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This lesson teaches students about calculating the volume of rectangular prisms using two different formulas: 1) length × width × height and 2) area of the base × height. Students work through examples calculating the volume of various rectangular prisms using both formulas. They learn that it does not matter which face is used as the base, as the volume will be the same. The lesson reinforces that volume can be expressed in multiple equivalent ways and emphasizes using the area of the base times the height.

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Lesson 12: From Unit Cubes to the Formulas for Volume Date: 5/27/15 170 © 2014 Common Core, Inc. Some rightsreserved. commoncore.org This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM 6•5Lesson 12 Lesson 12: From Unit Cubes to the Formulas for Volume Student Outcomes  Students extend the volume formula for a right rectangular prismto the formula 𝑉 = Area of base ∙ 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 Module5 of Grade 5 (Topics A and B). The word face, though referenced in the lastlesson, should betaught to students who may not know this meaning of it. A student-friendly definition and illustration can beposted 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)  Look at the rectangular prisms in thefirstexample. Write a numerical expression for the volume of each rectangular prism.  Answers provided below.  What do these expressions havein common?  They have the same dimensions for the lengths and widths.  What do these dimensions represent?  The area of the bases of the rectangular prisms.  Rewrite each of the numerical expressions to showwhat they have in common.  Answers provided below.  If we know volume for a rectangular prismas length times width times height, what is another formula for volume that we could use based on these examples?  Area of the base times the height.  What is the area of the baseof each of the rectangular prisms?  𝐴 = 𝑙 𝑤; 𝐴 = (15 𝑖𝑛. )(1 1 2 𝑖𝑛. ); 𝐴 = 22 1 2 in2  How would we use the area of the baseto determine the volumes? (Think about the unit cubes we have been using. The area of the base would be the firstlayer of unitcubes. What would the height represent?)  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 below.  How do the volumes of the firstand second rectangular prisms compare? The firstand third?  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. Scaffolding: You may want to useunit cubes to help students visualize the layers in this problem. MP.7 MP.8
Lesson 12: From Unit Cubes to the Formulas for Volume Date: 5/27/15 171 © 2014 Common Core, Inc. Some rightsreserved. commoncore.org This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM 6•5Lesson 12 Example 1 a. Write anumerical expression for thevolume ofeach oftherectangular prismsabove. ( 𝟏𝟓 𝒊𝒏. )(𝟏 𝟏 𝟐 𝒊𝒏. )(𝟑 𝒊𝒏.) ( 𝟏𝟓 𝒊𝒏. )( 𝟏 𝟏 𝟐 𝒊𝒏. )(𝟔 𝒊𝒏.) ( 𝟏𝟓 𝒊𝒏. )( 𝟏 𝟏 𝟐 𝒊𝒏. )(𝟗 𝒊𝒏.) b. What do all ofthese expressionshave in common? What do they represent? All oftheexpressions have ( 𝟏𝟓 𝒊𝒏. )( 𝟏 𝟏 𝟐 𝒊𝒏. ). This is thearea ofthebase. c. Rewrite the numerical expressionsto show what they have in common. ( 𝟐𝟐 𝟏 𝟐 𝒊𝒏 𝟐)(𝟑 𝒊𝒏. ) ( 𝟐𝟐 𝟏 𝟐 𝒊𝒏 𝟐)(𝟔 𝒊𝒏.) ( 𝟐𝟐 𝟏 𝟐 𝒊𝒏 𝟐)(𝟗 𝒊𝒏.) d. If we know volume for arectangular prism aslength timeswidth timesheight,what isanotherformulafor volume that we coulduse based on these examples? (Area ofthebase)(height), or area ofthebasetimes height. e. What isthe areaofthe base for all ofthe rectangular prisms? ( 𝟏𝟓 𝒊𝒏. )( 𝟏 𝟏 𝟐 𝒊𝒏. ) = 𝟐𝟐 𝟏 𝟐 𝒊𝒏 𝟐 f. Determinethe volumeofeach rectangular prism using eithermethod. ( 𝟏𝟓 𝒊𝒏. )( 𝟏 𝟏 𝟐 𝒊𝒏. )( 𝟑 𝒊𝒏. )= 𝟔𝟕 𝟏 𝟐 𝒊𝒏 𝟑 or ( 𝟐𝟐 𝟏 𝟐 𝒊𝒏 𝟐)( 𝟑 𝒊𝒏. ) = 𝟔𝟕 𝟏 𝟐 𝒊𝒏 𝟑 ( 𝟏𝟓 𝒊𝒏. )( 𝟏 𝟏 𝟐 𝒊𝒏. )( 𝟔 𝒊𝒏. )= 𝟏𝟑𝟓 𝒊𝒏 𝟑 or ( 𝟐𝟐 𝟏 𝟐 𝒊𝒏 𝟐)( 𝟔 𝒊𝒏. ) = 𝟏𝟑𝟓 𝒊𝒏 𝟑 ( 𝟏𝟓 𝒊𝒏. )( 𝟏 𝟏 𝟐 𝒊𝒏. )( 𝟗 𝒊𝒏. )= 𝟐𝟎𝟐 𝟏 𝟐 𝒊𝒏 𝟑 or ( 𝟐𝟐 𝟏 𝟐 𝒊𝒏 𝟐)( 𝟗 𝒊𝒏. ) = 𝟐𝟎𝟐 𝟏 𝟐 𝒊𝒏 𝟑 g. How do the volumesofthe first and second rectangular prismscompare? The volumesofthe first and third? 𝟏𝟑𝟓 𝒊𝒏 𝟑 = 𝟔𝟕 𝟏 𝟐 𝒊𝒏 𝟐 × 𝟐 𝒊𝒏.; 𝟐𝟎𝟐 𝟏 𝟐 𝒊𝒏 𝟑 = 𝟔𝟕 𝟏 𝟐 𝒊𝒏 𝟐 × 𝟑 𝒊𝒏. The volume ofthesecond prism is twicethat ofthefirst becausetheheightis doubled. Thevolumeofthe third prism is threetimes as muchas thefirst because theheight is triplethefirst prism’s height.
