Overview: In this summary, the teacher contrasts two potential diagrams for representing the shadow puppet problem.  In the diagrams that most students have drawn to work on the problem, the light source is aligned horizontally with the bottom of the shadow puppet and the bottom of the shadow, creating a right triangle.  The teacher asks students to consider this diagram in light of the context of the problem and points out that the right-triangle diagram would require the person holding the light source and puppet to be on the ground.  The teacher argues that a more realistic set-up would be to locate the light source so that it is horizontally in the middle of the shadow puppet and that shadow.  This would imply that light would extend both above and below the shadow puppet, creating an obtuse triangle.  With the diagram of obtuse triangles, the teacher points out that everything in the diagram extends from the light source, prompting a potential discussion of the center of dilation.

Prior knowledge: The teacher establishes a connection with the mathematical practice of setting up a diagram to represent the situation.  Selecting an appropriate representation for a problem in mathematics is part of the practice of reasoning and sense making.  The teacher attempts to highlight the merit of the obtuse-triangle representation compared to the right-triangle model, particularly because the obtuse-triangle representation better reflects how the light source would be positioned in real life.  By drawing upon the mathematical practice of representation the situation, the teacher seems to create an opportunity to connect with the school mathematics concept of center of dilation and scale factor.  The teacher hints at this idea, saying, “everything extends from point A.”  However, the teacher does not establish any new school mathematics concepts, but instead suggests they will continue their discussion the following day.

Other points of interest:  The teacher seems to want to direct students away from working with right triangles in this problem.  From our observations of students’ work in a focus group, we have seen that geometry students were most inclined to set up right-triangle diagrams to solve the shadow puppets problem.  However, to introduce the concept of dilations in similarity, a teacher may want to use a triangle that is not a right triangle, in order to make the center of dilation and scale factor more salient.  In addition, the teacher may be able to establish a metaphor with the light source and the puppet to establish a new prototypical image (González, 2013; Presmeg, 1992) of similar triangles.  If the teacher can use the light source and shadow puppet as a metaphor for dilations in the future, then students may recognize similar triangles and dilations without needing to justify the similarity relationship every time (González, 2013).  Recognizing similar triangles and center of dilation in the obtuse-triangle set-up can potentially reduce the cognitive load of future tasks for students.