Teachers Manual: Grade 4 Fractions (Unit 11, 4B, pp.38-51)

Page501. Goals of unit

Studentswillbeabletousefractionstoexpressmeasurements.Inaddition,studentswillunderstandtheconceptofproperfractions,mixednumbersandimproperfractionsanddeepentheirunderstandingofthemeaningoffractions.Studentswillunderstandhowtoaddandsubtractlike‐denominatorfractions.

Interest

Studentstrytoexpressusingfractionstheamountleftoverfrommeasuringwithaunit.

ThinkingStudentsnoticethattheycanexpressthesamequantityusingvariousfractionalunits.

ExpressionStudentschangeimproperfractionsbackandforthtomixednumbersorwholenumbers.Also,theyareabletoaddandsubtractfractionswithlikedenominators.

KnowledgeStudentsunderstandtheconceptofproperfractions,mixednumbers,andimproperfractions.Also,theyknowhowtoaddandsubtractfractionsthathavelikedenominators.

2. Major points of unit 1) Area

Inthisunit,mixednumbersareintroduced,usingthemtoexpresstheamountleftoverfrommeasuringwithaunit.Improperfractionsareintroducedandunderstandingofthemeaningoffractionsisdeepenedbygraspingmixednumbersandimproperfractionsas“howmanyofaunitfraction.”Understandingmixednumbersandimproperfractionswillestablishthefoundationforfractioncalculationssuchasaddition‐subtraction,andmultiplication‐divisionoffractionswithlikedenominators.

2) Equivalent fractions

Here,studentswillunderstandfractionsasnumbersbyshowingfractionsonanumberlineandbyconsideringequivalentfractionsandtherelativesizeoffractionswithdifferentdenominators.Sofar,fractionshavebeenunderstoodbystudentsasawaytoexpressaquantityofsomething,butduringthisunit,studentswillgainamoreabstractunderstandingoffractionsandidentifythemasnumberslikewholenumbersanddecimalnumbers.

3) Addition and subtraction of fractions Inthirdgrade,studentslearnedsimpleadditionandsubtractionoffractions

thathavelikedenominators.Butthemaingoalinintroducingfractioncalculationwastohelpstudentsunderstandthecompositionofproperfractions[thatnon‐unitfractionsarecomposedofunitfractions].(Thecalculationsumswerelessthan1.)Illustrationsandnumberlineswereimportantmaterialstohelpstudentsunderstandthebasicideaoffractioncalculations. Inthisunit,studentswilldeveloptheirskillinadditionandsubtractionbasedonknowledgetheylearnedinthirdgrade.Fractioncalculationseemsdifferentfromthatofwholenumbersanddecimalnumbersbutstudentscanseethatthesamebasicprinciplesfromtheirpriorstudyofwholenumbersanddecimalsapplyiftheygrasptheideaofunitfractions.It’simportanttobuildmasteryofadditionandsubtractoflike‐denominatorfractionsontheunderstandingthatitisthesamebasicprincipleofcalculation[asforwholenumbersanddecimals].3. Teaching and evaluation plan Subunit Per Goal LearningActivities MainEvaluationPoints1.Howtoexpressinfractions(page38‐43,4periods)

123

Studentsunderstandhowtoexpressinmixednumbersanamountexceedingameasure‐mentunit.Studentsunderstandtheconceptof“properfraction”and“mixednumber.”a)Students

a)Studentsgraspthetopicofthisunitthroughtheprioractivityofexpressingthevolumeof

milkas

€

23L.

b)Studentslearnhowtoexpressinafractionanamountofjuiceexceedingameasurementunit.c)Studentsunderstandhowtowriteandread

1

€

34L.

a)Studentsexpress1

€

34

onanumberline.b)Studentslearnthedefinitionsof“properfraction”and“mixednumber.”a)Studentsthinkabout

(Interest)Studentstrytoexpressanamountthatislargerthanthemeasurementunit.(Expression)Studentstrytoexpressanamountthatislargerthanthemeasurementunitinamixednumber.(Knowledge)Studentsunderstandhowtoexpressanamountlargerthanameasurementunit.(Knowledge)Studentsunderstandtheconceptof“properfraction”and“mixednumber.”(Thinking)Studentsnotice

4

understandtheconceptof“improperfraction.”b)Studentsunderstandthefractionsthatareequivalenttowholenumbers.Studentsunderstandhowtochangemixednumbersintoimproperfractionsorimproperfractionsintomixednumbers.

howmanymthree

€

13m,

four

€

13mare.

b)Studentsunderstandthedefinitionof“improperfraction.”c)Studentscomparethedenominatorandnumeratorinfractionsthatareequivalenttowholenumbers.a)Studentsthinkabout

howtochange2

€

13into

animproperfraction.b)Studentsthinkabout

howtochange

€

73intoa

mixednumber.

thatimproperfractions,likewholenumbers,expresshowmanyofaunit.(Knowledge)Studentsunderstandtheconceptofimproperfractions.(Expression)Studentsareabletochangemixednumbersintoimproperfractions.(Knowledge)Studentsunderstandhowtochangemixednumbersintoimproperfractions.

2)Equi‐valentfractions(page44‐45,1period)3)Additionandsubtractionoffractions(page46‐

11

a)Studentsunderstandequivalentproperfractions.b)Studentsunderstandhowtocomparefractionswithlikenumerators.Studentsunderstandandcanperformadditionof

a)Studentsexpressthecoloredpartsasfractions.b)Usingnumberlines,studentsfindfractionsthatareequivalent.c)Usingnumberlines,studentscomparetherelativesizeoffractions.a)Studentsthinkofamathsentencetoexpressthecombinedareaofcardboardpieces

(Thinking)Studentsnoticethattheyshouldconsiderthesizeofthedenominatorwhentheycomparetherelativesizeoflike‐numeratorfractions.(Expression)Studentsareabletocomparetherelativesizeoflike‐numeratorfractions.(Thinking)Studentsgraspadditionoflike‐denominatorproperfractions(thatresultinimproperfractions)as“how

505periods)

234

like‐denominatorproperfractions(whentheanswerisanimproperfraction).Studentsunderstandandcanperformsubtractionofproperfractionsfrommixednumbers(Thewholenumberpartis1,theanswersareproperfractions).StudentsunderstandandcanperformadditionofmixednumberstooneanotherStudentsunderstandandcanperformsubtractionofamixednumberfromanothermixednumber.

of

€

35m²and

54m².

b)Studentsthinkabout

howtocalculate

€

35+54 .

c)Studentsreviewhowtoaddfractionswithlikedenominators.a)Studentsthinkofamathsentencetoexpresshowmanykgwillbeleftifyouuse

54kgofsugarfrom1

€

25kg

ofsugar.b)Studentsthinkabout

howtocalculate1

€

25‐54 .

c)Studentswillreviewhowtocalculatefractionsthathavethesamedenominators.a)Studentswillthinkabouthowtocalculate

2

€

35+1

€

15.

a)Studentswillthinkabouthowtocalculate

254 ‐1

€

35.

manyofaunitfraction”;theynoticetheycanthinkaboutitlikeadditionofwholenumbers.(Thinking)Studentsnoticethattheycanusepreviouslylearnedideasaboutadditionoffractionswhentheysubtractaproperfractionfromamixednumber(whosewholenumberpartis1)andgetaproperfraction.(Thinking)Studentsnoticethattheycanaddmixednumbersbasedontheirknowledgeofadditionofproperfractions.(Express)Studentsareabletosubtractmixednumbers.(Knowledge)Studentsunderstandhowtosubtractmixednumbers.

5 Studentsreviewandpracticethecontentsofthisunit.

“Practice”section.

4)Check(page511period)

Studentswillreviewthecontentsofthisunit.

“Check”and“Challenge”sections.

Page52About the curriculum Fractionsareintroducedinthirdgrade.Atthatpoint,studentsshouldunderstandhowtoexpressanamountlessthanameasurementunit.Inthisunit,studentslearnmixednumbers.Lesson 1 How to express fractions (page.38-43, 4 periods)

(The first period) Goals: Studentslearnhowtoexpressanamountgreaterthanameasurementunit. Preparation:1Lsquaregraduatedcontainer,enlargementofthepictureinthetextbook.1. Looking at the scene in the textbook on page 38, students become interested in how to express amounts of milk and juice.

(KeyQuestion)Howmanylitersistheamountofmilk?∙Studentsgraspthat1Lisdividedinto3equalpartsandthattheycanexpresstheamountwithfractions.

∙Studentsgraspthat,when1Lisdividedinto3equalparts,eachpartis

€

13L,

and2of

€

13Lmakes

€

23L.

(KeyQuestion)Howmanylitersistheamountofjuice?∙Studentsgraspthatthejuiceismorethan1Lbutlessthan2L.∙Confirmthatstudentsarethinkingabouthowtoexpressanamountgreaterthan1Lusingafraction.

Page53(Sample questions) 1. Let’s express the amounts with fractions.

1) Thecombinedquantitiesof5Land

€

14L.

2) Thecombinedquantitiesof2mand

€

35m.

3) Thecombinedquantitiesof3kgand

€

34kg.

4) Thecombinedquantitiesof

€

25kgand1kg.

2. Students read question 1 on page 38 and understand the content.

∙Teacherswriteonblackboardorpaper“?”frompage39(“Let’sinvestigatehowtoexpressnumbersthataregreaterthan1infractions!”).(KeyQuestion)“Let’sthinkabouthowtoexpressanumberthatisgreaterthan1infractions.”(Thinking)Studentsthinkabouthowtoexpressanamountthatisgreaterthanameasurementunit,bythinkingaboutdecimalnumbers.(Notebooks/Observation)(ExtraSupport)Ifnostudentmakesaconnectionbetweendecimalnumbersandfractionswhenthinkingabouthowtoexpress1Landtheremainder,teacherscanask“Howdidyouexpressitindecimalnumbers?Helpstudentstounderstandthattheamountofjuiceismorethan1L,sotheycancombine1Lwithpartleftoverfrommeasuring1L.(KeyQuestion)“Howmuchistheamountleftover[frommeasuringwithaunit]?(ExtraSupport)Ifthereareanystudentswhoarestrugglingtoexpressasafractiontheamountleftoverfrommeasuringwithaunit,teacherscanshowtheillustrationofthesquaregraduatedcontaineronpage38andask,“Howmanypiecesis1Ldividedinto?”

∙Understand3piecesof

€

14Lis

€

34Landmakesuretheamountofjuiceincludes

1Land

€

34L.

Page54About definitions

Themeaningofmathematicaltermsshouldbemadeclear.Asentenceorequationthatexplainsamathematicaltermiscalledadefinition.Thistextbookdefinesthemeaningof“properfraction”and“mixednumber”onpage40,andthendefines“improperfraction”onpage42.

3. Students learn how to express and read the combined quantity

of 1L and

€

34L.

