ReflectionsonPractice
LessonPlanfor[3rdYear,Polynomialfactorisation]
Forthelessonon[20/03/2015]
At[PresentationSecondarySchool,WexfordTown],[DeirdreDeegan’s]class
Teacher:[DeirdreDeegan]
Lessonplandevelopedby:[SeanRossiter,MoO’Brien,DeirdreDeegan]
1. TitleoftheLesson:Usingarraystofactorisepolynomials
2. Briefdescriptionofthelesson:Studentswillusethearraymodeltohelpthemfactorise
polynomialsofdegree2anddegree3.
3. AimsoftheLesson:
I’dlikemystudentstodeveloptheirabilitytoworkinagroupandtousecooperativethinkingin
formingsolutions.
I’dlikestudentsofallabilitiestobeengagedwithandchallengedbythelessoncontent.
I’dlikestudentstoexploretheareamodelasamethodfordivisionofcubicexpressions.
I’dlikestudentstoextendtheirunderstandingoftheareamodelforitsuseindivisionofquadratic
expressions.
4. LearningOutcomes:
Asaresultofstudyingthistopicstudentswillbeableto:
 usetheareamodeltodividequadraticexpressionsoftheform1
bylinearexpressions
oftheform
whereb,c,d&earepositiveintegers.
bylinearexpressions
 usetheareamodeltodividequadraticexpressionsoftheform
oftheform
wherea,b,c,d&eareintegers.
 explainverballythethinkingneededtofillouttheareamodel.
 explainverballytheneedfora3 2arrayfordivisionofacubicexpressionbyalinear
expression.
bylinear
 usetheareamodeltodividecubicexpressionsoftheform1
expressionsoftheform
,whereb,c,d,e&farepositiveintegers.
bylinear
 usetheareamodeltodividecubicexpressionsoftheforma
expressionsoftheform
,whereaisapositiveintegerandb,c,d,e&farepositiveintegers.
5. BackgroundandRationale
Thedistributivepropertyofmultiplicationissomethingthatmanystudentsfindeasybutothersfind
difficulttounderstand,torememberandtoapply.Theideaofdistributionisencounteredinfirstyear,
firstwithnumbersandthenwithalgebraicexpressions.Theabilitytoapplythedistributiveproperty
ofmultiplicationisimportantinallbranchesofmathsandstudentslackingthisabilityoftenstruggle
tosolvemanybasicmathsproblems.Thearraymodelhasbeendemonstratedtobeaneffectivetool
forhelpingstudentstoapplythedistributivepropertyofmultiplication.Ourstudentsarepresented
withthismethodinfirstyear,alongwiththemoretraditionalapproachofapplyingthedistributive
law.Studentsarenotforcedtouseonemethodoveranotherbutareencouragedtochoosethe
methodwhichmakesmostsensetothem.
Oneadvantageoftheareamodelisthatitalsooffersanalternativeapproachtocarryingoutalgebraic
longdivision.Algebraiclongdivisionisaskillthatmanystudentsfinddifficulttomaster.Evenwhen
thetechniqueispresentedalongsideanumericexample(toallowstudentsrecognisethesame
process),manystudentsstillfinditdifficulttounderstand,torememberandtoapply.Thisproblemis
exasperatedbytoday’sstudentsbeingoverlyreliantoncalculatorsfromayoungage.Becauseofthis
manyofthemhavelittleexperienceofnumericlongdivisioninthefirstplace.
Inthislessonwehopetopresentstudentswithanunderstandableandusablemethodforperforming
algebraicdivisionusingtheareamodel.Wehopethatbygivingstudentsasolidunderstandingof
multiplicationusinganareamodelthattheywillbeabletoapplysomesimpleproblemsolvingto
carryoutthereverseprocessofdivision.
