Look at the container of popcorn that your teacher has given your group. Guess how many pieces of popcorn it holds without touching it.

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1 Exploring Volume SUGGESTED LEARNING STRATEGIES: Guess and Check, Quickwrite, Debrief, Vocabulary Organizer Look at the container of popcorn that your teacher has given your group. Guess how many pieces of popcorn it holds without touching it. 1. Record your guess below and explain the strategy you used. ACTIVITY Containers like these are commonly called estimation jars. a. Have you ever played a game like this before? If so, describe where it was and the purpose? b. Is there any way of knowing the exact answer? Explain. Volume is a measure of the space inside a figure such as a cube, a ball, or a cylinder. 3. How does this game relate to volume? 4. How would knowing how to find volume of a figure make it easier to make a reasonable prediction? Would you be able to find the exact number in the jar every time? Explain. See if you can discover a formula for finding the volume of your group s popcorn container. ACADEMIC VOCABULARY Volume is the amount of space occupied by a threedimensional figure. It is measured in cubic units, such as cubic inches (in. 3 ). ACADEMIC VOCABULARY A solid is a 3-dimensional geometric figure with dimensions of length, width, and height. 5. The 2-dimensional drawing at right, representing the popcorn container, is a solid. Label each dimension on the drawing. Unit 5 Geometry 311

2 ACTIVITY 5.8 Exploring Volume MATH TERMS Attributes are characteristics or qualities of something. SUGGESTED LEARNING STRATEGIES: Group Presentation, Look for a Pattern, Vocabulary Organizer, Use Manipulatives The popcorn container is a type of solid called a prism. Look at the figures in the table to help remember what a prism is. Prisms Not Prisms 6. In your own words, describe the attributes of a prism. 7. Why is the popcorn container called a rectangular prism? Circle all rectangular prisms in the chart above Question 6. WRITING MATH Cubic units can be written as units 3. To find the amount of 3-dimensional space that is filled with popcorn, you need to measure the volume of the popcorn container. You use cubic units to measure volume. Look at the blocks your teacher has given you. 8. Why is one of these blocks called a cubic unit? 9. Measure the dimensions of the block. Now give the cubic unit a more specific name and explain your reasoning. 312 SpringBoard Mathematics with Meaning TM Level 1

3 Exploring Volume ACTIVITY 5.8 SUGGESTED LEARNING STRATEGIES: Think/Pair/Share, Create Representations, Use Manipulatives, Look for a Pattern, Group Presentation 10. Name at least two other cubic units that could be used to fill a 3-dimensional space. Since you do not have enough cubic units to fill the popcorn container as a way of finding its volume, you can build smaller prisms and look for a pattern. 11. Use the cubic-unit blocks to build rectangular prisms with the dimensions given in the table. Count the blocks to determine the volume of each prism, and record your findings. Length Width Height Volume Figure Figure Figure Figure Use the data in the table to describe a pattern that can be used for finding volume of any rectangular prism. 13. You can write formulas to represent these patterns. a. Write a formula for volume, V, to represent the pattern you have found. Use l for length, w for width, and h for height. b. Write a formula for volume, V, relating the area of the base, B, to the height, h. Compare this formula to the one you wrote in Part a. 14. Use both formulas from Question 13 to find the volume of a rectangular prism with a length of 4 units, width of 5 units, and a height of 2 units. Use blocks to check your answer. Unit 5 Geometry 313

4 ACTIVITY 5.8 Exploring Volume SUGGESTED LEARNING STRATEGIES: Create Representations, Debriefing, Use Manipulatives, 15. It is possible to make different rectangular prisms with a total of 12 cubic units. a. Make as many different prisms as you can. Make a table below to record your results. b. Use either formula from Questions 13 and 14 to confirm that your dimensions are accurate. 16. Use a ruler to measure. a. Find the volume of your popcorn container in cubic centimeters. b. Will this tell you the number of popcorn pieces? Explain why or why not. 17. This time find the volume of the popcorn container using popcorn as the unit. Your teacher will provide you with a handful of popcorn. l = w = h = V = 18. How does volume in cubic centimeters relate to volume in popcorn units? 314 SpringBoard Mathematics with Meaning TM Level 1

