Since the regular polygons in a tessellation must fill the plane at each vertex, the interior angle must be an exact divisor of 360 degrees. This works for the triangle, square, and hexagon, and you can show working tessellations for these figures. For all the others, the interior angles are not exact divisors of 360 degrees, and therefore those figures cannot tile the plane.

Directions:

1.  Start with a shape that is known to tessellate… in our case, a square.

2.  Using your pencil, outline a shape on the bottom side of the square. (Step 1 below.)

3.  Cut out the outlined slice and translate it to the top of the square. Tape the slice to the bottom so that there are no spaces between the slice and the square. (Step 2 below.)

4.  Now, using your pencil, outline a shape on the left side of the square. (Step 3 below.)

5.  Cut out the outlined slice and translate it to the right side of the square. Tape the slice to the right side so that there are no spaces between the slice and the square. (Step 4 below.)

6.  At this point you are ready to create your tessellation. Place your created tessellation tile at the top left of your blank paper. Trace the tile onto the paper.

7.  Translate your tile to the left and continue to trace your tile.

8.  After you have filled the entire paper with tracings of the tile, you may begin to design your tile and create a picture tessellation. Be creative! J

Online resources:

1.  http://www.tessellations.org/

2.  http://mathforum.org/sum95/suzanne/whattess.html

3.  http://www.math.csusb.edu/courses/m129/angles.html


Name ______Period ______Date ______

Tessellation - Paragraph (5 sentences or more):

Explain the design of your tessellation in your own words. Why did you decide to create this tessellation?

What makes it meaningful to you? What are the Geometry terms you have learned connected to this activity?

CATEGORY / 4 / 3 / 2 / 1 / Your score
Use of Class Time / Used time well during each class period. Focused on getting the project done. Never distracted others. / Used time well during each class period. Usually focused on getting the project done and never distracted others. / Used some of the time well during each class period. There was some focus on getting the project done but occasionally distracted others. / Did not use class time to focus on the project OR often distracted others.
Required Elements / The finished project includes all required elements. / All but 1 of the required elements are included in the finished project. / All but 2 of the required elements are included in the finished project. / Only 1 part of the finished project was submitted.
Paragraph / The written paragraph is the indicated length or more, grammatically correct, and provides a thorough explanation of the unique student tessellation. / The written paragraph is the indicated length, provides an explanation of the unique student tessellation, and contains very few grammatical errors. / The written paragraph is not complete (5 sentences) and very little explanation of the tile design is included. / The written paragraph is not complete (5 sentences) and no explanation of the tile design is included.
Attractiveness / The tessellation is exceptionally attractive in terms of design, layout, and neatness. / The tessellation is attractive in terms of design, layout and neatness. / The tessellation is acceptably attractive though it may be a bit messy. / The tessellation is distractingly messy or very poorly designed. It is not attractive.
Originality & Creativity / The design of the tile used in the tessellation reflects an exceptional degree of student creativity in their creation and/or display. / The design of the tile used in the tessellation reflects student creativity in their creation and/or display. / The design of the tile used in the tessellation was based on the designs or ideas of others. / No graphics made by the student are included.

Total Score = ______

Percent = ______

What is tessellation?

·  Tessellation covers the surface of a plane in a symmetrical way without overlapping or

·  leaving gaps.

·  A tessellation is created when a shape is repeated over and over again covering a

·  plane without any gaps or overlaps.

·  Another word for a tessellation is a tiling.

A regular polygon is a figure with congruent sides and congruent angles. A regular tessellation means a tessellation made up of congruent regular polygons. Only three regular polygons tessellate in the Euclidean plane: triangles, squares or hexagons. We can't show the entire plane, but imagine that these are pieces taken from planes that have been tiled. Here are examples of

a tessellation of triangles /
a tessellation of squares /
a tessellation of hexagons /

When you look at these three samples you can easily notice that the squares are lined up with each other while the triangles and hexagons are not. Also, if you look at 6 triangles at a time, they form a hexagon, so the tiling of triangles and the tiling of hexagons are similar and they cannot be formed by directly lining shapes up under each other - a slide (or a glide!) is involved.

Why are these shapes the only shapes that can be tessellated? Well, you can work out the interior measure of the angles for each of these polygons:

Shape
triangle
square
pentagon
hexagon
more than six sides / Angle measure in degrees
60
90
108
120
more than 120 degrees