# Thinking Outside of the Box

## Surface Area and Volume of Rectangular Prisms

Cacey Wells, Lindsey Link, Michell Eike, Samaya Williams | Published: March 23rd, 2023 by K20 Center

• Subject Mathematics
• Course Geometry
• Time Frame 85-95 minutes
• Duration 2-3 class periods
• ICAP Resource Individual Career Academic Plan (ICAP)

### Summary

In this lesson, students explore concepts of perimeter, area, surface area, and volume while engaging in hands-on construction of origami boxes. Students collaborate to discover the formulas for volume and surface area. Then students listen to a civil engineer and learn how he uses volume and surface area in his job before solving a real-world problem.

### Essential Question(s)

What is the relationship between surface area and volume? How are surface area and volume used in the real world?

### Snapshot

Engage

Students make predictions about how the area of a square piece of paper once it is folded might compare to the surface area and volume of a box.

Explore

Students explore the relationships between perimeter, area, surface area, and volume of origami boxes.

Explain

Students critically think about and discuss the relationships between perimeter, area, surface area, and volume.

Extend

Students learn how a civil engineer uses volume and surface area then solve a real-world problem.

Evaluate

Students reflect on their learning by using the Muddiest Point strategy.

### Materials

• Lesson Slides (attached)

• Constructing Origami Boxes handout (attached; one per student; print front/back)

• Exploring Origami Boxes handout (attached; one per student; print front/back)

• Copy paper (2 per student, each cut into 8.5” square)

• 1 cm cubes (approx. 200 per pair) or 1 inch cubes (approx. 20 per pair)

• Rulers (one per student)

• Dice (1 for teacher use)

• Pencils

• Paper

• Individual dry erase boards (optional; one per student)

• Dry erase markers (optional; one per student)

• Chart paper (optional; one per class)

### Engage

15 Minute(s)

Introduce the lesson using the attached Lesson Slides. Slide 3 displays the lesson’s essential questions. Slide 4 identifies the lesson’s learning objectives. Review each of these with your class to the extent you feel necessary.

Display slide 5 and distribute one sheet of square paper to each student. Guide students to use this paper for measuring and recording their thinking; alternatively, have students use an individual dry erase board for recording.

Share the Think-Pair-Share strategy with your students and ask the prompts: How big is your paper? In what ways can you measure it?

Ask students to think independently about finding the area and perimeter of their square paper. Have them record their thinking and answers on that square sheet of paper.

Display slide 6 and have students pair up to share their work and their answers with their partner. Transition to slide 7 and have a few student pairs share with the class.

Next, show slide 8 and pose the following question: Given the area and perimeter, how much can this paper hold?

Encourage students to be creative to answer this question. They can fold, cut, tear, etc. the paper to try to create a box and estimate its volume. Give students approximately 5 minutes to complete this task.

Have students record their thinking on a piece of notebook paper or on an individual dry erase board.

If time allows, ask a few volunteers to share their thoughts with the whole class.

### Explore

25 Minute(s)

Show slide 9 and distribute the Constructing Origami Boxes handout. Then give each student a new square sheet of paper. Direct each student to create their own origami box following the directions on the handout.

Once students are almost done constructing their boxes, distribute rulers and cubes to students to use as manipulatives to help measure the volume of their box in the next step of the activity.

Display slide 10 and give each student a copy of the attached Exploring Origami Boxes handout. Direct students’ attention to the Making Predictions section of the handout. Have them answer the following questions and record their answers on their handouts:

1. How much volume do you think your box can hold? How can you measure this without a formula?

2. How much surface area do you think your box has?

3. How could you measure this?

Direct pairs to continue following the directions on their handout to complete the activity. Here students are using rulers and cubes to help them measure the volume and surface area of their boxes. Then they are to use their answers to questions 3-7 to complete the Developing Formulas table.

### Explain

20 Minute(s)

Show slide 11 and ask students to consider the following question: "What relationships do you see between the answers you put in your table?"

Introduce the Critical Thinking Cube strategy to the class and explain that you want them to think about this question as they complete the next activity.

Preview the activity by explaining that you will roll a dice and share out the number. Students will then have about 2 minutes to talk with their partner about the corresponding prompt. Call on pairs to share with the whole class. For clarity, explain to the class that if a 3 is rolled, they will discuss prompt #3 (which they will see on the following slide) with their partner for 2 minutes and then share with the whole class.

Transition to slide 12 and share the following prompts:

1. Describe: Describe how you found the volume of your box.

2. Apply: Write a formula to find the volume of any box.

3. Compare: Compare volume to surface area.

4. Analyze: What are the dimensions of volume? Of surface area?

5. Synthesize: Write a formula to find the surface area of any box (with a lid).

6. Argue: "Is there a relationship between area, perimeter, volume, and surface area.” Justify your answer.

During the whole class share out, record the common big idea of the class for each prompt on the board or a designated spot on the chart paper. The goal is for students to understand the concepts of volume and surface area and to be able to find the volume and surface area of any rectangular prism and later any prism.

After discussing prompt #2, show slide 13 and help students see the connection between the two formulas for volume. Explain to students that the volume of any prism is the area of the base times the height.

After discussion prompt #5, show slide 17 and help students understand this likely new formula for surface area. Take time to discuss the definition of the lateral faces. Consider having students label the lateral faces of their origami boxes.

### Extend

20 Minute(s)

Show slide 22 and ask the class, “What are some careers you can think of that would use surface area, volume or both in their job?” Use the 30-second timer on the slide to give all students a chance to think about all the careers that may use surface area and volume in their job. Once the timer expires, have students quickly share their ideas with the whole class.

Show slide 23 and introduce the “Civil Engineer and Rectangular Prisms” video, which features an interview with Bobby Williams, a civil engineer talking about his career and how he uses surface area and volume in his job.

Move to slide 24 and pass out the attached Engineering the Bridge Cap handout to your students. This task represents a real-world problem posed by the civil engineer in the video interview. Depending on your preference or your students’ needs, you may consider having your students work with a partner or independently.

After students finish, transition to slide 29 and pose the question: If the emblem mold has a diameter of 5.2 feet, do we need a redesign? Be sure to have students justify their conclusions.

### Evaluate

5 Minute(s)

Go to slide 30. Have students reflect on the lesson and their overall understanding of the content using the Muddiest Point strategy. Have students answer the following questions:

• Crystal Clear: What do you think is the easiest part of finding volume and surface area?

• Muddiest Point: What do you think is the most confusing part of finding volume and surface area?

You can collect responses in a variety of ways depending on your class. Sticky notes, pieces of paper, or digital posts are a few examples.