Lesson 2

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.

If students working with the pentagon and octagon add other shapes to make a more complicated tessellation, remind them that a regular tessellation uses copies of a single shape.

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 one vertex and checked to see if there was extra space leftover.)
• “Why does the pentagon not work to tessellate the plane?” (3 fit together at one vertex, but there is extra space, not enough for a fourth.)
• “Why does the octagon not work?” (2 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 hence) 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 (MP2) in this activity. 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.

In the previous task, equilateral triangles, squares, and hexagons appeared to make regular tessellations of the plane. Tell students that the goal of this activity is to use geometry to verify that they do.

Refer students to regular polygons printed in the previous activity for a visual representation of an equilateral triangle.

Students may know that an equilateral triangle has 60-degree angles but may not be able to explain why. Consider prompting these students for the sum of the three angles in an equilateral triangle.

Students may not see a pattern of hexagons within the triangle tessellation. Consider asking these students what shape they get when they put 6 equilateral triangles together at a single 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 six triangles meeting at a vertex?” (The angles total 360 degrees, which is a full circle.)
• “How does your tessellation with triangles relate to hexagons?” (You can group the triangles meeting at certain vertices into hexagons, which tessellate the plane.)
• “Are there other tessellations of the plane with triangles?” (Yes. You can make infinite rows of triangles that 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.

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 two could meet at a vertex, but this isn’t possible since the angles have to add up to 360 degrees.)

There are only three 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)