Headstart - Math Coursebook 4

COURSEBOOK

Aligned with NEP 2020 With HEADSTART APP

MATHS COURSEBOOK

Aligned with NEP 2020

Headstart Maths Coursebook - Class 4

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ISBN 978-81-967554-9-2

First Edition

Preface

Dear Parents, Educators, and Guardians,

In the ever-evolving landscape of education, the National Education Policy (NEP) 2020 is a pivotal milestone, emphasizing the critical role of ages 3 to 6 in shaping a child’s mental faculties. Moving away from the traditional method of memorization, it highlights the significance of key learning goals, places a greater emphasis on multidisciplinary education, and aims to nurture the creative talents of every learner.

Aligned with the visionary NEP 2020, we proudly introduce the “HeadStart Programme” by Infinity Learn— an innovative educational initiative meticulously designed to fortify your child’s foundation during their most formative years. Tailored for Grades 1 to 5, this program aims to instil a passion for learning and establishes a formidable academic base.

The HeadStart Program unfolds a tailored educational experience, commencing with Grades 1 and 2, where the focus lies on making numbers (Math) comprehensible and fostering exploratory learning in Environmental Studies (EVS). Progressing into Grades 3, 4, and 5, the program expands to encompass critical mathematical thinking, exploration of Science, and a nuanced understanding of society, environment, and global citizenship in Social Studies.

Equipped with thoughtfully crafted course books, home reinforcement workbooks, and a cutting-edge learning app, the HeadStart Program leverages both traditional and digital tools to ensure a comprehensive educational experience.

What is the HeadStart Advantage?

We firmly believe that enhancing creativity, logical and critical thinking in the early years will pave the way for a robust foundation in Mathematics and Science during the middle years. This, in turn, positions learners for success in senior-grade challenges, differentiating them in high-stakes exams such as JEE, NEET, and CUET.

For educators, we provide meticulously designed lesson plans and in-class videos, ensuring uniform and effective teaching methodologies. Our mission encapsulates the question ‘Baccha Seekha ki Nahi’—did the child learn?—as we are dedicated to ensuring that each child’s learning journey is not merely a progression but a meaningful and enriching experience.

We invite you to join us in this exciting new chapter of your child’s education. The “HeadStart Programme” is more than a curriculum; it’s a pathway to nurturing a lifelong passion for learning. Let’s provide our children with the optimal start in their educational journey.

Warm regards,

The Infinity Learn Team

Large Numbers

My Study Plan

5-digit and 6-digit numbers in words and numbers and their place values

Successor and predecessor

Comparison of numbers

Forming the greatest and smallest numbers

Rounding off numbers

International place value system

Roman numerals

Let’s Recall

We have learnt about 4-digit numbers earlier. Here is an example of a four digit number: 3426. Let us look at the place value chart of this number.

Here, at ones place we have 6.

At tens place we have 2.

At hundreds place we have 4.

At thousands place we have 3.

Now, let us answer some questions.

The greatest four digit number is __ __ __ 9.

The smallest four digit number is __ __ __ 0.

Compare the following numbers using < / >.

a. 5505 5550

b. 3161 3106

c. 5445 5544

d. 7451 1756

Let’s Learn

5-Digit Numbers

Look at the given abacus. It represents the number 9,999.

Thinking Zone

I am a number, big and tall, Made of nines, four in all.

Flip me around, and I don’t change.

Find the answer but, don’t exchange.

What number am I?

On adding 1 to 9999, it becomes 9999 + 1, that is 10,000.

10000 is the smallest 5-digit number. We read it as ‘ten thousand’. 99999 is the greatest 5-digit number. We read it as ‘ninety-nine thousand nine hundred ninety-nine’.

Placing commas

According to the Hindu-Arabic numeral system, we place a comma to show the thousands place. For example, 5834 is written as 5,834. Here, the comma is after the thousands place and before the hundreds place.

The commas help us recognise bigger numbers and also help us say these number names easily.

So, ten thousand is written as 10,000, and ninety-nine thousand nine hundred ninety-nine is written as 99,999.

5-digit numbers on place value chart

Each digit in a number has a place that shows us its value. The place value of a digit is equal to the product of the digit and the value of the place it occupies. We can write the expanded form of any number using this concept.

For example, 4326 = 4 × 1000 + 3 × 100 + 2 × 10 + 6

Similarly, 56734 = 5 × 10000 + 6 × 1000 + 7 × 100 + 3 × 10 + 4

Here, 56734 is read as fifty-six thousand seven hundred and thirty-four.

The place value chart for 56734 can be made as follows:

Here, we can see that the place value chart has the ten thousands place. Every five-digit number will have a ten thousands place. Let us see some examples based on this.

Example 1:

Show the place value of number 11,000 in place value chart. Also write the number in words.

Solution:

The number 11,000 can be seen in the abacus as shown.

Now, we can make a table of the place values as follows.

In words, we can say that the number is eleven thousand.

Example 2:

Show the place value of number 10,467 in place value chart. Also, write the number in words.

Solution:

The number 10,467 can be seen in the abacus as shown below.

Now, we can make a table of the place value as follows.

In words, we can say that the number is ten thousand four hundred sixty-seven.

In figure or digits, we say that the number is 10,467.

TTh

TTh

The greatest 5-digit number is 99,999. Such numbers are called palindromes. These numbers remain the same when reversed. One more such number is 12321. Try to make more such palindromes.

Let’s Learn

6-Digit Numbers

We know that 99,999 is the greatest 5-digit number. 99,999 is shown in the abacus below.

On adding 1 to 99,999, it becomes 99,999 + 1, that is one hundred thousand or one lakh.

100000 is the smallest 6-digit number.

TTh Th H T O

TTh L Th H T O

We read it as ‘one hundred thousand’ or ‘one lakh’.

999999 is the greatest 6-digit number.

We read it as ‘nine lakh ninety-nine thousand nine hundred ninety-nine’.

A new place value called ‘lakhs’ is added to the place value chart for 6-digit numbers.

Just as we put commas after the thousands place, we also put a comma after the lakhs place.

So one lakh is written as 1,00,000 and nine lakh ninety-nine thousand nine hundred ninety-nine is written as 9,99,999.

When reading the number aloud, start with the lakhs place, then move on to the ten thousands place, followed by the thousands, hundreds, tens and the ones place.

Example:

Show the place value chart of number 3,76,467.

Solution:

In words, we can say that the number is three lakh seventy-six thousand four hundred sixty-seven.

TTh

1. Identify and write the number represented by the given abacus.

2. Write the number names of the following numbers and write them in the place value chart. a. 21534

Let’s Learn

Successor and Predecessor

Look at the number 1,25,475 on the number line.

Predecessor of 1,25,476

Successor of 1,25,475

1,25,475 is followed by 1,25,476 on the number line.

1,25,476 is called the successor of 1,25,475.

The successor of any number is the number that comes just after the number.

The successor of any number can be found by adding 1 to it.

The predecessor of a number is the number that comes just before it.

We can find the predecessor of a number by subtracting 1 from it.

Here 1,25,475 is the predecessor of 1,25,476.

Let’s Practise - 2

1. Find the predecessor of:

a. 2,354 __________

c. 9,99,999_________

2. Find out the successor of:

a. 4,208 ________

c. 8,77,666_________

Let’s Learn

Comparison of Numbers

b. 46,030 __________

d. 2,43,540__________

b. 9,791 _________

d. 7,56,940__________

Comparing numbers helps us find which one is greater and which one is smaller.

As we know:

• The > sign stands for ‘greater than’

• The < sign stands for ‘less than’

• The = sign stands for ‘equal to’

Let’s compare two numbers, 97,389 and 97,388, and represent them using their respective digits in a place value chart:

The number 97,389 is formed using the digits 9, 7, 3, 8 and 9.

The number 97,388 is formed using the digits 9, 7, 3 and 8.

Step 1: Write the digits of the two numbers in a place value chart as shown below.

Step 2: Look at the digits in the thousands and ten thousands place. They are same.

Step 3: Look at the digits in the hundreds place. They are the same.

Step 4: Look at the digits in the tens place. They are the same.

Step 5: Look at the digits in the ones place. Here, 9 > 8.

So, the number 97,389 is greater than the number 97,388 or 97,389 > 97,388.

Let’s Practise - 3

1. Fill in the correct sign <, > or =.

Let’s Learn

Forming Greatest and the Smallest Numbers

We can arrange the numbers in increasing or decreasing order to form the greatest and the smallest number with the given digits. Let us understand this with the help of a few examples.

Example 1:

Form the greatest 5-digit number with digits 8, 6, 5, 2, and 1 used only once.

Solution:

To form the greatest 5-digit number with digits 8, 6, 5, 2, and 1 used only once, we arrange the digits in decreasing order.

8 > 6 > 5 > 2 > 1

So, the required number is 86521.

Example 2:

Form the smallest 6-digit number with digits 8, 7, 6, 4, 2, and 1 used only once.

Solution:

To form the smallest 6-digit number with digits 8, 7, 6, 4, 2, and 1 used only once, we arrange the digits in increasing order.

1 < 2 < 4 < 6 < 7 < 8

So, the required number is 124678.

Example 3:

Form the smallest 6-digit number with the digits 4, 7, 8, 6, 1, and 0 used only once.

Solution:

To form the smallest 6-digit number with the digits 4, 7, 8, 6, 1, and 0 used only once, we arrange the digits in increasing order.

0 < 1 < 4 < 6 < 7 < 8

But the digit 0 at the beginning of a number has no value. To make it the smallest 6-digit number, place the digit 0 after the second-smallest digit, that is, 1.

So, the smallest number that can be formed using the digits 4, 7, 8, 6, 1, and 0 only once is 104678.

Let’s Practise - 4

1. Form the greatest and the smallest number by using the given digits only once.

Digits

2, 8, 0, 5, 3

2, 4, 8, 3, 7, 3

4, 7, 3, 8, 9, 1

8, 6, 9, 0, 9

Smallest Number Greatest Number

Thinking Zone

Write the pin code of your address, such as 110001. Then, use those digits to make the largest and smallest 6-digit numbers. Write the numbers in words and find their successors and predecessors.

Let’s Learn

Rounding Off

Rounding off is a way of estimating.

Rounding off means changing a given number to the nearest tens, hundreds or thousands.

Rounding off helps us in performing calculations easily.

Let us see the following rules to help us understand rounding off.

Rules for rounding off

Rule 1

Let us say we need to round off the number 11489 to the nearest tens place. Here, we will consider the digit at the ones place which is ‘9’. Now, if the ones place is less than five, then replace the ones place with zero.

Also, if the ones digit is 5 or greater than 5 then, the tens place is increased by one, and the ones place is changed to zero. In 11489, 9 is greater than 5, so we will add 1 to the tens place and the ones place is changed to 0.

After rounding off

Rule 2

When rounding off a number to the nearest hundreds place, if the digit to the right of the hundreds place is 5 or greater than 5, then add 1 to the hundreds place. The tens and the ones digits become 0. If the tens place is less than 5, then simply make the ones and the tens digits 0.

Here, in 11,789, 8 is greater than 5. So, we will add 1 to the hundreds place, and the tens and ones place are taken to be 0.

After rounding off

Rule 3

When rounding off a number to the nearest thousands place, if the digit to the right of the thousands place is 5 or greater than 5, then add 1 to the thousands place. The hundreds, the tens and the ones digits become 0. If the hundreds place is less than 5, then simply make the ones, tens and hundreds place digits 0.

Here, in 11,443, 4 is the hundreds place digit and is less than 5. So, the thousands place digit, 1, will remain as it is, and the hundreds, tens, and ones place are taken to be 0.

Let’s Practise - 5

1. Round off the following numbers to the nearest tens.

a. 98,115 _______

c. 45,342 ________

b. 21,263 ________

d. 12,824 ________

2. Round off the following numbers to the nearest hundreds.

a. 38,018 ________

c. 54,846 ________

b. 23,456 ________

d. 49,845

3. Round off the following numbers to the nearest thousands.

a. 1,23,243

c. 3,44,354 _______

Let’s Learn

b. 2,32,345 _______

d. 43,559 _______

International Place Value System

India uses the Indian Place Value System, also known as the Hindu-Arabic Place Value System. The system used internationally is the International Place Value System.

The Indian place value has ones, tens, hundreds, thousands, ten thousands, and lakhs.

Similarly, the international place value has ones, tens, hundreds, thousands, ten thousands, hundred thousands, and millions.

We can see this in the example given below:

Similar to the Indian Place Value System, the International System also uses commas to understand and read the numbers easily. In the International System, we use commas after the thousands place and then after the millions place.

For instance, in the International System, the number 5259113 is written as 5,259,113 and is read as five million two hundred fifty-nine thousand one hundred thirteen.

Let’s Practise - 6

1. Write the following numbers in words in the International System.

a. 168,729

d. 8,359,498

b. 845,367

e. 4,805,029

c. 5,765,379

f. 927,540

2. Write the following numbers in place value chart according to the International System.

a. Seventy million eight hundred twenty-six thousand four hundred forty-two.

b. Fifty-seven million ninety-six thousand four hundred twenty-nine.

Roman Numerals

Roman numeral is an ancient numeral system that uses combinations of letters to represent numbers.

Key Roman numerals

I - Represents the number 1

V - Represents the number 5

X - Represents the number 10

L - Represents the number 50

C - Represents the number 100

D - Represents the number 500

M - Represents the number 1000

Rules for writing Roman numerals

• The letters I, X, C, and M can be repeated up to three times in a row.

• The letters V, L, and D cannot be repeated; they can only appear once.

• Smaller numerals placed before larger numerals are subtracted from the larger ones. For example, IV represents 4 (V - I), IX represents 9 (X - I), and XL represents 40 (L - X).

• Smaller numerals placed after larger numerals are added to the larger ones. For example, VI represents 6 (V + I), XI represents 11 (X + I), and LX represents 60 (L + X).

• Combine the Roman numerals following the rules mentioned above to form larger numbers. For instance, 37 is represented as XXXVII (10 + 10 + 10 + 5 + 1 + 1).

Example 1:

Write 1421 using Roman numerals.

Solution:

Step 1: Start with the largest possible Roman numeral on the left and work your way to the right, from the thousands to the ones place.

Here, begin with thousands place, which is represented by M.

Step 2: Move to the hundreds place where C represents 100 and D represents 500. Here, to represent 400, we can combine C and D or subtract C from D. Hence, CD represents 400.

Step 3: Move to the tens and ones place where X represents 10 and L represents 50.

To represent 21, we will use XXI.

Hence, putting it all together, 1421 can be represented as MCDXXI in Roman numerals.

Fact Zone

Romans didn’t have a symbol for the number zero. Because of this, their math was a little harder to do compared to what we do today. It became much easier when people in Europe started using the Hindu-Arabic numeral system, which includes the number ‘0’

Let’s Practise - 7

1. Write the Roman numerals for the following numbers.

a. 1212

c. 1398

b. 2148

d. 1879

Let’s Sum Up

Numerical and word form: Representing numbers like 54,321 as ‘fifty-four thousand three hundred twenty-one’ and vice versa.

Place values: Recognizing the value of each digit in a number, e.g., in 6,824, the 6 is in the thousands place.

Successor and predecessor: Finding the next (successor) or previous (predecessor) number, e.g., the successor of 378 is 379.

Comparison of numbers: Using symbols to compare, e.g., 5,892 > 4,761.

Forming numbers: Arranging digits to create the largest (e.g., 9,874) or smallest (e.g., 1,235) numbers, also rounding off (e.g., rounding 4,689 to the nearest hundred), and understanding Roman numerals (e.g., IV for 4).

Life Skills

Write down the average cost of a two bedroom flat in your city. Also talk to your parents and grandparents and write down the average prices before 20 years. Compare how much the prices have increased in the last 20 years.

Cross-Curricular Connections

Social Studies:

Find out the population of any two metropolitan cities and compare their population numbers.

21st Century Skills

Find your area’s postal code. Look at the numbers in nearby areas’ postal codes. What do the digits in a postal code mean? Which numbers change when you go to different parts of your city? Which ones change when you go to a different city? Find out.

Extend Your Knowledge

In the Indian Number System:

One Crore: 1 crore is equal to 10,000,000. In the International Number System, this is equivalent to 10 million.

One Billion: 1 billion is equal to 1,000,000,000. In the Indian Number system, it is also referred to as 100 crores.

Tip for the Parent

• Make children aware of what the average cost of certain things is, for example, a smart television or a mobile phone is in the range of 20 to 30 thousand. A car is between 5 to 8 lakhs and so on.

Addition 2

My Study Plan

Addition of large numbers

Properties of addition

Addition in real life

Estimating sums

Let’s Recall

In the previous class, we have learned about the addition of 3-digit numbers and 4-digit numbers.

Let’s do a quick warm-up by solving the below word problem.

I collected `1,555, and my brother Aarav has collected `1,612 for a charity. How much money have we collected in total?

Thinking Zone

Fill the blanks with equal numbers to get solution equal to 6000.

a. __________ + __________ = 6000

b. __________ + __________ + __________ = 6000

c. __________ + __________ + __________ + __________ = 6000

Let’s Learn

Addition of Large Numbers

Addition combines two or more numbers to find their total.

Addition without regrouping

Let’s solve the addition problem given below.

Example 1: Add 13642 and 15312.

Solution:

Step 1: Arrange the digits of the given numbers in the correct column as shown below.

Step 2: Add the digits in the ones place (rightmost column) of both numbers.

Step 3: Similarly, add the digits in the tens, hundreds, thousands, and ten thousands places of both numbers.

Therefore, 13642 + 15312 = 28954.

Addition with regrouping

Example 2: Add 25632 and 28329.

Solution:

Step 1: Arrange the digits of the given numbers in the correct column as shown below.

Step 2: Add the digits in ones place. So, 9 + 2 = 11, which is a 2-digit number. Now, regroup 11 as 1 tens and 1 ones. Write 1 under the ones column and carry over 1 to the tens place.

Step 3: Similarly, add the tens, hundreds, thousands, and ten thousands places.

Therefore, 25632 + 28329 = 53961.

Word Zone

Carry Over (in addition): Moving extra to the next column.

Addition of three numbers

Example 3: Add 24136, 33423, and 37440.

Solution:

Step 1: Arrange the digits of the given numbers in the correct column as shown below.

Step 2: Add the digits in ones place.

Step 3: Similarly, add tens, hundreds, thousands, and ten thousands place.

Therefore, 24136 + 33423 + 37440 = 94999.

Fact Zone

The two phrases, “twelve plus one” and “eleven plus two”, both have 13 letters.

Let’s Practise - 1

1. Add the following.

Let’s Learn

Properties of Addition

Property 1:

The sum of two numbers remains the same even if we change the order of the numbers. This is called commutative property or order property of addition.

Example:

Thus, 4141 + 1301 = 5442.

1301 + 4141 = 5442.

Hence, 4141 + 1301 = 1301 + 4141.

Property 2:

While adding three or more numbers, the numbers can be grouped in any way. The sum always remains the same. This is called the grouping property of addition.

Example:

(2241 + 5189) + 2246 =

2241 + (5189 + 2246) =

Thus,

(2241 + 5189) + 2246 = 9676. 2241 + (5189 + 2246) = 9676.

Hence, (2241 + 5189) + 2246 = 2241 + (5189 + 2246).

Property 3:

When we add 1 to a number, the sum is the successor of the number. For example,

Thus, 3983 + 1 = 3984.

Hence, 3984 is the successor of 3983.

Property 4:

If 0 is added to any number, the sum is the number itself. For example,

Thus, 2735 + 0 = 2735.

Thinking Zone

Sarah is working on a math project where she needs to add the populations of four different cities. The populations are 2,345, 1,789, 4,567, and 3,210. To make her task more challenging, she decided to add the populations in a way that each sum includes at least two cities. How many different ways can Sarah add the populations to get unique sums, and what are those sums?

Word Zone

Successor: The next number when counting

Let’s Practise - 2

1. Fill in the blanks:

a. 9837 + __ = 9837.

b. The successor of 2533 is ______.

c. 16253 + 32354 = _____ + 16253

d. 3987 + 3287 + 9473 = 3287 + _________ + 3987

2. Find the sum of the following.

a. (2375 + 4221) + 5362

b. (3943 + 4765) + 5324

Let’s Learn

Estimating the Sum

To estimate sum,

(i) Round off the numbers separately to the required place.

(ii) Then add them as normal numbers.

Example 1: Add 3227 and 4153 together through estimation to the nearest tens.

Solution:

Step 1: Round off numbers to the nearest 10:

If the digit in the ones place is 5, 6, 7, 8, or 9, replace the ones place digit with 0 and add 1 to the tens place digit.

If the digit in the ones place is 0, 1, 2, 3, or 4, replace the ones place digit with 0, and the tens place digit remains unchanged.

So, the number 3227 round off to 3230.

And the number 4153 round off to 4150.

Step 2: Add these rounded numbers together.

Therefore, 3227 + 4153 = 7380 (approximately).

Example 2: Add 25412 and 14366 together through estimation to the nearest hundreds.

Solution:

Step 1: Estimating numbers to the nearest 100:

If the digit in the tens place is 5, 6, 7, 8, or 9, replace the tens and ones place digits with 0 and add 1 to the hundreds place digit.

If the digit in the tens place is 0, 1, 2, 3, or 4, replace the tens and ones place digit with 0, and the hundreds place digit remains unchanged.

So, 25412 round off to 25400.

14366 round off to 14400.

Step 2: Add these rounded numbers together.

Therefore, 25412 + 14366 = 39800 (approximately).

Let’s Practise - 3

1. Solve.

a. Estimate the sums of 5783 and 2253 to the nearest tens.

b. Estimate the sums of 7853 and 965 to the nearest tens.

c. Estimate the sums of 22783 and 14466 to the nearest hundreds.

d. Estimate the sums of 13633 and 30970 to the nearest hundreds.

