Class 8 NCERT bridge course Answers Activity W 4.2 square numbers through a pattern!

 Activity W 4.2  - Square numbers through a pattern! 

Teacher can give either printed sheets of the following number pattern to students or draw the number pattern on the blackboard. 

Procedure 

Observe the following number pattern: 

The Pattern

• 1

• 1 + 3 = 4

• 1 + 3 + 5 = 9

• 1 + 3 + 5 + 7 = 16

• 1 + 3 + 5 + 7 + 9 = 25

These sums are:
1,     4,     9,     16,     25 — which are perfect square numbers!

1. Write next 5 rows in the same pattern:

1+3+5+7+9+11=36

1+3+5+7+9+11+13=49

1+3+5+7+9+11+13+15=64

1+3+5+7+9+11+13+15+17=81

1+3+5+7+9+11+13+15+17+19=100

These numbers are square numbers: $6^2, 7^2, 8^2, 9^2, 10^2$.

2. Add the numbers of each row and write the result. 

Row	Numbers	Sum
1	1	1
2	1 + 3	4
3	1 + 3 + 5	9
4	1 + 3 + 5 + 7	16
5	1 + 3 + 5 + 7 + 9	25
6	1 + 3 + 5 + 7 + 9 + 11	36
7	1 + 3 + 5 + 7 + 9 + 11 + 13	49
8	1 + 3 + 5 + 7 + 9 + 11 + 13 + 15	64
9	1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17	81
10	1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19	100

3. Observe these numbers and name the type of these numbers.

They are square numbers!
$1^2, 2^2, 3^2, 4^2, 5^2, 6^2, 7^2, 8^2, 9^2, 10^2$.

4. Write these numbers in other possible ways:

• As squares: $1^2, 2^2, 3^2, 4^2, 5^2, \dots$
  

• As repeated additions of odd numbers.

• As dot patterns in square shapes.

5. Draw the result of each row on the grid sheet: keeping in mind that 1 box on grid is equal to 1 unit square. 

• Each sum forms a square on the grid — for example:

  • Sum = 1 → $1\times 1$

  • Sum = 4 → $2\times 2$

  • Sum = 9 → $3\times 3$

  • Sum = 16 → $4\times 4$

  • Sum = 25 → $5\times 5$

  • and so on.

Reflection and Discussion 

What difference are you observing in these various square boxes on the grid sheet?

The squares grow larger as the row number increases — each time the area grows by the next odd number.

What pattern have you observed?

The pattern is:
Sum of the first $n$ odd numbers  = n².

Q: Can you tell the sum of consecutive first 10 odd numbers?
A: Sum = 10² = 100

 How do you calculate the sum without writing and adding the numbers actually? 

Q: How do you calculate the sum without writing and adding the numbers actually?
A: Use the formula : Sum = n²

Write the rule or formula to find the sum of n consecutive odd numbers?

Q: Write the rule or formula to find the sum of $n$ consecutive odd numbers?
A: Sum of first n odd numbers=n².

Extended Learning and Exploration 

Teacher can give various number patterns like square number pattern, triangular number pattern, Virahanka/fibonacci number. 

 Students have to discover the rule of assigned number patterns.

similar patterns like:

• Triangular numbers: $1, 3, 6, 10, 15...$

• Fibonacci numbers: $1, 1, 2, 3, 5, 8...$

Class 8 NCERT bridge course Answers Activity W 4.1 zigzag puzzle

Activities for Week  4 

Activity W 4.1: 

Teacher may ask students to play this puzzle. 

This is a zigzag puzzle with numbers 1, 2, 3 and 4. 

Your objective is to navigate through the grid, starting from the number 1 in the top-left corner. 

You must follow the numbers in sequential order, ensuring that each number is visited exactly once. 

 The path can move in horizontal, vertical, or diagonal directions but cannot cross itself.

let’s tackle this puzzle step by step!

We start at the 1 in the top-left corner (marked "START").

Step 1: Start at 1 (Row 1, Column 1).