Lesson 12: From Unit Cubes to the Formulas for Volume Date: 5/27/15 172 © 2014 Common Core, Inc. Some rightsreserved. commoncore.org This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM 6•5Lesson 12  What do you think would happen to the volume if we turn this prismon its sideso that a different face is the base? (Have students calculatethe area of the base times the height for this new prism. To help students visualizewhatis happening with this rotation,you could use a textbook or a stack of index cards and discuss how this prismis similarand/or differentto the rectangular prisms in part(a).)  Answers will vary. Some students may see that the volume will be the same no matter which face is the base. Area of the base = (3 𝑖𝑛. ) (1 1 2 𝑖𝑛. ) Area of the base = 4.5 𝑖𝑛2 Volume =Area of the base × height Volume = (4 1 2 𝑖𝑛2) (15 𝑖𝑛. ) Volume = 67 1 2 𝑖𝑛3  How does this volume compare with the volume you calculated usingthe other face as the base?  The volumes in both solutions are the same.  What other expressions could weuse to determine the volume of the prism?  Answers will vary. Some possible variations are included below.  15 in. × 1 1 2 in. × 3 in. 15 in. × 3 in. × 1 1 2 in. 3 in. × 15 in. × 1 1 2 in. 45 in2 × 1 1 2 in.  We noticethat 3 in.× 15 in. × 1 1 2 in.and 45 in2 × 1 1 2 in.areequivalent and both represent the volume. How do they communicate different information?  The first expression (3 in. × 15 in. × 1 1 2 in.) shows that the volume is the product of three edge lengths. The second (45 in2 × 1 1 2 in.) shows that the volume is the product of the area of the base times the height. Example 2 (5 minutes) Example 2 The base ofa rectangular prism hasan areaof 𝟑 𝟏 𝟒 in2. The height ofthe prism is 𝟐 𝟏 𝟐 in. Determinethevolume ofthe rectangular prism. 𝑽 = 𝑨𝒓𝒆𝒂 𝒐𝒇 𝒕𝒉𝒆 𝒃𝒂𝒔𝒆 × 𝒉𝒆𝒊𝒈𝒉𝒕 𝑽 = ( 𝟑 𝟏 𝟒 𝒊𝒏 𝟐)( 𝟐 𝟏 𝟐 𝒊𝒏. ) 𝑽 = ( 𝟏𝟑 𝟒 𝒊𝒏 𝟐)( 𝟓 𝟐 𝒊𝒏. ) 𝑽 = 𝟔𝟓 𝟖 𝒊𝒏 𝟑 𝟏𝟓 𝐢𝐧. 𝟑 𝐢𝐧. 𝟏 𝟏 𝟐 𝐢𝐧. MP.7
Lesson 12: From Unit Cubes to the Formulas for Volume Date: 5/27/15 173 © 2014 Common Core, Inc. Some rightsreserved. commoncore.org This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM 6•5Lesson 12  Do we need to know the length and the width to find the volume of the rectangular prism?  No, if we know the area of the base, we do not need the length and width. The length and width is used to calculate the area, and we are already given the area. Exercises1–6 (20 minutes) The cards areprinted out and used as stations or hung on the classroomwallsso thatstudents can move from question to question. Copies of the questions can be found at the end of the lesson. Multiplecopies of each question can be printed so that a small number of students visiteach question at a time. Students should spend about three minutes at each station where they will showtheir work by firstwritinga numerical expression,and then use the expression to calculatethe volume of the rectangular prismdescribed. They will usethe rest of the time to discusstheanswers,and the teacher can answer any questions students have about the lesson. 1. Draw a sketch of the figure. Then calculatethe volume. Rectangular Prism Area of the base= 4 3 8 ft2 Height = 2 1 2 ft. 𝑉 = 𝐴𝑟𝑒𝑎 𝑜𝑓 𝑏𝑎𝑠𝑒 × ℎ𝑒𝑖𝑔ℎ𝑡 𝑉 = (4 3 8 𝑓𝑡2) (2 1 2 𝑓𝑡) 𝑉 = ( 35 8 𝑓𝑡2) ( 5 2 𝑓𝑡) 𝑉 = 175 16 𝑓𝑡3 2. Draw a sketch of the figure. Writethe length, width, and height in feet. Then calculatethe volume. Rectangular Prism Length is 2 1 2 times as longas the height. Width is 3 4 as longas the height. Height = 3 ft. 𝐿𝑒𝑛𝑔𝑡ℎ = 3 𝑓𝑡 × 2 1 2 = 15 2 𝑓𝑡 𝑊𝑖𝑑𝑡ℎ = 3 𝑓𝑡 × 3 4 = 9 4 𝑓𝑡 𝑉 = 𝑙 𝑤 ℎ 𝑉 = ( 15 2 𝑓𝑡) ( 9 4 𝑓𝑡) (3 𝑓𝑡) 𝑉 = 405 8 𝑓𝑡3 3. Write two different expressions to represent the volume and explain whateach one represents. Answers will vary. Some possible solutions include (4 2 3 𝑚) ( 1 3 𝑚) (1 1 8 𝑚) and ( 14 9 𝑚2) (1 1 8 𝑚) The first expression shows the volume as a product of the three edge lengths. The second expression shows the volume as a product of a base (4 2 3 𝑚) ( 1 3 𝑚) times the height. 𝟒 𝟐 𝟑 𝐦 𝟏 𝟏 𝟖 𝐦 𝟏 𝟑 𝐦 MP.7
Lesson 12: From Unit Cubes to the Formulas for Volume Date: 5/27/15 174 © 2014 Common Core, Inc. Some rightsreserved. commoncore.org This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM 6•5Lesson 12 4. Calculatethe volume. 𝑉 = 𝐴𝑟𝑒𝑎 𝑜𝑓 𝑏𝑎𝑠𝑒 × ℎ𝑒𝑖𝑔ℎ𝑡 𝑉 = ( 4 3 𝑓𝑡2) ( 3 10 𝑓𝑡. ) 𝑉 = 12 30 𝑓𝑡3 𝑉 = 2 5 𝑓𝑡3 5. Calculatethe volume. 𝑉𝑜𝑙𝑢𝑚𝑒 = 𝐴𝑟𝑒𝑎 𝑜𝑓 𝑏𝑎𝑠𝑒 × ℎ𝑒𝑖𝑔ℎ𝑡 𝑉𝑜𝑙𝑢𝑚𝑒 = (13 1 2 𝑖𝑛2) (1 1 3 𝑖𝑛. ) 𝑉𝑜𝑙𝑢𝑚𝑒 = 108 6 𝑖𝑛3 𝑉𝑜𝑙𝑢𝑚𝑒 = 18 𝑖𝑛3 6. Challenge! 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 prismis 5 ft. 𝐿𝑒𝑛𝑔𝑡ℎ = 5 𝑓𝑡. 𝑊𝑖𝑑𝑡ℎ = 5 𝑓𝑡.÷ 3 = 5 3 𝑓𝑡. 𝐻𝑒𝑖𝑔ℎ𝑡 = 5 3 𝑓𝑡.× 3 2 = 5 2 𝑓𝑡. 𝑉 = 𝑙 𝑤 ℎ 𝑉 = (5 𝑓𝑡. ) ( 5 3 𝑓𝑡. )( 5 2 𝑓𝑡. ) 𝑉 = 125 6 𝑓𝑡3 𝐀𝐫𝐞𝐚 = 𝟏𝟑 𝟏 𝟐 𝐢𝐧 𝟐 𝟏 𝟏 𝟑 𝐢𝐧. 𝐀𝐫𝐞𝐚 = 𝟒 𝟑 𝐟𝐭 𝟐 𝟑 𝟏𝟎 𝐟𝐭.