6. Research
UnderSection4.6oftheJuniorCertificatemathssyllabusstudentsareexpectedto:
1. multiplyexpressionsoftheform
o
o
where
2. divideexpressionsoftheform
o o 3. factoriseexpressionsoftheform
o
,
where ∈ o
where , ∈ o
where , , ,
o
where , ∈ where , ∈ o
o
o
where ∈ ,
, , , , ∈ where , , , , ,
∈ arevariable
∈ 7. AbouttheUnitandtheLesson
1. Studentswillstartbybeingaskedtousetheareamodeltomultiplyapairoflinearexpressions.
Thistaskaimstoremindstudentsoftheareamodelformultiplicationofbracketedterms.
Whilethestudentspracticethetechnique,theteacherwillhighlighttheimportantfeaturesof
theareamodelwhichstudentsneedtounderstandbeforetheycanhopetouseitfordivision.
ThisactivitycoversthefirstlearningoutcomedetailedinSection6.
2. Studentsarethenpresentedwithaquadraticexpressionandoneofitsfactors.Theyareasked
tofindthemissingfactorbyworkingbackwardsusingtheareamodel.Asstudentsbecome
comfortablewithusingtheareamodelfordivisiontheyarechallengedtofindthefactorsof
moredifficultquadraticexpressions.Thisactivitycoversfirstpartofthesecondlearning
outcomedetailedinSection6.
3. Studentsarepresentedwiththedivisionofacubicexpressionbyalinearexpression.Thefirst
thingtheyaretaskedwithisdeterminingthesizeofarrayneeded.Withteachersupportthey
arethenchallengedtodescribetheprocessinastep‐by‐stepway.Tosolidifystudents’learning
theyareaskedtocompleteamatchingactivitybasedondivisionofacubicexpressionbya
linearexpression.Thisactivitycoverstheremainingpartofthesecondlearningoutcome
detailedinSection6.
8. FlowoftheUnit:
#oflesson
periods
Lesson
1
2
Evaluationofandoperationsonalgebraicexpressions‐revisionand
extensionofsecondyearmaterial

Terms,coefficientsandexpressions

Generatingalgebraicexpressionsfromsimplecontexts

Evaluatingexpressions

Addingandsubtractingalgebraicexpressions

Multiplyingtermsandexpressions,andusingtheassociativeand
distributivepropertiestosimplifyexpressions
2x40min.
1x40min.
3

Dividingaquadraticexpressionbyalinearexpression

Dividingacubicexpressionbyalinearexpression
9. FlowoftheLesson
TeachingActivity
1.Introduction
Theteacherexplainswhattheaimsoftoday’s
lessonare.
1.Tofactorisequadraticexpressions.
2.Todivideacubicexpressionbyalinear
expression.
Theteacherexplainsthatwearegoingtorevise
someimportantpriorknowledge.
Studentsarepresentedwiththeexpression
1
2 andremindedthatthereare
variouswaysofexpandingthebrackets.
Studentsareaskedtoconsiderhowtousethearea
modeltoexpandthepairofbrackets.
Theteacherasksindividualstudentstodescribe
howtousetheareamodeltoexpandthebracket
pair.
Theteacherasksstudentswhereeachbracketed
termshouldgoonthediagram.
Theteacherasksstudentshowtocompletethe
fourentriesinthearray.
Theteacherasksstudentshowtousethefour
entriestowritedowntheiranswerintheform:
.
Theteacherhighlightsthefactthatwhenthearray
isusedinthisway,liketermsendupalongoneof
thediagonals,thehighestordertermendsupin
thefirstspaceandthenumbertermendsupinthe
finalspace.
2.PosingtheTask
Theteacherwritesanewproblemontheboard:
6
8
2 andasksstudentsifit
wouldbepossibletousetheareamodeltoanswer
thisquestion.
Theteacherasksindividualstudentstodescribe
howthearraycouldbesetuptocompletethis
division.
Withthesupportofstudents,heteacher
demonstrateshowthearraycanbecompletedto
findthequotient.
2x40min.
(ofwhichthe
firstlessonis
theresearch
lesson)
PointsofConsideration
Theteacherwritesthelearningoutcomesonthe
boardforstudentstosee.
