Volume of Right Prisms Objective To provide experiences with using a formula for the volume of right prisms.


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1 Volume of Right Prisms Objective To provide experiences with using a formula for the volume of right prisms. epresentations etoolkit Algorithms Practice EM Facts Workshop Game Family Letters Assessment Management Common Core State Standards Curriculum Focal Points Interactive Teacher s Lesson Guide Teaching the Lesson Ongoing Learning & Practice Differentiation Options Key Concepts and Skills Use a formula to calculate the volumes of prisms. [Measurement and Reference Frames Goal 2] Define and classify prisms according to common properties. [Geometry Goal 2] Key Activities Students examine a model of a prism to verify a formula for finding the volume of right prisms, and then use the volume formula to calculate the volumes of right prisms. Key Vocabulary prism face Materials Math Journal 2, pp B Study Link 9 8 Math Masters, pp. 282 and 283 Class Data Pad demonstration materials: metric ruler, foam board, knife, tape (optional) Playing Polygon Capture Student Reference Book, p. 328 Math Masters, pp (optional); p. 497 Polygon Capture Pieces and Property Cards (Math Journal 1, Activity Sheets 3 and 4) Students practice identifying attributes of polygons. Ongoing Assessment: Recognizing Student Achievement Use Math Masters, page 497. [Geometry Goal 2] Math Boxes 9 9 Math Journal 2, p. 326 Student Reference Book, p. 158 Students practice and maintain skills through Math Box problems. Study Link 9 9 Math Masters, p. 284 Students practice and maintain skills through Study Link activities. READINESS Analyzing Prism Nets for Triangular Prisms Math Masters, pp. 285 and 286 scissors tape or glue Students build a triangular prism from a pattern. ENRICHMENT Building Nets Math Masters, pp. 287 and 288 scissors tape or glue Students explore the properties of prisms by building nets for prisms. ELL SUPPORT Building a Math Word Bank Differentiation Handbook, p. 142 Students define and illustrate the term prism. EXTRA PRACTICE 5Minute Math 5Minute Math, pp. 214 and 215 Students calculate the volumes of rectangular prisms. Advance Preparation For Part 1, you will need two pieces of foam board, each at least 4 inches by 10 inches. Use the templates on Math Masters, pages 282 and 283 to draw and cut out a triangular prism and a parallelogram prism. With a pen or marker, draw the dashed lines (representing the heights of the bases). Draw a line segment connecting the midpoints of two sides of the triangle. Teacher s Reference Manual, Grades 4 6 pp , Lesson
2 Getting Started Mathematical Practices SMP1, SMP2, SMP3, SMP4, SMP5, SMP6, SMP8 Content Standards 5.MD.3a, 5.MD.5b, 5.MD.5c Mental Math and Reflexes Have students show thumbs up if the statement is true and thumbs down if the statement is false. Suggestions: Prisms are 3dimensional shapes. Thumbs up; true A rectangular prism is formed by eight flat surfaces. Thumbs down; false The amount of space enclosed by a 3dimensional shape is measured in cubic units. Thumbs up; true The volume of a prism is the area of its base. Thumbs down; false Math Message Are all of the cube structures shown on Study Link 9 8 rectangular prisms? Be prepared to explain your answer. Study Link 9 8 FollowUp Have partners compare answers and resolve any differences. 1 Teaching the Lesson Math Message FollowUp WHOLECLASS DISCUSSION Ask volunteers to explain which of the cube structures on Study Link 98 are rectangular prisms and why. The structure in Problem 1 is a rectangular prism because all six surfaces are formed by rectangles. The other three structures are not rectangular prisms because their faces are not all rectangles. The bases are polygons, but not rectangles. Remind students that geometric solids are hollow, but thinking of cube structures can help when measuring volume. All of the cube structures are prisms because each is a closed 3dimensional figure made up of polygonal regions the flat surfaces, or faces and each has two congruent and parallel faces. Ask students what the two congruent and parallel faces are called. Bases of the prism The other faces are formed by rectangles and connect the two bases. Ask volunteers to explain how a cube structure can be used to find the volume of a prism. Use their responses to emphasize the following points: You can think of each prism as being made up of layers of centimeter cubes with the same number of cubes in each layer. The number of layers is equivalent to the height of the prism. The number of cubes in the first layer is the same as the number of squares in the base (the area of the base). If you know the number of cubes in the first layer, then you can find the volume of the entire prism by multiplying the number of cubes in the first layer by the number of layers (the height of the prism). 754 Unit 9 Coordinates, Area, Volume, and Capacity
