Investigate Understand Volume

Transcription

1 LESSON 11.6 Progress to Algebra Investigate Understand Volume FOCUS COHERENCE RIGOR LESSON AT A GLANCE F C R Focus: Common Core State Standards 5.MD.C.3b Recognize volume as an attribute of solid figures and understand concepts of volume measurement. A solid figure which can be packed without gaps or overlaps using n unit cubes is said to have a volume of n cubic units. 5.MD.C.4 Measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and improvised units. MATHEMATICAL PRACTICES (See Mathematical Practices in GO Math! in the Planning Guide for full text.) MP3 Construct viable arguments and critique the reasoning of others. MP5 Use appropriate tools strategically. MP6 Attend to precision. F C R Coherence: Standards Across the Grades Before Grade 5 After 4.MD.A.3 5.MD.C.3b 6.G.A. 4.G.A. 5.MD.C.4 F C R Rigor: Level 1: Understand Concepts...Share and Show ( Checked Items) Level : Procedural Skills and Fluency...On Your Own, Practice and Homework Level 3: Applications...Think Smarter and Go Deeper Learning Objective Count unit cubes that fill a solid figure to find volume. Language Objective Students work in teams to analyze how you can use unit cubes to find the volume of a rectangular prism. Materials MathBoard, Rectangular Prism Net A (see eteacher Resources), centimeter cubes FC R For more about how GO Math! fosters Coherence within the Content Standards and Mathematical Progressions for this chapter, see page 635J. About the Math Professional Development Progress to Algebra Teaching for Depth In this lesson, students move beyond using unit cubes as building blocks to counting them to determine the volume of rectangular prisms. About the Math Students fill a rectangular prism with centimeter cubes, counting both the number of cubes in each layer and the Professional Development number of layers. Students are slowly building towards the volume formula, one small step at a time. They discover that the volume of a rectangular prism can be found by multiplying the length and width of the first (or base) layer by the number of layers, which is the prism s height. Students also make the transition from unit cubes to cubic units of real measurements. They understand the unit size may vary and volume can be measured in different-sized cubic units. Professional Development Videos 669A Chapter 11 Interactive Student Edition Personal Math Trainer Math on the Spot Video HMH Mega Math

2 Daily Routines Common Core Problem of the Day 11.6 Bruce builds the solid figure below using unit cubes. How many unit cubes does he use? 8 unit cubes Vocabulary cubic unit, volume Interactive Student Edition Multimedia Glossary e Vocabulary Builder Materials Word Definition Map (see eteacher Resources) Word Definition Map Have students fill in a word definition map for volume. Encourage students to write the definition in their own words and to use an illustration with their definition. For What is it like? students can describe how to find volume by filling a rectangular prism with cubes. 1 ENGAGE with the Interactive Student Edition Essential Question How can you use unit cubes to find the volume of a rectangular prism? Making Connections Invite students to tell you what they know about volume. What does the volume of an object tell you? the amount of space that the object takes up What information helps you find the volume? length, width, and height Learning Activity What is the problem the students are trying to solve? Connect the story to the problem. What is the volume of each crate? one cubic yard How many layers of crates are there? How many crates are in each layer? 6 What is the problem asking you to find? the volume of all the crates Literacy and Mathematics Choose one or more of the following activities. Have students describe in their own words what it means to talk about the volume of an object. Have students look at the illustration of the bundle of crates and write a short description of the shapes they see. How can you use unit cubes to find the volume of a rectangular prism? 669B

