5-1

TheMathGoals

• RecognizebothtypesofquestionsthatwillappearontheSATmathsections(StandardMultiple-ChoiceandStudent-ProducedResponse).

• KnowthedirectionsforeachtypeofSATmathproblem.

• KnowthetimingandpacingneededtocompletetheSATmathsections.

• MastertheEssentialsofMathematicsnecessaryforSATmathsuccess.

• ApplytheoffensiveGUARDandtheABCDefensestrategiesforSMCs.

• ApplytheRPSoffenseanddefenseonSPRs.

• KnowtherolecalculatorsplayinSATsuccess.

• DevelopconfidenceabouttheSATmathsections.

• Achieveascaledscoreof_______witharawscoreof_______.

MATH

5-2

SATMathMiniPretest Directions Forquestions1-5,solvetheproblemandenteryouranswerchoiceintheovals.Forquestions6-10,solvetheproblemandenteryouranswerinthegrids.1.Markonlyonecircleinanycolumn.2.Someproblemsmayhavemanycorrectanswers,butgridonlyone.3.Mixednumbersmustbegriddedasimproperfractionsordecimals.4.Decimalanswersmaybetruncatedorrounded,butmustfillthegrid.

1 …… ……………… If (2b + 3) – (4b – 5) = 6, what is the value of (4b – 5) ? A) 5

B) 1

C) -1

D) -5

2 …… ………………

Which of the following points is a solution to the system of equations consisting of the line l (above) and the parabola y = x2 – 4x + 3? A) (3, 0)

B) (2, -1)

C) (1, 0)

D) (-1, 1)

3 …… ………… …… Which of the following expressions is equivalent to (3x + 2)2 – (2x – 3)2? A) 5x2 – 5

B) 5x2 + 13

C) 5x2 + 24x – 5

D) 5x2 + 24x + 13

4 …… …… ……… …

x + 2y = -11 x + 3y = -14

Given the system of equations above, what is the value of (2x + 5y) ? A) 5

B) 31

C) -17

D) -25

5 …… …………… …

Solve 2" + 3 = 3 2 . A) 7.5

B) 1.5

C) -4.5 and 1.5

D) -10.5 and 7.5

5-3

Directions Forquestions6–10,solvetheproblemandenteryouranswerinthegrids.1.Markonlyonecircleinanycolumn.2.Someproblemsmayhavemanycorrectanswers,butgridonlyone.3.Mixednumbersmustbegriddedasimproperfractionsordecimals.4.Decimalanswersmaybetruncatedorrounded,butmustfillthegrid.

6 …… ………………

Find the value of 8'() .

7 …… … ……………

If the circle O above has radius 35, and the length of *+ is between 5 and 6, what is a possible integer value of x ?

8 …… ……….……………

For what value of k does the expression ,-./-.0

+ 1-.0

+ ,-.3-.0

= 4 ? 9 …… ……..………………

If a2 + 7a = 44, find the value of the expression (a – 4). 10 …… …….. ………………

If 5 = −1, and (5 + bi)(5 – bi) = 29, what is a positive value of b?

5-4

THESATSCOUTINGREPORT

TheStartingLineupTheSATwillpresenttwomathsections,oneof20questionstobecompletedin25minuteswithoutacalculatorandtheotherof38questionstobecompletedin55minutes.Thespecificquestiontypesmayvaryfromtesttotest,butthetestmakerwillselectmanyquestionssimilartopast(P)SATs.AccordingtotheCollegeBoard,youcancountonthesamelineupforthequestionsoneachofthetwomathsections:

Section3(20questions-nocalculatorallowed)

• 15StandardMultiple-Choice(#1-15,Part1ofthe25-minutesection)

• 5Student-ProducedResponse(#16-20,Part2ofthe25-minutesection)

Section4(38questions-calculatorallowed)

• 30StandardMultiple-Choice(#1-30,Part1ofthe55-minutesection)

• 8Student-ProducedResponse(#31-38,Part2ofthe55-minutesection)

Inadditiontothisspecificinformationabouttheformatofthequestions,youcancountonthefollowingdistributionofmathquestionsonanytest:

• HeartofAlgebra:19questions– 8inSection3,11inSection4(linearequationsandsystems)

• PassporttoAdvancedMath:16questions– 9inSection3,7inSection4(manipulatingcomplexequations)

• ProblemSolvingandDataAnalysis:17questions– 0inSection3,17inSection4(quantitativeliteracy)

• AdditionalTopicsinMath:6questions– 3inSection3,3inSection4(geometryandtrigonometry)

5-5

ACloserLookattheOppositionYoucanalsocountonthefollowingdistributionofdifficultiesamongthe58mathquestionsoneveryformoftheredesignedSAT:

• 5-7easyquestions(about10%)

• 33-38mediumquestions(about60%)

• 15-18difficultquestions(about30%)

Eachblockofquestiontypes(StandardMultiple-ChoiceandStudent-ProducedResponse)willbeginwitheasierquestionsandbecomeprogressivelymoredifficult.ThismeansthatinSection3,the“nocalculator”section,question#1iseasyandtheitemsbecomemoredifficultupthrough#15.However,thefirstSPRquestion,#16,willbemucheasierthanquestion#15,thelastSMCquestioninthatsection.SimilarlyinSection4,the“calculatorallowed”section,question#1iseasyandtheitemsbecomeprogressivelymoredifficultthrough#30,butthefirstSPRquestion,#31willbeeasierthan#30,thelastSMCquestion.

The Edge … SAT 1600 graduates know where to find all of the easy questions.

ACommonFormationYoushouldalsobeawarethatalmosteverySATmathquestionisaskedinprose(i.e.,insteadof“Solve.”,theSATwillask“Forwhatvaluesofthevariablewilltheexpressionabovebezero?”.Unpreparedstudentsmaybecomefrustratedwithwhatappearatfirstglancetobemostly"wordproblems."Testtakerswiththeedgeknowthatupto35%ofSATmathquestionsarereallyshort“wordy”problems.Althoughtheseshortquestionsmaybeconceptual,manytimestheyarenotdifficult.SAT1600graduateswillbeabletoidentifytheeasiestproblemsbythenumberandlocationofthequestion.Moreimportant,youwillknowthatmanyofthequestionsposedwithwordsareNOTwordproblems,butjustquestionsaskedinproseform. MentalPreparationSinceyouwillhavetoanswer58questionsinonly80minutes,youwillneedtoworkmoreforspeedandlessfortheapprovalofyourmathteacher.Yourmathteachermaywantthoroughandwell-documentedroutestoproblemsolutions,regardlessofthedegreeofdifficulty.TheSAT,ontheotherhand,rewardsonlyresults,notprocesses.Youwillneedtoknowanduseeveryadvantagethatyoupossiblycan,evenifitbreakstheheartofyouralgebrateacher.

5-6

AsanSAT1600grad,youradvantageswillincludeknowingthefollowing:

• thetestformatandtimesforeachsection. • thenumberofquestionsineachsection. • theprecisedirectionsforeachsection. • thetypesofquestionstoexpectineachsection. • whatkindofcalculatortobringandwhentouseit. • wheretofindtheeasiestproblems. • theessentialmathcontentoftheSAT. • offensivetacticstofindrightanswers. • defensivetacticstoavoidwronganswers. • specifictacticsforeachtypeofquestionineachsection. • howtogridstudent-producedresponses. • aplanfortestsuccess.PhilosophyoftheOppositionAccordingtotheCollegeBoard,SATmathsectionswilltestyourmathematicalreasoningandtheabilitytoapplythemathematicalconcepts,skills,andpracticesthataremostusefulacrossabroadrangeofcollegemajorsandcareers.Asaresult,therearethreeareasofemphasis:

1) essentialalgebra,advancedalgebra,andsomegeometry-basedskills2) problemsolvinganddataanalysisinthecontextofrealproblems3) efficientperformanceofimportantmathtasks,usingacalculatorwhenappropriate.

Morethan90%ofthecontentwillbeevenlydistributedacrossthreecategoriesreferredtoastheHeartofAlgebra(linearequationsandinequalities),PassporttoAdvancedMath(manipulationandanalysisofrationalexpressions),andProblemSolvingandDataAnalysis(coreconceptsofsolvingproblemsincludingratios,proportions,rates,andsimplestatistics).TheremainderofthecontentwillbecategorizedasAdditionalTopicsinMath(triangles,circles,congruence,similarity,andrighttriangles,includingquestionsabouttrigonometry).SomeSATquestionswillaskforananswersimilartoquestionsinyourmathtextbooks,butsomewilltestyourabilitytothinkoriginally(likeatestauthor),createorinterprettherepresentationofaproblem,considerandconverttheunitsinvolved,orusedifferentalgebraicpropertiesofexpressionsandequations.Youmayalsobeaskedquestionswithanunusualtwist(e.g.,thevalueofr+6ora–b,whichstatementisfalse,whichvalueisnotasolution,whichistheminimumvaluetomeetthecriteria,etc.).Thetestdesignrewardssustainedattentiontocoremathconceptsandskills,andminimizesanyadvantagefrombriefexperienceswithmathematicaltopics.Ifyouareapoormathstudent,takeadvantageofyournewlyfoundparity.Ifyouareagoodmathstudent,enjoythechallenge.Inanyevent,donotworryaboutyourmathematicsbackground.TheSATisatestonwhicheveryonecandowellwithproblemsolvingskillsandthemasteryofacorefocusedamountofessentialmathematicalcontent.

5-7

StandardFormationsThefirstedgethatSAT1600graduateshaveisthatpriortotakingtheSAT,youwillhaveathoroughunderstandingofthedirectionsforbothStandardMultiple-ChoicequestionsandStudent-ProducedResponses.Otherstudentsmaywastevaluabletimedecipheringthedirectionsduringthetest.Youshouldknowthedirectionsforeachsectionandyoushouldbeabletoaccuratelyparaphraseandexplainallofthetestdirectionswiththetestbookletclosed.HighTechTrainingGraphingcalculatorsarerecommendedforstudentuseontheSAT.Theyarenotrequired,andcanonlybeusedforSection4,thelongermathsection.AnycalculatorcanprobablyhelponsomeoftheStandardMultiple-ChoiceproblemsandsomeoftheStudent-ProducedResponses.

Typesofcalculatorsyoucanuse

ontheSAT:

• graphingcalculator• scientificcalculator• businesscalculator• 4-functioncalculator

TypesofcalculatorsyoucanNOTuseontheSAT:

• laptopcomputers • pocketorganizers

• phoneswithcalculatorcapabilities • calculatorswithprintingcapability

Somestudieshaveindicatedapositivecorrelationbetweenstudentcalculatorusageandstudentperformanceonstandardizedtests.TheSATrecognizesthatacalculatorisatoolandspecificallydesignedonesectionofthetesttobecompletedwithoutone.Forthecalculatorsectionofthetest,itisexpectedthatyouhavetheabilitytodeterminewhenitisappropriatetousethecalculator.ThetestmakerrecommendscalculatorusagebystudentsforsomeproblemsontheSATcalculator-allowedsection.HavingexaminedmathquestionsonseveralSATs,weagreeandhavesummarizedcalculatorrecommendationsbelow.

• Bringandusethemostpowerfulcalculatorwithwhichyouarefamiliar. Agraphingcalculatorispreferred. • Befamiliarwiththecalculatoryouchoosetobring. Useofanunfamiliarcalculatormaycauseasmanyproblemsasitsolves. • PracticewiththecalculatoryouwilluseontheSAT. Practicemakesforspeedandaccuracyonbothmathandcalculatorskills. • Donotbuyanexpensivecalculatorjusttouseforthetest. Ifyouareunfamiliarwithgraphingcalculators,knowthatnoitemsrequireacalculator. • Decidehowtosolveaproblem;thendecideifacalculatorwillbehelpful. Calculatorusagemayinterferewithpacingandmayevenloweryourscore.

5-8

Tosummarize,forthefirstofthetwoSATmathsections,calculatorsarenotallowed.Fortheothersection,powerfulgraphingcalculatorsareallowedandrecommended,butnotrequired.NoSATmathtestitemrequiresacalculatortoobtainthecorrectresponse.Therewillundoubtedlybequestionsonwhichyouwillfindithelpfultouseacalculator,howevertherearealsoitemswherecalculatorusemayimpairyourpacemorethanitimprovesyouraccuracy.Nomatterhowreliantyouareonacalculatorforbasiccalculations,becarefulenteringdata,andbecertainthatyourcalculatorofchoicehasfreshbatteriesandafreshchargeforthetest.PlaySelectionAnalysisTohelplocatetheeasiestproblemsandtoprovideagraphicrepresentationofthestructureofthetwomathsectionsoftheSAT,lookatthecomputeranalysisofitemdifficultyconstructedfromSATtestdata.Questionshavebeenassignedadegreeofdifficultyfrom1to5,with1indicatingtheeasyquestionsand5indicatingextremelydifficultquestions.

SATItemDegreeofDifficultyMathAnalysis

Section3(nocalculator) Section4(calculatorallowed) SMC SPR SMC SMC(cont.) SPR

1.2.0 16.2.5 1.1.0 16.3.0 31.3.0

2.2.0 17.3.0 2.1.0 17.1.5 32.3.0

3.2.5 18.3.5 3.1.5 18.2.5 33.3.5

4.3.0 19.4.0 4.1.5 19.3.5 34.2.5

5.3.0 20.4.5 5.1.5 20.3.5 35.4.0

6.3.0 6.2.0 21.3.5 36.4.0

7.3.5 7.2.0 22.3.5 37.4.5

8.3.5 8.2.0 23.4.0 38.4.5

9.4.0 9.2.5 24.4.0

10.4.0 10.2.5 25.4.0

11.4.0 11.2.5 26.4.0

12.4.0 12.1.5 27.4.0

13.4.0 13.2.5 28.4.0

14.4.5 14.3.0 29.4.5

15.4.5 15.3.0 30.4.5

5-9

DrillonFundamentalsTheSAThasspecific,well-documentedmathcontentinitsquestions.Youcancorrectlyanswermorethan80%ofthemathquestionswithmasteryoftheessentials.Correctlyanswering70%(40questions)ofthepredictableSATcontentisenoughtoobtainascaledscoreof600-700.SolvethefollowingproblemsthatarerepresentativeofthemostcommontypesofSATmathquestions.Doyourfiguringrightonthepage,andifyouhavetime,transferyouranswerstotheanswergrids.Checkyouranswersattheendofthissection.BelowyouwillfindasampleemptySPRgridandacompletedgridwitharesponsethefollowingsamplequestion:800+800=

1 1 1 1

2 2 2 2

3 3 3 3

4 4 4 4

5 5 5 5

6 6 6 6

7 7 7 7

8 8 8 8

9 9 9 9

0 0 0

YoumaybeginworkingtheDrillonFundamentalsproblemsonthefollowingpagewhentoldtodosobyyourteacher.Althoughacalculatorshouldnotbenecessaryforitems#1-#5,youmayuseacalculatorforthequestionsintheDrillonFundamentals.

ItmaybeinterestingtoviewasummaryofthecomplexdirectionsoftheSPRsectionasyouwaittobeginworking: •Recordyouranswerbeforegridding.(optional)

•Gridaccuratelybecauseonlythegridsarescored.

•Gridonlyoneanswertoeachquestion.

•Rewriteanymixednumbersasimproperfractionsordecimals.

•Anydecimalresponsesmustbeasaccurateaspossible.

•Markonlyoneovalineachcolumn.

5-10

Directions Forquestions1-5,solvetheproblemandenteryouranswerchoiceintheovals.Forquestions6-10,solvetheproblemandenteryouranswerinthegrids.1.Markonlyonecircleinanycolumn.2.Someproblemsmayhavemanycorrectanswers,butgridonlyone.3.Mixednumbersmustbegriddedasimproperfractionsordecimals.4.Decimalanswersmaybetruncatedorrounded,butmustfillthegrid.

1 … …………………… If y = kx, where k is a constant, and y = 24 when x = 3, what is the value of y when x = 4 ?

A) 3 B) 8 C) 12 D) 32 2 … ………………… European pediatricians approximate the height h , in centimeters, of boys between the age of 2 and 5, in terms of their age a, in years, using the model

h = 7.8a + 72

Based on this model, what is the estimated change in a boy’s height each year (in centimeters)? A) 7.8 B) 15.6 C) 23.4 D) 36.0 3 nu ………………… The table summarizes the number of annual smart phone contracts initiated between 2008 and 2014.

Which best describes the relationship between time and the number of annual smart phone contracts?

A) Increasing linear B) Decreasing linear C) Exponential growth D) Exponential decay

4 … ………… ……… The teacher of an SAT preparation class surveyed each of 36 total juniors from two separate sections, asking students for their score on the math portion of the PSAT. The mean PSAT math score was 52.4 with a 19.1 margin of error. How could another teacher conduct the survey to get a lower margin of error for the mean? A) Ask 18 students randomly selected from all eight sections of test prep students

B) Ask 18 students randomly selected from all juniors at the school

C) Ask 90 students randomly selected from all eight sections of test prep students

D) Ask 90 students randomly selected from all juniors at the school

5 … ………………… Given the quadratic function f and the cubic function g, graphed below, which interval contains the value of x for which f(x) + g(x) = 0?

A) -2.5 ≤ x ≤ -1.5 B) -1.5 ≤ x ≤ -0.5 C) 0..5 ≤ x ≤ 1.5 D) 1.5 ≤ x ≤ 2.5

5-11

Directions Forquestions6-10(calculatorallowed),solvetheproblemandenteryouranswerinthegrid(directionsbelowsummarizedabove).1.Itissuggestedthatyoucopyyouransweratthetopanswergridasaguide.2.Fillinthegridcarefully,ascreditisbasedonlyoncirclesfilledinonthegrid.3.Markonlyonecircleinanycolumn.4.Noquestionhasonlynegativesolutions. 5.Althoughsomeproblemsmayhavemanycorrectanswers,gridonlyone.6.Mixednumbersmustbegriddedasimproperfractionsordecimalequivalents,7.Decimalanswersmaybetruncatedorrounded,butmustfilltheentiregrid.

6 ………… ………………..…………………

The number of attempts to complete a level in a video gaming competition for 30 semifinalists is shown in the graph above The median number of attempts was 5.5. What is the absolute value of the difference between the mean and mode? 7 … …… A scientific model predicts the stabilization of CO2 concentrations at a level that would create an annual global temperature increase of 0.045°F. At this rate, how many years would it take for the temperature of the Earth to increase 1.8°F?

8 … …… "8 − 8", = −15"

If x>0, what is a positive solution to this equation? 9 … … … ……… Ramona and Adrian both work part-time each week to save money for college. Last week Ramona worked 10 hours less than Adrian. If they worked a combined total of 51 hours, how many hours did Adrian work last week?

10 … … …………

In a unit circle O, a central angle AOB has a measure

of :; radians. The area of the sector formed by

ÐAOB is what fraction of the area of the circle?

