 ___ ___ ____ ____ 5
To make the largest 6-digit number, we will use the largest 1-digit number = 9.
5 9 ___ ____ 9 5
As we need to use 3 digits, we will use the second largest 1-digit number = 8. Hence, the number will be 598895.
Case Study
1. Population of Macao = 7,04,149
Place values of 4 = 4000 and 40 Difference = 4000 – 40 = 3960 Thus option b is correct.
2. On writing the populations in Rahul’s data in descending order, we get 7,87,424 > 7,04,149 > 5,35,064 > 5,21,021 or Bhutan > Macao > Malta > Maldives. So, the most populated country in Rahul’s data is Bhutan.
3. On writing the populations in Megha’s data in ascending order, we get 3,75,318 < 6,23,236 < 8,13,834 < 9,36,375 Thus, the least populated country in Megha’s data is Iceland.
4. The populations can be arranged in descending order as: 3,75,318 < 5,21,021 < 5,35,064 < 6,23,236 < 7,04,149 < 7,87,424 < 8,13,834 < 9,36,375 The countries can be arranged as per the population in ascending order as: Iceland < Maldives < Malta < Suriname < Macao < Bhutan < Guyana < Fiji
5. The population of Fiji = 9,36,375 The population of Fiji to the nearest 1000 = 9,36,000
Chapter 2
Let’s Warm–up
1. a. 692 – 30 436
b. 355 + 345 662
c. 556 – 120 700
d. 666 – 66 740
e. 722 + 18 600
2. a. 455 – 5 = 450
b. 987 + 13 = 1000
c. 525 + 5 = 530
d. 879 – 9 = 870
e. 910 – 10 = 900
Do It Yourself 2A
1.
2.
6. Animals rescued last year = 1000
Animals rescued this year = 1000 + 1145 = 2145
2145 animals were rescued this year.
Total number of animals rescued = 2145 + 1000 = 3145 animals
7. Cars produced in 2021 = 45,821
Cars produced in 2022 = 45,821 + 1208 = 47,029
47,029 cars were produced in 2022.
8. Number of tulips that bloomed at first = 80,000
Number of tulips that bloomed later = 8574
Total number of tulips that were displayed = 80,000 + 8574 = 88,574
4. 57,801 – 13,456 = 9 7 10 11 5 7 8 0 1 – 1 3 4 5 6 4 4 3 4 5 So, 44,345 should be added to 13,456 to get 57,801.
5. 17,890 – 1829 = 8 10 1 7 8 9 0 – 0 1 8 2 9 1 6 0 6 1 So, 16,061 should be subtracted from 17,890 to get 1829.
6. 12,345 – 2335 = 1 2 3 4 5 – 0 2 3 3 5
1 0 0 1 0 So, 10,010 is 2335 less than 12,345.
7. Total tree species = 16,000 Tree species that store carbon = 1600
Total trees that do not store carbon = Total tree species − Tree species that store carbon =16,000 − 1600 = 14,400
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. Rahul thinks of a number X. Rahul’s friend, Amit, thinks of a number Y which is 1234 more than Rahul’s number X. So, Y = 1234 + X
2. The weight of the chocolates sold (in pounds) = 17,645
The weight of candies sold (in pounds) = 24,891
Their estimated total weight = 18,000 + 25,000 = 43,000 pounds, which is more than 40,000 pounds. Hence, the estimate is right.
Case Study
1. Number of trees planted each week = 2750
Number of trees planted in two weeks = 2750 + 2750 = 5500
Thus, option c is correct.
2. Number of trees planted each week = 2750
Number of trees planted in 4 weeks = 2750 + 2750 + 2750 + 2750 = 11,000
Number of trees cut each week = 1250
Number of trees cut in 4 weeks = 1250 + 1250 + 1250 + 1250 = 5000
Difference between the number of trees planted and cut in 4 weeks = 11,000 – 5000 = 6000
Thus, option b is correct.
3. At the start of the project, the forest had 2750 trees. After 4 weeks, the forest had 6000 trees. So, the statement that the forest has fewer trees than before the project started, is false.
4. Number of trees planted each week = 2750
Number of trees planted in 4 weeks = 11,000
Number of trees cut each week = 1250
Number of trees cut in 4 weeks = 5000
Total number of trees in the forest after 4 weeks = 11,000 − 5000 = 6000
Increase in number of trees planted each week = 500
So, after the increase, the total number of trees planted each week = 2750 + 500 = 3250
Number of trees planted in 4 weeks after the increase = 3250 + 3250 + 3250 + 3250 = 13,000
After the increase, the total number of trees in the forest after 4 weeks = 13,000 − 5000 = 8000
So, the forest will have 8000 − 6000 = 2000 more trees than it started with.
This change helps the forest support both wildlife and human communities, improves the air and soil quality and regulates water and climate.
Chapter 3
Let’s Warm–up
1. 8 + 8 + 8 + 8 + 8 = 8 × 4. False
2. 23 × 0 = 0. True
3. 30 is the product of 10 and 5. False
4. 7 multiplied by 11 is 77. True
5. There are 9 petals in each flower. We have 9 such flowers. Therefore, we have 72 petals in all. False
Do It Yourself 3A
1. a. 233 × 2 = 466
b. 622 × 4 = 2488
c. 2001 × 7= 14,007
d. 4011 × 9 = 36,099
2. a. 313 × 3
313 = 300 + 10 + 3
3 3 × 300 = 900
900 + 30 + 9 = 939
b. 802 × 9
802 = 800 + 0 + 2
9 9 × 800 = 7200
7200 + 0 + 18 = 7218
c. 1002 × 2
1002 = 1000 + 0 + 0 + 2
2000 + 0 + 0 + 4 = 2004
d. 2908 × 4
2908 = 2000 + 900 + 0 + 8
4 4 × 2000 = 8000 4 ×
8000 + 3600 + 0 + 32 = 11,632
3. a. 2
1 9 3 × 3 5 7 9 b. 2
4. 3000 = 3 hearts
200 = 2 green rectangles
10 = 1 star
So, 3000 + 20 + 10 = 3210
3 2 1 0 × 2
6 4 2 0 × 2 = 6420
5. Answers may vary. Sample answer:
Rohit bought 345 pencils each costing 8 rupees. How much did Rohit pay for the pencils?
The product of 375 and 34 should have 0 in the ones place. 5 multiplied by 0 is 0, but the product of 375 and 340 is 1,27,500 which is not equal to 1,29,000.
3 7 5
× 3 4 0
0 0 0 0
+ 1 5 0 0
+ 1 1 2 5 1 2 7 5 0 0
4 multiplied with 5 gives us 20, which has 0 in its ones place.
375 × 344 = 1,29,000
3 7 5
× 3 4 4
1 5 0 0
+ 1 5 0 0
+ 1 1 2 5
1 2 9 0 0 0
Hence, the digit at ones place is 4.
4. Water consumed by 1 person in 1 day = 200 L
Water consumed by 4 people in 1 day = 4 × 200 L = 800 L
Water consumed by 4 people in 365 days = 365 × 800 L
= 2,92,000 L
So, the family consumes 2,92,000 litres of water in a year.
5 4 3 6 5
× 8 0 0
0 0 0
+ 0 0 0 0 0
2 9 2 0 0 0
2 9 2 0 0 0
Do It Yourself 3D
1. Entry fee at the club = ₹423
Number of tourists = 9
Amount paid = 423 × 9
They will pay ₹3807.
2. Number of days Jupiter has in a year = 4333
Number of years = 5
Number of days in 5 Jupiter years = 4333 × 5
There will be 21,665 days in 5 years.
3. Number of calories Riti burns in a day = 427
Number of days in January = 31
Number of calories burnt in January = 427 × 31
Riti will burn 13,237 calories in January.
4. Number of members = 28
Budget of tickets = ₹1,00,000
Cost of 1 plane ticket = ₹3879
Total cost of the tickets = 3879 × 28
The total cost is ₹1,08,612.
Hence, the total cost of tickets won’t be in their budget.
5. a. Number of seats = 755
Number of shows for Horror Show = 32
Number of people who watched the Horror Show = 755 × 32
Hence 24,160 people watched the Horror Show.
b. Number of shows for ’Fun with Mary’ = 45
Total number of seats = 755
Number of seats unused = 250
Number of seats used = 755 – 250 = 505
Total number of people if all seats were used = 505 × 45
Total number of people who watched ’Fun with Mary’ is 22,725.
6. Bundle of notes with Karan = ₹200
Number of notes with Karan = 86
Number of notes Karan gave to Ramesh = 24
Notes left with Karan = 86 − 24 = 62
Amount left with Karan = 200 × 62
There are ₹12,400 left with Karan.
7. Number of pens purchased in February = 350
Number of pens purchased in November = 265
Total pens in 2017 = 350 + 265 = 615
There are 615 pens in all.
Cost of 1 pen = ₹9
Amount spent on 615 pens = 615 × 9
The total cost of the pens is ₹5535.
Challenge
1. Number of rows of crates = 213
Number of crates in each row = 187
2 1 3
× 1 8 7
1 4 9 1
+ 1 7 0 4 0
+ 2 1 3 0 0
3 9 8 3 1
Number of boxes of items in each crate = 126
3 9 8 3 1
× 1 2 6
2 3 8 9 8 6
+ 7 9 6 6 2 0
+ 3 9 8 3 1 0 0
5 0 1 8 7 0 6
Number of defective boxes = 1 10
Number of defective boxes 5,01,871 =
Number of non-defective boxes
= Total number of boxes – number of defective boxes = 50,18,706 – 5,01,871 = 45,16,835 boxes
So, 4,516,835 boxes are not defective.
Do It Yourself 3E
1. a. 235 × 13
Rounding-off 235 and 13 to the nearest 10:
235 = 240 and 13 = 10
240 × 10 = 2400
b. 582 × 84
Rounding-off 582 and 84 to the nearest 10.
582 = 580
84 = 80
580 × 80 = 46,400
c. 809 × 96
Rounding-off 809 and 96 to the nearest 10: 809 = 810
96 = 100
810 × 100 = 81,000
d. 409 × 962
Rounding-off 409 and 962 to the nearest 10:
409 = 410
962 = 960
410 × 960 = 3,93,600
e. 849 × 167
Rounding-off 849 and 167 to nearest 10.
849 = 850
167 = 170
850 × 170 = 1,44,500
f. 655 × 845
Rounding-off 655 and 845 to the nearest 10:
655 = 660
845 = 850
660 × 850 = 5,61,000
2. a. 169 × 74
Rounding off to the nearest 100:
169 = 200
74 = 100
200 × 100 = 20,000
b. 518 × 96
Rounding off to the nearest 100:
518 = 500
96 = 100
500 × 100 = 50,000
c. 874 × 228
Rounding off to the nearest 100:
874 = 900
228 = 200
900 × 200 = 1,80,000
3. a. 109 × 54
Rounding off to the nearest 10
109 = 110
54 = 50
110 × 50 = 5500
Rounding off to the nearest 100
109 = 100
54 = 100
100 × 100 =10,000
b. 444 × 777
Rounding off to the nearest 10
444 = 440
777 = 780
440 × 780 = 3,43,200
Rounding off to the nearest 100
444 = 400
777 = 800
400 × 800 = 3,20,000
c. 976 × 862
Rounding off to the nearest 10
976 = 980
862 = 860
980 × 860 = 8,42,800
Rounding off to the nearest 100
976 = 1000
860 = 900
1000 × 900 = 9,00,000
4. Number of athletes = 110
Distance to be covered = 43 km
Estimated distance to be covered = 40 km
Total distance = 110 × 40 = 4400 km
So, the estimated distance is 4400 km.
