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    Mathematics SL guide 17

    Syllabu

    s

    Syllabuscontent

    Top

    ic1

    Algebra

    9

    hours

    Theaimofthistopicistointroducestudentstosomebasicalgebraicconce

    ptsandapplications.

    Content

    Furtherguidance

    L

    inks

    1.1

    Arithmeticsequencesandseries

    ;sumoffinite

    arithmeticseries;geometricsequencesandseries;

    sumoffiniteandinfinitegeometricseries.

    Sigmanotation.

    Technologymay

    beusedtogenerateand

    displaysequencesinseveralways.

    Linkto2.6,exponentialfunctions.

    I

    nt:Thechesslegend(SissaibnDahir).

    I

    nt:Aryabhattaissometimesconsideredthe

    fatherofalgebra.Comparewith

    al-Khawarizmi.

    T

    OK:HowdidGaussaddupinteg

    ersfrom

    1

    to100?Discusstheideaofmathe

    matical

    intuitionasthebasisforformalproof.

    T

    OK:Debateoverthevalidityofthenotionof

    infinity:finitistssuchasL.

    Kronecker

    considerthatamathematicalobjectdoesnot

    existunlessitcanbeconstructedfromnatural

    n

    umbersinafinitenumberofsteps

    .

    T

    OK:WhatisZenosdichotomyp

    aradox?

    H

    owfarcanmathematicalfactsbe

    from

    intuition?

    Applications.

    Examplesinclud

    ecompoundinterestand

    populationgrowth.

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    Content

    Furtherguidance

    L

    inks

    1.2

    Elementarytreatmentofexpone

    ntsand

    logarithms.

    Examples:

    3 4

    16

    8

    ;

    16

    3

    log

    8

    4

    ;

    log

    32

    5l

    og

    2

    ;

    4

    3

    12

    (2

    )

    2

    .

    A

    ppl:Chemistry18.1

    (Calculation

    ofpH).

    T

    OK:Arelogarithmsaninvention

    or

    d

    iscovery?(Thistopicisanopport

    unityfor

    t

    eacherstogeneratereflectionon

    thenatureof

    m

    athematics.)

    Lawsofexponents;lawsoflogarithms.

    Changeofbase.

    Examples:

    4

    ln

    7

    log

    ln

    4

    7

    ,

    25

    5 5

    log

    log

    log

    1

    25

    3

    125

    25

    2

    .

    Linkto2.6,

    loga

    rithmicfunctions.

    1.3

    Thebinomialtheorem:

    expansionof(

    ),

    n

    a

    b

    n

    .

    Countingprinciplesmaybeusedinthe

    developmentof

    thetheorem.

    A

    im8:Pascalstriangle.

    Attributingtheorigin

    o

    famathematicaldiscoverytothe

    wrong

    m

    athematician.

    I

    nt:Theso-calledPascalstrianglewas

    k

    nowninChinamuchearlierthanPascal.

    Calculationofbinomialcoefficientsusing

    Pascalstriangleand

    n r

    .

    n r

    s

    houldbef

    oundusingboththeformula

    andtechnology.

    Example:finding

    6 r

    f

    rom

    inputting

    6n

    r

    y

    C

    X

    and

    thenreadingcoefficientsfrom

    thetable.

    Linkto5.8,

    bino

    mialdistribution.

    Notrequired:

    formaltreatmentofpermutation

    sandformula

    forn

    rP

    .

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    ic2

    Functions

    andequations

    24

    hours

    Theaimsofthistopicaretoexplore

    thenotionofafunctionasaun

    ifyingthemeinmathematics,an

    dtoapplyfunctionalmethodstoavarietyof

    mathematicalsituations.Itisexpectedthatextensiveusewillbemadeof

    technologyinboththedevelopm

    entandtheapplicationofthistop

    ic,ratherthan

    elaborateanalyticaltechniques.

    Onexa

    minationpapers,questionsmay

    besetrequiringthegraphingof

    functionsthatdonotexplicitly

    appearonthe

    syllabus,andstudentsmayneedtoch

    oosetheappropriateviewingwindow.

    Forthosefunctionsexplicitlymentioned,questionsmayalsobeseton

    compo

    sitionofthesefunctionswiththelinearfunctiony

    ax

    b

    .

    Content

    Furtherguidance

    L

    inks

    2.1

    Conceptoffunction

    :

    (

    )

    f

    x

    fx

    .

    Domain,range;image(value).

    Example:for

    2

    x

    x

    , domainis

    2

    x

    ,

    rangeis

    0

    y

    .

    Agraphishelpf

    ulinvisualizingtherange.

