PatternsA Relations Approach
to Algebra
Junior Certificate SyllabusLeaving Certificate Syllabus
© Project Maths Development Team 2011 www.projectmaths.ie 2
© Project Maths Development Team 2011 www.projectmaths.ie 3
Contents
Chapter 1: Relationship to syllabuses, introduction and unit learning outcomes . . . . . . . . . . . . . . . . . 5
Chapter 2: Comparing Linear Functions . . . . . . . . . . . . . . . . 11
Chapter 3: Proportionaland non proportional situations . . . . . . . . . . . . . . . . . . . . . . . 19
Chapter 4: Non constant rates of change-quadratic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
Chapter 5: Exponential growth . . . . . . . . . . . . . . . . . . . . . . . . 41
Chapter 6: Inverse Proportion . . . . . . . . . . . . . . . . . . . . . . . . 53
Chapter 7: Moving from linear to quadratic to cubic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
Chapter 8: Appendix containing some notes and suggested solutions . . . . . . . . . . . . . . . . . . . . 65
© Project Maths Development Team 2011 www.projectmaths.ie 4
5
Chapter 1: Relationship to syllabuses, introduction and
unit learning outcomes
Patterns a Relations Approach to Algebra
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Junior Certificate Mathematics: Draft Syllabus
Topic Description of topic Students learn about
Learning outcomes Students should be able to
4.1 Generatingarithmeticexpressionsfromrepeatingpatterns
Patterns and the rules that govern them; students construct an understanding of a relationship as that which involves a set of inputs, a set of outputs and a correspondence from each input to each output.
• usetablestorepresentarepeating-patternsituation
• generaliseandexplainpatternsandrelationshipsinwordsandnumbers
• writearithmeticexpressionsforparticulartermsinasequence
4.2 Representingsituationswithtables,diagramsandgraphs
Relations derived from some kind of context – familiar, everyday situations, imaginary contexts or arrangements of tiles or blocks. Students look at various patterns and make predictions about what comes next.
• usetables,diagramsandgraphsastoolsforrepresentingandanalysinglinear,quadraticandexponentialpatternsandrelations(exponentialrelationslimitedtodoublingandtripling)
• developandusetheirowngeneralisingstrategiesandideasandconsiderthoseofothers
• presentandinterpretsolutions,explainingandjustifyingmethods,inferencesandreasoning
4.3 Findingformulae
Ways to express a general relationship arising from a pattern or context.
• findtheunderlyingformulawritteninwordsfromwhichthedataisderived(linearrelations)
• findtheunderlyingformulaalgebraicallyfromwhichthedataisderived(linear,quadraticrelations)
4.4 Examiningalgebraicrelationships
Features of a relationship and how these features appear in the different representations.Constant rate of change: linear relationships.Non-constant rate of change: quadratic relationships.Proportional relationships.
• showthatrelationshavefeaturesthatcanberepresentedinavarietyofways
• distinguishthosefeaturesthatareespeciallyusefultoidentifyandpointouthowthosefeaturesappearindifferentrepresentations:intables,graphs,physicalmodels,andformulasexpressedinwords,andalgebraically
• usetherepresentationstoreasonaboutthesituationfromwhichtherelationshipisderivedandcommunicatetheirthinkingtoothers
• recognisethatadistinguishingfeatureofquadraticrelationsisthewaychangevaries
• discussrateofchangeandthey-intercept,considerhowtheserelatetothecontextfromwhichtherelationshipisderived,andidentifyhowtheycanappearinatable,inagraphandinaformula
• decideiftwolinearrelationshaveacommonvalue(decideiftwolinesintersectandwheretheintersectionoccurs)
• investigaterelationsoftheformy =mxandy =mx+c• recogniseproblemsinvolvingdirectproportionand
identifythenecessaryinformationtosolvethem
Patterns a Relations Approach to Algebra
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Leaving Certificate Mathematics: Draft Syllabus
Students learn about
Students working at FL should be able to
In addition, students working at OL should be
able to
In addition, students working at HL should be able to
3.1 Numbersystems
• recogniseirrationalnumbersandappreciatethatR≠Q
• revisittheoperationsofaddition,multiplication,subtractionanddivisioninthefollowingdomains:
• Nofnaturalnumbers
• Zofintegers
• Qofrationalnumbers
• Rofrealnumbers
andrepresentthesenumbersonanumberline
• appreciatethatprocessescangeneratesequencesofnumbersorobjects
• investigatepatternsamongthesesequences
• usepatternstocontinuethesequence
• generaterules/formulaefromthosepatternsdevelopdecimalsasspecialequivalentfractionsstrengtheningtheconnectionbetweenthesenumbersandfractionandplacevalueunderstanding
• consolidatetheirunderstandingoffactors,multiples,primenumbersinN
• expressnumbersintermsoftheirprimefactors
• appreciatetheorderofoperations,includingbrackets
• expressnon-zeropositiverational
numbersintheformax10n,
wheren∈Nand1≤a<10andperformarithmeticoperationsonnumbersinthisform
• generaliseand
explainpatterns
andrelationships
inalgebraicform
• recognisewhether
asequence
isarithmetic,
geometricor
neither
• findthesumto
ntermsofan
arithmeticseries
• verifyandjustifyformulaefromnumberpatterns
• investigategeometricsequencesandseries
• provebyinductionsolveproblemsinvolvingfiniteandinfinitegeometricseriesincludingapplicationssuchasrecurringdecimalsandfinancialapplications,e.g.derivingtheformulaforamortgagerepayment
• derivetheformulaforthesumtoinfinityofgeometricseriesbyconsideringthelimitofasequenceofpartialsums
Patterns a Relations Approach to Algebra
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A functions based approach to algebra
Theactivitiesbelowfocusontheimportantrolethatfunctionsplay
inalgebraandcharacterisetheopinionthatalgebraicthinkingisthe
capacitytorepresentquantitativesituationssothatrelationsamong
variablesbecomeapparent.Emphasisingcommoncharacteristics
helpsustothinkoffunctionsasobjectsofstudyinandofthemselves
andnotjustasrulesthattransforminputsintooutputs.Students
examinefunctionsderivedfromsomekindofcontexte.g.familiar
everydaysituations,imaginarycontextsorarrangementsoftilesor
blocks.Thefunctions-basedapproachenablesstudentstohaveadeep
understandinginwhichtheycaneasilymanoeuvrebetweenequations,
graphsandtables.Itpromotesinquiryandbuildsonthelearner’sprior
knowledgeofmathematicalideas.
Strand 4: Algebra
Thisstrandbuildsontherelations-basedapproachofJunior
Certificatewherethefivemainobjectiveswere
1. tomakesenseoflettersymbolsfornumericquantities
2. toemphasiserelationshipbasedalgebra
3. toconnectgraphicalandsymbolicrepresentationsofalgebraicconcepts
4. tousereallifeproblemsasvehiclestomotivatetheuseofalgebraandalgebraicthinking
5. touseappropriategraphingtechnologies(graphingcalculators,computersoftware)throughoutthestrandactivities.
Studentsbuildontheirproficiencyinmovingamongequations,
tablesandgraphsandbecomemoreadeptatsolvingrealworld
problems.
JuniorCertificateDraftSyllabuspage26,LeavingCertificateDraftSyllabuspage28.NationalCouncilforCurriculumandAssessment(2011),MathematicsSyllabus,JuniorCertificate[online]available:http://www.ncca.ie/en/Curriculum_and_Assessment/Post-Primary_Education/Project_Maths/Syllabuses_and_Assessment/Junior_Cert_Maths_syllabus_for_examination_in_2014.pdf
NationalCouncilforCurriculumandAssessment(2011),MathematicsSyllabus,LeavingCertificate[online]available:http://www.ncca.ie/en/Curriculum_and_Assessment/Post-Primary_Education/Project_Maths/Syllabuses_and_Assessment/Leaving_Cert_Maths_syllabus_for_examination_in_2013.pdf
AccessedSeptember2011.
Patterns a Relations Approach to Algebra
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In this unit:1. Studentslinkbetweenwords,tables,graphsandformulaewhen
describingafunctiongiveninareallifecontext.
2. Studentscometounderstandfunctionsasasetofinputsandasetofoutputsandacorrespondencefromeachinputtoeachoutput.
3. Studentsidentifywhatisvaryingandwhatisconstantforafunction.
4. Studentswillunderstandthetermsvariableandconstant,dependentandindependentvariableandbeabletoidentifytheminreallifecontexts.
5. Studentswillbuildfunctionalrulesfromrecursiveformulae.
6. Studentsfocusonlinearfunctionsfirst,identifyingaconstantrateofchangeasthatwhichcharacterisesalinearfunction.Studentswillidentifythisasconstantchangeinthey -valuesforconsecutivex-values.Theywillidentifythischangeinyperunitchangeinxastheslopeofthestraightlinegraph.
7. Studentswillseethatlinearfunctionscanhavedifferent‘startingvalues’i.e.valueofywhenx=0.Theywillidentifythisasthey–intercept.
8. Studentswillseehowtheslopeandyinterceptrelatetothecontextfromwhichthefunctionisderived.
9. Studentswillexploreavarietyoffeaturesofafunctionandexaminehowthesefeaturesappearinthedifferentrepresentations–
a. Isthefunctionincreasing,decreasing,orstayingthesame?Ifthefunctionisincreasingtheslopeispositive,ifdecreasingtheslopeisnegativeandifconstantithaszeroslope.
b.Isthefunctionincreasing/decreasingatasteadyrateoristherateofchangevarying?Ifitisincreasingataconstantratethisischaracteristicofalinearfunction.
10.Studentswillinvestigatereallifecontextswhichleadtoquadraticfunctionswheretherateofchangeisnotconstantbuttherateofchangeoftherateofchangeisconstant.Studentswillidentifythisasconstantchangeofthechangesintheyvaluesforconsecutivexvalues
11.Studentswillexamineinstancesofproportionalandnonproportionalrelationshipsforlinearfunctions.Studentswillknowthecharacteristicsofaproportionalrelationshipwhenitisgraphed–itislinearandpassesthroughtheorigin(no‘start-up’value,i.e.noy-intercept)andthe‘doublingstrategy‘[ifxisdoubled(orincreasedbyanymultiple)thenyisdoubled(orincreasedbythesamemultiple)].
Patterns a Relations Approach to Algebra
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12.Studentsinvestigatecontextswhichgiverisetoquadraticfunctionsthroughtheuseofwords,tables,graphsandformulae.