Theteacherwritesthequadraticexpressionon
theboard.
Theteacherprojectsanemptyarrayonthe
board.
Theteacherfillsinthedifferentpartsofthe
arrayasstudentsdescribetheprocess.
Canstudentsdescribehowtousethearray
correctly?
Theteachercirclestheliketermsonthe
diagonal.
Theteacherwritesouttheanswertothe
.
questionintheform:
Theteachermaysupportstudentsintheir
thinkingbyremindingthemthatdivisionisthe
inverseprocessofmultiplication.
Theteachergivesstudentssometimetodiscuss
howthiscouldwork.
Itisimportanttosupportandencourage
studentswhoaregivingfeedbacktotheclass.
Dostudentsunderstandwherethedivisoris
placedinthearray?
Dostudentsunderstandthatthedividendis
placedinthefourspacesinthearray?
Canstudentscommunicatethatthehighest
orderterminthedividendwillgointhefirst
spaceofthearraywhilethenumbertermwillgo
inthelastspace?
Dostudentsunderstandthatthe termofthe
dividendissplitacrossthetworemaining
spaces?
3.AnticipatedStudentResponses
Studentsshouldhavelittledifficultyidentifying
Theteachermayneedtoleadthediscussionso
wherethedivisorisplacedinthearray.
thatstudentsthinkabouttheimportantaspects
Studentsmayhavedifficultyunderstandingthat
ofusinganarraytodividetwoexpressions.
thedividendmustgointhefouremptyspacesin
Theteachermayneedtoaskstudentsspecific
thearray.
questionssuchas“Whatmustgointhefirst
Studentsshouldhavelittledifficultyidentifying
spaceofthearray?”followedby“Whatdoesthis
meanforthefirstterminouranswer?”,followed
thatthe termofthedividendshouldgointhe
by“Whatotherinformationcanwenowfillinto
firstspaceofthearray.
Studentsmayhavedifficultyidentifyinghowtouse thearray?”andsoon.
Theteachershouldhighlightthefactthatthe
thearraytofindthequotient.
arraymodelcanbeusedtocheckthatthe
Studentsmaystrugglewiththeideaofusingthe
sumofthe termsonthediagonaltocompletethe answeriscorrect.
quotient.
Theteacherdistributescopiesoftheworksheet
Studentsmaynotseethattheycanchecktheir
toeachstudent.
answerusingtheremainingentryinthearray.
Theteachercirculatestheroomtocheckthatall
StudentsaredirectedtocompleteQuestion1–3
studentsareabletocompletetheworksheet.
ontheirworksheetindividually.Theyaretoldto
Studentswhofinishquicklyareencouragedto
comparetheiranswerstothoseoftheirpartner’s
attemptmoredifficultexamples.
andtocheckeachsolutionifdifferencesexist.
4.ComparingandDiscussing
Didallstudentscompletetheworksheet?
Theteachertakeseachquestionfromthe
Didallstudentsanswerallquestionscorrectly?
worksheetandasksstudentstoholduptheir
Canstudentsdescribetheprocesstheyusedto
answersontheirshow‐meboards.
answerthesequestions?
Theteacherasksindividualstudentstodescribe
Theteacherwritesupeachstudent’sdescription
howtheyusedthearraytoanswereachquestion. ontheboard.
Theteacheremphasizesthelocationofthe
divisor,dividendandquotientandtheexistence
ofliketermsalongonediagonal.
5.Posingthetask
Canstudentsapplytheirpriorknowledgeto
Theteacherpresentsstudentswithanother
solvethisproblem?
2
15
5 and
divisionproblem:
Canstudentssplitthe 2 termcorrectlyalong
asksthemtoattemptsolvingitontheirshow‐me
thediagonal?
boards.
Aftersometimetheteacherasksstudentstohold Theteachercirculatesaroundtheroomtohelp
studentswhoarehavingdifficultieswiththe
uptheiranswers.
task.
Theteacherasksindividualstudentstodescribe
eachstepoftheprocessofdividingthequadratic
expressionbythelinearexpressionandwritesup
thesolutionontheboard.
