1. Kyle stacks 30 sheets of paper as shown to the right. Each sheet weighs about 5 g. How can you find the weight of the whole stack?

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1 Prisms and Cylinders Answer Key Vocabulary: cylinder, height (of a cylinder or prism), prism, volume Prior Knowledge Questions (Do these BEFORE using the Gizmo.) [Note: The purpose of these questions is to activate prior knowledge and get students thinking. Students who do not already know the answers will benefit from the class discussion.] 1. Kyle stacks 0 sheets of paper as shown to the right. Each sheet weighs about 5 g. How can you find the weight of the whole stack? Multiply the number of sheets (0) times the weight of each sheet (5 g). 2. The stack of paper accidently gets nudged and tilted a little to the side. Does this change the weight of the stack? No. Explain. The weight of each sheet is always the same, so the total weight is the same no matter how the sheets of paper are arranged. Gizmo Warm-up A prism is a closed, three-dimensional figure like the one shown to the right. Prisms are made of flat, polygonal surfaces called faces. Two parallel faces are called bases. A cylinder (like a can) is also a closed, three-dimensional figure, but its bases are circles, and it has a curved lateral surface. In the Prisms and Cylinders Gizmo, you can explore the volume (cubic units inside) of a dynamic prism or cylinder. To resize a figure, either drag the sliders, or click on the number in the text field next to a slider, type a new value, and hit Enter. 1. In the Gizmo, be sure Rectangle (under Shape of Base) and Drag to rotate are selected. The figure has rectangular bases, like the one above, so it s called a rectangular prism. A. Drag the Height slider back and forth. How does the prism change? The prism gets taller as the height increases. B. The height of the prism is actually a distance. What distance is it? (Fill in the blank.) The height of a prism is the perpendicular distance between the two bases. 2. Drag the Base length and Base width sliders. How does the prism change? The areas of both bases increase as the base length and/or base width increases.

2 Activity A: Volume of prisms Be sure Rectangle is selected from the Shape of Base dropdown menu. Be sure Drag to rotate is turned on. 1. In the Gizmo, set the Height of the prism to 1 unit, the Base length to 9 units, and the Base width to 6 units. A. Find the area of the base. Show your work. 9 6 = 54 units 2 Turn on Show area of base to check your answer. B. Select Show volume. What is the volume of this prism? 54 units C. Explain why the units used for area of the base and volume of the prism are different. Area is the number of square units (units 2 ) inside a flat, two-dimensional figure. Volume is the number of cubic units (units ) inside a three-dimensional figure. D. Fill in the first row of the table below for the prism above. Then, create 4 more rectangular prisms of your choice, and fill in the rest of the table (including the units). Height (h) Base length (l) Base width (w) Base area (B) Volume (V) 1 unit 9 unit 6 unit 54 units 2 54 units vary. vary. vary. vary. vary. vary. vary. vary. E. Study the table above to try to figure out how the volumes were calculated. Then, below, write two different formulas for finding the volume (V) of a rectangular prism. In the first, use base area (B). In the second, use length (l) and width (w). V = Bh F. Explain why both of the formulas you wrote above will work. Then, experiment with a variety of rectangular prisms to check the formulas. The area of the base (B) equals length (l) times width (w). If you substitute lw for B in the first formula, V = Bh, you get the second formula,. (Activity A continued on next page)

3 Activity A (continued from previous page) 2. Turn off Show volume and Show area of base. Set Height, Base length, and Base width to all be equal to each other. Sketches will vary. Sample sketch: A. This is a special type of prism called a cube. Sketch your cube in space to right. Label all dimensions (height, base length, and base width). B. Find the volume of your cube. Show your work. Then select Show volume to check. depend on the sketch. [For the cube above, V = = 27 units.] C. Experiment with a variety of cubes in the Gizmo and find their volumes. Suppose s is the length of the edge of each cube. Use s to write a formula for the volume of a cube. V = s s s = s. Turn off Show volume and Show area of base. Set Height to 4 units, Base length to 5 units, and Base width to 7 units. A. Find the volume of this prism. V = = 140 units Then check your answer in the Gizmo. B. Select Drag to skew. Drag the prism to tilt it to one side. A tilted prism is called an oblique prism (as opposed to a right prism, which is not tilted). How does the volume of this oblique prism compare to that of the right prism with the same dimensions? The volumes are the same. Experiment with other prisms to see if this is always true. C. How is an oblique prism similar to a tilted stack of papers? At every level, an oblique prism has the same cross-sectional area as the corresponding right prism. 4. Turn off Show volume. Select Triangle from the Shape of Base dropdown menu. Turn on Drag to rotate and Show area of base. Set Height to units and Base edge to 7 units. A. What is the area of the base of this triangular prism? units 2 B. What do you think the volume of this prism is? 6.66 units Explain why. You can find the volume of a rectangular prism by multiplying the area of the base times the height, so that should also be true for a triangular prism. Explore other triangular prisms to verify that V = Bh always works for them too.

4 Activity B: Volume of cylinders Be sure Drag to rotate is selected. 1. In the Gizmo, select Circle under Shape of Base to make a cylinder. Set the cylinder s Height to 1 unit and the Radius to 5 units. A. Find the exact area of the base. (Write the area with a in it, not as a long decimal.) 5 5 = 25 units 2 Turn on Show area of base to check. B. Select Show volume. What is the volume of the cylinder? 25 units C. Fill in the first row of the table below. Then, in the Gizmo, create 4 more cylinders of your choice, and record your results (with units). Express area and volume using. Height (h) Radius (r) Base area (B) Volume (V) 1 unit 5 units 25 units 2 25 units vary. vary. vary. vary. vary. vary. vary. vary. D. With the help of the table above, write two different formulas for finding the volume (V) of a cylinder. In the first, use base area (B). In the second, use radius (r). V = Bh E. Experiment with a variety of cylinders to verify your formulas. Then explain why they both work. The area of the base (B) is r 2. If you substitute r 2 for B in the formula, V = Bh, you get the second formula,. 2. Set Height to 8 units and Radius to 6 units. The cylinder you created is a right cylinder (straight up and down). Select Drag to skew. Drag an edge of the cylinder to tilt it to one side and make it oblique. How do the volumes of the oblique and right cylinders compare? They are equal. Both volumes are 28 units. Experiment with other cylinders to see if this is always true.

5 Activity C: Using volume Be sure Drag to rotate is selected. Solve each problem. Show all of your work. Then, if possible, check your answers in the Gizmo. 1. Find the volume of the prism. cm 4. Find the volume of the cylinder in terms of. 4 cm 6 cm 7 cm 4 cm V = 7 4 V = = 84 cm = 16 6 = 96 cm 2. An oblique triangular prism has a height of 5 in. and a base area of 10.4 in. 2. Find the volume of this prism. V = = 52 in. 5. An oblique cylinder has a diameter of 7 ft and a height of 4 ft. What is the volume of this cylinder in terms of? The radius is half of the diameter, so the radius is 7 2, or.5 ft. V = = = 49 ft. The base of a rectangular prism is 4 m long and m wide. If the prism has a volume of 72 m, what is its height? 72 = 4 h 72 = 12h = 12h 12 6 = h The height is 6 m. 6. Find the radius of a cylinder with a height of 8 m and a volume of 12 m. 12 = r 2 12 = 16 = r 2 4 = r 2 r The radius is 4 m.

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