Integrating Art And Math: Tessellations And Symmetry
Ed 3601 Art Unit PlanIntegrating Art and Math:Tessellations and SymmetryA Lesson Plan for Grades 5 and 6Melissa Martin“For me it remains an open question whether [this work]pertains to the realm of mathematics or to that of art.”– M.C. Escher
1Introduction and RationaleThis unit plan is focused on integrating grades 5 and 6 art with mathematics. It isimportant to try and integrate subjects together to help make students’ learning moremeaningful. They will understand how these different subjects apply to each other in thereal world. Art also makes learning math much more interesting and fun for a class. It isa creative way to show students different mathematical theories and principles. It isbeneficial for visual and hands-on learners who are not able to understand elements ofmathematics written down on paper. Once they are able to create artistic examples ofmathematical properties, then they will achieve a better understanding of what they areexpected to know in their math class.This unit is focused on tessellations and symmetry. It is related to the Shape andSpace strand of mathematics. It is also categorized under Component 4 of the artcurriculum that concentrates on Main Forms and Proportions. Tessellations can easily beused in lower grades as well as junior and senior high classes. Tessellations are excellentexamples of how to illustrate mathematical properties of symmetry and geometric shapes.In grades 5 and 6, students are beginning to look at symmetrical objects and trying tounderstand their properties. This topic in math requires a lot of hands-on activities toshow the students how symmetry is applied to objects. Tessellations allow the studentsto study the symmetry of these objects and create beautiful designs and patterns.In order to integrate art history into this unit, the students will observe works byM.C. Escher. His drawings show how he was able to visualize different theories ofmathematics by using his creativity to produce magnificent works of art. His drawings oftessellations will show students that the possibilities of applying symmetry and pattern toobjects are endless. This can inspire students to try creating their own tessellations andexperiment with different methods of symmetry. By giving students visualdemonstrations and examples of a theme in art, this will help them to better understandwhat that theme is and how they can go about creating it.This unit plan also integrates math and art with computer technology. Thecomputer program TesselMania allows students to manipulate a geometric shape andthen copy it into a tessellation pattern on the computer screen. This is an excellentmethod of teaching students how to use computers to create works of art. It alsoreinforces the students’ memory of what they learned about symmetry and geometricshapes.Once the students have created a graphic design of a tessellation on the computer,they will have a chance to extend this project by using a 3D medium. Clay tiles allow thestudents to imprint designs using a variety of tools. They can take the design they createdon the computer and imprint it onto their clay tile. The students can also glaze their tilein different colours to enhance the effect of the tessellation pattern in the clay. Once theclay tiles are fired, the students can look at the tessellations they created on paper, usingthe computer, and on a 3D medium. This will show students how art projects can bestudied through the use of different methods and materials. They can study the
2similarities and differences between the projects and determine which ones are better orworse for this type of activity.My goal with this unit plan is to show how math can be creative and artistic. Ialso want this unit to encourage students to be more optimistic about learning math byincorporating hands-on art activities. This unit plan is meant to show students howsymmetry is incorporated into artwork. It will also teach them how to create symmetricalshapes and develop them into a tessellating pattern that can be carried on into differenttypes of media.Scope and Sequence: Program of StudiesThe following are the strands and learner outcomes that this unit applies to for artand math in grades 5 and 6. It was obtained from the program of studies for elementaryart and math.Mathematics – Grade 5Strand: Shape and Space (Transformations)General Outcome: Describe motion in terms of a slide, a turn or a flip.Specific Outcomes:21. Recognize tessellations created with regular and irregular shapes in the environment.22. Cover a surface, using one or more tessellating shapes.23. Create tessellations, using regular polygons.24. Identify planes of symmetry by cutting solids.Art – Grade 5Component 4: MAIN FORMS AND PROPORTIONS: Students will modify forms byabstraction, distortion and other transformations.Mathematics – Grade 6Strand: Shape and Space (Transformations)General Outcome: Create patterns and designs that incorporate symmetry, tessellations,translations, and reflections.Specific Outcome:19. Create, analyze and describe designs, using translations (slides) and reflections(flips).