    I

    nt:Thedevelopmentoffunctions,

    Rene

    D

    escartes(France),GottfriedWilh

    elmLeibniz

    (

    Germany)andLeonhardEuler(Switzerland).

    Compositefunctions.

    (

    )

    (

    (

    ))

    f

    g

    x

    f

    g

    x

    .

    T

    OK:Iszerothesameasnothing

    ?

    T

    OK:Ismathematicsaformallanguage?

    Identityfunction.

    Inversefunction

    1

    f

    .

    1

    1

    (

    )(

    )

    (

    )(

    )

    f

    f

    x

    f

    f

    x

    x

    .

    Onexaminationpapers,studentswillonlybe

    askedtofindthe

    inverseofaone-to-onefunction.

    Notrequired:

    domainrestriction.

    2.2

    Thegraphofafunction;itsequation

    ()

    y

    fx

    .

    A

    ppl:Chemistry11.3.1

    (sketching

    and

    i

    nterpretinggraphs);geographicsk

    ills.

    T

    OK:Howaccurateisavisualrep

    resentation

    o

    famathematicalconcept?(Limitsofgraphs

    i

    ndeliveringinformationaboutfun

    ctionsand

    p

    henomenaingeneral,relevanceofmodesof

    r

    epresentation.)

    Functiongraphingskills.

    Investigationofkeyfeaturesof

    graphs,suchas

    maximumandminimumvalues

    ,intercepts,

    horizontalandverticalasympto

    tes,symmetry,

    andconsiderationofdomainandrange.

    Notethedifferenceinthecommandterms

    drawandske

    tch.

    Useoftechnologytographava

    rietyof

    functions,includingonesnotsp

    ecifically

    mentioned.

    Ananalyticapproachisalsoexpectedfor

    simplefunctions,includingallthoselisted

    undertopic2.

    Thegraphof

    1

    (

    )

    y

    f

    x

    asthereflectionin

    theliney

    x

    ofthegraphof

    ()

    y

    fx

    .

    Linkto6.3,

    loca

    lmaximumandminimum

    points.

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    Content

    Furtherguidance

    L

    inks

    2.3

    Transformationsofgraphs.

    Technologyshouldbeusedtoinvestigatethese

    transformations.

    A

    ppl:Economics1.1

    (shiftingofsupplyand

    d

    emandcurves).

    Translations:

    (

    )

    y

    f

    x

    b

    ;

    (

    )

    y

    f

    x

    a

    .

    Reflections(inbothaxes):

    ()

    y

    fx

    ;

    (

    )

    y

    f

    x

    .

    Verticalstretchwithscalefactorp:

    (

    )

    y

    pf

    x

    .

    Stretchinthex-directionwithscalefactor

    1 q:

    y

    f

    qx

    .

    Translationbythevector

    3 2

    d

    enotes

    horizontalshiftof3unitstotheright,and

    verticalshiftof2down.

    Compositetransformations.

    Example:

    2

    y

    x

    usedtoobtain

    2

    3

    2

    y

    x

    by

    astretchofscale

    factor3inthey-direction

    followedbyatranslationof

    0 2

    .

    2.4

    Thequadraticfunction

    2

    x

    ax

    bx

    c

    :its

    graph,y-intercept(0,

    )c.

    Axiso

    fsymmetry.

    Theform

    (

    )(

    )

    x

    ax

    p

    x

    q

    ,

    x-intercepts(

    ,0)

    p

    and(,

    0)

    q

    .

    Theform

    2

    (

    )

    x

    a

    x

    h

    k

    ,ve

    rtex(,

    )

    h

    k

    .

    Candidatesaree

    xpectedtobeabletochange

    fromoneformtoanother.

    Linksto2.3,

    transformations;2.7,quadratic

    equations.

    A

    ppl:Chemistry17.2

    (equilibrium

    law).

    A

    ppl:Physics2.1

    (kinematics).

    A

    ppl:Physics4.2

    (simpleharmonicmotion).

    A

    ppl:Physics9.1

    (HLonly)(proje

    ctile

    m

    otion).

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    Content

    Furtherguidance

    L

    inks

    2.5

    Thereciprocalfunction

    1

    x

    x

    ,

    0

    x

    :its

    graphandself-inversenature.

    Therationalfunction

    ax

    b

    x

    cx

    d

    andits

    graph.

    Examples:

    4

    2

    ()

    ,

    3

    2

    3

    hx

    x

    x

    ;

    7

    5

    ,

    2

    5

    2

    x

    y

    x

    x

    .

    Verticalandhorizontalasympto

    tes.

    Diagramsshouldincludeallasymptotesand

    intercepts.

