Introduction

The students will be attempting to see what types of regular polygon shapes will fit together to cover a surface without overlapping or leaving gaps. In addition, they will try to conjecture a theory on how angles play an important role in deciding what combinations of regular polygons fit together to cover a plane.

The Materials

1. Each group will receive cutouts of the various regular polygons.

The Task

1. The students, in groups of two, will be asked to surround a point, without leaving any gaps, using regular triangles.

2. The students will attempt to surround a point using squares

3. The students will attempt to surround a point using hexagons.

4. The students will attempt to surround a point using pentagons.

5. The students will attempt to surround a point using regular polygons of larger sizes.

6. The students will attempt to surround a point using the various combinations at http://forum.swarthmore.edu/sum95/suzanne/whattess.html

7. The students will try to see which of the eight figures can cover the plane and which ones cannot.

Assessment

The groups will write up their theories on why certain regular polygons can surround a point without leaving gaps while others cannot. Also, the groups will describe their ideas on covering a plane with polygons that are not regular. Finally, the groups will discuss their views on why certain combinations of regular polygons can cover the plane and others cannot.

Guiding Questions

1. Why are triangles, squares, and hexagons able to surround a point when a pentagon cannot?

2. Can you make a rule for polygons that will be able to surround a point without any gaps?

3. Of the eight figures which ones can cover the plane? Are there figures that you are not sure about? What makes you so sure that certain ones can cover the plane?

Conclusion

At the end of this lesson students will better understand the mathematical and logical complexities within the world of tessellations.