ENGAGE. Daily Routines Common Core. Essential Question How can you find the volume of rectangular prisms with fractional edge lengths?

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1 LESSON.6 Algebra Volume of Rectangular Prisms FOCUS COHERENCE RIGOR LESSON AT A GLANCE F C R Focus: Common Core State Standards Learning Objective 6.G.A. Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V = l w h and V = b h to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems. Use formulas to find the volume of rectangular prisms with fractional edge lengths. Also 6.EE.A.c Student pairs discuss and present to another pair how you can find the volume of rectangular prisms with fractional edge lengths. MATHEMATICAL PRACTICES (See ematical Practices in GO! in the Planning Guide for full text.) MP Reason abstractly and quantitatively. MP5 Use appropriate tools strategically. MP6 Attend to precision. F C R Coherence: Language Objective Materials Board Standards Across the Grades Before Grade 6 Aer 5.MD.C.5 6.G.A. 7.G.B.6 F C R Rigor: Level : Understand Concepts...Share and Show ( Checked Items) Level : Procedural Skills and Fluency...On Your Own, Practice and Homework Level : Applications...Think Smarter and Go Deeper F C R For more about how GO! fosters Coherence within the Content Standards and ematical Progressions for this chapter, see page 595J. About the Professional Development If Students Ask Having previously learned about capacity, students might ask about the difference between volume and capacity. Explain that volume usually refers to the amount of space an object occupies, while capacity usually refers to the amount of a substance a container can hold. Units of volume are always cubic units, such as cubic inches (). Note that every unit of volume is based on a unit of length, such as inch or centimeter. Units of capacity are not based on units of length; examples are cup (c) and milliliter (ml). However, volume and capacity are really measures of the same physical attribute threedimensional size as evidenced by these equivalencies in the metric system: cm = ml and,000 cm = L. So, it is mathematically correct to use units of volume for capacity. Professional Development Videos 69A Chapter Interactive Student Edition Personal Trainer on the Spot Video Animated Models HMH Mega

2 Daily Routines Common Core Problem of the Day.6 Alex drew this diagram of a bench he is planning to build. What is the area of the shaded region? 8 Interactive Student Edition Multimedia Glossary e Vocabulary Builder 0 Bubble Map Use a bubble map to review terms associated with volume. Have students draw a bubble map with the word volume in the center. In the surrounding bubbles, include units used in measures of volume as well as figures that have volume. ENGAGE with the Interactive Student Edition Essential Question How can you find the volume of rectangular prisms with fractional edge lengths? Making Connections Have students tell what they know about volume. What does volume measure? the space contained in a three-dimensional object What type of units do you use to measure volume? cubic units Learning Activity Tell students: A box is in the shape of a rectangular prism with a length of 6 units, a width of units, and a height of units. The box is filled with wooden blocks that are cubes. If each wooden block is one cubic unit, how could you use the blocks to find the volume of the box? Count the cubes that fill the box. What is the volume of the box? 7 cubic units Literacy and ematics Have each student write a word problem in which unit cubes are used to measure the volume of a rectangular prism whose dimensions are given in units. Have student pairs exchange problems. cm volume rectangular prism How can you find the volume of rectangular prisms with fractional edge lengths? cubic units cube km Lesson.6 69B