3 Refer to Study Link 98, Problem 1, and pose the following question: What would the volume of the prism be if it were 5 centimeters high? _ 2 centimeters high? 37.3 Have volunteers demonstrate their solution strategies on the board. NOTE Slanted prisms are a second type of rectangular prism. They are distinguished from right prisms because all the lateral faces are not all rectangles and the lateral edges are not perpendicular to the bases. Slanted prisms are defined by transformations of a polygonal base. Triangular Prism Base Template This template is a pattern for the base of a triangular prism. Place the template on a sheet of foam board. Using a pencil or ballpoint pen, mark the position of the six dots by piercing the template. Remove the template and connect the points with solid or dashed lines, the same as on the template. Cut out the prism along the outer solid line, using a serrated knife or saw, making sure that your cuts are made perpendicular to the base. Right prisms Slanted prisms Math Masters, p. 282 Verifying the Volume Formula for Prisms (Math Masters, pp. 282 and 283) WHOLECLASS Algebraic Thinking Show students the two prisms that you cut out of foam board. Ask volunteers to describe the shapes. Expect that students might respond that the shapes are 3dimensional figures, geometric solids, and/or prisms. Explain that prisms are named by the shapes of the bases. Ask: What are the names of these two prisms? Triangular prism and parallelogram prism Ask students how we know that the triangular faces on the triangular prism are the bases. Because the bases of a prism must be parallel and congruent Write the formula V = B h on the Class Data Pad. Tell students that this formula can be used to calculate the volume of any prism regardless of its base shape. The volume of any prism is the product of the area of its base times its height. Write this definition under the formula. Have volunteers use the following activity to verify the volume formula: 1. Use a metric ruler to measure the base and height of one of the parallelogram prism faces. This face will be one of the bases. Use these measures to calculate the area of the base (B). Parallelogram Prism Base Template This template is a pattern for the base of a parallelogram prism. Place the template on a sheet of foam board. Using a pencil or ballpoint pen, mark the position of the five dots by piercing the template. Remove the template and connect the points with solid or dashed lines, the same as on the template. Cut out the prism along the solid line, using a serrated knife or saw, making sure that your cuts are made perpendicular to the base. py g g p Math Masters, p. 283 Lesson
4 Date Volume of Prisms The volume V of any prism can be found with the formula V B h, where B is the area of the base of the prism, and h is the height of the prism from that base. Find the volume of each prism Volume cm 3 Volume cm cm height 8 cm base 10 cm 2 12 cm 80 9 in. 80 height base 20 cm 2 2. Measure the height (h) of the parallelogram prism. Record the area of the base (B) and the height (h) of the prism on the board. 3. Calculate the volume (V) of the prism, and record it on the board. Cut the prism into two parts along the dashed height line. Move the triangular piece to the other end to form a rectangular prism. (See Figure 1.) Tape the two pieces together if you wish. Ask students if the volume of the prism has changed. No. Cutting and rearranging the prism has not changed its volume. 7 in Volume cm 3 Volume in ft base = 15 ft cm Volume ft 3 Volume cm 3 Figure 1 Math Journal 2, p. 324 Date Volume of Prisms continued cm base = 3 2 Volume cm 3 Volume cm in. 11 in. 8 cm Volume in 3 Volume cm 3 4. Use a metric ruler to measure the base and height of one of the rectangular prism faces. This face will be one of the bases. Use these measures to calculate the area of the base (B). 5. Measure the height (h) of the rectangular prism. Record the area of the base (B) and the height (h) of the rectangular prism on the board. 6. Calculate the volume (V) of the prism, and record it on the board. Explain that when the parallelogram prism was cut and rearranged, the volume did not change. Because the formula V = B h applies to any prism, using this formula should result in the same volume for the parallelogram prism and the rectangular prism. Ask: When applied to the parallelogram prism and the rectangular prism, did the formula V = B h result in the same volume? Yes Repeat this procedure with the triangular prism. (See Figure 2.) This time cut the prism into three sections. First cut through the line connecting the midpoints of two sides. This creates a trapezoidal prism and a smaller triangular prism. Then cut the resulting triangular prism through the dashed height line, creating two smaller triangular prisms. Rotate and reattach these small triangular prisms at the ends of the trapezoidal prism to form a rectangular prism. Tape the pieces together if you wish m base = 8.3 in 2 base = 14 m Volume in 3 Volume m 3 Math Journal 2, p. 325 Figure Unit 9 Coordinates, Area, Volume, and Capacity
