Lesson 3

In this activity, students experiment with copies of a triangle (no longer equilateral) and discover that it is always possible to build a tessellation of the plane. A key in finding a tessellation with copies of a triangle is to experiment with organizing copies of the triangle, and then reasoning that two copies of a triangle can always be arranged to form a parallelogram. Students may not remember this construction from the sixth grade, but with copies of the triangle to experiment with, they will find the parallelogram or a different method. These parallelograms can then be put together in an infinite row, and these rows can then be stacked upon one another to tessellate the plane.

Students may struggle to put together copies of their triangle in a way that can be continued to tessellate the plane.

• Ask these students to put together two copies of the triangle.
• If they have made a parallelogram, ask them what kind of quadrilateral they have made.
• If they have not made a parallelogram, ask them if there is a different way they can combine two copies of the triangle.

Invite several students to share their tessellations for all to see.

Consider asking the following questions to help summarize the lesson:

• “Were you able to make a tessellation with copies of your triangle?” (Most students should respond yes.)
• “How did you know that you could continue your pattern indefinitely to make a tessellation?” (Any parallelogram can be used to tessellate the plane as they can be placed side by side to make infinite “rows” or “columns” and then these rows or columns can be displaced to fill up the plane.)

Share some of the tessellation ideas students come up with and relate them back to previous work, that is the tessellation of the plane with rectangles and parallelograms.

The previous activity showed how to make a tessellation with copies of a triangle. A natural question is whether or not it is possible to tessellate the plane with copies of a single quadrilateral. Students have already investigated this question for some special quadrilaterals (squares, rhombuses, regular trapezoids), but what about for an arbitrary quadrilateral? This activity gives a positive answer to this question. Pentagons are then investigated in the next lesson, and there we will find that some pentagons can tessellate the plane while others can not.

In order to show that the plane can be tessellated with copies of a quadrilateral, students will experiment with rigid motions and copies of a quadrilateral.

This activity can be made more open ended by presenting students with a polygon and asking them if it is possible to tessellate the plane with copies of the polygon.

Students may need to be reminded of the definition of a trapezoid: one pair of sides are contained in parallel lines.

If the figures are not traced accurately, it may be difficult to determine if the pattern, using 180-degree rotations, can be continued. Ask these students what they know about the sum of the three angles in a triangle and in a quadrilateral.

Invite some students to share their tessellations.

Discussion questions include:

• “Were you able to tessellate the plane with copies of the trapezoid?” (Yes, two of them can be out together to make a parallelogram, and the plane can be tessellated with copies of this parallelogram.)
• “What did you notice about the quadrilateral and the 180-degree rotations?” (They fit together with no gaps and no overlaps and leave space for four more quadrilaterals.)
• “How do you know that there are no overlaps?” (The sum of the angles in a quadrilateral is 360 degrees. At each vertex in the tessellation, copies of the four angles of the quadrilateral come together.)

Revisit the question from the start of the activity, “Can any quadrilateral be used to tessellate the plane?” Invite students to share if their answer has changed and explain their reasoning.

All triangles and all quadrilaterals give tessellations of the plane. For the quadrilaterals, this was complicated and depended on the fact that the sum of the angles in a quadrilateral is 360 degrees. Regular pentagons that do not tessellate the plane have been seen in earlier activities. The goal of this activity is to study some types of pentagons that do tessellate the plane. Students make use of structure (MP7) when they relate the pentagons in this activity to the hexagonal tessellation of the plane, which they have seen earlier.

This activity can be made more open ended by presenting students with a polygon and asking them if it is possible to tessellate the plane with copies of the polygon.

Students may struggle tracing the rotated hexagon. Ask them what happens to the segments making angle A when the hexagon is rotated about A by 120 (or 240) degrees.

Students may wonder why the hexagon that they make by putting three pentagons together is a regular hexagon. Invite these students to calculate the angles of the hexagon.

Students may be successful in building a tessellation in the first question. The following questions guide them through a method while also asking for mathematical justification. Students who are successful in the first question can verify that their tessellation uses the strategy indicated in the following, and they will still need to answer the last two questions.

Invite some students to share their tessellations.

Some questions to discuss include: 

• “Does the hexagon made by three copies of the pentagon tessellate the plane?” (Yes.)
• “How do you know?” (I checked experimentally, or I noticed that all of the angles in the hexagon are 120 degrees.)
• “Why was it important that the two sides of the pentagons making the 120 angles are congruent?” (So that when I rotate my pentagon, those two sides match up with each other perfectly.)
• “What is special about this pentagon?” (Two sides are congruent, three angles measure 120 degrees. . . )