Overview:  This Explore is organized into three parts corresponding to students’ work on the three parts of the pottery problem.  Elijah, Jeff, Taylor, and Leah are working together.  Throughout the three parts of the problem, Elijah uses perpendicular lines from given vertices to construct the reflections of those vertices.  Elijah uses his ID card as a straightedge, and he uses a compass to construct line segments of equal length.  On the third part of the problem, Elijah identifies a different line of symmetry than the other students in the group.  On all three parts of the problem, Jeff uses the artistic context to create patterns that look appealing to him.  In parts I and II of the problem, Taylor draws the reflections of the given figures by visually estimating how the reflection looks over the line of reflection.  In parts I and II of the problem, Leah uses translations instead of reflections to create the pattern.  She makes the translations by tracing the given image onto tracing paper, sliding the tracing paper, and tracing again.

Prior knowledge:  Elijah seems to use prior knowledge of the school mathematics concept of perpendicular bisector in the context of reflections.  Elijah seems either to already know, or to develop an understanding through working on the problem, that a pre-image and its image under a reflection are connected by a line segment for which the line of reflection is the perpendicular bisector.  Jeff uses prior knowledge of the context of the problem, that the problem calls for an “artistic creation.”  Taylor uses prior knowledge of the mathematical practice of visually estimating the image of a figure under a reflection.  Leah seems to use prior knowledge of school mathematics, specifically knowledge of transformations.  In parts I and II of the problem, Leah uses translations instead of reflections.

Other points of interest:  Elijah’s conception is similar to that of Miguel in Explore 3.  The difference between the two students is in the tools they use.  Miguel uses a ruler as a straight edge and to measure line segments of equal length.  Elijah uses his ID card as a straight edge, and he uses a compass to construct congruent line segments without having to measure.  Leah’s’ use of translations has different results in parts I and II of the problem.  Because the leaf design has internal reflective symmetry, the images of Leah’s translations look the same as if she had reflected the leaf.  However, because the figure in part II does not have internal symmetry, when Leah translates the leaf she does get a pattern with reflective symmetry.  It is not apparent in the vignettes whether Leah recognizes this difference.  In part II of the problem, the teacher approaches the group to assess what each person has been doing.  The teacher pushes students to consider whether the figures should all be facing the same direction, or whether they should be facing different directions.  In this way, the teacher seems to try to provoke students to consider whether their patterns have reflective symmetry.  The teacher seems not to want to disregard the solutions students have produced, because the students understand that they can all have different solutions.  At the same time, the teacher may want to provoke students to use reflections so that the concept of perpendicular bisector can become more salient.