DRAFT
Mathematical Models with Applications
Fine Arts Module
Geometry
DRAFT
About the Charles A. Dana Center’s Work in Mathematics and Science
The Charles A. Dana Center at the University of Texas at Austin works to support education
leaders and policymakers in strengthening Texas education. As a research unit of UT Austin’s
College of Natural Sciences, the Dana Center maintains a special emphasis on mathematics and
science education. We offer professional development institutes and produce research-based
mathematics and science resources for educators to use in helping all students achieve academic
success. For more information, visit the Dana Center website at www.utdanacenter.org.
This material is based upon work supported in part by the National Science Foundation under
Cooperative Agreement #ESR-9712001, with additional support from the Charles A. Dana Center
at The University of Texas at Austin. Any opinions, findings, conclusions, or recommendations
expressed in this material are those of the author(s) and do not necessarily reflect the views of the
National Science Foundation or The University of Texas at Austin.
Permission is given to any person, group, or organization to copy and distribute this publication, for
noncommercial educational purposes only, so long as the appropriate credit is given. This
permission is granted by the Charles A. Dana Center, a unit of the College of Natural Sciences at
The University of Texas at Austin.
December, 2002
Developers
Dr. Mary Hannigan, Project Director
Tarrant County College
828 Harwood Road
Hurst, Texas
Carolyn Foster
AP Strategies
Dallas, Texas
(formerly of Plano Senior High School)
Editor
Diane McGowan
Charles A. Dana Center
Assistant Editors
Hee Joon Kim, Charles A. Dana Center
Joseph A. Bean, Charles A. Dana Center
DRAFT
Geometry Unit Table of Contents
For Teachers
Introduction
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Sequencing
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Unit Overview
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Suggested and Required Materials
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Prerequisites
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TEKS Objectives
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TAKS Objectives
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Symmetry of Design
Teacher Notes
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Required Materials
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Optional Materials
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Vocabulary
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Procedures
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Resources. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Transparencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
Student Activities
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Using Transformations to Create Designs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .24
Strip Designs
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Analyzing Strip Designs
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Finite Designs
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
Symmetry Project
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
Answers to Activities
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Kaleidoscopes
Teacher Notes
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Instructional Time Required for this Section
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Required Materials
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Optional Materials
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Procedures
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Evaluation
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Resources
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Student Activity
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Answers to Activity
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Anamorphic Art
Teacher Notes
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Instructional Time Required for this Section
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Required Materials
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Optional Materials
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Vocabulary
Procedures
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Evaluation
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Resources
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Transparencies.
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Student Examples of Anamorphic Art. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Student Activity
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National Council of Teachers of Mathematics Article and Activity . . . . . . . . . . . . . .
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DRAFT
Geometry Unit Introduction
This unit investigates the geometric structure behind designs. While students may be familiar with
geometric transformations, they likely have not studied how those transformations both create
designs (other than tessellations) and are used to classify these designs.
During the course of this unit, students will analyze border and finite designs to discover the
existence of the transformations within the designs and learn to classify designs by the
combinations of transformations that exist. They will also explore the connections between
kaleidoscopes and finite designs.
The unit concludes with an investigation of a non-rigid transformation, a distortion, used to create
interesting works of art. This artistic style is known as anamorphic art.
Sequencing
The recommended sequence for this unit is:
Symmetry of Design
Kaleidoscopes
Anamorphic Art
Each of the sections could be taught without the other two; however all three sections together
create a collection of activities that connect geometry to fine arts topics. The Finite Designs activity
in the Symmetry of Design unit sets up the ideas for studying kaleidoscopes, so should precede
that section.
The Geometric Music activity from the Music Unit could be placed in this unit if desired. If
students have limited experience with transformations prior to the music unit, the Geometric Music
activity could be placed into the Symmetry of Design section following the Analyzing Strip
Designs activity. In that location, students will have reviewed rigid transformations before being
asked to work with them in the Geometric Music activity.
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DRAFT
Unit Overview
Some group work is desirable in this unit. Students can work in groups of 2-3 to complete the
activities in the Symmetry of Design section. In the Kaleidoscopes section, students can again
work together in groups of 2-4 during the theoretical part of the activity and may continue to work
together to help each other build kaleidoscopes, but working independently is certainly an option.
Students can again help each other by working in small groups to learn the ideas behind
anamorphic art but the concluding activity will have students working largely on their own.
Students will need to bring their own materials to put in their kaleidoscope. They should be told to
bring materials at the beginning of the unit so that everyone will have it there by the time it is
needed. Students should begin collecting toilet paper rolls at the beginning of the unit to use as the
cylinder for their kaleidoscopes.