3 LESSON 11.6 EXPLORE Investigate MATHEMATICAL PRACTICES 669 Chapter 11 Hands MP5 Use appropriate tools strategically. Discuss the meaning of volume, and then work though the Investigation with students. Be sure students understand that the unit cubes need to fit together without any gaps. What is the volume of a rectangular prism built with unit cubes? cubic units In Step B, how can you figure out how many centimeter cubes are in the first layer? Possible answers: Count them all, use repeated addition, or multiply the length by the width: 5 5 = 5. In Step C, how can you figure out how many centimeter cubes fill the prism? Possible answers: Use repeated addition to add the number of centimeter cubes in each layer; = 15; or multiply the number of centimeter cubes used in each layer, 5, by the number of layers, 5. So, 5 5 = 15. Why do you give the answer in cubic centimeters? Possible answer: Because I am using centimeter cubes to measure the volume of the prism, and each centimeter cube has a volume of 1 cubic centimeter. MP6 Attend to precision. What special prism do the cubes in your model make? How do you know? They make up a cube. I know because the shape is long, wide, and tall. ELL Strategy: Develop Meanings On Volume can mean how loud a sound is. Volume can also mean the number of unit cubes needed to fill a space. Volume is measured in cubic units. Build a rectangular prism that is 4 units long, units wide, and 3 units high. To find the volume, we multiply the number of unit cubes in the base by the number of layers. How many unit cubes are in the base? 4 = 8 How many layers are in this prism? 3 How can we find its volume? Multiply the number of unit cubes in the base by the number of layers: 8 3 = 4 Have partners tell each other how to find the volume of a rectangular prism. Progress to Algebra 5.MD.C.3b Recognize volume as an attribute of solid figures and understand concepts of volume measurement. DO A solid NOT EDIT--Changes figure which can must be packed be made without through gaps File or info overlaps using n unit cubes is said to have a volume of n cubic units. CorrectionKey=A 5.MD.C.4 Measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and improvised units. DO NOT EDIT--Changes must be made through "File info" CorrectionKey=A Name Understand Volume The volume of a rectangular prism is equal to the number of unit cubes that make up the prism. Each unit cube has a volume of 1 cubic unit. Find the volume of the prism. 1 unit cube 5 1 cubic inch Step 1 Count the number of unit cubes in the bottom layer of the prism. There are There are Name Understand Volume Essential Question How can you use unit cubes to find the volume of a rectangular prism? connect You can find the volume of a rectangular prism by counting unit cubes. Volume is the measure of the amount of space a solid figure occupies and is measured in cubic units. Each unit cube has a volume of 1 cubic unit _MNLESE34415_C11L06.indd unit cubes that make up the length of the first layer. unit cubes that make up the width of the first layer. There is 1 unit cube that makes up the height of the first layer. So, altogether, there are 8 unit cubes that make up the bottom layer of the prism. Step Count the number of layers of cubes that make up the prism. The prism is made up of 3 layers of unit cubes. Step 3 Find the total number of cubes that fill the prism. Multiply the number of layers by the number of cubes in each layer unit cubes Each unit cube has a volume of 1 cubic inch. So, the volume of the prism is 4 3 1, or 4 cubic inches. Use the unit given. Find the volume. 1. Volume 5 Chapter Resources Investigate cu. 6 cm Each cube 5 1 cu cm Differentiated DO NOT EDIT--Changes Enrich must be made through 11.6 "File info" 1 CorrectionKey=A Instruction Reteach cm 3 cm 4 cm Volume 5 cu Reteach The rectangular prism above is made up of _ 8 unit cubes and has a volume of _ 8 cubic units. Materials rectangular prism net A centimeter cubes A. Cut out, fold, and tape the net to form a rectangular prism. B. Use centimeter cubes to fill the base of the rectangular prism without gaps or overlaps. Each centimeter cube has a length, width, and height of 1 centimeter and a volume of 1 cubic centimeter. How many centimeter cubes make up the length of the first layer? the width? the height? length: _ 5 width: _ 5 height: _ 1 How many centimeter cubes are used to fill the base? _ 5 C. Continue filling the rectangular prism, layer by layer. Count the number of centimeter cubes used for each layer. How many centimeter cubes are in each layer? _ 5 Name Volume of Irregular Figures Use the unit given. Find the volume Hands How many layers of cubes fill the rectangular prism? _ 5 How many centimeter cubes fill the prism? _ 15 So, the volume of the rectangular prism is _ 15 cubic centimeters. Reteach 11.6 On Each cube 5 1 cu cm Volume 5 84 cu cm Volume cu ft. 4. Enrich Volume 5 68 cu in. Each cube 5 1 cu yd Volume 5 7 cu yd 5. Explain how you found the volume of the figure in Exercise 4. Possible explanation: I found the volume of a prism that is 8 units long, 3 units wide, and 4 units high. From that volume, I subtracted the volume of a figure that is 4 units long, 3 units wide, and units high. The subtracted volume represents the hole in the prism. Chapter Resources Measurement and Data 5.MD.C.3b, 5.MD.C.4 MATHEMATICAL PRACTICES MP3, MP5, MP6 Chapter Enrich /0/14 :33 AM 5_MNLEAN34390_C11R06.indd 15 5/0/14 1:59 AM 5_MNLEAN34390_C11E06.indd 16 5/0/14 1:58 AM

