Lesson Plan

## Calculating Surface Area and Volume in the Real World

### Objectives

The lesson focuses on attainment of procedural knowledge and conceptual understanding, related to surface area and volume. Students will:

• draw nets from rectangular prisms, in order to solve surface area problems.
• calculate the surface area and volume of rectangular prisms and cubes.
• solve real-world problems, involving surface area and volume, using a variety of problem-solving strategies.

#### Essential Questions

• How can recognizing repetition or regularity assist in solving problems more efficiently?
• How are spatial relationships, including shape and dimension, used to draw, construct, model, and represent real-world situations or solve problems?
• How can the application of the attributes of geometric shapes support mathematical reasoning and problem solving?
• How can geometric properties and theorems be used to describe, model, and analyze situations?

### Vocabulary

• Congruent Figures: Figures that have the same size and shape. Congruent angles have the same measure; congruent segments have the same length.
• Cylinder: A solid that has two parallel, congruent bases (usually circular) connected with a curved side.
• Net: A two-dimensional shape that can be folded to create a three-dimensional figure.
• Prism: A three-dimensional solid that has two congruent and parallel faces that are polygons. The remaining faces are rectangles. Prisms are named by their bases.
• Surface Area: The sum of the areas of all of the faces of a three-dimensional figure.
• Volume: The amount of space found within a solid.

90–120 minutes

### Prerequisite Skills

Prerequisite Skills haven't been entered into the lesson plan.

### Related Materials & Resources

The possible inclusion of commercial websites below is not an implied endorsement of their products, which are not free, and are not required for this lesson plan.

### Formative Assessment

• View
• Observations during Activity 1: Real-World Needs will aid in determining how well students understand the concepts.
• Use the Calculating Surface Area and Volume table to assess student understanding.
• Observations during the Applying Surface Area and Volume activity will aid in recognizing which students may need more learning opportunities.
• Assess student comprehension using the exit ticket.

### Suggested Instructional Supports

• View
Scaffolding, Active Engagement, Modeling, Formative Assessment W: Begin the lesson by telling students you will be calculating surface area and volume, and applying this to solve real-world problems. H: Draw students into the lesson by using the link provided to show surface area and volume for various solids. Have the students complete the table individually. E: Provide real-world situations for which students must use either surface area or volume in order to solve. Solutions can include nets, drawings, and calculations as necessary. R: Have students discuss the trade-offs of more volume versus more surface area of a solid. Have them find which conditions make one or the other more advantageous. Ask groups to share their conclusions. E: Have students complete the exit ticket to assess their comprehension of the lesson. T: Using the Extension section, the lesson may be tailored to meet the needs of the students. The Routine section supplies ideas that may be implemented throughout the school year to keep students’ skills sharp when calculating volume and surface area. The Small Group section explains various ways to provide additional support to students who may benefit from more learning opportunities. The Expansion section suggests activities that will challenge students who are ready to move beyond the requirements of the standard. O: This lesson teaches students how to calculate volumes and surface areas, and explores some relationships between the two different concepts.

### Instructional Procedures

• View

In this lesson, students use an investigative approach to learn about rectangular prisms. Students use nets to help in their understanding of surface area and volume of prisms. The lesson is purposely organized from a student-centered perspective in order to give students real experience calculating surface areas and volumes using multiple methods and several different prism examples.

“What is the difference between two dimensions and three dimensions? You can answer this general question with some specific measurements of familiar objects, cubes, and rectangular prisms. We already know how to find the area of a rectangle or a square. Now we can apply that knowledge to the area of the faces of a cube or rectangular prism. Since every cube is a rectangular prism and every rectangular prism has six faces, finding the surface area is all about adding the areas of all six faces in a systematic way. Finding the volume is easier, because we need only one arithmetic operation: length × width × height.”

In this lesson, students will develop skill in calculating surface area and volume, determining when each measurement is appropriate, and applying necessary procedures for solving real-world surface area and volume problems. They will use their results to justify the best choice in a given situation, and perform in-depth thinking to find ways to expand a problem.

