# Plean ceachta 10 Réasúnaíocht chomhréireach – tuiscint iomasach a thabhairt do scoláirí trí réiteach fadhbanna.

```ReflectionsonPractice
LessonPlanfor[SecondYear,Proportionalreasoning]
Forthelessonon[25/03/2015]
Teacher:[Mr.O’Leary]
1. TitleoftheLesson:Ratio and proportion - giving students an intuitive understanding through
problem-solving
2. Briefdescriptionofthelesson:Wewantstudentstounderstandproblemswhichinvolve
proportionalreasoning.Wewantstudentstolearnmethodsforsolvingsuchproblems
themselves.Aspartofthisprocesswewantstudentstorecognizetheconceptofcalculatingaunit
quantityandhowthisunderstandingcanbeappliedtomanydifferenttypesofproblems
includingpercentages.Thelessonisonehourinduration.
3. AimsoftheLesson:
Long‐range/thematicgoals:
I’dlikemystudentstodevelopapositivedispositiontowardsmathematics.
I’dlikemystudentstolearntobecomeeffectiveproblemsolvers.
I’dlikemystudentstobemoreindependentlearners.
I’dlikemystudentstoworkeffectivelyingroupsandtobecomfortableexplainingtheirideastoother
students.
I’dlikemystudentstorecognizethattheycandoalotofmathematicsthemselves.
I’dlikemystudentstounderstandtheimportanceoftheirthinkingprocessasopposedtobeingreally
I’dlikemystudentstoenjoytheirmathematics.
I’dlikemystudentstodevelopanattitudeofperseverancetowardstheirmathematics.
Short‐termgoals
I’dlikemystudentstoapplyproportionalreasoning.
I’dlikemystudentstorecognisethattherearedifferentwaysinwhichtotackleproblemsbasedon
proportionalreasoning.
I’dlikemystudentstorecognisetheimportanceofunitquantityinsolvingproportionalproblems.
I’dlikemystudentstounderstandhowtheoperationsofmultiplicationanddivisionfitinto
proportionalreasoning.
I’dlikemystudentstorecogniseproblemsinvolvingpercentagesgreaterthan100%.
4.LearningOutcomes
Asaresultofstudyingthistopicstudentswillbeableto:
 Understandtheconceptofproportionalreasoning
quantity.
 Relatetheprocessofdivisiontofindingthecostofaunit.
 Relatetheoperationofmultiplicationtofindingthecostofseveralunits.
 Selectsuitablestrategiesforsolvingproblemsrelatingtoproportionalreasoning.
 Applyproportionalreasoningtoproblemsbasedaroundpercentages.
4. BackgroundandRationale
Manystudentsexperiencedifficultyinunderstandingtheconceptswhichunderlieproblems
associatedwithproportionalreasoning.Proportionalreasoningisacoreskillinmathematicsandis
usefulinallbranchesofmathematics.Atsecondlevelstudentsencounterproportionalreasoningall
thetime,infractions,inpercentages,inspeed,inareaandvolumeandmanymoresituations.
Hereisasimpleexampleofaproportional‐reasoningproblem:Thepriceofaticketis€88which
includesa10%bookingfee.Whatisthecostoftheticketalone?
Studentsoftensolvetheseproblemsbyfollowingroutine,withoutmuchunderstandingofwhythey’re
doingwhatthey’redoing.Accordinglywhentheyencountersuchproblemsindifferentsituationsthey
havelimitedabilitytodealwiththem.Wethinkthatbypresentingproportional‐reasoningproblems
inasimplebutspecificwaywecanletstudentsfigureouthowsolvesuchproblemsandthatby
makingsomeimportantconnectionswecanshowstudentsthatthistypeofreasoningisimportantin
lotsofdifferentcontexts.Thehopeisthatbydoingsostudentswillunderstandandretaintheskillfor
longer,theywillseethattheycandomathsthemselvesandtheywillbeexposedtotheideathata
singleapproachcanbeusedinmanydifferentsituations.
