SOME FUNNY ANSWERS

 WRITE 10 VEGETABLE NAMES.

I AM NON-VEGETARIAN.

SOLVE 

11x = π

x =  π / 11

Solve  

X² = 25

x = 5

PROVE TH MID POINT THEOREM

Class 8 NCERT bridge course Answers Activity W 4.3 pictorial patterns

 Activity W 4.3  Pictorial patterns

Students may be asked to extend the following pictorial patterns further for two steps. 

Express each of these as a numerical pattern as directed. 

1. Stacked Squares

Count the number of small squares in each case and write it. 1, 4, ... 

Extend the sequence till 10 terms. 

ANSWER: 

Number Pattern:

1, 4, 9, 16, 25, 36, 49, 64, 81, 100

Do you find any pattern? 

ANSWER: 

Pattern Observed:

These are square numbers — the number of squares increases by the next odd number each time.

Formula: Number of squares=n² where  n is the position in the sequence.

2. Stacked Triangles

Count the number of small triangles in each case and write it. 

ANSWER: 

1,4,9

 

Extend the sequence till 10 terms.

ANSWER: 

Number Pattern:

1, 4, 9, 16, 25, 36, 49, 64, 81, 100

 Do you find any pattern? 

Pattern Observed:

• This is a square number pattern.

• Formula: Tn​=n²

Where $T_n$ is the number of small triangles in the nth figure.

3. Koch Snowflake

 To get from one shape to the next shape in the Koch Snowflake sequence, one replaces each line segment ‘—’ by a ‘speedbump’ +. 

As one does this multiple times, the changes become tinier with very extremely small line segments.

 Extend it by three more steps. 

 How many total line segments are there in each shape of the koch snowflake? 

 Starting with an equilateral triangle (Step 0).

At each step, each line segment is replaced by 4 smaller segments.

Step	Formula	Total Line Segments
0	$3 \times 4^0 = 3$	3
1	$3 \times 4^1 = 12$	12
2	$3 \times 4^2 = 48$	48
3	$3 \times 4^3 = 192$	192
4	$3 \times 4^4 = 768$	768
5	$3 \times 4^5 = 3072$	3072

What is the corresponding number sequence?

ANSWER:

Corresponding Number Sequence:   3,12,48,192,768,3072,12288,…

• Each new step multiplies the number of line segments by 4.

• Formula:

$$\text{Total segments at step } n = 3 \times 4^n.$$

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!