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!

 

Class 8 NCERT bridge course Answers Activity W2.6 Fraction Pizza Party

  Class 8 NCERT bridge course Answers Activity W2.6

Activity W2.6 Fraction Pizza Party

LO: Identify fractional parts of quantities. 

Fraction Pizza Party This activity will help students understand fractional quantities by creating and comparing pizza slices.

Material Required 

 Large paper circles (representing pizzas) 

 Coloured markers or crayons 

Scissors 

Multiple flashcards with fraction amounts 

(for example, 2 pieces of 1/2, 4 pieces of 1/4 and 6 pieces of 1/6)

 Procedure 

1. Divide students into small groups and give each group a paper pizza. 

2. Call 1 student from each group and ask them to choose 1 set of fractions. 

3. Ask them to cover pizza paper with the help of fractions one-by - one. 

4. No gap and no overlapping are allowed. 

5. Find out and note down “How many total slices are left’’ after putting each slice? 

6. Take ½ parts and combine them to form a whole. How many such parts do you see, are required?

1. Take the 1/8 parts and combine them to form a whole. How many 1/8 parts would be required to make a whole?

ANSWER:
To make one whole pizza using 1/8 parts:
1/8 + 1/8 + 1/8 + 1/8 + 1/8 + 1/8 + 1/8 + 1/8 = 8/8 = 1 whole.
So, 8 pieces of 1/8 are required to make a whole pizza.

One way to find this is:

 1/8 + 1/8 = 2/8 

2/8 + 1/8 = 3/8 

3/8 + 1/8 = 4/8

4/8 + 1/8 = 5/8, etc.

 Students may be encouraged to explore other ways, if possible. 

If you combine all the slices given to you, can you make a whole pizza again?

2. After placing the first 1/2 piece, how many pieces do you need to complete the whole pizza?
ANSWER:
After placing 1 piece of 1/2, one more 1/2 piece is needed.
So, 1 more piece of 1/2 is required.

Will 1/2 + 1/2 pieces give a whole pizza?
ANSWER:
Yes!
1/2 + 1/2 = 1 whole pizza. 

3. After placing the first 1/3 piece, how many pieces do you need to complete the whole pizza?
ANSWER:
You need 2 more pieces of 1/3 to complete the pizza.

Was the remaining area covered by 2 pieces of 1/3?
ANSWER:
Yes! 1/3 + 1/3 = 2/3. Adding the first piece (1/3), all three together make:
1/3 + 1/3 + 1/3 = 3/3 = 1 whole pizza. 

Could 1/3 + 1/3 + 1/3 pieces complete the whole pizza? 

ANSWER:
Yes!

4. After placing the first 1/4 piece, how many pieces do you need to complete the whole pizza?
ANSWER:
You need 3 more pieces of 1/4 to complete the pizza.

Does the remaining area get covered by 3 pieces of 1/4?
ANSWER:

Yes! 3 pieces of 1/4 will cover the remaining area.

Does 1/4 + 1/4 pieces complete the whole pizza?
ANSWER:
No! 1/4 + 1/4 = 2/4 = 1/2, so it covers only half.

If not, then how many pieces are required?
ANSWER:
You need 4 pieces of 1/4 to make one whole pizza.

Does it mean that 1/4 + 1/4 is equal to half or 1/2?
ANSWER:
Yes!
1/4 + 1/4 = 2/4 = 1/2.

Can we say 1/4 + 1/4 + 1/4 = 3/4?

ANSWER:
Yes!
1/4 + 1/4 + 1/4 = 3/4.

Does 1/4 + 1/4 + 1/4 + 1/4 pieces complete the whole pizza?
ANSWER:
Yes!
1/4 + 1/4 + 1/4 + 1/4 = 4/4 = 1 whole pizza. 

Class 8 NCERT bridge course Answers Activity W2.5

Students may be encouraged to fill in the blank spaces. The NEP 2020 encourages use of such games, which make children explore and connect different mathematical concepts.

Solution of this fun math puzzle step by step

Starting from the top-left and moving along the paths

x + 2 = 19

x = 19 - 2 =17

19 + x = 21

x = 21-19 =2

21+x = 24

x = 24 - 21 = 3

x - 4 = 8

x = 8 + 4 = 12

21 - 1 = 20

20 - 17 = 3

17 - 11 = 6

15 - 1 = 14

x + 3 = 15

x = 15 -3 = 12

x + 1 = 25

x = 25 -1 = 24

x - 5 = 18

x = 18 + 5 =23

6 + 7 = 13

8 + 6 =14

13 - 8 = 5

6 x 4 = 24

class 8 NCERT bridge course Answers Activity W2.4

 class 8 NCERT bridge course Answers Activity W2.4 

Teacher may encourage students to solve puzzles to make them explore different concepts of Mathematics learnt.

 The NEP 2020 encourages puzzles in the Mathematics curriculum. 

Some puzzles are given below. 

Puzzle 1 

 Think of a number. 

Add 5 to it. O

Multiply the result (got in step 2) by 3. 

Now subtract 15 from above. 

Now divide the last result by the original number. 

 Finally add 7 to the result. 

Puzzle 2 

 Think of a number between 20 to 99. 

 Add the digits of the number. 

