Lesson 11: Volume with Fractional Edge Lengths and
Lesson 11 NYS COMMON CORE MATHEMATICS CURRICULUM 6β’5 Lesson 11: Volume with Fractional Edge Lengths and Unit Cubes Student Outcomes ο§ Students extend their understanding of the volume of a right rectangular prism with integer side lengths to right rectangular prisms with fractional side lengths. They apply the formula π = π β π€ β β to find the volume of a right rectangular prism and use the correct volume units when writing the answer. Lesson Notes This lesson builds on the work done in Grade 5, Module 5, Topics A and B. Within these topics, students determine the volume of rectangular prisms with side lengths that are whole numbers. Students fill prisms with unit cubes in addition to using the formulas π = πβ and π = π β π€ β β to determine the volume. Scaffolding: Students start their work on volume of prisms with fractional lengths so that they can continue to build an understanding of the units of volume. In addition, they must continue to build the connection between packing and filling. In the following lessons, students move from packing the prisms to using the formula. One way to do this would be to have students make a conjecture about how many cubes will fill the prism, and then use the cubes to test their ideas. The sample activity provided at the end of the lesson will foster an understanding of volume, especially in students not previously exposed to the Common Core standards. Classwork Fluency Exercise (5 minutes): Multiplication of Fractions II Sprint: Refer to the Sprints and the Sprint Delivery Script sections in the Module Overview for directions to administer a Sprint. Use unit cubes to help students visualize the problems in this lesson. Provide different examples of volume (electronic devices, loudness of voice), and explain that although this is the same word, the context of volume in this lesson refers to threedimensional figures. Opening Exercise (3 minutes) Please note that although scaffolding questions are provided, this Opening Exercise is an excellent chance to let students work on their own, persevering and making sense of the problem. Opening Exercise Which prism will hold more π π’π§. × π π’π§. × π π’π§. cubes? How many more cubes will the prism hold? MP.1 π π’π§. ππ π’π§. π π’π§. ππ π’π§. π π’π§. π π’π§. Lesson 11: Date: Volume with Fractional Edge Lengths and Unit Cubes 2/5/15 © 2014 Common Core, Inc. Some rights reserved. commoncore.org This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 154 Lesson 11 NYS COMMON CORE MATHEMATICS CURRICULUM 6β’5 Students discuss their solutions with a partner. ο§ How many 1 in. × 1 in. × 1 in. cubes will fit across the bottom of the first rectangular prism? οΊ ο§ How did you determine this number? οΊ ο§ ο§ ο§ Both rectangular prisms hold the same number of cubes in one layer, but the second rectangular prism has more layers. How many more layers does the second rectangular prism hold? οΊ ο§ The second rectangular prism will hold more cubes. How did you determine this? οΊ ο§ I will need 12 layers because the prism is 12 in. tall. Which rectangular prism will hold more cubes? οΊ ο§ 40 cubes will fit across the bottom. How many layers will you need? οΊ MP.1 There are 6 inches in the height; therefore, 6 layers of cubes will fit inside. How many 1 in. × 1 in. × 1 in. cubes will fit across the bottom of the second rectangular prism? οΊ ο§ Answers will vary. Students may