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10AuthoringMathematicalQuestions
10.1AuthoringMathematicalQuestions
MapleT.A.providescross-platformsupportforlearningcontentthatincludesnumericorsymbolicmathematicsex-
pressions,andchemistryequationsandexpressions.
FundamentalCapabilities
• IntegratedmathematicsparserandMathSyntaxforexpressions
• Maple-gradedexpressions
• Gradedfree-responsemathematicsquestionscontainingnumericresponses,equations,formulas,physicalunits,
andcombinationsofnumbers,formulas,andunits
• MathMLsupport
• InlineexpressionrenderingfromgraphingcalculatorsyntaxtoformattedMathMLexpressions
• Systemevaluationofequivalentsymbolic,numeric,equation,andunitexpressions
• Automaticrecognitionofscientificnotationintheanswerfieldandinstudentresponses
• Controloverprecision,significantdigits,andmarginoferror/toleranceinquestions
• SupportforquestionsusingMaple
• SupportforquestionsthatuseplotsgeneratedbyMaple
MathematicsQuestionTypes
MapleT.A.offers14varietiesofmathematicalandscientificfreeresponsequestiontypes(includingformula,numeric,
andquestionsthatusetheMaplemathematicalsoftwaretogradeandplotstudentresponses).
SymbolicandEquationQuestionTypes
• Maple-gradedFormula(page161)
• MapleSyntax(page162)
• Formula(page156)
• RestrictedFormula(page170)
• FormulawithUnits(page159)
• ConstantsofIntegrationQuestions(page155)
• Equation(page156)
• MultiFormula(page164)
• FormulaList(page158)
• VectorsofFormulas(page170)
• GraphPlottinginLaTeXQuestions(page279)
NumericQuestionTypes
• Numeric(page166)(includingnumberswithphysicalunits)
• LabelingGraphicsinLaTeXQuestions(page283)
• Matrix(page164)
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10.2BasicMathSyntaxintheSystem
• MathematicalExpressionSyntax(page202)
• VariableNamesSyntax(page202)
• OperatorandFunctionSyntax(page202)
WritingMathematicalExpressions
Thebasicsyntaxforenteringmathematicalformulasorexpressionsinthesystemenablesyoutoquicklyenterexpressions
usingsymbolicmath.Youcantypeformulasusingstandardmathematicalnotation(similartothatusedforagraphing
calculator)and,ingeneral,thesystemcorrectlyinterpretsit.
Forexample,thefollowingformulaisacceptable.
(x^2-2x+1) 2sin(x) ) (x^2+1)e^(-x^2)
Note:Ifaproductincludesoneormorevariables,alwaysuseanasterisk"*".Forexample,specify2*x*yor2*$A.
ForaMaplequestion,youmustalwaysincludeanasterisk(*).
(x^2-2*x+1)*2*sin(x)*(x^2+1)*e^(-x^2)
Themostcommonmistakeistoforgetparentheses"()".Forexample,theexpression:
1/(x+1)
isdifferentfrom:
1/x+1
whichthesysteminterpretsas:
Alternatively,youcanuseMathMLexpressions,whicharesupportedforbothdisplayandcontentwithinthesystem.
VariableNames
Youcanuseanyletterforavariablename.
Note:Thegradingsystemiscase-sensitivewithrespecttovariablenames.Soifthecorrectansweris:
(t+1)^2
thentheresponse
T^2 + + 2T T +1
isgradedwrong,but
t^2 + + 2t t +1
iscorrect.
Note:BecertainthatthecaseofletterslikexandXorvandVinanswersmatchthecaseofthevariablesinquestions.
OperatorandFunctionSyntax
Formultiplication,useanasterisk"*".Alternatively,simplywrite:
• Twoletterswithaspace(forexample,xy)
202 • 10AuthoringMathematicalQuestions
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• Aletterandanumberwithorwithoutaspace(forexample,2xor2 x).Notethattwoletterswithouta
spacerepresentsasinglevariablename.
• ForMaple-gradedquestions,youmustincludeanasterisk*,forexample2*xandnot2x,x*yandnot
xy,orx*yandnotxy,whichwillbemarkedasincorrect.
For:
• Exponentiation,usethecaret"^".
