 Manipulative 1: Golden section. Created with GeoGebra. 

The golden section is the ratio that divides a line segment
into two parts such that the ratio of the length of the whole to the length
of the larger segment is the same as the ratio of the length of the larger segment
to the smaller segment: .
The golden section is also called the
golden ratio, golden mean
and divine proportion. The value of the golden
section is represented by the
Greek letter
φ (phi). The value of the golden section is
A golden rectangle is a rectangle where the ratio of the length
of the sides is φ. A golden triangle is an
isosceles triangle
where the ratio of the length of the legs to the length of the base is
φ. The golden triangle was used by Euclid to
approximate the value of φ.
Derivation of the golden section
The value of the golden section is derived using the quadratic formula. Table 1 shows the derivation.
Step  Formula  Description 
1   Start with the definition of the golden section. 
2   Cross multiply. 
3   Apply the distributive property of multiplication over addition and subtraction. 
4   Use the additive property of equality to add a^{2} to both sides of the equation. 
5   Use the substitution property of equality to substitute x for a. 
6   Apply the quadratic formula. 
7   Simplify the multiplication 
8   Simplify the addition inside the parenthesis 
9   Simplify the square root. 
10   Use the distributive property of multiplication over addition and subtraction to pull b out of the fraction. 
11   Multiply the numerator and denominator of the fraction by 1. 
12   Since a negative number does not make sense here, change the ± to +. 
13   Use the substitution property of equality to substitute a back in for x. 
14   Use the multiplication property of equality to multiply both sides of the equation by 1/b.
 15   Since definition of the golden section states φ=a/b, use the substitution property of equality to substitute &phi for a/b. 
Table 1  Derivation of the golden section. 
Representations of φ
Many different ways to represent φ have been found.
One way to represent φ is the repeating fraction
