Robert Calotta
Grade Level: 46
Time Required: 2  3 weeks
Goal:
Design integrates geometric
shapes to create what we see. One popular form of design generates
symmetrical art. This unit asks students to observe, react and create
symmetrical designs found in their immediate and distant environment and
culture.
Objectives: PREACTIVITIESThe students should be familiar
with how to calculate the area and perimeter of a rectangle, square and
triangle. They should be experienced with using a ruler, protractor and
compass making linear and angular measurements. They should be familiar
with the common names and characteristics of three to eight sided polygons Students willIdentify four basic forms of
symmetryRecognize symmetry in everyday
objects and life formsRecognize symmetrical patterns
in cultural artCreate symmetrical designs Standards:
Mathematics
Mathematical
Reasoning  Students use mathematical reasoning to analyze mathematical situations,
make conjectures, gather evidence, and construct an argument.
Modeling  Students use mathematical modeling/multiple representation to provide a
means of presenting, interpreting, communicating, and connecting
mathematical information and relationships.
Patterns  Students use patterns and functions
to develop mathematical power, appreciate the true beauty of
mathematics, and construct generalizations that describe patterns simply
and efficiently
ELA
Speaking and Listening –
Students develop arguments with effective use of details and evidence
that reflect a coherent set of criteria
ARTS
Students compare the
ways ideas, themes and concepts are communicated through the visual arts
in other disciplines and the various [they] are manifested within the
discipline.
Materials: SPECIAL NEEDSStudents having visual or
spatial difficulty may need larger pattern blocks. Other assistance for
students with math disabilities is offered at http://forum.swarthmore.edu/social/math.disabled.html Materials for the Student:
pencil, ruler, protractor, compass, pattern block set and dot papernotebook, drawing and graph
papercomputer with Internet accesscalculator (optional)Materials for the Teacher:Overhead projectorTransparent pattern blocks and
graph paper
water base markers
LESSONS:
WEEK ONE  WHAT IS SYMMETRY
INTRODUCTION – The Mathematics
of Symmetry A WHOLE CLASSROOM ACTIVITYThe student cuts out shapes of
pattern blocks from a sheet of paper. Folding each in half leads to the
discovery that both halves are identical in shape and size. From this
observation comes a discussion that leads to a definition for the line
of symmetry and the symmetrical form of reflection.To extend this lesson go to: http://geom.umn.edu/~demo5337/s97a/reflect.html Now, introduce the students to
the other three basic forms of symmetry: translation, rotation and glides. The following site supports as a whole class presentation;
examples of each symmetrical form http://forum.swarthmore.edu/geometry/rugs/symmetry/basic.html Follow up using each of the
different transparent pattern blocks on the overhead and with the students
following along with their desk sets. Students observe the result of each
symmetrical form, replicate it and then trace each symmetrical form into
their notebooks.The following definitions are
found on the following two sites and are well designed for use with the
whole classRotational Symmetry http://geom.umn.edu/~demo5337/s97a/rotate.htmlGlide or Slide Symmetry http://geom.umn.edu/~demo5337/s97a/students.html Enrichment Assignment: Students
go through the alphabet to identify letters that have a line of symmetry.
Those struggling can be helped with the use of a pocket mirror. Results
can be tested by students at http://geom.umn.edu/~demo5337/s97a/letters.htm Next, combine two shapes to
again demonstrate symmetry in its four forms. As students gain experiences
and confidence seeing and identifying the form of symmetry, present them
with one pattern design and ask that they create with their desk sets the
corresponding symmetrical shape. ASSIGNMENT #1As a concluding project the
student will create a design using 3 or more pattern blocks to demonstrate
the ability to translate, rotate, reflect and glide. The results will be
traced or sketched onto paper. Some students may need or prefer pattern
block dot paper to assist with drawing the results.RUBRIC100% Student demonstrates each
form correctly75% Student demonstrates 3
of 4 forms correctly50% Student demonstrates 2
of 4 forms correctly25% Student demonstrates 1
of 4 forms correctly Next, the students design a
symmetrical shape with the use of the blocks and the Cartesian Graph.
The teacher first models this activity on the overhead projector with
transparent graph pattern and pattern blocks. Select two identical pattern
blocks and place them on graph paper side by side using the yaxis as the
line of symmetry. Students them observe how the corners of the pattern
block match up with the x,y coordinate points. A discussion follows that
leads to observing the pattern of symmetrical reflective points; (3,4) is
symmetrical with (3,4) along the yaxis; likewise (3,4) is symmetrical
with (3, 4) along the xaxis. The ability to use graph paper to create a
symmetrical design on a Cartesian graph will later be used to complete
online activities. ASSIGNMENT #2The students design a Cartesian
graph. Students number the x and y axis on graph paper using positive and
negative numbers (110 should be sufficient). The student outlines one
half of a selected block on one side of the x or yaxis. The vertices are
labeled with the coordinates. Those coordinates are then “reflected” to
the other half of the graph and the points are connected completing the
symmetrical design. If done correctly the block should fit into the drawn
design. The teacher checks the work for accuracy. The student now combines 46
pattern blocks to create one image that reflects along either the x or
yaxis to the other side. Each of the corners (vertices) of one
symmetrical side is then clearly labeled with the exact coordinates. The
student then removes the blocks and identifies and labels the reflective
points. When finished the student puts the blocks back into place to see
if the corners of the blocks match the points on the graph. Upon completion of their project
have the student calculate the area and perimeter of the
entire symmetrical design. Initially, the students may complain that they
did not learn the formulas for the area of these unusual shapes. Some
discussion will be needed about what they did learn about area of
rectangles and triangles. Then a few questions referring to the original
shapes of the pattern blocks should get some to realize that they need to
"breakdown " the design into familiar and manageable rectangle and
triangle shapes. This will then allow them to find the area of each
respective shape and then total them for the grand sum, the complete area.