5-12

PlaySelectionEssentialsNumberSense

r OrderofOperationsr MultiplicationofSignedNumbersr Fractions(RationalNumbers)r Percentsr RatioandProportion

HeartofAlgebrar Polynomials(SubstitutionandSimplification)r DistributivePropertiesr AbsoluteValuer LinearEquationsandInequalitiesr LinearFunctions,FunctionNotation,CompositionofFunctionsr Slope-InterceptFormofLinearEquationsandInequalitiesr SystemsofLinearEquationsandInequalitiesr TablesandGraphs

PassporttoAdvancedMathr ProductsandQuotientsofPowers,PowersofPowersr SimplificationofPolynomialsandRationalExpressionsr RootsofPolynomialsandSolvingRationalEquationsr SolvingRadicalEquationsr EquationsofParabolasandCirclesandtheirGraphsr SolvingQuadraticEquations

ProblemSolvingandDataAnalysisr MotionProblemsr WeightedAveragesr DescriptiveStatistics(Mean,Median,Mode)r StandardDeviation,MarginofErrorandConfidenceIntervalsr RandomSamplingandAssignmentr ConditionalProbabilityr PositiveandNegativeAssociations,ExponentialGrowthandDecayr GraphicalRelationships(minima/maxima,symmetry,asymptotes,transformations)

AdditionalTopicsinMathr Angles,Triangles,andPythagoreanTheoremr PolygonAreaandPerimeterr SolidGeometryr CirclePropertiesr CongruenceandSimilarityr TrigonometricFunctionsandRadianMeasurer ComplexNumberArithmeticr CoordinateGeometry

5-13

EssentialAdvice NowthatyouhaveseentherangeofmathcontentontheSAT,youshouldbecertainthatyouarefluentwithproportionalreasoning.Beawarethatyouwillneedtounderstandandinterpretfunctionnotation.Youshouldbeabletosimplifypolynomialexpressionsandequations,andfind(orconfirm)thevaluesforvariablesthatmakeanequationorasystemofequationstrue.Dataanalysisitemswillrequireyoutoreadandinterprettablesandgraphs,findmeasuresofcentraltendency,andtomakeprobabilisticpredictions.Theaveragecollege-boundstudentgetslessthanhalfofthe58mathquestionscorrect.However,thatisnotbecausetheSATcontentisdifficult.Itisnot.Ifyoucancorrectlyanswer70%(40questions)of58mathquestions,yourscorewillbeinthe650range.Testtakerswiththeedgecananswermorethan80%(46-47questions)ofthequestionsandscoreabove700withmasteryofthemathessentials.Unlikemanyothermathexams,60%,70%and80%areverygoodscores.SAT1600studentswhomastertheaccompanyingessentials,andemployproblemsolvingskillsandourstrategiescanreachthis80%levelandbeyond.Pre-GamePepTalk ThecontentoftheSATispredictable.TheessentialmathematicalknowledgeneededtoanswerSATquestionsisprovidedintheEssentialsofMathematicssectionofSAT1600.AlthoughitmaynotbepossibletopredictorpracticeeverypossibletypeofSATmathquestion,inthenextseveralweeksyoucanstillimproveyourSATskillsandscoresbytakingouradvice:

• Worklotsof(redesigned)SATandPSATproblems. • DecideonacalculatorandpracticewithitonSMCsandSPRs. • Knowthecommontypesofproblemswell. • ApplyoffensiveGUARDstrategiesonStandardMultiple-Choiceitems. • ApplyABCDefensivestrategiesonStandardMultiple-Choiceitems. • ApplyoffensiveanddefensivestrategiesonStudent-ProducedResponses. • Practicegriddinganswersandwritinganswersinagriddableform. • ReadandusethehintsinanyfreeSATmaterialsfromtheCollegeBoard.

• Followthegameplanthatwepracticewithyou.

The Edge... Calculator competence and confidence are born of experience. Begin practicing with your calculator as soon as possible.

AnswerstoDrillonFundamentals

1. D) 32 2. A) 7.8 3. C) exponential growth 4. C) 90 from all 8 sections 5. B) -1.5 ≤ x ≤ -0.5 6. 0.7 7. 40 years 8. 3 or 5 9. 30.5 hours 10. 1/16, 0.062, 0.063

5-14

ESSENTIALSOFMATHEMATICS(Üpointsoutbigideas,þcheckscomprehension,rmarksproblemstodo,¦specifiesSATformat)

NUMBERSENSE OrderofOperationsÜ RATIONALPEOPLE EvaluatePowersandRadicals,then DON'TMISS performMultiplicationsandDivisions,then SIMPLEANSWERS performAdditionandSubtractions.

þ 3 + (2 + 52) / 3 = 3 2 53

3 2 253

3 273

3 9 122

++

= ++

= + = + =( ) ( )

r 1 + 2 ´ 3 – 4 ÷ 5 = ________

r (-6) (-8) (-10) = ________ Fractions(RationalNumbers)

Ü zca

zc

za ±

=± Ü bdbcad

dc

ba ±

=±

þ 236

342

34

32

==+

=+ þ 157

15512

35)15()34(

31

54

=-

=´

´-´=-

Ü

þ

Ü

þ

r 8,+ 1

< = ______ r

/,× 18 = ______ r

><÷ 8

1 = ______

abcd

acbd

× =

1234

38

× =

ab

cd

abdc

adbc

÷ = × =

1523

1532

310

÷ = × =

5-15

Percents

Ü p ab

ab

p a bp= × Û = Û =100100

100

þ

þ

þ

r <;= = _______% r 10 is 125% of_______ r 20 is_______% of 4000

¦ On the redesigned version of the SAT, 44.7% of the content of the calculator portion

of the math test is categorized as Problem Solving and Data Analysis. If there are 17 of these items, how many items are on the calculator portion of the test?

A) 17 B) 21 C) 31 D) 38

RatioandProportion

Ü )(

:nm

mnm

+Û

þ If the ratio of boys to girls in a class of 24 students is 3:5, then 8

3

)53(

3=

+ of the class are boys.

.98

72

18

243

1

24

8

324

8

3 is 24 of

8

3==

´

´=´=´ The class has 9 boys and 15 girls.

r If the pass:fail ratio in an algebra class is 3:1, how many students fail in a class of 28? ______

Ü cbdad

c

b

a×=×Û=

þ 18327227

18

3

2×=×Û= þ

524

131

413521 =Û×=×

35 100

3 100 5= Û × =p p

300 5 3005

60= Þ = = Þ =p p % 35

60%

94 100

9 100 4= Û × =p p

900 4 9004

225= Þ = = Þp p 9 is 225% of 4

61000 100

6 100 1000= Û × =p p

600 1000 6001000

0 6= Þ = = Þp p . 6 is 0.6% of 1000

5-16

HEARTOFALGEBRAPolynomialSubstitutionÜ Polynomialsareexpressionsusedasplaceholdersforquantitieswhosevaluesmayvary.

Whenthereisasimplerelationshipbetweentwoquantitiestheyusethesamevariable.(e.g. x and x + 3). Iftherelationshipislessclear,twovariablesareused (e.g. b and c). þ The expression x + 3 means 3 more than a number. The value of (x + 3) is 8 when x is 5.

þ The value of the expression x2 + 1 when x = 3 is 10.

r Evaluate 2)1( +nn when n = 8. ____

r If x = 2 and y = 3, then x2 – 2xy + y2 = ? ____

r Find the value of -16t2 + 34t + 5 when t = 2. ____

PolynomialEquivalents

Ü ))((22 yxyxyx -+=- þ 9)3)(3( 2 -=-+ aaa

Ü 222 )())((2 yxyxyxyxx -=--=+- þ 22 )5()5)(5(2510 -=--=+- bbbbb

Ü 222 )())((2 yxyxyxyxx +=++=++ þ 3612)6)(6()6( 22 ++=++=+ ccccc

Ü 2222 but,)( xxxx ¹-=- þ 36)6( 2 =- þ 3662 -=-

r Factor x2 – 36 . ____________________

r Expand (y + 1)2. __________________

r Factor z2 – 6z + 9. ________________

r Simplify -32 – (-3)

2. ________________

5-17

DistributiveProperties

Ü Parentheses Really Make Doing Arithmetic Simple

Ü Distributive property(multiplicationoveraddition/subtraction): a(b ± c) = ab ± ac þ 5(2.9+0.2) = 5(2.9) + 5(0.2) = 14.5+1 = 15.5 þ 3(2x-5) = 3×2x - 3×5 = 6x – 15

r 3(2x + 1) =________ r 5(3x – 7) =________

Ü Distributiveproperty(divisionoveraddition/subtraction): cb

ca

cba

±=±

þ =+=+=+

412

41

48

418 2¼ þ 32

39

36

396

-=-=- aaa

r 262 +a = ________ r

aaa-3 =________

Ü Distributiveproperty(powersovermultiplication/division):

(A×B)D = AD ∙ BD; GH

D= G

I

HI

þ ( )2 6 2 6 8 216 17283 3 3´ = ´ = ´ = þ 33

33 6444bbb

==÷øö

çèæ

r (2x)3 = ________ r 2

23÷øö

çèæ =________

Ü Distributive property (radicals over multiplication/division): a b a b ab

ab

× = × =;

þ 488 399 zzz ×=×= þ 6.0106

10036

1003636.0 ====

r 205 × ________ r 36

8a = ________

AbsoluteValue

Ü | x | = îíì

³<-0 if

0 ifxxxx

þ | -4 | ´ | 6 | = [-(-4)] ´ (6) = (4) × (6) = 24

þ If | 2x + 1 | = 5, then îíì

=Þ=Þ=+Þ=+-=Þ-=Þ-=+Þ=+-2425125)12(3625125)12(

xxxxxxxx , so x = -3 or 2.

r Solve | z + 1 | = 3. ________

5-18

SolvingLinearEquationsandInequalities

Thereareseveralbasicsimilarprocessesusedinsolvinglinearequationsandinequalities.Eachoftheprocessesrequirescompliancewithafewsimplerules.Ü AdditivePropertyofEquality/Inequality–

[adds(subtracts)thesamevalueto(from)bothsidesofanequation/inequality]Ü MultiplicativePropertyofEquality–

[multiplies(divides)bothsidesofanequationbythesamenon-zerovalue]Ü MultiplicativePropertiesofInequality–

[multiplies(divides)bothsidesofaninequalitybysomepositivevalueormultiplies(divides)bothsidesofinequalitybysomenegativevaluereversingtheorder]Thestepsinsolvingalinearequation/inequalitytypicallyinvolve1)simplifyingtheexpressions,2)collectinglikevariabletermsononeside,3)simplifyingtheliketermscreated,4)multiplyingbythereciprocalofthecoefficientofthevariable,reversingtheorderifnecessary.

2 + 3x + 3 = 4x – 3 + x, þ Tosolve 2 + 3x + 3 = 4x – 3 + x,

1) Simplifytheexpressions 5 + 3x = 5x – 3, 2) Collecttheterms 5 + 3x + (-5) + (-5x) = 5x – 3 + (-5) + (-5x) 3) Simplifytheterms 0 + (-2x) = -8 + 0 4) Multiplybythecoefficient’sreciprocal (-½)(-2x) = -8 (-½) so x = 4

þ Tosolve 3(x + 2) < 6x + 10 - x, 1) Simplifytheexpressions 3x + 6 < 5x + 10, 2) Collecttheterms 3x + 6 + (-6) + (-5x) < 5x + 10 + (-6) + (-5x) 3) Simplifytheterms (-2x) + 0 < 4 + 0 4) Multiplybythereciprocalandreversetheorder -2x (-½) > 4 (-½) so x > -2

r Solve (x + 5) + 4x = 9 + (x + 8). r Find the solution: 2x + (3 – 5x) > 3(9 + x)

m The monthly membership fee for an internet video service $9.95. The cost of

viewing network television online is included in the monthly membership fee, but there is an additional fee of $1.25 to rent each movie online. For one month, Jaclyn’s membership and movie rental fees were $14.95. How many movies did Jaclyn rent online that month?

A) 1 B) 2 C) 3 D) 4

5-19

Functions(DomainandRange)

Ü Afunctionisarelationshipdefinedbetweentwovariablessuchthatforall x values,they valueisunique.

Ü Thedomainofafunctionisthesetofall x valuesforwhichthey valueisdefined.

Ü Therangeofafunctionisthesetofall y valuesthatpairwiththedefined x values.

þ The domain of the set of points on the semicircle is the set of points such that -4 £ x £ 4. The range of the set of points on the semicircle is the set of points such that 0 £ y £ 4.

r If the domain of the function defined above is limited to x values such that -4 £ x £ 0, what inequality describes the range of the function? ________.

FunctionNotationandCompositionofFunctions Ü Functions,typicallydenotedbyalowercaseitalicizedletter(e.g., f or g)definethe

relationshipbetweentwovariables(e.g.,x and y)whereeachelementofthedomaincorrespondstoexactlyoneelementoftherange.

Ü Considerthefunctionfthatdefinestherelationshipbetweenx and y tobe y = x + 3.

Thisfunctionwouldtypicallybewritteninthegeneralform (e.g., f(x) = x + 3) butitcouldalsobewritteninmorespecificforms(e.g., f(1) = 4, f(-1) = 2, f(0) = 3, f(-4) = -1, etc.)

Ü Thecompositionoffunctionsalsodefinesarelationshipbetweentwovariablesxandywhereeachinputcorrespondstoexactlyoneoutput,howevertheoutputisdeterminedbythefunctionsinseries.Thefunctionswouldeachbewritteninthegeneralform.

f(x) = 2x – 3 g(x) = x2 + 1

andthegeneralformofthecompositionoffunctionswouldbewrittenas f(g(x)).

þ If f(x) = 2x – 3 and g(x) = x2 + 1, then g(f(4)) = [(2·4 – 3)2 + 1] = 26.

r If f(x) = 2x – 3 and g(x) = x2 + 1, then what is the value of f(g(-1)) ?

r The table shows several values of the linear function f. What equation defines f ? ________________________

5-20

Slope-interceptFormofLinearFunctionsÜ Linearfunctions(functionswhosegraphsarelines)arecommonlywrittenintheform y = mx + b, or f(x) = mx + b where m istheslopeand b isthe y-intercept.

þ The graph of the linear equation y = 2x + 3 has a slope m = 2 and y-intercept b = 3. Ü Theslope, m, ofalinearfunctionistheratioofthechangeinthevariables x and y.

Givenanytwopoints (x1, y1) and (x2, y2), theslopeofthelinedeterminedis m = 12

12

xxyy

--

þ The slope of the line through the points (1, 2) and (3, 6) is m = 224

1326

==-- .

Ü The y-intercept, b, ofalineisthe y-coordinate ofthepointwhereitcrossesthe y–axis. Givenaline,the y-intercept b canbefoundbyfindingthepoint (0, b) onthe y–axis. þ The y-intercept of the line passing through (2, 7) and (1, 5) is 3. The definition of slope says that

from any point on this line, that moving down 2 and left 1, gives another point on the line. Since (1, 5) is 2 down and 1 left from (2, 7), then the point 2 down and 1 left from (1, 5) is also on the line. The point 2 down and 1 left from (1, 5) is (0, 3), so the y-intercept of the line is 3.

r What is the slope of the line f(x) = 4x – 3 ? _____ What is the y-intercept? _____

r What is the slope of the line through (2, 4) and (4, 3) ? _____ What is the y-intercept? _____

SolvingSystemsofLinearEquationsandInequalitiesÜ Asystemoflinearequationsiscomposedoftwoequationsintwovariables.Although

eachlinehasaninfinitenumberorpoints,twodistinctlineswillhave(atmost)onepointincommon.Findingthepointisreferredtoassolvingthesystemofequations.

x – 3y = -5 [or x = 3y – 5] -x + 2y = 3

Ü Therearethreecommonmethodstosolvesystemsoftwoequationsintwovariables: 1)Inspection–graphtheequationstoseewheretheyintersect. 2)Combination–add(multiplesof)theequationstoeliminateavariable. 3)Substitution–substituteanequivalentexpressionfromoneequationintotheother.

Solvingthesystemoflinearequationsabovegivesthefollowingresults:

Combination(addingthepairabovetofindy,andsubstitutinginanequationtofindx)

-y=-2,y=2…sox – 3(2) = -5, x – 6 = -5, x = -5 + 6, and x = 1

Substitution(substitutingthevalueofxinthefirstequationintothesecond) -(3y – 5)+ 2y = 3, -3y + 5+ 2y = 3, -3y + 2y = 3 – 5, -y = -2,…x = 1

5-21

PASSPORTTOADVANCEDMATHExponents(PositiveIntegerExponents) Ü Exponentsaresymbolswrittenassuperscriptstotherightofacorrespondingnumber,

calledthebase.Exponentsindicatethenumberoftimesthebaseismultipliedbyitself.

þ 63 = 6 ´ 6 ´ 6 þ x2 = x × x þ z1 = z

r 23 = ________ r 32 = ________ r 41 = ________

ProductsofPowers,QuotientsofPowers

Ü ( ) ( )nmn

mnmnm x

x

xxxx -+ ==× ,

þ

r 42 aa × = ________ r 4

8

aa = ________ r =× 2

121aa ________

PowersofPowers

Ü þ ( )b b b2 3 2 3 6= =×

r ( )342x = ____ r ( ) 216x = ____ r ( ) 23 --x = ____

Exponents(Zero,Negative,andRationalExponents)

Ü Theexponentrule,QuotientsofPowers,allowsforthedefinitionofnegativeandzeroexponents.Similarly,therulePowersofPowersdefinesrationalexponents.

þ 10)33(3

3=== - aa

aa þ

bbb

bb 11)43(4

3=== -- þ 122

1)( cc = so cc =2

1

þ 10 =a if a ¹ 0 þ nn

bb 1

=- , nn

bb -=

1 þ n mnm

cc = = mn c )(

r 03 = ____ r 16- = ____ r 21

16 = ____ r 328 = ____ r 24- = ____ r 2

19- = ____

a a a aa

a3 2 55

23× = =,

( )b bx y xy=

5-22

SimplifyingRationalAlgebraicExpressions

Ü zca

zc

za ±

=± Ü bdbcad

dc

ba ±

=±

þ 2)4()4(2

482

45

432

=++

=++

=+

+++

ww

ww

www þ

92

93

93

31

31

222 -=

-

++

-

-=

-+

+ xx

xx

xx

xx

Ü

þ mn

mnn

m 263

63 222

==×

Ü

þ sr

rsr

sr

rrs

r==×=÷

22

211

r Simplify 33

3 ++

+ xxx ________

r Simplify 39

33

--

- yyy ________

r Simplify 23

21

++

- zz ________

r Simplify 22 +

-- w

www ________

r Simplify a

a815

34 3

´ ________

r Simplify 456

49

bb÷ ________

¦ If 4

3=

b

a and 5

4=

c

b then =ac

A) 35

B) 1516

C) 1615

D) 53

abcd

acbd

× =

ab

cd

abdc

adbc

÷ = × =

5-23

PolynomialRootsandSolutionsofRationalEquationsandInequalities

Ü Ifapolynomialisdivisibleby (x – k) then k isarootofthepolynomial.

þ -)J>-(.//-J>

(-J8) = ", − 3" + 2, so 3 is a solution to "8 − 6", + 11" − 6 = 0.

Ü Solvingrationalequations/inequalitiesissimilartosolvinglinearequations.The multiplicative property of equality/inequality is used to convert equations with

rational expressions to related linear equations/inequalities. 1) simplify the expressions in the equation, 2) isolate a rational term on one side of the equation and other terms on the other 3) simplify any like terms created, 4) multiply both sides of the equation by the denominator of the rational term (reversing the order of the inequality if necessary) 5) Repeat as needed until there are no algebraic fractions remaining. 6) Solve the remaining equation AND check the solutions for validity.

þ To solve 423

-=+-

mm , multiply both sides of the equation by (m + 2) to obtain )2(43 +-=- mm .

