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## Investigating Tessellations Filling 2-D Space

Students explore tessellating, the arrangement of shapes that can fill a plane with no overlaps or gaps.

### Overview

Students explore the arrangement of shapes that can fill a plane with no overlaps or gaps. This is called tessellating. Students watch a video segment from Cyberchase in which the CyberSquad uses tessellating to overcome challenges to reach Hacker. Students are also introduced to the terms "congruence" and "similar."

2-5

1 hour

### Media Resources

Squares QuickTime Video
Triangles QuickTime Video
Hexagons QuickTime Video

### Part I: Learning Activity

1. Read the following to your students, "On handout 1 you will find a number of shapes for you to cut out. Imagine that these represent tiles you have purchased from the local hardware store. Figure out which of these tiles can be used to make a solid floor with no overlaps and no gaps. If we can make a pattern in which a simple shape is used to cover an area with no gaps or overlaps, we call that pattern a "tessellation."

2. After you decide which shapes or combination of shapes can work, distribute Handout 1: Variety of Shapes.

3. Ask students to cut out all the shapes and try to tessellate them. Note: To save time, cut out the shapes for each group in advance.

4. Tell the students that they will watch a video segment in which the CyberSquad is racing to the top of The Pyramid of Petuluma to stop Hacker from bringing Gigabyte back to life. To get there they must cross a river and then cross a stream of burning lava. Finally, they must find a way to block the sun to stop Hacker. Ask the students to watch each video and record the different shapes that the CyberSquad uses to tessellate. Discuss the term "regular" with the students. A regular figure is one which all sides are of equal length. Provide students with some examples on the board.

5. Play the Squares QuickTime Video.

6. Play the Squares QuickTime Video.

7. Play the Hexagons QuickTime Video.

8. Discuss handout 1. Ask the students if all the shapes on the handout are regular figures.

9. Distribute Handout 2: Congruence and Similarity.

10. Tell students that when the length of the sides and the angles of two shapes are identical we call the shapes congruent. One way to tell is to see if one of the shapes fits exactly on top of the other. When two objects have the same shape only one is larger or smaller than the other we call the objects similar.

11. Discuss handout 2. Pay particular attention to the relationship between congruence and similarity. Note the fact that all congruent figures are similar to each other, but two figures can be similar without being congruent.

### Part II: Assessment

Assessment: Level A (proficiency): Students are asked to tessellate a right triangle.

Assessment: Level B (above proficiency): Students are asked to tessellate an equilateral triangle and create a larger triangle, which is similar to the original.

## Educational Standards

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