Class 8 NCERT bridge course Answers Activity W 3.3 Difference between Volume and Area

 Activity W 3.3 

Students will get an opportunity to explore the difference between Volume and Area, through this activity.

 This will be done by comparing how much space different containers occupy and how much they can hold

Material Required 

Two flat trays. 

 Two deep containers (different shapes). 

 Rice or small beans or sand. 

 Small cups (to be taken as measuring cups). 

(The teacher can try arranging one deep container with a base approximately the same as one of the trays.)

Procedure 

Step 1: Observation

 Show students the flat trays and deep containers (of different shapes). 

 Ask: 

Q: What do you notice about these things (trays and containers)?

ANSWER: The trays are flat and wide, while the containers are deep and can hold more things inside.

 Q: How are the trays different from the deep containers?

A: Trays are shallow and only have surface space, while deep containers have depth and can store more material.

Q: Which ones do you think will hold more rice/sand? Why?

A: The deep containers will hold more rice or sand because they have more depth and space inside.

Step 2 : Measuring Area

 Sprinkle a thin layer of rice or sand on the flat trays until the surface is fully covered. 

(The teacher needs to ensure that there is only one layer of the rice or sand on the trays.) 

 Use measuring cups to measure how much rice or sand each tray holds. 

 Ask:

Q: Did the rice or sand cover the entire tray?

A: Yes, the rice or sand covered the entire surface in a single, thin layer.

Q: Without changing the shape the rice or sand takes on the tray, would the tray be able to ‘hold’ more of it?(We cannot make a heap of rice or sand.) 

A: No, because a tray only has a flat surface and no depth, so it cannot hold more without spilling.

Q: How can we measure the space covered by the rice or sand?

A: We can measure the area by using square units (like cm²) to cover the surface.

Q: Can you think of other examples where we use area in daily life?(for example, carpets, tiles, painting walls)?

A: Yes — laying carpets, painting walls, tiling floors, covering tables, or making posters.

Step 3: Measuring Capacity (Volume)

Measuring Capacity 

 Fill the deep containers with rice or sand to the top. 

 Use measuring cups to measure how much rice or sand each container holds. 

 Ask: 

 Q: Why do the deep containers hold more rice or sand than the trays?

A: Because the containers have depth, which allows them to store more in the same surface area.

Q: What changed when we measured volume instead of area?
A: Instead of just covering the surface, we now measured the full space inside the container, including depth.

 Q: How many layers of rice or sand would we need to make the trays hold the same amount as the deep containers? Estimate.

A: It would take many thin layers (depending on the depth of the container), for example, if the container is 5 cm deep and the rice layer is 0.5 cm thick, it would take around 10 layers.

 Q: How is volume different from area?

A:

• Area measures the flat surface space.

• Volume measures the total space inside an object.

Step 4: 

Making Real-world Connections 

Discuss examples from real life: 

 Q: Why do water bottles and storage boxes have depth?

A: Because depth allows them to hold more water or items — volume is important for storage.

Q: Would a swimming pool be useful, if it only had area but no depth?

A: No, because without depth, it couldn’t hold water for swimming.

Q: Do architects and builders use volume when designing rooms and buildings? How?
A: Yes, they calculate volume to make sure there is enough space for air, furniture, and people inside a room.

Reflections 

 Summarise that area measures surface coverage, while volume measures the total space inside an object or a container. 

SUMMARY

Area measures the surface coverage of an object — it tells us how much flat space is covered, like when we lay a carpet or paint a wall.

Volume measures the total space inside an object or a container — it tells us how much material or liquid it can hold, like water in a bottle or rice in a jar.

In simple words:
👉 Area = Surface covered
👉 Volume = Space filled inside

 Encourage students to think about other objects, where area and volume play a role in real life. 

 Let students predict, which objects in the classroom have a large area but small volume and vice versa. 

 Let students predict situations, where Area is used and situations, where Volume is used. Participation of Special Children— 

Use lightweight containers and trays that are easy to handle. 

 If students have difficulty lifting or pouring, provide pre-measured cups of rice or sand so they can still participate. 

 Arrange the activity on lower tables or surfaces so all students, including those using wheelchairs, can comfortably access the materials.

Pair students so they can work together, allowing those with mobility challenges to observe, instruct and record findings, while their peers assist with pouring and measuring.

CONCLUSION:

• Area is about measuring how much surface something covers (like the top of a tray or the floor for a carpet).

• Volume is about measuring how much space an object can hold inside (like filling a bottle, a bucket, or a container).

• Objects with the same area can have different volumes, and vice versa.
• Area is used when covering something flat, like painting walls or laying tiles.
•  Volume is used when filling something, like a water bottle, swimming pool, or a storage box.

When I sprinkled rice or sand on the flat tray, it only covered the surface — that is area. But when I filled the deep containers, they could hold much more rice because of their depth — that is volume.

Class 8 NCERT bridge course Answers Activity W 3.2 Let’s Brainstorm

 Activity W 3.2 Let’s Brainstorm 

 Students may be asked to solve the following puzzles either in groups or individually.

 They may explain their strategy of obtaining the result. 

Let’s Brainstorm 

Puzzle 1: Symbol Grid

1. The grid below has symbols that contain a whole number value less than 10. 

Each symbol has its own value. 

The numbers you see at the end of each row and column are the sums of the figures’ values for that row or column.

Can you find out the value of each symbol ? 

