In this activity, students make sense of different methods for calculating the surface area of a figure. They then think about generalizing the methods to figure out if they would work for any prism. This activity connects to work they did with nets in a previous grade and builds upon strategies students might have to calculate surface area. Using the net helps students use the structure of a prism to make sense of the surface area (MP7). They do not need to generalize a formula for surface area at this time.

As students work on the task, monitor for students who understand the different methods and can explain if any of them will work for any other prisms.

Note: It is not important for students to learn the term “lateral area.”

Students may think that Andre’s method will not work for all prisms, because it will not work for solids that have a hole in their base and, therefore, more lateral area on the inside. Technically, those solids are not prisms, because their base is not a polygon. However, students could adapt Andre’s method to find the surface area of a solid composed of a prism and a hole.

The purpose of this discussion is to show multiple methods of calculating surface area and when each method could work. Select previously identified students to share their reasoning. If not brought up in students' explanations, display the image for all to see and point out to students that the length of the “1 big rectangle” is equal to the perimeter of the base.

Students may have trouble generalizing which method would work for any prism. Here are some questions for discussion:

• “Which of the students’ methods will work for finding the surface area of this particular prism?” (all 3)
• “Which of the students’ methods will work for finding the surface area of any prism?” (Noah’s and Andre’s)
• “Which of the students’ methods will work for finding the surface area of other three-dimensional figures that are not prisms?” (only Noah’s)

If not mentioned by students, be sure that students understand:

• Noah’s method will always work, but it can be inefficient if there are a lot of faces.
• Elena’s method will not always work because the rectangles will not always be the same size, but we can notice that some shapes are the same and, therefore, not have to work out all of them individually.
• Andre’s method does always work even if the rectangles have different widths. The length of the rectangle will be the same as the perimeter of the base, and the width of the rectangle will be the height of the prism.
• Prisms can always be cut into three pieces: two bases and one rectangle whose length is the perimeter of a base and whose width is the height of the prism. This can be more efficient than the other methods because students need to calculate only two areas (since the two bases will be identical copies).
• This method works only for prisms. For other shapes, such as pyramids, Noah’s method of finding all the faces individually or Elena’s method of combining those faces into identical copy groups will work. Solids with holes, such as the triangular prism with a square hole, can use a variation of Elena’s method: two congruent triangles with holes for the bases, one rectangle for the outside side faces, and another rectangle for the faces that form the hole.

Explain to students that they will have the opportunity in the next activity to practice using any of these strategies.

In this activity, students are presented with a figure that was used in a previous lesson to explore volume. Here, they explore that figure’s surface area and compare different methods of doing so based on their work in the previous task. Students work with a partner to share the task of investigating two methods to calculate the surface area.

As students work on the task, listen for students who find similarities and differences between the method they used and the one their partner used.

The purpose of this discussion is to compare and contrast the two methods for finding the surface area of a prism. Select previously identified students to share the discussion they had with their partner. Here are some possible discussion questions:

• “How did you find the area of the base?”
• “How did you find any other areas that you needed, in order to solve the problem?”
• “When you used using the first method (calculating the area of all the faces), how many different shapes did you need to calculate the area of?”
• “When you used the second method (using the perimeter of the base), how many different shapes did you need to calculate the area of?”
• “Which method do you prefer for this problem? Why?”
• “Do you think you will prefer the same method for every problem? Why or why not?”
• “What would make you change methods?”
• “Do you need to know all of the measurements in the picture to solve for surface area?” (No, you need to know just the perimeter and area of the base and the height of the figure.)
• “Could you solve for volume with the measurements given in the picture? If so, are there any unnecessary measurements? If not, what else would you need to know?” (Yes, you can solve for the volume with the given measurements, and there are no unnecessary measurements.)

If not brought up in students’ explanations, explain to students that the first method requires finding the area of 6 different shapes (there are 7 faces, but the two bases are the same). While the calculations using this method were simple, there were more pieces. The second method requires visualizing the solid in a different way, but we needed to find only the area of two different pieces (the long rectangle and the base).