Aims:

1. Students will create and analyze patterns and designs using congruence, symmetry, translation, rotation, and reflection.

Standards Addressed:

1. Understands concepts of symmetry, congruence, and similarity .
2. Draws lines of symmetry, similar, and congruent figures .
3. Understands and use properties of symmetry, similarity, and congruence.

4. Builds new mathematical knowledge through problem solving and showing relationships between figures.

5. Develops an understanding of translations (slides), rotations (turns), and reflections (flips).

Materials:

computer with Internet capabilities, printer, tangram pieces, graph paper, mirrors, drawing application such as Kidpix (published by Broderbund) or AppleWorks(published by Apple),

Tesselmania (published by MECC).

Vocabulary:

Dictionary.com or the geometric transformation vocabulary page may be used to define these words:

Motivation:

1. Distribute tangram pieces. (Students can cut out their own tangrams from this pattern).

Have students name and describe each shape in terms of the number of sides and angles.

Which shapes tessellate? have students demonstrate how the shapes can tessellate.

Development:

1. Students log onto: The Four Types of Symmetry in the Plane
written by Dr. Susan Addington to answer these questions:

• What is a reflection?

• What is a translation?

• What is a rotation?

• What is a glide reflection?

• What does symmetry mean?

To reinforce what a flip or reflection is, have students hold a small mirror next to each tangram piece one at a time. Students practice flipping each tangram piece and trace the original tangram piece and the reflection. Identify the line of symmetry in each reflection by labeling it on the graph paper drawings.

2. To reinforce what a slide or translation is, students trace each tangram piece on graph paper, slide each piece and trace (being careful not to rotate each piece).

3. To reinforce what a rotation is, have students trace each tangram piece on graph paper.

Then rotate each piece 90 degrees, 180 degrees, and 270 degrees and draw each rotation.

4. Design a jigsaw puzzle on a drawing application using the 7 tangram pieces. Print two copies of the puzzle. Cut out the shapes on one printout. Leave the other as a solution page.

(It would be a good idea to laminate the solutions). Have students exchange their jigsaw puzzles and solve.

5. Students, in cooperative groups (triangle, parallelogram, square), examine the polygons in the following figures and predict which shapes a congruent (same size and shape).

image credit: http://standards.nctm.org/document/chapter4/geom.htm#bp3

Students check each pair for congruency by placing each pair thought to be congruent together.

If the sides and edges match exactly, there is congruency. Students then explain what transformations took place to from one to get from one polygon in the congruent sets to the other polygon in the set. Students demonstrate the transformations on a flannel board.

6. Students log onto What is a Tessellation to answer these questions:

• What is a tessellation?

• What is the derivation of the word tessellation?

• What is a regular shape?

• What are some regular polygons that can tessellate?

• Where can we find examples of a tessellation in our school?

Students choose one tangram piece, use rotations, translations, and reflections to create a tessellation on graph paper.

Summary:

Students describe by writing in their own words and illustration (using a drawing application) what a reflection, translation, and rotation is. See sample.

Students create their tessellations on the computer with Tesselmania, and describe in words the transformations that took place to create their tessellations.

Examine Andrew Crompton's and M. C. Escher's drawings to view examples of tessellations in art.

Evaluation:

An assessment of students understanding of the concepts in this lesson can be found at in the section entitled: "Assessment through Observations and Conversations."

Related Web Sites:

1. Tessellations in rugs may be viewed at

http://mathforum.org/geometry/rugs/symmetry/grids.html

2. Suzanne Alejandre provides activities on ClarisWorks that involve tessellations.

The concept of symmetry is reinforced.
http://mathforum.org/sum95/suzanne/cwtess.html

3. Suzanne Alejandre provides activities on Hyperstudio that involve tessellations,

rotations, and reflections.

http://mathforum.org/sum95/suzanne/colortips.html

4. Cox Education Learning Station offeres a sixth grade unit on symmetry, rotation,

flips, and slides.

http://216.239.39.100/search?q=cache:yWOE1VZLkdcC:www.coxednet.org/

edu2000-slp.pdf+GEOMETRY+SYMMETRY&hl=en

5. Tiling activities involving symmetry may be found at this site.

http://scienceu.com/geometry/articles/tiling/tilings.html

6. Interactive tiling of triangles to form patterns will be found here.

http://ScienceU.com/geometry/articles/tritile/tiling.html

7. Puzzles inolving symmetry can be found here.

http://m759.freeservers.com/puzzle.html 

Related Materials:

Julie Kanazawa, Teaching Mathematics with the Internet, Classroom Connect,

(El Segunda, CA, 1998) pages 78-81. This is a lesson on patterns and tesselations.

Follow Up:

In Lesson 4, students use tangram pieces to reinforce the concept of area.