G6 m5-c-lesson 13-t

Lesson 13: The Formulas for Volume
Date: 5/27/15 186
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NYS COMMON CORE MATHEMATICS CURRICULUM 6•5Lesson 13
Lesson 13: The Formulasfor Volume
Student Outcomes
 Students develop, understand,and apply formulas for findingthe volume of right rectangular prisms and
cubes.
Lesson Notes
This lesson is a continuation of Lessons 11,12, and Module 5, Topics A and B from Grade 5.
FluencyExercise (5 minutes)
Multiplication and Division Equation with Fractions WhiteBoard Exchange
Classwork
Example 1 (3 minutes)
Example 1
Determinethe volumeofacube with sidelengthsof 𝟐
𝟏
𝟒
cm.
𝑽 = 𝒍 𝒘 𝒉
𝑽 = ( 𝟐
𝟏
𝟒
𝒄𝒎)( 𝟐
𝟏
𝟒
𝒄𝒎)( 𝟐
𝟏
𝟒
𝒄𝒎)
𝑽 =
𝟗
𝟒
𝒄𝒎 ×
𝟗
𝟒
𝒄𝒎 ×
𝟗
𝟒
𝒄𝒎
𝑽 =
𝟕𝟐𝟗
𝟔𝟒
𝒄𝒎 𝟑
Have students work through the firstproblem on their own and then discuss.
 Which method for determining the volume did you choose?
 Answers will vary. Sample response: I chose to use the 𝑉 = 𝑙 𝑤 ℎ formula to solve.
 Why did you choose this method?
 Explanations with vary according to the method chosen. Sample response: Because I know the length,
width, and height of the prism, I used 𝑉 = 𝑙 𝑤 ℎ instead of the other examples.
MP.1
Scaffolding:
Providea visual of a cubefor
students to label. If needed,
begin with less complex
numbers for the edge lengths.
𝑉 = (9cm)(9 cm)(9 cm)
𝑉 = 729 cm3
9 cm
9 cm
9 cm

Date: 5/27/15 187
𝐀𝐫𝐞𝐚 =
𝟏𝟑
𝟐
𝐟𝐭 𝟐
𝟓
𝟑
𝐟𝐭
Example 2 (3 minutes)
Example 2
Determinethe volumeofarectangular prism withabase areaof
𝟕
𝟏𝟐
ft2 and aheight of
𝟏
𝟑
ft.
𝑽 = 𝑨𝒓𝒆𝒂 𝒐𝒇 𝒃𝒂𝒔𝒆 × 𝒉𝒆𝒊𝒈𝒉𝒕
𝑽 = (
𝟕
𝟏𝟐
𝒇𝒕 𝟐)(
𝟏
𝟑
𝒇𝒕. )
𝑽 =
𝟕
𝟑𝟔
𝒇𝒕 𝟑
 What makes this problem different than the firstexample?
 This example gives the area of the base instead of just giving the length and width.
 Would it be possibleto use another method or formula to determine the volume of the prismin this example?
 I could try fitting cubes with fractional lengths. However, I could not use the 𝑉 = 𝑙 𝑤 ℎ formula
because I do not know the length and width of the base.
Exercises1–5 (27 minutes)
In the exercises,students will explorehow changes in the lengths of the sides affectthe volume. Students can use any
method to determine the volume as longas they can explain their solution. Students work in pairs or small groups.
(Pleasenote that the relationshipsbetween the volumes will be more easily determined if the fractions are left in their
original formwhen solving. If time allows,this could bean interestingdiscussion pointeither between partners,groups,
or as a whole class when discussingthe results of their work.)
Exercises 1–5
1. Use the rectangular prism toanswer thenextset ofquestions.
a. Determinethe volumeofthe prism.
𝑽 = 𝑨𝒓𝒆𝒂 𝒐𝒇 𝒕𝒉𝒆 𝒃𝒂𝒔𝒆× 𝒉𝒆𝒊𝒈𝒉𝒕
𝑽 = (
𝟏𝟑
𝟐
𝒇𝒕 𝟐)(
𝟓
𝟑
𝒇𝒕. )
𝑽 =
𝟔𝟓
𝟔
𝒇𝒕 𝟑
Scaffolding:
 The wording half as long
may confuse some
students. Explain that half
as longmeans that the
original length was
multiplied by one half. A
similar explanation can be
used for one third as long
and one fourth as long.
 Explain to students that
the word doubled refers to
twice as many or
multiplied by two.