4 1. Describe the relationship among the number of centimeter cubes you used to fill each layer, the number of layers, and the volume of the prism.. Find the volume of the prism if each cube represents 1 cu cm, 1 cu in., and 1 cu ft. 670 Draw Conclusions Possible description: The number of centimeter cubes in each layer, 5, multiplied by the number of layers, 5, is equal to the total number of centimeter cubes used to fill the prism, 15. So, its volume is 15 cubic centimeters. MATHEMATICAL PRACTICE 3 Apply If you had a rectangular prism that had a length of 3 units, a width of 4 units, and a height of units, how many unit cubes would you need for each layer? How many unit cubes would you need to fill the rectangular prism? You would need 1 unit cubes for each layer; 4 unit cubes to fill the rectangular prism. Make Connections To find the volume of three-dimensional figures, you measure in three directions. For a rectangular prism, you measure its length, width, and height. Volume is measured using cubic units, such as cu cm, cu in., or cu ft. Which has a greater volume, 1 cu cm or 1 cu in.? Explain. 1 cu in.; Possible explanation: 1 inch is greater than 1 centimeter, so 1 cubic inch would have a greater volume. MATHEMATICAL PRACTICE 3 units 6 units units _ 36 cu cm _ 36 cu in. _ 36 cu ft 1 cu cm 6 Would the prism above be the same size if it were built with centimeter cubes, inch cubes, or foot cubes? Explain. 1 cu in. No; Possible explanation: The prisms would have the same number of cubes and be the same shape. Feet are greater units than inches or centimeters, so the prism made with foot cubes would be greater than the prisms made with inch or centimeter cubes. Draw Conclusions MP3 Construct viable arguments and critique the reasoning of others. Help students make the connection that the volume can be found by multiplying the number of cubes in a layer by the number of cubes in the height. How can you figure out the number of unit cubes in a layer without adding? Multiply the number of unit cubes in a row, or the length, by the number of rows, or the number of unit cubes in the width. In Exercise 1, how can you figure out the number of unit cubes needed to fill the rectangular prism without adding? Possible answer: Multiply the number of cubes in each layer by the number of layers, since each layer has the same number of unit cubes. Make Connections In rectangular prisms built with the same number of cubes, students make the connection that if the unit of one prism is greater than the other, it means the volume is also greater. How can you compare the volumes of rectangular prisms built with the same number of unit cubes, but the unit cubes are of different measures? Possible answer: The rectangular prisms composed of larger unit cubes will have greater volumes. MP6 Attend to precision. Why is it important to give the measurement unit when stating a measurement? Possible answer: Giving the name of the unit indicates the size of the object. Advanced Learners Materials index cards, centimeter cubes Logical / Mathematical Individual / Partners Have students write a volume problem such as the following on an index card and its solution on the back: Deirdre builds a rectangular prism with 75 centimeter cubes. The base of her prism has 15 centimeter cubes. What is the height of her prism? 5 centimeters Have students exchange and solve each other s problems, checking their answers on the back of the index card. COMMON ERRORS Error Students may incorrectly find the volume as the measure of one layer of the rectangular prism. Example Volume = 18 cu ft Springboard to Learning Remind students that the number of cubes in each layer times the number of layers is the total number of cubes in the rectangular prism, or its volume. 670

