Grade&Level/Course:&!Grade!5!
Lesson/Unit&Plan&Name:&!Volume: A Foundation in Unit Cubes!
&
Rationale/Lesson&Abstract:&Students will decompose rectangular prisms to recognize volume as
an attribute of solid figures. They will build an understanding of concepts of volume
measurements in relationship to unit cubes. Students will also derive the volume formula; then
apply the formula to real world situations involving volume. They will then use their
understanding of decomposing prisms and the volume formula to recognize volume as additive:
the volume of solids comprised of 2 non-overlapping rectangular prisms.
*This lesson/unit should be completed after students have been introduced to square units and
cubic units.
!
Timeframe:&Four 60 minute lessons&
&
Common&Core&Standard(s):&
&
5.MD.3: Recognize volume as an attribute of solid figures and understand concepts of volume
measurement.
a) A cube with side length 1 unit, called a “unit cube,” is said to have “one cubic unit” of
volume, and can be used to measure volume.
b) A solid figure, which can be packed without gaps or overlaps using n unit cubes, is said to
have a volume of n cubic units.
5.MD.4: Measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and
improvised units.
5.MD.5: Relate volume to the operations of multiplication and addition and solve real world and
mathematical problems involving volume.
a) Find the volume of a right rectangular prism with whole-number side lengths by packing it
with unit cubes, and show that the volume is the same as would be found by multiplying the edge
lengths, equivalently by multiplying the height by the area of the base. Represent threefold
whole-number products as volumes, e.g., to represent the associative property of multiplication.
b) Apply the formulas V = l ×w ×h and V = B ×h for rectangular prisms to find volumes of right
rectangular prisms with whole-number edge lengths in the context of solving real world and
mathematical problems.
c) Recognize volume as additive. Find volumes of solid figures composed of two nonoverlapping right rectangular prisms by adding the volumes of the non-overlapping parts,
applying this technique to solve real world problems.
!
Instructional&Resources/Materials:&
• Student!resource!masters!(1!set!per!student)!
• Notebooks!
• Pencils!
• Scissors!
• Glue!
Page 1 of 15
MCC@WCCUSD 03/14/2014
Lesson PART I: Determine Volume of Prisms by Analyzing Layers and Counting Unit Cubes
***Distribute*student*resource*pages.**Have*students*cut*out*(suggest*cutting*prior*to*lesson)*and*paste*graphics*into*their*notebooks.*This*
will*allow*students*to*focus*on*the*concept*rather*than*the*difficulty*of*drawing*prisms.*
Directions: Given a rectangular prism, determine the volume by decomposing the figure into layers and cubic units.
Example #1
!
!
!
!
!
!
!
!
!
=!!!!
!
!
1!cm!
!
!
!
!
!
!
!
!
=!!
!
!
!
!
!
!
!
!
2!cm!
2!cm!
3!cm!
We Try! 2!cm!
What!type!of!prism!is!this?!
[right!rectangular!prism]!
Observe!the!prism.!!What!do!
you!notice?!!
3!cm!
label*dimensions,*create*list*of*
attributes*
What!does!it!mean!to!
decompose?![to!break!
apart/down]!
2!
layers!
2!cm!
3!cm!
What!do!you!think!will!happen!if!the!
prism!is!rotated?![record!student!
predictions]!
If!we!decompose!it,!will!it!have!the!same!
volume?![record!student!predictions]!
Decompose*and*compare*predictions*to*
results*as*a*class.*
2!cm!
Can!we!decompose!this!figure!
another!way?!!
Can!we!decompose!this! !
prism?![Yes]!
=!
How?![By!layers,!
columns!&!rows,!
blocks]!
2!cm!
How!many!layers?![2]!
3!
layers!
1!cm!
Will!it!have!the!same!volume?!
How!do!you!know?!
Test*student*decomposition*
strategies*and*analyze*
reasoning*for*predictions.*
Can!we!decompose!
further?![Yes]!
What!is!the!smallest!
12!cubic!cm!
piece!we!can!
(12!cm 3 )!
decompose!the!prism!
into?![unit!cubes]!
2!cm!
12!unit!cubes!
=!
(12!cm 3 )!
How!many!cubes!in!one!
layer?![6]!!
How!many!altogether?!
[12!unit!cubes]! Page 2 of 15
MCC@WCCUSD 03/14/2014
!
3!layers!
You Try!
!
! 3!in!
!
!
!
!
3!in!
!
!
!
!
!
4!in!
