Adding Volumes of Solid Figures

Transcription

1 Project Adding Volumes of Solid Figures Objectives To find the volumes of solid figures composed of two non-overlapping rectangular prisms. etoolkit Algorithms Practice EM Facts Workshop Game Family Letters Assessment Management Common Core State Standards Curriculum Focal Points Interactive Teacher s Lesson Guide Doing the Project Recommended Use During or after Unit 9 Key Concepts and Skills Use unit cubes to calculate volume. [Measurement and Reference Frames Goal 2] Use a formula to calculate the volumes of rectangular prisms. [Measurement and Reference Frames Goal 2] Explore properties of solid figures. [Geometry Goal 2] Write number sentences with variables to model volume problems. [Patterns, Functions, and Algebra Goal 2] Key Activities Students use centimeter cubes and formulas to explore the volumes of solid figures composed of two rectangular prisms. Key Vocabulary unit cube Extending the Project Mathematical Practices SMP1, SMP2, SMP3, SMP4, SMP5, SMP6, SMP8 Content Standards 5.MD.3a, 5.MD.3b, 5.MD.4, 5.MD.5a, 5.MD.5b, 5.MD.5c Materials Math Journal 2, Activity Sheet 8 Math Masters, p. 410 transparent tape per group of students: about 100 centimeter cubes Students find the volumes of rectangular prisms to find the total approximate volume of Willis Tower. Materials Math Masters, p. 410A Project 9 Adding Volumes of Solid Figures 990A

2 Student Page Date Time Rectangular Prism Patterns 1 Doing the Projects Box A Exploring the Volume of Solid Figures (Math Journal 2, Activity Sheet 8) SMALL-GROUP ACTIVITY Box B Math Journal 2, Activity Sheet 8 Provide each group of students with about 100 centimeter cubes. In each group, students should display the two open boxes that they constructed in Lesson 9-8 from Activity Sheet 8. If the constructed boxes are no longer available, distribute copies of Activity Sheet 8 as needed. Have students tape together Boxes A and B to form one solid figure. Specify that they should put the prisms together so that two faces are together and edges line up where possible. Note that students may tape the prisms together in different ways. The figure below is one example. Ask: Suppose each box has a lid so that each is a rectangular prism. Is the figure formed by the two rectangular prisms also a rectangular prism? Explain. Sample answer: No. A rectangular prism is formed by six faces that are rectangles. The new figure that is formed has more than six rectangular faces. Mention that a three-dimensional shape that is not of a specific type is simply called a solid figure. Ask: How could you use the centimeter cubes to find the volume of the solid figure you formed? Sample answer: Fill each box with centimeter cubes. Add or count the cubes used in all. What is the volume of the solid figure? 69 cubic centimeters 990B Project 9 Adding Volumes of Solid Figures

3 Ask students to attach the two prisms in different ways (such as one above the other or one next to the other) and compare the volume of each new solid that has been formed. Students should recognize that the volume is the same no matter how the prisms are attached. Ask a volunteer to explain why this is so. Sample answer: The two prisms have the same volumes no matter how they are put together, so the sum of the two volumes is the same. Explain that in this case we can consider each dimension of a centimeter cube as being 1 unit long. Mention that a cube with side lengths of 1 unit is called a unit cube. Explain that in general, when a solid figure can be packed without gaps and overlaps using n unit cubes, the volume of the solid figure is n cubic units. Building a Solid Figure to Find Volume (Math Masters, p. 410) SMALL-GROUP ACTIVITY Have students use the centimeter cubes to build the two rectangular prisms in each of Problems 1 3 on Math Masters, page 410. Students find the volume of each prism and then find the volume of the solid figure formed by the two prisms. Encourage students to use a formula (either V = B h or V = l w h) to find the volumes of the rectangular prisms. Students should use the cubes to check their results. When students complete Problem 4, they should conclude that you can find the volume of a solid figure formed by two rectangular prisms by adding the two volumes. So, finding the volume of a solid figure that includes more than one rectangular prism is an additive process. You may want to explain that if a solid figure has overlapping parts, as in the solid figure below, you add the volumes of the non-overlapping parts to the volumes of the parts that overlap. Project Master Name Date Time PROJECT 9 Building a Solid Figure to Find Volume For Problems 1 3, do the following: a. Use centimeter cubes to build each rectangular prism. b. Find the volume of each rectangular prism. c. Find the volume of the solid figure formed by the two rectangular prisms Math Masters, p. 410 Rectangular Prism A Rectangular Prism B by Prisms A and B Rectangular Prism C Rectangular Prism D by Prisms C and D Rectangular Prism E Rectangular Prism F by Prisms E and F 4. Explain how to find the volume of a solid figure made from two rectangular prisms, one with dimensions 3 cm by 4 cm by 5 cm and one with dimensions 2 cm by 5 cm by 4 cm. Sample answer: I would use the formula V = l w h to find the volume of each rectangular prism and then add the volumes. So, the total volume is = , or 100 cm 3. 2 Extending the Project Finding the Volume of Willis Tower in Chicago (Math Masters, p. 410A) INDEPENDENT ACTIVITY Mention that at 1,450 feet tall (excluding antennas), Willis Tower in Chicago (formerly named Sears Tower) is the tallest building in the United States. When it was built in 1973, it was the tallest building in the world. Project 9 990C