Lesson 12: From Unit Cubes to the Formulas for Volume Date: 5/27/15 175 © 2014 Common Core, Inc. Some rightsreserved. commoncore.org This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM 6•5Lesson 12 Extension(3 minutes) Extension A company iscreating arectangular prism that must have avolumeof 𝟔ft3. The company also knowsthat the areaofthe base must be 𝟐 𝟏 𝟐 ft2. How can you use what you learnedtoday about volume todeterminetheheight ofthe rectangular prism? I know that thevolumecan becalculated by multiplying theareaofthebasetimes theheight. So, ifI needed theheight instead, I would do theopposite. I woulddividethevolumeby thearea ofthebaseto determinetheheight. 𝑽 = 𝑨𝒓𝒆𝒂 𝒐𝒇 𝒃𝒂𝒔𝒆 × 𝒉𝒆𝒊𝒈𝒉𝒕 𝟔 𝒇𝒕 𝟑 = ( 𝟐 𝟏 𝟐 𝒇𝒕 𝟐)( 𝒉) 𝟔 𝒇𝒕 𝟑 ÷ 𝟐 𝟏 𝟐 𝒇𝒕 𝟐 = 𝒉𝒆𝒊𝒈𝒉𝒕 𝟐 𝟐 𝟓 𝒇𝒕. = 𝒉𝒆𝒊𝒈𝒉𝒕 Closing(2 minutes)  How is the formula 𝑉 = 𝑙 ∙ 𝑤 ∙ ℎ related to the formula 𝑉 = Area of the base ∙ height?  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 Date: 5/27/15 176 © 2014 Common Core, Inc. Some rightsreserved. commoncore.org This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM 6•5Lesson 12 Name Date Lesson 12: From Unit Cubes to the Formulas for Volume Exit Ticket 1. Determine the volume of the rectangular prismin two different ways. 2. The area of the base of a rectangular prismis 12 cm2 and the height is 3 1 3 cm. Determine the volume of the rectangular prism. 3 4 ft. 3 4 ft. 3 8 ft.
Lesson 12: From Unit Cubes to the Formulas for Volume Date: 5/27/15 177 © 2014 Common Core, Inc. Some rightsreserved. commoncore.org This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM 6•5Lesson 12 𝟑 𝟒 𝐟𝐭. 𝟑 𝟒 𝐟𝐭. 𝟑 𝟖 𝐟𝐭. Exit Ticket Sample Solutions 1. Determinethe volumeofthe rectangular prism. 𝑽 = 𝒍 ∙ 𝒘 ∙ 𝒉 𝑽𝒐𝒍𝒖𝒎𝒆 = 𝑨𝒓𝒆𝒂 𝒐𝒇 𝒃𝒂𝒔𝒆 ∙ 𝒉𝒆𝒊𝒈𝒉𝒕 𝑽 = ( 𝟑 𝟒 𝒇𝒕. )( 𝟑 𝟖 𝒇𝒕. )( 𝟑 𝟒 𝒇𝒕. ) 𝑽𝒐𝒍𝒖𝒎𝒆 = ( 𝟗 𝟑𝟐 𝒇𝒕 𝟐)( 𝟑 𝟒 𝒇𝒕. ) 𝑽 = 𝟐𝟕 𝟏𝟐𝟖 𝒇𝒕 𝟑 𝑽𝒐𝒍𝒖𝒎𝒆 = 𝟐𝟕 𝟏𝟐𝟖 𝒇𝒕 𝟑 2. The areaofthe base ofa rectangular prism is 𝟏𝟐 𝐜𝐦 𝟐 and the heightis 𝟑 𝟏 𝟑 𝐜𝐦. Determinethevolumeofthe rectangular prism. 𝑽𝒐𝒍𝒖𝒎𝒆 = 𝑨𝒓𝒆𝒂 𝒐𝒇 𝒃𝒂𝒔𝒆 ∙ 𝒉𝒆𝒊𝒈𝒉𝒕 𝑽𝒐𝒍𝒖𝒎𝒆 = ( 𝟏𝟐 𝒄𝒎 𝟐 )( 𝟑 𝟏 𝟑 𝒄𝒎) 𝑽𝒐𝒍𝒖𝒎𝒆 = 𝟏𝟐𝟎 𝟑 𝒄𝒎 𝟑 𝑽𝒐𝒍𝒖𝒎𝒆 = 𝟒𝟎 𝒄𝒎 𝟑 Problem Set Sample Solutions 1. Determinethe volumeofthe rectangular prism. 𝑽 = 𝒍 𝒘 𝒉 𝑽 = ( 𝟏 𝟏 𝟐 𝒎)( 𝟏 𝟐 𝒎)( 𝟕 𝟖 𝒎) 𝑽 = 𝟐𝟏 𝟑𝟐 𝒎 𝟑 2. The areaofthe base ofa rectangular prism is 𝟒 𝟑 𝟒 ft2 and the heightis 𝟐 𝟏 𝟑 ft. Determine thevolume ofthe rectangular prism. 𝑽𝒐𝒍𝒖𝒎𝒆 = 𝑨𝒓𝒆𝒂 𝒐𝒇 𝒃𝒂𝒔𝒆 × 𝒉𝒆𝒊𝒈𝒉𝒕 𝑽𝒐𝒍𝒖𝒎𝒆 = ( 𝟒 𝟑 𝟒 𝒇𝒕 𝟐)( 𝟐 𝟏 𝟑 𝒇𝒕. ) 𝑽𝒐𝒍𝒖𝒎𝒆 = ( 𝟏𝟗 𝟒 𝒇𝒕 𝟐)( 𝟕 𝟑 𝒇𝒕. ) 𝑽𝒐𝒍𝒖𝒎𝒆 = 𝟏𝟑𝟑 𝟏𝟐 𝒇𝒕 𝟑 𝟕 𝟖 𝐦 𝟏 𝟐 𝐦 𝟏 𝟏 𝟐 𝐦
Lesson 12: From Unit Cubes to the Formulas for Volume Date: 5/27/15 178 © 2014 Common Core, Inc. Some rightsreserved. commoncore.org This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM 6•5Lesson 12 3. The length ofarectangular prism is 𝟑 𝟏 𝟐 timesaslong asthe width. The height is 𝟏 𝟒 ofthe width. The widthis 𝟑cm. Determinethe volume. 𝑾𝒊𝒅𝒕𝒉 = 𝟑 𝒄𝒎 𝑳𝒆𝒏𝒈𝒕𝒉 = 𝟑 𝒄𝒎× 𝟑 𝟏 𝟐 = 𝟐𝟏 𝟐 𝒄𝒎 𝑯𝒆𝒊𝒈𝒉𝒕= 𝟑 𝒄𝒎 × 𝟏 𝟒 = 𝟑 𝟒 𝒄𝒎 𝑽𝒐𝒍𝒖𝒎𝒆 = 𝒍 𝒘 𝒉 𝑽𝒐𝒍𝒖𝒎𝒆 = ( 𝟐𝟏 𝟐 𝒄𝒎)( 𝟑 𝒄𝒎)( 𝟑 𝟒 𝒄𝒎) 𝑽𝒐𝒍𝒖𝒎𝒆 = 𝟏𝟖𝟗 𝟖 𝒄𝒎 𝟑 4. a. Write numerical expressionsin two differentways, and explain what each reveals. ( 𝟏𝟎 𝟏 𝟐 𝒊𝒏. )( 𝟏 𝟐 𝟑 𝒊𝒏. )(𝟔 𝒊𝒏.) represents theproduct ofthreeedgelengths. ( 𝟑𝟓 𝟐 𝒊𝒏 𝟐)( 𝟔 𝒊𝒏)represents the product ofthebasearea times theheight. Answers will vary. b. Determinethe volumeofthe rectangular prism. ( 𝟏𝟎 𝟏 𝟐 𝒊𝒏. )( 𝟏 𝟐 𝟑 𝒊𝒏. )( 𝟔 𝒊𝒏. ) = 𝟏𝟎𝟓 𝒊𝒏 𝟑 or ( 𝟑𝟓 𝟐 𝒊𝒏 𝟐)( 𝟔 𝒊𝒏. ) = 𝟏𝟎𝟓 𝒊𝒏 𝟑 5. An aquarium in the shape ofarectangular prism hasthe following dimensions: length = 𝟓𝟎cm, width= 𝟐𝟓 𝟏 𝟐 cm, height = 𝟑𝟎 𝟏 𝟐 cm. a. Write numerical expressionsin two differentways, and explain what each reveals. ( 𝟓𝟎 𝒄𝒎)( 𝟐𝟓 𝟏 𝟐 𝒄𝒎)( 𝟑𝟎 𝟏 𝟐 𝒄𝒎)represents theproduct ofthethreeedgelengths. (𝟏,𝟐𝟕𝟓 𝒄𝒎 𝟐 )( 𝟑𝟎 𝟏 𝟐 𝒄𝒎)represents thebase area times theheight. b. Determinethe volumeofthe rectangular prism. ( 𝟏,𝟐𝟕𝟓 𝒄𝒎 𝟐)( 𝟑𝟎 𝟏 𝟐 𝒄𝒎) = 𝟑𝟖,𝟖𝟖𝟕 𝟏 𝟐 𝒄𝒎 𝟑 𝟏 𝟐 𝟑 in. 𝟏𝟎 𝟏 𝟐 in. 𝟔 in.