• Euler'sConstant,2.718...,usethelettere.(Theexponentialfunctionise^x.)ForMaple-gradedquestions,use
theexpfunction:exp(1)andexp(x)foreande^x.
• Scientificnotation,usetheuppercaseletterE.(Forexample,0.078is7.8E-2.)
• Thetrigonometricconstant,3.141...,usepi.ForMaple-gradedquestions,usePi.IntheEquationEditor
(MathMLEditor),selectPifromthepalette.
• Commonmathematicalfunctions,usethestandardabbreviatednames(forexample,sin,cos,andtan).
• Theargumentsoftrigonometricfunctions,remembertheyaremeasuredinradians.
• Thesquarerootfunction,usesqrt(x).(Alternatively,usex^(1/2)orx^0.5.)InTextentrymodethesquare
rootfunctionissqrt(x).(Alternatively,youcanenterx^(1/2)orx^0.5.)IntheMathMLEditor,select
thesquarerootsymbolfromthepalette.
• Theinversetrigonometricfunctions,usethestandardabbreviatednames(arcsin(x),arccos(x),and
arctan(x)).
Note:ThevariableeshouldnotbeassignedasaglobalvariableinMaplecode.Itcanbeused,however,asalocalvariable.
SeeAlso:
EditingwiththeEquationEditor(page210)
MathematicalFunctions&Operations(page224)
Maple-gradedQuestionType-Overview(page159)
10.3UnderstandingtheMathCapabilitiesintheSystem
TheMapleT.A.systemhaspowerfulquestiontypesandcapabilitiesdesignedfortechnicalandscientificeducators.
Itgradesresponsesthatcontainsymbolicornumericanswers.Therearemanyrestrictionsandextensionstothetype
ofanswersthatinstructorsaccept,whichdeterminethevarietyofquestiontypesinthesystem.
Learnmoreaboutthesystemmathcapabilities.
• Howarequestionswithalgebraicandnumericmathauthored?AuthoringMathematicalQuestions(page201)
• Whyaretheresomanyvarietiesofmathematicsquestions?MathQuestionTypes-Overview(page152)
• HowcanIcontrolmarginoferrorandtolerance?SettingaMarginofErrorinNon-numericQuestions(page206)
• CanthesystemdisplayWYSIWYGmathematics?MathQuestionTypes-Overview(page152)
• Doesthesystemaccept"graphingcalculator"stylemathentry?UsingFormulaSyntax(page209)
• Whatsyntaxconstraintsareplacedontheprocessingofmathexpressions?BasicMathSyntaxintheSystem(page202)
• Whatfunctionsandoperatorsaresupported?MathematicalFunctions&Operations(page224)
• Whatunitsdoestheparserunderstand?DefaultTableofPhysicalUnitEquivalents(page225)
• DoesthesystemsupportMathML?MathMLintheQuestions(page218)
• DoesthesystemsupportLaTeX?LaTeXAuthoringOverview(page267)
10.3UnderstandingtheMathCapabilitiesintheSystem • • 203
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10.4StudentResponsesandGrading
Thissectionprovidesdetailedinformationrelatedtogradingofstudentresponses.
StudentResponses
Onanassignment,studentscanopttoentertheirresponsesinSymbolmodeorTextmode.Thissettingappliestoany
questiontypethatsupportsbothmodes.Forotherquestiontypes,aspecializedinputfieldispresentedtothestudents.
Formoreinformationontheinputfieldforaspecificquestiontype,seethesectiononthatquestiontypeinQuestion
Types(page151).ForMaplesyntaxquestions,theinstructorspecifiesSymbolmodeorTextmodeforresponsesinthe
questionitself.Formoreinformation,seeMaple-gradedQuestionsintheQuestionEditor(page99).
ControllingAnswerFormatinNumericQuestions
Note:YoucanusetheQuestionEditor'son-screendialogboxestosettheanswerformat.Formoreinformation,seeSettingAnswer
FormatinNumericQuestions(page86).
Studentresponsesdonotneedtobeformattedidenticallytoyouranswertobecorrect.Youcanspecifynotationsthat
thestudentisallowedtouse.