    2.6

    Exponentialfunctionsandtheir

    graphs:

    x

    x

    a

    ,

    0

    a

    ,

    ex

    x

    .

    I

    nt:TheBabylonianmethodofmu

    ltiplication:

    2

    2

    2

    (

    )2

    a

    b

    a

    b

    ab

    .SulbaSutr

    asinancient

    I

    ndiaandtheBakhshaliManuscrip

    tcontained

    a

    nalgebraicformulaforsolvingqu

    adratic

    e

    quations.

    Logarithmicfunctionsandtheir

    graphs:

    log

    a

    x

    x

    ,

    0

    x

    ,

    ln

    x

    x

    ,

    0

    x

    .

    Relationshipsbetweenthesefun

    ctions:

    ln

    e

    x

    x

    a

    a

    ;log

    x

    a

    a

    x

    ;

    loga

    x

    a

    x

    ,

    0

    x

    .

    Linksto1.1,geo

    metricsequences;1.2,

    lawsof

    exponentsandlogarithms;2.1,

    inverse

    functions;2.2,g

    raphsofinverses;and6.1,

    limits.

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    Content

    Furtherguidance

    L

    inks

    2.7

    Solvingequations,bothgraphicallyand

    analytically.

    Useoftechnologytosolveavarietyof

    equations,includingthosewherethereisno

    appropriateanalyticapproach.

    Solutionsmaybereferredtoasrootsof

    equationsorzerosoffunctions.

    Linksto2.2,

    fun

    ctiongraphingskills;and2.3

    2.6,equationsin

    volvingspecificfunctions.

    Examples:

    4

    5

    6

    0

    e

    sin

    ,

    x

    x

    x

    x

    .

    Solving

    2

    0

    ax

    bx

    c

    ,

    0

    a

    .

    Thequadraticformula.

    Thediscriminant

    2

    4

    b

    ac

    andthenature

    oftheroots,

    thatis,

    twodistinct

    realroots,

    two

    equalrealroots,norealroots.

    Example:Findk

    giventhattheequation

    2

    3

    2

    0

    kx

    x

    k

    hastwoequalrealroots.

    Solvingexponentialequations.

    Examples:

    1

    2

    10

    x

    ,

    1

    1

    9

    3

    x

    x

    .

    Linkto1.2,exponentsandlogarithms.

    2.8

    Applicationsofgraphingskillsandsolving

    equationsthatrelatetoreal-lifesituations.

    Linkto1.1,geometricseries.

    A

    ppl:Compoundinterest,growthanddecay;

    p

    rojectilemotion;brakingdistance;electrical

    c

    ircuits.

    A

    ppl:Physics7.2.77.2.9,

    13.2.5,

    13.2.6,

    1

    3.2.8

    (radioactivedecayandhalf-life)

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    ic3

    Circularfu

    nctionsandtrig

    onometry

    16

    hours

    Theaimsofthistopicaretoexploreth

    ecircularfunctionsandtosolve

    problemsusingtrigonometry.

    On

    examinationpapers,radianmeasureshouldbe

    assumedunlessotherwiseindicated.

    Content

    Furtherguidance

    L

    inks

    3.1

    Thecircle:radianmeasureofangles;lengthof

    anarc;areaofasector.

    Radianmeasure

    maybeexpressedasexact

    multiplesof,

    ordecimals.

    I

    nt:SekiTakakazucalculatingtoten

    d

    ecimalplaces.

    I

    nt:Hipparchus,MenelausandPto

    lemy.

    I

    nt:Whyarethere360degreesinacomplete

    turn?LinkstoBabylonianmathematics.

    T

    OK:Whichisabettermeasureo

    fangle:

    r

    adianordegree?Whatarethebestcriteria

    b

    ywhichtodecide?

    T

    OK:Euclidsaxiomsasthebuild

    ingblocks

    o

    fEuclideangeometry.

    Linktonon-Euclidean

    g

    eometry.

    3.2

    Definitionofcos

    andsinintermsofthe

    unitcircle.

    A

    im8:WhoreallyinventedPythagoras

    theorem?

    I

    nt:Thefirstworktoreferexplicitlytothe

    s

    ineasafunctionofanangleisthe

    A

    ryabhatiyaofAryabhata(ca.510).

    T

    OK:Trigonometrywasdevelope

    dby

    s

    uccessivecivilizationsandculture

    s.Howis

    m

    athematicalknowledgeconsidere

    dfroma

    s

    ocioculturalperspective?

    Definitionoftana

    ssin

    cos

    .

    Theequationof

    astraightlinethroughthe

    originis

    ta

    n

    y

    x

    .