Let’s Learn

Stories Based on Addition

In our daily life, we often need to add things together. When you read stories with numbers and see words like ‘total,’ ‘together,’ ‘in all,’ ‘altogether,’ or ‘sum of,’ it means you should use addition to find the answer.

Example 1: A shopping mall has 12,436 items on one floor and 13,295 items on another. How many items does the mall have in total?

Solution:

Number of items on one floor of a mall = 12,436

Number of items on another floor of a mall = 13,295

Total number of items in the mall = 12436 + 13295

Word Zone

Altogether: Total when everything is added

Together: Things in one place or as a team

Therefore, the total number of items in the mall is 25,731.

Example 2: The population of two small villages are 34,456, and 22,198. Find the total population of the two villages combined.

Solution:

Population of one village = 34,456

Population of another village = 22,198

Total population = 34456 + 22198

Therefore, the total population of two villages combined is 56,654.

Let’s Practise - 4

1. A library has 11,367 fiction books and 12,798 non-fiction books. How many books does the library have in total?

2. In an election between two candidates, one candidate spent ` 25,347 for campaigning and ` 41,568 on other expenses. How much money did he spend in election in total?

3. In a book fair, 14,789 people visited during weekdays while 35,676 people visited during weekends. How many people visited the book fair during the week, in total?

4. Roy purchased a house for ` 97,041 and spent an additional ` 19,873 in renovations. Find the total amount Roy spent on the house.

Let’s Sum Up

Addition is the process of combining two or more numbers to find their total.

The sum of two numbers remains the same even if we change the order of the numbers. This is called the order property of addition or commutative property.

While adding three or more numbers, the numbers can be grouped in any way. The sum always remains the same. This is called the grouping property of addition.

When we add 1 to a number, the sum is the successor of the number. If 0 is added to any number, the sum is the number itself.

Life Skills

You are organising a community event to support a local charity. You’ve collected donations from various sources: `52,345 from online donations, `29,876 from sponsorships, and `16,789 from in-person contributions. What is the total amount of money raised for the charity event?

Cross-Curricular Connections

Social Studies:

In a city, the total number of female voters is 44,297 and male voters is 48,732. Find the total number of voters in the city.

21st Century Skills

You are part of a team tracking recycling efforts at school. In the first week, Class A collected 30,428 recyclable items, Class B collected 22,769, and Class C collected 34,135.

How many recyclable items did all three classes collect together in the first week?

Extend Your Knowledge

When you multiply the sum of two numbers by a third number, the result is the same as adding the products of each number multiplied by the third number separately.

Example:

(4 + 6) × 5 = 10 × 5 = 50

(4 × 5) + (6 × 5) = 20 + 30 = 50

So, (4 + 6) × 5 = (4 × 5) + (6 × 5).

Tip for the Parent

Encourage your child to solve real-life problems that involve the sum of large numbers.

Subtraction

My Study Plan

Subtraction of large numbers with and without regrouping

Verifying subtraction by addition

Properties of subtraction

Estimating the difference

Subtraction in real life

Let’s Recall

Asim asked his father for `1150 to purchase a gift for his sister’s birthday. If the gift he bought cost ` 1120, how much money does he have remaining?

The remaining money = ` _______.

Thinking Zone

While finding the difference of two numbers 2742 and 9873, which way would you use from the following options and why?

a. 2742 – 9873

b. 9873 – 2742

Let’s Learn

Subtraction of Large Numbers

Subtraction is the process of taking away a smaller number from a bigger number.

In subtraction, the bigger number is known as the minuend, while the number being subtracted is called the subtrahend. The result of the subtraction is referred to as the difference.

For example:

Subtraction without regrouping

Let us solve the subtraction problem given below.

Example 1: Subtract 33424 from 65976.

Solution: 65976 – 33424 = ________

Step 1: Arrange the digits of the given numbers in the correct column as shown below.

Remember, the greater number must be written above the smaller number.

Step 2: First, subtract the digits in the ones place.

Step 3: Similarly, subtract the digits in the tens, hundreds, thousands, and ten thousands places, respectively.

Therefore, 65976 – 33424 = 32552.

Let us solve another subtraction problem given below.

Example 2: Subtract 12442 from 25957.

Solution: 25957 – 12442 = ________

Step 1: Arrange the digits of the given numbers in the correct column as shown below.

Remember, the greater number must be written above the smaller number.

Step 2: First, subtract the digits in the ones place.

Step 3: Then subtract the digits in the tens, hundreds, thousands, and ten thousands places, respectively.

Therefore, 25957 – 12442 = 13515.

Subtraction With Regrouping

Example 3: Subtract 42449 from 56287.

Solution: 56287 – 42449 =________

Step 1: Arrange the digits of the given numbers in the correct column as shown below.

Remember, the greater number must be written above the smaller number.

Step 2: Subtract the digits in ones place. But, since 7 is less than 9, we cannot take away 9 ones from 7 ones. So, we borrow 1 ten from the digit in the tens place.

After we borrow 1 ten from 8 tens, we are now left with 7 tens. We know that 1 ten = 10 ones.

So, the number in the ones place becomes 10 + 7 = 17 ones. Now let us subtract 9 from 17 at the ones place.

Step 3: Subtract the digits at the tens place.

Step 4: Subtract the digits in hundreds place. But since 2 is less than 4, we cannot take away 4 hundreds from 2 hundreds. Similarly, we borrow 1 thousand from the digit in the thousands place.

After we borrow 1 thousand from 6 thousands, we have 5 thousands. We know that 1 thousand = 10 hundreds.

So, the number in the hundreds place becomes 10 + 2 = 12 hundreds. Now, let us subtract 4 from 12 at the hundreds place.

Step 5: Subtract the digits at the thousands place.

Step 6: Subtract the digits at the ten thousands place.

Therefore, 56287 – 42449 = 13838.

Example 4: Subtract 23544 from 36453.

Solution:

36453 – 23544 = _______

Step 1: Arrange the digits of the given numbers in the correct columns as shown below.

Remember, the greater number must be written above the smaller number.

Step 2: Subtract the digits in ones place. But, since 3 is less than 4, we cannot take away 4 ones from 3 ones. So, we borrow 1 ten from the digit in the tens place.

After we borrow 1 ten from 5 tens, we are now left with 4 tens. We know that 1 ten = 10 ones.

So, the number in the ones place becomes 10 + 3 = 13 ones. Now, let us subtract 4 from 13.

Step 3: Subtract the digits at the tens place.

Step 4: Subtract the digits in hundreds place. But, since 4 is less than 5, we cannot take away 5 hundreds from 4 hundred.

So, we borrow 1 thousand from the digit in the thousands place. After we borrow 1 thousand from 6 thousands, we have 5 thousands left. We know that 1 thousands = 10 hundreds.

So, the number in the hundreds place becomes 10 + 4 = 14 hundreds. Now, let us subtract 5 from 14 at the hundreds place.

Step 5: Subtract the digits at the thousands place.

Step 6: Subtract the digits at the ten thousands place.

Therefore, 36453 – 23544 = 12909.

Checking Subtraction Using Addition

To check whether we have subtracted two numbers correctly or not, we add the difference to the subtrahend. If we get minuend, that is, the greater number as the sum, then the answer is correct.

For example,

Now, let us check the result.

After the addition of 3244 and 3243, we get the number 6487, which is the minuend.

Hence, verified.

Subtraction is the inverse operation of addition. This means that if you start with a number, add another number, and then subtract the same number, you will end up back where you started.

For example,

10 + 9 = 19

19 – 9 = 10

Hence, performing addition and subsequent subtraction with the same number gives the initial number.

Let’s Practise - 1

1. Solve the following subtractions.

Properties of Subtraction

Property 1: If we subtract 0 from a number, the difference is the number itself.

For example, 16487 − 0 = 16487.

Property 2: If we subtract 1 from a number, we get the predecessor of the number.

For example, 28576 − 1 = 28575.

Here, 28575 is the predecessor of 28576.

Property 3: If we subtract a number from itself, the difference is 0.

For example, 13144 − 13144 = 0.

Let’s Practise -2

1. Fill in the blanks.

a. Predecessor of 35164 = _____

b. Predecessor of ____ = 29372

c. 13498 - 0 = _________

d. 36253 - 36253 = _____

Let’s Learn

Estimating the Difference

To estimate the difference,

(i) Round off the numbers separately to the required place.

(ii) Then subtract them as regular numbers.

Example 1: Subtract 11357 from 11593 through estimation to the nearest tens.

Solution:

Step 1: Round off numbers to the nearest tens:

If the digit in the ones place is 5, 6, 7, 8, or 9, replace it with 0 and add 1 to the tens place digit.

If the digit in the ones place is 0, 1, 2, 3, or 4, replace it with 0, and the tens place digit remains unchanged.

So, 11357 rounds off to 11360, and 11593 rounds off to 11590.

Step 2: Subtract these rounded numbers.

Therefore, 11593 – 11357 = 230 (approximately).

Example 2: Subtract 14482 from 17346 through estimation to the nearest hundreds.

Solution:

Step 1: Estimating numbers to the nearest hundreds:

If the digit in the tens place is 5, 6, 7, 8, or 9, replace the digits in the tens and ones places with 0 and add 1 to the digit in the hundreds place.

If the digit in the tens place is 0, 1, 2, 3, or 4, replace the digits in the tens and ones places with 0, and the digit in the hundreds place remains unchanged.

So, 17346 rounds off to 17300, and 14482 becomes 14500.

Step 2: Subtract these rounded numbers.

Therefore, 17346 – 14482 = 2800 (approximately).

Let’s Practise - 3

1. Solve the following problems.

a. Estimate the difference of 9863 and 9613 to the nearest tens.

b. Estimate the difference of 14521 and 13768 to the nearest tens.

c. Estimate the difference of 42592 and 41740 to the nearest hundreds.

d. Estimate the difference of 64864 and 48372 to the nearest hundreds.

Let’s Learn

Subtraction in Real Life

In our daily lives, we often need to subtract smaller numbers from greater numbers.

When you read stories with numbers and come across words like ‘difference’, ‘less’, ‘subtract,’ or ‘how much is left’, you should use subtraction to find the answer. Let us look at some examples now.

Example 1: A toy gun costs ` 13805. A child has only ` 11480. How much more money does he need to buy the toy?

Solution:

Cost of the toy gun = ` 13805

Money child has = ` 11480

More money needed = Cost of toy gun − Money the child has

More money needed = ` 13805 – 11480

Therefore, more money needed = `2,325.

Example 2: In an election between two candidates, one candidate defeated her rival by 15290 votes. If she received 32458 votes, find out the number of votes received by the losing candidate.

Solution:

Total number of votes received by winning candidate = 32458

The winning candidate defeated the rival by = 15290

Total votes received by the losing candidate = 32458 – 15290

Therefore, the losing candidate received 17168 votes.

1. Solve the following problems.

a. A classroom has a total supply of 17,657 crayons and markers. If there are 13,364 crayons, then how many markers does the classroom have in total?

b. Seema had `19,343 in wallet. She spent `14,569 on clothes and groceries, how much money is she left with?

c. You had 13708 candies, and you gave away 11625 to the needy. How many candies do you have left in total?

d. You had 18544 stickers, and you used up 13329 stickers. How many stickers do you have now?

Let’s Sum Up

Subtraction is the process of taking away a smaller number from a bigger number.

In subtraction, the bigger number is known as the minuend, while the number being subtracted is called the subtrahend. The result of the subtraction is referred to as the difference.

To check whether we have subtracted two numbers correctly or not, we add the difference to the subtrahend. If we get the minuend, that is, the greater number as the sum, the answer is correct.

If we subtract 0 from a number, the difference is the number itself.

If we subtract 1 from a number, we get the predecessor of the number.

If we subtract a number from itself, the difference is 0.

To estimate the difference,

■ Round off the numbers separately to the required places.

■ Then subtract them as regular numbers.

In our daily life, we often need to subtract smaller numbers from greater numbers. When we read stories with numbers and come across words like ‘difference’, ‘less’, ‘subtract’, or ‘how much is left’, you should use subtraction to find the answer.

Skills `11,350

You have saved `13,500 to buy a bicycle. The bicycle you want to buy costs `11,350. How much money will you have after buying the bicycle?

Cross-Curricular Connections

Social Studies:

You have learned that India got its Independence in 1947. Now, if the year is 2024 and you want to know how many years have passed since then, what math can you do?

21st Century Skills

You and your team are working on a project to reduce food waste at school. Last month, you collected 11,250 kilograms of food scraps for composting. This month, you collected 1,975 kilograms. How much less food waste did you collect this month compared to last month?

Extend Your Knowledge

We have learned about numbers like 1, 2, 3, and 4 that are greater than 0. But there are also numbers less than 0, known as negative numbers, and they have a minus sign in front of them.

For example: -1, -2, -3, -4, …. are examples of negative numbers.

Tip for the Parent

Encourage your child to solve real-life problems involving subtraction of large numbers, such as finding the difference in the cost of two products.

Multiplication

My Study Plan

Multiplication of large numbers

Properties of multiplication

Multiplying numbers by 10, 100, and 1000

Lattice multiplication method

Estimating the product

Multiplication in real-life situations

Let’s Recall

We learned the multiplication of 2-digit, 3-digit, and 4-digit numbers with 1-digit numbers in the previous grade. Let’s refresh it by solving the following word problem.

Bobby is going to a book fair organized by his school. He bought four sets of books, costing ₹245 each. Find the total that he spent to buy the book set.

Let’s Learn

Multiplication is the mathematical operation of combining two or more numbers to find a total or a result called the product. The specific number being multiplied in a given problem is called the multiplicand. The multiplier is the number by which the multiplicand is multiplied. Example:

Multiplication of Large Numbers

Multiplication of a larger number can be done in ways such as with or without regrouping.

Multiplication of a 4-digit number by a 2-digit number (without regrouping)

Multiplication without regrouping refers to the multiplying numbers when the product of each place value is less than 10. In this case, there is no need to carry or regroup any values to the next place value.

Consider a 4 digit number say, 4,423 and multiply it by 2 digit number such as 21. Here is how we can do it step by step:

Step 1

Arrange the digit in the multiplicand and multiplier correctly.

Step 2

Step 3

Multiply the multiplicand by the digit in the ones place of the multiplier, that is 4,423 by 1.

Step 4

Multiply the multiplicand by the digit in the tens place of the multiplier, that is 4,423 by 2

Combine the products by adding 4423 from step 2 and 8846 from step 3.

Therefore, 4,423 × 21 = 92,883.

Multiplication of a 4-digit number by a 2-digit number (with regrouping)

Multiplication with regrouping occurs when the product of multiplying two digits together in a specific place value exceeds 9. When this happens, the excess value is carried over or regrouped to the next place value.

Multiplying a 4-digit number by a 2-digit number, say 2,234 by 34.

Here are the steps to be followed:

Step 1

Step 2

Arrange the digits in the multiplicand and multiplier correctly.

Step 3

Multiply the multiplicand by the digit in the ones place of the multiplier, that is 2,234 by 4.

Here, when the product exceed 9, it is necessary to carry forward the number in the tens place and hundred place to the nearby number as shown.

Step 4

Multiply the multiplicand by the digit in the tens place of the multiplier, that is 2,234 by 3.

Combine the products by adding 8936 from step 2 and 6702 from step 3.

Therefore, 2,234 × 34 = 75,956.

Multiplication of a 5-digit number by a 3-digit number

Multiplying a 5 digit number by a 3 digit number say, 54,218 by 413. Here is how we can do it step by step:

Step 1

Step 2

Arrange the digits in the multiplicand and multiplier correctly.

Step 3

Multiply the multiplicand by the digit in the ones place of the multiplier, that is 54,218 by 3.

Multiply the multiplicand by the digit in tens place of the multiplier, that is 54,218 by 1.

Step 4

Multiply the multiplicand by the digit in the hundreds place of the multiplier, that is 54,218 by 4.

Step 5

Combine the products by adding 1,62,654 from step 2, 54,218 from step 3, and 2,16,872 from step 4.

Therefore, 54218 x 413 = 2,23,92,034.

Multiplying an even number by 6 results in a product with the same last digit as the even number. For example, 4 × 6 = 24, and 8 × 6 = 48. Fact Zone

Let’s Practise - 1

1. Multiply the following numbers. a.

Let’s Learn

Properties of Multiplication

Multiplication properties are special rules or formulas that make it easier to solve math problems that involve multiplication.

The five properties of multiplication are:

1. Identity Property: The identity property of multiplication states that when a number is multiplied by 1, the product is always the number itself.

Example:

6528 × 1 = 6528

9079 × 1 = 9079

2. Zero Property: The zero property of multiplication states that when we multiply any number by 0, the result is always 0.

Example:

1827 × 0 = 0

5434 × 0 = 0

3. Commutative Property: The commutative property of multiplication states that we can multiply two numbers in any order, and the product will remain the same.

Example: 2 × 5 = 5 × 2 = 10

7 × 9 = 9 × 7 = 63

4. Associative Property: According to the associative property of multiplication, changing the grouping of numbers does not affect the product of numbers.

Example:

(4 × 6) × 3 = 4 × (6 × 3) = 72

(5 × 7) × 8 = 5 × (7 × 8) = 280

5. Distributive Property: The distributive property of multiplication states that multiplication can be distributed over addition as well as subtraction. This property helps us solve the expressions with brackets.

Example:

2 × (3 + 4) = (2 × 3) + (2 × 4) = 6 + 8 = 14

5 × (3 – 2) = (5 × 3) – (5 × 2) = 15 – 10 = 5

Let’s Learn

Multiplication by 10, 100, 1000

Multiplying a number by 10, 100, or 1000 is a way to make it larger by adding zeros to the end of it.

Multiplying by 10

When we multiply a number by 10, we make it ten times bigger. It is like adding a zero to the end of the number.

Example:

5 × 10 = 50 (5 became 50 by adding a zero).

7 × 10 = 70 (7 became 70 by adding a zero).

Multiplying by 100

When we multiply a number by 100, we make it a hundred times bigger. To do this, add two zeroes to the end of the number.

Example:

4 × 100 = 400 (4 became 400 by adding two zeroes).

9 × 100 = 900 (9 became 900 by adding two zeroes).

Multiplying

by 1000

When we multiply a number by 1000, we make it a thousand times bigger. This time, add three zeros to the end of the number.

Example:

3 × 1000 = 3000 (3 became 3000 by adding three zeroes).

7 × 1000 = 7000 (7 became 7000 by adding three zeroes).

Let’s Practise - 2

1. Fill in the blanks to solve the following.

a. 119 × 1 = _____

c. 113 × 43 = _____ × 113 = 4859

e. 5324 × _____ = 5324

g. 74 × _____ = 7400

2. Fill in the blanks.

a. 3 x ( 4 +5 ) = (_______ ) + ( 3 x5 ) = _____ + _______ = 27

b. 5 x ( 6 + 4 ) = ( ) + ( ) = 50

Let’s Learn

Lattice Multiplication

b. 354 × 0 = _____

d. 1243 × _____ = 0

f. 214 × 10 = _____

h. 562 × 1000 = _____

Lattice multiplication is a visual and systematic method for multiplying multi-digit numbers.

Multiplication of 49 by 28 through Lattice multiplication method.

Step 1: Set up the Lattice

Draw a grid with as many columns as the multiplicand and as many rows as the multiplier.

In this case, we have 2 digits for 49 and 2 digits for 28, so the grid will be 2 × 2.

Word Zone

Multi-digit numbers: Numbers with more than one digit, like 25 or 354

Step 2: Draw the diagonals

Draw a diagonal line through each box from the upper right corner to the lower left corner as shown.

Step 3: Write the Numbers

Set up the multiplication grid by placing one number’s digits at the top and the other number’s digits on the side.

For example, for 49 multiplied by 28, write 4 and 9 at the top and 2 and 8 on the right side of the grid.

Step 4: Multiply the Digits

Now, multiply the digits in each box where a row and a column meet. Start at the bottom-right box and work your way to the top-left box.

For example, in the bottom-right box, we would multiply 9 by 8, which equals 72. Put the tens digit on top and the ones digit on the bottom of each box.

Step 5: Add the Numbers

Now, add up the numbers in each diagonal line of boxes from bottom right to top left. So here we have 2

8 + 7 + 2 = 17 (Write 7 and carry 1)

1 + 8 + 3 + 1(carried) = 13 (Write 3 and carry 1)

0 + 1(carried) = 1

Step 6: Find the Answer

The final product is composed of the digits of the sums written outside the lattice.

We read the digits from the left side and then move towards the right at the bottom to obtain the final answer.

Therefore, 49 multiplied by 28 equals 1372.

Lattice multiplication, also known as the Italian method or Chinese method, is a method of multiplication that uses a lattice to multiply two multi-digit numbers.

Let’s Practise - 3

1. Answer the following using lattice multiplication.

a. 80 × 90 = ____

c. 45 × 25 = ____

b. 45 × 68 = ____

d. 40 × 83 = ____

Fact Zone

Let’s Learn

Estimating the Product

Estimating the product means finding a close or approximate answer when we multiply numbers.

To estimate the product, first, round off any numbers which are two digits or larger. For numbers with two digits, round off to the nearest tens place, for numbers with three digits, round off to the nearest hundreds place, etc. Then multiply the rounded off numbers together. This will give you the estimate of the product.

Rounding off method

Rounding off is a common method used to estimate numbers.

The general rule for rounding off

Step 1: If the number you are rounding off is followed by 5, 6, 7, 8, or 9, round off the number up.

Example: 107 rounded off to the nearest tens is 110, and to the nearest hundreds is 100.