(KeyQuestion)“Howdidyouexpressitusingdecimalnumbers?Howcanyouexpressitusingfractions?”(Interest)Studentstrytoexpressanamountgreaterthanonemeasurementunit,drawingontheirknowledgeofhowitisdonewithdecimalnumbers.(Notebooks/Observation/Comment)

4. Students work on the questions 1& 2 on page 40.

(Expression)Studentscanexpressanamountthatisgreaterthan1usingamixednumber.(Comment/Notebook)(ExtraSupport)Teachershelpstudentsclarifyeachaspectinturn,asneeded:“Whatisthemeasuredquantity?Howmanyunitsarethere?Howmuchistheamountleftoverfrommeasuringwiththeunit?∙Makesurestudentswritemeasurementunits(likeliterandmeter)totherightofthefractionalline,notnexttothedenominator.

2nd period Goals: Studentsunderstandtheconceptof“properfraction”and“mixednumber.”Preparation:Enlargementoftextbookillustration(p.40);Blackboardnumberline

1. Students understand that

€

1 34

can be expressed on a number

line and they grasp that mixed numbers are numbers.

(KeyQuestion)“Let’sshow1

€

34onanumberline.”

(Thinking)Studentsnoticethatmixednumberscanbeexpressedonnumberlineslikewholenumbers.(ExtraSupport)Helpstudentsnoticethatmixednumbersarenumbersbecausetheycanbeexpressedonthenumberlinelikewholenumbers.∙Readtheproperfractionsandmixednumbersshownbyarrowsonthenumberlinepage40.∙Categorizetheseintonumberslessthan1andnumbersgreaterthan1.

2.Students understand the definition of “proper fraction” and “mixed number.”

(Knowledge)Studentsunderstandtheconceptsofproperfractionandmixednumber.

Page55Structure of improper fractions

Itisimportanttounderstandunitfractionstounderstandthestructureof

fractions.Studentsshouldunderstand

€

23istwopiecesof

€

13astheyunderstood0.8is

eightpiecesof0.1.Throughtheseactivities,studentswillunderstand:1) “Denominators”expressaunitamount.(Itisthesameas“10’splace,100’s

place...”inwholenumbers.)2) “Numerators”expresshowmanyoftheunitareinthegivennumber,justas

eachnumeralinawholenumberrepresentshowmanyofthatplacevaluethereare.

3. Students work on question (1) page 41.

Usingthedefinitionsof“properfraction”and“mixednumber”identifythetypesoffractionsshownin(1)page41,inordertobecomeclearontheseterms.4. Students work on question (2) page 41.

(Expression)Likewholenumbers,fractionscanbecomparedinsize.

5. Work on question 3) page 41. 3rd period Goals: ∙Studentsunderstandtheconceptof“properfractions.”∙Studentsunderstandfractionsthatarethesamesizeaswholenumbers.Preparation:Enlargementoftextbookillustration(p.41);Blackboardnumberline

1. Students read question (3) page 41 and understand that they

need to think about how many

€

13m.

(KeyQuestion)Howmanyminlengtharethreeof

€

13m?Fourof

€

13m?

a)Usingthetapediagramandnumberline,havestudentsconfirmthatthree

€

13marethesameas1mandhavethemunderstandthatitcanbeexpressed

as

€

33m.

b)Inaddition,helpstudentsgraspthatfourmunitsarethesameas1

€

13m

andcanbeexpressedas

€

43m.

page56(Supplementary questions for p.42)

1) Let’schangeimproperfractionsintowholenumbers:

€

155,

€

183,

€

707,

€

1515,

€

444,

€

357,

€

426

• Let’sexpressimproperfractionswhosenumeratoris24andtheamountsarethesameas12,8,6,3,2,1.

2. Students understand the definition of improper fractions. (Thinking)Studentsnoticethatimproperfractions,likewholenumbers,expresshowmanyofsomeunitamount.(Notebook/Comment)(Knowledge)Studentsunderstandtheconceptofimproperfractions.(Observation/Comment)

3. Students work on question (1) at the bottom of page 42.

a)Makeclearthatinanimproperfractionthenumeratorisgreaterthanorequaltothedenominator.

4. Students work on question (2) page 42.

a)Studentsidentifyfractionsusingthedefinitionofimproperfraction,thusclarifyingtheconcept.

5. Students read question 4) and realize that, as in question 3,

they need to think about how many

€

14

’s are needed to make the

fraction. 6. Students consider the relationship between the numerator and denominator in fractions that are equal to the whole numbers 1, 2 and 3.

a)Studentsnoticethatwholenumberscanbeexpressedwithimproperfractions.

7. Students work on question (1), bottom of page 42. (ExtraSupport)Havestudentsnoticethenumeratorishowmanytimesthedenominator(howmanymultiplesofthedenominator)..

Page57About converting mixed numbers and improper fractions

Convertingmixednumberstoandfromimproperfractionsisfundamentaltoaddition‐subtractionandmultiplication‐divisionoffractions.Themostcommonmistakeinadditionandsubtractionoffractions(ournexttopicofstudy)ismistakes

inconversionbetweenmixednumbersandimproperfractions,sostudentsshouldthoroughlyunderstandconversion.Ifyoufocusonpracticingthisoperationwithoutdeepunderstanding,studentsaremorelikelytomakeerrorsontheoperation.Therefore,itisveryusefultomakeuseofnumberlinesandpicturesoffractions.4th period Goals:Studentsunderstandhowtochangemixednumbersintoimproperfractionsorimproperfractionsintomixednumbers.Preparation:Enlargementoftextbookillustration;Blackboardnumberline1. Students read question 5 page 43 and consider how to change

€

2 13 to an improper fraction. (Work Independently)

(Interest)Studentsareinterestedintherelationshipbetweenmixednumbersandimproperfractionsandinchangingonetoanother.

(KeyQuestion)“Howmany

€

13sdoyouneedtomake2

€

13?”

∙Studentsconsiderhowtochangethemixednumber2

€

13toanimproper

fraction.

1)2

€

13→2+

€

13

2)2includes6piecesof

€

13(3×2=6).Inaddition,thereis1pieceof

€

13.

Sothetotalis

!

7

3.(3×2+1=7)

3)2

€

13includes7piecesof

€

13.Sothetotalis

!

7

3.

2. Summarize how to find numerators of improper fractions when changing mixed numbers to improper fractions. 3. Students work on question 1) page 43.

(Expression)Studentscanchangemixednumberstoimproperfractions.

(ExtraSupport)Studentssometimeshavetheideathat

€

113is

€

113.

Soteacherstobesurethatstudentsunderstandhowtofindthenumeratoroftheimproperfraction.

4. Students think about how to change

!

7

3 into a mixed number.

(KeyQuestion)“Howmany

€

33sarein

!

7

3?”

(ExtraSupport)Referringtothenumberline,studentsthinkabouthow

many

€

33sarein

!

7

3.

5.Summarize how to find the whole number and numerator when converting an improper fraction to a mixed numbers. 6. Students work on question 1) page 43.

(Expression)Studentscanchangeimproperfractionsintomixednumbers.Page581st period About the size of fractions

Intheteachingofequivalentfractions,itiscentraltohelpstudentsunderstandthatmanydifferentfractionalunitscanbeusedtoexpressthesameamount.Sowhenconsideringfractionswithdifferentdenominatorsonthenumberlineandwhenconsideringtheirrelationship,itisimportanttomaketheconnectionbetweenthenumberlineandthesquareareaillustrationsothatstudentscanseetheamount.Inshort,teachersshouldteachequivalentfractionsusingboth1)and2)onpage44.Throughthislesson,studentswillunderstandequivalentfractionssuch

as

€

26=

€

13,

€

68=

€

34,

€

410=

€

25.

Goals: ∙Studentsunderstandequivalentproperfractions.∙Studentsunderstandhowtocomparefractionsthathavelikenumerators.Preparation:Enlargementoftextbookillustration;Blackboardnumberline

1. Students express the colored parts as fractions and find that there are many fractions that are equivalent.

(KeyQuestion)“Let’sexpressthecoloredpartsasfractions.”a)Studentsrealizethatallofthecoloredpartsarethesamesizefromthe

illustrationsinthetextbookandtheyunderstandthat21 canbeexpressed

withmanydifferentfractions(21 =

€

24=

€

510).

2. Students read question 2 page 44 and find fractions that are equivalent. (Work Independently)

(KeyQuestion)“Let’sfindfractionsthatareequivalentusingthenumberlinesbelow!”

a)Havestudentsrecognizethat21 =

€

24=

€

510andthatinaddition,thereare

otherfractionsthatareequivalent.

3. Students work on ★1 on page 45 and use the number line to

see that there are also equivalent fractions for

€

13 and for

€

23.

(Expression)Studentscanfindequivalentfractions.(ExtraSupport)Forstudentswhoarestrugglingwithfindingequivalentfractions,teacherscanadvisestudentstochecktheillustrationsonpage44.Studentswillunderstandwhatisequivalentbynotingwherethemarksonthenumberlineare.∙Afterstudentsfindequivalentfractions,teachershavestudentsexpress

theminequationssuchas

€

13=

!

26=

€

39,

€

23=

€

46=

€

69andhelpstudentsnoticethat

thesamevaluecanbeexpressedwithvariousfractions.Page594. Students read the question 3 page 45 and investigate the

relative sizes of fractions that have the same numerator.

(KeyQuestion)“Whichisbigger,21 or

€

14?Explainwhy.”

(Thinking)Studentsnoticethattheyshouldconsiderthesizeofdenominatorwhentheycomparetherelativeamountsoffractionsthathavelikenumerators.(ExtraSupport)Teacherhelpsstudentsunderstandthatthegreaterthedenominator,thesmallerthesizeofthefraction.

(KeyQuestion)“Whichisbigger,

€

23or

!

26?Andwhy?”

Buildingonwhattheylearnedaboutfractionswithanumeratorofone,studentswillunderstandthatforfractionswithanumeratoroftwo,also,thegreaterthedenominatorvalue,thesmallerthesizeofthefraction.(ExtraSupport)Forthestudentswhocannotfindtheanswer,teacherscaninstructthemtoreview★1and★2,thenadvisethem,“Howdidyoucomparefractionswhentheirnumeratorswere1?”

5. Summarize that for fractions that have the same numerator, the greater the value of the denominator, the smaller the size of the fraction. 6. Students work on (1) at the bottom of page 45. 7. Students review this sub-unit and write in their journals.

Page60Addition and Subtraction of fractions

Theaimofthisunitisforstudentstounderstandhowtoaddorsubtractfractions.Regardinginstructionforproblem1atthetopofpage46,teachershelp

studentsunderstandthat

€

35includesthree

€

15s,54 includesfour

€

15s.Sincetogether

thereareseven

€

15s(3+4=7)theproblemcanbesolvedas

€

35+54 =

€

75.Inthesame

way,question2onpage47,canbesolvedbythinkingaboutthenumberofunit

fractions“7‐4=3So

€

75‐54 =

€

35.”

