1. NYS COMMON CORE MATHEMATICS CURRICULUM 4 5Lesson 10
Lesson 10: Use the area model and divisionto showtheequivalence oftwo
fractions.
Date: 4/14/15
5.B.36
© 2014 Common Core, Inc. Some rightsreserved. commoncore.org
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
NOTES ON
MULTIPLE MEANS
OF REPRESENTATION:
While leading the Count byEquivalent
Fractions fluencyactivity, enunciate
the endingdigraph /th/ of fraction
names to help Englishlanguage
learners distinguishfractions from
whole numbers (e.g., fourths, not
fours).
Couple numbers on the boardwith
preparedvisuals, ifbeneficial.
Lesson 10
Objective: Use the area model and division to show the equivalence of two
fractions.
Suggested Lesson Structure
Fluency Practice (12 minutes)
Application Problem (8 minutes)
Concept Development (30 minutes)
Student Debrief (10 minutes)
Total Time (60 minutes)
Fluency Practice (12 minutes)
 Countby EquivalentFractions 3.NF.3 (4 minutes)
 FindEquivalentFractions 4.NF.1 (4 minutes)
 Draw EquivalentFractions 4.NF.1 (4 minutes)
Count by Equivalent Fractions (4 minutes)
Note: Thisfluency activity reinforcesModule5fraction
concepts.
T: Countby threesto24. Start at 0.
S: 0, 3, 6, 9, 12, 15, 18, 21, 24.
T: Countby 3 fourths to24 fourths. Startat 0 fourths.
(Write as studentscount.)
S:
0
4
,
3
4
,
6
4
,
9
4
,
12
4
,
15
4
,
18
4
,
21
4
,
24
4
.
T: 1 isthe same as howmany
fourths?
S: 4 fourths.
T: 2 isthe same as howmany
fourths?
S: 8 fourths.
T: 3 isthe same as howmanyfourths?
S: 12 fourths.
T: (Beneath
12
4
,write 3.) 4 is the same as how many fourths?
0
4
3
4
6
4
9
4
12
4
15
4
18
4
21
4
24
4
0
3
4
6
4
9
4
3
15
4
18
4
21
4
6

2. NYS COMMON CORE MATHEMATICS CURRICULUM 4 5Lesson 10
Lesson 10: Use the area model and divisionto showtheequivalence oftwo
fractions.
Date: 4/14/15
5.B.37
© 2014 Common Core, Inc. Some rightsreserved. commoncore.org
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
S: 16 fourths.
T: 5 isthe same as howmanyfourths?
S: 20 fourths.
T: 6 isthe same as howmanyfourths?
S: 24 fourths.
T: (Beneath
24
4
,write 6.) Count by3 fourthsagain. Thistime,saythe whole numberswhenyouarrive
at them. Start withzero.
S: 0,
3
4
,
6
4
,
9
4
, 3,
15
4
,
18
4
,
21
4
, 6.
Repeatthe process,countingby3 fifthsto30 fifths.
Find Equivalent Fractions (4 minutes)
Materials: (S) Personal white board
Note: Thisfluency activity reviewsLesson8.
T: (Write
3
4
=
×
×
=
8
. Pointto
3
4
.) Say the fraction.
S: 3 fourths.
T: On yourpersonal white boards, complete the numbersentence.
S: (Write
3
4
=
3 ×2
4 ×2
=
6
8
.)
Continue withthe followingpossible suggestions:
3
4
=
9
12
,
2
3
=
4
6
,
2
5
=
4
10
,
4
5
=
8
10
, and
3
5
=
9
15
.
Draw Equivalent Fractions (4 minutes)
Materials: (S) Personal white board
Note: Thisfluency activity reviewsLesson9.
T: (Projectamodel with2 out of 4 equal unitsshaded.) Draw the model,andwrite the fractionthat is
shaded.
S: (Draw a model with2out of 4 equal unitsshaded. Write
2
4
.)
T: (Write
2
4
=
÷
÷
= .) Compose the shadedunitsinto1 largerunitby circling.
Then, complete the numbersentence.
S: (Circle the shadedunitsinto1largerunit. Write
2
4
=
2 ÷2
4 ÷2
=
1
2
.)
Continue withthe followingpossible sequence:
3
9
=
1
3
,
4
8
=
1
2
,
2
8
=
1
4
,
5
10
=
1
2
, and
4
12
=
1
3
.

