In a school, students were told to design a series of patterns with triangles and squares. Daniel's group came up with the following series of patterns.

1. How many squares did Daniel's group need to form pattern 57?
2. How many triangles did Daniel's group need to form a pattern with 45 squares?

(a)
Patterns for the number of triangles and squares
Pattern 1: 1 triangle 5 squares
Pattern 2: 2 triangles 9 squares
Pattern 3: 3 triangles 13 squares

Formula:
Number of triangles = Pattern number

Pattern for the number of squares
Pattern 1: 1 + 1 x 4 = 5
Pattern 2: 1 + 2 x 4 = 9
Pattern 3: 1 + 3 x 4 = 13

Formula:
Number of squares = 1 + Pattern number x 4

Number of squares needed to form Pattern 57
= 1 + 57 x 4
= 229

(b)

1 + Pattern number x 4 = 45
Pattern number x 4 = 45 - 1 = 44
Pattern number = 44 ÷ 4 = 11

Pattern number = Number of triangles
Pattern number 11 = 11 triangles

Answer(s): (a) 229; (b) 11