Grade&Level/Course:&!Grade!5! Lesson/Unit&Plan&Name ...€¦ · a) A cube with side length 1 unit,...
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Page 1 of 15 MCC@WCCUSD 03/14/2014 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 non- overlapping 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
Transcript of Grade&Level/Course:&!Grade!5! Lesson/Unit&Plan&Name ...€¦ · a) A cube with side length 1 unit,...
Page 1 of 15 MCC@WCCUSD 03/14/2014
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 non-overlapping 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 2 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 !!!!!!!!!=!!!!!!!!!!!!!!=!!!!!!!!!!
!
2!cm!
2!cm!
3!cm!
1!cm!
2!cm!
3!cm!
2!
layers!
12!cubic!cm!
(12!cm 3 )!
What!type!of!prism!is!this?!
[right!rectangular!prism]!Observe!the!prism.!!What!do!you!notice?!!
label*dimensions,*create*list*of*attributes*
What!does!it!mean!to!
decompose?![to!break!apart/down]!
Can!we!decompose!this!
prism?![Yes]!
How?![By!layers,!
columns!&!rows,!blocks]!
How!many!layers?![2]!
Can!we!decompose!further?![Yes]!
What!is!the!smallest!
piece!we!can!decompose!the!prism!into?![unit!cubes]!
How!many!cubes!in!one!
layer?![6]!!
How!many!altogether?![12!unit!cubes]!
2!cm!
2!cm!
3!cm!
3!
layers!
1!cm!
2!cm!
2!cm!
12!unit!cubes!
(12!cm 3 )!
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.*
Can!we!decompose!this!figure!another!way?!!
Will!it!have!the!same!volume?!
How!do!you!know?!
Test*student*decomposition*
strategies*and*analyze*reasoning*for*predictions.*
=!
=!
We Try!
Page 3 of 15 MCC@WCCUSD 03/14/2014
!You Try! !!!!!!!!!!!!!!!!!!!!!!!!
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!are!the!dimensions!of!
the!prism?![4!in,!3!in,!3!in]!
How!many!layers?![3]!
!
∴The!volume!of!this!rectangular!
prism!is!36!cubic!units!or!36!cm 3 .!
4!in!
3!in!
3!in!
4!in!
1!in!
3!in!
3!layers!
1!layer!=!12!unit!cubes!
How!many!cubic!units!in!1!layer?![12!cm 3 ]!
How!many!cubic!units!in!2!layers?![24!cm 3 ]!
How!many!cubic!units!in!3!layers?![36!cm 3 ]!
What!would!the!first!layer!of!blocks!look!
like?!
5!blocks!
2!blocks!
1!block!!!!!!!1!layer!=!10!unit!cube!blocks!
1!layer!x!3!=!10!unit!cubes!x!3!
!!!!!3!layers!=!30!unit!cubes!
Decompose!
the!prism!into!cubes.!
Determine!total!number!of!cubes.!
∴The!volume!of!Billy’s!structure!will!
be!30!units 3 when!he!is!finished.!
Page 4 of 15 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. !
!
!
!
!
!
=!
!
!
!
!
!
!
=!
!
!
!
This!is!the!area!of!the!
base.!
This!is!the!height!or!number!of!layers.!
2!cm!
2!cm!
3!cm!
1!cm!
2!cm!
3!cm!
2!layers!
12!cubic!cm!
(12!cm 3 )!
12cm3 ? 3cm by 2cm by 2cm
12cm3 ? 3cm by 2cm by 2cm
12cm3 = (3cm × 2cm)× 2cm
12cm3 = (6cm2 )× 2cm
12cm3
= 12cm3
!
∴Volume!=!(Area!of!the!base!)!x!h!!!!!!!!!!!!!!!!!V!=!B!x!h!!!!!!!!!!!!!!!!!V!=!l!x!w!x!h!!
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 ?!!!
!
!
Page 5 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. !
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
4!in!
3!in!
3!in!
4!in!
1!in!
3!in!
3!layers!
1!layer!=!12!unit!cubes!
3!layers!=!?!unit!cubes!!!
V = B × hV = l × w( )× h
V = 4cm× 3cm( )× 3cm!
= (12cm2 )× 3cm ! = 36cm3
!