Lesson 12: From Unit Cubes to the Formulas for Volume Date: 5/27/15 179 © 2014 Common Core, Inc. Some rightsreserved. commoncore.org This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM 6•5Lesson 12 6. The areaofthe base in thisrectangular prismisfixed at 𝟑𝟔cm2. That meansfor the varying heightsthere will be variousvolumes. a. Complete thetableofvaluesto determinethe various heightsand volumes. Height in Centimeters Volume in Cubic Centimeters 𝟏 𝟑𝟔 𝟐 𝟕𝟐 𝟑 𝟏𝟎𝟖 𝟒 𝟏𝟒𝟒 𝟓 𝟏𝟖𝟎 𝟔 𝟐𝟏𝟔 𝟕 𝟐𝟓𝟐 𝟖 𝟐𝟖𝟖 b. Write an equation to representtherelationship in thetable. Be sure to definethevariablesused in the equation. Let 𝒙 be theheight oftherectangularprism incentimeters. Let 𝒚be thevolumeoftherectangular prism in cubic centimeters. 𝟑𝟔𝒙 = 𝒚 c. What isthe unit rate for thisproportional relationship? What doesit mean inthissituation? The unit rateis 𝟑𝟔. For every centimeter ofheight, thevolumeincreases by 𝟑𝟔cubic centimeters becausethearea ofthebaseis 𝟑𝟔cm2. In order to determinethevolume, multiply theheightby 𝟑𝟔. 7. The volume ofarectangular prism is 𝟏𝟔.𝟑𝟐𝟖cm3. The heightis 𝟑. 𝟏𝟒cm. a. Let 𝑨 represent the areaofthe base oftherectangular prism. Writean equation that relatesthevolume, the areaof the base, and the height. 𝟏𝟔.𝟑𝟐𝟖 = 𝟑.𝟏𝟒𝑨 b. Solve the equationfor 𝑨. 𝟏𝟔.𝟑𝟐𝟖 𝟑. 𝟏𝟒 = 𝟑.𝟏𝟒𝑨 𝟑. 𝟏𝟒 𝑨 = 𝟓.𝟐 The area ofthebaseis 𝟓.𝟐 cm2. 𝟏𝟐cm 𝟑cm
Lesson 12: From Unit Cubes to the Formulas for Volume Date: 5/27/15 180 © 2014 Common Core, Inc. Some rightsreserved. commoncore.org This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM 6•5Lesson 12 Station A Make a sketch of the figure. Then calculate the volume. Rectangular prism: Area of the base = 𝟒 𝟑 𝟖 ft2 Height = 𝟐 𝟏 𝟐 ft.
Lesson 12: From Unit Cubes to the Formulas for Volume Date: 5/27/15 181 © 2014 Common Core, Inc. Some rightsreserved. commoncore.org This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM 6•5Lesson 12 Station B Make a sketch of the figure. Write the length, the width, and height in feet. Then calculate the volume. Rectangular prism: Length is 𝟐 𝟏 𝟐 times as long as the height. Width is 𝟑 𝟒 as long as the height. Height = 𝟑 ft.
Lesson 12: From Unit Cubes to the Formulas for Volume Date: 5/27/15 182 © 2014 Common Core, Inc. Some rightsreserved. commoncore.org This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM 6•5Lesson 12 Station C Write two different expressionsto represent the volume, and explain what each expression represents. 𝟒 𝟐 𝟑 m 𝟏 𝟏 𝟖 m 𝟏 𝟑 m
Lesson 12: From Unit Cubes to the Formulas for Volume Date: 5/27/15 183 © 2014 Common Core, Inc. Some rightsreserved. commoncore.org This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM 6•5Lesson 12 Station D 𝐀𝐫𝐞𝐚 = 𝟒 𝟑 𝐟𝐭 𝟐 𝟑 𝟏𝟎 𝐟𝐭.
Lesson 12: From Unit Cubes to the Formulas for Volume Date: 5/27/15 184 © 2014 Common Core, Inc. Some rightsreserved. commoncore.org This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM 6•5Lesson 12 Station E 𝐀𝐫𝐞𝐚 = 𝟏𝟑 𝟏 𝟐 𝐢𝐧 𝟐 𝟏 𝟏 𝟑 𝐢𝐧.
Lesson 12: From Unit Cubes to the Formulas for Volume Date: 5/27/15 185 © 2014 Common Core, Inc. Some rightsreserved. commoncore.org This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM 6•5Lesson 12 Station F Challenge! 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.