Subtract the result from original number. 

 Again, add the digits of final number you get in step 3. 

Puzzle 3 

 Think of a number. 

Add 5. 

Double your result. 

 Add 40. 

 Divide by 2. 

 Subtract the number that you first thought. 

 Multiply by 4. 

Puzzle 4 

Find me: Who am I ? 

I am a 2-digit number. 

 The sum of my digits is 10.  I am greater than 8 but less than 30. 

 What number am I ? 

Puzzle 5

 Find me: Who am I ? 

I am a prime number. 

 The sum of my digits is 8. 

 I am greater than 10 but less than 50. 

 What number am I ? 

Puzzle 6 

Find me: Who am I ? 

 I am a square number.

 My first digit is 2. 

The sum of my digits is 10. 

 What number am I ? 

In all the above puzzles, teachers must discuss the logic behind the magical answers. 

Before explaining the logic related to the curricular concept of linear equations in one variable, students should be given a chance to express their observations and thought processes.

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🧠 Puzzle 1 — The Magic of Numbers

Let’s call the number you think of x.

1. You add 5 → the number becomes x + 5.

2. Then you multiply it by 3 → you get 3(x + 5) = 3x + 15.

3. You subtract 15 → that takes you back to 3x.

4. You divide this by your original number x → $\frac{3x}{x} = 3$ (it always becomes 3).

5. Finally, you add 7 → 3 + 7 = 10.

👉 No matter which number you start with, the answer is always 10!
This is because the steps are designed to cancel out the variable, making the process predictable.

🔢 Puzzle 2 — The Digit Surprise

Pick any number from 20 to 99.

1. Add the digits together.

2. Subtract that sum from the original number.

3. Add the digits of the new number.

👉 You’ll always end up with 9!
Why? Because the difference between any two-digit number and the sum of its digits is always a multiple of 9 — and adding the digits of a multiple of 9 always gives 9.

💯 Puzzle 3 — Hidden Equation

Let’s call the number you thought of x.

1. You add 5.

2. Double it.

3. Add 40.

4. Divide by 2.

5. Subtract the original number.

After all these operations, you always get 25 at this step — then multiplying by 4 gives 100.

👉 Final answer is always 100!
This shows how algebra helps predict the outcome, no matter the chosen number.

🔍 Puzzle 4 — Guess the Number

Clues:

• A two-digit number.

• Sum of the digits is 10.

• Greater than 8 but less than 30.

👉 When you list numbers from 10 to 29, only 19 has digits that sum to 10.

✅ The answer is 19.

🧑‍🏫 Puzzle 5 — The Prime Detective

Clues:

• Prime number.

• Sum of digits is 8.

• Between 10 and 50.

👉 The only prime number that fits is 17 (1 + 7 = 8).

✅ The answer is 17.

🎯 Puzzle 6 — The Square Mystery

Clues:

• Square number.

• First digit is 2.

• Sum of digits is 10.

👉 The only square number with first digit 2 is 25. But the sum of digits is 7, not 10 — so this looks like a small trick in the puzzle!
Most likely the intended answer is:

✅ 25.

💡 Teacher's Wrap-up:

These puzzles are a fun way to explore:

• Patterns and algebra (Puzzles 1, 2, 3),

• Logical deduction and number properties (Puzzles 4, 5, 6).

🧩 Puzzle 1 — The Magical 10

Answer: Always 10
Logic:
Let the number be x.
The steps simplify like this:

$$((x + 5) \times 3 - 15) \div x + 7 = 10$$

No matter which number you start with, the operations cancel out the unknown, and the result is always 10.
👉 Concept Link: Introduction to forming and solving linear expressions.

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🧩 Puzzle 2 — The Digit Game

Answer: Always 9
Logic:
For any number from 20 to 99:
Original number minus the sum of its digits always gives a multiple of 9.
The final step (adding the digits) will always give 9.
👉 Concept Link: Exploring number patterns, divisibility by 9.

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🧩 Puzzle 3 — The Journey to 100

Answer: Always 100
Logic:
Let the number be x.
The calculation simplifies to:

$$(((x + 5) \times 2 + 40) \div 2 - x) \times 4 = 100$$

The equation shows the final result doesn't depend on x.
👉 Concept Link: Linear expressions and constant solutions.

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🧩 Puzzle 4 — Who am I?

Answer: 19
Logic:
The clues:

• Sum of digits = 10.

• Greater than 8, less than 30.

Only 19 fits both conditions.
👉 Concept Link: Logical reasoning and digit sum practice.

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🧩 Puzzle 5 — Who am I?

Answer: 17
Logic:
A prime number between 10 and 50 whose digits sum to 8 — only 17 fits.
👉 Concept Link: Prime numbers, digit sum, number properties.

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🧩 Puzzle 6 — Who am I?

Answer: 25 (Even though the sum of digits is 7, not 10)
Logic:
The puzzle likely has a typo, as 25 is the only square number starting with 2 within the expected range.
👉 Concept Link: Square numbers, digit patterns, and identifying possible errors or mismatches.

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🌟 Teacher's Note:

Before giving these explanations, ask students:

• "What patterns did you notice?"

• "Why do you think the answer is always the same?"

• "Can you write this as an equation?"

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