What!are!the!dimensions!of!this!prism?!
How!do!we!find!the!area!of!the!base!(B)?!
How!can!we!use!that!to!find!the!volume?!
∴The!volume!of!this!prism!is! 36cm3 .!
Page 6 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. !
!
!
!
!
!
!
!
!
Quick!Write!!!What!is!the!formula!for!finding!volume?!!How!is!volume!related!to!multiplication!and!addition?!!
!
!
!
!
!
!
!
!
1!layer!=!4!unit!cubes!2!layers!=!?!unit!cubes!
2!ft!
2!ft!
2!ft!
2!ft!
1!ft!
2!ft!
V = B × hV = (l × w)× hV = (2cm× 2cm)× 2cmV = (4cm2 )× 2cmV = 8cm3
!
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 7 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.}!
Let’s!do!that!first.!!!What!is!the!length?![5!units]!!What!is!the!height?![4!units]!
What!is!the!width?![5!units]!!Do!we!have!enough!information!now!
to!determine!the!volume!of!this!prism?![Yes]!!
How!could!we!determine!the!volume?![Decompose!the!prism!or!use!the!formula]!
!Let’s!use!the!formula.!What!is!the!formula?![ V = B• h,V = l • w• h ]!
Does!it!matter!which!we!use?![no]!!In!which!unit!are!the!dimensions!
labeled?![unknown]!What!should!we!label!the!volume!then?![cubic!units]!
!!
V = B• hV = (5•5)• 4V = (25)• 4V = 100 cubic units!
4!yd!
12!yd!
3!yd!
V = l × w× hV = 4× 3×12V = 12×12V = 144 yd 3
!
What!do!we!notice!about!this!prism?!!
[It!does!not!show!the!cubic!units.]!
How!can!find!the!volume?![use!the!formula]!
16!mm!
2!mm!
1!mm!
V = B × hV = (16×1)× 2V = (16)× 2V = 32 units3
!
Page 8 of 15 MCC@WCCUSD 03/14/2014
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?!!!!!!!!!!!!!!!!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?!!!!!!!!!!!!!
9!in!
6!in!8!in!
V = l i w i hV = 9 i8 i 6V = 72 i 6V = 432 in3
!
Check72
6 432 72 ÷8 = 9
−4212
− 120
!
∴The!volume!of!Carol’s!box!is!432!in 3 .!
!Set!Up!
• bar!model!
• draw!picture!
• number!line!!
Solve!
• Equation!!
• Evaluate!or!simplify!
Check!
• Use!another!method!
• Use!the!inverse!operation!
Answer!the!Question!
• Write!your!answer!in!
a!complete!sentence.!!
12!in!
24!
in!
18!in!
V = B × hV = (24 i12)×18V = 288×18V = 5184 in3
!
Check24 288
×12 × 188 64
40 64040 1600
+200 80288 800
+ 20005184
!
∴Alvin’s!cooler!will!hold!5,184!cubic!inches!of!ice.!!
Page 9 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?!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
V = B × hV = (8× 4)× 2V = (32)× 2V = 64 meters3
!
!!!!!!!!!!!!!!!!!!
2m!
8m!
4m!
Check64 ÷ 2 = 32
32 ÷ 4 = 8!
∴The!maximum!volume!of!Deb’s!pool!is!64!cubic!meters!of!water.!
Page 10 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!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!
7!cm!
!5!cm!
3!cm!
5!cm!
1!cm!
5!cm!1!cm!
a!cm!
3!cm!
b!cm!
d!cm!
Whole!Class!!This!is!not!a!rectangular!prism.!How!
could!I!determine!the!missing!dimensions!of!this!figure?!Record*student*suggestions.!
Think/Pair/Share!
We!are!going!to!practice!decomposing!this!figure!into!smaller!rectangular!prisms.!!With!a!
neighbor,!decide!how!you!would!decompose!this!figure.!
!
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!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.!
!
!
!
I know:
b = 5cm d = 5cm -1cm a = 3cm + 7 cm
∴a = 10cmb = 5cmd = 4cm
!
Page 11 of 15 MCC@WCCUSD 03/14/2014
We!Try!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
4!in!
2!in!
2!in!
