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Abstract

This lesson allows students to examine the mathematical nature of art, tilings and tessellations. The 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 world culture.

Objectives

Upon completion of this lesson, students will have:

• been introduced to tessellations
• learned about several types of polygons
• examined tessellating patterns in the world around them

Standards

The activities and discussions in this lesson address the following NCTM standards:

Geometry

Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships

• precisely describe, classify, and understand relationships among types of two- and three-dimensional objects using their defining properties
• understand relationships among the angles, side lengths, perimeters, areas, and volumes of similar objects
• create and critique inductive and deductive arguments concerning geometric ideas and relationships, such as congruence, similarity, and the Pythagorean relationship

Apply transformations and use symmetry to analyze mathematical situations

• describe sizes, positions, and orientations of shapes under informal transformations such as flips, turns, slides, and scaling
• examine the congruence, similarity, and line or rotational symmetry of objects using transformations

Use visualization, spatial reasoning, and geometric modeling to solve problems

• use visual tools such as networks to represent and solve problems
• use geometric models to represent and explain numerical and algebraic relationships
• recognize and apply geometric ideas and relationships in areas outside the mathematics classroom, such as art, science, and everyday life

Links to other standards.

Student Prerequisites

• Arithmetic: Students must be able to:
  • understand the properties of polygons
  • be able to recognize types of symmetry after they are introduced
• Technological: Students must be able to:
  • perform basic mouse manipulations such as point, click and drag.
  • use a browser, such as Netscape, for experimenting with the activities.

Teacher Preparation

Students will need:

• Access to a browser
• pencil and paper
• Copies of supplemental materials for the activities:
  • Visual Patterns in Tessellations Worksheet

Key Terms

This lesson introduces students to the following terms through the included discussions:

• tessellation
• pattern
• symmetry
• optical illusion

Lesson Outline

1. 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:

   • Ask students what they know about tessellations. If needed, present the information in the introduction to tessellations discussion.

      

   • See if the students are familiar with symmetry, and describe to them the different types present in tessellations.

      

   • Finally, see what they already know about color and optical illusions and how they affect perception.

      

    

2. Objectives

   Let the students know what it is they will be doing and learning today. Say something like this:

    

   • Today, class, we will be talking more about tessellations.

      

   • We are going to use the computers to learn about them, but please do not turn your computers on or go to this page until I ask you to. I want to show you a little bit about the activity first.

      

3. Teacher Input

   You may choose to lead the students in a short discussion about tessellations in the world.

   Explain to the students how to do the assignment. You should model or demonstrate it for the students, especially if they are not familiar with how to use our computer applets.

    

   • Open your browser to the Tessellate activity in order to demonstrate it to the students.

      

   • Show students how to bend the edges or corners of the polygons to form a new shape.

      

   • Select colors for the new shape, and click the "tessellate" button to show students the pattern.

      

   • Choose another kind of polygon from the pull down menu and show students how to change shapes.

      

   • If you choose to, you may pass out the Visual Patterns in Tessellations Worksheet.

      

    

4. Guided Practice

   Try designing one more tessellation, letting the students direct your moves. Ask students to suggest a pattern from nature or art that tessellates, such as a honeycomb for bees.

   • If your class seems to be having a little trouble with understanding tessellations, do another example together.

      

     • Explain that if they start a design and it doesn't work out, clicking the "reset" button will clear the screen so they can begin again.

        

     • Each time you tessellate the pattern, review with the students which polygon you started with, what types of symmetry are present in the finished tessellation, and experiment with a variety of colors to see the different effects.

        

    

5. Independent Practice

    

   • Allow the students to work on their own and to complete the worksheet, should you choose to provide one. Monitor the room for questions and to be sure that the students are on the correct web site.

      

   • Another option for independent practice is to challenge students with creating alphabet letter shapes for their initials that will tessellate.

      

   • Again, for each pattern, remind the students to be aware of which polygon they started with, what types of symmetry are present in the original shape and the finished tessellation, and how the use of color changes how the design is perceived.

      

    

6. 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 Outlines

This lesson can be rearranged in several ways if there is only one available computer:

• Use the computer to model the tessellations, and have the class complete the worksheet together with you.

   

• Print out the information from the color and optical illusions discussions. Let groups of two or three students complete the worksheet using the computer while you present the other information to the remaining students, and rotate through groups of students. A Fine Arts teacher may be able to contribute color wheels and additional materials about warm and cool colors.

   

Extensions

This lesson may be followed by either of these lessons:

• Plane Symmetry
• Geometry in Tessellations

1. Think of three different objects in the world around you that have tessellating patterns.

    

   • Item 1:

      

   • Item 2:

      

   • Item 3:

      

    

2. Try drawing each of these patterns using the Tessellate! activity. Choose a starting polygon shape, then bend the lines until they look like the design you are re-creating, then click the "tessellate" button.

    

3. Which starting polygon shapes did you use for your three tessellations?

    

   • Item 1 polygon:

      

   • Item 2 polygon:

      

   • Item 3 polygon:

      

    

4. Which types of symmetry does each tessellation use?

    

   • Item 1 uses __________________________

      

   • Item 2 uses __________________________

      

   • Item 3 uses __________________________

      

5. Now choose one of your designs and experiment further by changing the colors.

    

   • Do you notice a difference when you use colors with a strong contrast rather than colors with little contrast? Describe the difference:

      

   • Observe your tessellations using warm colors (reds and oranges). Then try using cool colors (greens and blues). What differences are there?

      

   • Now mix a warm color with a cool color in the tessellation. How does this combination look compared to using two warm or two cool colors?