3Art – Grade 6Component 4: MAIN FORMS AND PROPORTIONS: Students will modify forms byabstraction, distortion and other transformationsUnit OverviewLesson 1: Exploring Symmetrical ShapesIn this lesson the students will use the Mira Math Kit to look at symmetricalshapes in their symmetry booklets from math class. They will see how the Mira reflects amirror image of different objects. This lesson is meant to introduce the students to theconcept of symmetry and allow them to experiment with it by using Miras to draw thereflected images and shapes. With these tools they can begin to create geometric patterns(i.e. mosaics) and understand how symmetry is used to create artistic patterns.Lesson 2: Tessellation TechniquesThis lesson introduces the topic of tessellations. Students will learn the definitionand different characteristics of tessellations by observing different examples. They willlearn the four techniques for tessellating shapes: translating (sliding), nibbling (cutting),rotating (turning), and reflecting (flipping). The students will be able to associate thetechnique of reflecting with what they learned about symmetry. The class willexperiment with these four techniques and illustrate them in their symmetry booklets tobe used as a reference for subsequent lessons.Lesson 3: Tessellations of PolygonsThis lesson reviews the introduction of tessellations. Students will observeexamples of tessellations of geometric shapes (polygons). Using geometric gridhandouts, the students will learn how these shapes can be manipulated to createtessellating patterns. They will achieve this by incorporating the four techniques oftessellations that they learned in the previous lesson. Once the students have enoughpractice using the grids, they can draw geometric tessellations by freehand.Lesson 4: Tessellations of Curved Shapes (Escher Style Tessellations)This lesson will apply the art history component by looking at tessellation worksby M.C. Escher. Escher’s works began with basic geometric shapes. Then hetransformed these shapes by using the four tessellation techniques to create curved shapesand objects. These objects would then be tessellated into different orientations(positions) to create interlocking shapes with no spaces or overlapping objects. Thestudents will recognize these patterns as works of art as well as illustrations ofmathematical principles (i.e. symmetry). The students will follow these same steps tocreate their own tessellating objects on a piece of paper.
4Lesson 5: Creating Tessellation Shapes on ComputerThis lesson will consist of a tutorial of the computer program TesselMania. Theclass will incorporate what they learned about tessellations and symmetry onto thecomputer. Students will learn the basic steps of the four tessellation techniques in theprogram. The students will follow the TesselMania tutorial that is included in theprogram to help guide them in creating their own tessellation shape.Lesson 6: Creating Tessellation Patterns on ComputerThe class will review what they learned about TesselMania in the previous lessonby briefly going over the program’s tutorial again. They will be able to look at otherstudents’ examples of tessellations online to help them visualize what they will becreating. The students can make more objects and learn how to create a tessellation ofthese shapes. They will also learn how to apply colour to their design. Once the studentshave finished their tessellations they can print out their designs.Lesson 7: Creating a Tessellation Pattern with ClayThis lesson will transfer the design of tessellations onto a 3 Dimensional medium.The students will make a clay tile and imprint the tessellation pattern they printed off thecomputer onto the clay. This lesson will allow students to become more creative withtheir tessellations as well as introduce them to the techniques of clay.Lesson 8: Glazing and Firing Clay Tessellation PatternsThis lesson extends the use of clay by allowing students to glaze and fire theirtiles. The students will learn the technique of glazing. This will allow the students toincorporate the use of colour in their tessellation pattern to create interesting and originalworks of art. Once the clay tiles have been fired, the students can look at theirtessellations created on paper, computer, and clay. This will show the students thedifferent effects each medium has on the patterns of tessellations.
