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: $1, 1, 2, 3, 5, 8...$