Overview:  The teacher gives students a new task to work on at the conclusion of their work on the pottery problem.  On the board, the teacher draws an irregular polygon and a line of reflection that is slanted, neither vertical nor horizontal.  The teacher invites a student to the board to reflect the polygon over the line.  Taylor solves the problem at the board.  Taylor makes lines from the vertices of the given figure perpendicular to the mirror line, but Taylor translates the figure across the line rather than reflecting the figure over the line.  The teacher erases Taylor’s image and draws a reflection, but students do not see how the teacher’s solution is different from Taylor’s.

Prior knowledge:  It seems that, by now, students have prior knowledge of the school mathematics concept of perpendicular bisectors in the context of working on a problem about reflective symmetry.  The teacher says to students, “Now you know something about mirror lines,” although it is not clear if the class has already talked about mirror lines together or if the teacher assumes that students discovered the relationship between the mirror line and the perpendicular bisector through their work on the problem.  The teacher seems to be attempting to extend students’ knowledge of the mirror line beyond the prototypical orientations (vertical or horizontal) so that students can apply this knowledge in a novel set-up.

Other points of interest:  When Taylor constructs a translation across what should be the mirror line, the teacher erases Taylor’s solution rather than presenting a new solution next to Taylor’s.  This may make the teacher’s work more difficult, because after the teacher presents the reflection of the figure, students do not see the difference between Taylor’s solution and the teacher’s.  Students’ difficulties may be due to the fact that Taylor used perpendicular lines in her solution, as did the teacher.  Taylor’s use of the perpendicular lines may have been an opportunity for the teacher and students to discuss the difference between a translation and a reflection and why the perpendicular bisector is only especially relevant for reflections.  The teacher’s solution on the board does not explicitly relate to the students who used paper folding to solve the problem, and the teacher disregards the process of measuring the line segments with a ruler.  These choices may make it difficult for students who used those strategies to relate to the solution the teacher presents on the board.