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The Inquiry-Oriented Linear Algebra (IOLA) project promotes a research-based, student-centered approach to the teaching and learning of introductory linear algebra at the university level. Based on the instructional design theory of Realistic Mathematics Education, the IOLA curricular materials build from a set of experientially real tasks that allow for active student engagement in the guided reinvention of key mathematical ideas through student and instructor inquiry. The online instructional support materials include resources such as rationales for task design, implementation suggestions, and examples of typical student work. In this talk, I will share some IOLA tasks and associated examples of student reasoning, as well as some guiding principles for inquiry-oriented instruction.