The goal of this activity is to introduce a regular tessellation of the plane and conjecture which shapes give regular tessellations. Students construct arguments for which shapes can and cannot be used to make a regular tessellation (MP3). The focus is on experimenting with shapes and noticing that in order for a shape to make a regular tessellation, we need to be able to put a whole number of those shapes together at a single vertex with no gaps and no overlaps. This greatly limits what angles the polygons can have and, as a result, there are only three regular tessellations of the plane. This conjecture will be demonstrated in the other two activities of this lesson.

In the digital version of the activity, students use an applet to decide if regular polygons create tessellations. The applet allows students to work with many copies of each polygon without tracing. The digital version may be preferable if time is limited.

To help students think more about what shapes do and do not tessellate and why, ask:

• “Which polygons appear to tessellate the plane?” (square, equilateral triangle, hexagon)
• “How did you decide?” (I put as many together as I could at 1 vertex and checked to see if there was extra space left over.)
• “Why does the pentagon not work to tessellate the plane?” (Three fit together at one vertex, but there is extra space, not enough for a fourth.)
• “Why does the octagon not work?” (Two fit together, but there is not enough space for a third.)

During the discussion, fill out the table, indicating that it is possible to make a tessellation with equilateral triangles, squares, and hexagons, but not with pentagons or octagons.

The goal of this activity is to verify, via angle calculations, that equilateral triangles and regular hexagons can be used to make regular tessellations of the plane. Students have encountered the equilateral triangle plane tessellations earlier in grade 8 when working on an isometric grid. In order to complete their investigation of regular tessellations of the plane, it remains to be shown that no other polygons work. This will be done in the next activity.

Students are required to reason abstractly and quantitatively in this activity (MP2). Tracing paper indicates that six equilateral triangles can be put together sharing a single vertex. Showing that this is true for abstract equilateral triangles requires careful reasoning about angle measures.

The goal of this activity is to show that only triangles, squares, and hexagons give regular tessellations of the plane. The method used is experimentation with other regular polygons. The key observation is that the angles on regular polygons get larger as we add more sides, which is a good example of observing structure (MP7). Since three is the smallest number of polygons that can meet at a vertex in a regular tessellation, this means that once we pass six sides (hexagons), we will not find any further regular tessellations. The activities in this lesson now show that there are three and only three regular tessellations of the plane: triangles, squares, and hexagons.

In the digital version of the activity, students use an applet to decide if other regular polygons create tessellations. The applet allows students to work with many copies of each polygon without tracing. The digital version may be preferable if time is limited.