Lesson 3
Tessellating Polygons
3.1: Triangle Tessellations (15 minutes)
Optional activity
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.
Launch
Assign different triangles to different students or groups of students. Provide access to tracing paper if using the print materials. If using the digital materials, the activity can be done in the applet.Assign different triangles to different students or groups of students. Provide access to tracing paper.
If students finish early, consider asking them to work on building a different tessellation or coloring their tessellation.
Supports accessibility for: Memory; Organization
Student Facing
Your teacher will assign you one of the three triangles. Your goal is to find a tessellation of the plane with copies of the triangle.
Student Response
For access, consult one of our IM Certified Partners.
Launch
Assign different triangles to different students or groups of students. Provide access to tracing paper if using the print materials. If using the digital materials, the activity can be done in the applet.
If students finish early, consider asking them to work on building a different tessellation or coloring their tessellation.
Supports accessibility for: Memory; Organization
Student Facing
Your teacher will assign you one of the three triangles. You can use the picture to draw copies of the triangle on tracing paper. Your goal is to find a tessellation of the plane with copies of the triangle.
Student Response
For access, consult one of our IM Certified Partners.
Anticipated Misconceptions
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.
Activity Synthesis
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.
Design Principle(s): Cultivate conversation; Maximize metaawareness
3.2: Quadrilateral Tessellations (20 minutes)
Optional activity
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.
Launch
Begin the activity with, “Any triangle can be used to tile the plane (some of them in many ways). Do some quadrilaterals tessellate the plane?” (Yes: squares, rectangles, rhombuses, and parallelograms.) Next, ask “Can any quadrilateral be used to tessellate the plane?” Give students a moment to ponder, and then poll the class for the number of yes and no responses. Record the responses for all to see. This question will be revisited in the Activity Synthesis.
Provide access to tracing paper.
Supports accessibility for: Visualspatial processing; Organization
Design Principle(s): Optimize output (for explanation); Maximize metaawareness
Student Facing
 Can you make a tessellation of the plane with copies of the trapezoid? Explain.

Choose one of the other two quadrilaterals. Next, rotate the quadrilateral 180 degrees around the midpoint of each side. What do you notice?
 Can you make a tessellation of the plane with copies of the quadrilateral from the previous problem? Explain your reasoning.
Student Response
For access, consult one of our IM Certified Partners.
Launch
Begin the activity with, “Any triangle can be used to tile the plane (some of them in many ways). Do some quadrilaterals tessellate the plane?” (Yes: squares, rectangles, rhombuses, and parallelograms.) Next, ask “Can any quadrilateral be used to tessellate the plane?” Give students a moment to ponder, and then poll the class for the number of yes and no responses. Record the responses for all to see. This question will be revisited in the Activity Synthesis.
Provide access to tracing paper.
Supports accessibility for: Visualspatial processing; Organization
Design Principle(s): Optimize output (for explanation); Maximize metaawareness
Student Facing
 Can you make a tessellation of the plane with copies of the trapezoid? Explain.

Choose and trace a copy of one of the other two quadrilaterals. Next, trace images of the quadrilateral rotated 180 degrees around the midpoint of each side. What do you notice?
 Can you make a tessellation of the plane with copies of the quadrilateral from the previous problem? Explain your reasoning.
Student Response
For access, consult one of our IM Certified Partners.
Anticipated Misconceptions
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 180degree rotations, can be continued. Ask these students what they know about the sum of the three angles in a triangle and in a quadrilateral.
Activity Synthesis
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 180degree 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.
3.3: Pentagonal Tessellations (20 minutes)
Optional activity
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.
Launch
Ask students:
 “Can you tessellate the plane with regular pentagons?” (No.)
 “Can you think of a type of pentagon that could be used to tessellate the plane?” (A square base with a 454590 triangle on top, for example.)
Arrange students in groups of two.
Print version: Provide access to tracing paper.
Supports accessibility for: Language; Organization
Student Facing

Can you tessellate the plane with copies of the pentagon? Explain. Note that the two sides making angle \(A \) are congruent.
Pause your work here.

Take one pentagon and rotate it 120 degrees clockwise about the vertex at angle \(A\), and trace the new pentagon. Next, rotate the pentagon 240 degrees clockwise about the vertex at angle \(A\), and trace the new pentagon.

Explain why the three pentagons make a full circle at the central vertex.
 Explain why the shape that the three pentagons make is a hexagon (that is, the sides that look like they are straight really are straight).
Student Response
For access, consult one of our IM Certified Partners.
Launch
Ask students:
 “Can you tessellate the plane with regular pentagons?” (No.)
 “Can you think of a type of pentagon that could be used to tessellate the plane?” (A square base with a 454590 triangle on top, for example.)
Arrange students in groups of two.
Print version: Provide access to tracing paper.
Supports accessibility for: Language; Organization
Student Facing

Can you tessellate the plane with copies of the pentagon? Explain why or why not. Note that the two sides making angle \(A\) are congruent.

Take one pentagon and rotate it 120 degrees clockwise about the vertex at angle \(A\), and trace the new pentagon. Next, rotate the pentagon 240 degrees clockwise about the vertex at angle \(A\), and trace the new pentagon.

Explain why the three pentagons make a full circle at the central vertex.
 Explain why the shape that the three pentagons make is a hexagon (that is, the sides that look like they are straight really are straight).
Student Response
For access, consult one of our IM Certified Partners.
Anticipated Misconceptions
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.
Activity Synthesis
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. . . )
Design Principle(s): Optimize output (for explanation); Cultivate conversation