At some point a student may discover that you only need to calculate the
area and perimeter of onehalf of a symmetrical shape and then multiply by
two for the grand total. If not, a teacher initiated question such as, "
How can the fact that this is a symmetrical design cut your calculating
efforts in half?" provides a moment of reflection. RUBRIC #2Creating the Cartesian Graph
with 01 errors 25 pts.Creating the Cartesian Graph
with 23errors 20 pts.Creating the Cartesian Graph
with 4+ errors 15 pts. Creating the Symmetrical Image
with 46 blocks 25 ptsCreating the Symmetrical Image
with one block out of place 20 ptsCreating the Symmetrical Image
with 2+ blocks out of place 15 pts Labeling points of one half of
image on graph with 01 errors 25 pts.Labeling points of one half of
image on graph with 23 errors 20 ptsLabeling points of one half of
image on graph with 4+ errors 15 pts Labeling the other half of image
on graph with 01 errors 25 pts.Labeling the other half of image
on graph with 23 errors 20 ptsLabeling the other half of image
on graph with 4+ errors 15 ptsREFERENCEGlossary of terms http://forum.swarthmore.edu/geometry/rugs/resources/glossary.html
WEEK TWO 
IT'S ALL AROUND US
This week's activities will move
the students from the math of symmetry to its application. OBSERVING SYMMETRY – a teacher
led, whole class presentationA lesson plan is found at: http://forum.swarthmore.edu/geometry/rugs/resources/act/observe.htmlStudents look for examples of
symmetrical shapes and patterns found in the school, their homes and
nature.Typically students will find
such patterns on walls and floor coverings, fabrics, and jewelry. They are
to also look through print media for examples of symmetry and identify the
line of symmetry. GOING ONLINEThe following three activities
can be organized as an individual or small group activity considering
factors such as time, computer resources and student/teacher skills. Students go online to view
examples of symmetry in commercial and cultural design and nature. The
student will be involved each day with a particular theme. Each activity
asks the students some questions to respond to in written or discussion
format. There is also the opportunity to create a symmetrical design based
on the theme. ACTIVITY  SYMMETRICAL CIRCLESCircle symmetry is evident in
many items from the automobile hubcap http://styleline.com/ to the Pennsylvania Dutch Hex artwork. The
student will use previously learned skills, a protractor and compass to
create symmetry using rotational and reflective form. They are also
expected to write an interpretative essay. Using the analogy of a pizza pie
with six identical slices reminds students to recall the concept of
reflective and rotational symmetry. Also going to http://geom.umn.edu/~demo5337/s97a/rotate.html will help students
to visualize the concept. Hex Signs of the Pennsylvania
Dutch people http://folkart.com/~latitude/hex/hexx.htm puts cultural meaning
behind the art. Students use this site to learn how these people combined
symmetry and form to design this folk art. There is a link to help
interpret the meaning of the colors, shapes and images. Students once
again identify the line of symmetry and its form. After viewing the
examples the student designs a rubric to define degrees of best to worst
designs using criteria for correct and effective use of symmetry design,
symbols and colors. Finally the student selects the two best and least
favorite designs based on the rubric criteria. ASSIGNMENT #3Each student creates either his
or her own hex design with reflective and rotational symmetry integrating
symbolic design and color. The selection of color and symbols need to
reflect at least five characteristics of the student’s perception of self. It is conceivable to create the
hex design using actual dimensions on the website. In addition to using
the protractor and compass the student may use other “inventive” tools.
Folding the paper circle into “pizza slices” will enable many to better
judge how much rotation will give the best look. At this point students
may recall their younger “snow flaking” cut out skills and wish to employ
this to create a template to trace. The student submits a written
assessment of how the design measures up to his/her criteria for best
deign. RUBRIC #3
Design has rotational and
reflective symmetry 50 pts
Design has rotational or
reflective symmetry only 25 ptsDesign is lacking symmetry 0
pts.