Step 2: If the number you are rounding off is followed by 0, 1, 2, 3, or 4, round off the number down.

Example: 103 rounded off to the nearest tens is 100, and to the nearest hundreds is also 100.

Now, let us understand the process of estimation through some more examples.

Example 1: Estimate the product of 17 × 29.

Solution:

Here are the steps to estimate the product of 17 and 29 using rounding off:

Step 1: Round off the first number:

Round off 17 to the nearest tens, which is 20. This is because 7, which is the ones place digit is more than 4. So, we round up to 20.

Step 2: Round off the second number:

Round off 29 to the nearest tens, which is 30. Again, in 29, the ones place digit 9 is more than 4. So, we round up to 30.

Step 3: Multiply the rounded off numbers:

Multiply the rounded off numbers together, so 20 × 30 = 600.

So, by rounding off 17 to 20 and 29 to 30, we can estimate that the product of 17 and 29 is approximately 600.

Example 2: Estimate the products of 331 and 267 by rounding off to the nearest hundreds.

Solution:

Here are the steps to estimate the product of 331 and 267 using rounding off:

Step 1: Look at the first number, 331. We want to round off it to the nearest hundreds. Since 331 is closer to 300 than 400, we round down to 300.

Step 2: Look at the second number, 267. Since 267 is closer to 300 than 200, we round up to 300.

Step 3: Multiply the rounded off numbers together. 300 × 300 = 90,000.

So, the estimated product of 331 and 267 is 90,000 when rounded off to the nearest hundreds.

Let’s Practise - 4

1. Estimate the following product.

a. 23 × 87 by rounding off to the nearest tens.

b. 148 × 14 by rounding off to the nearest tens.

c. 321 × 287 by rounding off to the nearest hundreds.

Thinking Zone

Look at the sequence: 3, 6, 9, 12, 15. Can you identify the multiplication pattern? What would be the next three numbers in the sequence?

Let’s Learn

Real Life Application

Multiplication is useful in real life for tasks like calculating total quantities of items, finding the area of objects or fields, scaling or resizing objects, and understanding rates over time, such as distance traveled. It is a helpful skill to understand numbers and solve everyday problems.

Example 1: A factory produces 1,172 boxes of crayons each day. If each box contains 8 crayons, how many crayons does the factory produce daily?

Solution: The factory produces 1,172 boxes of crayons each day.

Each box contains 8 crayons.

To find the total number of crayons the factory produces in one day, we need to multiply the number of boxes by the number of crayons in each box.

Therefore, total number of crayons produced by factory in one day = 1,172 boxes x 8 crayons per box = 9,376 crayons.

Example 2: A school choir is performing at a charity concert. The school auditorium can accommodate 601 person, and there is an entry fee of ₹15 per person. How much funds can the school raise through this choir in total?

Solution: Number of person to be accommodated = 601

Cost of entry ticket per person = ₹15

So, the total amount to be generated for charity = Number of person × Cost of entry ticket = 601 × 15 = ₹9,015

So, the school can raise a total of ₹9,015 through this choir performance at the charity concert.

Word Zone

Choir: A group of people who sing together in churches, schools, etc.

Concert: A music show where artists perform for an audience

Accommodate: Providing space or room for someone or something

1. A pizza store has 894 pizzas in stock. If each pizza costs ₹50, what is the total cost of all the pizzas?

2. A bookstore has 6,721 books in its inventory. If each book costs ₹25, what is the total value of all the books in the store?

3. A toy car factory produced 3,750 cars in a month. If each toy car is sold at ₹14, how much is generated through the sale of these cars in total?

Let’s Sum Up

Multiplication is a fundamental mathematical operation that involves combining a multiplicand and a multiplier to calculate a product or total.

It is done in two ways: with and without regrouping.

The properties of multiplication include identity property, zero property, commutative property, associative property, and distributive property.

A number is multiplied by 10, 100, or 1000 by adding zeroes to its end.

The lattice multiplication method is used to multiply multi-digit numbers by using a grid.

Rounding off is a common method to estimate a product.

Multiplication is used in real life to calculate item quantities, distance travelled, and so on.

Life Skills

In a classroom, there are 6 rows of desks and each row has 4 desks. How can students use multiplication to determine the total number of desks in the classroom?

Cross-Curricular Connections

Social Studies:

Check the receipt your father received for the items he bought. Consider the price of each item and its quantity bought to calculate the total amount spent on the purchases.

21st Century Skills

Imagine you want to make a colourful banner for a school event about recycling. Each row of the banner will have 6 different recycled materials on it, and you plan to have 8 rows in total. How many recycled materials will you need for the entire banner?

Also, if each recycled material costs ₹13, how can you find the total cost?

Extend Your Knowledge

When we multiply a number by its reciprocal, the result is always 1.

For example, let us use the number ‘6.’

The reciprocal of 6 is 1 6 .

Now, when we multiply 6 by 1 6 , the answer is 1.

Tip for the Parent

Help your child understand multiplication by using it in everyday situations. For example, ask them to figure out how many days are in a week and a month or how much money they will need to buy 5 packets of pencils.

Incorporate games that involve multiplication: board games, card games, or online activities designed to reinforce multiplication skills. This can be both educational and entertaining.

Division 5

My Study Plan

Division as equal sharing

Terms associated with division

Division using long division method

Properties of division

Division by 10, 100, and 1000 by shortcut method

Divisions in real life

Estimating the quotient

Let’s Recall

Rahul had a total of 50 toffees that he wanted to share equally with his 5 friends on his birthday. How many toffees did each friend receive?

Fill in the blank.

Each friend received ____________ toffees.

Thinking Zone

Think and find the missing digit.

Let’s Learn

Basics of Division

Division is one of the four basic operations in mathematics, along with addition, subtraction, and multiplication. It is the process of separating a quantity into equal parts.

Division as equal sharing

Division means sharing things equally. When we share things equally, it means we have divided them.

For example, if we have to divide 20 sweets into 4 separate bowls, then each bowl will have 5 sweets.

This can be written as: 20 ÷ 4 = 5

Word Zone

Equally: In amounts or parts that are the same

20

Terms associated with division

4 5

20 4 5

In the above division sentence:

a. ‘÷’ is the sign or symbol used for division.

b. The number that we want to divide is called the dividend. Here, the dividend is 20.

c. The divisor is the number by which the dividend is to be divided or the number of parts into which the dividend is to be shared. Here, 4 is the divisor.

d. The quotient is the result or the answer. Here, 5 is the quotient.

Division using multiplication tables

We can also divide numbers using multiplication tables.

Example 1: Divide 40 by 5.

Solution: Recall the table of 5 until you get 40 as an answer.

5 × 1 = 5

5 × 2 = 10

5 × 3 = 15

5 × 4 = 20

5 × 5 = 25

5 × 6 = 30

5 × 7 = 35

5 × 8 = 40

Therefore, 40 ÷ 5 = 8

This means that there are 8 fives in 40.

Let’s Practise - 1

1. Divide the following using multiplication tables.

a. 72 ÷ 9 = _________________

b. 64 ÷ 8 = _________________

c. 48 ÷ 6 = _________________

d. 36 ÷ 4 = _________________

e. 35 ÷ 5 = _________________

f. 63 ÷ 7 = _________________

g. 81 ÷ 9 = _________________

h. 72 ÷ 6 = _________________

Let’s Learn

Division of Large Numbers

If the numbers are large and we want to divide them quickly, we use the long division method.

Let us learn it by looking at the following examples.

Example 1: Divide 325 by 5.

Solution:

Dividend: 325, Divisor: 5

Step 1: Write the numbers in their correct place, as shown. 325 5 Divisor Dividend Divisor Dividend

Step 2: Since 3 at the hundreds place is less than the divisor 5, we consider the tens digit too.

Divide 32 by 5.

We know that 5 × 6 = 30 and 32 - 30 = 2. So, 32 ÷ 5 gives quotient = 6 and remainder = 2.

Step 3: Bring down 5 from the ones place of the dividend.

Divide 25 by 5.

25 ÷ 5 gives quotient = 5 and remainder = 0.

Hence, 325 ÷ 5 = 65

Example 2: Divide 4881 by 3.

Solution:

Dividend: 4881, Divisor: 3

Step 1: Write the numbers in their correct place, as shown.

Step 2: Since 4 at thousands place is more than the divisor 3, we divide 4 by 3.

4 ÷ 3 gives quotient = 1 and remainder = 1.

Step 3: Bring down 8 from the hundreds place of the dividend.

Divide 18 by 3.

18 ÷ 3 gives quotient = 6 and remainder = 0.

Step 4: Bring down 8 from the tens place of the dividend.

Divide 8 by 3.

8 ÷ 3 gives quotient = 2 and remainder = 2.

Step 5: Bring down 1 from the ones place of the dividend.

Divide 21 by 3.

21 ÷ 3 gives quotient = 7 and remainder = 0.

Hence, 4881 ÷ 3 = 1627

Verification

We can verify the result of division by the following relation.

Divisor × Quotient + Remainder = Dividend

In the previous example, Divisor = 3, Quotient = 1627, and Remainder = 0

On applying the formula, Divisor × Quotient + Remainder

We get, 3 × 1627 + 0 = 4881 = Dividend.

So, our calculation is correct.

Example 3: Divide 4932 by 12.

Solution:

Dividend = 4932, Divisor = 12

Step 1: Write the numbers in their correct place, as shown.

Step 2: Since 4 at the thousands place is less than the divisor 12, we consider the hundreds place digit too.

Divide 49 by 12.

49 ÷ 12 gives quotient = 4 and remainder = 1.

Step 3: Bring down 3 from the tens place of the dividend.

Divide 13 by 12.

13 ÷ 12 gives quotient = 1 and remainder = 1.

Step 4: Bring down 2 from the ones place of the dividend.

Divide 12 by 12.

12 ÷ 12 gives quotient = 1 and remainder = 0.

Hence, 4932 ÷ 12 = 411

Verification

Here, Divisor = 12, Quotient = 411, and Remainder = 0

On applying the formula, Divisor × Quotient + Remainder

We get, 12 × 411 + 0 = 4932 = Dividend

So, our calculation is correct.

Let’s Practise - 2

1. Divide the following numbers using long division method.

÷ 5 = ____

4076 ÷ 3 = ____ c.

÷ 24 = ____ a.

Let’s Learn

Properties of Division

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

Example :2285 ÷ 1 = 2285 Quotient

b.

5904

730

Property 2 - When we divide a number by itself, we get 1 as the quotient.

Example : 1943 ÷ 1943 = 1

Divisor 1943 - 1943 0000

Quotient

Dividend

Remainder 1943 1

Property 3 - 0 divided by any number gives the quotient 0.

Example:

0 ÷ 1070 = 0

0 ÷ 2178 = 0

Property 4 - Division by 0 is undefined.

Example:

1776 ÷ 0 = undefined

2812 ÷ 0 = undefined

Let’s Learn

Division by 10, 100, and 1000

Division by 10

When you divide a number by 10, the digit at the ones place becomes the remainder, and the number formed by the other digits is the quotient.

For example,

1. 3109 ÷ 10: Quotient = 310 and Remainder = 9

2. 1546 ÷ 10: Quotient = 154 and Remainder = 6

3. 1246 ÷ 10: Quotient = 124 and Remainder = 6

4. 4673 ÷ 10: Quotient = 467 and Remainder = 3

Division by 100

When you divide a number by 100, the digits at the tens and ones places become the remainder, and the number made up of the remaining digits is the quotient.

For example,

1. 1926 ÷ 100: Quotient = 19 and Remainder = 26

2. 4957 ÷ 100: Quotient = 49 and Remainder = 57

3. 2242 ÷ 100: Quotient = 22 and Remainder = 42

4. 9634 ÷ 100: Quotient = 96 and Remainder = 34

Division by 1000

When you divide a number by 1000, the digits at the hundreds, tens, and ones places become the remainder, and the number made up of the remaining digits is the quotient.

For example,

1. 4226 ÷ 1000: Quotient = 4 and Remainder = 226

2. 9957 ÷ 1000: Quotient = 9 and Remainder = 957

3. 6242 ÷ 1000: Quotient = 6 and Remainder = 242

4. 9424 ÷ 1000: Quotient = 9 and Remainder = 424

Let’s Practise - 3

1. Divide the following numbers.

a. 1284 ÷ 10 =

b. 3424 ÷ 100 =

c. 2468 ÷ 1000 = d. 4531 ÷ 10 =

2. Fill in the blanks.

a. 1542 ÷ 1542 = ______

c. 2468 ÷ _____ = 2468

b. 0 ÷ 1898 = ________

d. 4514 ÷ ______ = 1

Let’s Learn

Division in Real Life

In our daily lives, we often need to divide things into groups or share them equally. When you read stories with numbers and see words like ‘share,’ ‘divide,’ ‘equally,’ ‘among,’ or ‘each,’ it means you should use division to find the answer.

Let us look at different examples.

Example 1: In a library, 1272 books are to be equally shared among 8 children. How many books will each child get?

Solution:

Total number of books = 1272

Number of children = 8

Number of books each child will get = 1272 ÷ 8 = 159

Hence, each child will get 159 books.

Example 2: You have 3228 mangoes and you want to pack them in boxes where each box contains 25 mangoes. How many boxes would be required? How many mangoes will be left?

Solution:

Number of mangoes = 3228

Number of mangoes in each box = 25

Total number of boxes = 3228 ÷ 25 25 129 3228

Quotient

Remainder

Here quotient = 129 and remainder = 3

Therefore, 129 boxes would be required and 3 mangoes would be left over.

Estimating the Quotient

Estimating the quotient refers to finding a close or approximate answer while dividing numbers. This is very useful in everyday life when we need to make quick calculations.

Example 1: Estimate the quotient of 1584 ÷ 18.

Solution: To estimate the quotient follow the below steps.

Step 1: Round the numbers to make them simpler. In our example, 1584 ÷ 18, we can round 1584 to 1600 and 18 to 20. It is easier to work with rounded numbers.

Step 2: Now, we divide the rounded numbers. 1600 ÷ 20 is much easier to calculate, and it equals 80. So, our estimate is 80.

Step 3: After we have estimated, we can check it with the actual division. In this case, if we divide 1584 ÷ 18, we will find that the actual answer is 88. So, our estimate was pretty close!

1. Solve the following questions.

a. There are 1742 pencils which need to be placed in 12 boxes with an equal number of pencils in each box. How many pencils will be in each box? Will there be any leftovers?

b. There are 8061 ice cream cones which need to be shared equally among 12 children. How many ice cream cones will each child get? Will there be any leftovers?

c. There are 1055 pizza slices which need to be divided equally for 15 plates. How many slices will be on each plate? Will there be any leftovers?

2. Estimate the quotient and compare it with the actual answer.

a. 896 ÷ 28 = Estimated:____, Actual:___

b. 410 ÷ 19 = Estimated:_____, Actual:__

Let’s Sum Up

Division means sharing things equally.

Long division method is used to divide larger numbers easily.

The result of division can be verified by using the formula, Divisor × Quotient + Remainder = Dividend.

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

When we divide a number by itself, we get 1 as the quotient.

0 divided by any number gives the quotient 0.

Division by 0 is undefined.

When you divide a number by 10, the digit at the ones place becomes the remainder, and the number formed by the other digits is the quotient.

When you divide a number by 100, the digits at the tens and ones places become the remainder, and the number made up of the remaining digits is the quotient.

When you divide a number by 1000, the digits at the hundreds, tens, and ones places become the remainder, and the number made up of the remaining digits is the quotient.

Estimating the quotient refers to finding a close or approximate answer while dividing numbers.

Life Skills

Vaibhav has 1,666 pencils to share with 14 children at the orphanage on his birthday. How many pencils will each child get? Think of a kind act you can do for your next birthday.

Cross-Curricular Connections

Social Studies:

In a remote village in Maharashtra, there are 720 people. The village got a new well for clean water. If each family, with 5 people in it, gets an equal share of water, how many families will get the clean water?

21st Century Skills

You are planning a donation event. You have 864 food packets to pack in 6 boxes. You will be packing the same number of food packets in each box. How many food items should go in each box? How can you use technology or the internet to get more people to donate food for your event?

Extend Your Knowledge

A multiplication fact gives us two division facts. This is related to the inverse relationship between multiplication and division.

For example,

Multiplication Fact: 5 × 3 = 15

This means that if you have 5 groups, each containing 3 items, the total is 15 items.

Division Facts:

First Division Fact: 15 ÷ 5 = 3

If you have 15 items and want to divide them into groups of 5, you will have 3 items in each group.

Second Division Fact: 15 ÷ 3 = 5.

Similarly, if you have 15 items and want to divide them into groups of 3, you will have 5 items in each group.

So, the multiplication fact 5×3=15 is related to the division facts 15 ÷ 5 = 3 and 15 ÷ 3 = 5.

Tips for the Parent

• Encourage your child to practice the long division method for larger numbers.

• Incorporate division into daily activities, such as splitting a pizza into equal slices or distributing cookies among family members.

Factors and Multiples

My Study Plan

• Multiples of numbers

• Common multiples of two or more numbers

• Factors of numbers

• Common factors of numbers

• Prime and composite numbers

• HCF and LCM

• Tests of divisibility

Let’s Recall

Answer the following questions:

1. What is your birth date?

2. Write the multiplication table based on the day of your birth month.

3. Solve the following multiplications:

a. 23 × 4 = _______

b. 59 × 7 = ________ c. 98 × 5 = _____

4. Solve the following divisions:

a. 90 ÷ 5 = _______

b. 128 ÷ 7 = ______ c. 396 ÷ 12 = ______

5. An aircraft can carry 6935 kg of luggage in a single flight. If it took 325 such flights, calculate the total weight of luggage that it carried.

6. A packet of crayons contains 25 crayons. How many packets are required if 6500 crayons are to be packed equally?

Let’s Learn

Multiples

Aayu is the captain of his team and is playing relay race on school’s sports day. He needs to group students in sets of 3, but counting them one by one seems time-consuming.

Skip counting is a way to count faster by adding the same number each time. Let’s skip count by 3: 3, 6, 9, 12, 15, 18, 21, and so on.

When we skip count by 3, we are actually finding the multiples of 3. So, here, each number in the sequence is a multiple of 3 because we are adding 3 to the previous number each time.

Thus, we can say that the multiples of 3 are 3, 6, 9, 12, 15, 18, 21, and so on.

Similarly, we can skip count using other numbers to find their multiples. For example, skip counting by 5 gives us the multiples of 5 as 5, 10, 15, 20, 25, and so on.

A multiple of a number is found by multiplying that number with any whole number. For example, 14 is a multiple of both 2 and 7, as 2 × 7 = 14.

We can use a multiplication table to identify multiples of a number and even find common multiples between two numbers.

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

Interesting multiples facts

• The product of two numbers is a multiple of each of the two numbers. For example, when we multiply 3 by 5, we get 15, which means 15 is a multiple of both 3 and 5.

• Every number is a multiple of 1. For example, when we multiply 1 by 7, we get 7; when we multiply 1 by 100, we get 100.

• Every number is a multiple of itself. For example, when we multiply 4 by 1, we get 4; when we multiply 25 by 1, we get 25.

• Every multiple of a non-zero number is either greater than or equal to that number. For example, multiples of 6 are 6, 12, 18, 24, and so on. All these multiples are either equal to or greater than 6.

• Any multiple of a number is divisible by that number. For example, 40 is a multiple of 8, and it is divisible by 8.

• There is no last multiple of a number; and the number of multiples of any number is unlimited. For example, multiples of 9 are 9, 18, 27, 36, and the list goes on infinitely.

Finding multiples

We can find the multiples of a number by multiplying it with 1, 2, 3, 4, and so  on.

For example, the first five multiples of 7 are: 1 × 7 = 7 2 × 7 = 14 3 × 7 = 21 4 × 7 = 28 5 × 7 = 35

Hence, the first five multiples of 7 are 7, 14, 21, 28 and 35.

Check whether one number is a multiple of another number

To check whether the given number is a multiple of another number, divide the greater number by the smaller number. If the remainder is 0, then the greater number is a multiple of the smaller number.

Let us have a look at the following examples:

Example: Is 48 a multiple of 4?

Solution: If 48 is a multiple of 4, it should be divisible by 4.

48 ÷ 4 gives a quotient of 12 and a remainder of 0.

Hence, 48 is a multiple of 4. 1 2 4 4 8 – 4 0 8 – 8 0

Example: Is 53 a multiple of 3?

Solution: If 53 is a multiple of 3, it should be divisible by 3.

53 ÷ 3 gives a quotient of 17 and a remainder of 2.

Hence, 53 is not a multiple of 3.

Common multiples

Common multiples are numbers that are multiples of two or more given numbers. They help us to identify the numbers divisible by multiple values.

Let’s find the common multiples of 3 and 4:

Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, ...

Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, ...

Here, the number 12 is a common multiple of both 3 and 4 as it appears in the list of multiples for both numbers.

• We know that 2, 4, 6, 8, ... are even numbers. To find the even multiples of a number, we multiply the number by 2, 4, 6, 8, 10, … .

For example, the even multiples of 5 are 10, 20, 30, 40, 50, … .

• We know that 1, 3, 5, 7, ... are odd numbers. To find the odd multiples of a number, we multiply the number by 1, 3, 5, 7, 9, … .

For example, the odd multiples of 5 are 5, 15, 25, 35, 45, … .

Let’s Practise – 1

1. Answer the following questions:

a. Find the first five multiples of 3.

b. Find the first seven multiples of 7.

c. Write the first five multiples of 4 and 5 and circle the common multiples.

d. Write the first seven multiples of 7 and 9 and circle the common multiples.

e. Find three even and odd multiples of 9.

Fact Zone

Thinking Zone

I am a multiple of 15 and have 3 digits. I am an odd number and a multiple of 9. Which number am I?