    Exactvaluesoftrigonometricratiosof

    0,

    ,

    ,

    ,

    6

    4

    3

    2

    andtheirmultipl

    es.

    Examples:

    sin

    ,

    cos

    ,

    tan

    210

    3

    2

    4

    3

    2

    .

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    Furtherguidance

    L

    inks

    3.3

    ThePythagoreanidentity

    2

    2

    cos

    sin

    1

    .

    Doubleangleidentitiesforsine

    andcosine.

    Simplegeometricaldiagramsand/or

    technologymay

    beusedtoillustratethedouble

    angleformulae(andothertrigonometric

    identities).

    Relationshipbetweentrigonometricratios.

    Examples:

    Givensin,

    fin

    dingpossiblevaluesoftan

    withoutfinding

    .

    Given

    3

    cos

    4

    x

    ,andx

    isacute,

    findsin2x

    withoutfinding

    x.

    3.4

    Thecircularfunctionssin

    x,co

    sxa

    ndtan

    x:

    theirdomainsandranges;amplitude,

    their

    periodicnature;andtheirgraphs.

    A

    ppl:Physics4.2

    (simpleharmonicmotion).

    Compositefunctionsoftheform

    (

    )

    sin

    (

    )

    f

    x

    a

    b

    x

    c

    d

    .

    Examples:

    (

    )

    tan

    4

    f

    x

    x

    ,

    (

    )

    2cos

    3(

    4)

    1

    f

    x

    x

    .

    Transformations.

    Example:

    sin

    y

    x

    usedtoobtain

    3sin

    2

    y

    x

    byastretchofscalefactor3inthey-direction

    andastretchofscalefactor

    1 2

    inthe

    x-direction.

    Linkto2.3,

    tran

    sformationofgraphs.

    Applications.

    Examplesinclud

    eheightoftide,motionofa

    Ferriswheel.

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    Content

    Furtherguidance

    L

    inks

    3.5

    Solvingtrigonometricequationsinafinite

    interval,bothgraphicallyandanalytically.

    Examples:2sin

    1

    x

    ,0

    2

    x

    ,

    2sin

    2

    3cos

    x

    x

    ,

    o

    o

    0

    180

    x

    ,

    2tan

    3(

    4)

    1

    x

    ,

    x

    .

    Equationsleadingtoquadraticequationsin

    sin

    ,

    cos

    or

    tan

    x

    x

    x.

    Notrequired:

    thegeneralsolutionoftrigonom

    etricequations.

    Examples:

    2

    2sin

    5cos

    1

    0

    x

    x

    for0

    4

    x

    ,

    2sin

    cos2

    x

    x

    ,

    x

    .

    3.6

    Solutionoftriangles.

    Pythagorastheorem

    isaspecialcaseofthe

    cosinerule.

    A

    im8:Attributingtheoriginofa

    m

    athematicaldiscoverytothewrong

    m

    athematician.

    I

    nt:Cosinerule:Al-KashiandPyt

    hagoras.

    Thecosinerule.

    Thesinerule,

    includingtheambiguouscase.

    Areaofatriangle,

    1

    sin

    2

    ab

    C.

    Linkwith4.2,scalarproduct,notingthat:

    2

    2

    2

    2

    c

    a

    b

    c

    a

    b

    a

    b

    .

    Applications.

    Examplesinclud

    enavigation,problemsintwo

    andthreedimen

    sions,includinganglesof

    elevationandde

    pression.

    T

    OK:Non-Euclideangeometry:anglesum

    on

    a

    globegreaterthan180.

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    ic4

    Vectors

    16

    hours

    Theai

    mofthistopicistoprovideanelementaryintroductiontovectors,includingbothalgebraicandgeom

    etricapproaches.

    Theuseofdynamicgeometry

    softwa

    reisextremelyhelpfultovisualizesituationsinthreedimensions.

    Content

    Furtherguidance

    Links

    4.1

    Vectorsasdisplacementsinthe

    planeandin

    threedimensions.

    Linktothree-dimensionalgeometry,x,y

    andz-

    axes.

    Appl:Physics1.3.2

    (vectorsumsa

    nd

    differences)Physics2.2.2,

    2.2.3

    (v

    ector

    resultants).

    TOK:Howdowerelateatheoryt

    othe

    author?Whodevelopedvectorana

    lysis:

    JWG

    ibbsorOHeaviside?

    Componentsofavector;colum

    n

    representation;

    1 2

    1

    2

    3

    3v v

    v

    v

    v

    v

    v

    i

    j

    k.

    Componentsare

    withrespecttotheunit

    vectorsi,jandk

    (standardbasis).

