An Inquiry-Oriented Approach to Determinants

Abstract

In introductory linear algebra classrooms, determinants are often taught in a formulaic way that obscures their rich connections to graphical interpretations of linear transformations. As an alternative, in this paper, we introduce an innovative task sequence exploring the determinant conceptualized as a measure of the distortion of space. The task sequence is part of the Inquiry-Oriented Linear Algebra (IOLA) curricular materials, which build from experientially real tasks that allow for active student engagement in the guided reinvention of key mathematical ideas through student and instructor inquiry. The task sequence begins with students developing a conceptualization of the 2 × 2 matrix determinant as a ratio between the (signed) area of an image object and the area of its pre-image object transformed by the associated linear transformation. They then use this conceptualization to construct for themselves the 2 × 2 determinant formula and the determinant of a matrix inverse. Finally, students explore 2D and 3D GeoGebra applets to reinvent and explain core theorems related to the determinant involving linear dependence and switching/scaling rows and columns. Student work examples are provided throughout the paper.