Polynomials and Polynomial Functions Lesson Plans
Polynomials and Polynomial Functions Lesson Plans
Polynomials and Polynomial Functions Lesson Plans
LessonPlan:IntroductiontoPolynomialUnit
Aim
:Icanexplaintheunitproblemandwhatapolynomialinstandardformis,andevaluate
polynomialsforagivenvalue.
ContentStandards
:HSFIF.A.2A2.A.41
MaterialsandResources
:
1. Notebooks
2. Cardboardboxes
3. Tape
4. Scissors
5. Rulers
KeyVocabulary
:
Polynomial
Volume
Motivation
:
Studentswillmakeafirstattempttosolveaproblemusingtheirownintuition.Thiswill
contrastwiththeendoftheunitassessmentwhichwillinvolvewritingapolynomial
equationtosolvethesameproblemmathematically.Thepuzzle/competitionoftrying
tocreatetheboxwithoutmathcoulddrawstudentsin.
ProceduralChallenge
:
StudentsmayfinishtheEngagementActivityquicklyandstruggletostayengaged.
Buildingtheboxaccuratelycouldprovedifficult.
ConceptualChallenge
:
Studentsmayhavetroublerememberinghowtocalculatevolumeofaprism.
Implementation
:
EngagementActivity:
1. Studentsarepresentedtheunitproblem(adaptedfrom
McDougallLittellAlgebra2
):If
wecutsquareswithsidesoflengthxoutofacardboardrectangle(seediagram
below),whatvalueofxwillmaximizethevolumeofaboxmadefromthesamepiece
ofcardboard?Seethediagrambelow:
1. Studentsworkinpairstochooseavalueforx,maketheboxusingcardboard
rectangles,rulers,scissors,andtape,andmeasurethevolume.
2. StudentsputtheirdataintoaspreadsheetatthefrontoftheroominExcel.
3. Ifstudentsfinishearly,theycanchooseanothervalueforx,helpothergroups,record
answersorhelpmanagematerials.Theycouldalsodecoratetheirboxes.
Summary/Debrief:
1. Theteachercallsthestudentsbacktogetherandgoesoverthefindings.Hopefully
valuesof
x
intheExcelspreadsheetontheSmartBoardwillyieldbiggervolumesfor
thebox,thentheywilldropoff.Ifnot,theteachercanjustcongratulatethegroupthat
hadthehighestvolume.
2. Theteacherthendiscussesthefactthatthestudentsweresolvingaproblemjustby
guessingorusingintuition.Maybesomestudentstriedtousealgebra,andmaybe
theyweresuccessful.Theteacherelicitsreflectionsonwhatitwaslike,howstudents
madeguesses,etc.
MiniLesson
3. Finally,theteacherintroducesthenotionofapolynomial,andthestudentsdefinethe
termintheirnotes(amonomial=avariableoranumberpolynomial=amonomialor
asumofmonomials).Theteacheralsointroducesthenotionofstandardform.The
teacherthenexplainsthatbytheendoftheunit,thestudentsshouldbeabletosolve
thesameproblemusingpolynomialsasatooltomaximizethevolumeofthebox.
Assessment:Exitslipputthispolynomialinstandardform:
4x + 15 + 12 + 7x2 9x4 + x3
HW:Worksheet(seeattached)
Name:
Date:
PolynomialFunctions
Putthefollowingpolynomialsinstandardformandevaluatethemforthegivenvalueofx.
1. f (x) = 8 x2 evaluateforx=2
3. h(x) = x4 + 6 evaluateforx=2
4. f (x) = x + 21 x4 43 x3 + 10 evaluateforx=4
Name:
Date:
EvaluatingPolynomials
II.LessonPlan:PolynomialArithmetic
Prerequisiteskills
:Understandingliketermsandcomfortmanipulatingpolynomials
Aim
:Wecanadd,subtract,andmultiplypolynomials.
ContentStandards
:HSAAPR.A.1A2.N.3
MaterialsandResources
:
SmartBoard
Notebooks
Sortcards
KeyVocabulary
:
Liketerms
Motivation
:
Adding,subtracting,andmultiplyingpolynomialsislikecleaningupthekitchenputthe
foodinthefridge,thedishesinthesink,trashinthetrashcan,andwipeupthe
counter.