NumberStyle
UsethenumStylefieldtosettheacceptableformattingforstudentresponses.Listtheacceptablenumberstyles(in
aspace-separatedlist)asthevalue.
Theavailablenumberstylesare:
• thousands-responsescontainingcommas(,)asaseparatoraregradedcorrect
• scientific-responsesspecifiedusingscientificnotation,forexample,2.0E2,aregradedcorrect
• dollars-responsescontainingaleadingdollarsign($)aregradedcorrect
• arithmetic-responsesspecifiedusingthearithmeticoperators,thatis,+,-,*,/,(),and^,aregradedcorrect
Forexample:
numStyle = thousands dollars@
allowsstudentstocorrectlyenterresponsesusingcomma(,)separatorsorleadingdollarsigns($).
NegativeNumberStyle
UsethenegStylefieldtosettheacceptableformattingfornegativestudentresponses.SetnegStyletooneof
minus,paren,orboth.
• minus-Negativeresponsesmustbeenteredusinganegativesign(-).
• paren-Negativeresponsesmustbeenteredusingparentheses(()).
• both-Negativeresponsescanbeenteredusinganegativesign(-)orparentheses(()).Note:IfyousetnumStyle
toincludearithmetic,thenyoucannotallowparenthesestobeusedtoindicateanegativeresponse.Thatis,
negStylemustbesettominus.
SeeAlso:
Numeric(page166)
SettingAnswerFormatinNumericQuestions(page86)
204 • 10AuthoringMathematicalQuestions
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ControllingAnswerTolerance
Thereisnotgeneralagreementonhowtogradenumericanswerswithrespecttothenumberofsignificantdigits.For
example,ifthecorrectanswerto3significantdigitsis2.77,someinstructorsgradetheresponse2.77021incorrect.
Otherinstructorsgrade2.77021correctbecausetheythinkthattheinclusionofextradigitsdoesnotmakearesponse
incorrect.
InMapleT.A.youcontrolhownumericresponsesaregraded.Forexample,youcan:
• Acceptonly2.77(andnot2.77021).Formoreinformation,seeSettingthePrecision.
• Acceptallanswersthatagreewiththecorrectanswerto3(oranotherpositiveinteger)significantfigures(that
is,anyresponsebetween2.765and2.775iscorrect).Formoreinformation,seeSettingaMarginofError.
ForMaple-gradedquestions,seeSignificantDigitsinMapleQuestions(page206).
SettingthePrecision
Forthenumericquestiontype,youcanspecifythatthestudentresponseexactlymatchthecorrectanswer,orspecify
thatinadditiontomatchingthevalueofyourcorrectanswerthestudentresponsehaveaspecifiednumberofsignificant
digits.
Studentresponsesthatarecorrectuptothespecifiednumberofsignificantdigitsreceive50%credit.
ForinformationonsettingtheprecisionforanumericquestionintheQuestionEditor,seeSettingAnswerTolerance
inNumericQuestions(page86).Forrelatedplain-textscriptfileexamples,seeAnswerToleranceExample
Script(page305).
Forotherformulaquestions,unlessamarginoferrorisspecified,thevalueofthestudentresponsemustmatchthat
ofthecorrectansweruptothetolerancespecifiedbythetolerancefield,whichisbydefaultisthesystemfloating-
pointlimit,1.0E-9.
SettingaMarginofError
Youcanspecifyamarginoferrorinoneoftwoways:
• Percentage: ± x%.Thisrequiresthestudentresponsetobewithinaspecifiedpercentageofthecorrect
answer.
• Absolute: ± x.Thisrequiresthestudentresponsetobewithinaspecifiednumberofthecorrectanswer.
ForinformationonsettingamarginoferrorforanumericquestionintheQuestionEditor,seeSettingAnswerTol-
eranceinNumericQuestions(page86).
Forinformationonsettingamarginoferrorforotherquestiontypes,seeSettingaMarginofErrorinNon-numeric
Questions(page206).
Forrelatedplain-textscriptfileexamples,seeAnswerToleranceExampleScript(page305).
Note:Itisrecommendedthatyoudevelopapolicyforthetoleranceinresponsesatthebeginningofaproject,sothateverynumeric
questionusessimilarpolicies.