Suggested and Required Materials
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student activity sheets, one per student
transparency film (approximately one quarter sheet per student) or tracing paper (one or
two sheets per student)
overhead pen (if using transparency film), one per student
transparency of Examples of Strip Designs (located in the Resources section of the
Symmetry of Design Teacher Notes)
transparency of the homework problems from each of the Symmetry of Design activities
transparency of the Strip Design Possibilities table located in the Resources section of the
Symmetry of Design Teacher Notes
transparency of the Mandala Design located in the Resources section of the Symmetry of
Design Teacher Notes
transparency of the Mandala located in the Resources section of the Symmetry of Design
Teacher Notes
ruler, one per student
dot paper; isometric and rectangular grids
transparencies of both isometric and rectangular grids
colored pencils
reflective surface such as a mirror or other reflector (like a Mira) to assist students in
visualizing reflections
examples of strip designs such as wallpaper borders, some quilt borders, photos of friezes
in architecture, patterned ribbons, etc. (see the Resources section of the Symmetry of
Design Teacher Notes for World Wide Web sites with other examples)
examples of finite designs such as mandalas, other artwork (some Escher), logos, photos
of car wheels or hubcaps, etc. (see the Resources section of the Symmetry of Design
Teacher Notes for World Wide Web sites with other examples)
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 the Kaleidoscope Teacher Notes for more
information on mirrors)
parallel reflections worksheet, one per group
intersecting reflections worksheet; one per group
angles worksheet, one per group
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DRAFT
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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 materials to view in their mirrors
protractors, one per group of 2-3 students
1”x3” reflective surfaces, three per student (see the Resources section of the
Kaleidoscope 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
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
student activity sheets, one per student
The Secret of Anamorphic Art, article/activity from The Mathematics Teacher (see the
Resources section of the Anamorphic Art Teacher Notes for more information)
rectangular grid (located in the Resources section of the Anamorphic Art Teacher Notes);
one per student
polar grid (located in the Resources section of the Anamorphic Art Teacher Notes), two
grids per student
mirrored cylinder (see Resources section of the Anamorphic Art Teacher Notes for more
information); one per student
colored pencil or crayons
transparency of Texas on rectangular grid
transparency of polar grid
transparency of distorted outline of Texas to use in grading (located in the Resources
section of the Anamorphic Art Teacher Notes)
computer clip art or other small designs
Prerequisites
Students should have a familiarity with translations, reflections, and rotations that are originally
addressed in the elementary mathematics TEKS.
TEKS Objectives
(1)
The student uses a variety of strategies and approaches to solve both routine and non-routine
problems. The student is expected to:
(A) compare and analyze various methods for solving a real-life problem;
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DRAFT
(B) use multiple approaches (algebraic, graphical, and geometric methods) to solve
problems from a variety of disciplines; and
(C) select a method to solve a problems, defend the method, and justify the reasonableness
of the results.
(9) The student uses algebraic and geometric models to represent patterns and structures. The
student is expected to:
(A) use geometric transformations, symmetry, and perspective drawings to describe
mathematical patterns and structure in art and architecture.
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DRAFT
TAKS Objectives
Exit Level
Objective 6:
G(c)(1)
The student will demonstrate an understanding of geometric
relationships and spatial reasoning.
Geometric patterns. The student identifies, analyzes, and describes patterns that
emerge from two- and three-dimensional geometric figures.
(B) The student uses the properties of transformations and their compositions to make
connections between mathematics and the real world in applications such as
tessellations or fractals.
(C) The student identifies and applies patterns from right triangles to solve problems,
including special right triangles (45-45-90 and 30-60-90) and triangles whose
sides are Pythagorean triples.
Objective 10:
(8.14)
The student will demonstrate an understanding of the mathematical
processes and tools used in problem solving.
Underlying processes and mathematical tools. The student applies Grade 8
mathematics to solve problems connected to everyday experiences, investigations in
other disciplines, and activities in and outside of school. The student is expected to
(A) identify and apply mathematics to everyday experiences, to activities in and
outside of school, with other disciplines, and with other mathematical topics;
(B) use a problem-solving model that incorporates understanding the problem, making
a plan, carrying out the plan, and evaluating the solution for reasonableness; and
(C) select or develop an appropriate problems-solving strategy from a variety of
different types, including drawing a picture, looking for a pattern, systematic
guessing and checking, acting it out, making a table, working a simpler problem,
or working backwards to solve a problem.
(8.15)
Underlying processes and mathematical tools. The student communicates about
Grade 8 mathematics through informal and mathematical language, representations, and
models. The student is expected to
(A) communicate mathematical ideas using language, efficient tools, appropriate units,
and graphical numerical, physical, or algebraic mathematical models.
(8.16)
Underlying processes and mathematical tools. The student uses logical reasoning
to make conjectures and verify conclusion. The student is expected to
(A) make conjectures from patterns or sets of examples and nonexamples; and
(B) validate his/her conclusions using mathematical properties and relationships.
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