!
!
!
What!are!the!dimensions!of!
!
the!prism?![4!in,!3!in,!3!in]!
!
!
How!many!layers?![3]!
!
!
!
!
!
3!in!
1!layer!=!12!unit!cubes!
1!in!
4!in!
How!many!cubic!units!in!1!layer?![12!cm 3 ]!
∴ The!volume!of!this!rectangular!
How!many!cubic!units!in!2!layers?![24!cm 3 ]!
prism!is!36!cubic!units!or!36!cm 3 .!
How!many!cubic!units!in!3!layers?![36!cm 3 ]!
!
!
!
You Try! Billy is playing with blocks. He lays out a layer of blocks that is 2 by 5 blocks. Then he tells his friend that he plans to add 2
more layers just like the bottom layer. What will the volume of Billy’s structure be once he finishes adding layers?
!
!!!!!!!!!!!!!!!Billy’s!1st!layer!
!
!
!
!
!
!
What!would!
the!first!layer!of!
blocks!look!
like?!
!!!!!!!!!
1!block!
!!!!!!1!layer!=!10!unit!cube!blocks!
1!layer!x!3!=!10!unit!cubes!x!3!
5!blocks!
!!!!!3!layers!=!30!unit!cubes!
2!blocks! Decompose!
the!prism!into!
cubes.!
Determine!total!
number!of!cubes.!
Page 3 of 15
∴ The!volume!of!Billy’s!structure!will!
be!30!units 3 when!he!is!finished.!
MCC@WCCUSD 03/14/2014
Lesson!PART!II:!!Relating!Counting!Cubic!Units!to!the!Volume!Formula!
Example!1!Given a rectangular prism, use the its dimensions to derive the volume formula. !
2!cm!
!
!
?
3cm by 2cm by 2cm
12cm 3
?
3cm by 2cm by 2cm
12cm
!
2!cm!
!
3
!
This!is!the!
area!of!the!
base.!
3!cm!
!
=!
2!
layers!
!
!
1!cm!
!
!
2!cm!
This!is!the!height!or!
number!of!layers.!
12cm 3
=
(3cm × 2cm) × 2cm
12cm 3
=
(6cm 2 ) × 2cm
3!cm!
!
Partner!Discussion!
What!is!the!relationship!between!
the!dimensions!of!the!prism!and!
the!total!number!of!unit!cubes!or!
volume?!
List*student*predictions*on*the*
board.*
Guiding!Questions!for!group!
discussion!
• What!do!you!notice!about!the!
units!on!the!left!and!the!units!
on!the!right!of!the!equation?!
• How!can!we!manipulate!the!
numbers!to!get! 12cm3 ?!!!
!
!
12cm
!
=!
12!cubic!cm!
!
(12!cm 3 )!
!
!
3
=
12cm 3
∴Volume!=!(Area!of!the!base!)!x!h!
!!!!!!!!!!!!!!!!V!=!B!x!h!
!!!!!!!!!!!!!!!!V!=!l!x!w!x!h!
!
Page 4 of 15
MCC@WCCUSD 03/14/2014
We!Try!!Given a rectangular prism, use the formula derived in the previous example to find the volume. !
!
3!layers!
!
!
3!in!
!
!3!in!
!
!
4!in!
!
!
!
!
!
V = B×h
3!in!
1!layer!=!12!unit!cubes!
3!layers!=!?!unit!cubes!!!
1!in!
V = ( l × w) × h
V = ( 4cm × 3cm) × 3cm
4!in!
= (12cm2 ) × 3cm !
!
= 36cm3
!
!
!
!
What!are!the!dimensions!of!this!prism?!
∴ The!volume!of!this!prism!is! 36cm3 .!
How!do!we!find!the!area!of!the!base!(B)?!
How!can!we!use!that!to!find!the!volume?!
!
!
Page 5 of 15
MCC@WCCUSD 03/14/2014
You!Try!!Given a rectangular prism, use the formula derived in the previous examples to find the volume. !
!
2!ft!
V = B×h
V = (l × w) × h
2!ft!
V = (2cm × 2cm) × 2cm
2!ft!
1!ft!
1!layer!=!4!unit!cubes!
2!layers!=!?!unit!cubes!
2!ft!
!
!
V = (4cm2 ) × 2cm
!
V = 8cm3
!
!
!
!
2!ft!
!
Quick!Write!!!What!is!the!formula!for!finding!volume?!!How!is!volume!related!to!multiplication!and!addition?!!