Actual product = 110 × 43
So, the actual distance is 4730 km.
Thus, the actual distance is greater than the estimated distance.
5. Number of stands in the stadium = 17
Approximate number of stands (rounded off to 10) = 20
Number of seats in the stadium = 238
Approximate number of seats (rounded off to 10) = 240
Approximate number of bottles required = 240 × 20 = 4800
Approximately, they will put out 4800 bottles.
Challenge
1. Number of books = 39 rounded off to 40
Cost of each book = ₹83 rounded off to ₹80
Estimated amount = 40 × ₹80 = ₹3200
Actual amount = 39 × ₹83 = ₹3237
8 3 × 3 9
7 4 7 + 2 4 9
3 2 3 7
The estimated amount will not be enough as the actual amount is more than the estimated amount.
The first two common multiples of 10 and 15 are: 30 and 60.
8. The multiples of 8 are 8, 16, 24, 32 and 40. The multiples of 9 are 9, 18, 27, 36 and 45.
27 is a multiple of 9, but it is not a multiple of 8. So, 27 is not a common multiple of 8 and 9.
9. The number of stickers in the packet = 50 Every 5th sticker is a special glittery sticker, so it is a multiple of 5. The glittery sticker will be the stickers that are on the following numbers: 5, 10, 15, 20, 25, 30, 35, 40, 45, 50.
10. Raj goes to meet his grandparents on every 4th day.
The dates on which Raj goes to meet his grandparents = 4, 8, 12, 16, 20, 24 and 28
Raj goes to the animal shelter on every 6th day. The dates on which Raj goes to the animal shelter = 6, 12, 18, 24 and 30
The dates on which he will go to both the places = 12 and 24
On 12th and 24th, he will go to meet his grandparents and the animal shelter.
Challenge
1. The multiples of 11 less than 160 are 11, 22, 33, 44, 55, 66, 77, 88, 99, 110, 121, 132, 143 and 154
The multiples of 7 are 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 77, 84, 91, 98, 105, 112, 119, 126, 133, 140, 147 and 154
The even multiples of 7 are 14, 28, 42, 56, 70, 84, 98, 112, 126, 140, 154, ...
The even common multiple of 7 and 11 is 154.
So, the number is 154
2. Answers may vary. Sample answer:
Sum of the numbers in each row and each column = 15 (Multiple of 5)
6 1 8
7 5 3
2 9 4 Do It Yourself 5B 1.
1 ×
Thus, the factors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24.
2. a. 14
Multiplying by 1: 1 × 14 = 14
Multiplying by 2: 2 × 7 = 14
So, the factors of 14 are 1, 2, 7 and 14.
b. 21
Multiplying by 1: 1 × 21 = 21
Multiplying by 3: 3 × 7 = 21
So, the factors of 21 are 1, 3, 7 and 21.
c. 36
Multiplying by 1: 1 × 36 = 36
Multiplying by 2: 2 × 18 = 36
Multiplying by 3: 3 × 12 = 36
Multiplying by 4: 4 × 9 = 36
Multiplying by 6: 6 × 6 = 36
So, the factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18 and 36.
d. 39
Multiplying by 1: 1 × 39 = 39
Multiplying by 3: 3 × 13 = 39
So, the factors of 39 are 1, 3, 13 and 39.
e. 40
Multiplying by 1: 1 × 40 = 40
Multiplying by 2: 2 × 20 = 40
Multiplying by 4: 4 × 10 = 40
Multiplying by 5: 5 × 8 = 40
So, the factors of 40 are 1, 2, 4, 5, 8, 10, 20 and 40.
f. 42
Multiplying by 1: 1 × 42 = 42
Multiplying by 2: 2 × 21 = 42
Multiplying by 3: 3 × 14 = 42
Multiplying by 6: 7 × 6 = 42
So, the factors of 42 are 1, 2, 3, 6, 7, 14, 21 and 42.
g. 48
Multiplying by 1: 1 × 48 = 48
Multiplying by 2: 2 × 24 = 48
Multiplying by 3: 3 × 16 = 48
Multiplying by 4: 4 × 12 = 48
Multiplying by 6: 6 × 8 = 48
So, the factors of 48 are 1, 2, 3, 4, 6, 8, 12, 16, 24 and 48.
h. 50
Multiplying by 1: 1 × 50 = 50
Multiplying by 2: 2 × 25 = 50
Multiplying by 5: 5 × 10 = 50
So, the factors of 50 are 1, 2, 5, 10, 25 and 50.
3. a. 9
Dividing by 1: 9 ÷ 1 = 9
Dividing by 3: 9 ÷ 3 = 3
So, the factors of 9 are 1, 3 and 9.
b. 12
Dividing by 1: 12 ÷ 1 = 12
Dividing by 2: 12 ÷ 2 = 6
Dividing by 3: 12 ÷ 3 = 4
So, the factors of 12 are 1, 2, 3, 4, 6 and 12.
c. 15
Dividing by 1: 15 ÷ 1 = 15
Dividing by 3: 15 ÷ 3 = 5
So, the factors of 15 are 1, 3, 5 and 15.
d. 30
Dividing by 1: 30 ÷ 1 = 30
Dividing by 2: 30 ÷ 2 = 15
Dividing by 3: 30 ÷ 3 = 10
Dividing by 5: 30 ÷ 5 = 6
So, the factors of 30 are 1, 2, 3, 5, 6, 10, 15 and 30.
4. To check if 18 is a factor of 126, we can divide 126 by 18.
18 1 2 6 7 – 1 2 6
0 Leaves 0 as the remainder. So, 18 is a factor of 126.
5. The factors of the numbers between 5 and 15 are as follows:
6: 1, 2, 3 and 6
7: 1 and 7
8: 1, 2, 4 and 8
9: 1, 3 and 9
10: 1, 2, 5 and 10
11: 1 and 11
12: 1, 2, 3, 4, 6 and 12
13: 1 and 13
14: 1, 2, 7 and 14
So, the number between 5 and 15 with the greatest number of factors is 12.
6. a. 14, 20
The factors of 14 are 1, 2, 7 and 14.
The factors of 20 are 1, 2, 4, 5, 10 and 20.
The common factors of 14 and 20 are 1 and 2.
b. 16, 18
The factors of 16 are 1, 2, 4, 8 and 16.
The factors of 18 are 1, 2, 3, 6, 9 and 18.
The common factors of 16 and 18 are 1 and 2.
c. 35, 50
The factors of 35 are 1, 5, 7 and 35.
The factors of 50 are 1, 2, 5, 10, 25 and 50.
The common factors of 35 and 50 are 1 and 5.
d. 54, 64
The factors of 54 are 1, 2, 3, 6, 9, 18, 27 and 54.
The factors of 64 are 1, 2, 4, 8, 16, 32 and 64.
The common factors of 54 and 64 are 1 and 2.
7. a. 6 b. 8 c. 12 d. 3
a. On dividing 78 by 6, Q = 13, R = 0. So, 6 is a factor of 78.
On dividing 96 by 6, Q = 16, R = 0. So, 6 is a factor of 96.
So, 6 is a common factor of 78 and 96.
b. On dividing 78 by 8, Q = 9, R = 6. So, 8 is not a factor of 78.
On dividing 96 by 8, Q = 12, R = 0. So, 8 is a factor of 96.
So, 8 is not a common factor of 78 and 96.
c. On dividing 78 by 12, Q = 6, R = 6. So, 12 is not a factor of 78.
On dividing 96 by 12, Q = 8, R = 0. So, 12 is a factor of 96.
So, 12 is not a common factor of 78 and 96.
d. On dividing 78 by 3, Q = 26, R = 0. So, 3 is a factor of 78.
On dividing 96 by 3, Q = 32, R = 0. So, 3 is a factor of 96.
So, 3 is a common factor of 78 and 96.
8. a. 1 is the common factor of 11 and 13. 11 and 13 have no common factors. False
b. 1 is the common factor of all the numbers.
0 is a common factor of all the numbers. False
c. The factors of 15 are 1, 3, 5, 15.
The factors of 25 are 1, 5, 25.
So, 1 and 5 are the common factors of 15 and 25. 15 and 25 have a total of 3 common factors. False
d. 41 is a prime number while 49 is not a prime number. The factors of 49 are 1, 7 and 49. (It has more than 2 factors)
Both 41 and 49 are prime numbers. False
9. The factors of 14 are 1, 2, 7 and 14.
The factors of 45 are 1, 3, 5, 9, 15 and 45.
45 has more factors than 14.
So, Raj is wrong.
10. She can arrange the eggs in 5 ways, which are as follows: 2 × 8 = 16 8 × 2 = 16 1 × 16 = 16 16 × 1 = 16 4 × 4 = 16
Hence, Tina can arrange the eggs in five ways.
11. The total numbers of biscuits = 72
The different possible arrangements of biscuits in the packets are as follows:
Number of Packets Biscuits in Each Packet Total
12. Answers may vary. Sample answer:
A teacher has 18 pencils and 24 erasers. She wants to distribute them among the 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. Answers may vary. Sample answer:
a. 3 + 7 + 11 + 13 = 34
b. 4 + 8 + 10 + 12 = 34
2. The number of apple saplings with Bhawar Lal = 36
He can distribute the saplings equally by giving the saplings in the factors of 36.
The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18 and 36.
The number of orange saplings with Bhawar Lal = 48 He can distribute the saplings equally by giving the saplings in the factors of 48.
The factors of 48 are 1, 2, 3, 4, 6, 8, 12, 16, 24 and 48.
The common factors of 36 and 48 are 1, 2, 3, 4, 6 and 12.
The greatest common factor of 36 and 48 is 12. He can distribute the saplings to a maximum of 12 children, where each child will get 3 apple saplings and 4 orange saplings.
3. a. The factors of 50 are 1, 2, 5, 10, 25 and 50.
b. The factors of 66 are: 1, 2, 3, 6, 11, 22, 33 and 66.
c. The factors of 72 are: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36 and 72.
d. The factors of 88 are: 1, 2, 4, 8, 11, 22, 44 and 88.
e. The factors of 98 are: 1, 2, 7, 14, 49 and 98.
f. The factors of 120 are: 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60 and 120.
g. The factors of 156 are: 1, 2, 3, 4, 6, 12, 13, 26, 39, 52, 78 and 156.
h. The factors of 180 are: 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, 90 and 180.
4. a. The multiples of 4 are as follows: 4 × 1 = 4 4 × 2 = 8 4 × 3 = 12 4 × 4 = 16 4 × 5 = 20
4 × 6 = 24 4 × 7 = 28 4 × 8 = 32
The multiples of 4 that are smaller than 30 are: 4, 8, 12, 16, 20, 24 and 28.
b. The multiples of 8 are as follows: 8 × 3 = 24 8 × 4 = 32 8 × 5 = 40 8 × 6 = 48 8 × 7 = 56
8 × 8 = 64 8 × 9 = 72 8 × 10 = 80
The multiples of 8 that are greater than 24, but smaller than 80 are: 32, 40, 48, 56, 64 and 72
c. The multiples of 7 are as follows:
7 × 1 = 7 7 × 2 = 14 7 × 3 = 21 7 × 4 = 28
7 × 5 = 35 7 × 6 = 42 7 × 7 = 49 7 × 8 = 56
7 × 9 = 63 7 × 10 = 70 7 × 11 = 77 7 × 12 = 84
The multiples of 7 that are divisible by 2 are: 14, 28, 42, 56, 70 and 84.
