Relation to Number Sequences

Class 6 Maths · Chapter 1: Patterns in Mathematics · NCERT Ganita Prakash section 1.6

Learn

Shape sequences are often linked to number sequences. Count the sides of the regular polygons: 3, 4, 5, 6 … the counting numbers starting at 3. The corners give the same numbers, because every side ends at a corner where the next side begins.

Count the lines in the complete graphs: 0, 1, 3, 6, 10 … the triangular numbers! Why? A new point must join every point already there. In K5 the new point adds 4 new lines (coral).

0 + 1 + 2 + 3 + 4 = 10 lines

Stacked squares have 1, 4, 9, 16 little squares: the square numbers, because the big square has n rows of n little squares.

4 × 4 = 16

Stacked triangles also give 1, 4, 9, 16! Count each row: 1, 3, 5, 7 triangles. These are the odd numbers, and odd numbers add up to square numbers.

1 + 3 + 5 + 7 = 16 = 4 × 4

Koch snowflake: each line segment turns into 4 smaller segments at every step. So the segments go 3, 12, 48, 192 … that is 3 times the powers of 4.

3 → 12 → 48 → 192 (× 4 each step)

Explore: stack small squares and triangles to make bigger ones, and grow a snowflake.

1. Make a big square out of 16 small squares.
2. Make a big triangle out of 9 small triangles.
3. Grow the Koch snowflake until it has 48 sides.

Remember

• Sides of regular polygons: 3, 4, 5 … (counting numbers from 3).
• Lines in complete graphs: 0, 1, 3, 6, 10 … (triangular numbers).
• Stacked squares and stacked triangles: 1, 4, 9, 16 … (square numbers).
• Koch snowflake segments: 3, 12, 48 … (3 × powers of 4).

Quick check

6 points, every point joined to every other point. How many lines?

1. 12
2. 15
3. 18
4. 30

Think about it

Count the corners of each regular polygon. Do you get the same sequence as the sides? Why?

Practice

Practise with fresh questions at three levels (easy, medium, hard), some answered hands-on, with hints and worked answers. Get 4 of your last 5 right to finish the topic.