Mathematics: Class 4 — Chapter 5: Sharing and Measuring

5.1 Parts and Wholes

Fractions represent parts of a whole. When an object is divided into equal parts, each part is a fraction of the whole.

Half: When an object is divided into two equal parts, each part is called a half. We write half as $\frac{1}{2}$.
Quarter: When an object is divided into four equal parts, each part is called a quarter. We write quarter as $\frac{1}{4}$.

Ikra and Samina share a single drawing sheet:

Ikra proposes dividing it in half ($\frac{1}{2}$) or into two quarters ($2 \times \frac{1}{4}$).
Two quarters ($\frac{2}{4}$) is equal to one half ($\frac{1}{2}$).

5.2 Many Ways to Make Halves and Quarters

There are multiple ways to divide shapes (like rectangles, circles, and squares) into equal halves or quarters by drawing lines or folding.

Matching fractional parts (like half of a triangle, circle, or rectangle) to their corresponding complete shapes.

5.3 Ding Dong Bell!! (Sharing and Unit Fractions)

As more guests arrive, a whole dhokla is shared among more people, making each individual share smaller:

2 people: Each gets half ($\frac{1}{2}$). Two halves make a whole: $\frac{1}{2} + \frac{1}{2} = 1$.
3 people: Each gets one-third ($\frac{1}{3}$). Three thirds make a whole: $\frac{1}{3} + \frac{1}{3} + \frac{1}{3} = 1$.
4 people: Each gets one-fourth or a quarter ($\frac{1}{4}$). Four quarters make a whole: $\frac{1}{4} + \frac{1}{4} + \frac{1}{4} + \frac{1}{4} = 1$.
5 people: Each gets one-fifth ($\frac{1}{5}$). Five fifths make a whole: $5 \times \frac{1}{5} = 1$.

When a whole is equally divided into a larger number of parts, each part gets smaller:

$$\frac{1}{2} > \frac{1}{3} > \frac{1}{4} > \frac{1}{5} > \frac{1}{6} > \dots$$

5.4 My Flower Garden (Fractions of a Whole)

Representing parts of a garden using fifths ($\frac{1}{5}$):

Mogra in $\frac{1}{5}$ (one-fifth) part of the garden.
Rose in $\frac{2}{5}$ (two-fifths) part of the garden ($\frac{1}{5} + \frac{1}{5}$).
Rose in $\frac{3}{5}$ (three-fifths) part of the garden ($\frac{1}{5} + \frac{1}{5} + \frac{1}{5}$).
Rose in $\frac{4}{5}$ (four-fifths) part of the garden.
Rose in $\frac{5}{5}$ (five-fifths) or the complete garden.

5.5 Dream Dosa Designer!! (Fractions of a Circle)

Designing dosas with different toppings represented as fractions of a circle:

A dosa divided into 4 equal parts: each topping can be $\frac{1}{4}$ (e.g., Classic potato, Spicy onion, Chilly paneer, Tangy tomato).
A dosa divided into 8 equal parts: toppings can be represented as eighths (e.g., $\frac{3}{8}$ Classic potato, $\frac{1}{8}$ Chilly paneer, $\frac{4}{8}$ Tangy tomato).

5.6 Fractions of a Collection

Fractions can also represent parts of a group or collection of objects:

Diyas: Finding $\frac{1}{4}$ of 8 diyas. Dividing 8 into 4 equal parts gives 2 diyas in each part. So, $\frac{1}{4} \text{ of } 8 = 2$.
Cookies: Sharing 12 cookies among different numbers of children:
- Among 3 children: Each gets $\frac{1}{3}$ of 12 = 4 cookies.
- Among 6 children: Each gets $\frac{1}{6}$ of 12 = 2 cookies.
- Among 2 children: Each gets $\frac{1}{2}$ of 12 = 6 cookies.
- Among 4 children: Each gets $\frac{1}{4}$ of 12 = 3 cookies.
Hairbands: Finding $\frac{1}{3}$ of 9 friends = 3 friends.
Pots: Finding $\frac{1}{5}$ of 15 pots = 3 pots.

5.7 Equivalent Fractions

Fractions that represent the same value or part of a whole are called equivalent fractions.

Folding a paper into 3 equal parts and coloring 1 part represents $\frac{1}{3}$.
Folding it in half again divides the paper into 6 equal parts, and the colored part becomes $\frac{2}{6}$.
Further folding shows: $$\frac{1}{3} = \frac{2}{6} = \frac{3}{9} = \frac{4}{12}$$
Similarly, starting with halves: $$\frac{1}{2} = \frac{2}{4} = \frac{3}{6} = \frac{4}{8}$$

Using a fraction chart to compare and find equivalences:

Two $\frac{1}{4}$ pieces are equal to one $\frac{1}{2}$ piece: $$\frac{1}{4} + \frac{1}{4} = \frac{2}{4} = \frac{1}{2}$$
$\frac{2}{3}$ is greater than $\frac{1}{2}$.
Ten pieces of $\frac{1}{10}$ make a complete whole: $$10 \times \frac{1}{10} = 1$$
Three pieces of $\frac{1}{6}$ ($\frac{3}{6} = \frac{1}{2}$) are equal to four pieces of $\frac{1}{8}$ ($\frac{4}{8} = \frac{1}{2}$).