5 Finding the Volumes of Prisms (Math Journal 2, pp. 324 and 325) PARTNER Algebraic Thinking Have students complete journal pages 324 and 325. Date 9 9 Adding Volumes of Solids Find the volume of each solid figure. First find the volumes of the two rectangular prisms that make up each solid. Use either of these two formulas: V = B h or V = l w h. Then add the two volumes to find the volume of the entire solid. In Problems 1 4, you can count the cubes to check your work. 1. Volume of a prism: cubic units Volume of a prism: 2 cubic units Volume of entire solid: 10 cubic units 8 Adding Volumes of Solids (Math Journal 2, pp. 325A and 325B) WHOLECLASS 2. Volume of a prism: cubic units Volume of a prism: Volume of entire solid: cubic units cubic units To solve problems on journal pages 325A and 325B, students find the volumes of rectangular prisms and then use those results to find the volumes of solid figures composed of two nonoverlapping rectangular prisms. Write on the board. Ask: What does it mean if a rectangular prism has the dimensions 3 2 4? Sample answer: The length is 3 units, the width is 2 units, and the height is 4 units. Does the notation suggest how you can find the volume? Yes Explain your answer. Sample answer: The notation suggests that you multiply the three dimensions together to find the volume. Have students look at the prism in Problem 1, Math Journal 2, page 325A. Explain that this figure can be viewed as two rectangular prisms put together. Ask: How could you find the volume of this solid using the formula for finding the volume of a rectangular prism? Sample answer: Use the formula to find the volumes of the two smaller prisms, and then add the two volumes together. Have students complete journal pages 325A and 325B. The solids on page 325A show stacks of cubes so students can count cubes to verify their results. The solids on page 325B provide the dimensions of the prisms but do not show the cubes Volume of a prism: cubic units Volume of a prism: 5 cubic units Volume of entire solid: 41 cubic units Volume of a prism: cubic units Volume of a 3 1_ prism: cubic units Volume of entire solid: 74 cubic units Math Journal 2, p. 325A 325A325B_EMCS_S_MJ2_G5_U09_ indd 325A Date 3/22/11 12:42 PM 9 9 Adding Volumes of Solids continued 5. Volume of a 12 cm 2 cm prism: cm 3 Volume of a 8 cm prism: cm Volume of the entire solid: cm cm 12 cm 2 cm Volume of one ream of paper: in 3 Volume of two reams of paper: in 3 Each ream of paper measures 8 1_ 2 in. by 11 in. by 2 in. 1,320,000 2,268,000 3,588, Volume of 100 ft 60 ft 220 ft prism: ft 3 Volume of 210 ft 60 ft 180 ft prism: ft 3 Volume of entire solid: ft 3 60 ft 40 ft 100 ft 210 ft 180 ft Try This 8. 2 m Shallow End 1.5 m 50 m 25 m Deep End Estimate the volume of the water in this pool by doing the following: a. Underestimate the volume by using the dimensions from the shallow end. 50 m by 25 m by 1.5 m: m 3 b. Overestimate the volume by using the dimensions from the deep end. 50 m by 25 m by 2 m: m 3 c. Find the average of the two volumes to estimate the volume of the pool. 2,187.5 m 3 1,875 2,500 Math Journal 2, p. 325B 325A325B_EMCS_S_MJ2_G5_U09_ indd 325B 4/6/11 11:02 AM Lesson
6 Date 9 9 Math Boxes 1. a. Plot the following points on the grid: ( 4, 1); ( 3,1); (1,3); (2,1); ( 2, 1) b. Connect the points with line segments in the order given above. Then connect ( 4, 1) and ( 2, 1). What shape have you drawn? pentagon 2. Find the diameter of the circle. Choose the best answer. 48 units 66 units 44 units 11 units 22 units What transformation does the figure show? Circle the best answer. A translation preimage image B reflection C rotation D dilation Ongoing Learning & Practice Playing Polygon Capture (Student Reference Book, p. 328; Math Masters, pp ) PARTNER Students practice identifying properties of polygons by playing Polygon Capture. Students first played this game in Unit 3 and stored the sets of Polygon Capture pieces and cards. If additional sets are needed, copy Math Masters, pages Have students cut out the pieces and cards a. What is the perimeter of the rectangle? 30 units 6 units 9 units b. What is the area? 54 square units Math Journal 2, p _EMCS_S_G5_MJ2_U09_ indd Two cups of flour are needed to make 24 oatmeal cookies. How many cups of flour are needed to make... a. 4 dozen cookies? 4 cups b. 6 dozen cookies? 6 cups c. 120 cookies? 