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G6 m5-c-lesson 12-t

  • 1.
    Lesson 12: FromUnit Cubes to the Formulas for Volume Date: 5/27/15 170 © 2014 Common Core, Inc. Some rightsreserved. commoncore.org This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM 6•5Lesson 12 Lesson 12: From Unit Cubes to the Formulas for Volume Student Outcomes  Students extend the volume formula for a right rectangular prismto the formula 𝑉 = Area of base ∙ 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 Module5 of Grade 5 (Topics A and B). The word face, though referenced in the lastlesson, should betaught to students who may not know this meaning of it. A student-friendly definition and illustration can beposted 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)  Look at the rectangular prisms in thefirstexample. Write a numerical expression for the volume of each rectangular prism.  Answers provided below.  What do these expressions havein common?  They have the same dimensions for the lengths and widths.  What do these dimensions represent?  The area of the bases of the rectangular prisms.  Rewrite each of the numerical expressions to showwhat they have in common.  Answers provided below.  If we know volume for a rectangular prismas length times width times height, what is another formula for volume that we could use based on these examples?  Area of the base times the height.  What is the area of the baseof each of the rectangular prisms?  𝐴 = 𝑙 𝑤; 𝐴 = (15 𝑖𝑛. )(1 1 2 𝑖𝑛. ); 𝐴 = 22 1 2 in2  How would we use the area of the baseto determine the volumes? (Think about the unit cubes we have been using. The area of the base would be the firstlayer of unitcubes. What would the height represent?)  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 below.  How do the volumes of the firstand second rectangular prisms compare? The firstand third?  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. Scaffolding: You may want to useunit cubes to help students visualize the layers in this problem. MP.7 MP.8
  • 2.
    Lesson 12: FromUnit Cubes to the Formulas for Volume Date: 5/27/15 171 © 2014 Common Core, Inc. Some rightsreserved. commoncore.org This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM 6•5Lesson 12 Example 1 a. Write anumerical expression for thevolume ofeach oftherectangular prismsabove. ( 𝟏𝟓 𝒊𝒏. )(𝟏 𝟏 𝟐 𝒊𝒏. )(𝟑 𝒊𝒏.) ( 𝟏𝟓 𝒊𝒏. )( 𝟏 𝟏 𝟐 𝒊𝒏. )(𝟔 𝒊𝒏.) ( 𝟏𝟓 𝒊𝒏. )( 𝟏 𝟏 𝟐 𝒊𝒏. )(𝟗 𝒊𝒏.) b. What do all ofthese expressionshave in common? What do they represent? All oftheexpressions have ( 𝟏𝟓 𝒊𝒏. )( 𝟏 𝟏 𝟐 𝒊𝒏. ). This is thearea ofthebase. c. Rewrite the numerical expressionsto show what they have in common. ( 𝟐𝟐 𝟏 𝟐 𝒊𝒏 𝟐)(𝟑 𝒊𝒏. ) ( 𝟐𝟐 𝟏 𝟐 𝒊𝒏 𝟐)(𝟔 𝒊𝒏.) ( 𝟐𝟐 𝟏 𝟐 𝒊𝒏 𝟐)(𝟗 𝒊𝒏.) d. If we know volume for arectangular prism aslength timeswidth timesheight,what isanotherformulafor volume that we coulduse based on these examples? (Area ofthebase)(height), or area ofthebasetimes height. e. What isthe areaofthe base for all ofthe rectangular prisms? ( 𝟏𝟓 𝒊𝒏. )( 𝟏 𝟏 𝟐 𝒊𝒏. ) = 𝟐𝟐 𝟏 𝟐 𝒊𝒏 𝟐 f. Determinethe volumeofeach rectangular prism using eithermethod. ( 𝟏𝟓 𝒊𝒏. )( 𝟏 𝟏 𝟐 𝒊𝒏. )( 𝟑 𝒊𝒏. )= 𝟔𝟕 𝟏 𝟐 𝒊𝒏 𝟑 or ( 𝟐𝟐 𝟏 𝟐 𝒊𝒏 𝟐)( 𝟑 𝒊𝒏. ) = 𝟔𝟕 𝟏 𝟐 𝒊𝒏 𝟑 ( 𝟏𝟓 𝒊𝒏. )( 𝟏 𝟏 𝟐 𝒊𝒏. )( 𝟔 𝒊𝒏. )= 𝟏𝟑𝟓 𝒊𝒏 𝟑 or ( 𝟐𝟐 𝟏 𝟐 𝒊𝒏 𝟐)( 𝟔 𝒊𝒏. ) = 𝟏𝟑𝟓 𝒊𝒏 𝟑 ( 𝟏𝟓 𝒊𝒏. )( 𝟏 𝟏 𝟐 𝒊𝒏. )( 𝟗 𝒊𝒏. )= 𝟐𝟎𝟐 𝟏 𝟐 𝒊𝒏 𝟑 or ( 𝟐𝟐 𝟏 𝟐 𝒊𝒏 𝟐)( 𝟗 𝒊𝒏. ) = 𝟐𝟎𝟐 𝟏 𝟐 𝒊𝒏 𝟑 g. How do the volumesofthe first and second rectangular prismscompare? The volumesofthe first and third? 𝟏𝟑𝟓 𝒊𝒏 𝟑 = 𝟔𝟕 𝟏 𝟐 𝒊𝒏 𝟐 × 𝟐 𝒊𝒏.; 𝟐𝟎𝟐 𝟏 𝟐 𝒊𝒏 𝟑 = 𝟔𝟕 𝟏 𝟐 𝒊𝒏 𝟐 × 𝟑 𝒊𝒏. The volume ofthesecond prism is twicethat ofthefirst becausetheheightis doubled. Thevolumeofthe third prism is threetimes as muchas thefirst because theheight is triplethefirst prism’s height.
  • 3.