5. a. 4 and 8
The factors of 4 are: 1, 2 and 4.
The factors of 8 are: 1, 2, 4 and 8.
The common factors of 4 and 8 are: 1, 2 and 4.
b. 6 and 10
The factors of 6 are: 1, 2, 3 and 6.
The factors of 10 are: 1, 2, 5 and 10.
The common factors of 6 and 10 are: 1 and 2.
c. 9 and 15
The factors of 9 are: 1, 3 and 9.
The factors of 15 are: 1, 3, 5 and 15.
The common factors of 9 and 15: 1 and 3.
d. 12 and 15
The factors of 12 are: 1, 2, 3, 4, 6 and 12.
The factors of 15 are: 1, 3, 5 and 15.
The common factor of 12 and 15 are: 1 and 3.
e. 25 and 60
The factors of 25 are: 1, 5 and 25.
The factors of 60 are: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30 and 60.
The common factors are: 1 and 5.
f. 28 and 42
The factors of 28 are: 1, 2, 4, 7, 14 and 28.
The factors of 42 are: 1, 2, 3, 6, 7, 14, 21 and 42.
The common factors of 28 and 36 are: 1, 2, 7 and 14.
g. 36 and 81
The factors of 36 are: 1, 2, 3, 4, 6, 9, 12, 18 and 36.
The factors of 81 are: 1, 3, 9, 27 and 81.
The common factors are: 1, 3 and 9.
h. 41 and 87
The factors of 41 are: 1 and 41.
The factors of 87 are: 1, 3, 29 and 87.
The common factor is 1.
6. a. 1 and 6 are factors of 7. No because factors of 7 are 1 and 7.
b. 61 is a prime number. Yes beacuse factors of 61 are 1 and 61 only.
c. 1 is the smallest and only factor of 31. No because factors of 31 are 1 and 31.
d. 2 and 4 are factors of 8. Yes because factors of 8 are 1, 2, 4, 8.
e. 6 and 9 are factors of 54. Yes because factors of 54 are 1, 2, 3, 6, 9, 18, 27 and 54.
f. Both 91 and 93 are prime numbers. No because factors of 91 are 1, 7, 13 and 91 and factors of 93 are 1, 3, 31 and 93.
7. The church 1 bell rings every 60 minutes, so the intervals between the ringing are in the multiples of 60.
The bell will ring after intervals of: 60, 120 and 180 minutes.
The church 2 bell rings every 45 minutes, so the intervals between the ringing are in the multiples of 45.
The bell will ring after intervals of: 45, 90, 135 and 180 minutes. The common interval is 180 minutes.
So, after 180 minutes, both the church bells will ring together.
8. Number of roses in a group: 5
The multiples of 5 are: 5, 10, 15, 20 and 25.
Number of lilies in a group: 4.
The multiples of 4 are: 4, 8, 12, 16 and 20.
As we can see, their common multiple is 20.
Therefore, Megha should purchase 20 flowers of each type.
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?
Literacy rate of Kerala out of 100 = 472 502 × = 94 100
7. Fraction of day for which panda sleeps = 7 12
Fraction of the day for which the panda eats = 1 3
Fraction of the day for which the panda does the rest of the activities = 1 12
A day has 24 hours, so let us convert all the fractions with a denominator of 24.
7 12 = 7 12 × 2 2 = 14 24
1 3 = 1 3 × 8 8 = 8 24
1 12 = 1 12 × 2 2 = 2 24
Let us shade a grid to represent the number of hours dedicated to each activity.
8. Answers may vary. Sample answer:
Amir is pouring juice into glasses. He uses 3 5 of a litre 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. Help Amir find two equivalent fractions for 3 5 of a litre of juice?
Challenge
1. 6 11 , 12 22 , 18 33 , 26 44 , 30 55 , 36 66
The denominators are 11, 22, 33, 44, 55, 66. The numbers are growing by 11 for each fraction. The numerators are 6, 12, 18, 26, 30, 36. Thee numbers are usually growing by 6 for each fraction except for 26 44 . So, the fraction 26 44 is not part of the pattern.
Do It Yourself 6C
1. a. Since the fractions 1475 , , , 9999 have the same denominator, they are like fractions.
b. Since the fractions 26109 ,,, 11111111 have the same denominator, they are like fractions.
c. Since the fractions 131412 ,,, 15151515 have the same denominator, they are like fractions.
d. Since the fractions 4444 ,,, 814712 have the different denominator, they are unlike fractions.
2.
Therefore, 4 9 > 3 7 Therefore, 9 16 > 7 15
3. a. The denominators of all the fractions are the same, so, the fraction with the smallest numerator is the smallest.
3 9 , 6 9 , 1 9 , 5 9 , 2 9 , 7 9
b. The numerators of all the fractions are the same, so, the fraction with the greatest denominator is the smallest.
2 7 , 2 5 , 2 3 , 2 4 , 2 8 , 2 10
c. The denominators of all the fractions are the same, so, the fraction with the smallest numerator is the smallest.
2 7 , 6 7 , 3 7 , 5 7 , 7 7 , 4 7
d. The numerators of all the fractions are the same, so, the fraction with the greatest denominator is the smallest.
5 9 , 5 8 , 5 7 , 5 11 , 5 6 , 5 10
4. a. 843617 ,,,,, 999999
The denominators of all the fractions are the same.
Arranging the numerators in ascending order.
1 < 3 < 4 < 6 < 7 < 8
The ascending order of the fractions is: 134678 999999 <<<<<
b. 555555 ,,,,, 671191310
The numerators of all the fractions are the same. Arranging the denominators in descending order.
13 > 11 > 10 > 9 > 7 > 6
The ascending order of the fractions is: 555555 131110976 <<<<<
c. 482937 ,,,,, 101010101010
The denominators of all the fractions are the same. Arranging the numerators in ascending order.
2 < 3 < 4 < 7 < 8 < 9
The ascending order of the fractions is: 234789 101010101010 <<<<< .
d. 222222 ,,,,, 4107963
The numerators of all the fractions are the same.
Arranging the denominators in descending order.
10 > 9 > 7 > 6 > 4 > 3
The ascending order of the fractions is: 222222 1097643 <<<<< .
5. Since the denominators of the fractions are the same, let us compare the numerators.
92 > 87 > 84 > 75
So, 92878475 100100100100 >>>
Thus, the descending order is 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 swan 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. Fraction of a cake eaten by Rajiv = 5 8
According to the question, 5 8 9 >
Let us find the equivalent fractions for the fractions such that the denominators for both the fractions are the same.
5 8 = 5 × 9 8 × 9 = 45 72
9 = × 8 9 × 8 = × 8 72
So, according to the question, 45 > × 8
The maximum value for the same can be 5.
Thus, Arti ate 5 slices or 5 9 of the cake.
Do It Yourself 6D
1. a. 2 3 is a proper fraction since the numerator is smaller than the denominator.
b. 15 8 is an improper fraction since the numerator is greater than the denominator.
c. 2 3 6 is a mixed number since it consists of a whole number part and a proper fraction.
d. 16 9 is an improper fraction since the numerator is greater than the denominator.
e. 6 2 10 is a mixed number since it consists of a whole number part and a proper fraction.
f. 3 7 is a proper fraction since the numerator is smaller than the denominator.
5. Fraction of cake Susan had =
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. 177321 2 33339 × === ×
We are multiplying the numerator and the denominator by 3. 155315 1 444312 × === ×
Thus, 15 12 is equivalent to 1 1 4 .
Do It Yourself 6E
2.
3. Length of cloth with Ria = 1 5 8 Length of cloth used = 7 8
Remaining cloth = 1 5
Ria is left with 3 4 m of cloth.
4. Part of the packet of cookies with Manya
Part of the packet of cookies Manya ate = 7 8
Part of the packet of cookies Manya left with = 10
Manya is left with 3 8 part of the packet of cookies.
5. Length of the jogging track = 2 2 8 = 2 × 8 + 2 8 = 18 8
Distance travelled by Sunil on Sunday = 2 3 6 = 2 × 6 + 3 6 = 15 6
Farther length Sunil ran on Sunday than Saturday = 15 6 –7 6 = 8 6 = 1 2 6 Sunil ran 1 2 6 = 1 13 km farther on Sunday than Saturday.
7. Answers may vary. Sample answer: Lily baked a batch of cookies and decided to divide then 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. Total fraction of pizza with all of them = 7 1 12 = 19 12
Fraction of pizza with Jeeshan = 3 12
Fraction of pizza with Raghav = 2 × 3 12 = 6 12
Fraction of pizza with Animesh = Fraction of pizza with Priya
Total fraction of pizza with Animesh and Priya = 1936 121212 = 1936 12 = 10 12
Fraction of pizza with Animesh = 10 12 ÷ 2 = 5 12
Fraction of pizza with Priya = 5 12
Thus, the fraction of pizza with Jeeshan = 3 12 , Raghav = 6 12 , Animesh = 5 12 and Priya = 5 12
Chapter Checkup
1. Figures may vary. Sample figures: a. 5 9 shows 5 parts shaded out of 9 equal parts.
b. 7 10 shows 7 parts shaded out of 10 equal parts.
c. 3 8 shows 3 parts shaded out of 8 equal parts.
d. 4 7 shows 4 parts shaded out of 7 equal parts.
2. a. 1 6 of 18 flowers
1 6 of 18 flowers is 3 flowers (18 ÷ 6 = 3)
c. 1 4 of 36 boxes
1 4 of 36 boxes is 9 boxes (36 ÷ 4 = 9)
3. Answer may vary. Sample answer:
a. 5 6
b. 1 3 of 27 cakes
1 3 of 27 cakes is 9 cakes (27 ÷ 3 = 9)
d. 1 5 of 50 balloons
1 5 of 50 balloons is 10 balloons (50 ÷ 5 = 10)
First equivalent fraction of 5 6 = 5 × 2 6 × 2 = 10 12
Second equivalent fraction of 5 6 = 5 × 3 6 × 3 = 15 18
Third equivalent fraction of 5 6 = 5 × 4 6 × 4 = 20 24
1. a. Every point on the boundary of circle is at the same distance from the centre.
b. All the radii of a circle are equal in length.
c. A circle can have an infinite number of diameters.
d. The length of the boundary of a circle is called its circumference
e. A circle has only one centre.
2.
Centre
Diameter
3. a. Diameter = 10 cm
Radius
Radius of the circle = Diameter ÷ 2 = 10 ÷ 2 = 5 cm
Option iii is the correct answer.
b. A diameter divides a circle into 2 equal parts. Option ii is the correct answer.
c. The relation between the radius (R) and diameter (D) of a circle is R = 2 D
Option iv is the correct answer.
4. Not drawn to scale. Sample figures:
O b.
5. The diameter of the planet Mars is = 6780 km
Radius = Diameter ÷ 2 = 6780 ÷ 2 = 3390 km
6. Diameter of the original frisbee = 18 cm
Radius of the original frisbee = 18 2 = 9 cm
Radius of the frisbee to be made = 9 cm − 2 cm = 7 cm
Not drawn to scale. Sample figure:
7 cm O
Challenge
1. A lies on the circumference of Circle 2.
B lies on the circumference of Circle 1.
The distance between A and B is equal to the radius of Circle 1 because B is on Circle 1’s circumference.
Similarly, the distance between B and A is also equal to the radius of Circle 2 because A is on Circle 2’s circumference.
The distance between A and B is the same as radii of the circles.