Attheend,drawconclusionsaboutcalculationmethods.Themostimportant

pointofinstructionhereistohavestudentsnoticetheunitfraction

€

15andto

discoverthatiftheyuseittheycancalculatefractionsinthesamewayaswholenumbers(3+4=7).Whenthistopicisfirstintroduced,thereareseveralcommon

mistakessuchas

€

35+54 =

€

710,

€

75‐54 =

€

30,

€

75‐54 =3.Soteachersshouldtrytodeepen

understandingofthecompositionoffractionsas“howmanyofaunitfraction.”

3. Addition and subtraction of fractions (pages 46-50. 5 periods) 1st period Goals: Studentswillunderstandandperformadditionofproperfractionswiththesamedenominator(whentheanswerisanimproperfraction).Preparation:Enlargementoftextbookillustration1. Students discuss the scene in the textbook and become interested in the question. 2. Students read question 1 on page 46 and understand the contents.

(KeyQuestion)“Whatdoyouknowinthisquestion?Whatareyoutryingtofindout?”

3. Students think about how many m of cardboard they used altogether.(Self-solving)

(KeyQuestion)“Whatkindofmathsentencedoyouneedtofindtheanswer?”

(ExtraSupport)Forthestudentswhoarestrugglingwithmakingamathsentenceforthisquestion,Teacherscanasksomethinglike,“ifyouchange

€

35mto3m,

54mto4m,whatkindofmathsentencecanyoumake?”

4. Students grasp the goal of this unit (to understand how to add

proper fractions) by thinking about how to calculate

€

35

+45. (Work

Independently)

(Keyquestion)“Howdoyoucalculate

€

35

+45?”

(Interest)Studentsnoticethattheycancalculateitinthesamewayaspreviousadditioncalculationsiftheymakeuseoftheconceptofunitfractions.(Notebooks/Comments)(ExtraSupport)Referringto★2teachershelpstudentsunderstandthattheyneedtoknowhowmanyoftheunitfractionstherearealtogether.

5. Students present and discuss their calculation methods. a)Studentsrealizethattheycancalculateeasilyiftheymakeuseofthe

knowledgeof

€

15(unitfraction).

6. Summarize how to add fractions that have the same denominator. 7. Students work on question (1), bottom of page 46. Page61How to express the answer of calculation

1) Atelementaryschool,whenstudentsarestillgraspingthesizeoffractions,

whentheanswertoacalculationisanimproperfractionsuchas

€

107,asa

generalrulewehavestudentschangetheimproperfractiontoamixed

number(1

€

37)becauseitiseasierforstudentstounderstandthesizeasa

mixednumber.(However,leavingtheansweras

€

107ismathematically

correct.Soteacherscanacceptitasacorrectanswer,dependingontheiragreementwiththestudents.)Butinwordproblems,teachersshouldguidestudentstochangeimproperfractionstomixednumbersinordertounderstandthesizeoffractions.

2) Iftheansweris3and

€

75,studentsneedtochangeitto4

€

25because3

€

75is

neitheramixednumbernoranimproperfraction.(Teachershavetohelpstudentsdistinguishthisfromthewholeexpressedasanimproperfraction,asin(1).)

3) Note,ifanswersbecome

!

26or

€

36,studentsdonothavetosimplifythemto

€

13

or21 becausetheystudysimplificationinfifthgrade.

2nd period Goals:Studentsunderstandandcanperformsubtractionofproperfractionsfrommixednumbers(Thewholenumberpartis1,theanswerisaproperfraction).Preparation:Enlargementoftextbookillustration.1.Students read question 2 page 47, understand the content and write a math sentence.

(Keyquestion)“Let’swriteamathsentenceforhowmanykgwillbeleft.”

2. Students think about how to calculate 1

€

25‐54

.

(Keyquestion)“Let’sthinkabouthowtocalculate1

€

25‐54 .Canyoucalculateit

inthesamewayasaddition?”(Thinking)Studentswillnoticethatwhensubtractingfractionsthathavethesamedenominators,theycandrawontheirpriorlearningaboutadditionoffractions,andusethesamekindofthinking.(Notebooks/comment)(ExtraSupport)Teachershelpstudentsrecallthatinadditiontheyconsideredhowmanyunitfractionsthereare.

3. Students will present how to calculate 1

€

25‐54

.

a)Teachersshouldmakestudentschange1

€

25to

€

75inordertocalculate1

€

25‐

54 inthesamewayasaddition.Usingtheideaofunitfraction,theanswerof

1

€

25(

€

75)‐ 54 canbeelicitedbyjustsubtractingnumerators.(7‐4)

4. Summarize how to subtract fractions that have the same denominator.

5. Students work on question 1) page 47. Page62Calculate whole parts and fractional parts separately

Whenweaddwholenumbers,weaddthetensplacetothetensplaceandtheonesplacetotheonesplace.Thismeansthatnumbersthatrepresentthesameunitscanbeaddedtoeachother.Itisimportanttohelpstudentsunderstandhowtoaddmixednumberstomixednumbersusingillustrationsorsomethingsimilar.Thenstudentsshouldcomparetheoperationlearnedinthislessonwithwholenumberadditionsothattheynoticethatmixednumbersandwholenumbershavethesameprinciplesofaddition.3rd period Goals:Studentsunderstandhowtoaddmixednumberstooneanotheranddemonstrateit.Preparation:Enlargementoftextbookillustration.1. Students read problem 3 page 48 and understand the context of this lesson. Then they will think about how to calculate

2

€

35+1

€

15. (Work Independently)

(KeyQuestion)“Let’sthinkabouthowtocalculate2

€

35+1

€

15.”

(Interest)Studentswilltrytorelatemixednumberadditiontothatofproperfractions(Notebooks/Comments)(ExtraSupport)Teachershelpstudentsattendto★1andexplainthesimilaritybetweencalculatingwithmixednumbersandproperfractions.

2. Students present and discuss their calculation methods.

(Thinking)Studentsnoticethattheycanaddmixednumbersbasedontheirknowledgeofadditionofproperfractions.

3. Summarize addition of mixed numbers. ∙Makesurestudentsknowhowtoaddmixednumbers.Onemethodistoseparatemixednumbersintowholenumberpartsandfractionalparts.Theothermethodistochangemixednumberstoimproperfractions.

4. Students will work on 1), 2) page 48. a)Forproblem1(3)nearthebottomofp.48,Havestudentsconsiderwhosemethodisbetter,Makoto’sorNaoko’s?b)Fortheproblemsin2atthebottomofp.48,thefractionalpartsbecomeimproperfractions,sostudentsneedtonoticethattheymustbeconvertedtowholenumbers.Problemsconvertingtomixednumberswillberevealedbytheseproblems,sotherelationshipbetweenmixednumbersandimproperfractionscanbetaughtagaincarefullyforstudentswhomakemistakes.

Page63Supplementary question

4) Therewasa6

€

38mtape.Youusedsomemetersso3

€

78mwasleft.Howmany

metersdidyouuse?4th period Goals: Studentsunderstandhowtosubtractmixednumbersfromoneanotheranddemonstratethosecalculations. Preparation:Enlargementoftextbookillustration.1. Students read question 4 page 49 and think about how to

calculate 254 ‐1

€

35.

(KeyQuestion)“Whatisthedifferencebetween254 ‐1

€

35andprevious

subtractions?”(Possiblereactions)a.Bothofthenumbersintheproblemaremixednumbers.

(KeyQuestion)“Canyoucalculate254 ‐1

€

35usingthemethodsusedfor

additionwithmixednumbers?”(Interest)Studentstrytoperformmixednumbersubtraction,referringbacktoadditionwithmixednumbersorsubtractionwithimproperfractions.(Notebooks/Comments)(ExtraSupport)Teachershelpstudentsattendto★1andaskhowthecalculationmethodsofthetwostudentsdrawonpreviouslearningaboutfractioncalculation.

2. Summarize how to calculate. ∙Summarizesubtractionmethods.Onemethodistoseparatemixednumbersintothewholenumberpartsandthefractionalparts.Theotheristochangemixednumberstoimpropernumbers.

3.Students work on problem (1) at the bottom of page 49. Teachershavestudentsthinkaboutwhichcalculationmethodisbetterfor

parts(2)and(3)ofproblem1.4. Students work on (2)

Sincethefractionalpartcan’tbesubtracted,havestudentsgraspthattheminuendneedstobeconverted.

Page64Relationships between whole numbers, decimal numbers and fractions

Somestudentsgraspwholenumbers,decimalnumbers,andfractionsasverydifferentthings.Thisisespeciallytrueoffractionsbecausetheydonotfollowthebase‐10structureofwholenumbersanddecimalnumbers.

Thoughadditionandsubtractionoffractionsthathavethesamedenominatorsisthefocusinthisunit,studentsareexpectedtoestablishthefoundationtounderstandwholenumbers,decimalnumbersandfractionsasnumbersbecauseallofthemarecalculatedbasedonunitamounts.5th period Goals: Studentswillapplyandpracticethecontentsofthisunitincluding:

1) Problemsthatbuildunderstandingofthemeaningofproperfractions,mixednumbersandimproperfractions.

2) Problemsthatbuildunderstandingoftherelationshipbetweenproperfractions,mixednumbersandimproperfractions.

3) Problemsonadditionandsubtractionoffractionsthathavethesamedenominators.

Math Story Goal:Withunderstandingofthecompositionofwholenumbers,decimalnumbers,andfractions,studentswillcomprehendthatadditionandsubtractionproblemswiththesamedenominatorscanbecalculatedinthesamewayaswholenumbersanddecimalnumbers.page653.Check (page 51,1 period) (Goals of 1st period)

Checkhowmuchstudentsunderstandthisunit.1.Problemsthatassessunderstandingofthemeaningof“properfraction”,“mixednumber,”and“improperfraction”andoftheconversionbetweenmixednumbersandimproperfractions.2.Problemsthatassessunderstandingofrelativesizeoffractions.3.Problemsthatassessadditionandsubtractionthatoflikedenominatorfractions.

Challenge (page 51 no period) Goals: Bymakingmagicsquares,studentsdemonstrateandpracticeadditionandsubtractionoffractionsthathavethesamedenominators.

“Magic Squares”

Here,studentsextendtheirpreviouslearninginordertocalculatefractionadditionandsubtractionproblemswith3terms.Forthesecalculations,

1) Calculatethefirst2terms,thenoperateonthethirdtermandtheanswerfromthefirstcalculation.Intheprocessofthoseoperations,evenifthe

answersbecomeimproperfractionssuchas

€

117,wegoonthenextoperation

withoutchangingittomixednumbers.2) Ifyouneedtodoonlyaddition,youcanadd3termsatonceasyoudowhen

youadd2terms.Inthatcase,youhavetomakesuretochangeimproperfractionstomixednumbersattheendofthecalculation.Whenyouarecalculatinginthisway,youshouldseparatefractionsintothewholenumberpartandthefractionalpartandthencalculate.Ifthefractionalpartsofthoseanswersbecomeimproperfractionstheyshouldbechangedtomixednumbersaslearnedinthisunit.