determine how many cubes will fill the bottom layer of the prism and then decide how many layers are needed. Students who are English language learners may need a model of what βlayersβ means in this context. How many layers of 1 in. × 1 in. × 1 in. cubes will fit inside the rectangular prism? οΊ ο§ 40 cubes will fit across the bottom. It holds 6 more layers. How many more cubes does the second rectangular prism hold? οΊ The second rectangular prism has 6 more layers than the first, with 40 cubes in each layer. οΊ 6 × 40 = 240 more cubes. What other ways can you determine the volume of a rectangular prism? οΊ We can also use the formula π = π β π€ β β. Example 1 (5 minutes) Example 1 A box with the same dimensions as the prism in the Opening Exercise will be used to ship miniature dice whose side lengths have been cut in half. The dice are π π π’π§. × π π π’π§. × π π π’π§. cubes. How many dice of this size can fit in the box? Scaffolding: Students may need a considerable amount of time to make sense of cubes with fractional side lengths. π π’π§. π π’π§. An additional exercise has been included at the end of this lesson to use if needed. ππ π’π§. Lesson 11: Date: Volume with Fractional Edge Lengths and Unit Cubes 2/5/15 © 2014 Common Core, Inc. Some rights reserved. commoncore.org This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 155 Lesson 11 NYS COMMON CORE MATHEMATICS CURRICULUM ο§ How many cubes could we fit across the length? The width? The height? οΊ ο§ 6β’5 Two cubes would fit across a 1-inch length. So, I would need to double the lengths to get the number of cubes. Twenty cubes will fit across the 10-inch length, 8 cubes will fit across the 4-inch width, and 12 cubes will fit across the 6-inch height. How can you use this information to determine the number of 1 2 in. × 1 2 in. × 1 2 in. cubes it will take to fill the box? ο§ οΊ I can multiply the number of cubes in the length, width, and height. οΊ 20 × 8 × 12 = 1920 of the smaller cubes. How many of these smaller cubes will fit into the 1 in. × 1 in. × 1 in. cube? οΊ ο§ How does the number of cubes in this example compare to the number of cubes that would be needed in the Opening Exercise? οΊ οΊ ο§ new old 1920 240 = 8 1 If I fill the same box with cubes that are half the length, I will need 8 times as many. 1 The volume of the box is of the number of cubes that will fit in it. 8 What is the volume of 1 cube? οΊ π= π= ο§ = How is the volume of the box related to the number of cubes that will fit in it? οΊ ο§ Two will fit across the length, two across the width, and two for the height. 2 × 2 × 2 = 8. Eight smaller cubes will fit in the larger cube. 1 2 1 8 in. × 1 2 in. × 1 2 in. in3 What is the product of the number of cubes and the volume of the cubes? What does this product represent? 1 οΊ 1920 × = 240 οΊ The product represents the volume of the original box. 8 Example 2 (5 minutes) Example 2 A π π π’π§. cube is used to fill the prism. π How many π’π§. cubes will it take to fill the prism? π π What is the volume of the prism? π π’π§. π How is the number of cubes related to the volume? π π’π§. π π π’π§. π Lesson 11: Date: Volume with Fractional Edge Lengths and Unit Cubes 2/5/15 © 2014 Common Core, Inc. Some rights reserved. commoncore.org This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 156 Lesson 11 NYS COMMON CORE MATHEMATICS CURRICULUM ο§ 6β’5 How would you determine, or find, the number of cubes that fill the prism? οΊ One method would be to determine the number of cubes that will fit across the length, width, and height. Then, I would multiply. 