Therefore, the radii of both circles must be equal.
So, the radii of both circles have the same length.
Chapter Checkup
1. a. i) ray b. iii) point c. iv) 12 d. iii) 8 e. iii) light from a torch
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.
Polygon
Name Rectangle Triangle Pentagon Decagon
Number of Sides 4 3 5 10
4. a. Diameter = 4 cm, Radius = Diameter 2 = 4 2 = 2 cm
Measurements may vary. Sample figure is given.
b. Radius = 4 cm 4 cm O
c. Radius = 5 cm
d. Diameter = 6 cm
Radius = Diameter 2 = 6 2 = 3 cm 3 cm O
5. AB is straight line, and it is a shortest route.
So, Mary should take the route AB.
6. Diameter of the jar = 16 cm
Radius of the jar = 16 2 = 8 cm
Radius of the lid to be bought = 8 cm + 2 cm = 10 cm
7. Radius of the circular park = 8 m
As the stationary shop is on the other side of the circle, the shortest distance is 8 m + 8 m = 16 m
Challenge
1. AB = 12 cm OP = radius of the small circle = 4 cm
Since OP, OB, and OD are the radii of the same circle, OP = OB = OD = 4 cm
Distance between points A and D = AB – OB – OD = 12 cm – 4 cm – 4 cm = 8 cm – 4 cm = 4 cm
Therefore, AD = 4 cm
2. If horse tied to a post starts grazing everywhere that it can reach, the final shape of the area that has no grass left will be the circle of radii 10 m.
3. 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.
Distance between the centre of the field to its boundary = Radius of the circular field = 16 m
Diameter = 2 × Radius = 32 m
So, the assertion is true.
The reason is false, since the diameter is twice the radius and not half the radius.
Thus, option c is correct.
Case Study
1. The shape of the track used in the Track and Field Relay Race is circle.
Option b is correct.
2. In the concentric circles, the radii of the circles increases as you go from inner circle to outer circle, and so is the circumference.
Therefore, to maintain the same distance, teams start at different positions along the concentric circles.
Option b is correct
3. Measurements may vary. Sample figure is given:
4. Answer may vary
Challenge
1. a. Number of candles seen in the top view = 14
b. Number of candles seen in the front view = 5
c. Number of candles seen in the left-side view = 7
Do It Yourself 8B
1. The nets of a cube:
2. Figures b, c and f are not the nets of a cuboid.
Do It Yourself 8C
1. School Meera 1 2 Meera turns left 2 times
2. a. Suhani’s house is on the second road. True
b. If Kavita steps out of her house on the first road, the bank will be to the left of her house. False because the bank will be in the front of her house.
c. The post office is the nearest place to the factory. False because post office is in the second road whereas factory is in first road.
d. The restaurant is in front of the park. True
e. To reach the factory, one has to go to the second road. True
3. a. If Rani is on mall road facing Rose street, the police station will be to her right
b. Sam’s house is on MG road.
c. The restaurant is in front of Rani’s house.
d. Supermarket is centre of all the roads.
e. Sam’s house in front of Brooke park.
4. Suraj can use different routes to reach bank which are as follows:
4. Answers may vary. Sample answer:
5.
The given shape has three rectangular faces and two triangular faces.
Thus, the net of the given shape is shown as:
Challenge
1. Opposite faces on a dice add up to 7. So, 5 + 2 = 7
1 + 6 = 7
3 + 4 = 7
Thus, the net of the cube with dots to make a dice is given as:
a. 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.
b. To reach the bank Suraj will take the right turn and move straight. He will take the third right turn and move straight, then take another right turn and move straight, and he will stop in front of bank.
c. To reach the bank, Suraj will take the right turn and move straight. He will take a second right turn and move straight, then take a left turn and will stop in front of the bank.
d. To reach the bank, Suraj first took a left turn and moved straight, then took another left turn and moved straight. He will take a left turn and move straight, and then the bank will be there after crossing 2 cuts.
5. The truck will not cross the Farm Road to reach the burning house as Farm Road is not connected to the burning house.
4. Side C will be opposite to the purple side when the net is folded to make a box.
5. Net of the following figures. Answers may vary. Sample answers:
6. Answers may vary.
1. Mohan wants to reach the bus stop from his house by following the given directions:
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.
Representation may vary. Sample representation:
6. Answers may vary. Sample answer: Opposite sides of the cube have the same colour.
Checkup
Top view of the objects:
3. Objects that look the same when looked at from the side or front views.
7. a. The mall is between John’s house and Rohit’s house.
b. The park is nearest to the school.
c. The grocery store is nearest to Kavya’s house.
d. If Rohit is on First Road with the mall to his left, he needs to take a right turn to reach the Pizza house.
8. a. Rita’s house is farthest from the school.
b. Mina’s house is not opposite to Anand Garden.
c. 4 roads meet at the Central Chowk.
9. The front view of the structure is shown.
10. Answer may vary. Sample answer:
Fancy Store Footwear shop
Bakery House School
2. Given below are three views of a cube, and 6 faces of the same cube.
The net for the given cube is drawn as: Answers may vary Sample answer:
Case Study
1. Nathula Tsomgo Lake
Police Check Post
Chogyal Palden Thedyp Memorial Park Ropeway Point Assembly House
2. a. Nathula is in the left direction from Tsomgo Lake. True b. Chogyal Palden Thedyp Memorial Park is between the ropeway point and the assembly house. False
3. A flower pot in Chogyal Palden Thedyp Memorial Park is shaped like a cube. The net of the pot is shown as:
2. a. The pattern consists of 1 triangle followed by 1 circle, then 1 triangle followed by 2 circles. So, each time 1 circle is increasing. Thus, the missing shapes in the sequence will be 3 circles.
b. The pattern consists of 2 diamond shapes 1 triangle, then 4 diamond shapes followed 1 triangle.
So, each time 2 diamond shape is increasing.
Thus, the missing shapes in the sequence will be 6 diamond shapes.
c. The pattern consists of 1 circle followed by 2 parallelograms, then 1 circle followed by 3 parallelograms. So, each time 1 parallelogram is increasing.
Thus, the missing shapes in the sequence will be 1 circle and 4 parallelograms.
d. The pattern consists of 1 rectangle followed by 1 pentagon, then 2 rectangles followed by 1 pentagon. So, each time 1 rectangle is increasing.
Thus, the missing shapes in the sequence will be 1 pentagon and 4 rectangles.
3. a. The pattern is the figure is moving in the clockwise direction.
4. David is standing at the police check post facing the assembly house. The ropeway point is on David’s left side.
Chapter 9
Let’s Warm–up
1. ,
2. 1, 11, 111, 1111, 11111, 111111
3.
4. A, BB, CCC, DDDD, EEEEE, FFFFFF
Do It Yourself 9A
1. a. The pattern consists of a square followed by a star.
b. The pattern consists of a green heart followed by two red hearts.
c. The pattern consists of two sad emojis followed by three smiling emojis.
d. The pattern consists of one blue cylinder followed by two green cylinders.
b. The pattern is the figure is moving in the clockwise direction.
c. The pattern is the figure is moving in the anticlockwise direction.
d. The pattern is that figure is moving in a anticlockwise direction.
4. a. In the pattern, two shapes are added each time.
b. In the pattern, 1 green rectangle is removed from the first line each time.
c. In the pattern, one circle is added on top and one circle is added on the side each time.
d. In the pattern, one innermost square is removed each time.
5. Answer may vary. Sample answer: Growing pattern.
Challenge 1.
By rotating the figure as shown by the arrows:
We will get a figure that replaces the question mark:
Do It Yourself 9B
1. a. 20 is added to the previous number each time. 10, 30, 50, 70, 90, 110
b. 13 is added each time to the previous number. 13, 26, 39, 52, 65, 78
c. We are subtracting 10 from the previous number each time.
84, 74, 64, 54, 44, 34
d. We are adding +1, +2, +3, +4, +5, +6 each time.
1, 2, 4, 7, 11, 16, 22
3. Answer may vary. Sample answer:
Challenge
1. Answer may vary. Sample answer:
Do It Yourself 9C
1. Tessellations are the patterns that are created by fitting identical shapes (tiles) together without any gaps or overlaps to cover a flat surface.
Say yes to the pattern, if it is tessellating.
a Yes, since it has no gaps or overlaps. b.
No, since it has gaps. c.
Yes, since it has no gaps or overlaps.
No, since it has gaps.
1 parallelogram, 5 triangles, 1 square
4. Answers may vary. Sample answer: Tessellating pattern.
Challenge
1. No, this block cannot be used to create a tessellating pattern.
d. READ and answer the question (18-5-1-4) Answer may vary. Sample answer: I love to read exciting stories before bedtime. 2.
16 17 18 19 20 21 22
a. KEEP IT UP—11-5-5-16 9-20 21-16
b. SAVE WATER—19-1-22-5 23-1-20-5-18
c. PLANT TREES—16-12-1-14-20 20-18-5-5-19
d. FANTASTIC WORK—6-1-14-20-1-19-20-9-3 23-15-18-11
e. REDUCE REUSE RECYCLE—18-5-4-21-3-5 18-5-21-19-5 18-5-3-25-3-12-5
Challenge
1. CAMEL is coded as 25106
So, C = 2, A = 5, M = 1, E = 0, L = 6 LION is coded as 6793.
So, L = 6, I = 7, O = 9, N = 3
Hence, N = 3, A = 5, M = 1, E = 0 Thus, NAME is coded as 3510.
Do It Yourself 9E
We can modify it as shown to make it a tessellating block. The tessellating pattern would be—
Do It Yourself 9D
1.
3. A B C D E F G H I J K L M N O P Q R S T U V W X Y Z
A, B, C, D, E, H, I, K, M, O, T, U, V, W, X and Y are symmetrical.
A, M, T, U, V, W, Y has vertical line of symmetry.
B, C, D, E, K has horizontal line of symmetry.
a. Go to (19-3-8-15-15-12)
Answer may vary. Sample answer: School is where we learn new things, make friends and have fun every day.
b. Find the (17-21-9-26) Answer may vary. Sample answer: A quiz is a fun way to test what you have learnt and see how much you know.
c. Click quiz now (1-20-20-5-13-16-20)
SCHOOL QUIZ ATTEMPT
Answer may vary. Sample answer: To attempt something means to try your best to do it, even if it is challenging.
H, I, O, X has both vertical and horizontal line of symmetry.
4. a. b. c. d.
Challenge 1.
2. This polygon has 11 lines of symmetry as shown. These lines are line of symmetry because each of these lines divides the polygon into symmetrical halves:
Chapter Checkup
1. a. The pattern consists of a plus sign followed by a cross sign.
b. The pattern consists of a two downward curve followed by a upward curve.
c. The pattern is that figure is moving in a clockwise direction.
d. The pattern is that mathematical symbols in figure is moving in a anticlockwise direction.
2. a. b.
Yes, because the line of symmetry divides the figure into two identical halves.
c. d.
Yes, because the line of symmetry divides the figure into two identical halves.
The other half of each symmetrical shape.
3. a. 110, 130, 150, 170, 190, 210, 230 20 is added to the previous number each time.
b. 111, 122, 133,144, 155, 166 11 is added to the previous number each time.
c. 890, 780, 670, 560, 450, 340, 230
We are subtracting 110 from the previous number each time.
d. 140, 131, 122, 113, 104, 95, 86 We subtracted 9 from the previous number each time.
