Patterns A Relations Approach to Algebra - · PDF filePatterns A Relations Approach to Algebra Junior Certificate Syllabus Leaving Certificate Syllabus

PatternsA Relations Approach

to Algebra

Junior Certificate SyllabusLeaving Certificate Syllabus

© Project Maths Development Team 2011 www.projectmaths.ie 2


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


5

Chapter 1: Relationship to syllabuses, introduction and

unit learning outcomes

Patterns a Relations Approach to Algebra


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



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



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.



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)].



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



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.



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?



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?



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.



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?



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.



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



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).



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?



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



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



1. Whenyoulookatthesetableshowcanyoutellifthevariablesareinproportion?

2. Modelthe4situationsrepresentedinthetablesbydrawinggraphsforeachtable.

3. Whenyoulookatlineargraphswherethevariablesareproportionalwhatdoyounoticeaboutthegraphs?

4. Whenyoulookatlineargraphswherethevariablesarenotproportionalwhatdoyounoticeaboutthegraphs?

5. Writeaformularelatingthevariablesforeachofthesetables.

6. Intheformulaethereisanamountyoumultiplybyandanamountyouadd.Howarethesequantitiesrepresentedinthegraphs?

7. Giveareallifecontextforeachofthesetables.

Next activity on graph matching:

JCHL:Matchingallthedifferentrepresentationssimultaneously

JCOL:Matchingrepresentationsinappropriatepairs



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





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.





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).



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



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.



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.



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?



4. Predictwhatagraphofthesituationwilllooklike.Plotagraphtocheckyourprediction.


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



Example 3:StaircaseTowersJCHL

Lookatthestaircasesbelow.Makeatablerepresentingtherelationship

betweenthetotalnumberoftilesandthenumberoftowers.Make

observationsaboutthevaluesinthetable.Whatwouldagraphlook

like?Woulditbelinear?Howdoyouknow?Makeagraphtocheckyour

prediction.

Allowstudentstocontinuethepatternthemselvesandmaketheirown

observations.Thefollowingquestionscouldbeusedaspromptquestions

ifnecessary.

1. Howmanytilesdoyouaddontomakestaircase2fromstaircase1?



4. Completethetablebelow

Numberof

towers

Numberoftiles

1 1

2 3

3 6

4

5

6

7

8

9

10



5. Willagraphforthistablebelinear?Explain,usingthetablebelow.


1

23

45

67

1

32

63

1

6. Arethechangesconstant?Arethechangesofthechangesconstant?



We need to develop a formula for the number of tiles required for the nth tower .

Comparethepatternforthe‘growingrectangles’withthatofthe

staircasetowers.

1. Whatistheformulaforthenumberoftilesinthenthrectangleintermsofn?

2. Hencewhatistheformulaforthenumberoftilesinthenthstaircaseintermsofn?



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


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’?



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




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.



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?



14.Expresstheareaintermsofthelengthandperimeteroftherectangle.Checkthattheformulaworksbysubstitutinginvaluesoflengthfromthetable.Extension:Whatiftherectanglehadafixedarea–whatistherelationshipbetweenthelengthandwidth?(Seeinverseproportion).

41

Chapter 5: Exponential growth



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.



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)



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/



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.)



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.



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?



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.



10.Doyouexpectactualdatavaluestomatchdatavaluesobtainedusingtheformula?Explain.



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?



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?



Example 2: LCOLDuringavisittothehospital,Janereceivesadoseof

radioactivemedicationwhichdecaysorlosesitseffectivenessatarate

of20%perhour.

1. Ifshereceives150mgofthemedicationinitially,approximatelyhowmanymilligramsofthemedicationremaininherbodyafterthefirst6hours?

2. Drawupatableshowingtheamountofradioactivemedicationinherbodyfor1upto6hours,increasingby1houreachtime.

3. Plotagraphofmgofmedicationleftinherbodyagainsthourselapsed.

4. IsitpossibletoreducetheamountofradioactivemedicationinJane’sbodyto0?Explain.

5. Whataretheimplicationsofthisforthedecayofradioactivewastematerial?(Someradioactivematerialstakemorethan20,000yearsforhalfofittodecay.Seemathematicaltablesforhalflivesofradioactiveelements.)



53

Chapter 6: Inverse Proportion



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 .



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?



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



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 .



4. Plotagraphtoshowtheaboverelationship.

5. Checkifvaluesforperimeterpredictedbytheformulaagreewithvaluespredictedbythegraph.

Completethetablebelowfortotalsurfaceareaofthecube.

Edge length /cm Surface area of the cube /

cm2

1

2

3

4

5

6

7



3. Writeaformulaforthetotalsurfaceareaintermsoftheedgelength.


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



8. Writeaformulafortheperimeterofthebaseintermsoftheedgelength.


10.Checkifvaluesforperimeterpredictedbytheformulaagreewithvaluespredictedbythegraph.



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



(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?



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.



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.



65

Chapter 8: Appendix containing some notes and

suggested solutions



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



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


GraphsforStudentActivityPage31GraphsforStudentActivityPage31




Are

ain

m2







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









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



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.



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




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



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



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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