Factors

When two numbers are multiplied together, they are called factors of the product. For example, 3 and 4 are factors of 12 because when we multiply 3 and 4, we get 12.

Factors of a number divide the number completely without leaving any remainder. So, when we divide 12 by 3 or 4, the quotient is a whole number, and the remainder is  0.

Apart from 3 and 4, there are other factors of 12.

Some of the ways to get the product 12 are:

1 × 12

b. 12 × 1

a.

c. 2 × 6

d. 6 × 2

e. 4 × 3

f. 3 × 4

Interesting factor facts

• Every number has 1 as a factor. When we multiply any number by 1, the product is the number itself. Therefore, 1 is a factor of every number.

For example, 1 × 7 = 7 and 1 × 9 = 9.

• Every number is a factor of itself. When we multiply any number by 1, the product is the number itself. Therefore, every number is a factor of itself. For example, 1 × 8 = 8. Thus, both 8 and 1 are factors of 8.

• Unlike multiples, a number has a limited number of factors. For example, 1, 3, 5, and 15 are all the factors of 15, and there are no more factors for 15.

• A factor of a number is always less than or equal to that number but never greater than it. For example, 1, 3, 5 and 15 are all the factors of 15. Any number greater than 15 can never be a factor of 15.

• 1 is the only number that has only one factor. All other numbers have at least 2 factors. For example, 1 and 3 are factors of 3, and 1 and 5 are factors of 5.

• 1 is the smallest factor of any number.

• When a number is divided by one of its factors, the remainder is always zero.

Finding factors

We can find the factors of a number by two different methods:

• By multiplication: Express the given number as a product of two numbers.

• By division: Divide the given number by 1, 2, 3, and so on.

All the numbers that divide the given number without leaving a remainder will be its factors.

Let us find the factors of 24.

By multiplication:

1 × 24 = 24

2 × 12 = 24

3 × 8 = 24

4 × 6 = 24

Hence, the factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24.

Try out all the possible combinations of multiplication.

By division: 24 ÷ 1 = 24

24 ÷ 6 = 4

÷ 8 = 3

÷ 12 = 2

Hence, the factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24.

Check whether one number is a factor of another

÷ 24 = 1

Example: Is 4 a factor of 72?

To check whether one number is a factor of another, divide the greater number by the smaller number. If it is exactly divisible, then the smaller number is a factor of the greater number. If it is not exactly divisible, then it is not a factor. 1

Solution:

Example: Is 7 a factor of 85?

Solution:

Since the remainder is 0, 4 is a factor of 72.

Common factors

Since the remainder is not 0, 7 is not a factor of 85.

To find the common factors of two numbers, follow these steps:

• Write the factors of the first number.

• Write the factors of the second number.

• Highlight the factors that are common to both the numbers.

Let us find the common factors of 6 and 15.

Factors of 6 are 1, 2, 3, and 6.

Factors of 15 are 1, 3, 5, and 15.

We can see that 1 and 3 are the common factors of 6 and 15.

Let’s Practise - 2

1. Fill in the blanks:

a. The number which is the factor of all numbers is ___ .

b. The factors of the number 25 are 1, 5, and ___.

2. Every number has minimum two factors, 1 and the number itself. State true or false.

3. Find the factors through multiplication.

a. 21 b. 16 c. 42 d. 36

4. Find the factors through division.

a. 14 b. 28 c. 30

5. Find the common factors of the following:

a. 24 and 36 b. 35 and 70

Prime and Composite Numbers

Prime numbers

17

Prime numbers are natural numbers greater than 1 with only two distinct positive divisors: 1 and the number itself. For example, 2, 3, 5, 7, 11, 13, 17, and so on are prime numbers.

Properties of prime numbers:

• 2 is the smallest and the only even prime number.

• All other prime numbers greater than 2 are odd numbers.

• Prime numbers have no other factors apart from 1 and themselves.

• 2 and 5 are the only prime numbers that end with 2 and 5, respectively. All the other numbers ending with 2 or 5 are never prime.

Composite

numbers

Composite numbers are natural numbers greater than 1 that have more than two positive divisors. For example, 4, 6, 8, 9, 10, 12, and 15 are composite numbers.

Properties of composite numbers:

• Composite numbers can be represented as the product of two or more prime numbers.

• The number 1 is neither prime nor composite, as it has only one positive divisor, which is 1. It is called a unique number.

Example: Find out if the number 17 is a prime or a composite number.

Solution: The number 17 is a prime number because it has only two distinct positive divisors: 1 and 17.

Sieve of Eratosthenes

Eratosthenes, a Greek mathematician found a simple method to find prime numbers between 1 and 100 by removing numbers that are not prime.

To do so, draw a 10 * 10 grid on a paper and write numbers 1 to 100 on it as shown below.

Now, follow these steps:

1. Cross out the number 1, as it is neither prime nor composite.

2. Cross out all the multiples of 2, except 2.

3. Cross out all the multiples of 3, except 3.

4. Cross out all the multiples of 5, except 5.

5. Cross out all the multiples of 7, except 7.

After crossing all the numbers asked in the above steps, we are left with the following numbers:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, and 97.

These are the prime numbers between 1 and 100.

All the crossed out numbers, except 1 are composite numbers. A number can have both prime and composite factors.

Prime Factorisation

The factors of a number which are prime numbers are called prime factors. All composite numbers can be written as the product of their prime factors. Two methods of prime factorisation are factor tree method and division method.

Factor tree method

Example: Write prime factorisation of 48.

Solution:

Step 1: Start with the smallest prime factor of 48.

Step 2: Continue to find the smallest prime factor at every step.

Step 3: Stop when the last row has only prime numbers.

Step 4: Express the numbers as a product of all the prime factors.

Prime factorisation of 48 = 2 × 2 × 2 × 2 × 3

Division method

Example: Write prime factorisation of 84.

Solution:

Step 1: Divide with the smallest prime factor of 84.

Step 2: Continue to divide by the smallest prime factor at every step.

Step 3: Stop when you get the quotient as 1.

Express the numbers as a product of all the prime factors. For example, prime factorisation of 84 = 2 × 2 × 3 × 7.

• Every composite number can be expressed as a product of its prime factors.

• Two prime numbers whose difference is 2 are called twin primes. For example ‘3 and 5’ and ‘11 and 13’ are twin primes.

• Two numbers are said to be co-prime if they have no common factor, other than 1. For example, 5 and 7 are co-prime numbers.

[Note: Co-prime numbers need not be prime.]

Let’s Practise - 3

1. Find out if the following numbers are prime or composite: a. 25 b. 37 c. 44 d. 100

2. Find the prime factorisation of the following numbers: a. 24 b. 72 c. 18 d. 112

3. Write two pairs of prime numbers which have a difference of 2. What are these numbers called?

Highest Common Factor

HCF of two or more numbers is the greatest number among all their common factors.

Factors of 42: 1, 2, 3, 6, 7, 14, 21, 42

Factors of 28: 1, 2, 4, 7, 14, 28

Common factors of 42 and 28: 1, 2, 7 and 14

The greatest common factor is 14.

Thus, the HCF of 42 and 28 is 14.

There are two methods of finding HCF of two or more numbers.

1. Prime factorisation method

2. Common division method

HCF by prime factorisation method

Example: Find the HCF of 24 and 36 by prime factorisation method.

Solution:

Prime factorisation of 24 = 2 × 2 × 2 × 3

Prime factorisation of 36 = 2 × 2 × 3 × 3

HCF of 24 and 36 = 2 × 2 × 3 = 12

HCF by common division method

Example: Find the HCF of 24, 48, and 72 by common division method.

Solution:

Step 1: Divide the numbers by the smallest common prime factor.

Step 2: Continue dividing by the smallest prime factor.

Step 3: Stop when the quotients cannot be divided by a common prime factor.

Step 4: Multiply all the divisors.

HCF of 24, 48, and 72 = 2 × 2 × 2 × 3 = 24

HCF of two or more numbers is the product of their common prime factors. In other words, the HCF of two or more numbers is the greatest number that divides all the numbers.

1. Find the HCF of the following numbers using prime factorisation method.

a. 54, 72 b. 72, 108 c. 135, 180 d. 32, 80, 96

2. Find the HCF of the numbers using division method.

a. 36, 45 b. 54, 114 c. 120, 240 d. 405, 783, 513

3. Identify the co-prime numbers.

a. 11 and 13 b. 12 and 32 c. 8 and 21 d. 37 and 55

Lowest Common Multiple (LCM)

The LCM of two or more numbers is the smallest number among all their common multiples.

Example: Find LCM of 4 and 8.

Solution:

Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 40, ...

Multiples of 8: 8, 16, 24, 32, 40, ... 2 24, 48, 72 2 12, 24, 36 2 6, 12, 18 3 3, 6, 9 1, 2, 3

of 8

Common multiples of 4 and 8 are 8, 16, 24, 32, 40, ...

The smallest common multiple is 8.

The LCM of 4 and 8 is 8.

There are two methods of finding the LCM of two or more numbers.

1. Prime factorisation method

2. Common division method

LCM by prime factorisation method

Example: Find the LCM of 18 and 60 using the prime factorisation method.

Solution:

Prime factorisation of 18 = 2 × 3 × 3

Prime factorisation of 60 = 2 × 2 × 3 × 5

Multiply the common factors only once by the uncommon factors.

LCM of 18 and 60 = 2 × 2 × 3 × 3 × 5 = 180

LCM by common division method

Example: Find the LCM of 12, 60, and 72 by division method.

Solution:

Step 1: Divide by the smallest prime factor which divides at least two of the given numbers.

Step 2: Continue dividing by the smallest prime factor.

Step 3: Stop when neither of the given numbers is divisible by the same numbers.

Step 4: Now, multiply the divisors and the quotients.

LCM of 12, 60 and 72 = 2 × 2 × 3 × 5 × 6 = 360

2 12, 60, 72

2 6, 30, 36

3 3, 15, 18 1, 5, 6

Let’s Practise - 5

1. List the first three common multiples of the numbers and find their LCM.

a. 6, 8 b. 12, 15 c. 12, 18 d. 5, 10, 15

2. Find the LCM of the following numbers by prime factorisation method.

a. 48, 72 b. 30, 49 c. 12, 30, 32 d. 36, 54, 60

3. Find the LCM of the following numbers by division method.

a. 48, 56 b. 14, 24 c. 24, 18 d. 96, 72, 64

Relationship between HCF and LCM

The product of the HCF and LCM of two numbers is equal to the product of the two numbers themselves.

Let us consider the numbers 12 and 18.

HCF of 12 and 18 is 6.

LCM of 12 and 18 is 36.

According to the relationship, 12 × 18 = 6 × 36.

Thus,

HCF × LCM = Product of the given numbers

Word Problems

Example: A shopkeeper sold caps at 72 and 84 on two different days. What could be the maximum price of a cap?

Solution: Find the HCF of 72 and 84 to get the maximum price of a cap.

72 = 2 × 2 × 2 × 3 × 3

84 = 2 × 2 × 3 × 7

HCF of 72 and 84 = 2 × 2 × 3 = 12

Thus, the maximum price of a cap is 12.

Example: Find the least number which is exactly divisible by 36, 54, and 60 leaving a remainder 3 in each case.

Solution:

The least number which is exactly divisible by 36, 54, 60 is the LCM of 36, 54, and 60. The required number will be 3 more than the LCM.

So, LCM of 36, 54, and 60 = 2 × 2 × 3 × 3 × 3 × 5 = 540.

Thus, the required number is: 540 + 3 = 543.

2 36, 54, 60

2 18, 27, 30

3 9, 27, 15

3 3, 9, 5 1, 3, 5

Let’s Practise - 6

1. Find the HCF and LCM of the numbers 8 and 48. Verify if their product is equal to the product of the two numbers.

2. Find the greatest number which exactly divides 72 and 120.

3. Namita made groups of 4, 5 and 3 shells, and each time, one shell was left. Find the minimum number of shells Namita had.

Tests of Divisibility

Tests of divisibility are rules used to determine whether a given number is divisible by another number, without performing the actual division. These tests are useful in quickly identifying factors or multiples of a number.

Here are some common tests of divisibility:

Divisibility by 2:

A number is divisible by 2 if its units digit (ones digit) is even; that is, it ends with 0, 2, 4, 6, or 8. For example, 846 is divisible by 2 because its units digit is 6 (even).

Divisibility by 3:

A number is divisible by 3 if the sum of its digits is divisible by 3. For example, 246 is divisible by 3 because the sum of its digits is 2 + 4 + 6 = 12, which is divisible by 3.

Divisibility by 4:

A number is divisible by 4 if the number formed by its last two digits is divisible by 4. For example, 1,828 is divisible by 4, because the number formed by its last two digits (28) is divisible by 4.

Divisibility by 5:

A number is divisible by 5 if its units digit is 0 or 5. For example, 615 is divisible by 5 because its units digit is 5.

Divisibility by 6:

A number is divisible by 6 if it is divisible by both 2 and 3. For example, 1512 is divisible by 6, because it is divisible by both 2 and 3.

Divisibility by 8:

A number is divisible by 8 if the number formed by its last three digits is divisible by 8. For example, 7776 is divisible by 8 because the number formed by its last three digits (776) is divisible by 8.

Divisibility by 9:

A number is divisible by 9 if the sum of its digits is divisible by 9. For example, 4095 is divisible by 9 because the sum of its digits is 4 + 0 + 9 + 5 = 18, which is divisible by 9.

Divisibility by 10:

A number is divisible by 10 if its units digit is 0. For example, 760 is divisible by 10 because its units digit is 0.

The divisibility rule for 3 is connected to the rule for 9. When we add up the digits of a number and the sum is divisible by 3, then the original number is also divisible by 3.

Fact Zone

Let’s Practise - 7

1. Check the divisibility of the following. Write (Yes/No) in the blank space provided.

a. 2419 is divisible by 3 _______

c. 7934 is divisible by 5 _______

e. 8929 is divisible by 9 _______

Let’s Sum Up

b. 3230 is divisible by 10

d. 8424 is divisible by 6

f. 4742 is divisible by 4

Factors: The numbers that completely divide the given number, resulting in a remainder of zero. A number can have finite number of factors.

Example: Factors of the number 12 are: 1, 2, 3, 4, 5, 6 and 12

Multiples: The numbers obtained by multiplying the given number by a natural number. A number can have infinite number of multiples.

Example: Multiples of 6 are:

Cross-Curricular Connections

Social Studies

Write the name of the Greek mathematician who developed an interesting and simple method for identifying prime and composite numbers. Explore and discuss how else the Greek society and culture contributed to the modern civilisation.

Life Skills

Ravi earns 25 as pocket money every week. He decides to save 7 from his weekly pocket money. Determine whether Ravi will have exactly 210 saved with him at any time. By the way, do you also have the habit of saving money?

Extend Your Knowledge

21st Century Skills

Circle the prime numbers. 5 12 34 73 78 41 55 75 21 78 45 55 59 74

Find out the dictionary meanings of ‘multiples’ and ‘factors’ and explore how these words convey different meanings in different contexts.

Tip for the Parent

Play Mental Math quiz games with your children based on multiples and factors.

Fractions 7

My Study Plan

Fraction

Proper, improper, and mixed fractions

Equivalent fraction

Like and unlike fractions

Addition and subtraction of like fractions

Fractions in real life

Let’s Recall

Jasmine’s grandmother baked some cookies for Jasmine and her four siblings and asked her to distribute them equally between all.

How many cookies did Jasmine get if there were a total of 25 cookies?

Thinking Zone

Colour the shapes that are divided into equal parts.

Let’s Learn

Fraction

When an object or a group of objects is divided into equal parts, each part is called a fraction of the whole.

Example: is divided into five equal parts.

So, represents a part or fraction of the whole pizza.

Writing a fraction

A fraction has two parts — numerator and denominator.

The parts are separated by a dividing line. The part above the dividing line is called the numerator and the part below the dividing line is called the denominator.

Word Zone

Whole: Complete

For example, 1 7 is a fraction.

Numerator

Denominator 1 7

We can read it as ‘one upon seven’ or ‘one-seventh’ or ‘one by seven.’

Numbers like 2 4 and 4 6 are called fractional numbers.

Let us take an example of two friends dividing a pizza into two equal parts.

Can you find how many parts will each friend get?

Yes, one part.

So, each friend’s share of pizza is one part out of two equal parts.

Fraction of pizza each friend gets =

Number of part seachfriendget Totalparts s

Hence, fraction of each friend’s pizza = 1 2 .

This is called one-half.

Word Zone

One-half: One part out of two equal parts of an object

Similarly, Example 1:

(i) Fraction for one slice = 1 3

(ii) Fraction for one slice = 1 4

This is called one-third.

This is called one-fourth or quarter.

Fraction of the flower which is white in colour = 1 4 or one-fourth.

Fraction of the flower which is red in colour = 3 4 or three-fourth.

1. Complete the table given below. One has been done for you.

Let’s Learn

Proper, Improper, and Mixed Fractions

Fractions are categorised into various types based on the numbers present in the numerator and denominator. The different types of fractions are:

Proper fraction

A proper fraction is a fraction where the numerator is smaller than the denominator.

For example, 1 2 is a proper fraction because the numerator (1) is smaller than the denominator (2).

= 1 2

Improper fraction

An improper fraction is a fraction where the numerator is equal to or greater than the denominator.

For example, 5 4 is an improper fraction because the numerator (5) is greater than the denominator (4).

It represents more than the whole thing that is having 1 whole and a little extra.

Mixed fraction

A mixed fraction is a combination of a whole number and a proper fraction. It represents a quantity more than one whole along with some smaller parts.

For example, 1 1 8 is a mixed fraction.

The whole number part (1) represents one whole, and the fractional part 1 8 represents an additional part that is smaller than a whole.

1 8 1

Converting improper fraction into mixed fraction

To change an improper fraction into a mixed fraction:

Step 1: We divide the numerator by the denominator. The number we get from the division as quotient becomes the whole number part of the mixed number.

Step 2: The remainder becomes the numerator of the proper fraction and the denominator remains the same. Write the whole part and the proper fraction.

Step 3: Combine the whole number and the proper fraction to create the mixed fraction.

Example 1: Convert 15 4 into a mixed fraction.

Solution: 15 4

Here, numerator = 15 and denominator = 4

Step 1: Divide numerator by denominator. 15 ÷ 4 3 15 4 -12 3

Here, Quotient = 3, Remainder = 3

Step 2: Write the whole part and the proper fraction.

Whole part = 3, Proper fraction = 3 4

Step 3: Combine the whole part and the proper fraction.

Hence, 15 4 = 3 3 4 .

Converting mixed fraction into improper fraction

To change a mixed fraction into an improper fraction:

Step 1: We multiply the whole number part by the denominator.

Step 2: The number we get from the multiplication is added to the numerator of the fraction. This becomes the numerator of the fraction.

Step 3: The denominator stays the same as before.

Example 2: Convert 1 3 4 into improper fraction.

Solution: 1 3 4

Here, whole part = 1, numerator = 3, and denominator = 4

Step 1: Multiplying the whole part and the denominator.

1 × 4 = 4

Step 2: Adding the numerator and the number we get in Step 1. 4 + 3 = 7

Step 3: Writing improper fraction.

The number we get in Step 2 becomes the numerator, and the denominator remains the same.

Hence, 1 3 4 = 7 4 .

Let’s Practise - 2

1. Write improper fractions for the coloured parts.

2. Convert the following into mixed fractions. a. 13 4 = b. 34 7 = c. 14 3 = d. 11 5 =

3. Convert the following into improper fractions.

2 3 7 =

Thinking Zone

Find the similarity between these figures.

If we place the figures one over the other, we will find them equal. Do you agree?

Equivalent Fraction

The fractions for the coloured part for the figures are as follows:

1:

2:

3:

Can we say all these represent the same part of the whole?

Yes, they represent the same value.

Figure 1: 1 2

Figure 2: 2 4 = 1 2

Figure 3: 4 8 = 1 2

Therefore, 1 2 = 2 4 = 4 8

The fractions that represent the same part of the whole are called equivalent fractions. They are equal in value.

Forming equivalent fractions

To find equivalent fractions, we can multiply or divide the numerator and the denominator of a fraction by the same non-zero number other than 1.

Figure

Figure

Figure

Example 1: Find four equivalent fractions of 2 3 .

Solution:

Multiplying the numerator and the denominator by 2, 3, 4, and 5 one by one.

So, the fractions 4 6 6 9 8 12 ,, , and 10 15 are all equivalent fractions of 2 3 .

Checking equivalent fractions

To check for equivalent fractions, we can cross multiply the numerator of the first fraction with the denominator of the second fraction and the denominator of the first fraction with the numerator of the second fraction of the compared fractions.

Consider two fractions, a b c d and a b c d {cross multiplying}

If fractions are equivalent, then a × d = c × b.

If fractions are not equivalent, then a × d ≠ c × b.

Example 2: Check whether the fractions 4 6 6 9 and are equivalent or not.

Solution:

To check whether the fractions are equivalent or not, we will apply cross multiplication. 4 × 9 = 36 6 × 6 = 36 36 = 36

Since the products are equal, the fractions 4 6 6 9 and are equivalent.

Reducing fractions to their lowest form

A fraction is in its simplest form when its numerator and denominator do not have any common factor except 1.

For example: 5 7 , here 5 and 7 have only one common factor, i.e. 1.

To make a fraction simpler:

Use the Highest Common Factor (HCF): Divide the numerator and the denominator directly by their HCF.

Example 1: Reduce 16 36 to its lowest form.

Solution:

Step 1: Find HCF of numerator and denominator.

Step 2: Now dividing both numerator and denominator by HCF.

Thus, 4 9 is the lowest form of 16 36 . 1. Find two equivalent fractions for the given fractions.