13.Whendealingwithgraphsofquadraticfunctionsstudentsshouldcontrastthequadraticwiththelinearfunctionsasfollows:
• thegraphsarenonlinear
• thegraphsarecurved
• thechangesarenotconstant
• thechangeofthechangesisconstant
• thehighestpoweroftheindependentvariableis2
14.Studentsshouldinvestigatesituationsinvolvingexponentialfunctionsusingwords,tables,graphsandformulaeandunderstandthatexponentialfunctionsareexpressedinconstantratiosbetweensuccessiveoutputs.
15.Studentsshouldseeapplicationsofexponentialfunctionsintheireverydaylivesandappreciatetherapidrateofgrowthordecayshowninexponentialfunctions.
16.Studentsshoulddiscoverthatforcubicfunctionsthe‘thirdchanges’areconstant.
17.Studentswillunderstandthatx3increasesfasterthanx2andtheimplicationsofthisinnatureandindesign.
18.Studentswillinvestigatesituationswhereinverseproportionappliesandcontrastthesewithlinear,quadratic,andexponentialrelationships.Studentswillseethattheproductofthevariablesisaconstantinthesesituations.
19.Studentswillinterpretgraphsincasesof‘relationswithoutformulae’.
11
Chapter 2: Comparing Linear Functions
Patterns a Relations Approach to Algebra
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In this activity, students
• Identifypatternsanddescribedifferentsituationsusingtablesand
graphs,wordsandformulae
• Identifyindependentanddependentvariablesandconstants
• Identify‘startamount’inthetableandformulaandasthe
y-interceptofthegraph
• Identifytherateofchangeofthedependentvariableinthetable,
graph(astheslope)andformula
• Identifylinearrelationshipsashavingaconstantchangebetween
successiveyvalues(outputs)
• Knowthatparallellineshavethesameslope(samerateofchangeof
ywithrespecttox)
• Connectincreasingfunctionswithpositiveslope,decreasing
functionswithnegativeslopeandconstantfunctionswithslopeof
zero.(Itisimportantthatstudentsrealisethattables,graphsand
algebraicformulaearejustdifferentwaystorepresentasituation.)
Thefollowingdatashowsthemeasuredheightsof4different
sunflowersonaparticulardayandtheamounttheygrewin
centimetreseachdayafterwards.
Sunflower a b c d
Startheight/cm 3 6 6 8
Growthperday/cm 2 2 3 2
Student Activity – comparing linear functions
(Thisactivitymaybeintroducedwithstudentsatany
stage-JCOL,LCFL,LCOL)GRAPHPAPERREQUIRED
Theactivitiesgivestudentsanopportunitytolinkthisstrandwith
co-ordinategeometryandcanbetakenbeforeorafterit.Through
thestudyofpatternsofchange,theycangetamoremeaningful
understandingforsuchcharacteristicsasslopes,intercepts,
equationofaline,etc.
Patterns a Relations Approach to Algebra
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Investigatehowtheheightofeachsunflowerchangesovertime.Is
thereapatterntothegrowth?Ifyoufindapatterncanyouuseitto
predicthowtallthesunflowerswillbeinsay30days,orinanynumber
ofdays?Whatdifferentrepresentationscouldyouusetohelpyour
investigation?
Representthisinformationinatablesimilartotheonebelow,showing
theheightofthesunflowersoverasixdayperiodforeachsituation(a),
(b),(c)and(d).
Timeindays Heightincm Change
0
1. Whereisthe‘startheight’foreachplantseeninthetables?
2. Whereistheamounttheplantgrowsbyeachdayseeninthetables?
3. Whatdoyounoticeaboutthechangebetweensuccessiveoutputsforeachthetables?(4 and 5 may be omitted as the main aim of the activity is for students to recognise the features of a linear relationship in a table and in a graph.)
4. Foreachofthesituationsandtables(a),(b),(c),and(d)identify2valueswhichstaythesameand2valueswhichvary.
SituationandTable Varying Stayingthesame
a
b
c
d
Wecallthevalueswhichvary‘variables’andthevalueswhichstaythe
same‘constants’.
Wehaveidentified2variables.Whichvariabledependsonwhich?
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Wecallonevariablethedependentvariableandwecalltheother
variabletheindependentvariable.
5. Whichistheindependentvariableandwhichisthedependentvariableineachcaseabove?
Plotting graphs of the situations represented in table 1:
Usinggraphpaper,onthefirstsetofaxes,graphsituationsaandb.
Whichvariabledoyouthinkshouldgoonwhichaxis?Discuss.
Weusuallyplottheindependentvariableonthex-axisandthe
dependentvariableonthey-axis.
Wecanthinkofthexvaluesastheinputsandtheyvaluesasthe
outputs.
1. Whatobservationscanyoumakeaboutgraphsofaandb?
2. Howiseachobservationseeninthesituation?
3. Howiseachobservationseeninthetable?
4. Arethesunflowersaandbeverthesameheight?Explain.
y - intercept and slope5. Wherearethestartheightsseeninthegraphs?
Wecallthesevaluesthey -interceptsofthegraphs.
6. Wherearethegrowthratesseeninthegraphs?(Asthedayincreasesby1whathappenstotheheight?)Wecallthisvaluetheslopeofthegraph–thechangeinyperunitchangeinx.
7. Ifthenumberofdaysincreasesby3,whatdoestheheightincreaseby?Whatistheaverageincreaseinheightperday?Isthisthesameastheslope?
8. Canyoucomeupwithawayofcalculatingslope?
9. Goingbacktothesunflowerexample,whatdoestheslopecorrespondto?
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Formulae in words and symbols: The following can be applied to Sunflowers (a) and (b) JC Syllabus 4 .3 – HL in bold
4.3 FindingFormulae
Ways to express a general relationship arising from a pattern or context
• findingtheunderlyingformulawritteninwordsfromwhichthedataisderived(linearrelations)
• findtheunderlyingformulaalgebraicallyfromwhichthedataisderived(linear,quadraticrelations)
Usingthetableandthegraph(whicharejustdifferentrepresentations
ofthesamesituation):
1. Describeinwordstheheightoftheplantin(a)onaparticularday(e.g.day4)intermsofits‘startheight’anditsgrowthrateperday.
2. Describeinwordstheheighthoftheplantin(a)onanydayd,intermsofits‘startheight’andtheamountitgrowsbyeachday.
3. Writeaformulafortheheighthoftheplantin(a)onanydayd,intermsofits‘startheight’andtheamountitgrowsbyeachday.
4. Identifythevariables(dependentandindependent)andtheconstantsintheformula.
Formulaeinwordsandsymbols:Sunflower(b)
5. Whereisthe‘startheight’seenintheformulaeforaandb?Whereistheamountgrowneachdayseenintheformulaforaandb?
6. Identifywhereeachvariableandconstantintheformulaeappearsinthegraphs.
Possible Homework or class group work:Investigatethepattern
ofsunflowergrowthinsituationsbandcusingtables,graphsand
formulaeasyoudidforsituationsaandb.
Next class:Investigatethepatternofsunflowergrowthinsituationsc
anddusingtables,graphsandformulaeasyoudidforsituationsaand
bandbandc.
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Intersecting graphs c and d
1. Afterhowmanydaysarethesunflowersinsituations(c)and(d)thesameheight?
2. Whydoyouthinksunflower(c)overtakessunflower(d)eventhoughsunflower(d)startsoutwithagreater‘startheight’thansunflower(c)?
Increasing functions and positive slopes
1. Asthenumberofdaysincreasedwhathappenedtotheheightofthesunflowerineachsituation?
2. Whatsignhastheslopeofthegraphineachsituation?
Line with constant slope
Monicadecidedtoplantaplasticsunflowerwhoseheightwas30cm.
Representthisinformationinatable,agraphandanalgebraicformula
forday0today5.
1. Whatshapeisthegraph?
2. Contrastthesituationwiththeprevioussunflowersituations.Whyisthegraphthisshape?
3. Whereisthe30cminthetable?Whereisthe30cminthegraph?
4. Howmuchdoesthesunflowergroweachday?Whereisthatinthetable?Whereisitinthegraph?
5. Asthenumberofdaysincreasedwhathappenedtotheheightofthesunflower?
6. Whatistheslopeofthegraph?
7. Whatformulawoulddescribetheheightofthesunflower?
8. Whereisthe30cmintheformula?
9. Whatarethelimitationstothemodelpresentedforsunflowergrowth?Itisimportantforstudentstorealiseearlyonthatitisimpossibletomirroraccuratelymanyreallifesituationswithamodelandtheymustalwaysrecognisethelimitationsofwhatevermodeltheyuse.
Line with a negative slope:
Youhave€40inyourmoneyboxonSunday.Youspend€5onyour
luncheachdayfor5consecutivedays(takeMondayasday1).
Asthevariableonthex -axisincreaseswhathappenstothevariable
onthey -axis?
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Issunflowergrowtharealisticsituationfornegativeslope?
Canyouthinkofotherreallifesituationswhichwouldgiverisetolinear
graphswithnegativeslopes?
Summary of this Student Activity:
Theheightofeachplantdependsonhowolditis.Wehaveaninput
(timeindays)andanoutput(heightoftheplant)whichdependson
theinput.Wesaythattheoutput(inthiscaseheightoftheplant)isthe
dependentvariableandtheinput(time)istheindependentvariable.
Whenweusexandy,xistheindependentvariableandyisthe
dependentvariable.
Weseethata,banddhavedifferentstartingheightsandbandchave
thesamestartingheights.Thestartingheightappearsinthetableas
theheightonday0andinthegraphastheyvaluewhenthexvalue
(numberofdays)iszero.Intheformulaitappearsonitsownasa
constant.
Weseethatthereisaconstantrateofchangeeachdayfortheplant’s
heightineachsituation.Inthetableitappearsasaconstantchange
forsuccessiveyoutputswhichcharacteriseslinearrelationships.Inthe
graphitappearsastheslopeoftheline–constantchangeinyforeach
unitchangeinx.Intheformulaforaparticularsituationthisconstant
rateofchangeismultipliedbytheindependentvariable,inthiscase
thenumberofdays.Whentwoplantshavethesamerateofchangein
theirgrowthperdaytheirgraphsareparallel,i.e.theyhavethesame
slope.
Whentwoplantshavedifferentgrowthratesperdaythegraphswill
intersectandtheplantwiththehighergrowthratewilleventually
overtaketheplantwiththelowergrowthrateeveniftheonewith
lowergrowthratestartsatagreaterheight.
Student Activity – draw a graph from the situation without a table of values JCOL
Asinthepreviousactivities,thereareobviouslinkstoco-ordinate
geometryandtofunctions/graphs.
