Chapter2 FractionsandProblemSolving
2.1SolvingProblemsandExplainingSolutions
2.2DefiningandReasoningAboutFractions
1.AnnasaysthatthedarkblockspicturedinFigure2.1can’trepresent 1 4 because thereare6darkblocksand6ismorethan1but 1 4 issupposedtobelessthan 1.Whatshouldyouclarifyinordertointerpretthedarkblocksas 1 4 ?
2.WhenMartinwasaskedtosaywhatthe5inthefraction 2 5 means,Martinsaid thatthe5isthewhole.Explainwhyitisnotcompletelycorrecttosaythat“5 isthewhole.”Whatisabetterwaytosaywhatthe5inthefraction 2 5 means?
3.IfHarryneeds 4 5 ofaliterofdragonsnottomakeafullbatchofpotionbuthe onlyhas 3 5 ofaliterofdragonsnot,thenwhatfractionofabatchofpotioncan Harrymake(assuminghehasenoughoftheotheringredients)?
(a)Makeamathdrawingtohelpyousolvetheproblemandexplainyour solution.Useourdefinitionoffractioninyourexplanationandattendto thewhole(unitamount)thateachfractionis of
(b)Describethedifferentwholesthatoccurinpart(a).Discusshowone amountcanbedescribedwithtwodifferentfractionsdependingonwhat thewholeistakentobe.
4.TherectangleofXsbelowis 4 3 ofanotherrectangleofXs.Showtheoriginal rectangleandexplainhowtodetermineit.Useourdefinitionoffractioninyour explanation.
Copyright c 2018PearsonEducation,Inc.
5.TherectangleofXsbelowis 3 5 ofanotherrectangleofXs.Showtheoriginal rectangleandexplainhowtodetermineit.Useourdefinitionoffractioninyour explanation. XXXXXXXXXX XXXXXXXXXX XXXXXXXXXX XXXXXXXXXX XXXXXXXXXX XXXXXXXXXX
6.TherectangleofXsbelowis 3 5 ofanotherrectangleofXs.Show 2 5 oftheoriginal rectangleandexplainhowtodetermineit.Useourdefinitionoffractioninyour explanation. XXXXXXXXXX XXXXXXXXXX
7.If 3 4 ofacupofafoodgivesyouyourdailyvalueofpotassium,thenwhat fractionofyourdailyvalueofpotassiumisin1cupofthefood?
Makeamathdrawingtohelpyousolvetheproblemandexplainyoursolution. Useourdefinitionoffractioninyourexplanationandattendtothewhole(unit amount)thateachfractionis of
8.SeeFigure2.2.
(a)Howmanyoftheshortstripdoesittaketomakethelongstripexactly? Useourdefinitionoffractiontoexplainyouranswer.
Copyright c 2018PearsonEducation,Inc.
Short strip:
Long strip:
Figure2.2:Ashortstripandalongstrip.
(b)Howmanyofthelongstripdoesittaketomaketheshortstripexactly? Useourdefinitionoffractiontoexplainyouranswer.
9.(a)(SeeFigure2.3.)Thestripbelowrepresents 8 5 ofaunitamount.Using ourdefinitionoffractions,explainhowtodrawanotherstriptorepresent theunitamount.
This strip represents 8/5 of a unit amount:
Draw a strip that represents the unit amount:
Figure2.3:Fractionstrips.
(b)Inpart(a),howmanyofthegray 8 5 stripdoesittaketomakethestrip youdrewexactly?Explain.
10.(a)(SeeFigure2.4.)Thestripbelowrepresents 3 8 ofaunitamount.Using ourdefinitionoffractions,explainhowtodrawanotherstriptorepresent theunitamount.
(b)Inpart(a),howmanyofthegray 3 8 stripdoesittaketomakethestrip youdrewexactly?Explain.
11.(a)(SeeFigure2.5.)Thestripbelowrepresents 4 37 ofaunitamount.Using ourdefinitionoffractions,explainhowtodrawanotherstriptorepresent 15 37 oftheunitamount.
