Ratios and proportions are important concepts that occur in real life, but they are difficult to learn and complicated to teach (Lamon 2007). In general, proportional reasoning in the middle grades is an area in need of attention because of its connection to later concepts (Ojose 2015). In this article, we explain the meanings of ratios, proportions, and equivalent ratios and then provide useful methods to determine equivalent ratios using students' examples. Because multiple definitions exist, it is necessary to determine definitions to avoid possible confusion. Johnson (2010) defines a ratio as a pair of positive nonzero real numbers such that there are a units for every b units, written a:b; read a to b; and represented as a fraction, a/b. We follow definitions suggested by the writers of the Common Core (CCSSI 2010): Fraction representation is known as the associated unit rate or the value of a ratio A:B and is found by dividing B into A, where “equivalent ratios have the same unit rate” (McCallum, Zimba, and Daro 2011). Using both contextual and noncontextual proportion tasks, we saw various ways that students found the equivalency of ratios as well as misconceptions they have.

## Footnotes

Edited by Jennifer Eli, jeli@math.arizona.edu, The University of Arizona, Tucson, and George J. Roy, roygj@mailbox.sc.edu, University of South Carolina, Columbia.

## Contributor Notes

Ricardo Martinez, ricardom@iastate.edu, is a graduate student in the School of Education at Iowa State University in Ames. He is interested in critical pedagogy within the teaching of mathematics that works toward equitable mathematics problem solving.

Ji Yeong I, jiyeongi@iastate.edu, is an assistant professor at Iowa State University who has interests in preparing teachers to support English language learners and investigating teacher beliefs toward culturally diverse students in learning mathematics.

(Corresponding author is Martinez ricardom@iastate.edu)
(Corresponding author is I jiyeongi@iastate.edu)
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