Overview:  The teacher attempts to establish a connection between students’ work on the pottery problem and the existence of seven different frieze patterns in mathematics.  The teacher points out at the start of the summary that the key mathematical concept students should learn is perpendicular bisector.  With that, the teacher transitions to say that students’ patterns were an example of frieze patterns, and the teacher asks students to guess how many unique frieze patterns there are.  The teacher shows the seven different frieze patterns, and he asks students to identify which pattern their work is an example of.  The teacher then returns to the idea of perpendicular bisector, and he points out that the mirror line is the perpendicular bisector of the segment connecting a point and its reflection.

Prior knowledge:  The teacher attempts to make a connection with the mathematical practice of using different symmetries to make patterns.  By introducing the idea of frieze patterns, the teacher seems to suggest that the work students have done on the problem is one example of a much larger mathematical phenomenon, and one that his implications for art and design.  The teacher also establishes the school mathematics concept of perpendicular bisectors.  The concept of “mirror lines” seems to be the concept that the teacher uses to bridge the knowledge of the frieze patterns with knowledge of the perpendicular bisector.  At the end of the Summary, the teacher recalls a new element of students’ prior knowledge of school mathematics by telling students that they will need to use distance in a problem about perpendicular bisector.

Other points of interest:  By introducing frieze patterns, the teacher may be trying to show students how their work on the problem was part of the larger discipline of mathematics.  However, it is not clear that the teacher has made this connection very obvious.  The question of how many unique frieze patterns there are is not a question that draws upon the knowledge students may have gained through their work on the problem.  Although the teacher is excited that there are only seven unique frieze patterns, students may not have the relevant prior knowledge (or not have had enough time to think about it) to be impressed by the idea of only seven unique patterns.  Also, at the end of the Summary, the teacher introduces a new task in which students will need to think about perpendicular bisector in terms of distance.  Based on Liza’s comment at the end, it seems that students may be struggling to keep up with all the shifts in the teacher’s ideas.