Abstract
This lesson allows students to examine tessellations and their geometric properties. This activity and discussions may be used to develop students' understanding of polygons and symmetry as well as their ability to analyze patterns and explore the role of mathematics in nature and our culture.
Objectives
Upon completion of this lesson, students will:
- have been introduced to tessellations
- have learned about polygons
- have identified types of symmetry in tessellations
- have examined tessellations in the world around them
Standards Addressed
Grade 3
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Geometry
- The student demonstrates conceptual understanding of similarity, congruence, symmetry, or transformations of shapes.
- The student demonstrates a conceptual understanding of geometric drawings or constructions.
Grade 4
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Geometry
- The student demonstrates conceptual understanding of similarity, congruence, symmetry, or transformations of shapes.
- The student demonstrates a conceptual understanding of geometric drawings or constructions.
Grade 5
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Geometry
- The student demonstrates conceptual understanding of similarity, congruence, symmetry, or transformations of shapes.
- The student demonstrates a conceptual understanding of geometric drawings or constructions.
Grade 6
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Geometry
- The student demonstrates conceptual understanding of similarity, congruence, symmetry, or transformations of shapes.
Grade 7
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Geometry
- The student demonstrates conceptual understanding of similarity, congruence, symmetry, or transformations of shapes.
Grade 8
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Geometry
- The student demonstrates conceptual understanding of similarity, congruence, symmetry, or transformations of shapes.
Grade 9
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Geometry
- The student demonstrates an understanding of geometric relationships.
- The student demonstrates conceptual understanding of similarity, congruence, symmetry, or transformations of shapes.
- The student demonstrates a conceptual understanding of geometric drawings or constructions.
Grade 10
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Geometry
- The student demonstrates an understanding of geometric relationships.
- The student demonstrates conceptual understanding of similarity, congruence, symmetry, or transformations of shapes.
- The student demonstrates a conceptual understanding of geometric drawings or constructions.
Third Grade
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Geometry
- Reason with shapes and their attributes.
Fifth Grade
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Operations and Algebraic Thinking
- Analyze patterns and relationships.
Geometry
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Congruence
- Experiment with transformations in the plane
- Understand congruence in terms of rigid motions
Grades 6-8
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Geometry
- Apply transformations and use symmetry to analyze mathematical situations
- Use visualization, spatial reasoning, and geometric modeling to solve problems
Grades 9-12
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Geometry
- Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships
- Apply transformations and use symmetry to analyze mathematical situations
- Use visualization, spatial reasoning, and geometric modeling to solve problems
Geometry
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Data Analysis and Probability
- Competency Goal 3: The learner will transform geometric figures in the coordinate plane algebraically.
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Geometry and Measurement
- Competency Goal 2: The learner will use geometric and algebraic properties of figures to solve problems and write proofs.
Technical Mathematics I
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Geometry and Measurement
- Competency Goal 2: The learner will measure and apply geometric concepts to solve problems.
Technical Mathematics II
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Geometry and Measurement
- Competency Goal 1: The learner will use properties of geometric figures to solve problems.
Integrated Mathematics III
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Geometry and Measurement
- Competency Goal 2: The learner will use properties of geometric figures to solve problems.
3rd Grade
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Geometry
- The student will demonstrate through the mathematical processes an understanding of the connection between the identification of basic attributes and the classification of two-dimensional shapes.
6th Grade
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Geometry
- The student will demonstrate through the mathematical processes an understanding of shape, location, and movement within a coordinate system; similarity, complementary, and supplementary angles; and the relationship between line and rotational symmetry.
5th Grade
-
Geometry
- The student will demonstrate through the mathematical processes an understanding of congruency, spatial relationships, and relationships among the properties of quadrilaterals.
7th Grade
-
Geometry
- The student will demonstrate through the mathematical processes an understanding of proportional reasoning, tessellations, the use of geometric properties to make deductive arguments. the results of the intersection of geometric shapes in a plane, and the
- The student will demonstrate through the mathematical processes an understanding of proportional reasoning, tessellations, the use of geometric properties to make deductive arguments. the results of the intersection of geometric shapes in a plane, and the relationships among angles formed when a transversal intersects two parallel lines.
4th Grade
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Geometry
- 4.17.c The student will investigate congruence of plane figures after geometric transformations such as reflection (flip), translation (slide) and rotation (turn), using mirrors, paper folding, and tracing.
