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 prism that was used in a previous lesson to calculate volume. Here, they calculate the surface area of the prism. This provides students with the opportunity to work,  within a given context, with complex shapes to find surface area.

Monitor for students who use these different strategies:

• Decomposing the heart into shapes to calculate the area.
• Calculating the area of a square around the heart and removing the negative space.

In each approach, students will be decomposing the figure into smaller areas to do calculations.

Students who are familiar with actual heart-shaped boxes of chocolate may want to double the lateral area to represent the way the top and bottom pieces nest together.

The goal of this discussion is to connect different methods of calculating the surface area of the box. Focus on connecting a few different strategies and how they arrive at the same answer.

Display, for all to see, 2–3 approaches/representations from previously selected students. Use Compare and Connect to help students compare, contrast, and connect the different approaches. Here are some questions for discussion:

• “What do the approaches have in common? How are they different?”
• “Did anyone solve the problem the same way, but would explain it differently?”
• “Are there any benefits or drawbacks to one representation compared to another?”

This activity reinforces work that students have done in previous activities with regard to surface area and volume. Students work with a contextual problem to determine the surface area and volume of an object.

The purpose of this discussion is to clarify when to calculate volume and when to calculate surface area, as well as acknowledging that there are times that a partial surface area is needed. Here are some questions for discussion:

• “Are there other ways to calculate the area of the base?”
• “How did you know when to calculate surface area or volume for this problem?”