Overview:  Ryder, Galen, Tivona, and Harriet all have essentially the same diagram on their worksheets.  They have aligned the light source horizontally with the shadow puppet to create a right triangle.  The students have marked given distances, including the 18-inch height of the puppet, the distance of 24 inches from the puppet to the wall, and the distances of 16 inches from the top of the puppet to the top of the shadow.  Harriet suggests using the Pythagorean theorem, and Ryder takes up this suggestion and begins working on computations.  The students need to find the length of the hypotenuse of the right triangle in order to determine the distance from the light source to the puppet, but they do not have enough information to finish this computation.  In the meantime, Tivona recognizes a pair of similar triangles in the diagram.  The students work together to justify that the triangles are similar, and then they find the scale factor between the two triangles to determine the distance from the light source to the puppet.

Prior knowledge:  The students use prior knowledge of different concepts in school mathematics.  First, when the students recognize that they have a right triangle, they attempt to use the Pythagorean theorem.  Tivona is persistent about recognizing similar triangles in the diagram, so when students realize the Pythagorean theorem is not a viable strategy for solving the problem, they rely on their prior knowledge of similarity.  The students use knowledge of the sum of interior angles of a triangle, complementary angles, and vertical angles to justify that they have similar triangles.  Finally, they use knowledge of finding the scale factor between two similar triangles to determine the missing length.

Other points of interest:  Although students are all working with the same diagram, there are initially two competing strategies for solving the problem.  Ryder and Harriet promote the method of using the Pythagorean theorem, while Tivona suggests using similar triangles.  To make the shadow puppets problem a novel task for students (Doyle, 1988; Herbst, 2003), the teacher has not been explicit about what mathematical concepts or procedures students should use to work on the problem.  Because of that, the students use a variety of concepts and procedures that may not be typically treated within a single lesson in a geometry class.