Teachersshouldfocushereonpracticeadditionandsubtractionof3terms.Butcalculationsmoredifficultandcomplexthan3termsneednotbeintroducedtostudents.

Research Volume of Teachers’ Manual1. Goals of this unit (see 4B teachers’ manual unit goals, p. 1)

Studentswillbeabletousefractionstoexpressmeasurements.Inaddition,studentswillunderstandtheconceptofproperfractions,mixednumbersandimproperfractionsanddeepentheirunderstandingofthemeaningoffractions.Studentswillunderstandhowtoaddandsubtractlike‐denominatorfractions.

(Seeadditionalgoalsrelatedtoassessmentpointsshownonthefirstpageoftheaccompanying4Bteachers’manualvolume.)2. The structure and development of the curriculum over grades 3-6 3rd grade (unit 16)

1) Meaningofdividinganamount.2) Expressmeasurementamountwithfractions.(properfractions)3) Expressfractionsonanumberlineandunderstandtherelativesizeof

fractionswithlikedenominators.(properfractions)4) Addandsubtractfractionswithlikedenominators.

4th grade (this unit) 1) Theconceptofproperfractions,mixednumbers,andimproperfractions.2) Expressfractionsonnumberlinesandunderstandrelativesizeand

equivalentfractions.3) Convertimproperfractions,wholenumbers,andmixednumbers.4) Addandsubtractfractionswithlikedenominators.(properfractionsand

mixednumbers)5th grade (unit 10)

1) Relativesizeandequivalenceforfractionsthathavedifferentdenominators.2) Characteristicsoffractions,fractionreduction,leastcommondenominator.3) Addandsubtractfractionswithunlikedenominators.

The 5th grade (unit 11) 1) Meaningoffractions(quotientofdivisionmeaning)2) Relationshipbetweenfractions,decimalnumbersandwholenumbers,and

howtoconvertbetweenthem.

3. Interpretation of Teaching Materials Inthirdgrade,studentshavelearnedthattoexpressanamountlessthana

measurementunit,theycandividethemeasurementunitintosomenumberofequalpartsanduseaproperfractiontoexpressit.

Inaddition,studentshavelearnedthatfractionscanbeexpressedonanumberline.Usinganumberline,studentsgraspedthestructureoffractionsusingtheideaofunitfractionsandpracticedsimpleadditionandsubtractionoffractionswithlikedenominators.

Inthirdgrade,studentslearnedabouttheadditionandsubtractionofsimplefractionswithlikedenominators,buttheemphasiswasondeepeningunderstandingofthestructureofproperfractionsandunderstandingthattheycanbeaddedandsubtractedlikewholenumbers.Therefore,inthirdgrade,theystudiedfractioncalculationsforamountslessthan1usingpicturesandnumberlines.

Inthisunit,mixednumbersareintroduced,andinadditiontousingthemtoexpressameasurementamount,studentswillplacethemonanumberlineandcomparethesizeoffractionswithdifferentdenominators.Fractionsthathadbeenconsideredasmeasurementamountsandmissingpartsuntilnowwillbemademoreabstract,sothattheycanbeseenasnumbers,likedecimalsandintegers.

Inaddition,studentswilllearnimproperfractionsandhowtoconvertbetweenimproperfractionsandmixednumbers.Thisprovidesafoundationforadditionandsubtractionoffractionsthathavethesamedenominatorsandunderstandingmultiplicationanddivisionoffractionsthatcomeslateringrade6.

Throughthisunit,studentswillbecomemorefamiliarwithvariousfractioncalculations.Additionandsubtractionoffractionsseemdifferentfromthatofwholeanddecimalnumbersbutinfactbothofthemrelyontheideaofadditionofunits.Therefore,itisimportanttounderstandtheconceptofunitfractions.

Whenacalculationresultsinanimproperfraction,itismathematicallycorrecttoleaveitinthatform.However,becauseitiseasierforstudentstounderstandthesizeoffractionswhentheyareshownasmixednumbers,weencourageteacherstohavestudentsroutinelyconvertanswersintomixednumbersfromimproperfractions.Thistextbookusesfractionswithdenominatorsthatarelessthan10becausewewantstudentstounderstandadditionandsubtractioncalculationmethodsandtoconnectwhattheylearninthisunittodailylife.3. Instruction of this unit How to Express Quantities as Fractions Sub unit 1 (page 38-43)

Tohelpstudentsunderstandtheconceptofproperfractions,mixednumbersandimproperfraction,thisunitfollowsthesesteps:

1) Throughmeasuringanamountofjuice,studentslearntouseamixednumbertoexpressanamountthatincludesafractionalpartofameasurementunit.

2) Afterlearningthatmixednumbersexpressanamount,studentsgobeyondthattolearnthatmixednumbersarealsonumbers,becausetheylearnthatunitfractionscanbeexpressedonanumberline.

3) Basedontheknowledgeof2),studentslearnthedefinitionofproperfractionsandmixednumbers.

4) Usingthenumberlineasanaid,studentsunderstandthestructureoffractionsbyaccumulating2,3,4…ofaunitfraction,andinthiswaystudentsunderstandfractionslargerthan1asaccumulationsofunitfractions,andusethistounderstandthedefinitionofimproperfractions.

5) Studentswillexpressimproperfractionsonanumberlineandtrytofindfractionsthatarethesamevalueofwholenumbers.Thisknowledgehelpsstudentsconvertbetweenimproperfractionsandwholenumbers.

6) Basedon5),studentswillunderstandhowtoconvertbetweenmixednumbersandimproperfractions.

Equivalent Fractions, the second sub unit (page 44-45)

Studentswillconsiderequivalentfractionsandrelativesizeoffractionsbyshowinghowfractionswithvariousdenominatorscanbeexpressedonanumberlinebyfindingthesameunitamount.

1) Studentswillinvestigateequivalentfractionsthathavedifferentdenominators.

2) Studentswillinvestigaterelativesizeoffractionsthathavedifferentdenominators.

Throughthoseactivities,studentswillunderstandthatfractionsexpressnumbers,whichcanbecomparedbyrelativesize,justaswholenumbersanddecimalnumbersexpressnumbers.Addition and Subtraction of Fractions, the third sub unit (page 46-50) Additionandsubtractionoffractionswiththesamedenominatorscanbedonejustaswithwholenumbers.Studentswilllearnaboutthiswiththestepsbelow.

1.Additionofproperfractions(includescarrying)2.Subtractionofproperfractionfrommixednumber(wholenumberportionofmixednumberis1;includesregrouping);subtractionofproperfractionsfromeachother.3.Additionofmixednumbers(withandwithoutcarrying)4.Subtractionofmixednumbers(withandwithoutregrouping).

6. Explanation of Unit 11 and Notes for Instruction Meaning of fractions

Fractionsareintroducedinthirdgradetoexpressanamountthatislessthanameasurementunit.Hereinfourthgrade,studentswilllearnhowtoexpressanamountlargerthanameasurementunitinfractionsandwilllearnproperfractions,mixednumbers,andimproperfractions.

Sofar,fractionshavebeentaughtasthewayofexpressingsizesofanamount.Here,fractionsareintroducedasthewayofexpressingthesizesofdiscreteandcontinuousquantitiesaswholenumbersanddecimalnumbers.

Wholenumbersareusedtoexpressnotonlythesizeofanamountsuchas“3people,”“3papers”and“3cm”butalsotherelationshipbetween2amountssuchas“3times.”Likewise,fractionshaveseveralmeanings,listedbelow.Note,thesemeaningsarenoteasilyseparatedbutratherareverycloselyrelated.Wecategorizethemheretohelpwithorderingtheideasforinstruction.

1) Fractions which express the size of amounts (grade 3,4) Ifyouwanttoexpresshowmanypapersyouhave,howmanypeopleare

here(discretequantities)youcanusewholenumberstoexpressthem.But,ifyou

wanttoexpresssomeamountofwater(

€

2 34L)orlength(

€

23m),whicharecalled

continuousquantities,youhavetochooseaunitamounttomeasurethem.Theremaybesomeamountleftoverfrommeasuringwiththatunit.Decimalnumbersandfractionswereoriginallycreatedtoexpressthesecontinuousquantities.

2) Fractions as quotient of division (grade 5 /unit 11)

€

23expressesthequotientof2÷3.Thisconceptwillbeintroducedinunit11in

gradefive.BetweenwholenumbersAandB,theanswersofA+B,A‐B(A>B),A×Barealwayswholenumbers.ButtheanswerforthemathsentenceA÷B,isnotgenerallyawholenumber;itprobablyhasaremainder.But,ifyouexpresstheanswerwithafraction,A÷Bisalwayspossible.Thisisanimportantfunctionoffractionsasnumbers.

3) To express ratio (grade 5)

Themeaningof

€

23inthiscaseistherelativevalueof2and3:when3is

regardedas1,then2canberegardedas

€

23.Thisinterpretationoffractionsis

introducedfollowingthetreatmentoffractionasquotientinGrade5,unit11,“FractionsandDecimalNumbers.”Fractionsasratioscanbethoughtofinthesamewayas“3timesasmuch”forintegers.4) Part-Whole Fraction (grade 3)

“

€

23of12”or“

€

23of3m”arecalled“part‐wholefractions.”(grade3)Part‐whole

fractionsexpresstwopartsofanamountthatisdividedinto3equalparts.Herefractionsareviewedasoperationsanddonotexpressnumbers.

Ifstudentsstudypart‐wholefractionstoomuch,theywillunderstand

€

23as

“÷3×2”anditwillconfusestudents’understandingoffractionsasnumbers.Therefore,4)isexplainedonlyinthecontextofteachersintroducinghowtoexpressthefractionalamountleftoverfrommeasuringwithaunit.To understand fractions as numbers

Evenifstudentsexpressthelengthofatapeas

€

35m,wecannotbecertainthat

theyunderstandfractionsasnumbers.Somestudentsmaybethinkingaboutthe

conceptofmeasurementfraction(

€

35mis3piecesof1mdividedinto5equalparts)

anddonotunderstandfractionsasnumbers.Therefore,teachershavetomakesurestudentsunderstandfractionsasnumbers.

Studentslearnedtothinkaboutwholenumbersbyreplacingconcretematerialssuchasapplesorpaperswithabstractmaterials(picturesofsquaresandtapediagrams),thencomparingthemonthenumberline.

Thesameapproachcanalsobeusedtohelpstudentsunderstandfractionsasnumbers.Thistextbookunitfirstintroducesfractionsusinganamountofwater,thenreplacesthiswithapictureofasquare,andfinallymovestoexpressingtheamountonanumberline.

Onanumberline,allnumbers(0,1,2….)areplacedatequalintervals,andcontinuousquantities,discretequantities,andunitsareallrepresentedbynumbers.Byplacingfractionsonanumberline,studentsclarifythatfractionsarenumbersandareabletoprogressbeyondthinkingoffractionsaspiecesofawhole.

