Lesson 12 NYS COMMON CORE MATHEMATICS CURRICULUM 6β’5 Lesson 12: From Unit Cubes to the Formulas for Volume Student Outcomes ο§ Students extend the volume formula for a right rectangular prism to the formula π = Area of base β height. They understand that any face can be the base. Lesson Notes This lesson is a continuation of the ideas in Lesson 11 and the lessons in Grade 5 Module 5 Topics A and B. The word face, though referenced in the last lesson, should be taught to students who may not know this meaning of it. A student-friendly definition and illustration can be posted on the wall (along with definitions of edge(s) and vertex/ vertices). Here is a link to a useful illustration: http://www.11plusforparents.co.uk/Maths/shape8.html. Classwork Example 1 (10 minutes) ο§ Look at the rectangular prisms in the first example. Write a numerical expression for the volume of each rectangular prism. οΊ ο§ What do these expressions have in common? οΊ ο§ ο§ MP.8 We could use area of the base times the height. π΄ = π π€; π΄ = (15 in. ) (1 1 1 in. ); and π΄ = 22 in2 2 2 How would we use the area of the base to determine the volume? (Think about the unit cubes we have been using. The area of the base would be the first layer of unit cubes. What would the height represent?) οΊ ο§ Answers provided below. What is the area of the base of each of the rectangular prisms? οΊ ο§ They represent the area of the bases of the rectangular prisms. If we know volume for a rectangular prism as length times width times height, what is another formula for volume that we could use based on these examples? οΊ ο§ You may want to use unit cubes to help students visualize the layers in this problem. Rewrite each of the numerical expressions to show what they have in common. οΊ ο§ Scaffolding: They have the same dimensions for the lengths and widths. What do these dimensions represent? οΊ MP.7 Answers provided below. We would multiply the area of the base times the height. The height would represent how many layers of cubes it would take to fill up the rectangular prism. Sample answers are on the next page. How do the volumes of the first and second rectangular prisms compare? The first and third? οΊ The volume of the second prism is twice that of the first because the height is doubled. The volume of the third prism is three times that of the first because the height is tripled. Lesson 12: From Unit Cubes to the Formulas for Volume This work is derived from Eureka Math β’ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from G6-M5-TE-1.3.0-10.2015 171 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 12 NYS COMMON CORE MATHEMATICS CURRICULUM 6β’5 Example 1 a. Write a numerical expression for the volume of each of the rectangular prisms above. (ππ π’π§. ) (π b. π π’π§. ) (π π’π§. ) π (ππ π’π§. ) (π π π’π§. ) (π π’π§. ) π (ππ π’π§. ) (π π π’π§. ) (π π’π§. ) π What do all of these expressions have in common? What do they represent? π π All of the expressions have (ππ π’π§. ) (π π’π§. ). This is the area of the base. c. Rewrite the numerical expressions to show what they have in common. (ππ d. π π π’π§ ) (π π’π§. ) π (ππ π π π’π§ ) (π π’π§. ) π (ππ π π π’π§ ) (π π’π§. ) π If we know volume for a rectangular prism as length times width times height, what is another formula for volume that we could use based on these examples? We could use (ππ«ππ π¨π ππ‘π πππ¬π)(π‘ππ’π π‘π), or area of the base times height. e. What is the area of the base for all of the rectangular prisms? (ππ π’π§. ) (π f. g. π π π’π§. ) = ππ π’π§π π π Determine the volume of each rectangular prism using either method. (ππ π’π§. ) (π π π π’π§. ) (π π’π§. ) = ππ π’π§π π π or (ππ π π π π’π§ ) (π π’π§. ) = ππ π’π§π π π (ππ π’π§. ) (π π π’π§. ) (π π’π§. ) = πππ π’π§π π or (ππ π π π’π§ ) (π π’π§. ) = πππ π’π§π π (ππ π’π§. ) (π π π π’π§. ) (π π’π§. ) = πππ π’π§π π π or (ππ π π π π’π§ ) (π π’π§. ) = πππ π’π§π π π How do the volumes of the first and second rectangular prisms compare? The volumes of the first and third? πππ π’π§π = ππ πππ π π π’π§ × π π π π π π’π§ = ππ π’π§π × π π π The volume of the second prism is twice that of the first because the height is doubled. The volume of the third prism is three times as much as the first because the height is triple the first prismβs height. Lesson 12: From Unit Cubes to the Formulas for Volume This work is derived from Eureka Math β’ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from G6-M5-TE-1.3.0-10.2015 172 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 12 NYS