Date: 5/27/15 188
b. Determinethe volumeofthe prism ifthe height ofthe prism isdoubled.
𝑯𝒆𝒊𝒈𝒉𝒕× 𝟐 = (
𝟓
𝟑
𝒇𝒕.× 𝟐) =
𝟏𝟎
𝟑
𝒇𝒕.
𝑽 = 𝑨𝒓𝒆𝒂 𝒐𝒇 𝒃𝒂𝒔𝒆× 𝒉𝒆𝒊𝒈𝒉𝒕
𝑽 = (
𝟏𝟑
𝟐
𝒇𝒕 𝟐)(
𝟏𝟎
𝟑
𝒇𝒕. )
𝑽 =
𝟏𝟑𝟎
𝟔
𝒇𝒕 𝟑
c. Compare the volumeofthe rectangular prismin part (a)with the volume oftheprism in part (b). What do
you notice?
When theheight oftherectangularprism is doubled, thevolumeis also doubled.
d. Complete and use thetablebelow to determine the relationshipsbetween theheight and volume.
Height in Feet Volume in Cubic Feet
𝟓
𝟑
𝟔𝟓
𝟔
𝟏𝟎
𝟑
𝟏𝟑𝟎
𝟔
𝟏𝟓
𝟑
𝟏𝟗𝟓
𝟔
𝟐𝟎
𝟑
𝟐𝟔𝟎
𝟔
What happened to thevolumewhentheheight wastripled?
The volumetripled.
What happened to thevolumewhentheheight wasquadrupled?
The volumequadrupled.
What conclusionscan you make when thebase areastaysconstant and only theheight changes?
Answers will vary but should includetheidea of a proportionalrelationship. Eachtimetheheight is
multiplied by a number, theoriginalvolumewill bemultiplied by thesameamount.
2. a. If 𝑨 representsthe areaofthe base and 𝒉 representstheheight,writean expression that representsthe
volume.
𝑨𝒉
b. If we double theheight, write an expressionfor the new height.
𝟐𝒉
c. Write an expression that representsthe volumewith the doubled height.
𝑨𝟐𝒉
d. Write an equivalent expressionusing thecommutative and associativepropertiesto show the volumeis
twice the original volume.
𝟐(𝑨𝒉)
MP.2
MP.7

Date: 5/27/15 189
𝟗 𝐟𝐭.
𝟐 𝐟𝐭.
𝟑 𝐟𝐭.
3. Use the cube to answer thefollowing questions.
a. Determinethe volumeofthe cube.
𝑽 = ( 𝟑𝒎)( 𝟑𝒎)( 𝟑𝒎)
𝑽 = 𝟐𝟕 𝒎 𝟑
b. Determinethe volume ofacube whose sidelengthsare halfaslong asthe side lengthsofthe original cube.
𝑽 = (
𝟑
𝟐
𝒎)(
𝟑
𝟐
𝒎)(
𝟑
𝟐
𝒎)
𝑽 =
𝟐𝟕
𝟖
𝒎 𝟑
c. Determinethe volumeifthe sidelengthsare one fourth aslong asthe original cube’ssidelengths.
𝑽 = (
𝟑
𝟒
𝒎)(
𝟑
𝟒
𝒎)(
𝟑
𝟒
𝒎)
𝑽 =
𝟐𝟕
𝟔𝟒
𝒎 𝟑
d. Determinethe volumeifthe sidelengthsare one sixth aslong asthe original cube’ssidelength.
𝑽 = (
𝟑
𝟔
𝒎)(
𝟑
𝟔
𝒎)(
𝟑
𝟔
𝒎)
𝑽 =
𝟐𝟕
𝟐𝟏𝟔
𝒎 𝟑
OR
𝑽 =
𝟏
𝟖
𝒎 𝟑
e. Explain the relationship between the sidelengthsand thevolumesofthe cubes.
If each ofthesides arechanged by thesamefractional amount ( 𝟏
𝒂
)oftheoriginal,then the volumeofthe
new figurewill be ( 𝟏
𝒂
)
𝟑
oftheoriginal volume. For example, ifthesides are
𝟏
𝟐
as long, then thevolumewillbe
( 𝟏
𝟐
)
𝟑
=
𝟏
𝟖
as much.
4. Check to see ifthe relationship you foundin Exercise 1isthe same for rectangular prisms.
a. Determinethe volumeofthe rectangular prism.
𝑽 = ( 𝟗 𝒇𝒕. )( 𝟐 𝒇𝒕. )( 𝟑 𝒇𝒕. )
𝑽 = 𝟓𝟒 𝒇𝒕 𝟑
𝟑 𝐦

