*Result*: Students' productive use of equivalence transformations.

*Equivalence transformations – that is, transformations that produce an object that is equivalent to the original – are a unifying conceptual thread in K-16 mathematics. Though researchers have already established that productive reasoning about equivalence transformations hinges on an awareness that the transformed objects are equivalent to the given object, research (a) has not yet explored the various ways in which students might attend to equivalence, and (b) has primarily examined equivalence transformations on only one type of object, leaving open the question of what commonalities might be present in students' reasoning across transformations of multiple types of objects. In this study, we present our analysis of task-based clinical interviews with university students. This paper's primary contribution to the literature involves the description and illustration of three common, unified ways in the students productively reasoned about the equivalence of the objects they produced with transformations. Our findings extend the theoretical scope of an existing equivalence framework and suggest that these ways of reasoning can inform efforts to help students overcome the widespread reports of difficulties they experience. We conclude with a discussion of the theoretical implications for research on equivalence transformations across K-16 mathematics. • Conducted interviews with undergraduates to examine their reasoning about equivalence transformations. • As equivalence transformations are so pervasive in, it is critical to identify through-lines in students' reasoning. • We set forth useful criteria for productive activity with equivalence transformations. • We illustrate and describe three unified ways of reasoning about equivalence and identify their affordances. [ABSTRACT FROM AUTHOR]*