    Lesson 12: FromUnit Cubes to the Formulas for Volume Date: 5/27/15 172 © 2014 Common Core, Inc. Some rightsreserved. commoncore.org This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM 6•5Lesson 12  What do you think would happen to the volume if we turn this prismon its sideso that a different face is the base? (Have students calculatethe area of the base times the height for this new prism. To help students visualizewhatis happening with this rotation,you could use a textbook or a stack of index cards and discuss how this prismis similarand/or differentto the rectangular prisms in part(a).)  Answers will vary. Some students may see that the volume will be the same no matter which face is the base. Area of the base = (3 𝑖𝑛. ) (1 1 2 𝑖𝑛. ) Area of the base = 4.5 𝑖𝑛2 Volume =Area of the base × height Volume = (4 1 2 𝑖𝑛2) (15 𝑖𝑛. ) Volume = 67 1 2 𝑖𝑛3  How does this volume compare with the volume you calculated usingthe other face as the base?  The volumes in both solutions are the same.  What other expressions could weuse to determine the volume of the prism?  Answers will vary. Some possible variations are included below.  15 in. × 1 1 2 in. × 3 in. 15 in. × 3 in. × 1 1 2 in. 3 in. × 15 in. × 1 1 2 in. 45 in2 × 1 1 2 in.  We noticethat 3 in.× 15 in. × 1 1 2 in.and 45 in2 × 1 1 2 in.areequivalent and both represent the volume. How do they communicate different information?  The first expression (3 in. × 15 in. × 1 1 2 in.) shows that the volume is the product of three edge lengths. The second (45 in2 × 1 1 2 in.) shows that the volume is the product of the area of the base times the height. Example 2 (5 minutes) Example 2 The base ofa rectangular prism hasan areaof 𝟑 𝟏 𝟒 in2. The height ofthe prism is 𝟐 𝟏 𝟐 in. Determinethevolume ofthe rectangular prism. 𝑽 = 𝑨𝒓𝒆𝒂 𝒐𝒇 𝒕𝒉𝒆 𝒃𝒂𝒔𝒆 × 𝒉𝒆𝒊𝒈𝒉𝒕 𝑽 = ( 𝟑 𝟏 𝟒 𝒊𝒏 𝟐)( 𝟐 𝟏 𝟐 𝒊𝒏. ) 𝑽 = ( 𝟏𝟑 𝟒 𝒊𝒏 𝟐)( 𝟓 𝟐 𝒊𝒏. ) 𝑽 = 𝟔𝟓 𝟖 𝒊𝒏 𝟑 𝟏𝟓 𝐢𝐧. 𝟑 𝐢𝐧. 𝟏 𝟏 𝟐 𝐢𝐧. MP.7
  • 4.
    Lesson 12: FromUnit Cubes to the Formulas for Volume Date: 5/27/15 173 © 2014 Common Core, Inc. Some rightsreserved. commoncore.org This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM 6•5Lesson 12  Do we need to know the length and the width to find the volume of the rectangular prism?  No, if we know the area of the base, we do not need the length and width. The length and width is used to calculate the area, and we are already given the area. Exercises1–6 (20 minutes) The cards areprinted out and used as stations or hung on the classroomwallsso thatstudents can move from question to question. Copies of the questions can be found at the end of the lesson. Multiplecopies of each question can be printed so that a small number of students visiteach question at a time. Students should spend about three minutes at each station where they will showtheir work by firstwritinga numerical expression,and then use the expression to calculatethe volume of the rectangular prismdescribed. They will usethe rest of the time to discusstheanswers,and the teacher can answer any questions students have about the lesson. 1. Draw a sketch of the figure. Then calculatethe volume. Rectangular Prism Area of the base= 4 3 8 ft2 Height = 2 1 2 ft. 𝑉 = 𝐴𝑟𝑒𝑎 𝑜𝑓 𝑏𝑎𝑠𝑒 × ℎ𝑒𝑖𝑔ℎ𝑡 𝑉 = (4 3 8 𝑓𝑡2) (2 1 2 𝑓𝑡) 𝑉 = ( 35 8 𝑓𝑡2) ( 5 2 𝑓𝑡) 𝑉 = 175 16 𝑓𝑡3 2. Draw a sketch of the figure. Writethe length, width, and height in feet. Then calculatethe volume. Rectangular Prism Length is 2 1 2 times as longas the height. Width is 3 4 as longas the height. Height = 3 ft. 𝐿𝑒𝑛𝑔𝑡ℎ = 3 𝑓𝑡 × 2 1 2 = 15 2 𝑓𝑡 𝑊𝑖𝑑𝑡ℎ = 3 𝑓𝑡 × 3 4 = 9 4 𝑓𝑡 𝑉 = 𝑙 𝑤 ℎ 𝑉 = ( 15 2 𝑓𝑡) ( 9 4 𝑓𝑡) (3 𝑓𝑡) 𝑉 = 405 8 𝑓𝑡3 3. Write two different expressions to represent the volume and explain whateach one represents. Answers will vary. Some possible solutions include (4 2 3 𝑚) ( 1 3 𝑚) (1 1 8 𝑚) and ( 14 9 𝑚2) (1 1 8 𝑚) The first expression shows the volume as a product of the three edge lengths. The second expression shows the volume as a product of a base (4 2 3 𝑚) ( 1 3 𝑚) times the height. 𝟒 𝟐 𝟑 𝐦 𝟏 𝟏 𝟖 𝐦 𝟏 𝟑 𝐦 MP.7
  • 5.
    Lesson 12: FromUnit Cubes to the Formulas for Volume Date: 5/27/15 174 © 2014 Common Core, Inc. Some rightsreserved. commoncore.org This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM 6•5Lesson 12 4. Calculatethe volume. 𝑉 = 𝐴𝑟𝑒𝑎 𝑜𝑓 𝑏𝑎𝑠𝑒 × ℎ𝑒𝑖𝑔ℎ𝑡 𝑉 = ( 4 3 𝑓𝑡2) ( 3 10 𝑓𝑡. ) 𝑉 = 12 30 𝑓𝑡3 𝑉 = 2 5 𝑓𝑡3 5. Calculatethe volume. 𝑉𝑜𝑙𝑢𝑚𝑒 = 𝐴𝑟𝑒𝑎 𝑜𝑓 𝑏𝑎𝑠𝑒 × ℎ𝑒𝑖𝑔ℎ𝑡 𝑉𝑜𝑙𝑢𝑚𝑒 = (13 1 2 𝑖𝑛2) (1 1 3 𝑖𝑛. ) 𝑉𝑜𝑙𝑢𝑚𝑒 = 108 6 𝑖𝑛3 𝑉𝑜𝑙𝑢𝑚𝑒 = 18 𝑖𝑛3 6. Challenge! 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 prismis 5 ft. 𝐿𝑒𝑛𝑔𝑡ℎ = 5 𝑓𝑡. 𝑊𝑖𝑑𝑡ℎ = 5 𝑓𝑡.÷ 3 = 5 3 𝑓𝑡. 𝐻𝑒𝑖𝑔ℎ𝑡 = 5 3 𝑓𝑡.× 3 2 = 5 2 𝑓𝑡. 𝑉 = 𝑙 𝑤 ℎ 𝑉 = (5 𝑓𝑡. ) ( 5 3 𝑓𝑡. )( 5 2 𝑓𝑡. ) 𝑉 = 125 6 𝑓𝑡3 𝐀𝐫𝐞𝐚 = 𝟏𝟑 𝟏 𝟐 𝐢𝐧 𝟐 𝟏 𝟏 𝟑 𝐢𝐧. 𝐀𝐫𝐞𝐚 = 𝟒 𝟑 𝐟𝐭 𝟐 𝟑 𝟏𝟎 𝐟𝐭.
  • 6.