6 will fit across the length, 4 across the width, and 15 across the height. 6 × 4 × 15 = 360 cubes ο§ How are the number of cubes and the volume related? οΊ The volume is equal to the number of cubes times the volume of one cube. The volume of one cube is 360 cubes × ο§ 1 64 in3 = 360 64 1 4 in. × 1 in. × 4 40 in3 = 5 64 1 4 in. = 1 64 in3 . 5 in3 = 5 in3 8 What other method can be used to determine the volume? π=ππ€β οΊ 1 3 π = (1 in.) (1 in.) (3 in.) 2 π= π= ο§ 3 in. × 2 45 8 4 1 1 in. × 15 4 in. 5 in3 = 5 in3 8 Would any other size cubes fit perfectly inside the prism with no space left over? οΊ We would not be able to use cubes with side lengths of 1 2 1 2 3 3 1 in., in., or spaces left over. However, we could use a cube with a side length of 8 in. because there would be in. without having spaces left over. Exercises (20 minutes) Students will work in pairs. Exercises 1. Use the prism to answer the following questions. a. Calculate the volume. π π½ = πππ π π π π½ = (π ππ¦) ( ππ¦) (π ππ¦) π π π ππ π π π½= ππ¦ × ππ¦ × ππ¦ π π π π½= b. π ππ¦ π πππ ππ ππ¦π or ππ ππ¦π ππ ππ If you have to fill the prism with cubes whose side lengths are less than π ππ¦, what size would be best? The best choice would be a cube with side lengths of c. π π ππ¦ π π ππ¦ π π π ππ¦. How many of the cubes would fit in the prism? ππ × π × π = πππ cubes Lesson 11: Date: Volume with Fractional Edge Lengths and Unit Cubes 2/5/15 © 2014 Common Core, Inc. Some rights reserved. commoncore.org This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 157 Lesson 11 NYS COMMON CORE MATHEMATICS CURRICULUM d. Use the relationship between the number of cubes and the volume to prove that your volume calculation is correct. The volume of one cube would be π π ππ¦ × π π ππ¦ × π π ππ¦ = Since there are πππ cubes, the volume would be πππ × 2. 6β’5 π ππ ππ¦π . π πππ ππ ππ¦π = ππ¦π or ππ ππ¦π . ππ ππ ππ Calculate the volume of the following rectangular prisms. a. π π½ =πππ π π π π½ = (π ππ¦) ( ππ¦) (π ππ¦) π π π ππ π π π½= ππ¦ × ππ¦ × ππ¦ π π π ππ ππ π π½= ππ¦ or π ππ¦π ππ ππ π ππ¦ π π ππ¦ π π π ππ¦ π b. π π π 3. π π’π§. π π½=πππ π π π π½ = (π π’π§.) (π π’π§.) (π π’π§.) π π π ππ ππ ππ π½= π’π§. × π’π§. × π’π§. π π π π, πππ π ππ π π½= π’π§ or ππ π’π§ ππ ππ π π’π§. π π π’π§. π A toy company is packaging its toys to be shipped. Some of the very small toys are placed inside a cube-shaped box with side lengths of π π’π§. These smaller boxes are then packed into a shipping box with dimensions of π π π ππ π’π§. × π π’π§. × π π’π§. π π a. How many small toys can be packed into the larger box for shipping? ππ × π × π = ππππ toys b. Use the number of toys that can be shipped in the box to help determine the volume of the box. One small box would have a volume of π π π’π§. × π π π’π§. × π π π’π§. = π π π’π§π . Now, I will multiply the number of cubes by the volume of the cube. ππππ × π π ππππ π π’π§ = π’π§ = πππ π’π§π π π Lesson 11: Date: Volume with Fractional Edge Lengths and Unit Cubes 2/5/15 © 2014 Common Core, Inc. Some rights reserved. commoncore.org This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 158 Lesson 11 NYS COMMON CORE MATHEMATICS CURRICULUM 4. 