5 Exploring Volume ACTIVITY 5.8 SUGGESTED LEARNING STRATEGIES: Discussion Group, Look for a Pattern, Vocabulary Organizer, Quickwrite 19. Your teacher will tell you how much popcorn is in each container. Compare the actual number to your answer to Question 17? Explain why they are the same or different. 20. Refer to your original guess. How close were you to the actual amount and by how much? Who had the closest guess in your group? Now play the estimation jar game again. 21. Look at the second popcorn container your teacher has filled. The drawings below relate to this container. 3-D View (not drawn to scale) Top View In a triangle, a height is the distance from a vertex to the line containing the opposite side. This distance is the length of the perpendicular line segment from the vertex to the line containing the opposite side. In a prism, or other three-dimensional figure with parallel bases, the height is the distance between the parallel bases. This distance is the length of the line segment perpendicular to both bases and with endpoints on those bases in in in. a. What is the name of this container? Justify your thinking. b. Guess how many pieces of popcorn are in the container. Write your guess and your name on a sticky note and post it when your teacher asks you to do so. c. Describe your estimation strategy. Unit 5 Geometry 315

6 ACTIVITY 5.8 Exploring Volume SUGGESTED LEARNING STRATEGIES: Look for a Pattern, Create Representations, Use Manipulatives, Debriefing, Vocabulary Organizer, Think/Pair/Share 22. Determine the formula for finding the volume of a rectangular prism and explain how you could use it to develop a formula for finding the volume of the second popcorn container. 23. Find the volume of the second popcorn container in each unit. a. popcorn units b. cubic centimeters c. cubic inches 24. Your teacher will tell you the actual volumes. How close were your answers? CONNECT TO AP In calculus, you will learn how to compute the volume of an irregular solid like the vase shown below. 25. An estimation game can be played with a glass jar in the shape of a cylinder. a. Compare and contrast cylinders and prisms. b. Is a cylinder a type of prism? Explain. 26. Use what you know about finding the volume of a prism to develop a formula for finding the volume of a cylinder. Explain your reasoning. 316 SpringBoard Mathematics with Meaning TM Level 1

7 Exploring Volume ACTIVITY 5.8 SUGGESTED LEARNING STRATEGIES: Think/Pair/Share, Quickwrite, Self Revision/Peer Revision 27. Apply your formula to find the missing variable in each problem below. Figures are not drawn to scale. a. b. c. r = 2.3 cm d = 200 in. r = 15 m h = 7.6 cm h = 90 in. h = V = V = V = 19,792 m Look at the estimation jar your teacher has displayed. Consider what you have learned about finding volume. a. Make an educated estimate of the number of items in the estimation jar: b. Describe your estimation strategy. c. Write your estimate and your name on a sticky note and post it when your teacher asks you to do so. 29. How does knowing the formulas for volume help you to make better estimates when playing the estimation jar game? Unit 5 Geometry 317

8 ACTIVITY 5.8 Exploring Volume CHECK YOUR UNDERSTANDING Write your answers on notebook paper. Show your work. 1. What is the difference in unit measures when calculating area and volume? 2. Find the volume of a rectangular prism with a height of 5 cm, length of 7 cm, and width of 8 cm. 3. Find the missing dimension of each rectangular prism below. a. b. 6.5 ft h 3.2 ft 173 mm V = ft 3 V = 6,099,980 m m 3 4. How much can the following container hold in cubic inches? w 215 mm 6. A glass company sells vases in 3 different styles: a. The dimensions of the rectangular prism are shown below. Find the volume. h = 10 in. w = 4 in. l = 6 in. b. What do the dimensions of the other vases need to be so that they hold about the same amount of water as the rectangular prism vase? Show your work. 3 in. 8.3 in. 2.7 in. 5. A can of breadcrumbs has a diameter of 4 in. and a height of 10.5 in. What is the volume of the can? base = radius = 2.7 in. altitude = 4 in. height = height = 10 in. 7. MATHEMATICAL REFLECTION Compare and contrast the formulas for finding volume of rectangular prisms, triangular prisms, and cylinders. 318 SpringBoard Mathematics with Meaning TM Level 1