COMMON CORE MATHEMATICS CURRICULUM ο§ 6β’5 What do you think would happen to the volume if we turn this prism on its side so that a different face is the base? (Have students calculate the area of the base times the height for this new prism. To help students visualize what is happening with this rotation, you could use a textbook or a stack of index cards and discuss how this prism is similar to and different from the rectangular prisms in part (a).) οΊ Answers will vary. The volume is the same no matter which face is the base. 15 in. 1 Area of the base = (3 in. ) (1 in. ) 2 Area of the base = 4.5 in2 Volume = Area of the base × height Volume = (4 1 2 in ) (15 in. ) 2 1 1 Volume = 67 in3 2 ο§ The volumes in both solutions are the same. What other expressions could we use to determine the volume of the prism? οΊ MP.7 ο§ 3 in. How does this volume compare with the volume you calculated using the other face as the base? οΊ ο§ 1 in. 2 Answers will vary. Some possible variations are included below. 1 15 in.β× 1 in.β× 3 in. 2 1 15 in.β× 3 in.β× 1 in. 2 1 3 in.β× 15 in.β× 1 in. 2 1 2 45 in × 1 in. 2 1 2 1 2 We notice that 3 in.β× 15 in.β× 1 in. and 45 in2 × 1 in. are equivalent, and both represent the volume. How do they communicate different information? οΊ 1 2 The first expression (3 in.β× 15 in.β× 1 in.) shows that the volume is the product of three edge 1 2 lengths. The second (45 in2 × 1 in.) shows that the volume is the product of the area of the base and the height. Lesson 12: From Unit Cubes to the Formulas for Volume This work is derived from Eureka Math β’ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from G6-M5-TE-1.3.0-10.2015 173 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 12 NYS COMMON CORE MATHEMATICS CURRICULUM 6β’5 Example 2 (5 minutes) Example 2 π π π π The base of a rectangular prism has an area of π π’π§π . The height of the prism is π π’π§. Determine the volume of the rectangular prism. π½ = ππ«ππ π¨π πππ¬π × π‘ππ’π π‘π π π π½ = (π π’π§π ) (π π’π§. ) π π ππ π π π½=( π’π§ ) ( π’π§. ) π π ππ π π½= π’π§ π ο§ Do we need to know the length and the width to find the volume of the rectangular prism? οΊ The length and width are needed to calculate the area of the base, and we already know the area of the base. Therefore, we do not need the length and width. Exercises (20 minutes) The cards are printed out and used as stations or hung on the classroom walls so that students can move from question to question. Copies of the questions can be found at the end of the lesson. Multiple copies of each question can be printed so that students visit each question in small groups. Students should spend about three minutes at each station where they should show their work by first writing an equation and then using the equation to calculate the volume of the rectangular prism described. They should use the rest of the time to discuss the answers, and the teacher can answer any questions students have about the lesson. Card a. Draw a sketch of the figure. Then, calculate the volume. Rectangular Prism 3 8 Area of the base = 4 ft 2 1 2 Height = 2 ft. b. Draw a sketch of the figure. Write the length, width, and height in feet. Then, calculate the volume. Rectangular Prism Sample Response π = Area of baseβ× height 3 2 1 ft ) (2 ft. ) 8 2 35 2 5 π=( ft ) ( ft. ) 8 2 175 3 π= ft 16 1 15 Length = 3 ft.β× 2 = ft. 2 2 3 9 Width = 3 ft.β× = ft. 4 4 π = (4 π=ππ€β 1 2 Length is 2 times as long as the height. 15 9 ft. ) ( ft. ) (3 ft. ) 2 4 405 3 π= ft 8 π=( 3 Width is as long as the height. 4 Height = 3 ft. Lesson 12: From Unit Cubes to the Formulas for Volume This work is derived from Eureka Math β’ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from G6-M5-TE-1.3.0-10.2015 174 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 12 NYS COMMON CORE MATHEMATICS CURRICULUM c. Write two different expressions to represent the volume, and explain what each one represents. π MP.7 π d. π π¦ π π π¦ π π π¦ π π (4 2 1 m) ( m), shows the volume as a product of the 3 3 base area times the height. 4 3 π = ( ft 2 ) ( ft. ) 3 10 12 3 π= ft 30 2 π = ft 3 5 ππ«ππ = Calculate the volume. Answers will vary. Some possible solutions include 2 1 1 14 1 (4 m) ( m) (1 m) and ( m2 ) (1 m) . 