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Unit 1, Lesson 1

Tiling the Plane

Grade 6

1.1

Warm-up

Instructional Routines

Which Three Go Together?

Materials

To Copy (from Blackline Masters)

6–12 Blank Math Community Chart

To Gather

Chart paper Sticky notes

Activity Narrative

This is the first Which Three Go Together routine in the course. In this routine, students are presented with four items or representations and asked: “Which three go together? Why do they go together?”

Students are given time to identify a set of three items, explain their rationale, and refine their explanation to be more precise or find additional sets. The reasoning here prompts students to notice common mathematical attributes, look for structure (MP7), and attend to precision (MP6), which deepens their awareness of connections across representations.

This Warm-up prompts students to compare four geometric patterns. It gives students a reason to use language precisely and allows the teacher to hear how students use terminology in describing geometric characteristics. Comparing the patterns also urges students to think about shapes that cover the plane without gaps and overlaps, which supports future conversations about the meaning of “area.”

Before students begin their work, consider establishing a small, discreet hand-signal that students can display when they have an answer that they can support with reasoning. This might include a thumbs-up or a certain number of fingers that indicates the number of responses that they have. Using a signal is a quick way to see if students have had enough time to think about the problem. A subtle signal keeps students from being distracted or rushed by seeing hands being raised around the class.

Students may choose to describe the patterns in terms of:

Colors (blue, green, yellow, white, or no color).
Size of shapes or other measurements.
Geometric shapes (squares, pentagons, hexagons, or other polygons).
Relationships of shapes (whether each side of the polygons meets the side of another polygon, what polygon is attached to each side, whether there is a gap between polygons, and so on).

Activity Synthesis

Invite each group to share one reason why a particular set of three go together. Record and display the responses for all to see. After each response, ask the class if they agree or disagree. Because there is no single correct answer to the question of which three go together, attend to students’ explanations, and ensure that the reasons given are correct.

During the discussion, ask students to explain the meaning of any geometric terminology they use (names of polygons or angles, parts of polygons, “area”) and to clarify their reasoning as needed. For example, a student may say that Patterns A, B, and C each have shapes with different side lengths, but all the shapes in Pattern D have the same side lengths. Ask how they know that is the case, and whether this is true for the white (or non-filled) regions in Pattern D.

Explain to students that covering a two-dimensional region with copies of the same shape or shapes such that there are no gaps or overlaps is called "tiling" the plane. Patterns A, B, and C are examples of tiling. Tell students that they will explore more tilings in upcoming activities.

Math Community
Tell students that today is the start of planning the type of mathematical community they want to be a part of for this school year. The start of this work will take several weeks as the class gets to know one another, reflects on past classroom experiences, and shares their hopes for the year.

Display and read aloud the question “What do you think it should look like and sound like to do math together as a mathematical community?” Give students 2 minutes of quiet think time and then 1–2 minutes to share with a partner. Ask students to record their thoughts on sticky notes and then place the notes on the sheet of chart paper. Thank students for sharing their thoughts and tell them that the sticky notes will be collected into a class chart and used at the start of the next discussion.

After the lesson is complete, review the sticky notes to identify themes. Make a Math Community Chart to display in the classroom. See the blackline master Blank Math Community Chart for one way to set up this chart. Depending on resources and wall space, this may look like a chart paper hung on the wall, a regular sheet of paper to display using a document camera, or a digital version that can be projected. Add the identified themes from the students’ sticky notes to the student section of the “Doing Math” column of the chart.

1.2

Activity

Standards Alignment

Building On

Addressing

Building Toward

Instructional Routines

Materials

To Gather

Geometry toolkits

Activity Narrative

This activity asks students to compare the amounts of the plane covered by two tiling patterns, with the aim of supporting two big ideas of the unit:

If two figures can be placed one on top of the other so that they match up exactly, then they have the same area.
A region can be decomposed and rearranged without changing its area.

Students are likely to notice that in each pattern:

The same 3 polygons (triangles, rhombuses, and trapezoids) are used as tiles.
The shapes are arranged without gaps and overlaps, but their arrangements are different.
A certain set of smaller tiles form a larger hexagon. Each hexagon has 3 trapezoids, 4 rhombuses, and 7 triangles.
The entire tiling pattern is composed of these hexagons.

Expect students to begin their comparison by counting each shape, either in the entire pattern or in a portion that repeats. For students to effectively compare how much of the plane is covered by each shape, they need to be aware of the relationships between the shapes. For example, two green triangles can be placed on top of a blue rhombus so that they match up exactly. This means that two green triangles cover the same amount of the plane as one blue rhombus covers. (Students don’t need to use the word “area” in their explanations. At this point, such phrases as “they match up” or “two triangles make one rhombus” suffice.)

Geometry toolkits are introduced here, giving students an opportunity to use tools strategically (MP5). Students could, for instance, use tracing paper to trace shapes, use a straightedge to extend lines, or use scissors to cut out shapes.

Monitor for the different ways in which students make comparisons. Here are some approaches students may take, from more common to less common:

Compare the numbers of shapes in the entire pattern—56 green triangles, 32 blue rhombuses, and 24 red trapezoids. This doesn’t yet tell us which shape covers more of the plane.
Compare the numbers of shapes in a larger hexagon—7 green triangles, 4 rhombuses, and 3 trapezoids. There are fewer shapes to count, but these numbers alone also don’t tell us which shape covers more of the plane.
Compare the numbers of shapes and their relationships. Examples: One rhombus matches up exactly with 2 triangles. One trapezoid matches up exactly with 3 triangles. In a larger hexagon, there are 7 green triangles, 4 rhombuses that match up with 8 triangles, and 3 trapezoids that match up with 9 triangles. (Students may also relate the areas of shapes in the entire pattern this way. See student responses for additional examples.)

The routine of Anticipate, Monitor, Select, Sequence, Connect requires a balance of planning and flexibility. The anticipated approaches might not surface in every class, and there may be reason to change the order in which strategies are presented. While monitoring, keep in mind the learning goal and adjust the order to ensure that all students have access to the first idea presented (whether that be a common misconception or a different approach).

In the digital version of the activity, students use an applet to explore the shapes that comprise the pattern. The applet allows students to see the patterns on a triangular grid and to frame the repeating larger hexagons. It also allows students to isolate a larger hexagon and to move and rotate individual shapes within it. These features might help students in quantifying and relating the shapes.

Activity Synthesis

The purpose of this discussion is to help students connect their comparisons of shapes to the idea of area.

Invite previously selected students to share their answers and explanations. Sequence the discussion of strategies in the order listed in the Activity Narrative. Be sure to discuss the idea of comparing shapes by placing them on top of one another and seeing if or how they match. Consider demonstrating this idea using the applet.

Make it explicit that when students are asked, “Which shape covers more of the plane?”, they are being asked to compare the areas covered by the different shapes.

Connect the discussion to the learning goals by asking questions, such as:

“How does the area of the trapezoid compare to the area of the triangle?” (The area of the trapezoid is three times the area of the triangle.)
“How does the area of the rhombus compare to the area of the triangle?” (The area of the rhombus is twice the area of the triangle.)
“Is it possible to compare the area that all of the rhombuses cover in Pattern A to the area that all the triangles cover in Pattern B? If yes, how? If no, why not?” (Yes, we can count the number of rhombuses in A and the number of triangles in B. Because 2 triangles have the same area as does 1 rhombus, divide the number of rhombuses in A by 2. Then compare the result to the number of triangles in B.)