Part1:DividingFractionsUsingVisualRepresentations
Todividefractions,rememberthatdivisioncanberepresentedbyrepeatedsubtraction,justlike
multiplicationcanberepresentedbyrepeatedaddition.
Here’ssometerminologythatwillbeusedthroughoutthislesson.
Section1.1.DividingWholeNumbersbyFractionsUsingObjectsandNumberLines
Modeldividingawholenumberbyafractionusingbothobjectsandnumberlines.
Foreachproblem,writeawordproblemthatdescribesthesituation.
A.3 ÷ #$ = Wordproblem:
'
C.4 ÷ = *
B.4 ÷ =
(
Wordproblem:
D.3 ÷ =
+
,
Wordproblem:
'
Wordproblem:
1
Section1.2.DividingFractionsbyFractionsWhenThereisNORemainder
Modeldividingamixednumber(awholenumberplusafraction)byafractionusingobjectsandnumberlines.
+
'
E.3 ÷ = *
*
'
,
(
(
F.2 ÷ = +
*
G.1 ÷ = '
,
+
(
*
/
H.3 ÷ =
Section1.3.DividingFractionsbyFractionsWhenThereISaRemainder
Dividethesefractionsusingnumberlinesandfractionbarstomodeltheproblems.
+
+
I.2 ÷ = *
'
+
+
,
'
J.3 ÷ =
+
'
K.3 ÷ = '
*
+
*
'
,
L.2 ÷ =
2
Part2:DescribingDivisionwithFractions
Section2.1.DescribingDividingFractions
Inyourownwords,describehowvisualrepresentations(objectsandnumberlines)canbeusedtomodel
divisionwithfractions.Inyourdescription,answerthequestion,“Howdoesdivisionbyfractionswork?”
Makesuresomeoneelsecouldreadyourdescriptionandbeabletomodeldivisionwithfractions.
Section2.2.ButIsn’ttheQuotientAlwaysSmallerWhenWe’reDividing?
Whatisthequotientwhenyoudivide10by2?______
Whenyoudivide5by10?______
Arethesequotientslargerorsmallerthanthedividendsanddivisors?
Canyouthinkofanexamplewhenyoudivideonewholenumberbyanotherandgetaquotientthatislarger
thanthedividendordivisor?
LookatthequotientsforproblemsAthroughL.Whatdoyounotice?
Writeanexplanationforyourobservation.
3
Section2.3:WritingandSolvingaWordProblemThatUsesDividingwithFractions
Thinkaboutsomethingyou’veseenordonethatinvolvesdividingsomethingintoparts,whereboththe
dividendandthedivisorarefractions.
A.Writeawordproblemtodescribethesituation.
B.Writethedivisionproblemusingfractions.
C.Drawobjectstomodelthesituation.
D.Writethedivisionproblemagainwiththequotient.
4
Part3:SolvingDivisionProblemsWhentheDivisor
isLargerThantheDividend
Section3.1.DivisorLargerThanDividend
InalloftheproblemsinParts1and2,thedividendhasbeenlargerthanthedivisor.However,thedivisorcan
*
alsobelargerthanthedividend,suchas3 ÷ 10,whichcanbewrittenas .
+2
Wordproblemswithfractionsthatcanbemodeledwithadivisionprobleminwhichthedivisorislargerthan
thedividendareshownbelow.
1
You’regoingtoshare½poundofchocolatewith3friends,soeach
÷3
2
persongetsanequalsizeportion.Howmuchwilleachofyouget?
3
1
Howwideisarectangularstripoflandwithlengthof1#$milesand
÷1
4
2
anareaof3/4squaremile?(Remember,A=LW,orW=A/L)
2 3
÷
Howmany3/4-cupservingsarein2/3ofacupofyogurt?
3 4
Modelingthefirstproblemiseasy!Thinkaboutwhenyoudivide12by3,or12 ÷ 3.Youcanrepeatedly
subtractgroupsof3,resultingin4groupsofsize3(quotitiveorrepeatedsubtraction:youknowhowmany
areineachgroupandyou’vegottofindhowmanygroups).Oryoucandivide12into3groups,eachofsize4
(partitiveorequalsharing:youknowthenumberofgroupsbutneedtofindhowmanypartsareineach
group).Previously,wedidthefirstthing:repeatedlysubtractinggroupsofthesamesizetofindouthowmany
groups.Let’strysomeproblemswherewedivideintoequalsizegroups.Usebothfractionbarsandnumber
linestomodeltheseproblems.