!
!
!
!
!
!
!
Sample!Student!Answer:!
There are 2 formulas for finding volume: V = B × h or V = l × w × h . They both
need the same information and are really the same. Volume is related to
multiplication in that you can find the area of the base and multiply it by
the number of layers or height. It is related to addition in that you can
find the number of cubic units in each layer and find their sum.
!
Page 6 of 15
MCC@WCCUSD 03/14/2014
Lesson!PART!III:!Using!the!Volume!Formula!
!
Directions:!Calculate!the!volume!of!each!rectangular!prism.!
Example!1!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!We!Try!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!You!Try!!!
What!do!we!notice!about!this!prism?!!
!
[The!dimensions!are!not!labeled.}!
!
!
2!mm!
Let’s!do!that!first.!!!
!
12!yd!
What!is!the!length?![5!units]!!
!
What!is!the!height?![4!units]!
!
What!is!the!width?![5!units]!
!
!
!
3!yd!
Do!we!have!enough!information!now!
!
4!yd!
to!determine!the!volume!of!this!
!
V = l × w× h
!
prism?![Yes]!
! V = B• h
!
V = 4 × 3× 12
! V = (5• 5) • 4
How!could!we!determine!the!volume?!
V = 12 × 12
!
[Decompose!the!prism!or!use!the!
V = (25) • 4
V = 144 yd 3
!
formula]!
!
! V = 100 cubic units
!
! !
Let’s!use!the!formula.!
!
What!is!the!formula?!
!
[ V = B • h,V = l • w• h ]!
What!do!we!notice!about!this!prism?!!
!
Does!it!matter!which!we!use?![no]!
[It!does!not!show!the!cubic!units.]!
!
!
!
How!can!find!the!volume?![use!the!
In!which!unit!are!the!dimensions!
!
formula]!
labeled?![unknown]!
!
What!should!we!label!the!volume!
!
then?![cubic!units]!
!
!
!
!
Page 7 of 15
MCC@WCCUSD 03/14/2014
1!mm!
16!mm!
V = B×h
V = (16 × 1) × 2
V = (16) × 2
V = 32 units3
!
Example!2!Carol!is!constructing!a!box!to!hold!her!bookmark!collection.!!It!has!a!length!of!9!in,!a!width!of!8!in,!and!a!height!of!6*in.!!What!is!
the!volume!of!her!box?!
!
V = l i wi h
Check
!
∴ The!volume!of!Carol’s!box!is!432!in 3 .!
!
V = 9i8i6
72
!
V = 72 i 6
6!in!
6 432
72 ÷ 8 = 9
!
3
8!in!
V = 432in
!
− 42
!
Answer!the!Question!
!
9!in!
12
• Write!your!answer!in!
!
− 12
!
a!complete!sentence.!
Solve!
!Set!Up!
0
!
!
• Equation!!
•
bar!model!
!
!
• Evaluate!or!
Check!
• draw!picture!
!
simplify!
• Use!another!method!
• number!line!
!
• Use!the!inverse!
!
!
operation!
!
We!Try!!!Alvin!wants!to!determine!how!much!ice!it!takes!to!fill!his!cooler.!The!cooler!has!a!length!of!24!inches,!width!of!12!inches,!and!height!
of!18!inches.!!How!much!ice!will!his!cooler!hold?!
Check
V = B×h
!
∴ Alvin’s!cooler!will!hold!5,184!
24
288
24!
!
V = (24 i12) × 18
cubic!inches!of!ice.!!
in!
×12
×
18
!
12!in!
V = 288 × 18
!
8
64
V = 5184 in3
!
40
640
18!in!
!
!
40
1600
!
+200
80
!
!
288
800
!
+ 2000
!
5184
!
!
Page 8 of 15
MCC@WCCUSD 03/14/2014
You!Try!!!Deb’s!pool!is!in!the!shape!of!a!rectangular!prism.!!It!has!a!length!of!8!meters,!a!width!of!4!meters,!and!a!height!of!2!meters.!!What!
is!the!maximum!volume!of!water!that!Deb’s!pool!can!hold?!
!
!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
!
4m!
2m!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
V = B×h
V = (8 × 4) × 2
!
V = (32) × 2
V = 64 meters3
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
Check
64 ÷ 2 = 32
∴ The!maximum!volume!of!Deb’s!pool!is!64!
cubic!meters!of!water.!
32 ÷ 4 = 8
!
8m!