3 LESSON.6 EXPLORE Unlock the Problem MATHEMATICAL PRACTICES To introduce the lesson, have students watch the Real World video. Remind students that they can use the formula V = l w h to find the volume of a rectangular prism regardless of whether the dimensions are whole numbers, mixed numbers, or fractions. When students get to Step, ask: MP Reason abstractly and quantitatively. Would a 5 by by _ rectangular prism have the same volume as a by _ by 5 rectangular prism? Expla Yes; to find the volume, you multiply the length times width times height. Since you can multiply factors in any order without changing the product, the volumes of the prisms are the same. MP Construct viable arguments and critique the reasoning of others. Is it possible for two solid figures to have the same surface areas, but different volumes? Expla Talk Use what students already know about appropriate units of linear measure to discuss appropriate units for measuring area. ELL Strategy: Rephrase Draw a rectangular prism and label these measurements: B = 6 ; h = _ Have students use a sentence frame to explain what it means: The volume is the measure of the number of unit cubes takes to fill the prism. Explain how to find the volume of this prism, using V = Bh. Have pairs rephrase the explanation using sentence frames as needed: The formula means volume = area of one base height. First, we changed the mixed _ number ½ into a fraction:. Then we multiplied the area of the base, _ 6, by the height,. The volume is 9 cubic inches. 69 Chapter it 6.G.A. Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V = l w h and V = b h to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems. Image Credits: (t) Studio Eye/Corbis Name Algebra Volume of Rectangular Prisms You can find the volume of a prism by using the formula V 5 Bh. V stands for volume, B stands for the area of the base, and h stands for the height. For a rectangular prism, any face can be the base, since all faces are rectangles. _ Find the volume of the rectangular prism. Step Because the length, width, and height are all equal, you can use a special formula. Find the volume... Name Volume of Rectangular Prisms Essential Question How can you find the volume of rectangular prisms with fractional edge lengths? You can use the formula V = l w h to find the volume of a rectangular prism when you know the length, width, and height of the prism. V 5 Bh 5 l w h V 5 s _ cm 5 m _ cm m m _ cm _.. _ _ Step Substitute _ 6 5 for s. V 5 s 5 ( _ ) 5 ( 5_ ) V 5 5_ yd 5_ 5_ 5 58 yd yd 5_ 0 Differentiated Enrich.6 Instruction yd Step Find the area of the base. Step Multiply the area of the base by The base is _ by _. the height. A 5 l w V 5 Bh _ A 5 _ 5 8 _ V _ So, the volume of the rectangular prism is. Find the volume of the cube. So, the volume of the cube is 5 5_ 8 yd. 50 m _ cm Chapter Resources Unlock the Problem An obento is a single-portion meal that is common in Japan. The meal is usually served in a box. A small obento box is a rectangular prism that is 5 inches long, inches wide, and _ inches high. How much food fits in the box? Find the volume of a rectangular prism You can use the formula V = l w h to find the volume of a rectangular prism when you know the length, width, and height of the prism. STEP Sketch the rectangular prism. _ 8 Lesson.6 Reteach 5 yd 5 8 yd _ 6,000 _ yd yd -5 Reteach STEP Identify the value for each variable. The length l is 5 The width w is _ The height h is STEP Evaluate the formula. Write the formula. V = l w h Replace l with 5, w with V = _ 5 _, and h with _. V = _ 50 Multiply. So, _ 50 of food fits in the box. Reteach.6 Name 5 Aquarium Volumes on Display A pet store is selling aquariums. You have been asked to find the volumes of the aquariums in cubic inches. Aer finding the volumes, answer the questions that follow. 5-Gallon Aquarium 5-Gallon Aquarium 8,7 $5 Volume: 55-Gallon Aquarium $85,5 Volume:. A customer says he has a rectangular aquarium with a length of 7, a width of 9, and a height of 9 What is the volume of the aquarium?, Lesson.6 Enrich $ ,5 Volume: 00-Gallon Aquarium 8.8 7,89.6 Volume: 0.75 $75. The gallon sizes above are approximate. To which size is the customer s aquarium closest? 55-gallon. Stretch Your Thinking What is the volume of the 5-gallon aquarium in cubic feet? Round to the nearest tenth..6. Which of the four aquariums has the best value, based on the advertised gallon sizes? Explain your answer. The 55-gallon aquarium has the best value because it has the lowest unit cost at $.55 per gallon. Chapter Resources Talk ALGEBRA Lesson.6 Geometry 6.G.A. Also 6.EE.A.c MATHEMATICAL PRACTICES MP, MP, MP, MP Underline the sentence that tells you what you are trying to find. Circle the numbers you need to use. Problem Types: Equal Groups Unknown Product, Area Unknown Product The side lengths are given in inches, so the volume is given in cubic inches. MATHEMATICAL PRACTICES Reasoning How do you know what units to use for the volume of the box? Chapter 69-6 Enrich

4 connect You know that the volume of a rectangular prism is the product of its length, width, and height. Since the product of the length and width is the area of one base, the volume is also the product of the area of one base and the height. Example Find the volume of the prism. STEP Identify the value for each variable. The height h is The area of the base B is _ 9 STEP Evaluate the formula. Write the formula. Volume of a Prism Volume = area of one base height Replace B with _ 9 _ and h with _. 9 Write the mixed number as a fraction greater than. Multiply and write the product as a mixed number. 0 _ So, the volume of the prism is. V = Bh V = Bh V = V = 9 9 V = 8 = 0 9 CONNECT Help students understand that a rectangular prism has three pairs of parallel, congruent, rectangular faces, and any of these pairs can be considered the bases. Students should also understand that this is not true for other types of prisms, such as triangular prisms. Example MP Model with mathematics. Explain to students that when working with a prism, the area of a base is represented by B. Explain that it is important to use uppercase B for the area of a base and lowercase b for the length of a base of a parallelogram or a triangle. Example Point out that a cube is a special case of a rectangular prism, and there are three expressions for its volume: V= Bh = lwh = s. Write the formula. The area of the square base is s. The height of a cube is also s, so V = Bh = s. _ Substitute _ for s. Write the mixed number as a fraction greater than. Then use repeated multiplication. Simplify. Example Find the volume of the cube. 7_ So, the volume of the cube is. 8 V = s V = ( ) V = 7 ( ) = ( ) ( ) ( ) V = _ = Advanced Learners Logical / ematical Individual / Partners Have students solve the following problems. Find possible dimensions of a rectangular prism that has a volume of 85, with at least one fractional edge length, and draw its net. Possible dimensions: length 5 _, width 7, height 0 Check students nets. Find the dimensions and surface area of a cube that has a volume of _ 8. Draw a net of the cube. Every edge is _ long; the surface area is _. Check students nets. Determine a volume for a rectangular prism and give it to a partner to determine possible dimensions of the prism. Check students work Lesson.6 60

