Solution: Figure It Out 1.5

Section 1.5: Patterns in Shapes

Q1. Can you recognise the pattern in each of the sequences in Table 3?

Answer:

1. Regular Polygons:
   • Sequence of Sides: $3, 4, 5, 6, 7 \dots$ (Counting numbers starting at 3).
2. Complete Graphs:
   • Sequence of Lines: $1, 3, 6, 10, 15 \dots$ (Triangular numbers).
3. Stacked Squares:
   • Sequence of small squares: $1, 4, 9, 16 \dots$ (Square numbers).
4. Stacked Triangles:
   • Sequence of small triangles: $1, 4, 9, 16 \dots$ (Square numbers).
5. Koch Snowflake:
   • Sequence of sides: $3, 12, 48 \dots$ ($3 \times 4^n$).

Q2. Try and redraw each sequence… Can you draw the next shape?

Answer:

• Regular Polygons: The next shape is a Regular Hendecagon (11 sides).
• Complete Graphs: The next graph is K6 (6 points). It will have 15 lines.