3. NYS COMMON CORE MATHEMATICS CURRICULUM 4 5Lesson 10
Lesson 10: Use the area model and divisionto showtheequivalence oftwo
fractions.
Date: 4/14/15
5.B.38
© 2014 Common Core, Inc. Some rightsreserved. commoncore.org
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
NOTES ON
MULTIPLE MEANS
OF REPRESENTATION:
There are multiple ways of showinga
given fractionusinganareamodel. Area
models may, therefore, lookdifferent
from student to student. Allowstudents
to share how theyhave drawndifferent
area models andbe accepting ofthose
that are mathematicallycorrect.
Application Problem (8 minutes)
Nuri spent
9
12
of hismoneyona bookand the rest of hismoneyon a pencil.
a. Expresshowmuchof hismoneyhe spentonthe pencil
infourths.
b. Nuri startedwith$1. How muchdidhe spendonthe
pencil?
Note: ThisApplicationProblem connectsTopicA and Lesson9
by findingthe otherfractional partof the whole andexpressing
equivalentfractions. Usingwhatstudentsknow aboutmoney,
ask whyit ispreferable toanswerinfourthsratherthan
twelfths. Studentsconnectfourthstoquarters of a dollar.
Revisitthis probleminthe Debrief toexpresshow muchmoney
was spentonthe bookin fourths.
Concept Development (30 minutes)
Materials: (S) Personal white board
Problem1: Simplifyafractionbydrawingto findacommon
factor,and relate itto division.
T: Draw an area model thatrepresents
10
12
.
T: If we wantto compose an equivalentfraction,whatdo
we do?
S: We make equal groups.  We divide the numerator
and the denominatorbythe same number.  We
shoulddivide by10. We dividedbythe same number
that wasin the numeratoryesterday.
T: Can I divide boththe numeratoranddenominatorby10?
S: No.
T: Discusswithyourpartnerhow to determinethe largestpossibleunit.
S: We can try to make groupsof 2, then3, then4, until we have the largest
numberof unitsina groupwithno remainder.  We can onlymake equal
groupsof 2. The othernumbersdon’tdivide evenlyintoboththe
numeratoranddenominator.
T: Showme. (Allowtime forstudentstocompose anareamodel.) What
happenedtothe numberof shaded units?
S: There were 10 unitsshaded,andnowthere are 5 groupsof 2 unitsshaded!

4. NYS COMMON CORE MATHEMATICS CURRICULUM 4 5Lesson 10
Lesson 10: Use the area model and divisionto showtheequivalence oftwo
fractions.
Date: 4/14/15
5.B.39
© 2014 Common Core, Inc. Some rightsreserved. commoncore.org
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
T: Considerthe unitfractions
1
12
and
1
6
. What do younotice abouttheirdenominators?
S: 6 isa factorof 12.
T: What aboutthe numerators10 and 5?
S: 5 isa factorof 10.
T: List the factorsof 10 and 12.
S: The factors of 12 are 1, 2, 3, 4, 6, and12. The factors of 10 are 1, 2, 5, and 10.
T: 1 and 2 are factors of both. We knowthen that we can make equal groupsof 2. Equal groupsof 1
bringus back to the original fraction.
Problem2: Draw an area model of a numbersentencethatshowsthe simplificationofafraction.
T: (Project
6
10
=
6 ÷2
10 ÷2
=
3
5
.)
T: Draw an area model toshow howthisnumbersentence istrue.
S: The numeratorand denominatorare bothbeingdividedby2. I will circle groupsof 2.  I know2 is
a factor of 6 and 10, so I couldmake groupsof 2.  There are 3 shadedgroupsof 2 and5 total
groupsof 2.  That’s
3
5
.
Problem3: Simplifyafractionbydrawingto finddifferent commonfactors,andrelate ittodivision.
T: Withyour partner,draw an area model torepresent
8
12
. Rename
8
12
usinglargerfractional units.
You may talkas youwork. (Circulate andlisten.)
S: I can circle groupsof 2 units.  2 isa factorof 8 and 12.  There are 6 groupsof 2 units.
 Fourgroups are shaded. That’s
4
6
.
T: What happenswhenIuse 4 as a commonfactor insteadof 2? Turn and talk.
S: Four isa factor of both8 and 12. It works.  We can make largerunitswith groupsof 4.  Thirds
are largerthan sixths.
8
12
=
2
3
.  We have fewerunits, butthey’re bigger.
T: Expressthe equivalentfractionsas twodivision number
sentences.
S: (Write
8
12
=
8 ÷4
12 ÷4
=
2
3
and
8
12
=
8 ÷2
12 ÷2
=
4
6
.)
T: What can you conclude about
2
3
and
4
6
?
S: Theyare bothequivalentto
8
12
.