5LESSON 1: EXPLORING SYMMETRICAL SHAPESSubject: Art and MathematicsGrade: grades 5-6Focus: Symmetry and PatternsUnit: Tessellations and SymmetryTopic: Exploring Symmetrical ShapesTime: 50 minutesTeaching Strategies Used: demonstration, hands-on learning, discussionLearning Objectives: Students will use Miras to explore symmetrical shapes and patterns. Students will use Miras to help draw their own symmetrical shapes.Materials/Resources:- pencil, markers, pencil crayonspapereraserrulersMira Math Kit (pass out before class starts)Symmetry Booklet (pass out before class starts)Introduction (Opening) (5 minutes)Introducing the Concept of Symmetry- Introduce the lesson by asking the class if they know what symmetry means.Definition: Symmetry is an exact correspondence in position or form about a given point,line, or plane.*To put it more simply, symmetry is when a shape shows a mirrored reflection along aline that splits the shape.- Explain the definition of symmetry to the class and illustrate on the board a shape that issymmetrical along a line (i.e. an equilateral triangle with a vertical line cutting throughthe center).- Explain to the students about the line of symmetry (the line that splits the shape into 2mirrored reflections).- Show some other shapes on the board that may or may not have symmetry. Ask thestudents if these shapes are symmetrical. If so, where? Have the students come up anddraw the line(s) of symmetryUsing the Miras and Symmetry Booklets-Demonstrate to the students how to use the Mira to show reflections of shapes. Have thestudents experiment with their Miras using the booklets.
6-The students can begin to draw symmetrical shapes using their Miras.-Ask the students about the shapes in their booklets (i.e.: are they symmetrical?)Skill Development/Concept (Body) (35 minutes)Drawing Designs of Geometric Shapes-Using rulers and pencils, have the students draw geometric shapes. Then with theirMiras, have them draw the reflected shape.-Students can create patterns with these shapes.Drawing a Symmetrical Pattern-Pass out a piece of white paper to the students-Have the students fold the paper twice into 4 sections-In the top left square, the students can draw their own, original pattern using a pencil.-Once they have completed their pattern, they can use their Mira to draw the reflectedpatterns in the other three sections of their paper. They will use the paper folds as theirline of symmetry.-Once the students have completed their drawing, they can colour their pattern usingcontrasting colours that will heighten the effect of their symmetrical designs.Closure and Evaluation (10 minutes)-Ask the students if they found this exercise too easy or too difficult.-Allow students to come up to the front of the class and show their picture and allow therest of the class to critique the work.-Ask the students what they can see in the picture, how well do the colours go together,etc.Evaluation Criteria for Lesson 1:-Student demonstrated use of Miras to create a symmetrical pattern-Student used an original colour scheme that helped to further demonstratethe symmetrical pattern-Student experimented with the Mira and booklet as well as drawing shapes5 marks5 marks3 marks
7-Student participated in discussions and critique2 marksTOTAL: 15 marksLESSON 2: TESSELLATION TECHNIQUESSubject: Art and MathematicsGrade: grades 5-6Focus: Learning How to Manipulate Shapes for TessellationsUnit: Tessellations and SymmetryTopic: Tessellation TechniquesTime: 50 minutesTeaching Strategies Used: demonstration, hands-on learning, discussionLearning Objectives: Students will learn the definition and characteristics of tessellations Students will learn the different techniques for tessellating shapes (translationslides, nibbling, rotating, and reflecting)Materials/Resources:- pencil- paper- eraser- rulers- pencil crayons- overhead and slides-markerstessellation gridsexamples of tessellationsoverhead projector (for demonstration)Miras and Symmetry Bookletsscissors and glueIntroduction (Opening) (5 minutes)Review of Symmetry Concepts- Reintroduce the concepts of symmetry to the class by having them quickly go over theMiras and Symmetry (definition of symmetry, lines of symmetry, etc.)- Have the students take out their symmetry drawings from yesterday and have them usethe Miras to show how they reflected the pattern to cover the whole page.- Review the different types of geometric shapes (polygons) and ask the students whichones are symmetrical (i.e. triangle, rhombus, rectangle, square, parallelogram, etc.)-Have the students sketch some symmetrical shapes and use their Miras to see if they aresymmetrical or not.Introduce the Concept of Tessellations- Explain to the students what a tessellation is:Definition: A tessellation is a pattern of interlocking shapes with no space and nooverlaps.-Show examples of geometric tessellations so that the students will visually understandhow to tessellate shapes:
