Lesson 14

The purpose of this warm-up is for students to recognize important parts of solids in anticipation of computing volume and surface area. The figure used in the next activity is introduced in this warm-up as a way for students to start thinking about parts of solids and how we use them to compute surface area or volume.

Arrange students in groups of 2. Display the prism assembled from the blackline master for all to see. Give students 1 minute of quiet think time followed by time to discuss their ideas with a partner. Follow with a whole-class discussion.

Select students to share their responses. Ask students to think about units that do not make sense to use for measurements (feet, miles, yards, etc).  Invite students to share their explanations of why these units do not make sense to use.

In this activity, students make sense of three different methods for calculating the surface area of a figure. Three different methods are described to students and they are asked to determine which one they agree with (if any) (MP3). 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.

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.”

Arrange students in groups of 2. Display the prism assembled previously in the warm-up for all to see. Ask students: “how might we find surface area of this prism?” Invite students to share their ideas. Give students 1 minute of quiet think to read Noah’s method for calculating surface area followed by time discuss whether they agree with Noah or not. Repeat this process for the remaining two methods. Once all three methods have been discussed give students 1–2 minutes of quiet work time to answer the rest of the questions in the task statement.

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, these 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.

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 guiding questions:

• “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 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 not have to work them all out 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 only need to calculate two areas (since the two bases will be identical copies).
• This method only works 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 on Elena’s method: two congruent triangles with holes for the bases, one rectangle for the outside side faces, and another rectangle for the faces forming 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 its surface area and compare different methods from 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.

Arrange students in groups of 2. Tell students that they might recognize this figure from a previous lesson, but today they are going to compare two different methods for calculating its surface area.Give students 1–2 minutes of quiet work time followed by time to trade their work with a partner to compare answers and methods. Follow with a whole-class discussion.

Select previously identified students to share the discussion they had with their partner. To highlight the difference between the two methods, ask:

• “How did you find the area of the base?”
• “How did you find any other areas you needed to solve the problem?”
• “How many different shapes did you need to calculate the area of when using the first method (calculating area of all the faces)?”
• “How many different shapes did you need to calculate the area of when using the second method (using perimeter of base)?”
• “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 just need to know 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?”

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 only needed to find the area of two different pieces (the long rectangle and base).