Studentsdeepentheirunderstandingoffractionsasnumbersthroughrepresentingthemonthenumberline,comparingtheirrelativesizes,andaddingandsubtracting.

Instruction on addition and subtraction of fractions

Tounderstandadditionandsubtractionoffractions,firstofall,studentsshouldunderstandthatfractionsaremadeupofunitfractions.Inaddition,theyhavetounderstandequivalentimproperfractionsandmixednumbersandhowtoconvertbetweenthem.Therefore,teachersshouldhavestudentspracticethesecalculationsafterteachingtheseideas.

Foundation of addition and subtraction of fractions

Whenstudentsarecalculatingfractions,someofthemarelikelytomake

mistakessuchas“

€

35

+15

=410.”Studentsseethedenominatorsandwanttoaddthem,

justastheyaddthenumberineachplacevalueinthecaseofadditionofintegers.Forthisreason,itisbestforteacherstoprovideacontextofmeasurement

(e.g.,length)whenintroducingadditionoffractions,andaconcreteexamplesuchas

“howmanymetersare

€

35mand

€

15mintotal?”Itisimportantforinstructiontofocus

ontheideathat“

€

35is3piecesof

€

15and

€

15is1pieceof

€

15.Sotogetherthereare

3+1=4piecesof

€

15andtheansweris

€

45,ifyouregard

€

15astheunitamount.”Unit

fractionshelpstudentsunderstandthatfractionalcalculationsaredoneinthesamewayasforwholenumbers.

2) Calculation with mixed numbers that require carrying or regrouping

Whenyouaddorsubtractmixednumbers,youaddorsubtractwholenumberpartsandfractionalpartsseparately.Thisideaisveryusefulbecauseitlikecalculatingwithwholenumberswithtwoplaces.Therefore,whenteachersare

teachingadditionandsubtractionofmixednumbers,itiseffectiveforthemtohavestudentsrecallhowtheylearnedaboutadditionandsubtractionoftwo‐digitwholenumbers.

However,manystudentsmakemistakeswhencarryingorregroupinginadditionandsubtractionofmixednumbers.Thesemistakesseemtooccurbecausefractionalcalculationscannotbeperformedasautomaticallyaswholenumbersordecimalnumbers,whichbothusethebasetensystem.Therefore,studentsshouldexplaintheirthinking,ratherthanjustcalculatequickly,whenpracticingcalculations.Page481.Goal

Studentsunderstandhowtoexpressanamountlargerthanameasurementunitwithmixednumbers.

2. Assessment Criteria

(Interest)Studentsexpressanamountlargerthanameasurementunitwithfractions.(Thinking)Studentscanthinkabouthowtoexpressinfractionsanamountlargerthanameasurementunit,byconsideringhowthisisdoneindecimalnumbers.(Expression)Studentsexpressanamountlargerthanameasurementunitwithmixednumbers.(Knowledge)Studentsunderstandhowtoexpressanamountlargerthanameasurementunitwithmixednumbers.

3. Teaching Point Relation of mixed numbers

Inthisunit,studentsshouldunderstandhowtoexpressanamountlargerthanameasurementunitusingknowledgeofproperfractionsfromgrade3,wheretheyexpressedanamountsmallerthanameasurementunitbyrelatingtodecimalnumbers.

4. Lesson Lesson and Key Questions (K)

Learning Activities and Reactions

Points to Emphasize*; Assessment & Extra Support +

1.Students look at the scene on page 38 and think about how many L of juice there are. Students get interested in the illustration.

Students understand the situation in the textbook: To measure milk, it has been put in a graduated 1L container divided into 3 equal parts, and the quantity

*Teachers show a carton of milk and students get interested in the textbook illustration. *Fractions are introduced in

(K) ”What is happening in the illustration in the textbook?” (K) “How many L of milk are there?”

of milk is 2 parts (Possible reactions) 1) Students try to measure how many L by pouring milk into the 1L square graduated containers.

2) There are 2 parts of

€

13

L.

So the quantity is

€

23

L.

grade 3. The purpose of introduction here is to remind students that fractions express how many unit fractions of an amount divided into some number of equal parts.

2. Students will look at the scene on page 38 and discuss the quantity of juice. They understand the lesson’s purpose. (K) ”How many L is the amount of juice?” (K) “Today, let’s think together about how to express fractions that are greater than 1. ”

Students think about how to express the amount of juice in liters. (Possible reactions) 1) Juice is more than 1L but less than 2L. 2) It is about 1.8L 3) It is 1L plus and amount less than 1L. Students write the goal of this lesson in their notebooks: “Let’s think about how to express fractions that are greater than 1.”

*Teachers show the illustration in the textbook and students discuss the quantity of juice. Students grasp that there is more than 1L of juice. *Students grasp the lesson’s purpose.

3.Students read the question and understand the meaning of it. They express the part less than 1L as a fraction (independent work). (K) “First of all, let’s express as a fraction the amount that is leftover from measuring 1L.

Based on the previous lesson, students express the left over part as a fraction by themselves. (Possible reactions) Students express the left over part as a fraction.

1)

€

34

L

2) 1L that has been divided into 4 equal parts and there

are 3 of them. So it is

€

34

L.

3 )We learned it in grade 3.

It is

€

34

L.

* Teacher draws a square graduated container for juice like that in the textbook, on drawing paper posted on the blackboard, or on projector. * Textbooks are closed. + For students who are struggling to express the left over part as a fraction, teachers can use a copy of the illustration on the textbook and ask them questions like “how many equal parts is 1L divided

into.”

4.Students learn how to express and read the total of

1L and

€

34

L.

(K) “With decimal numbers, you can express 1.3 by combining 1 and 0.3. Likewise, you can express

the amount here as

€

1 34

by

combining 1L and

€

34

L. ”

Students discuss how to express the total of 1L and

€

34

L.

(Possible reactions) 1) With decimal numbers, we expressed the combined amount of 1L and 0.3L as

1.3L. So this is

€

1 34

L.

Students learn how to express mixed numbers and write the amount of juice in mixed numbers in their notebooks.

(Interest) Students relate fractions to decimal numbers, and try to express an amount that is greater than 1 unit in fractions. Students confirm from the picture that the volume of juice is one liter plus 3 of 4 equal parts of one liter.

5, Summary of how to express a fraction greater than 1 and practice.

Students understand how to express as a fraction an amount that is greater than 1. Students see that it is not just for liters, and they confirm how to express, for example, length (2m and

€

34

m) and weight (3kg and

€

15

kg) in mixed numbers.

(Possible reactions)

1) It is

€

2 34

m.

2) It is

€

315

kg.

Summary that, as for decimals, an amount that is greater than 1 can be expressed by combining the whole number part and the fractional part. Students do practice questions 1) and 2) page 40. Students discuss the

(Knowledge) Understand how to express an amount that is greater than 1 measurement unit in a mixed number. Teachers confirm that students can express an amount greater than 1 measurement unit as a fraction, and introduce definition. (Thinking) Students make a connection to decimal numbers to express an amount larger than a measurement unit. (Expression) Students express an amount larger than a measurement unit using mixed numbers.

benefits of being able to express an amount that is greater than 1 as a fraction.

5. Example of black board organization Howmanylitersofjuicearethere?Goal ‐Let’sthinkabouthowtoexpressfractionsgreaterthan1.What is the volume left over after measuring one liter?

‐Itis3outof4equalpiecesof1L.So,itis

€

34L.

How can we express 1L and

€

34L?

‐Indecimalnumbers,itwasexpressedas1.3Lbycombining1Land0.3L.Summary

‐Wecombine1Land

€

34Landexpressitas

€

1 34L.

Page501. Goal

Studentsunderstandtheconceptofproperfractionsandmixednumbers.2. Assessment Criteria

(Interest)Studentsexpressmixednumbersonanumberline.(Thinking)Studentsnoticethatmixednumberscanbeexpressedonanumberlinelikewholenumbers.(Expression)Studentsexpressmixednumbersonanumberlineandcomparerelativesize.(Knowledge)Studentsunderstandtheconceptofproperfractionsandmixednumbers.

3. Teaching Point Express mixed numbers on a number line

Helpstudentsbecomeawarethat

€

1 34,studiedinthepreviouslesson,canbe

expressedonanumberline,likewholenumbersanddecimalnumbers.Definition of proper fraction and mixed number

Properfractionsarefractionswhosenumeratorsaresmallerthantheirdenominators.Mixednumbersarefractionsexpressedbycombiningawholenumberandaproperfraction.Byplacingproperfractionsandmixednumbersona

numberline,studentswillbecomefurtherawareofthedifferencesbetweenproperfractionsandmixednumbersanddeepentheirunderstandingofthem.4. Lesson Lesson and Key Questions (K)

Learning Activities and Reactions

Points to Emphasize; Assessment & Extra Support

1.Students understand that

€

1 34

can be expressed on a

number line and they understand mixed numbers as numbers. (Independent work) (K)“If the size of this square is 1, how do you

show

€

1 34

?”

(K)“Then where is the mark

for

€

1 34

on the number

line?” (K) “What fractions are indicated by a, b, c, d?"

Students express

€

1 34

on a

number line.

First, express

€

1 34

with the

square area illustration. Possible Reactions:

1)

€

1 34

is the size of 1 square

plus 3 out of 4 equal pieces of 1 square.

Express

€

1 34

on a number

line. Possible Reactions: 1) The marks on the number line divide 1 into 4 equal

parts, so

€

1 34

is three parts

after 1. Students read proper fractions and mixed numbers associated with points on the number line from textbook p.40

1) a-

€

14

2) b-

€

34

3) c-

€

2 24

Textbooks are closed. Teachers show the square and have students think

about how to express

€

1 34

.

(Interest) Students will express mixed numbers on a number line. Teachers show a number line and based on the structure of mixed numbers, put the mark on the number line for the whole numbers part. Then put the mark for the fractional part. (Expression) Students can express mixed numbers on a number line. Teachers have students understand the size of each scale on the number line from 0 to 1. It is useful for students to be able to read a fractional amount on number line.

(K) “Among these fractions, which fractions are less than 1? Which are greater than 1?” 2.Students will understand the definition of proper fractions and mixed numbers. (K) “Are proper fractions smaller than 1 or greater than 1? How about mixed numbers?” 3.Students work on 1) page 41. “Let’s find the mixed numbers.”

4) d-

€

3 14

Students categorize the fractions 1)-4) above and

€

1 34

into fractions that are

less than 1 and into fractions that are greater than 1. Possible reactions:

·fractions less than 1:

€

14

,

€

34

·fractions more than 1:

€

1 34

,

€

2 24

,

€

3 14

Students learn the definition of proper numbers and mixed numbers. “Proper fraction:” A fraction whose numerator is less than the denominator. “Mixed number:” A fraction that is made up of a whole number and a proper fraction. Students learn the fractions that are smaller than 1 are called proper fractions and the fractions that are greater than 1 are called mixed numbers. Students work on 1) page 41. Students use the definitions of proper fractions and mixed numbers to judge the fractions and to deepen their understanding of the concepts.

