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.