Solving the linear equation m – 3 = -4m + (-8) yields 5m = -5, so m = -1, which checks out.

r Solve xx 25

211=+ . ____ r Solve y

yy

yy 2

23

292

=-+

+-- . _____ r Solve

593

54 zz

+-< ._____

SolvingRadicalEquations

Ü Solvingradicalequationsisalsosimilartosolvinglinearequations.1)simplifytheexpressionsintheequation,2)isolatearadicaltermononesideoftheequationandothertermsontheother3)simplifyanyliketermscreated,4)squarebothsidesoftheequation,5)Repeatasneededuntiltherearenoradicaltermsremaining.6)SolvetheremainingequationANDcheckthesolutionsforvalidity. þ If 183 =x , then 318=x = 6 so x = 62 = 36. Checking, 18)6(3363 == , so 36 is correct.

þ If 114 =++ z , then 31 -=+z , and z + 1 = 9, so z = 8. Checking, 1184 =++ yields 4 + 3 = 1. Since 7 ¹ 1, there is no solution.

r Solve for x: 312 =-x ________ r Solve for x: xx =-12 ________

¦ Solve 1953 =-x . A) 4 B) 8 C) 16 D) 64

¦ Solve 34 +=- x . A) -19 B) 13 C) 19 D) there is no solution

5-24

StandardFormofEquationsofParabolasandCircles,andtheirGraphs

Ü Quadraticequations[functions], f(x) = ax2 + bx +c or y = ax2 + bx +c areusuallygraphedfromthestandardform f(x) = a(x – h)2 + k or y = a(x – h)2 + k.

Ü Quadraticgraphsareparabolaswhoseshapeanddirectiondependonthevalueofa.

Ü Quadraticshavegraphsthatareparabolicwiththevertexoftheparabolalocatedat(h, k).

Ü Quadraticequationswiththevariablesreversed(i.e., x = y2) arenotfunctions.

þ The graph of y = x2 can be written as y = 1x2 or y = 1(x – 0)2 + 0. The vertex of the parabola is (0, 0).

` þ The graph of y = 3x2 can be written y = 3x2 or y = 3(x – 0)2 + 0. The vertex of the parabola is (0, 0).

þ The graph of y = -2x2 can be written as y = -2 (x – 0)2 + 0. The vertex of the parabola is (0, 0).

þ The graph of y = ½ x2 can be written as y = ½ (x – 0)2 + 0. The vertex of the parabola is (0, 0).

þ The graph of y = (x – 8)2 is written

y = 1(x – 8)2 + 0. The graph is shifted right 8 units from the graph of y = x2. The vertex is (8, 0).

þ The graph of the parabola shifted to the right 6 units and up 4 units from the parabola y = x2 is written

y = (x – 6)2 + 4.

5-25

þ The graph of y = x2 + 3 is written

y = 1(x – 0)2 + 3. The graph is shifted up 3 units from the graph of y = x2. The vertex is (0, 3).

þ The graph of the parabola shifted to the right 8 units and down 2 units from the parabola y = -x2 is written

y = -(x – 8)2 - 2.

Ü Equationsofcircles, x2 + y2 + Dx + Ey + F areusuallygraphed

fromthestandardform (x – h)2 + (x – k)2 = r2.

Ü Circlesinstandardformhavegraphsthatarecenteredat(h, k) withradius r .

þ The graph of x2 + y2 = 1 is a unit circle. This can be written in standard form as (x – 0)2 + (y – 0)2 = 12 The center is (0, 0) with radius 1.

þ The graph of x2 + y2 – 4x + 2y + 1 = 0 is a circle. This can be written in standard form as x2–4x+4 + y2+2y+1=4 = (x – 2)2 + (y + 1)2 = 22

The center of the circle is (2, -1) with radius 2.

r What is the equation of the parabola below shifted to the left 6 and

down 2 from the graph of the parabola y = x2 ?

r What is the equation of the circle centered at (-3, 2) with radius 3?

5-26

SolvingQuadraticEquationsinStandardForm

Ü Althoughquadraticequationsaretypicallywrittenintheform y = a(x – h)2 + k foreaseingraphing,twocommonmethodsforsolvingquadraticequationsrequirethemtobewritteninthestandardform, ax2 + bx +c = 0.

Ü Whenwrittenintheform ax2 + bx +c = 0, solutionstoquadraticequationsmaybefoundby

factoringthequadratictrinomialintotwobinomials,(x–r1)and(x–r2).Sincetheproductofanytwofactorscanonlybezerowhenoneofthefactorsiszero,thesolutions,r1andr2,caneasilybecalculated.Thesesolutions,r1andr2arealsoreferredtoasrootsorzeroes.

Ü Whenwrittenintheform ax2 + bx +c = 0, thesolutionsalsomaybecalculateddirectlyby

thequadraticformula, a

acbbx

242 -±-

= . ThismaynotbenecessaryontheSAT.

þ The sum of the two solutions of a quadratic equation is J,H,M

= − HG

.

þ The product of the two solutions of a quadratic equation is H(J(H(J1GN),G (,G)

= 1GN1GG

= NG

.

þ If x2 – 16 = 0, then since (x + 4)( x – 4) = 0, either x = -4 or x = 4.

þ If x2 + x – 6 = 0, then since (x + 3)( x – 2) = x2 + x – 30, (x + 3)( x – 2) = 0. Since either

(x + 3) = 0 or ( x – 2) = 0, x = -3 or x = 2 (-3 and 2 are the roots/zeroes of the equation).

r What is the smallest solution of x2 – 2x = 15?

r What is the sum of the solutions of x2 – 3x = 10?

r What is the product of the solutions of 4x2 – 25 = 0 ?

¦ What are the x-intercepts of the graph of x2 – x = 12 ?

A) -3 and -4

B) -3 and 4

C) 3 and -4

D) 3 and 4

5-27

PROBLEMSOLVINGANDDATAANALYSIS Motion(Amount)Problems

Ü Inmotionoramountproblems,distance(amount)istheproductoftimeandtherate.

d = rt or (a = rt) þ The rate of a driver who drives 2 hr at 55 mph and 3 hr at 65 mph is

=

r How long does it take to drive 225 miles driving 75mph for 100 miles and 50mph for 125 miles? ____

DescriptiveStatistics

Ü Descriptivestatisticsisthecollectionandanalysisofdatafromapopulationsampletoestimateparametersoftheentirepopulation(mostpreciseifsampleisrandom).

Ü Mean,median,andmodearethecommonmeasuresofcentraltendencyofapopulation. þ The mean of a set of data is the average of the numbers (i.e., the sum of the numbers divided by the size of the group …

MO.M(.M)P.⋯.MID

).

þ The average (arithmetic mean) of 26, 23, and 17 is 223

66

3

172326==

++

þ The average of 5a + 3, 6a – 5, and 10a – 1 is

þ The median of a set of data is the middle number when the numbers are ordered. (The median is the mean of the two center numbers if there are two center numbers.)

þ The median of 22, 26, 31, 15, and 28 is 26.

þ The median of 22, 26, 33, 13, 24 and 28 is 25250

22624

==+

þ The mode of a set of data is the number that occurs most often. (There may be more than one mode.)

þ The mode of 20, 19, 26, 28, 27, 26, and 22 is 26.

r What is the mode of 18, 20, 33, 25, 20, 24, and 19? ________

r What is the mean of 21, 31, 27, and 15? ________

r What is the average of 4a + 7 and 1 – 6a ?________

r What is the median of 18, 20, 33, 25, 22, and 19? ________

r What is the median of 24, 19, 30, 23, and 16? ________

d dt t

r t r tt t

1 2

1 2

1 1 2 2

1 2

2 55 3 652 3

++

=++

=× + ×

+110 195

53055

61+= = mph

( )5 6 10 3 5 13

21 33

7 1a a a a a+ + + - -=

-= -

5-28

StatisticalMethodsandTerminology

Ü Inadditiontothemeasuresofcentraltendencyofapopulation(mean,median,andmode),statisticalmethodsalsoquantifytheamountofvariationinapopulation.Datapointsfarfromthecentermaybeerrorsormaybeveryimportant.Thesepointsareknownasoutliersandalthoughtheyaffectthemean,theyhavelittleeffectonthemedianormode.

Ü Onemeasureofvariationisknownasstandarddeviation,ameasureofhowfardatapointsvaryfromthemean(e.g.,chaptertestscoresof69,70,and71telladifferentstorythanscoresof40,70,and100…(thesecondsetofscoreshasamuchhigherstandarddeviation).

Ü Estimatesofaparameterofapopulationmadefromcollectionandanalysisofsampledataaremostaccuratewhenthesampleisrandom.Inarandomsample,eachmemberischosenbyrandomselectionandeachmemberhasanequalchanceofbeingselected.

Ü Statisticalestimatesareverylikelytobereasonable,butareunlikelytobeexactlycorrect.Anadditionalcomponentofstatisticsisquantifyingthevariabilityofanestimate.

Ü Themeasuresthatquantifytheaccuracyofanestimateareknownasmarginoferrorandconfidencelevel.Althoughtherearedifferingstatisticalconfidencelevels,theSATwillalwaysusethecommonconfidencelevelof95%.This95%confidencelevelwillbecoupledwithacorrespondingmarginoferror.

þ Whenwesaythat50randomlyselectedfemalesfromasophomoreclassof500femaleshaveameanverticaljumpof33.6cmwitha2.3cmmarginoferror,thismeansthatfor95%ofpossiblesamples,themeanwillbeintheconfidenceintervalbetween31.3and35.9cm.(Youcanbe95%confidentthatthemeanverticaljumpisbetween31.3cmand35.9cm.)

þForthe95%confidencelevel,themarginoferrorisdependentonthesamplesizeandthevariabilityofthedata.Thestandarddeviationispositivelyassociatedwiththemarginoferror(i.e.,theyriseandfalltogether),butthemarginoferrorisnegativelyassociatedwithsamplesize(i.e.,assamplesizeincreases,themarginoferrordecreases).

þConfidenceintervalsapplytoanentirepopulation.A95%confidenceintervaldoesnotimplythat95%ofthesophomorefemaleshaveaverticaljumpbetween31.3and35.9cm.

m Ten families, one with an annual income in excess of $100 million, move into a small town. Which will probably not change substantially during the analysis of the town’s income data in next census?

A) Mean income B) Median income C) Outliers D) Standard Deviation

m The mean SAT math score for 25 random students is 510 with a 13 point margin of error. What does this mean?

A) 95% of scores are between 510 and 523

B) 95% of scores are between 497 and 523

C) 95% of similar random samples have a mean between 510 and 523

D) 95% of similar random samples have a mean between 497 and 523

5-29

WeightedAverages

Ü

þ Three scores of 17 and two scores of 12 average to

r If 18 students scored 80% and 12 scored 100%, the average score was ________ .

PositiveandNegativeAssociation,ExponentialGrowthandDecay

Ü Twoquantities, x and y, haveapositiveassociationifasoneincreasestheotherincreases.Thebestfitlinecanberepresentedbyanequation y = mx +b wheremispositive.

Ü Thetwoquantities, x and y, varydirectlyiftheyincreaseanddecreaseproportionally.Thebestfitlinecanberepresentedbytheequationy = kx, where k isaconstant(ofvariation).

Ü Twoquantities, x and y, haveapositiveexponentialassociationifasoneincreasestheotherincreasesexponentially.Thebestfitcurveisoftheformy = mxt wheretispositive.

þ If x and y vary directly, and y = 12 when x = 2, then k = (6), and when x = 3, y = 18 = 3×(6).

Ü Twoquantities, x and y, haveanegativeassociationifasoneincreasestheotherdecreases.Thebestfitlinecanberepresentedbyanequation y = mx +b wheremisnegative. Thesetwoquantities, x and y, varyinverselyifoneincreasesastheotherdecreaseswhilethepairmaintainsaconstantproduct(xy = k or y = kx-1). Thepairmayalsohaveanegativeexponentialassociationifasoneincreasestheotherdecreases(decays)exponentially.Thebestfitcurveisoftheformy = mxt wheretisnegative.

pa qb rc zkp q r z+ + + ++ + + +

......

3 17 2 123 2

51 245

755

15× + ×+

=+

= =

5-30

ProbabilityandConditionalProbability

Ü Theprobabilityofaneventisdefinedasthenumberoffavorableoutcomesdividedbythetotalnumberofpossibleoutcomes.Thismeansthattheprobabilityisbetween0and1.Ifaneventisimpossible,itsprobabilityis0;ifaneventiscertain,itsprobabilityis1. þ Picking a black marble from a bag with a black marble and a white marble has probability ½.

þ Picking a red marble from a bag with two blue, one green, and one red marble has probability ¼.

Ü Conditionalprobabilityinvolvestwoevents.Theconditionalprobabilityistheprobabilityoftheoccurrenceofaneventgiventhattheothereventhasalreadyoccurred.

þ When a single die is rolled, the probability that the number is greater than 3, given that it is odd, exemplifies conditional probability. One event is that the face shows an odd number. The other event is that the face shows a number greater than 3. The conditional probability is ⅓ because the the condition of being odd can be met in 3 ways (1, 3, or 5) and one of those (5) is greater than 3.

GraphicalRepresentationofData

Ü Forthepurposesofanalysis,datamayberepresentedinseveralcommonways.Amongthecommonmethodsarebargraphs,linegraphs,andscatterplots.Scatterplotsareoftenusedtogeneratealinearmodelforthepointsknownasalineofbestfit.

þ Thethreegraphsaboveeachshowadifferentrepresentationofthesamesetofdatapoints:{(1,3),(2,2),(3,6),(4,7),(5,2),(6,6),(7,7),(8,4)}.

m A set of 33 experimental data points were collected and graphed below.

Which of the equations best fits the data?

A) y = 10 B) y = x + 10 C) y = -x D) y = -x + 10

m A set of 36 experimental data points were collected and graphed below.

Which of the equations best fits the data?

A) y = 10 B) y = -x + 10 C) y = 10"J/., D) y = -

O.(

/S

5-31

ADDITIONALTOPICSINMATHAngles

Ü Anglesarerayswithacommonvertex.Anglemeasuresrangefrom0°to360°.

þ mÐUVW » 45 + 30 = 75

r a »_______

r b »_______

r c »_______

5-32

AreaSquare Rectangle Triangle Circle

Ü A = Ü A = !w Ü A =

Ü A = p

¦ What is the area of the obtuse triangle above? A) 10 B) 12 C) 20 D) 24 Perimeter(Circumference)

Square Rectangle Circle

Ü P = 4s Ü P = 2! + 2w Ü C = 2pr = pd

r What is the perimeter of a 5 by 7 rectangle? ________ ¦ What is the perimeter of a square if a circle \ inscribed in the square has a circumference of 8p?

A) 32p B) 16p C) 32 D) 16

s2 bh2

r 2

5-33

SolidGeometry

Cube Cylinder Box(rectangular) Cone

Ü V = s3 Ü V = pr2h Ü V = lwh Ü V = pr2h/3 þ A cube of side length s = 2 has volume V = (2)(2)(2) = 8 ¦ What is the volume of a right circular cone of height 4 if the circular face has an area of 9p?

A) 36p B) 18p C) 12p D) 4p Circles

Ü Themeasureofacentralangleinacircleisequaltothemeasureofitsinterceptedarc.

Ü Themeasureofaninscribedangleisequaltohalfthemeasureofitsinterceptedarc.

þ If ÐXOY is 88°, then TUis 88° þ If ÐWPZ is 59°, then VWis 118° þ If TUis 88°, then ÐXOY is 88° þ If VWis 118°, then ÐWPZ is 59°

¦ In circle O, ÐCAD and ÐCBD are inscribed in the circle.

If the length of chord AD is greater than the length of chord BC, which of the following statements is true?

A) the measure of ÐCAD is greater than the measure of ÐCBD B) the measure of ÐCBD is less than the measure of ÐCAD C) the measure of ÐCAD is equal to the measure of ÐCBD D) there is not enough information to determine the relationship between mÐCAD and m ÐCBD

s

s s

h

r

h

l w

h

r

5-34

Triangles

Ü mÐA + mÐB + mÐC = 180 Ü mÐA + mÐC = mÐCBD Ü a £ b £ c Û mÐA £ mÐB £ mÐC

\

þ mÐC = mÐCBD - mÐA = 100 - 30 = 70

r If the sides of a triangle measure 4, 7, and c, then ______ < c < 11. RightTriangles

Ü Inarighttrianglewithleglengthsofaandb,andahypotenuseoflength c, a2 + b2 = c2

Ü If a2 + b2 = c2, thenatriangleisarighttriangle. þ A triangle with side lengths 8, 15, and 17 is a right triangle because 82 + 152 = 172.

Ü If a, b, and c arerighttriangleleglengths,then an, bn, and cn arerighttriangleleglengths.

þ Since a triangle with side lengths 3, 4, and 5 is a right triangle, so are triangles of side lengths 6, 8, and 10; 9, 12, and 15; 12, 16, and 20 etc.

þ (XZ)2 = (YZ)2 - (XY)2

= 100 - 36 r n = ________ = 64, so XZ = 8

5-35

Polygons:CongruenceandSimilarity

Ü Thesumoftheinterioranglesofapolygonwithnsidesis180(n–2).Ü Thesumoftheexterioranglesofapolygonwithnsidesis360.

þ The sum of the interior angles of a pentagon is 180(5 – 2) = 180×3 = 540

þ The sum of the exterior angles of a pentagon is 360.

r What is the sum of the interior angles of an isosceles trapezoid? ________ r What is the sum of the exterior angles of an isosceles trapezoid? ________

Ü CPCPC-CorrespondingPartsofCongruentPolygonsareCongruent

þ Since DABC @ DDEF, AB = DE and mÐC = mÐF.

r If DABC @ DDEF, and mÐB = 30 and mÐF is 35, what is mÐD? ________

5-36

Ü CSSPP -CorrespondingSidesofSimilarPolygonsareProportionalÜ CASPC-CorrespondingAnglesofSimilarPolygonsareCongruent

þ In the two similar hexagons, if the perimeters are 18 and 24, the side corresponding to a side of length 3 on the small hexagon has length 4.

r Two hexagons are similar. The perimeter of small hexagon is 18, and the perimeter of the large hexagon is 24. What is the interior angle sum of the large hexagon if the small hexagon has an angle sum of 720? ________

ComplexNumbers

Ü Acomplexnumberisanumberwrittenintheforma+biwhereaandbarerealnumbersandiisasolutionoftheequationx2=-1.

Ü aisreferredtoastherealpartandbisreferredtoastheimaginarypart.Ü Complexnumberscanbeaddedandsubtractedandmultipledlikebinomialswherethe

variableisiandi2=-1.

þ a+bi +c+di = (a + c) + (b + d)i

þ a+bi–(c+di) = (a – c) + (b – d)i

þ (a+bi)× (c+di) = ac+adi+bci+bdi2 = ac + (ad+bc)i + bd(-1) = (ac-bd) + (ad+bc)i

þ If i2 = -1, the sum of the complex numbers 1 +2iand3+5i is 4 + 7i

5-37

CoordinateGeometry

Ü The slope of the line connecting two points, (x1, y1) and (x2, y2) is 12

12

xxyy

m--

=

þ The slope of the segment from (4,3) to (8,11) is 248

48311

==-- .

Ü The midpoint of two points, (x1, y1) and (x2, y2) is ÷ø

öçè

æ ++2

,2

2121 yyxx . [midpoint formula]

þ The midpoint of the segment from (8,11) to (16,4) is ( ) ( ) ( )5.7,12215,

224

2411,

2168

==++ .