Itisimportantthatstudentsareconfidentatsketching/drawinglinear
graphswithouttheneedtoformallysetoutatable,plotpointsand
thendrawthegraph.
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Eventually,theyshouldbeabletousethisknowledge/skillwhen
presentedwithasimilarsetofconditions.
(Discussionpoint:Isthelinearrelationshipacontinuousoneorjust
discretevalues?Isjoiningthepointsalwaysappropriate?)
Draw graphs of linear functions without making a table of
values, by comparing them to the graphs you have already
made .
1. OnthesameaxesyouusedforStudent Activity 1(situationsaandb)sketchagraphofthefollowingsunflowergrowthpattern:startingheight5cm,grows2cmeachday.Comparethisgraphtothegraphsofsituationsaandb?Explainthecomparison.
2. OnthesameaxesyouusedforStudent Activity 1(situationsbandc)sketchagraphofthefollowingsunflowergrowthpattern:startingheight6cm,grows4cmeachday.Comparethisgraphtothegraphsofsituationsbandc?Explainthecomparison.
3. OnthesameaxesyouusedforStudent Activity 1(situationscandd)sketchagraphofthefollowingsunflowergrowthpattern:startingheight6cm,grows2cmeachdayComparethisgraphtothegraphsofsituationscandd?Explainthecomparison.
Situations a and b
a: y = 2x + 3b: y = 2x + 6
25
20
15
10
5
0 1 2 3 4 5 6 7 8 9
Situations b and c
b: y = 2x + 6c: y = 3x + 6
25
20
15
10
5
0 1 2 3 4 5 6 7 8 9
Situations c and d
c: y = 2x + 8d: y = 3x + 6
25
20
15
10
5
0 1 2 3 4 5 6 7 8 9
19
Chapter 3: Proportionaland non proportional
situations
Patterns a Relations Approach to Algebra
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Example 1:Thefollowingtableshowsapatternofgrowthofa
fastgrowingplant.
Timeindays Heightincm
0 2
1 7
2 12
3 17
4 22
5 27
6 32
1. Whatinformationdoesthetablegiveaboutthepatternofgrowth?
2. Canyoupredicttheheightoftheplantafter12days?Canyoudothisindifferentways?
3. Ifthenumberofdaysisdoubledwilltheheightalsodouble?
4. After4daysistheheightdoublewhatitwasafter2days?Explainyouranswer.After6daysistheheightdoublewhatitwasafter3days?Explainyouranswer.Didthe‘doublingstrategy’work?
5. Whatistheformulafortheheightoftheplantintermsofthenumberofdayselapsedandthestartupvalue(usewordsfirst,thenusesymbols)?
6. Canyoupredictfromtheformulawhetherornotthe‘doublingstrategy’willwork?
Whenthe‘doublingstrategy’doesnotworkwearedealingwitha
nonproportionalsituation.Whenthe‘doublingstrategy’worksitisa
proportionalsituation.Thevariablesareproportionaltoeachother.
Student Activity: Proportional and non proportional situations JCOL
Inthefollowingexamplesstudentsexamineinstancesof
proportionalandnonproportionalrelationshipsforlinear
functions.Studentsarriveataknowledgeofthecharacteristics
ofaproportionalrelationshipwhenitisgraphed–itislinear
andpassesthroughtheorigin,hasno‘start-up’value,i.e.no
y-interceptandthe‘doublingstrategy‘works.Inotherwords,if
xisdoubled(orincreasedbyanymultiple)thenyisdoubled(or
increasedbythesamemultiple).
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Example 2:Thistablerepresentsacontextwhereeachfloorina
buildinghas5rooms.
Numberoffloors Totalnumberofrooms
0 0
1 5
2 10
3 15
4 20
5 25
6 30
How many rooms in total would be in a building which has 12
floors?
1. Find3differentwaystosolvethisproblem.
2. Ifyoudoublethenumberoffloorswhathappenstothetotalnumberofroomsinthebuilding?
3. Ifyoutreblethenumberoffloorswhathappenstothetotalnumberofroomsinthebuilding?
4. Doesthe‘doublingstrategy’work?
5. Doesthe‘treblingstrategy’work?
6. Arethevariablesproportionaltoeachother?
7. Whatistheformularelatingthenumberofroomstothenumberoffloors(usewordsfirst,thenusesymbolsforHL)?
8. Canyoupredictfromtheformula,whetherornotthe‘doublingstrategy’willwork?
Plot graphs for the above 2 tables from example 1 and example 2
9. Whatisthesameaboutthetwographs?Whatisdifferentaboutthetwographs?
10.Forwhichgraphwillthe‘doublingstrategy’work?Explain.
11.Whichgraphrepresentsasituationwherethevariablesareproportionaltoeachother?
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Example 3:InHomeEconomicsstudentsmade
gingerbreadmen.Theyused2raisinsfortheeyesand1
raisinforthenoseoneachone.Howmanyraisinsdidthey
usefor5gingerbreadmen?
1. Representthisinformationinatableshowingthenumberofraisinsusedfor0to5gingerbreadmen.
2. Statethenumberofraisinsusedinwords.
3. Statethenumberofraisinsusedinsymbols.
4. Plotagraphtomodelthesituation.
5. Doesthe‘doublingstrategy’workhere?Explainfromthegraph,tableandsymbols.
Example 4:Growing‘worm’
Thediagramsshowanewborn,1dayold,and2dayold‘worm’made
fromtriangles.
1. Drawupatableforthenumberoftrianglesusedfora0to5dayoldworm.
2. Whatistherelationshipbetweentheageofthewormandthenumberoftrianglesused(usewordsfirstandthensymbols)?
3. Howmanytrianglesina4dayoldworm?
4. Howmanytrianglesinan8dayoldworm?Doesthe‘doublingstrategy’work?
5. Arethevariablesproportionaltoeachother?Explainusingthetableandformula.
6. Plotagraphtomodelthesituation.
7. Howdoesthegraphshowwhetherornotthevariablesareproportionaltoeach
Patterns a Relations Approach to Algebra
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Example 5:Howmanybluetilesareneededfornyellowtiles?
(JCHLandmaybeappropriateforOLalso.)
Thisactivityhasgreatpotentialandcanbeusedwithstudentsatany
stage.Studentsmustbeallowedtomakethepatternsthemselves.
Thereisagreatopportunitytoreinforcetheconceptofequalitywhen
studentsgeneralisetherelationshipdifferently.Studentscanmakethe
patternsthemselveswithblocksordrawthem.Byphysicallymakingthe
patternstudentswillstarttoseetherelationships.
1. Representinatablethenumberofbluetilesneededfor0to5yellowtiles.
2. Howmanybluetilesareneededfor4yellowtiles?(accessibleforOL)
3. Whatistherelationshipbetweenthenumberofbluetilesandthenumberofyellowtiles?Expressthisinwordsinitially.(accessibleforOL)
4. Considerhowotherstudentsexpressthisrelationship.Arealltheexpressionsequal?Howcanyouverifythis?
5. Doesthedoublingstrategyworkhere?
6. Ifnotwhynot?
7. Drawagraphshowingtherelationshipbetweenthenumberofyellowandthenumberofbluetiles.Canyouexplainfromthetableandthegraphwhetherornotthe‘doublingstrategy’works?
8. Arethevariablesproportionaltoeachother?Explain.
Class Discussion on proportional and non proportional situations
0 0
1 5
2 10
3 15
4 20
5 25
6 30
0 6
1 8
2 10
3 12
4 14
5 16
6 18
0 2
1 7
2 12
3 17
4 22
5 27
6 32
0 0
1 3
2 6
3 9
4 12
5 15
6 18
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1. Whenyoulookatthesetableshowcanyoutellifthevariablesareinproportion?
2. Modelthe4situationsrepresentedinthetablesbydrawinggraphsforeachtable.
3. Whenyoulookatlineargraphswherethevariablesareproportionalwhatdoyounoticeaboutthegraphs?
4. Whenyoulookatlineargraphswherethevariablesarenotproportionalwhatdoyounoticeaboutthegraphs?
5. Writeaformularelatingthevariablesforeachofthesetables.
6. Intheformulaethereisanamountyoumultiplybyandanamountyouadd.Howarethesequantitiesrepresentedinthegraphs?
7. Giveareallifecontextforeachofthesetables.
Next activity on graph matching:
JCHL:Matchingallthedifferentrepresentationssimultaneously
JCOL:Matchingrepresentationsinappropriatepairs
Patterns a Relations Approach to Algebra
© Project Maths Development Team 2011 www.projectmaths.ie 25
Match up the stories to the tables, graphs, and formulae and fill in
missing parts .
(Thiscouldformthebasisforgrouporclassdiscussion.Notetheseandother
examplesarereferredtoonthenextpage.Labeltheaxesofthegraph.)
Story Table Graph Formula Linear/ non linear,
proportional/non proportional?
Justify
Changing euro to cent.
No. of
cars
€/w
1 450
2 500
3 550
4 600
5 650
Y = 45x + 20
John works for €12 per hour. The graph and table show how much money he earns for the hours he has worked.
y=100/x
Salesman gets paid €400 per week plus an additional €50 for every car sold.
m/kg
t/mins
1 65
2 110
3 155
4 200
5 245
6 290
y=100x
A cookbook recommends 45 mins per kg to cook a turkey plus an additional 20 minutes.
y=12x Linear and proportional because it’s a line (constant change)through (0,0)
Henry has a winning ticket for a lottery for a prize of €100. The amount he receives depends on how many others have winning tickets.
h A/€
1 12
2 24
3 36
4 48
5 60
Patterns a Relations Approach to Algebra
© Project Maths Development Team 2011 www.projectmaths.ie 26
Story Table Graph Formula Linear/ non linear,
proportional/non proportional?
Justify
Henry is one of the winners of a prize of €100. The amount he receives depends on how many others have winning tickets.
n €
1 100
2 50
3 33.33
4 25
5 20
6 16.67
y = 100/xCan you write this formula another way?
Not linear.(This relationship represents an inverse proportion. This is dealt with later on in the document.)For each couple (x,y) in this situation what is the value of the product xy?
A cookbook recommends 45 minutes per kg to cook a turkey plus and additional 20 minutes.
m/k g
t/k mins
1 65
2 110
3 155
4 200
5 245
6 290
y = 20+45x Linear but not proportionalThe graph is a line but it does not pass through (0, 0)
John works for €12 per hour. The graph and table show how much money he earns for the hours he has worked.
h €
0 0
1 12
2 24
3 36
4 48
5 60
y = 12x Relationship is linear and proportional as its graph is a line through (0,0).