8Skill Development/Concept (Body) (40 minutes)Introduce the Techniques for Tessellations*pass out geometric grids to the students for each of these steps1. Translation SlidesDefinition:SLIDE TRANSLATION: Tessellating a shape across a surface, without flipping it orchanging the position of the corners.-Using the overhead projector, demonstrate to the students how a geometric shape canslide to another position on the paper.*the numbers in the corners of the shape are to indicate the orientation of the shape-Ask the students to cut out a shape on their tessellation grid handout and slide it toanother location on their paper-Have the students glue this grid on a blank page in their symmetry booklets and label thepage “Translation Slides” (it may be helpful if they write down the definition as well)-allow the students to slide more shapes and glue them into their booklets2. Nibbling-This is the process where an ordinary geometric shape is transformed into an irregularshapeDefinition:NIBBLING: One side of a geometric shape (from corner to corner) is cut into a patternand that new shape slides to the opposite side.-Demonstrate on the overhead projector how to nibble (cut) one side of a geometric shapeand then translate (slide) that piece over to the opposite side of that shape (example onthe following page).-Allow the students to experiment with nibbling and sliding geometric shapes with theirgeometric grids.
9-Have the students glue an example of nibbling on a blank page in their symmetrybooklets and label the page “Nibbling”-The students should show each step of nibbling in their symmetry booklet asdemonstrated
103. RotatingDefinition:ROTATING: A geometric shape can be turned around at a specific point to change theorientation of that shape.-Demonstrate on the overhead projector how to rotate an object 90ϒ, 180ϒ, 270ϒ, and 360ϒ-Show students how to rotate a geometric shape at different points (i.e. in the center ofthe shape, at the corners, etc.)These triangles have been rotated around the center to create a tessellation.121234344321These squares have been rotated around the corner to create a tessellation.-Allow the students to cut out geometric shapes and rotate them from their grids.Have the students glue an example of rotating on a blank page in their symmetry bookletsand label the page “Rotating”.-The students should show each step of rotating in their symmetry booklet asdemonstrated
114. ReflectingDefinition:REFLECTING: Flipping an object on one side (or point) so that it will show a mirroredreflection of that shape*This tessellation technique demonstrates how these geometric shapes are symmetricalabout a line of symmetry-Demonstrate on an overhead projector how to reflect an object on different sides andpointsline of symmetryline of symmetry-Allow the students to cut out geometric shapes from their grids and reflect them.-Have the students glue an example of reflecting shapes on a blank page in theirsymmetry booklets and label the page “Reflecting”.-The students should show each step of reflecting in their symmetry booklet asdemonstrated.-It may be helpful for the students to use the Miras to reflect the shapes.*If there is extra time left over the students can experiment with these four techniques intheir symmetry booklets and can begin to create tessellation patterns.Closure and Evaluation (5 minutes)-Review with the students what the four techniques for creating tessellations are:translating (sliding), nibbling (cutting), rotating (turning), and reflecting (flipping)-Ask the students if they found this exercise too easy or too difficult.-Allow students to come up to the front of the class and show their examples from theirsymmetry booklets to ensure that they understood how to record them in their booklets.-Students must hand in their booklets at the end of class to receive marks for this lesson.Evaluation Criteria for Lesson 2 (examples of Tesellation Techniques in symmetrybooklets):-Student provided a good example of translating a geometric object3 marks-Student provided a good example of nibbling a geometric object3 marks-Student provided a good example of rotating a geometric object3 marks-Student provided a good example of reflecting a geometric object3 marks-Student’s examples are a good reference for tessellating shapes3 marksTOTAL:15 marks