Let’s Learn

Like and Unlike Fractions

Fractions with the same denominator are called ‘like fractions.’

For example: 3 7 , 5 7 , and 6 7

Fractions with different denominators are called ‘unlike fractions.’

For example: 3 4 , 5 3 , and 6 5

Converting unlike fractions into like fractions

To convert unlike fractions into like fractions:

Step 1: Find the least common multiple (LCM) of the denominators of the given fractions.

Step 2: Multiply the numerator and the denominator of each fraction by a number so that the denominator becomes equal to the LCM.

Example 1: Change 2 3 and 3 5 into like fractions.

Solution:

Step 1: Find the LCM of the denominators.

The LCM of 3 and 5 is 15.

Step 2: Change the denominator of the first fraction to 15.

To do this, divide 15 by the denominator of the first fraction.

15 ÷ 3 = 5

Now, multiply the numerator and denominator by 5.

25 35 × × = 10 15

Step 3: Change the denominator of the second fraction to 15.

To do this, divide 15 by the denominator of the second fraction and multiply the numerator and denominator by the result.

15 ÷ 5 = 3

Now, multiply the numerator and denominator by 3.

33 53 × × = 9 15

Thus, 10 15 and 9 15 are like fractions.

1. Convert the given fractions into like fractions.

Let’s Learn

Addition

of Like Fractions

To add like fractions,

Step 1: Add their numerators and keep the denominator the same.

Step 2: Reduce the number to its simplest form.

Step 3: If the answer is an improper fraction, convert it into a mixed fraction, if required.

Example: Add 3 4 6 4 and .

Solution:

Step 1: Adding numerators and keeping the denominator same.

Step 2: The fraction is already in its simplest form.

Step 3: Converting it into a mixed fraction.

Divide numerator by denominator.

Subtraction of Like Fractions

To subtract like fractions,

Step 1: Subtract their numerators and keep the denominator the same.

Step 2: Reduce the number to its simplest form.

Step 3: If the answer is an improper fraction, convert it into a mixed fraction, if required.

Example: Subtract 4 9 from 5 9 .

Solution:

Step 1: Subtracting numerators and keeping denominator same.

Step 2: The fraction is already in its simplest form.

Step 3: The fraction is a proper fraction.

Hence, 1 9 is the answer.

Let’s Practise - 5

1. Solve the following.

2. Solve the following.

Let’s Learn

Fractions in Real Life

We can find fractions in our real life.

Example 1: You and your friends had a pizza. If you ate 1 1 3 slices and your friend ate 1 2 3 slices, what fraction of the pizza did you both eat together?

Solution:

Fraction of pizza I ate = 1 1 3 = 4 3

Fraction of pizza my friend ate = 1 2 3 = 5 3

Fraction of pizza we ate together = 4 3 + 5 3 =

Now, 9 3 is reduced to 3.

Thus, the total fraction of pizza we ate together = 9 3 or 3 slices.

Example 2: Pihu had 3 2 3 m ribbon. She cuts off 1 2 3 m ribbon.

Find the length of ribbon left with her.

Solution:

Total length of ribbon Pihu had = 3 2 3 m = 11 3 m.

Length of ribbon she cut off = 1 2 3 m = 5 3 m.

Length of ribbon left = 11 3 m –5 3 m = 11

Hence, 2 m of ribbon is left with Pihu.

1. Jitu walked 5 3 7 km during the weekend and 9 4 7 km during the weekdays.

How much did he walk through the week?

2. Aneeqa has 1 1 6 kg of rice and she donated 2 6 kg rice. How much rice is left? Let’s Practise - 6

Let’s Sum Up

When an object or a group of objects is divided into equal parts, each part is called a fraction of the whole.

A fraction has two parts numerator and denominator.

Numerator and denominator are separated by a dividing line (or fraction bar).

A proper fraction is a fraction where the numerator is smaller than the denominator.

An improper fraction is a fraction where the numerator is equal to or greater than the denominator.

A mixed fraction is a combination of a whole number and a proper fraction. It represents a quantity more than one whole along with some smaller parts.

The fractions representing the same part of the whole are called ‘equivalent fractions.’ They are equal in value.

Fractions with the same denominator are called ‘like fractions.’

Fractions with different denominators are called ‘unlike fractions.’

Life Skills

While having a picnic with your friends, two of them brought the same amount of sweets to share.

The first friend shared 3 4 of their sweets with you, and the second friend shared 2 4 of theirs. Find out the total fraction of sweets you have now.

Cross-Curricular Connections

Science:

In a science experiment, you are studying the composition of air. You find that the air is composed of 12 20 nitrogen, 5 20 oxygen, and the rest is other gases. If you want to represent the fraction of other gases, would you add or subtract 12 20 5 20 and ? Explain your reasoning and calculate the resulting fraction to represent the composition of other gases in the air.

21st Century Skills

Imagine your friend and you are painting a wall. You finished painting half of the wall, and you also helped your friend by completing one-fourth of their work.

a. How would you represent your work in fractions?

b. How much more work you did than your friend?

Extend Your Knowledge

Adding and Subtracting Fractions with Unlike Denominators:

For adding and subtracting fractions with unlike denominators, we need to convert the unlike fractions to like fractions by making their denominators same.

Let us understand this by adding and .

Step 1: Find the least common multiple (LCM) of the denominator. Here, the LCM of 3 and 7 is 21.

Step 2: Convert the given fraction to like fractions by making their denominators same.

Step 3: Now, add the numerator of the converted fraction and keep the denominator same.

Hence, the sum of 1 3 and 2 7 is 13 21 .

Subtraction of fractions with different denominators can be carried out in a similar manner.

Tip for the parent

Encourage your child to practise fractions in everyday situations such as sharing snacks. Use everyday objects to show how parts make up a whole to reinforce their understanding of the concept.

Geometry

My Study Plan

• Point, line, ray, and a line segment

• Measuring a line segment

• Drawing a line segment

• Open and closed figures

• Polygon and its types

• Circle

• Solid shapes

• Nets and cubes

Let’s Recall

In grade 3, you learned about the different types of shapes, such as squares, rectangles, triangles, and so on.

Let us do a quick review.

1. Look at the rangoli that Aayu made for Diwali and identify the shapes he used in it. How many of each shape did he use?

a. Number of triangles: ______

b. Number of rectangles: _______

c. Number of squares:__________

d. Number of circles: ______

2. Match the object to its shape.

Object

Shape

3. Aayu spends his summer holidays making things. Today, he made a paper plane by folding a sheet of paper. The paper plane is in the shape of a _____________________.

a. Triangle

b. Rectangle

c. Circle

d. Square

Thinking Zone

Think about the last time you saw someone cutting a round birthday cake into slices. They probably cut along the radii of the cake, right? A circle has an infinite number of radii. Think of other examples where we use this concept in our daily lives and discuss with your teacher.

Let’s Learn

You know that circles, squares, rectangles, and triangles are the basic shapes in geometry. There are many other shapes and concepts in geometry. Let us learn about some of them.

Point

A point is a tiny dot (.) made with a sharp pencil on paper. A point shows a definite position. It has no length, breadth, or thickness. We use dots to show points and name them using capital letters such as A, B, P, and Q, as shown below:

A B P Q

Points

Line

A line is a straight path of a set of points that extends in both directions. It has no endpoints. It is drawn with an infinite double-headed arrow indicating that it is infinitely long. In a diagram, if A and B are two points on the line, we denote this as AB and read this as ‘line AB’.

A B Line

Word Zone

Definite: Clear

Dots: Tiny round marks

Line Segment

Line segment is a straight path that has two endpoints. It lies between two fixed points, so it has a finite length.

A

Line segment

B

The line segment shown above has two endpoints, A and B. We denote this as AB and call it ‘line segment AB’.

Ray

A ray is a line that extends infinitely in only one direction. It starts at a point, known as the initial point, and extends endlessly in one direction, but does not have an endpoint. A ray does not have a fixed length, so we cannot measure it.

A B

Ray

The ray you see above starts at point A and extends endlessly in one direction. We denote it as AB, and read it as ‘ray AB’.

Many road signs have a red slanting line drawn across. The red slanting line across the symbol means that something is ‘not allowed’. Another word for ‘not allowed’ is prohibited. Fact Zone

Do Not Enter

Let’s Practise - 1

1. Write the name of each of the following figures.

d.

2. Aayu is playing the role of Arjun in a drama competition. He has to be able to shoot an arrow at a target. Which of the following does an arrow look like?

a. Point

b. Line

c. Line segment

d. Ray

Let’s Learn

Measuring a Line Segment

Measuring a line segment involves finding the distance from the starting point to the end point. We do this easily using a ruler.

Example 1: Measure the length of line segment AB.

Line Segment AB

Solution:

Step 1: Place the ruler straight along the line with point A at the zero mark.

Step 2: Look at the other end (point B) of the line, the reading of the scale at B gives the length of the line segment AB in cm.

Here, it reads 5 cm.

Thus, the length of line segment AB is 5 cm.

Let’s Learn

Drawing a Line Segment

We can also draw a line segment using a ruler.

Example 1: Draw a line segment of 5 cm.

Solution:

Step 1: Place the ruler on the paper and mark point A where it starts at 0.

Step 2: Mark another point B where the ruler reads 5 cm.

Step 3: Use your pencil to draw a line from point A to point B.

Now, you have a 5 cm long line segment called AB.

Let’s Practise - 2

2. Draw the following line segments.

a. 7 cm

b. 2 cm

c. 9 cm

1.

a. b.

Let’s Learn

Open and Closed Figures

Think of a square, a circle, or a rectangle. When you trace, their edges will start and end at the same point.

These shapes that start and end at the same point are called closed shapes.

Closed Shapes

Now, do you know what open shapes are? Shapes that do not start and end at the same point are called open shapes.

Open Shapes

Let’s Learn

Polygons

Some closed shapes are made of straight lines. We call them polygons. A polygon is a flat, closed shape made up of three or more line segments. These line segments are called the sides of the polygon. The point where the two sides meet is called a vertex.

Polygons are divided into different types depending on how many sides they have.

Here are some examples of polygons.

A three-sided polygon is called a triangle.

Vertex

Side

Triangle

A four-sided polygon is called a quadrilateral.

Quadrilateral

A five-sided polygon is called a pentagon. Vertex

Vertex

Side

Pentagon

A six-sided polygon is called a hexagon. Vertex

Side

Side

Hexagon

Let’s Practise - 3

1. Colour all the closed figures.

2. Aayu is excited about Makar Sankranti. Makar Sankranti is a harvest festival that is celebrated in January. Children and adults celebrate by flying kites. Aayu has a new kite. Can you identify which type of polygon a kite is?

a. Hexagon

b. Triangle

c. Pentagon

d. Quadrilateral

3. Pihu has made a Diwali lantern. The shape of the blue coloured part of the lantern is a _____________.

Circle

A circle is a round shape made with one curved line. It has a point in the middle called the centre. In the figure ‘O’ is the centre of the circle. Let us learn some important terms related to circles.

Circumference: This is the length around the edge of the circle.

Radius: This is a line segment that joins the centre of the circle to any point on the circumference. In the figure, ‘OQ’ is the radius of the circle.

Chord: This is a line that connects two points on the circumference of the circle. In the figure, ‘AB’ is the chord of the circle.

Diameter: This is a line segment that joins any two points on the circumference while also passing through the centre of the circle. In the figure, ‘XY’ is the diameter.

The diameter of the circle splits the circle into two equal parts called semicircles. The diameter is twice as long as the radius. That’s because it is like two radii joined together at the centre!

Diameter = 2 × radius

In other words, the radius is half the length of the diameter.

Radius = Diameter 2

Parts of a Circle

Example 1: Find the diameter of a circle whose radius is 5 cm.

Solution:

Diameter = 2 x radius = 2 x 5 cm = 10 cm.

Example 2: Find the radius of a circle whose diameter is 16 cm.

Solution:

Radius = Diameter ÷ 2 = 16 ÷ 2 = 8 cm.

Let’s Learn

Drawing a Circle

Try drawing a perfectly round circle on a sheet of paper. Were you successful? Most of us find it difficult to draw perfect circles every time.

But if we have the right tools, we can do this easily. To draw a circle, we will need a pencil, a ruler, and a pair of compasses. Collect these tools and get ready to learn.

Example 1: Use a pair of compasses to draw a circle of radius 5 cm.

Solution:

Step 1: Place a sharpened pencil into your compasses.

Step 2: Extend the compasses on a ruler so that the arms are separated by a distance of 5 cm.

Step 3: Secure the sharp point of the compasses in your notebook or paper.

Step 4: Hold the arm with the needle gently and rotate the arm with the pencil to draw a smooth circle.

Your circle is ready!

Let’s Practise - 4

1. Draw circles with the following radius.

a. 4 cm

b. 5 cm

2. Aayu has made pizza for himself and his friend. He wants to cut the round pizza into two equal parts. Along which line of the circular pizza should he cut the pizza?

a. Chord

b. Diameter

c. Radius

Let’s Learn

Solid Shapes

Shapes that we can hold or carry are called solid shapes. They are also known as three dimensional shapes or 3-D shapes. 3-D shapes have three dimensions that is length, breadth and height. They can have faces, corners (or vertices), and sides (or edges).

The surface of a solid shape is called a face.

The line where two faces meet is called an edge or a side.

The point where the edges meet is called a vertex or a corner.

3D Shape Edges Vertices

Look at these shapes.

3D Shape

Cube

This is a cube. It has 6 equal faces, 8 vertices, and 12 edges. All its faces are identical.

Cuboid

This is a cuboid. It has 6 faces, 8 vertices, and 12 edges. Its opposite faces are identical.

Cone

This is a cone. It has 1 curved edge, 1 curved face, 1 flat face, and 1 vertex.

Cylinder

This is a cylinder. It has 2 curved edges, 2 flat faces, and 1 curved face. It has no vertices.

Real-life example

Dice

This dice is an example of a cube.

Pencil Box

The pencil box is an example of a cuboid.

Party Cap

The party cap is an example of a cone.

Cola Can

The can of cola is an example of a cylinder.

Sphere

This is a sphere. It is round and has 1 face. It does not have any vertices or edges.

Let’s Learn

Nets and Cubes

A tennis ball is a perfect example of a sphere.

Now that you know what 3D shapes are, let us learn about the nets of 3D shapes. A cube is a three-dimensional shape that looks like a box. It has six equal square faces, just like the sides of a dice. When you put a cube together, it looks like a solid, 3D object.

When we take this cube apart and lay it flat on a piece of paper, we get a ‘net’ of the cube. A net of a cube is like a map that shows all the pieces we need to fold and put together to make a cube. A net of a cube has six squares because a cube has six faces. These squares are connected along their edges, just like the faces of a cube are connected. Each square in the net represents one face of the cube.

Tennis Ball

Faces of a Cube

Let’s Practise - 5

1. Aayu is confused about the shapes. He has named them wrong. Write the correct name of each shape in the blank space.

Shapes Aayu’s answers Correct answers

2. Fill in the blanks.

a. A cube is a ________dimensional shape.

b. A cube has _______equal square faces.

c. We get the _______ of a cube when we take a cube apart and lay it flat on a piece of paper.

d. The net of a _______ has 6 rectangles.

Many of the foods we eat can be broken into basic 3D shapes. Look at an ice-cream cone. The ice-cream cone is clearly a cone, while the scoop of ice-cream on top is a sphere. A sphere is the 3D shape of a circle.

Fact Zone

Let’s Sum Up

• A point is a tiny dot created with a sharp pencil, representing a definite position.

• A line consists of connected dots, forming a straight path with no endpoints.

• Line segments are straight paths with two fixed endpoints.

• Rays are lines that extend infinitely in only one direction.

• Shapes that start and end at the same point are classified as closed shapes.

• Shapes that don’t start and end at the same point are referred to as open shapes.

• Polygons are flat, closed shapes made up of three or more line segments. The meeting point of the two sides is called a vertex. Examples of polygons include triangles, quadrilaterals, pentagons, and hexagons.

• Circles are round shapes formed by a curved line, with a central point called the centre. The other terms related to circle include circumference, radius, diameter, and chord.

• Radius is half the length of the diameter.

• Solid shapes, or 3-D shapes, can be held or carried. They have three dimensions: length, breadth, and height.

• Examples of 3-D shapes include cube, cuboid, cone, cylinder, and sphere.

Cross-Curricular Connections

Art and Craft:

Draw a rangoli design using as many geometric shapes as you can. Use different coloured crayons or colour pencils to make your rangoli beautiful.

A tangram is a Chinese puzzle game consisting of 7 pieces. Make your own tangram by tracing these shapes on a piece of cardboard and cutting them out.

Try arranging the pieces to make the shapes of different animals. Can you put the pieces back into a square, without looking at this picture?

21st Century Skills

Make a poster on the different 3D shapes you have learnt about. Include pictures of real-life objects of each type of 3D shape. Use old newspapers, comic books and magazines to find the pictures.

Extend Your Knowledge

An angle is made up of two rays that begin at a common point known as vertex.

Vertex

Angles ranges from 0 to 180°.

There are four types of angles.

Right Angle: An angle that is exactly 90°.

Acute Angle: An angle that is less than 90°.

Obtuse Angle: An angle that is more than 90° but less than 180°.

Straight Angle: An angle that is exactly 180°.

Tip for the Parent

• Encourage your child to explore objects around the house and identify different types of shapes.

Perimeter and Area 9

My Study Plan

• Perimeter of 2D shapes, including squares and rectangles, with given side lengths

• Area of 2D shapes, such as squares and rectangles, using appropriate formulas and units

Let’s Recall

In grade 3, you learnt that the length around the border of a shape or figure is called its perimeter and the space enclosed by a shape is its area.

Let’s do a quick warm-up of these two concepts.

1. During the summer holidays, Pihu enjoys walking around her garden in the evenings. Her garden is square in shape and each side measures 15 m. If Pihu takes one round of her garden, what distance does she cover? (Hint: find the perimeter of the garden keeping in mind its shape.)

Pihu’s Garden

2. If each of the squares is equal to 1 square unit, find the area for the given rectangles.

= 1 square unit

(Hint: count the number of squares in each shape.)

Area: __________ square unit

Area: __________ square unit

Consider how knowing the area or perimeter of a shape can be useful in your daily life. For instance, if you’re decorating the lid of the box with a gold lace border, you need to find the perimeter of the lid and measure the length of the lace to ensure it’s enough. Now, come up with another example from your daily life that involves measuring area or perimeter. How does this knowledge benefit you in practical situations?

Using Perimeter in Everyday Life

Thinking Zone

Let’s Learn

You know how to calculate the perimeter by adding the sides. Now, let us learn some quicker ways to calculate perimeter.

PERIMETER

Diwali is one of the biggest festivals in India. During Diwali, Pihu decorates her rangoli with a marigold garland. Can you figure out the length of the garland she used?

Rangoli

Solution:

Length of each side of rangoli = 25 cm

We can find the length of the marigold garland around the rangoli by adding the lengths of all its sides.

Garland length = 25 cm + 25 cm + 25 cm + 25 cm. This is repeated addition. We can also find length of marigold by multiplication.

Garland length = 4 × 25 cm = 100 cm

Thus, the length of the garland required is the perimeter of the rangoli.

The perimeter is the total length of the outline of a closed shape.

Example 1: Find the perimeter of the given shape, if each side of the shape is 5 cm long.

Solution:

Perimeter = Sum of all sides

= 5 + 5 + 5 + 5 = 20 cm

Example 2:: Find the perimeter of the given shape.

Solution:

Perimeter = Sum of all sides

= 9 + 6 + 14 + 12 = 41 cm

Perimeter of a rectangle

Look at this rectangle whose length and breadth are represented as ‘L‘and ‘B’ respectively.

The perimeter of the rectangle is: L +B + L +B = 2 L + 2B

We can simplify the perimeter of a rectangle into the formula: 2 (L+B)

Perimeter of a Rectangle (P) = 2 (Length + Breadth)

Example 1: Find the perimeter of the given rectangle using the formula.

Solution:

Here, L = 5 cm and B = 3 cm

Perimeter of a rectangle = 2 (L + B) = 2 (5 + 3) = 2 x 8 = 16 cm

Hence, the perimeter of the given rectangle is 16 cm.

Perimeter of a square

Look at the square with sides represented by ‘S’. s s s s

The perimeter of the square is: S + S + S + S = 4 x S

Hence, the formula for the perimeter of a square = 4 x side

Perimeter of a Square (P) = 4 × Side

Let’s practise using the formulas we just learned.

Example 1: Find the perimeter of a square whose sides measure 3 m each.

Solution:

Here, S = 3 m

Using the formula, we have:

Perimeter of the square = 4 x side = 4 x 3 = 12 m

Hence the perimeter of square is 12 m.

Perimeter in Real Life

Now, let’s see how we use the concept of perimeter in everyday life.

Example 1: A farmer has a field that measures 20 m long and 10 m wide. If the farmer walks around his field twice, then what distance does he cover?

Solution:

Length of the field = 20 m

Breadth of the field = 10 m

Perimeter of the field = 2 × (L + B) = 2 × (20 + 10) m = 2 × 30 m = 60 m

Hence, the perimeter of the field is 60 m. Since the farmer walks twice around the field, the distance covered by him = 2 × 60 m = 120 m

So, the farmer covers 120 m when he walks twice around the field.

Example 2: A square-shaped tabletop has to be decorated with ribbons along the edges. Find the length of the ribbon needed if one side of the tabletop is 20 cm.

Solution:

Side of the square-shaped tabletop, S = 20 cm

Length of the ribbon = Perimeter of the tabletop

Perimeter of the tabletop = 4 × S = 4 × 20 cm = 80 cm

Hence, the required length of the ribbon is 80 cm.

