This activity has students explore and construct several different types of tessellations. The first part of the activity uses only regular polygons to create pure and semi-pure tessellations; therefore, you will need custom tools for the constructions of an equilateral triangle, a square, and a regular hexagon. The second part extends the notion of tessellations to create non-regular polygon and "Escher-like" tessellations.

Mathematics: Students will use translations, rotations, and reflections to create tessellations. Moreover, the students will investigate the following theorems:

• Equilateral triangles, squares, and hexagons are the only three regular polygons that create pure tessellations.
• There are eight different semi-regular tessellations.
• Any scalene triangle or quadrilateral will create a pure tessellation.

Mathematical Thinking: In this activity, students will be asked to conjecture about the construction of possible tessellations.

Technology: This activity uses several Sketchpad commands and features such as: Custom Tools, Translate, Rotate, Reflect, Mark Center, Mark Mirror, Mark Vector, and Mark Angle.

The following sketches illustrate the three regular polygons that tessellate the plane, and five of the eight semi-pure regular tessellations described in the activity.