Integrating Art and Math: Tessellations and Symmetry

Ed 3601 Art Unit Plan
Integrating Art and Math:
Tessellations and Symmetry
A Lesson Plan for Grades 5 and 6
Melissa 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
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Introduction and Rationale
This unit plan is focused on integrating grades 5 and 6 art with mathematics. It is
important to try and integrate subjects together to help make students’ learning more
meaningful. They will understand how these different subjects apply to each other in the
real world. Art also makes learning math much more interesting and fun for a class. It is
a creative way to show students different mathematical theories and principles. It is
beneficial for visual and hands-on learners who are not able to understand elements of
mathematics written down on paper. Once they are able to create artistic examples of
mathematical properties, then they will achieve a better understanding of what they are
expected to know in their math class.
This unit is focused on tessellations and symmetry. It is related to the Shape and
Space strand of mathematics. It is also categorized under Component 4 of the art
curriculum that concentrates on Main Forms and Proportions. Tessellations can easily be
used in lower grades as well as junior and senior high classes. Tessellations are excellent
examples 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 to
understand their properties. This topic in math requires a lot of hands-on activities to
show the students how symmetry is applied to objects. Tessellations allow the students
to 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 by
M.C. Escher. His drawings show how he was able to visualize different theories of
mathematics by using his creativity to produce magnificent works of art. His drawings of
tessellations will show students that the possibilities of applying symmetry and pattern to
objects are endless. This can inspire students to try creating their own tessellations and
experiment with different methods of symmetry. By giving students visual
demonstrations and examples of a theme in art, this will help them to better understand
what that theme is and how they can go about creating it.
This unit plan also integrates math and art with computer technology. The
computer program TesselMania allows students to manipulate a geometric shape and
then copy it into a tessellation pattern on the computer screen. This is an excellent
method of teaching students how to use computers to create works of art. It also
reinforces the students’ memory of what they learned about symmetry and geometric
shapes.
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 the
students to imprint designs using a variety of tools. They can take the design they created
on the computer and imprint it onto their clay tile. The students can also glaze their tile
in different colours to enhance the effect of the tessellation pattern in the clay. Once the
clay tiles are fired, the students can look at the tessellations they created on paper, using
the computer, and on a 3D medium. This will show students how art projects can be
studied through the use of different methods and materials. They can study the
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similarities and differences between the projects and determine which ones are better or
worse for this type of activity.
My goal with this unit plan is to show how math can be creative and artistic. I
also want this unit to encourage students to be more optimistic about learning math by
incorporating hands-on art activities. This unit plan is meant to show students how
symmetry is incorporated into artwork. It will also teach them how to create symmetrical
shapes and develop them into a tessellating pattern that can be carried on into different
types of media.
Scope and Sequence: Program of Studies
The following are the strands and learner outcomes that this unit applies to for art
and math in grades 5 and 6. It was obtained from the program of studies for elementary
art and math.
Mathematics – Grade 5
Strand: 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 5
Component 4: MAIN FORMS AND PROPORTIONS: Students will modify forms by
abstraction, distortion and other transformations.
Mathematics – Grade 6
Strand: 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).
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Art – Grade 6
Component 4: MAIN FORMS AND PROPORTIONS: Students will modify forms by
abstraction, distortion and other transformations
Unit Overview
Lesson 1: Exploring Symmetrical Shapes
In this lesson the students will use the Mira Math Kit to look at symmetrical
shapes in their symmetry booklets from math class. They will see how the Mira reflects a
mirror image of different objects. This lesson is meant to introduce the students to the
concept of symmetry and allow them to experiment with it by using Miras to draw the
reflected 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 Techniques
This lesson introduces the topic of tessellations. Students will learn the definition
and different characteristics of tessellations by observing different examples. They will
learn the four techniques for tessellating shapes: translating (sliding), nibbling (cutting),
rotating (turning), and reflecting (flipping). The students will be able to associate the
technique of reflecting with what they learned about symmetry. The class will
experiment with these four techniques and illustrate them in their symmetry booklets to
be used as a reference for subsequent lessons.
Lesson 3: Tessellations of Polygons
This lesson reviews the introduction of tessellations. Students will observe
examples of tessellations of geometric shapes (polygons). Using geometric grid
handouts, the students will learn how these shapes can be manipulated to create
tessellating patterns. They will achieve this by incorporating the four techniques of
tessellations that they learned in the previous lesson. Once the students have enough
practice 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 works
by M.C. Escher. Escher’s works began with basic geometric shapes. Then he
transformed these shapes by using the four tessellation techniques to create curved shapes
and objects. These objects would then be tessellated into different orientations
(positions) to create interlocking shapes with no spaces or overlapping objects. The
students will recognize these patterns as works of art as well as illustrations of
mathematical principles (i.e. symmetry). The students will follow these same steps to
create their own tessellating objects on a piece of paper.
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Lesson 5: Creating Tessellation Shapes on Computer
This lesson will consist of a tutorial of the computer program TesselMania. The
class will incorporate what they learned about tessellations and symmetry onto the
computer. Students will learn the basic steps of the four tessellation techniques in the
program. The students will follow the TesselMania tutorial that is included in the
program to help guide them in creating their own tessellation shape.