Let’s Practise - 1

1. The perimeter of a square is 64 cm. What is the length of a side of this square?

2. Calculate the perimeter of a rectangle with length 6 cm and width 5 cm.

Let’s Learn

Area

Now, let us learn about another important concept, Area.

Area is the space covered by a shape. It is measured in square units. It represents the amount of surface enclosed by a flat object. It can be determined by counting squares on graph paper.

For example, look at the pictures below.

Number of squares covered by = 6 squares

Hence, the area of is 6 square units.

Similarly, the number of squares covered by = 9 squares.

Hence, the area of = 9 square units.

Standard unit of area

You have learned that we measure things using units. For example, we measure the length of an object using cm or m. When we want to find the weight of an object, say a ladoo, we use grams or kilograms. Similarly, we also use units to measure area.

The unit of area depends on the unit of length used for measuring the sides of a shape.

If the lengths are in millimetres (mm), the area will be in square millimetres (mm²).

If the lengths are measured in centimetres (cm), the area will be in square centimetres (cm²). Areas of small surfaces such as tabletops are measured in sq. cm. Similarly, if the lengths are measured in metres (m), the area will be in square metres (m²). Areas of bigger surfaces such as a playground or a wall are measured in sq. m.

Area of a rectangle

Look at the rectangle with sides ‘L’, and ‘B’.

Since the space occupied by a rectangle is L x B, its area would be length x breadth.

Area of a rectangle = Length × Breadth

Note: If the lengths of the sides are not in the same units, convert the lengths to the same units before finding the area of a shape.

Now let’s look at an example.

Example 1: Find the area of the given rectangle.

Solution:

Here, length = 6 cm and breadth = 5 cm

We know that the formula for the area of a rectangle = length x breadth = 5 x 6 = 30 sq. cm.

Hence, the area of the rectangle is 30 sq. cm.

Area of a square

We know that a square is a polygon with 4 equal sides. It is also a special type of rectangle, where all four sides are equal. s s s s

Using the formula for the area of a rectangle, we get:

Area of a square = length × breadth = side x side

Area of a Square = Side × Side

Now let’s look at some examples to see how we use this formula.

Example 1: Find the area of the given square.

Solution:

Here, side = 5 m

Hence, the area of the square with sides 5 m = 5 × 5 = 25 sq. m.

Area in real life

How do we use the concept of area in real life? In many ways!

Example 1: Find the area of a field whose length is 15 m and breadth is 12 m.

Solution:

Length of the field = 15 m

Breadth of the field = 12 m

From the measures of the sides, we know that the field is a rectangle. So, we use the formula for the area of a rectangle.

Area of the field = length × breadth = 15 m × 12 m = 180 sq. m.

Hence, the area of the field is 180 sq. m.

Example 2: A square sheet of paper has a length of 10 cm. Find the area of the sheet of paper.

Solution:

Side of the square sheet of paper, a = 10 cm

Area of the sheet = side × side = 10 cm × 10 cm = 100 sq. cm.

Hence, the area of the square sheet of paper is 100 sq. cm.

Let’s Practise - 2

1. Find the area of a rectangular New Year’s greeting card with length 16 cm and width 14 cm.

2. Calculate the area of a square-shaped book cover with sides of length 8 cm.

3. The area of a rectangular field is 120 sq. m. If the breadth of the field is 10 m, find its length.

A circle does not have any sides, but it still has a perimeter. The perimeter of a circle is called the circumference. Fact Zone Perimeter of Circle = Circumference of Circle

Let’s Sum Up

• Perimeter is the total length around the edges of a shape.

• We calculate the perimeter of a shape by adding the lengths of all its sides.

• Perimeter of a rectangle = 2 (Length + Breadth)

• Perimeter of a square = 4 × Side

• Area is the space covered by a shape. It is expressed in square units.

• Area of a rectangle = Length × Breadth

• Area of a square = Side × Side

Life Skills

You have to make a chart for your classroom. You want to decorate the border of the chart paper using colourful tape. If the breadth of the chart paper is 35 cm and its length is 24 cm, calculate the length of the tape required to decorate all four sides of the chart paper.

Social Studies:

Explore India’s map: Identify the largest and smallest states, and pinpoint the Union Territory with the largest land area.

21st Century Skills

Design a school playground with a rectangular field (50 ft by 80 ft) and a square courtyard (side length 40 ft). Calculate their areas and find the total playground area. If you split the playground into two equal spaces, what would be the area of each shape? Explain with drawings.

Extend Your Knowledge

A parallelogram is a four-sided shape where opposite sides are equal in length and parallel (non-intersecting) to each other. The area of a parallelogram is found by multiplying the length of its base (one side) by its height (the perpendicular distance between the base and its opposite side).

Tips for the Parent

• Share with your child how we use the concepts of perimeter and area at home. For example, when we want to paint a wall, we need to know the area of the wall so we can buy enough paint.

• Encourage and help children measure the lengths of objects like books, picture frames, bathroom tiles, etc., and calculate the area and perimeter.

Area of Parallelogram = Base x Height

Measurement

My Study Plan

Metric system

Conversion of metric units

Addition and subtraction of metric measures

Measurements in real life

Let’s Recall

Circle the unit to measure the given objects.

Height of a small plant l m g

Capacity of a box of milk ml mg mm

Weight a bag of cement kg kl km

Length of a table m l g

Capacity of a Water bottle g l m

Weight of a bag of apples

Thinking Zone

Imagine you have an apple, a water bottle, and a bat. Comment about the different parameters that you can measure for these objects. What tools could you use for the measurements? Share your answers with your teacher.

Let’s Learn

Metric System

The metric system is used worldwide for measurement. It is used for counting and understanding various things. Let us look how to measure using the metric system.

Length

Length is the measurement of an object from one end to the other along the longest side. It helps us to identify the size of an object. When the size of an object is measured horizontally, it is known as length and when measured vertically, it is known as the height of the object. And when we measure how far is one place from another then it is called the distance.

For example, measuring the distance between school and home, or the length of a piece of cloth.

Millimetre (mm), decimetre (dm), centimetre (cm), metre (m), and kilometre (km) are units used to measure lengths and distances.

Weight

Weight tells how heavy or light an object is. Some objects can be very heavy, like a big rock, while others are very light, like a feather.

To find out how heavy or light an object is, we measure its weight. We use balance and weights to weigh things.

1kg

In an electronic weighing balance, we have only one pan and we can see the weight directly on the screen.

Milligrams (mg), grams (g) and kilograms (kg) are the units used to measure weight.

Capacity

Capacity is a measure of how much quantity a container can hold. For example, amount of water or juice in a bottle.

Millilitre (ml) and litre (l) are units used to measure the capacity or volume of an object in the metric system. The millilitre is a smaller unit, often used for smaller quantities, while the litre is a larger unit used for bigger quantities.

Three countries, including the US, Myanmar, and Liberia do not officially use the metric system. Fact Zone

Let’s Practise - 1

1. Tick the correct answer.

a. _______________ is used for measuring weight.

i. Metre

ii. Litre

b. Kilometre is a unit of _______________.

i. Distance

ii. Weight

iii. Gram

iii. Volume

c. Which one of the following is not a metric unit?

i. Gram

2. Match the following.

ii. Metre

Column I Column II Weight metre

iii. Feet

Conversion of Metric Units

To convert the units, we should first understand the relationship between them.

Relationship between units of length – metres, millimetres, centimetres, and kilometres

One centimetre is equal to 10 millimetres.

1 cm = 10 mm

One metre is equal to one hundred centimetres or one thousand millimetres.

1 m = 100 cm

1 m = 1000 mm

One kilometre is equal to one thousand metres.

1 km = 1000 m

Relationship between units of weight – grams and kilograms

One gram is equal to 1000 milligram.

1 g = 1000 mg

One kilogram is equal to one thousand grams.

1 kg = 1000 g

Relationship between units of capacity – litres and millilitres

One litre is equal to one thousand millitres.

1 l = 1000 ml

Conversion in the metric system can be done in two ways: one is to change from a larger unit to a smaller one, and the other is to change from a smaller unit to a larger one.

Conversion of larger units into smaller units

To convert larger units into smaller units, we perform multiplication.

Example 1: Convert 4 km into m.

Solution: We know that 1 km = 1000 m

So, to convert 4 kilometres to metres we will multiply it by 1000

4 km = 4 × 1000 = 4000 m

Example 2: Convert 9 kg into grams

Solution: We know that 1 kg = 1000 g

So, to convert 9 kilograms to grams, we will multiply it by 1000

9 kg = 9 × 1000 = 9000 g

Example 3: Convert 6 m and 40 cm into cm.

Solution: Here we will do two things,

Step 1: Convert 6 m into cm.

We know that 1 m = 100 cm.

So, to convert 6 m to cm we will multiply it by 100.

6 m = 6 × 100 = 600 cm.

Step 2: Add centimetre values to get the final answer.

600 + 40 = 640 cm.

Conversion of smaller units into larger units

To convert smaller units into larger units we perform division.

Example 1: Convert 300 cm into m.

Solution: We know that 1 m = 100 cm

So, to convert 300 centimetres to metres we will divide it by 100

300 cm = 300 ÷ 100 = 3 m

Example 2: Convert 6000 ml to litres.

Solution: We know that 1 litre = 1000 millilitres.

So, to convert 6000 millitres to litres we will divide it by 1000 6000 ml = 6000 ÷ 1000 = 6 litres.

Example 3: Convert 1320 metres to kilometres.

Solution: We know that 1km = 1000 metres.

So, to convert 1320 metres into kilometres we will divide it by 1000 1320 m = 1320 ÷ 1000 = 1.32 km

Example 4: Convert 5000 g into kg.

Solution: We know that 1 kg = 1000 g.

So, to convert 5000 g to kg we will divide it by 1000

5000 g = 5000 ÷ 1000 = 5 kg

Let’s Practise - 2

1. Fill in the blanks with the appropriate number.

a. 1 km = _______ m

c. 1000 g = _______ kg

2. Convert as instructed.

a. 4 kg into g.

c. 1458 m into km.

b. 1 l = _______ ml

d. 1 m = _______ mm

b. 5 m and 6 cm into cm.

d. 3098 ml into l.

Let’s Learn

Addition and Subtraction of Metric Measures

We can easily add and subtract two metric measures of the same type. For example, we can add and subtract two metric units of weight, two metric units of length and so on. We cannot add or subtract metric units of different types. For example, we cannot add 1 metre to 1 litre.

To add/subtract metric measurements, follow these steps.

Step 1. Write units beside each number as given (if multiple units are involved).

Step 2. Arrange digits by their place value.

Step 3. Carry over or borrow if necessary.

Step 4. Add/subtract the numbers as regular whole numbers.

Let us look at some examples.

Example 1: Add 22 km 500 m and 23 km 400 m.

Solution:

Step 1: Arrange the numbers column-wise by their place value.

Step 2: Add the digits column-wise from the right side.

Therefore, 22 km 500 m + 23 km 400 m = 45 km 900 m.

Example 2: Add 2 kg 750 g and 3 kg 425 g.

Solution:

Step 1:

Step 2:

Arrange the numbers column-wise by their place value.

Add the digits columnwise from the right side and carry forward to the left of the column if necessary.

Therefore, 2 kg 750 g + 3 kg 425 g = 6 kg 175 g.

Example 3: Subtract 30 kg 250 g from 31 kg 435 g.

Solution:

Step 1:

Arrange the numbers columnwise by their place value.

Step 2: Subtract the digits column-wise from the right side and borrow the number from the left of the column if necessary.

Therefore, 31 kg 435 g – 30 kg 250 g = 1 kg 185 g.

Example 4: Subtract 7 km 415 m from 9 km 22 m.

Solution:

Step 1: Arrange the numbers columnwise by their place value.

Step 2: Subtract the digits column-wise from the right side and borrow the number from the left of the column if necessary.

Therefore, 9 km 22 m – 7 km 415 m = 1 km 607 m.

Let’s Practise - 3

13 kg 350 g + 12 kg 450 g

76 km 350 m - 23 km 950 m

Let’s Learn

Measurements in Real Life

We learn to put numbers together and take them away. Now, let us use that to solve problems we might face in everyday life!

Example 1: Lily brought 6 kg 800 g of tomatoes on Monday and 2 kg 250 g of tomatoes on Saturday. What is the total weight of tomatoes she bought in the week?

1. Solve.

2. Solve.

Solution:

Weight of tomatoes bought on Monday = 6 kg 800 g

Weight of tomatoes bought on Saturday = 2 kg 250 g

We will add 6 kg 800 g and 2 kg 250 g to find the total weight of tomatoes bought by Lily.

So, Lily bought 9 kg 50 g of tomatoes in the week.

Example 2: Chutki has a piece of cloth that is 45 m 25 cm long. She needs 25 m 55 cm to stitch a shirt. How much cloth will be left after Chutki uses it?

Solution:

Length of cloth = 45 m 25 cm

Cloth required to stitch a shirt = 25 m 55 cm

Length of remaining cloth = 45 m 25 cm – 25 m 55 cm

So, 19 m 70 cm of the cloth will be left.

Example 3: Aayu walks 1 km 670 m in the morning and 2 km 540 m in the evening. How much distance does he walk through the day?

Solution:

Distance walked in the morning = 1 km 670 m

Distance walked in the evening = 2 km 540 m

Total distance covered = 1 km 670 m + 2 km 540 m

So, Aayu walks 4 km 210 m all through the day.

Let’s Sum Up

The metric system is a helpful way to measure things.

The units used to measure length are metres, centimetres, millimetres and kilometres.

1 cm = 10 mm

1 m = 100 cm

1 m = 1000 mm

1 km = 1000 m

The units used to measure weight are milligrams, grams and kilograms.

1 g = 1000 mg

1 kg = 1000 g

The units used to measure capacity are litres and millilitres.

1 l = 1000 ml

We convert smaller units to larger units by division, and larger units to smaller units by multiplication.

When adding or subtracting, the same units are used.

Life Skills

Bittu drinks 1 litre 20 ml of water in the morning, 815 ml of water in the afternoon and 930 ml of water in the evening. What is the total amount of water that Bittu drinks?

Cross-Curricular Connections

English:

What is the combined weight of the two novels, if volume-I weighs 920 g and volume-II weighs 1 kg 33 g.

Social Studies:

The height of the famous statue of a freedom fighter in a city is 383 cm. How much taller is it compared to his original height, which is 173 cm?

21st Century Skills

1. Write these distances in kilometres and metres.

John’s house to mum’s office 4000 m

Grandfather’s house to the park 2500 m

Grandfather’s house to mum’s office 1700 m

2. Find the total distance in kilometres and metres.

a. John leaves his house, passes his mum’s office and goes to his Grandfather’s house.

b. John leaves his house, passes the park and goes to his Grandfather’s house, if the distance between John’s house to park is 2 km 700 m.

Extend Your Knowledge

Multiplication of units of measurement:

Multiply 2 km 56 m by 4.

Step 1: Convert 2 km 56 m into m

2 km 56 m = 2 × 1000 + 56 = 2000 + 56 = 2056 m

Step 2: Now, multiply 2056 by 4

Step 3: Convert m to km and m.

8224 m = 8000 + 224 = 8 km 224 m (Since 1000 m = 1 km)

So, when 2 km 56 m is multiplied by 4, it gives 8 km 224 m.

Tip for the Parent

• Connect measurements to daily activities like travelling, home projects, and shopping to show real-life applications. Discuss why accurate measurements matter. This hands-on approach helps children understand the importance of length, weight, and volume in practical situations.

Time

My Study Plan

• Units of time

• A.M. and P.M.

• 24-hour clock

• Addition and subtraction of units of time

• Time duration in hours, minutes, and days

• Days, weeks, months, and years

Let’s Recall

Look at the pictures carefully and fill in the blanks. Write down the time using ‘A.M.’ and ‘P.M.’

Aayu Left for School

Aayu left for school at ______.

Aayu returned from school at ______.

Aayu stayed in school for ______ hours.

Aayu Returned from School

Thinking Zone

How do you think people long ago used a sundial to determine the time? Discuss with your teacher.

Let’s Learn

Units of Time

Time in a day is measured using a clock.

A clock has 12 numbers and two hands: a short one called the hour hand and a long one called the minute hand.

The hour hand takes 12 hours to go around the clock two times in one day, while the minute hand takes 60 minutes to go around the clock once in an hour.

Some clocks have a very thin hand called the second hand, which goes around the clock once in one minute because there are 60 seconds in one minute.

Let us examine the relationships between these time units:

• 1 year = 365 days

• 1 week = 7 days

• 1 day = 24 hours

• 1 hour = 60 minutes

• 1 minute = 60 seconds

These relationships help us understand how different units of time relate to each other.

Conversion of Units of Time

Let us look at how we can convert time from one unit to another.

Word Zone

Sundial: A clock that uses the position of the sun to indicate the time

Hours to minutes:

Imagine it took 2 hours to complete an activity. To convert 2 hours into minutes, consider how many minutes are in each hour.

We know that,

1 hour = 60 minutes.

So, 2 hours = 2 × 60 = 120 minutes.

Minutes to hours

Now, let us say you have 90 minutes. To convert this to hours, think about how many full hours are in 90 minutes.

We can divide 90 minutes into 60 minutes and 30 minutes. Since there are 60 minutes in an hour, so we can say 90 minutes is the same as 1 hour and 30 minutes.

Now, let us look at some examples.

Example 1: Convert 7 hours into minutes.

Solution:

1 hour = 60 minutes

7 hours = 60 × 7 = 420 minutes.

Example 2: Convert 3 hours 45 minutes into minutes.

Solution:

To convert 3 hours 45 minutes into minutes, we can multiply the number of hours by 60 (because there are 60 minutes in 1 hour) and then add the remaining minutes.

1 hour = 60 minutes

So, 3 hours = 3 × 60 = 180 minutes

Now, add the remaining 45 minutes. 180 minutes + 45 minutes = 225 minutes

Therefore, 3 hours 45 minutes is equal to 225 minutes.

One hour is equal to 60 minutes, and 60 minutes is equals to 60 times 60 seconds, which equals

3600 seconds. Fact Zone

Let’s Practise - 1

1. Convert the following.

a. 5 hours into minutes.

b. 4 hours 30 minutes into minutes.

c. 2 hours into minutes.

2. Match the following with their correct pairs.

Let’s Learn

A.M. or P.M.

A day has 24 hours. When it is 12 o’clock at night, we call it midnight. When it is 12 o’clock during the day, we call it noon.

From midnight until 12 noon, we say ‘A.M.’ in the time, like 5:00 A.M. From 12 noon until midnight, we say ‘P.M.’ in the time, like 5:00 P.M.

However, we do not use ‘A.M.’ or ‘P.M.’ for noon and midnight.

Let’s Learn

24 Hour Clock

We usually use a clock with 12 hours on it. But in trains, planes, and the army, a clock with 24 hours is used.

In the 24-hour clock, there are no ‘A.M.’ or ‘P.M.’ Instead, it displays a four-digit number, where first two digits represent the hours, and the last two digits represent the minutes.

It is a way of telling time that uses the numbers 00:00 to 23:59 to represent the hours and minutes in a day. In the 24-hour clock, the day starts at midnight (00:00) and goes through the hours of the day until it reaches midnight again. This is different from the 12-hour clock, which repeats the same 12 hours twice a day.

For example:

6:00 A.M. in the 12-hour clock is 06:00 in the 24-hour clock.

3:30 P.M. in the 12-hour clock is 15:30 in the 24-hour clock.

Conversion of 12-hour clock into 24-hour clock

To convert a time from a 12-hour clock to a 24-hour clock, we follow these steps:

Morning (A.M.)

If it is a time in the morning (A.M.), keep the same hours.

For example, 7:30 A.M. in a 24-hour clock is 07:30.

Afternoon/Evening (P.M.)

If it is a time in the afternoon or evening (P.M.), add 12 to the hours. For example, 4:45 P.M. in a 24-hour clock is 16:45 (4 + 12 = 16).

Conversion of 24-hour clock into 12-hour clock

To convert a time from a 24-hour clock to a 12-hour clock, we follow these steps:

Morning (00:00 to 11:59)

If the time is between 00:00 and 11:59, just use the same hours. For example, 09:30 in a 12-hour clock is also 09:30 A.M..

Afternoon/Evening (12:00 to 23:59)

If the time is between 12:00 and 23:59, subtract 12 from the hours. For example, 17:45 in a 12-hour clock is 5:45 P.M. (17 - 12 = 5).

Practise - 2

1. Convert the following into 24-hours format.

a. 7:15 A.M.

b. 9:30 P.M.

2. Convert the following into 12-hours format.

a. 02:30 hours

b. 13:15 hours

Let’s Learn

Addition of Time

To add units of time together, we can follow the steps below:

Step 1: Align units

When adding time, align the units. For example, add seconds to seconds, minutes to minutes, and so on.

Step 2: Carry over

If the sum of seconds is 60 or more, carry over the extra seconds to the minutes. If the sum of minutes is 60 or more, carry over the extra minutes to the hours.

Example:

Let us say we want to add 2 hours and 30 minutes to 3 hours and 45 minutes.

Start with minutes:

30 + 45 = 75 minutes.

Since 75 minutes is more than 60, carry over 1 hour to the hours place, and we have 15 minutes left.

Now add the hours:

2 hours + 3 hours + 1 hour (from the carry-over) = 6 hours.

So,

2 hours 30 minutes + 3 hours 45 minutes = 6 hours 15 minutes.

Let’s Learn

Subtraction of Time

To subtract units of time, we can follow the steps below:

Step 1: Align units

When subtracting time, align the units. For example, subtract seconds from seconds, minutes from minutes, and so on.

Step 2: Borrowing

If the top number (minuend) is smaller than the bottom number (subtrahend), we will need to borrow from the next higher unit.