TheteacherasksstudentstocompleteQuestion4
toQuestion6oftheirworksheet,againworking
individuallyandthencheckingtheiranswerswith
theirpartner.
6.AnticipatedStudentResponses
Theteacherneedstosupportstudentswhohave
Studentscorrectlyusetheareamodeltocomplete difficultieswithintegeroperations.
thedivisionexercises.
Studentswhocompletetheworkquicklymay
Somestudentsmyfindusingintegersdifficultdue
tonotknowinghowtocorrectlyadd/subtract
integersormultiplyintegers.
7.
Theteachertakeseachquestionontheworksheet
andasksstudentstoholduptheiranswersontheir
show‐meboards.
Theteacherasksindividualstudentstoexplain
howtheysolvedeachquestionusingthearea
model.
Theteacherasksstudentswhythisactivitywas
harderthanthelast.
8.Posingthetask
Theteacherwritesthefollowingdivisiononthe
board:
5
7
3
3 andasks
studentswhatisdifferentaboutthisproblem
comparedtotheotherstheyhavedone.
Theteacherasksstudentstodrawtheoutlineof
thearrayneededtosolvethisproblem.
Theteacherasksstudentstoholduptheirshow‐
meboards.
Theteacherasksstudentswhytheyneedabigger
arraythistime?
Theteacherasksstudentstofillinasmuch
informationontheirarray.
Theteacherexplainstheprocessofdividingacubic
expressionbyalinearexpression,questioning
individualstudentstoexplaindifferentstepsinthe
process.
Theteacherasksstudentsifthereisawayinwhich
theycanchecktheiranswer.
Theteacherpresentsanotherdivisiononthe
6
3
10
2 andasks
board:
studentstosolvethisontheirshow‐meboards.
Theteacherasksstudentstoshowtheirsolutions
ontheirshow‐meboards.
Theteacherasksindividualstudentstoexplain
howtheysolvedthisproblem.
Theteacherdistributesamatchingactivitytoeach
groupoffourstudents.
Studentsareinstructedtosolvethefourdivision
problemsintheactivitybyplacingthevarious
termsinthecorrectspacesineacharray.
attemptmorechallengingquestions.
Ifstudentshavedifficultieswithspecific
questionstheteachermaywritethesolutionon
theboard.
Dostudentsunderstandthatthepresenceof
integersinthedivisoranddividendmakethese
questionsmoredifficult?
Dostudentsrecognizethatdifficultywith
integersmayhindertheirprogressinother
areasofmaths?
Dostudentsrecognizethattheyarenow
dividingacubicexpressionbyalinear
expression?
Dostudentsunderstandthattheyneeda2 3
array?
Canstudentsexplainwhytheyneeda2 3
array?
Dostudentsunderstandthatwhentheydividea
cubicexpressionbyalinearexpressiontheywill
getaquadraticexpression?
Canstudentsfillinsomeinformationonthe
array?
Dostudentsunderstandthatonediagonalwill
holdthe termswhileanotherdiagonalwill
holdthe terms?
Canstudentssplitthe termsandthe terms
correctlytoallowthemtocompletethearray?
Dostudentsrecognizethatbyfillinginthefinal
entryofthearraytheycanchecktheiranswer?
Canstudentsapplytheareamodeltoworkwith
acubicexpressionwithintegercoefficients?
Theteachercirculatestheroomtoobserve
studentinteractionandtoquestionstudents
abouttheirreasoning.
Canstudentsapplytheirknowledgeofthearea
modeltocompletethematchingactivity?
Dostudentsworkwelltogetherasateam?
Dostudentscheckeachother’swork?
Dostudentsspeakupwhentheythinkthatpart
ofthegroup’sworkisincorrect?
Dostudentsexplaintheirreasoningasthey
completethematchingactivity?
9.AnticipatedStudentResponses
Studentsmayfinditdifficulttoimaginethearray
neededtocompletethisproblem.
