Geometric Reflections: Why Should They not be Just a Flip?

Grade 8 students were given a task that had pairs of shapes that had been transformed (i.e., preimages and images). Students were to identify which of the three rigid transformations--reflections, rotations, or translations--were used to map the preimage to the image. Students were also asked to give a justification for their choice of transformation. Using only a single transformation, the answer is that the preimage has been rotated to create the given image. However, Ava, a student in the class, identified a pair of shapes as representing both a rotation and a reflection. How many times have students answered a question similarly? Why do students think that this is a reflection? Is there something within the mathematics of reflections that students do not understand that causes them to respond that figures are reflections? In this article, the authors explore this misconception, give possible reasons why students struggle to differentiate between a reflection and rotation, and share ideas for teachers to elicit and discuss these misconceptions with students.