5 DO NOT EDIT--Changes must be made through File info CorrectionKey=A 3 EXPLAIN Name Share and Show MATH BOARD Use the unit given. Find the volume. MATH Share and Show MATH BOARD The first problems connect 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 the Quick Check on the MathBoard cm Each cube 5 1 cu cm 4 cm 3 cm Volume =_ 48 cu _ cm. 4 in. in. 3 in. Volume =_ 4 cu _ in in. Quick Check 3 1 Rt I ft 6 ft Volume = _ 36 cu _ ft 4 in. 5 in. Volume = _ 60 cu _ in. If Then a student misses the checked exercises Differentiate Instruction with Reteach 11.6 Personal Math Trainer 5.MD.C.3b, 5.MD.C.4 RtI Tier 1 Activity (online) Compare the volumes. Write <, >, or =. 5. in. in. in. 4 in. 8 in. 4 in. _ 3 cu in. = _ 3 cu in ft 8 ft 9 ft _ 108 cu ft > _ 80 cu ft ft Chapter 11 Lesson _MNLESE34415_C11L06.indd 671 3/13/14 1:59 PM PROBLEM TYPE SITUATIONS Multiplication and Division Equal Groups Unknown Product Exercises: 7, 9 Equal Groups Number of Groups Unknown Exercise: Chapter 11

6 MATHEMATICAL PRACTICES M 7. MATHEMATICAL PRACTICE 3 Verify the Reasoning of Others Gerardo says that a cube with edges that measure 10 centimeters has a volume that is twice as much as a cube with sides that measure 5 centimeters. Explain and correct Gerardo s error. A cube with sides of holds 5 layers of 5 centimeter cubes, which is 15 cu cm. A cube with sides of 10 cm could hold 10 layers of 100 centimeter cubes, which is 1,000 cu cm. Possible explanation: Although 10 is twice as much as 5, a cube has 3 dimensions, so the volume of the larger cube will be or or 8 times as great. 8. SMARTER Pia built a rectangular prism with cubes. The base of her prism has 1 centimeter cubes. If the prism was built with 108 centimeter cubes, what is the height of her prism? The height of her prism is 9 centimeters. 9. DEEPER A packing company makes boxes with edges each measuring 3 feet. What is the volume of the boxes? If 10 boxes are put in a larger, rectangular shipping container and completely fill it with no gaps or overlaps, what is the volume of the shipping container? 7 cu ft; 70 cu ft WRITE Math Show Your Work 4 ELABORATE Problem Solving Applications MATHEMATICAL PRACTICES SMARTER Problem 8 requires students to determine the height of the rectangular prism given the volume of one layer and the total volume. 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 SMARTER Carlton used 1-centimeter cubes to build the rectangular prism shown. 67 Find the volume of the rectangular prism Carlton built. _ 60 cubic centimeters 4 cm 3 cm SMARTER In this multistep problem, students must be able to use unit cubes to find the volume of a rectangular prism. For students who have difficulty seeing how to find the volume of the prism, ask them how many unit cubes are needed to build each level of the prism. This represents the area of the base. Then guide students to multiply the area of the base by the number of levels to find the volume of the prism. Students will apply this concept later in the chapter when using the formula for the volume of a rectangular prism. DIFFERENTIATED INSTRUCTION Activities What s in the Box? Students complete blue Activity Card 1 by finding volume using unit cubes. Differentiated Centers Kit INDEPENDENT ACTIVITIES Literature City of the Future Students read about the solid figures a class uses to construct a model of a city. 5 EVALUATE Formative Assessment Essential Question Using the Language Objective Reflect Have students work in teams to analyze and discuss to answer the essential question. How can you use unit cubes to find the volume of a rectangular prism? Possible answer: I can find the number of unit cubes that it takes to fill the base of the rectangular prism without any gaps or overlaps, and then multiply that number by the number of layers that make up its height. Then I can write the volume in cubic units. Math Journal WRITE Math Explain how to find the volume of a rectangular prism in cubic inches that is 4 inches long, 3 inches wide, and inches high. Include a drawing in your solution. 67

