12.1: The Faces of Geometry (5 minutes)
In this warm-up, students practice visualizing and drawing the faces of several solids. This will be helpful in upcoming activities as they categorize solids based on features of their choosing, and as they build solids from nets as a foundation for developing the formula for the volume of a pyramid.
Arrange students in groups of 2. Give students quiet work time and then time to share their work with a partner.
Three solids are shown.
Draw all the surfaces of each solid.
Students may struggle to draw the cone surface that’s in the shape of a sector of a circle. Ask them to consider snipping the cone in a straight line along this face and unrolling it.
Here are questions for discussion.
- “What are the names of these solids?” (rectangular pyramid, triangular prism, cone)
- “What is the same and different about the surfaces of the prism and the pyramid?” (Each of these solids has 5 faces, and the faces are all triangles and rectangles. The prism has a triangle for a base and rectangles for the other faces, while the pyramid has a rectangle for a base and triangles for the other faces.)
- “Which is the only surface that’s not a polygon?” (The cone has one “face” shaped like a wedge from a circle.)
12.2: Card Sort: Sorting Shapes (10 minutes)
A sorting task gives students opportunities to analyze representations, statements, and structures closely and make connections (MP2, MP7). In this task, students sort solids based on features of their choosing. The structures students identify will allow them to extend the adjectives right and oblique to pyramids and cones.
Monitor for different ways groups choose to categorize the solids, but especially for categories that distinguish between right and oblique solids, and between solids that have an apex (pyramids and cones) and those that do not (prisms and cylinders).
As students work, encourage them to refine their descriptions of the solids using more precise language and mathematical terms (MP6).
Arrange students in groups of 2 and distribute pre-cut slips. Tell students that in this activity, they will sort some cards into categories of their choosing. When they sort the solids, they should work with their partner to come up with categories.
Design Principle(s): Optimize output (for comparison); Maximize meta-awareness
Your teacher will give you a set of cards that show geometric solids. Sort the cards into 2 categories of your choosing. Be prepared to explain the meaning of your categories. Then, sort the cards into 2 categories in a different way. Be prepared to explain the meaning of your new categories.
Are you ready for more?
The platonic solids are a special group of solids with some specific criteria:
- The faces are all congruent and are all the same regular polygon.
- They are convex, meaning that the faces only meet at their edges.
- The same number of faces meet at every vertex.
- Draw a platonic solid constructed with faces that are squares.
- Draw a platonic solid constructed with faces that are triangles.
- Draw a different platonic solid constructed with faces that are triangles.
Select groups to share their categories and how they sorted their solids. Choose as many different types of categories as time allows, but ensure that one set of categories distinguishes between right and oblique solids, and another distinguishes between solids with an apex versus those without. Attend to the language that students use to describe their categories, giving them opportunities to describe their solids more precisely. Highlight the use of terms like hexagonal, perpendicular, and circular.
If students use phrasing such as: “The pyramids and cones get smaller while the cylinders and prisms do not,” encourage them to use the language of cross sections. A sample response might be: “Cross sections taken parallel to the base of prisms and cylinders are congruent throughout the solid. On the other hand, cross sections taken parallel to pyramid and cone bases are similar to each other, but are not congruent.”
Tell students that we can use the categories they created to define some characteristics of solids. A pyramid is a solid with one face (called the base) that’s a polygon. All the other faces are triangles that all meet at a single vertex, called the apex. A cone also has a base and an apex, but its base is a circle and its other surface is curved.
Just like prisms and cylinders can be right and oblique, so can cones and some pyramids. For a cone, imagine dropping a line from the cone’s apex straight down at a right angle to the base. If this line goes through the center of the base, then the cone is right. Otherwise, the cone is oblique. Pyramids with bases that have a center, like a square, a a pentagon, or an equilateral triangle, can also be considered right or oblique in the same way as cones.
For example, the cone on Card E from this activity is a right cone because its apex is centered directly over the center of its base. However, the pyramid on Card G has its center shifted; if we drop a height line straight down at right angles to the plane of the base, the line doesn’t hit the center of the pyramid’s base. This pyramid is oblique.