Step 2: Find 2

From the start position, the closest 2 is diagonally down-right to (Row 2, Column 2).

Step 3: Find 3

From (2,2), move diagonally down-right again to (3,3), which is a 3.

Step 4: Find 4

From (3,3), move right to (3,4) where you’ll find 4.

Step 5: Find the next 1

From (3,4), move diagonally down-left to (4,3) where there's a 1.

Step 6: Find 2

From (4,3), move left to (4,2) for 2.

Step 7: Find 3

From (4,2), move up to (3,2) for 3.

Step 8: Find 4

From (3,2), move diagonally down-left to (4,1) for 4.

Step 9: Find 1

From (4,1), move down to (5,1) for 1.

Step 10: Find 2

From (5,1), move right to (5,2) for 2.

Step 11: Find 3

From (5,2), move right to (5,3) for 3.

Step 12: Find 4

From (5,3), move right to (5,4) for 4.

Step 13: Find 1

From (5,4), move diagonally down-left to (6,3) for 1.

Step 14: Find 2

From (6,3), move left to (6,2) for 2.

Step 15: Find 3

From (6,2), move down to (7,2) for 3.

Step 16: Find 4

From (7,2), move right to (7,3) for 4.

Step 17: Find 1

From (7,3), move right to (7,4) for 1.

Step 18: Find 2

From (7,4), move down to (8,4) for 2.

Step 19: Find 3

From (8,4), move left to (8,3) for 3.

Step 20: Find 4

From (8,3), move left to (8,2) for 4.

Step 21: Find 1

From (8,2), move down to (9,2) for 1.

Step 22: Find 2

From (9,2), move right to (9,3) for 2.

Step 23: Find 3

From (9,3), move right to (9,4) for 3.

Step 24: Find 4

From (9,4), move right to (9,5) for 4.

Step 25: Find 1 (Final — END!)

From (9,5), move right to (9,6) for the final 1 — marked END!

Puzzle solved!

  Class 8 NCERT bridge course Answers Activity W 3. 6 SOLVE THE GRID

 Students may be motivated to work on this puzzle. This will help them link different mathematical concepts

Fill in the missing numbers: 

1. The missing values are the whole numbers between 1 and 16. 

2. Each number is only used once. 

3. Each row is a math equation. 

4. Each colu6ath equation. 

Remember that multiplication and division are performed before addition and subtraction

• Third row, middle cell = 11.

• Last row result = 38.

• First column result = 9.

• Last row first number = 9.

• Bottom total = 19 for the last column.

• One row equals 98 (that’s high — so likely multiplication-heavy).

Another way

find more ways but follow the rules

Class 8 NCERT bridge course Answers Activity W 3. 5 Fun riddle

 Activity W 3.5 Fun riddle 

Students may be encouraged to solve this riddle and should be asked to explain their strategy to solve it. This can give them an idea about solving linear equations. If the following equations are true: Then solve these

1️⃣ 🌱 + 🌸 = 🦋
2️⃣ 🦋 - 🌱 = 🌸
3️⃣ 🦋 - 🌸  = 🌱

4️⃣ 🌸+ 🌱  - 🌸 =🌱

Let’s assign:

• 🌱 = p

• 🌸 = f

• 🦋 = b

Step 1: Solve the first two equations.

From Equation 1: p + f = b

From Equation 2: b - p = f

Step 2: Substitute Equation 1 into Equation 2.

Substitute b = p + f into Equation 2:

(p + f) - p = f
f = f

 This is always true, so the values depend on the third equation.

Step 3: Use the third equation.

From Equation 3:
b + f - p = p

Substitute b = p + f:

(p + f) + f - p = p
p + f + f - p = p
2f = p

Now substitute back:

If p = 2f,
then from Equation 1:

2f + f = b
so, b = 3f

So the values are:

• 🌱 (p) = 2f

• 🌸 (f) = f

• 🦋 (b) = 3f

Now solve the bottom part!

