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.
Student Lesson in Spanish
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.
Activity
None
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?
Student Response
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Building on Student Thinking
Activity Synthesis
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.
13.2
Activity
Standards Alignment
Building On
Addressing
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.
This activity serves two goals: to uncover the defining features of prisms and pyramids as well as to introduce nets as two-dimensional representations of polyhedra.
Students first analyze prisms and pyramids and try to define their characteristics. Next, they learn about nets and think about the polygons needed to compose the nets of given prisms and pyramids. They then use their experience with the nets of prisms and pyramids to sharpen and refine their definitions of these polyhedra.
As students discuss the features of prisms and pyramids, encourage them to use the terms face, edge, and vertex (vertices) in their descriptions.
Launch
Arrange students in groups of 3–4. Tell students that there are 3 questions in this activity and that the class will pause for a discussion after responding to each question and before moving on to the next one.
Display images of the prisms and pyramids in the activity, or display and pass around physical representations of those polyhedra, if available. Tell students that Polyhedra A–F are all prisms and Polyhedra P–S are all pyramids.
For the first question:
Give students 2–3 minutes of quiet think time and 1–2 minutes to discuss their observations in their groups.
Solicit students' ideas about characteristics that distinguish prisms and pyramids. Record students' responses in two columns—one for prisms and the other for pyramids. It is not important that the lists are complete at this point.
Next, tell students that we are going to use nets to better understand prisms and pyramids. Explain that a net is a two-dimensional representation of a polyhedron.
Display a cube assembled for the Warm-up, as well as a cutout of an unfolded net (consider removing the flaps). Demonstrate how the net with squares could be folded and assembled into a cube, or use this book of digital applets, https://ggbm.at/rcu3Ka3j, created in GeoGebra by the GeoGebra DocuTeam. Point out how the number and the shape of the faces on the cube correspond to the number and the shape of the polygons in the net.
For the second question:
Give students 1 minute of quiet think time and 1 minute to discuss their response in their groups.
Tell students that they will now verify their answer. Give each group one of the three nets from the first page of the blackline master. Ask them to try to assemble a triangular pyramid from their net.
Invite groups to share with the class whether it can be done. Discuss why Net 3 cannot be assembled into Pyramid P (two of the triangles would overlap).
For the last question, tell students that they will create a net of another prism or pyramid:
Assign each group a prism or a pyramid from the task statement (except for Prism B and Pyramid P).
Give each group a set of pre-cut polygons from the last two pages of the blackline master.
Tell students to choose the right kind and number of polygons that make up their polyhedron. Then, arrange the polygons so that, when taped and folded, the arrangement is a net and could be assembled into their prism or pyramid. Encourage them to think of more than one net, if possible.
MLR2 Collect and Display. Circulate, and listen for and collect the language that students use as they talk about characteristics of prisms and pyramids. On a visible display, record words and phrases such as: “faces,” “edges,” “vertex,” “parallel,” “rectangular (or triangular) faces,” and names of polygons. Invite students to borrow language from the display as needed and update it throughout the lesson. Advances: Speaking, Conversing
Representation: Develop Language and Symbols. Use virtual or concrete manipulatives to connect symbols to concrete objects or values. Provide prisms and pyramids for students to view or manipulate. These hands-on models help students identify characteristics of polyhedra and support net building. Supports accessibility for: Visual-Spatial Processing, Conceptual Processing
Activity Synthesis
Select groups to share their arrangements of polygons. If time permits and if possible, have students tape their polygons and fold the net to verify that it could be assembled into the intended polyhedron. Discuss:
“What do the nets of prisms have in common?” (They all have rectangles. They have a pair of polygons that may not be rectangles.)
“What do the nets of pyramids have in common?” (They all have triangles. They have one polygon that may not be a triangle.)
“Is there only one possible net for a prism or a pyramid?” (No, the polygons can be arranged in different ways and still be assembled into the same prism or pyramid.)
Explain the following points about prisms and pyramids:
A prism has two parallel, identical faces called bases and a set of rectangles connecting the bases.
Prisms are named for the shape of the bases. For example, if the base of a prism is a pentagon, then the prism is called a “pentagonal prism.”
A pyramid has one face called the base that can be any polygon and a set of faces that are all triangles. Each edge of the base is shared with an edge of a triangle. All of these triangles meet at a single vertex.
Pyramids are named for the shape of their base. For example, if the base of a pyramid is a square, then the pyramid is called a “square pyramid.”
13.3
Activity
Optional
Standards Alignment
Building On
Addressing
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.
This optional activity gives students the experience of assembling polyhedra from nets. Counting the vertices, edges, and faces of a polyhedra helps to reinforce their understanding of the vocabulary. You will need the same blackline master as the one provided for the Warm-up.
Students are likely to need assistance in assembling their polyhedra. Circulate, and support students as needed.