Date: 5/27/15 190
b. Determinethe volumeifall ofthe sidesare halfaslong asthe original lengths.
𝑽 = (
𝟗
𝟐
𝒇𝒕. )(
𝟐
𝟐
𝒇𝒕. )(
𝟑
𝟐
𝒇𝒕. )
𝑽 =
𝟓𝟒
𝟖
𝒇𝒕 𝟑
OR
𝑽 =
𝟐𝟕
𝟒
𝒇𝒕 𝟑
c. Determinethe volumeifall ofthe sidesare one third aslong asthe originallengths.
𝑽 = (
𝟗
𝟑
𝒇𝒕. )(
𝟐
𝟑
𝒇𝒕. )(
𝟑
𝟑
𝒇𝒕. )
𝑽 =
𝟓𝟒
𝟐𝟕
𝒇𝒕 𝟑
OR
𝑽 = 𝟐 𝒇𝒕 𝟑
d. Is the relationship between the sidelengthsand thevolumethesame asthe one that occurred in Exercise 1?
Explain your answer.
Yes, therelationship that was found intheproblem with thecubes still holds truewith this rectangular prism.
When I found thevolumeofa prism with sidelengths that wereone-thirdtheoriginal,thevolumewas
( 𝟏
𝟑
)
𝟑
=
𝟏
𝟐𝟕
theoriginal.
5. a. If 𝒆 representsa side length ofthe cube,create an expressionthat showsthe volumeofthe cube.
𝒆 𝟑
b. If we divide the sidelengthsby three,create an expressionfor the new edgelength.
𝟏
𝟑
𝒆 or
𝒆
𝟑
c. Write an expression that representsthe volume ofthe cube with one thirdtheside length.
( 𝟏
𝟑
𝒆)
𝟑
or ( 𝒆
𝟑
)
𝟑
d. Write an equivalent expressionto show that thevolumeis
𝟏
𝟐𝟕
ofthe original volume.
𝑽 = (
𝟏
𝟑
𝒆)
𝟑
𝑽 = (
𝟏
𝟑
𝒆)(
𝟏
𝟑
𝒆)(
𝟏
𝟑
𝒆)
𝑽 = (
𝟏
𝟗
𝒆 𝟐)(
𝟏
𝟑
𝒆)
𝑽 =
𝟏
𝟐𝟕
𝒆 𝟑
MP.2
&
MP.7

Date: 5/27/15 191
Closing(2 minutes)
 How did you determine which method to use when solvingtheexercises?
 If I were given the length, width, and height, I have many options for determining the volume. I could
use 𝑉 = 𝑙 𝑤 ℎ. I could also determine the area of the base first and then use 𝑉 = Area of the base ×
height. I could also use a unit cube and determine how many cubes would fit inside.
 If I was given the area of the base and the height, I could use the formula 𝑉 = Area of the base ×
height, or I could also use a unit cube and determine how many cubes would fit inside.
 What relationships did you noticebetween the volume and changes in the length, width, or height?
 Answers will vary. Students may mention that if the length, width, or height is changed by a certain
factor, the volume will be affected by that same factor.
 They may also mention that if all three dimensions are changed by the same factor, the volume will
change by that factor cubed. For example, if all the sides are
1
2
as long as the original, the volume will
be (
1
2
)
3
as large as the original.
Exit Ticket (5 minutes)

