“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. [IS.4 - Struggling Learners]
In order to review the concepts learned in Lesson 1, you will present two animated activities, called Interactives, from Annenberg Media:
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1. Explore and Play with Surface Area
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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.
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2. Find the Volume of a Rectangular Prism
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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.” [IS.5 - Struggling Learners]
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“Now, fill in the other four rows. Remember that length times width is the area of the base. ”
Rectangular Prism Dimensions: Surface Area and Volume
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Area of the Base
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Length
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Width
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Height
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Surface Area
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Volume
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4
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6
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2
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88
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48
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3
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1
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4
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|
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6
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2
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2
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|
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9
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4
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6
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|
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5
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3
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2
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|
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- Key:
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38
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12
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56
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24
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228
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216
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62
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30
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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 (e.g., amount of wrapping paper needed to wrap a gift).
Review: Drawing a Net to Solve (optional)
Example 1
“We will find the surface area of the rectangular prism shown by first drawing a net.”
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“We can find the surface area using two steps:”
1. Find the area of each face [IS.6 - All Students]
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a. Area of the top or bottom
7 x 2 = 14
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b. Area of either side
2 x 4 = 8
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c. Area of the front or back
7 x 4 = 28
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2. Sum the areas of all 6 faces
2(14) + 2(8) + 2(28) = 28 + 16 + 56
= 100
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Example 2
“We will find the surface area of the rectangular prism shown by first drawing a net.”
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“We can find the surface area using two steps:”
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1. Find the area of each face
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a. Area of the top or bottom
3 x 2 = 6
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b. Area of either side
2 x 9 = 18
- c. Area of the front or back
3 x 9 = 27 -
2. Sum the areas of all 6 faces
2(6) + 2(18) + 2(27) = 12 + 36 = 54
- = 102
Two Problems for Students
Give students the following two problems to solve.
Directions: Draw a net for the given rectangular prisms, label the dimensions, and solve for the surface area.
- 1.
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2.
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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 worksheet (M-7-2-2_Surface Area and Volume.doc). Have students complete the table.
Surface Area and Volume Word Problems [IS.7 - All Students]
Example Scenario 1:
A gourmet cheese company is designing a new plastic wrap to cover their 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?” (Students should respond that 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:”
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So, the least amount of plastic wrap needed to cover the box is 288 in².
Example Scenario 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?” (Students should respond that 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:”
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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 Scenario 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? (Students should respond that volume gives the 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? (Students should respond that they will be finding a missing dimension, the height.) [IS.8 - All Students]
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V = lwh
120 = (10)(4)h
120 = 40
h
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.
1. Find the surface area and volume of the classroom.
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2. If one gallon of paint covers 350 ft², how many gallons of paint are needed to cover the classroom?
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3. 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.
Activity 3: Comparing Buildings
Have students work with a partner. [IS.9 - All Students]
Look at the two buildings below. Which building requires more building materials? Which building design 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 looking to get more space for your money, which would you choose?
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Answer: The building shaped as a cube requires more building materials, but also has more space.
Building 1 (30 ft by 30 ft by 30 ft)
Surface Area equals 5,400 ft²
Volume equals 27,000 ft³
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This one will cost you more money to build ($1000), but you get 3,000 more cubic feet of space.
Building 2 (30 ft by 20 ft by 40 ft)
Surface Area equals 5,200 ft²
Volume equals 24,000 ft³
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This one will cost you less money to build, but you get 3,000 less cubic feet of space.
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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
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Area of the Base
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Length
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Width
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Height
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Volume
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4
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6
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48
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5
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2
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90
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6
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2
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96
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| |
4
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6
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168
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5
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3
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180
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- Key:
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Debate Activity
Divide the class into two groups. Have them discuss the trade-off between more volume or 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-7-2-2_Exit Ticket.doc).
Extension:
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Use the following strategies to tailor the lesson to individual needs and interests:
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Routine: This lesson includes much student interaction through work with partners and groups, as well as classroom discussion. With such involved and higher-level activities as the classroom surface area and volume activity and debate over surface area and volume, students have ample opportunity to discuss, verbalize, reason, and justify their ideas related to these concepts. Extra assistance can be provided to students having difficulty by integrating handmade nets with one-on-one discussions with the teacher or a classmate. Also, the classroom surface area activity, debate, and word-problem activity are certainly beneficial to students having difficulty.
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Estimation: Bring a collection of three or four empty dry food boxes of different shapes, such as cereal, pasta, or cracker boxes. 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.
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Small Groups: The group activities can be used to identify students having difficulty and then offer or plan additional assistance. Use the estimation activity in small groups and encourage students to share their strategies for estimating.
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Expansion: Challenge students to write a real-world problem involving cylinders and surface area and/or volume.
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Technology: This lesson involves use of animated activities for demonstrative and interactive purposes.
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 student-centered in order to give students real experience calculating surface areas and volumes using multiple methods and several different prism examples.