Changing euro to cent.
euro cent
0 0
1 100
2 200
3 300
4 400
5 500
y = 100x Relationship is linear and proportional because its graph is a line through (0,0). The table shows constant change for successive outputs where the input x is a natural number.
Patterns a Relations Approach to Algebra
© Project Maths Development Team 2011 www.projectmaths.ie 27
Story Table Graph Formula Linear/ non linear,
proportional/non proportional?
Justify
Fixed wage + commission
Cars sold
€
0 400
1 450
2 500
3 550
4 600
5 650
y = 400+ 50x
Linear but not proportional.The graph is a line but it does not pass through (0,0).
Patterns a Relations Approach to Algebra
© Project Maths Development Team 2011 www.projectmaths.ie 28
Examples of proportional situations:
• Changingeurotocent–ifyouhavenoeuroyouhavenocents,if
youdoublethenumberofeuroyoudoublethenumberofcent,if
youtreblethenumberofeuroyoutreblethenumberofcent
Numberofcent=100xnumberofeuro.
• Changingmilestokms.
• Ifyougetpaidbythehourthemorehoursyouworkthemore
moneyyougetinyourwages.
• Thecircumferenceofacircleisproportionaltoitsdiameter(alsoto
itsradius).
• Ifyoudrive(walk,runorcycle)ataconstantspeedthedistance
travelledisproportionaltothetimetaken.
Give2otherexamplesofproportionalsituations.
Examples of non proportional:
• Therecommendedtimeittakestocookaturkeyis45minutes
perkilogramplusanadditional20minutes:totalcookingtimein
minutes=45xnumberofkg+20minutes.
• Youareacarsalesmanandyougetpaidafixedwageof€400per
week+€50foreverycarsold.
• WhenyoupayyourESBbillyoupayafixedstandingchargeplusa
fixedamountperunitofelectricityused.
• Thecostsforaretailersellingshoes,forinstance,arethecostsof
shoespurchasedplusfixedoverheadcosts(rent,heating,wages,
etc.)
Give2otherexamplesofnonproportionalsituations
What is the same and what is different about the following relationships?(i) Toconvertmetrestoinchestheapproximateconversionfactoris1
metreequals39.37inches.
(ii) ToconverttemperaturesfromdegreesCelsiustodegreesFahrenheitmultiplythenumberofdegreesCelsiusby1.8(9/5)andadd32.
29
Chapter 4: Non constantrates of change-
quadratic functions
Patterns a Relations Approach to Algebra
© Project Maths Development Team 2011 www.projectmaths.ie 30
Example 1: Growing Squares Pattern . Draw the next two patterns of growing squares .
Wewishtoinvestigatehowthenumberoftilesineachpatternis
relatedtothesidelengthofeachsquare.Identifytheindependentand
dependentvariables.
Student Activity: Non constant rates of change - quadratic functions JC OL
Wesawthatforlinearrelationships,whichgivestraightline
graphs,therewasaconstantrateofchangeasseenbyconstant
changeintheoutputsforconsecutiveinputvalues.Areallratesof
changeconstant?Wewilllookforpatternsindifferentsituations
andinvestigatetheirratesofchange.Studentswillinvestigate
contextswhichgiverisetoquadraticfunctionsthroughtheuseof
tables,graphsandformulae.
Whendealingwithgraphsofquadraticfunctionsstudentsshould
contrastthequadraticwiththelinearfunctionsasfollows:
• thegraphsarenon-linear
• thegraphsarecurved
• therateofchangeisnotconstant
• therateofchangeoftherateofchangeisconstanti.e.
constantchangeofthechanges
• thehighestpoweroftheindependentvariableis2.
Patterns a Relations Approach to Algebra
© Project Maths Development Team 2011 www.projectmaths.ie 31
Complete the table
Side length of each square Number of tiles to complete each square
1 1
2 4
3 9
4
5
6
7
8
9
10
1. Howisthenumberoftilesusedtomakeeachsquarerelatedtothesidelengthofthesquare?Writetheanswerinwordsandsymbols.Whatdifferencedoyounoticeaboutthisformulaandtheformulaeforlinearrelationships?
2. Lookingatthetable,istherelationshiplinear?Explainyouranswer.
3. Predictwhatagraphofthissituationwilllooklike.Willitbeastraightline?Explainyouranswer.Plotagraphtocheckyourprediction.
4. Whatshapeisthegraph?Istherateofchangeofthenumberoftilesconstantasthesidelengthofeachsquareincreases?Explainusingboththetableandgraph.
5. Whyisthegraphnotastraightline?Howcanyourecognise
whetherornotagraphwillbeastraightlineusingatable?
Note:Studentsshouldcomeupwiththelasttwocolumnsinthe
tablebelowthemselveswhenlookingforapatternintheoutputsand
throughquestioning.Itisnotenvisagedthattheywillbelabelledas
belowinitiallyforthem.
Side length Number of tiles Change Change of changes
1
23
45
67
89
1
49
3
52
6. Wesawthatthechangesinthetablewerenotconstant.Isthereapatterntothem?Calculatethechangeofthechanges.
Patterns a Relations Approach to Algebra
© Project Maths Development Team 2011 www.projectmaths.ie 32
7. Whenthechangeofthechangesisconstantwecallthisrelationshipbetweenvariablesaquadraticrelationship.
8. Give3characteristicsofaquadraticrelationshipfromthe‘growingsquares’example.
Example 2: Growing Rectangles JCOL
Completethenexttwopatternsinthissequenceofrectangles.
Lookatthepatternofgrowingrectangles.Makeatableforthenumber
oftilesineachrectangleforrectanglesofheightsfrom1to10.Make
observationsaboutthevaluesinthetable.
Whatwouldagraphforthistablelooklike?Willitbelinear?Howdo
youknow?Makeagraphtocheckyourprediction.
Complete the table
Height of rectangle Width of rectangle Number of tiles to make each rectangle
1 2 2
2 3
3
4
5
6
7
h
1. Canyousee1groupof2(or2groupsof1)inthefirstdiagram,2groupsof3(or3groupsof2)intheseconddiagram,etc?
2. Thelastrowofthetablegivesaformulaforthenumberoftilesinarectangleofheighth.Writedowntheformulaintermsoftheheightofeachrectangle.Callthisformula(i)Checkthattheformulaworksforvaluesalreadycalculated.
3. Willagraphforthistablebelinear?Explainusingthetablebelow.Isthechangeofthechangesconstant?Whattypeofrelationshipisindicatedbythepatternofthechanges?ExplainWhatdoyounoticeaboutthechangeofthechanges?
Patterns a Relations Approach to Algebra
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4. Predictwhatagraphofthesituationwilllooklike.Plotagraphtocheckyourprediction.
5. Whatshapeisthegraph?
6. Doesthisrelationshiphavethe3characteristicsidentifiedinthelastexampleforaquadraticrelationship?Explain.
Relate growing rectangles to growing squares JC OL (with
scaffolding)
Makethebiggestsquarepossiblefromeachofthe5rectanglesyou
havepreviouslydrawn.Shadetheremainingareaineachrectangle.
Drawthenexttworectanglesinthepatternbelow.
Completethetableforthe
numberoftilesineach
rectangleintermsofthe
heightofeachrectangle.
Height Number of tiles Area of the rectangle = area of square + area of a rectangle
1 2 12+1
2 6
3 12
4
5
6
7
h
1. Whatistheformulaforthenumberoftilesinarectangleofheighthintheabovesequence,usingthepatternseenintherighthandcolumnwithspecificnumbers?Callthisformula(ii)
2. Showthatthetwoformulae(i)and(ii)derivedforthenumberoftilesintheeachrectangleareequivalentexpressionsusing(a)substitutionand(b)thedistributivelaw.
Height of rectangle Number of tiles Change Change of changes
1
23
45
67
h
2
64
Patterns a Relations Approach to Algebra
© Project Maths Development Team 2011 www.projectmaths.ie 34
Example 3:StaircaseTowersJCHL
Lookatthestaircasesbelow.Makeatablerepresentingtherelationship
betweenthetotalnumberoftilesandthenumberoftowers.Make
observationsaboutthevaluesinthetable.Whatwouldagraphlook
like?Woulditbelinear?Howdoyouknow?Makeagraphtocheckyour
prediction.
Allowstudentstocontinuethepatternthemselvesandmaketheirown
observations.Thefollowingquestionscouldbeusedaspromptquestions
ifnecessary.
1. Howmanytilesdoyouaddontomakestaircase2fromstaircase1?
2. Howmanytilesdoyouaddontomakestaircase3fromstaircase2?
3. Howmanytilesdoyouaddontomakestaircase4fromstaircase3?
4. Completethetablebelow
Numberof
towers
Numberoftiles
1 1
2 3
3 6
4
5
6
7
8
9
10
Patterns a Relations Approach to Algebra
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5. Willagraphforthistablebelinear?Explain,usingthetablebelow.
Side length Number of tiles Change Change of changes
1
23
45
67
1
32
63
1
6. Arethechangesconstant?Arethechangesofthechangesconstant?
7. Predictwhatagraphofthesituationwilllooklike.Plotagraphtocheckyourprediction.
8. Whatshapeisthegraph?
We need to develop a formula for the number of tiles required for the nth tower .
Comparethepatternforthe‘growingrectangles’withthatofthe
staircasetowers.
1. Whatistheformulaforthenumberoftilesinthenthrectangleintermsofn?
2. Hencewhatistheformulaforthenumberoftilesinthenthstaircaseintermsofn?
Patterns a Relations Approach to Algebra
© Project Maths Development Team 2011 www.projectmaths.ie 36
3. Thestaircasepatternshowingthenumbers1,3,6,10,...couldalsobeshownusingtheabovepattern.Whatdoyouthinkthesenumbersarecalled?(Lookattheshapesformed).
Contrast the rates of change for ‘growing rectangles’ and ‘staircase towers’ – use tables and graphs JC HL
Growing rectangles
Side length Number of tiles Change Change of changes
1
23
45
67
2
64
126
2
8
910
20
30
810
2
Staircase TowersSide length Number of tiles Change Change of changes
1
23
45
67
1
32
63
1
Comparehowthevaluesofyincreaseforthesamechangeinxin
‘growingrectangles’and‘staircasetowers’.Focusonhowthenumbers
changeinthefirsttwocolumns
1. Whatisthesameaboutthegraphsandthetablesfor‘growingrectangles’and‘staircasetowers’?
2. Whatisdifferentaboutthegraphsandthetablesfor‘growingrectangles’and‘staircasetowers’?
Patterns a Relations Approach to Algebra
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3. Howisthenon-linearnatureofthegraphrelatedtothetablesandthecontext?
