Investigate Explore Surface Area Using Nets

Transcription

1 LESSON 11.2 Investigate Explore Surface rea Using Nets OCUS COHERENCE RIGOR LESSON T GLNCE C R ocus: Common Core 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. Use nets to recognize that the surface area of a prism is equal to the sum of the areas of its faces. MTHEMTICL PRCTICES (See Mathematical Practices in GO Math! in the Planning Guide for full text.) MP1 Make sense of problems and persevere in solving them. MP2 Reason abstractly and quantitatively. MP3 Construct viable arguments and critique the reasoning of others. MP4 Model with mathematics. Language Objective C R Coherence: Standards cross the Grades Before Grade 6 fter 5.MD.C.5 6.G..4 7.G.B.6 C R Rigor: Level 1: Understand Concepts...Share and Show ( Checked Items) Level 2: Procedural Skills and luency...on Your Own, Practice and Homework Level 3: pplications...think Smarter and Go Deeper Student pairs explain to another pair the relationship between a net and the surface area of a prism. Materials MathBoard, Grid Paper (1-cm) (see eteacher Resources), ruler, scissors C R or more about how GO Math! fosters Coherence within the Content Standards and Mathematical Progressions for this chapter, see page 595J. bout the Math Professional Development MP2 Reason abstractly and quantitatively. Interactive Student Edition s students study geometry, it is important for them to move from the concrete to the abstract in their reasoning. Personal Math Trainer These two lessons, when taken together, allow students to apply the patterns they identify during concrete explorations and use them to represent surface area symbolically. sk questions such as these: What do the grid squares on your net represent? Possible answer: Square units such as square inches or square centimeters What does the total number of grid squares represent? the total surface area of the prism Professional Development Videos 603 Chapter 11 Math on the Spot Video itools: Geometry

2 Daily Routines Common Core Problem of the Day 11.2 What is the area of the figure? 66 m 2 6 m 4 m 1 ENGGE with the Interactive Student Edition Essential Question What is the relationship between a net and the surface area of a prism? Making Connections Have students use the i Tools Solid igures activity to review threedimensional figures. sk: How many faces does the rectangular prism have? 6 4 m 8 m Vocabulary surface area Interactive Student Edition Multimedia Glossary e 3 m How would you find the area of one face of the figure? Multiply the length by the width of the face. Learning ctivity Have students work in pairs to repeat the i Tools Solid igures activity to review other solid figures. sk: What are solid figures that have a square face? rectangular prism, cube, square pyramid Literacy and Mathematics Have students create a chart describing four different solid figures. The chart should include: a sketch of the solid figure luency Builder Common Core luency Standard 6.NS.B.3 Materials eteacher Resources pages TR the number of faces a sketch of the net of the solid Then have students create real-world scenarios that involve objects shaped like one of the solid figures. Operations Have students find the product or quotient What is the relationship between a net and the surface area of a prism? Pages in Strategies and Practice for Skills and acts luency provide additional fluency support for this lesson. 603B

