Maths Notes Class 6 Chapter 1 Patterns in Mathematics Ganita Prakash

What is Mathematics?

• Definition: Mathematics is the search for patterns and the explanations for why those patterns exist.
• Patterns in Daily Life: We see patterns in nature, homes, schools, and even in the motion of celestial bodies like the sun, moon, and stars.
• Importance: Understanding patterns has led to scientific advancements like launching satellites and curing diseases.
• Mathematics as Art and Science: It combines creativity and logic, and finding patterns helps in applications beyond their original context.

Patterns in Numbers

• Number Sequences: A basic concept in mathematics, focusing on whole numbers.
• Key Number Sequences:
  • Counting numbers: 1, 2, 3, 4, 5, 6, 7,…
  • Odd numbers: 1, 3, 5, 7, 9, 11, 13,…
  • Even numbers: 2, 4, 6, 8, 10, 12, 14,…
  • Triangular numbers: 1, 3, 6, 10, 15, 21, 28,…
  • Square numbers: 1, 4, 9, 16, 25, 36, 49,…
  • Cube numbers: 1, 8, 27, 64, 125, 216,…
  • Virahānka numbers: 1, 2, 3, 5, 8, 13, 21,…
  • Powers of 2: 1, 2, 4, 8, 16, 32, 64,…
  • Powers of 3: 1, 3, 9, 27, 81, 243, 729,…

Visualising Number Sequences

• Pictorial Representation: Visuals help understand the sequences better.
• Examples:
  • Square numbers are represented by square grids.
  • Triangular numbers are visualized by arranging dots in triangular shapes.
  • Cubes, odd, even, and counting numbers can also be depicted using diagrams.

Relations Among Number Sequences

• Example of Adding Odd Numbers:
  • Adding odd numbers gives square numbers:
    • 1=1
    • 1+3=4
    • 1+3+5=9
  • This pattern continues forever, and can be explained with pictures.
• Adding Counting Numbers Up and Down:
  • Adding sequences like 1+2+3+2+1=9 gives square numbers.

Patterns in Shapes

• Shape Sequences: Shapes, like numbers, follow patterns. Examples include:
  • Stacked Triangles
  • 
  • Stacked Squares
  • 
  • Regular Polygons (like triangles, squares, pentagons, etc.)
  • Koch Snowflake: A fractal pattern made by replacing straight lines with zig-zag patterns.

Relations to Number Sequences

• Example: The number of sides in regular polygons follows the counting numbers: 3 (triangle), 4 (square), 5 (pentagon), and so on.
• Stacked Squares and Triangles: The number of little shapes inside larger shapes can be counted to form number sequences.