7 Name Understand Volume 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 unit given. Find the volume cm 7 cm Each cube = 1 cu cm Volume = _ 105 cu _ cm 3. ft 4 ft 7 ft Each cube = 1 cu ft in. Each cube = 1 cu in. Each cube = 1 cu cm 3 in. in. Volume = _ 48 cu _ in. COMMON CORE STANDARD 5.MD.C.3b 5.MD.C.4 Geometric measurement: understand concepts of volume and relate volume to multiplication and to addition. Volume = _ 56 cu _ ft Volume = _ 15 cu _ cm 5. Compare the volumes. Write <, >, or =. 4 ft ft Each cube = 1 cu ft _ 60 cu ft 5 _ 60 cu ft 6 ft Each cube = 1 cu ft Problem Solving 6. A manufacturer ships its product in boxes with edges of 4 inches. If 1 boxes are put in a carton and completely fill the carton, what is the volume of the carton? 7. Matt and Mindy each built a rectangular prism that has a length of 5 units, a width of units, and a height of 4 units. Matt used cubes that are 1 cm on each side. Mindy used cubes that are 1 in. on each side. What is the volume of each prism? 768 cubic inches Matt: 40 cu cm, Mindy: 40 cu in. Chapter PROFESSIONAL DEVELOPMENT Mathematical Practices in Your Classroom CCSS.Math.Practice.MP5 Use appropriate tools strategically. The hands-on approach used in this lesson with centimeter cubes can help students visualize what volume means. They identify how many centimeter cubes are needed to fill the base and then how many layers of cubes are needed to fill the rectangular prism. By adding the number of cubes in each layer or by multiplying the number of cubes in the base by the number of layers, students can find the total number of cubes, or the volume of a rectangular prism. Ask questions such as the following to help students understand that the total number of cubes filling the rectangular prism represents its volume: Why does putting unit cubes into a rectangular prism help find its volume? Possible answer: We are finding the amount of space a solid figure occupies using unit cubes as a measurement unit. The total number of unit cubes needed to fill the rectangular prism is the amount of space it occupies. Why is it necessary to find the height of a rectangular prism in unit cubes as well as the number of unit cubes in a layer? Possible answer: A rectangular prism has three dimensions. So, while the number of cubes in the base may tell you two dimensions, what the area of the base is, you need the height of the rectangular prism, or the number of layers, to find the number of cubes that make up all three dimensions, or the volume. 673 Chapter 11

8 Lesson Check (5.MD.C.3b, 5.MD.C.4) 1. Elena packed 48 cubes into this box. Each cube has edges that are 1 centimeter. How many layers of cubes did Elena make?. What is the volume of the rectangular prism? 6 ft 8 ft Each cube = 1 cu ft 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. 3 layers 40 cubic feet Spiral Review (5.MD.A.1, 5.G.A.1, 5.G.B.3, 5.G.B.4) 3. Juan made a design with polygons. Which polygon in Juan s design is a pentagon? 4. What ordered pair describes the location of point P? A B C D y P x Figure B (3, 4) 5. What is the least number of acute angles that a triangle can have? Karen bought 3 pounds of cheese to serve at a picnic. How many ounces of cheese did Karen buy? 48 ounces FOR MORE PRACTICE GO TO THE Personal Math Trainer 674