Tessellations

Students explored which shapes tessellated the surface and why.

Students cut out 6 regular polygons from oaktag and them used these 6 shapes to decide which of the shapes could tessellate the surface. After discovering only three regular polygons could tessellate the surface the students decided that the angles of the polygon helped them decide if the polygon could be used. The pentagon could not tessellate the surface because each angle was 105 degrees and 105 is not a factor of 360.

Then the students looked for semi-pure tessellations and learned that there were various combinations that could be used. It was difficult to use the pentagon and the decagon in the tessellations. The rest were easily used: the square, the triangle, the hexagon and the octagon.

During our one week unit on transformations the students learned to how a shape could be transformed to a new position without changing its shape or size. On the first day the students used patty paper to explore translations, reflections, and rotations. Then the students used the Geometer's Sketchpad to explore how to replicate these translations, reflections, and rotations on a computer.

Then the students used these transformations to create the regular polygons: triangles, squares, pentagons, hexagons, heptagons, octagons, and decagons.

On the following day the students explored using these three transformations to tessellate these regular polygons across the plane. The students found that they could only tessellate the plane with an equilateral triangle, a square, a rectangle, a parallelogram, and the hexagon. After looking at the figures more closely the students found that the these polygons could tessellate the plane because the angle in each figure was a factor of 360. And if a figure was to cover all the space about a point the angles had to fit about the 360 degrees.

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But it was not possible to tessellate the plane with pentagons or octagons.

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	The students learned that this last tessellations with an octagon was called a semi-pure tessellations because two regular polygons could be used to tessellate the plane: an octagon and a square because the angles were 135 degrees, 135 degrees, and 90 degrees or a total or 360 degrees.
	Then for two days the students worked with their knowledge of tessellations and transformations to create Escher type tessellations using translations on a square, rhombus, or hexagon. This can lead to some interesting tesselations.