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Tessellations (Grades 5-8)

Master Teacher

Sarah Lee

Overview

In this lesson, students will identify and construct figures that tessellate. They will investigate which regular polygons tessellate and how to modify them to make other tessellating figures. Students will explore how naturally occurring tessellations have been imitated in man-made tessellation patterns. The video will portray some careers which implement tessellations.

SC Math/Science Standards Met

Math Achievement Standards

Patterns, Relationships, and Functions/Middle Grades
Geometry and Spatial Sense/Middle Grades

Science Achievement Standards

Process Skills
Area I—Living Things
Strand 1: B

Learning Objectives

The students will be able to:

identify which regular polygons tessellate
design a tessellation using a transformation
identify examples of tessellation patterns in architecture, nature, and in other man-made objects
identify man-made tessellations that imitate the tessellation patterns that occur in nature

Materials

For each student:

index card
scissors
tape
paper
Activity 1: pattern blocks or regular polygon patterns
pencil and colored pencils/markers

For each group of four:

Activity 2: pictures of tessellations, such as beehives, fish scales, schools of fish, crystal or mineral structures, pineapples, brick designs, bucky-ball, Escher art, etc.
butcher paper
glue

ITV Series

Math Vantage: Patterns, Lesson 4, "Tessellations/Transformations"

Previewing Activities

1. Introduce the lesson with the definition of a tessellation. The word "tessella" is a Greek word for a small, square piece of stone or tile used in mosaics. Since a mosaic extends over a given area without leaving any region uncovered, the geometric meaning of the word "tessellate" is "to cover the plane with a pattern in such a way as to leave no region uncovered."

2. Provide examples of tessellation patterns. Display examples of M. C. Escher's work, a soccer ball, etc. You may also show examples of tessellations created by other students.

3. Now have students investigate which regu- lar polygons tessellate. Explain that a polygon is a many-sided shape. A regular polygon is one in which all sides and angles are equal. These are triangles, squares, pentagons, hexagons, heptagons, and octagons. Distribute pattern blocks or patterns of regular polygons for students to cut out. Instruct students to try to make a tessellation pattern of each figure by tracing around the pattern many times on a piece of paper. Emphasize no gaps or overlaps. Ask students to identify which regular polygons tessellated.

Focus for Viewing

To give students a specific responsibility while viewing, ask them to watch for examples of tessellation patterns and their use in some careers. The second portion of the video will also reveal which regular polygons tessellate and how to create other shapes that tessellate.

Viewing Activities

Before beginning the video, ask students to answer the following questions in their journals as they watch the video:

What is a tessellation?
List three examples of tessellation patterns in the video.
Which polygons tessellate?

Start the video at its beginning and pause after the girl states that the aqua-colored pentagons can't form a tessellation and the pentagons fall down the screen. Discuss answers to the above questions. Ask students to name any regular polygons that weren't demonstrated in the video and ask whether the figure tessellated.

Distribute the materials for students to make a tessellation shape using a transformation. Each student needs scissors, tape, a pencil, an index card, and paper. Instruct students to watch the process of changing a rectangle into a fun-damental shape that tessellates. Play the next portion of the video beginning where the girl talks about the tessellation pattern on her tie. Stop the video after the girl blows out one side of the rectangle and sucks in the same shape from the other side. Have students cut out a "nibble" from one side of their index cards. Be sure they cut into one corner and exit an adjacent corner. This "nibble" is then slid to the opposite side of the index card and the two pieces are taped together. Be careful that the piece is not flipped during the sliding process. Allow time for students to trace around their shapes to make a tessellation pattern on paper. You may also have them color their tessellations.

Ask students to list jobs that involve tessellations. Allow them to share their lists and make a class list. Then watch the rest of the video and ask students to raise their hands when the video discusses a job not on the class list.

Post-Viewing Activity

1. Tessellations in nature, architecture, and other objects

Distribute tessellation pictures and butcher paper to students in groups of four.
Group directions: Divide the butcher paper into two columns. Label one column "Naturally Occurring Tessellations" and the other column " Man-Made Tessellations."
Sort the tessellation pictures into the two categories and glue them on the
chart.
Answer the question, "Which of the man-made tessellations are imitations
of the naturally occurring tessellations?"
Display the charts around the room and have the groups share their findings with the entire class. Discuss the question posed above.

Action Plan

1. Students may bring other examples of tessellations into the classroom to share with the class.

2. Students may further investigate transformations by completing a tessellation that is colored or made from two contrasting colors of construction paper.

3. Students could also search the Internet for other information on tessellations and visit other schools' Web sites where student creations are displayed. Students may also submit their tessellations to be posted on a Web page.

Possible Internet sites:

Extensions

Math: Students may create other tessellation patterns using the rotation and/or reflection transformations.
Art: Students may research artists who use tessellation patterns in their works, such as M. C. Escher. Collect examples for a tessellation bulletin board.
Technology: Students may construct computer-generated tessellations.
Science: Students could find other naturally occurring or man-made tessellation examples in books, magazines, photographs, or in the world around them.
Language Arts: Students may write a story or poem for one of Escher's works or for one of their own creations.

Activity Sheets (PDF)