Page 9 of 15
MCC@WCCUSD 03/14/2014
Lesson!PART!IV:!Recognize!Volume!as!Additive!!
!
Directions:!Find!the!missing!dimensions!of!the!rectangular!prism.!
!
Example!1!!!
!!
a!cm!
!
5!cm!
1!cm!
!
!5!cm!
!
1!cm!
!
7!cm!
b!cm!
!
!
d!cm!
!
3!cm!
!
! Whole!Class!!
I know:
! This!is!not!a!rectangular!prism.!How!
3!cm!
! could!I!determine!the!missing!
b = 5cm
! dimensions!of!this!figure?!Record*
d = 5cm -1cm
! student*suggestions.!
!
a = 3cm + 7 cm
! Think/Pair/Share!
! We!are!going!to!practice!
! decomposing!this!figure!into!smaller!
∴ a = 10cm
! rectangular!prisms.!!With!a!
! neighbor,!decide!how!you!would!
b = 5cm
! decompose!this!figure.!
d = 4cm
!
!!
!
!
!
Page 10 of 15
Think!Aloud!
When!I!decomposed!it,!I!notice!that!it!is!
missing!some!dimensions.!!I’m!going!to!label!
the!unknown!dimensions!with!variables.!
Think/Pair/Share!
5!cm!
How!could!we!solve!for!the!unknown!
dimensions?!Record*student*suggestions.*
!
Think!Aloud!
Let’s!start!by!recording!what!we!know.!
• I!know!that!the!edge!measuring!b!is!
parallel!and!congruent!to!the!edge!
measuring!5!cm,!so!b*must!equal!5!cm.!
• I!know!that!d*+!1cm!=!5cm,!so!5cm!c!1cm!
will!tell!me!what!d!equals.!
•
!
Determining!the!length!of!the!edge!
measuring!a!is!more!challenging.!!This!
edge!is!parallel!to!two!other!edges;!the!
edge!of!the!thin!part!of!the!figure!and!
the!edge!of!the!base!of!the!figure.!!
Together,!their!lengths!comprise!the!
length!of!edge*a.!That!means!that!3cm!+!
7!cm!will!equal!a.!
!
!
!
MCC@WCCUSD 03/14/2014
We!Try!!!
!
2!in!
!
!
!
!
!
5!in!
!
!
!
4!in!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
I know:
2!in!
x#in!
y=2
w=2
x in + y in = 5 in
w#in!
2!in!
2!in!
y!in!
2!in!
4!in!
Because I know y,
I can solve for x.
2!in!
x in + y in = 5 in
x in + 2 in = 5 in
(3 in) + 2 in = 5in
∴ x = 3in
y = 2in
w = 2in
Page 11 of 15
I know the edge that
measures y = 2 because it
is parallel to the height of
the bottom prism.
I know that the edge
measuring w = 2 because
it is parallel to the base of
the bottom prism.
I know that x + y = 5 in
and that y = 2 so I can use
what I know about
solving for an unknown to
determine x.
MCC@WCCUSD 03/14/2014
3!yd!
You!Try!!!!
3!yd!
I know:
3!yd!
!
3!yd!
!
m = 12 – 2
!
m#
!
m#
p = 3 yd
! 12!yd!
!
n#
n = 15 – 3
!
n#
!2!yd!
!
!
∴ m = 10 yd
!
2!yd!
p#
!
n = 12 yd
15!yd!
p#
!
15!yd!
p = 3yd
!
Directions:!!Find!the!volume!of!the!composite!figure.!
!
TOTAL!
Example!2!!
Volume!of!Figure!1! Volume!of!Figure!2!
VOLUME!
!
V = l × w× h
V
=
l
×
w
×
h
2!in!
2!in!
2!in!
!
16in3 + 12in3
V
=
4
×
2
×
2
!
V
=
2
×
2
×
3
!
3
=
28in
x#in!
3!in!
V = 8× 2
!
V = 4×3
w#in!
2!in!
!
5!in!
V = 16in3
V = 12in3
2!in!
!
!
!
!
2!in!
∴ The!volume!of!the!composite!
4!in!
!
y!in!
2!in!
2!in!
2!in!
figure!is!28!inches!centimeters.!
!
I!notice!this!is!the!same!figure!from!our!
2!in!
2!in!
!
4!in!
4!in!
previous!example.!!I!know!I!need!to!solve!
!
There!are!now!two!rectangular!prisms.!!I!can!
for!the!missing!dimensions.!!I!will!need!to! Now!that!it!is!decomposed!I!will!need!