    Algebraicandgeometricapproachestothe

    following:

    Applicationstosimplegeometricfiguresare

    essential.

    thesumanddifferenceoftw

    ovectors;the

    zerovector,thevectorv;

    Thedifferenceo

    fva

    ndwi

    s

    (

    )

    v

    w

    v

    w

    .Vectorsumsanddifferences

    canberepresentedbythediagonalsofa

    parallelogram.

    multiplicationbyascalar,

    k

    v;parallel

    vectors;

    Multiplicationb

    yascalarcanbeillustratedby

    enlargement.

    magnitudeofavector,v;

    unitvectors;basevectors;i,

    jandk;

    positionvectorsOA

    a

    ;

    AB

    OB

    OA

    b

    a.

    Distancebetwee

    npointsAandBisthe

    magnitudeofA

    B

    .

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    Furtherguidance

    L

    inks

    4.2

    Thescalarproductoftwovectors.

    Thescalarprodu

    ctisalsoknownasthedot

    product.

    Linkto3.6,cosi

    nerule.

    Perpendicularvectors;parallelv

    ectors.

    Fornon-zerovectors,

    0

    v

    w

    isequivalentto

    thevectorsbeingperpendicular.

    Forparallelvect

    ors,

    k

    w

    v,

    v

    w

    v

    w

    .

    Theanglebetweentwovectors.

    4.3

    Vectorequationofalineintwo

    andthree

    dimensions:

    t

    r

    a

    b.

    Relevanceofa

    (position)andb

    (direction).

    Interpretationof

    t

    astimeandb

    asvelocity,

    withb

    representingspeed.

    A

    im8:Vectortheoryisusedfortr

    acking

    d

    isplacementofobjects,

    including

    forpeaceful

    a

    ndharmfulpurposes.

    T

    OK:Arealgebraandgeometrytwoseparate

    d

    omainsofknowledge?(Vectoralgebraisa

    g

    oodopportunitytodiscusshowgeometrical

    p

    ropertiesaredescribedandgenera

    lizedby

    a

    lgebraicmethods.)

    Theanglebetweentwolines.

    4.4

    Distinguishingbetweencoincidentandparallel

    lines.

    Findingthepointofintersection

    oftwolines.

    Determiningwhethertwolinesintersect.

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    ic5

    Statisticsa

    ndprobability

    35

    hours

    Theaimofthistopicistointroducebas

    icconcepts.

    Itisexpectedthatmo

    stofthecalculationsrequiredwillbedoneusingtechnology,

    bute

    xplanationsof

    calculationsbyhandmayenhanceunderstanding.

    Theemphasisisonunde

    rstandingandinterpretingtheresu

    ltsobtained,

    incontext.Statisticaltableswillno

    longer

    beallowedinexaminations.Whilemanyofthecalculationsrequire

    dinexaminationsareestimates,i

    tislikelythatthecommandterms

    writedown,

    find

    andcalculatewillbeused.

    Content

    Furtherguidance

    L

    inks

    5.1

    Conceptsofpopulation,sample,random

    sample,

    discreteandcontinuous

    data.

    Presentationofdata:frequencydistributions

    (tables);frequencyhistogramswithequalclass

    intervals;

    Continuousand

    discretedata.

    A

    ppl:Psychology:descriptivestatistics,

    r

    andomsample(variousplacesintheguide).

    A

    im8

    :Misleadingstatistics.

    I

    nt:TheStPetersburgparadox,Ch

    ebychev,

    P

    avlovsky.

    box-and-whiskerplots;outliers.

    Outlierisdefine

    dasmorethan1.5

    IQR

    from

    thenearestquartile.

    Technologymaybeusedtoproduce

    histogramsandbox-and-whiskerplots.

    Groupeddata:useofmid-interv

    alvaluesfor

    calculations;intervalwidth;upp

    erandlower

    intervalboundaries;modalclass.

    Notrequired:

    frequencydensityhistograms.

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    inks

    5.2

    Statisticalmeasuresandtheirin

    terpretations.

    Centraltendency:mean,median,mode.

    Quartiles,percentiles.

    Onexamination

    papers,datawillbetreatedas

    thepopulation.

    Calculationofm

    eanusingformulaand

    technology.

    Studentsshouldusemid-interval

    valuestoestimatethemeanofgroupeddata.

    A

    ppl:Psychology:descriptivestatistics

    (

    variousplacesintheguide).

    A

    ppl:Statisticalcalculationstoshowpatterns

    a

    ndchanges;geographicskills;sta

    tistical

    g

    raphs.

    A

    ppl:Biology1.1.2

    (calculatingm

    eanand

    s

    tandarddeviation);Biology1.1.4

    (comparing

    m

    eansandspreadsbetweentwoor

    more

    s

    amples).

