Overview:  Allen, Khalil, Paige, and Alejandra are standing up by the wall next to their desks, and they are working on modeling the shadow puppet situation with a flashlight and some rulers.  Allen holds the flashlight, while Paige holds a pair of rulers that have been taped together to represent the 18-inch long shadow puppet.  Khalil is standing by the wall to measure the length of the shadow.  Alejandra uses a ruler to measure the distance between Paige (holding the puppet) and Allen (holding the light source).  First, the students direct Paige to move back and forth, until they recall that the shadow must remain a fixed distance from the wall.  Then the adjust Allen until the shadow measures the appropriate length.  Allen is concerned that the shadow on the wall is very faint, and he argues that they might be able to make the shadow more pronounced if they were to move the light closer to the puppet.

Prior knowledge:  Allen, Khalil, Paige, and Alejandra use prior knowledge of the mathematical practice of modeling to solve the shadow puppet problem.  With the tools provided to them, students use empirical evidence to determine the appropriate distances between the light source, the shadow puppet, and the wall.  The students in the group also seem to have some prior knowledge of the context of the problem, because they make predictions about which direction they will need to move the puppet and/or the light source to make a larger shadow projected onto the wall.  Allen seems to bring prior knowledge of light and shadows to his work on the problem.  Allen is concerned throughout the episode that the shadow is “really blurry,” and that they may be able to fix that issue by moving the flashlight closer to the shadow puppet.  Allen’s comments seem to reflect a misconception about light and shadows that increasing the distance between the light source and an object will create a diffused shadow (Eshach, 2003).  Allen’s understanding of light and shadows becomes especially salient for the practice of modeling the problem.

Other points of interest:  In this Explore we see a much different approach to solving the shadow puppet problem than more typical solutions involving diagrams, ratios, and proportions.  Because students had tools available—such as the flashlight—they have a different entry point towards solving the problem.  This Explore illustrates the idea that students’ understandings in geometry are shaped in important ways by the tools they use (National Council of Teachers of Mathematics [NCTM], 2012).  This is true for more typical geometry tools, such as compasses or dynamic geometry software, but it is also true when students have novel tools to model a situation.