Algebra Surface Area of Prisms

Transcription

1 LESSON.3 lgebra Surface rea of Prisms FOUS OHERENE RIGOR LESSON T GLNE F R Focus: ommon ore State Standards Learning Objective 6.G..4 Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. pply these techniques in the context of solving real-world and mathematical problems. Find the surface area of prisms. Language Objective lso 6.EE..c Students draw and label a step-by-step sequence to explain how to find the surface area of a prism. MTHEMTIL PRTIES (See ematical Practices in GO! in the Planning Guide for full text.) MP Reason abstractly and quantitatively. MP4 Model with mathematics. MP8 Look for and express regularity in repeated reasoning. F R oherence: Materials oard, ruler Standards cross the Grades efore Grade 6 fter 5.M..5 6.G..4 7.G..6 F R Rigor: Level : Understand oncepts...share and Show ( hecked Items) Level : Procedural Skills and Fluency...On Your Own, Practice and Homework Level 3: pplications...think Smarter and Go eeper F R For more about how GO! fosters oherence within the ontent Standards and ematical Progressions for this chapter, see page 595J. bout the Professional evelopment ack rawing a Net Materials Left a box or other object in the shape of a rectangular prism Top Front The following activity can help students visualize the net of a solid figure when given a diagram of the figure. 0 cm Right 0 cm 30 cm Label all six faces of a box: front, top, right, back, bottom, and left as shown in Figure. lso label the side lengths. ottom Figure Point to the front; have students draw and label it. Point to the top. Then point alternately to the front, the top, and the edge where they are joined. Explain that students need to imagine flattening the top and front faces and keeping them joined at the edge. Tell students to add the top ontinue the process, having students add one new face at a time until they have the complete net, as shown in Figure. Left Top 0 cm Front Right ack 30 cm ottom 0 cm 0 cm 0 cm 30 cm Figure Professional evelopment Videos 609 hapter 0 cm

2 aily Routines ommon ore Problem of the ay.3 farmer digs an irrigation ditch in his field along. What is the area of the shaded region? 600 m 50 m 40 m Interactive Student Edition Multimedia Glossary e 34 Fluency uilder 30 m ommon ore Fluency Standard 6.NS..3 Materials eteacher Resources page TR38 40 ecimals Have students find the product or quotient. ENGGE with the Interactive Student Edition Essential Question How can you find the surface area of a prism? Making onnections Have students use i Tools to find areas of two-dimensional figures. sk: How could you use what you know about finding areas of rectangles to find the surface area of a rectangular prism? raw a net and find the sum of the areas of the rectangular faces of the prism. Learning ctivity Have students use i Tools to find areas of other two-dimensional figures. sk: How do you find the area of a parallelogram? Multiply its base times its height. Literacy and ematics Present the following scenario: student says that the surface area of a rectangular prism can always be found by making a net, finding the area of one of its faces, and multiplying by 6. sk students if they agree and to explain their answers. Have students write a How to paper that explains how to correctly use a net to find the surface area of a rectangular prism , , Pages in Strategies and Practice for Skills and Facts Fluency provide additional fluency support for this lesson. How can you find the surface area of a prism? Lesson.3 609