Date: 5/27/15 192
Name Date
Lesson 13: The Formulasfor Volume
Exit Ticket
1. A new company wants to mail out samples of its hair products. The company has a samplebox that is a rectangular
prismwith a rectangular basewith an area of 23
1
3
in2. The height of the prismis 1
1
4
in.
Determine the volume of the samplebox.
2. A different samplebox has a height that is twice as longas the original. Whatis the volume of this samplebox?
How does the volume of this samplebox compare to the volume of the samplebox in Problem 1?

Date: 5/27/15 193
𝐀𝐫𝐞𝐚 =
𝟑𝟎
𝟕
𝐜𝐦 𝟐
𝟏
𝟑
𝐜𝐦
Exit Ticket Sample Solutions
1. A new company wantsto mail out samplesof its hair products. The company has asample box that isa rectangular
prism with a rectangular base with an areaof 𝟐𝟑
𝟏
𝟑
in2. The height oftheprism is 𝟏
𝟏
𝟒
in. Determinethe volumeof
the sample box.
𝑽 = ( 𝟐𝟑
𝟏
𝟑
𝒊𝒏 𝟐)( 𝟏
𝟏
𝟒
𝒊𝒏. )
𝑽 =
𝟕𝟎
𝟑
𝒊𝒏 𝟐
×
𝟓
𝟒
𝒊𝒏.
𝑽 =
𝟑𝟓𝟎
𝟏𝟐
𝒊𝒏 𝟑
OR
𝑽 =
𝟏𝟕𝟓
𝟔
𝒊𝒏 𝟑
2. A different sample box hasaheight that istwice aslong asthe original. What isthe volumeofthissample box?
How doesthe volume ofthissamplebox compare tothe volumeofthe samplebox in Problem 1?
𝑽 = ( 𝟐𝟑
𝟏
𝟑
𝒊𝒏 𝟐)( 𝟐
𝟏
𝟐
𝒊𝒏. )
𝑽 = (
𝟕𝟎
𝟑
𝒊𝒏 𝟐)(
𝟓
𝟐
𝒊𝒏. )
𝑽 =
𝟑𝟓𝟎
𝟔
𝒊𝒏 𝟑
OR
𝑽 =
𝟏𝟕𝟓
𝟑
𝒊𝒏 𝟑
By doubling theheight, we havealso doubledthevolume.
Problem Set Sample Solutions
1. Determinethe volumeofthe rectangular prism.
𝑽 = (
𝟑𝟎
𝟕
𝒄𝒎 𝟐)(
𝟏
𝟑
𝒄𝒎)
𝑽 =
𝟑𝟎
𝟐𝟏
𝒄𝒎 𝟑
OR