4. A B C D E F G H I J K L M 1 3 5 7 9 11 13 15 17 19 21 23 25
O P Q R S T U V W X Y Z 27 29 31 33 35 37 39 41 43 45 47 49 51
a. BEST WISHES → 3-9-37-39 45-17-37-15-9-37
b. PLANT A TREE → 31-23-1-27-39 1 39-35-9-9
c. SAVE PAPER → 37-1-43-9 31-1-31-9-35
d. RECYCLE → 35-9-5-49-5-23-9
5. a. b. c.
No, because the line of symmetry does not divide the figure into two identical halves.
Line of symmetry on each shape.
No, because the line of symmetry does not divide the figure into two identical halves.
e. f. g. h.
6. a. b. c.
7. A circle have an infinite number of lines of symmetry.
8. Answer may vary.
Challenge
1. Each row has flowers equal to the sum of flowers in the two preceding rows.
b. 10 kg 500 g = (10 × 1000) g + 500 g = 10000 g + 500 g = 10,500 g
c. 5 kg 10 g = (5 × 1000) g + 10 g = 5000 g + 10 g = 5010 g
d. 15 kg 25 g = (15 × 1000) g + 25 g = 15,000 g + 25 g = 15,025 g
6. Weight of a puppy at 3 months = 4 kg
We know that 1 kg = 1000 g
So, 4 kg = 4 × 1000 = 4000 g
Thus, a puppy’s weight in grams is 4000 g.
7. The total weight of the onions sold by the vegetable seller = 145 kg 500 g
The price of 1 kg of onions = ₹30
The price of 145 kg of onions = ₹30 × 145 = ₹4350
We know that 1 kg = 1000 g. So, 1 2 kg = 1000 2 g
1 2 kg = 500 g; 500 g = 1 2 kg
The price of 500 g or 1 2 kg of onions = ₹30 ÷ 2 = ₹15
Thus, the total amount of money earned by the vegetable seller for 145 kg 500 g of onions = ₹4350 + ₹15 = ₹4365
8. Number of pancakes made from 500 g of flour = 65
Number of pancakes made from 1 g of flour = 6513 = 500100
We know that 1 kg = 1000 g
So, 5 kg = 5 × 1000 g = 5000 g
×
Number of pancakes made from 5000 g flour = 13 5000 100
9. Weight of Sunita = 9 kg 200 g
= 650 pancakes
Weight of Sunita’s father = 9 times of Sunita’s weight = 9 × (9 kg 500 g) = 9 × 9 kg and 9 × 500 g = 81 kg 4500 g
We know that 1 kg = 1000 g
So, 4500 g = 4000 g + 500 g = (4000 ÷ 1000) kg + 500 g = 4 kg 500 g
Thus, weight of Sunita’s father = 81 kg + 4 kg + 500 g = 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 of bag in grams?
Challenge
1. Weight of each box = 600 g
Number of boxes = 6
Weight of 6 boxes = 600 g × 6 = 3600 g
Total number of children = 36
Since, all the marbles in those 6 boxes have been divided equally among 36 children.
Thus, weight of marbles that each child gets = 3600 g ÷ 36 = 100 g
Do It Yourself 10C
1. a. Litre
b.
The amount of liquid in the measuring jug = 1 4 L
We know that 1 L = 1000 mL
1 4 of 1000 mL = 1000 ÷ 4 = 250 mL
The amount of liquid in the measuring jug = 1000 mL
We know that 1000 mL = 1 L
Thus, there is 1 L liquid in the measuring jug.
2. Remember, 1000 mL = 1 L
a. 3500 mL = 3000 mL + 500 mL = (3000 ÷ 1000) L + 500 mL = 3 L 500 mL
If 1000 mL = 1 L, then 1000 2 mL = 1 2 L = 500 mL = 1 2 L
So, 3 L 500 mL = 3 1 2 L
b. 5000 mL = 5000 ÷ 1000 L = 5 L
c. 10,500 mL = 10,000 ml + 500 mL = (10,000 ÷ 1000) L + 500 mL = 10 L 500 mL
If 1000 mL = 1 L, then 1000 2 mL = 1 2 L = 500 mL = 1 2 L
So, 10 L 500 mL = 10 1 2 L
d. 17,000 mL = 17000 ÷ 1000 L = 17 L
3. Remember, 1000 mL = 1 L
a. 7650 mL = 7000 mL + 650 mL
= (7000 ÷ 1000) L + 660 mL
= 7 L 650 mL
b. 8235 mL = 8000 mL + 235 mL
= (8000 ÷ 1000) L + 235 mL
= 8 L 235 mL
c. 9250 mL = 9000 mL + 250 mL
= (9000 ÷ 1000) L + 250 mL
= 9 L 250 mL
d. 11300 mL = 11,000 mL + 300 mL = (11,000 ÷ 1000) L + 300 mL
= 11 L 300 mL
4. The amount of tea 1 cup can hold = 100 mL
The amount of tea needed to fill in 6 cups = 100 mL × 6 = 600 mL
So, 600 mL of tea is needed to fill 6 cups of tea.
5. The number of bottles Coco needs to fill = 3
The capacity of each bottle = 2 L
The total capacity of 3 bottles = 2 L × 3 = 6 L
So, Coco would use 6 L of water to fill all three bottles.
6. The number of small packs with 250 mL shampoo each = 4
The total amount of shampoo in these packs = 250 mL × 4 = 1000 mL
We know that 1 L = 1000 mL
So, there is 1 L shampoo in 4 packs of 250 mL shampoo.
Similarly, the number of bottles with 1 L shampoo each = 2
So, the total amount of shampoo in 2 bottles = 2 L
Thus, the total shampoo in all packs and bottles = 1 L + 2 L = 3 L
7. a. The number of glasses of lemonade JJ and her sister have sold so far = 8
The capacity of each glass = 250 mL
The total capacity of 8 glasses sold = 250 mL × 8 = 2000 mL
We know that 1000 mL = 1 L. So, 2000 mL = 2 L
Thus, JJ and her sister have sold 2 L of lemonade so far.
b. The number of glasses of lemonade JJ and her sister have sold so far = 8
The earnings from 1 glass of lemonade = ₹25
The earnings from 8 glasses of lemonade = ₹25 × 8 = ₹200
Thus, JJ and her sister have earned ₹200 so far by selling lemonade.
Challenge
1. The oil vendor can measure 4 L of oil with 3 L and 5 L jars:
a. Fill 3 L jar, pour into 5 L jar (leaving 2 L space).
b. Empty 3 L jar.
c. Refill 3 L jar, pour into 5 L jar until full (1 L left in 3 L).
d. Discard oil from 5 L jar.
e. Pour 1 L from 3 L jar into 5 L jar.
f. Refill 3 L jar, pour into 5 L jar until empty (fills to 4 L).
Chapter Checkup
1. a. Remember, 100 cm = 1 m
i. 205 cm
205 cm = 200 cm + 5 cm
= (200 ÷ 100) m + 5 cm = 2 m 5 cm
ii. 507 cm
507 cm = 500 cm + 7 cm
= (500 ÷ 100) m + 7 cm = 5 m 7 cm
iii. 764 cm
764 cm = 700 cm + 64 cm
= (700 ÷ 100) m + 64 cm = 7 m 64 cm
iv. 894 cm
894 cm = 800 cm + 94 cm
= (800 ÷ 100) m + 94 cm = 8 m 94 cm
b. Remember, 1000 m = 1 km
i. 1205 m
1205 m = 1000 m + 205 m
= (1000 ÷ 1000) km + 763 m = 1 km 205 m
ii. 5763 m
5763 m = 5000 m + 763 m
= (5000 ÷ 1000) km + 763 m = 5 km 763 m
iii. 6049 m
6049 m = 6000 m + 49 m
= (6000 ÷ 1000) km + 49 m = 6 km 49 m
iv. 7777 m
7777 m = 7000 m + 777 m
= (7000 ÷ 1000) km + 777 m = 7 km 777 m
2. Remember, 1 kg = 1000 g
a. 5065 g = 5000 g + 65 g = (5000 ÷ 1000) kg + 65 g = 5 kg 65 g
b. 4600 g = 4000 g + 600 g = (4000 ÷ 1000) kg + 600 g = 4 kg 600 g
c. 7450 g = 7000 g + 450 g = (7000 ÷ 1000) kg + 450 g = 7 kg 450 g
d. 10,500 g = 10,000 g + 500 g = (10,000 ÷ 1000) kg + 500 g = 10 kg 500 g
3. Remember, 1 kg = 1000 g
a. 2 kg 500 g = (2 × 1000) g + 500 g = 2000 g + 500 g = 2500 g
b. 4 kg 600 g = (4 × 1000) g + 600 g = 4000 g + 600 g = 4600 g
c. 5 kg 750 g = (5 × 1000) g + 750 g = 5000 g + 750 g = 5750 g
d. 12 kg 500 g = (12 × 1000) g + 500 g = 12,000 g + 500 g = 12,500 g
4. Remember, 1 L = 1000 mL
a. 7200 mL = 7000 L + 200 mL
= (7000 ÷ 1000) L + 200 mL
= 7 L 200 mL
b. 8660 mL = 8000 L + 660 mL
= (8000 ÷ 1000) L + 660 mL
= 8 L 660 mL
c. 16,250 mL = 16,000 L + 250 mL
= (16,000 ÷ 1000) L + 250 mL = 16 L 250 mL
d. 17,600 mL = 17,000 L + 600 mL = (17,000 ÷ 1000) L + 600 mL = 17 L 600 mL
5. We know that 1 L = 1000 mL
So, my mother has 1000 mL of milk.
The amount of milk mother gave me = 350 mL
The amount of milk mother gave to my brother = 175 mL
Total milk consumed by me and my brother = 350 mL + 175 mL = 525 mL
Thus, the amount of milk left = 1000 mL – 525 mL = 475 mL
6. The capacity of 1 jar of oil = 3 L
The capacity of 2 jars of oil = 3 L + 3 L = 6 L
We know that 1 L = 1000 mL
So, 6 L = 6 × 1000 mL = 6000 mL
The capacity of the smaller jars = 500 mL
The number of 500 mL jars that can be filled with 6000 mL of oil = 6000 ÷ 500 = 12
Thus, Tara can fill 12 jars of 500 mL with 2 jars of 3 L cooking oil in each jar.
7. The price of 1 kg of sugar = ₹60
a. The price of 2 kg of sugar = ₹60 × 2 = ₹120
b. The price of 5 kg of sugar = ₹60 × 5 = ₹300
c. The price of 12 kg of sugar = ₹60 × 12 = ₹720
The price of 500 g of sugar = ₹60 ÷ 2 = ₹30
The total price of 12 kg and 500 g of sugar = ₹720 + ₹30 = ₹750
d. The price of 25 kg of sugar = ₹60 × 25 = ₹1500
The price of 500 g of sugar = ₹60 ÷ 2 = ₹30
The total price of 25 kg and 500 g of sugar = ₹1500 + ₹30 = ₹1530
8. Weight on earth is 6 times the weight on the moon.
Weight on earth = 60 kg
So, the weight on the moon = 60 kg ÷ 6 = 10 kg
On the moon, you would feel much lighter. The moon has a weaker pulling force, so things weigh less there.
9. Each rock weighs 100 grams, and there are 5 rocks, so the total weight of the rock is 100 grams × 5 = 500 grams. If each handful of sand has about 20 grams, then it takes about 500 20 = 25 handfuls to equal the weight of the rocks.
