Describing Reflections
When
things are placed in front of a mirror, we see a reflection. In geometry, a shape in the plane can be
reflected across a line of reflection or a mirror in the plane to give an
image. In this activity you will learn
properties of reflections. |
Students should be placed
into groups of three or four and the following materials should be provided for
each group:
• 1 Mirror
• 1 Plane Shaped Polygon
• The Geometers Sketchpad software
Begin
the exercise by asking students to place the plane shaped polygon in front of
the mirror and then describe what they see. Students should express the
reflection (the image) as appearing to be the plane-shaped-polygon (the
pre-image) that was flipped over the mirror.
Some may go on to explain that the image appears to be at a distance
behind the mirror equal to the pre-image’s distance from the mirror. With this preliminary understanding of a
reflection, the following questions should be posed and further guided
explorations on The Geometers Sketchpad should be conducted to help answer
these questions.
1. What
information must be given to describe a reflection?
2. How can you describe the process of reflecting an
object or shape?
The first step is to understand
how a point
is reflected. Click
here for a GSP activity
Upon completion of this
exercise, have students write down the most important things that they learned
while conducting this investigation.
They will build upon this exercise to establish a definition of the
reflection of a plane
shaped polygon within the plane.
Click here for a GSP activity
At this point students should
have enough experience to enable them to answer the previously posed questions.
1. What
information must be given to describe a reflection?
2. How can you describe the process of reflecting a
plane shaped object or shape?
Once the students have come
up with their conclusions, discuss the definitions amongst the class and
finally present and compare with the following formal definition.
A transformation of a plane is a mapping from the
plane to itself such that
a. no two
points have the same image
b. each point
in the plane is the image of a point in the plane
If g is a
line in a plane, a reflection with axis
g is a transformation of the plane where each point P has image P’ such that
a.
if P is not on g, then g is the perpendicular bisector of PP’
b.
If P is on g, then P is a fixed point and P=P’.
This information provides a
base that is uses full in determining optimal angle settings for department
store three-paneled mirrors.