[Elementary Lesson Plans] [Middle School Lesson Plans] [High School Lesson Plans] [Lesson Plans by Teacher Name] [Main Page]

 

 

MAISA and the REMC Association of Michigan

Best Practices in Technology Integration

Plan

 

Title: Terrific Tessellations

Subject(s): Math: transformational geometry, tessellations, symmetry, patterns.

Intended Grade Level(s): 4 - 6

Download PDF version

Description:

  • Tessellation: A tiling, made up of the repeated use of a shape that completely fills a plane without gaps or overlaps.Terrific Tessellations is a geometry unit that allows students to experience the excitement and beauty of mathematics beyond numerical calculations. This unit integrates art and math followed by a technology component. Students explore tessellations that are found in nature and everyday objects. They look at art connections in designs of fabric, quilts, and mosaics. The life and artwork of M. C. Escher is highlighted through examples of his work and related web sites. Students continue to explore transformational geometry through the use of pattern blocks and geometric shapes, and then they begin to create their own tessellations with paper and pencil. The terms translation, rotation, and glide reflection are introduced, demonstrated and developed. Finally, the software program TesselMania! is introduced to the whole class. This software actually shows students how a design was made and through a slideshow they can watch it tessellate. Students then work in pairs to explore tessellations and to create new tessellations combining their previous hands-on knowledge of tessellations and the power of TesselMania! A final project allows students to create designs that could be used for wrapping paper, posters, or clothing designs.

    • Curriculum Benchmarks:

     

    Materials/Hardware/Software:

     

    Activities/Procedures:

  • Preparatory Activities:
    1.  
    2. As a class, discuss the meaning of tessellation or tiling of the plane. Generate a list of tessellations that can be found in nature or everyday objects such as: kitchen or bathroom tiles, honeycomb, patio bricks. Look at items (or pictures) of wallpaper, fabric, quilts, stained glass windows, mosaics.
    3. Study the life and artwork of M.C. Escher. Check out books and websites that have collections of his work. Escher's work can be found on posters, T-shirts, giftwrap, ties, and mugs.
    4. Use pattern blocks or geometric figures cut from paper or cardboard to clarify the concept of tessellations. Give each student one polygon and have them place it in the middle of a blank piece of paper (or a transparency) and trace around it. Then, lining the polygon up with any side of the traced image, have them trace the polygon again, repeating this process until they cover the entire page.
    5. Working in pairs, have students investigate which polygonal shapes will tessellate and which ones won't. Discuss characteristics of tessellating figures.

    Activity # 1: Introduction to Tessellations: TranslationsMaterials: Teacher: overhead projector, blank transparencies, copy of rectangle, markers, scissors, tape. Student: paper, pencil crayons or colored pencils, scissors, tape, and copy of rectangle. Rationale: Students who are unfamiliar with tessellations will benefit from a hands-on experience with translations before they begin to use the software. Procedure:

    1. Review the fact that certain polygons, such as triangles, quadrilaterals, and hexagons will tessellate the plane. Explain that in this lesson they will learn a method of modifying a polygon so it will still tile the plane.
    2. Distribute copies of a rectangle 2" by 3". Have students cut out the rectangle and use a light color to shade one side. Make a copy of the student rectangle on a colored transparency and cut it out. Recall with students that the rectangle will tessellate the plane.
    3. Place your rectangle on the overhead projector and select any side. With a marker, draw a free-form line from one corner of the chosen side to the other. Students do the same. Keep the line fairly simple because they will be cutting on the drawn line. Suggest that the students flip the rectangle over so that they are drawing on the blank side.

    4.  

    5. Cut along the line that has been drawn from one corner to the other. Remind students that they need to make a smooth cut so that they end up with only two pieces and no leftover scraps. Every piece of the rectangle must be retained.
    6. On the overhead, demonstrate how to take the piece just cut and slide it across the rectangle to the opposite side. Make sure the edges are lined up and tape the two pieces together.
    7. This type of move is called a "translation" or "slide." A translation is a type of geometric transformation; a transformation is use to change or transform the original figure.
    8. Have students perform a translation with their piece and tape it in the appropriate position. Make sure that the blank sides of the two pieces are facing up.
    9. Have students perform another translation on the remaining pair of sides. Do the same with your modified rectangle.
    10. Once the two translations are complete, trace your modified shape in the middle of a transparency. Show that it tessellates by tracing several more copies of your shape. Lead your students to identify the fact that translations are used both to transform the initial shape and to tessellate the plane.
    11. Have students trace their shapes on paper to create a unique tessellation. Summarize the terms "tessellation," "transformation," and "translation."
    12. Have students color their tessellation creations and try translation with other polygons.