3 LESSON EXPLORE Investigate Materials Grid Paper (1-cm) (see eteacher Resources), ruler, scissors) Students construct a rectangular prism to investigate finding surface area. How do you draw a net for a rectangular prism? Possible answer: Draw all of the faces of the solid figure as if the figure had been unfolded. What is the area of each square on the centimeter grid paper? 1 square centimeter, or 1 cm² MP1 Make sense of problems and persevere in solving them. How are the dimensions of the prism 8 cm, 6 cm, and represented in the rectangles of the net? There are two rectangles that are 8 cm by, two that are 8 cm by 6 cm, and two that are 6 cm by. Draw Conclusions MP8 Look for and express regularity in repeated reasoning. Describe how to find the surface area of any rectangular prism. ind the area of each face and add all the areas. 603 Chapter 11 Hands or Exercise 2, ask students to describe the shape of each face and remind students that they already know a rule for finding the area of a rectangle. ELL Strategy: Cooperative Grouping On Pair fluent students with English Language Learners. Give each pair a rectangular prism and grid paper. Together, draw a net of your prism. Tell each other the steps you will take. Then count the unit squares to find the surface area. Have volunteer pairs show the class their prism and net and tell its surface area. Encourage fluent students to help their partners apply such terms as dimensions, edge, face, opposite, congruent, area, and surface area. 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. Image Credits: (t) Nikreates/lamy Name Name Explore Surface rea Using Nets Essential Question What is the relationship between a net and the surface area of a prism? connect The surface area of a solid figure is the sum of the areas of all the faces or surfaces of the figure. Surface area is measured in square units. You can use a net to help you find the surface area of a solid figure. Explore Surface rea Using Nets The net of a solid figure shows you all of the faces or surfaces of the figure. net can help you find the surface area of a figure. ind the surface area of the rectangular prism. Step 1 Make a net of the rectangular prism. The prism has 6 rectangular faces, so the net has 6 rectangles. Step 2 ind the area of each face of the prism. irst Way: Count the unit squares on each rectangle to find its area. Second Way: Calculate the area of each rectangle by multiplying length 3 width. 4 in. E C B D 3 Differentiated 2 Enrich Instruction : 8 unit squares B: 8 unit squares C: 4 unit squares D: 4 unit squares E: 2 unit squares : 2 unit squares Step 3 dd the areas of all the rectangular faces. 28 unit squares 28 square inches So, the surface area of the rectangular prism is 28 square inches (in. 2 ). Use the net to find the surface area of the prism. 1. a. ind the area of each face. C E D Chapter Resources Investigate Materials centimeter grid paper, ruler, scissors box is shaped like a rectangular prism. The box is 8 cm long, 6 cm wide, and high. What is the surface area of the box? ind the surface area of the rectangular prism.. Draw a net of the prism on centimeter grid paper. B. Cut out the net. C. old the net to confirm that it represents a rectangular prism measuring 8 cm by 6 cm by. D. Count the unit squares on each face of the net. Draw Conclusions B : B: C: D: E: : b. dd: 1 B 1 C 1 D 1 E c. The surface area is cm So, the surface area of the box is cm Explain how you used the net to find the surface area of the box. Possible answer: I counted the unit squares on each face and then found the sum of the unit squares on all 6 faces. Reteach 1 in Reteach Name Nets, Nets, Nets Draw and label two nets for the rectangular prism. Then use the nets to find the surface area. 1. length 9 cm width height Surface area: 2. length 4 in. width 6 in. height Surface area: 3. length 7.3 m width 3.2 m height Surface area: Chapter Resources 2. Describe how you could find the area of each face of the prism without counting unit squares on the net. 6 in. 6 cm Possible answers shown. 3.2 m 6 in. 9 cm 88 in in. 7.3 m m 2 Geometry 6.G..4 MTHEMTICL PRCTICES MP1, MP2, MP6 3.2 m 8 cm 6 in. 9 cm 7.3 m 4 in. Enrich 3.2 m 6 in. Hands On Make sure you include all surfaces in the net of a three-dimensional figure, not just the surfaces you can see in the diagram of the figure. Possible answer: You could find the product of the length and the width of each face. Reteach 11.2 See below. Problem Type: Put Together/Take part Total Unknown Chapter m 11-8 Enrich

4 Make Connections You can also use the formula for the area of a rectangle to find the surface area of the box. ind the surface area of the box in the Investigate, which measures 8 cm by 6 cm by. STEP 1 Label the rectangles in the net through. Then label the dimensions. 8 cm 6 cm B C D 6 cm STEP 2 ind the area of each face of the prism. Think: I can find the area of a rectangle by multiplying the rectangle s length times its. width Record the areas of the faces below ace : 4 8 = 3 2 ace B: _ cm 2 ace C: _ cm 2 ace D: _ 2 2 ace E: _ 3 2 ace : _ 48 cm 2 E Problem Types: rea Unknown Product, Put Together/Take part Total Unknown 6 cm 8 cm Make Connections Some students may find it helpful to label the net and then cut it into six rectangles before finding the area of each face. What is the formula for finding the area of a rectangle? = l w What do the variables in the formula for area of a rectangle represent? represents area, l represents length, and w represents width. MP2 Reason abstractly and quantitatively. or Exercise 3, suggest that students look at the faces of a box. Then ask, Which faces are congruent? What do you know about the areas of congruent shapes? Math Talk Use Math Talk to focus on students understanding of how to find surface area of a prism. Students should realize that they can use either method to find the surface area, but using multiplication can be faster than counting unit squares. STEP 3 dd the areas to find the surface area of the prism The surface area of the prism is _ 208 cm 2. MTHEMTICL PRCTICE 2 Use Reasoning Identify any prism faces that have equal areas. How could you use that fact to simplify the process of finding the surface area of the prism? Two faces have areas of 48 cm 2, two have areas of 3 2, and two have areas of 2 2. I could find the surface area by doubling 48, 32, and 24 and finding their sum. 4. Describe how you could find the surface area of a cube. Possible answer: ind the area of one face and multiply by 6. Math Talk MTHEMTICL PRCTICES 6 Compare What do you notice about the surface area you found by adding the areas of the faces and the surface area you found by counting grid squares? Explain. They are equal. Possible explanation: inding the areas of the faces using the formula for the area of a rectangle is an equivalent but faster method than finding the areas by counting grid squares. DEEPER MP7 Look for and make use of structure. Write the number sentence Surface rea = + B + C + D + E + on the board. Tell students that this number sentence represents the surface area of a rectangular prism, where, B, C, D, E, and represent the areas of the six faces of the prism. You know that faces and E have the same area, faces B and D have the same area, and faces C and have the same area. How can you write a number sentence for the surface area of the prism using only the variables, B, and C? Possible answer: Surface rea = 2 + 2B + 2C dvanced Learners Kinesthetic / Visual Partners Materials Grid Paper (1-cm) (see eteacher Resources) Draw two rectangular prisms, as shown. Explain to students that the two prisms are blocks of wood with exactly the same dimensions, and that a carpenter is gluing them together, as shown by the arrow. 10 cm Have partners work together to draw the new prism formed by gluing together the two smaller prisms. Then have students draw a net of the new prism and find the surface area. 136 cm² COMMON ERRORS Error Students may confuse the area of a rectangle with its perimeter. Example The area of a 6 cm by rectangle is = 20 cm². Springboard to Learning Remind students that the area of a figure is the measure of the number of unit squares it takes to cover the figure. Perimeter is the total distance around the figure. 604