3 LESSON.3 EXPLORE Materials 3-dimensional models of a rectangular prism and a triangular prism Unlock the Problem MTHEMTIL PRTIES 6.G..4 Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. pply these techniques in the context of solving real-world and mathematical problems. Name Surface rea of Prisms Essential Question How can you find the surface area of a prism? You can use a net to find the surface area of a solid figure, such as a prism. LGER Lesson.3 Geometry 6.G..4 lso 6.EE..c MTHEMTIL PRTIES MP, MP5, MP6 MP4 Model with mathematics. raw the prism on the board, and help students identify the pairs of congruent faces. Label the figure as shown. EEPER & & E & F in. 5 in. Unlock the Problem lex is designing wooden boxes for his books. Each box measures 5 in. by in. by efore he buys wood, he needs to find the surface area of each box. What is the surface area of each box? See below. in. in. E 5 in. What is the shape of each face? rectangle What are the dimensions of each face? 5 in. by in. (), 5 in. by (), and in. by () 5 in. MP8 Look for and express regularity in repeated reasoning. The dimensions of a rectangular prism are 8 cm by 4 cm by 0 cm. What are the dimensions of the faces of a rectangular prism with these dimensions? Two faces are 8 cm by 4 cm, two faces are 8 cm by 0 cm, and two faces are 4 cm by 0 cm. ELL Strategy: Rephrase Give each pair of students a cube. Together, draw a net of your cube. Let s say that each side measures inches. Have students discuss with a partner how to find the surface area of the cube. Then have them summarize how they found the surface area. Use sentence frames to help students report: cube has 6 faces. Each face has the same area. To find the area of one face, we multiplied in. in. and got 4 in.². Then we multiplied 4 in.² 6 and got 4 in.². Use the to check that students understand the relationships among the parts of the net of a rectangular prism. 609 hapter Name lgebra Surface rea of Prisms You can find the surface area of a figure by adding the lateral surface area to the sum of the areas of the bases. Step raw a net. Note any faces that have equal areas.. 5 in. 8 in in. m STEP 7 in. m 5 m ase : 5 _ cm 8 cm 6 cm 8 ft 8 ft 8 ft in. 6 in. 8 in. 6 in. in. in. bh = _ E 6 in. 3 ifferentiated Enrich.3 Instruction 8 in. 6 in. Step oth triangular bases have the in. same area. ase E: 5 4 in. Step 3 Find the areas of the rectangular faces. Face : 5 lw in. Face : 5 lw in. Face : 5 lw Step 4 dd the areas: E in. So, the surface area of the triangular prism is 336 square inches (in. ). 760 in. 434 m hapter Resources F Find the area of each lettered face. Face : 0 = Face : 5 0 = _ 50 in. Face : _ _ = _ in. Face : _ _ = _ in. Face E: _ _ = _ in. Face F: _ _ = _ in. STEP Find the sum of the areas of the faces. 900 in. So, the surface area of each box is. 900 in. Problem Types: rea Unknown Product, Put Together/Take part Total Unknown Reteach cm 384 ft Lesson.3 Reteach -9 Reteach Name Use Surface rea to Make Food Packages You ve been hired to design packaging for the following food items. Match each box with its description. Then find the surface area of the indicated material. Item: Green Energy Tea ags Item: ountry abin orn The tea is sold in a cube-shaped box with The box has length 4.5 in., width 9. in., sides that are 9.5 in. long. The box is fully and height 5 in. The box is wrapped in two lined with tissue paper. layers of freezer paper. ox: ox: Tissue paper area: Item: Tasty Flakes and Raisins ereal The top and bottom of the cereal box is by.5 in. The box is.3 in. high. The cardboard is coated with dye, except on the bottom. ox: yed area: 54.5 in.. Stretch Your Thinking Tissue paper for the tea boxes costs $0.0 per square inch. How much would it cost to line a dozen boxes, to the nearest penny? Total paper area: Item: V.I.P. Select hocolates Two boxes are stacked and tied with a ribbon. Each box is fully wrapped in decorative paper. ottom box: 8 in. 3 6 in. 3 3 in. Top box: 5 in. 3 3 in. 3 in. ox: Total paper area: Lesson.3 Enrich,007.6 in in. 4 in. $64.98 hapter Resources in. Possible answer: The areas of opposite faces are equal. To find the area of each pair of opposite faces, find the area of one face and double it. MTHEMTIL PRTIES escribe What do you notice about the opposite faces of the box that could help you find its surface area? hapter 609. Green Tea decided they do not want tissue paper on the top of the box. Using the surface area you found, explain how to find the new surface area. Possible answer: Subtract 90.5 in. from the surface area to get 45.5 in. -0 Enrich E

4 Example triangular prism. Example One Way Use a net. STEP Find the area of each face. Use a net to find the surface area of the The surface area equals the sum of the areas of the three rectangular faces and two triangular bases. Note that the bases have the same area. area of bases and E: = bh = area of face : = lw = 5 0 = _ 50 Find the surface area of the cube. The surface area of the cube is 50 cm. in. 3 in. 5 in. 5 in. 3 in. in. 5 in. ll of the faces are squares with a side length of, so the areas of all the squares are the same. rea of one face: = _ 5 _ 5 = STEP Find the sum of the areas of all _ 6 faces _ + _ + _ + _ + _ + _ = 50 cm nother Way Use a formula. _ 5 = _ 30 area of face : = lw = _ _ 0 = _ 0 area of face : = lw = _ 3 _ 0 = _ 30 Surface area: _ 30 + _ 50 + _ 0 + _ 30 = _ 360 So, the surface area of the triangular prism is. 360 in. The triangular bases are congruent. You can also find the surface area of a cube using the formula S = 6s, where S is the surface area and s is the side length of the cube. Write the formula. Replace s with 5. Simplify. S = 6 s S = 6 (_ 5 ) 5 50 S = 6 ( _ ) = _ E 5 in. MTHEMTIL PRTIES 6 Explain Why was the area of one triangular base multiplied by? Example Show an actual model to help students visualize the faces of a triangular prism. fter students read Example, ask: How do you know the triangular bases have the same area? Every triangular prism has a pair of congruent triangles as its bases. What are the lengths of the sides that form the right angle of the triangle? 5 in. and in. Use the to check that students understand the relationships among the parts of the net of a triangular prism. MP6 ttend to precision. How are the two triangles of the net alike? The triangular bases are congruent. Example Show an actual model to help students visualize the faces of the cube. ll of the faces of a cube are squares. How do you find the area of a square? multiply the side length by itself Look at the formula for the surface area of a cube. Why is there a 6 in the formula? cube has six congruent faces. 60 dvanced Learners Logical / ematical Individual / Partners Write these problems on the board, and challenge students to solve them: The surface area of a cube is 384 in. Find the dimensions of the cube and explain your method. raw and label a net of the cube. 8 in. by 8 in. by 8 in.; Possible explanation: I divided 384 by 6 to find the area of one face. The area of one face is 64 in. I used guess and check to find that 8 in. 8 in. = 64 in. heck student s work. The surface area of a rectangular prism is 08 in. Find possible dimensions of the prism and explain your method. raw and label a net of the prism. Possible answer: 8 in. by 4 in. by 6 in.; I divided 08 by to get 04. Then I used guess and check to find the dimensions of three rectangles whose areas total 04 in. heck students work. OMMON ERRORS Error Students sometimes neglect to include all necessary surfaces when finding the surface area of a solid figure. Example OMMON Students might ERRORS neglect to include a surface in one of the examples. Springboard to Learning When finding surface area, encourage students to write the number of surfaces and identify all surfaces for every solid figure they work with. For the Examples on this page, ask: How many surfaces does the prism in Example have? Identify them. 5; triangular bases and 3 rectangular faces How many surfaces does the cube in Example have? 6 Lesson.3 60