+
N. ÷ 3 =
M.#$ ÷ 3 = ,
Wordproblem:
Wordproblem:
5
*
O. ÷ 2 = ,
'
P. ÷ 3 =
*
Wordproblem:
Wordproblem:
Section3.2:DivisorLargerThanDividendWhenBothareFractions
+
*
+
+
R. ÷ =
Q. ÷ = '
,
,
'
Wordproblem:
Wordproblem:
6
'
(
S. ÷ =
*
/
'
*
*
,
T. ÷ =
Wordproblem:
Wordproblem:
Section3.3.AretheQuotientsLargerorSmallerThantheDividendsandDivisors?
ArethequotientsinthetheproblemsinPart3largerorsmallerthanthedividendsanddivisors?
Writeanexplanationforyourobservation.
7
Part4:DividingFractionsUsingProcedures
Section4.1.DividingFractionsUsingtheStandardAlgorithm
Itistedioustousedrawingsandnumberlinestodividefractions.Also,asfractionsgetmorecomplexlikethe
onesshownbelow,fractionbarsornumberlineswon’twork.Weneedtofigureoutproceduresfordividing
fractions.YouneedtounderstandtheseproceduresandhowtheyworkinAlgebrawhenyousimplify
expressionsandperformoperationswithmorecomplexfractionsliketheonesbelow.
(2x 2 + 5x + 3) ÷ (x + 1) =
2x + 5x + 3
x +1
2
4a + 12a + 9
9a − 25
÷
4a 2 + 8a + 3 6a 2 + 13a + 5
2
2
25 y + 1
+
12
4
5 8y + 4
−
18
36
Followinstructionsinthetoprowofthetableonthenextpagetofindtheanswerstothedivisionproblems
usingtheStandardAlgorithm.Checktoseeifyouranswersarethesameastheanswersyougotforproblems
AthroughLinPart1.
Section4.2.DiscoveringaShortcutforDividingFractions
Lookattheoriginalproblemsinthefirstcolumnofthetableandtheproblemsasrewrittenincolumn4.
Canyouseearelationshipbetweenthesetwoequivalentrepresentationsoftheproblem?Describethe
relationshipbetweentheproblemsincolumn1andtheequivalentrepresentationsincolumn4.
Writeashortdescriptionforashortcutyoucanusetodividefractions.
8
Column1
Column2
Original
Rewritewith
problems
theproblem
improper
fractions
A.3 ÷ #$ =
+
3
1
1
2
3 1
÷ =
1 2
Column3
Multiplybya
fractionthat
equals1to
simplifythe
complexfraction
3 2
⎛ 21 ⎞
1(1)
⎜⎝ 2 ⎟⎠ = 1 =
1
Column4
Column5 Column6
Writethe
Writethe Writethe
multiplication
answer
answeras
problem
amixed
(withoutthe
number
denominator=1)
3⎛ 2⎞
⎜ ⎟
1 ⎝ 1⎠
=
6
= 6
1
= =
=
=
=
=
D.4 ÷ =
=
=
=
+
'
*
*
=
=
=
'
,
(
(
=
=
=
G.1 ÷ =
+
*
'
,
=
=
=
+
(
*
/
=
=
=
= =
=
=
=
=
=
=
=
=
=
=
=
=
=
=
B.3 ÷ =
,
'
C.4 ÷ =
*
'
(
E.3 ÷ =
F.2 ÷ =
H.3 ÷ =
+
+
*
'
+
+
,
'
+
'
'
*
+
*
'
,
I.2 ÷ =
J.3 ÷ =
K.3 ÷ =
L.2 ÷ =
Rewrite
using
alternate
division
notation
9
Column1
Column2
Original
Rewritewith
problems
theproblem
improper
fractions
Rewrite
using
alternate
division
notation
M.#$ ÷ 3 =
+
N. ÷ 3 =
,
*
O. ÷ 2 =
,
'
P. ÷ 3 =
*
+
*
'
,
+
+
,
'
'
(
*
/
'
*
*
,
Q. ÷ =
R. ÷ =
S. ÷ =
T. ÷ =
Column3
Multiplybya
fractionthat
equals1to
simplifythe
complexfraction
Column4
Column5 Column6
Writethe
Writethe Writethe
multiplication
answer
answeras
problem
amixed
(withoutthe
number
denominator=1)
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
10
Part5:AnotherWaytoThinkAboutDivisionwithFractions
Here’sanotherwaytohelpyouthinkaboutdividingfractions.Which,ifany,ofthesequestionshavea
differentanswer?
• Howmany3saretherein6?
• Howmanygroupsof3tensaretherein6tens?
• Howmanygroupsof3fivesaretherein6fives?
• Howmanygroupsof3tenthsaretherein6tenths?
• Howmanygroupsof3@saretherein6@s?
• Howmanygroupsof3gronksaretherein6gronks?
