Overview:  With this summary the teacher tells students that what they’ve really been working on is the mathematical concept of dilations.  The teacher gives students the definition of dilation as a transformation determined by a center of dilation and a scale factor.  The teacher draws a diagram and compares the light source from the shadow puppets problem to the center of dilation.  The teacher tells students that the scale factor in the problem was 34/18.  Khalil asks why they need to call the triangles a dilation instead of calling them similar.  The teacher responds to Khalil by saying that they are studying a new type of transformation in which the shape of an object stays the same but the size changes.

Prior knowledge:  The teacher attempts to connect students knowledge of the context of the problem with the school mathematics concept of dilations.  The teacher uses students’ placement of the light source, shadow, and puppet, to illustrate the ideas of a center of dilation and scale factor.  The scale factor is given by the ratio of the height of the shadow to the height of the shadow puppet.  In this summary, we do not know whether all students in the class set up a diagram to represent the context of the problem in the same way the teacher did.  The teacher seems to assume that all students have used basically the same diagram to represent the context, and she uses that as a starting point to introduce the mathematical concept.

Other points of interest:  The teacher seems to be establishing a metaphor between the shadow puppet problem and a prototypical diagram to represent a dilation (González, 2013).  This metaphor becomes apparent when the teacher asks, “where in the shadows problem did we have a center of dilation?” and Sabrina responds, “like the flashlight.”  It seems that the teacher and students establish a metaphor that the center of a dilation is like the flashlight, and the scale factor is like the beams of light coming from the flashlight.  In Summary 1 we can infer from the teacher’s work with the diagrams on the board that the teacher may want to establish a metaphor from the shadow puppet problem to the mathematical concept of dilations.  In this Summary, the teacher makes this connection much more explicit for students and does not consider alternative diagrams the students may have used.  Khalil raises an important question at the end of Summary 3 when he asks why the language of dilation is necessary when it represents an idea very much like similarity.  The teacher tries to emphasize that “dilation” refers to a mathematical transformation (as opposed to “similarity”, which refers to a property of figures).  The teacher seems to want to bring more focus to viewing similarity in terms of transformations, perhaps to align with the focus on transformations in the Common Core State Standards for Mathematics (NGAC, 2010).