Teachers help students learn the definitions having students underline or circle the definitions in their textbooks or write them in their notebooks. (Knowledge) Students learn the concept of “proper fractions” and “mixed numbers.” (Knowledge) Students understand that mixed numbers can be expressed as the combination of the whole number part and the proper fraction part. (Thinking) As for integers and decimal numbers,

4.Students do practice problems.

Students work on questions 2) and 3) on page 41. Students understand that fractions can be compared like whole numbers and decimal numbers. Students can express an amount greater than 1 with mixed numbers

students realize they can judge the relative size of fractions.

5. Example of black board

Let’s express

€

1 34

on a number line!

Illustrationofnumberlineandpartitionedrectanglefromtextbookp.40Fractionsthatcorrespondtopointsa,b,c,d

a:

€

14b:

€

34c:

€

2 24d:

€

3 14

Fractions that are smaller than 1

€

14,

€

34

Fractions that are greater than 1

€

1 34

,

€

2 24,

€

3 14

Summarize Properfraction Afractionwithanumeratorsmallerthanthe

denominator Afractionsmallerthan1Mixednumber Afractionthatexpressesanintegerandaproper fractioncombined Afractionlargerthan1Page521. Goal Studentswillunderstandtheconceptofimproperfractions.Studentswillunderstandfractionsthatareequalinsizetowholenumbers.2. Assessment Criteria

(Interest)Studentswillconsidertherelationshipofthenumeratoranddenominatorinfractionsthatareequalinsizetowholenumbers.(Thinking)Studentswillnoticethatimproperfractions,likewholenumbers,expresshowmanyofaunit.

(Expression)Studentsareabletochangeimproperfractionstowholenumbers(Knowledge)Studentswillunderstandtheconceptofimproperfractions.

3. Teaching Point Express improper fractions on a number line.

Studentshavelearnedhowtoexpressproperfractionsandmixednumbersonanumberlineinthepreviouslesson.Here,theywilllearntherelationshipbetweenwholenumbersandfractionsbyexpressingonanumberlinefractions

suchas3or4piecesof

€

13;or2,3or4piecesof

€

14.

The definition of improper fractions Studentswilldefinefractionsthathavethesamenumeratorsand

denominatorsandfractionswithalargernumeratorthandenominatorasimproperfractions.Teachersshouldhelpstudentsclarifytheirunderstandingofthemeaningofimproperfractionsmoreclearlybydistinguishingthemfromproperfractionsandmixednumbers.4. Lesson Lesson and Key Questions (K)

Learning Activities and Reactions

Points to Emphasize; Assessment & Extra Support

1.Students think about how

many m three

€

13

m, four

€

13

m are. (Independent

work) (K) “How many m are three

€

13

m, four

€

13

m?”

(K) “Let’s think about how

to express 1m and

€

113

m in

different ways.”

Students investigate how

many m three

€

13

m, four

€

13

m

are. (Possible reactions)

1) Three

€

13

m are the same

value as 1m and four

€

13

m

are the same value as

€

113

m.

Students compare the tape diagram and number line and try to find another way to express the points. 1) 1m can be expressed as

€

33

m.

2)

€

113

m can be expressed as

To get students’ interest, teachers show a tape diagram, divided into 3 equal parts and a number line. Teachers write the question on the black board or paper. Extra Support: Comparing the tape diagram and number line, help students realize they need to think about how

many

€

13

’s

(Thinking) Students notice that improper fractions, like whole numbers, are made up of so many units. Students notice that whole numbers and mixed numbers

2.Students learn the meaning of “improper fraction.” (K) “Are improper fractions larger or smaller than 1?” 3.Students do practice problems. 4.Students will investigate fractions made up of

€

43

m.

Students learn the definition of improper fraction. Improper fraction: A fraction whose numerator is the same as or greater than its denominator. Through pointing out on number line, students deepen understanding of the definition of improper fraction. (possible reactions) 1) It is more than 1 2) It is equivalent to or more than 1. Students work on question 1) page 42. Make sure numerators are larger than denominators. Students work on question 2) page 42. Make sure students judge fractions by the definition of improper fractions and clarify the concept of improper fractions. Students express proper and

can be expressed by improper fractions. Teachers have students underline the definition of proper fractions and improper fractions in their textbooks or write them in their notebook to help them understand the definitions clearly. (Knowledge) Students understand the concept of “improper fraction.” Teachers have students confirm the size of improper fractions visually by making use of number lines. Through ample use of the number line, students deepen their understanding of the definition of improper fractions, and understand the relationship to proper fractions and mixed numbers. Teachers show the number line scaled in fourths. Teachers make sure students

2,3,4…

€

14

’s, etc..

(K) “Let’s write proper and improper fractions made up

of 2, 3, 4….

€

14

’s.”

(K) “Let’s compare improper fractions which are equal to whole numbers and each other.” (K) “Let’s try 1) on page 42.”

improper fractions made up

of 2, 3, 4…

€

14

’s, etc..

Possible Responses

1)

€

24

,

€

34

,

€

44

,

€

54

….

2) It can go on forever. Students investigate the relationship between improper fractions and whole numbers. 1) Numerators can be divided evenly by denominators. 2) They are equal or 2 times, 3 times, etc.. Students work on 1) page 42.

know how many

€

14

’s there

are. (Interest) Students consider the relationship of numerator to denominator in fractions equivalent to whole numbers. Extra Support: Help students think about how many times as big the numerator is compared to the denominator. (Expression)Students change improper fractions to whole numbers.

5. Example of black board

Howmanymarethree

€

13m?Four

€

13m?

Summarize Improperfraction: Afractionwhosenumeratoristhesameasorgreaterthanits

denominator. Itisequalto1orgreaterthan1.

Let’swriteproperandimproperfractionsmadeupof2,3,4,…..

€

14’s.

Goal

Let’sinvestigatetherelationshipbetweenwholenumbersandimproperfractions.

Summarize

Animproperfractionisequaltoawholenumberifithasthesamenumeratorasdenominator.Orthenumeratoris2timesor3timesthedenominator.Page541. Goal

Studentswillunderstandhowtochangemixednumbersandimproperfractionsintoeachother.

2. Assessment Criteria

(Interest)Studentsareinterestedintherelationshipbetweenmixednumbersandimproperfractions.Theyarewillingtochangethemfromonetoanother.(Expression)Studentsareabletochangemixednumbersandimproperfractionsbackandforth.(Knowledge)Studentsunderstandhowtochangemixednumbersandimproperfractionsintoeachother.

3. Teaching Point Changing mixed numbers into improper fractions

Bythinkingabouthowmanyofaunitfractionthereare,studentsnoticethattheycanchangemixednumbersintoimproperfractions.Studentsneedtopayattentiontothedenominatorandlearntoconvertmixednumbersbymakinguseoftherelationshipbetweenimproperfractionsandwholenumbersstudiedinthepreviouslesson.

Changing improper fractions to mixed numbers

Usingtheirknowledgeofunitfractionsstudentsbecomefamiliarwithchangingimproperfractionsintomixednumbers.Teachersneedtoconfirmthatstudentsrealizethatwhenthedenominatorandnumeratorareequalthefractionisequalto1,andtoseethatthewholenumberportionofthemixednumberisdeterminedbyhowmany1’sthereare.4. Lesson Lesson and Key Questions (K)

Learning Activities and Reactions

Points to Emphasize; Assessment & Extra Support

1.Students work on problems in which they change mixed numbers into improper fractions. (Independent work) (K) “Let’s think about how

to change

€

2 13

into an

improper fraction.”

(K) “How many

€

13

s do you

need to make

€

2 13

?”

Students think about how to

change

€

2 13

into an

improper fraction. Possible Reactions: 1) (Using a number line) 1

is 3 pieces of

€

13

, so it is

€

33

.

2) (Using an area illustration) We can say the same things. Students think about how

many

€

13

s they need to make

Textbooks are closed. Teachers show a number line and have students think

about how to express

€

2 13

.

Students notice that they needs a number line scaled

in

€

13

’s.

(Interest) Students will be interested in the relationship between mixed numbers and improper fractions and

(K) “ Is there a way to calculate?” 2.Summarize how to change mixed numbers into improper fractions. 3.Students work on practice problems. (K) “Let’s try the problems using what you have learned in this lesson.” 4.Students take up the challenge of changing improper fractions to mixed numbers. (K) “What do you have to

know to change

€

73

into a

€

2 13

.

1) (Using a number line) 7

pieces of

€

13

are

€

73

.

2) (Using an area

illustration) 7 pieces of

€

13

are

€

73

.

3) 1 is the same amount as

€

33

. So 2 pieces of

€

33

and

one piece of

€

13

total

€

73

.

4) (From the previous

lesson) 2 is expressed as

€

63

in fraction. So 2 and

€

13

total

€

7

3.

Students think about how to change mixed numbers into improper fractions by calculation. (Possible reactions) 1) We can multiply denominators and the whole number part of mixed numbers, then add numerators. 2) We can find the numerator by 3×2+1=7. Students learn how to find the numerator when changing mixed numbers into improper fractions. Students work on question 1) on page 43. Students will think about

willing to change them into one another. Through discussion, teachers have students grasp how to change mixed numbers to improper fractions. Teachers have students write in their notebooks to strengthen their understanding. (Expression) Students will be able to change mixed numbers and improper fractions. (Interest) Students will be interested in the relationship between mixed numbers and improper fractions and

mixed number?”

(K) “Let’s change

€

73

into a

mixed number.” (K) “ Do you know how to find the answer by calculation? 5. Students summarize how to change improper fractions into mixed numbers. 6. Students work on problems.

how to change

€

7

3 into a

mixed number.

1) How many

€

33

’s do you

need to make

€

73

?

2) How many 1’s and how much left over is there in

€

73

?

Students think about how to

change

€

7

3 into mixed

numbers. (Possible reactions)

1) It would be

€

2 13

. Because

€

7

3 includes 2 pieces of

€

33

.

2) It would be

€

2 13

. Because

€

7

3 includes 2 pieces of 1(

€

33

)

and left over is

€

13

.

Students learn how to change improper fractions into mixed numbers. Students work on problem 1) page 43.

willing to change them into one another. Teachers show a number line and have students scale

the number line into

€

13

’s

and express

€

7

3 on the

number line. Comparing to the number line, students think about

how many

€

33

(1)’s exist in

€

73

.

(Knowledge) Students will understand the relationship between improper fractions, mixed numbers, and whole numbers. (Expression) Students will be able to change mixed numbers and improper fractions.

5. Example of black board organization (contains illustrations not reproduced here)

Let’sthinkabouthowtochange

€

2 13intoimproperfractions.

Summary

Changemixednumbersintoimproperfractions

Let’sthinkabouthowtochange

€

7

3intomixednumbers.

∙Howmany1’s(

€

33)doyouneedtomake

€

7

3?