Ü The distance between points (x1, y1) and (x2, y2) is 212

212 )()( xxyy -+- . [distance formula]

þ The distance between the points (8,11) to (4, 3) is 2)113(2)84( -+- = 6416 + = 80 = 54 .

r What is the slope of the line through (16, 4) and (8, 11)? ________________ r What is the midpoint of the segment from (4, 3) to (8, 11)? ________________ r What is the distance between the points (16, 4) and (4, 3)? _________________

Trigonometry

ÜInarighttriangle,

sin(q) = opposite leg length/hypotenuse length cos(q) = adjacent leg length/hypotenuse length tan(q) = opposite leg length/adjacent leg length

sin2(q) + cos

2(q) = 1, b×sin A = a×sin B

ÜInaunitcircle, sin(q) = cos(90 - q), cos(q) = sin(90 - q)

Ü90°=:,radians,180°=pradians,360°=2pradians

r If q < 180°, for which radian value does tan(q) = -1?

5-38

¦ Which of the following is equivalent to sin(π/10)?

A) -cos(π/10)

B) -sin(π/5)

C) cos(2π/5)

D) -sin(2π/5)

¦ The number of radians in a 630-degree angle can be written as nπ, where n is a constant. What is the value of n?

A) 1.5 B) 2.5 C) 3.5 D) 4.5

SAT“Theorems”forParallelandPerpendicularLinesandFiguresDrawntoScale

ÜAxiom:Figuresthatseemtobeinthesameplanearecoplanar.

ÜAxiom:Figuresthatappearthree-dimensionalarethree-dimensional.

ÜTheorem:Iftwolinesintersect,thenanglesthatlookcongruentarecongruent. Corollary:Adjacentanglesthatappearsupplementaryaresupplementary.

ÜTheorem:Ifparallellinesarecutbyatransversal,anglesthatlookcongruentarecongruent.Corollary:Anglesthatappearsupplementaryaresupplementary.

ÜTheorem:Ifaquadrilateralisaparallelogram,anglesthatlookcongruentarecongruent.

Corollary1:Anglesthatappeartobebisectedarebisected.Corollary2:Anglesthatlooklikerightanglesarerightangles.Corollary3:Segmentsthatlookcongruentarecongruent.Corollary4:Segmentsthatappeartobebisectedarebisected.

þ The diagonals of a square bisect the interior angles of the square.

þ The diagonals of a square are perpendicular to each other.

þ The diagonals of a rectangle are congruent and bisect each other.

¦ If the figure above is a rectangle, which of the following statements must be true?

A) Ð2 @ Ð4 @ Ð6 B) Ð4 @ Ð5 @ Ð8 C) Ð4 @ Ð5 @ Ð9 D) Ð4 @ Ð8 @ Ð12

5-39

ImprobableAlgebra

axioms syntheticdivision matrices determinants standardformoflinearequations quadraticinequalities jointvariation combinedvariation logarithms infinitegeometricseries factorials combinations permutations binomialtheorem remaindertheorem Descartes'ruleofsigns

ImprobableGeometry

proofs definitions planes locus constructions apothem secants orthocenter incenter circumcenter centroid kites geometricprobability Ceva'sTheorem Euler'sTheorem Heron'sTheorem Brahmagupta'sTheorem non-Euclideangeometry analyticgeometry

5-40

AnswerKeytoEssentialsofMathematicsProblems

NUMBERSENSE OrderofOperations r 1 + 2 ´ 3 – 4 ÷ 5 = 6.2

MultiplicationofSignedRealNumbers r (-6)(-8)(-10) = -480

Fractions(RationalNumbers) r 8

,+ 1

<= ,8

/S 23/10

r /,× 18= ,

8 2/3

r ><÷ 8

1= ;

< 8/5

Percents r = 62.5% 62.5%

r 10 is 125% of 8 8

r 20 is 0.5% of 4000 0.5%

¦ 17 is 44.7% of 38 D) 38

RatioandProportion r 3:1 = 21:7 7

58

5-41

HEARTOFALGEBRA

PolynomialSubstitution r

2)1( +nn when n = 8 36

r x2 – 2xy + y2 if x=2, y=3 1

r -16t2 + 34t + 5 if t = 2 9

PolynomialEquivalents r x2– 36 = (x + 6) (x – 6)

r (y + 1)2 = y2 + 2y + 1

r z2 – 6z + 9 = (z – 3) (z – 3)

r -3

2 – (-3) 2 = -18

DistributiveProperties r 3(2x + 1) = 6x + 3

r 5(3x – 7) = 15x – 35

r 262 +a = a + 3

r aaa-3 = 2

r 2" 8 = 8x3

r 8,

,= 9/4

r 205 × 10

r GX

;

) a2/2

AbsoluteValuer | z + 1 | = 3 -4, 2

SolvingLinearEquationsandInequalities r (x + 5) + 4x = 9 + (x + 8) 3

r 2x + (3 – 5x) > 3(9 + x) x < -4

m 4 movies D) 4

Functions(DomainandRange) r Range of the function 0 ≤ y ≤ 4

FunctionNotationandComposition r f(g(-1)) = 1

r f is defined by f(x) = 4x – 9

Slope-interceptFormofLinearFunctions r slope of f(x) = 4x-3 is 4

r y-intercept of line -3

r slope through (2,4) and (4,3) -1/2

r y-intercept of the line 5

5-42

PASSPORTTOADVANCEDMATH

Exponentsr 23 = 8 r 32 = 9 r 41 = 4

ProductsandQuotientsofPowers r 42 aa × = a6

r 4

8

aa = a4

r =× 21

21aa a

PowersofPowers r ( )342x = 8x12

r ( ) 216x = x3

r ( ) 23 --x = x6

Exponents(Zero,Negative,andRational) r 03 1

r 16- 1/6

r 21

16 4

r 328 4

r 24- 1/16

r 21

9- 1/3

SimplifyingRationalExpressions

r 33

3 ++

+ xxx = 1

r 3

93

3-

-- yyy = 3

r 23

21

++

- zz =

)2)(2()1(4-+

-zz

z

r 22 +

-- w

www =

)2)(2(4

-+ www

r a

a815

34 3

´ = 25 2a

r 456

49

bb÷ =

815 3b

¦ 4

3=

b

a and 5

4=

c

b , =ac A) 5/3

PolynomialRoots&SolvingRationalEquations

r Solve xx 25

211=+ . 3

r Solve yyy

yy 2

23

292

=-+

+-- . 1.5

r Solve 593

54 zz

+-< . z > 3

SolvingRadicalEquations r Solve 312 =-x 5

r Solve xx =-12 1

m Solve 1953 =-x D) 64

m Solve 34 +=- x D) no solution StandardFormofEquationsofParabolas…r shifted parabola y=(x+6)2-2

r circle equation (x+3)2+(y-2)2=32

r smallest solution -3

r solution sum 3

r solution product -6.25

m x-intercepts (roots) B) -3 and 4

5-43

PROBLEMSOLVING&DATAANALYSIS

Motion(Amount)Problems

r 1:20@75mph, 2:30@50 mph 23/6 hr

DescriptiveStatistics

r mode of 18,20,33,25,20,24,19 20

r mean of 21, 31, 27, 15 23.5

r average of 4a+7 and 1-6a -a + 4

r median of 18,20,33,25,22,19 21

r median of 24,19,30,23,16 23

StatisticalMethodsandTerminology ¦ least affected B) median income

¦ margin of error D) 95% of means

WeightedAverages r 12´100 and 18´80 averages to 88

PositiveandNegativeAssociation,ExponentialGrowthandDecay

ProbabilityandConditionalProbability

GraphicalRepresentationofData¦ best fit line D) y = -x + 10

¦ best fit curve C) y = 10x-1.2

ADDITIONALTOPICSINMATH

Angles

r Approximate measures a≈30, b≈120, c≈45

Area

¦ The area of the triangle B) 12

Perimeter(Circumference)

r Rectangle perimeter is 24

¦ Perimeter of the square C) 32

SolidGeometry

¦ Volume of cone C) 12π

Triangles

r sides 4, 7, c 3 < c < 11

Circles

r inscribed angles C) mÐCAD = mÐCBD

RightTriangles

r triangle side length n 8

Polygons:CongruenceandSimilarityr Trapezoid interior angles 360°

r Trapezoid exterior angles 360°

r mÐD = 180 – 30 – 35 = 115

r CASPC – angle sum is 720

CoordinateGeometryr slope through (16,4) & (8,11) is -7/8

r midpoint is (6, 7)

r distance is √145

Trigonometry

r tan θ is -1 at 3π/4

¦ sin(p/10) = C) cos(2p/5)

¦ 630° = np radians , n = C) 3.5

SATTestAxioms

¦ Any rectangle has angles B) Ð4@Ð5@Ð8

5-44

EssentialsofMathematicsClassroomPractice

DIRECTIONS

Solvetheseproblemsbyworkinginthespaceavailableoneachpage.Selectthebestanswerforeachquestionandfillinthematchingovalontheanswersheet.

NOTES• Youmayuseyourcalculator.• Allexpressionsusedrepresentrealnumbersunlessotherwiseindicated.• Figuresaredrawntoscaleunlessotherwiseindicated.• Allfiguresprovidedarecoplanarunlessotherwiseindicated.• Thedomainofafunctionfisthesetofallrealsforwhichf(x)isrealunlessotherwiseindicated.• Youmayusethegeometricreferenceinformationprovidedifyouchoose.

REFERENCE

Thereare180°(anglemeasures)inatriangle.Thereare360°(ofarc)inacircle.Thereare2πradians(ofarc)inacircle.

1 ………………………………… …

If 8H= /

GJ/ , which of the equations below

expresses b in terms of a?

A) b = 3a + 3

B) b = 3a + 1

C) b = 3a – 1

D) b = 3a – 3

2 …………………… …………… … At a local fast food restaurant, the two most popular lunch items are french fries and cheeseburgers. Each order of fries has 70 more calories than a cheeseburger. If 3 cheeseburgers and an order of fries have 1150 calories, how many calories does a cheeseburger have? A) 270

B) 305

C) 340

D) 375

5-45

3 ………………………………… … Which of the following complex numbers is equivalent to ,.8Y

Y?

A) 3 + 2i B) 3 – 2i C) -3 + 2i D) -3 – 2i 4 ………………………………… … If f(x) = x-1/2 , which of these expressions is equivalent to f(x) when x > 0?

A) - "

B) - x2

C) "J/

D) J/-(

5 ………………………………… …

1"+3"=15

An office supply store is using two printers (a regular printer and a high speed printer that prints three times as fast) to complete a large print job. Working together, the printers can complete the job in 5 hours. In the equation above that represents the situation, what is the meaning of 8

- ?

A) Time (in hours) for the fast printer to

finish one-fifth of the job

B) Portion of the job completed by the slow printer in 3 hours

C) Portion of the job completed by the fast printer in 1 hour

D) Time (in hours) for the fast printer to finish the job working alone

6………………………………… … Which of the following expressions is equivalent to sin(:

8)?

A) −cos(:8)

B) −sin(:8)

C) cos :>

D) sin(<:>)

7 ? )… …… …………………… The graph of y = x2 + 4x + 3 is a parabola in the xy-plane with a vertex at (-2, -1) and x-intercepts of -3 and -1. Which of the following equivalent equations shows the coordinates of the vertex of the parabola? A) y = (x – (-2))2 – 1 B) y = x(x – (-2)) + 2(x – 1) + (-1) C) y = x(x + 4) – 2 – 1 D) y = (x – 3) (x – 1)

8 … …… ……………………

Acknowledged for performance

Performance not Acknowledged

Took prep course 12 20

Did not take prep course 4 24

The table above summarizes the results of 60 students who took the NMSQT. If a student is randomly selected for an interview from those acknowledged for performance, what is the probability that the student did not take a prep course? A) 0.75 B) 0.25 C) 0.10 D) 0.025

5-46

9 …………………(1 …

Which scatterplot indicates a negative association between the variables? (Note: A negative association between two variables occurs when the values of one variable increase as the values of the other decrease and vice versa.)

B)

C)

D

10 … … …

The scatter plot above shows changes in annual average temperature in degrees Fahrenheit at selected sites between 1990 and 2010.

Which of the following is the best interpretation of the y-intercept of the best fit line for the graph? A) The annual temperature at selected sites

did not increase before 1990

B) There was no annual temperature change at selected sites during 1990

C) The average temperature of selected

sites in 1990 was 0 degrees D) The average temperature of selected

sites in 1990 is used as a baseline

Ä STOP Ä

5-47

EssentialsofMathematicsClassroomPracticeAnswerKey

1. D) b = 3a – 3

2. A) 270 calories

3. B) 3 – 2i

4. C) "J/

5. A) Time (hours) for the fast printer to finish one-fifth of the job

6. C) cos :>

7. A) y = (x – (-2))2 – 1

8. B) 0.25

9. A)

10. D) The average temperature of selected sites in 1990 is used as a baseline

5-48

STANDARDMULTIPLE-CHOICE

TheScoutingReport

PowerHitters.ThefastwaytoSATmathsuccessistoworkeachproblemasquicklyaspossibleandlookforthecorrectansweramongtheresponses.Althoughmostcollege-boundstudentscorrectlyanswerlessthanhalf(22-26)ofthe58questionsinthetwomathsections,yourgoalistocorrectlyanswerasmanyquestionsasyoucan.Todoso,youmustbeanexpertwiththeessentials.TheLineupCard.Therewillbeexactly45StandardMultiple-ChoicequestionsontheSAT.Thefirst(nocalculator)sectionwillhave15multiple-choicequestionsandthesecond(calculatorallowed)sectionwillhave30multiple-choicequestions.Onbothsections,questionsareorderedinincreasingdegreeofdifficulty.YoucancountonanapproximatelyevendistributionofquestionsfromtheHeartofAlgebra,PassporttoAdvancedMath,andProblemSolvingandDataAnalysissectionswithanoccasionalAdditionalTopicsinMathproblemfromGeometryorTrigonometry.

Youwillprobablyencounter3-5questions(1inthefirstsectionand3-4moreinthecalculatorsection)thatwillrequireyoutoreadanduseinformationintabularform.Approximately3-5additionalitemswillcontaindiagrams(1or2inthefirstsectionand2or3inthecalculatorsection)whereindividualproblemsareaccompaniedbygraphicsincludingdotandscatterplots,barandlinegraphs,coordinategrids,quadrants,etc.Therewillprobablybeatleastonediagramclearlyidentifiedasnotdrawntoscale.EachSMCquestionwillbefollowedby4responses(onlyoneofwhichisthecorrectanswer).GamePreparations.Sometestitemswilllookliketheycamerightfromalgebraandgeometryexamsyouhavealreadytakeninschool.Manyoftheotheritemswillnot,buttheywilllookliketheitemswetrainonbothinclassandonthehomeworkouts.TheOfficialStandardMultiple-ChoiceRules.Eachofthe45StandardMultiple-Choicequestions,#1-15inthefirstsectionand#1-30inthecalculatorsectionisfollowedbyfourresponses.Youaretochoosetheonecorrectanswertoeachquestion.Questionsstartouteasyandbecomeprogressivelymoredifficult.Inbothsections,thetestitemsareALWAYSprecededbythesameinformation.

• Solvetheproblemsusingthetestpageforscratch.• Findtherightanswer,andmarkitontheanswersheet.• Notethatallnumbersusedarerealnumbersunlessspecifiedotherwise.• Thedomainforafunctionfisassumedtobethesetofrealsforwhichf(x)isreal.• Notethatfiguresaredrawntoscale(unlessspecifiedNOTdrawntoscale).• Notethatallfiguresarecoplanar(unlessclearlythree-dimensional).• Youmayusethegeometricreferenceinformationprovidedifyouchoose.• Youmaynotuseacalculator(Section 3) …orYoumayuseacalculator(Section 4)

5-49

Whatyoushouldalsobeawareofisthat:

• youmaynotuseanyscratchpaper. • youshouldnotwriteontheanswersheet. • SATproblemsseldominvolveimaginarynumbers • mostofthefiguresprovidedareextremelyaccurate. • figuresnotdrawntoscaleareintendedtoconfuseyou. • SATproblemsinfrequentlyrequire3-dimensionalskills. • youshouldnotneedthereferenceinformation(althoughitiscorrect).

• youmaychoosetonotuseyourcalculatoronmuchofSection2

TheStandardMultiple-ChoiceOffensiveGUARD

UsethemnemonicphraseOffensiveGUARDtohelpyouremembergeneralstrategiesthatyoucanusetoanswerStandardMultiple-Choicequestionsdirectly(regardlessoftheirdegreeofdifficulty).Thesestrategieswillapplytoquestions#1-#15inthefirst“nocalculator”sectionand#1-#30inthesecondsectionthatallowscalculatoruse.

G LANCE at the answers U SE figures provided A NSWER every question R EAD through every problem D RAW visual representations

GLANCEattheanswers(andquestion)beforereadingaproblem.GLANCEattheanswers(andquestion)beforeyoubegintoreadaprobleminordertohelpyoufocusonwhatyouwillbeaskedtodo.Inthefollowingexample,trytoanticipatethequestionbylookingattheanswerchoices. ex 1. ………………………….……

A) 36° B) 40° C) 45° D) 60°

ex 2. ………………….…………… …

A) 235c + 57a ≥ 985 B) 235c + 57a > 985 C) 235a + 57c ≥ 985 D) 235a + 57c ≥ 985

Willtheexamplesinvolvearithmetic,algebra,orgeometry?_____________________

Whatwillthequestionsask?______________________________________________Whatinformationwillyoubegiven?________________________________________

5-50

Inexample1,theuseofdegreesintheanswertellsyouthatthisisprobablyageometryprobleminvolvinganglesizesinvolvinglines,atriangle,aquadrilateral,oracircle.Youwillbegivenenoughinformationtodeterminetheanglesizeslogically.Theproblemwillprobablynotbeaccompaniedbyadiagram,butifitis,itwillusuallybedrawntoscale.Also,inthequestionsaswellastheanswers,angleswillprobablybelessthan180°.Inexample2,youshouldrecognizethatthefocusisonunderstandingalgebraconcepts.Youmayoftenencounteralgebraproblemsthataskyoutointerpretthemeaningofexpressionsortoidentifyequationsorinequalitiesthatmodelasetofcircumstances.Youmayalsobeaskedtofindasolutionusingthemodel.

HerearetheactualSATproblems(usetheanswersheetattheendofthissection):

1 ………………………………… … The number of sides, n, of a regular polygon is related to the measure A, in degrees, of an exterior angle of the polygon by the formula nA = 360. If a regular polygon has n > 7, what is the largest measure for an exterior angle?

A) 36°

B) 40°

C) 45°

D) 60°

2 …………………………………… … … A cup of almond milk contains 235 mg of calcium and a cup of carrot juice has 57 mg. If the recommended daily student calcium intake is 985 mg, which inequality depicts the number of cups of almond milk a and carrot juice c are needed to meet the daily calcium requirement?

A) 235c + 57a ≥ 985 B) 235c + 57a > 985 C) 235a + 57c ≥ 985 D) 235a + 57c > 985

AsyouGLANCEattheanswers,youmayalsofinditeasytolocatethequestioninthelastfewlinesoftheproblem.GLANCEatthequestionifyoucan.Whenitdiffersfromwhatyouwouldexpect,thedistracterssometimesincludethecorrectanswertothequestionthatyouwereexpecting.Assuch,itisimperativethatyouanswerthespecificquestionthatisasked.TesttakerswiththeedgeGLANCEattheanswers(andquestion)tohelpfocusonwhatisrelevantbeforetheybegintoreadaproblem.

Activity:GLANCEattheanswers(andquestion)priortoreadingthisproblem.