6β’5 A rectangular prism with a volume of π cubic units is filled with cubes. First, it is filled with cubes with side lengths π π unit. Then, it is filled with cubes with side lengths of unit. π π π π a. How many more of the cubes with -unit side lengths than cubes with -unit side lengths will be needed to π π of fill the prism? π π There are π cubes with -unit side lengths in π cubic unit because the volume of one cube is π π π cubic units. π Since we have π cubic units, we would have π × π = ππ total cubes with -unit side lengths. π π There are ππ cubes with -unit side lengths in π cubic unit because the volume of one cube is π π π cubic units. ππ Since we have π cubic units, we would have π × ππ = πππ total cubes with -unit side lengths. πππ β ππ = πππ more cubes b. Why does it take more cubes with π unit side lengths to fill the prism? π π π π < . The side length is shorter for the cube with a -unit side length, so it takes more to fill the rectangular π π π prism. 5. Calculate the volume of the rectangular prism. Show two different methods for determining the volume. Method 1: π½=πππ π π π π½ = (π π¦) ( π¦) (π π¦) π π π π π π π½ = ( π¦) ( π¦) ( π¦) π π π ππ π π½= π¦ ππ π π½=π π¦π ππ π Method 2: Fill the rectangular prism with cubes that are The volume of the cubes is π ππ π π π¦× π π π¦× π π π π π¦ π π¦. π π¦ π π π¦ π π¦π . We would have π cubes across the length, π cubes across the width, and ππ cubes across the height. π × π × ππ = πππ cubes total πππ cubes × π π π¦π = π π¦π ππ ππ Lesson 11: Date: Volume with Fractional Edge Lengths and Unit Cubes 2/5/15 © 2014 Common Core, Inc. Some rights reserved. commoncore.org This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 159 Lesson 11 NYS COMMON CORE MATHEMATICS CURRICULUM 6β’5 Closing (2 minutes) ο§ When you want to find the volume of a rectangular prism that has sides with fractional lengths, what are some methods you can use? οΊ One method to find the volume of a right rectangular prism that has fractional side lengths is to use the volume formula π = ππ€β. οΊ Another method to find the volume is to determine how many cubes of fractional side lengths are inside the right rectangular prism, and then find the volume of the cube. To determine the volume of the right rectangular prism, find the product of these two numbers. Exit Ticket (5 minutes) Lesson 11: Date: Volume with Fractional Edge Lengths and Unit Cubes 2/5/15 © 2014 Common Core, Inc. Some rights reserved. commoncore.org This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 160 6β’5 Lesson 11 NYS COMMON CORE MATHEMATICS CURRICULUM Name Date Lesson 11: Volume with Fractional Edge Lengths and Unit Cubes Exit Ticket Calculate the volume of the rectangular prism using two different methods. Label your solutions Method 1 and Method 2. 2 1 Lesson 11: Date: 3 cm 8 1 cm 4 5 cm 8 Volume with Fractional Edge Lengths and Unit Cubes 2/5/15 © 2014 Common Core, Inc. Some rights reserved. commoncore.org This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 161 Lesson 11 NYS COMMON CORE MATHEMATICS CURRICULUM 6β’5 Exit Ticket Sample Solutions Calculate the volume of the rectangular prism using two different methods. Label your solutions Method 1 and Method 2. Method 1: π=π₯π°π‘ π π π π π = (π ππ¦) ( ππ¦) (π ππ¦) π π π ππ π π π= ππ¦ × ππ¦ × ππ¦ π π π πππ π= ππ¦π πππ π π ππ¦ π π ππ¦ π π ππ¦ π Method 2: Fill shape with π π ππ¦ cubes. ππ × π × ππ = πππ cubes Each cube has a volume of π = πππ × π π ππ¦ × π π ππ¦ × π π ππ¦ = π πππ ππ¦π . π πππ πππ ππ¦π = ππ¦π = ππ¦π πππ πππ πππ Problem Set Sample Solutions 1. Answer the following questions using