ANSWER:

Let’s name the shapes:

• 🟦 Pentagon = P

• 🔺 Triangle = T

• ⭐ Star = S

• ➕ Plus = C

And solve!

Row equations:1️⃣ 

1. P+T+P+S=15

2. C+S+P+S=23

3.T+C+T+?=16

4. S+T+?+P=14

Column equations:

1. P+C+T+S=20

2. T+S+C+T=24

3. P+P+T+?=7

4. S+S+?+P=17

SOLVE

From Row 3 and Column 3, the missing symbol must be the same in both — let’s call it X for now.

Let’s focus on easy rows:

Row 3:  T+C+T+X=16

Row 4:   S+T+X+P=14

Col 3:  P+P+T+X=7

Let’s assume X = 0 to test:

From Col 3:   𝑃+P+T=7

From Row 3:  T+C+T=16

So  2T+C=16.

From Col 1: P+C+T+S=20.

And from Row 1: 2P+T+S=15.

From Row 3:  2T+C=16

From Col 3:  2P+T=7

From Row 4: S+T+P=14 (since X=0)

Now solving this small system:

From (2)  2P+T=7 → T=7−2P

Substitute into (1):

2(7−2P)+C=16

14−4P+C=16

C=4P+2.

Now substitute into Col 1:

P+C+T+S=20.

Substitute 

C=4P+2 and 

T=7−2P:

P+(4P+2)+(7−2P)+S=20

Simplify:

P+4P+2+7−2P+S=20

3P+9+S=20

S=11−3P.

If  P=1: 

S=11−3(1)=8

T=7−2(1)=5

C=4(1)+2=6

Now check Row 2:

C+S+P+S=23

6+8+1+8=23 

SOLUTION:

• 🟦 Pentagon P=1
• 🔺 Triangle T= 5
• ⭐ Star  S = 8
• ➕ Plus C = 6

Puzzle 2: Symbol Equation

2. Here, you are given two representations, where symbols have been used. 

Each symbol represents a numeric value. Find the value of each symbol.

SOLUTION:

Given the equations:

• • 🟦 Blue Square = S

  • 🔺 Orange Triangle = T

  • 🟡 Yellow Circle = C

  • ⭐ Star = R

EQUATION 1:  S+S+S=15 --> 3S = 15 --> S = 5

EQUATION 2:  T+T+S=13 -->  2T + 5 = 13 --> 2T = 8 --> T = 4

EQUATION 3 : T+C+S=15 --> 4 + C + 5 = 15 --> C = 15 - 9=6

EQUATION 4: C+C+S=?

EQUATION 5: C + T + T = ?

EQUATION 6 : C + T = 8

EQUATION 7: C + T = 4

EQUATION 8 : R + S =12 --> R + 5 = 12 --> R = 12 -5 =7

EQUATION 9 : R + S = S

EQUATION 10 : S + S = R --> 2S = R

SOLUTION:

🟦 Blue Square (S)

= 5

🔺 Orange Triangle (T)

= 4

🟡 Yellow Circle (C) =

6

⭐ Star (R)

= 12 - 5 = 7

🟡 + 🟡 + 🟦 = 6 + 6 + 5 = 17

🟡 + 🔺+ 🔺 = 6 + 4 + 4 = 14

Puzzle 3: Make 5+5+5 = 550 True

3. Make the following equation true by drawing/putting/writing a single line.

SOLUTION : 

 Just draw a slanted line on the first "+" to turn it into 4:

545 + 5 = 550! 

Puzzle 4:Roman Numeral Trick 

What should be added to IX to make six?

Add S in front of IX to form SIX.
So the answer is: Add ‘S’! 

Teachers may try to find some more such puzzles that will engage students in the process of exploration

Puzzle 1: Symbol Sums

Symbol	Meaning
🍎 Apple = ?
🍌 Banana = ?
🍇 Grapes = ?

Equations:

1. 🍎 + 🍎 + 🍎 = 18

2. 🍌 + 🍎 + 🍌 = 16

3. 🍇 + 🍇 + 🍎 = 20

Find the value of each fruit!

Solution:

1. 🍎 + 🍎 + 🍎 = 18 → 🍎 = 6

2. 🍌 + 6 + 🍌 = 16 → 2🍌 = 10 → 🍌 = 5

3. 🍇 + 🍇 + 6 = 20 → 2🍇 = 14 → 🍇 = 7

Final Answer:
🍎 = 6, 🍌 = 5, 🍇 = 7.

Puzzle 2: Number Logic

Symbol	Meaning
🐾 Paw = ?
🐟 Fish = ?
🦴 Bone = ?

Equations:

1. 🐾 + 🐾 + 🐟 = 22

2. 🐟 + 🦴 + 🦴 = 14

3. 🐾 + 🦴 = 13

Find the value of 🐾, 🐟, 🦴.

Solution:

From (3):
🐾 + 🦴 = 13 → 🦴 = 13 - 🐾.

Substitute into (2):
🐟 + 2(13 - 🐾) = 14
Simplify and solve using substitution or trial.
For example:
If 🐾 = 8, 🦴 = 5.

Now check in (1):
8 + 8 + 🐟 = 22 → 🐟 = 6.

 So final values:
🐾 = 8, 🐟 = 6, 🦴 = 5.

Puzzle 3: Matchstick Equation

Make the equation correct by moving 1 matchstick:

6 + 4 = 9  

Solution:
Move one stick from "6" to make it "5":

5 + 4 = 9 

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:

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

​
 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:

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

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

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

​
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.