3 …..…………………… …

A) 0.005d + 0.03v B) 0.005d – 0.03v C) 0.03d + 0.005v D) 0.03d – 0.005

v

3 ……………………..……………………… … An author receives a payment of $0.03 each time a short story is downloaded and $0.005 each time it is viewed. Which expression is the dollar amount earned if d stories are downloaded and v stories are viewed? A) 0.005d + 0.03v B) 0.005d – 0.03v C) 0.03d + 0.005v D) 0.03d – 0.005v

5-51

USEthefiguresprovidedinfindinganswers.Figuresareprovidedtoassistyouinansweringquestions.Althoughonlyabout10-15%ofthemathquestions(3to6multiple-choicequestions)willbeaccompaniedbyfigures,theseareoftenUSEfulfigures.Figuresprovidedarealmostalwaysdrawntoscale,makingitpossibletomeasureanswersdirectlyfromthefigure.Ontheotherhand,whenfiguresareNOTdrawntoscale,itisoftenobviousandthefigureswillalwaysbelabeled"NOTdrawntoscale."Inthosecases,thefigureissometimesintendedtodeceive,butascalefigurecanbedrawnandUSEdtomakeasolutionobvious.MostformsoftheSATcontainatleastonefigurethatislabeledNOTdrawntoscale.WheneverafigureislabeledNOTdrawntoscale,therearecertainthingsthatyouwillneedtoassumeaboutthefigureandcertainassumptionsthatyoucannotmake.Certainlyanypointthatlookslikeitisonalinethatextendsfromalinesegmentcanbeassumedtobeonthesameline,andanypointsthatlookliketheyareonacurvecanbeassumedtobeonthecurve,butwhenusingafigurenotdrawntoscale,onlytheorderandrelativelocationsofpointsandanglescanbeassumed.

Considerthefollowingfigure:

Figure NOT drawn to scale

WhenanSATquestionmakesreferencetoafigurethatisnotdrawntoscale,itconveysinformationabouttherelativeorderandpositionofpointsandangles,butitdoesnotprovideinformationabouttherelativemagnitudeofthesizesofsegmentsandangles.Youmayassumeallofthefollowing:

• WXY,WYZ,andWXZaretriangles• Y isbetweenX andZ. • X,Y,andZarecollinear• mÐXWY <mÐXWZ • mTU <m TW • AreaofDWXZ >areaofDWXY • AreaofDWXZ >areaofDWYZ

Youmaynotassumeanyofthefollowing:

• DWXZ isarighttriangle• DWXZ isisosceles• mÐXWY <mÐXYW • mÐXYW <mÐZYW • mÐYWZ <mÐYZW • mTU >m UW • AreaofDWXY >areaofDWYZ

5-52

4 …………………………… ……

Note: This figure is NOT drawn to scale.

In the figure, lines m and n are parallel and lines s and t are parallel. If the measure of Ð1 is 45°, what is the measure of Ð2 ? A) 120° B) 135° C) 150° D) 155°

Inexample4,youshouldknowthatparallellinescutbyatransversalcreateanglepairsthataddto180.Ifnecessary,youcaneasily(re)drawthefiguresothatthe45°angleisapproximatelytherightsize.BecauseyouarelookingtofindthemeasureofÐ2inafigurenotdrawntoscale,becarefulnottochooseanswerchoiceC),theapproximatesizeoftheangleinthediagram.ThefewSATproblemsinvolvingthemeasuresofsegmentsoranglescansometimesbeansweredcorrectlywithminimalknowledgeofalgebraandgeometry.YoucanUSEafiguretoseeormeasurecorrectanswersdirectly.FiguresareprovidedtobeUSEd,andinthecaseswhentheyarenot,theyareclearlymarked.Thisisnotavisualizationtest.StudentswiththeedgeUSEfiguresinordertofreetheirbrainsforthinking.

Activity:USEthefiguretosolvetheproblem.

5 …………………… … ……

A square with area 8 is inscribed in a circle. What is the radius of the circle? A) 2 B) 22 C) 4 D) 24

5-53

ANSWEReveryquestion. SinceyourSATscoreplaysaroleincollegeacceptanceandthescoreisdeterminedbythenumberofquestionsthatyouANSWERcorrectly,youshouldANSWEReveryquestion.YourscoreislimitedbyhowmanyquestionsyouANSWER.TheaveragemathrawscoreontheSATisusuallyonlyabout22-26.Notethattheonlywaytogetarawscoreof58istoANSWERall58questions.YouhavenothingtoloseandsomethingtogainbyANSWERingeveryquestion.Evenonquestionsthatyoudonotunderstand,yourrawscorewilltypicallyimproveby1pointforeveryfourSMCquestionsthatyouANSWER.Supposethatyouhadaprettytypicalperformanceandcorrectlyanswered10questionsonthefirstmathsection.Youwerecorrecton8ofthe15SMCquestionsandhadcorrectlygridded2ofthefinal5SPRquestionswhentimewascalled.Youweredisappointedthatyoudidnotfinishthoselastthreequestions,butsincetheyareoftenverytoughproblems,yourationalizebysayingthatbyworkingtoofastyouprobablywouldhavemissedthem.Here’swhereyouwentwrongandwhattodofromnowon. Sinceyoudidnotfinishthefirstsection,youneedtobedeterminedtofinishthe38questionsinthesecondmathsectionin55minutes.Youmayfindthatyouareabletocorrectlyanswer14ofthefirst21(easier)questionsin38minutes.Asyoustruggletofinishquestion22,yourealizethatyouarenowfacingsomeofthemostdifficultproblemsonthetest(8difficultstandardmultiple-choiceproblemsandthefinal8student-producedresponses).Youalsorealizethatyouhaveonly15minutesremaining.Whentoldtodosobyyourinstructor,quicklyANSWERthesefinal8 multiple-choicequestionsofthisimaginarysecondsectioninthegridbelow.Compareyouranswerstoyourinstructor’skey,calculateyourrawscore,anddiscussthestrategy.

#correctSMCbeforeANSWERactivity: 14#wrongSMC(blanks)beforeactivity: 8#wrongSPR(blanks)beforeactivity: 8

#correctSMCafterANSWERactivity:14+___=_____#wrongSMC(blanks)afteractivity: 0#wrongSPR(blanks)afteractivity: 8

Notetheimportanceofconsideringeveryquestion.Ifyoudonothaveenoughtimeforthelast8SPRquestions(#31-38),youwillgetnothingforthem.However,ifyoudon’thavetimeforthelast8SMCquestions(#23-30),youcanblindlyANSWERSMCquestionsandmoveontotheSPRstohavethebestchancetocorrectlyANSWEReveryquestion.

5-54

Activity:BesuretoANSWERbothquestionsregardlessoftheirdegreeofdifficulty. 6 …………………… …

3x + b = 5x – 7 3y + c = 5x – 7

In the equations above, b and c are constants. If b is c minus /

, , which of the

following is true?

A) x = y – /1

B) x = y + /1

C) x = y – />

D) x = y + />

7 …………………… …

Which equation below is written in the form that identifies the coordinates of the roots of x2 + 2x – 15 = 0? A) y = (x – 1)2 – 16

B) y = x(x +2) – (3·5)

C) y = (x – (-3)) (x – 5)

D) y = (x – 3) (x – (-5))

8 …………………… … ……

A clock gains s seconds every t hours. How many minutes does the clock gain each day? A) s

t24

B) 25st

C) 52ts

D) 24ts

9 …………………… … ……

During a baseball season, players batting exclusively right-handed had a batting average of 0.251 and those batting exclusively left-handed had an average of 0.273. Which of the following must be true about the batting average m for the combined group of hitters? A) m = 0.262

B) m < 0.262

C) m > 0.262

D) 0.250 < m < 0.274

The Edge … There is a minimal association between problem length and problem difficulty.

5-55

READthrougheveryproblemcarefully.ApproximatelyhalfoftheSATmathitemsareapplicationproblemsandhalfarenot.Almosthalfoftheapplicationproblemsfocusoncomprehensionmorethancalculation(i.e.,anitemmayask“Whatisthemeaningof3xintheequation?”).Thehalfofthemathtestitemsthatarenotapplicationproblemsmayalsofocusonfluencymorethancomputation(i.e.,aquestionmayaskfor“thevalueof6x+1”or“theminimumvalueofthefunction”).Asaresult,manySATitemslooklike"wordproblems."Sinceeasyitemsmayappeartobe"wordproblems,"itisimportanttoREADallofeachproblemcarefully.Unpreparedstudentsmaywronglyassumethatoneoftheseisadifficultwordproblem:

•aneasyalgebraicsimplification •aneasyquestioninproseform •alongproblemwithmultipleeasysteps

AlthoughsomeSATmathitemsmayaskuncommonquestions,theapplicationproblemsareoftenquitestraightforward.ByREADingthrougheachproblemcarefully,youwillfindseveralproblemsaskedwithwordsarequiteeasy.YoumayalsofindmanySATquestionsthatarefarmoreconceptualthantraditionalmathhomeworkproblems.

10 …………… …… As the result of a recent merger of two semiconductor companies, the new company has increased its production capacity to the point that it can now manufacture 14,000 microprocessors each week. The increased production will also require more stringent quality control testing. To maintain the current quality levels, 3 of every 350 microprocessors will need to be randomly selected for testing. At this rate, how many microprocessors will be tested each week?

A) 120 B) 140

C) 240 D) 400

11 ………………… … Of the students currently enrolled at Dewmoor High School, approximately 4% of the seniors and 6% of the juniors were selected to be National Merit Semifinalists when they tested as juniors. If there are 377 seniors and 335 juniors at Dewmoor, which is closest to the total number of juniors and seniors who were selected as Semifinalists? A) 15

B) 20 C) 35

D) 71

Torecap,halfofthemathitemswillappearasapplicationsandtheremainingnon-applicationitemsmaybeaskedinprosetomakethemappeartobe"wordproblems."Unpreparedstudentsmayquicklysurrenderforthewrongreason,whiletesttakerswiththeedgeknowthatapplicationproblemsmayfocusonasimpleunderlyingmathconceptandthatsomequestionsaskedwithwordsmaynotbewordproblems,buteasymathproblemsaskedwithwords.

5-56

Example10maylooklikeacomplexcomputerprobabilityanalysis,butSAT1600gradsREADthroughaproblemcarefullyenoughtorecognizethatthequestioninvolvesonlyasimpleproportionalrelationship.Sampling3outof350isthesameassampling6outof700or12outof1400,etc.Example11alsolooksdifficultunlessyouREADthroughitcarefully.Althoughthestudentsdescribedintheproblemmayhaveveryhighacademicability,thequestionrequiresonlythebasicmathabilitiestocalculatepercentagesandaddintegers.Despitesomeoftheoccasional(andperhapsintentional)misdirectiononSATmathquestions,READingthrougheveryproblemcarefullywillhelpyoutofindeasyproblemsindifficultdisguises.SAT1600graduatesREADthrougheveryproblemcarefullyatleastonce. Activity:BesuretoREADthroughtheeachquestioncarefullybeforeanswering. 12 …… ……………

A research study on the grade point averages of American students was designed to verify if there is an association between 9th grade GPA and college GPA for the population of recent college graduates educated in the United States. Survey responses were obtained from a random sample of 3141 recent United States college graduates and the analysis found clear and convincing evidence that there is a positive association between a student’s 9th grade GPA and college GPA. Which of these conclusions is well supported by the survey data? A) There is a positive association between 9th grade GPA and college GPA for recent college graduates in the United States.

B) There is a positive association between 9th grade GPA and college GPA for recent college graduates in the world. C) Using GPA as defined by the study, an increase in the college GPA is caused by an increase in the 9th grade GPA for recent college graduates in the United States. D) Using GPA as defined by the study, an increase in the 9th grade GPA is caused by an increase in the college GPA for recent college graduates in the United States.

13 …… ………. …

After the 2010 census, a local high school board recognized that the student population was increasing and created a model for predicting average class size until the 2020 census. The mean number of students per classroom s is estimated by the equation s = 0.46t + 17.32 where t represents the time in years since the last census. In the context of the problem, which phrase is the best interpretation of the 0.46 coefficient of t ? A) The estimated annual increase in the mean number of students per classroom B) The estimated confidence interval of the mean number of students per classroom C) The estimated mean number of students per classroom in 2010 D) The estimated mean number of students per classroom in 2020

5-57

DRAWvisualrepresentationsofproblems. Yourbrainhastwodistinctlydifferentabilities,theabilitytovisualize(memorize),andtheabilitytoreason(calculate).Althoughyoumaybeproficientatboth,itisdifficultforthehumanbraintodobothsimultaneously.Thepopularityofcomputersandsmartphonesistheresultoftheirabilitytoexceedhumancapacitiestostore(memorize)andprocess(calculate)informationalmostsimultaneously.Donotdecreaseyourreasoningabilitybyunnecessarilytryingtomentallystorediagrams,tables,graphs,chartsornumbers.DRAWandlabelfigures.ThemoreyouDRAW,themoreclearlyyoucanthink. 14 ………………… ……. …

The trapezoid AECD of area 1.5 is formed from square ABCD by connecting the midpoint E of side AB to point C. Which is the best estimate for the side length of the square? A) 1.0 B) 1.4 C) 1.7 D) 2.0

15 …….………… … A square of area 2 is inscribed in a circle. What is the radius of the circle? A) 1

B) √2

C) √3

D) 2

Example14wouldbeconsideredadifficultproblembySATstandards,butwithadiagram,theanswerbecomesobvious.OncethesquareandtrapezoidareDRAWn,theareaofthesquarecanbeseentobeabout(exactly)2,butdefinitelymorethan1.5andmuchlessthan3.Thesidelengthofthesquareisthenmorethan1.2andlessthan1.7.Inexample15,acircleDRAWnaroundasquarewithasidelengththatisabout1.4,showsthattheradiusofthecircleismuchshorterthanthesidelengthofthesquare.AFinalWordaboutOffense AnswerstomanySATmathproblemscanbeobtainedinmultipleways.Notonlyshouldyouglanceattheanswers,readproblemscarefully,useanddrawgeometrydiagrams,andanswereveryquestion,youshouldpracticetheoffensiveGUARDstrategiesonthehomeworkouts.considereveryquestionsinceitisalsofairplaytodrawnumberlines,pictures,charts,tables,andgraphs.Onceyouhavesomethingonpaper,youmightbeabletolimitthedomain,workasimplerproblem,workalgebraasarithmetic,orfindapattern.SAT1600gradsuseanyavailableinformationtofindanswersdirectlywithouttediouscalculations.Eveniftheoffensecannotfindthecorrectanswerdirectly,thenextsectionwillintroduceyoutodefensivestrategiestoincreaseyourscore.

Answers to Standard Multiple-Choice GUARD examples and Activities

1) C 45° 2) C 235a+57c ≥ 985 3) C 0.03d + 0.005v 4) B 135° 5) A 2 6) D x = y + 1/6 7) D y = (x - 3)(x – (-5)) 8) B 2s/5t 9) D 0.250 < m < 0.274 10) A 120 11) C 35 12) A US Gr 9-college GPA 13) A annual increase 14) B 1.4 15) A 1

5-58

StandardMultiple-Choice

CompletingtheLineupandOffensiveGUARDStrategiesPlaybook Format_____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ Content_____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ Directions _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ DegreeofDifficulty 1. If x – 1 = 3k and k = 3 what is the value x = ? 2. For i = Ã-1 what is the (7 + 3i) + (-8 + 9i) = ?

Additions_____________________________________________________________________ _____________________________________________________________________ GUARD_____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________

5-59

StandardMultiple-ChoiceOffensiveGUARDHomeworkoutDIRECTIONSForpracticequestions#1-15,solvetheproblemusinganyavailablespaceforscratch.DecidewhichisthebestanswerfollowingthequestionandfillinthematchingovalonthePractice1answersheet(20minutes).

NOTESTheSATwouldalsorequireyoutoproduceandgridanswersto#16-20.1.Calculatoruseisnotpermittedonthissection.2.Allvariablesandexpressionsrepresentrealnumbers…unlessindicatedotherwise.3.Figuresprovidedinthistestaredrawntoscale…unlessindicatedotherwise.4.Allfiguresarecoplanar…unlessindicatedotherwise.5.Thedomainofafunctionfisthesetofallrealsforwhichf(x)isrealunlessotherwiseindicated.

REFERENCE

Thereare180°(anglemeasures)inatriangle.Thereare360°(ofarc)inacircle.Thereare2πradians(ofarc)inacircle. 1 ……………… ……

If -./8= _ and k = 3, what is the

value of x ?

A) 8

B) 10

C) 12

D) 17

2 ……………… ……

If f(x) = 2.5x + c where c a constant, what is the value of f(-2) given that f(4) = 8?

A) 8 B) 3 C) -2 D) -7

3. ……………… …… 2(y+2) = x

-`

= 4

If the ordered pair (x. y) is a solution to the system of equations above, what is the value of y ? A) 2

B) 4

C) 8

D) 16

5-60

4 ……………… …… Karen is a cell phone screen repair technician. Each day, she receives a box of phones needing repairs. The number of phones left to repair at the end of each hour can be estimated with the equation P = 37 – 5h, where P is the number of phones left and h is the number of hours she worked that day. What is the meaning of the value 37 in this equation?

A) Karen will complete the repairs within 37 hours. B) Karen starts each day with 37 phones to fix. C) Karen repairs phones at a rate of 37 every 5 hours. D) Karen repairs phones at a rate of 37 per hour. 5 ……………… …… Which of the following is equivalent to ( x2y – 5xy2 + 3y2) – (–x2y + 3y2 – 3xy2) ?

A) 0 B) -8xy2 + 6y2

C) 2x2y – 2xy2

D) 2x2y – 8xy2 + 6y2 6 ……………… ……

In the xy-plane above, line k and line l are parallel. What is the value of p ?

A) 4 B) 5 C) 8 D) 10

7 ……………… ……

If Ga

(Pb(

G(ab= a16 and x > y > 1, what is

the value of x – y ?

A) 2 B) 4 C) 8 D) 16 8 ……………… ……

The line y = kx – 6 where k is a constant, is graphed in the xy-plane. If (c, d) is a point on the line where c ≠ 0 and d ≠ 0, what is the slope of the line in terms of c and d ?

A) c.>N

B) N.>c

C) cJ>N

D) NJ>c

9 ……………… ……

kx – 2y = 3 3x – 4y = 5

In the system of equations above, k is a constant and x and y are variables. For what value of k will the system of equations have no solution?

A) -9/5

B) -3/2

C) 3/2

D) 9/5

5-61

10 ……………… ……

If d.8dJ8

= 9, what is the value of v ?

A) ///S

B) 8,

C) ,3/S

D) /<1

11 ……………… ……

y = 2x + 5 y = (2x + 3)(x – 1)

How many ordered pairs (x, y )satisfy the system of equations above?

A) 0 B) 1 C) 2 D) 3 12 ……………… …… A line in the xy-plane passes through the origin and has a slope of

/> . Which of the

following points lies on the line?

A) (0, -6) B) (-1, -6) C) (-6, -6)

D) (-12, -2)

13 ……………… …… What is the sum of all values of n that satisfy 3n2 – 18n + 12 = 0? A) -6 B) -3√5 C) 3√5 D) 6 14 ……………… …… If 350 grams of a substance experiences exponential decay at a 25% annual rate, which of these functions f models the amount of the substance, in grams, remaining after t years have elapsed?

A) f ( t) = 350 (0.75)t

B) f ( t) = 350 (0.25)t

C) f ( t) = 0.75 (350)t

D) f ( t) = 0.25·(350)t

15 ……………… …… If (ax + 2)(bx + 3) = 20x2 + cx + 6 for all values of x, and (a + b) = 9, what are the two possible values for c ?