Copyright c 2018PearsonEducation,Inc.
This strip represents 3/8 of a unit amount:
Draw a strip that represents the unit amount:
This strip represents 4/37 of a unit amount:
Draw a strip that represents 15/37 of the unit amount:
Figure2.5:Fractionstrips.
(b)Inpart(a),howmanyofthegray 4 37 stripdoesittaketomakethestrip youdrewexactly?Explain.
12.(a)(SeeFigure2.6.)Thestripbelowrepresents 4 65 ofaunitamount.Using ourdefinitionoffractions,explainhowtodrawanotherstriptorepresent 11 65 oftheunitamount.
(b)Inpart(a),howmanyofthegray 4 65 stripdoesittaketomakethestrip youdrew?Explain.
13.ThestripinFigure2.7showsacanalthatis 16 100 ofamilelong.Usingour definitionoffractions,explainhowtodrawanotherstriptoshowacanalthat is 7 100 ofamilelong.
14.ThediagraminFigure2.8showsamapofMr.McGregor’sgarden,whichconsistsoftwoplotsofdifferentareas.Eachplotisdividedinto3partsofequal
Copyright c 2018PearsonEducation,Inc.
This strip represents 4/65 of a unit amount:
Draw a strip that represents 11/65 of the unit amount:
area.Mr.McGregorplantedpeasinthetwopartsthatareshaded.WhatfractionofMr.McGregor’s(entire)gardenisplantedwithpeas?Explainwhyyour answeriscorrectbyusingourdefinitionoffractionwithoutfurthersubdividing theplots.
15.Plot 3 4 onthenumberlineinFigure2.9andexplainwhythislocationfitswith ourdefinitionoffraction.
16.Plot 5 4 onthenumberlineinFigure2.9andexplainwhythislocationfitswith ourdefinitionoffraction.
Copyright c 2018PearsonEducation,Inc.
17.Accordingtoourtext,whatdoes 5 3 mean?
18.Explainwhat 7 4 meansaccordingtoourdefinition.Discusshowtomakesense ofit.
19.Makedrawingsanddiscusswhat 7 5 meansaccordingtothedefinitionoffraction weareusing.
20.Studentsoftenhavetroubleinterpretingimproperfractionssuchas 7 4 andthey sometimesthinkofafractionasapairofnumbersratherthanasasingle number.Useourdefinitionoffractiontointerpret 7 4 .Explainhowtounderstand 7 4 asanumberjustlikeanyothernumber,suchas3.
21.Studentsoftenhavetroubleinterpretingimproperfractionssuchas 4 3 andthey sometimesthinkofafractionasapairofnumbersratherthanasasingle number.Discusshowtointerpret 4 3 accordingtoourdefinitionoffractions.As partofyourdiscussion:
• explainhowtosee 4 3 asasinglenumber(notapairofnumbers);
• explainhowtoplot 4 3 onthenumberlineinFigure2.10.
Figure2.10:Plotthefraction 4 3 .
22.(a)Givetwodifferentfractionsthatyoucanlegitimatelyusetodescribethe shadedregioninFigure2.11.Foreachfraction,explainwhyyoucanuse thatfractiontodescribetheshadedregion.
(b)WriteanunambiguousquestionabouttheshadedregioninFigure2.11 thatcanbeansweredbynamingafraction.Explainwhyyourquestionis notambiguous.
Copyright c 2018PearsonEducation,Inc.
23.YoushowedJohnnythepictureinFigure2.12tohelpJohnnyunderstandthe meaningofthefraction 5 3 .ButJohnnydoesn’tgetit.Hesaysthepictureshows 5 6 ,not 5 3
5 3
(a)IsJohnnyrightthatthepicturecanbeinterpretedasshowing 5 6 ?Explain briefly.
(b)Whatmustyouclarifyinordertointerpretthepictureasshowing 5 3 ?
(c)Whatisanotherwaytoshow 5 3 toJohnny,otherthanwithpartsofobjects?