• Howmanygroupsof3anythingsaretherein6anythings(aslongasbothrefertothesameunit)?
Thepointofthesequestionsisthattheunitsoftheproblemdonotmatter.Iftheunitsarethesameobjects,
they“disappear”byformingafractionthatequals1whenyoudivide.Thismeansthatif6/10isdividedby
3/10,3/10goesinto6/10thesamenumberoftimesas3goesinto6,or3/12goesinto6/12,or3/20goesinto
6/20,or3/29goesinto6/29.
UsethestandardalgorithmfromSection4todivide6/29by3/29.
Describewhathappened.
Thissuggestsanewalgorithmfordividingwithfractions:Todividetwofractions,findacommondenominator
sothedenominatorsformafractionthatequals1,andthendividethenumerators.Let’slookatsomeofthe
problemsweworkedearlierthatyousolved.Rewriteequivalentfractionsthathavecommondenominators.
Then,knowingthedenominatorsformafractionthatequals1whendividing,dividethenumerators.
+
'
U.3 ÷ = *
*
W.3 ÷ #$ = '
,
(
(
V.2 ÷ =
X.4 ÷ =
'
*
#
Y. ÷ 2 = Z.$ ÷ 3 =
,
+
+
+
+
AA.3 ÷ =
AB.2 ÷ =
,
'
*
'
+
*
'
(
AC. ÷ = AD. ÷ =
'
,
*
/
'
*
+
'
AE. ÷ = AF.3 ÷ =
*
,
*
(
Inyourownwords,writeadescriptiontotellwhythismethodworkstodividefractions.
*
11
Part6:SolvingWordProblemsUsingFractionswithDivision
Weneedtodividefractionsbyfractionsinmanysituations,asshowninthefollowingproblems.Foreach
problem,drawadiagramornumberlinesshowinghowyoucanuserepeatedsubtractiontomodeldividing
fractions.Remembertouseastraightedgeanddrawnumberlinescarefully.Then,setupthedivisionproblem
tofindtheanswer.Writeanswersasmixednumbersifquotientisnotawholenumber.
Forproblems1-5,usethestandardalgorithmtodividefractionsandfindtheanswertotheproblem.
+
+
1) Alawnmowertankholds gallonofgas.Ian’s5-galloncontainerhas3 gallonsinit.Howmanytimeswill
'
'
hebeabletofillthelawnmower?
2) Tarikiscuttingstringers,theboardsinhousewallsthatrunhorizontallybetweenstudstoaddstability.
+
Eachstringeris14”or1 ft.HowmanystringerscanTarikcutoutofeach8ft.stud?
3
3) Amal’smotherisplanninghisbirthdaypartyandwantstoservepizza.Eachpizzaiscutinto8equalsize
*
pieces.Ifeachchildeatsthreepieces,or ofapizza,howmanychildrencanbefedwith4pizzas?
4
+
4) Mariahas256yardoffabric.Shewantstomakeplacemats.Eachplacematneeds yard.Howmany
*
placematscanMariamake?
+
'
5) Jaronhas5 gallonsofpaintandneedstopaint8chairs.Eachchairneeds gallonofpaint.Howmany
'
*
chairscouldJaronpaint?Willhehaveenoughforthe8chairs?
Forproblems6-9,usetheshortcutyoudiscoveredinPart4tofindtheanswer.
*
6) Angelodrawscaricatures.Eachcaricaturetakesanaverageof18minutes,or ofanhour.IfAngelo
+2
+
rentsaboothatthefairfor5 hours,howmanycaricatureswillhebeabletodraw?
'
7) Annhas22/3yardofribbon.Sheneeds4pieces,allthesamelength.Howlongshouldshecuteachpiece
ofribbon?
*
+
8) Manuelismakingbannersforaclient.Eachbanneris2 feetlong.Thereare21 feetofpaperlefton
,
'
thebannerroll.Howmanybannerscanhemake?
9) JudyisinGreatBritainandwantstohelpherfriendmakepancakesforafundraisingevent.Judy’sfavorite
'
recipecallsfor cupofbuttermilkforeachbatchofpancakes.ButtermilkintheUKcomesin
*
+
1litercontainers.Alitercontainsapproximately4 cups.Howmanybatchesofpancakescantheymake
,
iftheyhave2litersofbuttermilk?
10) WriteyourownwordproblemforProblemEinPart1.
11) WriteyourownwordproblemforProblemGinPart1.
12) WriteyourownwordproblemforProblemJinPart1.
12
Math Vocabulary Notebook: Fraction Bars
6/10/14, 5:22 PM
Fraction Bars
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Math Vocabulary Notebook: Fraction Bars
6/10/14, 5:22 PM
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Fraction Strips