NCERT Class 7 Ganita Prakash 1 Chapter 03 A Peek Beyond the Point PDF Download

Chapter 3: A Peek Beyond the Point

3.1 The Need for Smaller Units

Sonu's mother was fixing a toy. She was trying to join two pieces with the help of a screw. Sonu was watching his mother with great curiosity. His mother was unable to join the pieces. Sonu asked why. His mother said that the screw was not of the right size.

She brought another screw from the box and was able to fix the toy. The two screws looked the same to Sonu. But when he observed them closely, he saw they were of slightly different lengths.

Sonu was fascinated by how such a small difference in lengths could matter so much. He was curious to know the difference in lengths. He was also curious to know how little the difference was because the screws looked nearly the same.

In the following figure, screws are placed above a scale. Measure them and write their length in the space provided.

[Figure: Screws placed above scales for measurement, See in your textbook]

Which scale helped you measure the length of the screws accurately? Why?

What is the meaning of \( 2\frac{7}{10} \) cm (the length of the first screw)?

As seen on the ruler, the unit length between two consecutive numbers is divided into 10 equal parts. To get the length \( 2\frac{7}{10} \) cm, we go from 0 to 2 and then take seven parts of \( \frac{1}{10} \). The length of the screw is 2 cm and \( \frac{7}{10} \) cm. Similarly, we can make sense of the length \( 3\frac{2}{10} \) cm.

We read \( 2\frac{7}{10} \) cm as two and seven-tenth centimeters, and \( 3\frac{2}{10} \) cm as three and two-tenth centimeters.

Can you explain why the unit was divided into smaller parts to measure the screws?

Measure the following objects using a scale and write their measurements in centimeters (as shown earlier for the lengths of the screws): pen, sharpener, and any other object of your choice.

Teacher's Note

When you write a mixed number like \( 2\frac{7}{10} \) cm, remember that the whole number part (2) and the fractional part (\( \frac{7}{10} \)) are added together. The whole number always comes first, and tells you how many complete units you have. This is different from just looking at position on a ruler - you must read both the whole centimetres and the tenths carefully.

Write the measurements of the objects shown in the picture:

[Figure: Objects with screws measured on scales, See in your textbook]

As seen here, when exact measures are required we can make use of smaller units of measurement.

3.2 A Tenth Part

The length of the pencil shown in the figure below is \( 3\frac{4}{10} \) units, which can also be read as 3 units and four one-tenths, i.e., \( (3 \times 1) + \left(4 \times \frac{1}{10}\right) \) units.

[Figure: Pencil and other objects measured on rulers, See in your textbook]

\[ \frac{1}{10} + \frac{1}{10} + \frac{1}{10} + \frac{1}{10} + \frac{1}{10} + \frac{1}{10} + \frac{1}{10} + \frac{1}{10} + \frac{1}{10} + \frac{1}{10} \]

\[ = 10 \text{ times } \frac{1}{10} = 10 \times \frac{1}{10} = 1 \text{ unit} \]

This length is the same as 34 one-tenths units because 10 one-tenths units make one unit.

Teacher's Note

Notice that we can write \( 3\frac{4}{10} \) as "34 one-tenths" or 34 tenths. This is because \( 3 \times 10 = 30 \) tenths from the whole units, plus 4 more tenths, giving \( 30 + 4 = 34 \) tenths total. When converting a mixed number to pure tenths, multiply the whole number by 10 and add the numerator.

Key Points

• We divide measurement units into smaller parts when we need to measure objects more accurately than whole units allow.
• One unit divided into 10 equal parts gives us \( \frac{1}{10} \) or one-tenth as the smaller unit.
• A mixed number like \( 3\frac{4}{10} \) means 3 complete units plus 4 tenths, and is the same as 34 tenths altogether.
• Reading a scale with tenths divisions requires careful observation: count the whole units first, then count the smaller parts.

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