Visual Patterns in Tessellations

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:

  • Introduced to tessellations
  • Learned about several types of polygons
  • Examined tessellating patterns in the world around them

Standards Addressed

Grade 3

  • 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

  • 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

  • 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

  • Geometry

    • The student demonstrates conceptual understanding of similarity, congruence, symmetry, or transformations of shapes.

Grade 7

  • Geometry

    • The student demonstrates conceptual understanding of similarity, congruence, symmetry, or transformations of shapes.

Grade 8

  • Geometry

    • The student demonstrates conceptual understanding of similarity, congruence, symmetry, or transformations of shapes.

Grade 9

  • Geometry

    • The student demonstrates an understanding of geometric relationships.
    • The student demonstrates a conceptual understanding of geometric drawings or constructions.

Grade 10

  • Geometry

    • The student demonstrates an understanding of geometric relationships.
    • The student demonstrates a conceptual understanding of geometric drawings or constructions.

Third Grade

  • Geometry

    • Reason with shapes and their attributes.

Fourth Grade

  • Operations and Algebraic Thinking

    • Generate and analyze patterns.

Geometry

  • Congruence

    • Experiment with transformations in the plane
    • Understand congruence in terms of rigid motions

Grades 6-8

  • Geometry

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

Grades 9-12

  • Geometry

    • Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships
    • Use visualization, spatial reasoning, and geometric modeling to solve problems

Geometry

  • Data Analysis and Probability

    • Competency Goal 3: The learner will transform geometric figures in the coordinate plane algebraically.

Technical Mathematics I

  • Geometry and Measurement

    • Competency Goal 2: The learner will measure and apply geometric concepts to solve problems.

Technical Mathematics II

  • Geometry and Measurement

    • Competency Goal 1: The learner will use properties of geometric figures to solve problems.

7th Grade

  • Algebra

    • The student will demonstrate through the mathematical processes an understanding of proportional relationships.

  • 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.

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 for experimenting with the activities.

Teacher Preparation

Key Terms

optical illusion

A drawing or object that appears to have an effect that it does not really have, such as when a flat painting seems to have three-dimensional depth

pattern

Characteristic(s) observed in one item that may be repeated in similar or identical manners in other items

symmetry

The correspondence in size, form, or arrangement of parts on a plane or line. In line symmetry, each point on one side of the line has a corresponding point on the opposite side of the line (picture a butterfly, with wings that are identical on either side). Plane symmetry refers to similar figures being repeated at different but regular locations on the plane

tessellation

A tessellation is a repeated geometric design that covers a plane without gaps or overlaps

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 Outline

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.