Challenge
1. Meira has 4 litres (4000 mL) of hand sanitiser. The demand is:
Four 500 mL bottles = 4 × 500 mL = 2000 mL
Five 200 mL bottles = 5 × 200 mL = 1000 mL
Twelve 100 mL bottles = 12 × 100 mL = 1200 mL
Total demand = 2000 mL + 1000 mL + 1200 mL = 4200 mL.
Since 4000 mL < 4200 mL, she does not have enough sanitiser to meet the full demand. She is short by 4200 mL − 4000 mL = 200 mL.
2. 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. The assertion is true because a 1 kilogram bag of rice (1000 grams) is indeed heavier than a 500 gram bag of sugar. The reason is also true and correctly explains the assertion, as it clarifies the weight conversion showing why 1 kilogram is greater than 500 grams.
So, the answer is option a. Both A and R are true, and R is the correct explanation of A.
Case Study
1. One hasta is equivalent to 24 angulas.
2. 1 hasta = 24 angulas
So, 7 hasta = 24 × 7 = 168 angulas. Hence, option a is correct.
3. The height of a sand castle = 2 hastas
1 hasta = 24 angulas
So, 2 hasta = 24 × 2 = 48 angulas
4. If the width of a sand castle is 3 hastas, it is equal to 72 angulas. True
Height of a sand castle = 3 hastas
1 hasta = 24 angulas
So, 3 hasta = 24 × 3 = 72 angulas
5. 24 angulas = 1 hasta
So, 1 angulas = 1 24 hasta
1 Dhanus or Danda = 96 angulas
So, 96 angulas = 96 24 = hasta
⇒ 1 Dhanus or Danda = 4 hasta
Chapter 11
Let’s Warm-up
1. 1 cm = 10 mm. So, 25 cm = 250 mm
2. 1 m = 100 cm. So, 3 m = 300 cm
3. 1 m = 100 cm. So, 6 m 30 cm = 630 cm
4. 1 km = 1000 m. So, 5 km = 5000 m
5. 1 km = 1000 m. So, 6 km 40 m = 6040 m
Do It Yourself 11A
1. On measuring the length of each figure, the thread over figure Y is longer than the thread over figure X. So, the perimeter of figure Y is longer.
2. a.
Perimeter = 18 cm
3. a. Perimeter = 4 cm + 4 cm + 4 cm +
b. Perimeter = 3 cm + 2
2 cm + 3 cm = 20 cm
c. Perimeter = 14 m + 9 m + 6 m + 5 m + 4 m + 4 m + 8 m
d. Perimeter = 5 cm +
e. Perimeter = 20 m + 30
= 135 m
f. Perimeter = 8 cm + 8 cm + 6 cm + 6 cm + 8 cm = 36 cm
4. a. Perimeter = 5 cm + 4 cm + 2 cm + 1 cm + 3 cm + 3 cm + 6 cm + Length of unknown side
32 cm = 24 cm + Length of the unknown side
The length of the unknown side = 32 cm – 24 cm = 8 cm
b. Perimeter = 30 m + 12 m + 17 m + 20 m + 17 m + Length of the unknown side
128 m = 96 m + Length of the unknown side
The length of the unknown side = 128 m – 96 m = 32 m
c. Perimeter = 11 cm + 7 cm + 11 cm + 6 cm + 6 cm + Length of the unknown side
46 cm = 41 cm + Length of the unknown side
The length of the unknown side = 46 cm – 41 cm = 5 cm
5. a. 4 cm
23 mm = 70 mm
6. To find the length of the sewing thread around the pillow cover, we will find the perimeter of the pillow cover.
Perimeter of the pillow cover
= 30 cm + 45 cm + 30 cm + 45 cm = 150 cm
Perimeter of 3 pillow covers = 150 × 3 = 450 cm
So, the length of the sewing thread required to sew 3 pillow covers is 450 cm.
7. The length of each side of the tiles in each pattern = 25 cm
Number of tiles in each pattern = 6
a. The perimeter of the tiles in each pattern
= 25 cm + 25 cm + 25 cm + 25 cm = 100 cm
The perimeter of the pattern = 100 × 6 = 600 cm
b. The number of sides of the tiles on the outer side of the pattern = 14
The perimeter of the pattern = 25 cm × 14 = 350 cm
The other way the tiles can be joined is:
Arrangement may vary
Challenge
1. Answer may vary. Sample answer:
Shape
Squares
Full
Half squares ( ) 4
More than half squares ( ) 0 0
Less than half squares ( ) 0 0
Total area (approx.) - 30 sq. units e.
c. Area of playing field in School A = 21 sq. units
Area of playing field in School B = 19 sq. units
Since 21 is more than 19, so School A has a bigger playing field.
4. The number of the full squares (before eating) = 20
Full
Half
More
Less than half squares ( ) 7 0
Total area (approx.) -
3. Vegetable patch
a. The area of the playing field in School A = 21 sq. units
b. The area of the vegetable patch in School B = 6 sq. units
( ) 12
Half squares ( ) 0 0
More than half squares ( ) 1 1
Less than half squares ( ) 4 0
Total area (approx.) - 13 sq. units
The area of the chocolate bar he ate = The area of the full chocolate bar – area of chocolate bar left = 20 – 13 = 7 sq. units
b. The area of the chocolate bar before eating = 20 sq. units
5. Answers may vary. Sample answer:
All these six rectangles have area equal to 12 sq. units. b. The perimeter of the rectangle with sides 12 and 1 = 12 + 12 + 1 + 1 = 26 units
The perimeter of the rectangle with sides 6 and
2 = 16 units
The perimeter of the rectangle with sides 4 and 3 =
3 = 14 units
So, the perimeter is not the same as the area.
Chapter Checkup 1.
Perimeter = 18 units
Shape a has smaller boundary.
2.
a. Figure Q and figure S have the same area but different perimeters.
b. Figure R and figure S have the same perimeter but different areas.
c. Figure P and figure T have the same area and perimeter. 6. a.
Shape
The perimeter of figure A is 10 cm.
The perimeter of figure B is 14 cm.
The perimeter of figure C is 14 cm.
Figure B and figure C have the same perimeter.
3. a. Perimeter = 3 cm + 2 cm + 2 cm + 1 cm + 4 cm = 12 cm
b. Perimeter = 7 m + 9 m + 6 m + 12 m = 34 m
c. Perimeter = 3 m + 4 m + 6 m + 4 m + 3
Shape Squares
Full
More
Less
Total
c.
Shape Squares
Full
Less
Shape
Shape
Both of the shapes have an area of 8 sq. units. (Drawings may vary)
Shape
10. Length of the boundary = 120 m
Breadth of the boundary = 150 m
Length of fencing = Perimeter of boundary
Perimeter of the boundary = 120 m + 150 m + 120 m + 150 m = 540 m
Length of fencing available = 150 m
Remaining length of the fence required = 540 m – 150 m = 390 m
11. Length of the rectangular field = 200 m
Breadth of the rectangular field = 170 m
Distance covered by Suhani in one round = 200 m + 170 m + 200 m + 170 m = 740 m
Distance covered in two rounds = 2 × 740 m = 1480 m
Length of the square field = 150 m
Distance covered by Rishabh in one round = 150 m + 150 m + 150 m + 150 m = 600 m
Distance covered in three rounds = 3 × 600 m = 1800 m
Hence, Rishabh covered more distance by 1800 – 1480 = 320 m
Challenge
1. Answers may vary. Sample answers:
Chapter 12
Let’s Warm-up
1. 1 hour = 60 minutes
6 hours = 6 × 60 = 360 minutes
6 hours is the same as 360 minutes.
2. 1 minute = 60 seconds
1 second = 1 ÷ 60 minutes
360 seconds = 360 ÷ 60 = 6 minutes
360 seconds is equal to 6 minutes.
3. 1 hour = 60 minutes and 1 minute = 60 seconds
So, 60 minutes = 60 × 60 = 3600 seconds or 1 hour = 3600 seconds
2 hours = 2 × 3600 = 7200 seconds
There are 7200 seconds in 2 hours.
4. 1 hour = 60 minutes
So, 40 < 60
40 minutes is less than 1 hour.
5. 1 hour = 60 minutes
2 hours = 2 × 60 = 120 minutes
So, 120 < 130 2 hours is less than 130 minutes.
Do It Yourself 12A
1. a. Evening 5 o’ clock – 5:00 p.m.
All these have a perimeter of 20 units. (Drawings may vary)
5 rectangles of perimeter 20 units:
Rectangle 1: Length = 1 unit, Width = 9 unit;
Rectangle 2: Length = 2 unit, Width = 8 unit;
Rectangle 3: Length = 3 unit, Width = 7 unit;
Rectangle 4: Length = 4 unit, Width = 6 unit;
Rectangle 5: Length = 5 unit, Width = 5 unit
2. Answer may vary. Sample answer:
Case Study
1. Square metre of carpet needed = Area of the living room.
Area of the living room = Number of squares covering the area = 40 square metres
Hence, option d is correct.
2. Length of electrical wire required = Perimeter of the kitchen = 7 + 4 + 7 + 4 = 22 metres
Hence, option b is correct.
3. Area of bedroom 1 = 49 square metres
Area of bedroom 2 = 28 square metres
Area of bedroom 3 = 31 square metres
Since, 48 > 31 > 28
As the area of bedroom 1 has the largest area, so it should be given to Mr Sharma’s parents.
4. Garden, Bedroom 2 and Kitchen cover the same area and perimeter
Perimeter = 22 metres
Area = 28 square metres
5. Answer may vary.
b. At 10:00 in the morning – 10:00 a.m.
c. At 04:30 in the afternoon – 04:30 p.m.
d. At 10:00 in the night – 10:00 p.m.
2. a. 8:20
b. 11:47
c. 1:28
d. 4:44
3. Akhil goes for practice at 11:30 in the morning. This time would be written as 11:30 a.m.
4. Time one hour before the time given:
a. 12:30 p.m.—11:30 a.m.
b. 03:15 a.m.—02:15 a.m.
c. 12:59 a.m.—11:59 p.m.
d. 07:44 p.m.—06:44 p.m.
b. 4:12
c. 10:24
d. 12:14
6. Isha went to Gurudwara at 10:30 a.m.
The time after 2 hours: 1 hour 1 hour
10:30 a.m. 11:30 a.m. 12:30 p.m.
Isha came back at 12:30 p.m.
Challenge
1. The first clue eliminates options a, b, d, f and h, as they are before 8:30 a.m. or after 2:30 p.m.
The second clue eliminates option g, as it has an even number of minutes.
The fourth clue eliminates option e, as it is closer to 5:30 a.m. than to 5:30 p.m.
Thus, the correct time is c, 1:33 p.m. which satisfies all the criteria.
Do It Yourself 12B
1. a. 03:28 p.m.
For p.m. time, add 12 to the hour value: 3 + 12 = 15
Write the minutes as they are: 15:28
Replace p.m. with hours: 15:28 hours
b. 11:56 p.m.
For p.m. time, add 12 to the hour value: 11 + 12 = 23
Write the minutes as they are: 23:56
Replace p.m. with hours: 23:56 hours
c. 12 midnight
Time after 23:59 is read as 00:00 and not 24:00.
So, 00:00.
d. 11:59 p.m.
For p.m. time, add 12 to the hour value: 11 + 12 = 23
Write the minutes as they are: 23:59
Replace p.m. with hours: 23:59 hours
2. a. 22:40 hours
The first two digits of the time are 22. Since it is more than 12, it is p.m. time.
