The authors who are mathematics teacher educators, have found that classifying, composing, and transforming shapes (in particular, rotations and reflections) are areas of difficulty for adults as well as for children. However, these are also some of the most important geometric ideas. They are fundamental topics in the K-8 Geometry and Measurement Standards for many reasons, including their powers to elicit conjectures, attend to limitations of generalizations, and develop a solid understanding of area and perimeter. This article describes a task that has helped their students gain a deeper understanding of (and appreciation for) these concepts. Interesting to children and adults alike, the task addresses the third of the eight important Standards for Mathematical Practice (SMP 3) by providing an opportunity for forming and testing conjectures and critiquing the arguments made by others (NGA Center and CCSSO 2010). The authors first provide an overview of the Flipping Out and Turning Up task. Then they will describe the potential of this task in terms of generating conjectures and discussion about classifying and transforming shapes. Throughout, they share their ideas about how to modify the task to meet a variety of instructional contexts and preferences while still maintaining the mathematical potential of the task.