SeeAlso:
Tutorial:SettingMarginofToleranceinNon-numericQuestions(page235)
lsu(n,x)(page245)
decimal(n,x),sig(n,x),int(x)(page242)
10.4StudentResponsesandGrading • • 205
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SettingaMarginofErrorinNon-numericQuestions<
Note:UsetheQuestionEditor'son-screendialogboxesforsettingthemarginoferrorinnumericandnumeric-with-unitquestions.
Formoreinformation,seeSettingAnswerFormatinNumericQuestions(page86).
Foranintroductiontotheconceptofmarginoferror,seeControllingAnswerTolerance(page205).
Theexactanswerisnotalwaysrequired.InMapleT.A.,youcanspecifyamarginoferror,withinwhichstudentre-
sponsesaregradedcorrect,foranymathematicalquestiontype.Youcanspecifyamarginoferrorforanswersthatare
staticordefinedusingalgorithmicvariables.
Use"?"intheanswerfieldtosetamarginoferror.The?(plusorminus)operatorrepresents"+/-"inordinarymath
notation.Youcanspecifyanabsolutetolerance.Forexample,toaccept2.5 +/- 0.25,thatis,anyresponsebetween
2.25and2.75(inclusive),use2.5 ? 0.25.Youcanalsospecifyapercentagetolerance.Toaccept5.12 +/-
5%,thatis,anyresponsebetween4.864and5.376(inclusive),use5.12(1 ? 0.05).
Note:The?operatorcannotbeusedinanalgorithmicvariabledefinition(orotheralgorithmicstatement).
FornumericandMaple-gradedquestions,youcannotusethe"?"operator.Forinformationonspecifyingamargin
oferrorforanumericquestion,seeSettingAnswerToleranceinNumericQuestions(page86).Tospecifyamargin
oferrorinaMaple-gradedquestion,useMaplecode.
Considerthefollowingwhenworkingwithmarginsoferror:
• Precedenceofthe?Operator:The?operatorisofthesameprecedenceas"+"and"-".Forexample,2*3 ?
0.5isthesameas6 ? 0.5and2*(3 ? 0.5)isthesameas6 ? 1.
• Units:Becauseexpressionswithunitshavedifferentdimensionsfrompurenumbers,youmustspecifyadimen-
sionedmarginoferrorforadimensionedanswer.Tospecify"2kgwithatoleranceof0.1kg",enter2kg ?
0.1kgor(2 ? 0.1)kg.Tospecify"50cmwithin5%",enter(1 ? 0.05)50cm.
Important:Theexpression50cm ? ? 2isinvalid.Itisamixedexpression.Theleft-handsidehasdimensionlength.
Theright-handsideisdimensionless(apurenumber).
• The?operatorisnotavailabletostudents:Astudentcannotspecifyamarginoferror.Theaboveissuesdo
notariseforstudents.
Formoreexamplesandinformation,seeTutorial:SettingMarginofToleranceinNon-numericQuestions(page235).
SeeAlso:
AnswerToleranceExampleScript(page305)
SignificantDigitsinMapleQuestions
TheMapleDigitsenvironmentvariablecontrolsthenumberofdigitsthatMapleuseswhencalculatingwithsoftware
floating-pointnumbers.
• ThedefaultvalueofDigitsis10.
• ThevalueofDigitsischangedbyusingtheassignmentoperator.Forexample,Digits:= 20;
UsingtheevalfCommandtoSettheValueoftheDigitsVariable
Theparametern,whichcanbegivenasasecondargument(evalf(expr, n)),setsthevalueofDigitsfora
singlecomputation.
Toroundtheexactanswer,assigntheexactanswertoadummyvariable,andthenusetheevalf(expr,n)command
intheevalbcommandtocomparethestudentresponsetotheroundedanswer.
206 • 10AuthoringMathematicalQuestions
Example
Calculatethequadraticmeanofthefollowinglistofdata:[695,607,511,588,710,500,682,515,709,478,497,657,
652,634,488,615,714,579,589,493,584,484,503,711,622,695].Theanswerrequires5significantdigits.