3 3 8 9 8 The first expression shows the volume as a product of the three edge lengths. The second expression, π = Area of base × height Calculate the volume. π ππ. ππ e. π π ππ π π = Area of base × height π = (13 π π’π§. π 6β’5 π= 1 2 1 in ) (1 in. ) 2 3 108 3 in 6 π = 18 in3 ππ«ππ = ππ π π π’π§ π Lesson 12: From Unit Cubes to the Formulas for Volume This work is derived from Eureka Math β’ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from G6-M5-TE-1.3.0-10.2015 175 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 12 NYS COMMON CORE MATHEMATICS CURRICULUM f. Challenge: Length = 5 ft. Determine the volume of a rectangular prism whose length and width are in a ratio of 3: 1. The width and height are in a ratio of 2: 3. The length of the rectangular prism is 5 ft. Width = 5 ft. ÷ 3 = 6β’5 5 ft. 3 5 3 5 Height = ft.β× = ft. 3 2 2 π=ππ€β 5 5 π = (5 ft. ) ( ft. ) ( ft. ) 3 2 125 3 π= ft 6 Extension (3 minutes) Extension A company is creating a rectangular prism that must have a volume of π ππ π. The company also knows that the area of π π the base must be π ππ π . How can you use what you learned today about volume to determine the height of the rectangular prism? I know that the volume can be calculated by multiplying the area of the base times the height. So, if I needed the height instead, I would do the opposite. I would divide the volume by the area of the base to determine the height. π½ = ππ«ππ π¨π πππ¬π × π‘ππ’π π‘π π π ππ π = (π ππ π ) π π π π π π ππ ÷ π ππ = π π π π ππ. = π π Closing (2 minutes) ο§ How is the formula π = π β π€ β h related to the formula π = Area of the base β height? οΊ When we multiply the length and width of the rectangular prism, we are actually finding the area of the base. Therefore, the two formulas both determine the volume of the rectangular prism. Exit Ticket (5 minutes) Lesson 12: From Unit Cubes to the Formulas for Volume This work is derived from Eureka Math β’ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from G6-M5-TE-1.3.0-10.2015 176 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 12 NYS COMMON CORE MATHEMATICS CURRICULUM Name 6β’5 Date Lesson 12: From Unit Cubes to the Formulas for Volume Exit Ticket 1. Determine the volume of the rectangular prism in two different ways. 3 ft. 4 3 ft. 4 2. 3 ft. 8 The area of the base of a rectangular prism is 12 cm2 , and the height is 3 1 cm. Determine the volume of the 3 rectangular prism. Lesson 12: From Unit Cubes to the Formulas for Volume This work is derived from Eureka Math β’ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from G6-M5-TE-1.3.0-10.2015 177 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 12 NYS COMMON CORE MATHEMATICS CURRICULUM 6β’5 Exit Ticket Sample Solutions 1. Determine the volume of the rectangular prism in two different ways. π½ =πβπβπ π π π½ = ππ«ππ π¨π πππ¬π β π‘ππ’π π‘π π π π π π½ = ( ππ. ) ( ππ. ) ( ππ. ) π½= ππ ππ π πππ π½=( π½= π ππ. π π π ππ π ) ( ππ. ) ππ π ππ ππ π πππ π ππ. π 2. π ππ. π π π The area of the base of a rectangular prism is ππ ππ¦π, and the height is π ππ¦. Determine the volume of the rectangular prism. π½ = ππ«ππ π¨π πππ¬π β π‘ππ’π π‘π π π½ = (ππ ππ¦π ) (π ππ¦) π πππ π π½= ππ¦ π π½ = ππ ππ¦π Problem Set Sample Solutions 1. Determine the volume of the rectangular prism. π½=πππ π½ = (π π½= π π π π¦) ( π¦) ( π¦) π π π π π¦ π ππ π π¦ ππ π π¦ π π π π¦ π 2. π π π π The area of the base of a rectangular prism is π ππ π, and the height is π ππ. Determine the volume of the rectangular prism. π½ = ππ«ππ π¨π πππ¬πβ× π‘ππ’π π‘π π½ = (π π π π ππ ) (π ππ. ) π π ππ π π π½=( ππ ) ( ππ. ) π π π½= πππ π ππ ππ Lesson 12: From Unit Cubes to the Formulas for Volume This work is derived from Eureka Math β’ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from G6-M5-TE-1.3.0-10.2015 178 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 12 NYS COMMON CORE MATHEMATICS CURRICULUM 3. π π The length of a rectangular prism is π times as long as the width. The height is π π 6β’5 of the width. The width is π ππ¦. Determine the volume. ππ’πππ‘ = π ππ¦ πππ§π ππ‘ = π ππ¦β× π πππ’π π‘π = π ππ¦β× π ππ = ππ¦ π π π π = ππ¦ π π π½=πππ ππ π π½=( ππ¦) (π ππ¦) ( ππ¦) π π π½= πππ ππ¦π π 4. π π’π§. π ππ π’π§. π a. π π π’π§. π Write numerical expressions to represent the volume in two different ways, and explain what each reveals. (ππ π π ππ π’π§. ) (π π’π§. ) (π π’π§. ) represents the product of three edge lengths. ( π’π§π ) (π π’π§. ) represents the π π π product of the base area times the height. Answers will vary. b. Determine the volume of the rectangular prism. (ππ 5. π π ππ π π’π§. ) (π π’π§. ) (π π’π§. ) = πππ π’π§π or ( π’π§ ) (π π’π§. ) = πππ π’π§π π π π π π An aquarium in the shape of a rectangular prism has the following dimensions: length = ππ ππ¦, width = ππ ππ¦, and height = ππ a. π ππ¦. π Write