In order to review the concepts learned in Lesson 1, you will present two animated activities, called Interactives, from Annenberg Media:

1. Explore and Play with Surface Area

http://www.learner.org/interactives/geometry/area_surface.html

Using a “data-show” projector connected to a computer, play the animation and demonstrate how to enter values for each face of the prism and determine whether an entry is correct or incorrect. Depending on the number of available computers in the classroom, students should be placed into groups and given an opportunity to experiment with the activity and work with at least three different rectangular prisms.

2. Find the Volume of a Rectangular Prism

http://www.learner.org/interactives/geometry/area_volume.html

Again play the animation and demonstrate how to enter values for number of cubes, number of layers, and volume of the rectangular prism. The same groupings and instructional strategy should apply to this activity.

“Take a few minutes to find the surface area and volume of the rectangular prisms listed in the table. We will look at the first one together.”

“Now, fill in the other four rows. Remember that the area of the base can be found multiplying the width by the length.”

Rectangular Prism Dimensions: Surface Area and Volume

 Area of the Base (length × width) Length Width Height Surface Area Volume 4 6 2 88 48 3 1 4 38 12 6 2 2 56 24 9 4 6 228 216 5 3 2 62 30

Activity 1: Real-World Needs

Provide a list of real-world situations that involve either surface area or volume. Ask students to determine which measurement should be used. Some examples are listed below:

• Amount of wrapping paper needed to wrap a gift (SA)
• Packaging of a food container (SA) and amount of food inside (V)
• Water in a fish tank or swimming pool (V)
• Amount of paint needed to paint the walls in a room or the outside faces of a figure (SA)

Optional Review: Drawing a Net to Solve

Example 1

“We will find the surface area of the rectangular prism shown by first drawing a net.”

“We can find the surface area using two steps:”

1. Find the area of each face

a.   Area of the top or bottom       (7 × 2 = 14)

b.   Area of either side                  (2 × 4 = 8)

c.   Area of the front or back        (7 × 4 = 28)

2. Sum the areas of all 6 faces            (2(14) + 2(8) + 2(28) = 28 + 16 + 56 = 100)

Example 2

“We will find the surface area of the rectangular prism shown by first drawing a net.”

“We can find the surface area using two steps:”

1.   Find the area of each face

a.   Area of the top or bottom       (3 × 2 = 6)

b.   Area of either side                  (2 × 9 = 18)

c.   Area of the front or back        (3 × 9 = 27)

2.   Sum the areas of all 6 faces          (2(6) + 2(18) + 2(27) = 12 + 36 + 54 = 102)

Calculating Surface Area and Volume

Draw a table with two rectangular prisms and include columns for dimensions, surface area, and volume, such as the one on the Calculating Surface Area and Volume table (M-6-4-2_Surface Area and Volume and KEY.docx). Have students complete the table.

Surface Area and Volume Word Problems

Example 1

A gourmet cheese company is designing a new plastic wrap to cover its newest product, packaged in a box 12 inches by 6 inches by 4 inches. What is the least amount of plastic wrap needed to cover the box?

“We must first determine what the least amount of plastic wrap needed relates to, in measurement terms. Any ideas?” (The surface area will reveal the area of the outside of the box. This area is the same as the least amount of plastic wrap needed to cover the box.) “Correct. We are simply looking for the surface area here. We can calculate the surface area as follows:

SA = 2lw + 2lh + 2wh

SA = 2(12 × 6) + 2(12 × 4) + 2(6 × 4)

SA = 2(72) + 2(48) + 2(24)

SA = 144 + 96 + 48

SA = 288

“So, the least amount of plastic wrap needed to cover the box is 288 in².”

Example 2

Sarah receives a box of dark chocolate-covered strawberries as a gift. The box has dimensions of 6 inches by 4 inches by 2 inches. If each strawberry fits into a 1-cubic-inch space, how many strawberries does Sarah have in her box?

“As before, we need to determine what we are being asked. What type of measurement are we looking at here?” (The volume gives information related to the amount of space within a container.) “Correct. We are looking for the volume, in this case. We can calculate the volume as follows:

V = lwh

V = 6 × 4 × 2

V = 48

“If each strawberry occupies 1 cubic inch of space and the volume is 48 cubic inches, then the box contains 48 strawberries.”