Summary Changeimproperfractionsintomixednumbers.Page561. Goal ‐Studentswillunderstandtheequivalentrelationshipbetweenproperfractions.‐Studentswillunderstandhowtocomparetherelativesizeoffractionswhosenumeratorsarethesame.2. Assessment Criteria

(Interest)Studentswillconsidertherelativesizeandequivalenceoffractionsmakinguseofanumberline.(Thinking)Studentswillunderstandthattheycancomparetherelativesizeoffractionsthathavethesamedenominatororthesamenumeratorbythinkingaboutunitfractions.(Expression)Studentscancomparevariousfractionsonanumberlineandtheycancomparetherelativesizeoffractionsthathavethesamenumerator.(Knowledge)Studentswillunderstandequivalentfractionsusinganareaillustration.

3. Teaching Point Fractions expressed on a number line

Makenumberlinesalignedat0,withintervalsof

€

12,

€

13,

€

14,

€

15,

€

16,

€

17,

€

18,

€

19,

€

110.

Havestudentsgraspthesizeofthefractionsandequivalentfractionsusingthesedifferentnumberlinesalignedatzero.

4. Lesson Lesson and Key Questions (K)

Learning Activities and Reactions

Points to Emphasize; Assessment & Extra Support

1.Students will find

fractions equivalent to

€

12

.

(K) “Let’s express the colored parts as fractions.”

Students express the colored parts as fractions. Possible Reactions:

1) They are

€

12

,

€

24

,

€

510

.

2) All of them are half of the square but they are expressed as various fractions.

Using a number line, teachers explain that all of the colored parts are the

same size and

€

12

can be

expressed as various fractions such as

2.Student read and understand question 2 on page 44. (K) “What fractions are expressed by the intervals on each number line?” (K) “Among these, are there any fractions that are equal in size?” (K) “Let’s find the fractions that are equal in

size to

€

13

or

€

23

.”

3.Students read question 3 on page 45 and compare fractions to each other using a number line.

(K) “Which is bigger,

€

12

or

€

14

? And why?”

(K) “List fractions whose numerators are 1 in decreasing order.”

Students look for the fractions that are equivalent using a number line. Students look for the fractions that are equivalent

to

€

13

or

€

23

using a number

line. Students compare the size of

€

12

and

€

14

.

Possible Reactions:

1)

€

12

is greater than

€

14

on a

number line.

2)

€

12

is 1 out of 2 equal

pieces of 1.

€

14

is 1 out of 4

equal pieces of 1. So

€

12

is

greater than

€

14

.

Students compare the relative size of fractions whose numerators are 1.

1)

€

12

is the greatest one.

€

13,

€

14,

€

15,

€

16,

€

17,

€

18,

€

19,

€

110

is the

decreasing order. Students compare the size of

€

23

and

€

26

.

€

12

=

€

24

=

€

510

.

(Knowledge) Students will understand equivalent fractions referring to area illustrations. After finding equivalent fractions, teachers have students express them as math sentences such as

€

15

=210

or

€

13

=26

to

understand that various fractions can express the same amount. Teachers need to introduce equivalence and relative size of fractions on the number line but do not need to go into great depth. (Interest) Students consider relative size and equivalence of fractions using number lines. (Expression) Students can compare the size of fractions on a number line. And they can compare the relative size of fractions that have the same numerator. Teachers work on �1 using a number line and have students understand the

(K) “Which is bigger,

€

23

or

€

26

? And why?”

(K) “List fractions whose numerators are 2 in decreasing order.” 4. Summarize the contents of this lesson. 5.Students work on exercise.

(Possible reactions) 1) I see from the number line

that

€

23

is greater than

2) When 1 is divided into 3

equal pieces,

€

23

is 2 of the

pieces; when 1 is divided

into 6 equal pieces,

€

26

is 2 of

the pieces. Both of them have 2 pieces but the unit

fraction

€

13

is greater than

unit fraction

€

16

so

€

23

is

greater than

€

26

.

Students compare the size of fractions whose numerators are 2.

1)

€

23

is the greatest one.

€

24,25,26, 27,28,29, 210

are in

decreasing order. Students summarize that there are various fractions that are equivalent. In addition, they understand that the size of fractions can be compared. Possible Reactions: 1) Among the fractions that have the same numerator, the greater the denominator, the smaller the size of the fraction. Students work on the problem 1) on page 45. Making use of the summary of today’s lesson, students

relative size of fractions whose numerators are 1. Teachers summarize 1 and expand the lesson. Students pick up all of the fractions whose numerators are 1 and understand the relative size using a number line. Teachers work on 1, 2, 3 and develop the lesson. Students understand that for fractions that have the same numerator, the greater the denominator, the smaller the size of the fractions. At this level, it is difficult for students to convince themselves of the contents of this lesson logically, so teachers should use a number line to help them understand visually.

compare the size of fractions whose numerators are 3 and 4.

(Expression) Students can compare the various fractions on a number line. In addition, they can compare the relative size of fractions that have the same numerator. (Thinking) Students will understand how to compare the relative size of fractions with the same denominator and numerator using unit fractions.

5. Example of Black Board Organization

Let’sthinkaboutthesizeoffractions.Equivalent fractions

€

12

=

€

24

=

€

510

€

13

=26,23

=46,25

=410, 35

=610

Let’s compare the size of fractions

∙

€

12,

€

13,

€

14…..decreasinginthisorder

∙

€

23, 24,25….decreasinginthisorder.

Summarize Forfractionsthathavethesamenumerator,thegreaterthedenominator,the

smallerthesizeofthefraction.Page581. Goal

Studentsunderstandandcanperformadditionofproperfractionswhosedenominatorsarethesame(theanswersareimproperfractions).

2. Assessment Criteria

(Thinking)Studentswillnoticethatadditionofimproperfractionsthathavethesamedenominator(theanswersareimproperfractions)canbedonein

thesamewayasadditionforwholenumbersiftheypayattentiontotheconceptofunitfractions(Expression)Studentscanaddproperfractionsthathavethesamedenominator.(Theanswersareimproperfractions.)(Knowledge)Studentsunderstandhowtoaddproperfractionsthathavethesamedenominator.(Theanswersareimproperfractions.)

3. Teaching Point 1)UnitamountStudentsareexpectedtonoticeadditionofproperfractions,whichtheylearnedingrade3,canbeappliedtocalculationsingrade4.Soteachershavetoconfirmtheconceptofunitfractions.4.Lesson Lesson and Key Questions (K)

Learning Activities and Reactions

Points to Emphasize; Assessment & Extra Support

1. Understand the contents of this lesson. (K) “What do you know about this number sentence? How would you answer it?” 2. Make a math sentence. (Independent work) (K) “Let’s think about what kind of math sentence is good to answer this

Referring to the illustration, students discuss the scene in the textbook and develop interest in solving it.

Shiori used

€

35

m² of

cardboard, and Kiyoshi used

€

45

m². How many m² of

cardboard did they use altogether? Possible reactions 1) The numbers used in the

question….Shiori used

€

35

m²

of cardboard, and Kiyoshi

used

€

45

m².

2) What we have to find out… How many m² of cardboard did they use altogether? Students think about a number sentence to answer how many m² of cardboard they used altogether.

Teachers show illustration so that students develop interest. Teachers motivate students to think about how to calculate proper fraction + proper fraction, a goal of this lesson. Teachers instruct them to underline the important amounts in this problem and to clearly identify what already known and what they have to find out. Textbooks are closed. For the students who are struggling with making number sentences, teachers can give them hints such as

question. Why?” 3.Think about how to calculate. (K) “How would you

calculate

€

35

+

€

45

?”

4.Summarize how to add fractions that have the same denominators. (K) “What is the common thing among those ways of solving the problem?” (K) “How do you do it when the answer is an improper fraction?” 5.Students work on practice problems.

Possible reactions:

1)

€

35

+

€

45

2) Use

€

35

m² and

€

45

m²

“altogether” so we can add them. Students understand the goal of this lesson. ·How to add fractions. Students think about how to calculate. (Possible reactions)

1)

€

35

is 3 pieces of

€

15

,

€

45

is

4 pieces of

€

15

. So

€

35

+

€

45

is 7

pieces of

€

15

. Therefore, the

answer is

€

75

.

2) We learned

€

35

+

€

15

=

€

45

in

grade 3. So we can make

€

35

+

€

45

=

€

75

.

3) Using the area of the

square, 7 pieces of

€

15

=

€

75

.

So the answer is

€

75

m².

4) I found the answer the

same way, but I changed

€

75

to

€

125

m².

5) I used a number line (shows illustration). Students summarize calculation process.

“if you replace

€

35

m² with 3

m²,

€

45

m² with 4 m², what

kind of number sentence can you make?” (Thinking) Students will notice that addition of improper fractions whose denominators are the same (the answers are improper fractions) can be operated on the same way as addition of whole numbers using the concept of unit fractions. All of the calculations can be done as in 2 on page 46. (Knowledge) Students understand how to add proper fractions that have the same denominator. (The answers are improper fractions.)

(Possible reactions)

1) How many

€

15

’s are there?

2) We can add 3 and 4 if we

notice the unit amount is

€

15

.

3) We can add numerators. 4) When adding fractions that have the same denominators, just add the numerators and leave the denominators as they are. When the answer is an improper fraction, students should change it into a mixed number or whole number. Students do question 1) on page 46.

(Expression) Students can add proper fractions that have the same denominators. (The answers are improper fractions.)

5. Example on Blackboard

Shioriused

€

35

m²ofcardboard,andKiyoshiused

€

45

m².Howmanym²of

cardboarddidtheyusealtogether?Math sentence

Addition

€

35

+

€

45

Let’sthinkabouthowtocalculate

€

35

+

€

45.

1) Thinkabouthowmany

€

15

’sexist.

2) Iaddedthenumeratorsinthesamewayasingrade3.

€

35

+

€

45

=

€

75

3) Finditoutusingareasquare.7piecesof

€

15=

€

75m²=

€

125m²

4) Finditoutusinganumberline.

Summary Whenaddingfractionsthathavethesamedenominator,justaddthe

numeratorsandleavethedenominatorasitis.

€

35

+

€

45

=

€

75=

€

125

Page601. Goal Studentsunderstandandcanperformsubtractionofproperfractionsfrommixednumbers.(Theintegerportionofthemixednumberis1andtheanswerisaproperfraction.)2. Assessment Criteria

(Interest)Studentswillunderstandhowtosubtractfractionsthathavethesamedenominators,relatingthewayofcalculationtowhattheylearnedbeforeforaddition.(Thinking)Studentswillnoticethatsubtractionofproperfractionsfrommixednumberscanbedonebythinkingabouthowmanyunitfractionsareineach,astheydidintheearlierstudyofaddition(Expression)Studentscansubtractfractionsthathavethesamedenominator.(Knowledge)Studentsunderstandhowtosubtractfractionsthathavethesamedenominators.