Studentsmayfindpowersofthreemoredifficultto
dealwith.
Studentsmaystrugglewithmatchingthe terms
alongonediagonalandthe termsonanother
diagonal.
Studentsmayfindthematchingactivitytooeasy.
Studentsmaygetconfusedbymultiplicationand
addition.
Studentsmayhavedifficultieswithnegative
integers.
10.ComparingandDiscussing
Theteacherasksstudentstoholduptheir
completedworkandcommendstheclassfortheir
exceptionalefforts.
11.Summingup
Theteacherasksstudentswhattheyhavelearned
todayandreinforcesthelearningoutcomeswritten
upontheboardatthestartofclass.
Studentsareaskedtocompletetheremaining
questionsontheirworksheetforhomework.
Theteachermaysuggestthatstudentscirclethe
diagonalforthe termsandthediagonalfor
the terms.
Studentswhohavecompletedthetaskshould
haveallcorrectanswersduetothefactthatit’sa
matchingactivity.
10. Evaluation
Therewillbethreeobserversinthelessonalongwiththeteacher.
Theteacherisexpectedtointeractwithstudentsduringgroupworksandtousesuitablequestioning
strategiestofindoutaboutstudents’thinking.Theteacherwilluseshow‐meboardstoidentifywhich
studentsareabletocompletethevariousactivitiesandwhichstudentsarehavingdifficulties.
Observer1willrecordthelessonflowandhoweachpartofthelessonprogressesinrelationtothe
predictedtiming.
Observer2willrecordwhichpartsofthelessonstudentsfounddifficultandwhichpartsstudents
foundeasy.Observer2willrecordwhatdifficultiesstudentshadandhowthesedifficultieswere
addressed.
Observer3willrecordexamplesofstudentworkbyphotograph.
Allobserverswillrecorddetailsofstudentinteractionincludinghowstudentsworkedtogetherasa
group,ifcertainstudentsweredisengagedwiththelesson,ifstudentsdemonstratedinitiativewhen
finishedataskandsoon.
Studentswillbeissuedwithapost‐lessonquestionnaire.
Teacherswillhaveapost‐lessonmeetingtoreflectontheirobservationsfromthelesson.
11. BoardPlan
Note:Colourboxesareusedineacharraytohelpstudentsrecognisethelocationofsimilarterms.
Thisisanimportantstepinstudentsbeingabletousetheareamodeltomultiplyanddivide.
12. Post‐lessonreflection
Generalfindings
 Thelessonwasaverypositiveexperienceforallstudents.Inthestudentsurveyallstudentsrated
thelessonas8+.
 Thelessonwaseasytofollowformoststudents.Studentsunderstoodwhattheyweretryingtodo
atalltimes.Studentswereaskedtoratethepaceofthelessonandtoratehowsuccessfultheywere
inusingtheareamodel.Hereisasummaryoftheiranswers:
16
14
12
10
8
6
4
2
0
Success in using area model
5
4
3
2
Too Fast
Fast
Just
Right
Slow
1
Too Slow
Studentswerefullyengagedwiththelessoncontent.
 Studentsfollowedinstructionverywell.
 Theteacher‐studentinteractionwasexcellentwithallstudentswillingtoaskquestions,answer
questionsandexplaintheirreasoning.
 Theuseofcolourtoidentifyliketermsinthearrayhelpedstudentsunderstandtheprocessof
multiplicationanddivisionusinganarray.Theuseofloopsbystudentstodosameyieldedsimilar
benefits.
 Somestudentshaddifficultiesfillinginthearrayswhereunderstandingofintegerswasrequired.
Theprogressofthesestudentsinthelessonwasslowedbytheirlackofabilitywithintegernumber
operations.
 Asmallnumberofstudentshaddifficultiesdifferentiatingbetweenmultiplicationandaddition
whenfillinginthearray.Thisconfusiontendedtohappenwhenstudentsweresplittingupthe termintotwoseparatepartsalongthediagonalofthearray.
