Mathematics: Class 3 — Chapter 11: Filling and Lifting

11.1 Capacity: Comparing and Measuring Volume

Capacity refers to the amount of liquid a container can hold. We can compare and measure capacity using both non-standard and standard units.

When comparing containers of different sizes:

A larger container has a greater capacity and holds more liquid.
A smaller container has a lesser capacity and holds less liquid.
To accurately compare different containers, we can pour their contents into identical, same-sized glasses and observe the liquid levels.

We can use smaller vessels (like a ladle, bowl, or glass) to measure the capacity of larger vessels (like a jug or bottle):

Example: Counting how many small bowls of water are needed to completely fill a glass or a bottle.
Utensils with less capacity can be used to fill a utensil of larger capacity, and vice versa. In both cases, the total quantity of liquid remains the same.

The standard unit for measuring capacity is the Litre (denoted as $L$).

A standard measuring bottle or cup is used to measure exact quantities of liquids like milk, water, or fuel.
Sub-units of Litre:
- Half Litre: Two half-litre measurements equal $1 \\text{ Litre}$ ($2 \\times \\text{half-litre} = 1 \\text{ Litre}$).
- Quarter Litre: Four quarter-litre measurements equal $1 \\text{ Litre}$ ($4 \\times \\text{quarter-litre} = 1 \\text{ Litre}$).

11.2 Weight: Comparing and Measuring Mass

Weight is a measure of how heavy an object is. We can compare weights directly or measure them using standard and non-standard units.

We can estimate weight by holding different objects in our hands:

The hand holding the heavier object tends to go down due to gravity.
We can compare common objects to determine which is heavier or lighter (e.g., a watermelon is heavier than an apple; a school bag is heavier than a lunch box).

A Pan-balance is a tool used to compare the weights of two objects or to measure weight using non-standard units:

We can balance an object (like a pencil or a ping pong ball) against a certain number of coins or erasers.
We can also use sand-filled matchboxes as uniform weights to measure and compare slightly heavier objects.

The standard unit for measuring weight is the Kilogram (denoted as $kg$).

Common household items, like a packet of salt or rice, often weigh exactly $1 \\text{ kg}$.
Objects can be classified as weighing more than $1 \\text{ kg}$ (e.g., a table, a filled water bottle) or less than $1 \\text{ kg}$ (e.g., a pencil, a balloon).
Sub-units of Kilogram:
- Half Kilogram ($1/2 \\text{ kg}$): Two half-kilogram packets equal $1 \\text{ kg}$.
- Quarter Kilogram ($1/4 \\text{ kg}$): Four quarter-kilogram packets equal $1 \\text{ kg}$.

Using a pan-balance without weights to solve weight puzzles:

Problem: Out of 3 similar-looking balls, one is heavier and two are equal in weight. How can you find the heavy ball using the pan-balance only once?
Solution: Place one ball on each pan of the balance. If the pans balance, the third (unweighed) ball is the heavier one. If they do not balance, the pan that goes down holds the heavier ball.