numerical expressions to represent the volume in two different ways, and explain what each reveals. (ππ ππ¦) (ππ π π ππ¦) (ππ ππ¦) represents the product of the three edge lengths. π π (π, πππ ππ¦π ) (ππ b. π ππ¦) represents the base area times the height. π Determine the volume of the rectangular prism. (π, πππ ππ¦π ) (ππ Lesson 12: π π ππ¦) = ππ, πππ ππ¦π π π From Unit Cubes to the Formulas for Volume This work is derived from Eureka Math β’ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from G6-M5-TE-1.3.0-10.2015 179 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM 6. Lesson 12 6β’5 The area of the base in this rectangular prism is fixed at ππ ππ¦π. As the height of the rectangular prism changes, the volume will also change as a result. a. b. Complete the table of values to determine the various heights and volumes. Height of Prism (in centimeters) Volume of Prism (in cubic centimeters) π ππ π ππ π πππ π πππ π πππ π πππ π πππ π πππ π ππ¦ ππ ππ¦ Write an equation to represent the relationship in the table. Be sure to define the variables used in the equation. Let π be the height of the rectangular prism in centimeters. Let π be the volume of the rectangular prism in cubic centimeters. πππ = π c. What is the unit rate for this proportional relationship? What does it mean in this situation? The unit rate is ππ. For every centimeter of height, the volume increases by ππ ππ¦π because the area of the base is ππ ππ¦π. In order to determine the volume, multiply the height by ππ. 7. The volume of a rectangular prism is ππ. πππ ππ¦π . The height is π. ππ ππ¦. a. Let π© represent the area of the base of the rectangular prism. Write an equation that relates the volume, the area of the base, and the height. ππ. πππ = π. πππ© b. Solve the equation for π©. ππ. πππ π. πππ© = π. ππ π. ππ π© = π. π The area of the base is π. π ππ¦π. Lesson 12: From Unit Cubes to the Formulas for Volume This work is derived from Eureka Math β’ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from G6-M5-TE-1.3.0-10.2015 180 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 12 NYS COMMON CORE MATHEMATICS CURRICULUM 6β’5 Station A Draw a sketch of the figure. Then, calculate the volume. Rectangular Prism π π Area of the base = π ππ π Height = π Lesson 12: π ππ. π From Unit Cubes to the Formulas for Volume This work is derived from Eureka Math β’ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from G6-M5-TE-1.3.0-10.2015 181 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 12 6β’5 Station B Draw a sketch of the figure. Write the length, width, and height in feet. Then, calculate the volume. Rectangular Prism π π Length is π times as long as the height. π Width is as long as the height. π Height = π ππ. Lesson 12: From Unit Cubes to the Formulas for Volume This work is derived from Eureka Math β’ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from G6-M5-TE-1.3.0-10.2015 182 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 12 NYS COMMON CORE MATHEMATICS CURRICULUM 6β’5 Station C Write two different expressions to represent the volume, and explain what each one represents. π π Lesson 12: π π¦ π From Unit Cubes to the Formulas for Volume This work is derived from Eureka Math β’ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from G6-M5-TE-1.3.0-10.2015 π π¦ π π π¦ π 183 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 12 NYS COMMON CORE MATHEMATICS CURRICULUM 6β’5 Station D Calculate the volume. π ππ. ππ ππ«ππ = Lesson 12: From Unit Cubes to the Formulas for Volume This work is derived from Eureka Math β’ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from G6-M5-TE-1.3.0-10.2015 π π ππ π 184 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 12 6β’5 Station E Calculate the volume. π π π’π§. π ππ«ππ = ππ Lesson 12: π π π’π§ π From Unit Cubes to the Formulas for Volume This work is derived from Eureka Math β’ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from G6-M5-TE-1.3.0-10.2015 185 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 12 NYS COMMON CORE MATHEMATICS CURRICULUM 6β’5 Station F Challenge: Determine the volume of a rectangular prism whose length and width are in a ratio of π:π. The width and height are in a ratio of π:π. The length of the rectangular prism is π ππ. Lesson 12: From Unit Cubes to the Formulas for Volume This work is derived from Eureka Math β’ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from G6-M5-TE-1.3.0-10.2015 186 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
© Copyright 2026 Paperzz