5. NYS COMMON CORE MATHEMATICS CURRICULUM 4 5Lesson 10
Lesson 10: Use the area model and divisionto showtheequivalence oftwo
fractions.
Date: 4/14/15
5.B.40
© 2014 Common Core, Inc. Some rightsreserved. commoncore.org
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
T: What istrue aboutdividingthe numeratoranddenominatorin
8
12
by2 or 4?
S: Two and4 are bothfactors of 8 and 12.  The largerthe factor used,the largerthe new fractional
unitswill be.
T: Interesting. Discusswhatyourclassmate said. “The largerthe factor, the largerthe new fractional
units.”
S: Whenwe dividedby2,we gotsixths, andwhenwe dividedby4,we got thirds. Thirdsare larger.
Four islargerthan 2. A larger factorgave a largerunit.  Whenthe factor islarger,it meanswe can
make fewerunitsbutlargerones.
Problem4: Simplifyafractionusingthe largestpossiblecommonfactor.
T: Discusswithyourpartnerhow to rename
8
12
withthe largestunitspossible withoutusinganarea
model.
S: Figure outthe greatestnumberof unitsthat can be placedinequal groups.  Divide the numerator
and denominatorbythe same number,justlike we’vebeendoing.  Findafactor of both 8 and 12,
and use itto divide the numeratorandthe denominator.
T: Expressthe equivalence usingadivisionnumbersentence.
S:
8
12
=
8 ÷2
12 ÷2
=
4
6
. Four and 6 are still botheven,sothatwasn’tthe largestfactor. 
8
12
=
8 ÷4
12 ÷4
=
2
3
. The
onlycommonfactor 2 and 3 have is1, so 4 must be the largestfactor that 8 and 12 have in common.
T: How can we knowwe expressedanequivalentfractionwiththe largestunits?
S: Whenwe make equal groups,we needtosee if we can make largerones.  Whenwe findthe
factors of the numeratorand denominator,we have topickthe largestfactor. Fouris largerthan 2,
so dividingthe numeratoranddenominatorby4 gets usthe largestunits.  WhenI found
4
6
, I
realized2and 4 are both even,soIdividedthe numeratoranddenominatoragain by2. Two and 3
only have a commonfactor of 1, so I knew Imade the largestunitpossible.  Dividingby2 twice is
the same as dividingby4. Justget itoverwithfasterand divide by4.
T: It’snot wrongto say that
8
12
=
4
6
. It is true. It’sjustthat, at times,itreallyis simplertoworkwith
largerunitsbecause itmeansthe denominatorisasmallernumber.
Problem Set (10 minutes)
Studentsshoulddotheirpersonal besttocompletethe ProblemSetwithinthe allotted10minutes. Forsome
classes,itmaybe appropriate tomodifythe assignmentbyspecifyingwhichproblemstheyworkonfirst.
Some problemsdonotspecifyamethodforsolving. Students shouldsolvethese problemsusingthe RDW
approach usedforApplicationProblems.

6. NYS COMMON CORE MATHEMATICS CURRICULUM 4 5Lesson 10
Lesson 10: Use the area model and divisionto showtheequivalence oftwo
fractions.
Date: 4/14/15
5.B.41
© 2014 Common Core, Inc. Some rightsreserved. commoncore.org
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Student Debrief (10 minutes)
LessonObjective: Use the area model anddivisionto
showthe equivalence of twofractions.
The StudentDebrief isintendedto invitereflectionand
active processingof the total lesson experience.
Invite studentstoreviewtheirsolutionsforthe Problem
Set. Theyshouldcheckworkby comparinganswerswitha
partnerbefore goingoveranswersasa class. Lookfor
misconceptionsormisunderstandingsthatcanbe
addressedinthe Debrief. Guide studentsina
conversationtodebrief the ProblemSetandprocess the
lesson.
Anycombinationof the questionsbelow maybe used to
leadthe discussion.
 In Problem2(b),didyoucompose the same units
as your partner? Are both of your answers
correct? Why?
 In Problem4(a–d),howisithelpful toknow the
commonfactors forthe numeratorsand
denominators?
 In Problem4,youwere askedto use the largest
commonfactor to rename the fraction:
4
8
=
1
2
.
By doingso,you renamed
4
8
usinglargerunits.
How isrenamingfractionsuseful?
 Do fractions alwaysneedtobe renamedtothe
largestunit? Explain.
 Why isit importanttochoose a common factor to
make largerunits?
 How can youtell that a fractioniscomposedof
the largestpossible fractional units?
 Whenyouare drawingan area model andcircling
equal groups, doall of the groupshave to appear
the same in shape? Howdo you knowthatthey
still showthe same amount?
 Explainhowknowingthe factorsof the
numeratorandthe factorsof the denominator
can be helpful inidentifyingequivalentfractions
of a largerunitsize.

