DRAFT
Teacher Notes
Kaleidoscopes
Teacher Notes
The Kaleidoscopes section of the geometry unit is an application of finite designs. In this activity,
the students will explore the mathematics behind kaleidoscopes and then build their own
kaleidoscope.
Students will investigate the idea that multiple reflections in intersecting lines create a rotation and
also revisit finite designs from the last section of the geometry unit. These ideas are the
mathematical basis behind kaleidoscopes. They will also explore how a change in the angle
measure of intersecting lines changes the number of images that occur in the finite design.
Students should be told in advance that they need to bring small materials (beads, candy, sequins,
rocks, ribbon, paper clips, marbles, seeds, small seashells, small leaves, small pasta, etc.);to view in
their kaleidoscope so that everyone will have something to put in the object chamber.
Instructional Time Required for this Section
One or two hours
Required Materials
•
•
•
•
•
•
•
•
•
•
•
•
•
•
student activity sheets, one per student
one reflector (a Mira or similar tool) per group of 2-3 students
three mirrors (approximately 4”x6”) per group of 2-3 students or one hinged mirror
assembly (with 4”x6” mirrors) and one mirror (approximately 4”x6”) per group of 2-3
students (see the Resources section of these Teacher Notes for more information on
mirrors)
Transparencies of student worksheets:
parallel reflections worksheet
intersecting reflections worksheet
angles worksheet
pattern blocks, small clip art, or other suitable small objects or designs for the students to
view while completing the angles part of the activity; enough for each group to have
enough “stuff” to view in their mirrors
protractors, one per group of 2-3 students
Building a Kaleidoscope
1”x3” reflective surfaces, three per student (see the Resources section of these Teacher
Notes for more information on materials to use for the mirrors)
scotch tape (several rolls for the class)
newsprint, one sheet per student (filler to keep the mirror system in place)
toilet paper roll, one per student
condiment cup with lid, one per student (the opaque or transparent plastic kind; Dixie is a
brand name; available at Sam’s Wholesale for example)
2” square (approximately) of transparency film or other transparent, reasonably stiff
plastic, one per student
overhead pens, several for the class
contact paper, one 4.5”x6” piece and one 3”x3” piece per student
Mathematical Models with Applications, Fine Arts Module
Geometry Unit, Kaleidoscopes
73
DRAFT
•
•
•
•
Teacher Notes
hole punch (single hole), one or two per class
hot glue guns, 3-4 per class with sufficient hot glue sticks
scissors, several pairs per class
materials to put in condiment cup for viewing (beads, candy, sequins, rocks, ribbon, paper
clips, marbles, seeds, small seashells, small leaves, small pasta, etc.); students should bring
a small handful for their kaleidoscope
Optional Materials
•
interesting designs to view between angles mirrors (such as fractal designs, geometric
designs, colorful magazine/catalog covers)
Procedures
Before the day of this lesson, practice building a Kaleidoscope using the directions on the activity
sheet.
Kaleidoscope Activity: Multiple Reflections
The first part of this activity is learner-oriented. The teacher’s role during this activity is to monitor
students’ progress and respond to additional questions that may arise.
Prior to distributing the first activity and associated materials, it may be necessary to review the use
of Miras or similar reflector and protractors depending on students’ experience with these tools.
Distribute the activity sheet and the three worksheets, Parallel Reflections, Intersecting Reflections,
and Angles. Allow time for students to work on the activities and summarize their conclusions.
Building a Kaleidoscope
Before class begins, set up the room like a lab; indicate one area where students will cut the
transparency pieces they need, one area where they will be using the hot glue guns, one area where
they will be using the hole punch, and one area where they will be using the scotch tape and
newsprint.
Distribute the activity and discuss the directions with the students. Distribute the supplies: mirrors,
toilet paper roll, contact paper, condiment cup. Building the mirror system and placing the eyepiece
square over the end of the toilet paper roll may need to be demonstrated for the students before they
begin. Allow the students to build their kaleidoscopes. Monitor their activities and respond to
questions.
As students finish building their kaleidoscopes, have them share their kaleidoscopes with each
other. It is interesting to see how the different combinations of material create unique
kaleidoscopes.
Ask the students whether they can identify the fundamental unit of the design created by the
kaleidoscope. Also ask whether what they see in their kaleidoscope is what they expected to see (in
terms of mathematical structure); do they see a tessellation or just a finite design?
Evaluation
Mathematical Models with Applications, Fine Arts Module
Geometry Unit, Kaleidoscopes
74
DRAFT
Teacher Notes
The first activity was used a homework grade in the pilot course. Building a kaleidoscope is the
major project of this section and was used as one-third of the unit test grade (with the Symmetry
Project). The kaleidoscope was evaluated on construction – ability to construct, accuracy in
following directions, and neatness in construction. Additional questions would need to be asked to
evaluate the mathematical content.