Point out that some mathematicians consider a cone to be a “circular pyramid,” others consider pyramids to be “polygonal cones,” and still others classify them in totally separate categories. Regardless of what we call them, the two kinds of solids share certain properties that will be explored in upcoming activities.
12.3: Building a Prism from Pyramids (15 minutes)
In this activity, students build a triangular prism out of 3 pyramids and make a conjecture about the volume of one of the pyramids. This activity creates a foundation for upcoming activities in which students will derive the formula for the volume of a pyramid.
Arrange students in groups of 3. Provide each group with scissors, tape, and 1 set of nets.
Tell students that they’ll be building pyramids, and that they should save the pyramids when they’re done for use in an upcoming activity.
Supports accessibility for: Organization; Attention
Your teacher will give your group 3 nets. Each student should select 1 of the 3 nets.
- Cut out your net and assemble a pyramid. The printed side of the net should face outward.
- Assemble your group’s 3 pyramids into a triangular prism. Each pair of triangles with a matching pattern will come together to form one of the rectangular faces of the prism. You will need to disassemble the prism in a later activity, so use only a small amount of tape (or no tape at all if possible).
- Make a conjecture about the relationship between the volume of the pyramid marked P1 and the volume of the prism.
- What information would you need to verify that your conjecture is true?
Don’t throw away your pyramids! You’ll use them again.
The goal of the discussion is to make observations about the relationships between the 3 pyramids and the prism. Here are some questions for discussion. It’s okay for the answers about triangle congruence to be informal.
- “Take a look at the pyramid marked P3. Which face would you consider its base? Is there only one possibility?” (Any face of this pyramid could be considered the base. No matter which face we choose to call the base, the remaining faces are all triangles. This is actually true for all 3 pyramids.)
- “Which faces of the prism are its bases?” (The faces marked P1 and P3 are the prism’s bases.)
- “Do the pyramids marked P1 and P3 have any congruent faces? If so, which are they, and how do you know?” (The faces marked P1 and P3 are congruent because they are the prism’s bases. The faces with the lines are also congruent, because together, they form a rectangle.)
- “Do the pyramids marked P2 and P3 have any congruent faces? If so, which are they, and how do you know?” (The gray-colored faces are congruent because together, they form a rectangle. The two faces that are unmarked are congruent to each other. When assembled into the prism, each line segment that forms the sides of the triangles is shared between the two shapes.)
To ensure the pyramids are available for an upcoming activity, collect the assembled pyramids or direct students to place them in a safe storage area.
Design Principle(s): Support sense-making; Optimize output (for justification)
The goal of the discussion is to consider what information would be needed to show that the volumes of the 3 pyramids are equal.
Display this image for all to see. Ask students, “How do the volumes of these 2 pyramids compare? How do you know?” Challenge them to use the language of cross sections. The bases of the pyramids have equal area, and the pyramids have the same height. Even though one pyramid’s apex is shifted compared to the other pyramid, the cross sections of the two pyramids have the same area at all heights. Therefore, the pyramids have the same volume.
Choose 2 of the 3 assembled pyramids and display them for all to see. Ask students, “What would we need to know in order to verify the volumes of these two pyramids are equal?” We would need to know the pyramids have bases with equal area, and that the heights of the pyramids are equal. Then, the pyramids’ cross sections would have equal area at all heights, and the pyramids would have equal volume.
12.4: Cool-down - How Many Faces? (5 minutes)
Student Lesson Summary
Pyramids and cones are different from prisms and cylinders in that they have just one base and an apex, or a single point at which the other faces of the solid meet.
Cones are like cylinders and prisms in that they can be oblique or right. If a line dropped from the cone’s apex at right angles to the base goes through the center of the base, then the cone is right. Otherwise, the cone is oblique. Pyramids that have a clear center in their bases can also be considered right or oblique.
We can use relationships between pyramids and prisms to build a formula for the volume of a pyramid. The image shows 3 square pyramids assembled into a cube. We’ll use similar thinking, but with triangular pyramids and prisms, to create a pyramid volume formula in an upcoming lesson.