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:

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