    Lesson 12: FromUnit Cubes to the Formulas for Volume Date: 5/27/15 175 © 2014 Common Core, Inc. Some rightsreserved. commoncore.org This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM 6•5Lesson 12 Extension(3 minutes) Extension A company iscreating arectangular prism that must have avolumeof 𝟔ft3. The company also knowsthat the areaofthe base must be 𝟐 𝟏 𝟐 ft2. How can you use what you learnedtoday about volume todeterminetheheight ofthe rectangular prism? I know that thevolumecan becalculated by multiplying theareaofthebasetimes theheight. So, ifI needed theheight instead, I would do theopposite. I woulddividethevolumeby thearea ofthebaseto determinetheheight. 𝑽 = 𝑨𝒓𝒆𝒂 𝒐𝒇 𝒃𝒂𝒔𝒆 × 𝒉𝒆𝒊𝒈𝒉𝒕 𝟔 𝒇𝒕 𝟑 = ( 𝟐 𝟏 𝟐 𝒇𝒕 𝟐)( 𝒉) 𝟔 𝒇𝒕 𝟑 ÷ 𝟐 𝟏 𝟐 𝒇𝒕 𝟐 = 𝒉𝒆𝒊𝒈𝒉𝒕 𝟐 𝟐 𝟓 𝒇𝒕. = 𝒉𝒆𝒊𝒈𝒉𝒕 Closing(2 minutes)  How is the formula 𝑉 = 𝑙 ∙ 𝑤 ∙ ℎ related to the formula 𝑉 = Area of the base ∙ height?  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)
  • 7.
    Lesson 12: FromUnit Cubes to the Formulas for Volume Date: 5/27/15 176 © 2014 Common Core, Inc. Some rightsreserved. commoncore.org This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM 6•5Lesson 12 Name Date Lesson 12: From Unit Cubes to the Formulas for Volume Exit Ticket 1. Determine the volume of the rectangular prismin two different ways. 2. The area of the base of a rectangular prismis 12 cm2 and the height is 3 1 3 cm. Determine the volume of the rectangular prism. 3 4 ft. 3 4 ft. 3 8 ft.
  • 8.
    Lesson 12: FromUnit Cubes to the Formulas for Volume Date: 5/27/15 177 © 2014 Common Core, Inc. Some rightsreserved. commoncore.org This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM 6•5Lesson 12 𝟑 𝟒 𝐟𝐭. 𝟑 𝟒 𝐟𝐭. 𝟑 𝟖 𝐟𝐭. Exit Ticket Sample Solutions 1. Determinethe volumeofthe rectangular prism. 𝑽 = 𝒍 ∙ 𝒘 ∙ 𝒉 𝑽𝒐𝒍𝒖𝒎𝒆 = 𝑨𝒓𝒆𝒂 𝒐𝒇 𝒃𝒂𝒔𝒆 ∙ 𝒉𝒆𝒊𝒈𝒉𝒕 𝑽 = ( 𝟑 𝟒 𝒇𝒕. )( 𝟑 𝟖 𝒇𝒕. )( 𝟑 𝟒 𝒇𝒕. ) 𝑽𝒐𝒍𝒖𝒎𝒆 = ( 𝟗 𝟑𝟐 𝒇𝒕 𝟐)( 𝟑 𝟒 𝒇𝒕. ) 𝑽 = 𝟐𝟕 𝟏𝟐𝟖 𝒇𝒕 𝟑 𝑽𝒐𝒍𝒖𝒎𝒆 = 𝟐𝟕 𝟏𝟐𝟖 𝒇𝒕 𝟑 2. The areaofthe base ofa rectangular prism is 𝟏𝟐 𝐜𝐦 𝟐 and the heightis 𝟑 𝟏 𝟑 𝐜𝐦. Determinethevolumeofthe rectangular prism. 𝑽𝒐𝒍𝒖𝒎𝒆 = 𝑨𝒓𝒆𝒂 𝒐𝒇 𝒃𝒂𝒔𝒆 ∙ 𝒉𝒆𝒊𝒈𝒉𝒕 𝑽𝒐𝒍𝒖𝒎𝒆 = ( 𝟏𝟐 𝒄𝒎 𝟐 )( 𝟑 𝟏 𝟑 𝒄𝒎) 𝑽𝒐𝒍𝒖𝒎𝒆 = 𝟏𝟐𝟎 𝟑 𝒄𝒎 𝟑 𝑽𝒐𝒍𝒖𝒎𝒆 = 𝟒𝟎 𝒄𝒎 𝟑 Problem Set Sample Solutions 1. Determinethe volumeofthe rectangular prism. 𝑽 = 𝒍 𝒘 𝒉 𝑽 = ( 𝟏 𝟏 𝟐 𝒎)( 𝟏 𝟐 𝒎)( 𝟕 𝟖 𝒎) 𝑽 = 𝟐𝟏 𝟑𝟐 𝒎 𝟑 2. The areaofthe base ofa rectangular prism is 𝟒 𝟑 𝟒 ft2 and the heightis 𝟐 𝟏 𝟑 ft. Determine thevolume ofthe rectangular prism. 𝑽𝒐𝒍𝒖𝒎𝒆 = 𝑨𝒓𝒆𝒂 𝒐𝒇 𝒃𝒂𝒔𝒆 × 𝒉𝒆𝒊𝒈𝒉𝒕 𝑽𝒐𝒍𝒖𝒎𝒆 = ( 𝟒 𝟑 𝟒 𝒇𝒕 𝟐)( 𝟐 𝟏 𝟑 𝒇𝒕. ) 𝑽𝒐𝒍𝒖𝒎𝒆 = ( 𝟏𝟗 𝟒 𝒇𝒕 𝟐)( 𝟕 𝟑 𝒇𝒕. ) 𝑽𝒐𝒍𝒖𝒎𝒆 = 𝟏𝟑𝟑 𝟏𝟐 𝒇𝒕 𝟑 𝟕 𝟖 𝐦 𝟏 𝟐 𝐦 𝟏 𝟏 𝟐 𝐦
  • 9.
