Mathematics: Class 2 — Chapter 8: Grouping and Sharing

8.1 Introduction to Grouping and Sharing

This chapter introduces the fundamental concepts of multiplication (as repeated addition) and division (as equal sharing and equal grouping) through real-life scenarios like counting vehicle wheels, sharing sweets, and arranging objects.

8.2 Repeated Addition and Multiplication

Multiplication is a faster way of adding the same number repeatedly. The symbol used for multiplication is $\times$.

Examples of Repeated Addition

Bicycles: There are $4$ bicycles. Each bicycle has $2$ wheels.
Total wheels = $2 + 2 + 2 + 2 = 8$ wheels.
This can be written as $4$ times $2$ is $8$, or $4 \times 2 = 8$.
Autorickshaws: There are $2$ autorickshaws. Each autorickshaw has $3$ wheels.
Total wheels = $3 + 3 = 6$ wheels.
This can be written as $2$ times $3$ is $6$, or $2 \times 3 = 6$.
Cars: There are $7$ cars. Each car has $4$ wheels.
Total wheels = $4 + 4 + 4 + 4 + 4 + 4 + 4 = 28$ wheels.
This can be written as $7$ times $4$ is $28$, or $7 \times 4 = 28$.
Butterflies: There are $3$ butterflies. Each butterfly has $2$ wings.
Total wings = $2 + 2 + 2 = 6$ wings.
This can be written as $3 \times 2 = 6$.
Octopuses: There are $2$ octopuses. Each octopus has $8$ legs.
Total legs = $8 + 8 = 16$ legs.
This can be written as $2 \times 8 = 16$.
Soldiers: There are $4$ lines of soldiers. Each line has $10$ soldiers.
Total soldiers = $10 + 10 + 10 + 10 = 40$ soldiers.
This can be written as $4 \times 10 = 40$.

8.3 Order in Multiplication (Commutative Property)

Changing the order of numbers in multiplication does not change the total product. This is demonstrated using groups of objects:

Bouquets:
$4$ groups of $3$ flowers = $4 \times 3 = 12$ flowers.
$3$ groups of $4$ flowers = $3 \times 4 = 12$ flowers.
Gulab Jamuns:
$4$ groups of $5$ gulab jamuns = $4 \times 5 = 20$ gulab jamuns.
$5$ groups of $4$ gulab jamuns = $5 \times 4 = 20$ gulab jamuns.
Flowers on Curtains:
$6$ groups of $4$ flowers = $6 \times 4 = 24$ flowers.
$4$ groups of $6$ flowers = $4 \times 6 = 24$ flowers.

8.4 Building Multiplication Tables

Multiplication tables can be constructed by combining smaller tables. This helps in understanding the relationships between different numbers.

Table of 4 from Table of 2

By adding the values of the Table of $2$ to itself, we get the Table of $4$:

$$\text{Table of 2: } 2, 4, 6, 8, 10, 12, 14, 16, 18, 20$$

$$\text{Table of 4: } 4, 8, 12, 16, 20, 24, 28, 32, 36, 40$$

Table of 6 from Table of 3

By adding the values of the Table of $3$ to itself, we get the Table of $6$:

$$\text{Table of 3: } 3, 6, 9, 12, 15, 18, 21, 24, 27, 30$$

$$\text{Table of 6: } 6, 12, 18, 24, 30, 36, 42, 48, 54, 60$$

Table of 7 from Tables of 3 and 4

By adding the corresponding values of the Table of $3$ and Table of $4$, we get the Table of $7$:

$$\text{Table of 3: } 3, 6, 9, 12, 15, 18, 21, 24, 27, 30$$

$$\text{Table of 4: } 4, 8, 12, 16, 20, 24, 28, 32, 36, 40$$

$$\text{Table of 7: } 7, 14, 21, 28, 35, 42, 49, 56, 63, 70$$

8.5 Equal Sharing (Introduction to Division)

Equal sharing means distributing a total number of items equally among a given number of groups or individuals.

Example: Sharing Gulab Jamuns

There are $8$ gulab jamuns to be shared equally between $2$ children:

First, share $1$ by $1$: Each gets $1$ ($2$ shared, $6$ left).
Take $1$ more: Each gets $2$ ($4$ shared, $4$ left).
Take $1$ more: Each gets $3$ ($6$ shared, $2$ left).
Take $1$ more: Each gets $4$ ($8$ shared, $0$ left).
Result: Each child gets exactly $4$ gulab jamuns.

8.6 Equal Grouping

Equal grouping involves finding how many groups of a specific size can be made from a total number of items.

Examples of Equal Grouping

Beads and Bracelets: Garima has $20$ star-shaped beads. She uses $5$ beads to make each bracelet.
Total bracelets made = $20 \div 5 = 4$ bracelets.
Bead Strings: How many strings of $7$ beads can be made from $21$ beads?
Number of strings = $21 \div 7 = 3$ strings.
Flower Bracelets: How many bracelets of $9$ flowers can be made from $54$ flowers?
Number of bracelets = $54 \div 9 = 6$ bracelets.
Roses in Vases: How many vases are needed to place $25$ roses if each vase holds $5$ roses?
Number of vases = $25 \div 5 = 5$ vases.
Candles in Boxes: Put $27$ candles equally into $3$ boxes.
Candles per box = $27 \div 3 = 9$ candles.
Buttons on Shirts: A tailor puts $6$ buttons on one shirt. How many shirts can be made with $30$ buttons?
Number of shirts = $30 \div 6 = 5$ shirts.
Sharing Bananas: Share $24$ bananas equally among $3$ monkeys.
Bananas per monkey = $24 \div 3 = 8$ bananas.