this rectangular prism: π π π π’π§. a. π π’π§. π π π’π§. π What is the volume of the prism? π½=πππ π π π’π§.) (π π’π§.) π π π π ππ π½ = ( π’π§.) ( π’π§.) ( π’π§.) π π π πππ π π½= π’π§ π π½ = ππ π’π§π π½ = (π π’π§.) (π Lesson 11: Date: Volume with Fractional Edge Lengths and Unit Cubes 2/5/15 © 2014 Common Core, Inc. Some rights reserved. commoncore.org This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 162 Lesson 11 NYS COMMON CORE MATHEMATICS CURRICULUM b. 6β’5 π π Linda fills the rectangular prism with cubes that have side lengths of π’π§. How many cubes does she need to fill the rectangular prism? She would need ππ across by π wide and ππ high. Number of cubes = ππ × π × ππ π π Number of cubes = π, πππ cubes with π’π§. side lengths c. How is the number of cubes related to the volume? ππ × ππ = ππππ The number of cubes needed is ππ times larger than the volume. d. Why is the number of cubes needed different from the volume? Because the cubes are not each π π’π§., the volume is different from the number of cubes. However, I could multiply the number of cubes by the volume of one cube and still get the original volume. e. π π Should Linda try to fill this rectangular prism with cubes that are π’π§. long on each side? Why or why not? π π π π π π Because some of the lengths are and some are , it would be difficult to use side lengths of to fill the prism. 2. Calculate the volume of the following prisms. a. π½=πππ π π ππ ππ¦ π ππ¦ π π ππ¦ π π π ππ¦) (π ππ¦) π π π π π½ = (ππ ππ¦) ( ππ¦) ( ππ¦) π π π, πππ π π½= ππ¦ π π½ = πππ ππ¦π π½ = (ππ ππ¦) (π b. π π π’π§. π ππ π π π’π§. π π π π’π§. π Lesson 11: Date: π½=πππ π π π π½ = (π π’π§. ) (π π’π§. ) (π π’π§. ) π π π ππ ππ ππ π½ = ( π’π§. ) ( π’π§. ) ( π’π§. ) π π π ππππ π π½= π’π§ ππ ππ π π½ = ππ π’π§ ππ Volume with Fractional Edge Lengths and Unit Cubes 2/5/15 © 2014 Common Core, Inc. Some rights reserved. commoncore.org This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 163 Lesson 11 NYS COMMON CORE MATHEMATICS CURRICULUM 3. A rectangular prism with a volume of ππ cubic units is filled with cubes. First, it is filled with cubes with 6β’5 π unit side π π unit side lengths. π π π How many more of the cubes with unit side lengths than cubes with unit side lengths will be needed to fill π π lengths. Then, it is filled with cubes with a. the prism? π π There are π cubes with -unit side lengths in π cubic unit because the volume of one cube is π π π cubic units. π Since we have ππ cubic units, we would have ππ × π = ππ total cubes with -unit side lengths. π π There are ππ cubes with -unit side lengths in π cubic unit because the volume of one cube is π π π cubic units. ππ Since we have ππ cubic units, we would have ππ × ππ = πππ total cubes with -unit side lengths. πππ β ππ = πππ more cubes b. Finally, the prism is filled with cubes whose side lengths are π π unit. How many unit cubes would it take to π π fill the prism? π π π cubic units. ππ π Since there are ππ cubic units, we would have ππ × ππ = πππ total cubes with side lengths of unit. π There are ππ cubes with -unit side lengths in π cubic unit because the volume of one cube is 4. A toy company is packaging its toys to be shipped. Some of the toys are placed inside a cube-shaped box with side π π π π lengths of π π’π§. These boxes are then packed into a shipping box with dimensions of ππ π’π§. × π π’π§. × π π’π§. a. How many toys can be packed into the larger box for shipping? π × π × π = π toys b. Use the number of toys that can be shipped in the box to help determine the volume of the box. π π π π π π π π One small box would have a volume of π π’π§. × π π’π§. × π π’π§. = ππ π’π§π . π π Now, I will multiply the number of cubes by the volume of the cube. π × ππ π’π§π = πππ π’π§π 5. A rectangular prism has a volume of ππ. πππ cubic meters. The height of the box is π. π meters, and the length is π. π meters. a. Write an equation that relates the volume to the length, width, and height. Let π represent the width, in meters. ππ. πππ = (π. π)(π. π)π b. Solve the equation. ππ. πππ = π. πππ π = π. π The width is π. π π¦. Lesson 11: Date: Volume with Fractional Edge Lengths and Unit Cubes 2/5/15 © 2014 Common Core, Inc. Some rights reserved. commoncore.org This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 164 Lesson 11 NYS COMMON CORE MATHEMATICS CURRICULUM 6β’5 Additional Exercise from Scaffolding Box This is a sample activity that fosters understanding of a cube with fractional edge lengths. It begins with three (two1 1 dimensional) squares with side lengths of 1 unit, unit, and unit, which leads to an understanding of three-dimensional 2 1 1 3 cubes that have edge lengths of 1 unit, unit, and unit. 2 ο§ 1 How many squares with -unit side lengths will fit in a square with 1 unit side lengths? 2 οΊ ο§ 3 1 Four squares with -unit side lengths will fit in the square with 1-unit side lengths. 2 1 What does this mean about the area of a square with -unit side lengths? 2 οΊ 1 1 2 4 The area of a square with -unit side lengths is of the area of a square with 1-unit side lengths, so it 1 has an area of square units. 4 Lesson 11: Date: Volume with Fractional Edge Lengths and Unit Cubes 2/5/15 © 2014 Common Core, Inc. Some rights reserved. commoncore.org This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 165 Lesson 11 NYS COMMON CORE MATHEMATICS CURRICULUM ο§ 1 How many squares with side lengths of units will fit in a square with side lengths of 1 unit? 3 οΊ ο§ 6β’5 1 Nine squares with side lengths of unit will fit in a square with side lengths of 1 unit. 3 1 What does this mean about the area of a square with -unit side lengths? 3 οΊ 1 1 3 9 The area of a square with -unit side lengths is of the area of a square with 1-unit side lengths, so it 1 has an area of square units. 9 ο§ Letβs look at what we have seen so far: Side Length (units) How many fit into a unit square? 1 1 1 2 4 1 3 9 Sample questions to pose: ο§ 1 Make a prediction about how many squares with -unit side lengths will fit into a unit square; then, draw a 4 picture to justify your prediction. οΊ 16 squares Lesson 11: Date: Volume with Fractional Edge Lengths and Unit Cubes 2/5/15 © 2014 Common Core, Inc. Some rights reserved. commoncore.org This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 166 Lesson 11 NYS COMMON CORE MATHEMATICS CURRICULUM ο§ 6β’5 1 How could you determine the number of -unit side length squares that would cover a figure with an area of 15 2 1 square units? How many -unit side length squares would cover the same figure? 3 οΊ 1 4 squares of -unit side lengths fit in each 1 square unit. So, if there are 15 square units, there will be 2 1 15 × 4 = 60 squares of -unit side lengths. 2 οΊ 1 9 squares of -unit side lengths fit in each 1 square unit. So, if there are 15 square units, there will be 3 1 15 × 9 = 135 squares of -unit side lengths. 