    Lesson 12: FromUnit Cubes to the Formulas for Volume Date: 5/27/15 178 © 2014 Common Core, Inc. Some rightsreserved. commoncore.org This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM 6•5Lesson 12 3. The length ofarectangular prism is 𝟑 𝟏 𝟐 timesaslong asthe width. The height is 𝟏 𝟒 ofthe width. The widthis 𝟑cm. Determinethe volume. 𝑾𝒊𝒅𝒕𝒉 = 𝟑 𝒄𝒎 𝑳𝒆𝒏𝒈𝒕𝒉 = 𝟑 𝒄𝒎× 𝟑 𝟏 𝟐 = 𝟐𝟏 𝟐 𝒄𝒎 𝑯𝒆𝒊𝒈𝒉𝒕= 𝟑 𝒄𝒎 × 𝟏 𝟒 = 𝟑 𝟒 𝒄𝒎 𝑽𝒐𝒍𝒖𝒎𝒆 = 𝒍 𝒘 𝒉 𝑽𝒐𝒍𝒖𝒎𝒆 = ( 𝟐𝟏 𝟐 𝒄𝒎)( 𝟑 𝒄𝒎)( 𝟑 𝟒 𝒄𝒎) 𝑽𝒐𝒍𝒖𝒎𝒆 = 𝟏𝟖𝟗 𝟖 𝒄𝒎 𝟑 4. a. Write numerical expressionsin two differentways, and explain what each reveals. ( 𝟏𝟎 𝟏 𝟐 𝒊𝒏. )( 𝟏 𝟐 𝟑 𝒊𝒏. )(𝟔 𝒊𝒏.) represents theproduct ofthreeedgelengths. ( 𝟑𝟓 𝟐 𝒊𝒏 𝟐)( 𝟔 𝒊𝒏)represents the product ofthebasearea times theheight. Answers will vary. b. Determinethe volumeofthe rectangular prism. ( 𝟏𝟎 𝟏 𝟐 𝒊𝒏. )( 𝟏 𝟐 𝟑 𝒊𝒏. )( 𝟔 𝒊𝒏. ) = 𝟏𝟎𝟓 𝒊𝒏 𝟑 or ( 𝟑𝟓 𝟐 𝒊𝒏 𝟐)( 𝟔 𝒊𝒏. ) = 𝟏𝟎𝟓 𝒊𝒏 𝟑 5. An aquarium in the shape ofarectangular prism hasthe following dimensions: length = 𝟓𝟎cm, width= 𝟐𝟓 𝟏 𝟐 cm, height = 𝟑𝟎 𝟏 𝟐 cm. a. Write numerical expressionsin two differentways, and explain what each reveals. ( 𝟓𝟎 𝒄𝒎)( 𝟐𝟓 𝟏 𝟐 𝒄𝒎)( 𝟑𝟎 𝟏 𝟐 𝒄𝒎)represents theproduct ofthethreeedgelengths. (𝟏,𝟐𝟕𝟓 𝒄𝒎 𝟐 )( 𝟑𝟎 𝟏 𝟐 𝒄𝒎)represents thebase area times theheight. b. Determinethe volumeofthe rectangular prism. ( 𝟏,𝟐𝟕𝟓 𝒄𝒎 𝟐)( 𝟑𝟎 𝟏 𝟐 𝒄𝒎) = 𝟑𝟖,𝟖𝟖𝟕 𝟏 𝟐 𝒄𝒎 𝟑 𝟏 𝟐 𝟑 in. 𝟏𝟎 𝟏 𝟐 in. 𝟔 in.
  • 10.
    Lesson 12: FromUnit Cubes to the Formulas for Volume Date: 5/27/15 179 © 2014 Common Core, Inc. Some rightsreserved. commoncore.org This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM 6•5Lesson 12 6. The areaofthe base in thisrectangular prismisfixed at 𝟑𝟔cm2. That meansfor the varying heightsthere will be variousvolumes. a. Complete thetableofvaluesto determinethe various heightsand volumes. Height in Centimeters Volume in Cubic Centimeters 𝟏 𝟑𝟔 𝟐 𝟕𝟐 𝟑 𝟏𝟎𝟖 𝟒 𝟏𝟒𝟒 𝟓 𝟏𝟖𝟎 𝟔 𝟐𝟏𝟔 𝟕 𝟐𝟓𝟐 𝟖 𝟐𝟖𝟖 b. Write an equation to representtherelationship in thetable. Be sure to definethevariablesused in the equation. Let 𝒙 be theheight oftherectangularprism incentimeters. Let 𝒚be thevolumeoftherectangular prism in cubic centimeters. 𝟑𝟔𝒙 = 𝒚 c. What isthe unit rate for thisproportional relationship? What doesit mean inthissituation? The unit rateis 𝟑𝟔. For every centimeter ofheight, thevolumeincreases by 𝟑𝟔cubic centimeters becausethearea ofthebaseis 𝟑𝟔cm2. In order to determinethevolume, multiply theheightby 𝟑𝟔. 7. The volume ofarectangular prism is 𝟏𝟔.𝟑𝟐𝟖cm3. The heightis 𝟑. 𝟏𝟒cm. a. Let 𝑨 represent the areaofthe base oftherectangular prism. Writean equation that relatesthevolume, the areaof the base, and the height. 𝟏𝟔.𝟑𝟐𝟖 = 𝟑.𝟏𝟒𝑨 b. Solve the equationfor 𝑨. 𝟏𝟔.𝟑𝟐𝟖 𝟑. 𝟏𝟒 = 𝟑.𝟏𝟒𝑨 𝟑. 𝟏𝟒 𝑨 = 𝟓.𝟐 The area ofthebaseis 𝟓.𝟐 cm2. 𝟏𝟐cm 𝟑cm
  • 11.
    Lesson 12: FromUnit Cubes to the Formulas for Volume Date: 5/27/15 180 © 2014 Common Core, Inc. Some rightsreserved. commoncore.org This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM 6•5Lesson 12 Station A Make a sketch of the figure. Then calculate the volume. Rectangular prism: Area of the base = 𝟒 𝟑 𝟖 ft2 Height = 𝟐 𝟏 𝟐 ft.
  • 12.
    Lesson 12: FromUnit Cubes to the Formulas for Volume Date: 5/27/15 181 © 2014 Common Core, Inc. Some rightsreserved. commoncore.org This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM 6•5Lesson 12 Station B Make a sketch of the figure. Write the length, the width, and height in feet. Then calculate the volume. Rectangular prism: Length is 𝟐 𝟏 𝟐 times as long as the height. Width is 𝟑 𝟒 as long as the height. Height = 𝟑 ft.
  • 13.
    Lesson 12: FromUnit Cubes to the Formulas for Volume Date: 5/27/15 182 © 2014 Common Core, Inc. Some rightsreserved. commoncore.org This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM 6•5Lesson 12 Station C Write two different expressionsto represent the volume, and explain what each expression represents. 𝟒 𝟐 𝟑 m 𝟏 𝟏 𝟖 m 𝟏 𝟑 m
  • 14.
    Lesson 12: FromUnit Cubes to the Formulas for Volume Date: 5/27/15 183 © 2014 Common Core, Inc. Some rightsreserved. commoncore.org This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM 6•5Lesson 12 Station D 𝐀𝐫𝐞𝐚 = 𝟒 𝟑 𝐟𝐭 𝟐 𝟑 𝟏𝟎 𝐟𝐭.
  • 15.
    Lesson 12: FromUnit Cubes to the Formulas for Volume Date: 5/27/15 184 © 2014 Common Core, Inc. Some rightsreserved. commoncore.org This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM 6•5Lesson 12 Station E 𝐀𝐫𝐞𝐚 = 𝟏𝟑 𝟏 𝟐 𝐢𝐧 𝟐 𝟏 𝟏 𝟑 𝐢𝐧.
  • 16.
    Lesson 12: FromUnit Cubes to the Formulas for Volume Date: 5/27/15 185 © 2014 Common Core, Inc. Some rightsreserved. commoncore.org This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM 6•5Lesson 12 Station F Challenge! 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.