Tiling the Plane
10 min
Teacher Prep
Setup
Required Preparation
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).
Launch
Arrange students in groups of 2–4. Display the four patterns for all to see. Give students 1 minute of quiet think time and ask them to indicate when they have noticed three patterns that go together and can explain why. Next, tell students to share their response with their group and then work together to find as many sets of three as they can.
Student Task
Which three go together? Why do they go together?
Sample Response
Sample responses:
A, B, and C go together because:
- All the shapes have color (or are shaded).
- All the colored shapes meet other colored shapes on all sides. There are no gaps between the shapes.
- They don’t have white (or non-shaded) regions.
- The shapes have different side lengths.
A, B, and D go together because:
- They all have pentagons.
A, C, and D go together because:
- They have the color blue.
- They have shapes with one or more right angles.
B, C, and D go together because:
- They have the color yellow.
Activity Synthesis (Teacher Notes)
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.
Standards
- 3.G.A·Reason with shapes and their attributes.
- 3.G.A·Reason with shapes and their attributes.
25 min
Knowledge Components
All skills for this lesson
No KCs tagged for this lesson
Tiling the Plane
10 min
Teacher Prep
Setup
Required Preparation
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).
Launch
Arrange students in groups of 2–4. Display the four patterns for all to see. Give students 1 minute of quiet think time and ask them to indicate when they have noticed three patterns that go together and can explain why. Next, tell students to share their response with their group and then work together to find as many sets of three as they can.
Student Task
Which three go together? Why do they go together?
Sample Response
Sample responses:
A, B, and C go together because:
- All the shapes have color (or are shaded).
- All the colored shapes meet other colored shapes on all sides. There are no gaps between the shapes.
- They don’t have white (or non-shaded) regions.
- The shapes have different side lengths.
A, B, and D go together because:
- They all have pentagons.
A, C, and D go together because:
- They have the color blue.
- They have shapes with one or more right angles.
B, C, and D go together because:
- They have the color yellow.
Activity Synthesis (Teacher Notes)
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.
Standards
- 3.G.A·Reason with shapes and their attributes.
- 3.G.A·Reason with shapes and their attributes.