Question 1 - Patterns in Shapes - Chapter 1 Class 6 - Patterns in Mathematics (Ganita Prakash)

Transcript

Question 1 Can you recognise the pattern in each of the sequences in Table 3?Let’s try one by one 1. Regular Polygons Each shape adds one side: Triangle (3) → Square (4) → Pentagon (5) → Hexagon (6) → Heptagon (7) → ... Pattern: Counting numbers starting from 3 i.e. 3, 4, 5, 6, 7, 8, 9, 10,… 2. Complete Graphs (K₂ to K₆) These are complete graphs — every pair of vertices is connected. K₂ has 2 vertices, 1 edge K₃ has 3 vertices, 3 edges K₄ has 4 vertices, 6 edges K₅ has 5 vertices, 10 edges K₆ has 6 vertices, 15 edges So, number of lines are 1, 3, 6, 10, 15… Which are triangular numbers 3. Stacked Squares Each shape adds one full row and column of small squares. Pattern: Square numbers (1, 4, 9, 16, 25, ...) 3. Stacked Squares Each shape adds one full row and column of small squares. This is literally stacking squares to make larger squares: 1² = 1 2² = 4 3² = 9 4² = 16 5² = 25 Pattern: Square numbers i.e. 1, 4, 9, 16, 25, ... 4. Stacked Triangles Each shape is made of small equilateral triangles forming a larger equilateral triangle. 1 triangle 3 small triangles 6 small triangles 10 small triangles 15 small triangles Pattern: 1, 3, 6, 10, 15, … So, its triangular numbers again 5. Koch Snowflake Each shape adds bumps on each line segment. At each step: You divide each edge into 3 parts Replace the middle third with two sides of an equilateral triangle Here, Number of lines = 3 × Powers of 4 Pattern: 3, 12, 48, 192,…