7. NYS COMMON CORE MATHEMATICS CURRICULUM 4 5Lesson 10
Lesson 10: Use the area model and divisionto showtheequivalence oftwo
fractions.
Date: 4/14/15
5.B.42
© 2014 Common Core, Inc. Some rightsreserved. commoncore.org
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Exit Ticket (3 minutes)
Afterthe StudentDebrief,instructstudentstocomplete the ExitTicket. A review of theirworkwillhelpwith
assessingstudents’understandingof the conceptsthatwere presentedintoday’slessonandplanningmore
effectivelyforfuture lessons. The questions maybe read aloudtothe students.

8. NYS COMMON CORE MATHEMATICS CURRICULUM 4 5Lesson 10 Problem Set
Lesson 10: Use the area model and divisionto showtheequivalence oftwo
fractions.
Date: 4/14/15
5.B.43
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Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Name Date
Each rectangle represents 1.
1. Compose the shadedfractionintolarger fractional units. Expressthe equivalentfractionsinanumber
sentence usingdivision. The firstone hasbeendone foryou.
a. b.
c. d.
2. Compose the shadedfractionsintolarger fractional units. Expressthe equivalentfractionsinanumber
sentence usingdivision.
a. b.
4
6
=
4 ÷ 2
6 ÷ 2
=
2
3

9. NYS COMMON CORE MATHEMATICS CURRICULUM 4 5Lesson 10 Problem Set
Lesson 10: Use the area model and divisionto showtheequivalence oftwo
fractions.
Date: 4/14/15
5.B.44
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Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
3. Draw an area model torepresenteachnumbersentence below.
a.
4
10
=
4 ÷2
10÷2
=
2
5
b.
6
9
=
6 ÷3
9 ÷3
=
2
3
4. Use divisionto rename eachfractiongivenbelow. Draw a model if thathelpsyou. See if youcan use the
largestcommonfactor.
a.
4
8
b.
12
16
c.
12
20
d.
16
20

10. NYS COMMON CORE MATHEMATICS CURRICULUM 4 5Lesson 10 Exit Ticket
Lesson 10: Use the area model and divisionto showtheequivalence oftwo
fractions.
Date: 4/14/15
5.B.45
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Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Name Date
Draw an area model to showwhythe fractionsare equivalent. Show the equivalence in anumbersentence
usingdivision.
4
10
=
2
5

11. NYS COMMON CORE MATHEMATICS CURRICULUM 4 5Lesson 10 Homework
Lesson 10: Use the area model and divisionto showtheequivalence oftwo
fractions.
Date: 4/14/15
5.B.46
© 2014 Common Core, Inc. Some rightsreserved. commoncore.org
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Name Date
Each rectangle represents 1.
1. Compose the shadedfractionintolarger fractional units. Expressthe equivalentfractionsinanumber
sentence usingdivision. The firstone hasbeendone foryou.
a. b.
c. d.
2. Compose the shadedfractionsintolarger fractional units. Expressthe equivalentfractionsinanumber
sentence usingdivision.
a. b.
4
6
=
4 ÷ 2
6 ÷ 2
=
2
3

12. Lesson 10: Use the area model and divisionto showtheequivalence oftwo
fractions.
Date: 4/14/15
5.B.47
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This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
NYS COMMON CORE MATHEMATICS CURRICULUM 4 5Lesson 10 Homework
3. Draw an area model torepresenteachnumbersentence below.
a.
6
15
=
6 ÷ 3
15 ÷3
=
2
5
b.
6
18
=
6 ÷3
18÷3
=
2
6
4. Use divisiontorename eachfractiongivenbelow. Draw a model if thathelpsyou. See if youcan use the
largestcommonfactor.
a.
8
10
b.
9
12
c.
8
12
d.
12
18