5 3 EXPLIN MTH Share and Show MTH OR The first problem connects to the learning model. Have students use the oard to explain their thinking. Use the checked exercises for Quick heck. Students should show their answers for Quick heck on the oard. Name Share and Show. ft ft ft ft 0 cm ft ft MTH OR. area of each face: _ _ = _ 4 ft ft ft 3. number of faces: _ 6 surface area = _ _ = _ ft 8 in If Then Then Quick heck a student misses the checked exercises 3 Rt I ifferentiate Instruction with Reteach.3 Personal Trainer 6.G..4 RtI Tier ctivity (online) 4. 6 cm 6 cm 8 cm 43 cm On Your Own 8 m 5. 4 in. 3 in. Possible answer: raw a net and find the sum of the areas of each rectangle in the net. S = 9 ft _ 55 in. MTHEMTIL PRTIES 5 Use Tools What strategy could you use to find the surface area of a rectangular prism with a length of 8 ft, a width of ft, and a height of 3 ft? On Your Own For Exercise 6, help students use the formula correctly. MP6 ttend to precision. What number will you substitute into the formula? 7 What is the first step in evaluating the formula? Find the second power of m 5 m 58 m MTHEMTIL PRTIE 7 in. 7 in. 7 in. _ 337 in. 6 ttend to Precision alculate the surface area of the cube in Exercise 5 using the formula S = 6s. Show your work. S = 6 ( _ 7 ) _ = 6(56 ) = 337 ; S = in. hapter Lesson 3 6 Use the to check that students understand how to find the surface area of a rectangular prism using the formula and using a net. dditional Example Equal Groups Unknown Product, Put Together/Take part Total Unknown Yash is making a candy box in the shape of a triangular prism. The dimensions are shown below. What is the surface area of Yash s candy box? 508 square cm 3 cm PROLEM TYPE SITUTIONS ddition and Subtraction Put Together/Take part Total Unknown Exercises: 7, 8, 9 Put Together/Take part ddend Unknown Exercise: 7 Multiplication and ivision Equal Groups Unknown Product Exercise: 8 0 cm cm rea Unknown Product Exercises: 7, 8, 9 6 hapter