4. Whatdothechangesincreasebyfor‘growingrectangles’?
5. Whatdothechangesincreasebyfor‘staircasetowers’?
6. Howisthedifferenceintheanswerstothelasttwoquestionsshowninthegraphsforeachsituation?
Whichsituationshowsthegreatestchangeofthechanges?
Example 4:ThegrowthofafantasycreaturecalledWalkasaurusJCHL
ThefollowingtableshowshowWalkasaurus’sheightchangeswith
time.
Age (years) Height (cm)
0 1
1 2
2 4
3 7
4 11
5 16
6 22
1. Willagraphforthistablebelinear?Explainusingatable.
Note:Thelasttwocolumnsofthetableneednotbegivento
students.Studentsshoulddrawontheskillstheyhavelearnedin
previousactivitiesandshouldnotneedtheseprompts.Theaimof
thisapproachistoaskquestionsthatgivestudentstheopportunity
tothinkandinvestigateinordertomakesenseofthefeaturesofa
quadraticrelationship.
Age (years) Height (cm) Change Change of changes
0
12
34
56
1
21
4
711
1622
21
2. Predictwhatagraphofthesituationwilllooklike.Plotagraphtocheckyourprediction.
Patterns a Relations Approach to Algebra
© Project Maths Development Team 2011 www.projectmaths.ie 38
3. Whatshapeisthegraph?ComparethepatternofyvaluesforWalkasaurus’sheightwiththepatternofyvaluesforthestaircasetowers.
4. Howisthepatternsimilar?
5. Howisthepatterndifferent?
6. BycomparingwiththestaircasetowerspatternwriteaformulaforWalkasaurus’sheighthintermsofhisagedinyears.
7. UsingthesameaxesandscalesplotthegraphsforthestaircasetowersandWalkasaurus’sheight?
Whatisthesameandwhatisdifferentaboutthetwographs?
Example 5:GravityandQuadraticsJCHL
Thisquestionissetincontextsothatstudentswillseetheapplication
ofquadraticsineverydaylife.
a. Ifacentisdroppedfromaheightof45m,itsheightchangesovertimeaccordingtotheformulah=45–4.9t2,wheretismeasuredinseconds.Usemathematicaltools(numericalanalysis,tables,graphs)todeterminehowlongitwilltakeforthecenttohittheground.Whatdoesthegraphofheightvs.timelooklike?Whatconnectionsdoyouseebetweenthegraphandthetable?
b. Supposeyouwantedthecenttolandafterexactly4seconds.Fromwhatheightwouldyouneedtodropit?Explainyouranswer.
c. SupposeyoudroppedthecentfromthetopoftheEiffelTower(300mhigh).Howlongwouldittaketohittheground?Whatdoesthegraphofheightvs.timelooklike?Whatconnectionsdoyouseebetweenthegraphandthetable?
JCHL Linear and quadratic:
i Investigatetherelationshipbetweenlengthandwidthforarectangleoffixedperimeter
ii Investigatehowchangeinthedimensionsofarectangleoffixedperimeteraffectsitsarea.
Givenarectanglewithafixedperimeterof24metres,ifweincrease
thelengthby1mthewidthandareawillvaryaccordingly.Useatable
ofvaluestolookathowthewidthandareavaryasthelengthvaries.
Patterns a Relations Approach to Algebra
© Project Maths Development Team 2011 www.projectmaths.ie 39
Length/m Width /m Area/m2
0 12 0
1 11 11
2 10
3
4
5
6
7
8
9
10
11
12
1. Predictthetypeofgraphyouexpecttogetifyouplottedwidthagainstlength?Explainyouranswerusingthevaluesofthechangebetweensuccessiveoutputs.
2. Whatisdifferentaboutthesechangesandchangesinpreviousexamples,e.g.sunflowergrowth?Plotthegraphofwidthagainstlengthtoconfirmyourpredictions.
3. Whatcanyousayabouttheslopeofthisgraph?
4. Explainwhattheslopemeansinthecontextofthissituation.
5. Expressthewidthintermsofthelengthandperimeteroftherectangle.
6. Whatarethevariablesandtheconstantsinthesituation?
7. Doesitmatterwhichyoudecidetocalltheindependentvariable?
8. Whatdoyounoticeaboutthevaluesofareainthetable–aretheyincreasing,decreasingoristhereanypattern?
9. Doestheareaappeartohaveamaximumorminimumvalue?Ifitdoeswhatvalueisit?
10.Predictthetypeofgraphyouexpecttogetifyouplottedareaagainstlength.Explainyouranswer.
11.Plotthegraphofareaagainstlengthtoconfirmyourpredictions.
12.Whatshapeisthisgraph?
13.Usethegraphofareaplottedagainstlength,forafixedperimeterof24,tofindwhichlengthofrectanglegivesthemaximumarea.DoesthisagreewithyourconclusionfromQ9?
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14.Expresstheareaintermsofthelengthandperimeteroftherectangle.Checkthattheformulaworksbysubstitutinginvaluesoflengthfromthetable.Extension:Whatiftherectanglehadafixedarea–whatistherelationshipbetweenthelengthandwidth?(Seeinverseproportion).
41
Chapter 5: Exponential growth
Patterns a Relations Approach to Algebra
© Project Maths Development Team 2011 www.projectmaths.ie 42
Example 1:Pocketmoneystory
Contrastadding2centeachday(multiplicationasrepeated
addition)withmultiplyingby2eachday(exponentiationasrepeated
multiplication).
IputthispropositionforpocketmoneytomyDadatthebeginningof
July.
“Ijustwantyoutogivemepocketmoneyforthismonth.AllIwantis
foryoutogiveme2conthefirstdayofthemonth,doublethatforthe
secondday,anddoublethatagainforthe3rdday...andsoon.Onthe
1stdayIwillget2c,onthe2ndday4c,onthe3rdday8candsoon
untiltheendofthemonth.ThatisallIwant.”
WasthisagooddealforDadorisitagooddealforme?
MakeatabletoshowhowmuchmoneyIwillgetforthefirst10days
ofthemonth.
Time/days Money/c0 2
1 4
2
3
4
5
6
7
8
9
10
Student Activity: Exponential growth
JCOLLCOL,LCHL,–Exponentialrelationslimitedtodoublingand
tripling
Studentsinvestigatesituationsinvolvingexponentialfunctions
usingwords,tables,graphsandformulaeandunderstandthat
exponentialfunctionsareexpressedinconstantratiosbetween
successiveoutputs.
Studentsshouldseeapplicationsofexponentialfunctionsintheir
everydaylivesandappreciatetherapidrateofgrowthanddecay
showninexponentialfunctions.
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© Project Maths Development Team 2011 www.projectmaths.ie 43
1. Usingthetablewouldyouexpectagraphofthisrelationshiptobelinear?Explain.
2. Usingthetablewouldyouexpectagraphofthisrelationshiptobequadratic?Explain.
3. Checkchanges,changeofthechanges,changeofthechangeofthechangesetc.
4. Whatdoyounotice?Isthereapatterntothedifferences?Ifsowhatisit?
5. Predictthetypeofgraphyouwillgetifyouplotmoneyincentagainsttimeindays.
6. Makeagraphtocheckyourprediction.Whatdoyounotice?
7. Canyoucomeupwithaformulafortheamountofmoneyyouwillhaveafterndays?
8. Usingyourcalculatorfindouthowmuchmoneyyouwouldgetonthe10thday,the20thday,the30thday,31stday?
9. Whatarethevariablesinthesituation?Whatisconstantinthissituation?
10.Whereisthefactorof2(thedoubling)inthetable?Whereisitinthegraph?
11.Contrastthissituationwithadding2ceveryday.Howmuchwouldyouhaveonday31?
12.HowwouldtheformulachangeifyourDadtrebledtheamountofmoney,startingbygivingyou2conthefirstdayasabove?Makeatableandcomeupwiththeformulaforthisnewsituation.
13.Whenpeoplespeakof‘exponentialgrowth’ineverydayterms,whatkeyideaaretheytryingtocommunicate?
y=abx
Final value = starting value multiplied by (growth factor) no. of intervals of time elapsed
(with the independent variable in the exponent).
Thistypeofgrowthiscalledexponentialgrowth.Thevariableisinthe
exponent.Thereisaconstantratiobetweensuccessiveoutputs.This
constantiscalledthegrowthfactor.Duringeachunittimeinterval
theamountpresentismultipliedbythisfactor.Canyouthinkofother
examplesofthingswhichgrowinthiswaye.g.doubleoveraconstant
periodoftime?(Thegrowthfactordoesnothavetobea‘doubling’.)
Computing:Byte=23bits,kilobyte=210bytes,Megabyte=220bytes
Populationgrowth,paperfolding,celldivisionandgrowth,compound
interest–(repeatedmultiplication)
Patterns a Relations Approach to Algebra
© Project Maths Development Team 2011 www.projectmaths.ie 44
Exponential growth in legend–the origins of the game of chess and the payment for its inventor
Legendtellsusthatthegameofchesswas
inventedhundredsofyearsagobytheGrand
VizierSissaBenDahirforKingShirhamofIndia.
TheKinglovedthegamesomuchthatheoffered
hisGrandVizieranyrewardhewanted.“Majesty,
givemeonegrainofwheattogoonthefirst
squareofthechessboard,twograinstoplace
onthesecondsquare,fourgrainstoplaceonthethirdsquare,eight
onthefourthsquareandsoonuntilallthesquaresontheboardare
covered”.TheKingwasastonished.“Ifthatisallyouwishforyou
poorfoolthenyoumayhaveyourwish”.Whenthe41stsquarewas
reached,overonemillionmilliongrainsofricewereneeded.Sissa’s
requestrequiredmorethanthekingdom’sentirewheatsupply.Inall,
thekingowedSissamorethan18,000,000,000,000,000,000grainsof
wheatwhichwasworthmorethanhisentirekingdom.Foolishking!
Square Rice on each square Pattern for rice on the square
Rice on the board
1 1 1 1
2 2 1x2 1+2
3 4 1x2x2 1+2+4
4 8
5
Exponential growth and computing power
SeethefollowinglinkformoreonthislegendandMoore’sLawdealing
withtheexponentialgrowthoftransistorsinintegratedcircuitsleading
tosmallerandfastercomputers.
http://www.authorstream.com/Presentation/Arkwright26-16602-wheat-
1-Exponential-Growth-Moores-Law-seedcount-presentation-002-
Entertainment-ppt-powerpoint/
Patterns a Relations Approach to Algebra
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Example 2:Howmanyancestorsdoyouhave?JCOL
1. Youhave2parents,4grandparents,andeightgreatgrandparents.Goingbackfurtherintoyourfamilytree,howmanyancestorsdoyouhave5generationsago,ngenerationsago?Makeatabletoshowthis.(30yearsonaveragepergeneration?)
