Multiple Ways to Solve Proportions

In this article by Ercole, Frantz and Ashline, the authors describe a variety of ways for students to develop proportional reasoning. They divide the methods into three categories of methods: unit-rate, factor of change methods, and, finally, transitional methods that help students move from an understanding of fractions to proportional reasoning.
Using the first method, the authors introduce the concept of reciprocal rates. For instance, if given the price of donuts by the dozen, student may be asked to determine the price in dollars for one donut or, conversely, the number of donuts that can be purchased for a dollar. In the second method, students either (a)look for the factor of change between the numerator and denominator in a non-unit-rate description or (b) look for a rate of change in the numerators or denominators of two non-unit-rates. Transitional methods allow students to look use their understanding of common denominators or repeated addition to come to an understanding of proportional reasoning. They make use of such things as ratio tables to help the number patterns become more visible.

The methods described in this article allow students multiple ways to make sense of the assigned problem. They also allow students to connect their learning to their past mathematical understanding and use their intuition. They learn to model real-life relationships using mathematics in a way in which helps them identify and make use of structure. Ratio tables help repeated reasoning more visible. Unit-ratios require they also communicate precisely, with appropriate unit labels. By learning all of these methods, students develop a deeper understanding of proportional reasoning and will likely develop procedural fluency.

Ercole, L., Frantz, M. and Ashline, G. (2011). Multiple ways to solve proportions. Mathematics Teaching in the Middle School 16(8), 482-490.