5. Research
TheJuniorCert.mathssyllabusmakesspecificreferencetoproblemsbasedonratioand
proportionality:
Havingexaminedanumberoftextbookswefeelthatnosingletextdealswiththissyllabuslearning
outcomeinawaythatpromotesdeepunderstandingoftheconcepts.
Thelessonwillhelpstudentsrecogniseproblemsbasedonproportionalreasoningandsolvesuch
problems.
Studentswillbeintroducedtothelessonusingasimplepricingproblemandencouragedtoestimate
Studentswillstartworkontwosimplematchingactivities.Thefirstactivityinvolvescardswith
certainmassesandprices.Usingproportionalreasoningstudentsshouldbeabletofindthemissing
designedtoexposestudentstomoredifficultexamplesofproportionalrelationshipssothattheymay
firstmatchingactivityistogetstudentsdescribingdifferentwaystoapproachproportionalproblems
andtothenapplythenumberoperationsofdivisionandmultiplicationtosolvetheproblems.
Thesecondmatchingactivityisagainbasedaroundpairsofcards.Thistimetherelationshipsare
expressednotingramsbutinpercentages.Undertheguidanceoftheteacherstudentswillseethat
thesearejustanotherexampleofaproportionalproblem.Underthisguidanceitishopedthat
studentswillapplytheirnewly‐acquiredknowledgetosolvetheseproblems.
Finallytheclasswillrevisittheintroductoryproblemandstudentswillbegiventheopportunityto
recognisethisasyetanotherproportionalproblemandwillthengetthechancetousetheir
understandingtosolvetheproblem.
7. FlowoftheUnit:
#oflesson
periods
Lesson
Ratioandproportion
1

Distinguishbetweenabsolutecomparisonandrelative
comparison

Seeratiosascomparingparttopartandfractionsascomparing
parttowhole,wherethequantitiesbeingcomparedhavethe
sameunits.

Seeratesastheratiooftwoquantitieshavingdifferentunits.

Appreciatetheimportanceoforderwhendealingwithratios

Findequivalentratios

Divideanumberintoagivenratio

Recogniseaproportionasastatementofequivalentratios5:2=
10:4orsetupaproportiontofindxasin5:2=8:x
2

Distinguishbetweenproportionalandnon‐proportional
situationsrecognisingthemultiplicativerelationshipthatexists
betweenthequantitiesinproportionalsituationsasseenin
tables,graphsandalgebraicexpressions
3

Useavarietyoftechniquesincludingtheunitarymethod,factorof
thesetechniquesareallrelated

Solveproblemsinvolvingproportionalreasoningindifferent
contexts

Howtodrawandinterpretscaleddiagrams
3x1hour
1x1hour
3x1hour
Thefirstof
whichisthe
researchlesson.
8. FlowoftheLesson
TeachingActivity
1.Introduction
Studentsarepresentedwiththefollowing
problem:Youaregivenatickettoaconcert.The
costoftheticketincludesa10%bookingfee.Ifthe
ticketcosts€88,howmuchdoestheticketcost?
Studentsarepresentedwithasimplematching
activitywhichconsistsofanumberofcardsof
differentsizeswithassociatedmasses(ingrams)
PointsofConsideration
Wewantstudentstobeengagedwiththelesson.
Wewantstudentstomakeanestimateofthe
ticketprice.
Dostudentsunderstandthattheticketprice
mustbelessthan€88?
Arestudents’guessesreasonable?
Dostudentshavearoughideaofthevalueof
10%inthiscontext?
Theteacherhandsoutthematchingactivityand
theaccompanyingworksheet.
Note:Theactivityhas10parts.Theteacher
shouldchoosewhichpartsaresuitablefortheir
costofdifferentmasses.
students.