Lesson 6: Creating Tessellation Patterns on Computer
The class will review what they learned about TesselMania in the previous lesson
by briefly going over the program’s tutorial again. They will be able to look at other
students’ examples of tessellations online to help them visualize what they will be
creating. The students can make more objects and learn how to create a tessellation of
these shapes. They will also learn how to apply colour to their design. Once the students
have finished their tessellations they can print out their designs.
Lesson 7: Creating a Tessellation Pattern with Clay
This 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 the
computer onto the clay. This lesson will allow students to become more creative with
their tessellations as well as introduce them to the techniques of clay.
Lesson 8: Glazing and Firing Clay Tessellation Patterns
This lesson extends the use of clay by allowing students to glaze and fire their
tiles. The students will learn the technique of glazing. This will allow the students to
incorporate the use of colour in their tessellation pattern to create interesting and original
works of art. Once the clay tiles have been fired, the students can look at their
tessellations created on paper, computer, and clay. This will show the students the
different effects each medium has on the patterns of tessellations.
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LESSON 1: EXPLORING SYMMETRICAL SHAPES
Subject: Art and Mathematics
Grade: grades 5-6
Focus: Symmetry and Patterns
Unit: Tessellations and Symmetry
Topic: Exploring Symmetrical Shapes
Time: 50 minutes
Teaching Strategies Used: demonstration, hands-on learning, discussion
Learning 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 crayons
paper
eraser
rulers
Mira 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 a
line that splits the shape.
- Explain the definition of symmetry to the class and illustrate on the board a shape that is
symmetrical along a line (i.e. an equilateral triangle with a vertical line cutting through
the center).
- Explain to the students about the line of symmetry (the line that splits the shape into 2
mirrored reflections).
- Show some other shapes on the board that may or may not have symmetry. Ask the
students if these shapes are symmetrical. If so, where? Have the students come up and
draw the line(s) of symmetry
Using the Miras and Symmetry Booklets
-Demonstrate to the students how to use the Mira to show reflections of shapes. Have the
students experiment with their Miras using the booklets.
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-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 their
Miras, 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 reflected
patterns in the other three sections of their paper. They will use the paper folds as their
line of symmetry.
-Once the students have completed their drawing, they can colour their pattern using
contrasting 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 the
rest 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 demonstrate
the symmetrical pattern
-Student experimented with the Mira and booklet as well as drawing shapes
5 marks
5 marks
3 marks
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-Student participated in discussions and critique
2 marks
TOTAL: 15 marks
LESSON 2: TESSELLATION TECHNIQUES
Subject: Art and Mathematics
Grade: grades 5-6
Focus: Learning How to Manipulate Shapes for Tessellations
Unit: Tessellations and Symmetry
Topic: Tessellation Techniques
Time: 50 minutes
Teaching Strategies Used: demonstration, hands-on learning, discussion
Learning Objectives:
∞ Students will learn the definition and characteristics of tessellations
∞ Students will learn the different techniques for tessellating shapes (translation
slides, nibbling, rotating, and reflecting)
Materials/Resources:
- pencil
- paper
- eraser
- rulers
- pencil crayons
- overhead and slides
-
markers
tessellation grids
examples of tessellations
overhead projector (for demonstration)
Miras and Symmetry Booklets
scissors and glue
Introduction (Opening) (5 minutes)
Review of Symmetry Concepts
- Reintroduce the concepts of symmetry to the class by having them quickly go over the
Miras and Symmetry (definition of symmetry, lines of symmetry, etc.)
- Have the students take out their symmetry drawings from yesterday and have them use
the 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 which
ones 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 are
symmetrical 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 no
overlaps.
-Show examples of geometric tessellations so that the students will visually understand
how to tessellate shapes:
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Skill Development/Concept (Body) (40 minutes)
Introduce the Techniques for Tessellations
*pass out geometric grids to the students for each of these steps
1. Translation Slides
Definition:
SLIDE TRANSLATION: Tessellating a shape across a surface, without flipping it or
changing the position of the corners.
-Using the overhead projector, demonstrate to the students how a geometric shape can
slide 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 to
another location on their paper
-Have the students glue this grid on a blank page in their symmetry booklets and label the
page “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 booklets
2. Nibbling
-This is the process where an ordinary geometric shape is transformed into an irregular
shape
Definition:
NIBBLING: One side of a geometric shape (from corner to corner) is cut into a pattern
and that new shape slides to the opposite side.
-Demonstrate on the overhead projector how to nibble (cut) one side of a geometric shape
and then translate (slide) that piece over to the opposite side of that shape (example on
the following page).
-Allow the students to experiment with nibbling and sliding geometric shapes with their
geometric grids.
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-Have the students glue an example of nibbling on a blank page in their symmetry
booklets and label the page “Nibbling”
-The students should show each step of nibbling in their symmetry booklet as
demonstrated
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3. Rotating
Definition:
ROTATING: A geometric shape can be turned around at a specific point to change the
orientation 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 of
the shape, at the corners, etc.)
These triangles have been rotated around the center to create a tessellation.