5 3 EXPLIN MTH Share and Show MTH BORD The first problem connects to the learning model. Have students use the MathBoard to explain their thinking. Use the checked exercises for Quick Check. Students should show their answers for Quick Check on the MathBoard. Name Share and Show MTH BORD Use the net to find the surface area of the prism. 1. B C D E ind the surface area of the rectangular prism. ace : _ 8 cm 2 ace B: _ 6 cm 2 ace C: _ 1 2 Surface area: _ cm cm 9 cm 7 cm ace D: _ 6 cm 2 ace E: _ 1 2 ace : _ 8 cm 2 1 Quick Check Rt I 10 cm cm cm 2 If Then You may suggest that students draw nets of the prisms shown in Exercises 2 4 before they attempt to find the surface area of the prisms. DEEPER a student misses the checked exercises Differentiate Instruction with Reteach 11.2 Personal Math Trainer 6.G..4 RtI Tier 1 ctivity (online) MP1 Make sense of problems and persevere in solving them. or Exercise 7, ask: How many square centimeters are there in one square meter? 10,000 cm 2 = 1 m 2 dditional Example Equal Groups Unknown Product, Put Together/Take part Total Unknown Lindy and Josh are making rectangular prisms for math class. Lindy s prism has a base that is by 6 cm and a height of 7 cm. Josh s prism has a base that is by 8 cm and a height of 6 cm. Whose rectangular prism will require more material to make? How much more? Lindy s prism; 10 square cm more Problem Solving pplications 5. cereal box is shaped like a rectangular prism. The box is 20 cm long by wide by 30 cm high. What is the surface area of the cereal box? 1,700 cm 2 6. MTHEMTICL PRCTICE 1 Darren is painting a wooden block as part of his art project. The block is a rectangular prism that is 1 long by 9 cm wide by high. Describe the rectangles that make up the net for the prism. Possible description: There are 6 rectangles. Two are 9 cm by, two are 1 by 9 cm, and two are 1 by. 7. DEEPER In Exercise 6, what is the surface area, in square meters, that Darren has to paint? m 2 PROBLEM TYPE SITUTIONS ddition and Subtraction Put Together/Take part Total Unknown Exercises: 5, 7, 8 Multiplication and Division Equal Groups Unknown Product Exercise: 7 rea Unknown Product Exercises: 5, 7, 8 Chapter 11 Lesson Chapter 11