For example, while subtracting minutes if the top number is smaller then borrow 1 hour and add 60 minutes to the top number.

Example:

Let us say we want to subtract 4 hours and 30 minutes from 7 hours and 15 minutes.

Start with minutes:

Since 15 is smaller than 30, borrow 1 hour (60 minutes) from the hours place, making it 6 hours.

Subtract the minutes: (60 + 15) – 30 = 75 – 30 = 45 minutes.

Now subtract the hours: 6 hours - 4 hours = 2 hours.

So,

Hence, 7 hours 15 minutes - 4 hours 30 minutes = 2 hours 45 minutes

1. Solve the following.

a. Add 2 hours 15 minutes + 3 hours 30 minutes.

b. Add 4 hours 30 minutes + 2 hours 45 minutes.

c. Subtract 1 hour 45 minutes from 3 hours 30 minutes.

d. Subtract 5 hours and 45 minutes from 8 hours and 15 minutes.

Time Duration in Hours and Minutes

The length of time it takes to complete something is called the duration of an activity. Activities can vary in duration; some are short, like making a cup of tea, while others are long, like sleeping.

Let us learn it through some real-life examples.

Example 1: Imagine you leave home at 7:10 A.M. and arrive at school by 7:30 A.M. How much time does it take you to reach school?

Solution:

To determine the time taken to reach school, you subtract the time you leave from the time you arrive.

Let us do it step by step:

Time you leave home: 7:10 A.M.

Time you arrive at school: 7:30 A.M.

Now, subtract the time you left from the time you arrived:

7:30 A.M. – 7:10 A.M.

Subtract the minutes first: 30 – 10 = 20 minutes

Subtract the hours: 7 – 7 = 0 hours

So, it takes you 20 minutes to reach school.

Example 2: Aayu left school at 10:30 A.M. and reached home one hour later. What time did he reach home?

Solution:

Aayu left school at 10:30 A.M. It means he started his journey at 10:30 A.M.

He reached home one hour later. So, we need to add 1 hour to 10:30 A.M.

Let us add the hours:

10:30 A.M. + 1 hour = 11:30 A.M.

Therefore, Aayu reached home at 11:30 A.M.

Let’s Practise - 4

1. If a movie starts at 9:40 P.M. and ends at 11:15 P.M., what is the duration of the movie?

2. If you start doing your homework at 6:45 P.M. and finish at 7:15 P.M., how much time did it take to complete your homework?

Let’s Learn

Conversion of Days, Weeks, Months, and Years

Understanding the conversion between days, weeks, months, and years is an important concept. Let’s take a closer look at it.

1. Days to weeks:

A week is a group of seven days.

• Conversion Rule: To convert days to weeks, divide the number of days by 7. Example: If you have 14 days, how many weeks is that? 14 ÷ 7 = 2 weeks.

2. Weeks to days:

• Conversion Rule: To convert weeks to days, multiply the number of weeks by 7.

Example: If you have 3 weeks, how many days is that? 3 × 7 = 21 days.

3. Weeks to months:

A month can be about 4 weeks, depending on the month.

• Conversion Rule: To convert weeks to months, divide the number of weeks by 4 (assuming 4 weeks).

Example: If you have 12 weeks, how many months is that? 12 ÷ 4 = 3 months.

4. Months to weeks:

• Conversion Rule: To convert months to weeks, multiply the number of months by 4.

Example: If you have 2 months, how many weeks is that? 2 × 4 = 8 weeks.

5. Months to years: A year is made up of 12 months.

• Conversion Rule: To convert months to years, divide the number of months by 12.

Example: If you have 24 months, how many years is that? 24 ÷ 12 = 2 years.

6. Years to months:

Conversion Rule: To convert years to months, multiply the number of years by 12.

Example: If you have 3 years, how many months is that? 3 × 12 = 36 months.

1. Fill in the gaps in each table.

Let’s Learn

Time Duration in Days

Time duration in days shows the length of time between two specific dates or events, expressed in terms of days.

For example:

• If you go on a vacation from June 1st to June 10th, the time duration of your vacation is 10 days.

• The time duration between your birthday on April 1st and your friend’s birthday on May 15th is 44 days.

The time taken by an event or an activity to reach completion is also known as elapsed time. The elapsed time can be in number of days, hours, or minutes.

We can calculate the elapsed time using the below formulae:

Duration = (Finish Date – Start Date) + 1

Let us understand it with an example.

Example:

Aayu’s house is getting painted. The painting started on March 10th. The painter says that the job will be over by March 29th. How many days will it take for the painter to complete the job?

Solution:

Painting started on: 10th March

Painting will finish on: 29th March

Time taken for painting:

Duration = (Finish Date – Start Date) + 1 = (29 - 10) + 1 = 19 + 1 = 20 days

So, it will take 20 days for the painter to complete the job.

1. Rishu practised playing the piano for 2 hours each day from Monday to Friday. How many hours did he practice in total during that week?

2. If Rajat starts reading a book on March 7th and finishes it on April 20th, how many days did it take him to read the book?

3. Sarah planted flowers in her garden on May 3rd, and they fully bloomed on June 15th. Calculate the number of days it took for the flowers to bloom.

Let’s Sum Up

• Time is a way we measure and understand the duration of events and activities.

• Time can be grouped into various units. The unit conversion chart table is given below as follows:

• We can visualize the conversion of time from one unit to another as follows.

• Formula to calculate the elapsed time: Elapsed time = (End date – Start date) + 1

We have different units of time, like seconds, minutes, hours, and days, to help us understand time better. Name some things you do in a minute. Name some things you do in an hour.

Cross-Curricular Connections

Science:

Aayu observed a caterpillar turning into a butterfly. He noticed that it took 7 days for the caterpillar to form a pupa and 12 days for the butterfly to emerge. How many days, in total, did Aayu observe the entire transformation process from caterpillar to butterfly?

Social Studies:

Imagine you are going on a vacation from New Delhi to Goa. It takes your family 3 days to drive from New Delhi to Goa, and you plan to spend 7 days relaxing in the Sun and playing in the water in Goa. How many days, in total, will your vacation last from the time you leave New Delhi until you return?

21st Century Skills

Imagine you have a time machine that can take you to any period in the past or future. If you could visit any time in history or see what the future holds, where would you go and why? Think about how people lived, what they wore, and what they did during that time. How do you think your life would be different in that time period?

Extend Your Knowledge

A leap year is a special year that has an extra day. It happens every four years to keep our calendar in sync with the Earth’s journey around the Sun. In a leap year, February has 29 days instead of its usual 28. This makes our year match up with the time it takes Earth to revolve around the Sun.

LEAP YEAR

FEBRUARY

Tips for the Parent

• Help your child allocate specific time slots for homework. This teaches them to manage their time efficiently.

• Encourage the use of timers for various activities. Set a timer for completing homework, chores, or playtime. This helps in developing a sense of time urgency and teaches them to allocate time appropriately.

Money

My Study Plan

• Units of money

• Operation on money

• Unitary method

• Money in real life

Let’s Recall

Look at the picture. Answer the following questions.

1. What fruits do you see in the fruit shop?

2. What is your favourite fruit?

3. If you want to buy your favourite fruit from the shop, what should you give to the shopkeeper?

Thinking Zone

Think about the last time you went shopping. Can you remember what you bought and how much it cost? Write down the amount you spent in both rupees and paise.

Units of Money

We need money to buy things. Money is available in two forms: (i) Coins (ii) Notes

Coins Currently in Use in Our Country

Notes Currently in Use in Our Country

Each coin or note has a specific value. This value is expressed in rupees or paise. The symbol for a rupee is '₹' and for paisa, it is 'p'.

Example:

Expressing money in words and figures

An amount of money can be written in two ways — in figures and in words.

For example:

The price of the football that Aayu wanted to buy from a mall is 205 rupees 50 paise.

In Words: Two hundred five rupees and fifty paise.

In Figures: ₹205.50

₹205.50 rupees paise

205 rupees 50 paise

When expressing money in figures, we use a dot (.) to separate rupees and paise. The number to the left of the dot indicates the amount of rupees, while the number to the right of the dot shows the amount of paise.

When the amount of paise is less than 10, that is, only a single digit, we add a zero to the left of the digit in order to make it two digits after the dot.

Example:

This amount can be read as 70 rupees and 5 paise.

Conversion of money

We can convert rupees into paise and paise into rupees.

Converting rupees into paise

To convert rupees into paise, we multiply the rupees by 100.

₹1 = 100 paise

Example 1: Convert ₹205 into paise.

Solution:

We know that 1 rupee = 100 paise

To convert ₹205 into paise, we multiply 205 by 100.

Hence, ₹ 205 = 205 × 100 p = 20500 p.

Example 2: Convert 121 rupees and 25 paise into paise.

Solution:

To convert the money in rupees and paise into paise, remove ₹ and the dot (.) and write 'p' for paise at the end.

Therefore, 121 rupees and 25 paise = ₹121.25 = 12125 p

Remove it

₹ 121 . 25 = 12125 p

Remove it write it

Alternative Method:

₹121.25 = ₹121 + 25p = 121 x 100 p + 25p = 12100 p + 25p = 12125p

Converting paise (p) into rupees (₹)

To convert paise into rupees, divide the number of paise by 100.

1 paisa = 1 100 rupee

Example: Convert 6000 paise into rupees.

Solution:

6000 paise = 6000 100 = ₹60.00

Alternative method:

To convert paise into rupees, count the last two digits from the right as paise, put a dot (.) before them, and write '₹’ at the beginning. Remove the 'paise' or 'p' symbol.

So, 6000p = ₹60.00 ₹ 60 . 00

Number of rupees

Number of paise

Put dot (.) after 2 digits from the right

6000 paise = 60 rupees

Example: Convert 5045 paise into rupees.

Solution:

5045 paise or 5045 p = ₹50.45

Alternative Method:

5045 p = 5000p + 45p = + 45p

= ₹50.00 + 45p = ₹50.45

Fact Zone

Many coins have interesting designs on them. For example, the United States has a penny with Abraham Lincoln’s face, and Canada has a coin with a picture of a maple leaf.

Let's Practise - 1

1. Convert the given amount of money from rupees into paise.

a. ₹229.05 = _______________________

b. ₹344.45 = _______________________

2. Convert the given amount of money from paise into rupees.

a. 2700 paise =___________

b. 9710 paise =___________

Let’s Learn

Operations on Money

There are four main operations that can be performed with money: addition, subtraction, multiplication, and division. 5000

Addition of money

When adding money, we add rupees to rupees and paise to paise.

Example 1:

Add ₹523. 50 + ₹41.75

Solution: ₹ p 1

5 2 3 . 5 0 + 4 1 . 7 5

5 6 5 . 2 5

 ₹523. 50 + ₹41.75 = ₹565.25 in figures and five hundred sixty-five rupees and twenty-five paise in words.

Example 2:

Add 152 rupees 63 paise and 123 rupees and 24 paise.

Solution:

152 rupees 63 paise = ₹152.63 123 rupees 24 paise = ₹123.24

₹ p 1 5 2 . 6 3 + 1 2 3 . 2 4

2 7 5 . 8 7

 ₹152.63 + ₹123.24 = ₹275.87 in figures and two hundred seventy-five rupees and eighty-seven paise in words.

Subtraction of money

When subtracting money, we subtract rupees from rupees and paise from paise.

To subtract two amounts of money, we simply subtract the digits in each column like ordinary numbers.

Example:

₹718.25 − ₹414.75

 ₹718.25 − ₹414.75 = ₹303.50 in figures and three hundred rupees and fifty paise in words.

Multiplication of money

We can multiply money by a given number just like ordinary numbers after arranging the money in ‘₹’ and ‘p’ in columns.

Example:

Multiply ₹134.25 by 4

 ₹134.25 × 4 = ₹537.00 in figures and five hundred thirty-seven rupees in words.

Division of money

Money can be divided by a given number, similar to ordinary numbers. To divide money by a number, convert rupees into paise and divide by the given number. After division put a dot at the second place from right.

Example:

Let us divide ₹94.50 by 7

Solution:

First, convert ₹94.50 into paise.

Since 1 rupee equals 100 paise, multiply ₹94.50 by 100, which gives 9450 paise.

₹94.50 × 100 = 9450 paise

Next, divide the total paise (9450) by 7

This gives a quotient of 1350.

Finally, convert the quotient back into rupees. Since 1 rupee equals 100 paise, 1350 paise equals ₹13.50

Therefore, ₹94.50 divided by 7 equals ₹13.50

1. Add the following.

2. Subtract the following.

3. Multiply the following.

a. ₹950.50 × 4 b. ₹572.50 × 6

4. Divide the following.

a. ₹350.50 ÷ 5 b. ₹272.50 ÷ 4 4) 272.50

5) 350.50

Let’s Learn

Unitary Method

The unitary method is a helpful tool for solving money problems by finding the value of one unit and applying it to calculate the value of other units or quantities.

Let us understand this through an example.

Example 1: A shopkeeper sells five candies for ₹10. How much would three candies cost? = ₹10 = ?

Solution:

Step 1: First, we find out the cost of one candy.

To do this, we divide the total cost of candies (₹10) by the total number of candies (5).

₹ 10 ÷ 5 = ₹ 2

Therefore, each candy costs ₹2.

Step 2: Now that we know the cost of one candy, we can find out how much three candies would cost.

To do this, multiply the cost of one candy by the total number of candies.

₹2 (cost of 1 candy) × 3 (total number of candies) = ₹6

Hence, the cost of three candies is ₹6.

Example 2:

15 litres of petrol cost ₹150. How much would 10 litres of petrol cost?

Solution:

Step 1: First, we find out the cost of one litre of petrol.

To do this, we divide the total cost (₹150) by the total quantity of petrol (15 litres).

₹ 150 ÷ 15 = ₹10

Therefore, one litre of petrol costs ₹10.

Step 2: Now that we know the cost of one litre petrol, we can use it to find the cost of 10 litres of petrol.

We do this by multiplying the cost of one litre petrol by the total quantity of petrol.

₹10 (cost of 1 litre of petrol) × 10 (total quantity of petrol) = ₹100

Therefore, 10 litres of petrol would cost ₹100.

Let's Practise - 3

Solve the following.

1. If the cost of 5 packets of crayons is ₹48.25. What is the cost of 3 packets of crayons?

2. If a dozen bananas cost ₹72, how much would 10 such bananas cost?

Let’s Learn

Money in Real Life

Example 1: Pihu bought a t-shirt, a frock, and a pair of socks at ₹155.50, ₹270.50, and ₹30.99 respectively. How much money did she pay to the shopkeeper in total?

Solution:

Price of t-shirt = ₹155.50

Price of frock = ₹270.50

Price of socks = ₹30.99

Total price to pay to the shopkeeper = ₹155.50 + ₹270.50 + ₹30.99 = ₹456.99

Example 2: Aayu gave ₹500 to the shopkeeper against the purchase of a pen, a pencil and a register at ₹115.50, ₹230.25 and ₹100.25, respectively. How much money did the shopkeeper return to Aayu?

Solution:

Total price to pay to shopkeeper = ₹

Amount given to shopkeeper = ₹500 Amount spent = ₹446. 00

+ ₹

+ ₹

= ₹

Amount returned by the shopkeeper = ₹500 – ₹446 = ₹54

Example 3: A packet of pen costs ₹85.25. How much would be the cost of five such packets?

Solution:

Cost of one packet = ₹85.25

Cost of five packets = ₹(85.25 × 5)

Hence, the cost of five packets of pen is ₹426.25

Example 4: If the cost of five packets of chips is ₹88.25, what is the cost of one packet of chips?

Solution:

Cost of 5 packets of chips = ₹88.25

Change ₹88.25 into paise, 88.25 × 100 = 8825p

Divide 8825 by 5.

Convert the quotient 1765p back into rupees. 1765p = ₹17.65

Thus, the cost of a packet of chips is ₹17.65

Let's Practise - 4

Solve the following.

1. Mona bought a pack of chocolate bars for ₹203.50 and a pack of candies for ₹414.75. How much did she spend?

2. Seema bought items worth ₹418.75. She gave a ₹500 note to the shopkeeper. How much should the shopkeeper return?

3. The cost of a candle is ₹102.50. What will be the cost of five candles?

4. A shopkeeper sells three packets of maggi noodles for ₹109.75. How much would one packet of maggi cost?

Let’s Sum Up

• We use money in our daily life.

• Money is available in two forms: coins and notes.

• Each coin or note has a certain value. This value is expressed in rupees or paise.

• 1 rupee = 100 paise

• 1 paisa = 1 100 rupee

• Addition, subtraction, multiplication, and division of money can be done the same way as they are done for ordinary numbers.

• The unitary method is a helpful tool for solving money problems by finding the value of one unit and applying it to calculate the value of other units or quantities.

Life Skills

Imagine your friend borrowed some money from you and promised to give it back soon. But now they want to buy a video game instead of returning your money. What would you do? Why?

Cross-Curricular Connections

Science:

If a coin is made of metals like copper and nickel, what are some properties of these metals that make them suitable for coins?

Social Studies:

How has the design of the Indian rupee changed over time? What do the symbols and images on the rupee represent?

Arts:

Design your own currency. What symbols or images would you use and why?

21st Century Skills

Imagine you have ₹500 for a fun day out with your friends. You want to go to a theme park, have lunch, and buy a memorable gift. Make a plan for how you will spend your money on these activities. Explain why you decided to allocate your money that way.

Extend Your Knowledge

Plastic Cards:

Plastic money is what most of us use a lot these days. These are cards we use to buy things. They are small, rectangular, and fit easily in our wallets. The cards are of two types:

Debit Card: Money comes straight from your bank account.

Credit Card: You borrow money from the bank to buy things and pay back later.

Tips for the Parent

• Let your child participate in activities such as grocery shopping. Give them a budget and ask them to help you choose items without exceeding the budget.

• Encourage them to save a portion of their pocket money or any money they receive.

13 Symmetry and Patterns

My Study Plan

• Symmetry in shapes

• Patterns in numbers and shapes

• Coding and decoding

• Tessellations

Let’s Recall

Let us do an activity.

STEPS

1. Take a piece of paper and fold it from the centre.

IMAGE

2. Draw a half circle as shown.

3. Cut the shape and unfold the paper.

Similarly, make other shapes by doing this activity.

Thinking Zone

1. Here are some papers in different shapes. Select the correct figure where a line has divided these shapes into two equal halves and then colour it. a. b.

Let’s Learn

Symmetry

Symmetry is when something looks the same on both sides. For example, look at this picture of a butterfly.

Now, see the red line passing through the middle of the butterfly. The line which divides a figure into two equal halves is known as the line of symmetry.

There are some objects that have a line of symmetry. Shapes or figures that can be divided into two equal halves by a line of symmetry are called symmetrical shapes.

When we put a mirror along the line of symmetry, we see the same shape on the other side of the mirror.

Some shapes have more than one line of symmetry. For example, rectangles and ellipses have two lines of symmetry.

Squares have four lines of symmetry,

A circle has multiple of lines of symmetry.

Also, there are some shapes that do not have a line of symmetry. Shapes that cannot be divided into two equal halves by a line of symmetry are called asymmetrical shapes.

Nature is filled with symmetry. Human bodies are symmetrical. Similarly, a lot of animals also have symmetrical bodies.

Let’s Practise -1

1. Draw a line of symmetry to divide the given shapes into two equal halves:

2. Tick the asymmetrical shapes.

Let’s Learn

Patterns

A pattern consists of a design or sequence that repeats. This repetition can involve numbers or shapes in the sequence.

Let us look at the following pattern of shapes and try to fill the blanks.

When we look carefully, we can see a pattern of shapes here.

Look at the first four shapes.

1 2 3 4

The pattern of shapes is a circle, rectangle, star, and diamond. Now, if we see the next four shapes, the same shapes are repeated. Hence, we can say that this is a pattern.

Hence, we can fill in the blanks as follows.

Therefore, we can say that, the pattern here is a circle, rectangle, star, and diamond, which keeps on repeating.

We can see many patterns everywhere. It can be a sequence of colours, shapes, numbers, or even sounds.

When we clap our hands two times and then tap our feet two times, and do it again and again, it makes a sound pattern.

Let us try it.

Clap Clap Tap Tap Clap Clap Tap Tap Clap Clap Tap Tap

Patterns with Numbers

We can even form patterns with numbers.

Here is a pattern with numbers: 1 1 2 1 2 3 1 2 3 4 1 2 3 4 5

Guess the next line in the above number pattern.

Rangoli Patterns

We can make different beautiful rangoli patterns using many dot patterns, as shown below.

Thinking Zone

Think and draw the pattern that will come in the empty space.

Patterns Around Us

There are several patterns around us. Patterns make everything look beautiful. Here are some things we use daily that have patterns.

• Patterns on clothes:

• Patterns in gates and furniture:

We can also form a pattern by rotating or repeating a shape. For example, here we are rotating the shape of an arrow( ) clockwise.

1. Identify and extend the pattern in the following series.

a. 4, 7, 10, 13, 16, __, __, __

b. 1, 6, 11, 16, 21, __, __, __

c.

d. Clap, Clap, Clap, Click, Clap, Clap, Clap, Click, ____, ____, ____, ____

e. A, B, C, C, B, A, A, B, C, C, B, A, A, B, C, C, B, A, ___, ___, ___, ___, ___, ___

2. Try to make a rangoli pattern in the box given below and colour it:

Word Zone

Clockwise: Moving or turning in the same direction as the hands on a clock ____, ____, ____

3. Draw a pattern that you see on the gates, wallpapers, or furniture around you.

Let’s Learn

Coding

Coding is a way to send secret messages.

In coding, each letter and number is usually replaced with another letter, number, or symbol to create a code.

