A real number, say 157/68, is typically represented by its decimal expansion, 2.3088235... However it also has another expansion called the continued fraction expansion given below. 157/68 = 2 + 1/(3+1/(4+1/5)). This is typically denoted [2, 3, 4, 5], and in some ways this expansion is more useful, as [2, 3, 4, 5] provides more information about 157/68 than 2.3088235... The theory of continued fractions is a branch on number theory with many applications in the field. In this paper, we investigate patterns that appear in the continued fraction expansions of real numbers. In particular, we look at the continued fraction expansion of some number x, and observe how it changes when we add or subtract small quantities.