5 If Then EXPLAIN MATH Share and Show MATH BOARD The first problem connects to the learning model. Have students use the Board to explain their thinking. Quick Check a student misses the checked exercises Rt I Differentiate Instruction with Reteach.6 Personal Trainer 6.G.A. RtI Tier Activity (online) Name Share and Show Find the volume. 5 0 On Your Own MATH BOARD. V = lwh. 5 0 _ 5 5 Find the volume of the prism. 8 6 V = _ _ _,97 V = 8 Find the cube of the length of an edge. 8 8 Talk _ _ 5,096 yd 7 5 MATHEMATICAL PRACTICES Describe the steps for finding the volume of a cube. 6 yd 0 yd yd On Your Own Talk Use Talk to check students understanding of methods they can use to find the volume of a cube. What are some different ways you can find the volume of a cube? For Exercise 5, help students to understand that they have been given an extra measurement that is not needed to solve the problem. They can multiply the area of a base (the right face of the prism) by 6 yd to find the volume. The measurement of yd is not needed. 6. DEEPER Wayne s gym locker is a rectangular prism with a width and height of _ inches. The length is 8 inches greater than the width. What is the volume of the locker?,70 5_ 8 7. SMARTER Abraham has a toy box that is in the shape of a rectangular prism. The volume is _. 5 _. 6 _. feet feet feet Chapter Lesson 6 6 DEEPER For Exercise 6, ask: Which dimension of the locker is not given in the problem? How can you find it? SMARTER Exercise 7 assesses a student s ability to find the volume of a rectangular prism with fractional edge lengths. Students who incorrectly answer this item may have difficulty multiplying fractions. It is important for students to make the connection that the volume of a rectangular prism is the product of its length, width, and height. Since the product of the length and width is the area of one base, the volume is also the product of the area of one base and the height. PROBLEM TYPE SITUATIONS Addition and Subtraction Compare Larger Unknown Exercise: 6 Multiplication and Division Area Unknown Factor Exercise: 8 Area Unknown Product Exercises: 6, 7, 9 Equal Groups Unknown Product Exercises: 6, 7, 9 Equal Groups Number of Groups Unknown Exercise: 9 6 Chapter

6 Aquariums Large public aquariums like the Tennessee Aquarium in Chattanooga have a wide variety of freshwater and saltwater fish species from around the world. The fish are kept in tanks of various sizes. The table shows information about several tanks in the aquarium. Each tank is a rectangular prism. Find the length of Tank. V = lwh 5,500 = l _ 0 _ 5 5,500 = l,050 5,500,050 = l _ 50 = l So, the length of Tank is. 50 cm Solve. Aquarium Tanks Length Width Height Volume Tank 50 cm 0 cm 5 cm 5,500 cm Tank m 8 m m 8 m Tank 8 m m 0 m,60 m Tank 7 cm 55 cm 0 cm 58,00 cm ELABORATE Connect to Science MATHEMATICAL PRACTICES SMARTER Exercise 9 requires students to use higher order thinking skills. Students must find the volume of the tank. Then they use the equivalencies given to find the capacity of the tank in liters, and finally, they use the ratio of 9 L of water to every fish to find the number of fish the tank can support. 8. Find the width of Tank and the height of Tank. WRITE 9. SMARTER To keep the fish healthy, there should be the correct ratio of water to fish in the tank. One recommended ratio is 9 L of water for every fish. Find the volume of Tank. Then use the equivalencies cm = ml and,000 ml = L to find how many fish can be safely kept in Tank m; 0 m 58,00 cm ; 5 fish MATHEMATICAL PRACTICE Use Reasoning Give another set of dimensions for a tank that would have the same volume as Tank. Explain how you found your answer. 8 m by 8 m by 6 m; Possible explanation: I found three numbers that when multiplied have a product of 8. Show Your Work Image Credits: (t) Comstock/Corbis on the Spot Video Tutor Use this video to help students model and solve this type of Think Smarter problem. on the Spot videos are in the Interactive Student Edition and at MP Reason abstractly and quantitatively. For Exercise 0, ask students to describe how they can find the volume of a rectangular prism. Ask: Are there other numbers whose product will be the same? 6 5 EVALUATE Formative Assessment DIFFERENTIATED INSTRUCTION Activities Volume of a Prism Differentiated Centers Kit Literature More Than a Guess INDEPENDENT ACTIVITIES Games What s My Volume? Essential Question Using the Language Objective Reflect Have student pairs discuss and present to another pair to answer the essential question. How can you find the volume of rectangular prisms with fractional edge lengths? Use the formula V = Bh, where B is the area of the base and h is the height, or the formula V = lwh, where l is the length, w is the width, and h is the height. Students complete online purple Activity Card 7 by choosing the possible dimensions of a prism when given the volume. Students read about finding the volume of a jar to guess how many marbles are filling the jar. Students practice finding the volume of threedimensional figures. Journal WRITE Explain how you can find the side length of a rectangular prism if you are given the volume and the two other measurements. Does this process change if one of the measurements includes a fraction? Lesson.6 6

