Class 8 NCERT bridge course Answers Activity W 3.1 Understanding Denseness of Fractions

 Activities for Week 3 

Activity W 3.1 Understanding Denseness of Fractions 

Objective:

To understand that there are endless fractions between any two given fractions.

Through this activity, students will get an idea about the denseness of fractions. 

That is, they will be able to know that they can find as many fractions as possible between any two fractions. 

This activity will also help to improve number sense and reasoning skills with fractions. 

Material Required 

Long rolls of paper strips 

 Scissors 

 NCERT Mathematical kits (if present in school) 

 Blank cards

Procedure

 Step 1 

Write two fractions, say, 1/4 and 1/2 on the board and the students may be asked to check, if there are fractions between them. 

Discuss that denseness of fractions means that there can be as many fractions as we want between the two fractions. 

Step 2: 

Hands-on Exploration 

 Take two copies of a paper strip

Ask the students to fold those strips in 2 equal halves. 

Take one of the strips and cut it into two equal parts with the help of scissors.  

Take one part and keep it on the other strip. 

 Take the remaining half and put it on the first half. 

 Continue this process until the students are unable to cut remaining part in to 2 equal parts

Discussion to Explore

1. What does this activity explain?

ANSWER:

This activity explains that fractions are dense — meaning, between any two fractions, no matter how close they are, there are always more fractions that can fit in between.

2. Can we divide these strips further more? If yes, then to what extent?

ANSWER:

 Yes, we can keep dividing the strip into smaller and smaller pieces endlessly — in theory, we can keep cutting the parts infinitely, because between any two fractions, there is always another fraction.

3. If half of a unit is 1/2, then what will be the half of 1/2?

ANSWER:

 Half of 1/2 is:

$\frac{1}{2} \div 2 = \frac{1}{4}$

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 So, the half of 1/2 is 1/4.

4. Does 1/4 lie in between 0 and 1/2?

ANSWER:

 Yes!
1/4 is greater than 0 but less than 1/2, so it lies between 0 and 1/2 on the number line.

5. How many fractions can lie between 2 fractions?

ANSWER:

Infinite fractions can lie between any two fractions.
No matter how close two fractions are, there will always be more fractions between them.

6. Ask students, if they see gaps between their fractions.

ANSWER:

 Yes, students will observe gaps between fractions on the strip or number line, which shows there is always room for another fraction in between.

7. Challenge: “Is there another fraction that can go between these parts of strips?”

ANSWER:

 Yes! Always.

For example, between 1/4 and 1/2, you can find:

$\frac{1}{4} + \frac{1}{2} \div 2 = \frac{3}{8}$

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And between 1/4 and 3/8, you can again find:

$\frac{1}{4} + \frac{3}{8} \div 2 = \frac{5}{16}$

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And so on... endlessly!

Fractions are dense — there is always another fraction between any two fractions, no matter how small the gap looks.

 Extension 

Students may be motivated to observe and generalise the above processs to find a fraction between two fractions 

A simple formula to find a fraction between two given fractions:

$$\text{New Fraction} = \frac{\text{Fraction 1} + \text{Fraction 2}}{2}$$

This gives a new fraction that lies exactly between the two.

You can also practice this on a number line or using the Math Kit for better understanding!

Conclusion:

Through this activity, I learned that fractions are dense. This means that between any two fractions, there are infinite fractions. No matter how close two fractions are, we can always find another fraction between them by using the formula:

$$\text{New Fraction} = \frac{\text{Fraction 1} + \text{Fraction 2}}{2}$$

This activity helped me understand that fractions can be divided into smaller and smaller parts, and there is no end to the number of fractions that can exist between any two numbers. Using strips, number lines, or the Math Kit makes this concept easier and fun to learn!