12LESSON 3: TESSELLATIONS OF POLYGONSSubject: Art and MathematicsGrade: grades 5-6Focus: Patterns of Geometric Shapes (Polygons) and TessellationsUnit: Tessellations and SymmetryTopic: Tessellations of PolygonsTime: 50 minutesTeaching Strategies Used: demonstration, hands-on learning, discussionLearning Objectives: Students will create geometric patterns with grids of polygon shapes. Students will use the four tessellation techniques (translating, nibbling, rotating,and reflecting) to create elaborate, symmetrical patterns.Materials/Resources:- pencil- paper- eraser- rulers- pencil crayons-markerstessellation gridsexamples of tessellationsoverhead projector (for demonstration)Miras and Symmetry BookletsIntroduction (Opening) (5 minutes)Review of Symmetry Concepts- Reintroduce the concepts of symmetry to the class by having them quickly go over theMiras and Symmetry (definition of symmetry, lines of symmetry, etc.).- Review the definition of tessellation: A tessellation is a pattern of interlocking shapeswith no space and no overlaps.-Review the four techniques for tessellating geometric shapes: translating (sliding),nibbling (cutting), rotating (turning), and reflecting (flipping).-Demonstrate on the board how different types of geometric shapes (polygons) can betransformed by using these techniquesSkill Development/Concept (Body) (35 minutes)Review of Tessellations-Show the class more examples of tessellations of geometric shapes (polygons). Ask thestudents what shapes they can see:
13-Using rulers and pencils, have the students draw geometric shapes.-Instruct the students to practice each of the four tessellation techniques with their shapes.They can use the Miras for reflecting these shapes.-Students can create patterns with these shapes.Demonstration of a Tessellation-Using an overhead projector and a transparency of a tessellation grid, demonstrate howto create a tessellation pattern of polygon shapes (divide these shapes into halves,quarters, etc. and colour them in to create a geometric pattern):-Pass out a tessellation grid to the students-Have the students create their own tessellation patterns using these grids (as seen in thedemonstration).-Pass out a tessellation grid to the students-Have the students create their own tessellation patterns by coloring these grids (as seenin the demonstration):
14Drawing Tessellations-Pass out a white sheet of paper to the students-Instruct the students to place the tessellation grid underneath the piece of paper.-Show by demonstration how to create a tessellation of polygons by using the tessellationgrid as a reference.-The students can draw their own patterns in pencil and then colour them in after thepattern in finished.*The use of the tessellation grid is optional. If students feel comfortable enough to drawwithout the grid, they may do so.*The students may use their Miras in this exercise.Closure and Evaluation (10 minutes)-Ask the students if they found this exercise too easy or too difficult.-Allow students to come up to the front of the class and show their picture and allow therest of the class to critique the work.-Ask the students what they can see in the picture, how well do the colours go together,etc.-Ask the students what polygons they see in each of the pictures.Evaluation Criteria for Lesson 3:-Student used creativity (original pattern and good use of geometric shapes)in creating his/her polygon tessellation-Student used an original colour scheme that helped to further demonstratethe symmetrical pattern-Student experimented and participated in the drawing exercises-Student participated in discussions and critiques5 marks5 marks3 marks2 marksTOTAL: 15 marks
15LESSON 4: TESSELLATIONS OF CURVED SHAPES(ESCHER STYLE TESSELLATIONS)Subject: Art and MathematicsGrade: grades 5-6Focus: Tessellations of Curved Shapes and Objects (Escher Style)Unit: Tessellations and SymmetryTopic: Tessellations of Curved Shapes (Escher Style Tessellations)Time: 50 minutesTeaching Strategies Used: demonstration, hands-on learning, discussionLearning Objectives: Students will study the tessellation artwork of M.C. Escher Students will create patterns of curved shapes from polygon grids. Students will create objects from these curved shapes (i.e. animals such as M.C.Escher’s horseman and reptiles)Materials/Resources:- pencil- paper- eraser- rulers- pencil crayons- Escher books-markerstessellation gridsexamples of tessellationsoverhead projector (for demonstration)Miras and Symmetry Bookletsscissors and glueIntroduction (Opening) (5 minutes)Review of Tessellations-Ask the students for the definition of a tessellation-Have students take out their tessellation pictures from last class and review how theywere created (use the overhead projector again to illustrate the steps)M.C. Escher- Introduce the topic of Escher-style tessellations by giving a brief history of Escher:-Maurits Cornelis Escher was born in Holland in 1898-He was a famous graphic artist who created unique works of art that exhibited awide range of mathematical theories-While he was still in school his family planned for him to follow his father'scareer of architecture, but poor grades and an aptitude for drawing and designeventually led him to a career in the graphic arts.-He did not become known as an accomplished artist until the 1950’s, but by 1956he had given his first important exhibition, was written up in Time magazine, andacquired a worldwide reputation.-Among his greatest admirers were mathematicians, who recognized in his workan extraordinary visualization of mathematical principles.