1️⃣ 🌱 + 🌱 + 🌸 = ?
= 2p + f
= 2(2f) + f = 4f + f = 5f

2️⃣ 🌱 - 🌸 = ?
= p - f = 2f - f = f

3️⃣ 🌱 + 🌸 - 🦋 = ?
= p + f - b
= 2f + f - 3f = 0

4️⃣ 🦋 - 🌱 = ?
= b - p = 3f - 2f = f

5️⃣ 🌸 + 🌱 - 🌱 = ?
= f + p - p = f

Final Answers:

1. 🌱 + 🌱 + 🌸 = 🦋 5f (depends on f)

2. 🌱 - 🌸 = 🌸 f

3. 🌱 + 🌸 - 🦋 = 0

4. 🦋 - 🌱 = 🌸

5. 🌸 + 🌱 - 🌱 = 🌸 f

We can also solve by assume values, like if 🌸 = 1, then:
🌱 = 2, 🦋 = 3 — and substitute!

TRY THIS!

🧠 Puzzle 1: Fruit Equation

🍎 + 🍎 + 🍌 = 12
🍌 + 🍇 = 8
🍎 + 🍇 = 10

Questions:
a) What is the value of each fruit?
b) 🍇 + 🍎 - 🍌 = ?

🧠 Puzzle 2: Animal Equation

🐶 + 🐱 = 14
🐱 + 🐰 = 10
🐶 - 🐰 = 6

Questions:
a) What is the value of 🐶, 🐱, and 🐰?
b) 🐶 + 🐱 + 🐰 = ?

🧠 Puzzle 3: Shape Equation

🔺 + 🔵 = 15
🔵 + 🟪 = 20
🔺 + 🟪 = 25

Questions:
a) What is the value of each shape?
b) 🔺 + 🔵 + 🟪 = ?

SOLUTIONS

🍎 Puzzle 1: Fruit Equation

Given:
1️⃣ 🍎 + 🍎 + 🍌 = 12
2️⃣ 🍌 + 🍇 = 8
3️⃣ 🍎 + 🍇 = 10

Solution:

From equation 1️⃣:
2🍎 + 🍌 = 12
So, 🍌 = 12 − 2🍎

Substitute into 2️⃣:
(12 − 2🍎) + 🍇 = 8
🍇 = 8 − (12 − 2🍎)
🍇 = 2🍎 − 4

Now substitute into 3️⃣:
🍎 + (2🍎 − 4) = 10
3🍎 = 14
🍎 = 14 ÷ 3 ≈ 4.67

But since the puzzle suggests whole numbers, let's double-check.
Actually, this puzzle seems set up for decimals or might have a typo — usually these are whole numbers.

👉 If you want, I can adjust the numbers slightly so they fit perfectly.

🐶 Puzzle 2: Animal Equation

Given:
1️⃣ 🐶 + 🐱 = 14
2️⃣ 🐱 + 🐰 = 10
3️⃣ 🐶 − 🐰 = 6

Solution:

From 1️⃣: 🐶 = 14 − 🐱
From 2️⃣: 🐰 = 10 − 🐱

Substitute into 3️⃣:
(14 − 🐱) − (10 − 🐱) = 6
14 − 🐱 − 10 + 🐱 = 6
4 = 6 — ❌ this is impossible!

So this puzzle needs correction — the values don't balance.

🔺 Puzzle 3: Shape Equation

Given:
1️⃣ 🔺 + 🔵 = 15
2️⃣ 🔵 + 🟪 = 20
3️⃣ 🔺 + 🟪 = 25

Solution:

From 1️⃣: 🔺 = 15 − 🔵
Substitute into 3️⃣:
(15 − 🔵) + 🟪 = 25
🟪 = 10 + 🔵

Now substitute 🟪 into 2️⃣:
🔵 + (10 + 🔵) = 20
2🔵 = 10
🔵 = 5

Now substitute 🔵 = 5 into:
🔺 = 15 − 5 = 10
🟪 = 10 + 5 = 15

✅ Final values:
🔺 = 10
🔵 = 5
🟪 = 15

Extra Question: 🔺 + 🔵 + 🟪 = 10 + 5 + 15 = 30