2. Whatexampleisthisthesameasnumerically?Howdoesitdifferfromit?(Thinkintermsoftimeintervals.)
Example 3:PaperfoldingJCOL
IfIfoldasheetofpaperinhalfonceIhave2sections.IfIfolditinhalf
againIhave4sections.WhathappensifIcontinuetofoldthepaper?
Makeatable,plotagraphandcomeupwithaformulatodescribethe
situation.Isitlinear,quadratic,exponentialornoneofthese?
Example 4:GrowthofbacteriaJCHL
1. Ifwestartwith1bacteriumwhichbygrowthandcelldivisionbecomes2bacteriaafteronehour–inotherwordsthenumberofbacteriadoubleseveryhour–howmanyofthesebacteriawouldtherebeattheendof1day(24hours)?
2. Whattypeofgrowthisthis–linear,quadratic,exponential,ornoneofthese?
3. Wouldthisgrowthincreaseindefinitely?Explain.(Exponentialgrowthwouldnotcontinueindefinitely–thebacteriawouldrunoutofforexamplenutrientsorspace.)
4. Certainbacteriadoubleinnumberevery20minutes.Startingwithasingleorganismwithamassof10-12g,andassumingtemperatureandfoodconditionsallowedittogrowexponentiallyfor1day,whatwouldbethetotalmassproduced?
5. Whatwouldbeitstotalmassafter2days?(Themassoftheearthisapproximately6x1024kg.)
Patterns a Relations Approach to Algebra
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Example 5:CompoundinterestandexponentialgrowthLCOL
€650isdepositedinafixedinterestratebankaccount.Theamount
(finalvalue)intheaccountattheendofeachyearisshowninthe
followingtable.
Endofyear 1 2 3 4 5
Finalvalue €676.00 €703.04 €731.16 €760.41 €790.82
1. Howcanyoutellfromthetableiftheaboverelationshipislinear,quadraticorexponential?Explain.
2. Ifyouplotagraphoffinalvalueagainsttimewhatdoesthegraphlooklikeforthislimitedrangeoftimes?Howmightplottingmorepointshelp?
3. Whatwouldbetheeffectofincreasingtheinterestrate?Makeatableshowingthefinalvaluesforthefirstfiveyearsusinganinterestrateof10%perannumcompoundinterest.Plotagraphforthisdata.Comparethegraphtothegraphproducedforthelowerinterestrate.
4. Whatformulaexpressesthefinalvalueaftertyearsintheabovecompoundinterestformulaiftheinitialvalueis€650?
5. Whatisthedangerofdrawingaconclusionaboutarelationshipfromaplotusingalimitednumberofpoints?
6. Fromjustthe2datapointsbelowwhataretwopossibleformulaewhichcouldrelatethexandyvalues?
x y
1 2
2 4
Population growth
Thetendencyofpopulationstogrowatanexponentialratewas
pointedoutin1798byanEnglisheconomist,ThomasMalthus,in
hisbook“An Essay on the Principle of Population”.Hesuggested
thatuncheckedexponentialgrowth(ofcoursepeopledieand
warandfaminestilloccur)wouldoutstripthesupplyoffoodand
otherresourcesandwouldleadtodiseaseandwars.Hewrotethat
“Population,whenunchecked,increasesinageometricalratio.
Subsistenceincreasesonlyinarithmeticratio”.
Ofcoursepopulationgrowthforanyparticularcountrydependsonthe
followingfactors:birthrate,deathrate,immigration,emigration,wars,
famine,diseasesandotherpossiblefactors.
Patterns a Relations Approach to Algebra
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Example 6:Growthofworld’spopulationLCOL
1. Theworld’spopulationreached6billionin1999.On07/01/2009theworld’spopulationwas6,755,987,239.http://www.census.gov/ipc/www/popclockworld.htmlOn07/01/10theworld’spopulationwas6,830,586,985.(Writeeachofthesefiguresto3significantfiguresifyouwishandusescientificnotation.)Approximatelyhowmanypeoplemorewereintheworldoneyearafter07/01/09?Whatfactordidtheworld’spopulationincreasebyintheyear?Thisisthegrowthfactor(roundedto2decimalplaces).
2. Assumingthisgrowthratecontinueswhatdoyoumultiplythepopulationin2010bytogetthepopulationin2011?
3. Makeatableofexpectedworldpopulationvaluesfrom2010to2015.Notehowmuchthepopulationincreasesbyeachyear.
4. Whattypeofgrowthisthis?
5. Writeaformulafortheexpectedvalueoftheworld’spopulationnyearsfrom2010?
6. Usingtrialanderrorandtheexpectedgrowthrateof1%peryear,howmanyyearsfrom2010wouldyouexpecttheworld’spopulationtohavedoubled?
7. Usingtrialanderrorandtheexpectedgrowthrateof2%peryear,howmanyyearsfrom2010wouldyouexpecttheworld’spopulationtohavedoubled?
8. (LCHL)Usinglogsworkouttheexactnumberofyearsfortheworld’spopulationtodoubleassumingaconstantgrowthrateof1%peryear.
http://www.learner.org/interactives/dailymath/population.html
TherateofEarth’spopulationgrowthisslowingdown.Throughout
the1960s,theworld’spopulationwasgrowingatarateofabout
2%peryear.By1990,thatratewasdownto1.5%,andbytheyear
2015,it’sexpectedtodropto1%.Familyplanninginitiatives,an
agingpopulation,andtheeffectsofdiseasessuchasAIDSaresome
ofthefactorsbehindthisratedecrease.
Evenattheseverylowratesofpopulationgrowth,thenumbers
arestaggering.By2015,despitealowexpected1%growthrate,
expertsestimatetherewillbe7billionpeopleontheplanet.By
2050,theremaybeasmanyas10billionpeoplelivingonEarth.
Cantheplanetsupportthispopulation?
Whenwillwereachthelimitofourresources?
Patterns a Relations Approach to Algebra
© Project Maths Development Team 2011 www.projectmaths.ie 48
Example 7:PopulationofSylvaniaLCHL
Thetableshowsthepopulationinthousandsofasmallmythical
countryforvariousyears
Year Population in thousands
1825 200
1850 252
1875 318
1900 401
1925 504
1950 635
1975 800
1. Usingthetableabove,checktoseeiftherelationshipbetweentimeandpopulationislinear,quadraticorexponentialornoneofthese?Explain.
2. Predictwhatagraphofthedatawilllooklike.
3. Plotagraphofthisdata.Whatdoesthegraphlooklike?
4. Abouthowlongdoesittakeforthepopulationtodouble?Explainhowyougottheanswerfromthetable.Explainhowyougottheanswerfromthegraph.
5. Whatwouldyouexpectthepopulationtobein2000,2050,and3000?Explainhowyougottheseanswersfrom(i)thetable(ii)thegraph
6. Hereare3differentformulaethatcanbeusedtocalculatepopulationinthissituation.
Explainwhatt,r,andnrepresent.
7. Explainhow fitsthetableandthesituationandhowitcanyieldthecorrectanswer.
8. Explainhow fitsthetableandthesituationandhowitcanyieldthecorrectanswer.
9. Explainhow fitsthetableandthesituationandhowitcanyieldthecorrectanswer.
10.Doyouexpectactualdatavaluestomatchdatavaluesobtainedusingtheformula?Explain.
Patterns a Relations Approach to Algebra
© Project Maths Development Team 2011 www.projectmaths.ie 49
Example 8LCHLIsthepopulationgrowthshowninthegraphbelow
exponential?
1650 1700 1750 1800 1850 1900 1950
500
1000
1500
2000
2500
3000
Year
World population in millions
1. Makeatableandinvestigateifthegrowthisexponential–doesitalwaysincreasebythesamefactorforsuccessiveequaltimeintervals?Calculatetheratiobetweenoutputvaluesforsuccessiveequaltimeintervals.
2. Calculatewhatconstantgrowthfactoroverthefirst100years(1650to1750)wouldhavegiventhesameresultinthatperiod.Usethisgrowthfactorforthenexttwointervalsof100years.Plotthepointsfor1850and1950.Whatcanyouconcludeaboutthisgrowthpattern?
Patterns a Relations Approach to Algebra
© Project Maths Development Team 2011 www.projectmaths.ie 50
Exponential Decay: Compoundeddepreciation(aninvestment
decreasesto0.80ofitsvalueattheendofeveryyear)andthedecayof
nuclearwasteareexamplesofexponentialdecay.Theyinvolverepeated
multiplicationbutinthiscasethegrowthfactorislessthan1.
Example 1 LC OLAlanis4metresawayfromawall.Hejumpstowards
thewallandwitheachjumphehalvesthedistancebetweenhimself
andthewall.
DrawupatableofvaluestoshowAlan’sdistancefromthewallwith
eachstep(jump).
Step Distance from the wall/m
0 4
1 2
2
3
4
5
6
1. Lookingatsuccessiveoutputvaluesinthetable,explainwhetherornottherelationshipislinear,quadratic,orexponential.
2. PlotagraphofAlan’sdistancefromthewallagainstthestepnumbertowardsthewall.
3. Howisthisgraphdifferentthanthegraphobtainedforcompoundinterest?
4. Howisthisgraphlikethegraphobtainedforcompoundinterest?
5. WillAlan’sdistancefromthewalleverbezero?
Patterns a Relations Approach to Algebra
© Project Maths Development Team 2011 www.projectmaths.ie 51
Example 2: LCOLDuringavisittothehospital,Janereceivesadoseof
radioactivemedicationwhichdecaysorlosesitseffectivenessatarate
of20%perhour.
1. Ifshereceives150mgofthemedicationinitially,approximatelyhowmanymilligramsofthemedicationremaininherbodyafterthefirst6hours?
2. Drawupatableshowingtheamountofradioactivemedicationinherbodyfor1upto6hours,increasingby1houreachtime.
3. Plotagraphofmgofmedicationleftinherbodyagainsthourselapsed.
4. IsitpossibletoreducetheamountofradioactivemedicationinJane’sbodyto0?Explain.
5. Whataretheimplicationsofthisforthedecayofradioactivewastematerial?(Someradioactivematerialstakemorethan20,000yearsforhalfofittodecay.Seemathematicaltablesforhalflivesofradioactiveelements.)