2!in!
5!in!
4!in!
2!in!
2!in!
y!in!
2!in!
x#in!w#in!
I know:
y = 2 w = 2 x in + y in = 5 in
Because I know y, I can solve for x. x in + y in = 5 in x in + 2 in = 5 in (3 in) + 2 in = 5in
∴ x = 3iny = 2inw = 2in
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.
Page 12 of 15 MCC@WCCUSD 03/14/2014
You!Try!!!!!!!!!!!!!!!!!Directions:!!Find!the!volume!of!the!composite!figure.!!Example!2!!!!!!!!!!!!!!!!
12!yd!
!2!yd!
15!yd!
3!yd!
3!yd!
m#
n#
p#
15!yd!
2!yd!
p#
m#
3!yd!3!yd!
n#
I know:
m = 12 – 2
p = 3 yd
n = 15 – 3
∴m = 10ydn = 12ydp = 3yd
4!in!
2!in!
2!in!
2!in!
5!in!
4!in!
2!in!
2!in!
y!in!
2!in!
x#in!w#in!
4!in!
2!in!
2!in!
2!in!
2!in!
3!in!2!in!
I!notice!this!is!the!same!figure!from!our!previous!example.!!I!know!I!need!to!solve!for!the!missing!dimensions.!!I!will!need!to!
decompose.!
Now!that!it!is!decomposed!I!will!need!to!think!about!what!I!know!and!label!each!missing!dimension.!
There!are!now!two!rectangular!prisms.!!I!can!find!their!volumes!and!then!combine!them!to!find!the!volume!of!the!composite!figure.!!
V = l × w× hV = 2× 2× 3V = 4× 3V = 12in3
!
V = l × w× hV = 4× 2× 2V = 8× 2V = 16in3
!
16in3 +12in3
= 28in3!
Volume!of!Figure!1! Volume!of!Figure!2!TOTAL!
VOLUME!
∴The!volume!of!the!composite!figure!is!28!inches!centimeters.!
Page 13 of 15 MCC@WCCUSD 03/14/2014
!You!Try!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
!p#
12!yd!
!2!yd!
15!yd!
3!yd!
3!yd!
n#
m#m#
n#
p#
3!!yd!3!yd!
15!yd!
2!yd!
3!yd!3!yd!
15!yd!
2!yd!
10!yd!
3!yd!
Volume!of!Figure!1!
V = l × w× hV = 3× 3×10V = 9×10V = 90yd 3
!
Volume!of!Figure!2!
V = B × hV = (15× 3)× 2V = 45× 2V = 90yd 3
!
Sum!of!Volumes!
90yd + 90yd= 180yd
!∴The!volume!of!the!composite!figure!is!180!
cubic!yards.!
Page 14 of 15 MCC@WCCUSD 03/14/2014
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 )!!!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!
#2!!A!group!of!scientists!are!experimenting!with!oil!and!water.!!They!are!using!a!rectangular!container!that!has!a!base!area!of! 12in2 !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.!!!!!!!!!!!!!!!
12!cm!
11cm!
9cm!
500ml!
1L!
6!in!
Page 15 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 )!!!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!
#2!!A!group!of!scientists!are!experimenting!with!oil!and!water.!!They!are!using!a!rectangular!container!that!has!a!base!area!of! 12in2 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.!!!!!!!!!!!!!!!
12!cm!
11cm!
9cm!
500ml!
1L!
Volume!of!Fish!Tank!
V = l × w× hV = 12× 9×11V = 108×11V = 1188cm3
!
!!!Unit!Conversion!
1cm3 = 1ml1188cm3 = 1188ml
1L = 1,000ml
!
Difference!in!Volume!
= 1188ml −1000ml= 188ml
!
∴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.!
6!in!Height!of!the!Water!Layer!!
total height − oil height= 8in− 6in= 2in
!
Volume!of!Water!
A = B × hA = (12in2 )× 2inA = 24in3
!
∴The!volume!of!water!in!the!rectangular!container!is!24!cubic!inches.!
STUDENT RESOURCES: RECTANGULAR PRISMS
STUDENT RESOURCES: PRISM LAYERS
STUDENT RESOURCES: UNIT CUBES (copy 2x per student)
