In this activity, students extend their understanding of prisms to see that even when the base is not a rectangle, they can still calculate the volume of a prism by multiplying the area of the base by the height of the prism.

Students use snap cubes or the digital applet to build a prism with a base that matches the shape from the blackline master. Each group needs 30 snap cubes for the first question and a total of 60 snap cubes for the second question. If there are not enough snap cubes, two groups of 3 students may combine together (after answering the first question) to form one group of 6 students.

If using snap cubes that measure inch, make copies of the first page of the blackline master, which has the slightly smaller shapes. If using snap cubes that measure 2 cm, make copies of the second page of the blackline master, which has the slightly larger shapes.

In the digital version of the activity, students use an applet to build a prism. The applet allows students to manipulate digital versions of snap cubes. Use the digital version if you do not have access to enough snap cubes.

The goal of this discussion is to clarify that the volume of a prism can be calculated by multiplying the area of the base by the height of the prism, even if the base is not a rectangle.

Select students to share their reasoning.

Here are some questions for discussion:

• “How do you know this figure is a prism?” (Cross-sections parallel to the base are identical copies.)
• “What is the area of the base of this figure?” (It is the number of cubes in one layer of the prism.)
• “How do you calculate the total number of cubes to make the prism?” (Multiply the number of cubes in one layer by the number of layers.)
• “What is the volume of this prism?” (The volume is the same as calculating the number of cubes to make the prism.)
• “If you find the area of the base, how do you use that information to calculate the volume of the prism?” (Multiply the area of the base by the height of the prism.)
• “How would the volume of the prism change if we changed the shape of the base but still used 27 cubes to build it?” (The volume would not change.)

If not mentioned by students, explain that calculating the total number of cubes to make the prism is the same as calculating the volume of the prism. We can find the area of the base of the prism and multiply that by the number of layers in the prism, which is the same as the height of the prism. The height of the prism is measured in units, the area of the base is measured in units² , and the volume of the prism is measured in units³.

The purpose of this activity is for students to get hands-on experience with polyhedra, recognizing whether or not a figure is a prism and, if it is, determining which face is the base of the prism. Once students determine which face is the base, they use a ruler marked in inches to measure the height of the prism. The area of each face is labeled on the shape so that students do not get bogged down with calculating the base area and can focus on using the area to find the volume.

In this activity, students critique a statement or response that is intentionally unclear, incorrect, or incomplete and improve it by clarifying meaning, correcting errors, and adding details (MP3). 

Instead of creating enough sets of polyhedra for every group to have one of every shape at the same time, consider having the students pass the shapes from one group to the next or rotate around to different stations so that fewer sets of shapes have to be constructed.

In the digital version of the activity, students use an applet to explore the polyhedra. The applet allows students to turn and view the shapes from any direction. Use the digital version if it would be easier to use devices than the shapes made from the nets. If students don't have individual access to the applet, displaying it for all to see would be helpful during the synthesis.

• Display this first draft: 
  “Figure A is not a prism because it is a cube.”
  Ask, “What parts of this response are unclear, incorrect, or incomplete?” As students respond, annotate the display with 2–3 ideas to indicate the parts of the writing that could use improvement.
• Give students 2–4 minutes to work with a partner to revise the first draft. 
• Select 1–2 individuals or groups to read their revised draft aloud slowly enough to record for all to see. Scribe as each student shares, then invite the whole class to contribute additional language and edits, to make the final draft even more clear and more convincing.

The purpose of this activity is for students to work backward from the volume to the height of a prism. Students see that for two prisms to have the same volume, the one with the smaller base has the taller height, and the one with the larger base has the shorter height. The grid helps students find the area of the base so they can focus their attention on what it means to have a prism made out of stacks of layers of the same base.

The goal of this discussion is to solidify the connection between volume, height, and area of the base of a prism. Select students to share their responses and reasoning. If not brought up in a student's explanation, explain that for the last problem, there is more than one possible correct answer. The smallest possible volume that involves all whole number side lengths is , but there is nothing in the problem that requires all the heights to be whole numbers.

Here are some questions for discussion:

• “How do you calculate the volume of a prism?” (Display the equation for all to see: .)
• “Since , how could we find the area of the base if we knew the volume and height of the prism?” (Display the equation for all to see: .)
• “If we keep the volume the same, what happens to the height when we increase the area of the base?” (It decreases.)
• “If we keep the height the same, what happens to the volume when we increase the area of the base?” (It increases.)