1. Firstcalculatetheexactnumericanswer.
2. Roundittotherequirednumberofsignificantdigits.Forexample:
exact_answer:=evalf(Statistics[Mean](<695, 607,
511, 588, 710, 500, 682, 515, , 709, 478, , 497,
657, 652, 634, 488, 615,
714, 579, 589, 493, 584, 484, 503, 711, 622, 695>)):
rounded_answer:=evalf(exact_answer,5):
evalb($RESPONSE=rounded_answer);
UsingMapleCodetoPreventCheatinginMapleSyntaxQuestions
Astheauthor,youcanverifywhetherastudentisusingaMaplecommandtocalculatetheanswer.
ThefollowingexamplescriptsillustratecodethatallowsandpreventspotentialcheatingbythestudentusingMaple
commands.
Examples
Thestudentisrequiredtofactor(x^2-1).Theexpectedcorrectresponseis(x-1)*(x+1).
Inthisexamplequestioncode,thestudentcouldeasilyusetheMaplefactorcommandtoobtaintheanswer.
question=Factor x^2-1.@
maple=evalb($RESPONSE=factor(x^2-1))@
type=maple@
mode=Maple@
allow2d=0@
name=Allow Cheating@
editing=useHTML@
Inthisexample,thecodeverifiesifthefactorcommandhasbeenusedintheanswer.Thestudentresponsewillbe
markedincorrectiftheMaplefactorcommandisused.
question=Factor(x^2-1)@
maple=evalb(0=StringTools[Search]("factor","$RESPONSE")) and d evalb($RESPONSE=factor(x^2-1))@
type=maple@
mode=Maple@
allow2d=0@
name=Prevent Cheating@
editing=useHTML@
IfyouareconcernedaboutstudentsusingMaplecommandstocalculateanswers,considerusingMaple-graded
Formulatocreateyourquestion.Studentsrespondwithalimitednumberofexpressionsandfunctions.Assuch,you
donotneedtoexplicitlycheckthatthestudentresponseisaMaplecommand.Fordetails,seetheSyntaxandStudent
ResponsesubsectionsofMaple-gradedFormula(page161).
SeeAlso:
Maple-gradedQuestionType-Overview(page159)
MapleSyntax(page162)
EnteringChemistryExpressions
Therearetwowaystoentertheanswertoaquestionofthechemistryquestiontype.
10.4StudentResponsesandGrading • • 207
TextMode
ThesystemdefaultmodeisTextMode,inwhichyouentersymbolicornumericmathexpressionsinasimplekeyboard
notation,muchlikeagraphingcalculator.
Table10.1:GuidelinesforEnteringChemistryExpressions
Examples
Rule
H_2O
Entersuperscriptsusingthe
caret"^"character,andsub-
scriptsusingtheunderscore"_".
SuperscriptsandSubscripts
H_2SO_4
->
Usethetext->,<-,<=>forar-
rows.
ArrowsinEquations
<-
<=>
*
Use"*"forthecenterdotoper-
ator.
OtherOperators
+
Usethe"+"signinequations.
Note:Nootheroperationsareallowedinequations.Youcanusethe"-"signtoindicateion
charges(seebelow.)
Ca^2+(aq)+(CO_3)^2-(aq)->CaCO_3(s)
Besuretoincludephysical
states(inparentheses)ifyour
equationrequiresthem.
PhysicalStates,
Ioncharges,andParentheses
Usethe"+"and"-"characters
forpolarityandioncharges.
Useparenthesestoclarifyinter-
pretationofgroupsofcharac-
ters.
SymbolMode
Forchemistryformulaquestions,youcanalsouseSymbolModeentry,whichallowsyoutoenterformulasina
WYSIWYG("Whatyouseeiswhatyouget")editingwindow.Youcanpickarrows,symbolsoroperatorsfroma
palette,andseeyourexpressionsappearasyoutype.SelectEquationEditorintheEditQuestionwindowofthe
MathematicalFormulaquestiontypetodisplaythesymbolpalette.
RulesforChemistryExpressionsandEquations
• Spacesareignoredinexpressions.
• Parenthesesmaybeusedtoclarifyphysicalstates,compoundgroupings,andapplicationofioncharges.Use
theformulapreviewlinktopreviewyourexpressions.