3. Teaching Point 1)Subtractionoffractionsthathavethesamedenominators.Itisimportantforstudentstofindtheirownwayofsubtracting,usingtheconceptofunitfractionsandoftheknowledgetheyhavealreadylearnedinaddition.4.Lesson Lesson and Key Questions (K)

Learning Activities and Reactions

Points to Emphasize; Assessment & Extra Support

1. Understand the contents in this lesson. (K) “What do you know about this number sentence? How would you answer it?”

Referring to the illustration, students discuss the scene in textbooks for this number sentence and develop interest in solving it.

There is

€

125

kg of sugar. If

you use

€

45

kg, how many kg

will be left? 1) The numbers used in this

question….There is

€

125

kg of

Teachers show illustration and motivate students to think about how to subtract fractions that have the same denominators Teachers instruct them to underline the important part of the number sentence and help students understand what is already known in this question and what they need to figure out.

2 Make math sentence. (Self-solving) “What kind of number sentence is good to finding out how many kg of sugar will be left?” 3.Think about how to calculate. “How would you calculate

€

125

-

€

45

?”

“Which part of the number

sentence

€

125

-

€

45

is similar to

what you learned before in

sugar. And you use

€

45

kg.

2) What do students have to answer… How many kg will be left? Students think about the number sentence.

1)

€

125

-

€

45

.

2) We can subtract

€

125

-

€

45

because we want to find

out the remainder. Students will understand the goal of this lesson. ·How to subtract fractions. ·Can you subtract fraction in the same way as for addition? Students think about how to

calculate

€

125

-

€

45

.

(Possible reactions)

1) If we change

€

125

- into

€

75

,

we can make

€

75

-

€

45

Then, based on

€

15

, we can

subtract 4 from 7 and the remainder is 3. So the

answer is

€

35

.

2) Based on

€

15

, we can do 7-

4=3. So the answer is

€

35

kg.

Teachers have students close their textbooks. For the students who are struggling with making number sentences, teachers can give hints such as “if

you replace

€

125

kg with7kg,

€

45

kg with 4kg, what kind of

math sentence can you make?” Teachers have students write the goal of this lesson down in their notebooks to confirm it. For students who are struggling with making a number sentence, teacher can give hints such as “you

can change

€

125

into

€

75

,” or

“how many

€

15

s do you need

to make

€

75

or

€

45

?” Students

will think about how to calculate it. (Interest) Students will subtract fractions that have the same denominators, relating this to the method they learned before for addition. (Thinking) Students will notice that the subtraction of proper fractions from mixed numbers whose whole parts are 1 and whose answers are proper fractions, can be done using

addition?” 4.Summarize how to subtract fractions that have the same denominators. “What is common about the calculation?” 5.Work on problems. “How would you subtract fractions when the number to be subtracted (subtrahend) is a mixed number?”

3) Use square area illustration. 4) Use a number line. Students discuss the similarities to what they learned before in addition. 1) To find answer based on

€

15

.

2) To add/subtract fractions

based on

€

15

.

Students summarize how to subtract fractions that have the same denominators. 1) In any calculation, we use

€

15

as an unit amount.

·When subtracting fractions that have the same denominators, just subtract the numerators and leave the denominators as they are. Students work on 1) page 47. 1) When subtracted numbers (subtrahends) are mixed numbers, we have to change them into improper fractions. Then we can subtract only numerators based on the concept of unit fractions.

the method they learned before for addition. All of calculations can be done in the same way as 3 on page 47. (Knowledge) Students will understand how to subtract fractions that have the same denominators. (Expression) Students will subtract fractions that have the same denominators.

5. Example of Black Board

Thereis

€

125kgofsugar.Ifyouuse

€

45kg,howmanykgwillbeleft?

Math Sentence

€

125‐

€

45

Let’sthinkabouthowtocalculate

€

125‐

€

45.

1)Ifyouchange

€

125into

€

75,

€

75‐

€

45

2)Ifyouusebase

€

15,7‐4=3.Therearethree

€

15.Sotheansweris

€

35kg.

3)Useanillustration.4)Useanumberline.What is the common way to add fractions?

∙Weusebase

€

15asunitquantityandthinkabouthowmany

€

15sdoweneedtomake

€

75and

€

45.

Summarize

∙Change

€

125into

€

75

Whensubtractingfractionsthathavethesamedenominators,justsubtractthenumeratorsandleavethedenominatorsastheyare.

€

75‐

€

45=

€

35.

€

35kg

Page621.Goal Studentswillunderstandhowtoaddmixednumbersandbefamiliarwithit.2. Assessment Criteria

(Interest)Studentswillrelatemixednumberstoproperfractionswhendoingadditionoperations.(Thinking)Studentswillnoticethattheycanaddmixednumbersastheyaddedproperfractions.(Expression)Studentscanaddmixednumbers.(Knowledge)Studentsunderstandhowtoaddmixednumbers.

3. Teaching Point Addition of mixed numbers.

Basedonadditionofimproperfractionsthathavethesamedenominators,studentswilllearnthetwowaysofaddition.Oneinvolvesseparatingmixednumbersintothewholenumberspartsandthefractionalpartsthenfindingthesumofthewholenumberspartsandthefractionalpartsseparatelyandfinallycombiningthosepartstogether.Theotheristochangethemixednumbersintoimproperfractionsthencalculate.

4. Lesson Lesson and Key Questions (K)

Learning Activities and Reactions

Points to Emphasize; Assessment & Extra Support

1.Think about how to calculate. (K) “Let’s think about how

to calculate

€

2 35

+115

.”

2.Summarize how to add mixed numbers. (K) “How would you calculate mixed numbers that have the same denominators?”

Students understand the goal of this lesson. ·How to add mixed numbers? Students think about how to calculate. (Possible reactions) 1) Find the sum of the whole number parts and the fractional parts, then combine them. 2) Change the mixed numbers into improper fractions and calculate. 3) Calculate using an area illustration. Students summarize how to add mixed numbers that have the same denominators. ·Combine the sum of the whole number parts and the fractional parts. ·Change the mixed numbers into improper fractions and calculate.

Teachers have students write the goal down in their notebooks to make it clear. For the students who are struggling with how to solve this, hand out a paper with the area illustration on page 48 so that students can relate mixed numbers to an area illustration. (Interest) Students will think about how to add mixed numbers, relating it to addition of proper fractions. (Thinking) Students will notice that addition of mixed numbers can be done like addition of proper fractions. Teachers check that students understand that the both calculations are based on addition of proper fractions. If you change mixed numbers into improper fractions to calculate, you have to change the answer into mixed numbers again. Many students are likely to make a mistake when they are changing the answer back into mixed numbers. But we do not have to change mixed numbers into improper fractions all the time. Teachers should teach the benefit of calculation, which is that the whole numbers parts and the fractional parts are

3.Work on exercise.

Students work on 1) page 48. ·Think about how to add whole numbers. Students work on 2) page 48. ·Students think about how to operate the answer of mixed numbers when its fractional parts become improper fractions.

added/subtracted separately. Teachers check that students understand that if the fractional parts become improper fractions in the answer, put appropriate numbers of unit fractions on whole numbers parts. (Expression) Students can add mixed numbers.

5. Example of Black Board

Let’sthinkabouthowtocalculate

€

2 35

+115.

1)Findthesumofthewholenumberpartsandthefractionalpartsthencombinethem.2)Changethemixednumbersintoimproperfractionsthencalculate.3)Findtheanswerusinganareaillustration.Summary 1)Findthesumofthewholenumberpartsandthefractionalpartsandcombinethem.2)Changethemixednumbersintoimproperfractionsandcalculate.Page641. Goal Studentswillunderstandhowtosubtractmixednumbersfrommixednumbersandcalculate.2. Assessment Criteria

(Interest)Studentswillthinkabouthowtosubtractmixednumbers,relatingittotheadditionofmixednumbersorsubtractionofproperfractions.(Expression)Studentscansubtractmixednumbers.(Knowledge)Studentswillunderstandhowtosubtractmixednumbers.

3. Teaching Point Subtraction of mixed numbers that have the same denominators.

Usingwhattheylearnedaboutadditionandsubtractionofmixednumbersthathavethesamedenominators,studentsshouldfindtheirownwayofsubtracting.Wecanseparatethemixednumbersintothewholenumberpartsand

thefractionalpartsthencalculate.Orwecanchangethemixednumbersintoimproperfractionsbeforecalculating.

4. Lesson Lesson and Key Questions (K)

Learning Activities and Reactions

Points to Emphasize; Assessment & Extra Support

1.Think about how to calculate. (K) “Let’s think about how

to calculate

€

2 45−135

.”

2.Summarize how to calculate. (K) “Let’s review how to

calculate

€

2 45−135

.”

3.Work on exercise.

Students understand the goal of this lesson. Students think about how to calculate. 1) Find the difference of the whole number parts and the difference of the fractional parts then combine them. 2) Change the mixed fractions into improper fractions then calculate. 3) Find the answer using a number line. Students summarize how to subtract mixed numbers that have the same denominators. 1) Find the difference of the whole number parts and the difference of the fractional parts and combine them. 2) Change the mixed fractions into improper fractions and calculate. Solve problem set (1) on p. 49. - Think about how to do it by adding the whole numbers Solve problem set (2) on

Teachers check that students understand that subtracted parts and parts subtracted from are mixed numbers and that is the difference between this lesson and previous lesson. For the students who are struggling with how to find the solution, hand out a paper with the area illustration on page 49, so that students can relate mixed numbers to an area illustration. (Interest) Students will think about how to subtract mixed numbers, relating them to previous learning about mixed numbers addition or proper fractions subtraction. Teachers check that students understand that all of the calculations are based on the rule of the previously learned subtractions. (Knowledge) Students will understand how to subtract mixed numbers. Teachers explain to students that they can calculate without changing mixed numbers into improper fractions.

p.49 -Think about how to do mixed number addition when the fractional parts add to an improper fraction.

Teachers have students notice that they can take the same way of previous subtraction, which subtract improper fraction from mixed number whose whole part is 1. (Expression) Students can subtract mixed numbers.

5. Examples on blackboard

Let’sthinkabouthowtocalculate

€

2 45−135.

1)Findthedifferenceofthewholenumberpartsandthedifferenceofthefractionalpartsandcombinethem.2)Changethemixedfractionsintoimproperfractionsandcalculate.Summary ∙Findthedifferenceofthewholenumberpartsandthedifferenceofthefractionalpartsandcombinethem.∙Changethemixednumbersintoimproperfractionsandcalculate.

Let’sthinkabouthowtocalculate

€

4 57− 2 67.

1)Findthedifferenceofthewholenumberpartsandthedifferenceofthefractionalpartsthencombinethem.Butwecannotsubtractfractionalpartsfromeachother.So:

Change

€

4 57into

€

3127(3+

€

157)

2)Changethemixedfractionsintoimproperfractions.

€

4 57− 2 67

Howtocalculate

€

5 −1 34?

1)Change5into

€

4 44.

2)Changemixednumbersintoimproperfractions.