 Whencheckingtheirwork(byexaminingthefinalterminthearray)somestudentsrandomly
changedthesignsofcertaintermssothatthecheckworked.Theyfailedtothencheckthatthe
othertermswerestillcorrectgiventhesechanges.Somestudentsabandonedbasicnumbersense
to“get”theirchecktowork.
 Studentsworkedwellindependentlyandthenwerehappytocomparetheirworktothatoftheir
partner.Incaseswhereanswersdidnotagreemoststudentsproceededtochecktheirworkand
discusswhaterrorhadoccurred.Asmallnumberofstudentssimplychangedtheiranswerto
matchthatoftheirpartnerwhenanswersdisagreed.
 Groupworkedprovedveryeffectiveinmostcasesasitallowedstudentstodiscusswhattheywere
learning.
Effectiveness of Group work
5
4
3
2
1
 Thematchingactivityprovedagreatsuccessandwasagreatwaytofinishthelesson.Students
workedwelltogetherandmanywereobservedcheckingtheirpartner’ssuggestionsforwhichterm
shouldgowhere.Asstudentsworkedonthematchingactivitytheyhadtodosomebasicalgebraic
multiplicationandadditionintheirheadswhichwasaworthwhileexerciseinitself.Studentsalso
hadtothinkabouttheplacementofdifferenttermsinthearraywhichcanonlyhavehelpedsolidify
theirlearning.Wehadthoughtbeforethelessonthatsomestudentmightnotfindthematching
activitysufficientlychallenging,howeverthisturnedoutnottobeso.Allstudentsworkeduptothe
endofclassontheactivityandwerefullyengaged.
Studentswereaskedaboutthehelpfulnessofthematchingactivity[onascaleof1–5]herearethe
results:
Effectiveness of matching activity
5
4
3
2
1
Recommendations
Ifweweretoteachthelessonthereareveryfewchangeswewouldmake:
 Itmightbeagoodideatospendsometimeintheleaduptothislessonpracticingbasicinteger
operations,particularlyforstudentswhohaveexperienceddifficultieswiththistopicinthepass.
 Thetimingofthelessonwasexcellentwithalotofcontentfittedin.Ifpossibleitwouldbeniceto
givestudentsalittlemoretimeinthinkingaboutsomeoftheimportantfeaturesoftheareamodel.
Forexample,givingstudentstimetodiscusswhyacubicexpressiondividedbyalinearexpression
requiresa3 2arraywouldbegood.
 Thislessonwaspresentedtoahigher‐levelthird‐yeargroup.Somestudentsfoundtheareamodel
somewhatdifficultduetolackofknowledgeofintegers.Itwouldbeinterestingtoseeifanarea
modelwouldbeaseffectivewithstudentsoflesserability,whereintegerproblemsaremorelikely.
Finalconclusions
 Theareamodelworkedverywellfordivisionofalgebraicexpressionsinagroupofhigher‐level
students.
 Theareamodelrequiresstudentstohavegoodknowledgeofintegeroperations.
 Studentsgetalotofpracticeofbasicalgebraoperationswhilepracticingtheareamodel.
 Theeffectivenessoftheareamodelforstudentsoflessermathematicalabilityisuntestedinthis
case.
Worksheet
Q.1.
Divide
Name:
Q.4.
Divide
x
x
‐1
+2
Q.2.
Divide
Q.3.
Divide
2
+5
Q.5.
Divide
x
‐1
2
Q.6.
Divide
+5
Note:OncefinisheddoExtra
Questions(a),(b)&(c)below
ExtraQuestions(inCopybook)
(a)
(b)
(c)
Q.7.
Divide
ExtraQuestions(inCopybook)
(d)
(e)
(f)
Q.11.
Divide
Q.8.
Divide
Q.9.
Divide
Q.12.
Divide
2
‐1
Q.10.
Divide
Q.13.
Divide
+5
x
2
‐1
Q.14.
Divide
+5
ExtraQuestions(incopybook):
13
3
14
20 24 56
80 1 3 4 4
x
Q.15.
Q.16.
Q.17.