    Activity # 2 Introduction to Tessellations: Midpoint and Vertex Rotations Materials and Rationale : Same as Activity #1, except instead of handout of a rectangle, prepare a handout of a right triangle. Procedure: (Follow the same basic format of Activity # 1)

    1. Demonstrate midpoint and vertex rotations using the overhead and handouts of a right triangle. Review the terms triangle, right triangle, hypotenuse and vertex.
    2. Have students perform a midpoint rotation with their piece and tape it in the appropriate position.
    3. Have students perform a vertex rotation with their piece and tape it appropriately.
    4. After the two rotations are complete, allow students to add details to their transformed triangle.
    5. Have students trace their shapes on paper to create a unique tessellation.
  • Activity #3 Introduction to Tessellations: Glide ReflectionsMaterials and Rationale : Same as Activity #1, except instead of handout of a rectangle, prepare a handout of a square.Procedure: (Follow the same basic format of Activity # 1)
    1. Demonstrate glide reflections using the overhead and handouts of a square. Review the terms quadrilateral and square.
    2. Students perform the glide with the piece that they just cut.
    3. Demonstrate how to take the piece that was just translated (glide) and reflect (flip) it. The combination of the glide and the reflection becomes the glide reflection.
    4. In a similar manner, have students perform another glide reflection on the remaining pair of sides.
    5. Trace the modified shape and continue to trace more copies of the shape on the transparency to demonstrate the tessellation to the students.
    6. Have students trace their shapes to create a unique tessellation.
  • Activity # 4 : Introduction to TesselMania!Teacher materials: One computer and large monitor or LCD display. Rationale: Seeing a demonstration of the software will help the students to get started with the program, to connect how it ties to their hands-on experiences with tessellations (Activities # 1 - 3), and see what options are available with the program.Procedure: 1. Open up the TesselMania! program. Following the logo screens, the start-up options will appear. Choose New to create a new document. Explain that the Open button is for opening a saved tessellation and Instructions provides a set of instruction for using the program. Go through the instructions on the screen. See below for samples of the introduction.2. After the introduction, select New and then "Translations". Demonstrate how to "see" a translation by clicking on the Show Me button. Click on the other transformational types, discuss what they are and the meaning of the arrows, and show students the examples. 3. Select "Translations" and the OK button. Demonstrate the options and use the Show Me button to see an animated example and Example button to see a sample tessellation. Point out that the Go Back button returns to the previous screen.4. To transform the shape, select the Arrow and demonstrate how it can be used to move or reshape the original polygon. Select the Tack to demonstrate how to add points to the polygon.5. After the transformation is complete, Illustrate the process of making an automatic transformation by clicking on Tile Magic to view an animated sequence showing how each transformation is carried out. 6. Demonstrate how to add details to the transformation with the paint tools. 7. Click on the Tessellate button to see the plane automatically tessellated with the new shape. Additional: 8. Click on the Tessellation Magic button to watch the shape tile the plane step-by-step. Click on the Metamorphic Magic button to display the animation of the final tessellation as it is developed from the original polygon. 9. Use the Edit Tile button to return to the single tile and explain how to alter the tile with more colors or details. Activity # 5 Using TesselMania!Student Materials: One computer per student of pairs of students.Rationale: This lesson allows student to begin exploring and become creative with tessellations by using transformation to gain familiarity with the computer program, translations, rotations, and glide reflections.Procedure:1. Students work individually or in pairs to create one tessellation to share with the class. Explain to the students that they will need to explain the type of transformation used to make the shape and how the shape was moved to fill the plane.2. Ask students to create a title for their finished design.3. If a projection system is available, have students display their tessellation to the class. The class can guess the shape of the original tile, and the type of transformation used to modify the tile, and the transformation used to tessellate the plane. The students can then use the Tile Magic button to demonstrate how the tile was created and the Tessellation Magic button to show how the plane was filled.Activity # 6: Design with TesselMania! Teacher Materials: Sample of items such as wallpaper, wrapping paper, fabric or web sites with tessellation patterns.Student Materials: Samples of tessellation patterns, one computer per student or pairs of students.Rationale: Students combine their creative talents to create designs that could be used for wallpaper, wrapping paper, fabric, posters, or clothing designs.Procedure: 1. Display teacher and student examples of tessellation patterns. Brainstorm uses to tessellation patterns. 2. Explain to students their goal of experimenting with TesselMania! to produce a design they could use on a product of their choice. 3. Give students opportunity to share their designs. 4. Have students compile a slide show of the various designs.

    • Assessment/Evaluation

    Terrific Tessellations Assessment Rubric

     

     

     

    Follow-up Activities:

     

    Resources:

    Books:
    Britton, Jill, and Dale Seymour, Introduction to Tessellations.
    Palo Alto: Dale Seymour Publications, 1989.
    Britton, Jill and Walter, Teaching Tessellating Art. Palo Alto: Dale Seymour Publications, 1992.
    Web Sites: There are many tessellation-related websites. Here are a few to start with:
    Download the TesselMania Demo ftp://forum.swarthmore.edu/software/demos/tesselmania.sea.hqx
     
    World of Escher http://lonestar.texas.net/~escher/
     
    HyperStudio & Tessellations in Color http://forum.swarthmore.edu/sum95/suzanne/colortess.html
     
    What is a Tessellation? http://forum.swarthmore.edu/sum95/suzanne/whattess.html

     

    Name: Linda McConville

    School District: St. Joseph Public Schools

    School: E.P. Clarke Elementary

    Address: 515 E. Glenlord Rd., St. Joseph, MI 49085