Theteachercirculatestheroomtocheckthat
Arestudentsabletouseproportionalreasoning
tocalculatethemissingprices?
Canstudentsexplaintheirapproachto
calculatingthemissingquantities?
Dostudentsrecognizetheimportanceof
divisionforfindingthevalueofasub‐multiple?
Dostudentsrecognizetheimportanceof
multiplicationforfindingvalueofamultiple?
Canstudentsverbalisetheapplicationof
divisionandmultiplicationtosolving
proportional‐reasoningproblems?
3.AnticipatedStudentResponses
Studentsmaynotbeabletouseproportional
Itisimportantthatstudentsareencouragedto
reasoning.
explaintheirapproach.
finditdifficulttoexplaintheirthinking.
Studentsmayfindthefirstnumberofproblems
(whichinvolvedivisiononly)easytodealwithbut andusingthistofindthecostofalarger
findthelatterproblemsmoredifficult(which
quantity.
involvedivision,followedmymultiplication).
Itisimportantthatstudentsmakethe
Studentsmayfindthecostofdifferentsub‐
connectionbetweendivisionandmultiplication
multiplesandusethistoworkoutthenew
andtheprocessesinvolvedinproportional
multiple.
reasoning.
toworkoutthenewmultiple.
problemismoredifficultthatanother?
4.ComparingandDiscussing
TheteacherrepresentsproblemA1ontheboard
Wewantstudentstofocusonthereasoning
behindsolvingtheproblem.
Wewantstudentstoexplaintheirapproachin
approachtosolvingtheproblem.
theirownterms(asopposedtofocusingsolely
ontheoperationsofmultiplicationand
TheteacherrepresentsproblemE5ontheboard
division).
Wewantstudentstodescribehowdivisionand
multiplicationfitintotheprocessofdealingwith
approachtosolvingthisproblem.
proportionalreasoningproblems.
Wewantstudentstorecognizetherelevance/
TheteacherrepresentsproblemI9ontheboard
importanceoffindingthecostofoneunit.
Wewantstudentstoidentifysituationswhereit
makessensetofindthecostofoneunit.
problemcomparedtothepreviousproblems?
ideasonhowtoapproachthisproblem.
Theteachersummarisestheapproachoffinding
methodwouldhaveworkedforthefirsttwo
thecostofoneunitandthenfindingthecostofX
problemspresentedontheboard.
units,beingcarefultoexplainwhythisapproachis Dostudentsunderstandwhatproportional
neededforthisexample.Theteacheridentifiesthis reasoningmeans?
methodastheunitarymethod.
Theteacherexplainstothestudentsthatwhatthey
aredoingisknownasproportionalreasoningand
explainswhatthismeans.
typeofproblemandapproachtosolvingitonly
applytoscenariosinvolvinggramsandeuro.
Theteacherpresentsstudentswiththeirsecond
matchingactivity.Thisoneisbasedon
percentages.
thisproblem.
Theteacherexplainsthatthisissimplyanother
exampleofproportionalreasoning.
TheteacherdrawsthefirstpairA1ontheboard
weknow.
calculatewhichmightbeuseful.
Theteacherwritesdetailsofstudents’approaches
ontheboard.
groups.
6.Anticipatedstudentresponses
andthepreviousone.
Studentsmightimmediatelyturnoffbecauseofa
dislike/fearofpercentages.
Studentsmighthavedifficultyverbalisingwhat
Studentsmaygiveseveraldifferentapproaches.
Forexampletheymightfindwhat50%isworth
andthenfindwhat100%isworth.Theymightfind
what10%isworthandthenfindwhat100%is
worth.
Studentsmightfindthepairsatthestartofthe
worksheetstraight‐forwardbutmayfindthe
othersquitedifficult.
Studentsmayhavedifficultycommunicatingtheir
approachtosolvingeachproblem.
hocmannerwithouttryingtounderstandwhat
theyaredoing.