16- *Escher had no mathematical training beyond the secondary level! As his workdeveloped, he drew great inspiration from the mathematical ideas he read about,often working directly from structures in plane and projective geometry, andeventually exploring the fundamentals of non-Euclidean geometries. He was alsofascinated with paradox and "impossible" figures.-In mathematics, Escher’s work encompasses two broad areas: the geometry ofspace, and the logic of space.-Show the class examples of Escher’s tessellation drawings. Show how he was able tostart with basic, geometric patterns and then create curved shapes and recognizableobjects.-Show how he was able to tessellate these objects by changing their orientation andinterlocking them together to create no negative space.-Illustrate on the board how to he used the four tessellation techniques to manipulategeometric shapes to create a curved shape to use in a tessellation:
17Skill Development/Concept (Body) (35 minutes)Manipulating Polygons-Pass out a sheet of white paper to the students. Instruct the students to draw a polygon(i.e.: triangle, square, rhombus, etc.) on the piece of paper and cut it out.-Then ask the students to cut out a shape (nibble) from one side of the polygon. *Makesure the students are not copying each other or the shape demonstrated on the board!-Have the students slide that shape over to the other side of the polygon. Then ask themto flip the shape and see how it changes the original polygon.-Instruct the students to glue this new shape into their Symmetry Booklets. This will helpthem to remember the steps for manipulating a polygon.Creating an Escher-Style Tessellation-Pass out another sheet of white paper to the class.-With the help of a Mira and tessellation grid, have the students repeat the curved shapethey created in the exercise. They can use the appropriate tessellation grid to make surethey are creating a straight pattern. *Demonstrate the steps on the board or on overhead-Have the students create rows of these objects.-There will be negative space between these rows. By definition a tessellation cannothave negative space or overlapping objects. Have the students create a new object to fillin these negative spaces (i.e. Escher’s Sky and Water I uses fish and birds).-Once the students have created their Escher-style tessellations they can colour theshapes.
18Closure and Evaluation (10 minutes)-Ask the students if they found this exercise too easy or too difficult.-Allow students to come up to the front of the class and show their picture and allow therest of the class to critique the work.-Ask the students what objects they can see in the picture, what basic polygons did thestudent start off with?-Do the colours work well with the drawing?Evaluation Criteria for Lesson 4:-Student used creativity in designing his/her tessellation (original pattern,transforming geometric shapes using the four tessellation techniques tocreate original objects from these shapes)-Student used an original colour scheme that helped to further demonstratetheir tessellation pattern-Student experimented and participated in the drawing exercises-Student participated in discussions and critique5 marks5 marks3 marks2 marksTOTAL: 15 marks
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-Symmetry Booklet (pass out before class starts) Introduction (Opening) (5 minutes) Introducing the Concept of Symmetry - Introduce the lesson by asking the class if they know what symmetry means. Definition: Symmetry is an exact correspondence
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