For p.m. time, subtract 12 from the first two digits: 22 – 12 = 10
Write down the minutes as they are: So, 22:40 hours by a 12-hour clock is 10:40 p.m.
b. 18:25 hours
The first two digits of the time are 18. Since it is more than 12, it is p.m. time. For p.m. time, subtract 12 from the first two digits: 18 – 12 = 6
Write down the minutes as they are: So, 18:25 hours by a 12-hour clock is 6:25 p.m.
c. 23:24 hours
The first two digits of the time are 23. Since it is more than 12, it is p.m. time.
For p.m. time, subtract 12 from the first two digits: 23 – 12 = 11
Write down the minutes as they are: So, 23:24 hours by a 12-hour clock is 11:24 p.m.
d. 13:03 hours
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, 13:03 hours by a 12-hour clock is 1:03 p.m.
3. To know the time in 24-hour format, follow the steps for changing from a.m. time — Keep the same hour value: 4
Write the minutes as they are: 4:30
Replace a.m. with hours: 04:30 hours
Adding 90 mins = 04:30 hours + 90 mins = 06:00 hours
4. Departure Time in 24-hour format = 16:55 hours
Departure Time in 12-hour format = 16:55 – 12 = 4:55 p.m. (Since the hours are greater than 12, we subtract 12 from the hours)
Arrival time in 24-hour format = 8:35 hours
Arrival time in 24-hour format = 8:35 a.m. (Since the hours are less than 12, we keep the hours the same)
5. The departure time of the flight—14:45 hours 14:45 hours in the 12-hours format = 2:45 p.m. (Since the hours are more than 12, it is p.m., and subtract 12 from the hours)
The boarding pass will be given 2 hours before departure. The time before 2 hours: 1 hour 1 hour
12:45 p.m. 1:45 p.m. 2: 45 p.m.
The boarding pass will be given at 12:45 p.m.
Challenge
1. If London is 5 hours 30 minutes behind and the call starts at 20:00 hours London time, we need to subtract 5 hours 30 minutes to find Samaira's local time.
20:00 hours = 8 p.m. (Since the hours are greater than 12, we subtract 12 from the hours)
8:00 p.m. in London − 5 hours 30 minutes = 2:30 p.m.
Thus, it will be 2:30 p.m. for Samaira when the call starts. It is a still day for Samaira.
1. a.
minutes = 5 hours 450
40 minutes
minutes = 9 hours 20 minutes
2. a. 1 hour = 60 minutes
= 11 hours 15 minutes
7 hours = 7 × 60 minutes = 420 minutes
7 hours = 420 minutes
b. 1 hour = 60 minutes
3 hours = 3 × 60 = 180 minutes.
3 hours > 115 minutes
c. 1 hour = 60 minutes
6 hours = 6 × 60 minutes = 360 minutes
6 hours = 360 minutes
d. 1 hour = 60 minutes
10 hours = 10 × 60 minutes = 600 minutes
10 hours > 360 minutes
3. a. 12:00 noon to 12:00 midnight = 12 hours
12:00 midnight to 12:30 = 30 minutes
Duration = 12 hours 30 minutes.
b. 05:06 p.m. to 10:06 p.m. = 5 hours
10:06 p.m. to 10:55 p.m. = 49 minutes
Duration = 5 hours 49 minutes
c. 14:25 hours to 20:25 hours = 6 hours.
20:25 to 20:45 = 20 minutes
Duration = 6 hours 20 minutes
d. 10:15 hours to 23:15 hours = 13 hours
23:15 hours to 23:30 hours = 15 minutes
Duration = 13 hours 15 minutes
4. The start time of the oath-taking ceremony = 9:40 a.m.
Duration of the ceremony = 2 hours 15 minutes
The ceremony ended at = 9:40 + 2 hours = 11:40 a.m.
11:40 a.m. + 15 minutes = 11:55 a.m.
5. The starting time of the doctor = 10:15 a.m.
The return time of the doctor = 1 p.m.
The time taken by the doctor: 10:15 a.m. 11:00 a.m. 12:00 noon 1:00 p.m. 45 minutes 1 Hour 1 Hour = 2 hours 45 minutes
The doctor spent 2 hours and 45 minutes with his patients.
6. The starting time of reading the book = 16:30
Duration of reading the book = 45 minutes
The time at which Nihit stopped reading: 45 minutes = 30 minutes + 15 minutes
30 minutes 15 minutes
16:30 hours 17:00 hours 17:15 hours
He stopped reading the book at 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. Time at which Neha starts studying = 4:00 p.m.
Time taken to complete maths homework = 30 minutes
Time taken to take English notes = 15 minutes
Time taken to revise all other subjects for the upcoming test = 70 minutes
Total time taken = 30 + 15 + 70 = 115 minutes
Since, 1 hour = 60 minutes
So, 115 – 60 = 55
Total time taken = 115 minutes = 60 minutes + 55 minutes = 1 hour 55 minutes
Time at which Neha stop studying is = 4:00 p.m. + 1 hour 55 minutes
= 5:55 p.m.
Therefore, Pooja will arrive before 6:00 p.m.
Do It Yourself 12D
1. a. There are 366 days in a leap year.
b. The day after Friday (03.03.23) is Saturday (04.03.23). The next day after Saturday is Sunday (05.03.23).
If 03.03.23 is a Friday, then the next Sunday will be on 05.03.23
c. February is a month with 28 or 29 days.
d. Remember 1 year = 12 months
2 years = 2 × 12 = 24
2 years = 24 months
2. a. 19 November 1996 = 19.11.1996
b. 15 August 1947 = 15.08.1947
c. 29 July 2023 = 29.07.2023
d. 28 February 2004 = 28.02.2004
3. a. Total days in June = 30
The days remaining in June = 0
The number of days in July = 23
The duration is 23 days
b. The total days in September = 30
The days remaining in September = 30 – 5 = 25 days
The days in October = 31 days
The days in November = 2 days
The duration = 25 + 31 + 2 = 58 days
c. 12.05.20 = 12 May
The total days in May = 31
The days remaining = 31 – 12 = 19 days
The total days in June = 10 days
The duration = 19 + 10 = 29 days
d. 07.06.23 = 7 June, 23 and 23.07.23 = 23 July, 23
The days in June = 30
The days remaining = 30 – 7 = 23 days
The days in July = 23
The duration = 23 + 23 = 46 days
4. New Year begins on 1 January. 10 December is a Friday.
10 + 7 = 17 December will be a Friday
17 + 7 = 24 December will be a Friday
24 + 7 = 31 December will be a Friday
1st January will be a Saturday.
5. 1 week = 7 days
2 weeks = 2 × 7 = 14 days
Jay takes the leave from 5 March
The days until Jay returns to school = 5 March + 14 days = 19 March
Jay will rejoin the school on 19 March.
6. The day of 25th December—Monday
Riya’s birthday = 3rd Monday after 25th December
The 1st Monday after 25th December = 25 + 7 = 1st January
The 2nd Monday after 25th December = 1 + 7 = 8th January
The 3rd Monday after 25th December = 8 + 7 = 15th January
Riya’s birthday is on 15th January.
7. Answers may vary. Sample answer: Anaʼs mother went 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. The manufacturing date of the chocolate = 12.12.2023 = 12th December 2023
The expiry date is 18 months from the date of manufacturing
18 months = 12 months + 6 months
12 months after 12th December 2023 = 12th December 2024
6 months after 12th December 2024 = 12th June 2025
The chocolate will expire on 12th June 2025 or 12.06.2025
Hence, Manas is right.
Chapter Checkup
1. a. 6:42
b. 7:29
2. a. This morning Emily woke up at 7 a.m.
b. She took 45 minutes to get ready, then it was 7:45 a.m.
c. She had her lunch at 12:30 p.m. in the cafeteria with her friends.
3. a. 2 hours after 4:30 in the morning – 6:30 a.m.
b. 3 hours after 8:45 in the evening – 11:45 p.m.
c. 1 hour after 10:00 at night – 11:00 p.m.
d. 4 hours after 1:20 in the afternoon – 5:20 p.m.
4. a. 06:30 a.m.
For a.m. time, keep the same hour value: 6
Write the minutes as they are: 6:30
Replace a.m. with hours: 06:30 hours
b. 07:55 a.m.
For a.m. time, keep the same hour value: 7
Write the minutes as they are: 7:55
Replace a.m. with hours: 07:55 hours
c. 01:03 p.m.
For p.m. time, add 12 to the hour value: 1 + 12 = 13
Write the minutes as they are: 13:03
Replace p.m. with hours: 13:03 hours
d. 09:15 p.m.
For p.m. time, add 12 to the hour value: 9 + 12 = 21
Write the minutes as they are: 21:15
Replace p.m. with hours: 21:15 hours
5. a. 14:20 hours
The first two digits of the time are 14.
Since it is more than 12, it is p.m. time.
For p.m. time, subtract 12 from the first two digits: 14 – 12 = 2
Write down the minutes as they are:
So, 14:20 hours by a 12-hour clock is 02:20 p.m.
b. 15:45 hours
The first two digits of the time are 15.
Since it is more than 12, it is p.m. time.
For p.m. time, subtract 12 from the first two digits: 15 – 12 = 3
Write down the minutes as they are:
So, 15:45 hours by a 12-hour clock is 03:45 p.m.
c. 21:12 hours
The first two digits of the time are 21.
Since it is more than 12, it is p.m. time.
For p.m. time, subtract 12 from the first two digits: 21 – 12 = 9
Write down the minutes as they are:
So, 21:12 hours by a 12-hour clock is 09:12 p.m.
d. 04:30 hours
The first two digits of the time are 04.
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 04.
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. 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.
Value of six 50 paise = 6 × 50 = 300 paise = ₹3
Total amount paid by Raju to the junk collector = ₹943 + ₹3 = ₹946
Hence Statement 1 is correct.
According to Statement 2, Total amount paid by Raju to the junk collector = 6 × ₹100 + 6 × ₹50 + 1 × ₹20 + 4 × ₹5 + 3 × ₹1 = ₹943
But, Raju paid ₹946 to the junk collector.
Hence Statement 2 is incorrect. The correct answer is option a.
2. The money paid by each friend = ₹98
The money paid by 5 friends = ₹98 × 5 = ₹490
The total money paid by 5 friends and Rahul = ₹600
So, the money paid by Rahul = ₹600 – ₹490 = ₹110
Case Study
1. Cost of a bulb = 14200 paise
To convert the paise into amount remove the word 'paise' and put a dot after counting 2 numbers from the right of the number.
The cost of the bulb in rupees will be ₹142.
2. Cost of one battery = ₹196.50
The cost of 2 batteries = ₹196.50 + ₹196.50 = ₹393.00 Hence, option d is correct.
3. S. No.
1. Small wooden board ₹385.00 1 ₹385.00
2. Switch
4. Total number of students = 30
Number of students in each group = 5
Number of groups = 30 ÷ 5 = 6
Cost of the experiment for each group = ₹808.50
Cost of the experiment for 6 groups = 6 × ₹808.50 = ₹4851.00 So, ₹4851.00 will be required to buy experiment materials for all the groups.
5. Answers may vary.
Chapter 14
Let’s Warm-up
1. There are 5 lions.
2. There is 1 cat.
3. There are 4 giraffes. This is 1 less than the lions
4. There are 3 elephants. This is 1 more than the crocodiles
5. There are 2 crocodiles. This is 1 more than the cats.
Do It Yourself 14A
1. The tally marks count which shows the number 25.
2. Health Drink Number of People
tea 15
juice 7
water 4
The most popular health drink is green tea. Hence, option b is correct.