7.Comparinganddiscussing
TheteachersketchespairC3ontheboard,and
whattheywanttofindoutandhowtheymightgo
Canstudentsidentifyothersituationswhere
proportionalreasoningmayapply?
Theteacherdistributescopiesofthesecond
worksheet.
Waittimeisimportanthere.Studentsneeda
problem.
Canstudentsmakestatementsoftheform“We
knowthat150%isworth€300”?
Canstudentsmakestatementsoftheform“We
wanttoknowhowmuch100%isworth”?
Canstudentsidentifyusefulsub‐multiplesof
150%?
Canstudentssuggestsuitablestrategiesfor
tacklingthisproblem?
Canstudentsmakestatementsoftheform“50%
mustbeworth€100so100%mustbeworth
€200”?
Itisimportantthatstudentsaregiventhe
activityandthepreviousone.Thisshouldbe
donebygettingstudentstoverbalisewhatthey
know,whattheywanttofindoutandhowthey
Ifdifferentapproachesarepresenteditis
importanttoemphasisethemeritsofeach
approachandshowthattheyareequivalent.
Canstudentsverbalisetheirideas?
CanstudentsexplainwhyY102isamore‐
difficultproblem?
Theteacheremphasisesthefactthatthisisa
proportionalproblem.
TheteachersketchespairY102ontheboardand
previousexample.
couldsolvethisproblem.
Theteacherreinforcestheideaoftheunitary
methodandwhatitmeansandwhereit’suseful.
studentsiftheyhaveanysuggestionsonhowthey
mightsolvethisproblem.
Theteacherrepresentsthecostoftheticketalone
asasquareontheboardandidentifiesthissquare
asrepresenting100%.Theteacheridentifiesthe
priceofthissquareaswhatwearelookingfor.
Theteacherdrawsthesamesquareagainandthis
percentageeachpiecerepresents.
Theteacherwritesin100%onthesquareand10%
ontheextrapiece.
marksin€88onthediagram.
theticketalone.
Dostudentsunderstandtheunitarymethodand
whereitisuseful?
Itisimportanttodrawrelativelyproportional
shapes.
Theteachermayidentifythe100%squareas
thepricewearelookingfor.
Differentapproachesaretobecommendedas
theyshowthatstudentsarethinkingfor
themselves.
Theunitarymethoddoesnothavetobeused
here.Forsomestudentsitmaymakemoresense
tofindthecostof10%andusethistocalculate
thecostof100%.
Theteachershouldtrytogetstudentstopresent
differentapproaches.
andhowitcomparestotheirestimatefromthe
startofthelesson.
9.Anticipatedstudentresponses
Studentsmayfindithardtounderstandthatthe
finalcostoftheticketis100%oftheactualcost.
Somestudentsmayworkoutwhat10%costsand
thenscalethisupto100%.
Somestudentsmayrequiresupporttoapplythe
unitarymethodtosolvingtheproblem.
10.Comparinganddiscussing
ontheirshow‐meboardsandtoholdthemup.
tackledtheproblem.
totheirguess.
5.Summingup
havelearnedtoday?
Theteacherrecapsontheconceptofproportional
reasoning.
Theteacherstressesthattherearemanywaysto
solveproportionalproblemsbutthattheunitary
methodwillworkinanysituation.
problemsfortheirhomework.
9. Evaluation
Therewillbethreeobserversinthelessonaswellastheteacherteachingthelesson.
Observer1willtakepicturesofstudentwork.
Observer2willrecordstudentbehaviourusinganassessmentwheel.Thiswillincludelookingfor
evidenceofstudentscommunicatingwitheachothereffectively,studentsunderstandingthemain
motivation.
Observer3willrecorddetailsoftheteacher‐studentinteraction.
10. BoardPlan
11. Post‐lessonreflection
Whatworkedwellinthelesson:
 Moststudentestimateswerereasonableinthattheywerelessthanthetotalcostoftheticket.
