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.

Students may know that an equilateral triangle has 60-degree angles but may not be able to explain why or how this connects to regular hexagons. If students are having trouble explaining their thinking, consider asking:

• “What do you know about the angles in a triangle?”
• “How does the angle measure of an equilateral triangle connect to how many you can fit together at 1 vertex?”

Consider asking the following questions to lead the discussion of this activity:

• “How did you find the angle measures in an equilateral triangle?” (The sum of the angles is 180 degrees, and they are all congruent so each is 60 degrees.)
• “Why is there no space between 6 triangles meeting at a vertex?” (The angles total 360 degrees, which is a full circle.)
• “How does your tessellation with triangles relate to hexagons?” (The triangles meeting at certain vertices can be grouped into hexagons, which tessellate the plane.)
• “Are there other tessellations of the plane with triangles?” (Yes, infinite rows of triangles can be placed on top of one another—and displaced relative to one another.)

Consider showing students an isometric grid, used earlier in grade 8 for experimenting with transformations, and ask them how this relates to tessellations. (It shows a tessellation with equilateral triangles.)

Point out that this activity provides a mathematical justification for the “yes” in the table for triangles and hexagons.

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.

Consider asking the following questions:

• “How many triangles meet at each vertex in a regular tessellation with triangles?” (6)
• “What about squares?” (4)
• “Hexagons?” (3)

• “Why can’t there be any regular tessellations with polygons of more than 6 sides?” (Only 2 could meet at a vertex, but this isn’t possible since the angles have to add up to 360 degrees.)

There are only 3 regular tessellations of the plane. Ask students if they have encountered these tessellations before and if so, where. For example:

• Triangles (isometric grid)
• Squares (checkerboard, coordinate grid, floor and ceiling tiles)
• Hexagons (beehives, tiles)