3 1 1 2 3 ο§ Now letβs see what happens when we consider cubes of 1-, -, and -unit side lengths. ο§ How many cubes with -unit side lengths will fit in a cube with 1-unit side lengths? 1 2 οΊ 1 Eight of the cubes with -unit side lengths will fit into the cube with a 1-unit side lengths. 2 Lesson 11: Date: Volume with Fractional Edge Lengths and Unit Cubes 2/5/15 © 2014 Common Core, Inc. Some rights reserved. commoncore.org This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 167 Lesson 11 NYS COMMON CORE MATHEMATICS CURRICULUM ο§ 6β’5 1 What does this mean about the volume of a cube with -unit side lengths? 2 οΊ 1 1 2 8 The volume of a cube with -unit side lengths is of the volume of a cube with 1-unit side lengths, so it 1 has a volume of cubic units. 8 ο§ 1 How many cubes with -unit side lengths will fit in a cube with 1-unit side lengths? 3 οΊ ο§ 1 27 of the cubes with -unit side lengths will fit into the cube with 1-unit side lengths. 3 1 What does this mean about the volume of a cube with -unit side lengths? 3 οΊ 1 3 27 The volume of a cube with -unit side lengths is it has a volume of ο§ 1 1 27 of the volume of a square with 1-unit side lengths, so cubic units. Letβs look at what we have seen so far: Side Length (units) How many fit into a unit cube? 1 1 1 2 8 1 3 27 Sample questions to pose: ο§ 1 Make a prediction about how many cubes with -unit side lengths will fit into a unit cube, and then draw a 4 picture to justify your prediction. οΊ 64 cubes Lesson 11: Date: Volume with Fractional Edge Lengths and Unit Cubes 2/5/15 © 2014 Common Core, Inc. Some rights reserved. commoncore.org This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 168 Lesson 11 NYS COMMON CORE MATHEMATICS CURRICULUM ο§ 6β’5 1 How could you determine the number of -unit side length cubes that would fill a figure with a volume of 15 2 1 cubic units? How many -unit side length cubes would fill the same figure? 3 οΊ 1 8 cubes of -unit side lengths fit in each 1 cubic unit. So, if there are 15 cubic units, there will be 120 2 cubes because 15 × 8 = 120. οΊ 1 27 cubes of -unit side lengths fit in each 1 cubic unit. So, if there are 15 cubic units, there will be 405 3 cubes because 15 × 27 = 405. Understanding Volume Volume ο§ Volume is the amount of space inside a three-dimensional figure. ο§ It is measured in cubic units. ο§ It is the number of cubic units needed to fill the inside of the figure. Cubic Units ο§ Cubic units measure the same on all sides. A cubic centimeter is one centimeter on all sides; a cubic inch is one inch on all sides, etc. ο§ Cubic units can be shortened using the exponent 3. 6 cubic centimeter = 6 cm3 ο§ Different cubic units can be used to measure the volume of space figuresβcubic inches, cubic yards, cubic centimeters, etc. Lesson 11: Date: Volume with Fractional Edge Lengths and Unit Cubes 2/5/15 © 2014 Common Core, Inc. Some rights reserved. commoncore.org This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 169 Lesson 11 NYS COMMON CORE MATHEMATICS CURRICULUM 6β’5 Number Correct: ______ Multiplication of Fractions IIβRound 1 Directions: Determine the product of the fractions. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 1 5 × 2 8 3 3 × 4 5 1 7 × 4 8 3 2 × 9 5 5 3 × 8 7 3 4 × 7 9 2 3 × 5 8 4 5 × 9 9 2 5 × 3 7 2 3 × 7 10 3 9 × 4 10 3 2 × 5 9 2 5 × 10 6 5 7 × 8 10 3 7 × 5 9 Lesson 11: Date: 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 2 3 × 9 8 3 8 × 8 9 3 7 × 4 9 3 10 × 5 13 2 7 1 × 7 8 1 5 3 ×3 2 6 7 1 1 ×5 8 5 4 2 5 ×3 5 9 2 3 7 ×2 5 8 2 3 4 ×2 3 10 3 1 3 ×6 5 4 7 1 2 ×5 9 3 3 1 4 ×3 8 5 1 2 3 ×5 3 5 2 2 ×7 3 Volume with Fractional Edge Lengths and Unit Cubes 2/5/15 © 2014 Common Core, Inc. Some rights reserved. commoncore.org This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 170 Lesson 11 NYS COMMON CORE MATHEMATICS CURRICULUM 6β’5 Multiplication of Fractions IIβRound 1 [KEY] Directions: Determine the product of the fractions. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 1 5 × 2 8 3 3 × 4 5 1 7 × 4 8 3 2 × 9 5 5 3 × 8 7 3 4 × 7 9 2 3 × 5 8 4 5 × 9 9 2 5 × 3 7 2 3 × 7 10 3 9 × 4 10 3 2 × 5 9 2 5 × 10 6 5 7 × 8 10 3 7 × 5 9 Lesson 11: Date: π ππ π ππ π ππ π ππ ππ ππ ππ ππ π π = ππ ππ ππ ππ ππ ππ π π = ππ ππ ππ ππ π π = ππ ππ ππ π = ππ π ππ π = ππ ππ ππ π = ππ ππ 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 2 3 × 9 8 3 8 × 8 9 3 7 × 4 9 3 10 × 5 13 2 7 1 × 7 8 1 5 3 ×3 2 6 7 1 1 ×5 8 5 4 2 5 ×3 5 9 2 3 7 ×2 5 8 2 3 4 ×2 3 10 3 1 3 ×6 5 4 7 1 2 ×5 9 3 3 1 4 ×3 8 5 1 2 3 ×5 3 5 2 2 ×7 3 π π = ππ ππ ππ π = ππ π ππ π = ππ ππ ππ π = ππ ππ ππ π =π ππ π πππ π = ππ ππ ππ πππ π =π ππ π πππ ππ = ππ ππ ππ πππ ππ = ππ ππ ππ πππ ππ = ππ ππ ππ πππ π = ππ ππ π πππ ππ = ππ ππ ππ πππ = ππ ππ πππ = ππ ππ ππ π = ππ π π Volume with Fractional Edge Lengths and Unit Cubes 2/5/15 © 2014 Common Core, Inc. Some rights reserved. commoncore.org This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 171 Lesson 11 NYS COMMON CORE MATHEMATICS CURRICULUM 6β’5 Number Correct: ______ Improvement: ______ Multiplication of Fractions IIβRound 2 Directions: Determine the product of the fractions. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 2 5 × 3 7 1 3 × 4 5 2 2 × 3 5 16. 17. 18. 5 5 × 9 8 5 3 × 8 7 3 7 × 4 8 2 3 × 5 8 3 3 × 4 4 7 3 × 8 10 4 1 × 9 2 6 3 × 11 8 5 9 × 6 10 3 2 × 4 9 4 5 × 11 8 2 9 × 3 10 Lesson 11: Date: 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 3 2 × 11 9 3 10 × 5 21 4 3 × 9 10 3 4 × 8 5 6 2 × 11 15 2 3 1 × 3 5 1 3 2 × 6 4 2 2 1 ×3 5 3 2 1 4 ×1 3 4 1 4 3 ×2 2 5 3 3×5 4 2 1 1 ×3 3 4 3 2 ×3 5 5 1 1 ×3 7 2 1 9 3 ×1 3 10 Volume with Fractional Edge Lengths and Unit Cubes 2/5/15 © 2014 Common Core, Inc. Some rights reserved. commoncore.org This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 172 Lesson 11 NYS COMMON CORE MATHEMATICS CURRICULUM 6β’5 Multiplication of Fractions IIβRound 2 [KEY] Directions: Determine the product of the fractions. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 2 5 × 3 7 1 3 × 4 5 2 2 × 3 5 ππ ππ 16. π ππ 17. π ππ 18. 5 5 × 9 8 5 3 × 8 7 3 7 × 4 8 2 3 × 5 8 3 3 × 4 4 7 3 × 8 10 4 1 × 9 2 6 3 × 11 8 5 9 × 6 10 3 2 × 4 9 4 5 × 11 8 ππ ππ 19. ππ ππ 20. ππ ππ 21. π π = ππ ππ 22. π ππ 23. ππ ππ 24. π π = ππ π 25. ππ π = ππ ππ 26. ππ π = ππ π 27. π π = ππ π 28. ππ π = ππ ππ 29. ππ π = ππ π 30. 2 9 × 3 10 Lesson 11: Date: 3 2 × 11 9 3 10 × 5 21 4 3 × 9 10 3 4 × 8 5 6 2 × 11 15 2 3 1 × 3 5 1 3 2 × 6 4 2 2 1 ×3 5 3 2 1 4 ×1 3 4 1 4 3 ×2 2 5 3 3×5 4 2 1 1 ×3 3 4 3 2 ×3 5 5 1 1 ×3 7 2 1 9 3 ×1 3 10 π π = ππ ππ ππ π = πππ π ππ π = ππ ππ ππ π = ππ ππ ππ π = πππ ππ ππ =π ππ ππ ππ π =π =π ππ ππ π ππ π =π ππ ππ ππ ππ π =π =π ππ ππ π ππ π π =π =π ππ ππ π ππ π = ππ π π ππ π =π ππ ππ ππ π =π π π ππ =π ππ πππ ππ =π ππ ππ π =π π Volume with Fractional Edge Lengths and Unit Cubes 2/5/15 © 2014 Common Core, Inc. Some rights reserved. commoncore.org This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 173
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