ProceduralChallenge
:
ConceptualChallenge
:
Recognizingliketermscanbechallenging
Implementation
:
DoNow:
Sortingmonomialsstudentsaredirectedtosortasetofcardsintofourpileshoweverthey
3
3
2
2
thinkmakessense.Themonomialsare4x
,8x
,9x
,2x
,15x,x,6,and24.Thestudentswill
hopefullyseethattheyshouldbesortedbydegree.Theteachercanhintiftheydonot.
MiniLesson:
Theteacherwilllectureandthestudentswilltakenotesonthefollowing:
Reminders
:
3+3+3=9isthesameas3(3)=9.
Also
:
3(3)+5(3)
9+15
24=8(3)
ie3(3)+5(3)=8(3)
Thisshowsthatwehave
added
thecoefficients.
Now,wegeneralizetheabovetovariables:
x+x+xisthesameas3x.
2x+3xisthesameas(x+x)+(x+x+x)=x+x+x+x+x=5x
2
2
2
and2x
+3x
=5x
aswell.
Allthisistosaythatwhenyouareaddingandsubtractingpolynomialsyouare
combininglike
terms
(variablesthatareraisedtothesamepowers).Youworkwiththe
coefficients
ofthe
monomials.
Wedothesamethingwhenweareaddingandsubtractingpolynomials.
Examplestobeworkedoutbytheteacherbothhorizontallyandvertically:
3
2
3
2
(2x
5x
+3x9)+(x
+6x
+11)
3
2
2
(3y
2y
7y)+(4y
+2y5)
Whenaddinghorizontally,theteacherwillmodelcolorcodingliketermssothatstudentsdo
notgetconfused.
Multiplyingpolynomials:
Everymonomialgetsmultipliedbyeveryothermonomialinadjacentterms.
Groupwork:
Studentsworktogetheronexamples,thenpresentthesetotherestoftheclass.
Summary:
Addingorsubtractingpolynomialscanlookintimidating,buttherearereallyonlytwosteps:
1.Identifyliketerms(variablesraisedtothesamepower)
2.Addorsubtracttheircoefficientsiftheyarevariables,oradd/subtractthenumbersifthey
areconstants.
HW
:Seeattached
III.LessonPlan:Zeros,Factors,Solutions,andXIntercepts
Prerequisiteknowledge
:Factorpolynomials
Aim
:Wecanexplaintheconnectionbetweenzeros,factors,solutions,andxinterceptsofa
graph,andusethesetofind
turningpoints
(ielocalminimaandmaxima).
ContentStandards
:HSAAPR.B.3HSFIF.C.7cA2.A.26
MaterialsandResources
:
SmartBoard
Studentnotebooksandcalculators
KeyVocabulary
:
Minimum
Maximum
Motivation
:Thisisalessonwhereseveraltopicsreallycometogetherinexcitingways.
ProceduralChallenge
:
Studentsmighthaveadifficulttimewithcalculatorfunctions
Challengingsketchingcomplexpolynomials
ConceptualChallenge
:
Thisisacomplicatednotiontograsponitsown.
Implementation
:
DoNow/Exploration:
Ontheirown,studentsuseacalculatortographthefunction:
f (x) = 61 (x + 3)(x 2)2 andsketchit.
Then,theyanswerthefollowingquestions:
1.Whatarethexinterceptsofthefunction?
2.Whatarethefactorsofthepolynomialabove?
3.Whatistheconnectionbetweenthefactorsoff(x)andthexintercepts?
4.Useyourcalculatorsfeaturestofindminimumandmaximum.Whatdotheselooklikeona
graph?
MiniLesson:
NotesondiscussionflowingfromtheDoNowactivity:
Zerosareplaceswherethegraphofafunctioncrossesthexintercept.
Wecanseezerosjustbylookingatafunctionthatisfactoredcompletely.
Localminimumandlocalmaximumarepeaksandvalleysthatarehigherorlowerthanany
othernearbypoints.Discussinsmallgroups:canstudentsthinkofsituationwheretheyd
needtoknowmaximaorminima?
Summary:
Zeros,solutions,roots,andxinterceptsarecloselyrelated
Minimaandmaximaareplaceswherethegraphofapolynomialturns(iegoesfrompositive
slopetonegativeslope)
ExitSlip:Circleandlabelmaxima,minima,androotsofapolynomialgraph.
HW
:Explainthathomeworkispracticerelatedtofindingrootsusingavarietyofstrategies.It
requirespullingtogetherworkfromseveraldays.
Name:
Date:
Homework:SolvingPolynomialEquations
Name:
Date:
HW:Findingpolynomialroots
Directions:Useallyourstrategies(factoring,quadraticformula)tosolvetheequationsbelow.
