Class 6 Maths Chapter 1: Patterns in Mathematics Notes

What is mathematics?

Mathematics is, in large part, the search for patterns, and for the explanations of why those patterns exist. Patterns are all around us: in nature, in our homes and schools, in the motion of the sun, moon and stars, and in shopping, cooking, throwing a ball, playing games, the weather and technology. Searching for patterns can be fun and creative, which is why mathematicians see mathematics as both an art and a science. Explaining why a pattern happens matters as much as finding it, because explanations can be used far beyond where they were found: patterns in the motion of stars and planets led to the theory of gravitation and to satellites and rockets to the Moon and Mars, and patterns in genomes help diagnose and cure diseases.

Number sequences

The branch of mathematics that studies patterns in whole numbers, 0, 1, 2, 3, 4, …, is number theory, and number sequences are its most basic patterns.

Sequence	First numbers	Rule
All 1’s	1, 1, 1, 1, …	every number is 1
Counting numbers	1, 2, 3, 4, …	add 1
Odd numbers	1, 3, 5, 7, …	add 2, starting at 1
Even numbers	2, 4, 6, 8, …	add 2, starting at 2
Triangular numbers	1, 3, 6, 10, 15, …	add 2, then 3, then 4, …
Squares	1, 4, 9, 16, 25, …	n × n
Cubes	1, 8, 27, 64, 125, …	n × n × n
Virahānka numbers	1, 2, 3, 5, 8, 13, …	add the two numbers before
Powers of 2	1, 2, 4, 8, 16, …	multiply by 2
Powers of 3	1, 3, 9, 27, 81, …	multiply by 3

Read more in number sequences.

Seeing sequences in pictures

Many sequences can be drawn. Triangular numbers are dots arranged in triangles, squares are dots in square grids, and cubes are little cubes stacked into bigger cubes. The same number can play different roles: 36 can be arranged both as a 6 × 6 square and as a triangle with rows of 1 to 8 dots. Hexagonal numbers, 1, 7, 19, 37, …, make hexagons around a centre dot; the next one is 61. Even powers of 2 can be pictured: a point, two points, a square of 4, a cube of 8, and so on. Read more in triangular and square numbers.

Sequences are related

Number sequences can be related in surprising ways. Adding up odd numbers gives squares: 1 = 1, 1 + 3 = 4, 1 + 3 + 5 = 9, 1 + 3 + 5 + 7 = 16. Why? A square grid of dots can be split into L-shaped layers of 1, 3, 5, 7, … dots, and such a picture can be drawn for a square of any size. So the first 10 odd numbers add up to 10 × 10 = 100, and the first 100 to 10,000. Adding counting numbers up and then down gives squares too: 1 + 2 + 3 + 4 + 3 + 2 + 1 = 16.

More relations from the chapter: adding up counting numbers gives triangular numbers; two consecutive triangular numbers add up to a square (10 + 15 = 25); 1 + 2 + 4 + 8 = 15, and adding 1 gives 16, the next power of 2; 6 × a triangular number + 1 gives a hexagonal number (6 × 3 + 1 = 19); and adding hexagonal numbers gives cubes (1 + 7 + 19 = 27).

Patterns in shapes

The branch of mathematics that studies patterns in shapes is geometry. Shapes can be in one, two or three dimensions, or even more. The chapter’s shape sequences are regular polygons (triangle, square, pentagon, hexagon, …: equal sides and equal angles), complete graphs (every point joined to every other), stacked squares, stacked triangles and the Koch snowflake. Read more in patterns in shapes.

Shapes meet numbers

Shape sequence	What we count	Number sequence
Regular polygons	sides (or corners)	3, 4, 5, 6, … (counting numbers from 3)
Complete graphs	lines	1, 3, 6, 10, 15, … (triangular numbers)
Stacked squares	little squares	1, 4, 9, 16, … (squares)
Stacked triangles	little triangles	1, 4, 9, 16, … (squares)
Koch snowflake	line segments	3, 12, 48, 192, … (3 × powers of 4)

Answers to every Figure it Out question are in the NCERT solutions for Chapter 1.