3. Appliance Tally
5.
a. There are 8 carrots.
b. There are 9 pumpkins and 10 capsicums. Total = 10 + 9 = 19
c. There are 6 potatoes.
d. The total number of vegetables = 8 + 10 + 9 + 6 = 33
b. Number of households segregate wet waste = 8 Number of households segregate e-waste = 2 8 – 2 = 6, so there are more households segregating wet waste compared to e-waste.
c. Number of households segregate sanitary waste = 3 If 3 more households start segregating sanitary waste, then the total number of households segregate sanitary waste = 3 + 3 = 6
d. E- waste has the least number of households segregating it.
e. Answer may vary.
6. Answer may vary. Sample answer: How many more tea pots are there than mixer grinders?
Challenge
1. 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.
5. a. Apples consumed in School 1 = 10 × 35 = 350 apples
Apples consumed in School 2 = 2 × 35 = 70 apples
Apples consumed in School 3 = 3 × 35 = 105 apples
Oranges consumed in School 1 = 4 × 45 = 180 oranges
Oranges consumed in School 2 = 2 × 45 = 90 oranges
Oranges consumed in School 3 = 6 × 45 = 270 oranges
In School 2 and School 3, the number of students who consumed oranges is more than those who consumed apples.
b. Number of students who consumed apples at School 1 = 10 × 35 = 350 students
Number of students who consumed apples at School 3 = 3 × 35 = 105 students
Difference = 350 − 105 = 245 students
The tally marks for 8 and 9 marks together = 3 + 3 = 6
The tally marks for 8 and 9 marks together can be given as |||||
So, the assertion is false.
The reasoning (R) states that each group of tally marks represents five items, which is correct. Hence, option d is correct.
Do It Yourself 14B
1. a. A pictograph is a way to represent data using images and symbols. True
b. A pictograph makes the data representation visually interesting and easy to understand. True
c. Pictographs cannot be used to compare two quantities. False
2. 6 × 2 = 12 students scored grade A and 7 × 2 = 14 students scored grade B.
So, a total of 12 + 14 = 26 students scored grades higher than grade C.
Hence, option c is correct.
3. a. City A = 7 × 25 cm = 175 cm
City D = 2 × 25 cm = 50 cm
b. City B = 10 × 25 cm = 250 cm
City E = 3 × 25 cm = 75 cm
Rainfall at City B > Rainfall at City E
4. Each symbol represents: 10 tigers
Total tigers in Bihar: 30 tigers
Number of symbols for Bihar = 30 ÷10 = 3 symbols
Total tigers in Andhra Pradesh: 70 tigers
Number of symbols for Andhra Pradesh = 70 ÷ 10 = 7 symbols
Total tigers in Chhattisgarh: 50 tigers
Number of symbols for Chhattisgarh = 50 ÷ 10 = 5 symbols
Total tigers in Rajasthan: 60 tigers
Number of symbols for Rajasthan = 60 ÷ 10 = 6 symbols
Key: = 10 Tigers
States
Bihar
Andhra Pradesh
Chhattisgarh
Rajasthan
Number of Tigers
c. In School 2, 2 × 35 = 70 students consumed apples, and 2 × 45 = 90 students consumed oranges.
Total = 70 + 90 = 160 students
6. Answer may vary. Sample answer: How many students consumed oranges in School 3?
Challenge
Fruits
Number of Fruits
Apples 4
Bananas 6
Cherries 8
Grapes 9
1. If the key shows 1 circle = 1 fruit, 1 fruit = 1 circle
Number of symbols for apples = 4
Number of symbols for bananas = 6
Number of symbols for cherries = 8
Number of symbols for grapes = 9
a. Key: = 1 fruit
Fruit
Apples
Bananas
Cherries
Grapes
Number of Fruits
b. If the key shows 1 circle = 2 fruit, 2 fruit = 1 circle 1 fruit = 1 ÷ 2
Number of symbols for apples = 4 ÷2 = 2
Number of symbols for bananas = 6 ÷ 2 = 3
Number of symbols for cherries = 8 ÷ 2 = 4
Number of symbols for grapes = 9 ÷ 2 = 4.5
Key: = 2 fruits
Apples
Bananas
Cherries
Grapes
Number of Fruits
c. If the key shows 1 circle = 4 fruits, Number of grapes = 9
So, 4 fruits = 1 circle
⇒ 1 fruit = 1 4 of a circle
⇒ 9 fruits = 9 × 1 4 circle
⇒ 9 fruits = 2 1 4 circle
Hence, you will draw two and one-fourth circles for grapes.
Do It Yourself 14C
1. a. LPG is used highest in number. LPG is used in the most houses.
b. 10 houses are using coal as fuel.
5. a. The lowest number of children participate from Grade 3. b. The grade from which only 60 children participate is Grade 4.
c. Number of children who participated from Grade 3 = 40 Number of children who participated from Grade 1 = 50 50 − 40 = 10 fewer children participated from Grade 3 than Grade 1.
d. Number of children who participated from Grade 5 = 120 Number of children who participated from Grade 2 = 90 120 − 90 = 30 more children participated from Grade 5 than Grade 2.
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. Total number of books with Sanjay = 6 + 8 + 10 + 2 = 26 books
2. a. A greater number of childern in class 4 like summer more than autumn. False
b. The most preferred season is summer. False
c. Spring is liked by more students than autumn. False
d. Winter is liked by more students than spring. True
3. a. Fraction of the children who like to play hockey = 3 4 Hence, option iv is correct.
b. Fraction of the children who do not like to play hockey = 31 1 44 −=
Hence, option iii is correct.
c. i. Total number of students = 60
Fraction of the children who like to play lawn tennis = 1 4
Number of children who like to play lawn tennis
= 1 ×60=15 4
Therefore, 15 children like to play lawn tennis.
ii. Total number of students = 60
Fraction of the children who do not like to play lawn tennis = 1− 1 4 = 3 4
Number of children who do not like to play lawn tennis
= 3 ×60=45 4
Therefore, 45 students do not like to play lawn tennis.
4. a. Africa continent is the second largest in terms of area.
b. Total number of parts = 200
Number of parts covered by Europe = 14
Fraction of the total area is covered by Europe = 147 = 200100
c. Total number of parts = 200
Number of parts covered by Australia = 10
Number of parts covered by South America = 24
Total number of parts covered by Australia and South America = 10 + 24 = 34
Fraction of the total area covered by Australia and South America = 3417 = 200100
d. Antarctica continent has an area less than that of South America but more than that of Europe.
5. Answer may vary. Sample answer:
Which continent is the smallest in terms of area?
Challenge
1. a. Fraction of people playing = 1 2
Number of people playing = 1 100=50 2 ×
Number of people skating = 15
Number of people walking = 10
Number of people cycling = 100 − 50 − 15 − 10 = 25
b. Number of people skating = 15 + 10 = 25
Fraction = 251 = 1004
c. Number of people cycling = 25
Fraction = 251 = 1004
Chapter Checkup
1. Number of paintings the painter sold in March = 4 × 5 = 20 paintings
Hence, option a is correct.
2. a. There are 9 elephants in the zoo of Assam.
b. Andhra Pradesh and Maharashtra has equal number of elephants.
3. a. 1 pumpkin image = 50 pumpkins
4 pumpkin images = 4 × 50 = 200 pumpkins
Since Madhav has 4 pumpkin images in front of him, he harvested 200 pumpkins.
b. There are 5 and a half pumpkins in front of Hari.
1 pumpkin image = 50 pumpkins
5 pumpkin images = 5 × 50 = 250 pumpkins
Half pumpkin = 50 2 = 25 pumpkins
5 and a half pumpkins images = 250 + 25 = 275 pumpkins
Number of pumpkins harvested by Hari = 275
4. Number of students in Grade 4 who prefer dancing = 30
Number of students in Grade 3 who prefer dancing = 25
Number of students in Grade 4 who prefer dancing than the students in Grade 3 = 30 – 25 = 5
5. a. 22 bicycles were sold in week 1.
b. Bicycle sold in week 1 = 22
Bicycles sold in week 4 = 15
Total bicycles sold in Week 1 and week 4 = 22 + 15 = 37
c. 22 + 29 + 16 + 15 + 25 = 107
Hence, 107 bicycles were sold in 5 weeks.
6. a. 10 children would like to go to the National Museum.
b. 15 children would like to go to the Adventure Island.
c. Number of children who would like to go to the Zoo = 20
Number of children who would like to go to the Rail Museum = 5
20 − 5 = 15
15 more children would like to go to the Zoo than to the Rail Museum.
d. 20 + 15 + 10 + 5 = 50
Hence, 50 Students were surveyed.
7. Fraction for chess = 10 40 = 1 4
Fraction for ludo = 20 40 = 1 2
Fraction for carrom = 5 40 = 1 8
8. Answer may vary. Sample answer: How many friends vote for ludo?
Challenge
1. 1 4 fraction of students goes to school by car.
Fraction of students who do not go to school by car = 1 – 1 4 = 4 4 –1 4 = 3 4
2. Cakes sold on Day 1 = 6
Cakes sold on Day 5 = 2 × 6 = 12
Cakes sold on Day 3 = 12
Cakes sold on Day 6 = 12 2 = 6
Total cakes sold over 6 days = 6 + 9 + 12 + 6 + 12 + 6 = 51
Case Study
1. Number of plastic bottles used at the School Fair = 300
Number of plastic bottles used at the Sports Day = 450 Total number of plastic bottles used at the School Fair and Sports Day = 300 + 450 = 750 Hence, option b is correct.
2. Number of plastic bottles used at the School Fair = 300 Number of plastic bottles used at the Sports Day = 450
Number of plastic bottles used at the Town festival = 600 Number of plastic bottles used at the Community Picnic = 350 Town Festival event used the most plastic bottles. Hence, option c is correct.
3. The Community Picnic used fewer plastic bottles than the Sports Day. True
4. The event that used 350 plastic bottles was the Community Picnic
5. The total number of plastic bottles used at all four events = 300 + 450 + 600 + 350 = 1700
The total number of plastic bottles used at all four events was 1700
6. Answer may vary.
About the Book
The Imagine Mathematics teacher manuals bridge the gap between abstract mathematics and real-world relevance, offering engaging activities, games and quizzes that inspire young minds to explore the beauty and power of mathematical thinking. These teacher manuals are designed to be indispensable companions for educators, providing well-structured guidance to make teaching mathematics both effective and enjoyable. With a focus on interactive and hands-on learning, the lessons in the manuals include teaching strategies that will ensure engaging lessons and foster critical thinking and problem-solving skills. The teaching aids and resources emphasise creating an enriched and enjoyable learning environment, ensuring that students not only grasp mathematical concepts but also develop a genuine interest in the subject.
Key Features
• Alignment with Imagine Mathematics Content Book: Lesson plans and the topics in the learners’ books are in sync
• Learning Outcomes: Lessons designed as per clear, specific and measurable learning outcomes
• Alignment to NCF 2022-23: Lessons designed in accordance with NCF recommendations
• Built-in Recaps: Quick recall of pre-requisite concepts covered in each lesson
• Supporting Vocabulary: Systematic development of mathematical vocabulary and terminology
• Teaching Aids: Resources that the teachers may need to facilitate the lesson
• Activity: Concise and organised lesson plans that outline each activity
• Extension Ideas: Analytical opportunities upon delivery of each lesson
• Detailed Solutions: Solutions to all types of questions in the Imagine Mathematics Content Book
• Digital Assets: Access to supplementary interactive resources
About Uolo
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