    I

    nt:Discussionofthedifferentfor

    mulaefor

    v

    ariance.

    T

    OK:Dodifferentmeasuresofce

    ntral

    t

    endencyexpressdifferentpropertiesofthe

    d

    ata?Arethesemeasuresinvented

    or

    d

    iscovered?Couldmathematicsmake

    a

    lternative,equallytrue,formulae?

    Whatdoes

    t

    histellusaboutmathematicaltruths?

    T

    OK:Howeasyisittoliewithstatistics?

    Dispersion:range,interquartile

    range,

    variance,standarddeviation.

    Effectofconstantchangestoth

    eoriginaldata.

    Calculationofstandarddeviation/variance

    usingonlytechn

    ology.

    Linkto2.3,

    tran

    sformations.

    Examples:

    If5issubtracted

    fromallthedataitems,then

    themeanisdecr

    easedby5,

    butthestandard

    deviationisunchanged.

    Ifallthedataite

    msaredoubled,

    themedianis

    doubled,

    butthe

    varianceisincreasedbya

    factorof4.

    Applications.

    5.3

    Cumulativefrequency;cumulat

    ivefrequency

    graphs;usetofindmedian,quartiles,

    percentiles.

    Valuesofthemedianandquartilesproduced

    bytechnologym

    aybedifferentfromthose

    obtainedfroma

    cumulativefrequencygraph.

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    L

    inks

    5.4

    Linearcorrelationofbivariated

    ata.

    Independentvar

    iablex,

    dependentvariabley.

    A

    ppl:Chemistry11.3.3

    (curvesof

    bestfit).

    A

    ppl:Geography(geographicskills).

    M

    easuresofcorrelation;geographicskills.

    A

    ppl:Biology1.1.6

    (correlationdoesnot

    i

    mplycausation).

    T

    OK:Canwepredictthevalueof

    xfromy,

    u

    singthisequation?

    T

    OK:Canalldatabemodelledby

    a(known)

    m

    athematicalfunction?Considerthereliability

    a

    ndvalidityofmathematicalmode

    lsin

    d

    escribingreal-lifephenomena.

    Pearsonsproductmomentcorrelation

    coefficientr.

    Technologyshouldbeusedtocalculater.

    However,handcalculationsofrmayenhance

    understanding.

    Positive,zero,negative;strong,weak,no

    correlation.

    Scatterdiagrams;linesofbestfit.

    Thelineofbest

    fitpassesthroughthemean

    point.

    Equationoftheregressionlineofyonx.

    Useoftheequationforpredictionpurposes.

    Mathematicalandcontextualinterpretation.

    Notrequired:

    thecoefficientofdetermination

    R2

    .

    Technologyshouldbeusedfindtheequation.

    Interpolation,ex

    trapolation.

    5.5

    Conceptsoftrial,outcome,equallylikely

    outcomes,samplespace(U)andevent.

    Thesamplespac

    ecanberepresented

    diagrammaticall

    yinmanyways.

    T

    OK:Towhatextentdoesmathem

    aticsoffer

    m

    odelsofreallife?Istherealways

    afunction

    t

    omodeldatabehaviour?

    TheprobabilityofaneventAis

    (

    )

    P(

    )

    (

    )

    nA

    A

    nU

    .

    ThecomplementaryeventsAan

    dA

    (notA).

    UseofVenndiagrams,treediagramsand

    tablesofoutcomes.

    Experimentsusi

    ngcoins,dice,cardsandsoon,

    canenhanceund

    erstandingofthedistinction

    between(experimental)relativefrequencyand

    (theoretical)pro

    bability.

    Simulationsmay

    beusedtoenhancethistopic.

    Linksto5.1,

    frequency;5.3,cumulative

    frequency.

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    inks

    5.6

    Combinedevents,P(

    )

    A

    B

    .

    Mutuallyexclusiveevents:P(

    )

    0

    A

    B

    .

    Conditionalprobability;thedef

    inition

    P(

    )

    P

    |

    P(

    )

    A

    B

    A

    B

    B

    .

    Independentevents;thedefiniti

    on

    P

    |

    P()

    P

    |

    A

    B

    A

    A

    B

    .

    Probabilitieswithandwithoutr

    eplacement.

    Thenon-exclusi

    vityofor.

    Problemsareoft

    enbestsolvedwiththeaidofa

    Venndiagramortreediagram,withoutexplicit

    useofformulae.

    A

    im8:Thegamblingissue:useof

    probability

    i

    ncasinos.Couldorshouldmathem

    aticshelp

    i

    ncreaseincomesingambling?

