Ideas in Geometry/Instructive examples/Continued Fractions, Section 3.2
3.2 #30

Handwritten version
√5 To start note that 3 > √5 > 2
So this means that √5 = 2+ (√52) Where 2 is the whole number part and (√52) is the fractional part of √5.
In a simple continued fraction our numerator is 1. In order to get 1 in the numerator we must use the reciprocal 1/(√52).
From this point we want to separate the wholenumber part and the fractional part and can do so by multiplying both the numerator and denominator by the conjugate (√5+2) 1 x (√5+2) = (√5+2) and (√52) x (√5+2) = (54)
When we put both the numerator and denominator together we get (√5+2)/1 or (√5+2)
Now we need to get our original fractional part from (√5+2) in order to continue our fraction and can do so by adding and subtracting 4 from (√5+2) 4+(√5+24)
Using the associative property we get the next part of our continued fraction 4+(√52)
Repeat these steps to get the continued fraction √5= 2+1/ (4+1/(4+1/(4+1/(4+1/....)