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A WORLD OF SYMMETRY: MATH - GEOMETRY

 

Robert Calotta            
Grade  Level:  4-6               
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:        PRE-ACTIVITIESThe 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 will-Identify 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 y-axis as the line of symmetry. Students them observe how the corners of the pattern block match up with the x,y co-ordinate points.  A discussion follows that leads to observing the pattern of symmetrical reflective points; (3,4) is symmetrical with (-3,4) along the y-axis; likewise (3,4) is symmetrical with (3, -4) along the x-axis. The ability to use graph paper to create a symmetrical design on a Cartesian graph will later be used to complete on-line activities.  ASSIGNMENT #2The students design a Cartesian graph. Students number the x and y axis on graph paper using positive and negative numbers (1-10 should be sufficient). The student outlines one half of a selected block on one side of the x or y-axis. 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 4-6 pattern blocks to create one image that reflects along either the x or y-axis 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 one-half 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 0-1 errors 25 pts.Creating the Cartesian Graph with 2-3errors 20 pts.Creating the Cartesian Graph with  4+ errors 15 pts. Creating the Symmetrical Image with 4-6 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 0-1 errors 25 pts.Labeling points of one half of image on graph with 2-3 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 0-1 errors 25 pts.Labeling the other half of image on graph with 2-3 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 ON-LINEThe 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 on-line 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://style-line.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

3-4 symbols and colors used 15 pts0-2 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 good-looking 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 one-half 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, 0-2 errors 25 ptsDesign uses color symmetry with 3-4 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 on-line 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.1-2 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.1-2 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 2-4 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://style-line.com/. May 6, 1999

 

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