    T

    OK:Ismathematicsusefultome

    asurerisks?

    T

    OK:Cangamblingbeconsideredasan

    a

    pplicationofmathematics?(Thisisagood

    o

    pportunitytogenerateadebateon

    thenature,

    r

    oleandethicsofmathematicsregardingits

    a

    pplications.)

    5.7

    Conceptofdiscreterandomvariablesandtheir

    probabilitydistributions.

    Simpleexample

    sonly,suchas:

    1

    P(

    )

    (4

    )

    18

    X

    x

    x

    for

    1,

    2,

    3

    x

    ;

    5

    6

    7

    P(

    )

    ,

    ,

    18

    18

    18

    X

    x

    .

    Expectedvalue(mean),E()Xf

    ordiscretedata.

    Applications.

    E(

    )

    0

    X

    indicatesafairgamewhereX

    representsthegainofoneoftheplayers.

    Examplesinclud

    egamesofchance.

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    inks

    5.8

    Binomialdistribution.

    Meanandvarianceofthebinom

    ial

    distribution.

    Notrequired:

    formalproofofmeanandvariance.

    Linkto1.3,

    bino

    mialtheorem.

    Conditionsunde

    rwhichrandomvariableshave

    thisdistribution.

    Technologyisusuallythebestwayof

    calculatingbinomialprobabilities.

    5.9

    Normaldistributionsandcurves

    .

    Standardizationofnormalvariables(z-values,

    z-scores).

    Propertiesofthenormaldistribution.

    Probabilitiesand

    valuesofthevariablemustbe

    foundusingtech

    nology.

    Linkto2.3,

    transformations.

    Thestandardizedvalue(z

    )givesthenumber

    ofstandarddeviationsfromthemean.

    A

    ppl:Biology1.1.3

    (linkstonorm

    al

    d

    istribution).

    A

    ppl:Psychology:descriptivestatistics

    (

    variousplacesintheguide).

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    Top

    ic6

    Calculus

    40

    hours

    Theaimofthistopicistointroducestudentstothebasicconceptsandtech

    niquesofdifferentialandintegral

    calculusandtheirapplications.

    Content

    Furtherguidance

    L

    inks

    6.1

    Informalideasoflimitandconv

    ergence.

    Example:0.3,0.3

    3,

    0.3

    33,...convergesto

    1 3

    .

    Technologyshouldbeusedtoexploreideasof

    limits,numericallyandgraphically.

    A

    ppl:Economics1.5

    (marginalco

    st,marginal

    r

    evenue,marginalprofit).

    A

    ppl:Chemistry11.3.4

    (interpretingthe

    g

    radientofacurve).

    A

    im8:ThedebateoverwhetherN

    ewtonor

    L

    eibnitzdiscoveredcertaincalculu

    sconcepts.

    T

    OK:Whatvaluedoestheknowledgeof

    limitshave?Isinfinitesimalbehaviour

    a

    pplicabletoreallife?

    T

    OK:Opportunitiesfordiscussing

    hypothesis

    f

    ormationandtesting,andthenthe

    formal

    p

    roofcanbetackledbycomparing

    certain

    c

    ases,

    throughaninvestigativeapp

    roach.

    Limitnotation.

    Example:

    2

    3

    lim

    1

    x

    x x

    Linksto1.1,infi

    nitegeometricseries;2.52.7,

    rationalandexponentialfunctions,and

    asymptotes.

    Definitionofderivativefromfirstprinciplesas

    0

    (

    )

    ()

    ()

    lim

    h

    f

    x

    h

    f

    x

    f

    x

    h

    .

    Useofthisdefin

    itionforderivativesofsimple

    polynomialfunc

    tionsonly.

    Technologycouldbeusedtoillustrateother

    derivatives.

    Linkto1.3,

    bino

    mialtheorem.

    Useofbothform

    sofnotation,

    d dy x

    and

    f

    x

    ,

    forthefirstderivative.

    Derivativeinterpretedasgradientfunctionand

    asrateofchange.

    Identifyinginter

    valsonwhichfunctionsare

    increasingordecreasing.

    Tangentsandnormals,andtheirequations.

    Notrequired:

    analyticmethodsofcalculating

    limits.

    Useofbothanalyticapproachesand

    technology.

    Technologycan

    beusedtoexploregraphsand

    theirderivatives.

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    Furtherguidance

    Links

    6.

    2

    Derivativeof

    (

    )

    n

    x

    n

    ,sinx,cosx,

    tanx

    ,

    ex

    andlnx

    .

    Differentiationofasum

    andarealmultipleof

    thesefunctions.

    Thechainruleforcompositefu

    nctions.

