Overview: The teacher goes over page 1 of the perspective drawing problem in this summary.  The teacher asks students how they determined whether the houses in the diagram were the same height, and Elena offers her solution of drawing a perspective line through a common point in the roofs of the two houses.  Because the perspective line intersected the vanishing point, Elena says she determined the houses were the same height.  The teacher illustrates this on the board.  The teacher then tells students that the relationships in the diagram are that corresponding parts are in proportion.  The teacher draws a line segment to illustrate that the bases of the houses and the heights of the houses are in proportion.  The teacher tells students that the vanishing point in the diagram is the center of dilation, and tells students a dilation is a transformation that makes objects bigger or smaller.  The teacher also introduces the vocabulary of scale factor to describe how objects become bigger or smaller, like the houses.  The teacher tells students that making a drawing in 1-point perspective is equivalent to drawing similar figures that share a common center of dilation.  The teacher uses a line segment through the trees to illustrate objects that are not drawn in perspective and are not similar.  The teacher closes by telling students that for homework they will need to make a drawing in 1-point perspective.

Prior knowledge: The teacher in this summary attempts to make a connection between the perspective drawing problem and students’ prior knowledge of the school mathematics concept of transformations.  The teacher addresses the similar figures with a transformations-based approach, describing the dilation between the houses, rather than from a static approach (Seago et al., 2013).  The teacher uses the mathematical practice of drawing perspective lines to illustrate the dilation that relates the two houses, and to illustrate that the trees are not related by a dilation.  The teacher also attempts to connect the new concept of dilation with the context of the problem by telling students that a 1-point perspective drawing is like drawing similar figures with a common center of dilation.

Other points of interest:  The teacher seems to want to illustrate the concepts of center and dilation and scale factor through a pattern of IRE (Mehan, 1979) or triadic dialogue (Lemke, 1990) with students.  Galen shifts the interaction to student questioning (Lemke, 1990) when he asks about the case where the scale factor is 1.  Galen’s question is valid, but it seems that the teacher is more interested in illustrating cases where the scale factor is greater than or less than 1.  Also, the teacher makes a connection between the 1-point perspective diagram and the mathematical concept of center of dilation, especially by labeling the vanishing point on the diagram as, “center of dilation.”  The teacher may be establishing a metaphor with the diagram to help students recognize dilations in future work (González, 2013).