Example 3

Jonas has 120-cubic-inch boxes that he needs to pack into a larger box. The box he has is 10 inches long and 4 inches wide. How tall does the box need to be?

“As before, we need to determine what we are being asked. What type of measurement are we looking for?” (The volume gives information related to the amount of space in a container.) “Correct. We are going to use the volume formula, but what is different this time?” (Instead of looking for the volume, we are going to use the volume to find the missing height.)

“V = lwh

120 = 10 × 4 × h

120 = 40h

3 = h

“The height of the box is 3 inches.”

Activity 2: Applying Surface Area and Volume

Students will work in groups of four or five to find the surface area and volume of the classroom. They will answer the following questions:

• If one gallon of paint covers 350 ft², how many gallons of paint are needed to cover the classroom walls and ceiling?
• How might you use your knowledge of the volume of the classroom for practical purposes related or unrelated to the painting scenario? Provide a detailed description and include at least two examples. (Some examples include: determining the maximum capacity of the room in adherence to public safety guidelines; ordering new furniture to fill the room; buying balloons to decorate the room for a class party; etc.)

Activity 3: Comparing Buildings

Have students work with a partner to answer the following questions.

• Look at the two buildings below. Which building requires more building materials? Which building design has more space? (The building shaped as a cube requires more building materials, but it also has more space.)

• You are an architect and work with clients to design buildings that fit their price and space criteria. It will cost the client an average of \$5 per square foot for the building material in either design. After looking at your results, if you are trying to get more space for your money, which would you choose?

(Building 1 (30 ft. by 30 ft. by 30 ft.):

Surface Area = 5,400 ft²; Volume = 27,000 ft³

Building 1 one will cost you more money to build (\$1,000), but you get 3,000 more cubic feet of space.

Building 2 (30 ft. by 20 ft. by 40 ft.):

Surface Area = 5,200 ft²; Volume = 24,000 ft³

Building 2 will cost you less money to build, but you get 3,000 less cubic feet of space.

Best Choice:

Building 1 is the best choice because writing a ratio of additional space to cost (3000/1000 or 3:1) shows the extra space is three times that of the cost for that additional space. That seems to be a pretty good deal.)

Activity 4: Finding Missing Dimensions

Rectangular Prism Dimensions: Missing Dimension and Volume

 Area of the Base Length Width Height Volume 4 6 2 48 5 2 9 90 6 2 8 96 7 4 6 168 5 12 3 180

Debate Activity

Divide the class into two groups. Have them discuss the tradeoff between more volume and more surface area. Ask students to consider what conditions make a larger volume more desirable than a smaller volume and greater surface area. Similarly, what conditions make a larger surface area more desirable than a smaller surface area and greater volume? Following the small group discussions, bring the discussion together with both groups to share the results.

Word-Problem Activity

Divide students into groups of three or four and have each group write a real-world problem that involves the need to calculate both surface area and volume. Students will then share their problems with the class.

Ask students to complete an exit ticket (M-6-4-2_Exit Ticket and KEY.doc).

Extension:

Use the following strategies to tailor the lesson to meet individual needs and interests:

• Routine: For the duration of time that students are studying the concepts of surface area and volume, a classroom discussion comparing and contrasting surface area and volume would provide students with ample opportunities to discuss, verbalize, reason, and justify their ideas related to these concepts. Throughout the school year, students may review the concepts dealing with nets and three-dimensional figures by creating handmade nets and then discussing the shape of the net, the resulting figure, its volume, and/or its surface area.
• Small Group: Students who may benefit from additional practice or instruction, may find the following activity helpful. Bring a collection of three or four empty dry food boxes of different shapes, such as cereal, pasta, or crackers. Ask students to rank order them by volume, then by surface area. Do not measure or allow students to measure the boxes, but rather have them estimate by visual inspection. After reaching some level of consensus, measure the boxes and compare the measurements to the students’ estimates. Ask students to describe their sense of which measure was harder to estimate: surface area or volume.

Some practice worksheets for calculating volume of rectangular prisms may be found here: http://www.mathatube.com/files/Volume_of_a_Rectangular_Prism_worksheet_1.pdf

### Related Instructional Videos

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Final 07/12/2013