 Moststudentsworkedwelltogetheringroups,cooperatedwitheachother,helpedeachother
anddiscussedtheirmaths.
 Moststudentsweremotivatedtotrytosolveproblems.
 Moststudentswerewillingtooffersuggestionsonhowtosolveaproblemortodescribehow
theysolvedtheproblemthemselves.









Somestudentssuggestedalternativeapproachestosolvingaproblemthantheonesuggested
byotherstudents.
Moststudentswereinterestedinlearningandwerecontinuouslyengagingwiththeteacherto
studentsinfeedbackandgavedifferentstudentstimetodescribetheirapproachtosolvingthe
theirapproach.
Somestudentsdemonstratedexcellentabilityinproportionalreasoning.Forexamplewhen
tacklingtheproblemoffindingthecostof100ggiventhat150gcost€30onestudent
explained“Because150gisworth€30,thismeansthat50gmustbeworth€10and100g
mustbeworth€20”.Withsomeencouragementthisstudentexplainedhisthinkingtotherest
ofhisgroup.
Theteacherencouragedstudentstodescribetheoverallapproachtheywereusingfromone
questiontothenext.Thishelpedstudentssolidifywhattheywerelearning.
Somestudentsrecognizedtheimportanceofdivisionandmultiplicationtothesolvingof
proportionalproblems.
approachtosolvingproportionalproblemswouldonlyworkformassandprice.Students
identifiedlotsofotherareaswherethismethodwouldbeuseful,includingsomeareasinother
subjectstheystudy.
Whatwewouldchangeinthelesson:
 Wetriedtodotoomuchwiththegroupofstudentsthiswastriedwith.Wefeltthatspending
moretimeontheactivityrelatingmassandpricemayhavehelpedmorestudentsdevelopa
deeperunderstandingofproportionalreasoning.Thescopeofthelessonverymuchdepends
onthestudentswhositinfrontofyou.Forsomestudentswewouldrecommendreducingthe
scopeofthelessonwhileforbetterstudentswewouldtrytocoverallthecontentpresentedin
thelessonplan.
mostimportantpartoftheactivity.
preventedthemfromengagingproperlywiththelesson.Thisshouldn’tbeanissuewithfuture
lessons.
 Wecouldhavespentalittlemoretimeatthestartexplainingthefirstactivity.Studentsdidn’t
realisethatthecardswerespecificallycreatedwithrelativesizestohelpthemunderstandthe
studentsiftheythoughtthemissingpricewouldbemoreorlessthantheonegiven,thisidea
couldhavebeenteasedout.
 Studentstendedtofocusonthenumberoperationswhichprovidedthemwithsolutions.For
multipliedby…”.Wereallywantstudentstodescribetheprocessineverydayterms.For
examplestatementsoftheform“IfoundthepriceofXgramsandthenIusedthistofindthe
 Somestudentstendedtolookforaneasysolutionasopposedtothinkingtheproblemthrough
methodas“knockingazerooffbothofthem”.Thishighlightsthedangerofstudents
attemptingtomovestraighttosomeformofnumberoperationwithoutunderstandingwhat
theyaredoing.




Studentstookalongtimesortingthepairsofcards.Thisreducedvaluablethinkingtime.
Perhapsthereisabetterwaytopresenttheactivities.Alternativesincludepresentingthepairs
ofcardsonasheetofpaper.Ontheflipsideitshouldbesaidthatstudentsseemedtoenjoy
matchingthecardsandthisseemedtogetthemmoreinvolvedintheactivity.
Studentsdidn’tnaturallywritedownanexplanationofhowtheysolvedeachproblem.Itis
importanttoconstantlyremindthemoftheimportanceofdoingso.
Studentsmayneedextrasupportwhenmovingfromthemass‐priceactivitytothe
percentages‐priceactivitytoletthemrecognizethattheunitarymethodmayalsobeusedhere.