Date: 5/27/15 194
2. Determinethe volumeofthe rectangular prism inProblem1 ifthe heightisquadrupled (multiplied by four). Then
determinethe relationship between thevolumesin Problem 1and thisprism.
𝑽 = (
𝟑𝟎
𝟕
𝒄𝒎 𝟐)(
𝟒
𝟑
𝒄𝒎)
𝑽 =
𝟏𝟐𝟎
𝟐𝟏
𝒄𝒎 𝟑
OR
𝑽 =
𝟒𝟎
𝟕
𝒄𝒎 𝟑
When theheight was quadrupled,thevolumewas also quadrupled.
3. The areaofthe base ofa rectangular prism can be represented by 𝑨, and the height isrepresentedby 𝒉.
a. Write an expression that representsthe volumeofthe prism.
𝑽 = 𝑨𝒉
b. If the areaofthe base isdoubled,writean expression that represents the volumeofthe prism.
𝑽 = 𝟐𝑨𝒉
c. If the height ofthe prism isdoubled,writean expression that representsthe volumeofthe prism.
𝑽 = 𝑨𝟐𝒉 = 𝟐𝑨𝒉
d. Compare the volumein parts(b)and (c). What do you noticeabout thevolumes?
The expressions in part (b)and part (c)areequalto eachother.
e. Write an expression for thevolumeofthe prism ifboth theheight and theareaofthe base are doubled.
𝑽 = 𝟐𝑨𝟐𝒉 = 𝟒𝑨𝒉
4. Determinethe volumeofacube with aside length of 𝟓
𝟏
𝟑
in.
𝑽 = ( 𝟓
𝟏
𝟑
𝒊𝒏. )( 𝟓
𝟏
𝟑
𝒊𝒏. )( 𝟓
𝟏
𝟑
𝒊𝒏. )
𝑽 =
𝟏𝟔
𝟑
𝒊𝒏.×
𝟏𝟔
𝟑
𝒊𝒏.×
𝟏𝟔
𝟑
𝒊𝒏.
𝑽 =
𝟒𝟎𝟗𝟔
𝟐𝟕
𝒊𝒏 𝟑
5. Use the information in Problem 4 to answerthe following:
a. Determinethe volumeofthe cube inProblem4 ifall ofthe side lengthsare cut inhalf.
𝑽 = ( 𝟐
𝟐
𝟑
𝒊𝒏. )( 𝟐
𝟐
𝟑
𝒊𝒏. )( 𝟐
𝟐
𝟑
𝒊𝒏. )
𝑽 =
𝟖
𝟑
𝒊𝒏.×
𝟖
𝟑
𝒊𝒏.×
𝟖
𝟑
𝒊𝒏.
𝑽 =
𝟓𝟏𝟐
𝟐𝟕
𝒊𝒏 𝟑

Date: 5/27/15 195
b. How could you determine thevolume ofthecubewiththe sidelengthscutin half using thevolumein
Problem 4?
Becauseeach sideis halfas long, I know that thevolumewill be
𝟏
𝟖
thevolumeofthecubein Problem 4. This is
becausethelength, the width,and theheightwereall cutin half.
𝟏
𝟐
𝒍 ×
𝟏
𝟐
𝒘 ×
𝟏
𝟐
𝒉 =
𝟏
𝟖
𝒍𝒘𝒉
𝟏
𝟖
×
𝟒,𝟎𝟗𝟔
𝟐𝟕
𝒊𝒏 𝟑
=
𝟓𝟏𝟐
𝟐𝟕
𝒊𝒏 𝟑
6. Use the rectangular prism toanswer thefollowing questions.
a. Complete thetable.
Length Volume
𝒍 = 𝟖 𝐜𝐦 𝟏𝟐 𝒄𝒎 𝟑
𝟏
𝟐
𝒍 = 𝟒 𝐜𝐦 𝟔 𝒄𝒎 𝟑
𝟏
𝟑
𝒍 =
𝟖
𝟑
𝐜𝐦 𝟒 𝒄𝒎 𝟑
𝟏
𝟒
𝒍 = 𝟐 𝐜𝐦 𝟑 𝒄𝒎 𝟑
𝟐𝒍= 𝟏𝟔 𝐜𝐦 𝟐𝟒 𝒄𝒎 𝟑
𝟑𝒍= 𝟐𝟒 𝐜𝐦 𝟑𝟔 𝒄𝒎 𝟑
𝟒𝒍= 𝟑𝟐 𝐜𝐦 𝟒𝟖 𝒄𝒎 𝟑
b. How did the volume change when thelength wasone third as long?
𝟒is one third of 𝟏𝟐. Therefore, when thelength is one third as long, thevolumeis onethird as much also.
c. How did the volume change when thelength wastripled?
𝟑𝟔is threetimes as much as 𝟏𝟐. Therefore, when thelength is threetimes as long, thevolumeis also three
times as much.
d. What conclusion can you make about therelationship between the volumeand the length?
When only thelength is changed, and thewidth and height stay thesame, thechangein thevolumeis
proportional to thechangein thelength.
𝟖 cm
𝟏
𝟏
𝟐
cm
𝟏 cm