24.Manystudentshavedifficultyunderstandingimproperfractions,suchas 7 4 Statewhat 7 4 meansaccordingtoourdefinition.Thendescribeawaythat mighthelpastudentunderstandthismeaning.
25.Discusswhyitcanbeconfusingtoshowanimproperfractionsuchas 7 3 with piecesofpieorthelike.Whatisanotherwaytoshowthefraction 7 3 ?
2.3ReasoningAboutEquivalentFractions
1.Usingtheexample 2 3 = 2 4 3 · 4 andamathdrawing,explainwhymultiplyingthenumeratoranddenominatorof afractionbythesamenumberresultsinthesamenumber(equivalentfraction). Givea“general”explanation,inthesensethattheexplanationwouldworkthe samewayifothernumbershadbeenused.Inyourexplanation,attendcarefully tothenumberandthesizeoftheparts.
Copyright c 2018PearsonEducation,Inc.
2.Usingtheexample
andamathdrawing,explainwhymultiplyingthenumeratoranddenominatorof afractionbythesamenumberresultsinthesamenumber(equivalentfraction). Inyourexplanation,discussthefollowing:
• whathappenstothenumberofpartsandthesizeoftheparts;
• howyourmathdrawing showsthatthenumeratoranddenominatorare eachmultipliedby3;
• howyourmathdrawing showswhythosetwofractionsareequal.
3.Usingamathdrawing,explainwhymultiplyingboththenumeratoranddenominatorof
by4producesthesamenumber(anequivalentfraction).Discusshowtosee multiplicationby4inboththenumeratoranddenominatorintermsofyour mathdrawing.Attendcarefullytopointsthatmightbedifficultforstudents.
4.Withoutusingmultiplicationby1,explainwhymultiplyingthenumeratorand denominatorofafractionbythesamenumberproducesanequivalentfraction. Usetheexample
toillustrate.
5.Usingtheexample
explainin twodifferentways whymultiplyingthenumeratoranddenominator bythesamenumberproducesanequivalentfraction.
6.Usingthefractions 1 3 and 3 4 ,describehowtogivetwofractionscommondenominators.Intermsofamathdrawing,whatareyoudoingwhenyougive fractionscommondenominators?
Copyright c 2018PearsonEducation,Inc.
7.Usingamathdrawing,explainwhydividingboththenumeratoranddenominatorof
by3producesthesamenumber(anequivalentfraction).Discusshowtosee divisionby3inboththenumeratoranddenominatorintermsofyourmath drawing.Attendcarefullytopointsthatmightbedifficultforstudents.
8.Simplify
anduseamathdrawingtoshowtheprocess.
9.Plot 5 6 , 5 4 ,and 4 3 onthenumberlineinFigure2.13insuchawaythateach numberfallsonatickmark.Lengthenthetickmarksofwholenumbers.
Figure2.13:ANumberLine
10.(SeeFigure2.13.)Plot 5 6 ,0.7,and 4 5 onthenumberlineinsuchawaythat eachnumberfallsonatickmark.Lengthenthetickmarksofwholenumbers (ifany).
11.PlaceequallyspacedtickmarksonthenumberlineinFigure2.14sothatyou canplotboth 1 7 and 1 5 attickmarks.Explainyourreasoning.
0 2/7
Figure2.14:Aninterval
12.(SeeFigure2.15.)Explainhowtoplaceequallyspacedtickmarksonthe numberlinebelowsothatyoucanplotboth 1 5 and 1 6 onthenumberline. Explainyourreasoning.
13.(SeeFigure2.16.)Placeequallyspacedtickmarksonthenumberlinebelowso thatyoucanplotboth 1 3 and 1 5 attickmarks.Explainyourreasoning.
Copyright c 2018PearsonEducation,Inc.
Figure2.15:Plot 1 5 and 1 6 . 0 2/3
Figure2.16:Plot 1 3 and 1 5 attickmarks.
14.(SeeFigure2.17.)Sofar,Torreyhasrun 3 4 ofamile,butthatisonly 2 5 ofher totalrunningdistance.