Wefoundthatassoonasstudentssawpercentages,manyofthemconsideredtheproblems
muchtoohardwhereasinrealitytheywerepracticallyidenticaltothefirstactivity.
Inthecomparinganddiscussingsessionitisimportantthatstudentsareencouragedtoexplain
whytheyusedtheapproachtheyused.Thisexplanationshouldnotimmediatelyfocuson
divisionandmultiplication.
Summary:
Wefoundthelessontobesuccessfulbutwithareaswhereitmaybeimproved.Wedesignedafairly
generallessoninproportionalreasoningbutpresentedittoagroupofstudentswhooftenstruggle
ofyouinagivenlesson.
groupworkisn’tjustsomethingforthebeststudentsinayeargroup.Itcanworkwellforallstudents
butonlyiftheworkisappropriatetothosestudents’level.
Itisimportantthatstudentshavesomeeverydayunderstandingofwhattheyaredoingbefore
applyingnumericaloperationstosolveaproblem.Ifstudentsfocusonproceduresandoperationstoo
soontheyarelikelytostrugglewithretainingwhattheyhavelearnedandwhensimilarproblemsare
CARD K
100 g
€60
CARD 11
117 g
€??
CARD G
150 g
€20
CARD 7
100 g
€??
CARD H
150 g
€60
CARD 8
100 g
€??
CARD L
120 g
€90
CARD 12
109 g
€??
CARD A
1000 g
€40
CARD B
1000 g
€20
CARD C
1000 g
€60
CARD 1
500 g
€??
CARD D
1000 g
€5
CARD E
1000 g
€80
CARD F
1000 g
€25
CARD 2
500 g
€??
CARD 3
250 g
€??
CARD 4
250 g
€??
CARD 5
300 g
€??
CARD 6
300 g
€??
CARD I
250 g
€80
CARD 9
100 g
€??
CARD J
250 g
€25
CARD 10
100 g
€??
CARD 7
100 %
€??
CARD 8
100 %
€??
CARD 9
100 %
€??
CARD G
167 %
€500
CARD I
115 %
€800
CARD H
113 %
€800
CARD J
130 %
€250
CARD 10
100 %
€??
CARD K
200 %
€45
CARD 11
100 %
€??
CARD L
200 %
€70
CARD 12
100 %
€??
CARD A
150 %
€300
CARD 1
100 %
€??
CARD 5
100 %
€??
CARD B
150 %
€600
CARD 2
100 %
€??
CARD 6
100 %
€??
CARD E
120 %
€600
CARD C
250 %
€800
CARD 3
100 %
€??
CARD D
250 %
€250
CARD 4
100 %
€??
CARD F
120 %
€900
Percentages Worksheet:
Find the cost of the quantity given on the smaller card
Card Letter
A
Card Number
1
Explanation:
Card Letter
B
Card Number
2
Explanation:
Card Letter
C
Card Number
3
Explanation:
Card Letter
D
Card Number
4
Explanation:
Card Letter
E
Card Number
5
Explanation:
Card Letter
F
Card Number
6
Explanation:
Card Letter
G
Card Number
7
Explanation:
Card Letter
H
Card Number
8
Explanation:
Card Letter
I
Card Number
9
Explanation:
Card Letter
J
Explanation:
Card Number
10
Card Letter
K
Card Number
11
Explanation:
Card Letter
L
Explanation:
Card Number
12
Grams Worksheet:
Find the cost of the quantity given on the smaller card
Card Letter Card Number
A
1
Explanation:
Card Letter
B
Card Number
2
Explanation:
Card Letter
C
Card Number
3
Explanation:
Card Letter
D
Card Number
4
Explanation:
Card Letter
E
Card Number
5
Explanation:
Card Letter
F
Card Number
6
Explanation:
Card Letter
G
Card Number
7
Explanation:
Card Letter
H
Card Number
8
Explanation:
Card Letter
I
Explanation:
Card Number
9
Card Letter
J