standard_description: Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.
problem_activity: Optional activity
In this activity, students compare the surface areas and volumes of three rectangular prisms given nets that are not on a grid. To do this, they need to be able to visualize the three-dimensional forms that the two-dimensional nets would take when folded.
In grade 5, students had learned to distinguish area and volume as measuring different attributes. This activity clarifies and reinforces that distinction.
Launch
Keep students in the same groups of 2–3. Tell students that this activity involves working with both volume and surface area. To refresh students' understanding of volume from grade 5, ask students:
“When we find the volume of a prism, what are we measuring?”
“How is volume different than surface area?”
“How might we find the volume of a rectangular prism?”
Reiterate that volume measures the number of unit cubes that can be packed into a three-dimensional shape and that we can find the number of unit cubes in a rectangular prism by multiplying the side lengths of a prism.
Give students 1–2 minutes to read the task statement and questions. Ask them to think about how they might go about answering each question and to be prepared to share their ideas. Give students a minute to discuss their ideas with their group. Then, ask groups to collaborate: each member should perform the calculations for one prism (A, B, or C). Give students 5–7 minutes of quiet time to find the surface area and volume for their prism and then additional time to compare their results and answer the questions.
Action and Expression: Internalize Executive Functions.
Provide students with a graphic organizer for recording measurements and calculations of surface areas and volumes of the boxes.
Supports accessibility for: Language; Organization
Student Facing
Here are the nets of three cardboard boxes that are all rectangular prisms. The boxes will be packed with 1-centimeter cubes. All lengths are in centimeters.
[IMG]
Compare the surface areas of the boxes. Which box will use the least cardboard? Show your reasoning.
Now compare the volumes of these boxes in cubic centimeters. Which box will hold the most 1-centimeter cubes? Show your reasoning.
Student Response
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Student Facing
Are you ready for more?
Figure C shows a net of a cube. Draw a different net of a cube. Draw another one.  And then another one. How many different nets can be drawn and assembled into a cube?
Student Response
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Anticipated Misconceptions
Students should have little trouble finding areas of rectangles but may have trouble keeping track of pairs of measurements to multiply and end up making calculation errors. Suggest that they label each polygon in the net and the corresponding written work and double-check their calculations to minimize such errors.
If students struggle to find the volume of their prism using information on a net, suggest that they sketch the prism that can be assembled from the net and label the edges of the prism.
Students may need a reminder that area is measured in square units and volume is measured in cubic units.
Activity Synthesis
Select a few students to share the surface area and volume of each prism. After each person shares, poll those who worked on the same prism for agreement or disagreement. Record the results on the board.
Invite students to share a few quick observations about the relationship between the surface areas and volumes for these three prisms, or between the amounts of material needed to build the boxes and the number of cubes that they can contain. Discuss questions such as:
“If these prisms are boxes, which prism—B or C—would take more material to build? Which can fit more unit cubes?” (B and C would likely take the same amount of material to build since their surface areas are the same. C has a greater volume than B, so it can fit more unit cubes.)
“Which prism—A or B—would take more material to build? Which can fit more unit cubes?” (A and B can fit the same number of unit cubes but, B would require more material to build.)
“If two prisms have the same surface area, would they also have the same volume? How do you know?” (No, prisms A, B, and C are examples of how two figures with the same volume may not have the same surface area, and vice versa.)
Students will gain more insights into these ideas as they explore squares, cubes, and exponents in upcoming lessons. If students could benefit from additional work on distinguishing area and volumes as different measures, do the optional lesson Distinguishing Between Surface Area and Volume.
Representing, Speaking: MLR8 Discussion Supports.
Use this routine to support the use of mathematical language when comparing surface area and volume. Give students 3–4 minutes to write a response to the following prompt: “If two prisms have the same surface area, their volume will
always/sometimes/never
be the same because . . .” Invite students to discuss their responses with a partner before selecting 1–2 students to share with the whole class. Listen for and call students' attention to how the use of examples and counterexamples can help justify their reasoning.
Design Principle(s): Optimize output (for justification); Support sense-making


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