Launch
Tell the class that they are going to assemble polyhedra from nets. Point out that the net has shaded and unshaded polygons. Display an example and explain that only the shaded polygons in the nets will show once the net is assembled. The unshaded polygons are "flaps" to make it easier to glue or tape the polygons together; they will get tucked behind the shaded polygons and are not really part of the polyhedron. Tell students that creasing along all of the lines first will make it easier to fold up the net and attach the various polygons together. A straightedge can be very helpful for making the creases.
Give each student two copies of a net so they can compare the assembled version with the unfolded net. Provide access to geometry toolkits and glue or tape. Ask students to build their figures and complete the question, and then to discuss their responses with another student who has the same polyhedron.
Activity
None
Your teacher will give you the net of a polyhedron. Cut out the net, and fold it along the edges to assemble a polyhedron. Tape or glue the flaps so that there are no unjoined edges.
How many vertices, edges, and faces are in your polyhedron?
Student Response
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Activity Synthesis
After students have conferred with another student and agreed on the number of vertices, edges, and faces of their polyhedron, tell the class they will now share their completed polyhedra and the unfolded version of the net with the class. Consider either asking students to pass their two items around, or to leave their the polyhedra and nets displayed while students circulate around the room to view others' work.
Action and Expression: Provide Access for Physical Action. Provide access to pre-cut nets to reduce barriers for students who need support with fine-motor skills and students who benefit from extra processing time.
Supports accessibility for: Fine Motor Skills, Organization, Visual-Spatial Processing
Lesson Synthesis
Review the features of prisms and pyramids by selecting 1 or 2 polyhedra used in the Warm-up. Ask students to explain (using the terminology they learned, if possible) why each one is or is not a prism or a pyramid. If it is a prism or pyramid, ask students to name it.
Revisit the working definition of polyhedra generated earlier in the lesson and ask students to see if or how it might be refined. Ask if there is anything they should add, remove, or adjust given their work with prisms, pyramids, and nets.
Highlight the following points about polyhedra. Ask students to illustrate each point using a figure or a net.
A "polyhedron" is a three-dimensional figure built from filled-in polygons. We call the polygons "faces." (The plural of polyhedron is polyhedra.)
Every "edge" of a polygon meets another polygon along a complete edge.
Each polygon meets one and only one polygon on each of the edges.
The polygons enclose a three-dimensional region.
Consider displaying in a visible place the key ideas from the students' list and from this discussion so that they key ideas can serve as a reference later.
Student Lesson Summary
A polyhedron is a three-dimensional figure composed of faces. Each face is a polygon and meets only one other face along a complete edge. The ends of the edges meet at points that are called vertices.
A polyhedronalways encloses a three-dimensional region.
The plural of polyhedron is polyhedra. Here are some drawings of polyhedra:
A prism is a type of polyhedron with two identical faces that are parallel to each other and that are called bases. The bases are connected by a set of rectangles (or sometimes parallelograms that aren’t rectangles).
A prism is named for the shape of its bases. For example, if the base is a pentagon, then it is called a “pentagonal prism.”
triangular prism
pentagonal prism
rectangular prism
A pyramid is a type of polyhedron that has one special face called the base. All of the other faces are triangles that all meet at a single vertex.
A pyramid is named for the shape of its base. For example, if the base is a pentagon, then it is called a “pentagonal pyramid.”
rectangular pyramid
hexagonal pyramid
heptagonal pyramid
A net is a two-dimensional representation of a polyhedron. It is composed of polygons that form the faces of a polyhedron.
A cube has 6 square faces, so its net is composed of six squares, as shown here.
A net can be cut out and folded to make a model of the polyhedron.
In a cube, every face shares its edges with 4 other squares. In a net of a cube, not all edges of the squares are joined with another edge. When the net is folded, each of these open edges will join another edge.
Standards Alignment
Building On
Addressing
Building Toward
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.
Look at the prisms. What are their characteristics?
Look at the pyramids. What are their characteristics?
Which of these nets can be folded into Pyramid P? Select all that apply.
Three figures labeled net 1, net 2, and net 3. Net 1 has four small triangles arranged horizontally to create a parallelogram, net two has four small triangles arranged to make a larger triangle, and net 3 has four small triangles which all meet at their vertices.
Your teacher will give your group some polygons and assign a polyhedron.
Decide which polygons are needed to compose your assigned polyhedron. List the polygons and how many of each are needed.
Arrange the cut-outs into a net that, if taped and folded, can be assembled into the polyhedron. Sketch the net. If possible, show a different net for the same polyhedron.
Student Response
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Building on Student Thinking
Building on Student Thinking
Students may have trouble getting an accurate count of faces, edges, and vertices. Suggest that they set the figure on the table and then separately count the amount on the top, bottom, and lateral sides of the figure. Or recommend that they label each face with a number or a name and keep track of the parts associated with each face, taking care not to double count edges and vertices.