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Geomety
- 4.17.c
5th Grade
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Geometry
- 5.15a The student, using two-dimensional (plane) figures (square, rectangle, triangle, parallelogram, rhombus, kite, and trapezoid) will recognize, identify, describe, and analyze their properties in order to develop definitions of these figures
6th Grade
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Geometry
- 6.14 The student will identify, classify, and describe the characteristics of plane figures, describing their similarities, differences, and defining properties.
- 6.15 The student will determine congruence of segments, angles, and polygons by direct comparison, given their attributes. Examples of noncongruent and congruent figures will be included.
Student Prerequisites
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Geometric:
Students must be able to:
- Recognize regular polygons, such as triangles, rectangles and hexagons
- Understand the difference between an edge and a corner
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Technological:
Students must be able to:
- perform basic mouse manipulations such as point, click and drag
- use a browser for experimenting with the activities
Teacher Preparation
- Access to a browser
- Pencil and Paper
- Regular Polygons Data Table
Key Terms
polygon
A closed plane figure formed by three or more line segments that do not cross over each other
tessellation
A tessellation is a repeated geometric design that covers a plane without gaps or overlaps
Lesson Outline
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Focus and Review
Remind students what has been learned in previous lessons that will be pertinent to this lesson and/or have them begin to think about the words and ideas of this lesson:
- Has anyone ever heard of M. C. Escher? (Escher was a famous artist who enjoyed twisting perceptions of reality. He was responsible for works such as Reptiles, Horseman and many more that incorporated the use of tessellations.)
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Objectives
Let the students know what it is they will be doing and learning today. Say something like this:
- Today, class, we are going to learn about tellellations.
- We are going to use the computers to learn about tessellations, but please do not turn your computers on until I ask you to. I want to show you a little about this activity first.
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Teacher Input
- Introduce the t=Tessellation Applet in order to familiarize students to the idea of tessellations and how they developed.
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Guided Practice
- Have the students explore which regular polygons tessellate and why. Start them by examining tessellations of regular polygons including number of sides and interior angle measurements by using a data table . Encourage students to determine a pattern among the polygons that they tessellate. Ask the students to predict which regular polygons will and will not tessellate and why. Follow-up by having the students write a concise definition for a regular polygon tessellation. Have them expand this definition to describe a tessellation made from non-regular polygons.
- After the students have determined which regular polygons tessellate, discuss the types of symmetry present in tessellations.
- Have the students build tessellations and identify the types of symmetry present. Give them a table to record the basic shape used to tile and the types of symmetry present in the basic unit and in the tessellated pattern.
- Discuss how angle measure, area, and perimeter apply to tessellations.
- Allow students time to practice their knowledge about tessellations. Have teams of students work together. Instruct one student on the team to create a tessellation. Have that student describe the tessellation to other students and see if the other students can recreate the tessellation without looking. The students should formalize their terminology and describe the tessellation in terms of angle measure, polygon shape, symmetry, area and perimeter.
- Lead a discussion about tessellations in the world. Ask students to identify tessellations that they see in their daily lives and in nature.
- Discuss the ways that we perceive patterns. Lead a discussion about optical illusions to demonstrate how we perceive patterns. Also discuss the use of color in tessellations. Suggest that the students change the colors in their tessellations to see what effect that has on how they perceive the pattern. They may want to record their observations in a journal.
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Independent Practice
- Ask the students to use the Tessellation Activity to build tessellations of patterns they see in art and nature. You may also ask students to stretch the regular polygons into the letters of the alphabet or the letters of their name and tessellate the pattern. Have them record which polygon is best used to shape each letter. Also have them record what type of symmetries are present in each tessellation.
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Closure
- You may wish to bring the class back together for a discussion of the findings. Once the students have been allowed to share what they found, summarize the results of the lesson.
Alternate Outline
This lesson can be rearranged in several ways. Here is an example of a shorter version:
- The lesson can begin by introducing the Tessellation applet to introduce students to the idea of tessellations and how they developed.
- Discuss the types of symmetry present in tessellations.
- Have the students build tessellations and identify the types of symmetry present. Give them a table to record the basic shape used to tile and the types of symmetry present in the basic unit and in the tessellated pattern.
Suggested Follow-Up
After these discussions and activity, the students should have practiced their ability to recognize symmetry in plane figures. Students can gain a deeper understanding of other pr inciples of geometry by exploring tessellations in the Geometry Lesson. The tessellation activity could also be used to explore spatial visualization and pattern recogni tion with the Visual Pattern Lesson.