A) 4 and 5 B) 8 and 15

C) 10 and 12 D) 22 and 23

Ä STOP Ä

5-62

StandardMultiple-ChoiceOffensiveGUARDAnswerGrids

SMCOffensiveGUARD

ClassPractice

SMCOffensiveGUARD

Homeworkout

5-63

StandardMultiple-ChoiceOffensiveGUARDHomeworkoutAnswerKey

1. A) 8

2. D) -7

3. A) 2

4. B) Karen starts each day with 37 phones to fix

5. C) 2x2y – 2xy2

6. D) 10

7. B) 4

8. A) (d +6)/c

9. C) 3/2

10. D) 15/4

11. C) 2

12. D) (-12, 2)

13. D) 6

14. A) f(t) = 350(0.75)t

15. D) 22 o r 23

5-64

TheStandardMultiple-ChoiceABCDefense SpeedandSuccessTherearetwodifferentwaystosucceedonStandardMultiple-Choicemathitems: • findtherightanswertothequestions • findandavoidincorrectresponsestothequestions.Althoughthefirstmethodisfaster,thesecondcanbeequallyeffective.ThisisbecausetheSATauthorsincorrectlyassumethatifastudentdoesnotpossesstheabilitytofindthecorrectanswertoaquestiondirectly,thenthestudentwillalsobeunabletodeterminetheincorrectanswersinordertofindthecorrectanswerindirectly.Thisseemingsmallmisconceptionmakesdefensivestrategyextremelyimportant.Wheneveryoucannotansweramultiple-choicequestiondirectly(thismayhappenalot),thereisarewardforyouifyoucanfindanyincorrectresponses.Notonlyshouldyouanswereveryquestion,buteachtimeyouansweraquestionthereisabonus…it’sabigbonuswhenyoucanfindtherightanswer…andsmallbonusesthataccumulatewheneveryoucanidentifyandavoidanywronganswers.Evenifyoudonotknowthecorrectanswertoaquestion,youcanraiseyourscorebyansweringquestionsforwhichyouhaveidentifiedwronganswers.Thatiswhatwemeanbyplayinggooddefense.

• blindguessingananswerchoicemakesyourchances_1_outof___or______%• eliminating1incorrectanswermakesyourchances_1_outof___or______%• eliminating2incorrectanswersmakesyourchances_1_outof___or______%• eliminating3incorrectanswersmakesyourchances_1_outof___or______%

Inmuchthesamewaythatcolor-blindpeoplecandetectwhichobjectamongseveralredobjectsiswhite,youcanraiseyourSATscore.Color-blindpeoplecanlearntodetecttheonewhiteobjectamongagroupofredobjectsbydeterminingitisnotred,andyoucanlearntodetecttheonecorrectresponsetoaquestionbyeliminatingwronganswersontheSAT.Thedifferencebetweencorrectlyanswering26totalquestionstoscore500,andcorrectlyanswering32questionsor34-38questionstoscore600(byfindingoneortwowrongchoices)onthetwomathsectionscanbeaseasyasABC. AvoidBadChoicesonDefenseEverySATmathStandardMultiple-Choicequestionhasacorrectanswerjustbelowit.Youneedonlyfigureoutwhichof4responses"getcut"tomaximizeyourscore.TherightanswerwillalwaysbeamongtheremainingresponseswhenAnyBadChoiceshavebeeneliminated,anditwillAlwaysBetheonlyChoiceremainingwhenAllBadChoiceshavebeenavoided.ThegeneraldefensivestrategyfortheSATiscalledABCDefenseandthestrategyistoAvoidBadChoicesonDefense.

5-65

ABCDefensiveStrategies YoucanAvoidBadChoicesonDifficultproblemsbyusingspecificdefensivestrategies.Inordertohelpyourememberthem,wewillrefertothemastheABCDefenses:

Approximate (withscalefiguresandnastynumbers)Backwork (withnumberanswers)Crunch'n'plug(withvariableanswers)Don't (whenallelsefails)

TohelpyouAvoidBadChoicesonDifficultquestions,readabouttheABCDefensesandthenapplythemtothefollowingexamplesandactivities.APPROXIMATEYoucanAvoidBadChoicesinalgebrawordproblems,geometry,graphsandcharts,coordinategeometry,andarithmeticcalculationsbyAPPROXIMATinganswers.ThisAPPROXIMATiontechniquecanbeusedeffectivelyforproblemsfrom

• HeartofAlgebra ___________________________________• PassporttoAdvancedMath ___________________________________• ProblemSolvingandDataAnalysis ___________________________________• AdditionalTopics(geometry) ___________________________________

APPROXIMATionisaverypowerfuldefensivetechnique(becauseanyanswerchoicesnotintheproximityofyourAPPROXIMATionareneverthecorrectanswer,andthecorrectanswerisalwaysintheproximityofyourAPPROXIMATion).NotonlycanyoueffectivelyAPPROXIMATEonupto40%ofthe45multiple-choicequestionsandalmosthalfofthequestionsinthecalculatorallowedsection,butupto30%ofthecalculatoritemsmayonlyaskyouforanAPPROXIMATEanswer.

Asyouapplydefensivestrategiestothefollowingexamplesandactivities,rememberthatthesestrategiesareonlytobeappliedtoproblemsthatyoucannotsolvedirectly.

5-66

1 …………… ……… On a particular machine that requires manual monitoring, an electronics manufacturer can make a maximum of 30 new components during each of the machine’s cycles that take 48 seconds. What is the maximum number of these components that the machine can produce during an 8 hour shift? A) 300

B) 11,520

C) 18,000

D) 46,080

2 …… ……… ………

If segment AB is parallel to segment CD, and segment AC intersects segment BD at X, what is the length of segment AC? A) 60 B) 180 C) 225 D) 360

Inproblem1,APPROXIMATEly30+componentscanbemadeinAPPROXIMATElyaminute.In60minutes,about(30+)(60)≈1800+componentscanbemade.SinceAPPROXIMATEly2000componentscanbemadeeachhour,APPROXIMATEly8×2000(or10×2000)componentscanbemadeduringashift.ThisisAPPROXIMATEly16,000(or20,000)units. A)1440 ismuchlessthan16,000 (avoid) B)11,520 islessthan16,000 (avoid) C)18,000 isAPPROXIMATEly16,000(or20,000) D)46,080 ismorethandouble16,000 (avoid)YoushouldbeabletoAvoidBadChoices(A,B,andD)withminimalcalculationifyouApproximatetoseethat(C)isthemostacceptableguess.NotethatrandomcalculationsandpartialsolutionsaswellasconceptualerrorsareAmongtheBadChoices. Inproblem2,theunknownlengthofthesegmentAXlookstobeAPPROXIMATElyhalfthelengthofsegmentXC.InadditiontobeingAPPROXIMATEly60,itcanalsobeeasilyseenthatthelengthofsegmentAXismorethan48andlessthan96.ThismeansthelengthofsegmentACisAPPROXIMATEly180andislimitedtolengthsbetween168and216.BothAPPROXIMATionsallowyoutoeasilyAvoidalloftheBadChoices. A)60 istheAPPROXIMATElengthoftheshortsegment (avoid) B)180 istheAPPROXIMATElengthofthesegmentAC C)225 islongerthanthecombinedlengthsofAXandXC (avoid) D)360 ismoretriplethelengthofsegmentXC (avoid)YoushouldbeabletoAvoidalloftheBadChoices(A,B,andD)byeithervisualinspectionorbyphysicallymeasuringAPPROXIMATElengthswithyourpenciloreraser.

5-67

APPROXIMATEtoseethat(C)isthemostacceptableguess.NoteonceagainthatbothpartialsolutionsandconceptualerrorsareAmongtheBadChoices.

Activity:BesuretoAPPROXIMATEtosolveandcheckthefollowingpracticeproblems.

3 …………… …

If the semicircle above has radius 3, and chord BC has length 4 and is parallel to diameter, what is the distance between the chord and the diameter AD ? A) √3

B) √5

C) π

D) 2π

4 …… ………..... …… …

The typical sizes of several commonly downloaded multimedia files are summarized in the table below.

Although file sizes are usually reported in megabytes or gigabytes, as in the table, download rates are usually quoted in megabits per second. If a gigabyte is defined as 1000 megabytes and a megabyte is composed of 8 megabits, what is the best estimate for the time it takes to download a 2-hour HD movie if the tested download rate is 4 megabits per second ? A) 2.5 minutes

B) 15 minutes

C) 2.5 hours

D) 10 hours

The Edge … Approximation is a powerful underused tool that can affirm reasonable responses with both numerical calculations and diagram measurements.

5-68

BACKWORKYoucanAvoidBadChoicesonthosealgebraquestionswithnumericalresponsesbyBACKWORKing(insideout)fromthepossibleanswersbacktothequestion. YoucanonlyuseBackworkeffectivelyonabout20%ofthe45multiple-choicequestions.SinceBackworkismosteffectiveusedearlyinbothsections,itismostlyusedasadefenseagainstmakingcalculationerrorsorother“silly”mistakes. 5 ………… … …

The equation -5t2 + 20t gives a rough estimate of the height h, in meters, of a ball t seconds after it is launched upward from the ground at a vertical velocity of 20 meters per second. About how many seconds will the ball stay in the air before it hits the ground? A) 4.0 B) 4.5 C) 5.0 D) 6.0

6 ………… … …

In the xy-plane, the line determined by the points (n, 6) and (24, n) passes through the origin. Which of these could be the value of n ? A) 0

B) -4

C) -8

D) -12

Inproblem5,trytoBACKWORKoutwardbeginningwith(BorC)becausewhenanswersarenumbers,__________________________,usuallyfromsmallesttolargest.Totakeadvantageofthis,startBACKWORKingwiththesmallerinneranswerchoice(B)4.5. A) 4.0 B) 4.5 (avoid) C) 5.0 (avoid) D) 6.0 (avoid) After4.5seconds,theheightoftheballis-5(20.25)+20(4.5)=-101.25+90=-11.25.Sincethisheightisbelow0,theballhasalreadyhittheground.Thisnotonlyconfirmsthat(B)isABadChoice,itmakesanynumbersabove4.5AllBadChoicesbecausetheyareevenlarger.Sincechoices(B,C,andD)arewrong,theansweris(A).WithBACKWORK,youcanoftenAvoid2or3BadChoicesatatime.Ifyouhadstartedwith(C)5.0,youwouldgeth=-125+100,eliminatingboth(CandD),andthenconfirmedthat(A)wascorrectbecausetheheightoftheballwouldbeen-5(16)+20(4)=-80+80=0.Inproblem6,evenifyoudonotimmediatelyrecallthepropertiesofdirectvariationassociatedwithlinesthatpassthroughtheorigin,ifyousubstitutevaluesfornintotheorderedpairs,graphtheorderedpairsonacoordinateplane,anddrawthesegments,

5-69

youcanuseBACKWORKtotestwhetherthesegmentsbetweenthepointspassthroughtheorigin.(Wewillfurtherdevelopthissubstitutionconceptinournextstrategy.) A) 0 (0,6)and(24,0) (avoid) B) -4 (-4,6)and(24,-6) (avoid) C) -8 (-8,6)and(24,-8) (avoid) D) -12 (-12,6)and(24,-12)

Justaswithnumbers,regardlessofwhereyouchoosetostart,eachtimeyoudrawasegment,youcanseewhetherornotitismovingyouclosertoasolution.BACKWORKingwillalwayseitherfindtherightansweroreliminateABadChoice(ortwo).AnotherbenefitofBACKWORKisthatinsomealgebraproblems(likeproblem7),youmaynotneedtouseanyalgebratoconfirmthesolutiontoanalgebraproblem.Activity:BesuretotryBackworktosolveandcheckthefollowingpracticeproblem. 7 ………… … …

−/," + /

1e = 7

.− /1" + /

,e = /

,

Which ordered pair (x, y) satisfies the system of equations above?

A) (118, 60)

B) (34, 96)

C) (-18, -8)

D) (-50, -72)

5-70

CRUNCH'N'PLUG Inproblemswithalgebraicexpressionsasanswerchoices,youcanAvoidBadChoicesifyouevaluate(CRUNCH)thenumbersinvolvedafteryousubstitute(PLUGin)arbitraryeasyvaluesforthevariablesintheproblem.MostofthenumberCRUNCHingcanbereducedtosimplearithmeticifyouchoosetoPLUGinvalueslike(1or)2or3.YoucanuseCRUNCH'N'PLUGeffectivelyonabout30%oftheSATquestions,butitisparticularlyeffectiveinthefirstnocalculatorallowedsection.Onthebackhalfofthe15SMCitemswithnocalculatorallowed,youmayalsobeabletoCRUNCH'N'PLUGyourwaytothecorrectansweruptohalfthetime. 8 ………… … ……… …

The expression<-J/-.1

is equivalent to which of the

expressions below?

A) <J/1

B) 5 − 14

C) 5 − /-.1

D) 5 − ,/-.1

9 ………… … ……… … A clock loses m minutes every n days. How many hours will the clock lose every two weeks?

A) ,1Dg

B) >SgD

C) 3D8Sg

D) 3g8SD

The Edge... Crunch 'n' Plug works when responses are letters (variables) in the same way that Backwork is used when the responses are numbers.

Inproblem8,sincevariablesrepresentgeneralvalues,avariablecanalsobethoughtofashavingaspecificvalue.ToAvoidBadChoices,justCRUNCHthenumberstofindavalueforeachexpressionafterPLUGgingineasyspecificvalues(werecommend1,2,3,etc.)foreveryoccurrenceofeachvariableinthequestion.Youcanthensolvetheproblemdirectlybymatchingthevalueoftheexpressiontothenumericalanswer(orBackworkfromtheanswertothenumericalformofthequestion).Inproblem8,letx=1(because0or1sometimesmakequestionsoranswerstrivial,wewilloftenchoose2or3asvaluestoPLUGtoplugin).Thismeansthat(5x–1)=4and(x+4)=5.AfterCRUNCHingthenumbersforthevaluesPLUGgedintotheexpressionsinthequestion,thenewspecific(andeasier)questionbecomes"Whichexpressionisequalto1

<?".

5-71

TheresponsescanbetestedbyPLUGginginthevaluex=1toeachandCRUNCHingthenumbers

A) 1415 =- (avoid)

B)419

415 =- (avoid)

C)524

515 =- (avoid)

D)54

5215 =-

Theanswerto"Whichnumberequals

54 ?"is(D),.YoucanAvoidtheBadChoice(A)sinceits

valueis1;andsincethevaluesof(BandC)arecloseto5,theyareAlsoBadChoices.Thevalueof(D)is 4

5,so(D)isthecorrectanswer.

Fortheperfectionist,itispossiblebutunlikely,thatwithspecificchoicesforPLUGgingandCRUNCHing,thevalueofmorethanoneexpressionwillprovideanacceptableresponse.ThegoodnewsisthatonceadistracterisfoundtobeABadChoiceforanychoiceofnumbers,itisABadChoiceforeverychoiceofnumbers.Ifyouhavetime,youcancheckallfouranswerchoiceseverytime(buteachtime,youwillfindthattheincorrectanswerswillstillbewrongandthecorrectanswerwillstillberight).Inexample9,sinceyouwanttotrackhowmanyhourstheclocklosesduringtwoweeks,youwanttochoosewhateversimplenumbersaremostconvenienttosubstituteasvaluesforthevariables,m(thenumberofminutes)andn(thenumberofdays).Supposethatyouchoosetoletm=60andn=7sothattheclocklosesonehoureachweek.Thismeansthattheclockwilllose2hoursin2weeks,andyouneedonlytoapplyCRUNCH'N'PLUGbyreplacingnwith7andmwith60ineachoftheexpressionsuntilyoufindanexpressionwiththevalueof2.Evaluating(D)gives(7´60)/(30´7)=420/210=2. Unfortunately,thereisnobestchoicetostartwithwhenCRUNCH'N'PLUGgingresponses,butyouwillsavetimeonceyourealizethatyoudonotneedtoCRUNCH'N'PLUGallfouranswers(oncethreearewrong,theotheroneisright).Infact,sinceeachresponseisequallylikelytobecorrect,youwilloftenfindtherightanswerafteryouCRUNCH'N'PLUGonlyoneortwooftheresponses.WithCRUNCH'N'PLUG,evenwhenaquestioninvolvesalgebra,noalgebraisneededtofindtherightanswer.HeartofAlgebraproblems,somePassporttoAdvancedMathproblems,andevensomeproblemsfromAdditionalTopicsinMath(geometry)canbeconvertedtoarithmeticproblemswhenyoucanuseCRUNCH'N'PLUG.YousimplyselectthenumberstoCRUNCHafteryouPLUGthemin.

5-72

Activity:Substitutevaluesforthevariablesinboththequestionsandtheanswerchoicesbelow,andthenevaluatetheexpressionstoeliminatewronganswers. 10 …… …. ……

Kyra and Phala each ordered a bowl of soup at a restaurant. The price of Kyra’s soup was $2 dollars more than the price of Phala’s soup. If the cost of Phala’s soup was x, and they split the bill after adding 10% sales tax and a 20% tip (they calculated their tip based on the total of the bill that included the sales tax), which expression represents the amount, in dollars, each of them paid?

A) 0.32x + 0.32

B) 0.5x + 0.16

C) 1.32x + 1.32

D) 2.64x + 1.32

11 … …… …

An online auction site uses the formula h = i

D.i to generate a seller’s rating, r,

based on the number of positive transactions , p, and negative transactions, n. Which of the following expresses the number of negative transactions in terms of the other variables?

A) n = j(iJ/)i

B) n = j(/Ji)i

C) n = i(jJ/)j

D) n = i(/Jj)j

The Edge... Defensive strategies are a powerful last resort, but the best defense is a good offense. Master the essentials, use the offensive GUARD, and use ABCDefenses when other tactics fail.

The Edge ...

CRUNCH'N'PLUGworkswhentheresponsesareletters(variables)tosubstituteandevaluate.

BACKWORKisappropriatewhentheresponsesarenumbersthatyoucangobackandseeiftheywork.

APPROXIMATEcanbeusedatanytimeincludingbeingusedjointlywithBackworkandoccasionallyusedjointlywithCrunch‘n’Plug.ItcanalsobeusedjointlywiththestrategiesoflastresortfromtheDON’Tlist.

5-73

DON'T YoucanalsoAvoidBadChoicesifyouDON'TcommitthemostcommonerrorsandifyouDON'TlookforanswerswherethetestmakeroftenhideswrongAnswers(BadChoices).DON’Tstrategiesareonlytobeusedasalastresort,andinsomecasesyoumayonlyfindoneopportunitytouseaDON’Tstrategyonthetest.However,ifallelsehasfailed,theDON’Tstrategiesarepowerful(sometimesexperimental)medicineandtheuseofastrategyfromtheDON’Tlistmay(ormaynot)getyouclosertothescaledscorethatisthedifferencebetweenwhetherornotyouareeligibleforascholarship. Indifficultproblemsthatinvolvepercentagesoraverages,aswellasproblemswithparticularanswerconfigurations,orproblemslocatedinaparticularareaofthetest,thefollowingDON’Tlistmaysaveyoufromyourself(anddistinguishyoufromothers).

