In this second lesson in the sequence of three optional lessons, students look at regular tessellations, which are made of regular polygons. Students construct arguments about which regular polygons can create regular tessellations (MP3) and reason about why others cannot (MP2). This reasoning requires an understanding of the structure of interior angles of regular polygons (MP7).

Through the lesson, students show in detail that triangles, squares, and hexagons give the only possible regular tessellations. Students begin by exploring triangles, squares, pentagons, hexagons, and octagons to determine which can create a regular tessellation. Then students focus on equilateral triangles and the angle measurements that guarantee equilateral triangles will tessellate. Finally, students experiment with shapes that have more sides to determine if any of those can create regular tessellations, bringing in their knowledge of angle measures to support their claims.