    Theproductandquotientrules.

    Linkto2.1,com

    positionoffunctions.

    Technologymay

    beusedtoinvestigatethechain

    rule.

    Thesecondderivative.

    Useofbothform

    sofnotation,

    2

    2

    d d

    yx

    and

    ()

    f

    x

    .

    Extensiontohigherderivatives.

    d dn

    nyx

    and

    (

    )

    n

    f

    x

    .

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    L

    inks

    6.3

    Localmaximumandminimumpoints.

    Testingformaximumorminimum.

    Usingchangeof

    signofthefirstderivativeand

    usingsignofthe

    secondderivative.

    Useoftheterms

    concave-upfor

    (

    )

    0

    f

    x

    ,

    andconcave-do

    wnfor

    (

    )

    0

    f

    x

    .

    A

    ppl:profit,area,volume.

    Pointsofinflexionwithzeroandnon-zero

    gradients.

    Atapointofinflexion,

    (

    )

    0

    f

    x

    andchanges

    sign(concavitychange).

    (

    )

    0

    f

    x

    isnot

    asufficientconditionfora

    pointofinflexion:forexample,

    4

    y

    x

    at(0,

    0).

    Graphicalbehaviouroffunction

    s,

    includingtherelationshipbetwe

    enthe

    graphsoff,

    f

    and

    f.

    Optimization.

    Bothglobal(forlarge

    x

    )andlocal

    behaviour.

    Technologycan

    displaythegraphofa

    derivativewitho

    utexplicitlyfindingan

    expressionforth

    ederivative.

    Useofthefirsto

    rsecondderivativetestto

    justifymaximum

    and/orminimumvalues.

    Applications.

    Notrequired:

    pointsofinflexionwhere

    (

    )

    f

    x

    isnotdefined:

    forexample,

    1

    3

    y

    x

    at(0,0).

    Examplesinclud

    eprofit,area,volume.

    Linkto2.2,grap

    hingfunctions.

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    6.4

    Indefiniteintegrationasanti-dif

    ferentiation.

    Indefiniteintegralof

    (

    )

    n

    x

    n

    ,sin

    x

    ,cosx,

    1 x

    andex.

    1

    d

    ln

    x

    x

    C

    x

    ,

    0

    x

    .

    Thecompositesofanyofthese

    withthelinear

    functionax

    b

    .

    Example:

    1

    ()

    cos(2

    3

    )

    ()

    sin(2

    3)

    2

    f

    x

    x

    fx

    x

    C

    .

    Integrationbyinspection,orsubstitutionofthe

    form

    (())'()d

    fgx

    g

    x

    x

    .

    Examples:

    4

    2

    2

    2

    1

    d

    ,

    sin

    d

    ,

    d

    sin

    cos

    x

    x

    x

    x

    x

    x

    x

    x x

    .

    6.5

    Anti-differentiationwithaboun

    darycondition

    todeterminetheconstantterm.

    Example:

    if

    2

    d

    3

    dy

    x

    x

    x

    and

    10

    y

    when

    0

    x

    ,then

    3

    2

    1

    1

    0

    2

    y

    x

    x

    .

    I

    nt:Successfulcalculationofthev

    olumeof

    thepyramidalfrustumbyancientE

    gyptians

    (

    EgyptianMoscowpapyrus).

    U

    seofinfinitesimalsbyGreekgeo

    meters.

    Definiteintegrals,

    bothanalyticallyandusing

    technology.

    (

    )d

    ()

    (

    )

    b a

    g

    x

    x

    gb

    g

    a

    .

    Thevalueofsom

    edefiniteintegralscanonly

    befoundusingtechnology.

    A

    ccuratecalculationofthevolume

    ofa

    c

    ylinderbyChinesemathematician

    LiuHui

    Areasundercurves(betweenthecurveandthe

    x-axis).

    Areasbetweencurves.

    Volumesofrevolutionaboutthex-axis.

    Studentsareexp

    ectedtofirstwriteacorrect

    expressionbeforecalculatingthearea.

    Technologymay

    beusedtoenhance

    understandingofareaandvolume.

    I

    nt:IbnAlHaytham:firstmathematicianto

    c

    alculatetheintegralofafunction,

    inorderto

    f

    indthevolumeofaparaboloid.

    6.6

    Kinematicproblemsinvolvingd

    isplacements,

    velocityvandaccelerationa.

    d ds

    v

    t

    ;

    22

    d

    d

    d

    d

    v

    s

    a

    t

    t

    .

    A

    ppl:Physics2.1

    (kinematics).

    Totaldistancetravelled.

    Totaldistancetravelled

    21

    d

    t t

    v

    t

    .