Explainhowtouseequivalentfractionstoplaceequallyspacedtickmarkson thenumberlinebelowsothatyoucan(i)plotTorrey’stotalrunningdistance atatickmarkand(ii)determineTorrey’stotalrunningdistanceinmiles.
miles 3/4 miles
Figure2.17:Torrey’srun
15.Recipeproblem:Areciperequires 4 5 ofaliterofjuice.Youhave 2 3 ofaliterof juice.Whatfractionoftherecipecanyoumake?
Explainhowtoreasonwithamathdrawingandequivalentfractionstosolvethe recipeproblem.Discusshowequivalentfractionshelpyousolvetheproblem.
16.Swimmingproblem:Youswam 2 5 ofakilometerbutthatwasonly 3 8 ofthetotal distanceyouwantedtoswiminyourworkout.Howmanykilometersdidyou wanttoswiminyourfullworkout?
Explainhowtoreasonwithamathdrawingandequivalentfractionstosolvethe recipeproblem.Discusshowequivalentfractionshelpyousolvetheproblem.
17.Thelinesegmentbelowis 2 3 unitslong.Describehowtoremoveoraddaportion tothislinesegmenttocreatealinesegmentthatis 3 5 unitslong.Explainhow
Copyright c 2018PearsonEducation,Inc.
CHAPTER 2 youknowyournewsegmentwillbethecorrectlength.
18.Susiesaysthatwhenyoudothesamethingtothetopandbottomofafraction yougetanequivalentfraction.IsSusieright,orisitpossibletodothesame thingtothetopandthebottomofafractionand not getanequivalentfraction?
19.Kenordered 4 5 ofatonofsand.Kenwantstoreceive 1 3 ofhisordernow(and therestofhisorderlater).WhatfractionofatonofsandshouldKenreceive now?
Makeamathdrawingtohelpyousolvetheproblem.Explainhowyourdrawing helpsyoutosolvetheproblem.Inyourexplanation,attendcarefullytotheunit amount(whole)thateachfractionis of.
20.Frankismakingarecipethatcallsfor 3 4 ofacupofketchup.Frankonlyhas 1 3 ofacupofketchup.AssumingthatFrankhasenoughoftheotheringredients, whatfractionoftherecipecanFrankmake?
Makeamathdrawingtohelpyousolvetheproblem.Explainhowyourdrawing helpsyoutosolvetheproblem.Inyourexplanation,attendcarefullytotheunit amount(whole)thateachfractionis of
21. Ketchupproblem:Youaremakingarecipethatcallsfor 3 4 ofacupofketchup. Youonlyhave 1 3 ofacupofketchup.Assumingthatyouhaveenoughofthe otheringredients,whatfractionoftherecipecanyoumake?
Explainhowtosolvetheketchupproblemwiththeaidofamathdrawing.In yourexplanation,attendcarefullytotheunitamountthateachfractionis of
22.ThestripsinFigure2.18showtherelativeamountsofcansthatBenandCharles collectedforthecan-a-thon.(Notethateachrectanglerepresentssomefixed numberofcans,butthisnumbermaybegreaterthan1.)
(a)Writeasentenceinwhichyouuseafractiontodescribehowtheamount ofcansthatBencollectedcompareswiththeamountofcanthatCharles collected.Explainbriefly.
(b)Writeasentenceinwhichyouuseafractiontodescribehowtheamount ofcansthatCharlescollectedcompareswiththeamountofcanthatBen collected.Explainbriefly.
Copyright c 2018PearsonEducation,Inc.
Charles’s amount: Ben’s amount:
Figure2.18:TheAmountsofCansthatBenandCharlesCollected 2.4ReasoningtoCompareFractions
1.Describethreegeneralmethodsfordeterminingwhichoftwofractionsisgreater. Illustratethethreemethodswiththefractions 3 5 and 5 8 .
2.Usingthefractions 2 3 and 3 5 toillustrate,explainclearlyandindetailwhywecan determinewhichoftwofractionsisgreaterbygivingthetwofractionscommon denominators.Whatistherationalebehindthismethod?Intermsofpictures, whatarewedoingwhenwegivethefractionscommondenominators?