IV.LessonPlan:Factorstrategies
Aim
:Wecanusestrategiestofactorpolynomialexpressions.
ContentStandards
:HSASSE.B.3aHSFIF.C.7cA2.A.26
MaterialsandResources
:
SmartBoard
Studentnotebooks
KeyVocabulary
:
Factor
Perfectsquare
Trinomial
Motivation
:
Polynomialsareoftendisguisedmeaning,theylookmoreintimidatingordifficultto
workwiththantheyactuallyare.Factoringhelpsusbreakdownpolynomialssothat
wecanseewhatsreallygoingon.
ProceduralChallenge
:
ConceptualChallenge
:
Factoringseemslikeitcouldcomedowntojustseeingitornot,andthestudents
mighthavedifficultyseeingintuitivelyhowtofactorpolynomials.
Implementation
:
DoNow/Exploration:
PartI:Factorthefollowing:
16
27
2
x
2
4x
+2x
2
2x
3x+20
2
x+8x+16
2
9x
1
PartII:Multiplythefollowing:
(x + y )(x2 xy + y 2 )
(a b)(a2 + ab + b2 )
Minilesson:
Studentstakenotesonthefollowing:
Factoringmeanslookingatanumberorpolynomialandfiguringoutwhatvariablesor
numbersweremultipliedtoproducethatpolynomial.
Strategiesforfactoring
Recognize(asintheDoNow)whenpolynomialsarethesumordifferenceof
twocubes.
Factorbygrouping
Findcommonfactorsin
parts
ofapolynomial,thenusethesetobreak
downthepolynomial.
Factorpolynomialsinquadraticform
2
Factoraquadratic(ax
+bx+c)outofapolynomial
Assessment:Inthefinaltenminutesofclass,studentscreatefactoringproblemsforeach
otherandexchangethem.
HW:Seeattached.
Name:
Date:
Homework:PolynomialArithmetic
V.LessonPlan:VolumeMaximizationProjectWrapUp
Aim
:Wecanwritehigherdegreepolynomialfunctions.
ContentStandards
:HSFBF.A.1A2.S.7
MaterialsandResources
:
Cardboard
Scissors
Tape
Rulers
Calculators
KeyVocabulary
:
Polynomial
RectangularPrism
Motivation
:
Studentswillsolvethechallengeposedtothematthebeginningoftheunit,
demonstratingwhattheyhavelearned.
ProceduralChallenge
:
Studentsmighthaveadifficulttimeputtingtogetherallthepiecesoftheproblemand
requirehintstofinishtheproject.
ConceptualChallenge
:
Studentsmayhavetroubleunderstandinghowtosetupthepolynomial.
Implementation
:
DoNow:
Studentscalculatethevolumeofarectangularprismwithpolynomialsforlength,
width,andheight.
MiniLesson:
Theteacherpresentstheunitquestionagain:givenafixedamountofcardboardthat
wewillcuttomakeabox,howcanwedecidehowmuchtocutoutofthecornersto
maximizethevolume?
Theteachercanscaffoldthecreationofthepolynomialbydemonstratinghowto
calculatethedimensionsofthebox(lengthofcardboard2xwidthofcardboard2x
x).
Theteachercanthendemonstratethattheequationwillneedtobepluggedintoa
calculatorandthatthestudentsmightneedcertainfunctionstofindthelocal
maximum.
Summary:
Studentsshouldgetthesameanswer,sotheycanshareanswersastheclassis
wrappingup.
Assessment:Teacherwillcirculatetoensurethatstudentsaremakingcorrectcalculations.
HW:Buildtheboxwiththemaximumvolume,andcreateaposter/displayexplainingthe
volumemaximizationproblemanditssolution.Rubricattached.
VolumeMaximizationPoster/ModelProjectRubric
Category
Criteria
PointsEarned/Possible
Math
Correctlysolvesthevolume
project
Workisshownandpresented
clearly
Volumefunctionisgraphed
correctlyontheposterwith
localmaximumcircled
/8
Model
Studentturnsinaneatmodel
ofthemaximizedvolumebox
Modelisaccurate
/4
Writing
Writingontheposterisclear
Doesnotcontain
capitalization,punctuationor
spellingerrors
/4
OverallEffort
Studentwasontaskinclass
/4
Posterdemonstratestime,
care,andattentiontodetail
Takentogether,theproject
conveysthatitisthestudents
bestwork
Total
/20