!
to!think!about!what!I!know!and!label!
find!their!volumes!and!then!combine!them!to!
decompose.!
!
each!missing!dimension.!
find!the!volume!of!the!composite!figure.!!
!
Page 12 of 15
MCC@WCCUSD 03/14/2014
!
You!Try!!!
!
! 3!yd!
!
!
!
12!yd!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
3!yd!
3!yd!
3!!yd!
3!yd!
3!yd!
m#
10!yd!
m#
n#
!2!yd!
15!yd!
n#
p#
p#
15!yd!
Volume!of!Figure!1!
Volume!of!Figure!2!
Sum!of!Volumes!
V = l × w× h
V = 3× 3× 10
!
V = 9 × 10
V = B×h
V = (15 × 3) × 2
90 yd + 90 yd
!
= 180 yd
V = 90 yd 3
V = 90 yd 3
V = 45 × 2
!
Page 13 of 15
2!yd!
2!yd!
!
15!yd!
∴ The!volume!of!the!composite!figure!is!180!
cubic!yards.!
MCC@WCCUSD 03/14/2014
3!yd!
Assessment:!
!
#1!!!This!mini!fish!tank!is!shaped!like!a!rectangular!prism.!!It!is!filled!to!the!top!with!water.!!Will!the!beaker!hold!all!of!the!water!from!the!fish!
tank?!If!not,!how!much!more!water!does!the!fish!tank!hold!than!the!beaker?!Justify!your!answer.!!( 1cm3 = 1ml )!
!
!
9cm!
11cm!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
12!cm!
!
!
!
1L!
500ml!
!
2
#2!!A!group!of!scientists!are!experimenting!with!oil!and!water.!!They!are!using!a!rectangular!container!that!has!a!base!area!of! 12in !and!a!height!of! 8in .!!
It!is!filled!with!both!oil!and!water.!They!notice!that!the!oil!and!water!separate!into!layers!with!the!oil!rising!to!the!top!of!the!container.!Using!this!
information!and!the!scientists’!diagram!below,!determine!what!is!the!volume!of!the!water.!!
!
!
!
6!in!
!
!
!
!
!
!
!
!
!
!
Page 14 of 15
MCC@WCCUSD 03/14/2014
Assessment!Answers!!
#1!!!This!mini!fish!tank!is!shaped!like!a!rectangular!prism.!!It!is!filled!to!the!top!with!water.!!Will!the!beaker!hold!all!of!the!water!from!the!fish!
tank?!If!not,!how!more!water!does!the!fish!tank!hold!than!the!beaker?!Justify!your!answer.!!( 1cm3 = 1ml )!
!
!
Volume!of!Fish!Tank!
9cm!
11cm!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
12!cm!
!
!
!
!
V = l × w× h
V = 12 × 9 × 11
!
V = 108 × 11
1L!
500ml!
V = 1188cm3
!
!!!Unit!Conversion!
Difference!in!Volume!
1cm 3 = 1ml
1188cm 3 = 1188ml
!
= 1188ml − 1000ml
!
= 188ml
1L = 1,000ml
∴ The!beaker!will!not!hold!all!of!the!water!from!the!fish!tank!because!
the!fish!tank!holds!1188!cubic!centimeters!and!the!beaker!holds!1!liter.!!
The!fish!tank!will!hold!188!ml!more!water!than!the!beaker.!
2
#2!!A!group!of!scientists!are!experimenting!with!oil!and!water.!!They!are!using!a!rectangular!container!that!has!a!base!area!of! 12in and!a!height!of! 8in .!!
It!is!filled!with!both!oil!and!water.!They!notice!that!the!oil!and!water!separate!into!layers!with!the!oil!rising!to!the!top!of!the!container.!Using!this!
information!and!the!scientists’!diagram!below,!determine!what!is!the!volume!of!the!water.!!
!
!
Height!of!the!Water!Layer!
Volume!of!Water!
! !
6!in!
! total height − oil height
A= B×h
!
!
A = (12in2 ) × 2in !
! = 8in − 6in
! = 2in
A = 24in3
!
!
! ∴ The!volume!of!water!in!the!rectangular!container!is!24!cubic!inches.!
!
!
!
Page 15 of 15
MCC@WCCUSD 03/14/2014
STUDENT RESOURCES: RECTANGULAR PRISMS STUDENT RESOURCES: PRISM LAYERS STUDENT RESOURCES: UNIT CUBES (copy 2x per student)