3.Showhowtousethecross-multiplyingmethodtodeterminewhichof 5 8 and 7 12 isgreater.
4.Usingthefractions 2 3 and 3 5 toillustrate,explainclearlyandindetailwhywe candeterminewhichoftwofractionsisgreaterbyusingthecross-multiplying method.Whatistherationalebehindthismethod?Whatarewereallydoing whenwecross-multiplyinordertocomparefractions?
5.Givetwodifferentmethodsforsolvingthefollowingproblem.Findtwodifferent fractionsinbetween 3 4 and 4 5 whosenumeratorsanddenominatorsareallwhole numbers.
6.Conradsaysthat 3 8 > 2 7 because3 > 2and8 > 7.Regardlessofwhetheror notConrad’sconclusioniscorrect,discusswhetherornotConrad’sreasoning isvalid.
7.Ronsaysthat 5 8 > 3 7 because5 > 3and8 > 7.
(a)DiscusswhetherornotRon’sreasoningisvalid(whetherornothisconclusioniscorrect).
(b)Describeanotherwaytoreason(legitimately)tocompare 5 8 and 3 7 withoutconvertingtodecimals,givingthefractionscommondenominators,or cross-multiplying.
Copyright c 2018PearsonEducation,Inc.
8.Minjusaysthatfractionsthatusebiggernumbersaregreaterthanfractionsthat usesmallernumbers.MakeuptwoproblemsforMinjutohelpherreconsider herideas.Foreachproblem,explainhowtosolveit,andexplainwhyyouchose thatproblemforMinju.
9.Usereasoningotherthanconvertingtodecimals,usingcommondenominators orcross-multiplyingtodeterminewhichof 19 94 and 19 107 isgreater.Explainyour reasoningclearlyandindetail.
10.Usereasoningotherthanconvertingtodecimals,usingcommondenominators orcommonnumerators,orcross-multiplyingtodeterminewhichof 4 19 and 5 17 is greater.Explainyourreasoningclearlyandindetail.
11.Usereasoningotherthanconvertingtodecimals,usingcommondenominators orcommonnumerators,orcross-multiplyingtodeterminewhichof 13 17 and 14 15 is greater.Explainyourreasoningclearlyandindetail.
12.Usereasoningotherthanconvertingtodecimals,usingcommondenominators orcommonnumerators,orcross-multiplyingtodeterminewhichof 38 39 and 45 46 is greater.Explainyourreasoningclearlyandindetail.
13.Usereasoningotherthanconvertingtodecimals,usingcommondenominators orcommonnumerators,orcross-multiplyingtodeterminewhichof 37 39 and 43 45 is greater.Explainyourreasoningclearlyandindetail.
14.Usereasoningotherthanconvertingtodecimals,usingcommondenominators orcommonnumerators,orcross-multiplyingtodeterminewhichof 21 43 and 41 81 is greater.Explainyourreasoningclearlyandindetail.
15.Usereasoningotherthanconvertingtodecimals,usingcommondenominators orcommonnumerators,orcross-multiplyingtodeterminewhichof 23 44 and 25 48 is greater.Explainyourreasoningclearlyandindetail.
16.Usereasoning otherthan convertingtodecimals,usingcommondenominators orcommonnumerators,orcross-multiplyingtodeterminewhichof 13 24 and 17 32 is greater.Explainyourreasoningclearlyandindetail.
17.Usereasoningotherthanconvertingtodecimals,usingcommondenominators orcommonnumerators,orcross-multiplyingtodeterminewhichof 24 23 and 26 25 is greater.Explainyourreasoningclearlyandindetail.
Copyright c 2018PearsonEducation,Inc.
18.Usereasoningotherthanconvertingtodecimals,usingcommondenominators orcommonnumerators,orcross-multiplyingtodeterminewhichof 7 15 and 11 19 is greater.Explainyourreasoningclearlyandindetail.