6 MTHEMTIL PRTIES OMMUNI E ONSTRUT RGUMENTS Unlock the Problem 4 ELORTE 7. SMRTER The Vehicle ssembly uilding at Kennedy Space enter is a rectangular prism. It is 8 m long, 58 m wide, and 60 m tall. There are four 39 m tall doors in the building, averaging 9 m in width. What is the building s outside surface area when the doors are open? a. raw each face of the building, not including the floor. 58 m 8 m 8 m 8 m 58 m 60 m b. What are the dimensions of the 4 walls? walls measure 8 m by 60 m and walls measure 58 m by 60 m d. Find the building s surface area (not including the floor) when the doors are closed. 54,764 m e. Find the area of the four doors. WRITE 6,4 m Show Your Work f. Find the building s surface area (not including the floor) when the doors are open. Unlock the Problem MTHEMTIL PRTIES SMRTER For Exercise 7, explain that the doors averaging 9 m in width means that students should use 9 m as the width of each door. on the Spot Video Tutor Use this video to help students model and solve this type of Think Smarter problem. 6 c. What are the dimensions of the roof? 8 m by 58 m 8. EEPER rectangular prism is _ ft long, _ ft wide, and 5_ ft high. What is the surface 3 6 area of the prism in square inches? 808 in. 38,640 m 9. SMRTER gift box is a rectangular prism. The box measures 8 inches by 0 inches by 3 inches. What is its surface area? 68 in. Image redits: (t) ourtesy, NS on the Spot videos are in the Interactive Student Edition and at EEPER For Exercise 8, ask How many square millimeters are in one square centimeter? 00 SMRTER Exercise 9 assesses a student s ability to find the surface area of a rectangular prism. Students who incorrectly answer this item may not realize that they can use net to find the surface area of the prism. It may be helpful for students to draw the net of the rectangular prism. IFFERENTITE INSTRUTION INEPENENT TIVITIES 5 EVLUTE Formative ssessment ctivities Estimating Units of Measure Students complete online orange ctivity ard 7 by choosing the appropriate unit of measure for different objects. ifferentiated enters Kit Literature Room Makeover: Serving the ommunity Students read about using surface area to find the amount of paint and tile needed for a room. Essential Question Using the Language Objective Reflect Have students draw and label a step-by-step sequence to answer the Essential Question. How can you find the surface area of a prism? I add the areas of all the faces of the figure. Journal WRITE Explain why a two-dimensional net is useful for finding the surface area of a three-dimensional figure. Lesson.3 6

7 Name Surface rea of Prisms Practice and Homework Lesson.3 OMMON ORE STNR 6.G..4 Solve real-world and mathematical problems involving area, surface area, and volume. Practice and Homework Use the Practice and Homework pages to provide students with more practice of the concepts and skills presented in this lesson. Students master their understanding as they complete practice items and then challenge their critical thinking skills with Problem Solving. Use the Write section to determine student s understanding of content for this lesson. Encourage students to use their Journals to record their answers.. cm 6 cm rea of and F = ( 5 ) = 0 cm rea of and = ( 6 ) = 4 cm rea of and E = ( 6 5 ) = 60 cm S..= 0 cm + 4 cm + 60 cm = 04 cm in. 9 ft 4 in. 3.5 in. 9 ft 9 ft 4. cm cm F 0 cm E 6 cm 8 cm 6 cm cm 8 in. 486 f t 336 c m Problem Solving 5. shoe box measures 5 in. by 7 in. by 4 _ in. What is the surface area of the box? 6. Vivian is working with a styrofoam cube for art class. The length of one side is 5 inches. How much surface area does Vivian have to work with? 408 in. 50 in. 7. WRITE Explain why a two-dimensional net is useful for finding the surface area of a three-dimensional figure. heck students work. hapter 63 OMMON MON ORE PROFESSIONL EVELOPMENT in ction iscuss Exercise 4. How many surfaces does the triangular prism have, and what shapes are they? How can you find the areas of the rectangles? What are the areas? Verona: Five; the two bases are triangles and the three other faces are rectangles. Tavon: Multiply the length times the width. The areas of the rectangles are 0 cm², 7 cm², and 96 cm². Oscar: Joe: What are the dimensions of the rectangles? cm by 0 cm, cm by 6 cm, and cm by 8 cm How is the width of each rectangle related to the triangle? Each width is a length of a side of the triangle. anetza: ngela: How can you find the area of each triangle? What is the area? Find the product of, 8, and 6. Each triangle has an area of 4 cm². What is the surface area of the triangular prism? How can you find it? 336 cm²; add times the area of the triangle plus the area of each rectangle. 63 hapter

8 Lesson heck (6.G..4). What is the surface area of a cubic box that contains a baseball that has a diameter of 3 inches? 54 square inches. piece of wood used for construction is inches by 4 inches by 4 inches. What is the surface area of the wood? 304 square inches ontinue concepts and skills practice with Lesson heck. Use Spiral Review to engage students in previously taught concepts and to promote content retention. ommon ore standards are correlated to each section. Spiral Review (6.EE..9, 6.G.., 6.G..4) 3. etergent costs $4 per box. Kendra graphs the equation that gives the cost y of buying x boxes of detergent. What is the equation? 4. trapezoid with bases that measure 8 inches and inches has a height of 3 inches. What is the area of the trapezoid? y = 4x 8.5 square inches 5. ity Park is a right triangle with a base of 40 yd and a height of 5 yd. On a map, the park has a base of 40 in. and a height of 5 in. What is the ratio of the area of the triangle on the map to the area of ity Park? 6. What is the surface area of the prism shown by the net? 64,96: 7 square units FOR MORE PRTIE GO TO THE Personal Trainer Lesson.3 64