7 Name Volume of Rectangular Prisms Practice and Homework Lesson.6 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. Find the volume.. V = lwh. 5 m 9 m m 7 V = 5 _ 9 _ V = m 6 5 m 8 m _ f t 6 m. 5. mm mm mm 7 _ i n. 9 _ 8 m m. 6. COMMON CORE STANDARD 6.G.A. Solve real-world and mathematical problems involving area, surface area, and volume. 6 0 _ 8 f t Problem Solving 7. A cereal box is a rectangular prism that is 8 inches long and _ inches wide. The volume of the box is 00 in.. What is the height of the box? 8. A stack of paper is 8 _ long by wide by high. What is the volume of the stack of paper? WRITE Explain how you can find the side length of a rectangular prism if you are given the volume and the two other measurements. Does this process change if one of the measurements includes a fraction? Check students work. Chapter 6 Cross-Curricular Worm composting is a great way to recycle food waste. To create a compost bin, many people use a box shaped like a rectangular prism to hold the soil, worms, and food waste. It is recommended that the container be at least 8 inches deep and have one square foot of area at the top of the box for each pound of food waste per week. If Jonathan s family produces 6 pounds of food waste per week, what is a possible volume for their compost container if the container is 8 inches, or, deep? 6 or SOCIAL STUDIES Boxes of cereal have been sold in the U.S. since 906. A package designer shows rectangular boxes of various dimensions to a sample of customers and finds that a box 6 cm wide and 5 cm tall is the most appealing. She needs the volume of the box to be,00 cm. What should be the thickness of the box? 5.5 cm A grocer gives a supplier shelf space that is 5 cm high, cm wide, and 0 cm deep. The supplier wants to fit as many,50- cm cereal boxes on the shelf as possible. The boxes are 0 cm high and 5 cm thick (deep). Assume that each box must be positioned with its front label upright and facing the customer. How many boxes can the supplier fit on the shelf? 6 Chapter

8 Lesson Check (6.G.A.). A kitchen sink is a rectangular prism with a length of 9 7_ 8 inches, a width of _ inches, and height of 0 inches. Estimate the volume of the sink.,000 cubic inches TEST PREP. A storage container is a rectangular prism that is 65 centimeters long and 0 centimeters wide. The volume of the container is 6,00 cubic centimeters. What is the height of the container? centimeters 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.G.A., 6.G.A., 6.G.A.). Carrie started at the southeast corner of Franklin Park, walked north 0 yards, turned and walked west 80 yards, and then turned and walked diagonally back to where she started. What is the area of the triangle enclosed by the path she walked?. The dimensions of a rectangular garage are 00 times the dimensions of a floor plan of the garage. The area of the floor plan is 8 square inches. What is the area of the garage? 9,600 square yards 80,000 square inches 5. Shiloh wants to create a paper-mâché box shaped like a rectangular prism. If the box will be inches by 5 inches by 8 inches, how much paper does she need to cover the box? 6 8 square inches 6. A box is filled with 0 cubes with a side length of _ unit. What is the volume of the box in cubic units? 7 _ cubic units FOR MORE PRACTICE GO TO THE Personal Trainer Lesson.6 6