10 cups /22/11 5:18 PM Ongoing Assessment: Recognizing Student Achievement Math Masters Page 497 Use the Record Sheet from Polygon Capture (Math Masters, page 497) to assess students ability to match polygons with their properties. On their first two draws of Polygon Capture, have students write the properties and list or draw all of the shapes that fit the properties. Students are making adequate progress if they have correctly matched properties and shapes. [Geometry Goal 2] Math Boxes 9 9 (Math Journal 2, p. 326; Student Reference Book, p. 158) INDEPENDENT Study Link Master Mixed Practice Math Boxes in this lesson are paired with Math Boxes in Lesson 96. The skill in Problem 5 previews Unit 10 content. Writing/Reasoning Have students do the following: Draw and label a preimage of a simple figure and then draw the image rotated 180 clockwise about a point. Have students refer to Student Reference Book, page 158. Sample answer: STUDY LINK 9 9 Volumes of Prisms The volume V of any prism can be found with the formula V = B h, where B is the area of the base of the prism, and h is the height of the prism from that base Preimage Image 72 Volume = cm 3 Volume = cm 3 70 Volume = in 3 Volume = cm cm in. 5 in. 8 in. 162 Study Link 9 9 (Math Masters, p. 284) INDEPENDENT Home Connection Students calculate the volume of various types of prisms. base = 15 in Volume = in 3 Volume = m 3 Practice Solve each equation r = ,577  t = 3, ,577  m = 3, d 68 = m base = 20 m Math Masters, p _497_EMCS_B_MM_G5_U09_ indd 284 2/22/11 6:06 PM 758 Unit 9 Coordinates, Area, Volume, and Capacity
7 3 Differentiation Options READINESS Analyzing Prism Nets for Triangular Prisms (Math Masters, pp. 285 and 286) SMALLGROUP Min Unfolding Geometric Solids If you could unfold a prism so that its faces are laid out as a set attached at their edges, you would have a flat diagram for the shape. Imagine unfolding a triangular prism. There are different ways that you could make diagrams, depending on how you unfold the triangular prism. Which of the following are diagrams that could be folded to make a triangular prism? Write yes or no in the blank under each diagram To explore the properties of triangular prisms, have students build a prism from a net and identify other nets that will form a triangular prism. This activity mirrors activities in Lesson 98, except that students are now working with triangular prisms instead of cubes. Read and discuss the introduction on Math Masters, page 286 as a group. Discuss students solutions, and then have them identify the prism surfaces as either faces or bases. Yes Yes No No ENRICHMENT Building Nets (Math Masters, pp. 287 and 288) PARTNER 30+ Min Math Masters, p. 286 To extend students ability to visualize, name, and describe geometric solids, have students build nets for prisms. Partners cut out the individual shapes on Math Masters, page 287 and follow the instructions on Math Masters, page 288 to form nets for rectangular, triangular, pentagonal, hexagonal, or octagonal prisms. Ask students to discuss any discoveries or curiosities they encountered during the activity. Consider having partners cut out and fold their completed nets to check the configurations. ELL SUPPORT Building A Math Word Bank (Differentiation Handbook, p. 142) SMALLGROUP 5 15 Min Using Faces and Bases The flat diagram formed from unfolding a prism so that its faces are laid out flat and attached at their edges is called a geometric net. For a given prism, there are different nets, depending on how you think about unfolding the prism. 1. Cut out the figures on Math Masters, page 287. You and your partner will use the figures to build nets for the prisms below. To provide language support for volume, have students use the Word Bank Template found on Differentiation Handbook, page 142. Ask students to write the term prism, draw pictures related to the term, and write other related words. See the Differentiation Handbook for more information. EXTRA PRACTICE 5Minute Math To offer students more experience calculating the volumes of rectangular prisms, see 5Minute Math, pages 214 and 215. SMALLGROUP 5 15 Min 2. Take turns to select, draw, and place figures to form a net for a prism. 3. The partner who places the figure that completes the net states the number of faces and the number of bases. For example, if the net for a cube were completed, the partner would say, 4 faces, 2 bases. This ends the round. 4. A partner can also block the completion of a net. In this case, the partner would put down a figure that would prevent completing the net in the following placement and say block. The blocked partner then has the opportunity to complete the net by placing two figures and stating the number of faces and bases. Again, this would end the round. Example: Student 1 Student 2 Student 1 Draw 1: Draw 2: Draw 3: Student 2 Student 1 Student 1 states, Draw 4: Draw 5: 3 faces, 2 bases. This ends the round. Math Masters, p. 288 Lesson
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