5+ symbols and colors used 25
pts
34 symbols and colors used 15
pts02 symbols and colors used 0
points
Rubric is accurately written
and applied 25 pts
Rubric is vague or vaguely
applied 20ptsRubric is ineffective and can
not be applied 15 pts ACTIVITY – RUG SYMMETRYAll around the world floor
coverings have employed the art of symmetrical patterning. Materials used
include stone, wood, mosaic tile and rugs. The students will use the
Cartesian Graph and graph paper to create their own symmetrical pattern
rug. Students are introduced to the concept of color symmetry as well as
design. The Navaho people have been
creating symmetrically designed rugs before the arrival of the Europeans.
Students will read a bit about the history of rugs at http://indiantraders.com/aboutart/textiles/navrug1.htm. They will
view several examples of Navajo rugs at. http://americantrails.com/at_rugs.html. Each rug can be enlarged
with a mouse click on its image. When viewing each rug a transparent sheet
of graph paper is placed over the screen. Demonstrate to the students that
each rug can be centered with the x and y axis dividing the rug into four
quadrants. Each quadrant has a symmetrical image of the other in one form;
reflective, glide, translation or rotation. Students are looking for
elements (symmetrical form, color, and design) that appeal to their sense
of a goodlooking rug pattern so as to create a rubric similar to the Hex
rubric. This time the student is given latitude to decide what descriptors
go into the rubric for symmetrical rug design. ASSIGNMENT #4Once a student has determined
what constitutes a personal best rug design, use onehalf inch or larger
graph paper to create a design that measures up to the descriptors of best
design and includes at least two forms of symmetry including glide or
translation. Creating the rug design begins
with the Cartesian graph with the option to leave out the positive and
negative number labels. This will then assist the student to design the
rug in each quadrant so that the symmetrical forms can be incorporated
accurately. RUBRIC #4Design has Glide or Translation
and at least one other symmetrical form 50 pts.Design has Glide or Translation
but missing one other symmetrical form 25 pts.Design is missing either Glide
or Translation 15pts Design uses color symmetry
accurately, 02 errors 25 ptsDesign uses color symmetry with
34 errors 20 ptsDesign uses color symmetry with
5+ errors 15 pts.
Rubric is accurately written
and applied 25 pts
Rubric is vague or vaguely
applied 20ptsRubric is ineffective to be
applied 15 pts ACTIVITY – SYMMETRY IN NATUREThis activity integrates with
the science curriculum topic of ecology of living things. Many living
things have symmetry in their shape and some have color symmetry as well.
Show the class the butterfly site, http://butterflywebsite.com/gallery/index.cfm to observe the symmetry
of color and shape of butterfly wings. ASSIGNMENT #5The student will use the
Internet to look for and select five different images each of animal and
plant life that possess symmetrical shape and/or color. Along with these
images research online or in the library the practical purposes for shape
and color in the environment. The student prepares a
PowerPoint presentation to point out the symmetrical features of each life
form and at least one specific purpose each design serves in the biome of
the respective life form. This assignment and information is to be
integrated into the science class lesson on biomes and ecosystems. RUBRIC #5 – Part One5 symmetrical images of each
plant and animal shown 50 pts.12 images missing or not
symmetrical 35 pts.3+ images missing or not
symmetrical 20 pts RUBRIC #5 – Part TwoAll 10 life forms is accompanied
by a specific purpose of its symmetrical part 50 pts.12 explanations missing or
incorrect 40 pts.3+ explanations missing or
incorrect 20 pts
WEEK THREE 
THE CULMINATION
TEACHER DESIGNED EXAMStudent will need to demonstrate
knowledge of four forms of symmetry, calculating area and perimeter of a
given symmetrical polygon shape and create a symmetrical design on a
Cartesian Graph along one axis as a line of symmetry. CLASSROOM DISCUSSIONStudents will divide themselves
into small groups of 24 people to address the essential questions. The
teacher will introduce the topic with a review of what was learned in week
two and ask the students to extend their experience beyond those
experiences to other elements of our culture and society. Some other
elements to consider include music, writing, architecture, health and
personal relationships.Each group will then explain to
the class their opinions and supporting facts.
ROLE OF TECHNOLOGY:
This lesson plan is designed to
build upon the knowledge students have of calculating area and perimeter.
The use of Internet sites reinforces and enriches these tasks and
demonstrates the application of these skills to the world around them. The
subject of symmetry is best demonstrated visually. The Internet provides
an array of sites and tools for the teacher as well as for the student in
a convenient place for anywhere, anytime learning to take place.
BIBLIOGRAPHY:
The Butterfly Website, The
Nature Store, Online http://butterflywebsite.com/gallery/index.cfm, Mar 11, 99Dorsey, Hanifl, Kannel, Thomson,
Observing Symmetry, Textile Museum, Online http://forum.swarthmore.edu/geometry/rugs/resources/act/observe.html,
June 6, 1996Folk art and Craft exchange,
Online http://folkart.com/~latitude/hex/hexx.htm, May 6, 1999The Geometry Center, University
of Minnesota, Online http://geom.umn.edu/, Feb. 5, 1999L7 Group, Indian Traders, Online http://indiantraders.com/aboutart/textiles/navrug1.htm, April 28,
99
Style Line Hubcaps, Style Line,
Online, http://styleline.com/. May 6, 1999
