Understanding fraction as a quantity has been identified as a key developmental understanding. In this study, students in Grades 5, 8, and 11 were asked to compare the areas of two halves of the same square--a rectangle and a right triangle. Findings from this study suggest that students who understand fraction as a quantity use reasoning related to a generalization, whereas students who understand fraction as an arrangement use reasoning related to visualization, computation, or characteristics of the specific shapes involved. Knowing the reasoning exhibited by students can inform both teachers and mathematics curriculum writers in the creation of and planning for instructional tasks. [For the complete proceedings, see ED658295.]