Date: 5/27/15 196
7. The sum ofthe volumesoftwo rectangular prisms, Box A and Box B, are 𝟏𝟒.𝟑𝟐𝟓cm3. Box A hasavolume of 𝟓.𝟔𝟏
cm3.
a. Let 𝑩 representthevolumeofBox B in cubic centimeters. Write an equationthat could beusedto determine
the volume ofBox B.
𝟏𝟒.𝟑𝟐𝟓 = 𝟓.𝟔𝟏+ 𝑩
b. Solve the equation to determine the volumeofBox B.
𝑩 = 𝟖. 𝟕𝟏𝟓cm3
c. If the areaofthe base ofBox B is 𝟏. 𝟓 cm2 write an equation that couldbe used to determine the height ofBox
B. Let 𝒉 represent theheight ofBox B in centimeters.
𝟖. 𝟕𝟏𝟓 = 𝟏. 𝟓𝒉
d. Solve the equation to determine the height ofBox B.
𝒉 = 𝟓.𝟖𝟏cm

Date: 5/27/15 197
White Board Exchange: Multiplication and Division Equations with Fractions
Progression of Exercises: Answers:
1. 5𝑦 = 35 𝑦 = 7
2. 3𝑚 = 135 𝑚 = 45
3. 12𝑘 = 156 𝑘 = 13
4.
𝑓
3
= 24 𝑓 = 72
5.
𝑥
7
= 42 𝑥 = 298
6.
𝑐
13
= 18 𝑐 = 234
7.
2
3
𝑔 = 6 𝑔 = 9
8.
3
5
𝑘 = 9 𝑘 = 15
9.
3
4
𝑦 = 10 𝑦 =
40
3
= 13
1
3
10.
5
8
𝑗 = 9 𝑗 =
72
5
= 14
2
5
11.
3
7
ℎ = 13 ℎ =
91
3
= 30
1
3
12.
𝑚
4
=
3
5
𝑚 =
12
5
= 2
2
5
13.
𝑓
3
=
2
7
𝑓 =
6
7
14.
2
5
𝑝 =
3
7
𝑝 =
15
14
= 1
1
14
15.
3
4
𝑘 =
5
8
𝑘 =
20
24
=
5
6
Fluency work such as this exerciseshould take5–12 minutes of class.
How to Conduct a White Board Exchange:
All students will need a personal white board, white board marker, and a means of erasingtheir work. An economical
recommendation is to placecard stock insidesheet protectors to use as the personal whiteboards and to cut sheets of
felt into small squaresto use as erasers.
It is bestto prepare the problems in a way that allows you to reveal them to the class oneat a time. For example, use a
flip chartor PowerPoint presentation; write the problems on the board and cover with paper beforehand, allowingyou
to reveal one at a time; or, write only one problem on the board at a time. If the number of digits in the problem is very
low (e.g., 12 divided by 3), it may also beappropriateto verbally call outthe problem to the students.

Date: 5/27/15 198
The teacher reveals or says the firstproblem in the listand announces,“Go.” Students work the problem on their
personal white boards,holdingtheir answers up for the teacher to see as soon as they have them ready. The teacher
gives immediate feedback to each student, pointingand/or makingeye contactwith the student and respondingwith an
affirmation for correctwork such as,“Good job!”, “Yes!”, or “Correct!” For incorrectwork, respond with guidancesuch
as “Look again!”,“Try again!”, or “Check your work!”
If many students have struggled to get the answer correct, go through the solution of that problem as a class before
moving on to the next problem in the sequence. Fluency in the skill has been established when the class isableto go
through each problem in quick succession withoutpausingto go through the solution of each problem individually. If
only one or two students have not been ableto get a given problem correct when the rest of the students are finished,it
is appropriateto move the classforward to the next problem without further delay; in this case,find a time to provide
remediation to that student before the next fluency exerciseon this skill is given.

G6 m5-c-lesson 13-t

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