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3
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3
4
4
3
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These 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 booklets
and label the page “Rotating”.
-The students should show each step of rotating in their symmetry booklet as
demonstrated
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4. Reflecting
Definition:
REFLECTING: Flipping an object on one side (or point) so that it will show a mirrored
reflection of that shape
*This tessellation technique demonstrates how these geometric shapes are symmetrical
about a line of symmetry
-Demonstrate on an overhead projector how to reflect an object on different sides and
points
line of symmetry
line 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 their
symmetry booklets and label the page “Reflecting”.
-The students should show each step of reflecting in their symmetry booklet as
demonstrated.
-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 in
their 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 their
symmetry 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 symmetry
booklets):
-Student provided a good example of translating a geometric object
3 marks
-Student provided a good example of nibbling a geometric object
3 marks
-Student provided a good example of rotating a geometric object
3 marks
-Student provided a good example of reflecting a geometric object
3 marks
-Student’s examples are a good reference for tessellating shapes
3 marks
TOTAL:
15 marks
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LESSON 3: TESSELLATIONS OF POLYGONS
Subject: Art and Mathematics
Grade: grades 5-6
Focus: Patterns of Geometric Shapes (Polygons) and Tessellations
Unit: Tessellations and Symmetry
Topic: Tessellations of Polygons
Time: 50 minutes
Teaching Strategies Used: demonstration, hands-on learning, discussion
Learning 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
-
markers
tessellation grids
examples of tessellations
overhead projector (for demonstration)
Miras and Symmetry Booklets
Introduction (Opening) (5 minutes)
Review of Symmetry Concepts
- Reintroduce the concepts of symmetry to the class by having them quickly go over the
Miras and Symmetry (definition of symmetry, lines of symmetry, etc.).
- Review the definition of tessellation: A tessellation is a pattern of interlocking shapes
with 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 be
transformed by using these techniques
Skill Development/Concept (Body) (35 minutes)
Review of Tessellations
-Show the class more examples of tessellations of geometric shapes (polygons). Ask the
students what shapes they can see:
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-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 how
to 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 the
demonstration).
-Pass out a tessellation grid to the students
-Have the students create their own tessellation patterns by coloring these grids (as seen
in the demonstration):
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Drawing 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 tessellation
grid as a reference.
-The students can draw their own patterns in pencil and then colour them in after the
pattern in finished.
*The use of the tessellation grid is optional. If students feel comfortable enough to draw
without 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 the
rest 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 demonstrate
the symmetrical pattern
-Student experimented and participated in the drawing exercises
-Student participated in discussions and critiques
5 marks
5 marks
3 marks
2 marks
TOTAL: 15 marks
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LESSON 4: TESSELLATIONS OF CURVED SHAPES
(ESCHER STYLE TESSELLATIONS)
Subject: Art and Mathematics
Grade: grades 5-6
Focus: Tessellations of Curved Shapes and Objects (Escher Style)
Unit: Tessellations and Symmetry
Topic: Tessellations of Curved Shapes (Escher Style Tessellations)
Time: 50 minutes
Teaching Strategies Used: demonstration, hands-on learning, discussion
Learning 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
-
markers
tessellation grids
examples of tessellations
overhead projector (for demonstration)
Miras and Symmetry Booklets
scissors and glue
Introduction (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 they
were 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 a
wide range of mathematical theories
-While he was still in school his family planned for him to follow his father's
career of architecture, but poor grades and an aptitude for drawing and design
eventually led him to a career in the graphic arts.
-He did not become known as an accomplished artist until the 1950’s, but by 1956
he had given his first important exhibition, was written up in Time magazine, and
acquired a worldwide reputation.
-Among his greatest admirers were mathematicians, who recognized in his work
an extraordinary visualization of mathematical principles.
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- *Escher had no mathematical training beyond the secondary level! As his work
developed, he drew great inspiration from the mathematical ideas he read about,
often working directly from structures in plane and projective geometry, and
eventually exploring the fundamentals of non-Euclidean geometries. He was also
fascinated with paradox and "impossible" figures.
-In mathematics, Escher’s work encompasses two broad areas: the geometry of
space, and the logic of space.
-Show the class examples of Escher’s tessellation drawings. Show how he was able to
start with basic, geometric patterns and then create curved shapes and recognizable
objects.
-Show how he was able to tessellate these objects by changing their orientation and
interlocking them together to create no negative space.
-Illustrate on the board how to he used the four tessellation techniques to manipulate
geometric shapes to create a curved shape to use in a tessellation:
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Skill 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. *Make
sure 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 them
to 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 help
them 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 shape
they created in the exercise. They can use the appropriate tessellation grid to make sure
they 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 cannot
have negative space or overlapping objects. Have the students create a new object to fill
in 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 the
shapes.
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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 the
rest of the class to critique the work.
-Ask the students what objects they can see in the picture, what basic polygons did the
student 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 to
create original objects from these shapes)
-Student used an original colour scheme that helped to further demonstrate
their tessellation pattern
-Student experimented and participated in the drawing exercises
-Student participated in discussions and critique
5 marks
5 marks
3 marks
2 marks
TOTAL: 15 marks
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