4 PROJECT 9 Project Master Name Date Time Finding the Volume of Willis Tower At 1,450 feet tall, Willis Tower in Chicago is the tallest building in the United States. It is composed of nine rectangular prisms known as tubes. The tubes are built in a 3-by-3 arrangement. Although the tubes are of various heights, each one has a square base that measures 75 feet on a side. 1. What is the area of the base of each tube? 5,625 ft 2 The table below shows the approximate heights of the tubes. Only two of them reach all the way to the top. 2. What formula could you use to find the volume Sample answer: of one tube? V = B h Willis Tower 3. Complete the table to find the volume of one tube at each given height. Then find the total volume of the tubes for each height. Approximate Number of Volume of One Total Volume of Height of Tubes at Tube at this Tubes at this Height Tube this Height Height (ft 3 ) (ft 3 ) 646 ft ft 2 1,200 ft 2 1,450 ft 2 3,633,750 3,780,000 6,750,000 8,156, Describe what you will do to find the total approximate volume of Willis Tower. Sample answer: I will find the sum of the total volumes of the tubes for each height. 5. The total volume of Willis Tower is about 48,273,750 ft 3. 10,901,250 7,560,000 13,500,000 16,312,500 The building consists of nine square tubes constructed in a 3-by-3 arrangement. Each tube measures 75 feet on a side. Ask students to explain how to find the dimensions of the entire base of Willis Tower. Sample answer: There are three tubes on a side, and each is 75 feet long. So each side is 225 feet long, making the dimensions of the base 225 ft by 225 ft. Ask: What is the area of the base? , or 50,625 ft 2 Explain that the nine tubes are of various heights, with some being the same height. For example, although all nine tubes extend up through the 49th story, only two tubes extend to the full height of 110 stories. So to find the volume of Willis Tower, you need to find the volumes of the different tubes and then add. Of course, multiplication could be used to find the total volume of tubes with the same dimensions. Math Masters, page 410A guides students in finding the approximate volume of Willis Tower. Math Masters, p. 410A 990D Project 9 Adding Volumes of Solid Figures

5 Name Date Time PROJECT 9 Building a Solid Figure to Find Volume For Problems 1 3, do the following: a. Use centimeter cubes to build each rectangular prism. b. Find the volume of each rectangular prism. c. Find the volume of the solid figure formed by the two rectangular prisms. 1. Rectangular Prism A Rectangular Prism B by Prisms A and B 2. Rectangular Prism C Rectangular Prism D by Prisms C and D 3. Rectangular Prism E Rectangular Prism F by Prisms E and F 4. Explain how to find the volume of a solid figure made from two rectangular prisms, one with dimensions 3 cm by 4 cm by 5 cm and one with dimensions 2 cm by 5 cm by 4 cm. Copyright Wright Group/McGraw-Hill 410

6 Name Date Time PROJECT 9 Finding the Volume of Willis Tower At 1,450 feet tall, Willis Tower in Chicago is the tallest building in the United States. It is composed of nine rectangular prisms known as tubes. The tubes are built in a 3-by-3 arrangement. Although the tubes are of various heights, each one has a square base that measures 75 feet on a side. 1. What is the area of the base of each tube? The table below shows the approximate heights of the tubes. Only two of them reach all the way to the top. 2. What formula could you use to find the volume of one tube? ft 2 Willis Tower 3. Complete the table to find the volume of one tube at each given height. Then find the total volume of the tubes for each height. Approximate Number of Volume of One Total Volume of Height of Tubes at Tube at this Tubes at this Height Tube this Height Height (ft 3 ) (ft 3 ) 646 ft 3 Copyright Wright Group/McGraw-Hill 672 ft 2 1,200 ft 2 1,450 ft 2 4. Describe what you will do to find the total approximate volume of Willis Tower. 5. The total volume of Willis Tower is about ft A