2.5ReasoningAboutPercent
1.Ifyourfulldailyvalueofpotassiumis3600milligrams,thenhowmanymilligramsis45%ofyourdailyvalueofpotassium?Showhowtosolvetheproblem withtheaidofamathdrawingorapercenttable.Explainyourreasoning.
2.IfthenormalrainfallforAugustis2.5inches,butonly1.75inchesofrainfellin August,thenwhatpercentofthenormalrainfallfellinAugust?Showhowto solvetheproblemwiththeaidofamathdrawingorapercenttable.Explain yourreasoning.
3.Ifthefullcapacityofatankis25litersandthetankisfilledwithonly15liters, thenwhatpercentfullisthetank?Showhowtosolvetheproblemwiththeaid ofamathdrawingorapercenttable.Explainyourreasoning.
4.If$85,000is40%ofthebudget,thenwhatisthefullbudget?Showhowto solvetheproblemwiththeaidofamathdrawingorapercenttable.Explain yourreasoning.
5.If 3 4 ofacupoffoodgivesyou40%ofthedailyvalueofcalcium,thenhowmany cupsofthefoodwillgiveyou100%ofthedailyvalueofcalcium?Explainhow tosolvethisproblemwithamathdrawing and apercenttable.Relatethe informationinthetableandthedrawing.
6.If 3 4 ofacupofjuicegivesyou100%ofyourdailyvalueofvitaminC,thenwhat percentofyourdailyvalueofvitaminCwillyougetin1fullcupofjuice?Show howtosolvetheproblemwiththeaidofamathdrawingorapercenttable. Explainyourreasoning.
7.If 2 5 ofaliterofjuicegivesyou100%ofyourdailyvalueofvitaminC,then whatpercentofyourdailyvalueofvitaminCwillyougetin 1 2 literofjuice? Showhowtosolvetheproblemwiththeaidofamathdrawingorapercent table.Explainyourreasoning.
8.If 2 5 ofaliterofjuicegivesyou100%ofyourdailyvalueofvitaminC,then howmuchofyourdailyvalueofvitaminCwillyougetin 1 2 ofaliterofjuice?
Copyright c 2018PearsonEducation,Inc.
Explainhowtosolvethisproblemwithamathdrawing and apercenttable. Relatetheinformationinthetableandthedrawing.
9.If 1 3 ofapoundofcheesegivesyou100%ofyourdailyvalueofcalcium,then whatpercentofyourdailyvalueofcalciumdoyougetin 3 4 ofapoundofcheese? Explainhowtosolvethisproblemwiththeaidofapercenttable and amath drawing.
10.Ifrunning 5 8 ofamileis100%ofyourworkout,thenwhatpercentofyour workoutisrunning 1 2 mile?Showhowtosolvetheproblemwiththeaidofa mathdrawingorapercenttable.Explainyourreasoning.
11.IfCompanyAsells30%asmanycarsasCompanyB,thenwhatareCompany B’scarsales,whentheyarecalculatedasapercentageofCompanyA’ssales? Useamathdrawingorpercenttabletohelpyousolvetheproblem.Useyour drawingortabletohelpyouexplainyouranswer.
12.Explainingyourreasoningclearlyandusemathdrawingsortablestohelpyou answerthefollowing:
a.Whatpercentof 1 5 is 2 5 ?
b.Whatpercentof 2 5 is 1 5 ?
13.Solveusingequivalentfractionswithoutcross-multiplying:Ashirthadcost$15, butnowitisonsaleata$6discount.Whatpercentisthediscount?
14.Solveusingequivalentfractionswithoutcross-multiplying:Acompany’sprofits were12%ofitsrevenues.Ifthecompany’sprofitswere$360,000,whatwereits revenues?
15.Solveusingequivalentfractionswithoutcross-multiplying:12%ofacompany’s 75employeeswalktoworkeveryday.Howmanyemployeeswalktoworkevery day?
16.Therewere70membersinaclub.Ofthe70members,60%weregirlsandthe restwereboys.Aftersomemoregirlsjoinedtheclub(andnoboysjoinedor left),theclubwas75%girls.Howmanygirlsjoinedtheclub?