Polyhedra
10 min
Teacher Prep
Setup
Required Preparation
Narrative
In this Warm-up, students analyze examples and counterexamples of polyhedra, observe their defining characteristics, and use their insights to sort objects into polyhedra and non-polyhedra. They then start developing a working definition of "polyhedron."
Prepare physical examples of polyhedra and non-polyhedra for students to sort. These examples should be geometric figures rather than real-world objects such as shoe boxes or canisters. If such figures are not available, make some ahead of time using the nets in the blackline master.
As students work and discuss, notice those who can articulate defining features of a polyhedron and invite them to share later.
Launch
Arrange students in groups of 3–4. Give students 1 minute of quiet time to study the examples and non-examples in the task statement. Ask them to be ready to share at least one thing that they notice and one thing that they wonder. Give the class a minute to share some of their observations and questions.
Next, give each group a physical set of three-dimensional figures. The set should include some familiar polyhedra, some unfamiliar ones, and some non-polyhedra.
Ask groups to sort the figures into polyhedra and non-polyhedra (the first question). If members disagree about whether a figure is a polyhedron, prompt them to discuss the disagreements with their group. When the group has come to an agreement, give them 2–3 minutes of quiet time to complete the second question.
Student Task
These five drawings represent polyhedra.
The next four drawings do not represent polyhedra.
-
Your teacher will give you some figures or objects. Sort them into polyhedra and non-polyhedra.
- What characteristics helped you distinguish the polyhedra from the other figures?
Sample Response
- No response required.
- Sample responses:
- Polyhedra are made from polygons.
- Polyhedra don’t have any unattached edges.
- Non-polyhedra sometimes have curved or round surfaces.
- Some non-polyhedra have a face that is not a polygon.
Activity Synthesis (Teacher Notes)
Invite students to share what they see as characteristics of polyhedra. Record their responses for all to see. For each one, ask the class if they agree or disagree. If they generally agree, ask if there is anything they would add or elaborate on to make the description clearer or more precise. If they disagree, ask for an explanation or a counterexample.
Students will have a chance to refine their definition of polyhedra later in the lesson—after exploring prisms and pyramids and learning about nets, so it is not important to compile a complete or precise set of descriptions or features.
Use a sample polyhedron or a diagram as shown here to introduce or reinforce the terminology surrounding polyhedra.
- The polygons that make up a polyhedron are called "faces."
- The places where the sides of the faces meet are called edges.
- The “corners” are called vertices. (Clarify that the singular form is "vertex" and the plural form is "vertices.")
Math Community
At the end of the Warm-up, display the Math Community Chart. Tell students that norms are expectations that help everyone in the room feel safe, comfortable, and productive doing math together. Using the Math Community Chart, offer an example of how the “Doing Math” actions can be used to create norms. For example, "In the last exercise, many of you said that our math community sounds like ‘sharing ideas.’ A norm that supports that is 'We listen as others share their ideas.’ For a teacher norm, ‘questioning vs telling’ is very important to me, so a norm to support that is ‘Ask questions first to make sure I understand how someone is thinking.’”
Invite students to reflect on both individual and group actions. Ask, “As we work together in our mathematical community, what norms, or expectations, should we keep in mind?” Give 1–2 minutes of quiet think time and then invite as many students as time allows to share either their own norm suggestion or to “+1” another student’s suggestion. Record student thinking in the student and teacher “Norms” sections on the Math Community Chart.
Conclude the discussion by telling students that what they made today is only a first draft of math community norms and that they can suggest other additions during the Cool-down. Throughout the year, students will revise, add, or remove norms based on those that are and are not supporting the community.
Standards
- 6.G.4·Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving real-world and mathematical problems.
- 6.G.A.4·Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving real-world and mathematical problems.
25 min
Knowledge Components
All skills for this lesson
No KCs tagged for this lesson
Polyhedra
10 min
Teacher Prep
Setup
Required Preparation
Narrative
In this Warm-up, students analyze examples and counterexamples of polyhedra, observe their defining characteristics, and use their insights to sort objects into polyhedra and non-polyhedra. They then start developing a working definition of "polyhedron."
Prepare physical examples of polyhedra and non-polyhedra for students to sort. These examples should be geometric figures rather than real-world objects such as shoe boxes or canisters. If such figures are not available, make some ahead of time using the nets in the blackline master.
As students work and discuss, notice those who can articulate defining features of a polyhedron and invite them to share later.
Launch
Arrange students in groups of 3–4. Give students 1 minute of quiet time to study the examples and non-examples in the task statement. Ask them to be ready to share at least one thing that they notice and one thing that they wonder. Give the class a minute to share some of their observations and questions.
Next, give each group a physical set of three-dimensional figures. The set should include some familiar polyhedra, some unfamiliar ones, and some non-polyhedra.
Ask groups to sort the figures into polyhedra and non-polyhedra (the first question). If members disagree about whether a figure is a polyhedron, prompt them to discuss the disagreements with their group. When the group has come to an agreement, give them 2–3 minutes of quiet time to complete the second question.
Student Task
These five drawings represent polyhedra.
The next four drawings do not represent polyhedra.
-
Your teacher will give you some figures or objects. Sort them into polyhedra and non-polyhedra.
- What characteristics helped you distinguish the polyhedra from the other figures?
Sample Response
- No response required.
- Sample responses:
- Polyhedra are made from polygons.
- Polyhedra don’t have any unattached edges.
- Non-polyhedra sometimes have curved or round surfaces.
- Some non-polyhedra have a face that is not a polygon.
Activity Synthesis (Teacher Notes)
Invite students to share what they see as characteristics of polyhedra. Record their responses for all to see. For each one, ask the class if they agree or disagree. If they generally agree, ask if there is anything they would add or elaborate on to make the description clearer or more precise. If they disagree, ask for an explanation or a counterexample.
Students will have a chance to refine their definition of polyhedra later in the lesson—after exploring prisms and pyramids and learning about nets, so it is not important to compile a complete or precise set of descriptions or features.
Use a sample polyhedron or a diagram as shown here to introduce or reinforce the terminology surrounding polyhedra.
- The polygons that make up a polyhedron are called "faces."
- The places where the sides of the faces meet are called edges.
- The “corners” are called vertices. (Clarify that the singular form is "vertex" and the plural form is "vertices.")
Math Community
At the end of the Warm-up, display the Math Community Chart. Tell students that norms are expectations that help everyone in the room feel safe, comfortable, and productive doing math together. Using the Math Community Chart, offer an example of how the “Doing Math” actions can be used to create norms. For example, "In the last exercise, many of you said that our math community sounds like ‘sharing ideas.’ A norm that supports that is 'We listen as others share their ideas.’ For a teacher norm, ‘questioning vs telling’ is very important to me, so a norm to support that is ‘Ask questions first to make sure I understand how someone is thinking.’”
Invite students to reflect on both individual and group actions. Ask, “As we work together in our mathematical community, what norms, or expectations, should we keep in mind?” Give 1–2 minutes of quiet think time and then invite as many students as time allows to share either their own norm suggestion or to “+1” another student’s suggestion. Record student thinking in the student and teacher “Norms” sections on the Math Community Chart.
Conclude the discussion by telling students that what they made today is only a first draft of math community norms and that they can suggest other additions during the Cool-down. Throughout the year, students will revise, add, or remove norms based on those that are and are not supporting the community.
Standards
- 6.G.4·Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving real-world and mathematical problems.
- 6.G.A.4·Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving real-world and mathematical problems.