Patterns a Relations Approach to Algebra
© Project Maths Development Team 2011 www.projectmaths.ie 52
53
Chapter 6: Inverse Proportion
Patterns a Relations Approach to Algebra
© Project Maths Development Team 2011 www.projectmaths.ie 54
Example 1:Sharingafixedamountofprizemoney
Imagineyouwonthelottoandtheprizewas€1,000,000.Youdon’t
knowhowmanyotherpeoplealsohadthewinningnumberssoyou
arespeculatingonhowmuchyouwillcollect.Ifyouaretheonlywinner
thenyoucollect€1,000,000.However,ifthereare2winnersyou
collect€500,000.
1. Makeupatableshowingnumberofwinnersandthecorrespondingamountwonperperson.
2. Isitalinearrelationship?Explain
3. Isitaquadraticrelationship?Explain.
4. Plotagraphofprizecollectedperwinneragainstnumberofwinners.
5. Describethegraph.
6. Isthislikeexponentialgrowth/decay,i.e.isthereaconstantratiobetweensuccessiveoutputsforequalchangesoftheindependentvariable?
7. Whatformularelateswinningsperperson,numberofwinnersandtotalvalueoftheprize?
8. Therearevariablesandaconstantintheequation.Identifythevariablesandtheconstant.
9. Whichvariableistheindependentvariable?
10.Whichvariableisthedependentvariable?(The resulting graph is called a hyperbola. In real life contexts we are only considering positive values of the independent variable.)
11.Asthenumberofwinnersdoubleswhathappenstotheamountofmoneyeachwinnergets?
12.Asthenumberofwinnerstrebleswhathappenstotheamountofmoneyeachwinnergets?
Student Activity: Inverse Proportion
For an input x and an output y, y is inversely proportional
to x, if there exists some constant k so that xy = k .
The constant k is called the constant of proportionality .
Patterns a Relations Approach to Algebra
© Project Maths Development Team 2011 www.projectmaths.ie 55
13.Asthenumberofwinnersquadrupleswhathappenstotheamountofmoneyeachwinnergets?
14.Whenthenumberofwinnersdecreaseswhathappenstotheamountofmoneyeachpersonwins?
15.Whenthenumberofwinnersbecomesverybigwhathappenstotheamountofmoneyeachpersonreceives
16.Willthegraphevertouchthexaxis?Explain.
17.Willthegraphevertouchtheyaxis?Explain.
Thistypeofrelationshipiscalledaninverse proportion.Theproduct
ofthevariablesisaconstant.
• Themorepeoplewhoshareapizzathesmallerthesliceofpizzaeachonereceives.Thesizeofthesliceisinverselyproportionaltothenumberofpeoplesharing.
• Lengthvariesinverselywithwidthforarectangleofgivenarea.A = l x w.
• Depthofafixedvolumeofliquidindifferentcylindersvariesinverselyastheareaofthebaseofthecylinder.V=Areaofbasexh
Example 2:Relationshipbetweentimespenttravellingafixeddistance
andspeedforthejourney
1. Considerasituationwherepeopletravelbetweentwopointsafixeddistanceapart.Assumethatdifferentpeopletravelwithdifferentconstantspeedsbetweenthetwopoints.Whathappenstothetimetakentocompletethejourneybetweenthetwopointsasthespeedincreases?
2. Isthereaspeedwhichwillgiveatimeof0?Explain.
3. Isthereavalueoftimecorrespondingtoaspeedof0?Explain.
4. Whatdothesetwoanswerstellyouaboutthegraph?
5. Makeupatableshowingspeedandthecorrespondingtimetaken.(Youcanchoosethedistancebetweenthetwopointsandthespeedandtheunitsused.)
6. Isitalinearrelationshipbetweenspeedandtime?Explain.
7. Isitaquadraticrelationship?Explain.
8. Plotagraphoftimetakenagainstspeed.
9. Describethegraph.
10. Isthislikeexponentialgrowth/decay–isthereaconstantratiobetweensuccessiveoutputsforequaltimeintervals?
Patterns a Relations Approach to Algebra
© Project Maths Development Team 2011 www.projectmaths.ie 56
11.Whatformularelatesthespeed,timetakenandthedistancebetweenthetwopoints?
12.Therearevariablesandaconstantintheequation.Identifythevariablesandtheconstant
13.Whichvariableistheindependentvariable?
14.Whichvariableisthedependentvariable?The resulting graph is called a hyperbola.
15.Asthespeeddoubleswhathappenstothetimetakenforthejourney?
16.Asthespeedtrebleswhathappenstothetimetakenforthejourney?
17.Asthespeedquadrupleswhathappenstothetimetakenforthejourney?
18.Whenthespeedhasalowvaluewhathappenstothetimetakenforthejourney?
19.Whenthespeedhasahighvaluewhathappenstothetimetakenforthejourney?
20.Willthegraphevertouchthexaxis?Explain.
21.Willthegraphevertouchtheyaxis?Explain.
22. Isthisanexampleofaninverseproportion?Explain.
23.Givetwoexamplesofinverseproportions.
http://www.articlesforeducators.com/dir/mathematics/cat_and_mouse.
asp
Ifittakes5cats5daystocatch5mice,howlongwillittake3catsto
catch3mice?
Ifaboyandahalfcanmowalawninadayandahalf,howlongwillit
take5boystomow20lawns?
Hint:Onlyvary2quantitiesatanyonetime–keepthethirdone
constant.
57
Chapter 7: Moving from linear to quadratic to cubic
functions
Patterns a Relations Approach to Algebra
© Project Maths Development Team 2011 www.projectmaths.ie 58
Example 1:Usingacubestudentsinvestigatelinear,
quadraticandcubicrelationships
Forablockwithedgelengthsof1unit,theperimeterof
thebaseis4units,thesurfaceareais6squareunitsand
thevolumeis1cubicunit.Whatwouldthevaluesbeforablockwith
edgelengthsof2unitsor3unitsor34unitsornunits?
1. Maketablesforperimeter,forsurfaceareaandforvolumeastheedgelengthsoftheblockincrease.Examinethetablestopredicttheshapeofthegraphforeachofthethreerelationships.Explainyourpredictions.Makethegraphsforperimetervs.edgelength,surfaceareavs.edgelengthandvolumevs.edgelengthandcomparethemwithyourpredictions.
Studentscompleteatablefortheperimeterofthebaseforcubeswith
differentedgelengths
Edge length /cm Perimeter of the base of the
cube /cm
1
2
3
4
5
6
7
1. Predicttheshapeofgraphfortheaboverelationship.
2. Explainyourprediction.
3. Writeaformulafortheperimeterofthebaseintermsoftheedgelength.
Student Activity:Moving from linear to quadratic to cubic functions . LC OL
The linear model, occurs when there is a constant rate of
change . When there is a consistent force for change, the
quadratic model often fits well . For example, gravity is a
constant force . Many movements under its influence are
essentially quadratic . Cubic models appear in locations such
as highway designs which require a smooth transition from
a straight line into a curve .
Patterns a Relations Approach to Algebra
© Project Maths Development Team 2011 www.projectmaths.ie 59
4. Plotagraphtoshowtheaboverelationship.
5. Checkifvaluesforperimeterpredictedbytheformulaagreewithvaluespredictedbythegraph.
Completethetablebelowfortotalsurfaceareaofthecube.
Edge length /cm Surface area of the cube /
cm2
1
2
3
4
5
6
7
1. Predicttheshapeofgraphfortheaboverelationship.
2. Explainyourprediction.
3. Writeaformulaforthetotalsurfaceareaintermsoftheedgelength.
4. Plotagraphtoshowtheaboverelationship.
Completethetablebelowforvolumeofthecube
Edge length/cm Volume/cm3 Change Change of thechanges1
23
45
67
1
87
2719
12
Change in the changes of
the changes
1
6. Predicttheshapeofgraphfortheaboverelationship.
7. Explainyourprediction.
8. Writeaformulafortheperimeterofthebaseintermsoftheedgelength.
9. Plotagraphtoshowtheaboverelationship.
10.Checkifvaluesforperimeterpredictedbytheformulaagreewithvaluespredictedbythegraph.
Patterns a Relations Approach to Algebra
© Project Maths Development Team 2011 www.projectmaths.ie 60
Relationshipbetweensurfaceareaandvolumeforacube.
Doyouthinkthevolumeandsurfaceareaofacubewilleverbeequal
numerically?
Doyouthinkthevolumewillbeeverbenumericallygreaterthanthe
surfacearea?
Checkbycompletingthefollowingtable.
Edge 1 2 3 4 5 6 7 8 9 n
Area of base
Total surface area
Volume
Surface AreaVolume
Atwhatpointistheratioofsurfaceareatovolumeequalto1?
Whenisvolumelessthansurfacearea?
Whenisvolumegreaterthansurfacearea?
Howcanyouexplaintherapidgrowthofvolumeandtheslower
growthofsurfacearea?
Example 2(LCHL)Designingthelargestbox–cuboidshaped–froma
rectangularsheetoffixeddimensions
Youaremakingboxesfrom
cardboardforpersonalised
Christmaspresents.Youwish
toputasmanysweetsas
possibleintotheboxesusing
rectangularbasedboxesbut
youarelimitedbythesize
ofthesheetsofcardboard
available.(Alternative:design
asuitcasetoholdmaximumvolume)
Startwitharectangularsheetofcardboard9cmx12cm.Fromeach
cornercutsquaresofequalsize.Foldupthefourflapsandtapethem
togetherforminganopenbox.Dependingonthesizeofthesquares
cutfromtherectangularsheet,thevolumeoftheboxwillvary.
Investigatehowthevolumeoftheboxwillvary.
12cm
9cm
Patterns a Relations Approach to Algebra
© Project Maths Development Team 2011 www.projectmaths.ie 61
(Thestudentwillengageinproblemsolvingusingtheirexperiences
frompreviousactivities.Oneoftheaimsofthisapproachistoempower
studentstousethemultiplerepresentationstohelpsolveproblems
andsomerepresentationsmaybemoreusefulthanothers.Beloware
suggestedpromptquestionswherestudentsarehavingdifficulty.)
