Lesson 1

This activity introduces students to patterns of polygons that cover the plane. Examples include:

• patterns using a single shape
• patterns using multiple shapes

Studying these patterns and understanding how and why they repeat to fill up the plane is an example of MP8, expressing regularity in repeated reasoning. In this case, the repeated reasoning is continuing to lay out the shapes in the same pattern.

Arrange students in groups of 2. Tell students that they will look at a set of images, and their job is to think of at least one thing they notice and at least one thing they wonder. Display the images for all to see. Ask students to give a signal when they have noticed or wondered about something. Give students 1 minute of quiet think time, and then 1 minute to discuss the things they notice with their partner, followed by a whole-class discussion.

Ask students to share the things they noticed and wondered. Record and display their responses for all to see. If possible, record the relevant reasoning on or near the images. After each response, ask the class if they agree or disagree and to explain alternative ways of thinking, referring back to the images each time.

Important ideas that can come up are the following:

• The shapes in a tessellation cover the plane (or would if the pattern continues) without gaps.
• Sometimes there is only one shape. Sometimes there are several shapes in the tessellation.
• With the square one, all of the squares are complete. In the other patterns, the shapes on the boundary are cut off. 

If time allows, ask:

• “How do I know what happens as the tessellation continues to grow off of the page?” (There is often a pattern, but we would have to indicate that the pattern continues.)
• “Are the colors of the shapes important?” (They are important for identifying patterns and making the pictures prettier, but they could be changed and the pattern would still be the same.)

The goal of this task is to introduce the notion of a tessellation and then carefully examine how to create a tessellation with each pattern block shape (square, rhombus, equilateral triangle, isosceles trapezoid). For hexagons, there is only one way to fit them together because there will be gaps unless three hexagons meet at each vertex. The other shapes offer much more flexibility, and students have an opportunity to use their artistic creativity.

Students look for and make use of structure (MP7), both when they try to put copies of the shape together to build a tessellation and when they examine whether or not it is possible to construct a different tessellation.

Tessellations do not need to be edge to edge. That is, pieces do not need to fit together with edges of the same lengths matching exactly.

Tessellations do not need to have symmetric, repeating patterns, though sometimes the shape forces it (as with the regular hexagons).

Important questions to address include:

• “Were you able to make different tessellations with your shape?” (It depends on the shape.)
• “If not, why not?” (Three hexagons have to come together at each vertex: once the first hexagon of the pattern is placed, everything else has no flexibility.)
• “If so how?” (Triangles can be built into hexagons, or they can make rows that can be translated and stacked on top of one another. Parallelograms can also be made into hexagons or rows. Trapezoids can be made into hexagons or rows. Rhombuses can be made into rows and translated. Squares can also be made into rows and translated.)
• “What does it look like to not define a tessellation?” (Two octagons can be put together sharing a vertex, but there is a gap that is not large enough for a third octagon.)

Make sure, as students share their ideas for tessellations, to use the language of rigid motions to describe the tessellations. For example, if a student has built a tessellation with parallelograms, choose two parallelograms and ask:

• “How can I use rigid motions to move one of these parallelograms into the position of the other?”
• “Are there other ways I could do this?”

The goal of this activity is to accurately describe a tessellation of the plane. While students do not need to use the words translation, rotation, or reflection, their understanding of rigid motions of the plane will play a key role in explaining (and interpreting) where to place each shape in a tessellation. Communicating a geometric pattern clearly in words fully engages students in attending to precision (MP6).

As a warm-up, each student will describe a tessellation while the partner identifies a picture of the tessellation. After a brief discussion of what language was most helpful, students then take turns describing a tessellation as their partner attempts to build the tessellation. They switch roles and then reflect on any misinterpretations that happened and how they may have been related to the language used to describe the tessellations.

Students work in pairs. Each pair requires a set of 2 cards for Part 2, one card for each student.

Stop students after they have completed Part 1 of the task for the first part of the discussion given in the synthesis.

For the discussion after students complete Part 1:

• “In what ways was describing the tessellation difficult?” (Finding words to communicate how the rectangles are aligned with one another.)
• “Did you use the words translate, rotate, or reflect?” (Instead of “translate” students may use words like “move.” Similarly, students may describe rotations with words like “turn.”)
• “What was challenging about describing or identifying the tessellation?”

For the discussion after students complete Part 2, focus on the use of the words translate, reflect, and rotate (or equivalents).

• “When you used a translation, did you specify the direction and how far?” (Maybe, but they may also use the language “put next to” or equivalent.)
• “When a rotation was involved, did you specify the number of degrees of the rotation?” (Probably not. They will more likely say “rotate until the sides match. . .” or “turn upside down.”)
• “When a rotation was involved, did you specify the center of rotation?” (Answers vary. Probably not.)
• “Did you use any reflections?” (Answers vary.)