6 606 MTHEMTICL PRCTICES What s the Error? B D E C M 8. Emilio is designing the packaging for a new MP3 player. The box for the MP3 player is by by. Emilio needs to find the surface area of the box. Look at how Emilio solved the problem. ind his error. STEP 1 Draw a net. ace : 3 2 = 6 cm 2 ace B: 3 5 = 1 2 ace C: 3 2 = 6 cm 2 ace D: 2 5 = 10 cm 2 ace E: 3 5 = 1 2 ace : 2 5 = 10 cm 2 Surface area: 6 2 STEP 2 ind the areas of all the faces and add them. ace : 3 2 = 6 cm 2 ace B: 3 5 = 1 2 ace C: 3 2 = 6 cm 2 ace D: 3 5 = 1 2 ace E: 3 5 = 1 2 ace : 3 5 = 1 2 Surface area: 7 2 Correct the error. ind the surface area of the prism. So, the surface area of the prism is or numbers 9a 9d, select True or alse for each statement. 9a. The area of face is 10 cm 2. True alse 9b. The area of face B is 10 cm 2. True alse 9c. The area of face C is 40 cm 2. True alse 9d. The surface area of the True alse prism is 66 cm 2. DIERENTITED INSTRUCTION B D E C B C D E 8 cm INDEPENDENT CTIVITIES 4 5 ELBORTE Problem Solving pplications What s the Error? llow students to use centimeter grid paper to make their own nets to check against Emilio s. or Exercise 8, ask what are the dimensions of the six faces of the rectangular prism? The dimensions of the six faces are by, by, by, by, by, and by. Math on the Spot Video Tutor Use this video to help students model and solve this type of Think Smarter problem. Math on the Spot videos are in the Interactive Student Edition and at Exercise 9 assesses a student s ability to find the surface area of a prism using a net. Students who select True for 9b may have incorrectly counted the squares in the net. Students who select True for 9d may not understand that the surface area is the sum of all the faces. This item allows students to demonstrate their visual understanding of surface area. EVLUTE ormative ssessment ctivities Estimating Units of Measure Students complete online orange ctivity Card 17 by choosing the appropriate unit of measure for different objects. Differentiated Centers Kit Literature Room Makeover: Serving the Community 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 student pairs to explain to another pair to answer the Essential Question. What is the relationship between a net and the surface area of a prism? The surface area of a prism is the sum of the areas of all of its faces. net shows all of the faces of a prism and makes it easier to see the faces and find the area of each face. Math Journal WRITE Math Explain in your own words how to find the surface area of a rectangular prism. 606

7 Name Explore Surface rea Using Nets Practice and Homework 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 Math section to determine student s understanding of content for this lesson. Encourage students to use their Math Journals to record their answers. Use the net to find the surface area of the rectangular prism. 1. B C : 6 squares B: 8 squares D E C: 6 squares D: 12 squares E: 8 squares : 12 squares 52 square units ind the surface area of the rectangular prism mm square units 4 in. 5. COMMON CORE STNDRD 6.G..4 Solve real-world and mathematical problems involving area, surface area, and volume. 3 ft 7 mm 3 mm 5 in. 1 in. 6.5 ft 2 ft 102 m m 2 58 i n ft 2 Problem Solving 6. Jeremiah is covering a cereal box with fabric for a school project. If the box is 6 inches long by 2 inches wide by 14 inches high, how much surface area does Jeremiah have to cover? 7. Tia is making a case for her calculator. It is a rectangular prism that will be 3.5 inches long by 1 inch wide by 10 inches high. How much material (surface area) will she need to make the case? 248 in in WRITE Math Explain in your own words how to find the surface area of a rectangular prism. Check students work. Chapter Extend the Math ctivity Surface rea of a Cube Investigate In this activity, students will solve a problem involving the surface area of a cube. Draw the cube on the board and give students this problem. Geoff wants to use felt to decorate the box shown. He has 700 in. 2 of felt. Does he have enough to cover the entire cube? Explain. No, the surface area of the cube is 726 in.², so Geoff does not have enough felt to cover the entire cube. Math Talk How did you find the surface area of the cube? Possible answer: I drew a net and used the net to find the area of each face. Then I added the areas of the 6 faces. ll 6 faces of a cube have the same shape and size. How could you use this information to develop a shortcut for finding the surface area of a cube? Possible answer: I could find the area of one face and multiply by 6. Summarize sk students to describe how they could determine whether they have enough fabric or other material to cover a solid figure. Possible answer: I can find the surface area of the solid and then compare that to the amount of material I have available. 607 Chapter 11

8 Lesson Check (6.G..4) 1. Gabriela drew a net of a rectangular prism on centimeter grid paper. If the prism is 7 cm long by 10 cm wide by 8 cm high, how many grid squares does the net cover? 412 squares TEST PREP 2. Ben bought a cell phone that came in a box shaped like a rectangular prism. The box is 5 inches long by 3 inches wide by 2 inches high. What is the surface area of the box? 62 in 2 Continue concepts and skills practice with Lesson Check. Use Spiral Review to engage students in previously taught concepts and to promote content retention. Common Core standards are correlated to each section. Spiral Review (6.EE.B.5, 6.EE.C.9, 6.G..1, 6.G..4) 3. Katrin wrote the inequality x + 56 < 533. What is the solution of the inequality? 4. The table shows the number of mixed CDs y that Jason makes in x hours. Mixed CDs Hours, x CDs, y Which equation describes the pattern in the table? x < 477 y = 5x 5. square measuring 9 inches by 9 inches is cut from a corner of a square measuring 15 inches by 15 inches. What is the area of the L-shaped figure that is formed? square inches 6. Boxes of Clancy s Energy Bars are rectangular prisms. How many lateral faces does each box have? 4 lateral faces OR MORE PRCTICE GO TO THE Personal Math Trainer 608