Mathematical Models with Applications, Fine Arts Module
Geometry Unit, Kaleidoscopes
75
DRAFT
Teacher Notes
Kaleidoscope Resources
Mirrored Surfaces
Clearly, glass mirrors are one option for the reflective surfaces in this activity. Front surface
mirrors are the best option because they give the truest images; however, they are more expensive
than regular mirrors. While glass mirrors (front surface or regular) are fine for the 4”x6” mirrors
needed in the first activity, their width (thinnest is generally 1/8”) and weight (and expense) are
drawbacks to using them in building the kaleidoscopes.
Mirrored acrylic is a good substitute for glass mirrors. ETA-Cuisenaire manufactures a hinged
mirror, Reflect-It Hinged Mirror, that is two thin pieces of acrylic hinged together. Despite the fact
that the mirrors are square, this tool works very nicely for the first activity. Creative Publications
offers a bucket (24) plastic 4”x6” mirrors. Mirrored acrylic is also available in various thickness
from plastics distributors. The thinnest is generally 1/8”. Most plastics distributors will charge to
cut the plastic but it is less expensive than glass. Since the thinnest acrylic is 1/8”, it is too thick to
work well in the kaleidoscope.
The best option for the kaleidoscope mirrors is mirrored plastic or mirrored styrene. This is thin
plastic with the reflective surface mounted on the front. Dale Seymour offers a package of cut-able
mirrors that are made from this type of material. The mirrors Dale Seymour offers are two-sided
so you “waste” one reflective side if you use their product. These mirrors are also a little thick to
cut easily. Other companies have similar products. Plastics distributors can order mirrored styrene
in large sheets. These sheets are easily cut with a paper cutter that makes this the most cost
effective option. If the plastics distributor is unaware of mirrored styrene, ask them to contact
Advanced Technology in Greensboro, North Carolina about mirrored styrene. Regal Plastics in
Fort Worth was quite helpful. The drawback to going this route is the quantity of styrene that you
must purchase.
If you are unable to locate mirrored styrene, sheets of mylar or butyrate mounted on poster board
or mat board. The butyrate should be mounted on white or pale blue. Sheets of mylar can
sometimes be found in hobby stores but can also be purchased at a plastics distributor. Sheets of
butyrate are generally available at craft/hobby stores; the best thickness is 0.15 millimeters. The
reflections are not as true as with mirrored plastic but are acceptable. If using mylar, use spray glue
(in aerosol cans) in a light coating and carefully smooth out wrinkles in the mylar. Mount the
mylar or butyrate before cutting the 1”x3” mirrors. Silver wrapping paper is another, less effective
but reasonable, option for a reflective surface in the kaleidoscopes.
There are other options, both cheaper and more convenient. Check the local craft supply and art
supply retailers.
ETA-Cuisenaire
Creative Publications
www.etacuisenaire.com
www.creativepublications.com
In addition to the worksheet masters that follow, you may find the following resources useful or
informative:
Baker, Cozy. 1993. Kaleidoscope Renaissance. Beachcliff Books: Annapolis, Maryland. ISBN:
0-9608-9304-0
Bennett, Carolyn. 1994. The Kids’ Book of Kaleidoscopes. Workman Publishing: New York.
Comes with materials to make a kaleidoscope. ISBN: 1-5630-5638-0
Mathematical Models with Applications, Fine Arts Module
Geometry Unit, Kaleidoscopes
76
DRAFT
Teacher Notes
Boone, Symbra. Spring 1995. “A Kaleidoscope of Activities” in The Elementary Mathematician.
COMAP: Lexington, Massachusetts.
Boswell, Thomas (ed.). 1992. The Kaleidoscope Book: A Spectrum of Spectacular Scopes to
Make. Sterling Publishing: New York. ISBN: 0-8069-8370-1
Kennedy, Joe and Diane Thomas. 1989. Kaleidoscope Math. Creative Publications: Sunnyvale,
California.
Newlin, Gary. Simple Kaleidoscopes: 24 Spectacular Scopes to Make. Sterling Publishing: New
York.
Newlin, Gary and Ledell Murphy. 1992. Cheap Thrills: 12 Exciting Low-Cost Kaleidoscope
Projects. Authors: Asheville, North Carolina.
If you are interested in purchasing kaleidoscopes, MathArtFun.com sells a few. There are quite a
few sites on the Internet that sell kaleidoscopes. There were some that lightly discussed the
mathematics, but most were strictly commercial. Because there were so many and they were
primarily commercial, they are not listed here.
Mathematical Models with Applications, Fine Arts Module
Geometry Unit, Kaleidoscopes
77