• Don'tdopercentswithcommonsense • Don'tdoaverageslikeanaveragekid • Don'tguesstheexteriornumbersinaseries • Don'tpickthechoiceindistantleftfield • Don'tgowithfourinarow(i.e.,AAAA,BBBB,CCCC,orDDDD) • Don'tguessthemaximum(minimum)inmaximum(minimum)questions* • Don'tguessnonanswersinthesecondhalf* • Don'ttakethefake(inthefinalfour)*

*PreviouslyconfirmedontheoldSAT,butstillexperimentalonthenewredesignedSAT

12 … … …

DON'Tdopercentswithcommon sense

Crystal bought a tablet computer at a store that gave a 10% discount off its original price. If the original price of the computer was x dollars, and there was a sales tax of 9% on the discounted price, which of the following expressions represents her final cost? A) 0.99x

B) -S.kk

C) (0.9x) (1.09)

D) -

S.k (/.Sk)

13 … … …

DON'Tdoaverageslikeanaveragekid

If a is the average of n and 6, b is the average of 2n and 6, and c is the average of 3n and 6, what is the average of a, b, and c in terms of n ? A) n + 3

B) 2n + 6

C) 3n + 9

D) 6n + 18

5-74

Example12isanexampleofapercentageproblemthatisfrequentlyansweredincorrectlybystudentswhoperformaneasymentalcalculationinsteadofthepropercalculation.Don'tdocommon(sense)calculationswithpercents.Althoughitmayseemthatreducingapriceby10%andthenincreasingthatamountby9%,willgetyouto99%oftheprice,…butthisisnotthecase.Donotselect(A),0.99x. Ifanumberisdecreasedby10%,itis90%oftheoriginalamount.Ifthatreducedamountisincreasedby9%,theincreaseisonlybasedontheslightlysmalleramount,or(0.9x)(1.09).Thisvalue,0.981x,isslightlylessthan0.99x.DON'Tattempttoanswerpercentquestionswithcommonsense.Example13isanotherexampleofaproblemthatstudentsfrequentlyanswerincorrectlybecausetheyperformaneasymentalcalculationinsteadofthepropercalculation.Itseemseasytofindtheaverageofa,b,andc,whichareaveragesofthreepairsofsimplebinomials;(n+6),(2n+6),and(3n+6).Noalgebraisneededtoaveragethethreebinomials…theiraverageisclearly2n+6.Although(2n+6)isclearlytheaverageofthethreebinomials,thisisnotthesolutiontotheproblempresented.Thevariablesa,b,andc,aredefinedtobeaveragesofthebinomials,notthebinomialsthemselves.Thismeansthatthevariablesa,b,andcareeachhalfthesizeofeachofthethreebinomials,andthustheaverageofa,b,andcishalfof(2n+6)orn+3.Don’tdosomethingeasyinsteadofsomethingcorrect.Don’tselect(2n+6).DON'Tdoaverageslikeanaveragekid.Hereareotherthingstowatchoutfor(thatonlyoccurabout40%asoftenastheyshould). 14 … … …… …

DON'Tbetonthefirstorlastinaseries A hospital stores bulk quantities of non-addictive medicines in large 5-decagram containers. How many 1-milligram doses can be obtained from each of the 5-decagram containers? (Note: 1 decagram = 10 grams , 1 gram = 1,000 milligrams) A) 50 B) 500 C) 5,000 D) 50,000

15 … … …

DON'Tpickthechoiceindistantleftfield The density of an object is defined to be its mass divided by its volume. If the density of aluminum is 2.7 grams per cubic centimeter, what is the volume, in cubic centimeters, of a chunk of aluminum with a mass of 54 grams. A) 0.05 B) 20 C) 51.3 D) 56.7

Ifyouhaveanydifficultywithaproblem,youshouldavoidthefirstandlastresponsesinanalgebraicorgeometricsequence(e.g.,1,3,5,7or2,4,8,16).Inexample14,itislesslikelythatanswerchoices(A)50or(D)50,000arethecorrectresponse.DON'Tguessthefirstorlastresponseinaseries.

5-75

Youshouldalsoavoidaresponsethatsfarremovedfromtheotherresponses.Itislesslikelythataresponsethatdiffersgreatlyfromtheothersiscorrect(e.g.,2,5,8,and300).Inexample15thismakesitlesslikelythat(A)0.05iscorrect.DON'Tguessthechoiceindistantleftfield.Onelastthingtoavoid,becauseitonlyhappensabout40%asoftenasitshould,isanswerchoicesthatrepeatthemselvesfourtimesinarow(i.e.,AAAA,BBBB,CCCC,orDDDD).Althoughrandomlygeneratedanswersheetswouldoccasionallyhavelongsequencesofthesameanswerinarow,wefindthisnottobetrue.Itisunusuallydifficulttofindexamplesofanyanswerkeyswithlongstringsofanyresponses-As,Bs,Cs,orDs.Asaresult,wedonotrecommendquicklyfillingintheanswersheetwiththesameanswerforeachquestion.Eventhoughthiswillprobablyearnyouapproximately25%ofthepointsforthoseitems,werecommendthatyou“guessaround”sothatyoualsohavethebestchanceofgettingmanyitemsright.Werecognizethatitisalsopossibletogetmanyitemswrong,butontheSAT,everyanswerchoiceisequallylikelytobecorrect.

Thesefinalpiecesofadviceare“experimental,”althoughtheyheldtrueonpreviousSATs. * …maximum/minimum …………… …

DON'Tguessthemaximum(orminimum) Of 80 people in a class, 25% are smokers and 80% are men. What is the minimum number of men in the room who are nonsmokers? A) 12 B) 16 C) 44 D) 60 [Don’t guess the minimum distracter, A)12]

* …nonanswer , fake………………….……… …

DON'Tguessanonanswerafterhalftime A clock that runs 10 minutes fast every hour is set to the correct time at noon. What is the correct time when the clock first indicates 7:00?. A) 5:50 B) 6:00 C) 8:10 D) cannot be determined [Don’t guess the nonanswer, D) cannot be determined. Don’t take the fake, A) 5:50 … many others will.]

Thefirstexampleisatypeofminimum(ormaximum)problemthatappearsontheSAT.Theseproblemscanappearasalgebra,analysis,oradditionaltopics.Inthepast,thecorrectanswertoadifficultminimum(ormaximum)problemwasrarelythesmallest(orlargest)answerchoice.Thesecondexperimentalexamplehasitsfinalanswerchoiceasaformof“cannotbedetermined.”OnSATproblemsinthepast,when“cannotbedetermined”wasoffered,itwassometimesofferedasaresponsetoadifficultproblemthatcouldbedetermined(butonlybyahighperformingtesttaker.)Don'ttakethefake(inthefinalfour)isyourfinaldefensivestrategy.ItisexperimentalbecauseofuncertaintyabouttherateatwhichSATitemsbecomemoredifficult.WhatiscertainisthefactthatasSATquestionsbecomemoreandmoredifficult,itismoreandmoreimportanttoavoidfallingforanyobviousdistracters.Althoughthismayseemsomewhatcounter-intuitive,thisstrategykeepsyoufrommakingmistakesthatotherstudentsmake.

5-76

Thisstrategycanincreaseyourscorebecauseyouwillanswerdifficultquestionscorrectlymoreoftenthanmostotherstudents.Youwillknowthataneasilyobtainedanswerisnottherightanswertoadifficultproblem.Itwillbetoyouradvantagetoguess…justaslongasyouknowthatyourguessisdifferentfromwhatmostpeoplewillguess.Thetestmakerassumesthatyouarenotsmartenoughtofindwronganswers,butyouare…youcanfindyours.Onceagain,upto80%andmoreofthepeopletakingthetesteventuallybegintomisstheitemsoncethetestitemsgetpastacertaindegreeofdifficulty.Formostpeople,thatmeanstheirperformancewouldimproveiftheychoseanswersdifferentfromwhatmoststudentscanobtainquicklyandeasily.Onthemostdifficultitems,besurethatyoudonotselectananswerthatmanyotherstudentswillchoose.YoucangeneralizethisstrategytotheentireSAT…ifyouarewillingtotrustyourinstinctswhenquestionsareeasiest,doubtyourinstinctsastheproblemsbecomemoredifficult,anddenyyourinstinctswhentheproblemsareextremelydifficult.Fortheearlyportionofamathsection,moststudentswillusuallyberight,andforthefinalportion,moststudentswillusuallybewrong.TheSATisdesignedsothatthepointwherethingsstarttogowrongwillvarywidelyfrompersontoperson,butitwillnotvarymuchfromtesttotest.Examinetheanswergridbelowfora(calculatorsallowed)SMCsectionwherethedarknessofeachovalcorrespondstothepercentageofstudentswhotypicallyanswereachofthequestionscorrectly.

Clearly,#1isthedarkestand#30isthelightest.Onthefinalquestions,manystudentsperformlittlebetterthanrandomguessing.Youcanbeconfidentthatbyattendingclass,doingthehomeworkoutsthatpracticeoffensiveanddefensivestrategies,andmasteringtheStudent-ProducedResponsestrategiestocome,youwillbringyourscorecloserandclosertoyourpeakperformanceontheSAT.

Answers to SMC ABCDefense Examples and Activities

1. C 18,000 2. B 180 3. B √5 4. C 2.5 hr. 5. A 4.0

6. D 12 7. C (-18, 8) 8. D 5 – 21/(x+4) 9. D 7m/30 10. C 1.32x+1.32

11. D n = p(1-r)/r 12. C (0.9x)(1.09) 13. A n + 3 14. D 50,000 15. B 20

* C 44 * B 6:00

5-77

StandardMultiple-Choice

CompletingtheABCDefensivePlaybookGuessingGains(45items)Blind________________________________________________________________1out________________________________________________________________2out________________________________________________________________3out________________________________________________________________ABCDefensesA_______________________________

B_______________________________

C_______________________________

D_______________________________

_________________________________

_________________________________

_________________________________

_________________________________

E_______________________________

When?___________________________

When?___________________________

When?___________________________

_________________________________

_________________________________

_________________________________

_________________________________

_________________________________

_________________________________

5-78

StandardMultiple-ChoiceABCDefenseHomeworkout

DIRECTIONSSolvetheseproblemsbyworkinginthespaceavailableoneachpage.Selectthebestanswerforeachquestionandfillinthematchingovalontheanswersheet.

NOTES• Youmayuseyourcalculator.• Allexpressionsusedrepresentrealnumbersunlessotherwiseindicated.• Figuresaredrawntoscaleunlessotherwiseindicated.• Allfiguresprovidedarecoplanarunlessotherwiseindicated.• Thedomainforafunctionfisthesetofrealsforwhichf(x)isrealunlessotherwiseindicated.• Youmayusethegeometricreferenceinformationprovidedifyouchoose.

REFERENCE

Thereare180°(anglemeasures)inatriangle.Thereare360°(ofarc)inacircle.Thereare2πradians(ofarc)inacircle.1 …………………………………… … The graph below shows Dylan’s distance from his base camp during a 3-hour hike.

At one point, he stopped for 30 minutes to have lunch. Based on the graph, which of the following is closest to the time he stopped to have lunch?

A) 12:50 P.M. B) 1:20 P.M. C) 1:50 P.M. D) 2:10 P.M.

2 ……………………………………… For 5 = −1 what is the sum of the complex numbers (6 + 2i) + (-9 + 8i) ? A) -3 + 10i B) -3 – 6i C) 15 + 10i D) 15 – 6i 3 ………………………………… … On Saturday morning, Aaron sent a text messages each hour for 3 hours, and Britta sent b text messages each hour for 2 hours. Which expression represents the total number of messages sent by Aaron and Britta on Saturday morning? A) 5ab B) 6ab C) 3a + 2b D) 2a + 3b

5-79

4 ………………………………… …

The average class size at Forest H. S. from 2010 to 2015 can be modeled by the equation s = -0.47n + 29.3, where n represents the number of years since 2010, and s represents the average class size. Which of the following best describes the meaning of the constant 29.3 in the equation?

A) The total number of students at the school in 2010

B) The average number of students per classroom in 2010

C) The estimated annual decrease in class size each year since 2010

D) The estimated difference between the class size in 2015 and in 2010

5 ………………………………… …

The scatter plot above shows changes in annual average temperature in degrees Fahrenheit at selected sites across two decades between 1990 an 2010. Based on the line of best fit to the data above, which of the following values is closest to the average temperature increase (°F) per decade?

A) 0.04

B) 0.05

C) 0.07

D) 0.15

6………………………………… … When moving up off the earth, the decrease in atmospheric pressure, P, can be estimated using the formula

P = 1013·(0.88)n

where n is the number of kilometers above the earth. What is the meaning of the 0.88 in the formula?

A) The atmospheric pressure at sea level is typically about 0.88

B) The atmospheric pressure at sea level is approximately 0.88

C) One kilometer above the earth, the atmospheric pressure is 0.88 less than the reading at sea level

D) One kilometer above the earth, the atmospheric pressure is 88% of the reading at sea level

7 ?)… …… ……………………

It is common in games of cards to deal all 52 cards in the deck into 4 hands of 13 cards each. The histogram shows the distribution of the number of spades, n, in 30 consecutive hands dealt to one player. What is the difference between the mean number and the median number of spades in the 30 hands? A) 0

B) 0.25

C) 0.5

D) 1.0

5-80

8 ……………… ……… … Jonas has a set of identical cylindrical drinking glasses, each in the shape of a right circular cylinder, with an inside diameter of 8 cm. He pours orange juice from a 4-liter jug into each glass until it is considered full. A glass is considered full if the height of the juice inside the glass is at least 14 cm. What is the largest number of glasses of orange juice he can fill from the 4-liter container? (Note: There are 1000 cubic cm in a liter.) A) 2 B) 4 C) 5 D) 6 9 ……………… … …… …

A triangle was modified by decreasing its height by 20 percent and then increasing its length by x percent. If the area of the triangle decreased 12 percent as a result of the modifications, what was the value of x ?

A) 10 B) 12 C) 15 D) 28 10 ……………………………………. The sum of three numbers is 585. One of the numbers, z, is 50% more than the sum of the other two numbers. What is the value of z? A) 195 B) 234 C) 351 D) 390

11 ………………………………. …

In the circle O above, segment AB is a diameter and DO is perpendicular to AB. If the length of arc AD is :

,, what is the

length of the radius of the circle?

A) 0.5 B) 1.0 C) 2.0 D) 4.0 12 ……………… … …… … If 4n – 3 ≥ 1, what is the least possible value of 4n + 3 ? A) 7 B) 4 C) 3 D) 1 13 … …… ……………………

If s < 0 < r, which of the following equations best fits the shape of the data shown in the scatterplot above?

A) y = rx + s B) y = sx + r C) y = rxs D) y = sxr

5-81

14 ……………… … …… … Which of the following equations is an equivalent form of the function f(x) = (x + 1) (x – 3) in which the minimum value of the function appears as a constant or coefficient? A) f(x) = x2 – 3 B) f(x) = x2 – 2x – 3 C) f(x) = (x – 1)2 – 4 D) f(x) = (x + 1)2 – 5

15 ……………… … … …… …

If k is a constant such that f(x) = k , and the function g(x) graphed above in the xy-plane has five real solutions when f(x) = g(x), which of the following could be the value of k ?

A) 2

B) 1

C) -1

D) -2

Ä STOP Ä

5-82

StandardMultiple-ChoiceABCDefenseAnswerGrids

SMCABCDefenseClassActivities

SMCABCDefenseHomeworkout

5-83

StandardMultiple-ChoiceABCDefenseHomeworkoutAnswerKey

1. C) 1:50 p.m.

2. A) -3 + 10i

3. C) 3a + 2b

4. B) The average number of students per classroom in 2010

5. C) 0.07

6. D) One km above earth, atmospheric pressure is 88% of the reading at sea level

7. A) 0

8. C) 5

9. A) 10

10. C) 351

11. B) 1.0

12. A) 7

13. C) y = rxs

14. C) f(x) = (x – 1)2 – 4

15. B) k = 1

5-84

Student-ProducedResponses

TheScoutingReport

DesignatedHitters.Thesurewaytosuccessistoknowhowtoworkeachoftheseproblemsandtorecordandcorrectlygridyourresponses.Moststudentswillonlybeabletocorrectlyanswer5or6ofthese13questions(2or3ofthe5questionsonthe“nocalculator”sectionand3or4of8questionsonthe“calculatorallowed”section).Yourgoalistospendsometimeonall13questionsandquicklyanswereachofthemtothebestofyourability.Todoso,youmustbeanexpertwiththeessentials.TheLineupCard.Therewillbe5questions(#16-20)inthe20-question“nocalculator”sectionand8questions(#31-38)inthe38-question“calculatorallowedsection.Thequestionsineachsectionwillgraduallyincreaseindegreeofdifficulty.Thefirstfivequestionsinthe“no-calculator”sectionwillbeginwithslightlyeasierproblemsthanthefinaleightquestionsinthe“calculatorallowed”section.Inbothsections,youcanexpecttoseesomealgebraanddataanalysisandpossiblygeometry.Thequestionsmayormaynotbeaccompaniedbydiagrams.Youmustfind,record,andgridananswertoeachquestion.Notonlyaretheremultiplewaystogridanygivenanswer,forsomequestionstheremaybemultiplecorrectanswers,eachofwhichcaneachbegriddeddifferently.ThePlaybook.Someproblemsmaylookfamiliartoyou,butotheritemswillbecreatedspecificallyfortheSAT.Eitherway,thequestionswilladdressthesamemathcontentasthemultiple-choiceitems,buttheresponseswillbedeleted.Wewillprovidebothtypesofquestionsduringclassandonhomeworkouts.Someofthestrategiesfortheseproblemsapplyspecificallytothesestudent-producedresponseproblems. TheOfficialStudent-ProducedResponseRules.Sincethefinal5problems(#16-20)onthefirst”nocalculator”mathsectionandthefinal8problems(#31-38)onthe”calculatorallowed”mathsectionwillnothaveanyresponsesprovided,thequestionsinthesesectionsrequireyoutogeneratetheanswers,recordthem(optionalbutrecommended),andenterthemintoaspecialanswergrid.Theseproblemswillalwaysbeprecededbythesamespecificinstructionsforgriddingresponsesandthesamesamplecompletedgrids.ThegeneraldirectionsprecedingtheStudent-ProducedResponse(SPR)itemssimplytellstudentstosolvetheproblemandgridtheanswers.Moststudentswillnotbeabletosolveall13oftheseSPRproblems.ManystudentswhodonotfollowourgameplanwillnotevengettoseveraloftheseSPRproblems.WhetheryougettoalloftheseSPRproblemsornot,unlessyoucancompletetheanswergridcorrectly,eventhequestionsyoucananswerwillnotaddtoyourscore.Detailedinstructionsforgriddingyourresponseswillalwaysincludethefollowing:

5-85

1. Recordyouranswerbeforegridding(optional,butrecommended).2. Markonlyoneovalineachcolumn(doublefillsreceivenocredit).3. Noproblemshavenegativeanswers.4. Giveeachquestiononlyoneanswer(evenifseveralanswersarecorrect).5. Rewriteanymixednumbers(2 12 mustbegriddedas2.5or5/2).6. Decimalresponsesmustbeasaccurateaspossible(eitherchoppedorrounded).

TheGridIron.Recordandgridasmanysolutionsaspossibletothesystemofinequalities:6x≥2and2x≤1.