When we find the code, and we solve it to find the original message, then this is called decoding. Take a look at this.

Using the above coding, let us find out the code for the word ALIEN. The code for A is J, for L is Z, for I is M, for E is L and for N is O. So, we can say that the word ALIEN is JZMLO.

In the same way, if we need to decode JZMLO, we can use the above coding and say that JZMLO means ALIEN.

Let’s Practise - 3

1. Answer the following questions using the table given below:

a. Write the code words for the given words.

i. TIGER

ii. CIRCLE

iii. TODAY

iv. HEALTH

b. Decode the code words given below.

i. WMHJXXL

ii. SLAOLDAJT

iii. DIHJSBLHHT

iv. NMIKCLO

Let’s Learn

Tessellation

Tessellations are patterns made by putting shapes together without any gaps or overlaps. For example, tiles on walls or floors form tessellations.

Let us look at the following examples: Here, the first two shapes fit together with no gaps or overlaps, so they make a tessellation. But look at the third shape; even though the circles don’t overlap, there are spaces in between them. So, the third shape doesn’t make a tessellation.

Now, let’s see more examples of tessellations.

Let’s Practise - 4

1. Tick the patterns that are examples of tessellations.

Thinking Zone

Draw the following shapes and colour them. Also, create a pattern using the following shapes only once.

Let’s Sum Up

• Symmetry is when something looks the same on both sides.

• Line of symmetry is the line which divides a figure into two equal halves.

• Symmetrical shapes are the shapes or figures that can be divided into two equal halves by a line of symmetry.

• Some shapes have more than one line of symmetry like, squares, rectangles, and circles.

• Asymmetrical shapes are shapes that cannot be divided into two equal halves by a line of symmetry.

• A pattern consists of a design or sequence that repeats. Patterns can be found in numbers, shapes, rangoli, furniture, clothes, music, etc.

• In coding, each letter and number is usually replaced with another letter, number, or symbol to create a code.

• When we find the code and solve it to find the original message, then this is called decoding.

• Tessellations are patterns made by putting shapes together without any gaps or overlaps. For example, tiles on walls or floors form tessellations.

Life Skills

Your parents are going to change the flooring of your room. There are 2 colours of tiles that they have bought- yellow and blue. Consider this as the floor of your room with square tiles on it. Now, create a pattern by colouring the square tiles yellow and blue.

Arts:

In music, we use codes and patterns to play different instruments. Complete the given musical pattern.

Sa, Sa, Re, Re, Ga, Ma, Sa, Sa, Re, Re, Ga, Ma, , , , , ,

Cross-Curricular Connections

21st Century Skills

Most of the old monuments, like The Taj Mahal, are perfectly symmetrical. Why are they made symmetrical? Discuss.

Extend Your Knowledge

Palindromes are special words or numbers that read the same both ways. For example, the word EYE. It reads the same forward and backwards.

EYE

EYE

Even some numbers can be palindromes like 464 or 616.

Tips for the Parent

Assist your child in creating a symmetrical drawing, and introduce them to some natural shapes that show symmetry.

Data Handling

My Study Plan

• Pictographs

• Bar graphs

Let’s Recall

In a school, the number of students, class-wise, is given below:

Complete the pictograph to represent the above data.

Class I

No. of Students

Class

Class II

Class

Class

Class

Scale: One represents 5 students

Scale: One represents

Answer the following questions:

a. Which class has the maximum number of students?

b. Which class has the minimum number of students?

c. If there are 23 boys in Class IV, then how many girls are there in the Class?

d. Which two classes have the same number of students?

e. What is the total number of students in classes I to V?

Write the names of two graphs used for showing data, apart from a pictograph.

Let’s Learn

Pictograph

A pictograph is a way to show numbers or information using pictures, icons, or symbols. It is a simple way to represent data. In a pictograph, the symbols are placed in a line to show the data.

Some points to be remembered while making a pictograph are:

• There is a key to show how many items each symbol represents.

• All symbols should be of the same size.

• Sometimes, a part of a symbol is used to represent a number or data.

For example,

If the symbol represents 4

Then the symbol represents 3

And the symbol represents 2

And the symbol represents 1

The steps to be followed while reading a pictograph are:

1. Start with the title present at the top because the title explains the graph’s purpose.

2. Look at the labels to understand what information is being compared.

3. Finally, read the key to find out how much each symbol represents.

The pictograph below shows the profit earned by a company each year from 2019 to 2022.

1. What was the profit earned by the company in the year 2020?

Answer: The profit earned by the company in the year 2020

= 8 pictures

= 8 × ₹1,00,000

= ₹8,00,000

2. What was the profit earned by the company in the year 2019?

Answer: The profit earned by the company in the year 2019

= 6 pictures + 1 2 picture

= 6 × ₹1,00,000 + 1 2 × ₹1,00,000

= ₹6,00,000 + ₹50,000

= ₹6,50,000

3. In which year did the company earn the least profit?

Answer: In 2021, the profit earned was the least.

Practise - 1

1. The pictograph shows the books collected for a book fair.

Books Collected at Sunnydale High School

Number of Books

Class 3

Class 4 Class 5 Key: = 20 books

Use the graph to answer the following questions:

a. Which class has collected the least number of books? __________

b. How many books are collected by class 4 students? __________

c. What is the total number of books collected by class 3? __________

d. How many more books are collected by class 3 students than class 5 students? __________

2. The following pictograph shows the sales of bouquets on five different days in a week.

Key: = 2 bouquets

On the basis of this pictograph, answer the following questions:

a. On which day of the week were the maximum number of bouquets sold?

b. On which day were the sales the least?

c. How many bouquets were sold on Monday and Tuesday together?

d. How many bouquets were sold in total?

e. On which two days were the bouquets sold equally?

Let’s Learn

Bar Graph

A bar graph is a visual way to show numerical data using rectangular bars.

The bars in the graph are either vertical (standing up) or horizontal (lying flat from left to right). The most common type is the vertical bar graph.

In both vertical and horizontal graphs, the height or length of the bars represents the corresponding value of the numerical data on a suitable scale.

Reading bar graphs

1. Start with reading the title of the bar graph at the top. It gives information about what the graph is measuring and comparing.

2. Read the labels below the X and Y axes of the bar graph. They represent the categories that are compared in the graph.

3. Read the scale of the graph which consists of numbers along the axes. It shows the exact value represented in the data.

Word Zone

Corresponding: Matching or connected with something

Here is a bar graph that shows a survey to determine people’s favourite movie types:

Favourite Type of Movie

1. How many people like action movies, as shown in the graph?

Answer: 5 people like action movies, as shown in the graph.

2. What is the least favourite movie type based on the bar graph?

Answer: Drama is the least favourite movie type based on the bar graph.

3. How many people in the survey like both comedy and SciFi movies?

Answer: Number of people who like comedy = 4

Number of people who like SciFi = 4

The total number of people who like both Comedy and SciFi = 4 + 4 = 8

Drawing bar graphs

A bar graph consists of two lines: horizontal and vertical. The horizontal line is called the X-axis, and the vertical line is called the Y-axis. If the bars are drawn vertically on the X-axis, then the height of the bars are shown along the Y-axis. If the bars are drawn horizontally on the Y-axis, then the length of the bars are shown along the X-axis.

It is also important to label the axes.

Some points to be remembered while drawing a bar graph are:

• The bars should have the same width.

• The height (or length) of each bar shows the count of the given data.

• The space between each bar should be the same.

Let’s take a look at the following example to understand how to draw a bar graph:

Example:

David went to the market and bought 5 apples, 3 mangoes, 2 watermelons, 3 strawberries, and 6 oranges. He now wants to create a bar graph to visually represent the data and see which type of fruit he bought the most.

Solution:

While drawing a bar graph, remember to mention these three things: labels on axes, title, and name of the axes.

Most bought fruits Types of

The above graph shows that David bought more oranges than any other fruit.

1. The following bar graph shows the number of children having a specific choice of fruits. Observe the graph and answer the following questions:

Number of children

a. How many students like apples? _____

b. Which is the most popular fruit? __________

c. How many more students like mangoes than bananas? _____

d. The number of students liking guava is _____ less than the number of students liking pineapple.

2. In the following table, the favourite colours of 20 students in a class are given.

on the bar graph

per the given data.

colour of 20 students

Apple Mango Banana Guava Pineapple Fruits

Red Blue Green Orange Violet

Let’s Sum Up

• A pictograph is a way to show numbers or information using pictures, icons, or symbols.

• Sometimes, a part of a symbol is used to represent a number or data. For example, if the whole symbol represents 4, then half of the symbol represents 2.

• A bar graph is a visual way to show numerical data using rectangular bars.

• The two ways of representing bar graphs are vertical and horizontal bar graphs. If the bars in the graph are vertical (standing up), then it is a vertical bar graph. If the bars in the graph are horizontal (lying flat), then it is a horizontal bar graph.

• In the bar graph, the horizontal line is called the X-axis, and the vertical line is called the Y-axis.

• While drawing a bar graph, mention these three things: labels on the axes, names of the axes, and the title.

Life Skills

Navya recorded the amount of time she spent on six activities over a day.

Draw a bar graph for Navya’s daily activities.

Cross-Curricular Connections

Science:

Here is a list that shows how fast the hearts of different animals beat. It tells us the number of beats per minute.

Represent the above data using a bar graph.

21st Century Skills

Team up with your friend and count how many vehicles pass by you in 10 minutes. Then, use the information to finish the pictograph given below.

Type of Vehicle

Bicycle

Bike Auto Car

Others

Number of Vehicle

Key: �� = 2

Extend Your Knowledge

Pie Chart:

A pie chart is like a special picture that shows information in a circle. Each slice of the pie tells us how much of something there is. To make a pie chart, we need a list of numbers and categories. Let’s check out an example with a pie chart that shows how much money Lilly spent at the funfair.

Tip for the Parent

• Encourage your child to create simple bar graphs using everyday activities. For instance, they can track the number of sunny and rainy days in a week and represent it in a bar graph.

ANSWER KEY

CH-01: Large Numbers

Let’s Practise – 1

1. (a) 34,783 (b) 1,28,792

2. (a) Twenty-one thousand five hundred thirtyfour

(b) Nine lakh eighty-seven thousand six hundred fifty-four

(c) Seventeen thousand eight hundred ninetyfour

(d) One lakh twenty-three thousand four hundred fifty-six

Let’s Practise – 2

1. (a) 2,353 (b) 46,029 (c) 9,99,998 (d) 2,43,539

2. (a) 4,209 (b) 9,792 (c) 8,77,667 (d) 7,56,941

Let’s Practise – 3

1. (a) < (b) < (c) > (d) > (e) < (f) >

Let’s Practise – 4

1.

Digits Smallest Number Greatest Number

2, 8, 0, 5,3 20,358 85,320

2, 4, 8, 3,7,3 2,33,478 8,74,332

4, 7, 3, 8,9,1 1,34,789 9,87,431

8, 6, 9, 0,9 60,899 99,860

Let’s Practise – 5

1. (a) 98,120 (b) 21,260 (c) 45,340 (d) 12,820

2. (a) 38,000 (b) 23,500 (c) 54,800 (d) 49,800

3. (a) 1,23,000 (b) 2,32,000 (c) 3,44,000 (d) 44,000

Let’s Practise – 6

1. (a) One hundred sixty-eight thousand seven hundred twenty-nine

(b) Eight hundred forty-five thousand three hundred sixty-seven

(c) Five million seven hundred sixty-five thousand three hundred seventy-nine

(d) Eight million three hundred fifty-nine thousand four hundred ninety-eight

(e) Four million eight hundred five thousand twenty-nine

(f) Nine hundred twenty-seven thousand five hundred forty

2.

TM M HT TTh

Let’s Practise – 7

1. (a) 1212→ MCCXII

(b) 2148 → MMCXLVIII

(c) 1398 → MCCCXCVIII

(d) 1879 → MDCCCLXXIX

CH-02: Addition

Let’s Practise – 1

1. (a) 88,679 (b) 89,186 (c) 97,490 (d) 97,905 (e) 87,895 (f) 47,797

Let’s Practise – 2

1. (a) 0 (b) 2,534 (c) 32,354 (d) 9,473

2. (a) 11958 (b) 14032

Let’s Practise – 3

1. (a) 8,030 (b) 8,820 (c) 37,300 (d) 44,600

Let’s Practise – 4

1. 24,165

2. ₹ 66,915

3. 50,465

4. ₹ 1,16,914

ANSWER KEY

CH-03: Subtraction

Let’s Practise – 1

1. (a) 233 (b) 36,361 (c) 19,352 (d) 26,951

Let’s Practise – 2

1. (a) 35,163 (b) 29,373 (c) 13,498 (d) 0

Let’s Practise – 3

1. (a) 250 (b) 750 (c) 900 (d) 16,500

Let’s Practise – 4

1. (a) 4,293 (b) ₹ 4,774 (c) 2,083 (d) 5,215

CH-04: Multiplication

Let’s Practise – 1

1. (a) 34,05,848 (b) 35,60,794 (c) 3,27,29,754

Let’s Practise – 2

1. (a) 119 (b) 0 (c) 43 (d) 0 (e) 1 (f) 2,140 (g) 100 (h) 5,62,000

2. (a) 3 × 4, 12, 15 (b) 5 × 6, 5 × 4

Let’s Practise – 3

1. (a) 7,200 (b) 3,060 (c) 1,125 (d) 3,320

Let’s Practise – 4

1. (a) 1,800 (b) 1,500 (c) 90,000

Let’s Practise – 5

1. ₹ 44,700

2. ₹ 1,68,025

3. ₹ 52,500

CH-05: Division

Let’s Practise – 1

1. (a) 72 ÷ 9 = 8 (b) 64 ÷ 8 = 8 (c) 48 ÷ 6 = 8

(d) 36 ÷ 4 = 9 (e) 35 ÷ 5 = 7 (f) 63 ÷ 7 = 9

(g) 81 ÷ 9 = 9 (h) 72 ÷ 6 = 12

Let’s Practise – 2

1. (a) Quotient =146 and Remainder = 0

(b) Quotient =1358 and Remainder =2

(c) Quotient =246 and Remainder = 0

Let’s Practise – 3

1. (a) Quotient =128 and Remainder = 4

(b) Quotient =34 and Remainder = 24

(c) Quotient =2 and Remainder = 468

(d) Quotient = 453 and Remainder = 1

2. (a) 1 (b) 0 (c) 1 (d) 4514

Let’s Practise – 4

1. (a) Pencils in each box = 145 and left over pencils = 2

(b) Number of ice-cream cone with each child = 671 and there will be 9 ice-cream cones leftover.

(c) Number of slices in each plate = 70 and there are 5 pizza slices left over.

2. (a) Estimated = 30, Actual = 32

(b) Estimated = 20, Actual = Quotient -21 and Remainder - 11

ANSWER KEY

CH-06: Factors and Multiples

Let’s Practise – 1

1. (a) 3, 6, 9, 12 and 15

(b) 7, 14, 21, 28, 35, 42 and 49

(c) Multiples of 4= 4, 8, 12, 16, 20

Multiples of 5= 5, 10, 15, 20, 25

Common multiple= 20

(d) Multiples of 7 = 7, 14, 21, 28, 35, 42, 49

Multiples of 9 = 9, 18, 27, 36, 45, 54, 63

Common multiple = No

(e) Even = 18, 36, 54

Odd = 9, 27, 45

Let’s Practise – 2

1. (a) 1 (b) 25

2. False

3. (a) 1, 3, 7, and 21

(b) 1, 2, 4, 8, and 16

(c) 1, 2, 3, 6, 7, 14, 21, and 42

(d) 1, 2, 3, 4, 6, 9, 12, 18, and 36

4. (a) 1, 2, 7, and 14

(b) 1, 2, 4, 7, 14, and 28

(c) 1, 2, 3, 5, 6, 10, 15, and 30

(d) 1 and 17

5. (a) 1, 2, 3, 4, 6, and 12

(b) 1, 5, 7, and 35

Let’s Practise – 3

1.

(a) Composite

(b) Prime

(c) Composite

(d) Composite

2.

(a) 2×2×2×3

(b) 2×2×2×3×3

(c) 2×3×3

(d) 2×2×2×2×7

3. 3 and 5; 11 and 13; Twin primes

Let’s Practise – 4

1. (a) 18 (b) 36 (c) 45 (d) 16

2. (a) 9 (b) 6 (c ) 120 (d ) 27

3. a, c and d

Let’s Practise – 5

1. (a) 24 (b) 60 (c ) 36 (d) 30

2. (a) 144 (b) 1470 (c ) 480 (d) 540

3. (a) 336 (b) 168 (c ) 72 (d) 576

Let’s Practise – 6

1. HCF = 8 ; LCM = 48

2. 24

3. 61

Let’s Practise – 7

1. (a) No (b) Yes (c ) No (d) Yes (e) No (f) No

CH-07: Fractions

Let’s Practise – 1

1. a. 2/3 b. 4/6 c. 6/9 d. 2/8 e. 8/10

Let’s Practise – 2

1. (a) 3¼

(b) 2 ½

(c) 2 1/3

ANSWER KEY

(d) 4 ¼

2. (a) 3 ¼ (b) 4 6/7 (c) 4 2/3 (d) 2 1/5

3. (a) 17/7 (b) 22/4 (c ) 19/7 (d) 33/8

Let’s Practise – 3

1.

(a) 2/18, 3/27

(b) 6/14, 9/21

(c) 4/10, 6/15

(d) 10/12, 15/18

2.

(a) 2/3 (b) 5/6 (c) 7/4 (d) 3/4

Let’s Practise – 4

1. (a) 10/15 and 12/15

(b) 5/8 and 2/8

(c) 8/14 and 11/14

(d) 4/10 and 6/10

(e) 3/12 and 4/12

(f) 8/12 and 9/12

Let’s Practise – 5

1. (a) 7/5 (b) 1 (c) 13/9 (d) 10/15

2. (a) ½ (b) 3/5 (c ) 4/11 (d) 3/8

Let’s Practise – 6

1. 15 km

2. 5/6 kg

CH-08: Geometry

Let’s Practise – 1

1. (a) Point (b) Line (c) Line segment (d) Ray

2. (d) Ray

Let’s Practise – 2

1. (a) 7cm (b) 11 cm

Let’s Practise – 3

2. (d) Quadrilateral

3. Hexagon

Let’s Practise – 4

2. (b) Diameter

Let’s Practise – 5

1. (a) Cuboid (b) Cone (c) Cylinder (d) Triangle (e) Rectangle

2. (a) three (b) six (c) net (d) cuboid

CH-09: Perimeter and Area

Let’s Practise – 1

1. 16 cm

2. 22 cm

Let’s Practise – 2

1. 224 sq. cm

2. 64 sq. cm

3. 12 m

CH-10: Measurement

Let’s Practise – 1

1. (a) Gram (b) Distance (c) Feet

2. Weight – Gram

Volume – Litre

Length - Metre

Let’s Practise – 2

1. (a) 1000 m (b) 1000 ml (c) 1 kg

(d) 1000 mm

2. (a) 4000 g (b) 506 cm (c) 1.458 km

(d) 3.098 l

Let’s Practise – 3

1. (a) 27 km 197 m (b) 3 kg 524 g

2. (a) 25 kg 800 g (b) 52 km 400 m

CH-11: Time

Let’s Practise – 1

1. (a) 300 minutes

(b) 270 minutes

(c) 120 minutes

2. (a) 1 day- 24 hours

(b) 1 year – 365 days

(c) 1 week – 7 days

(d) 180 min – 3 hours

Let’s Practise – 2

1. (a) 07:15 (b) 21:30

2. (a) 2:30 A.M. (b) 1:15 P.M.

Let’s Practise – 3

1. a. 5 hours 45 minutes

b. 7 hours 15 minutes

c. 1 hours 45 minutes

d. 2 hours 30 minutes

Let’s Practise – 4

1. 1 hour 35 minutes

2. 30 minutes

Let’s Practise – 5

1. a. 3 weeks – 21 days

6 weeks – 42 days

10 weeks – 70 days 11 weeks – 77 days

ANSWER KEY

1. b. 3 years – 36 months 3 years – 36 months 4.5 years – 54 months 5.5 years – 66 months

Let’s Practise – 6 1. 10 hours 2. 45 days 3. 44 days

CH-12: Money

Let’s Practise – 1

1. a. 22905 paise

b. 34445 paise

2. a. ₹ 27.00 b. ₹ 97.10

Let’s Practise – 2 1. a. ₹ 582.25 b. ₹ 663.24 2. a. ₹ 530.00 b. ₹ 419.25

3. a. ₹ 3802.00 b. ₹3435.00 4. a. ₹ 70.10 b. ₹ 68.125

Let’s Practise – 3 1. ₹ 28.95 2. ₹ 60

Let’s Practise – 4 1. ₹ 618.25 2. ₹ 81.25 3. ₹ 512.50 4. ₹ 36.58

ANSWER KEY

CH-13: Symmetry and Patterns

Let’s Practise – 2

1. a. 19, 22, 25

b. 26, 31, 36

c.

d. Clap, Clap, Clap, Click

e. A, B, C, C, B, A

Let’s Practise – 3

1.

(a) i. IMWLH

ii. KMHKZL

iii. IUAJT

iv. CLJZIC

(b) i. GIRAFFE

ii. WEDNESDAY

iii STRAWBERRY

iv. KITCHEN

CH-14: Data Handling

Let’s Practise – 1

1. (a) Class 4

(b) 40 books

(c) 80 books

(d) 20 books

2. (a) Monday

(b) Wednesday

(c) 21 bouquets

(d) 43 bouquets

(e) Tuesday and Friday

Let’s Practise – 2

1. (a) 4 (b) Pineapple (c) 1 (d) 1