Overview:  This Explore is organized into three parts corresponding to students’ work on the three parts of the pottery problem.  Miguel, Jonah, Gabby, and Liza work on the problem.  Throughout all three parts of the problem, Miguel identifies a reflection line and uses perpendicular bisectors from given vertices to reflect points from the diagram or part of the diagram.  It does not seem that Miguel is using a protractor or any formal construction to construct a line that is perpendicular to the reflection line.  Gabby is using paper folding with the tracing paper to make the reflections.  Because of the way Gabby is folding the paper, copies of her design show up on the front and the back side of the tracing paper.  In part III of the problem, Gabby decides to try something new and connects the open vertices of the given diagram.  Liza and Jonah have both traced the given diagram onto the tracing paper in part I of the problem, but neither student has done anything to solve the problem yet.  Liza and Jonah also do not have a solution to part II.  On part III of the problem, Jonah draws the line segment in the bottom right corner of the figure.  The teacher intervenes throughout the Explore only to encourage students to talk to one another and find out what each person is doing.

Prior knowledge:  Throughout the Explore, Miguel uses prior knowledge of school mathematics, specifically the knowledge that a point and it’s reflection are connected by a line segment that is perpendicular to the line of reflection and that is bisected by the line of reflection.  Gabby seems to use prior knowledge of the mathematical practice of paper folding in parts I and II of the problem.  In part III of the problem, Gabby uses prior knowledge of the context to make a figure that “looks like a fish or something,” possibly because her strategy for solving parts I and II did not transfer easily to part III of the problem.  Gabby’s comment at the end of the Explore about trying different ideas also indicates that Gabby is using knowledge that the task should be a creative task.  When Jonah draws in a line segment in part III of the problem, he uses a mathematical practice of visually perceiving what the reflected image should look like.  Jonah also seems to use some prior knowledge of school mathematics when he says in part III, “you can just measure this part and do the same here.”  It seems that Jonah is using knowledge that a pre-image and its reflection are congruent, which in the case of line segments means they will have equal length.

Other points of interest:  With Jonah’s partial solution in part III, he determines the unique line of reflection for the figure.  By filling in the line segment he does, Jonah eliminates other possible lines of reflection and determines that the line of reflection will run vertically up and down the given figure.  Jonah does not make any comment to make it explicit whether he recognizes where the line of reflection is in the figure.  In the case of Miguel, it is not clear whether he has learned about perpendicular bisector in the context of reflections at some time in the past, or whether Miguel is developing these ideas about perpendicular bisector as he works on the problem.