Overview:  In this block, Conrad, Eric, Raquel, and Natalia are working on Part II of the pottery problem.  Conrad has not decided how to solve the problem, and he asks the other students in the group how they are working on the problem.  Eric drew a reflection line directly onto the worksheet, and he is using visual perception to draw the reflection of the given figure over the line.  Natalia traced the given figure onto tracing paper, and she uses paper folding to get the reflection of the figure.  Raquel traced the figure onto the tracing paper, but she is using a 180 degree rotation to create the pattern.

Prior knowledge:  Natalia is using prior knowledge of the mathematical practice of paper folding to construct the reflections.  Eric is using prior knowledge of the mathematical practice of visually estimating the reflected image (perhaps similar to what an artist, like Mr. Catori from Launch 6, may do).  Raquel seems to be drawing on some prior knowledge of school mathematics, specifically knowledge of reflections and reflective symmetry, but she seems to be applying this knowledge in a way that does not lead to a correct solution to the problem.  When Raquel says, “this one is already done, so we don’t have to reflect it,” exemplifies a conception of reflective symmetry that we saw among high school geometry students when they worked on this problem.  Some students seemed to think that, because the given figure in Part II did contain any internal symmetry, they did not need to (or could not) create a pattern with reflective symmetry. 

Other points of interest:  In this Explore, Conrad challenges Eric by suggesting that his design is not evenly spaced on the paper.  Because Natalia’s pattern is more evenly spaced, Conrad concludes that they should use tracing to solve the problem, because they can be more exact.  Conrad’s comment could lead to at least two possible outcomes.  The practice of paper folding could provoke Conrad (and other students) to consider carefully how to evenly space the design along the paper, which could lead to a discussion about measuring the distance from a point to its reflection along the line perpendicular to the fold line.  Such a discussion would be very productive in terms of making the perpendicular bisector explicit.  Alternatively, Conrad could interpret this task as a task only of “tracing” in which he traces the given figure as precisely as possible onto the tracing paper several times.  This second alternative may reduce the mathematical practice of paper folding into an activity with little mathematical substance.  This scenario raises the point that it is important to consider how students in a class view the activity of tracing and paper folding, and how the mathematical significance of that activity can be made apparent.