1. Whentheboxisformedwhichdimensionoftheboxwillberepresentedbyx?
2. Ifwecutoutsquaresofsidexfromthecardboard,predictwhatvalueofxwillgiveyouthelargestvolumebox.
3. WriteaformulaforthevolumeVoftheboxintermsofthelengthoftheboxl,itswidthw,andheightx.
4. Useatabletotestoutvaluesofxtoseewhichvaluewillgivetheboxoflargestvolume.Checkthemusingtheformula.
x/cm l/cm w/cm V/cm
0.0
0.5
1.0
1.5
5. Betweenwhattwoxvaluesdoyouexpectthelargestvolumetobe?
6. DoyouexpectagraphofVagainstxtobelinear,quadraticorcubic?Explainyouranswer.PlotthegraphofVagainstx.
7. Fromthegraphapproximatelywhatvalueofxwillyieldthemaximumvolume?
Patterns a Relations Approach to Algebra
© Project Maths Development Team 2011 www.projectmaths.ie 62
Example 3Thepaintedcube(LCHL)
Youhaveacubeliketheoneshownontheright,butit
hasbeenthrownintoabucketofpaint.Theoutsideiscoveredwith
paintbuttheinsideisnot.Youbreakitapartintounitcubestoseehow
manyoftheunitcubeshave0facespainted,1facepainted,2faces
painted,3facespainted,4faces,5faces,or6facespainted.
Youarelookingforapatternbetweenthenumberofpaintedfaces
andtheoriginalsizeofthecube.Explorethisfora2x2x2cubeupto
a6x6x6cube.Foreachcube,countthenumberofblocksineachof
thecategories0facespainted,1facepainted,2facespainted,3faces
painted,4faces,5faces,or6facespainted.(Studentscancomeup
withthetablethemselvesastheyshouldbeusedtodoingthisasa
problemsolvingstrategy.)
Dimensions 0 faces painted
1 face painted
2 faces painted
3 faces painted
4 faces painted
5 faces painted
6 faces painted
Total number of unit blocks
2x2x2 0 0 0 8 0 0 0 8
3x3x3
4x4x4 24 0 64
5x5x5
6x6x6 96
nxnxn (n>=2)
n3
Verifytheequivalentexpressionforn3byaddingupalltheexpressions
incolumns2–8ofthelastrow.
Patterns a Relations Approach to Algebra
© Project Maths Development Team 2011 www.projectmaths.ie 63
Example 4(extensionforHL)
Findarelationshipbetweentotalsurfacearea(TSA)ofacubeandthe
perimeter(P)ofoneside.
Edge length/cm P of one side TSA
1 4 (1) = 4 6 (1)2 = 6
2 4 (2) = 8 6 (2)2 = 24
3
4
5
6
n
1. Predictwhatyouexpectagraphoftotalsurfaceareaplottedagainstperimeterofacubetolooklike.Explainyourprediction.
2. Usingtheformulaeforperimeterandtotalsurfacearea,deriveaformulafortotalsurfaceareaintermsofperimeter.
3. Usingtheformuladerivedcheckthatitcorrectlypredictsvaluesoftotalsurfaceareagiventheperimetervaluesinthetable.
Patterns a Relations Approach to Algebra
© Project Maths Development Team 2011 www.projectmaths.ie 64
65
Chapter 8: Appendix containing some notes and
suggested solutions
Patterns a Relations Approach to Algebra
© Project Maths Development Team 2011 www.projectmaths.ie 66
GraphsforStudentActivityPage18
Sketchedlineinblack
Situations a and b
a: y = 2x + 3b: y = 2x + 6c: y = 2x + 5
25
20
15
10
5
0 1 2 3 4 5 6 7 8 9
Situations b and c
a: y = 4x + 6b: y = 2x + 6c: y = 3x + 6
25
20
15
10
5
0 1 2 3 4 5 6 7 8 9
Situations c and d
d: y = 2x + 8c: y = 3x + 6b: y = 2x + 6
25
20
15
10
5
0 1 2 3 4 5 6 7 8 9
Patterns a Relations Approach to Algebra
© Project Maths Development Team 2011 www.projectmaths.ie 67
0.5 1.0 1.5 2.0 2.5
1
2
3
4
5
6
Growing Rectangles
7
8
9
10
11
0 3.0 3.5 4.0 4.5 5.0 6.05.5
Staircase Towers
2 4
2
4
6
8
10
12
Walkasarus
14
16
18
0 6 8 10
Staircase Towers
Graphsforgrowingrectangles,staircasetowers,andWalkasaurus
GraphsforStudentActivityPage16
GraphsforStudentActivityPage31GraphsforStudentActivityPage31
GraphsforStudentActivityPage34
Patterns a Relations Approach to Algebra
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Are
ain
m2
GraphsforStudentActivityPage38
GraphsforStudentActivityPage38
GraphsforStudentActivityPage47
Patterns a Relations Approach to Algebra
© Project Maths Development Team 2011 www.projectmaths.ie 69
GraphsforStudentActivityPage46
SpreadsheetforExample6CompoundInterest
1 676
2 703.04
3 731.16
4 760.41
5 790.82
6 822.4528
7 855.3509
8 889.5649
9 925.1475
10 962.1534
11 1000.64
12 1040.665
13 1082.292
14 1125.583
15 1170.607
16 1217.431
Patterns a Relations Approach to Algebra
© Project Maths Development Team 2011 www.projectmaths.ie 70
GraphsforStudentActivityPage48
GraphsforStudentActivityPage51
GraphsforStudentActivityPage54
Patterns a Relations Approach to Algebra
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GraphsforStudentActivityPage59
Sketchedlineinblack
Showing growth of surface area and growth of volume for a cube of side x
Implicationsforfastergrowthrateofvolumecomparedtosurfacearea
Surfaceareaisimportantasiteffectshowfastanobjectcoolsoff
(greaterareaequalsquickercooling.Machinesthatneedtogetridof
extraheathavelittlemetalfinsstucktothemtoincreasetheirsurface
area).
Babieshavealargesurfaceareacomparedtotheirvolumeandlose
heatquickly–hencetheymustbewrappedupverywelltoprevent
heatloss.Astheygrowtheratioofsurfaceareatovolumedecreases
(volumegrowsfasterthansurfacearea)andsotheydonotloseheatas
quickly.
Surfaceareaaffectshowquicklyadropletevaporates–greaterarea
givesfasterevaporation.
Surfaceareaaffectshowquicklyachemicalreactionproceeds–greater
areagivesfasterreaction.
Insectsbreathethroughtheirsurfacearea(x2)buttheirneedforoxygen
isinproportiontotheirvolume(x3)–hencetheycan’tbecomethesize
ofhumansastheirsurfaceareawouldnotsupplyenoughoxygenfor
suchalargevolume.
1 2 3 4
1
3
4
6
7
9
0 5 6 7
Patterns a Relations Approach to Algebra
© Project Maths Development Team 2011 www.projectmaths.ie 72
Why can’t giants exist?
Consideranadultwoman160cmtallandagiantwoman10timesas
tall.Howtallisthegiant?16m
Assumethegiantissimilartothewomanincompositionofbones,
fleshetc.Ifthewomanweighs50kg,whatdoesthegiantweigh?
(Weight,likevolume,increasesasheight3)
Giantweighs50x10x10x10=50,000kg
Thewoman’sweightissupportedbyhertwolegbones.Ifthecross
sectionalareaofeachofherlegbonesis10cm2,howmuchweightis
supportedbyeachcm2oflegbone?
50/20kg/cm2=2.5kg/cm2
Whatisthecrosssectionalareaofeachofthegiant’sbones?
10x102=1,000cm2
Howmuchweightissupportedbyeachcm2ofthegiantwoman’sleg
bones?
50,000/2000=25kg/cm2
Ifthegiant’sbonesarethesamedensityasthoseofthewomanof
averageheight,whatwillhappentothegiant’slegboneswhenshe
standsup?
Theywillbreakbecausetheyhave10timestheweighttocarryforthe
samesurfacearea.
Howcouldthisproblembesorted?Denserbonesforthegiant,orwalk
onallfourstodistributetheweight.
Patterns a Relations Approach to Algebra
© Project Maths Development Team 2011 www.projectmaths.ie 73
Making a box with the largest volume, given a rectangle of fixed dimensions
x width length Volume
0 9 12 0
0.5 8 11 44
1 7 10 70
1.5 6 9 81
2 5 8 80
2.5 4 7 70
3 3 6 54
3.5 2 5 35
4 1 4 16
4.5 0 3 0
5 -1 2 -10
5.5 -2 1 -11
6 -3 0 0
6.5 -4 -1 26
GraphsforStudentActivityPage60
Patterns a Relations Approach to Algebra
© Project Maths Development Team 2011 www.projectmaths.ie 74
Dimensions 0 faces painted
1 face painted
2 faces painted
3 faces painted
4 faces painted
5 faces painted
6 faces painted
Total number of unit blocks
2 x 2 x 2 0 0 0 8 0 0 0 8
3 x 3 x 3 1 6 12 8 0 0 0 27
4 x 4 x 4 8 24 24 8 0 0 0 64
5 x 5 x 5 27 54 36 8 0 0 0 125
6 x 6 x 6 64 96 48 8 0 0 0 216
nxnxn (n>=2)(n-2) 3 6(n-2)2 12(n-2) 8 0 0 0 n3
(n-2)3 = n3-6n2+12n-8
6(n-2)2 = 6n2-24n+24
12(n-2) = 12n-24
+8 = 8
Explanation:
Forthe2x2x2,nocubesarepaintedonallfaces
Thecubeswith3paintedfacesarealwaysonthecornersandthereare
8ofthose.
Thecubeswith2paintedfacesoccurontheedgesbetweenthe
corners.Thereare12ofthoseifitisa3x3cubeand12(n-2)ifann
byncube(takeawaythe2onthecorners).
Thecubeswith1paintedfaceoccurassquaresoneachofthe6faces
andtherewillbe6(n-2)2ofthose.
Thosewithnopaintedfacesareontheinsideandaregivenby(n-2)3
} Totaln3
ThePaintedCube,page62
Patterns a Relations Approach to Algebra
© Project Maths Development Team 2011 www.projectmaths.ie 75
Relationship between TSA and Perimeter of a face for a cube
P of one side TSA Couples
1cm 4(1) 6(1)2 (4, 6)
2cm 4(2) 6(2)2 (8, 24)
3cm 4(3) 6(3)2 (12, 54)
4cm 4(4) 6(4)2 (16, 96)
ncm 4(n) 6(n)2
RelationshipbetweenTSAandperimeterofacube.
Graphgoingthroughabovesetofcouples
SuggestedSolution,StudentActivitypage68
Patterns a Relations Approach to Algebra
© Project Maths Development Team 2011 www.projectmaths.ie 76
Theequationofthisgraphisy=0.375x2.
Ithenaskedthequestion“Isitpossibleforthestudentstoderivethe
equationofthequadraticfromthetablebeforethegraphisdrawn
withthecouples?”Thefollowingapproachworks.
1. WearetryingtofindanequationwhichlinksPerimeter(P)andTotalSurfaceArea(TSA).
2. Fromthetable(theTn)P=4n and TSA=6n2
Thisistheequationofthequadraticgraphy=0.375x2
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