5-86

Howmanyanswers…

didyougrid?_____didyouknow?________didtheclassgrid?____________

Howmanycorrectanswerswillthescoringcomputerbeabletorecognize?_________

TheSPROffense(RPS)

ReadthrougheveryproblemcarefullyProduceananswertothequestionaskedSwitchyouranswertogriddableform

READthrougheveryproblemcarefullyAlthoughtheearlySPRsmayaskquestionsinoneortwolines,theywilleventuallyinvolvemoretextandappearas"thoughtproblems."SinceitisimportantthatyouReadthrougheachproblemcarefully,youneedtoassurethatyouhavetimetoreadandconsideralloftheproblemsinbothsections.Incontrasttotherestofthetest,itisworthtakingtimetocomprehendthese8questions.SPRitemsmayberealapplicationsortheymayprovideeitherfamiliarorunfamiliarmathematicalsituations.ByReadingthrougheveryproblemcarefully,youwillfindthatSPRscanbeanalyzedandsolved. PRODUCEananswertoeveryquestionAnswereverySPRquestioninbothsections.You'llprobablyneedtogethalfoftheSPRscorrectifyouwantascoreabove500.Thereisapotentialrewardandabsolutelynopenaltyforguessing.Somequestionsmayhavehundredsofcorrectanswers.Besuretoproduceandgridananswer.Youcan'twinifyoudon'tplay.SWITCHanswerstogriddableform OneofthemostfrustratingexperiencesontheSATwouldbetoansweraquestioncorrectlyandthenbeunabletofindanequivalentgriddableform.Correctanswerscanbewritteninseveralwaysandsomequestionshavelotsandlotsofcorrectanswers.Practicewritinganswersongrids.Findseveralwaystogridthesameanswer.Youcanincreaseyourscorebyincreasingyourfluency.Practicerewritinganswersinagriddableformandpracticefillinginthegrids.Onthenextpage,trytoconverttheanswersalreadyobtainedtoagriddableform,andthenfillinthegrid.Timelimitis5minutes.

5-87

SPRGriddingExercise

5-88

AFinalWordAboutOffense RememberthatyouwillhavemorethanaminutetoanswereachSPR.Duringthattime,it'sstillfairplaytodoanyofthefollowing: • useanexistingdiagram • drawanduseyourowndiagram • considerthedomain • workasimplerproblem • usethecalculatorofyourchoiceforthesecondsectionSpendsometimesearchingfortheanswertoeverySPRquestion.Bereadyandwillingtouseyourcalculatoronthe“calculatorallowed”section.Itmaynothelpwithalloftheitems,butwithsomeitemsitwillprobablysaveyousometimeandaggravation.

The Edge... Know when to use your calculator. With no answers to pick from, it's much tougher to catch calculation errors. Be careful when entering numbers though. Remember that there are some things calculators just can't do. ________________________________________ ________________________________________ ________________________________________

AnswerKeytoSPRgriddingexercise:1. 1600 2. 61.5 3. 17/3or5.66or5.674. ungriddable 5. 1.57 6. ungriddable7. 32 8. 52.5 9. .987or.98810. .33only 11. 125 12. ungriddable13. 1357or1358 14. 8/.4or20 15. 3.14

5-89

TheSPRDefense

Defensivetechniquesaredifficulttofindwithsuchatransparentopponent,butsomeslightweaknesseshaveappeared.YoucanusemodifiedformsoftheABCDefenses.

• Approximate–Lookforanapproximateanswer.• Backwork–Noanswertoworkbackfrom?Makeaguessandcheckit.• Crunch‘n’Plug–Pluginvariablevalues,crunchthenumbers,seewhathappens.

ThereisevenaDON'TlistforSPRs.Itmaybesmall,butforanyofthe13SPRproblemsforwhichyoucan'tfindananswer,hereitis.

TheSPRDefensiveDON’TStrategies

DON'T read the SPR directions during the test. You don't have time to learn how to grid. DON'T accept negative answers to questions. Gridded numbers cannot be negative. DON'T accept large answers to questions. No numbers larger than 9999 can be gridded. DON'T chop/round decimals until the grid is full. Be willing to work with fractions. DON'T rely on figures that are not drawn to scale. They may be intended to confuse the issue. DON'T take the fake in the final four. Lots of people miss these items believing that they have answered them correctly. Will you?

BesuretouseeveryminuteofthetimeallottedforeachmathsectionandmakesurethatyouconsiderandrespondtoalloftheSPRsinbothsectionsbeforetimeiscalled.Byplayingdefensewhenothershaveresigned,youwillcontinuetooutscorethecompetition.Thisisoneplacewheregivingaproblemadditionalconsiderationcanmakethedifferencebetweenreceivingarewardandreceivingnothing.YourstrategicapproachearnedyoutherighttotakeadditionaltimeontheseSPRs.Usethetimetoyouradvantageandmakeeveryminutecount.

5-90

Student-ProducedResponsesClassPracticeProblemSet

Directions

ForSection1:#16-20,youmaynotuseacalculatortosolvetheproblems.Gridyouranswers.ForSection2:#31-38,youmayuseacalculatortosolvetheproblems.Gridyouranswers.

1.Markonlyonecircleinanycolumn.2.Someproblemsmayhavemanycorrectanswers,butgridonlyone.3.Mixednumbersmustbegriddedasimproperfractionsordecimals.4.Decimalanswersmaybetruncatedorrounded,butmustfillthegrid.

16 …… ………………

x + y = -4 x + 2y = -36

According to the system of equations above, what is the value of x? 17 …… ……………… -4x(-2x – 5) – 3(5 – 2x) = ax2 + bx + c

In the equation above, a, b, and c are constants with a > 0. If the equation above is true for all x, what is the value of b? 18 … . ………… …

Two isosceles triangles with a common vertex are shown in the figure below.

If 180 – z = 2y and y = 75, what is x ?

19 . ………………

A camp counselor wants to find a length, x, in feet, across the lake adjacent to the camp property as shown in the drawing above. The lengths represented by *l, +l, ml, and mn on the sketch were measured to be 1080 feet, 840 feet, 420 feet, and 480 feet, respectively. If *n and +mintersect at E, and ÐABC and ÐBCD are congruent, what is the value of x? 20 …… …………………

At a distance of 40 miles above the Earth’s surface, the temperature is 40° below zero. For every additional 10 miles away from Earth, the temperature steadily decreases by k° Fahrenheit (F), where k is a constant. If the temperature drops to -100° F at a distance of 48 miles from Earth, what is the value of k ?

5-91

Directions

Forquestions#31-38,solvetheproblemsandgridtheanswers.Youmayuseacalculator.

1.Markonlyonecircleinanycolumn.2.Someproblemsmayhavemanycorrectanswers,butgridonlyone.3.Mixednumbersmustbegriddedasimproperfractionsordecimals.4.Decimalanswersmaybetruncatedorrounded,butmustfillthegrid.

31 …….…… ……….. ……

Warren can visually inspect at least 48 bushels of apples per hour and can inspect at most 72 bushels of apples per hour. Based on this information, what is a possible amount of time, in minutes, that it could take Warren to inspect 288 bushels of apples?

32 …… ………… ……

The normal systolic blood pressure P, in millimeters of mercury, for an adult male x years old can be modeled by the equation P = 110 + 0.5x . According to this model, by how many millimeters of mercury will the normal systolic blood pressure increase for an adult male during a 5-year period?

33 …… ………………

A local college radio station programs the time slots for two 30-second commercials in every 15 minute block of air time. If the station operates 24 hours per day, every day of the week, what is the total number of commercial time slots the station can sell and program for one week. 34 … ……… ………………

f(x) = 2x2 – bx + 12

In the xy-plane, the point (3, -3) lies on the graph of the function above. What is the value of b ?

5-92

35 … ……… ………………

An underground fuel storage tank is in the shape of the right circular cylinder above. If the volume of the tank is 63π cubic meters, what is the lateral width, w, of the cylinder, in meters? 36 ……………… ………………

f(x)= " − 4 , − 2 " − 4 + 1 J,Find a value for which this function is undefined.

37 … ……… ……………… Latanya opened a premiere savings account that earns 3 percent interest compounded annually. Her initial deposit was $300, and she can calculate the value of her account three years from now, in dollars with the expression 300(x)3. What is the value of x in the expression? 38 ……… ……………… Latanya’s friend Kendrick found an account that earns 3.5 percent interest compounded annually. Kendrick made an initial deposit of $300 into his account at the same time Latanya made the $300 deposit into her account. After 8 years, how much more money will Kendrick’s initial deposit have earned than Latanya’s initial deposit? (Ignore the dollar sign when gridding your response to the nearest cent.)

Ä STOP Ä

5-93

Student-ProducedResponseClassPracticeProblemsAnswerGrids

5-94

Student-ProducedResponsesClassPracticeProblemsAnswerKey

16. x=28

17. b=26

18. x=105

19. 960ft

20. k=75

31. 240≤m≤360

32. 2.5millimeters

33. 1344timeslots

34. b=11

35. 7meters

36. x¹5

37. 1.03

38. $15.01

5-95

Student-ProducedResponsesHomeworkout

Directions

ForSection1:#16-20)Youmaynotuseacalculatortosolvetheproblems.Gridyouranswers.ForSection2:#31-38)Youmayuseacalculatortosolvetheproblems.Gridyouranswers.

1.Markonlyonecircleinanycolumn.2.Someproblemsmayhavemanycorrectanswers,butgridonlyone.3.Mixednumbersmustbegriddedasimproperfractionsordecimals.4.Decimalanswersmaybetruncatedorrounded,butmustfillthegrid.

16 …….…… …… …

x3 (6 – x2) – 4x = 4x

If x > 0, what is one possible solution to the equation above? 17 …….…… …… …

Note: Figure not drawn to scale

In the figure above, if cos x = 0.8, what is the value of sin y? 18 ……….…… …… …

The function f is defined by a polynomial for which some values of x and f (x) are shown in the table below.

If (x – 3) is a factor of the polynomial that defines the function f , what is the missing value in the table?

19 ……….…… …… …

In the figure above, O is the center of the circle, mÐAOB is :

D radians. What is the

value of n ? 20 …….…… …… …

If a = 3 2 and 3A = 3" what is the value of x ?

5-96

Directions

Forquestions#31-38,solvetheproblemsandgridtheanswers.Youmayuseacalculator.

1.Markonlyonecircleinanycolumn.2.Someproblemsmayhavemanycorrectanswers,butgridonlyone.3.Mixednumbersmustbegriddedasimproperfractionsordecimals.4.Decimalanswersmaybetruncatedorrounded,butmustfillthegrid.

31 ……….…… …… …

A geologist estimates the combined effects of global warming and the corresponding thermal expansion of the oceans will cause sea levels to rise approximately 0.075 inches each year. According to the geologist’s estimate, how long will it take, in years, for the water level to rise 2 feet? 32 …… ………… ……

If s seconds and 6 minutes is equal to half an hour, what is the value of s? 33 … ………… ……

In a study of bat migration habits, 240 male bats and 160 female bats have been tagged. If 100 more female bats are tagged, how many more male bats must be tagged so that 75% of the total number of bats in the study are male?

34 …….. ………………

The furlong, a medieval measure of length still used in horse racing, is equal to 220 yards. It is also equivalent to 40 smaller units called rods. Based on these relationships, 160 rods is equivalent to how many feet? (1 yard equal 3 feet) 35 ……….…… …… …

q = /, nv2

The dynamic pressure q generated by a fluid moving with velocity v can be found using the formula above, where n is the density of the fluid. An aeronautical engineer uses the formula to find the dynamic pressure of a fluid moving with velocity v and the same fluid moving with velocity 0.75 v. What is the ratio of the dynamic pressure of the faster fluid to the dynamic pressure of the slower fluid?

5-97

36 …… …… ………………

In the figure above, point O is the center of the circle, TU and UW are tangent to the circle at points X and Z, as shown. If the length of minor arc TW is 355, what is the circumference of the circle?

Questions 37-38 refer to the information below.

The stock price of one share in a certain company is worth $54 today. A stock analyst believes that the stock will lose 7% of its value each month for the next three months as the result of not filing financial forms in a timely manner. The analyst uses the equation

V = 360·(r)t to model the value, V, of the stock after t months. 37 …… …… ………….…………

What value should the analyst use for r?

38 … …… ………….…………

To the nearest dollar, what does the analyst believe the value of the stock will be at the end of three months if the financial forms are not yet filed? (Note: Disregard the $ sign when gridding your answer.)

Ä STOP Ä

5-98

Student-ProducedResponsesHomeworkoutAnswerGrids

5-99

Student-ProducedResponsesHomeworkoutAnswerKey

16. x=2orx=1.41

17. siny=4/5

18. 0

19. n=3

20. x=54

31. 320years

32. 1440seconds

33. 540malebats

34. 2640feet

35. 16/9

36. 1065

37. 0.93

38. $43.4

5-100

SATMATHGAMEPLAN TheGamePlanOngameday,knowallthetestdirections.Bereadyfortwosections–onemathsectionof20questionstobecompletedin25minuteswithnocalculatorallowed,andonemathsectionof38questionstobecompletedin55minuteswithacalculatorallowed.Bothofthetwosectionshaventwodifferenttypesofmathquestions.Knowhowmanyquestionstherewillbeofeachtype.Knowthatalgebrawillbeaskedaboutmostoften.Bereadytoanswerquestionsthatpresumeathoroughmasteryoftheessentials. OffenseRememberthegeneraloffensivestrategyistofindandcompletetheeasiestproblems.Knowwheretheeasiestproblemsarefoundinbothsections.

SMC-OnStandardMultiple-Choiceitems,remembertheoffensiveGUARD.

• Glanceattheanswers(andquestion)• Usethefiguresprovided• Answereveryquestion• Readthrougheachproblemcarefully• Drawvisualrepresentations

SPR-OntheStudent-ProducedResponseitems:

• Readthrougheachproblemcarefully• Producearesponse• Switchyouranswertogriddableform

DefenseOndifficultquestions,rememberthegeneraldefensivestrategyistoAvoidBadChoices.ThecorrectanswerisneverAnyoftheBadChoices.SMCRememberthespecificABCDefensivestrategiesformultiple-choiceitems:

• Approximate• Backwork• Crunch'n'Plug• Don't

5-101

SPROntheStudent-ProducedResponseitems,workwithyourcalculatorwhenallowed.

• Avoidnegativenumbers• Avoidlargepositivenumber• Gridformaximumdecimalaccuracy• Don'tfallforphonyfigures

Haveaplananduseeveryminuteavailable,playingoffenseordefense.Sinceansweringmanyquestionscorrectlyistheonlywaytoobtainahighscore,youmustanswereveryquestion.Acommonwaystudentslimittheirscoresistofailtoanswereveryquestion.Acorrectguesscountsthesameasacorrectanswer.Ablankresponsecountsfornothing.EvenwithnoknowledgeofthecontentoftheSAT,mostpeoplecanguesscorrectlyon11or12ofthe45StandardMultiple-Choicequestions.Onrareoccasions,somewillevengridaStudent-ProducedResponsequestioncorrectlybyguessing.Sincetheonlywayaquestionwilldefinitelynotraiseyourscoreisifyouleaveitblank,youshouldanswereveryquestion.TheFirstPeriod Tomaximizeyourscore,youwillneedtogothrougheachSATmathsectionthreetimes.Thefirsttime,findtheeasiestquestions,answerthemquicklyormoveontothenextquestion.Ifyoufindthreeconsecutivequestionsthatyoucannotanswerinlessthan45seconds,beginthenextpassthroughthesection.Putyouregoaside…it'sthreestrikesandyou'reontothesecondpass.Thefirsttimethroughyouhaveonlyoneobjective. • Findandanswertheeasyquestions.Circletheirnumbersinthetestbooklet. 1. …… ………………

If 3(x + 4) = 24, which is also true? A) (x + 4) = 8 B) (x + 12) = 8 C) (3x + 4) = 24 D) (3x + 12) = 8

5. …… ……… …

If 2 – x = x – 2 , then x – 2 = A) -4 B) -2 C) 0 D) 2

Afterthefirstpassthroughthetest,someplacesinyourtestbookletshouldlooksomethingliketheabove.

5-102

TheSecondPeriodThistimethrough,spendallbut5-8minutesofthetestingtimeworking75-90secondsperproblem.YouredgeisthatyoucanraiseyourscorebyAvoidingBadChoicesonproblemsthatyoucannotsolvequickly.AvoidingBadChoicesgetseasierasquestionsgetharderbecauseSAT1600gradsavoidtheurgetochooseeasilyobtained(wrong)answerstodifficultquestions.Yoursecondtimethrough,youhave2objectives:• Answerthequestionsyoucancompleteinlessthan90seconds.• AvoidBadChoicesonquestionsyoucannotfinishinlessthan90seconds,Mark

thequestionnumberwithadash(-)andcrossoutincorrectanswerswithanX.

Afterthesecondtimethroughthetest,someplacesinyourquestionbookletshouldlooksomethinglikethesectionabove. TheThirdPeriodWhen5to8minutesremain,youshouldquicklychecksomeveryimportantdetails.First,ifyouhavenotbegunthestudent-producedresponseproblems,quicklyfillinananswertoanyremainingstandardmultiple-choicequestions.BecausethelastfewSMCitemsarequitedifficult,youwillprobablynotbeabletofindtherightanswerquickly,butyoucaneliminateanyoftheanswerchoicesthatyoucanobtainquickly.Beforeyoucompletethestudent-producedresponseproblems,doaquickcheckofthe(circled)easyquestionstobesureyoumadenocarelesserrors.Sinceitprobablyonlytookafewminutestocompletethoseeasyproblems,thisshouldrequireminimaltime.IftimeremainsafteryoucompletetheSPRs,youshouldreturntocompletemoremultiple-choiceproblems.Thisthirdtimethroughthetest,youhave3objectives.

• MakesurethateveryStandardMultiple-Choicequestionisanswered.• Quicklychecktheeasiestquestions.• CompletetheStudent-ProducedResponseproblems.

5-103

Here'swhattheSMCportionoftheanswersheetmightlooklikewith8minutesremainingonthe25minutesection.Severalcriticalitemshavebeenhighlighted.Nowwhat?

Thefollowingarethecriticalitemswith8minutesremaining:

• Answerquestions#14and#15now• Forquestions#14and#15,donotselectanychoicethatquicklyseemsright• Aquickcheckoftheeasyproblemsshouldcatchthemistakein#4• Completeandgridthe5SPRproblems#16-20• Spendtheremainingtimetofinishquestions#14and#15• Anytimeleftshouldbeusedtofindmoreincorrectanswersto#10and#12

1. …… ………… If 3(x + 4) = 24, which is also true? A) (x + 4) = 8 B) (x + 12) = 8 C) (3x + 4) = 24 D) (3x + 12) = 8 10. …… …………

When x = 3, 3x5 + 3x

5 + 3x

5 =

A) (3×3)·(3

5)

B) (3×3)5

C) (3×3)·(3)15

D) (3×3×3)·(35)

12 …… ………… If x and z are both even integers, which expression is not an even integer? A) (x - 1) + (z + 1) B) (x - 1) · (z + 1) C) (x) × (z + 1) D) (z) × (x - 1) 13. …… ………… A 25% drop in attendance followed an increase of 20% in ticket prices. What was the percent decrease in revenue? A) 90% B) 25 % C) 10% D) 5%

Undertime

Ifyouhavetimeleft,continuetoplayDefenseonquestionswithchoiceseliminated.YouarefarmorelikelytoimproveyourscorebycontinuingtoAvoidBadChoicesthanbyattackinganewdifficultproblem.Forgetabouttheimpossibleproblems.Yourgoalistocorrectlyanswerasmanyproblemsaspossible.Useourstrategiesandourgameplanforeveryavailableminute.

5-104

TheMathGoals Haveyoulearned…

• torecognizebothtypesofquestionsontheSATmathsections?

• thedirectionsforeachtypeofproblem?

• thetimingandpacing?

• theEssentialsofMathematics?

• toapplytheoffensiveGUARDandtheABCDefensestrategiesforSMCs?

• toapplytheRPSoffenseanddefenseonSPRs?

• therolecalculatorsplay?

• todevelopconfidenceabouttheSATmathsections?

• toachieveascaledscoreof_______witharawscoreof_______?

MATH