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Mathematics: Class 6 — Chapter 1: Patterns in Mathematics

1.1 What is Mathematics?

Mathematics is, in large part, the search for patterns, and for the explanations as to why those patterns exist.

Such patterns exist all around us—in nature, in our homes and schools, in the motion of celestial bodies, and in everyday activities like cooking, shopping, and playing games.

The search for patterns and their explanations is both an art and a science, helping to propel humanity forward through applications like the theory of gravitation and genome analysis.

1.2 Patterns in Numbers

Among the most basic patterns in mathematics are patterns of whole numbers: $0, 1, 2, 3, 4, \dots$

The branch of Mathematics that studies patterns in whole numbers is called number theory.

Number sequences are the most basic and fascinating types of patterns studied by mathematicians.

Table 1: Examples of Number Sequences

  • All 1's: $1, 1, 1, 1, 1, 1, 1, \dots$
  • Counting numbers: $1, 2, 3, 4, 5, 6, 7, \dots$
  • Odd numbers: $1, 3, 5, 7, 9, 11, 13, \dots$
  • Even numbers: $2, 4, 6, 8, 10, 12, 14, \dots$
  • Triangular numbers: $1, 3, 6, 10, 15, 21, 28, \dots$
  • Squares: $1, 4, 9, 16, 25, 36, 49, \dots$
  • Cubes: $1, 8, 27, 64, 125, 216, \dots$
  • Virahānka numbers: $1, 2, 3, 5, 8, 13, 21, \dots$
  • Powers of 2: $1, 2, 4, 8, 16, 32, 64, \dots$
  • Powers of 3: $1, 3, 9, 27, 81, 243, 729, \dots$
1.3 Visualising Number Sequences

Many number sequences can be visualised using pictures or diagrams, which is a fruitful way to understand mathematical patterns.

1.4 Relations among Number Sequences

Number sequences can be related to each other in surprising ways.

Sum of Odd Numbers

Adding consecutive odd numbers starting from $1$ always results in a square number:

  • $1 = 1 = 1^2$
  • $1 + 3 = 4 = 2^2$
  • $1 + 3 + 5 = 9 = 3^2$
  • $1 + 3 + 5 + 7 = 16 = 4^2$
  • $1 + 3 + 5 + 7 + 9 = 25 = 5^2$

Pictorial Explanation: A square grid of dots can be partitioned into L-shaped layers (gnomons) containing consecutive odd numbers of dots.

Adding Up and Down

Adding counting numbers up to a peak and then back down also yields square numbers:

  • $1 = 1 = 1^2$
  • $1 + 2 + 1 = 4 = 2^2$
  • $1 + 2 + 3 + 2 + 1 = 9 = 3^2$
  • $1 + 2 + 3 + 4 + 3 + 2 + 1 = 16 = 4^2$
1.5 Patterns in Shapes

The branch of Mathematics that studies patterns in shapes is called geometry.

Shape sequences are sequences of geometric figures that follow a specific rule.

Table 3: Examples of Shape Sequences

  • Regular Polygons: Triangle, Quadrilateral (Square), Pentagon, Hexagon, Heptagon, Octagon, Nonagon, Decagon.
  • Complete Graphs: $K_2, K_3, K_4, K_5, K_6$ (shapes where every vertex is connected to every other vertex).
  • Stacked Squares: Grids of squares of increasing sizes ($1\times1, 2\times2, 3\times3, \dots$).
  • Stacked Triangles: Triangles stacked in rows of increasing sizes.
  • Koch Snowflake: A fractal shape formed by iteratively replacing line segments with a triangular