Section 3.1 Scaling Up, Scaling Down
When we think of proportional reasoning, one of the areas of mathematics where we can truly "see" it is geometry. The ancient Greeks used a very special proportion in their artwork as well as architecture, The Golden Ratio. This ratio seemed to be found in nature and was deemed pleasing to the eye. In fact, the Fibonacci sequence, \(\left\{1,1,2,3,5,8,13, \ldots \right\}\text{,}\) has the interesting characteristic that the successive ratios of adjacent terms, \(\frac{F_n}{F_{n-1}}\text{,}\) approach The Golden Ratio as \(n \rightarrow \infty\text{.}\) You may recall that the Golden Ratio occurs when the ratio of the larger of two values to the smaller, \(a\) and \(b\text{,}\) is equal to the ratio of the sum of the two values to the larger of the two values. In other words, when \(a \gt b \gt 0\text{,}\) we have \(\frac{a+b}{a}=\frac{a}{b}=\varphi =\frac{1+\sqrt{5}}{2}\) where \(\varphi\) is typically used to represent this ratio. Just like the Fibonacci sequence that is seen throughout nature, The Golden Ratio also seems to be present all around us.
Since ratios seem to describe many patterns we see around us, we should consider how the concept of ratio develops in students and try to identify an appropriate place in the curriculum to begin engaging our students with ideas related to ratio and proportion. When we examine the Common Core State Standards, the first place we see ratio finding its way into the curriculum is in Grade 6 (see Figure 3.1.1). While fractions are seen in Grade 3, the treatment of the concept of rational numbers as ratios has not yet reached the stage where ratio and proportion would be viewed as relational as we would see them within the context of scaled up or scaled down representations of relative amounts or unit rates.
