Mathematical Investigation

Three-paneled mirrors are used in clothing stores for the purpose of helping a customer when trying on clothes.  The shopper is interested in having an all around view of their body.  He or she wants to be able to see their front, each side, and their back.  Three-paneled mirrors are expensive and must be positioned properly in order to be fully utilized, particularly, so the customer can have a backside view.

This is the part of the investigation where students will simulate three-paneled mirrors using The Geometers

Sketchpad.  Preceding work on GSP, students should be broken up into groups and each group should be given a hinged set of three mirrors.  A mini version of a three-paneled mirror can be easily constructed by obtaining three pieces of mirror from a local hardware store. (Place cardboard on the back of each mirror, put tape around the edges and use tape to hinge the mirrors together)  Instruct students to open and close the hinged mirrors and make note of what is happening to the reflections as they are doing this. They will notice that as the mirrors are closed the reflections increase and as the mirrors are opened the reflections decrease.  In addition, have students figure out a way to obtain a view of the back of their head.  While playing with these mirrors, students should be thinking of themselves as a customer trying on clothing and noticing which angles are optimal.  There are two ideas that that must be embedded upon completion of these hands on mirror exercises.

1.  There are certain angles at which the mirrors will give unnecessary reflections of the pre-image.  These extra reflections may become distracting to a customer trying on clothing.

2.    The reflections of the front and each of the sides are easily obtained.  The image of the backside requires a rotation or as previously discovered a composition of two reflections.

With these ideas in mind, students will now work with simulations of the mirrors on GSP.  Click here for GSP exercise

Upon completion of the above exercise students should begin to speculate as to what they would recommend for optimal angle setting when considering the front and sides only.  In addition, they must provide the reasoning used in coming to these conclusions.

Now it is time for students to make the final consideration.  How do you best obtain an image of the back?  How does this compare to the optimal angle setting for the images of the front and sides?  Are you obtaining extra images that are not necessary?  What is the over all best recommendation?  Click here for GSP exercise

At this point students should write down, informally, their assumptions and reasons for the optimal angle setting of three-paneled mirrors.