• Ingradingequations,anycharactersyouencloseinparentheses(forexample,ioncharges,superscripts,subscripts,
etc.)areassociatedwiththeprecedingterm(notthenextterm).Example:Pb^(2+)isPb
2+
• Fornumbersthatarenotsuperorsubscriptedthatprecedeacompoundorelement,thegraderassociatesthe
numberwiththetermthatfollows.Example:2H_2Ois2H
2
O
SeeAlso:
Chemistry(page154)
208 • 10AuthoringMathematicalQuestions
PlottingaStudentResponse
MapleT.A.usesMapleplottingfeaturestographstudentresponsesorafunctionderivedfromastudentresponse,for
example,thedefiniteintegralofthestudentresponse.Mapleprovidesmanywaysofrepresentingdataandmathemat-
icalexpressionsgraphicallyusingplots.
PlottingastudentresponseisavailableonlywiththeMaple-gradedQuestionType.
PlottingaStudentResponse
WhenenteringcodeforplottingintheQuestionEditor,useonlyoneMapleplotcommand.Itmustbethelastcommand
entered.YoumustusethelongformnameforMaplepackagefunctions.
Example
ThefollowingisthesourcefileforaMaple-gradedFormulaquestionthatplotsthestudentresponse.
question=Enter an expression in x@
maple=true@
maple_answer=@
plot=plot($RESPONSE,x=0..10)@
type=formula@
mode=Maple@
name=2D Maple plot@
editing=useHTML@
ThefollowingisthesourcecodeforaMapleSyntaxquestionthatplotsthestudentresponseusingtheMapletubeplot
command.StudentsmustuseMaplesyntaxintheirresponse.
question=Enter a set t of f spacecurves s using g Maple syntax, , for r example, [cos(t),sin(t),0]@
maple=true@
maple_answer=@
plot=plots[tubeplot]($RESPONSE, t=0..4, radius=.05, numpoints=70)@
type=maple@
mode=Maple@
name=Tube Plot@
editing=useHTML@
Important:
YoumustusethelongformnameforMaplepackagefunctions.Youmusttestyourplotcodepriortopublishingassignments.
MapleT.A.doesnotdisplayerrormessagesiftheplotdoesnotworkcorrectly.
SeeAlso:
Maple-gradedQuestionsintheQuestionEditor(page99)
Maple-gradedQuestionType-Overview(page159)
Tutorial:Maple-basedAlgorithmicQuestionAuthoring(page263)
10.5UsingFormulaSyntax
IntheMapleT.A.system,youcancreatequestionsthatusemathematicalformulasorexpressions.IntheQuestion
Editor,youcanenterformulasusingstandardmathematicalnotation.Ingeneral,thesystemacceptsthesameformulas
asagraphingcalculator.Thus,thesystemaccepts:
x^2-2x+1
2sin(x)
(x^2+1)e^(-x^2)
Note:Themostcommonmistakeistoforgetparentheses"()".Forexample,1/(x+1)isdifferentfrom1/x+1,whichthesystem
interpretsas(1/x)+1.
10.5UsingFormulaSyntax • • 209
SeeAlso:
BasicMathSyntaxintheSystem(page202)
MathematicalFunctions&Operations(page224)
10.6UsingtheEquationEditortoCreateFormattedMathExpressions
Theequationeditor,showninFigure10.1,providesaneasywaytocreateformattedmathexpressions.
EditingwiththeEquationEditor
UsingtheEquationEditor,youcanaddsymbolicmathexpressionsandsymbolstothequestiontextandanswerregions
inallquestiontypesavailablethroughtheQuestionEditor.
Figure10.1:EquationEditor
Adding/EditinganExpressionUsingtheEquationEditor
1. ClickthequestiontextregionortheanswertextregionandselecttheSigmaicon,
,inthetoolbar.
2. IntheEquationEditordialog,right-click(Control-clickonMacintosh)the2-DeditorregiontolaunchtheEquation
Editorpalette.Youcanalsoaddanexpressionusingkeyboardshortcuts.
3. ClickOK.
KeyboardShortcuts
TheEquationEditoracceptskeyboardshortcuts.
210 • 10AuthoringMathematicalQuestions
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