Must two triangles with equal areas and equal perimeters also be congruent? This question was introduced in "Mathematics Teacher" ("MT")by Rosenberg, Spillane, and Wulf in their article "Heron Triangles and Moduli Spaces" (2008), which also described the authors' subsequent investigation of a particular moduli space of triangles. McCallum (2009) suggested that this was likely the only "MT" article to address a moduli space, a representation arising naturally across various branches of advanced mathematics. His comment, along with the original article, inspired this author to explore the prerequisite topics of equivalence relations and equivalence classes and their appearance in secondary school mathematics. In particular, the author describes an exploration of equivalence relations in the high school curriculum that builds explicit connections across various mathematical domains and provides teachers with a greater depth of understanding of the mathematics they teach.